Constitutive Modelling of Solid Continua [1st ed. 2020] 978-3-030-31546-7, 978-3-030-31547-4

Table of contents :
Front Matter ....Pages i-xii
Basic Equations of Continuum Mechanics (José Merodio, Raymond Ogden)....Pages 1-16
Finite Deformation Elasticity Theory (José Merodio, Raymond Ogden)....Pages 17-52
Thermomechanics (Manuel Doblaré, Mohamed H. Doweidar)....Pages 53-79
Viscoelastic Solids (Alan Wineman)....Pages 81-123
A Primer on Plasticity (David J. Steigmann)....Pages 125-153
Nonlinear Constitutive Modeling of Electroelastic Solids (Luis Dorfmann, Raymond Ogden)....Pages 155-186
A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies (Roger Bustamante, Kumbakonam Rajagopal)....Pages 187-230
Continuum Damage Mechanics—Modelling and Simulation (Andreas Menzel, Leon Sprave)....Pages 231-256
Theories of Growth (Marcelo Epstein)....Pages 257-284
Finite-Strain Homogenization Models for Anisotropic Dielectric Elastomer Composites (Morteza H. Siboni, P. Ponte Castañeda)....Pages 285-309
Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives (Raimondo Penta, Laura Miller, Alfio Grillo, Ariel Ramírez-Torres, Pietro Mascheroni, Reinaldo Rodríguez-Ramos)....Pages 311-356
Multiscale Homogenization for Linear Mechanics (Reinaldo Rodríguez-Ramos, Ariel Ramírez-Torres, Julián Bravo-Castillero, Raúl Guinovart-Díaz, David Guinovart-Sanjuán, Oscar L. Cruz-González et al.)....Pages 357-389

Citation preview

Solid Mechanics and Its Applications

José Merodio Raymond Ogden   Editors

Constitutive Modelling of Solid Continua

Solid Mechanics and Its Applications Volume 262

Founding Editor G. M. L. Gladwell, University of Waterloo, Waterloo, ON, Canada Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, email: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

More information about this series at http://www.springer.com/series/6557

José Merodio Raymond Ogden •

Editors

Constitutive Modelling of Solid Continua

123

Editors José Merodio Department of Continuum Mechanics and Structures, Escuela de Caminos, Canales y Puertos Universidad Politécnica de Madrid Madrid, Spain

Raymond Ogden School of Mathematics and Statistics University of Glasgow Glasgow, UK

ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-030-31546-7 ISBN 978-3-030-31547-4 (eBook) https://doi.org/10.1007/978-3-030-31547-4 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Complex problems in the mechanics of solid materials in a wide range of applications require detailed knowledge of the structure and properties of the materials, which are provided by constitutive models of the materials and their constituents at different length scales. Multiscale modelling is essential for improving the influence of material structure on macroscopic behaviour, such as the constitutive models used in the biomechanics of soft biological tissues, the multi-physics of the so-called smart materials and many applications that involve fluid–structure interactions and thermo-, electro- and/or magneto-mechanical coupling. For different materials and different areas of application within science and engineering, the solution and simulation of complex problems require the use of advanced numerical methods, but these methods can only be as good as the constitutive models on which they depend. Thus, the possibility of obtaining solutions using modern computational methods requires the development of sophisticated multiscale constitutive modelling. This motivates the genesis of this volume on the Constitutive Modelling of Solid Continua, which arose in discussions during the international workshop on Modelling of Nonlinear Continua held in the Centro Internacional de Encuentros Matemáticos, Castro Urdiales, Spain in June 2017 organized by the two of us. It was felt that there was a need for a collection of articles that describe the different types of behaviour that solids can undergo as a reference source from the perspective of theoretical continuum mechanics, covering, in particular, the constitutive developments of nonlinear elasticity, thermomechanics, viscoelasticity, plasticity, damage, growth, electromechanical interactions, porous material mechanics and the techniques of homogenization for treating composites and multiscale applications. It is the purpose of this volume to bring together such a collection, comprised of 12 chapters prepared by carefully selected leading experts. The content of each chapter is now discussed briefly. Chapter “Basic Equations of Continuum Mechanics” provides an outline of the main features of the equations and principles of nonlinear continuum mechanics, which form the foundations for the constitutive theories elaborated in the other v

vi

Preface

chapters, while chapter “Finite Deformation Elasticity Theory” takes this on board and develops the theory of nonlinear elasticity, in particular, the general framework for the description of constitutive equations, thus underpinning the basic principles of constitutive theory adopted in the remaining chapters. Chapter “Thermomechanics” deals with the basic principles of constitutive equations for theormomechanics within the framework of the laws of theromodynamics, with application to non-dissipative materials (ideal fluids and elastic solids) and dissipative materials, including damage mechanics, viscoelasticity and thermoviscoplasticity. Next, chapter “Viscoelastic Solids” describes the characteristic time-dependent features of viscoelastic material response and then goes on to examine the constitutive description of both linear and viscoelastic behaviours. Chapter “A Primer on Plasticity” provides a development of the modern theory of finite elastoplasticity, with emphasis on, in particular, the elastic and plastic decomposition of the deformation, dissipation and material symmetry. The coupled fields of electrostatics and nonlinear elasticity are the subject of chapter “Nonlinear Constitutive Modeling of Electroelastic Solids”, which develops the nonlinear theory of electroelastic interactions, including the governing equations and constitutive theory, along with applications to illustrative boundary-value problems. We note that the corresponding theory of nonlinear magnetoelastic interactions is not considered separately in this volume since the equations follow a parallel pattern and the reader is referred to the book by Dorfmann and Ogden [1] for detailed coverage of this topic. Chapter “A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies” contains a treatment of the relatively new topic of implicit constitutive relations, which began with the nonlinear (implicit) theory of elasticity and was then extended to incorporate electroelastic, magnetoelastic and theromoelastic couplings. The theory is exemplified by the solution of a limited number of boundary-value problems for specific implicit constitutive equations. Chapter “Continuum Damage Mechanics—Modelling and Simulation” focuses on general modelling aspects of continuum damage mechanics, including application to isotropic and anisotropic damage with a view to finite element simulation, illustrated with numerical examples. From damage mechanics, we move on to another application of constitutive theory in chapter “Theories of Growth” which requires a fundamental reformulation of the classical equations of continuum mechanics described in chapter “Basic Equations of Continuum Mechanics”, this being concerned with the processes of growth and remodelling and associated phenomena, including ageing. In the context of the finite deformation of composite materials, chapter “FiniteStrain Homogenization Models for Anisotropic Dielectric Elastomer Composites” is concerned with homogenization, that is determining the effective overall properties of composite materials based on the constitutive properties and geometry of their constituents. The method of homogenization is exemplified by application to dielectric elastomer composites with rigid dielectric inclusions on the basis of the nonlinear theory of electroelasticity discussed in chapter “Nonlinear Constitutive Modeling of Electroelastic Solids”.

Preface

vii

In chapter “Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives”, after a review of porosity and diffusion in biological tissues, a derivation of the equations of poroelasticity is provided using asymptotic homogenization based on the material microstructure with a view to eliciting the complex interplay between porosity and diffusion, including aspects of growth and remodelling. The final chapter, “Multiscale Homogenization for Linear Mechanics”, continues the theme of homogenization, in particular, asymptotic homogenization of heterogeneous materials within a multiscale framework, with applications to fibrous materials and wavy laminated composites. This volume covers topics that are in the mainstream of current research in solid mechanics and interactions with other fields and provides an overview of the state of the art in several areas, pointers to the literature and the needs for further developments. It is aimed primarily at researchers who have a reasonable background in continuum mechanics and some area of solid mechanics, and provides an entry into other areas to enable the scope of their knowledge in solid mechanics to be broadened. It will also provide a valuable source of background information for Ph.D. students embarking on research in challenging and exciting areas of solid mechanics and their interactions with other fields. The editors are, especially, grateful to all the authors of the chapters herein who have dedicated much time and expertise to the preparation of the individual chapters. Madrid, Spain Glasgow, UK August 2019

José Merodio Raymond Ogden

Reference 1. Dorfmann L, Ogden RW (2014) Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer, New York

Contents

Basic Equations of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . José Merodio and Raymond Ogden

1

Finite Deformation Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . José Merodio and Raymond Ogden

17

Thermomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manuel Doblaré and Mohamed H. Doweidar

53

Viscoelastic Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alan Wineman

81

A Primer on Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 David J. Steigmann Nonlinear Constitutive Modeling of Electroelastic Solids . . . . . . . . . . . 155 Luis Dorfmann and Raymond Ogden A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Roger Bustamante and Kumbakonam Rajagopal Continuum Damage Mechanics—Modelling and Simulation . . . . . . . . . 231 Andreas Menzel and Leon Sprave Theories of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Marcelo Epstein Finite-Strain Homogenization Models for Anisotropic Dielectric Elastomer Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Morteza H. Siboni and P. Ponte Castañeda

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Contents

Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Raimondo Penta, Laura Miller, Alfio Grillo, Ariel Ramírez-Torres, Pietro Mascheroni and Reinaldo Rodríguez-Ramos Multiscale Homogenization for Linear Mechanics . . . . . . . . . . . . . . . . . 357 Reinaldo Rodríguez-Ramos, Ariel Ramírez-Torres, Julián Bravo-Castillero, Raúl Guinovart-Díaz, David Guinovart-Sanjuán, Oscar L. Cruz-González, Federico J. Sabina, José Merodio and Raimondo Penta

Contributors

Julián Bravo-Castillero Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, CDMX, Mexico Roger Bustamante Department of Mechanical Engineering, Universidad de Chile, Santiago, Chile Oscar L. Cruz-González Aix-Marseille University, CNRS, Centrale Marseille, LMA, Marseille Cedex 13, France Manuel Doblaré Aragón Institute of Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain Luis Dorfmann School of Engineering, Tufts University, Medford, MA, USA Mohamed H. Doweidar Aragón Institute of Engineering Research (I3A), University of Zaragoza, Zaragoza, Spain Marcelo Epstein Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada Alfio Grillo Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Turin, Italy Raúl Guinovart-Díaz Departamento de Matemáticas, Facultad de Matemática y Computación, Universidad de La Habana, La Habana, CP, Cuba David Guinovart-Sanjuán Department of Mathematics, University of Central Florida, Orlando, FL, USA Pietro Mascheroni Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research (HZI), Braunschweig, Germany Andreas Menzel Department of Mechanical Engineering, Institute of Mechanics, Dortmund, Germany; Division of Solid Mechanics, Department of Construction Sciences, Lund University, Lund, Sweden

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Contributors

José Merodio Departamento de Mecánica de los Medios Continuos y T. Estructuras, E.T.S.I de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Madrid, CP, Spain; Department of Continuum Mechanics and Structures, Escuela de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Madrid, Spain Laura Miller School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow, UK Raymond Ogden School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, UK Raimondo Penta School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow, UK P. Ponte Castañeda Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA Kumbakonam Rajagopal Department of Mechanical Engineering, University of Texas A&M, College Station, Austin, TX, USA Ariel Ramírez-Torres Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Reinaldo Rodríguez-Ramos Departamento de Matemáticas, Facultad Matemática y Computación, Universidad de La Habana, Havana, CP, Cuba

de

Federico J. Sabina Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, CDMX, Mexico Morteza H. Siboni Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA Leon Sprave Department of Mechanical Engineering, Institute of Mechanics, Dortmund, Germany David J. Steigmann Department of Mechanical Engineering, University of California, Berkeley, CA, USA Alan Wineman University of Michigan, Ann Arbor, MI, USA

Basic Equations of Continuum Mechanics José Merodio and Raymond Ogden

Abstract This first chapter of the volume on Constitutive Modelling of Solid Continua sets out briefly the main concepts of general continuum mechanics without reference to specific material behaviour as a backdrop for the detailed descriptions of different types of material behaviour that are contained in the remaining chapters. The focus is therefore on the fundamental ideas of kinematics of a continuum (deformation and motion), the global balance equations governing the motion of a general continuum, stress and the energy balance equation and the derivation of the local governing equations, including mass conservation and the equation of motion. For the most part, detailed proofs are not provided and reference is made to standard texts for more details since this volume is aimed at researchers who have a reasonable background in continuum mechanics and wish to widen the scope of their knowledge in different areas of solid mechanics.

1 Deformation and Motion of a Solid Continuum We consider a solid body B points of which (material points) are in one-to-one correspondence with points of a region B in three-dimensional Euclidean point space, which is referred to as a configuration of B. The points of B are also referred to as particles. Let t ∈ I ⊂ R denote time, where I is an interval in R. Under the action of body and/or surface forces, B deforms and the configuration depends on time, and this dependence is indicated by the subscript t for each t ∈ I via the notation

J. Merodio Department of Continuum Mechanics and Structures, Escuela de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain e-mail: [email protected] R. Ogden (B) School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_1

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J. Merodio and R. Ogden

Fig. 1 Depiction of the reference configuration Br and current (deformed) configuration Bt . Material points in Br are identified by their position vectors X relative to the origin O of rectangular Cartesian axes Eα , α = 1, 2, 3, and, after deformation, points X have position vectors x relative to the same origin with coincident rectangular Cartesian axes ei , i = 1, 2, 3

−1

.

.

x

X Bt

Br e2 E2 E3

E1

e1

O, o

e3

Bt , which is referred to as the current or deformed configuration. As B deforms in time, the set of configurations {Bt : t ∈ I } describes a motion of the body B. We identify a specific configuration, denoted by Br , as a reference configuration, from which other configurations can be tracked in terms of the position vectors X of points in Br relative to an arbitrarily chosen origin O, as indicated in Fig. 1. Under the motion, the point X becomes the position vector x in Bt relative to the origin o, which here is taken to coincide with O. For many purposes, it is convenient to make use of rectangular Cartesian coordinate systems with basis vectors {Eα } and {ei } for the configurations Br and Bt , respectively, as indicated in Fig. 1, with X having material coordinates X α , α = 1, 2, 3, and x having spatial coordinates xi , i = 1, 2, 3. Thus, relative to the coincident origins O and o, we have (1) X = X α Eα , x = xi ei , wherein the Einstein summation convention over repeated indices is used. Note that Greek and Roman indices are used for the reference and current configurations, respectively. In Fig. 1, Eα and ei are shown to coincide; but in general, this need not be the case. The time-dependent deformation or motion of B from Br to Bt is defined by the (bijection) mapping function χ, with x = χt (X) ≡ χ(X, t) for all X ∈ Br , t ∈ I,

(2)

X = χ−1 t (x) for all x ∈ Bt and each t ∈ I.

(3)

which has inverse

Basic Equations of Continuum Mechanics

3

Note that χ(X, t) is required to have sufficient regularity for the requirements of the subsequent analysis, and this often entails χ(X, t) being once or twice continuously differentiable with respect to position and time. The physical properties of the body B are represented by scalar, vector and tensor fields. These may be expressed in either of the two equivalent ways: in the Lagrangian (referential or material) description relative to Br , with X and t as the independent variables, or in the Eulerian (spatial) description relative to Bt with x and t as independent variables. For example, a scalar function φ(x, t) defined on Bt in the Eulerian description can be considered in the Lagrangian description by using (2) to define the function Φ(X, t) through φ(x, t) = φ[χ(X, t), t] ≡ Φ(X, t). (4) Conversely, by using the inverse (3), we can reverse the process to obtain Φ(X, t) = Φ(χ−1 t (x), t) = φ(x, t).

(5)

The two representations are equivalent, and this applies similarly to vector and tensor functions.

1.1 Derivatives of the Motion We now examine both time and spatial derivatives of scalar, vector and tensor functions in both the Lagrangian and Eulerian settings, beginning with the definition of the material time derivative.

1.1.1

The Material Time Derivative

The material time derivative of a scalar, vector or tensor function is its rate of change at fixed X. It determines how that function changes for the material particle that was at X in the reference configuration. When applied to the deformation χ(X, t), it yields the velocity v of the point at time t, i.e. v ≡ x,t =

∂ χ(X, t), ∂t

(6)

where a subscript t following a comma represents the material time derivative. Similarly, the acceleration a of the particle is a ≡ v,t ≡ x,tt =

∂2 χ(X, t). ∂t 2

(7)

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J. Merodio and R. Ogden

For the functions given in (4), we have, on use of the chain rule, φ,t =

∂ ∂ Φ(X, t) = φ(x, t) + x,t · gradφ(x, t), ∂t ∂t

(8)

where grad denotes the gradient operator with respect to x, and hence, by (6), we have the equivalent descriptions ∂ ∂ Φ(X, t) = φ(x, t) + v · grad φ(x, t) ∂t ∂t       Lagrangian description

(9)

Eulerian description

of the material time derivative, within which the scalar functions φ and Φ may be replaced by vector or tensor functions in the appropriate Eulerian and Lagrangian representations.

1.1.2

Spatial Derivatives and the Deformation Gradient

Let Grad , as distinct from grad , denote the gradient operator with respect to X. Then, the deformation gradient is the non-singular two-point second-order tensor, denoted by F, defined by F(X, t) = Grad x ≡ Grad χ(X, t), (10) with J ≡ det F > 0. With respect to the two sets of Cartesian basis vectors, F has components Fiα , and F=

∂xi ei ⊗ Eα , ∂ Xα

Fiα =

∂xi , ∂ Xα

(11)

where ⊗ signifies the tensor product and xi = χi (X, t), i = 1, 2, 3. Locally, at X, infinitesimal line elements of material dX are transformed linearly by F under the deformation into line elements dx of the same material at x according to (12) dx = FdX, dxi = Fiα dX α , with the inverse transformation dX = F−1 dx and F−1 = grad X, in which X is given by (3), in components (F−1 )αi = ∂ X α /∂xi . While line elements are one-dimensional objects, area elements are two dimensional. Let d A denote an area element of a surface in Br which transforms into da in Bt under the deformation, and denoted by N and n associated unit normal vectors. The area elements are related by Nanson’s formula, which states that nda = J F−T Nd A,

(13)

Basic Equations of Continuum Mechanics

5

where F−T = (FT )−1 = (F−1 )T , the superscript T signifying the transpose (of a second-order tensor). A volume element dV at X transforms into dv at x according to the simple formula dv = J dV.

(14)

The transformations of the line, area and the volume elements are useful, in particular, for transforming between Eulerian and Lagrangian forms of integrals, as discussed in Sect. 1.2. Since, from (14), J relates volume elements under the deformation, J = 1 if there is no change in volume in a particular deformation, in which case the deformation is said to be isochoric. The idealization of incompressibility requires that all deformations be isochoric, and then the constraint J ≡ det F = 1

(15)

holds at every point of the (incompressible) material. Suppose that φ(x, t) is the Eulerian form of the scalar function considered in (4), with Lagrangian counterpart Φ(X, t). Then, the respective gradients are related by Grad Φ = FT gradφ = (grad φ)F,

∂Φ ∂φ = Fiα . ∂ Xα ∂xi

(16)

This formula also applies when the scalar is replaced by a vector or tensor, but requires definitions of the gradient of a vector and a tensor. For example, let a(x, t) and τ (x, t) be the Eulerian versions of a vector function and second-order tensor function, respectively. In component form, a = a p e p , τ = τ pq e p ⊗ eq , their gradients are defined by grad a =

∂τ pq ∂a p e p ⊗ eq , grad τ = e p ⊗ eq ⊗ er , ∂xq ∂xr

(17)

these being second- and third-order tensors, respectively. Note that these definitions are not unique. For example, grad a may be defined as the transpose of the form given above. When contracted grad a becomes the scalar div a = ∂a p /∂x p but there are two alternative definitions of the contraction of grad τ which lead to two different definitions of div τ . These are, in component form, divτ =

∂τq p ∂τ pq eq , div τ = eq . ∂x p ∂x p

(18)

These are identical when τ is symmetric, but otherwise care should be exercised in distinguishing between them when div τ is encountered.

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Other Measures of Deformation Two important symmetric tensors are associated with F. These are the right and left Cauchy–Green deformation tensors, defined respectively by C = FT F, B = FFT ,

(19)

for which det C = det B = J 2 . Let M be the unit vector in the direction of the line element dX. We now define the stretch λ(M) in this direction as the ratio of deformed to undeformed length of the line element, i.e. |dx|/|dX|. With dX = M|dX|, we obtain |dx|2 = (FM) · (FM)|dX|2 = (FT FM) · M|dX|2 = (CM) · M|dX|2 , and hence λ(M) =

|dx| = [M · (CM)]1/2 , |dX|

(20)

(21)

the stretch in the direction M at X. We require that 0 < λ(M) < ∞ for all unit vectors M, and λ(M) = 1 if the direction M is unstretched. Similarly, from Nanson’s formula (13), we obtain da 2 = J 2 (F−T N) · (F−T N) = J 2 N · (C−1 N) = N · (C∗ N), d A2

(22)

where C∗ = J 2 C−1 is the adjugate of C.

1.1.3

The Polar Decomposition Theorem

A valuable aid in the local analysis of deformation is the polar decomposition theorem, which, since det F > 0, enables F to be decomposed as F = RU = VR,

(23)

where R is a proper orthogonal tensor and U and V are positive definite, symmetric tensors. Each of these decompositions is unique, and it follows from (19) that C = U2 , B = V2 .

(24)

Since U is positive definite and symmetric, there exist positive eigenvalues, denoted by λi , and (unit) eigenvectors, denoted by u(i) , such that Uu(i) = λi u(i) , i = 1, 2, 3, and U has the spectral decomposition

Basic Equations of Continuum Mechanics

U=

7

3 

λi u(i) ⊗ u(i) .

(25)

i=1

The eigenvalues λi are the principal stretches of the deformation and u(i) are the (Lagrangian) principal directions. Note that the terminology principal stretch is tied to the general definition of stretch (21) through the connection λi = λ(u(i) ). The tensor V also has eigenvalues λi while its eigenvectors are denoted by v(i) , the Eulerian principal directions, and given by v(i) = Ru(i) , and it has spectral decomposition 3  V= λi v(i) ⊗ v(i) . (26) i=1

The tensors U and V are called the right and left stretch tensors, respectively.

Deformation Versus Strain There is a subtle distinction between measures of deformation and measures of strain, which is often overlooked in the literature. If the deformation is measured from the reference configuration Br , then in this configuration (second-order) tensor measures of deformation, such as F and C, reduce to the identity tensor I while strain measures vanish. Examples of strain tensors are E(m) =

1 m (U − I) m = 0, E(0) = ln U. m

(27)

For m = 2, we obtain the so-called Green or Green–Lagrange strain tensor E(2) =

1 2 1 (U − I) = (C − I), 2 2

(28)

often denoted by E. Note that vanishing of strain still allows the possibility of a pure rotational ‘deformation’ R.

1.1.4

Time Dependence of the Deformation Gradient

We now consider how the deformation gradient and related quantities change in time. From Eqs. (6) and (10), we obtain F,t ≡

∂ F(X, t) = Grad v = (gradv)F, ∂t

(29)

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where the Eulerian tensor grad v is the velocity gradient tensor, which is denoted by L, so that ∂vi L = gradv, L i j = . (30) ∂x j This yields the important formula F,t = LF.

(31)

From (31) and FF−1 = I, it follows that  −1  ∂ F ,t ≡ (F−1 ) = −F−1 L. ∂t

(32)

We also have the result J,t =

∂ (det F) = (det F)tr(F−1 F,t ) = J tr(L) = J div v, ∂t

(33)

which shows that div v is a measure of the rate at which volume changes during the motion. In particular, for an isochoric motion J = 1, J,t = 0 and hence div v = 0. The proof of (33) is fairly lengthy and can be found in standard texts such as [1–4].

1.2 Transformations of Integrals Between Configurations As a prelude to the discussion of global balance laws, we now summarize relevant forms of the divergence theorem and their consequences, and the time derivatives of integrals defined in Eulerian form.

1.2.1

The Divergence Theorem

We begin by noting the divergence theorem for scalar and second-order tensor functions φ and τ , respectively, for the configuration Bt and its boundary ∂Bt in the form gradφ dv = φn da, gradτ dv = τ ⊗ n da, (34) Bt

∂Bt

∂Bt

Bt

where, in component form, grad τ is given by (17)2 . In respect of the definition (18)1 , the latter becomes div τ dv = τ T n da, (35) Bt

∂Bt

while for the definition (18)2 , the transpose is omitted.

Basic Equations of Continuum Mechanics

9

By using the connection dv = J dV between the volume elements, Nanson’s formula, and an application of the divergence theorem in the reference configuration, we obtain T T −T J div τ dV = τ n da = Jτ F N dA = Div(J F−1 τ )dV. (36) ∂Bt

Br

∂Br

Br

This applies for an arbitrary Br , with boundary ∂Br , and hence, provided the integrands are continuous, we obtain an identity and its kinematical specialization for τ = I, namely (37) J div τ = Div(J F−1 τ ), Div(J F−1 ) = 0, where Div is the divergence operator in Br . Equivalently, by defining T = J F−1 τ , we have (38) Div T = J div (J −1 FT), div (J −1 F) = 0.

1.2.2

Time Derivatives of Integrals

In developing the equations governing the motion of a continuous body, it is often required to form time derivatives of various integrals defined over a moving region. The easiest way to evaluate such terms is by making use of the formulas (12)–(14) for line, surface and volume elements in order to map the integrals back from Bt to Br , that is to the corresponding fixed regions in the reference configuration. These are then evaluated by using the formulas (31)–(33) after which the integrals are mapped back to the configuration Bt , yielding the so-called transport formulas. We denote curves, surfaces and regions (volumes) within Bt by Ct , St and Rt , respectively. Then, the following results are easily established. First, for a scalar function φ, we have d dt d dt



φ dx =

(φ,t dx + φ L dx),

Ct

(39)

Ct



φ n da = St

{[φ,t + φ tr(L)] n − φ LT n} da,

(40)

St

d dt



φ dv = Rt

[φ,t + φ tr(L)] dv,

(41)

(a,t + LT a) · dx,

(42)

Rt

and for a vector function a d dt d dt



a · dx = Ct

Ct



a · n da = St

[a,t + tr(L) a − La] · n da, St

(43)

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J. Merodio and R. Ogden





d dt

a dv = Rt

[a,t + tr(L) a] dv,

(44)

Rt

and corresponding formulas may be given for a tensor function τ of any order. To illustrate the procedure, we derive the formula (40). By converting the integral over St to its preimage over Sr by means of Nanson’s formula (13) and taking the derivative inside the integral, and then using φ,t = ∂Φ/∂t, the transpose of (32), and (33), we obtain d dt



φ n da = St

Sr

 ∂  J ΦF−T N d A = ∂t



J [φ,t I + φtr(L)I − φLT ]F−T Nd A,

Sr

where I is the identity tensor, and on further use of Nanson’s formula, this gives the result on the right-hand side of (40).

2 Global and Local Equations Governing the Motion We now turn from kinematical considerations to the forces causing the motion, or just deformation when there is no time dependence, beginning with a statement of the global balance equations. These require the definitions of linear momentum and angular momentum, denoted by M(Rt ) and H(Rt ; o), respectively, for an arbitrary region Rt within the configuration Bt , H being related to a fixed origin o. Thus, M(Rt ) =

ρv dv, H(Rt ; o) =

ρ x × v dv,

Rt

(45)

Rt

where ρ(x, t) is the mass density of the material (mass per unit volume in Bt ). We also require expressions for the forces, denoted by F(Rt ), and moments (about o) of the forces, denoted by G(Rt ; o), acting on the region Rt . In the absence of intrinsic body and surface couples, these are



F(Rt ) =

ρb dv + Rt

∂Rt

t(n) da, G(Rt ; o) =

ρ x × b dv +

Rt

∂Rt

x × t(n) da,

(46) where b is the body force per unit mass and t(n), the so-called stress vector, is the surface force per unit area of the boundary ∂Rt of Rt , the dependence of which on the unit outward normal n is being indicated. These are embodied in the global balance equations dM = F, dt

dH = G, dt

(47)

Basic Equations of Continuum Mechanics

11

also known as Euler’s laws of motion, the first of which is Newton’s second law for a deformable continuum, while the second, unlike the situation in classical particle and rigid body mechanics, is independent of the first. The equations in (47) are now written explicitly as d dt d dt







ρv dv =

ρb dv +

Rt

Rt

t(n) da,







ρx × v dv =

ρx × b dv +

Rt

(48)

∂Rt

Rt

∂Rt

x × t(n) da.

(49)

These are the equations of global linear and angular momentum balance. In order to reduce these subsequently to local form, we shall make use of mass conservation, which we now discuss. The total mass m(Rt ) within the region Rt may be defined either in terms of the density ρ(x, t) per unit volume in Rt or, since mass must be conserved during the motion, in terms of the density ρr (X) per unit volume in Rr , the preimage of Rt in Br . Thus, the global form of the mass conservation equation is written as m(Rt ) =

ρ dv =

ρr dV.

Rt

(50)

Rr

Since dv = J dV and Rt is arbitrary, it follows that the equation of mass conservation has the local form (51) ρr = J ρ, so that ρ dv = ρr dV . Equation (51) can also be expressed in rate form by recalling that J,t = J div v, and that ρr is independent of t, so that the material time derivative of (51) gives ρJ,t + ρ,t J = 0. Hence, we obtain an alternative form of the local conservation of mass equation: ρ,t + ρdiv v = 0.

(52)

By converting the left-hand side of (48) to the reference configuration and then converting back to Rt after differentiation, we obtain d dt

ρv dv = Rt

d dt







ρr v dV =

ρr v,t dV =

Rr

Rr

ρv,t dv,

(53)

Rt

and similarly in respect of (49): d dt



ρx × v dv = Rt

ρx × v,t dv. Rt

(54)

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J. Merodio and R. Ogden

Equations (48) and (49) are then reduced to

ρ(v,t − b) dv = Rt



ρx × (v,t − b) dv =

t(n) da, ∂Rt

Rt

∂Rt

x × t(n) da.

(55) To obtain the local forms of these equations, we use a result due to Cauchy which asserts that t(n) depends linearly on n, as we now discuss. The proof of this result can be found in standard texts such as [2, 4, 5].

2.1 Cauchy’s Theorem and Cauchy Stress Let t(n) and b be a pair of surface and body forces defined on ∂Rt and Rt , respectively, during a motion. Cauchy’s theorem states that there exists a second-order tensor, called the Cauchy stress tensor, in general a function of position and time but independent of n, here denoted by σ, such that t(n) = σ T n,

(56)

at each point of (arbitrary) ∂Rt where the unit normal is n. Moreover, the local form of the linear momentum balance equation (55)1 is divσ + ρb = ρv,t ,

(57)

which is also referred to as the equation of motion. Once (57) is established, the balance of angular momentum equation (55)2 reduces to the symmetry of σ since couple stresses are assumed here to be absent. Thus, σ T = σ.

(58)

To obtain (57), we first substitute t(n) = σ T n into (55)1 and then apply the divergence theorem (35) to the surface integral, yielding

ρ(v,t − b) dv = Rt

σ n da =

div σ dv,

T

∂Rt

(59)

Rt

and then, since Rt is arbitrary, and provided the above integrands in the volume integrals are continuous, Eq. (57) follows. Next, substitute t(n) = σ T n into (55) and use (57) to give

x × (divσ) dv = Rt

∂Rt

x × (σ T n) da,

(60)

Basic Equations of Continuum Mechanics

13

which can be rewritten as  x ⊗ (div σ) dv = 

∂Rt

Rt

x ⊗ (σ T n) da,

(61)

where  is the alternating tensor. Note that when applied to a second-order tensor such as σ, we define the operation σ in component form by (σ)i = εi jk σ jk with the usual summation convention applying. Application of the divergence theorem to the right-hand side of the previous equation yields





x ⊗ (divσ) dv = 

 Rt



[div(σ ⊗ x)]T dv =  Rt

x ⊗ (div σ) dv +  Rt

σ dv, Rt

(62)

in which the identity grad x = I has been used. Thus, σ dv = 0,



(63)

Rt

and hence, since Rt is arbitrary, σ = 0 locally, i.e. σ is symmetric and (58) is established.

2.2 Energy Balance During a motion of the region Rt , the external forces (body forces and surface tractions) do work on the region, and the rate of working, or power, thus developed, denoted by P(Rt ), is given by



P(Rt ) =

ρb · v dv + Rt

∂Rt

t(n) · v da.

(64)

This results in the production of kinetic energy, denoted by K (Rt ), defined as K (Rt ) =

1 2

ρv · v dv,

(65)

Rt

and additionally stored and/or dissipated energy, denoted by S(Rt ), which is defined below. The following standard procedure leads to the energy balance equation. By using t(n) = σn in (64), we obtain, using the symmetry of σ followed by an application of the divergence theorem and use of (57),

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J. Merodio and R. Ogden

P(Rt ) =

Rt



Rt

= =

Rt

ρb · v dv + ρb · v dv +

∂Rt

t(n) · v da =

Rt

ρb · v dv +



(σn) · v da

[ρb · v + div (σv)] dv [ρb · v + (div σ) · v + tr (σL)] dv = [(ρb + div σ) · v + tr (σD)] dv ∂Rt

(σv) · n da =

∂Rt

Rt

R

t 1 ∂ [ρv · v,t + tr (σD] dv = ρ (v · v) dv + tr (σD) dv = 2 Rt ∂t Rt Rt ∂ 1 ρr (v · v) dV + tr (σD) dv = 2 Rr ∂t Rt 1 d = ρr (v · v) dV + tr (σD) dv, 2 dt Rr Rt



which yields the energy balance equation P(Rt ) =

d K (Rt ) + S(Rt ), dt

(66)

where S(Rt ) and D are defined by S(Rt ) =

tr(σD) dv, D = Rt

1 (L + LT ). 2

(67)

The term S(Rt ) gives the internal stress rate of working on the region Rt , and may be developed in terms of different tensor measures of stress, which are discussed in the following.

2.3 Stress Tensors and Conjugacy The surface force on the boundary ∂Rt is given in (46). Locally on an area element da where the unit normal is n, the traction is t(n)da = σnda, and by use of Nanson’s formula (13) this can be rewritten as J σF−T Nd A = ST Nd A, where ST is the first Piola–Kirchhoff stress tensor, the transpose S of which is known as the nominal stress tensor and defined as (68) S = J F−1 σ, and from the symmetry of σ, it follows that FS = ST FT . By using (37)1 , with τ replaced by σ and the connection (51), the equation of motion (57) can equally be expressed in terms of S as (69) DivS + ρr b = ρr v,t . The expression S(Rt ) can be written with respect to the reference configuration, and then, by use of the symmetry of σ together with (31) and (68), we obtain

Basic Equations of Continuum Mechanics





S(Rt ) =

tr(σD) dv = Rt

15

tr(σL) dv =

Rt

J tr (σL) dV =

Rr

tr(SF,t ) dV. Rr

(70) The integrand in the latter two integrals of (70) is the rate of working of the stresses per unit reference volume, also referred to as the stress power. Next, we note from (19)1 , (28), (31) and (67)2 that E(2) =

1 T 1 (F F − I), E,t(2) = (FT F,t + F,tT F) ≡ FT DF. 2 2

(71)

Hence, by defining T(2) = SF−T = J F−1 σF−T , the so-called second Piola–Kirchhoff stress tensor, which is symmetric, we obtain tr(SF,t ) = tr(T(2) E,t(2) ).

(72)

Each of the pairs (S, F) and (T(2) , E(2) ) are work conjugate variables since each of the expressions in (72) delivers the stress power (internal stress work-rate density). Similarly, pairs of conjugate (stress and strain) variables (T(m) , E(m) ) may be defined based on the definitions (27) although only a limited number of the stress tensors T(m) are of practical use. One example is that corresponding to m = 1, for which the stress tensor T(1) , known as the Biot stress tensor, is conjugate to the strain tensor E(1) = U − I and defined by T(1) =

1 (2) 1 (T U + UT(2) ) = (SR + RT ST ), 2 2

(73)

where the connections FT F = U2 and E,t(2) =

1 (UU,t + U,t U), tr(T(2) E,t(2) ) = tr(T(2) UU,t ) = tr(T(1) U,t ) 2

have been used. We thus have

tr(SF,t ) = tr(T(m) E,t(m) )

(74)

(75)

for any real value of m, and we emphasize that this definition is independent of the type of material considered, as is the equation of motion (57) or its equivalent (69). Different types of material response are distinguished by different forms of the dependence of (some measure of) the stress, and the expression for S(Rt ) is an important starting point for the analysis of this distinction.

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J. Merodio and R. Ogden

3 Summary of the Governing Equations and the Need for Constitutive Laws The equations that govern the motion of a continuum are the equations of mass conservation and the equation of motion ρ,t + ρdiv v = 0, divσ + ρb = ρv,t ,

(76)

which together provide four equations. With σ symmetric here and the body force b considered to be known, there are ten unknowns to be determined, i.e. the scalar ρ, the vector v (or equivalently the function χ) and the tensor σ. The deficit of six equations is provided by, for example, specifying σ in terms of the motion through various quantities, such as X, x, v, F, L. Such a specification contributes to a constitutive law, which enables different types of material response to be distinguished. The subsequent chapters of this volume detail such constitutive laws for an extensive range of the mechanical behaviours of solid materials.

References 1. Chadwick P (1999) Continuum mechanics: concise theory and problems. Dover Publications, New York 2. Gurtin ME (1981) An introduction to continuum mechanics. Academic Press, New York 3. Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, Chichester 4. Ogden RW (1997) Non-linear elastic deformations. Dover Publications, New York 5. Spencer AJM (2004) Continuum mechanics. Dover Publications, New York

Finite Deformation Elasticity Theory José Merodio and Raymond Ogden

Abstract This chapter provides the framework for the development of constitutive theories of solids by focusing on constitutive laws for nonlinearly elastic solids. These exemplify the general principles of constitutive theory that should be applied to all types of material behaviour, in particular, the notions of objectivity and material symmetry, including the important symmetries of isotropy, transverse isotropy and orthotropy based in part on deformation invariants. Details are given for the various general stress–deformation relations for each case of symmetry in respect of hyperelastic materials (which are characterized by a strain-energy function), with or without an internal constraint such as incompressibility, and these are illustrated by particular prototype models. The notion of residual stress (in an unloaded configuration) is discussed and the form of strain-energy function required to accommodate residual stress in the material response is developed.

1 General Theory of Elasticity For the basic notation required in this chapter, we refer to the chapter “Basic Equations of Continuum Mechanics” [20]. It is given that the deformation gradient relative to the reference configuration Br is F and the Cauchy stress tensor in the deformed configuration Bt is σ (when there is no time dependence, the subscript on Bt may be omitted). As in the chapter “Basic Equations of Continuum Mechanics” [20], attention is restricted here to symmetric σ. Then, a general homogeneous elastic material is characterized by the fact that σ depends only on F, i.e. it is independent of J. Merodio (B) Department of Continuum Mechanics and Structures, Escuela de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, Spain e-mail: [email protected] R. Ogden School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_2

17

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J. Merodio and R. Ogden

the path of deformation taken to reach F. This is expressed by writing the constitutive equation in terms of a symmetric tensor-valued function, here denoted by g, in the form σ = g(F), (1) g being defined on the space of deformation gradients (relative to Br ). Equation (1) characterizes a Cauchy elastic material, which need not be associated with the existence of a strain-energy function, but suffices for the development of the notions of objectivity and material symmetry without the need for the introduction of a strainenergy function, which will be delayed until Sect. 2 and remain the focus thereafter. The function g is referred to as the response function for the Cauchy stress of the material relative to Br . We emphasize that the form of g is specific to the configuration Br , and a change in reference configuration will in general entail a change in the form of g, as discussed in Sect. 1.2 below. Thus, σ at a point X ∈ Br depends only on the value of F at X and depends on X only through F (for a homogeneous material). For an inhomogeneous material, σ depends separately on X in addition. For the most part, however, we shall confine attention to homogeneous materials and hence explicit dependence on X is not needed. In the configuration, Br , we have F = I and from (1) the Cauchy stress becomes g(I), which vanishes if Br is stress free. More generally, if Br is residually stressed and the residual stress tensor is denoted by τ , then g(I) = τ .

(2)

The presence of residual stress in Br has a strong influence of the elastic response of the material relative to Br , and this will be examined in Sect. 4. In particular, we note that a residual stress τ is necessarily non-homogeneous and a residually stressed material is also non-homogeneous. However, for our initial development, we assume that there is no residual stress and hence g(I) = O,

(3)

where O is the zero tensor.

1.1 Objectivity An important consideration in deriving material constitutive equations is that the properties of the material should not be dependent on the observer. This is known as material frame indifference or more simply as objectivity, which is the term we shall use here. This concept applies to all scalar, vector and tensor functions associated with the material, and we now examine how such quantities must transform under a change of observer in order for this requirement to be met. Let s, v, T, respectively,

Finite Deformation Elasticity Theory

19

be scalar, vector and (second-order) tensor functions of x and t defined on Bt , i.e. they are Eulerian in character. Now consider the motion x = χ(X, t) with deformation gradient F relative to Br and superimpose on this a rigid-body motion x∗ = Q(t)x + c(t),

(4)

where Q(t) is a proper orthogonal tensor (representing a rotation) and the vector c(t) is a translation. This takes Bt to the configuration B∗t so that the functions s, v, T become s ∗ , v∗ , T∗ when defined on B∗t . The functions are said to be objective if, for all motions of the form (4), s ∗ = s, v∗ = Qv, T∗ = QTQT .

(5)

Instead of being considered as a superimposed rigid motion, Eq. (4) can be thought of as an observer transformation, a subtle distinction that need not concern us here. See, for example, [34] for a full discussion of observer transformations and material frame indifference. Note that the material time derivatives of v and T do not satisfy the objectivity requirements (5)2,3 . The deformation gradient F∗ associated with the motion (4) is related to F by F∗ = QF,

(6)

and, for an elastic material with response function g, the stress tensor, σ ∗ say, associated with the deformation gradient F∗ is σ ∗ = g(F∗ ) = g(QF).

(7)

For σ to be objective, it must satisfy σ ∗ = QσQT ,

(8)

and as a result the response function, g must then satisfy the restriction g(QF) = Qg(F)QT

(9)

for all rotations Q at each deformation gradient F. This restriction ensures that the constitutive law (1) is objective, i.e. the description of the material properties is independent of the superimposed rigid motions (equivalently, independent of observer). Note that J = det F satisfies (5)1 and acts as a pullback of the density ρ from Bt to its Lagrangian version ρr in Br according to ρr = J ρ, as in Eq. (51) in the chapter “Basic Equations of Continuum Mechanics” [20]. Lagrangian versions of v and T may also be formed by various pullbacks based on F, and objectivity can then be cast in Lagrangian form. For example, let vr = FT v be a Lagrangian vector. Then, it is easily seen that vr∗ = F∗T v∗ = vr , and similarly for the alternative pullback

20

J. Merodio and R. Ogden

definition vr = F−1 v, we obtain vr∗ = vr . For the Lagrangian tensor Tr = F−1 TF−T , we obtain T∗r = Tr . In each case, the Lagrangian quantities are unaffected by the transformation (4), and this provides a useful alternative definition of objectivity. It follows, in particular, that the material time derivatives of these Lagrangian objective quantities are themselves objective, unlike their Eulerian counterparts.

1.2 Material Symmetry Material symmetry is a property possessed by a material in a reference configuration, such as Br , and the symmetry is in general different for different choices of reference configuration. Here, we restrict attention to reference configurations that are stress free. The Cauchy stress σ in a configuration Bt is independent of the choice of reference configuration and we now consider the consequences of two different choices of reference configuration, say Br and Br . Let F and F be the deformation gradients from Br and Br , respectively, to Bt , and denote by F¯ the deformation gradient that takes points of Br to points of Br , as depicted in Fig. 1. Then ¯ F = F F.

(10)

Let g and g denote the response functions of the material relative to Br and Br , respectively. Then, (11) σ = g(F) = g (F ). In general, g and g are distinct so that the response of the material relative to Br is different from that relative to Br , but when the material has some symmetry, g and ¯ Then, g may coincide for a certain set of deformation gradients F. g(FF¯ −1 ) = g(F)

(12)

for all deformation gradients F and for all F¯ within the set, and Eq. (11) specializes to

Fig. 1 Depiction of paths of deformation from the reference configurations Br and Br to the configuration Bt with deformation gradients F and F , respectively, with F¯ being the deformation gradient from Br to Br

Finite Deformation Elasticity Theory

21

σ = g(F) = g(FF¯ −1 ).

(13)

Thus, σ may be calculated by using any (stress-free) reference configuration. The set of tensors F¯ for which (12) holds defines the symmetry of the material relative to Br . Now, we reserve the notation F¯ for the specific deformation gradient relating Br to Br and use the notation P instead of F¯ −1 in (13). Let G denote the set of P for which g(FP) = g(F) (14) for all deformation gradients F. The set G of such tensors forms a multiplicative group, which is referred to as the symmetry group of the material relative to Br . In general, the symmetry group changes with a change in reference configuration, and this is quantified by Noll’s rule, which we now discuss briefly.

1.2.1

Noll’s Rule

Now consider F¯ again to denote the deformation gradient from Br to Br , but not in general a member of the symmetry group G relative to Br . Then, Noll’s rule states that the symmetry group G relative to Br is related to G by ¯ F¯ −1 . G = FG

(15)

This is easily established by using (11) and (14). For P ∈ G, we have ¯ ¯ F¯ −1 F) ¯ = g (F FP ¯ F¯ −1 ), = g(F FP g (F ) = g(F) = g(FP) = g(F FP)

(16)

¯ F¯ −1 . It follows that P ∈ G if and wherein the final step F in (11) is replaced by F FP −1  ¯ ¯ only if FP F ∈ G , and this result is represented in the form (15). If F¯ is a member of the symmetry group G, then Noll’s rule specializes to G = G.

1.2.2

Isotropy and Transverse Isotropy

In the important special case for which G consists of the full proper orthogonal group, the material is said to be isotropic relative to Br , and then g is restricted according to g(FP) = g(F)

(17)

for all proper orthogonal P (and for each deformation gradient F). In this case, the response of a small specimen of material in Br is independent of its orientation in Br . A less restrictive example is that of transverse isotropy, for which there is locally a distinguished (or preferred) direction defined by a unit vector in Br , say M. Then (17) holds, but only for those proper orthogonal P such that PM = ±M, i.e. for arbitrary

22

J. Merodio and R. Ogden

rotations about M or rotations through π which take M to −M. The pushforward of M from Br to Bt , denoted by m (= FM) must be independent of the choice of reference configuration, and hence F M = FM, where F = FF¯ and M is the counterpart of M in Br . Hence, M = F¯ −1 M and thus M = ±M if F¯ is a member of the symmetry group. In order to pursue the concepts of isotropy and transverse isotropy, we require a number of results from tensor algebra, which are described in the following section.

1.3 Scalar and Tensor Functions of a Second-Order Tensor Consider a second-order tensor T. Its characteristic equation is det (T − tI) = 0, a cubic equation in t for which the three solutions are the eigenvalues of T. An important property that such a tensor satisfies is the Cayley–Hamilton theorem, which states that T satisfies its own characteristic equation. Thus, T3 − I1 (T)T2 + I2 (T)T − I3 (T)I = O,

(18)

where the scalar functions I1 (T), I2 (T), I3 (T) are the principal invariants of T defined by I1 (T) = trT,

I2 (T) =

1 [I1 (T)2 − tr(T2 )], 2

I3 (T) = det T,

(19)

which are respectively first, second and third order in the components of T for any choice of basis. They are examples of scalar invariants, which are important in the construction of constitutive equations, and a general definition of a scalar invariant is therefore now given in respect of symmetric tensors T.

1.3.1

Scalar Invariants

Suppose that α is a scalar function of a symmetric second-order tensor T. Then, α is said to be an isotropic function of T if it satisfies the restriction α(QTQT ) = α(T)

(20)

for all orthogonal tensors Q (whether proper or improper orthogonal). Such an isotropic function is also called a scalar invariant of T. This definition is clearly satisfied for the scalar invariants defined in (19), as, for example, from the properties of trace and the orthogonality of Q, tr (QTQT ) = tr(TQT Q) = trT, and likewise for I2 (T). Similarly, from the properties of determinant and orthogonality, det(QTQT ) = (det Q)(det T) det(QT ) = (det T)(det Q)2 = det T.

Finite Deformation Elasticity Theory

23

A key property is that α(T) is a scalar invariant if and only if it is expressible as a function of I1 (T), I2 (T), I3 (T), or equivalently in terms of three alternative independent invariants of T. Turning to transverse isotropy, suppose again that M is a unit vector and that α now depends on the two tensors T and M ⊗ M, the latter being indifferent to the sense of M. Then, α is said to be an isotropic function (or scalar invariant) of the pair of tensors if it satisfies the restriction α(QTQT , QM ⊗ QM) = α(T, M ⊗ M)

(21)

for all orthogonal tensors Q, where we note that QM = MQT . This restriction is satisfied if and only if α(T, M ⊗ M) is expressible as a function of the invariants I1 (T), I2 (T), I3 (T) and I4 (T), I5 (T) for a fixed M, the latter two being defined by I4 (T) = M · (TM) ≡ tr(TM ⊗ M),

I5 (T) = M · (T2 M) ≡ tr(T2 M ⊗ M). (22) These are sometimes referred to as quasi-invariants but here we just use the term invariants.

1.3.2

Second-Order Tensor Functions

Suppose that G is a symmetric second-order tensor function of the symmetric secondorder tensor T. Then, it is said to be an isotropic tensor function of T if G(QTQT ) = QG(T)QT

(23)

for all orthogonal Q. Special properties of G that follow from this definition are as follows, for proofs of which we refer to, for example, [25]: 1. The eigenvalues of G are scalar invariants of T. 2. The eigenvectors of G(T) are the eigenvectors of T, i.e. G(T) is coaxial with T. 3. G(T) has the representation G(T) = α0 I + α1 T + α2 T2 ,

(24)

where α0 , α1 , α2 are scalar invariants of T, i.e. they are expressible as functions of I1 (T), I2 (T), I3 (T). Extending the notion of an isotropic tensor function to the case of transverse isotropy with fixed unit vector M, a function G of the two tensors T and M ⊗ M is said to be an isotropic function of its arguments if G(QTQT , QM ⊗ QM) = QG(T, M ⊗ M)QT

(25)

24

J. Merodio and R. Ogden

for all orthogonal Q. A function G satisfying this property has the representation G(T, M ⊗ M) = α0 I + α1 T + α2 T2 + α4 M ⊗ M + α5 (TM ⊗ M + M ⊗ TM), (26) where α0 , α1 , α2 , α4 , α5 are functions of the invariants I1 (T), I2 (T), I3 (T) and I4 (T), I5 (T). We refer to the article [28] for full discussion of representations of this type. We now set the above representations in the context of elasticity theory.

1.4 Isotropic and Transversely Isotropic Elasticity From the definition (17) of isotropy, we have σ = g(F) = g(FP) = g(V)

(27)

for all proper orthogonal P and each deformation gradient F, with P then chosen to be RT to obtain the final expression on use of the polar decomposition F = VR. Then, by the objectivity condition (9), g(QF) = Qg(F)QT = Qg(V)QT ,

(28)

and, on applying objectivity also to (27) with P replaced by RT QT and use of F = VR again, we obtain (29) g(QF) = g(QFP) = g(QVQT ) for all proper orthogonal Q. Hence g(QVQT ) = Qg(V)QT

(30)

for all proper orthogonal Q, which shows that g(V) is an isotropic function of V following the definition (23). Thus, the response function g attracts the properties of the isotropic tensor function G defined in Sect. 1.3.2. In particular, for an isotropic elastic material, σ = g(V) is coaxial with V and has the representation σ = g(V) = α0 I + α1 V + α2 V2 ,

(31)

where α0 , α1 , α2 are invariants of V, i.e. functions of I1 (V), I2 (V), I3 (V), according to the definitions (19) with T replaced by V. For the transverse isotropy case, g depends on M ⊗ M as well as F, where M is now a unit vector in the reference configuration Br , and we write this explicitly as σ = g(F, M ⊗ M). Then, by objectivity, since M is defined in the reference configuration, it is therefore unaffected by a rotation in the deformed configuration,

Finite Deformation Elasticity Theory

g(QF, M ⊗ M) = Qg(F, M ⊗ M)QT

25

(32)

for all proper orthogonal Q. To establish the counterpart of (31), we use a slightly different procedure with the polar decomposition F = RU by choosing Q = RT to obtain (33) RT σR = g(U, M ⊗ M). Note that the tensors RT σR, U and M ⊗ M are all Lagrangian. Recalling the definition of the second Piola–Kirchhoff stress tensor T(2) given in Sect. 2.3 of the chapter “Basic Equations of Continuum Mechanics” [20], we obtain T(2) = J F−1 σF−T = J U−1 g(U, M ⊗ M)U−1

(34)

with J = det F = det U. This is a function of U and M ⊗ M, which may equivalently be considered as a function of the right Cauchy–Green tensor C = U2 defined in (19) in the chapter “Basic Equations of Continuum Mechanics” [20] and M ⊗ M. We write this as (35) T(2) = G(C, M ⊗ M). For G to be an isotropic function of C and M ⊗ M according to the definition (23), T(2) has the expansion T(2) = α0 I + α1 C + α2 C2 + α4 M ⊗ M + α5 (M ⊗ CM + CM ⊗ M),

(36)

where α0 , α1 , α2 , α4 , α5 are functions of the invariants I1 (C), I2 (C), I3 (C) and I4 (C), I5 (C). The Cauchy stress tensor is then given by J σ = α0 B + α1 B2 + α2 B3 + α4 m ⊗ m + α5 (m ⊗ Bm + Bm ⊗ m),

(37)

where B = RCRT is the left Cauchy–Green tensor (19) in the chapter “Basic Equations of Continuum Mechanics” [20], and this can be put in standard form by using the Cayley–Hamilton theorem (18) for B to give σ = β0 I + β1 B + β2 B2 + β4 m ⊗ m + β5 (m ⊗ Bm + Bm ⊗ m),

(38)

where β0 = J α2 , β1 = (α0 − α2 I2 )/J, β2 = (α1 + α2 I1 )/J, β4 = α4 /J, β5 = α5 /J, (39) I1 and I2 here being principal invariants of B, equivalently of C. On setting β4 = β5 = 0 and using the Cayley–Hamilton theorem for V, with B = V2 , the isotropic expansion (31) is recovered, with appropriate adjustments in the definitions of the coefficients α0 , α1 , α2 , which are different from those in (31).

26

J. Merodio and R. Ogden

These types of expansions are important for the more detailed discussion of isotropic and transversely isotropic hyperelastic constitutive equations that will be derived in Sects. 3.1 and 3.2. By specializing (38) to the stress-free reference configuration, for which (3) holds, we obtain β0 + β1 + β2 = α0 + α1 + α2 = 0, β4 + 2β5 = α4 + 2α5 = 0,

(40)

only the first of which is relevant in the specialization to isotropy. We now introduce the notion of hyperelasticity.

2 Hyperelasticity Recall, from the Sect. 2.2 of the chapter “Basic Equations of Continuum Mechanics” [20], that the energy balance equation can be written in the form 

 ρb · v dv +

∂Rt

Rt

t(n) · v da =

1 d 2 dt



 ρv · v dv + Rt

tr(σL) dv.

(41)

Rt

We now consider the situation in which there is no dissipation so that the second term on the right-hand side of (41) is associated purely with elastic energy, and we now examine this term in detail. The integrand in this term is the rate of change of elastic energy per unit volume of Rt and, per unit reference volume in Rr is J tr(σL). By using the formula F,t = LF from (31) and the definition (68) of the nominal tensor S in the chapter “Basic Equations of Continuum Mechanics” [20], this can be written as tr(SF,t ). If there exists a scalar function of F, say W , such that W,t = tr(SF,t ), where W,t is the material time derivative of W , then   ∂W F,t = tr(SF,t ), W,t ≡ tr (42) ∂F and provided the components of F,t are independent, this yields the (nominal and Cauchy) stress deformation relations S=

∂W ∂W , σ = J −1 F . ∂F ∂F

(43)

Note that here we are using the convention that in component form gives  Sαi =

∂W ∂F

 αi

=

∂W so that tr(SF,t ) = Sαi Fiα,t . ∂ Fiα

(44)

Finite Deformation Elasticity Theory

27

The function W represents the elastic energy stored in the material per unit volume in Rr , and is referred to as the stored energy function or strain-energy function. For a homogeneous material, W depends on X only through F, but for an inhomogeneous material it depends separately on F and X although in general this will not be shown explicitly. The total elastic energy stored in Rr is 

 W (F) dV = Rr

and

d dt

(45)

Rt







W (F) dV = Rr

J −1 W (F) dv,

W,t dV = Rr

tr(σL) dv.

(46)

Rt

The right-hand side of (41) then has the form d (kinetic energy + elastic stored energy). dt

(47)

The form of elasticity for which there exists a strain-energy function is known as hyperelasticity and sometimes referred to as Green elasticity after the English mathematician George Green (1793–1841) who developed ideas relating to the notion of a strain-energy function (see the article [30] by Spencer in which Green’s contributions to elasticity are discussed). In terms of W , the Cauchy stress response function g introduced in (1) is given by ∂W (F). (48) σ = g(F) = J −1 F ∂F Let h be the corresponding response function associated with the nominal stress S = J F−1 σ. Then (49) S = h(F) = J F−1 g(F). From the objectivity condition (9), it follows that the corresponding objectivity condition for h is (50) h(QF) = h(F)QT for all proper orthogonal Q. Similarly, the condition for isotropy (17) is h(FP) = PT h(F)

(51)

for all proper orthogonal P. From the polar decomposition theorem, it may then be deduced that for an isotropic material h(F) = RT h(V) = h(U)RT

(52)

28

J. Merodio and R. Ogden

and that h(U) is symmetric and equal to the Biot stress tensor T(1) defined in Eq. (73) in the chapter “Basic Equations of Continuum Mechanics” [20]. Thus, in this case, S has the polar decomposition S = T(1) RT but since T(1) is not in general positive definite, this decomposition is not unique, unlike the polar decompositions (23) in the chapter “Basic Equations of Continuum Mechanics” [20], as discussed in [25]. It is usual to assume that there is no energy stored in the reference configuration so that W (I) = 0. Then, for a stress-free reference configuration, we have W (I) = 0,

∂W (I) = O, ∂F

(53)

the latter specializing (3) for hyperelasticity.

2.1 Hyperelastic Objectivity and Material Symmetry For the scalar function W , objectivity requires that it has the same value for any observer and is therefore unchanged by a rigid-body motion that is superimposed after deformation. Thus, this simply requires that W (QF) = W (F)

(54)

for all proper orthogonal Q for any deformation gradient F. On the other hand, if the material is isotropic relative to Br , then W is not affected by any proper orthogonal P applied in Br before deformation, and this requires that W (FP) = W (F)

(55)

for all proper orthogonal P for any deformation gradient F. By using the polar decomposition F = VR and choosing P = RT in the latter equation, we obtain W (F) = W (V), and then (54) gives W (QFP) = W (FP) = W (F) = W (V) for all (independent) proper orthogonal P and Q, and on setting P = RT QT in the left-most term, we obtain (56) W (QVQT ) = W (V) for all proper orthogonal Q (and in this equation, Q is now allowed to be improper orthogonal). According to the definition (20), Eq. (56) shows that W is an isotropic scalar function of V and therefore inherits the properties of such a function given in Sect. 1.3.1. In particular, an isotropic strain-energy function is a function of the three principal invariants of V, i.e. I1 (V) = trV,

I2 (V) =

1 [(trV)2 − tr(V2 )], 2

I3 (V) = det V.

(57)

Finite Deformation Elasticity Theory

29

Alternatively, on use of the polar decomposition F = RU and Q = RT , objectivity yields W (F) = W (QF) = W (U). Then, by combining isotropy and objectivity, we obtain W (U) = W (QFP) = W (PT UP) on setting Q = PT RT . Thus, W is an isotropic function of U, this being equivalent to (56) since V = RURT . Equivalently, W is an isotropic function of C = U2 . For transverse isotropy, we consider W to be a function of C and M ⊗ M, and it is an isotropic function of these if W (PT CP, PT M ⊗ PT M) = W (C, M ⊗ M)

(58)

for all orthogonal P. This means that W can be considered as a function of the invariants of C defined, following (19) and (22), as I1 (C) = trC,

1 [I1 (C)2 − tr(C2 )], 2 I5 (C) = M · (C2 M).

I2 (C) =

I4 (C) = M · (CM),

I3 (C) = det C, (59)

Application of these and other invariants will be examined in detail in Sects. 3.1 and 3.2. We note that objectivity alone requires only that W be a function of U, equivalently a function of C or indeed any appropriate functions of U such as the strain tensors defined in (27) in the chapter “Basic Equations of Continuum Mechanics” [20]. Thus, with reference to conjugacy in Sect. 2.3 of the chapter “Basic Equations of Continuum Mechanics” [20], the general stress tensor T(m) can be written as T(m) =

∂W . ∂E(m)

(60)

In particular, for m = 2 and m = 1, we obtain the second Piola–Kirchhoff and Biot stress tensors as T(2) =

∂W ∂W ∂W ∂W , T(1) = . =2 = ∂E(2) ∂C ∂E(1) ∂U

(61)

Strictly, a different notation should be used for W for each different argument but for now, we just use the notation W for convenience.

2.2 Hyperelasticity with Internal Constraints Many elastic materials can be idealized by constraining the deformation gradient so that not all its components are independent. For example, rubber-like materials and many soft biological tissues can be treated as incompressible, while inextensibility is an approximate feature of strongly reinforced materials such as rubber with

30

J. Merodio and R. Ogden

embedded steel fibres. These are known as internal constraints, and before considering incompressibility and inextensibility explicitly, we discuss a single internal constraint defined by the scalar function C (for constraint) in the form C(F) = 0

(62)

for all deformation gradients F. Such a constraint must be unaffected by a superposed rigid motion, i.e. C must be an objective scalar function, and hence C(QF) = C(F)

(63)

for all proper orthogonal Q. In particular, the choice Q = RT yields C(F) = C(U). Recalling that for the strain-energy function W , we have from (42), W,t = tr(SF,t ), we note that because of the constraint, not all the components of F,t are independent so it cannot be deduced that S is given by (43)1 . However, we also note that   ∂C (64) F,t = 0 C,t = tr ∂F and the expression for S accommodates the constraint by introduction of a Lagrange multiplier, say q, which is arbitrary and independent of F but depends on x in general to give S and the corresponding Cauchy stress as S=

∂W ∂C ∂W ∂C +q , σ = J −1 FS = J −1 F + q J −1 F . ∂F ∂F ∂F ∂F

(65)

For the reference configuration Br to be stress free, we have ∂C ∂W (I) + q0 (I) = O, ∂F ∂F

(66)

where q0 is the value of q in Br .

2.2.1

Incompressibility

The incompressibility constraint is given by (15) in the chapter “Basic Equations of Continuum Mechanics” [20] and thus C and its derivative with respect to F have the forms ∂C = F−1 (67) C(F) = det F − 1 = J − 1 = 0, ∂F on setting J = 1 after differentiation. Thus, S=

∂W ∂W + qF−1 , σ = F + qI. ∂F ∂F

(68)

Finite Deformation Elasticity Theory

31

Note that in this case, q is often written as − p, with p having the interpretation as a hydrostatic pressure. The corresponding second Piola–Kirchhoff and Biot stress tensors are given by T(2) = 2

∂W ∂W + qC−1 , T(1) = + qU−1 , ∂C ∂U

(69)

with C = 2E(2) + I, U = E(1) + I.

2.2.2

Inextensibility

Referring to the definition of stretch in the direction M in Eq. (21) in the chapter “Basic Equations of Continuum Mechanics” [20], i.e. λ(M) = [M · (CM)]1/2 , the material is said to be inextensible in the direction M in Br if λ(M) = 1 for all deformation gradients F. On use of the definition C = FT F, this enables C and its derivative with respect to F to be given as C(F) = M · (CM) − 1 = 0,

∂C = 2M ⊗ FM, ∂F

(70)

for all F, and hence that S=

∂W + 2qM ⊗ FM, ∂F

Jσ = F

∂W + 2qm ⊗ m, ∂F

(71)

where m = FM. Also, we obtain T(2) = 2

∂W ∂W + 2qM ⊗ M, T(1) = + q(M ⊗ UM + UM ⊗ M). ∂C ∂U

(72)

Note that while C(U) = det U − 1 is an isotropic invariant of U, a general C(U) need not be a scalar invariant of U, but, for example, C(F) = M · (CM) − 1 is an isotropic invariant of C and M ⊗ M together.

2.2.3

Volumetric–Isochoric Separation

When working with nearly incompressible materials, it is useful to consider the multiplicative split of the deformation gradient into dilatational and distortional (isochoric) parts. The isochoric part of the decomposition is denoted by F¯ and is given by F¯ = J −1/3 F

(73)

with det F¯ = 1. The isochoric counterpart of the right Cauchy–Green deformation ¯ which we denote by C. ¯ tensor C = FT F is F¯ T F,

32

J. Merodio and R. Ogden

Instead of regarding W as a function of F, the volumetric and isochoric parts of F can be considered separately so that W is then a function of F¯ and J . On differentiating W with respect to F, we then obtain, without any constraint,     ∂W ¯ −1 ∂W ¯ −1 1 ∂W −1/3 ∂W ¯ +J =J F , − tr F F S= ¯ ¯ ∂F 3 ∂J ∂F ∂F

(74)

and the Cauchy stress is given by σ = σ¯ +

∂W I, ∂J

(75)

where σ¯ is the deviatoric part of σ so that tr σ¯ = 0, and J σ¯ = F¯

  ∂W ∂W 1 − tr F¯ I, 3 ∂ F¯ ∂ F¯

(76)

while ∂W/∂ J is the hydrostatic part of σ, i.e. 1 ∂W = trσ. ∂J 3

(77)

¯ as an independent variable The deviatoric part of σ may also be expressed with C ¯ which yields instead of F, J σ¯ = 2F¯

  ∂W ¯ T 2 ¯ ∂W I. F − tr C ¯ ¯ 3 ∂C ∂C

(78)

This formulation is particularly useful for computational purposes where W is decoupled additively into a deviatoric and a volumetric part (a volumetric–deviatoric split), the latter being used as a penalty term towards enforcing incompressibility. This is effected by forming ¯ + Wpen (J ), ¯ J ) = Wiso (C) W (C,

(79)

where Wiso and Wpen are the isochoric and penalty (volumetric) contributions to W . This can be considered as a slightly compressible or nearly incompressible model, for which J − 1 is ‘small’.

3 Stress–Deformation Relations Expanded For isotropic or transversely isotropic materials, the strain-energy function can be expressed in terms of invariants. This is also the case for other types of material symmetry. To enable explicit expressions for the various stress tensors to be obtained, we therefore consider W to be a function of N invariants I1 , I2 , . . . , I N involving the

Finite Deformation Elasticity Theory

33

right Cauchy–Green tensor C, which is for simplicity omitted from the arguments of I1 , I2 , . . . , I N , here: W = W¯ (I1 , I2 , . . . , I N ). Then the nominal stress and Cauchy stress tensors can be expanded as S=

N  i=1

N  ∂ Ii ∂ Ii −1 ¯ , σ=J F , Wi W¯ i ∂F ∂F i=1

(80)

where W¯ i = ∂ W¯ /∂ Ii , i = 1, 2, . . . , N . Within these expressions, the quantities W¯ i embody the material properties and can be made explicit for particular choices of W while ∂ Ii /∂F are largely associated with the kinematics of the deformation.

3.1 Isotropic Materials As noted in Sect. 2.1, for an isotropic hyperelastic material W can be considered as a function of the principal invariants of C so that W = W¯ (I1 , I2 , I3 ) with I1 (C) = trC,

I2 (C) =

1 [I1 (C)2 − tr(C2 )], 2

I3 (C) = det C = J 2 .

(81)

To apply (80) to this case, we require the derivatives of I1 , I2 , I3 with respect to F, which are given by ∂ I3 = 2I3 F−1 . ∂F

(82)

∂ W¯ = 2 W¯ 1 FT + 2 W¯ 2 (I1 FT − FT FFT ) + 2I3 W¯ 3 F−1 , ∂F

(83)

∂ I1 = 2FT , ∂F Hence S= and

∂ I2 = 2I1 FT − 2FT FFT , ∂F

σ = 2J −1 (W¯ 1 + I1 W¯ 2 )B − 2J −1 W¯ 2 B2 + 2J W¯ 3 I,

(84)

where we recall that B is the left Cauchy–Green tensor FFT . In the stress-free reference configuration, where F = I, J = 1 and I1 = 3, we thus have W¯ 1 + 2 W¯ 2 + W¯ 3 = 0.

(85)

By comparing (84) with (38) specialized to the case of isotropy, the coefficients in (38) become β0 = 2J W¯ 3 , β1 = 2J −1 (W¯ 1 + I1 W¯ 2 ), β2 = −2J −1 W¯ 2 for hyperelasticity.

(86)

34

J. Merodio and R. Ogden

In terms of the principal stretches, the invariants (81) are given by I1 = λ21 + λ22 + λ23 ,

I2 = λ22 λ23 + λ23 λ21 + λ21 λ22 ,

I3 = λ21 λ22 λ23 ,

(87)

with J = λ1 λ2 λ3 , we note that these are symmetric functions of the stretches. Thus, we may treat W as a symmetric function of the stretches as independent variables, written as W = W (λ1 , λ2 , λ3 ), with W (λ1 , λ2 , λ3 ) = W (λ1 , λ3 , λ2 ) = W (λ3 , λ1 , λ2 )

(88)

and the restrictions λ1 , λ2 , λ3 ∈ (0, ∞), and, assuming that W is measured from the reference configuration Br , W (1, 1, 1) = 0. By differentiating W (λ1 , λ2 , λ3 ) with respect to the stretches and using the expressions (87), we obtain J −1 λi

∂W = 2J −1 W¯ 1 λi2 + 2J −1 W¯ 2 λi2 (I1 − λi2 ) + 2J W¯ 3 , i = 1, 2, 3. ∂λi

(89)

Recalling the spectral decomposition of V given in (26) in the chapter “Basic Equations of Continuum Mechanics” [20], the corresponding spectral decompositions of both B and σ (which are coaxial) are B=

3 

λi2 v(i)

(i)

⊗v , σ =

i=1

3 

σi v(i) ⊗ v(i) ,

(90)

i=1

where σi , i = 1, 2, 3, are the principal Cauchy stresses. Thus, the right-hand side of (89) gives the principal components of (84), leading to the simple formula σi = J −1 λi

∂W , i = 1, 2, 3. ∂λi

(91)

If, as assumed so far, the reference configuration Br is stress free, then ∂W (1, 1, 1) = 0, i = 1, 2, 3. ∂λi

(92)

Since S = J F−1 σ, F−1 = U−1 RT , RT v(i) = u(i) and U−1 u(i) = λi−1 u(i) , it follows that S can be expressed in the form S=

3  i=1

ti u(i) ⊗ v(i) , ti = J λi−1 σi =

∂W . ∂λi

Moreover, from Sect. 2, we recall that for an isotropic material

(93)

Finite Deformation Elasticity Theory

35

S = h(F) = h(U)RT ≡ T(1) RT ,

(94)

in terms of the Biot stress tensor T(1) . Hence, from (93) and (94), we have the spectral decomposition 3  (1) T = ti u(i) ⊗ u(i) (95) i=1

for T(1) , which identifies ti , i = 1, 2, 3, as the principal Biot stresses.

3.1.1

Alternative Invariants

Other representations for the energy function and the stresses can be obtained on the basis of any set of three independent invariants that are symmetric functions of the stretches. We now illustrate this by considering the principal invariants of the stretch tensor U, denoted by i 1 , i 2 , i 3 and defined in terms of the principal stretches by i 1 = λ1 + λ2 + λ3 , i 2 = λ2 λ3 + λ3 λ1 + λ1 λ2 , i 3 = λ1 λ2 λ3 ≡ J.

(96)

In this case, we write the strain energy as W˜ (i 1 , i 2 , i 3 ) and calculate the principal Biot stresses ti as ti = where

∂ W˜ = W˜ 1 + (i 1 − λi )W˜ 2 + i 3 λi−1 W˜ 3 , ∂λi

∂ W˜ ∂ W˜ ∂ W˜ , W˜ 2 = , W˜ 3 = . W˜ 1 = ∂i 1 ∂i 2 ∂i 3

Hence, T(1) =

3 

ti u(i) ⊗ u(i) = W˜ 1 I + W˜ 2 (i 1 I − U) + i 3 W˜ 3 U−1 .

(97)

(98)

(99)

i=1

Then, by using the connection S = T(1) RT , we obtain S = W˜ 1 RT + W˜ 2 (i 1 RT − FT ) + i 3 W˜ 3 F−1 .

(100)

We also have the expansion S=

3  i=1

∂i i , W˜ i ∂F

and hence comparison of this with (100) yields the formulas

(101)

36

J. Merodio and R. Ogden

∂i 1 = RT , ∂F

∂i 2 = i 1 R T − FT , ∂F

∂i 3 = i 3 F−1 . ∂F

(102)

These formulas may alternatively be obtained by using the Cayley–Hamilton theorem for V and the connections I1 = i 12 − 2i 2 ,

I2 = i 22 − 2i 1 i 3 ,

I3 = i 32 ,

(103)

which are easily derived in terms of the stretches. See, in particular, the derivation given by Steigmann [31]. In terms of i 1 , i 2 , i 3 , the Cauchy stress tensor has the representation σ = i 3−1 (W˜ 1 + i 1 W˜ 2 )V − i 3−1 W˜ 2 V2 + W˜ 3 I,

(104)

which may be compared with (31) to provide expressions for the coefficients α0 , α1 , α2 therein. When evaluated in the stress-free reference configuration with i 1 = i 2 = 3, i 3 = 1, this yields (105) W˜ 1 + 2 W˜ 2 + W˜ 3 = 0. By linearizing (104) in the strain e = V − I, the isotropic linear elastic stress– strain equation σ = 2μe + λ(tre)I (106) is obtained, where λ and μ are the classical Lamé elastic constants. For the representations W (λ1 , λ2 , λ3 ), W˜ (i 1 , i 2 , i 3 ) and W¯ (I1 , I2 , I3 ), it is straightforward to establish the connections 2μ = Wii − Wi j = W˜ 1 + W˜ 2 = 4(W¯ 1 + W¯ 2 ),

(107)

for each i and j = i, and λ + 2μ = Wii = W˜ 11 + 4W˜ 12 + 2 W˜ 13 + 4W˜ 22 + 4W˜ 23 + W˜ 33 = 4(W¯ 11 + 4W¯ 12 + 2 W¯ 13 + 4W¯ 22 + 4W¯ 23 + W¯ 33 )

(108)

for each i, the subscripts all representing partial derivatives with respect to the appropriate arguments.

3.1.2

Incompressible Isotropic Hyperelastic Materials

The constraint of incompressibility can be expressed with respect to different choices of deformation variable. For example, i 3 ≡ J ≡ det F ≡ det U ≡ λ1 λ2 λ3 = 1,

I3 = 1,

(109)

Finite Deformation Elasticity Theory

37

which apply at each point X of the material. Because of this constraint, the dependence of the strain energy on the invariants discussed in Sect. 2.1 in the case of isotropy reduces to a representation in terms of two independent invariants alone, for example, I1 and I2 . We therefore write W¯ (I1 , I2 ). It follows from (68)2 , on use of (82)1,2 , that σ = 2(W¯ 1 + I1 W¯ 2 )B − 2 W¯ 2 B2 − pI,

(110)

where we have now replaced q by − p. Alternatively, and equivalently, in terms of the invariants i 1 and i 2 , with W˜ (i 1 , i 2 ), we have (111) σ = (W˜ 1 + i 1 W˜ 2 )V − W˜ 2 V2 − pI, in which p is different in general from the p in (110). Equally, with W (λ1 , λ2 , λ3 ), the modification of Eq. (91) for the principal Cauchy stresses is ∂W − p, i = 1, 2, 3, (112) σi = λi ∂λi and (90)2 becomes σ=

3  i=1

λi

∂W (i) v ⊗ v(i) − pI. ∂λi

(113)

In the stress-free reference configuration, the value of p, say p0 , turns out to be the same for each representation, and is given by p0 = 2 W¯ 1 + 4W¯ 2 = W˜ 1 + 2 W˜ 2 = Wi ,

(114)

independently of i ∈ {1, 2, 3}. The counterpart of the linear stress–strain relation (106) in the incompressible case is simply σ = 2μe − pI.

(115)

This involves only one material constant, the shear modulus μ (> 0), which is given in the three representations via 2μ = 4(W¯ 1 + W¯ 2 ) = W˜ 1 + W˜ 2 = Wii − Wi j + Wi ,

(116)

independently of i and j subject to j = i ∈ {1, 2, 3}. Rubber-like materials are generally treated as isotropic and incompressible and the literature contains an abundance of different strain-energy functions that model the elastic properties of these materials with varying degrees of success. We do not review the extensive catalogue of models but refer to the works [1, 11, 21, 24, 25] where many details of such models can be found although these references do not

38

J. Merodio and R. Ogden

provide an exhaustive list of available models. Here, we mention some basic models within the frameworks of each of the representations of W highlighted above. First, there are the widely used neo-Hookean and Mooney–Rivlin strain-energy functions defined, respectively, by 1 1 1 W¯ = μ(I1 − 3), W¯ = μ1 (I1 − 3) − μ2 (I2 − 3), 2 2 2

(117)

where the constant μ is the shear modulus of the material in the natural configuration, i.e. the single Lamé modulus of incompressible linear elasticity appearing in (116), while μ1 (≥ 0) and μ2 (≤ 0) are material constants such that μ1 − μ2 = μ (> 0). The Mooney–Rivlin model reduces to the neo-Hookean model when μ2 = 0. The associated Cauchy stress tensors are obtained from (110) as σ = μB − pI, σ = μ1 B − μ2 (I1 B − B2 ) − pI,

(118)

respectively. In the representation W˜ , there is the well-known Varga model, which is simply W˜ = 2μ(i 1 − 3),

(119)

where again the shear modulus μ satisfies (116). The neo-Hookean, Mooney–Rivlin and Varga models provide adequate representations of data from experiments on rubber-like materials for deformations of moderate magnitude, but they are not suitable for very large deformations. This brings us to a model based on the stretches, which does not suffer from this restriction. This has the form W =

N  μn αn (λ1 + λα2 n + λα3 n − 3), λ1 λ2 λ3 = 1, α n=1 n

(120)

where N is a positive integer and μn and αn are material constants such that μn αn > 0, n = 1, 2, . . . , N ,

N 

μn αn = 2μ,

(121)

n=1

the latter resulting from (116). Note that the neo-Hookean, Mooney–Rivlin and Varga models are all special cases of (120) From (112), the principal Cauchy stresses are calculated as σi =

N  n=1

μn λiαn − p, i = 1, 2, 3.

(122)

Finite Deformation Elasticity Theory

39

The only general restriction imposed on W thus far is objectivity while material symmetries impose specific further restrictions. Other than these aspects the predictions of the response based on W should be consistent with the data obtained for particular materials. Mathematical restrictions may also be required in order for the resulting equations to be well behaved, and this often leads to inequalities such as strong ellipticity, which will be examined in Sect. 5.

3.2 Fibre-Reinforced Materials 3.2.1

Transversely Isotropic Hyperelastic Materials

With a single preferred direction M, the invariants (59) are required to describe transverse isotropy, where M is regarded as the direction of an embedded fibre within an isotropic matrix. Thus, transverse isotropy is used to model fibre-reinforced materials with a single family of fibres whose direction M can vary with X ∈ Br , in which case the material properties are inhomogeneous. With the arguments omitted, the five invariants are I1 , I2 , I3 and I4 , I5 , the latter two being written also as I4 = M · (CM) = m · m,

I5 = M · (C2 M) = m · (Bm),

(123)

where we recall that m = FM is the pushforward of M from Br to Bt . For a hyperelastic material without internal constraints, the most general strainenergy function W for a transversely isotropic nonlinear elastic solid depends only on the five invariants listed above. We write this as W = W¯ (I1 , I2 , I3 , I4 , I5 ).

(124)

To calculate σ from (80)2 , in addition to the derivatives (82), we require ∂ I4 = 2M ⊗ FM, ∂F

∂ I5 = 2(M ⊗ FCM + CM ⊗ FM), ∂F

(125)

and (80)2 yields J σ = 2W¯ 1 B + 2W¯ 2 (I1 B − B2 ) + 2I3 W¯ 3 I + 2W¯ 4 m ⊗ m + 2 W¯ 5 (m ⊗ Bm + Bm ⊗ m),

(126) where we recall that the subscripts 1, . . . , 5 on W¯ represent differentiation with respect to I1 , . . . , I5 , respectively. The energy function is taken to vanish in Br (where I1 = I2 = 3, I3 = I4 = I5 = 1), which is also considered to be stress free. Hence, W¯ = 0, W¯ 1 + 2 W¯ 2 + W¯ 3 = 0, W¯ 4 + 2 W¯ 5 = 0 in Br .

(127)

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J. Merodio and R. Ogden

For an incompressible material, the corresponding expression for σ is σ = 2 W¯ 1 B + 2 W¯ 2 (I1 B − B2 ) + 2 W¯ 4 m ⊗ m + 2 W¯ 5 (m ⊗ Bm + Bm ⊗ m) − pI. (128) In the reference configuration Br , where I1 = I2 = 3, I4 = I5 = 1, the counterparts of (127) are W¯ = 0, 2 W¯ 1 + 4W¯ 2 − p0 = 0, W¯ 4 + 2 W¯ 5 = 0 in Br ,

(129)

p0 being the value of p in Br . If the nonlinear theory is required to be consistent with the classical linear theory in the small strain limit, the values of the first and second derivatives of W¯ in Br should be expressed in terms of the classical elastic constants. In terms of the Voigt constants ci j (see, for example, the classical work of Love [10] for a detailed description or the more recent volume by Ting [32]), the required relations for an unconstrained material are W¯ 11 + 4W¯ 12 + 4W¯ 22 + 2 W¯ 13 + 4W¯ 23 + W¯ 33 = c11 /4, W¯ 2 + W¯ 3 = (c12 − c11 )/4, W¯ 1 + W¯ 2 + W¯ 5 = c44 /2,

W¯ 14 + 2 W¯ 24 + 2 W¯ 15 + W¯ 34 + 4W¯ 25 + 2 W¯ 35 = (c13 − c12 )/4, W¯ 44 + 4W¯ 45 + 4W¯ 55 + 2 W¯ 5 = (c33 − c11 + 2c12 − 2c13 )/4,

(130) (131) (132) (133)

as given in [13], where c11 , c12 , c13 , c33 , c44 are the relevant five Voigt constants for the case of transverse isotropy in which the preferred direction is aligned with the x3 coordinate direction. The corresponding conditions for the incompressible case are W¯ 1 + W¯ 2 = (c11 − c12 )/4, W¯ 1 + W¯ 2 + W¯ 5 = c44 /2, ¯ ¯ W44 + 4W45 + 4W¯ 55 = (c11 + c33 − 2c13 − 4c44 )/4,

(134) (135)

which were given in [14], the number of constants reducing to three, which are the combinations c11 − c12 and c11 + c33 − 2c13 together with c44 .

3.2.2

The Reinforcing Model

A particular example of a transversely isotropic strain-energy function has the form W = W¯ (I1 , I2 , I3 , I4 , I5 ) = W¯ iso (I1 , I2 , I3 ) + W¯ fib (I4 , I5 ),

(136)

which decouples the isotropic contribution W¯ iso , representing the properties of an isotropic matrix material, from the contribution W¯ fib , which represents the properties of the reinforcing fibres that are embedded in the matrix and is referred to as a reinforcing model.

Finite Deformation Elasticity Theory

41

For illustration, we restrict W¯ fib to be a function of just I4 , which is written as F(I4 ), a particular case of which is the so-called standard reinforcing model defined by 1 (137) F(I4 ) = α(I4 − 1)2 2 [26, 33], where the constant α > 0 is a parameter that measures the strength of the reinforcement. It was noted in [12] that for an incompressible material, the term W¯ 4 contributes a traction component 2I4 W¯ 4 to the Cauchy stress in the deformed direction m = FM, which is tensile (compressive) if F  (I4 ) is positive (negative), i.e. (138) F  (I4 ) > 0 (< 0) for I4 > 1 (< 1), with F  (1) = 0 required to satisfy (129)3 . Note that F  (I4 ) → ∞ as I4 → ∞ so that in tension, the material becomes stiffer with increasing stretch (I4 > 1). However, for I4 < 1, care must be taken in interpreting F  (I4 ) since under compression fibres may buckle or kink, leading to failure of ellipticity (which will be discussed in Sect. 5). For collagen rich fibrous biological tissues, it is often reasonable to assume that the fibres do not support compression and then the corresponding models take F  (I4 ) = 0 for I4 ≤ 1; see, for example, [7] for a discussion of this point. The standard reinforcing model is a prototype that has been widely used by several authors to model the qualitative mechanical behaviour of fibre-reinforced materials (for example, [9, 14, 17]). There are also micromechanics-based approaches that enable the properties of the individual constituents to inform macromechanical strainenergy functions for composite materials that also have features similar to an isotropic function W¯ iso augmented with a standard reinforcing model (see [3] for the case of a single fibre family and [2] for two families of fibres, and the references therein).

3.2.3

Two Preferred Directions—Two Fibre Families

We now consider the effect of reinforcement with two families of fibres, and confine attention to incompressible materials. We take the associated preferred directions to be M and M in Br . In addition to the isotropic invariants I1 , I2 , the invariants I4 , I5 associated with M are now joined by corresponding invariants associated with M , which we denote by I6 and I7 , but there is also a coupling invariant, denoted by I¯8 , which involves both fibre families. These are defined by I6 = M · (CM ),

 I7 = M · C2 M ,



   I¯8 = M · M M · CM ,

(139)

the factor M · M being included to ensure that I¯8 is not affected by reversal of either M or M , i.e. it is fully invariant and has the value M · M in Br . However, it is often convenient to drop this latter factor and consider I8 instead, which is simply M · CM . Besides the notation m = FM, we shall use m = FM . Then, in addition to (123), we have

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I6 = m · m ,

I7 = m · (Bm ),

I8 = m · m .

(140)

For the considered incompressible material, the strain-energy function now depends on the seven variables I1 , I2 , I4 , I5 , I6 , I7 , I8 , which is the maximal set of independent deformation-dependent invariants for an incompressible material with two fibre families (see Spencer [28] for the general theory of invariants), and we write (141) W = W¯ (I1 , I2 , I4 , I5 , I6 , I7 , I8 ). However, since I8 is not strictly an invariant, W should depend on it via I82 , which is strictly invariant. In the reference configuration, where W¯ is taken to vanish, we have I1 = I2 = 3, I4 = I5 = I6 = I7 = 1 and I8 = M · M . The Cauchy stress tensor based on (141) is given by σ = − pI + 2 W¯ 1 B + 2W¯ 2 (I1 B − B2 ) + 2W¯ 4 m ⊗ m + 2 W¯ 5 (m ⊗ Bm + Bm ⊗ m) + 2 W¯ 6 m ⊗ m + 2W¯ 7 (m ⊗ Bm + Bm ⊗ m ) + W¯ 8 (m ⊗ m + m ⊗ m),

(142) wherein W¯ i = ∂ W¯ /∂ Ii , i = 1, 2, 4, . . . , 8. In the stress-free reference configuration, the conditions to be satisfied are then 2 W¯ 1 + 4W¯ 2 − p0 , W¯ 4 + 2 W¯ 5 = 0, W¯ 6 + 2 W¯ 7 = 0, W¯ 8 = 0,

(143)

which are the counterparts of (129) for the present case. Two special cases are of particular interest. First, if the two families of fibres are orthogonal in the reference configuration, so that M · M = 0, some simplification occurs since I8 is no longer independent of the other invariants. Indeed, it can be shown that (144) I82 = I2 + I5 + I7 + I4 I6 − I1 (I4 + I6 ) [15], and I8 can therefore be omitted from the list of arguments in (141), and the term in W¯ 8 in (142) can be omitted. The material then has orthotropic symmetry; see Spencer [29] for a discussion of this case. The material response is also orthotropic if the two families are not orthogonal but have the same mechanical properties, in which case the bisectors of the two directions form planes of orthogonal symmetry. The second special case is that of plane strain. For example, for a plane strain deformation in the (1, 2) plane with the fibre families both embedded in this plane, the invariants are no longer all independent but satisfy the connections I82 = I4 I6 − |M × M |2 (145) [15]. Then, the energy function, now denoted by Wˆ , is restricted to plane strain, and depends only on three independent invariants that depend on the deformation. Thus, we write Wˆ (I1 , I4 , I6 ), with separate dependence on M · M being implicit. I2 = I1 ,

I5 = (I1 − 1)I4 − 1,

I7 = (I1 − 1)I6 − 1,

Finite Deformation Elasticity Theory

43

The expression for the Cauchy stress tensor (restricted to its planar part) is then simplified to σ = − pI ˆ + 2 Wˆ 1 B + 2 Wˆ 4 m ⊗ m + 2 Wˆ 6 m ⊗ m ,

(146)

in which all scalars and tensors are the two-dimensional counterparts of those in (142). If we also have M · M = 0, then (145) becomes I82 = I4 I6 − 1 with the further simplification I4 + I6 = I1 − 1 so that Wˆ can then be reduced to dependence on I1 and only one of I4 and I6 . A particular example of (141) in the reduced and decoupled form W = W¯ iso (I1 ) + W¯ fib (I4 , I6 )

(147)

is often used for the modelling of the elastic properties of soft tissues, such as arteries (see [6] for a review), with W¯ iso (I1 ) taken to be the neo-Hookean model (117)1 and W¯ fib (I4 , I6 ) to have the exponential form k1 k1 {exp[k2 (I4 − 1)2 ] − 1} + {exp[k2 (I6 − 1)2 ] − 1}, W¯ fib (I4 , I6 ) = 2k2 2k2

(148)

with the same material parameters k1 and k2 for the two fibre families.

4 Residual Stress The presence of residual stresses in the reference configuration Br has a significant effect on the subsequent material response, as has been demonstrated, for example, in respect of arteries [8]. In this section, we therefore drop our prior assumption that Br is stress free and consider there to be a residual stress, denoted by τ , therein. We assume that there are no intrinsic couples in Br so that τ is symmetric. By definition [5, 23], assuming the absence of body forces, the residual stress satisfies the equilibrium equation and traction-free boundary conditions in Br , i.e. Div τ = 0 in Br , τ N = 0 on ∂Br ,

(149)

where ∂Br is the boundary of Br and N is the unit outward normal to ∂Br . By using the fact that Grad X = I, the identity tensor in Br , and then noting that Div(τ ⊗ X) = τ and applying the divergence theorem, it follows that  Br

τ dV = O

(150)

[5, 23], This implies that a non-trivial τ cannot be homogeneous, so τ necessarily depends on the position vector X in Br .

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The material is deformed elastically from Br into a new configuration Bt and the elastic properties of the material are characterized in terms of a strain-energy function W , defined by unit volume in Br . The elastic properties depend on the deformation gradient F through C = FT F and now also τ and we write W = W (C, τ ), which is automatically objective since both C and τ are Lagrangian tensors. Since τ is symmetric, it has a spectral decomposition, which we write as τ =

3 

τi Mi ⊗ Mi ,

(151)

i=1

where τi are the eigenvalues and Mi , i = 1, 2, 3, the unit eigenvectors of τ , the 3 Mi ⊗ Mi = I. The eigenvectors have similar roles to the prelatter satisfying i=1 ferred directions discussed in Sect. 3.2 and thus τ generates anisotropy of the elastic response relative to Br and here we therefore assume that there are no separate preferred directions present that generate additional anisotropy. Thus, similiar to the situation with fibre reinforcement, W is an isotropic function of C and τ . It therefore depends on the invariants of C and τ and the combined invariants of C and τ . In three dimensions, a complete independent set of such invariants consists of the principal invariants of C, i.e. I1 , I2 , I3 defined by (81), three invariants of τ , for example, trτ , tr(τ 2 ) and tr(τ 3 ), and four invariants involving both C and τ , which we take to be I4 = tr(Cτ ),

I5 = tr(C2 τ ),

I6 = tr(Cτ 2 ),

I7 = tr(C2 τ 2 ).

(152)

Note that here the notation I4 , . . . , I7 duplicates that used for fibre reinforcement since the invariants I4 , . . . , I7 defined in Sect. 3.2 can be recovered by special choices of τ . For example, by setting τ = M ⊗ M, we obtain I4 = M · (CM), as in (123)1 . Thus, W depends on the above seven deformation-dependent invariants and we write this as W = W¯ (I1 , . . . , I7 ) with dependence on the invariants of τ being left implicit. Then, on specializing the expansion (80) for the present set of invariants and using (82) together with the derivatives of I4 , . . . , I7 with respect to F, which are ∂ I5 ∂ I6 ∂ I7 ∂ I4 = 2τ FT , = 2(τ C + Cτ )FT , = 2τ 2 FT , = 2(τ 2 C + Cτ 2 )FT , ∂F ∂F ∂F ∂F (153) we obtain the Cauchy stress tensor σ from J σ = 2 W¯ 1 B + 2 W¯ 2 (I1 B − B2 ) + 2I3 W¯ 3 I + 2 W¯ 4 Σ + 2 W¯ 5 (ΣB + BΣ) + 2 W¯ 6 ΣB−1 Σ + 2 W¯ 7 (ΣB−1 ΣB + BΣB−1 Σ) (154) [4, 27], within which we have introduced the notation Σ = Fτ FT , which is Eulerian in character, for the pushforward of τ from Br to Bt , W¯ i , i = 1, . . . , 7, stands for ∂ W¯ ∂ Ii , and I is now the identity tensor in Bt .

Finite Deformation Elasticity Theory

45

For an incompressible material, (154) is replaced by σ = 2 W¯ 1 B + 2 W¯ 2 (I1 B − B2 ) − pI + 2 W¯ 4 Σ + 2 W¯ 5 (ΣB + BΣ) + 2 W¯ 6 ΣB−1 Σ + 2 W¯ 7 (ΣB−1 ΣB + BΣB−1 Σ),

(155)

with W = W¯ (I1 , I2 , I4 , . . . , I7 ), p again being a Lagrange multiplier. In the reference configuration Br , the invariants reduce to I1 = I2 = 3,

I3 = 1,

I4 = I5 = trτ ,

I6 = I7 = tr(τ 2 ),

(156)

τ = 2(W1 + 2W2 + W3 )I + 2(W4 + 2W5 )τ + 2(W6 + 2W7 )τ 2 ,

(157)

τ = (2W1 + 4W2 − p0 )I + 2(W4 + 2W5 )τ + 2(W6 + 2W7 )τ 2 ,

(158)

and (154) and (155) reduce to

respectively, and the restrictions W1 + 2W2 + W3 = 0, 2(W4 + 2W5 ) = 1, W6 + 2W7 = 0,

(159)

2W1 + 4W2 − p0 = 0, 2(W4 + 2W5 ) = 1, W6 + 2W7 = 0,

(160)

follow, for unconstrained and incompressible materials, respectively, W1 , . . . , W7 being evaluated in Br , i.e. for (156), and p0 is again the value of p in Br . A particular prototype example of W satisfying the above conditions was introduced in [27] for an incompressible material and is given by 1 1 1 ¯ 4 − trτ )2 + (I4 − trτ ), W¯ (I1 , I4 ) = μ(I1 − 3) + μ(I 2 2 2

(161)

where μ (> 0) and μ¯ are material constants. The associated Cauchy stress is σ = μB − pI + [1 + 2μ(I ¯ 4 − trτ )]Σ.

(162)

This was used in [18] with μ¯ = 0 to examine the effect of residual stress on the plane strain problem of azimuthal shear of a circular cylindrical tube by considering residual stress to have radial and azimuthal components τ R R and τΘΘ with respect to cylindrical polar coordinates R, Θ, Z in Br . The equilibrium equation and boundary conditions in (149) then specialize to dτ R R 1 + (τ R R − τΘΘ ) = 0 in A < R < B, τ R R = 0 at R = A, B, dR R

(163)

where A and B are the inner and outer radii of the tube in the reference configuration. The specific form

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J. Merodio and R. Ogden

τ R R = α(R − A)(R − B), τΘΘ = α[3R 2 − 2(A + B)R + AB],

(164)

satisfying (163), where α > 0 is a constant, was adopted for illustration both in [18] and later in considering the combined extension, inflation and torsion of a circular cylindrical tube in [16] with the same constitutive model.

5 Ellipticity and Strong Ellipticity As indicated at the end of Sect. 3.1, restrictions placed on the strain-energy function are as yet relatively limited. However, from both the mathematical and mechanical perspectives some restrictions arise naturally. These relate mainly to consideration of what restrictions are required for reasonable mechanical behaviour and well-behaved mathematical equations. In this section, we examine one important aspect of the governing equations which has both mathematical and mechanical significance. We begin by considering the equation of motion (69) in the chapter “Basic Equations of Continuum Mechanics” [20], but without body forces, which are not relevant to the following discussion. We write this as Div S = ρr v,t ≡ ρr x,tt ,

(165)

where S is given by (43)1 for an unconstrained material, while for an incompressible material we use (68)1 with q = − p. In component form, this can be written as Aαiβ j x j,αβ = ρr xi,tt

(166)

for an unconstrained material, and Aαiβ j x j,αβ − p,i = ρr xi,tt ,

(167)

for an incompressible material with det(xi,α ) = 1, where a subscript following a comma represents a derivative with respect to a spatial coordinate, such as the Lagrangian coordinate X α or the Eulerian coordinate xi , as appropriate for the context, and ∂2 W Aαiβ j = = Aβ jαi (168) ∂ Fiα ∂ F jβ are the components of the elasticity tensor A.

Finite Deformation Elasticity Theory

47

5.1 Incremental Motions We now represent small incremental quantities by a superposed dot, with an incremental motion superimposed on a finite motion x = χ(X, t) denoted by x˙ = χ(X, ˙ t). Then, by forming the increment of the equation of motion (165), we obtain Div S˙ = ρr x˙ ,tt ,

(169)

where S˙ = AF˙ is the increment in S linearized in the increment F˙ = Grad x˙ of F. In components, Eq. (169) may be written as Aαiβ j x˙ j,αβ + Bαiβ jγk x˙ j,β xk,αγ = ρr x˙i,tt ,

(170)

for an unconstrained material, where Bαiβ jγk are the components of the (sixth-order) tensor B of second-order moduli defined by Bαiβ jγk =

∂3 W . ∂ Fiα ∂ F jβ ∂ Fkγ

(171)

For the special case in which the incremental motion is superimposed on a homogeneous finite deformation x = χ(X) independent of time, Eq. (170) reduces to Aαiβ j x˙ j,αβ = ρr x˙i,tt ,

(172)

where the coefficients Aαiβ j are now constants. By updating this equation from the reference configuration to the underlying finitely deformed configuration, we obtain A0 piq j u j, pq = ρu i,tt ,

(173)

where u i , i = 1, 2, 3, are the components of u(x, t) = u(χ(X), t) = x˙ = χ(X, ˙ t),

(174)

which is the Eulerian counterpart of x˙ , and A0 piq j , the pushforward of Aαiβ j , is given by (175) J A0 piq j = F pα Fqβ Aαiβ j . The counterpart of Eq. (33) in the chapter “Basic Equations of Continuum Mechan˙ = divu, and hence the incremental incompressibility ics” [20] is J˙ = J tr(F−1 F) condition has the form div u = 0. The equation corresponding to (173) for an incompressible material is A0 piq j u j, pq − p˙ ,i = ρu i,tt , with u i,i = 0.

(176)

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J. Merodio and R. Ogden

There are many contributions to infinitesimal wave propagation in prestressed bodies in the literature based on these equations, and we refer to [22] for a list of references on surface waves in a half-space, plate waves and waves in a cylinder. Here, the focus is on the properties of the tensor A0 , the pushforward of A, and for this, we first of all consider the propagation of plane waves.

5.1.1

Incremental Plane Waves

We consider plane waves of the form u = f (n · x − ct)m,

(177)

where f is a twice continuously differentiable function of its argument n · x − ct, c is the wave speed and m and n are unit vectors, m being known as the polarization vector. If n is real, it defines the direction of propagation of a homogeneous plane wave, while if complex the wave is referred to as an inhomogeneous plane wave. On substitution of (177) into the equation of motion (173), we obtain Q(n)m = ρc2 m

(178)

on removal of the factor f  (n · x − ct), where the tensor Q, which depends on n, is defined in component form by Q i j ≡ [Q(n)]i j = A0 piq j n p n q ,

(179)

and is known as the acoustic tensor because of its connection with wave propagation. Similarly, for an incompressible material, the equation of motion (176) yields [Q(n) − n ⊗ Q(n)n]m = ρc2 m,

(180)

subject to the restriction m · n = 0, which follows from the incremental incompressibility condition div u = 0. Equations (178) and (180) determine possible values of the wave speed c and polarization vectors m for which plane waves can propagate in any given direction n, for unconstrained and incompressible materials, respectively. They are thus referred to as propagation conditions. For given n, the wave speeds are obtained as solutions of the characteristic equations ¯ =0 ¯ (181) − ρc2 I] det[Q(n) − ρc2 I] = 0, det[Q(n) ¯ of unconstrained material and incompressible materials, respectively, where Q(n) is ¯ ¯ ¯ ¯ defined as Q(n) = IQ(n)I with I = I − n ⊗ n. For an unconstrained material, three values of ρc2 are obtained from (181)1 , but for an incompressible material, because

Finite Deformation Elasticity Theory

49

of the constraint m · n = 0, there are only two solutions of (181)2 , with polarizations ¯ in the plane normal to n, and (180) can be written as Q(n)m = ρc2 m. From either (178) or (180), we obtain ρc2 = [Q(n)m] · m ≡ A0 piq j n p n q m i m j .

(182)

It follows that the wave speed is real, provided [Q(n)m] · m > 0 for a given n and associated m. This condition is satisfied if the strong-ellipticity condition [Q(n)m] · m ≡ A0 piq j n p n q m i m j > 0

(183)

is satisfied for all m = 0 and n = 0, subject to m · n = 0 under incompressibility. Thus, for strong ellipticity to hold, Q(n) is positive definite for all unit vectors n in ¯ the case of an unconstrained material, while for an incompressible material, Q(n) is positive definite in the plane normal to n. This provides one important consideration for the properties of the strain-energy function encoded in the elasticity tensor A0 . However, it cannot always be guaranteed that strong ellipticity holds, and failure of strong ellipticity is associated with the emergence of discontinuous deformation gradients, as we now discuss in the context of the equilibrium.

5.1.2

Equilibrium Considerations

When there is no time dependence, the equation of motion (172) becomes the equilibrium equation (184) Aαiβ j x j,αβ = 0. Consider a body for which the deformation gradient F relative to the reference configuration Br is continuous in Br , and that its gradient Grad F is also continuous except that across a surface Sr in Br it may be discontinuous. The surface Sr (equivalently its image S under the deformation) is known as a surface of weak discontinuity. The jump in Grad F across Sr is given by [[GradF]] = m ⊗ N ⊗ N,

(185)

[[x j,αβ ]] = m j Nα Nβ ,

(186)

or, in components, where [[•]] denotes the difference in the enclosed quantity evaluated on the two sides of Sr , N, with components Nα , is a vector normal to Sr and m, with components m i , is as yet an undetermined vector that provides a measure of the strength of the discontinuity. By forming the difference of the left-hand side of (184) on the two sides of Sr and using (186), we obtain

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J. Merodio and R. Ogden

Aαiβ j m j Nα Nβ = 0, A0 piq j m j n p n q = 0,

(187)

where, consistently with Nanson’s formula (13) in the chapter “Basic Equations of Continuum Mechanics” [20], n is proportional to F−T N and is normal to the image of Sr in the deformed configuration. Thus, Q(n)m = 0, and the existence of a surface of weak discontinuity requires that Q(n) be singular for some non-zero n. The values of n for which this holds are given by det Q(n) = 0.

(188)

Once n is determined from (188), the null eigenvectors m of Q(n) are obtained from Q(n)m = 0. The equations require minor adjustments in the case of incompressibil¯ ity, and, in particular, (188) is replaced by det Q(n) = 0, but the other details are omitted. A necessary condition, therefore, for the existence of a surface of weak discontinuity for a deformation gradient F is that the condition [Q(n)m] · m ≡ A0 piq j n p n q m i m j = 0

(189)

holds for one or more pairs of non-zero vectors m and n. This means, in particular, that the strong ellipticity condition (183) cannot hold. A weaker condition than (183) is that of ellipticity, which merely requires [Q(n)m] · m ≡ A0 piq j n p n q m i m j = 0

(190)

for all vectors m = 0, n = 0, again subject to m · n = 0 when incompressibility is considered. If this holds, then weak surfaces of discontinuity do not exist. While a weak discontinuity relates to a discontinuity in the derivative of the deformation gradient, a so-called strong discontinuity involves a discontinuity in the deformation gradient itself. For a strong discontinuity to emerge the loss of ellipticity condition (189) must also hold, and hence a strong surface of discontinuity is also a weak surface of discontinuity. Consider a line element dX lying in the surface Sr at X. Under the deformation it becomes dx lying along S at x and of course suffers no jump across S, so that [[F]]dX = 0. Thus, if M is a unit vector along dX then [[F]]M = 0, and hence, on taking the scalar product of this with the unit normal n to S, we obtain [[FT n]] · M = 0 for all tangent vectors to Sr at X. Thus, [[FT n]] = αN for some scalar quantity α. This result can also be obtained by using Nanson’s formula (13) from the chapter “Basic Equations of Continuum Mechanics” [20] in the form FT nda = J Nd A, with N normal to Sr and n normal to S, from which it follows that α = [[J ]]d A/da. We then deduce that [[F]] = m ⊗ N for some vector m, which measures the strength of the discontinuity, and α = m · n. In the case of an incompressible material [[J ]] = 0 and hence α = 0 and m · n = 0, i.e. m is tangential to S. For a weak discontinuity, the traction ST N across Sr is continuous since it depends on F. On the other hand, for a strong discontinuity the traction continuity condition

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51

[[ST N]] = 0 has to be satisfied in addition to [[F]] = m ⊗ N. For further discussion of strong discontinuities for incompressible materials in the context of fibre-reinforced materials, we refer to the review paper [19]. Of course, the form of A0 depends on the strain-energy function W and hence the properties of A0 determine the ellipticity status of W . If a deformation gradient F satisfies (190) or (183) for every pair of unit vectors m and n, but subject to m · n = 0 in the case of incompressibility, then the deformation x = χ(X) is said to be an elliptic deformation, respectively strongly elliptic deformation, for that W . Moreover, if all possible deformations for a particular W are elliptic (strongly elliptic), then the material characterized by W is referred to as an elliptic (strongly elliptic) material. An example of a strongly elliptic material is the neo-Hookean model (117)1 for which we have simply A0 piq j n p n q m i m j = μ(Bn) · n, as discussed in a plane strain setting in [12]. By contrast, if a deformation gradient F satisfies Eq. (189) for some pair of unit vectors m and n, then the deformation is said to be non-elliptic for the considered W . In this case, each solution n is normal to the corresponding weak or strong surface of discontinuity S.

References 1. Boyce MC, Arruda EM (2000) Constitutive models of rubber elasticity: a review. Rubber Chem Technol 73:504–523 2. deBotton G, Shmuel G (2009) Mechanics of composites with two families of finitely extensible fibers undergoing large deformations. J Mech Phys Solids 57:1165–1181 3. deBotton G, Hariton I, Socolsky EA (2006) Neo-Hookean fiber-reinforced composites in finite elasticity. J Mech Phys Solids 54:533–559 4. Destrade M, Ogden RW (2013) On stress-dependent elastic moduli and wave speeds. IMA J Appl Math 78:965–997 5. Hoger A (1985) On the residual stress possible in an elastic body with material symmetry. Arch Ration Mech Anal 88:271–290 6. Holzapfel GA, Ogden RW (2010) Constitutive modelling of arteries. Proc R Soc A 466:1551– 1596 7. Holzapfel GA, Ogden RW (2015) On the tension-compression switch in soft fibrous solids. Eur J Mech A/Solids 49:561–569 8. Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48 9. Kassianides F, Ogden RW, Merodio J, Pence TJ (2008) Azimuthal shear of a transversely isotropic elastic solid. Math Mech Solids 13:690–724 10. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover Publications, New York 11. Marckmann G, Verron E (2006) Comparison of hyperelastic models for rubber-like materials. Rubber Chem Technol 79:835–858 12. Merodio J, Ogden RW (2002) Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Arch Mech 54:525–552 13. Merodio J, Ogden RW (2003) Instabilities and loss of ellipticity in fiber-reinforced compressible nonlinearly elastic solids under plane deformation. Int J Solids Struct 40:4707–4727 14. Merodio J, Ogden RW (2005) Mechanical response of fiber-reinforced incompressible nonlinearly elastic solids. Int J Non-linear Mech 40:213–227

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15. Merodio J, Ogden RW (2006) The influence of the invariant I8 on the stress-deformation and ellipticity characteristics of doubly fiber-reinforced nonlinearly elastic solids. Int J Non-linear Mech 41:556–563 16. Merodio J, Ogden RW (2014) Extension, inflation and torsion of a residually stressed circular cylindrical tube. Contin Mech Thermodyn 28:157–174 17. Merodio J, Pence TJ (2001) Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation: I. Mech Equilib J Elast 62:119–144 18. Merodio J, Ogden RW, Rodriguez J (2013) The influence of residual stress on finite deformation elastic response. Int J Non-linear Mech 56:43–49 19. Merodio J, Ogden RW (2019) Instabilities associated with loss of ellipticity in fibre-reinforced nonlinearly elastic solids. To appear in Mantiˇc V (ed) Computational and experimental methods in structures: Vol 5, Mathematical methods and models in composites, 2nd edn. World Scientific, Singapore 20. Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). Springer 21. Ogden RW (1982) Elastic deformation of rubberlike solids. In: Hopkins HG, Sewell MJ (eds) Mechanics of Solids, The Rodney Hill 60th Anniversary Volume. Pergamon Press, Oxford, pp 499–537 22. Ogden RW (2001) Elements of the theory of finite elasticity. In: Fu YB, Ogden RW (eds) Nonlinear elasticity: theory and applications. Cambridge University Press, Cambridge, pp 1– 57 23. Ogden RW (2003) Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue. In: Holzapfel GA, Ogden RW (eds) Biomechanics of soft tissue in cardiovascular systems. CISM courses and lectures, Vol 441. Springer, Wien, pp 65–108 24. Ogden RW (1986) Recent advances in the phenomenological theory of rubber elasticity. Rubber Chem Technol 59:361–383 25. Ogden RW (1997) Non-linear elastic deformations. Dover Publications, New York 26. Qiu GY, Pence TJ (1997) Loss of ellipticity in plane deformations of a simple directionally reinforced incompressible nonlinearly elastic solid. J Elast 49:31–63 27. Shams M, Destrade M, Ogden RW (2011) Initial stresses in elastic solids: constitutive laws and acoustoelasticity. Wave Motion 48:552–567 28. Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physics I. Academic, New York, pp 239–253 29. Spencer AJM (1984) Constitutive theory for strongly anisotropic solids. In: Spencer AJM (ed) Continuum theory of the mechanics of fibre-reinforced composites. CISM courses and lectures, Vol 282. Springer, Wien, pp 1–32 30. Spencer AJM (2015) George green and the foundations of the theory of elasticity. J Eng Math 95:5–6 31. Steigmann DJ (2002) Invariants of the stretch tensors and their application to finite elasticity theory. Math Mech Solids 7:393–404 32. Ting TCT (1996) Anisotropic elasticity: theory and applications. Oxford Engineering Science Series, Oxford University Press, Oxford 33. Triantafyllidis N, Abeyaratne RC (1983) Instability of a finitely deformed fiber-reinforced elastic material. J Appl Mech 50:149–156 34. Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der Physik, Vol III/3. Springer, Berlin

Thermomechanics Manuel Doblaré and Mohamed H. Doweidar

Abstract In this chapter, we review the fundamental concepts and formulation of the thermomechanics of continuous media. First, we revise the expressions of the two first laws of thermodynamics for thermomechanical processes, that is, those with changes in temperature and strains as state-independent variables. These equations constitute the generalization of the energy balance equation that was presented, particularized for the isothermal case, in chapter “Basic Equations of Continuum Mechanics” [10]. Next, we introduce different thermodynamic potentials such as the internal energy density, the Helmholtz free energy density, and the dissipation density. This latter is expressed in terms of additional independent internal state variables that take into account history-dependent changes in the material’s internal microstructure. The associated thermodynamic fluxes are then defined as derivatives of the dissipation density with respect to the corresponding thermodynamic drivers (internal variables). The next section introduces the fundamental principles for simple (local, nongraded) materials. These allow establishing the general constitutive equations for the rest of the state- dependent variables, stress and entropy, from the expression of the chosen thermodynamic potential and the fulfillment of the second law of thermodynamics. These expressions are finally applied to several examples: two types of non-dissipative materials, such as ideal fluids and elastic solids in thermomechanical processes, and two dissipative cases: damage mechanics and nonlinear viscoelastic solids. We finish with the formulation and results of a complex thermomechanical process, the extrusion forming process of an aluminum billet under high temperature and viscoplastic behavior.

M. Doblaré (B) Aragón Institute of Engineering Research (I3A), University of Zaragoza, R+D Bldg, Block 5, first floor, Mariano Esquillor s/n, 50018 Zaragoza, Spain e-mail: [email protected] M. H. Doweidar Aragón Institute of Engineering Research (I3A), University of Zaragoza, Betancourt Bldg, Maria de Luna, 1, 50018 Zaragoza, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_3

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1 Notation and Fundamental Variables Following the notation in chapter “Basic Equations of Continuum Mechanics” [10], let Br ⊂ R3 denote the reference configuration of a continuum body B (open bounded set in R3 ), whose points are identified by their respective coordinates, X ∈ Br , with respect to a fixed but otherwise arbitrary reference frame {Eα } at time t = 0.1 Let χ t be a motion of Br ⊂ R3 in R3 , that is, a continuously differentiable one-to-one time-parameterized family of mappings from Br ⊂ R3 → Bt ⊂ R3 : χ t : Br → Bt ⊂ R3 , χ t (X) = x(X, t)

(1)

with Bt ⊂ R3 , the set of deformed configurations. In a deformable body, any spatial-dependent property can be described either by the evolution of its value along the trajectory, χ t (X) = x(X, t), of each material particle X (material or Lagrangian description), or by its value at each fixed location in space, occupied by different particles of the body, for each time instant (spatial or Eulerian description). In a thermomechanical process without changes in the material’s microstructure, the motion χ t and the absolute temperature θ (x, t) = Θ(X, t) > 0 are the only independent state variables that control the whole process. In this case, the only energy dissipation possible is due to thermal heat, without any internal mechanical dissipation. On the contrary, in materials with irreversible changes in the internal microstructure, additional mechanical dissipation happens. Those changes in the microstructure are described by additional fields (e.g., the volume fraction in porous media, fiber orientation in fibered materials, Nye’s tensor in dislocations),2 so additional balance equations (besides the classical variations of linear and angular momenta and energy conservation presented in chapter “Basic Equations of Continuum Mechanics” [10]) have to be introduced to describe the evolution of these additional microscopic fields. Another more classical approach to account for changes in the material’s microstructure is based on the introduction of a set of local internal variables, κ i (x, t), that are considered as non-observable state variables. Thus, no additional mechanical power associated with the internal variables develops, and no additional balance equation is used. On the contrary, the evolution of such internal state variables is defined by explicit phenomenological equations or from specific thermodynamic constraints. We shall consider this latter approach in the following, as well as the normal situation in thermomechanics, in which no source of energy provision other than mechanical work and thermal flux is considered.3

1 From

now on, we shall consider only Euclidean reference frames, without any restriction, but allowing a simpler notation. 2 Note that the material’s microstructure is associated with an intrinsic length scale, leading most times to size effects in the constitutive response. 3 Other possible energy sources (chemical, electrical, metabolical) are considered only in an indirect form through changes in the internal state variables.

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55

2 Energy Balance Principle and Entropy Inequality It is now possible to write the first principle of thermodynamics (energy conservation) for any open set χ t (Rr ) ⊂ Bt ⊂ R3 in the current configuration, with differential of volume dv = J dV , J = det F being the Jacobian of the deformation gradient F = Grad x, and dV the corresponding differential volume in the set Rr ⊂ Br ⊂ R3 , as d [K (Rt ) + S(Rt )] = P(Rt ) + Q(Rt ) in Rt = χ t (Rr ) ⊂ Bt ⊂ R3 for any Rr ⊂ Br , (2) dt

with K (Rt ) =

1 2

 ρ(x, θ, t)v(x, t) · v(x, t) dv

(3)

Rt

the kinetic energy, ρ(x, θ, t) being the material density and v(x, t) the material velocity at each spatial point x and time t ∈ I , with I ⊂ R the time interval of interest;  S(Rt ) =

ρe(x, θ, κ i , t) dv,

(4)

Rt

the internal energy, with e(x, θ, κ i , t) the internal energy per unit mass, depending on the microstructural state variables κ i , i = 1, ..., N 4 ;   P(Rt ) = ρb(x, t) · v(x, t) dv + t(x, t) · v(x, t) da, (5) ∂Rt

Rt

the external power (work per unit time) induced by the external forces per unit mass b,5 and per unit surface t = σ · n, acting on Rt or its surface ∂Rt , the latter with normal n and differential area da; finally, σ is the symmetric Cauchy stress tensor, as defined in chapter “Basic Equations of Continuum Mechanics” [10], and  Q(Rt ) =

 ρr (θ, x, t) dv +

Rt

∂Rt

j (θ, x, t) da,

(6)

the net heat supply per unit time, with r (θ, x, t) the volume heat supply per unit time and per unit mass, and j (θ, x, t) the heat influx per unit time and unit surface area. This latter may be also written as j = −q · n, with q(θ, x, t) the external flux vector per unit time and per unit area. Therefore, Eq. (2) may be rewritten as d dt

 Rt

    1 ρ e + v · v dv = ρ(b · v + r ) dv + [(σ n) · v + j] da, 2 Rt ∂Rt

(7)

variable κ i may have a different tensorial order (scalar, vector, or tensor) according to its definition. 5 In the following, we shall consider the usual case of body forces b independent of the displacements. 4 Each

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for any Rt ⊂ Bt , where we have removed the independent variables for clarity. Assuming now χ t as C1 -regular and the fulfillment of the balance principles of mass, linear and angular momenta stated in chapter “Basic Equations of Continuum Mechanics” [10], Eq. (7) can be localized into the equivalent differential form ρ e˙ + divq = σ : D + ρr,

(8)

where div stands for the spatial divergence, that is, the scalar div v = ∂vk /∂ xk , and σ : D denotes the double contraction between the two second-order tensors σ and D, defined as the scalar σi j Di j , e˙ = ∂e/∂t + v · grad e is the material derivative of e, and, finally, D is the spatial strain rate defined as the Lie derivative of the spatial Euler–Almansi strain tensor e: D = Lve =

 1  ˙ −1 FF + F−T F˙ T . 2

(9)

The proof of (8) from (7) is straightforward and may be found, for example, in [9]. Applying now the pull-back operation to (7) and taking into account the definition of the first Piola–Kirchhoff stress tensor S in chapter “Basic Equations of Continuum Mechanics” [10], the material counterpart of (7) is readily obtained as d dt



 ρ0 Rr

   1 ˆ · V + Jˆ) d A (10) E + V · V dV = ρ0 (B · V + R) dV + (T 2 Rr ∂Rr

for any Rr ⊂ Br , where ρ0 = ρ J is the initial density, Tˆ = S · N and Jˆ = −Q · N, with S = J F−1 σ and Q = J F−1 q, the Piola transformations of the Cauchy stress tensor, σ , and of the heat flux vector, q, respectively, and N the normal vector to the boundary ∂Rr . In local form, the material expression equivalent to (8) yields ρ0 E˙ + Div Q = S : F˙ + ρ0 R = T(2) : E˙ + ρ0 R,

(11)

where Div stands for the material divergence, that is, the scalar Div V = ∂ VK /∂ X K , and with T(2) = SF−T = J F−1 σ F−T we denote the second Piola–Kirchhoff stress tensor (also defined in chapter “Basic Equations of Continuum Mechanics” [10]) that corresponds to the convected form of the Cauchy stress tensor, and, finally, E˙ = FT DF is the material strain rate, i.e., the pull-back of the spatial strain rate D. To finish this first section, we introduce the expression of the second law of thermodynamics or entropy production inequality, also known as the Clausius–Duhem inequality. Under the same conditions as stated above, that is, only thermomechanical energy is considered, this may be written again in spatial form as d dt



 ρη dv ≥ Rt

Rt

r ρ dv + θ

 ∂Rt

j da, θ

(12)

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57

with η(x, θ, κ i , t) the entropy per unit mass.6 Again, the local form of the above equation may be obtained after assuming the continuous and bijective character of the motion and the fulfillment of the rest of balance principles. Thus, Γ˙ = ρ η˙ − ρ

q r + div ≥ 0, θ θ

(13)

which defines the net rate of entropy production in terms of the heat flux and the absolute temperature at a spatial point. The material versions of (12) and (13) follow after introducing the material versions of entropy and absolute temperature Υ (X, t) = η(x, t), Θ(X, t) = θ (x, t), to obtain    d R (Q · N) d A for any Rr ⊂ Br , ρ0 Υ dV ≥ ρ0 dV − (14) dt Rr Θ Θ Rr ∂Rr Γ˙ = ρ0 Υ˙ − ρ0

R + Div Θ



Q Θ

 ≥ 0.

(15)

Sometimes, these equations are rewritten after introducing the so-called Helmholtz free energy density function per unit mass in spatial and material forms: ψ = e − θ η, Ψ = E − ΘΥ.

(16)

It is easy to prove that the local versions (spatial and material) of (13) and (15) may be rearranged in terms of this function as ˙ = σ : D − ρ(ψ˙ + ηθ) ˙ − (q · grad θ )/θ D = σ : D − ρ(e˙ − θ η) ˙ − (q · gradθ )/θ ≥ 0,

(17)

˙ = S : F˙ − ρ0 (Ψ˙ + Υ Θ) ˙ − (Q · Grad Θ)/Θ JD = S : F˙ − ρ0 ( E˙ − Θ Υ˙ ) − (Q · Grad Θ)/Θ ≥ 0,

(18)

or, in convected form 6 When r = 0 and j = 0, that is, in adiabatic thermally isolated processes (no heat sources or sinks are present), this equation reduces to  d ρη dv ≥ 0, dt Rt

meaning that the total entropy of the system cannot decrease.

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˙ = T(2) : E˙ − ρ0 (Ψ˙ + Υ Θ) ˙ − (Q · Grad Θ)/Θ JD (2) ˙ ˙ ˙ = T : E − ρ0 ( E − Θ Υ ) − (Q · Grad Θ)/Θ ≥ 0,

(19)

˙ = θ Γ˙ the rate of energy dissipation. Expressions (17)–(19) are known as with D reduced dissipation inequalities. Other well-known thermodynamic functions such as the enthalpy h (spatial), H (material), or the Gibbs energy g (spatial), G (material), are related to the internal energy and the Helmholtz free energy via ρ h˙ = ρ e˙ − σ : D, ρ g˙ = ρ ψ˙ − σ : D,

˙ ρ0 H˙ = ρ0 E˙ − S : F˙ = ρ0 E˙ − T(2) : E, ˙ ρ0 G˙ = ρ0 Ψ˙ − S : F˙ = ρ0 Ψ˙ − T(2) : E.

(20) (21)

For purely mechanical processes, without heat flux and uniform and time-constant temperature, the inequalities (17)–(19) simplify to ˙ = σ : D − ρ ψ˙ = σ : D − ρ(e˙ − θ η) D ˙ ≥ 0, ˙ = S : F˙ − ρ0 Ψ˙ = S : F˙ − ρ0 ( E˙ − Θ Υ˙ ), JD = T(2) : E˙ − ρ0 Ψ˙ = T(2) : E˙ − ρ0 ( E˙ − Θ Υ˙ ) ≥ 0.

(22)

˙ = 0 is called reversible as distinct from irreversible processes, A process where D where the internal mechanical dissipation rate is positive, and the mechanical energy loss is transferred to thermal energy. Thus, an irreversible mechanical process turns, in general, into a thermomechanical process. Observe that the external work rate in these purely mechanical processes, represented in (22) by σ : D is “employed” in modifying the free energy density, while the rest is dissipated. With this, we establish the following relation and assume that it may be extended to any kind of thermodynamic process: ˙ ˙ i , x, κ˙ i , t), ˙ t) + D(κ W˙ = σ : D = V(F, x, F,

(23)

with V the so-called strain-energy density function (SEDF), equal to the free energy density (ρψ) in isothermal processes, and D the dissipation density function (DDF).

3 Fundamental Principles of Constitutive Models The number of unknowns appearing in the fundamental balance principles, revised in chapter “Basic Equations of Continuum Mechanics” [10], is higher than the number of available equations. Therefore, some additional equations are needed to formulate the problem correctly. The missing equations correspond to the so-called constitutive model that depends on the particular material under study. Considering as known data the external body forces, b(x, t), the external heat production, r (x, t), and the boundary conditions, needed to have a well-posed partial differential equation problem, we

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have five differential equations: three corresponding to balance of linear momentum, one for mass conservation and one more for energy conservation.7 With such equations, we can solve only five independent variables that are usually identified with the three components of the motion, x, the absolute temperature, θ (x, t), and the density, ρ(x, t). The time evolution of the latter can be obtained in terms of the motion by only solving the mass balance principle, so, we keep only four equations (balance of linear momentum and balance of energy) to solve the four fundamental unknowns of motion and temperature. The rest of the state functions have to be defined in terms of these four by means of the respective constitutive functions. Of course, the total deformation gradient, F, is obtained from the motion. However, only part of F contributes to the SEDF. This is called the elastic part of the deformation gradient, Fe . One possible reason is taking the kinematic reference state Br different from the origin of the SEDF.8 In this case, the corresponding kinematic reference state has to be stated explicitly by means of phenomenological “ad hoc” equations.9 Another possibility corresponds to irreversible microstructural changes induced by the considered thermomechanical external energy provision. This additional part of the total deformation gradient has to be included in the set of internal state variables κ i that contribute to the DDF. These internal state variable (thermodynamic drivers), κ i , have to be determined by additional conditions,10 whose expressions are established in terms of the so-called thermodynamic fluxes, σ κ i , that correspond to the thermodynamic complementary variables of each κ i , that is, those that appear multiplying the rate of each κ i in the expression of the dissipation rate. Therefore, the rest of state variables (stresses σ , heat flux q, internal energy e, entropy η, and the thermodynamic fluxes σ κ i ) will be defined from the unique definition of the SEDF and the DDF in terms of Fe and κ i , respectively. Aside from the physical experience and experimental observations, theoretical considerations for constitutive models rest upon the fulfillment of a set of general principles that are here considered as initial axioms and are revised briefly in the next lines.

7 Remember that balance of angular momentum, in standard non-micropolar continuum mechanics,

is equivalent to the symmetry of the Cauchy stress tensor, so only six independent functions will be considered for this tensor. 8 This means that such initial stored energy in the reference state is provided by an energy provision mechanism not considered explicitly in the equations. 9 For example, when internal growth is considered, the associated deformation gradient Fg or its rate F˙ g are described by a convenient expression. 10 For example, the need to remain on the surface of a certain (elastic) state domain, like in plasticity or damage, and/or additional thermodynamic assumptions like the fulfillment of the maximum dissipation principle.

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1. Causality. The state of the domain at time t must depend only of the past history up to time t. 2. Equipresence. If an independent variable enters in one constitutive equation, it should be present in all of them (until there is proof to the contrary).11 3. Determinism. The response at a material point X at time t is determined by the histories of motion and temperature of all points of the body up to such time t and the initial conditions only.12 4. Locality. Only thermomechanical histories in an arbitrarily small neighborhood of a point X affect the material response at X. Hence, the global history state functions can be approximated, in a small neighborhood of X of radius ε, by Taylor series in ε up to certain finite order.13 5. Memory. In the same way that locality affects the spatial expressions, memory affects time functions, so we usually assume that values of constitutive variables from a distant past do not affect constitutive laws now. Mathematically, this means that the global history state functions can be approximated, in a small time neighborhood of a time t of width ε, by Taylor series in ε up to certain finite order.14 6. Material frame-indifference. It is obvious that, since the constitutive functions must characterize the intrinsic properties of the material, they should be independent of the observer, so they have to be indifferent to the change of reference frame. 7. Physical admissibility. Constitutive equations have to be consistent with the entropy inequality for any thermodynamic process [3, 22]. 8. Maximum dissipation. This principle is assumed to be fulfilled in many constitutive models, playing also a crucial role in the variational formulation of many dissipative systems (see, e.g., [5, 8, 21] among many others).15 In particular, when only linear approximations of the Taylor expansion are used, any constitutive equation (for example, the convected expression of the stress tensor) can be written as (24) T(2) (X, t) = T(2) (Fe , Θ, κ i , X, t).

11 This

principle, although usually assumed, has been controversial and even some authors [15] ridiculed it, so it has to be taken with care. 12 Although some additional variables will appear in the following in the expressions of the state functions, e.g., internal state variables, their value at each time, according to this principle, must be obtained from the histories of motion and temperature at all points of the body up to such time t. 13 It is noteworthy that the fulfillment of locality is sometimes excessive. There exist materials whose constitutive behavior at one point also depends on the values of the configuration and temperature in a finite neighborhood of such a point, leading to a nonlocal theory that falls out the limits of this chapter (see, for example, [18]). 14 Again, this principle does not always hold, so it should be considered as an additional assumption. 15 This principle, although not directly derived from any fundamental universal law, and therefore, at a different level than the previous ones, is a direct consequence of the trend of any physical system to dissipate as much energy as possible and therefore to maximize entropy production.

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A material with constitutive equation (24) is called a simple material, being this class of materials general enough to include most cases of practical interest. For simple materials, the principle of material frame invariance takes the form ˆ i , X, t) = T(2) (Fe , Θ, κ i , X, t), T(2) (ΛFe , Θ, Λκ

(25)

for any orthogonal tensor Λ ∈ SO(3), representing a rigid rotation, and Λˆ appropriate tensorial operations depending on the tensorial character of the state variables, κ i , and derived from the rotation Λ.16 In the case of the stress, T(2) , it is easy to demonstrate that (25) implies that its dependence with respect to Fe is through the right Cauchy–Green tensor Ce = FeT Fe or through the Lagrange strain tensor Ee = (Ce − I)/2 with I the second-order material unit tensor. Therefore, and from now on, we shall write T(2) (Ee , Θ, κ i , X, t). The entropy principle (19), for a simple material, may then be written as 

   ∂Ψ e ∂Ψ e e ˙ (F , Θ, κ i ) + Υ (F , Θ, κ i ) Θ + S − ρ0 e (F , Θ, κ i ) : F˙ e − ρ0 ∂Θ ∂F ∂Ψ e 1 − ρ0 (F , Θ, κ i )κ˙ i − Q(Fe , Θ) · Grad Θ ≥ 0. (26) ∂κ i Θ This expression holds for any thermodynamic process satisfying the balance principles, so we finally have Υ =−

∂Ψ e ∂Ψ e (F , Θ, κ i ), S = ρ0 (F , Θ, κ i ), ∂Θ ∂F

(27)

with the constraints 1 Q(Fe , Θ, κ i ) · Grad Θ ≤ 0, Σ κ i κ˙ i ≥ 0 ∀ i = 1, . . . , N . Θ

(28)

where Σ κ i = −ρ0 ∂Ψ/∂κ i are the thermodynamic fluxes, each associated with a thermodynamic driver.17 The principle of maximum dissipation states that, for a given state (Fe , Θ, κ i , X, t) and for the rates κ˙ i to be the actual solution of the continuum problem, the dissipation rate has to achieve a maximum, i.e., ˙ e , κ i , Θ, X, t, λ˙ i ). ˙ e , κ i , Θ, X, t, κ˙ i ) = max D(F D(F λ˙ i

(29)

equations can be written for the rest of constitutive equations (Q, Σ κ i ), since E and Υ , as scalar fields, do not change under rotations. 17 Observe that in purely mechanical processes, where (23) holds, Σ κ i = ∂D/∂κ , ∀ i = 1, .., N , i ˙ = (∂D/∂κ i )κ˙ i ≥ 0. which implies that D 16 Similar

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M. Doblaré and M. H. Doweidar

Finally, this principle usually implies convexity of the rate of DDF with respect to the rate of the thermodynamic drivers [20]. For a simple material, this reads18 ˙ e , κ i , Θ, X, t, (κ˙ i )1 ] + (1 − α)D[F ˙ e , κ i , Θ, X, t, (κ˙ i )2 ] α D[F ˙ e , κ i , Θ, X, t, α(κ˙ i )1 + (1 − α)(κ˙ i )2 ]. > D[F

(30) (31)

4 Examples of Non-dissipative Materials In non-dissipative materials, there is no change in the constitutive behavior after deformation or time, so the unique independent variables are the elastic deformation gradient, which is equal to the total deformation gradient when the elastic reference configuration coincides with the undeformed one, Fe = F, and the temperature θ , being κ i = 0.

4.1 Ideal Fluids An ideal fluid is defined as a material such that the independent variables are the temperature and the density19 or, equivalently [17], the entropy η instead of temperature and the specific volume γ = 1/ρ instead of density. The rate of dissipation (17) may be written again in terms of the internal energy e as



∂e

1 ∂e

q · gradθ + σ : D − div v ≥ 0, (32) η ˙ − ρ θ− ∂η γ˙ =0 θ ∂γ η=0 ˙ where we have used the mass balance law ρ˙ = −ρdiv v and the immediate expression γ˙ = −ρ/ρ ˙ 2 = div v/ρ. If we now take into account that div v = trD = i : D, with i the second-order unit spatial tensor, and introduce the tensor s = −∂e/∂γ |η=0 ˙ i + σ , we can rewrite (32) to obtain

 ∂e

1 ρ θ− η˙ − q · grad θ + s : D ≥ 0. (33)

∂η γ˙ =0 θ Since the three terms in (33) can be modified arbitrarily without modifying the other two and ρ > 0, θ > 0, (32) is equivalent to that constitutive functions for T(2) , Υ, Σ κ i are determined by a single scalar function Ψ (Fe , Θ, κ i ) (equivalently E(Fe , Θ, κ i )) or, for purely mechanical problems, by the two functions V(Fe ), D(κ i ), while the constitutive function for the heat flux satisfying (28) has to be added explicitly. 19 Observe that the dependence on motion here is only through changes in density or, equivalently, through changes in volume. 18 Note

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63

θ−

∂e

≥ 0, q · grad θ ≤ 0, s : D ≥ 0. ∂η γ˙ =0

(34)

The second inequality recovers the well-known fact that heat flows from higher to lower temperature, while the third states that the deformation always dissipates mechanical energy, transforming it into heat. Also, we notice that the tensor s must change its sign with D, which means that for null velocity v = 0, s = O ⇒ σ = coincides, except for the sign, with the ∂e/∂γ |η=0 ˙ i, which indicates that ∂e/∂γ |η=0 ˙ internal mechanical pressure of the fluid. Thus p=−

∂e

, σ = − p i + s, ∂γ η=0 ˙

(35)

such that the tensor s may be identified with the deviatoric (viscous) stress tensor dependent on the strain rate tensor D. Finally, from the first inequality in (34) θ=

∂e

, ∂η γ˙ =0

(36)

which provides the link between the two independent variables, (θ, η). Then, we can write



∂e

∂e

p e˙ = η ˙ + γ˙ = θ η˙ − div v, (37)



∂η γ˙ =0

∂γ η=0 ˙

ρ

and including this in the energy balance equation (8) we finally get ρθ η˙ = −div q + s : D + ρr,

(38)

which corresponds to the thermomechanic heat transfer equation in ideal fluids. As an example, for ideal gases, the internal energy density is expressed as e(η, γ ) =

exp [(η − η0 )/Cv ] Cv , R γ β−1

(39)

with Cv the specific heat at constant volume, β = C p /Cv and R = C p − Cv , with C p the specific heat at constant pressure. Applying the equations above, we get from (35) and (36) θ=

∂e

e = or e = Cv θ,

∂η γ˙ =0 Cv

p=−

∂e

e = (β − 1) , ∂γ η=0 γ ˙

(40)

(41)

and using (40) we arrive at pγ = Rθ , that is, the typical state equation for perfect gases.

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4.2 Elastic Solids Under Coupled Thermoelastic Processes The solid elastic behavior is characterized by the following two conditions [3]: (i) The stress is a unique function of the deformation Fe = F and the temperature. (ii) For adiabatic processes (r = 0, q = 0), the material completely recovers to a “natural” reference state after removal of the applied forces, i.e., for every admis˙ is zero. The material is, therefore, sible process the mechanical dissipation rate D non-dissipative. Using now (27), we have S = ρ0

∂Ψ (F, Θ) ∂Ψ (F, Θ) , Υ =− , ∂F ∂Θ

E =Ψ −Θ

∂Ψ (F, Θ) . ∂Θ

(42)

˙ = E˙ − Θ Υ˙ , using as independent variables F By taking into account that Ψ˙ + ΘΥ and Υ , and following a similar process, we arrive to similar equations in terms of the internal energy density20 S = ρ0

∂ E(F, Υ ) ∂ E(F, Υ ) ∂ E(F, Υ ) , Θ= , Ψ = E −Υ . ∂F ∂Υ ∂Υ

(43)

Observe that, since the mechanical dissipation is now null, the SEDF for isothermal processes V(F) = ρ0 Ψ (F, Θ = Θ0 ) while, for isoentropic processes, V(F) = ρ0 E(F, Υ = Υ0 ), and hence ∂V . (44) S= ∂F In other more general thermodynamic processes, the existence of the strain energy density function is not clear, although the hyperelastic model for solids assumes its existence as an initial axiom, with the following restrictions usually added: • Although not essential, since the value of the energy is indeterminate and only energy differences have physical significance, the strain-energy function is usually assumed to be zero in the reference configuration V(F = 1) = 0. • Although there is a large body of literature dealing with nonvanishing initial stresses, we shall consider in the following, as occurs in most cases, that the reference configuration is stress free S(F = 1) = O. • V must satisfy the so-called growth condition V → +∞ if J = det F → +∞ or 0. • V must be quasiconvex [2].21 20

Note that, in this particular case, all constitutive functions are determined by a single scalar ˆ function Ψ = Ψˆ (F, Θ) (E = E(F, Υ )), except the heat flux, which satisfies the first inequality of (28). 21 Observe that the condition of polyconvexity usually requested for hyperelastic strain energy functions is not necessary, but, on the other hand, is a sufficient condition for quasiconvexity [2].

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When having initial thermal deformations Fθ in an elastic material under finite deformation, the total deformation gradient F may be expressed in a multiplicative form as22 F(X, Θ, t) = Fe (X, Θ, t)Fθ (X, Θ, t). (45) From this, it is immediate that 1  ˙e e −1 1 ˙ −1 (FF + F−T F˙ T ) = F (F ) + Fe F˙ θ (Fθ )−1 (Fe )−1 2 2  + (Fe )−T (Fθ )−T (F˙ θ )T (Fe )T + (Fe )−T (F˙ e )T = De + Dθ ,

D =

(46)

with  1  ˙ e e −1 F (F ) + (Fe )−T (F˙ e )T , 2  1  e ˙ θ θ −1 e −1 Dθ = F F (F ) (F ) + (Fe )−T (Fθ )−T (F˙ θ )T (Fe )T . 2 De =

(47) (48)

It is clear that De is the standard definition of the strain rate tensor for the elastic part, while Dθ is the “elastic” push-forward of the strain rate tensor for the thermal part from the intermediate configuration (obtained after applying thermal strains only) to the final configuration. Using now the push-forward and pull-back operations and the definitions of the Cauchy and second Piola–Kirchhoff stress tensors in terms of the first Piola–Kirchhoff stress tensor included in chapter “Basic Equations of Continuum Mechanics” [10], from (44) it is possible to write T(2) = 2

∂V ∂V 1 ∂V = , σ = F e FT . e e ∂C ∂E J ∂E

(49)

Therefore, the distribution of initial stresses associated with the thermal deformation gradient may be written as23 1 σ = θ Fθ J θ

22 For



∂Ψ (E, Θ) ∂E





θ



˙ E=E Θ=0

(Fθ )T ,

(50)

an isotropic material, for example, Fθ = λθ I, λθ = exp



θ θ0

 α(θ) dθ ,

with λθ the isotropic thermal stretch and α the coefficient of thermal expansion of the material. fulfills internal and boundary compatibility, but, in general, this is not the case. This means that, except for such a compatible situation, the above stresses are not, in general, in equilibrium. An intermediate step is usually solved to get equilibrium under zero external forces, leading to a different set of strains, that may not be fully compatible yet. This is a problem intrinsically associated to this phenomenological decomposition of the deformation gradient.

23 These stresses are null if and only if Fθ

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M. Doblaré and M. H. Doweidar

and after application of the external loads, the total stresses become



∂Ψ (E, Θ)

FT



∂E E=Etot Θ=0 ˙

 

1 e ∂Ψ (E, Θ)

= e F J eσ θ + FeT .

J ∂E e ˙ E=E Θ=0

1 σ = F J



(51)

Including this in the balance equation (8), we get ρ e˙ + divq = σ : De + σ : Dθ + ρr,

(52)

˙ we get or, taking into account that ρ e˙ = σ : De + ρ ∂e/∂θ |De =0 θ,

∂e

θ˙ + div q = σ : Dθ + ρr, ρ ∂θ De =0

(53)

which corresponds to the heat transfer equation in thermoelastic processes. Remembering now the definition of the specific heat, C p , for a solid

∂e

, Cp = ∂θ De =0

(54)

then, the heat transfer equation for an elastic material can be written finally as div q = −ρC p θ˙ + σ : Dθ + ρr,

(55)

and, including the Fourier relationship for the heat flux (q = −Kgrad θ ), we get the differential equation that governs the heat transfer process in coupled thermoelasticity div(Kgrad θ ) = ρC p θ˙ − σ : Dθ − ρr,

(56)

where it is observed that there is a coupling between the variation of the deformations with time and the variation of temperature. That equation, together with the equation of balance of linear momentum, constitute the basic equations of coupled thermoelasticity. The balance of linear momentum depends on the stress that now depends on temperature since the elastic strain depends on both, total and thermal strains, getting a two-way coupled problem.24 24 In

the presence of an adiabatic process (r = 0, q = 0), the heat transfer equation yields θ˙ =

σ : Dθ , ρC p

which defines the temperature rate in an isolated material due to the strain rate, and constitutes the so-called Kelvin formula.

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67

Finally, the rate of dissipation yields ˙ = σ : Dθ − ρC p θ˙ − ρθ η˙ − 1 q · grad θ ≥ 0, D θ

(57)

which has to be fulfilled in any thermoelastic process.

5 Examples of Dissipative Materials As commented above, dissipative behavior (even in non-history-dependent materials such as viscoelastic ones) is usually written in terms of history-dependent strainlike internal state variables κ i . In the following, we summarize the corresponding equations, without considering initial strains, for simplicity. As it is impossible to describe in detail all possible dissipative mechanisms (see other chapters in this volume), we shall only sketch here some particular cases.

5.1 Continuum Damage Mechanics The terminology damage mechanics has been used in many different ways but, from a constitutive point of view, it refers to a reduction of the stress–strain curve slope (stiffness) during unloading. This is physically related to progressive material degradation due to the gradual appearance of microcracks or microvoids [12]. A typical example is the well-known Mullins effect in rubber materials [13]. When a virgin rubber sample is stretched from the undeformed state to a certain deformation, the stress–stretch curve follows the so-called primary loading curve. The subsequent unloading is characterized by a softened behavior. After reloading, it follows the unloading curve until the previous maximum stretch. At this point, the loading path swings up and traces the primary curve again. In Continuum Damage Mechanics (CDM), the process of progressive deterioration is formulated in terms of the evolution of an internal variable with scalar or tensorial character. For isotropic damage, that is, when the microcrack distribution is assumed to be independent of the direction, and isothermal processes (the extension to thermo-damage processes is straightforward following the concepts introduced above), we postulate a representation of the free energy density Ψ for isotropic damage as [6] (58) Ψ (E) = (1 − κ)Ψ 0 (E), with κ the damage internal state variable (thermodynamic driver of the process) and Ψ 0 the free energy density for the undamaged (virgin) material. Using again standard arguments based on the Clausius–Duhem inequality

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M. Doblaré and M. H. Doweidar

˙ = −ρ0 Ψ˙ + T(2) : E˙ ≥ 0, JD

(59)

and, together with (58), we get

∂Ψ 0 (2) ˙ : E˙ + ρ0 Ψ 0 κ˙ ≥ 0. J D = T − (1 − κ)ρ0 ∂E

(60)

Equation (60) leads to the representation T(2) = ρ0 (1 − κ)

∂Ψ 0 (E) = (1 − κ)(T(2) )0 , ∂E

(61)

with (T(2) )0 = ρ0 ∂Ψ 0 (E)/∂E the second Piola–Kirchhoff stress tensor for the undamaged material, whereas the principle of positive dissipation leads to ˙ = Σ κ κ˙ ≥ 0, JD

Σ κ = −ρ0

∂Ψ = ρ0 Ψ 0 (E) ≥ 0 ∂κ

(62)

with Σ κ the thermodynamic flux associated with the internal variable κ. In order to complete the constitutive model, we have to determine the evolution equation for the damage variable κ. Two coupled damage mechanisms are considered as examples: (i) Mullins-type discontinuous damage, where damage accumulation occurs only within the first cycle of a strain-controlled loading process. Further strain cycles below the maximum effective strain energy reached will not contribute to damage; (ii) continuous damage accumulation within the whole strain history of the deformation process, which is also governed by the local effective strain energy [11]. The total damage is then described by the expression . κ = κ α (α) + κ β (β),

(63)

where κ α : R+ → R+ and κ β : R+ → R+ are monotonically increasing smooth functions with the properties κ α (0) = 0, κ β (0) = 0, κ α (α) + κ β (β) ∈ [0, 1] ∀α, β, α and β describing the discontinuous and the continuous damage, respectively. These new variables are related to the evolution of the damage thermodynamic flux Σ κ as follows. The discontinuous damage (Mullins-type) is assumed to be governed by the variable  . ρ0 Ψ 0 (E(s)) = max Ξ (E(s)). (64) α(t) = max s∈(−∞,t)

s∈(−∞,t)

Thus, α(t) is simply the maximum thermodynamic flux, equal to the square root of the virgin free energy density, which has been achieved within the loading history interval [0, t]. We define a damage criterion [19] Φ(E(t), Ξ ) =



ρ0 Ψ 0 (E(t)) − α(t) = Ξ (E(t)) − α(t) ≤ 0.

(65)

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69

The equation Φ(E(t), Ξ ) = 0 defines a damage surface in the thermodynamic flux space. Finally, the evolution of the damage parameter κ α is characterized by an irreversible equation such as  α

κ˙ =

¯ h(Ξ, α)Ξ˙ if Φ = 0 and N : E˙ > 0 0 otherwise.

(66)

This underlines the discontinuous character of this damaging effect. In fact, there is no damage accumulation if the thermodynamic force Σ κ lies inside the undamaged domain Dα := {Ξ ∈ R+ |Ξ − α(t) ≤ 0}. Here, N := ∂Φ/∂E represents the normal to the damage surface in the strain space, Ξ is defined at the current time t, and ¯ h(Ξ, α) is a given function that characterizes the damage evolution of the material, ¯ and that strongly depends on the application. A typical expression for h(Ξ, α) is      α(t) Ξ 1 α 1 − exp − α ¯ ¯ ) dΞ = κ∞ , h(Ξ, α) = α exp − α ⇒ κ α (Ξ ) = h(Ξ γ γ γα 0

(67) α describing the discontinuous damage for α(t). Thus, we have the constraint with κ∞ α ∈ [0, 1]. κ∞ On the other hand, continuous damage is assumed to be governed by the arclength of the respective driving damage thermodynamic flux: . β(t) =



t

|Σ˙ κ (s)| ds.

(68)

0

Thus, we have the simple evolution equation ˙ = |Σ˙ κ (t)|, β(t)

(69)

with the initial condition β(0) = 0. Therefore, β monotonically increases within the deformation process. Finally, κ β is assumed to have the form β

κ = β

β κ∞

  β 1 − exp − β , γ

(70)

with κ∞ describing the continuous damage for β(t). We refer to γ α and γ β as the damage saturation parameters.

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5.2 Nonlinear Viscoelasticity In this case, the internal state variables κ i correspond usually to stress-like functions associated with different dissipative mechanisms. We can, therefore, write Ψ = Ψˆ (Ee , κ i ).25 Considering, for example, the usual case of a decoupled strain energy density function, we can write Ψ (Ee , κ i ) = Ψ e (Ee ) −

N 

Ee : κ i + Ξ (κ i ).

(71)

i=1

With this,

N N  ∂Ψ (Ee , κ i )

∂Ψ e (Ee )  (2) e = − κ = (T ) − κi , T = i

∂Ee ∂Ee κ˙ i =0 i=1 i=1

N  ∂Ψ (Ee , κ i )

˙ JD =

˙ e κ˙ i . ∂κ (2)

i=1

i

(72)

E =0

In this case, the evolution equation for each κ i is defined as κ˙ i +

1 γi κ i = (T(2) )e , τi τi

(73)

mechanism and γi the with τi the so-called characteristic time for each viscoelastic N γi = 1. so-called relative moduli with the constraint γ∞ + i=1 Of course, at t → ∞, we have to achieve thermodynamic equilibrium, so γi lim κ i = 0, i = 1, . . . N ⇒ κ i = t→∞ τi

  t −s (T(2) )e ds. exp − τi −∞



t

(74)

Including (74) in (73) for t = ∞, we have κ˙ i = 0 ⇒ κ i = γi (T(2) )e .

(75)

On the other hand, from the expression of the dissipation, the thermodynamic fluxes (now strain rate-like variables) α i are defined as

∂Ψ (Ee , κ i )

∂Ξ αi = − = Ee − . (76)

∂κ i ∂κ i E˙ e =0

25 We

consider again the isothermal case without restriction, working with the appropriate strainenergy density function when required.

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71

Again, at t → ∞, we get thermodynamic equilibrium, which means that the thermodynamic fluxes have to be null, so lim α i = 0 ⇒ Ee =

t→∞

∂Ξ , ∂κ i

(77)

which defines Ξ (κ i ) as the Legendre transformation of the elastic strain energy density function V (= Ψ e for isothermal processes), such that Ξ=

N  (−γi Ψ e (Ee ) + κ i : Ee ).

(78)

i=1

Finally, substituting κ i in the expression of the second Piola–Kirchhoff stress tensor and integrating by parts, we get   t −s (T(2) )e (s) ds exp − τ i −∞ i=1 i=1    N t  t −s (T(2) )e (s) ds, = γ∞ (T(2) )e + γi exp − τ i −∞ i=1

T(2) = (1 −

N 

γi )(T(2) )e +

N 



γi

t

(79)

with value at equilibrium lim T(2) = γ∞ (T(2) )e .

t/τi →∞

(80)

5.3 Thermoviscoplasticity. Application to Aluminum Extrusion Extrusion is a manufacturing process to obtain long and straight semi-finished metallic products of constant section such as tubes, cables, or metal profiles. The basic principle is very simple: a block is introduced in a chamber and passed through a matrix with a prefixed cross shape after applying a high rear pressure by means of a plunger that produces a significant reduction of the section. The process can be carried out at room temperature or more usually at high temperatures, depending on the alloy and the extrusion method used (Fig. 1). Aspects such as the composition of the alloy, properties of the chamber, its internal surfaces and those of the block to be extruded, the reduction ratio, the extrusion rate, the temperature and the cooling scheme after leaving the press, all affect the final properties and microstructure of the resultant profile as well as its surface finishing. In particular, the control of the temperature and pressure during the process is essential: when the temperature increases, the pressure to be applied in the plunger reduces, so

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M. Doblaré and M. H. Doweidar

Fig. 1 Picture of the exit of the extrusion process for aluminum profiles

higher extrusion speeds can be reached; however, a too high extrusion temperature causes superficial imperfections, deviations in the straight shape and reduction in the strength. Much attention has been paid to the design of the trawls and the cooling of the matrix to get an appropriate temperature rise and distribution all along the process. This is a very complex coupled thermomechanical process with high nonlinearity in the metal constitutive behavior (temperature-dependent viscoplastic behavior coupled with heat transfer and temperature changes) and in the friction between the billet and the dye walls that, together with the deformation, induce additional heat. Elastic strains are, in general, much smaller than the plastic ones, except in the exit area where, depending on its geometry, and due to the strong reduction in the plastic deformations, they have great influence in the final result (spring-back effect). Also, the yield stress and the material viscosity are much dependent on the temperature. It is very important, therefore, to compute the correct distribution of temperature, strains, and stresses along the whole process inside and outside the press as well as to calculate the properties of the material for such temperatures. Different techniques can be applied to simulate the extrusion process, numerically. Most of them are based on the Finite Element Method (FEM) [7]. The first approximation is based on a Lagrangian formulation, in which the mesh moves together with the material. The biggest drawback that this technique presents is that the mesh is so distorted that it is necessary to perform many remeshing and projection steps, which is costly and ineffective, especially in three-dimensional problems. In addition and in many cases, the original billet is transformed into the extruded profile with thickness reduction ratios over 100 in a very short distance, so it is necessary to use very fine meshes with a huge number of elements, which again highly raises the computational cost and the frequency to perform remeshing. Another possibility is using an Eulerian approach as in fluids, eliminating all the problems associated with the distortion of the mesh. However, it is much more difficult to track the evolution of the material, since the mesh is not connected to it. Besides, it is very difficult to

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73

obtain an exact description of the free surface of the extrudate, which is an important result in the simulation of the extrusion process. A third alternative is the so-called Arbitrary Lagrangian–Eulerian (ALE) technique that combines the two previous formulations. The method consists of three steps: a loading step with an updated Lagrangian formulation, then a remeshing step and, finally, a projection of the results. The difference is that the new mesh is not arbitrary, but the original mesh is modified according to the speed of the material. With this, the meshes are smoother and the remeshing and projection step is simpler and more accurate. In the past few years, a new family of numerical methods has been developed, generically referred to as meshless methods. In these methods, the connectivity between the nodes is not explicitly defined but it is calculated at each time step (so it can change along the computation) thus uncoupling the material behavior tracked with the nodes as in the total Lagrangian approach and the interpolation step, avoiding mesh distortion. One of these methods is the Natural Neighbor Galerkin Method or Natural Element Method (NEM) [4] that has been applied to extrusion processes in [1]. The equation that governs the thermal problem is, as usual, the balance of energy (56), but now adding all mechanical dissipative effects (viscous and plastic). With this, (81) ρ e˙ = ρC p θ˙ = div(Kgrad θ ) + σ : Dne + ρr, where C p is the specific heat of the solid, K the heat conductivity tensor, and Dne the strain rate for nonelastic strains.26 Although, as commented, the temperature changes the density, the conductivity, and the specific heat, viscosity has much greater variation, so, in the following examples, we have considered as an approximation, constant values for density, specific heat and conductivity. Finally, the generation of heat, and consequently, the distribution of temperature depends on σ : Dne . One constitutive model usually used in these kinds of problems is the so-called Sellars–Tegart model [16] or its simplified version, known as Norton–Hoff model [1]. The first one is a viscoplastic model while the second neglects the plastic part, assimilating the metal behavior to that of a non-Newtonian fluid. Both neglect the elastic strains due to their very small value with respect to the viscoplastic strains, obtaining directly the stresses from the strain rate, viscosity and yield stress. The Sellars–Tegart model uses the standard Huber–Mises yield criterion but generalized in the sense that the yield stress depends now upon the strain rate and temperature. Therefore,  f (σi j , Di j ) = 0,

3 ¯ θ ) = 0, si j si j − σ f ( D, 2

(82)

26 Normally, it is assumed that around 90% of the energy mechanically generated becomes heat while the remaining 10% goes for generating plastic dislocations, generation of grain boundaries and phase changes [23]. Therefore, we can also write σ : Dne ≈ 0.9σ : D.

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M. Doblaré and M. H. Doweidar

 √ with si j = σi j − σkk δi j /3 the deviatoric stress tensor, σ¯ = 3si j si j /2 = 3J2 the Huber–Mises effective stress, with J2 the second invariant of the deviatoric stress tensor, and, finally, the yield stress σ f follows a Sellars–Tegart-type law ¯ θ ) = Sm arcsinh σ f ( D,

 

 1/m  D¯ Q/RT e , A

(83)

where Sm , m, and A are material parameters, Q is the activationenergy of the vp vp deformation process, R is the universal constant of gases, and D¯ = 2Di j Di j /3 is the equivalent viscoplastic strain rate. It can be observed that for null strain rate, the yields stress becomes null. To avoid this, an initial equivalent viscoplastic strain rate D¯ 0 is added to the model, getting finally ¯ θ ) = Sm arcsinh σ f ( D,

 

 1/m    D¯ 1 Q/RT ¯ D¯ 0 . (84) e , with D¯ 1 = max D, A

Other authors [7] propose another type of modification to account for the metal initial resistance (see Fig. 2): ¯ θ ) = Sm arcsinh σ f ( D,

 

 1/m  D¯ + D¯ 0 Q/RT e . A

(85)

Figure 2 shows how these modifications of the Sellars–Tegart law are negligible for high deformation velocities such as those usually happening in extrusion, while they do modify the results for low deformation velocities. Using now an associate flow rule vp

Di j = Di j = γ˙

∂ f (σi j , Di j ) 3si j , = γ˙ ∂σi j 2σ¯

(86)

it can be easily checked, using the expressions (82) and (86) that γ˙ is precisely the equivalent viscoplastic strain rate:      3D 3D γ ˙ 2 3 i j i j γ˙ γ˙ = si j si j = γ˙ . D¯ = 3 2σ¯ 2σ¯ σ¯ 2

(87)

On the other hand, and following the Perzyna model for viscoplastic solids [14], it is also possible to obtain the value of the equivalent stress as a function of the effective strain rate as < σ¯ − σ f > ⇒ σ¯ = σ f + η D¯ if σ¯ ≥ σ f , D¯ = γ˙ = η

(88)

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Fig. 2 Curves showing the modifications induced in the Sellars–Tegart model by the inclusion of an initial strain rate

with η ≥ 0 the fluidity parameter for viscoplastic materials, and  < • >=

• for • > 0 0 for • ≤ 0.

Introducing now this relationship in the expression (86), we obtain finally the constitutive equation for this material27 si j = 2

¯ θ) η D¯ + σ f ( D, Di j . 3 D¯

(89)

Adding now the hydrostatic component, we get the total stresses as 27 Note that this fluidity parameter corresponds to the viscosity, if we reduce this model to a generalized viscous one. On the contrary, if this fluidity parameter is negligible (taken as null), this model identifies with a purely plastic behavior, reducing to

si j =

¯ θ) 2σ f ( D, Di j . 3 D¯

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σi j = − pδi j + si j = − pδi j + 2

¯ θ) η D¯ + σ f ( D, Di j , 3 D¯

(90)

where the pressure has to be obtained by forcing the incompressibility condition for vp the viscoplastic strains Dkk = 0. Another widespread model for extrusion processes is the so-called called Norton– Hoff model [1]. As in the Sellars–Tegart one, this latter assumes the material to be incompressible. With this, the deviatoric stress tensor is expressed as √ ¯ n−1 Di j , si j = 2K ( 3 D)

(91)

where n is the strain rate sensitivity index, and K = K 0 (¯ε + ε¯ 0 )h is the consistency parameter, with h the hardening index of the material, ε¯ the equivalent strain result of the integration of D¯ along the process, and K 0 and ε¯ 0 two model parameters. It can be observed that the stresses depend both on the strain rate and on the strain itself and therefore on the whole process history. A particular case of this behavior is the so-called “Generalized Newtonian Fluid”,√which does not exhibit ¯ n−1 Di j , equivalent D) strain hardening, so h = 0, giving immediately si j = 2K √0 ( 3n−1 ¯ to a viscous fluid with no more than identifying K 0 ( 3 D) with the viscosity coefficient η. Adding again the hydrostatic pressure, we get the constitutive equation for the Norton–Hoff model: √ ¯ n−1 Di j − pδi j . σi j = 2K 0 ( 3 D)

(92)

The following example (see [1]) shows the extrusion of a cross-shaped billet from an initial cylinder with 50 mm diameter. The geometry of the dye is presented in Fig. 3, having the arms of the cross a thickness of 6 mm. Thanks to the symmetry of the section, considering a dye without geometric imperfections, it is only necessary to simulate an eighth part of the billet. Symmetry conditions were, therefore, imposed on the X = Y and X = 0 planes. The plunger was imposed with a speed of one meter per minute. Friction with the walls of the press and dye was discarded. The material used was aluminum Al6063, an alloy widely used in hot extrusion processes. We consider a thermoplastic behavior following a Sellars–Tegart law. The initial aluminum temperature was set to 700 ◦ C and all boundaries were simulated as adiabatic, since, for small lengths of extrusion, the heat dissipated by convection is very small in the exit area compared to that generated. The properties of the material are shown in Table 1 and a time increment of Δt = 0.025 s was used. In the following, we shall present the results obtained by using as the discretization approach the NEM, which, as commented above, allows to overcome most problems associated with the large geometric changes in the extrudate. Figure 4a shows the distribution of the strain rate at t = 1 s, directly related to the velocity distribution and Fig. 4b the Cauchy stress distribution at the same time. Figure 5a shows that the pressure distribution in the billet is practically constant (as results to be throughout the whole process). Finally, the temperature distribution is shown in Fig. 5b, where it is clearly observed that the heat is generated in the

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Fig. 3 Geometry of the cross-shaped extrusion dye used for the example Table 1 Properties of aluminum Al6063 for the Sellars–Tegart model Sm D¯ 0 Am

25 N mm−2 0.005 s−1 6 · 109 s−1

Q R m

1.4 · 105 J mol−1 8.314 J mol−1 K−1 5.4

Fig. 4 Distributions of the strain rate (a) and Cauchy stress (b) at t = 1 s

areas of highest strain rate while the temperature in the rest of the billet increases by conduction. The total temperature increase, in this case, was approximately eight degrees, lower than that expected due to not considering friction with the walls.

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Fig. 5 Distribution of pressures (a) and temperatures (b) in the aluminum billet at t = 1 s

References 1. Alfaro I, Yvonnet J, Cueto E, Chinesta F, Doblaré M (2006) Meshless methods with application to metal forming. Comput Meth Appl Mech Eng 195:6661–6675 2. Ball J (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 61:317–401 3. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13:167–178 4. Cueto E, Sukumar N, Calvo B, Martínez-Barca MA, Cegonino J, Doblaré M (2003) Overview and recent advances in natural neighbour Galerkin methods. Arch Comput Meth Eng 10:307– 384 5. Duvaut G, Lions JL (1972) Les Inéquations en Mécanique et en Physiques. Dunod, Paris 6. Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, Cambridge 7. Lof J, Blokhuis Y (2002) FEM simulations of the extrusion of complex thin-walled aluminium sections. J Mater Process Technol 122:344–354 8. Lubliner J (1984) A maximum-dissipation principle in generalized plasticity. Acta Mech 52:225–237 9. Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover Publications, New York 10. Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). Springer 11. Miehe C, Keck J (2000) Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation. J Mech Phys Solids 48:323–365 12. Montagut E, Kachanov M (1988) On modeling a microcracked zone by weakened elastic material and on statistical aspects of crack-microcrack interaction. Int J Fract 37:R55–R62 13. Mullins L (1947) Effect of stretching on the properties of rubber. J Rubber Res 16:275–289 14. Perzyna P (1966) Fundamental problems in visco-plasticity. Adv Appl Mech 9:243–377 15. Rivlin RS (1997) Red herrings and sundry unidentified fish in nonlinear continuum mechanics. In: Barenblatt GI, Joseph DD (eds) Collected papers of R.S. Rivlin. Springer, New York 16. Sellars CM, Tegart WJ McG (1972) Hot workability. Int Metall Rev 17:1–24 17. Serrin J (1959) Mathematical principles of classical fluid mechanics. In: Flügge S (ed) Encyclopedia of physics, vol VIII/1. Springer, Berlin, Heidelberg, pp 125–263

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18. Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184 19. Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Meth Appl Mech Eng 60:153–173 20. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York 21. Temam R (1985) Mathematical problems in plasticity. Gauthier-Villars, Paris 22. Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Handbuch der Physik, III(1). Springer, Berlin, Heidelberg, pp 225–793 23. Zhou J, Li L, Duszczyk J (2003) 3D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Lagrangian approach. J Mater Process Technol 134:282–297

Viscoelastic Solids Alan Wineman

Abstract Elastomers and soft biological tissues can undergo large deformations and exhibit time-dependent behavior that is characteristic of nonlinear viscoelastic solids. An overview of this subject is contained herein, beginning with a review of pertinent topics from linear viscoelasticity. After stating the general constitutive assumption for nonlinear viscoelastic solids, and then imposing restrictions imposed by consideration of superposed rotations and material symmetry, a number of specific forms that have been proposed in the literature are discussed. The emphasis is then confined to nonlinear single integral constitutive equations, specific cases being finite linear viscoelasticity and the Pipkin–Rogers constitutive equations. The latter contains, as a special case, the quasi-linear viscoelastic model used in the biomechanics of soft tissue. Representations for the Pipkin–Rogers model are provided for isotropy, transverse isotropy, and orthotropy. Uniaxial stretch histories for isotropic materials are used to show the deviation from linear behavior as nonlinear effects become important. A number of examples involving non-homogeneous deformations that have appeared in the literature are summarized.

1 Introduction Elastomers and soft biological tissues can undergo large deformations and exhibit time-dependent behavior, both of which are characteristics of nonlinear viscoelastic solids. Whereas the theories of linear viscoelastic solids and nonlinear elastic solids have well-defined constitutive structures, there is not yet a generally accepted constitutive framework for nonlinear viscoelastic solids. Findley et al. [9] and Lockett [23] provided summaries of the constitutive theories as of about 30 years ago, but with few applications. A later review article by Morman [26] described the status of nonlinear viscoelasticity as applied to rubbery materials with emphasis again on constitutive equations. Schapery [40] reviewed constitutive theories for fracture and A. Wineman (B) University of Michigan, Ann Arbor, MI 48109, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_4

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strength of nonlinear viscoelastic solids, while Drapaca et al. [7] reviewed mathematical issues underlying the formulation of constitutive equations for nonlinear viscoelastic solids. The past few years have seen increasing interest in the subject of nonlinear viscoelastic solids arising from research in the biomechanics of soft tissue and the engineering of polymeric and elastomeric structural components. Thus, this article has a twofold purpose. It provides an overview of the current state of constitutive equations as well as a representative sampling of solutions to boundary-value problems. This article contains a condensed version of material that appeared in the earlier overview of nonlinear viscoelastic solids in [54]. It also contains new material on nonlinear viscoelastic fiber-reinforced solids as well as reference to recent applications. Overviews of viscoelastic phenomena and linear viscoelasticity are presented in Sects. 2 and 3. The characteristics of stress relaxation, creep, and sinusoidal response as well as the mathematical forms of constitutive equations provide a benchmark for comparison with corresponding concepts for nonlinear response. Sections 4 and 5 introduce the general constitutive assumption for nonlinear viscoelastic materials along with restrictions imposed from the consideration of superposed rigid body rotations, material symmetry, and the assumption of incompressibility. Several phenomenological types of constitutive equations that have appeared in the literature are summarized in Sect. 6. This includes clock models, which account for time-dependent material processes that are affected by temperature or deformation. Attention is then restricted to nonlinear single integral constitutive equations. Their representations for isotropy are stated in Sect. 7 and for transverse isotropy and orthotropy, for a particular constitutive equation, in Sect. 8. Section 9 discusses homogeneous deformations briefly, while Sect. 10 focuses on features of uniaxial extension for an isotropic material represented by a nonlinear single integral constitutive equation. These are connected to the corresponding features for linear viscoelastic response. A representative sampling of references containing solutions to boundary-value problems involving large deformation of nonlinear viscoelastic solids is provided in Sect. 11.

2 Fundamental Viscoelastic Phenomena As the word viscoelasticity suggests, the mechanical response of interest involves aspects of elastic solid response and viscous fluid response. Since the motion of fluids involves flow or continuing deformation as time increases, it is necessary to discuss viscoelastic response to account explicitly for time as a physical parameter. A detailed comparison of viscoelastic response with that of elastic solids and viscous fluids that shows the importance of time as a physical parameter is presented in [55]. Both the current time t and a generic earlier time s ∈ [0, t] are important in describing viscoelastic response. The fundamental viscoelastic phenomena are presented in the context of onedimensional stress and strain states, the material being either in uniaxial extension or simple shear. Thus, let σ denote either a normal or shear stress and ε denote the

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corresponding normal or shear strain. Attention is confined to conditions when the material is initially undeformed and unstressed, that is, ε(t) = 0 and σ (t) = 0 at times t < 0. The terminology stress history or strain history is used to refer to the set of values for the stress σ (s) or the strain ε(s), respectively, for s ∈ [0, t].

2.1 Creep Let a specimen be subjected to a step stress history in which the stress is instantaneously changed at t = 0 to some value σ0 and then held fixed. The typical strain response consists of (i) an instantaneous change in strain at t = 0 followed by (ii) continued straining in time at a nonconstant rate and (iii) an asymptotic approach to some limiting value as time increases. This behavior is called creep. Let J (t, σ0 ) denote the strain ε(t) at time t ≥ 0 when the value of the stress is fixed at σ0 . Then, (i) ε(t) = 0 when t < 0, (ii) ε(t) jumps to the value J (0, σ0 ) at t = 0, and (iii) ε(t) = J (t, σ0 ) monotonically increases to the limit value denoted by J (∞, σ0 ) as t → ∞. The jump in strain from ε(t) = 0 when t < 0 to ε(0) = J (0, σ0 ) at t = 0 indicates instantaneous springiness or elasticity. The fact that the material reaches a finite limit value of strain as t → ∞ indicates solid behavior. If the strain were to increase without bound, it would indicate fluid behavior, which is not considered here. The stress–strain relations, σ0 versus J (0, σ0 ) and σ0 versus J (∞, σ0 ), respectively, describe instantaneous elastic response and the longtime or equilibrium elastic response. J (t, σ0 ) has a different dependence on time t and stress σ0 for each material, and is therefore considered to be material property called the creep function.

2.2 Stress Relaxation Let a specimen be subjected to a step strain history in which the strain is instantaneously changed to some value ε0 at t = 0 and then held fixed. The typical stress history required to produce this strain history consists of (i) an instantaneous change in stress at t = 0 followed by (ii) a gradual monotonic decrease of stress magnitude at a nonconstant rate and (iii) an asymptotic approach to some nonzero limiting value as time increases. The behavior is called stress relaxation. Let G(t, ε0 ) denote the stress σ (t) at time t ≥ 0 when the value of the strain is fixed at ε0 . Then, (i) σ (t) = 0 when t < 0, (ii) σ (t) jumps to the value G(0, ε0 ) at t = 0, and (iii) σ (t) = G(t, ε0 ) monotonically decreases in magnitude to the nonzero limit value denoted by G(∞, ε0 ) as t → ∞. The jump in stress from σ (t) = 0 when t < 0 to G(0, ε0 ) at t = 0 is another indication of instantaneous springiness or elasticity. The fact that a nonzero stress G(∞, ε0 ) is required to maintain the strain at ε0 is another indication that the material is a solid. If G(∞, ε0 ) = 0, then no stress would be required to hold the material in a strained state, a characteristic of

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the response of fluids. The stress–strain relations, G(0, ε0 ) versus ε0 and G(∞, ε0 ) versus ε0 , also describe, respectively, instantaneous elastic response and the longtime or equilibrium elastic response. G(t, ε0 ) has a different dependence on time t and strain ε0 for each material, and is therefore considered a material property called the stress relaxation function.

2.3 Constitutive Assumption The phenomena of creep and stress relaxation show that the mechanical response of a viscoelastic material depends on time. They also illustrate the duality of responses, strain is found under stress control conditions, and stress is found under strain control conditions. This further raises the issue of how to determine the strain response when the stress varies with time or the stress response when the strain varies with time. Experimental results provide further evidence of this time dependence and imply that the stress σ (t) at time t depends on the preceding strain history, ε(s), s ∈ [0, t], or that the strain ε(t) at time t depends on the preceding stress history, σ (s), s ∈ [0, t]. It was assumed earlier that σ (s) = ε(s) = 0, s ∈ (−∞, 0). Some histories may have a jump from the value of zero at t = 0− to a nonzero value at t = 0+ . In order to include the influence of this jump, the response is assumed to depend on the entire history for s ∈ (−∞, t]. The constitutive assumption expressing the stress at time t in terms of the strain history up to time t is denoted by   σ (t) = Gˆ ε(s)|ts=−∞ ; t .

(1)

Gˆ is called a response functional. The notation ε(s)|ts=−∞ indicates dependence on the entire strain history, including the influence of the jump at t = 0. The explicit dependence on t indicates that the material is aging, i.e., if the material is concrete or epoxy, then it is curing. The explicit dependence on the parameter t then represents the time since the material was created. Equation (1) represents the essential nature of viscoelasticity, the evolution of a strain history as time increases and determines the evolution of the corresponding stress history as time increases. There is a corresponding constitutive assumption for the strain in terms of stress history,   (2) ε(t) = Jˆ σ (s)|ts=−∞ ; t , with corresponding comments. This shows the dual nature of viscoelasticity. For every statement of stress in terms of strain history, there is a dual statement of strain in terms of stress history. A central issue in the modeling of viscoelastic materials is the determination of the mathematical form of the response functional Gˆ in (1) or Jˆ in (2).

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2.4 Aging In the remainder of this article, it is assumed that the material does not age. A discussion of linear aging viscoelastic materials parallel to that presented below is given in [44]. When the material is non-aging, it can be shown that the parameter t ˆ In addition, Eqs. (1) does not appear explicitly in the response functionals Gˆ and J. and (2) become, respectively,

and

  σ (t) = G ε(t − s)|∞ s=0

(3)

  ε(t) = J σ (t − s)|∞ s=0 .

(4)

In the histories appearing as arguments in (3) and (4), the time variable s is measured backwards from the current time t. Its physical dimension can be thought of as “seconds ago”.

3 Linear Viscoelastic Response An important assumption about viscoelastic response is that it is linear. The property of linearity of response consists of two conditions: scaling and superposition. These are discussed here only for the stress response to a strain history. Analogous comments apply to the strain response to a stress history.

3.1 Linearity Let ε1 (s), ε2 (s), s ∈ (−∞, t], be two strain histories whose stresses at time t are σ1 (t), σ2 (t), respectively. Let λ1 , λ2 be constants. The composite strain history ε(s) = λ1 ε1 (s) + λ2 ε2 (s), s ∈ (−∞, t],

(5)

is constructed by first multiplying strain history ε1 (s) by λ1 and ε2 (s) by λ2 at each time s (scaling). The results are added at each time s (superposition). If the stress for the strain history (5) is given by σ (t) = λ1 σ1 (t) + λ2 σ2 (t)

(6)

for all times t, constants λ1 , λ2 , and strain histories ε1 (s), ε2 (s), s ∈ (−∞, t], the response is said to be linear. In other words, linearity of response means that if a strain history ε(s) is scaled by constant λ, then the corresponding stress σ (t) is also

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scaled by λ and if two strain histories are superposed, then the corresponding stresses are also superposed. A significant corollary of linearity of response is that there is no interaction between the stress history responses to separate strain histories. This assumption is assumed to be reasonable when the magnitude of the strain has been small for all past times, i.e., |ε(s)|  1 for s ∈ (−∞, t]. It is important to note that the property of linearity of response does not refer to the shape of any material response curve. It refers to the method of constructing the stress response to a composite strain history by scaling and superposing the stress responses to the component strain histories. Scaling and superposition have convenient graphical interpretations that lead to important and useful tests for determining if the mechanical response of a material can be regarded as linear. For a discussion of these, see [55].

3.1.1

Consequences of Linearity

Let H (t) denote the Heaviside step function, i.e., H (t) = 0, t ∈ (−∞, 0), and H (t) = 1, t ∈ [0, ∞]. According to (3), the stress response to the step strain history ε(s) = ε0 H (s) is   (7) σ (t) = G ε0 H (t − s)|∞ s=0 . This is also the stress relaxation function G(t, ε0 ) introduced earlier   G(t, ε0 ) = G ε0 H (t − s)|∞ s=0 .

(8)

The variable t in G(t, ε0 ) denotes the result of mathematical operations on the time variable in the response functional G. When there is linearity of response, the scaling property implies that G satisfies the condition     ∞ (9) G ε0 H (t − s)|∞ s=0 = ε0 G H (t − s)|s=0 . Let the response to the unit step strain be denoted by   G(t) = G H (t − s)|∞ s=0 .

(10)

Equations (8), (9), and (10) show that when there is linearity of response, G(t, ε0 ) has linear dependence on ε0 , G(t, εo ) = ε0 G(t).

(11)

G(t) is called the stress relaxation modulus. It is assumed that G(0) > 0 and G(t) monotonically decreases to a nonzero positive limit denoted by G(∞) as t → ∞. By (7)–(11), the stress can be written as σ (t) = ε0 G(t).

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Gurtin and Sternberg [17], in their fundamental article on linear viscoelasticity, showed that scaling and superposition lead to the representation for (3) as a Stieltjes integral  σ (t) =

t

−∞

ε(t − s)dG(s).

(12)

In a similar manner, it follows from (4) that the strain response to the step stress history σ (s) = σ0 H (t) is   ε(t) = J σ0 H (t − s)|∞ s=0 = J (t, σ0 ),

(13)

where J (t, σ0 ) is the creep function introduced earlier. When there is linearity of response, the scaling property means that J satisfies the condition     ∞ J σ0 H (t − s)|∞ s=0 = σ0 J H (t − s)|s=0 .

(14)

Let the response to the unit step stress be denoted by   J (t) = J H (t − s)|∞ s=0 .

(15)

Then, by (13)–(15), the creep function J (t, σ0 ) has linear dependence on σ0 J (t, σ0 ) = σ0 J (t).

(16)

J (t) is called the creep compliance. It is assumed that J (0) > 0 and J (t) monotonically increases to a finite limit denoted by J (∞) as t → ∞. By (13)–(16), ε(t) = σ0 J (t). It is shown in [17] that scaling and superposition lead to the following representation for (4):  ε(t) =

t

−∞

σ (t − s) d J (s).

(17)

The constitutive equations (12) and (17) have been written in the form of Stieltjes convolutions [17] in order to account for jump discontinuities in their arguments. When there is a jump in the stress or strain histories at t = 0 and the histories are differentiable for t > 0, (12) and (17) can be written, respectively, as 

t

σ (t) = ε(t)G(0) +

˙ − s) ds ε(s)G(t

(18)

σ (s) J˙(t − s) ds,

(19)

0



and

t

ε(t) = σ (t)J (0) + 0

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in which the superposed dot indicates differentiation with respect to the argument and there has been a change of the integration variable. Other alternate forms of these relations can be obtained by an integration by parts and/or a change of the integration variable. A complete list of these forms is given in [55]. Note that the definitions of G(t) in (10) and J (t) in (15) provides no information about their specific forms as functions of t. These relations state only that G(t) and J (t) are the observed responses to a step strain or step stress history. They can be represented by any mathematical form that is found convenient for representing experimental data.

3.1.2

Mechanical Analogs

An approach used to develop constitutive equations for linear viscoelastic response involves mechanical analogs. These are mechanical devices formed by combining linear elastic springs and linear viscous dampers in series or parallel. The devices can be shown to exhibit a time-dependent response that is similar to that observed in viscoelastic materials, namely, creep under constant load and force relaxation under constant deformation. For this reason, these devices are treated as mechanical analogs of viscoelastic response. Since the springs and dampers are described by linear equations, as are the equations for the kinematics of deformation and force transmission used to model the mechanical analog, the mechanical response of a device is described by a linear relation between the overall force and deformation. These are interpreted as relations between stress and strain for a material and have the form dn σ dn−1 σ dσ + p0 σ + p + · · · + p1 n−1 n n−1 dt dt dt dn ε dn−1 ε dε = qn n + qn−1 n−1 + · · · + q1 + q0 ε, dt dt dt

pn

(20)

where p0 , q0 , p1 , q1 , . . . , pn , qn are constants to be determined by experiments. Equation (20) is valid only when the strain or stress histories are sufficiently smooth. When either has a jump discontinuity, as might occur at t = 0, (20) must be supplemented by appropriate jump relations. The appropriate relations between the initial conditions on the stress and strain and their first n − 1 derivatives, that are consistent with (20), were developed in [17]. A complete statement of the constitutive equation obtained from the use of mechanical analogs accounts for both the jump and the response after the jump. In other words, it consists of both an equation of the form (20) and a set of appropriate initial conditions. This point is often overlooked in applications. Any constitutive equation of form (20), along with the appropriate initial conditions, can be expressed in either the form (18) or (19). On the other hand, a constitutive equation of form (18) or (19) can be reduced to the form (20) if and only if the creep compliance and the stress relaxation modulus satisfy specific conditions. A detailed

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discussion of this point can be found in [17]. Constitutive equation (20) gives equal emphasis to the stress and the strain. It, therefore, can be used to obtain the dual constitutive equations and their inverses.

3.1.3

Relation Between G(t) and J(t)

When the step stress history σ (t) = σ0 , t > 0 and the corresponding creep strain ε(t) = σ0 J (t) are substituted into (18), the result is 

t

1 = J (t)G(0) +

˙ J (t − s)G(s) ds.

(21)

0

Similarly, when the step strain history ε(t) = ε0 , t > 0, and the corresponding stress relaxation response σ (t) = ε0 G(t) are substituted into (19), the result is 

t

1 = G(t)J (0) +

G(t − s) J˙(s) ds.

(22)

0

Equations (21) and (22) can be transformed into each other by an integration by parts. They establish alternate forms of a relation between G(t) and J (t). If J (t) is known, then (22) is a linear Volterra integral equation for G(t). Conversely, if G(t) is known, (21) is a linear Volterra integral equation for J (t). It is known that these equations have a unique solution. Thus, G(t) and J (t) can be regarded as inverses of each other [17], and corresponding to a given stress relaxation modulus G(t), there is a uniquely determined creep compliance J (t) and vice versa. The simplest model of a linear viscoelastic solid that exhibits all of the important response characteristics, i.e., instantaneous elastic response, longtime or equilibrium elastic response, and gradual stress relaxation, is the standard linear solid, also known as the three-parameter solid. Its stress relaxation modulus is given by G(t) = G ∞ + [G 0 − G ∞ ] exp(−t/τ R ),

(23)

where τ R is called the characteristic stress relaxation time, G 0 = G(0) and G ∞ is the limit of G(t) as t → ∞. The corresponding creep compliance, found by applying the Laplace transform to (21) or (22) is given by J (t) = J∞ + [J0 − J∞ ] exp(−t/τC ),

(24)

where τC is called the characteristic creep time, J0 = J (0) and J∞ is the limit of J (t) as t → ∞. It is shown in [55] that G 0 J0 = 1, G ∞ J∞ = 1 , and τC = (G 0 /G ∞ )τ R . Since stress relaxation implies G 0 /G ∞ > 1, τC > τ R . Suppose that G(t) and J (t) are known. For a given stress history, (18) is a linear Volterra integral equation for the corresponding strain history. Conversely, for a given strain history, (19) is a linear Volterra integral equation for the corresponding stress

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history. It is straightforward to show, using (21) or (22) and elementary operations of calculus, that (18) is the solution to (19) and vice versa. Stated differently, (18) and (19) are the inverses of each other. Thus, for non-aging, linear viscoelastic materials, the dual constitutive equations are also inverses.

3.1.4

Sinusoidal Strain Histories

An important strain history that is used to study viscoelastic materials is the sinusoidal strain history, (25) ε(t) = ε0 sin ωt, where ε0 is a constant such that |ε0 |  1. The corresponding stress history is obtained by substituting (25) into (18). It can be shown that the stress reaches a state of steady sinusoidal oscillations described by

or

  σ (t) = ε0 G  (ω) sin ωt + G  (ω) cos ωt

(26)

 1/2 sin(ωt + δ(ω)), σ (t) = ε0 G  (ω)2 + G  (ω)2

(27)

where tan δ(ω) = G  (ω)/G  (ω). G  (ω) and G  (ω) are functions of frequency ω and are expressed in terms of the stress relaxation modulus by G  (ω) = G ∞ + ω



∞

ΔG(s) sin ωs ds,

(28)

0

G  (ω) = ω



∞

ΔG(s) cos ωs ds.

(29)

0

In deriving these relations, the stress relaxation modulus G(t) has been decomposed into its longtime equilibrium value and time-dependent parts, G(t) = G ∞ + ΔG(t),

(30)

where ΔG(t) → 0 as t → ∞. It is seen from (26) or (27) that the stress varies sinusoidally with time at the  1/2 and same frequency ω as the strain, but with amplitude ε0 G  (ω)2 + G  (ω)2 phase difference δ(ω). G  (ω), the coefficient of the term in (26) in phase with the strain, is called the storage modulus. G  (ω), the coefficient of the term in (26) out of phase with the strain, is called the loss modulus. It can be shown using (29) that the phenomenon of stress relaxation implies that G  (ω) > 0. It can also be shown that the work done on the material per cycle is ε02 π G  (ω). G  (ω) and G  (ω) are an alternate set of material properties and methods have been developed to measure them. They are defined by (28) and (29) in terms of the

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Fourier transform of ΔG(s). Consequently, G(t) can be expressed in terms of G  (ω) and G  (ω) using the inverse Fourier transform, 2 G(t) = G ∞ + π or G(t) =

2 π

 0



∞

0 ∞

G  (ω) cos ωt dω, ω

G  (ω) sin ωt dω. ω

(31)

(32)

It is a common practice to use complex variables to describe the sinusoidal response of viscoelastic materials. Thus, instead √ of specifying the strain history (25), one specifies ε(t) = ε0 exp(iωt) where i = −1. The complex modulus is defined by (33) G ∗ (ω) = G  (ω) + iG  (ω). Then, (26) is written

σ (t) = ε0 G ∗ (ω)eiωt .

(34)

Since the strain history in (25) is the imaginary part of ε0 exp(iωt), the stress response is the imaginary part of (34). It is also possible to specify the sinusoidal stress history σ (t) = σ0 sin ωt,

(35)

the imaginary part of σ0 exp(iωt). The strain history is determined using (19),   ε(t) = σ0 J  (ω) sin ωt + J  (ω) cos ωt ,

(36)

where J  (ω) and J  (ω) are components of the complex compliance, J ∗ (ω) = J  (ω) + iJ  (ω).

(37)

The strain history in (36) is the imaginary part of σ0 J ∗ (ω) exp(iωt). J  (ω) and J  (ω) can be expressed in terms of the creep compliance J (t) by expressions that are analogous to (28) and (29). These are not presented here, but can be found in [55]. It can be shown that G ∗ (ω) and J ∗ (ω) satisfy G ∗ (ω)J ∗ (ω) = 1

(38)

for all frequencies ω. Using (25) and (26), time t can be eliminated between the stress and strain histories so that the stress σ can be expressed directly in terms of the strain ε,   σ 2 − 2G  (ω)σ (ω)ε + G  (ω)2 + G  (ω)2 ε2 = ε02 G  (ω)2 .

(39)

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This describes an ellipse whose properties  the enclosed area is  depend on ω: (i) ε02 π G  (ω); (ii) the ε-axis intercept is ε0 / G  (ω)2 + G  (ω)2 ]; (iii) the σ -axis inter 1/2 . cept is σ = ε0 G  (ω), and (iv) the maximum value of σ is ε0 G  (ω)2 + G  (ω)2 The ellipse approaches the straight line σ = G(0)ε as ω → 0 and the straight line σ = G ∞ ε as ω → ∞.

4 Kinematics of Deformation The concepts and mathematical objects that are used to describe the kinematics of a deforming viscoelastic body are summarized in this section. For their detailed development, see Atkin and Fox [1] or Spencer [43]. Let P denote a material element or particle of the body that is identified by its position vector X at some reference time t0 , which is used as its label. The region of three-dimensional space occupied by the vectors X for all particles of the body at time t0 defines the configuration of the body at time t0 . It is assumed that a viscoelastic solid body has been at rest in the same configuration for times t < 0, which is taken as its reference configuration. Let x(s) denote the position of particle P at a typical time s ∈ (−∞, t]. The motion of the body is described by the relations x(s) = X, s ∈ (−∞, 0);

x(s) = χ(X, s), s ∈ [0, t].

(40)

The region of three-dimensional space occupied by the position vectors x(s) of all particles at time s defines the configuration of the body at time s. This motion is assumed to be one-to-one and invertible so that the label X of a particle can be expressed in terms of its position x(s) at time s X = χ −1 (x(s), s), s ∈ [0, t].

(41)

Let (41) be evaluated at time t and then substituted into (40). This introduces a description of the motion relative to the current configuration, X = χ (χ −1 (x(t), t), s) = χˆ (x(t), t, s).

(42)

The velocity and acceleration of particle P at time t are given by x˙ (X, t) =

∂ 2 χ (X, t) ∂χ(X, t) , x¨ (X, t) = , ∂t ∂t 2

(43)

where the superposed dot denotes the partial derivative with respect to time holding the particle label X fixed. Relation (41) evaluated at time t can be used to change the independent spatial variable in (43) from X to x(t) giving

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v(x, t) = x˙ (χ −1 (x, t), t), a(x, t) = x¨ (χ −1 (x, t), t).

(44)

The deformation gradient F(s) = ∂x(s)/∂X relates the configuration of the neighborhood of a particle at time s to that in its reference configuration. For the motion in (40), the history of the deformation gradient is F(s) = I, s ∈ (−∞, 0);

F(s) =

∂x(s) , s ∈ [0, t], ∂X

(45)

in which dependence on X has been omitted for notational simplicity. F(s) contains the information about the rotation and distortion of the neighborhood of a material particle as it goes from its reference configuration to its neighborhood in the configuration at time s. The ratio of the volume of the neighborhood of a particle at time s to that in the reference configuration is given by det F(s). It is assumed that det F(s) > 0, s ∈ [0, t].

(46)

The polar decomposition of F(s) is F(s) = R(s)U(s) = V(s)R(s), s ∈ [0, t],

(47)

where U(s), V(s), and R(s) satisfy R(s)R(s)T = R(s)T R(s) = I, U(s) = U(s)T , V(s) = V(s)T .

(48)

The tensor C(s) = F(s)T F(s) = U(s)2 ,

(49)

the right Cauchy–Green strain tensor, appears in constitutive equations in viscoelasticity. Another tensor that appears is the left Cauchy–Green tensor, B(t) = F(t)F(t)T = V(t)2 .

(50)

Some approaches to nonlinear viscoelasticity make use of the relative description of motion obtained by substituting (41), evaluated at t, into (40), giving x(s) = χ (x(t), t, s).

(51)

The deformation gradient associated with this description of the motion is Ft (s) =

∂x(s) , ∂x(t)

(52)

the relative deformation gradient. Ft (s) is expressed in terms of the deformation gradient F(s) by

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Ft (s) = F(s)F(t)−1 , s ∈ [0, t].

(53)

It follows from (46) that det Ft (s) > 0. Ft (s) contains information that gives the rotation and deformation of the neighborhood of a particle in the configuration at time s relative to its neighborhood in the configuration at time t. Thus, as in (47), the polar decomposition of Ft (s) is Ft (s) = Rt (s)Ut (s) = Vt (s)Rt (s), s ∈ [0, t],

(54)

where Rt (s), Ut (s) , and Vt (s) satisfy relations analogous to those in (48). Some constitutive equations are formulated in terms of the relative right Cauchy–Green strain tensor (55) Ct (s) = Ft (s)T Ft (s). Other approaches also make use of the velocity gradient at time t, L = ∂v/∂x, which is related to the deformation gradient by ˙ −1 . L = FF

(56)

5 Constitutive Equations for Nonlinear Viscoelastic Solids For nonlinear viscoelastic solids, the constitutive assumption states that the Cauchy stress σ , internal energy e, and specific entropy η at time t depend on the histories of the deformation gradient F, temperature θ , and temperature gradient ∇x θ . Thermodynamic arguments show that the stress, internal energy, and specific entropy do not depend on the temperature gradient ∇x θ . A thorough development of this result can be found in Noll [33] and the fundamental treatise by Truesdell and Noll [45] and will not be included here. Instead, the emphasis is on a presentation of the tensorial structure of the constitutive equations due to physical considerations. Temperature, being a scalar, plays no role in determining this tensorial structure and will not be explicitly indicated. As in the case of linear viscoelasticity, it is assumed that the solid is in its reference configuration for t < 0, i.e., F(t) = I, t < 0. It is further assumed that the material does not age and the stress at the current time t depends on the history of the deformation gradient, that is, on all values of F(s), s ∈ (−∞, t], thereby, allowing for jump discontinuities at t = 0. The constitutive assumption expressing this dependence is denoted by   (57) σ = F F(t − s)|∞ s=0 , a generalization of (3). F is called a tensor-valued response functional. There are three main sources of restrictions on F : (a) the influence of superposed rigid body motions; (b) material symmetry; (c) restrictions due to thermodynamics. For present

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purposes, the only restrictions considered here are those due to the influence of superposed rigid body motions and material symmetry. A dual constitutive assumption that is a generalization of (4) can also be made. Because of space limitations, attention is restricted to that in (57).

5.1 Influence of Superposed Rigid Body Motions Consider the motion in (40). Suppose that the body undergoes a second motion x(s) = χ ∗ (X, s) that is obtained from the first by a superposed rigid body translation d(s) followed by a rigid body rotation, χ ∗ (X, s) = Q(s)[χ(X, s) − d(s)],

(58)

Q(s)Q(s)T = Q(s)T Q(s) = I,

(59)

where for s ∈ (−∞, t]. It is assumed that the superposed rigid body motion affects the stress at time t only by its orientation at time t. This leads to the condition that     ∞ T F Q(t − s)F(t − s)|∞ s=0 = Q(t)F F(t − s)|s=0 Q(t)

(60)

for any rotation history Q(s), s ∈ [0, t] satisfying (59). When combined with the polar decomposition of F(s) in (47), this leads to the statement that (57) has the form   T σ = R(t)Fˆ U(t − s)|∞ s=0 R(t) .

(61)

Equation (61) is usually restated in the form,   T σ = F(t)G C(t − s)|∞ s=0 F(t) ,

(62)

where G is a new response functional. It is straightforward to show that (62) satisfies (60).

5.2 Material Symmetry The concept of material symmetry arises from the fact that a material has some physical microstructure in its reference configuration, such as a crystalline structure or a randomly oriented macromolecular network. Consider a sample of material in its reference configuration and its microstructure. Suppose there is a transformation

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of this reference configuration to a new configuration such that the material appears to have the same microstructure as before. Let both the original and transformed configurations then be subjected to the same homogeneous deformation history with deformation gradient F(s). The underlying microstructures, which appear to be the same in their respective reference configurations, are distorted in the same way. The stresses are assumed to be the same at each time t and these configurations are said to be mechanically equivalent. A transformation of the original reference configuration to one that is mechanically equivalent is a linear transformation denoted by H. One restriction on H is that it produce no volume change and this is expressed as the condition that | det H| = 1. In addition, for most equivalent microstructures of interest, H is a rotation or a reflection and satisfies (63) HHT = HT H = I. Symmetries of a material are described by specifying the set of transformations H that lead to equivalent microstructures. These form a mathematical entity called a material symmetry group. The material symmetries commonly used to describe nonlinear viscoelastic materials are isotropy, transverse isotropy, and orthotropy. For each transformation H of a material symmetry group, the response functional F in (57) must satisfy the restriction,     ∞ F F(t − s)|∞ s=0 = F F(t − s)H|s=0 .

(64)

Material symmetry restrictions can be imposed on the response functionals Fˆ and G by substituting (61) and (62) into (64) giving

and

 T   T  ∞ ˆ HFˆ U(t − s)|∞ s=0 H = F H U(t − s)H|s=0

(65)

 T   T  ∞ HG C(t − s)|∞ s=0 H = G H C(t − s)H|s=0 .

(66)

Note that this restriction is imposed by transformations of the reference configuration and is independent of the subsequent deformation history. In other terms, the proper statement of a material symmetry is that a material is isotropic, transversely isotropic or orthotropic with respect to its reference configuration. The statement is sometimes made that a deformation causes a material to become anisotropic. A more precise statement is that there is a material symmetry group associated with the current configuration that is determined from the symmetry group associated with the reference configuration and the current deformation gradient using Noll’s rule [33]. This new material symmetry group contains the transformations of the expected anisotropy as a subgroup.

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5.3 Constraints The possible motions of a body may be limited by constraints such as incompressibility or inextensibility in certain directions. Such constraints impose restrictions on the constitutive equations. Discussion here is limited to the constraint of incompressibility. In many polymeric and biological materials, the volume change during deformation is observed to be very small. By (46), this leads to an idealized material model for which any possible motion must satisfy the constraint det F(s) = 1, s ∈ [0, t].

(67)

Motions that satisfy (67) are said to be isochoric. Consideration of the restrictions from the thermodynamics for materials with constraint (67) leads to a modified form for constitutive equation (57),   σ (t) = − pI + F F(t − s)|∞ s=0 ,

(68)

in which p is an arbitrary scalar and F is a different response functional. The restriction imposed by consideration of the influence of superposed rigid body motions must still be satisfied so that F in (68) still must satisfy (60). Equations (61) and (62) then become   T σ (t) = − pI + R(t)Fˆ U(t − s)|∞ s=0 R(t) ,

(69)

  T σ (t) = − pI + F(t)G C(t − s)|∞ s=0 F(t) ,

(70)

in which det U(s) = det C(s) = 1. Similarly, material symmetry considerations imply that the response functionals Fˆ and G in (69) and (70) satisfy (65) and (66), respectively.

6 Some Proposed Constitutive Equations for Nonlinear Viscoelastic Solids In Sect. 3.1.1, it was pointed out that a representation for the response functional G in (3) was developed by making use of the assumed property of linearity of response, resulting in its integral representation in (12). For nonlinear viscoelastic solids, there is no generally accepted well-defined property of the response functional Fˆ in (61) or G in (62) that leads to a representation. A number of specific representations for the response functionals Fˆ and G have appeared in the literature based on different assumptions and these are summarized in the book by Lockett [23] and the recent review article by Drapaca et al. [7]. The latter also summarizes the mathematical issues used in the development of the representations. Attention is restricted, in

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this chapter, to a presentation of the mathematical forms of these representations, as restricted by consideration of superposed rigid body motions. The additional restrictions due to considerations of material symmetry are presented in the next section.

6.1 Rate and Differential Type Constitutive Equations One class of constitutive equations generalizes (20) to a relation between the stress and its first n time derivatives and the deformation gradient and its first m time derivatives, all evaluated at the current time t   dn σ dF d2 F dm F dσ d2 σ , 2 , . . . , n ; F, , 2 , . . . , m = 0, R σ, dt dt dt dt dt dt

(71)

where R is a function of m + n + 2 arguments. When subjected to the restrictions imposed by the considerations of superposed rigid body motions, the constitutive equation has the form [33]   k    d dj  R1 RT σ R, RT j Rt (s)T σ Rt (s) s=t R; U, RT U (s) R =0 t ds ds k s=t

(72)

in which the indexed variables can appear for all j = 1, . . . , n and k = 1, . . . , m. Such constitutive equations are said to be of rate type and  dj  Rt (s)T σ Rt (s) s=t j ds

(73)

is called the jth invariant stress rate. Equation (72) can be solved, in concept, for the stress in terms of the deformation history or for the deformation in terms of the stress history. Thus, it contains, in effect, both the dual constitutive equations and their inverses. As in the case of (20), (72) must be supplemented by a set of conditions relating the stress and kinematical variables at a jump discontinuity. Because R1 can be a nonlinear function of its arguments, the development of these conditions is more complicated than in the case of linear visceoelaticity. Such conditions were developed in the context of a particular constitutive theory by Prusa and Rajagopal [36]. A special case of (72) is explicit in the stress and does not depend on the stress rates  j    d T ˜ Ut (s) R RT . (74) σ = RW U, R ds j s=t This constitutive equation is said to be of differential type. It is assumed to be useful when the stress depends on F(s) for values of s near the current time t, i.e., on the

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recent past. F(s) can then be approximated by a Taylor series about the current time t. Retaining the first n terms of its power series leads to the presence of the arguments  R

T

dj Ut (s) ds j

 R,

j = 1, 2, . . . , n.

(75)

s=t

The remaining constitutive equations to be presented have been developed using the form in (62). For this reason, there will be no further reference to the form in (61).

6.2 Green–Rivlin Multiple Integral Constitutive Equations Consider the Green–St. Venant strain tensor defined by E(s) =

1 (C(s) − I). 2

(76)

Note from (45) and (49) that E(s) = O, s ∈ (−∞, 0). Let E(s) be introduced into (62), which then becomes   T σ = F(t)G1 E(t − s)|∞ s=0 F(t) .

(77)

Green and Rivlin [16] assumed that the response functional G1 is continuous in E(s) in a sense described in [7]. By expressing E(s), s ∈ [0, t] as a Fourier series and then using the Stone–Weierstrass approximation theorem, Green and Rivlin obtained a representation for G1 as a series of multiple integrals

 +

t −∞



t

 t   G1 E(t − s)|∞ = K1 (t − s1 )dE(s1 ) s=0 −∞  t  t K2 (t − s1 , t − s2 )dE(s1 )dE(s2 ) + −∞ −∞  t K3 (t − s1 , t − s2 , t − s3 )dE(s1 )dE(s2 )dE(s3 ) . . . .

−∞

(78)

−∞

This is written in the same form as in (12), i.e., in terms of Stieltjes convolutions, in order to allow for a jump discontinuity in E(s). K1 (t − s1 ), K2 (t − s1 , t − s2 ), K3 (t − s1 , t − s2 , t − s3 ) are tensor-valued functions of their time arguments of order four, six, and eight, respectively. For example, the index notation version of the integrand of the double integral is [K2 (t − s1 , t − s2 )]i jklmn (dE(s1 ))kl (dE(s2 ))mn . They have relaxation function-like properties in that they monotonically decrease to some limit.

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The dual assumption can also be made. As discussed in [23], it has the form   E(t) = J1 (F(s)−1 σ (s)F(s)−T )|t0− .

(79)

J1 has a multiple integral series representation analogous to (78) in which E(s) is replaced by F(s)−1 σ (s)F(s)−T . In applications, only the truncation of (78) or the series representation of (79) up to triple integrals has been considered.

6.3 Finite Linear Viscoelasticity Coleman and Noll [5] developed a constitutive equation based on the assumption of fading memory, i.e., the current stress depends more on recent deformations than past deformations. They assumed that deformation of the current configuration with respect to the reference configuration can be large and that the deformation of recent configurations relative to the current configuration changes slowly, in a sense made precise in [5]. This led to a Taylor series-like approximation to (62), the leading terms of which are   t   T K1 [C(t), t − s] F(t) (Ct (s) − I)F(t) ds F(t)T , σ = F(t) k1 [C(t)] + −∞

(80) where k1 [C] is a second-order tensor function of C and K1 [C, s] is a fourth-order tensor function of s and C. The integrand in (80) is linear in the tensor F(t)T (Ct (s)− I)F(t), i.e., the index  notation version of the integrand is K1 [C(t), t − s]i jkl F(t)T (Ct (s) − I)F(t) kl . K1 [C, s] has the property, made precise in [5], that it monotonically decays to zero as s increases. Dependence on the finite strain tensor C(t) expresses the notion that deformation of the current configuration with respect to the reference configuration can be large. The linear dependence of the integrand on Ct (s) − I arises from the assumption that the deformation occurs slowly. This constitutive equation is referred to as finite linear viscoelasticity (FLVE). There are several useful observations about this constitutive equation. First, suppose that the material has always been in its reference configuration, then x(t) = X, t ≥ 0, and F(t) = C(t) = I, Ct (s) = I, s ∈ (−∞, t]. Equation (80) reduces to σ = k1 (I). If it is assumed that the material is stress free in its reference configuration, then k1 (I) = O. If the material is incompressible, then it is assumed that k1 (I) = k1 I for some scalar k1 . Second, let the deformation be small for s ∈ (−∞, t] and introduce u = x − X, the displacement from the reference configuration and the displacement gradient H = ∂u/∂X. If |tr(HT H)| is small for s ∈ [0, t], then C(s) ≈ I + e(s), where e(s) = (H(s) + HT (s))/2, the infinitesimal strain tensor. It was shown in [5] that with this result, the linearization of (80) reduces to the constitutive equation for linearized viscoelasticity.

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Third, let the deformation undergo a step change from t = 0− to t = 0+ , then x(0+ ) = X, F(0+ ) = I, and the relative deformation gradient becomes F0+ (s) =

∂x(s) ∂X = = F−1 (0+ ), s ∈ (−∞, 0). ∂x(0+ ) ∂x(0+ )

(81)

Then, F(0+ )T (C0+ (s) − I)F(0+ ) = I − C(0+ ) and (80) becomes   σ = F(0+ ) k1 [C(0+ )] +

∞

  K1 [C(0+ ), x] I − C(0+ ) dx F(0+ )T .

(82)

0

Since the expression in braces is a function of C(0+ ), this is a constitutive equation for nonlinear elasticity associated with instantaneous response. Fourth, let the body approach a fixed deformation as t → ∞. It was also shown in [5] that the integral → 0 as t → ∞ and σ → F(∞)k1 [C(∞)]F(∞)T , which is a different constitutive equation for nonlinear elasticity associated with the large time equilibrium state. In these last two cases, the constitutive equation for FLVE reduces to that for a nonlinear elastic material, but with properties at t = 0+ differing from those in the limit as t → ∞. In other terms, the material could exhibit one kind of response at t = 0+ and then, evolve during its deformation history to a different type of response in the limit as t → ∞. If the material is assumed to be incompressible, then det F(s) = 1, det C(s) = 1, det Ct (s) = 1, s ∈ (−∞, t]. Equation (80) now includes the reaction term − pI due to the constraint, as shown in (70). The dual form of this constitutive equation in which the deformation is expressed in terms of the stress history does not appear to have been considered in the literature and is not considered here.

6.4 Pipkin–Rogers Constitutive Theory Pipkin and Rogers [34] developed a constitutive theory for nonlinear viscoelastic solids based on a set of assumptions about the response to step strain or stress histories. The response functional G in (62) has the form of a series in which the first term gives the best approximation to measured mechanical response using single step strain histories. The next level of approximation uses the response to double step strain histories, and so on. The leading term of the series is   σ = F(t) K2 [C(t), 0] +

t 0

∂ K2 [C(s), t − s]ds F(t)T ∂(t − s)

or the alternate form obtained by an integration by parts

(83)

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  σ = F(t) K2 [C(0), t] + 0

t

∂ ∂C(s) K2 [C(s), t − s] ds F(t)T . ∂C(s) ∂s

(84)

At a fixed value of C, K2 [C, s] is assumed to decrease monotonically with s to a nonzero limit denoted by K2 [C, ∞]. This, in effect, incorporates the notion of fading memory into the Pipkin–Rogers (PR) constitutive theory. Note that (83) is analogous in form to (18). Equation (84) is analogous to one of the alternate forms listed in [55]. Several observations about this constitutive equation can be made that are analogous to those in Sect. 6.3. First, if the material does not deform from its reference configuration, then (83) reduces to σ = K2 [I, t]. If it is assumed that the material is stress free in its reference configuration, then K2 [I, t] = O. If the material is incompressible, then K2 [I, t] = k2 I for some scalar k2 . Second, let the deformation be small for s ∈ [0, t], then, as in Sect. 6.3, C(s) ≈ I + e(s) and the linearization of (83) reduces to the constitutive equation for linearized viscoelasticity. Third, suppose the deformation undergoes a step change from t = 0− to t = 0+ , then σ = F(0+ )K2 [C(0+ ), 0]F(0+)T , a constitutive equation for nonlinear elasticity associated with instantaneous response. Fourth, let the body approach a fixed deformation as t → ∞. As in linear viscoelasticity, under reasonable assumptions, the stress reaches an equilibrium state and σ = F(∞)K2 [C(∞), ∞]F(∞)T , a constitutive equation for nonlinear elasticity associated with large time equilibrium response. As before, it is observed that the material could exhibit one kind of response at t = 0+ and then, evolve during its deformation history to a different type of response in the limit as t → ∞. If the material is assumed to be incompressible, then the motion must be such that det F(s) = det C(s) = 1, s ∈ [0, t]. Equation (83) now includes the reaction term − pI due to the constraint, as shown in (70). Pipkin and Rogers discussed the dual to (83). Although the dual formulation gives an expression that is convenient for modeling the results of creep experiments, it is less convenient for use in the equation expressing the balance of linear momentum. Consequently, only (83) and its version for incompressibility are considered here.

6.5 Quasi-linear Viscoelasticity The special case of (83) or (84) when K2 [C, s] is separable into a function of C and a function of s, i.e., (85) K2 [C, s] = Ke [C]G(s) has become known as quasi-linear viscoelasticity. If Ke is normalized so that G(0) = 1, then, (83) becomes

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  t ∂G(t − s) ds F(t)T . σ = F(t) Ke [C(t)] + Ke [C(s)] ∂(t − s) 0

(86)

If the material is assumed to be incompressible, then a term − pI must be added to (86). The terminology quasi-linear viscoelasticity arises because Ke [C] can be thought of as a nonlinear measure of strain. The expression in braces in (86) is linear in this nonlinear strain measure and has a form analogous to (18). This constitutive equation, proposed by Fung [14], is used to represent the mechanical response of a variety of biological tissues. It is also convenient for developing analytical results that illustrate qualitative features of nonlinear viscoelastic behavior that could be expected when using more complicated constitutive equations.

6.6 Constitutive Theories Utilizing the Clock Concept It is well established in linear viscoelasticity that stress relaxation is affected by temperature [55]. This effect is accounted for by introducing a new time-like variable ξ , called the material time, intrinsic time, or reduced time. It arises from the notion that stress relaxation occurs in a polymer as its macromolecular structure goes through a sequence of reconfigurations. The reduced time ξ represents the time during this sequence as seen by the material and differs from the laboratory time t. That is, the material follows its own clock for stress relaxation that can run faster or slower than the laboratory clock. The increment of material time dξ is related to the increment of laboratory time dt by dt , (87) dξ = a(T (t), T0 ) in which T0 is a reference temperature, T (t) is the current temperature, and a(T (t), T0 ) is a material property called the time–temperature shift function. The properties of this function are (1) a(T (t), T0 ) > 0 thereby ensuring that dt > 0 gives dξ > 0; (2) when T (t) > T0 , then a(T (t), T0 ) < 1 and dξ/dt > 1 so that the material time increment is larger than the laboratory time increment (the material clock moves faster); (3) when T (t) < T0 , then a(T (t), T0 ) > 1 and dξ/dt < 1. The material time increment is now smaller than the laboratory time increment (the material clock moves slower). The current material time ξ is related to the current laboratory time t by 

t

ξ(t) = 0

dx , a(T (x), T0 )

a relation often referred to as defining a temperature clock.

(88)

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A. Wineman

There is experimental evidence that the time dependence of response is also affected by strain. This has led to the notion of a strain clock, whose definition is analogous to the temperature clock,  ξ(t) = 0

t

dx . a(C(x)) ˆ

(89)

Knauss and Emri [18] and Shay and Caruthers [41], based on considerations from polymer science, assumed that a(C(x)) ˆ depends on volumetric deformation. McKenna and Zapas [25] interpreted results on torsion of PMMA as indicating that a(C(x)) ˆ should depend on shear deformation. Experiments by Popelar and Liechti [35] led them to express a(C(x)) ˆ in terms of both volumetric and shear deformation. Recently, Caruthers et al. [4] proposed a constitutive theory that expresses aˆ in terms of the configurational energy of the molecular structure. This led to an expression for aˆ in terms of the deformation history C(x  ), x  ∈ [0, x]. The strain clock concept is a subject of ongoing research. The clock is introduced into the constitutive equation by replacing the constitutive assumption (57) with      ∞ σ (t) = F F(ξ(t) − ξ(s))|∞ s=0 = F F(ξ − ξ )|ξ  =0 .

(90)

The restrictions due to superposed rigid body motions and material symmetry can be imposed on constitutive equations of the form (90). The dependence of aˆ on C(s) arises from the former restriction. The latter restriction implies that aˆ depends on the appropriate invariants of C(s). The Green–Rivlin, finite linear viscoelasticity and Pipkin–Rogers models can still be obtained by appropriate assumptions, but with the argument t − s replaced by ξ(t) − ξ(s). As an example, when the clock concept is incorporated into the Pipkin–Rogers constitutive equation in (84), it becomes   σ = F(t) K2 [C(0), ξ(t)] +

t 0

∂ ∂C(s) K2 [C(s), ξ(t) − ξ(s)] ds F(t)T . ∂C(s) ∂s (91)

7 Constitutive Equations for Isotropic Materials The forms for the constitutive equations presented in Sect. 6 reduce the problem of finding material symmetry restrictions on the response functional G in (62) to that of finding material symmetry restrictions on the tensor-valued functions in (72), (74), (78), (80) , and (83). Each of these is a tensor-valued function A(M1 , M2 , . . . , M N ) of a set of tensors Mi , i = 1, 2, . . . , N . The material symmetry condition (66) imposed on the function A has the form HA(M1 , M2 , . . . , M N )HT = A(HM1 HT , HM2 HT , . . . , HM N HT ).

(92)

Viscoelastic Solids

105

The method for determining the form of A satisfying (92) has been presented in the review article by Spencer [42]. There are two classes of isotropy: (a) proper (or hemihedral) isotropy whose material symmetry group contains all rotational transformations, and (b) full (or holohedral) isotropy, whose material symmetry group contains all rotations and a central reflection transformation. Note that the constitutive theories presented in Sect. 6 involve only second-order tensors. In this case, there is no distinction between proper or full isotropy because (66) is identically satisfied by central reflection transformations.

7.1 A Special Result for Isotropic Materials There is an interesting result for isotropic nonlinear viscoelastic solids that does not depend on the form of the response functional. Using the requirement that (60) holds for all rotation histories, Q(s), s ∈ [0, t], and the isotropy condition (64) holds for all rotation transformations H, Noll [33] showed that the constitutive equation can be written in the form   σ (t) = Gˆ B(t); Ct (t − s)|∞ s=0 ,

(93)

where B(t) was defined in (50) and Ct (s) was defined in (55). The response functional Gˆ satisfies     T ˆ ∞ Gˆ HT B(t)H; HT Ct (t − s)H|∞ s=0 = H G B(t); Ct (t − s)|s=0 H

(94)

for all orthogonal transformations H. Gˆ is said to be an isotropic functional. If, in addition, the material is incompressible and isotropic, the constitutive equation (93) now includes the term − pI.

7.2 Rate and Differential Type Constitutive Equations The rate type constitutive equation in (72) becomes   (1) (2) (n) R2 σ , σ , σ , . . . , σ ; A1 A2 , . . . , Am = 0.

(95)

The tensors Ak , known as Rivlin–Ericksen tensors [38], are defined recursively by A1 = L + LT , Ak+1 =

DAk + Ak L + LT Ak , Dt

(96)

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A. Wineman (n)

where L was introduced in (56). σ is a stress rate that is defined recursively by (0)

σ = σ,

(n)

(n) D σ (n) + σ L + LT σ . σ = Dt

(n+1)

(97)

R2 is an isotropic tensor-valued function whose general form is that of a matrix polynomial in its arguments. Its details depend on the material under consideration, and is hence a material property. The general form is not presented here because ratetype constitutive equations are rarely used in the description of viscoelastic solids, although they are used for viscoelastic fluids. This type of constitutive equation will not receive further consideration. The constitutive equation (74) for isotropic materials of differential type becomes σ = R3 [A1 , A2 , . . . , Am ; B] ,

(98)

where R3 is matrix polynomial in it arguments. The details of R3 depend on the material under consideration and is a material property. Materials modeled by (98) are referred to as Rivlin-Ericksen materials [38]. This constitutive equation has been used to study limited aspects of the mechanics of viscoelastic solid and also will not receive further discussion.

7.3 Green–Rivlin Multiple Integral Constitutive Equations The form of each integrand in (78) can be constructed by identifying it with A. Each integrand has a representation as a matrix polynomial in the strain increments dE(si ). This constitutive theory received a great deal of attention when first introduced. An extensive discussion of experimental and analytical work based on this theory is provided in the reference by Findley et al. [9]. Most of the experimental work on determining material properties make use of the dual form (78) because it is experimentally more feasible to apply step stresses and measure the creep response. There is little current interest in the model for several reasons. The triple integral truncation in (78) is adequate for strains of about 0.1. However, larger strains require integrals of higher multiplicity. This rapidly increases the number of experiments and functions of time to be measured as well as the cost of the numerical evaluation of the integrals. This theory will not receive further discussion in later sections.

7.4 Finite Linear Viscoelasticity The material symmetry condition (64) or (66), when applied to the general form of the constitutive equation (80) for finite linear viscoelasticity must be satisfied for all times

Viscoelastic Solids

107

t and deformation histories, as represented by all C(t) and all Ct (s), s ∈ (−∞, t]. This leads to the condition that k1 and K1 must satisfy the conditions, HT k1 [C(t)]H = k1 [HT C(t)H]

(99)

  HT K1 [C(t), t − s] F(t)T (Ct (s) − I)F(t) H   = K1 [HT C(t)H, t − s] HT F(t)T (Ct (s) − I)F(t)H ,

(100)

and

for all rotation transformations H and times t. According to [42], k1 [C] has the form k1 [C] = β0 [I (C)]I + β1 [I (C)]C + β2 [I (C)]C2 ,

(101)

where I (C) denotes the set of invariants associated with isotropy, I1 (C) = trC,

I2 (C) =

 1 (trC)2 − tr(C2 ) , 2

I3 (C) = det(C).

(102)

When (101) is substituted into (80), Fk1 [C]FT can be expressed in terms of B by using (50) and noting that Iα (C) = Iα (B). With the use of the Cayley–Hamilton ˜ where theorem, Fk1 [C]FT = k[B], ˆ k[B] = αˆ 0 [I (B)]I + αˆ 1 [I (B)]B + αˆ 2 [I (B)]B2 .

(103)

  A corresponding result gives that F(t)K1 [C(t), t − s] F(t)T (Ct (s) − I)F(t) F(t)T ˆ can be expressed as K[B(t), t − s](Ct (s) − I). Equation (80) can then be put in the form of (93), ˆ σ = k[B(t)] +



t

ˆ K[B(t), t − s](Ct (s) − I)ds.

−∞

(104)

In order to discuss the response to step changes in deformation, (104) is usually written in the alternate form obtained by an integration by parts, ˜ σ = k[B(t)] +



dCt (s) ˜ ds. K[B(t), t − s] ds −∞ t

(105)

˜ ˆ k[B] has the same tensorial form as k[B], but with αˆ i [I (B)] replaced by α˜ i [I (B)]. The integrand of (105) is given by

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A. Wineman

dCt (s) ˜ K[B(t), t − s] = ds +

  dCt (s) dCt (s) α φα (t − s) Bα + B ds ds α=0

2

2

2

φαβ (t − s)Bα tr[Bβ

α=0 β=0

dCt (s) ]. ds

(106)

The scalar coefficients φα and φαβ are functions of t − s and the invariants I (B). At a fixed B, φα and φαβ are monotonically decreasing functions of their the argument t − s and approach zero in the limit as t − s → ∞. If the material is assumed to be incompressible and isotropic, (104) and (105) are modified by the addition of the term − pI as in (70). Since I3 (B) = 1, the scalar coefficients αi now depend only on I1 (B) and I2 (B).

7.5 Pipkin–Rogers Constitutive Theory The material symmetry condition (64) or (66), when applied to the general form of the PR constitutive equation in (83) must be satisfied for all times t and deformation histories C(s), s ∈ [0, t]. This implies HT K2 [C, t]H = K2 [HT CH, t], t ≥ 0.

(107)

For isotropy, this is to be satisfied for all rotation transformations H. As this has the same form as (99), the same discussion used to arrive at (101) can be used here to show that K2 [C, t] = α0 [I (C), t]I + α1 [I (C), t]C + α2 [I (C), t]C2 .

(108)

The properties of the scalar coefficients αi [I (C), t] are induced by those of K2 [C, t] described in Sect. 6.4. They are relaxation properties that depend on the deformation and monotonically decrease with t to some nonzero limit αi [I (C), ∞]. Let (108) be substituted into (83), then σ = F(t)Π (t)F(t)T ,

(109)

Π (t) = α0 [I (C(t)), 0] I + α1 [I (C(t)), 0] C(t) + α2 [I (C(t)), 0] C2 (t)  t  ∂ α0 [I (C(s)), t − s] I + α1 [I (C(s)), t − s] C(s) + 0 ∂(t − s) + α2 [I (C(s)), t − s] C2 (s) ds. (110)

Viscoelastic Solids

109

For an incompressible material, (109) is modified by the addition of the term − pI. I (C) now represents (I1 (C), I2 (C)) since deformations are restricted by (67) to satisfy the constraint I3 (C) = 1. The terms outside the integral can be expressed in terms of B(t) by use of (50) and the observation that Iα (C) = Iα (B), α = 1, 2, 3. However, the integrand cannot be expressed only in terms of B(t) because it depends on C(s) for all times s ∈ [0, t]. It is possible to express F(t)C(s)F(t)T in terms of B(t) and Ct (s) by use of (50), (53), and (55). There seems to be no particular advantage in doing so and, therefore, it is not done here.

8 Constitutive Equations for Transversely Isotropic and Orthotropic Materials Materials such as fiber-reinforced composites or biological tissue have often been modeled as nonlinear elastic transversely isotropic solids. If the constituents exhibit viscoelasticity, then a nonlinear viscoelastic transversely isotropic solid would be appropriate. Restrictions due to transverse isotropy and orthotropy on the functions appearing in the rate and differential constitutive equations of Sect. 6.1, the Green–Rivlin constitutive equation of Sect. 6.2 and the FLVE constitutive equation of Sect. 6.3 lead to very complicated expressions. For the PR constitutive equation (83), the expressions are more tractable. For this reason, the rest of this section will be limited to the PR constitutive equation for incompressible materials.

8.1 Transverse Isotropy The general material symmetry restriction on the PR constitutive equation implies that K2 [C, t] must satisfy (107) for all material symmetry transformations describing transverse isotropy. In [37], this restriction was imposed under the assumption that K2 [C, t] can be derived from a potential. A more general approach is taken here. Let L be a unit vector in the reference configuration that denotes the fiber direction and hence the axis of transverse isotropy. According to [42], K2 [C, t] has the form K2 [C, t] = α0(TI) [I (TI) (C), t]I + α1(TI) [I (TI) (C), t]C + α2(TI) [I (TI) (C), t]C2 +α3(TI) [I (TI) (C), t]L ⊗ L + α4(TI) [I (TI) (C), t](L ⊗ CL + CL ⊗ L), (111) where I (TI) (C) denotes the set of invariants I1 (C), I2 (C), I4 (C) I5 (C), the first two being defined in (102) and I4 (C) = L · (CL),

I5 (C) = L · (C2 L).

(112)

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A. Wineman

The coefficients αi(TI) [I (TI) (C), t] are scalar deformation dependent relaxation properties for transverse isotropy. They decrease monotonically with t to some nonzero limit αi(TI) [I (TI) (C), ∞]. Comments analogous to those in Sects. 6.3 and 6.4 can also be made here, but are omitted because of space limitations. However, there is an additional comment that is appropriate to the application of this constitutive theory to fiber-reinforced materials. Just as for elastic fiber-reinforced materials, the first three scalars in (111) could be associated with the matrix and the last two with the fibers. In this regard, it should be noted that each scalar coefficient in (111) can be a different function of all the invariants I1 [C], I2 [C], I4 [C], I5 [C], and time t. This has two implications: (1) the invariants associated with the fibers I4 [C], I5 [C] can affect the properties of the matrix and vice versa; (2) the matrix and fiber constituents can have different viscoelastic responses. A general discussion of the various assumptions regarding the scalar coefficients αi(TI) [I (TI) (C), t] can be found in [48].

8.2 Orthotropy For an orthotropic material, the preferred directions in the reference configuration are defined by the orthogonal unit vectors L1 , L2 , and L3 = L1 × L2 . K2 [C, t] must satisfy (107) for the material symmetry transformations, stated in [42], that take L1 , L2 , and L3 into materially equivalent directions. A representation for K2 [C, t] for orthotropy, developed using the method described in [42], is given by K2 [C, t] = α0(O) [I (O) (C), t]I + α1(O) [I (O) (C), t]C + α2(O) [I (O) (C), t]C2 + α3(O) [I (O) (C), t]L1 ⊗ L1 + α4(O) [I (O) (C), t](L1 ⊗ CL1 + CL1 ⊗ L1 ) + α5(O) [I (O) (C), t]L2 ⊗ L2 + α6(O) [I (O) (C), t](L2 ⊗ CL2 + CL2 ⊗ L2 ) + α7(O) [I (O) (C), t](L1 ⊗ L2 + L2 ⊗ L1 ) + α8(O) [I (O) (C), t](L1 ⊗ CL2 + CL2 ⊗ L1 ),

(113)

where I (O) (C) denotes the set of invariants associated with orthotropy, which contains I1 (C) and I2 (C) in (102) and I3 (C) = L1 · (CL1 ),

I4 (C) = L1 · (C2 L1 ),

I5 (C) = L2 · (CL2 ),

I6 (C) = L2 · (C L2 ),

I7 (C) = L1 · (CL2 ),

I8 (C) = L1 · (C2 L2 ).

2

(114)

The coefficients αi(O) [I (O) (C), t] in (113) are scalar deformation dependent relaxation properties for orthotropy. Results analogous to those developed in the previous sections for isotropy and transverse isotropy can be presented here, but are omitted.

Viscoelastic Solids

111

9 Homogeneous Deformations Attention is now limited to applications using the nonlinear single integral type constitutive equations, i.e., the FLVE and PR models. Experimental programs to determine the material property functions in these models usually attempt to produce homogeneous deformations in specimens, that is, motions of the form in (45), where F(s) is independent of X. The most common are triaxial stretch histories for which ⎛

λ1 (s) F(s) = ⎝ 0 0

0 λ2 (s) 0

⎞ 0 0 ⎠ , s ∈ [0, t]. λ3 (s)

(115)

and simple shear histories for which ⎛

1 F(s) = ⎝ 0 0

K (s) 1 0

⎞ 0 0 ⎠ , s ∈ [0, t]. 1

(116)

The stretch ratio λi (s) or shear K (s) can have a jump discontinuity at s = 0 and then vary arbitrarily for s ∈ [0, t]. Uniaxial and biaxial stretch histories are special cases of (115) that are often used in experiments. Owing to space limitations, this presentation is limited to uniaxial stretch histories and the incompressible PR constitutive theory for isotropic materials because of its convenience for developing certain results. Corresponding treatments for shear histories are readily carried for both the PR and FLVE constitutive equations.

10 Uniaxial Stretch Histories Uniaxial stretch is the special case of triaxial stretch when there is only one nonzero stress component. As in other areas of solid mechanics such as elasticity and plasticity, an understanding of the response in uniaxial stretch is essential to an understanding of the material behavior. A number of aspects of uniaxial nonlinear viscoelastic response are presented here and then connected to their counterparts in linear viscoelasticity introduced in Sect. 3. Let the reference configuration of an isotropic nonlinear viscoelastic solid be a block with edges along the axes of a Cartesian coordinate system. On using (115) in the incompressible version of (83), it is found that the only nonzero stresses are σii (t), i = 1, 2, 3. These are related to the stretch ratio histories λi (s), s ∈ [0, t], i = 1, 2, 3 by a system of nonlinear Volterra integral equations. This is supplemented by the incompressibility condition λ1 (s)λ2 (s)λ3 (s) = 1, s ∈ [0, t]. Consider uniaxial extension along the X 3 -axis. For notational convenience, let λ3 (t) = λ(t) and σ33 (t) = σ (t). For uniaxial extension σ11 (t) = σ22 (t) = 0, t > 0.

112

A. Wineman

If the history λ3 (s) = λ(s), s ∈ [0, t] is specified, then equations for i = 1 and i = 2 of (83) , along with λ1 (s)λ2 (s)λ3 (s) = 1, s ∈ [0, t], become a system of nonlinear Volterra integral equations for p(s), λ1 (s) , and λ2 (s), s ∈ [0, t]. Once these are known, the equation for i = 3 of (83) is used to determine σ (t), t > 0. On the other hand, If the stress history σ (t), t > 0 is specified, (83) along with λ1 (s)λ2 (s)λ3 (s) = 1, becomes a system of nonlinear Volterra integral equations for p, λ1 (t), λ2 (t), λ(t), t > 0. These can be solved using the numerical method described in [22, 54]. The numerical solution implies that λ2 (t) = λ1 (t), t ≥ 0. With this result, the condition λ1 (t)λ2 (t)λ3 (t) = 1 gives λ1 (t) = λ(t)−1/2 .

(117)

The constitutive equation for uniaxial response can now be stated as     1 1 α0 [I (C(t)), 0] + α1 [I (C(t)), 0] λ(t)2 + σ (t) = λ(t)2 − λ(t) λ(t)   1 + α2 [I (C(t)), 0] λ(t)4 + λ(t) + λ(t)2     t ∂ 1 α0 [I (C(s)), t − s] λ(t)2 − + λ(t) 0 ∂(t − s)   1 + α1 [I (C(s)), t − s] λ(t)2 λ(s)2 − λ(t)λ(s)   1 ds, (118) + α2 [I (C(s)), t − s] λ(t)2 λ(s)4 − λ(t)λ(s)2 in which I (C(s)) denotes the invariants I1 (C(s)) = λ(s)2 +

2 , λ(s)

I2 (C(s)) = 2λ(s)4 +

1 . λ(s)2

(119)

This equation is the focus of the remainder of this paper.

10.1 Small Strain Limit Let (118) be expressed in terms of the strain by substituting the relation λ(s) = 1 + ε(s). Let it also be assumed that εm = max |ε(s)|, s ∈ [0, t] is small and expand the right-hand side of (118)  in a Taylor series in ε(s). The approximation to (118) including terms through O εm2 is

Viscoelastic Solids

113

   t  t dG(t − s) dG(t − s) ε(s)ds + ε(t) ε(t)G(0) + ε(s)ds 0 d(t − s) 0 d(t − s)  t ˆ d G(t − s) ˆ + + ε(t)2 G(0) ε(s)2 ds, 0 d(t − s)

σ (t) = ε(t)G(0) +

where

ˆ G(t) = 3α1 [I (I), t] + 6α2 [I (I), t], G(t) = 3α2 [I (I), t]

(120)

(121)

and the condition K2 (t) = O is used in the stress-free state . Thus, when εm  1 and only first-order terms are retained, (120) reduces to the constitutive equation for linear viscoelasticity (18).

10.2 Stress Relaxation Let λ(s) = λ0 = 1, s ∈ [0, t], where λ0 is a constant. By (119), I1 (C(s)) = λ20 +

2 , λ0

I2 (C(s)) = 2λ40 +

1 . λ20

(122)

If the notation α˜ i (λ0 , s) = αi [I (C(s), s] is introduced when the invariants are given by (122), the stress in (118) reduces to  σ (t) = α˜ 0 (λ0 , t)

λ20

1 − λ0



  1 4 + α˜ 1 (λ0 , t) λ0 − 2 λ0   1 6 + α˜ 2 (λ0 , t) λ0 − 3 = G(t, λ0 ). λ0

(123)

The right-hand side is a stretch-dependent stress relaxation function for uniaxial extension. Setting λ0 = 1 + ε0 in (123) gives G(t, ε0 ) introduced in Sect. 2.2. Various forms for G(t, ε0 ) have appeared in the literature: (1) a simple separation of variables product form as in the quasi-linear viscoelastic constitutive equation (86) (124) G(t, λ0 ) = f e (λ0 )G(t); (2) a summation of product terms G(t, λ0 ) =

k=N

k=1

f k (λ0 )G˜ k (t);

(125)

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A. Wineman

(3) stretch-dependent characteristic time, initial, and longtime values,  G(t, λ0 ) = G ∞ (λ0 ) + [G 0 (λ0 ) − G ∞ (λ0 )] G

 t . τ (λ0 )

(126)

10.3 Creep Consider the step stress history σ (t) = σ0 , t ≥ 0 where σ0 is a constant. Let the creep response to this step stress history be denoted by J (t, σ0 ), the creep function introduced in Sect. 2.1. This is found by solving the nonlinear Volterra integral equation (118), the counterpart here of (18). Owing to the general nonlinear dependence of (118) on λ(s) and the fact that there are many possible forms for the material property functions, it is unlikely that an analytical solution for J (t, σ0 ) can be found. Instead, J (t, σ0 ) will have to be determined numerically using the methods described in [22, 54]. It is possible, in linear viscoelasticity, to derive analytical relations (21) and (22) between the creep and stress relaxation properties from (18) or (19). Corresponding relations between J (t, σ0 ) and the material properties in (118) could be stated, but are omitted here. It is expected, based on linear viscoelasticity, that if the scalar coefficient αi monotonically decreases to a nonzero limit as t → ∞, then J (t, σ0 ) will monotonically increase to a finite limit J (∞, σ0 ).

10.4 Sinusoidal Oscillations About the Reference State Let the block be subjected to the sinusoidal stretch history λ(s) = 1 + ε0 sin ωs, s ∈ [0, t]. When this stretch history is substituted into (118), it is assumed that the stress reaches a state of steady oscillations. The mathematical issues in showing this are not discussed here. When ε0  1, the response is linear and is given by (26). When ε0 is larger, the nonlinear dependence of (118) on λ(s) causes the σ versus t plot to be periodic but not sinusoidal. To this end, let the strain history ε(s) = ε0 sin ωs be substituted into (120). The stress becomes   ε2  ˆ σ (t) = ε0 G  (ω) sin ωt + G  (ω) cos ωt + 0 G  (ω) + G(∞) 2    − cos 2ωt G  (ω) + Gˆ  (2ω) + sin 2ωt G  (ω) + Gˆ  (2ω) . (127) The terms in (127) that are linear in εo appear in (26) for linear viscoelastic response. The material functions of frequency in (127) are calculated from their corresponding functions of time using (28) and (29).

Viscoelastic Solids

115

Equation (127) shows that as the terms in ε02 become larger, the stress becomes modified by the addition of terms in sin 2ωt and cos 2ωt. In general, when ε0 increases further so that terms in ε0n must be included, the stress becomes further modified by the addition of terms in sin nωt and cos nωt. Thus, an indicator of nonlinearity is the appearance of these higher frequency terms and the associated change of the shape of the σ versus t plot. As these higher frequency terms become more significant, stress and strain are no longer related by (39). The stress versus strain plot changes from an ellipse to some other shape of closed curve. Both squared off and S-shaped curves have been observed in experiments.

10.5 Small Deformation Superposed on Finite Axial Stretch Let a bar be subjected to a uniaxial stretch history described as follows: a step stretch is applied at t = 0 and held constant until the stress has relaxed to its longtime value. The stretch then has a small perturbation. The stretch history is given by λ(s) = λ0 (1 + η(s)), s ∈ [0, t],

(128)

ˆ − T ∗ ), s ∈ [T ∗ , t]. η(s) = 0, s ∈ [0, T ∗ ]; η(s) = η(s

(129)

where |η(s)|  1 and

The stress relaxation response to the underlying step stretch history λ(s) = λ0 , s ∈ [0, t] is given by (123). Let T ∗ be a large enough time when the stress is very close to its longtime limit. The response to the perturbed stretch history is obtained by substituting (128) and (129) into (118), expanding terms in Taylor series and retaining only the terms linear in the perturbation η(s). Letting t = tˆ + T ∗ , where tˆ denotes a time measured from T ∗ , the stress becomes σ (t) = G(∞, λ0 ) + σ˜ (λ0 , t),

(130)

where     1 1 2 4 α0 (λ0 , ∞) + 2λ0 + 2 α1 (λ0 , ∞) 2λ0 + λ0 λ0     tˆ 1 ∂ M(λ0 , tˆ − s) + 2λ60 + 3 α2 (λ0 , ∞) + η( ˆ tˆ)M(λ0 , 0) + η(s) ˆ ds (131) λ0 ∂(tˆ − s) 0

ˆ tˆ) σ˜ (λ0 , t) = η(

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and   1 2 ∗ 2 M(λ0 , t) = 2 λ0 − α0 (λ0 , t) λ0    1 + 2λ40 + 2 α1 (λ0 , t) + 2 λ20 − λ0    2 6 + 4λ0 + 3 α2 (λ0 , t) + 2 λ20 − λ0 with αi∗ (λ0 , t) =

  1 1 2λ40 − 2 α1∗ (λ0 , t) λ0 λ0   1 1 6 λ0 − 3 α2∗ (λ0 , t), (132) λ0 λ0

∂αi 1 ∂αi (λ0 , t) + (λ0 , t). ∂ I1 λ0 ∂ I 2

(133)

The last two terms in (131) have the same form as (18) and hence represent a superposed linear viscoelastic response with a stress relaxation modulus M(λ0 , t) that depends on the underlying stretch λ0 . Let the perturbation be the sinusoidal deformation η(s) ˆ = ε0 sin ωs. When time tˆ becomes large, (131) becomes     1 1 α0 (λ0 , ∞) + 2λ40 + 2 α1 (λ0 , ∞) σ˜ (λ0 , tˆ) = ε0 sin ωtˆ 2λ20 + λ0 λ0      1 + 2λ60 + 3 α2 (λ0 , ∞) + ε0 M  (λ0 , ω) sin ωtˆ + M  (λ0 , ω) cos ωtˆ , (134) λ0 with  ∞ ∂ M(λ0 , s) cos ωs ds, M  (λ0 , ω) = M(λ0 , 0) + ∂s 0  ∞ ∂ M(λ0 , s) M  (λ0 , ω) = − sin ωs ds. ∂s 0

(135) (136)

This result shows the influence of the underlying stretch on the response to superposed small amplitude vibrations. In particular, the underlying stretch affects M  (λ0 , ω). Thus, recalling the discussion in Sect. 3.1.4, the underlying stretch affects the damping characteristics and work done per cycle.

11 Large Non-homogeneous Deformations of Nonlinear Viscoelastic Solids Nonlinearity in viscoelastic response occurs when there is large deformation and/or nonlinear material properties. This section describes a variety of examples of large non-homogeneous deformations of nonlinear viscoelastic solids that have appeared

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in the literature. It is not intended that an exhaustive summary of such examples be provided here. The purpose is to provide a representative listing of nonlinear viscoelastic problems that have been solved and thereby show that such solutions are quite feasible.

11.1 Controllable Deformations of Incompressible Isotropic Solids It has been shown in [2, 3, 10] that there are five families of non-homogeneous motions that are possible in any incompressible isotropic solid, whatever the form of the constitutive equation. Although the motions are independent of the constitutive equation, the stresses are not. For each family of motions, the stresses are such that the equations of motion can be satisfied and an expression for the scalar p associated with the incompressibility constraint can be determined. The families, termed controllable, are described in [54], each with its deformation gradient. The relevant kinematical quantities for use in any constitutive equation of interest can then be calculated. The families are 1. 2. 3. 4. 5.

Bending, stretching, and shearing of a rectangular block. Straightening, stretching, and shearing of a sector of a hollow cylinder. Inflation, torsion, extension, and shearing of an annular wedge. Inflation of a sector of a spherical shell. Inflation, bending, extension, and azimuthal shearing of an annular wedge.

The third family of motion contains two important special cases, the combined extension, inflation, and torsion of a hollow circular cylinder and the combined tension and torsion of a solid cylinder. Carroll [2] discussed these cases for a general isotropic incompressible solid. The latter motion was used by Yuan and Lianis [57] as part of an experimental program to determine the specific form of the FLVE constitutive equation for a particular material. A detailed example of combined tension and torsion of a solid cylinder using the Pipkin–Rogers constitutive equation for incompressible materials is given in [54].

11.2 Finite Deflection of Viscoelastic Beams Rogers and Lee [39] solved for the finite deflections of a viscoelastic cantilever beam loaded by a time-dependent concentrated force at its free end. The initial beam thickness was assumed to be small compared to its length so that the strains through the thickness caused by relative rotation of the cross sections would be small. In their formulation, the length of the central axis does not change and points on the central axis undergo large displacements while cross sections rotate and remain perpendicular to the deformed central axis.

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With reference to (47), this kinematical description implies: (1) the rotations R(t) are large; (2) U(t − s) is approximated by U(t − s) = I + ε(t − s), where ε is the small strain tensor; and (3) F in (57) is approximated by the constitutive equation (18) for linearized viscoelasticity. This allows the use of the assumptions of classical beam theory to calculate the moment–curvature relation with respect to axes attached to a rotating beam segment. Combining the small strain moment–curvature relation for linearized viscoelasticity written with respect to axes rotating with a beam segment and the equilibrium equations in the current configuration results in a partial differential–Volterra integral equation. This equation, with appropriate boundary conditions was solved numerically by a method described in [39]. Muliana [30] considered a more general version of the nonlinear viscoelastic cantilever beam by accounting for length changes of the central axis, distributed transverse loads, and concentrated force and moments at the tip. The equations expressing equilibrium in the current configuration, resultant force and moment–displacement relations and the constitutive equation written with respect to axes rotating with a beam segment were solved numerically as a system.

11.3 Nonlinear Viscoelastic Membranes A number of examples involving large deformations of viscoelastic membranes have appeared in the literature. For the examples mentioned here, the membrane material is incompressible and isotropic and is described by either the FLVE (105) or the PR (109) and (110) nonlinear single integral constitutive equation. When such a constitutive equation is combined with the equations for quasi-static motion, the resulting equation has a differential operator in the spatial variable and the Volterra integral operator in the time variable—a nonlinear partial differential–Volterra integral equation. The following examples describe methods for the numerical solution of the boundary-value problems. Large in-plane radial axisymmetric deformations of an initially plane annular membrane were treated in [49, 50]. In [49], the inner boundary was traction free and either tractions or displacements were specified at the outer boundary. In [50], the inner boundary was fixed, the outer boundary was traction free, and the membrane deformed due to centrifugal force while spinning. Several problems have been solved for out-of-plane deformations of initially plane membranes clamped along a circular boundary. In [8, 51], the membrane was inflated to a surface of revolution by lateral pressure. In another application [47], a tubular membrane was attached to rigid discs at its ends. The membrane was deformed by internal pressure and axial forces were applied to the discs. A general discussion of axisymmetric problems involving nonlinear viscoelastic membranes is given in [53]. Recent solutions involving contact are the indentation of an initially plane membrane by a spherical indenter in [31] and compression between two parallel plates of a spherical membrane enclosing an incompressible fluid [32].

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11.4 Automotive Related Applications Vehicle suspension systems make use of bushings, which are essentially hollow cylinders of elastomeric material contained between an inner metal rod and an outer metal sleeve. The sleeve and rod are connected to components of the suspension system. They undergo relative motions along and about their common centerline as well as along and about axes perpendicular to the centerline. The relation between the forces and moments applied to the rod and sleeve and their relative motion is used in the engineering of suspension systems. Models have recently been developed that relate forces and relative displacements along the common centerline (axial mode) [19], moments and relative rotations about the common centerline (torsional mode) [20] and their coupled response [21] using the FLVE constitutive equation for the bushing material. The connection between the histories of axial force and moment and the histories of the displacements is complicated and computationally expensive to implement. The model was thus used in a method developed in [21] to construct a force–displacement level constitutive equation for nonlinear viscoelastic response having the Pipkin–Rogers framework. Elastomeric engine mounts are used in vehicles for noise and vibration isolation. Morman et al. [27] considered a cylindrical engine mount that has rigid plates bonded to its end surfaces. Normal forces on the end plates compress the block while the lateral surface remains traction free. The NLVE constitutive equation was used to model the material. The related problem of the nonuniform extension of a nonlinear viscoelastic slab was treated independently in [6]. The PR constitutive equation was used to model the material. Numerical methods of solution similar to those in the previous examples were provided. Small amplitude vibrations superposed on a finite deformation described in Sect. 10.5 occur in many applications. Thus, Goldberg and Lianis [15] carried out calculations using the constitutive equation developed in [24] for FLVE as well as other models, performed experiments involving small amplitude oscillations on finite stretch, and compared the results with predictions of the models. For use in automotive applications, Morman et al. [28] and Morman and Nagtegaal [29] extended the ideas illustrated here to general small amplitude vibrations superposed on large deformations for the FLVE constitutive equation developed in [24] and incorporated the results in a finite element analysis.

11.5 Other Topics Although there has been research into the development of constitutive equations that use the notion of the strain clock discussed in Sect. 6.6, there have been few studies in the literature that explore its implications. This is probably due to the fact that the computational effort required to use such constitutive equations is large. The constitutive equation in [4] was developed under the auspices of Sandia National

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Laboratory, which has extensive experimental and computational facilities. Most applications in this laboratory are project oriented and few results have been presented in the open literature. However, some work has appeared. In [56], a constitutive equation for a compressible nonlinear viscoelastic material was developed that incorporates a strain clock into the Pipkin–Rogers framework. The clock was chosen to have a simple dependence on shear and volume strains so that its essential features could be explored and yet be amenable to computation. It was then used to study a block that was subjected to a homogeneous deformation consisting of shear superposed on triaxial extension. The dimensional, volume changes, and shear response in the absence of normal tractions was studied. In [46], the constitutive equation was used to study circular shear of a cylinder. The very time dependence of viscoelastic materials means that they dissipate energy. The influence of viscoelasticity on vibration damping was studied in [11–13]. A mass was attached to an elastomeric spring modeled by the FLVE constitutive equation. With the inertia term for the mass, the equation of motion is an integrodifferential equation. It was replaced by a system of first-order differential equations which was then analyzed. The problems of technical relevance that have appeared in the technical literature have been formulated with a semi-inverse method similar to that used in nonlinear elasticity. In nonlinear elasticity, the spatial dependence of the deformation is represented by an assumed expression containing parameters that determine the magnitude of the deformation. In nonlinear viscoelasticity, the deformation is embedded in a motion by letting the parameters in this expression be functions of time. Thus, as observed in [50, 53], boundary-value problems in nonlinear elasticity suggest corresponding problems in nonlinear viscoelasticity. These problems, as in the examples cited here, lead to partial differential–Volterra equations that can be solved by the methods in the references that have been mentioned. This should not be construed as suggesting that nonlinear viscoelasticity is just a straightforward extension of nonlinear elasticity. Nonlinear viscoelasticity incorporates the same interesting phenomena as nonlinear elasticity. However, the time-dependent behavior of nonlinear viscoelastic solids adds a layer of new and interesting phenomena to be investigated. In this regard, an important topic in nonlinear elasticity is the bifurcation of solutions. The corresponding idea in nonlinear viscoelasticity is the branching of a solution at some time during its history. This was pointed out in the context of the inflation of nonlinear viscoelastic spherical membranes in [52]. It was shown that there might be a time when the solution for the deformation branches into several solutions. This event was shown to depend on the pressure history and the material parameters. Such a study would determine the conditions for a branching time to exist, the branching time, the solution branches, and a criterion for the selection of the appropriate branch followed by the material. Most of the emphasis in the literature has been on isotropic materials. There appears to have been few studies of the response of anisotropic nonlinear viscoelastic solids. This topic is of interest in the study of fiber-reinforced materials and biological tissue. The PR constitutive theory for transversely isotropic materials has been used [48] to study phenomena in the biaxial stretching of sheets.

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Biological tissues generally exhibit nonlinear viscoelastic behavior. Indeed, the constitutive equation for quasi-linear viscoelasticity in Sect. 6.5 has been extensively used to model the response of a variety of such materials. A summary of its applications is more appropriately the subject of a separate article.

References 1. Atkin RJ, Fox N (1980) An introduction to the theory of elasticity. Longman Group, London, England 2. Carroll MM (1968) Finite deformations of incompressible simple solids I. isotropic solids. Q J Mech Appl Math 21:147–170 3. Carroll MM (1967) Controllable deformations of incompressible simple materials. Int J Eng Sci 5:515–525 4. Caruthers JM, Adolf DB, Chambers RS, Shirkande P (2004) A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers. Polymer 45:4577–4597 5. Coleman B, Noll W (1961) Foundations of linear viscoelasticity. Rev Mod Phys 33:239–249 6. Dai F, Rajagopal KR, Wineman AS (1992) Non-uniform extension of a non-linear viscoelastic slab. Int J Solids Struct 29:911–930 7. Drapaca CS, Sivaloganathan S, Tenti G (2007) Non-linear constitutive laws in viscoelasticity. Math Mech Solids 12:475–501 8. Feng WW (1992) Viscoelastic behavior of elastomeric membranes. J Appl Mech 59:529–535 9. Findley WN, Lai JS, Onaran K (1989) Creep and relaxation of nonlinear viscoelastic materials. Dover Publications, New York 10. Fosdick RL (1968) Dynamically possible motions of incompressible, isotropic simple materials. Arch Ration Mech Anal 29:272–288 11. Fosdick R, Ketema Y, Yu JH (1998) Vibration damping through the use of materials with memory. Int J Solids Struct 35:403–420 12. Fosdick R, Ketema Y, Yu JH (1998) A non-linear oscillator with history dependent force. Int J Non-linear Mech 33:447–459 13. Franceschini G, Flori R (2001) Vibrations of a body supported by shear mountings of incompressible material with memory. Int J Eng Sci 39:1013–1031 14. Fung YC (1981) Biomechanics: mechanical properties of living tissues. Springer, New York 15. Goldberg W, Lianis G (1968) Behavior of viscoelastic media under small sinusoidal oscillations superposed on finite strain. J Appl Mech 35:433–440 16. Green AE, Rivlin RS (1957) The mechanics of non-linear materials with memory. Arch Ration Mech Anal 1:1–21 17. Gurtin ME, Sternberg E (1962) On the linear theory of viscoelasticity. Arch Ration Mech Anal 11:291–356 18. Knauss WG, Emri IJ (1987) Volume change and the nonlinearly thermo-elastic constitution of polymers. Polym Eng Sci 27:86–100 19. Lee SB, Wineman A (1999) A model for non-linear viscoelastic axial response of an elastomeric bushing. Int J Non-linear Mech 34:779–793 20. Lee SB, Wineman A (1999) A model for nonlinear viscoelastic torsional response of an elastomeric bushing. Acta Mech 135:199–218 21. Lee SB, Wineman A (2000) A model for non-linear viscoelastic coupled mode response of an elastomeric bushing. Int J Non-linear Mech 35:177–199 22. Linz P (1985) Analytical and numerical methods for volterra equations. Society for industrial and applied mathematics. Philadelphia 23. Lockett FJ (1972) Nonlinear viscoelastic solids. Academic, New York 24. McGuirt CW, Lianis G (1970) Constitutive equations for viscoelastic solids under finite uniaxial and biaxial deformations. Trans Soc Rheology 14:117–134

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25. McKenna GB, Zapas LJ (1979) Nonlinear behavior of poly (methylMethacrylate) in torsion. J Rheology 23:151–166 26. Morman KN Jr (1985) Rubber viscoelasticity-a review of current understanding. In: Proceedings of second symposium analysis design rubber parts. Batelle Institute Limited, London, England 27. Morman KN Jr, Djiauw LK, Kilgoar PC, Pett RA (1978) Stress and dynamic analysis of a bonded, non-linear viscoelastic cylindrical block. Society of Automotive Engineers Technical Paper No. 770599 28. Morman KN Jr, Kao BG, Nagtegaal JC (1981) Finite element analysis of viscoelastic elastomeric structures vibrating about non-linear statically stressed configurations. Society of Automotive Engineers Technical Paper No. 811309: 29. Morman KN Jr, Nagtegaal JC (1983) Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids, Part I: theoretical development. Int J Num Methods Eng 198:1079–1103 30. Muliana A (2015) Large deformations of nonlinear viscoelastic and multi-responsive beams. Int J Non-linear Mech 71:152–164 31. Nguyen N, Wineman A, Waas A (2012) Indentation of a nonlinear viscoelastic membrane. Math Mech Solids 18:24–42 32. Nguyen N, Wineman A, Waas A (2013) Contact problem of a non-linear viscoelastic spherical membrane enclosing incompressible fluid between two rigid plates. Int J Non-linear Mech 50:97–109 33. Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Ration Mech Anal 2:197–226 34. Pipkin AC, Rogers TG (1968) A non-linear integral representation for viscoelastic behaviour. J Mech Phys Solids 16:59–72 35. Popelar CF, Liechti KN (2004) A distortion-modified free volume theory for nonlinear viscoelastic behavior. Mech Time-Depend Mater 7:89–141 36. Prusa V, Rajagopal KR (2011) Jump conditions in stress relaxationand creep experiments of Burgers type fluids-a study in the application of Colombeau algebra of generalized functions. Z Angew Math Phys 62:707–740 37. Rajagopal KR, Wineman AS (2009) Response of anisotropic non-linearly viscoelastic solids. Math Mech Solids 14:490–501 38. Rivlin RS, Ericksen JK (1955) Stress-deformation relations for isotropic materials. J Ration Mech Anal 4:323–425 39. Rogers TG, Lee EH (1962) On the finite deflection of a viscoelastic cantilever. In: Proceedings of the U.S. national congress of applied mechanics, vol 4, pp 977–987 40. Schapery RA (2000) Nonlinear viscoelastic solids. Int J Solids Struct 37:359–366 41. Shay RM Jr, Caruthers JM (1986) A new viscoelastic constitutive equation for predicting yield in amorphous solid polymers. J Rheology 30:781–827 42. Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physics I. Academic, New York, pp 259–353 43. Spencer AJM (1980) Continuum mechanics. Longman Group, London, England 44. Stouffer DC, Wineman AS (1972) A constitutive equation for linear, aging, environmental dependent properties. Acta Mech 13:30–53 45. Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der Physik, Vol III/3. Springer, Berlin 46. Waldron WK Jr, Wineman A (1996) Shear and normal stress effects in finite circular shear of a compressible non-linear viscoelastic solid. Int J Non-linear Mech 31:345–369 47. Wineman A (1979) On the simultaneous elongation and inflation of a tubular membrane of BKZ fluid. J Non-newtonian Fluid Mech 6:111–125 48. Wineman AS. Branching of stretch histories in biaxially loaded nonlinear viscoelastic fiberreinforced sheets. Math Mech. Solids, in press. https://doi.org/10.1177/1081286518755231 49. Wineman AS (1972) Large axially symmetric stretching of a nonlinear viscoelastic membrane. Int J Solids Struct 8:775–790

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50. Wineman AS (1972) Large axisymmetric stretching of a nonlinear viscoelastic membrane due to spinning. J Appl Mech 39:946–952 51. Wineman AS (1976) Large axisymmetric inflation of a nonlinear viscoelastic membrane by lateral pressure. Trans Soc Rheology 20:203–225 52. Wineman AS (1978) Bifurcation of response of a nonlinear viscoelastic spherical membrane. Int J Solids Struct 14:197–212 53. Wineman A (2007) Nonlinear viscoelastic membranes. Comput Math Appl 53:168–181 54. Wineman A (2009) Nonlinear viscoelastic solids-a review. Math Mech Solids 14:300–366 55. Wineman AS, Rajagopal KR (2000) Mechanical response of polymers, an introduction. Cambridge University Press, Cambridge 56. Wineman AS, Waldron WK Jr (1995) Yield-like response of a compressible nonlinear viscoelastic solid. J Rheology 39:401–423 57. Yuan H-L, Lianis G (1972) Experimental investigation of nonlinear viscoelasticity in combined finite torsion-tension. Trans Soc Rheology 16:615–633

A Primer on Plasticity David J. Steigmann

Abstract This chapter surveys fundamental aspects of the modern theory of finite elastoplasticity. We emphasize the now-ubiquitous decomposition of the deformation into elastic and plastic parts, the central roles played by dissipation and material symmetry, and a framework for the modeling of scale-dependent work hardening in crystalline and isotropic materials.

1 Introduction Our aim in this chapter is to outline, as concisely and clearly as possible, the modern theory for finite elastic–plastic deformations. This is intended for readers having a background in nonlinear continuum mechanics and seeking an introduction to the subject without having to wade through a vast, disorganized and often contradictory literature. Thus, we endeavor to synthesize basic ideas that have achieved a degree of permanence in the subject. Because of the vast amount of rather disjoint material in this field, it is advisable to focus attention on the truly revealing sources. In this regard, we strongly recommend [4–6, 11, 16, 23]. Most of the modern development of plasticity theory rests on a multiplicative decomposition of the deformation gradient F = ∇χ —where χ (x, t) is the deformation and ∇ is the referential gradient (with respect to position x in a reference configuration)—into a part H that generates stress, and a part G that furnishes a local map from a reference configuration to a stress-free natural state of the material; thus, F = HG [26]. The latter states are undistorted in the sense of a perfect crystal lattice; accordingly, for solids the symmetry group of the stress response function is contained in the orthogonal group. Importantly, the theory does not require H or G to be gradients of vector fields. This crucial feature facilitates the theoretical descripD. J. Steigmann (B) Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_5

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tion of material defects, or inhomogeneities [11, 23, 31]. This notion is the basis of Noll’s theory of materially uniform bodies [23], which has substantially advanced the theoretical treatment of a wide range of phenomena encompassing plasticity, growth, diffusion, and thermoelasticity. In Noll’s treatment, the stress is determined by the history of H = FK, where K (= G−1 ) is a tensor field that effectively serves to embed material points into a prescribed reference configuration. Noll’s framework also furnishes a natural setting for the systematic treatment of functionally graded materials [12], insofar as the constitutive functions, referred to the selected reference configuration and regarded as functions of the history of F, naturally inherit an explicit dependence on position x in a reference configuration, κr say, via the field K(x, t). This allows for spatial variation of material properties, relative to κr , resulting from a physical process underlying the emergence and development of the field K. The standard theory presumes the existence of a stress-free natural state of the material. However, it is not generally possible to have a state of vanishing stress at all points of the body. Typically, there is a distribution of residual stress due to the presence of various defects. These induce local lattice distortions in the case of crystalline metals, for example, which in turn generate elastic strain and a consequent distribution of stress. This is typically the case even when the body is entirely unloaded, i.e., when no body forces are applied and the boundary tractions vanish. Nevertheless, it is possible, in principle, to remove the mean stress via an equi¯ of the librium unloading process. In particular, in equilibrium the mean value, T, Cauchy stress T is given by [3] vol(κt )T¯ =

1 2

 ∂κt

(t ⊗ y + y ⊗ t)da +

1 2

 κt

ρ(b ⊗ y + y ⊗ b)dv,

(1)

where t and b, respectively, are the boundary traction and body force, ρ is the mass density and y is the position of a material point in the current configuration κt of the body. Accordingly, T¯ = O if the entire body is unloaded. In view of the Mean Value Theorem for continuous functions, there exists y¯ ∈ κt ¯ Therefore, T(¯y, t) = O for some y¯ ∈ κt if the body is unloaded such that T(¯y, t) = T. and in equilibrium. Let (2) d(κt ) = sup |y − z| . y, z ∈κt

This is the diameter of κt . Then for every y ∈ κt we have T(y, t) → T(¯y, t) as d(κt ) → 0. Accordingly, the local value of the stress can be made arbitrarily small as the diameter of the body is made to shrink to zero. Of course, it is not possible to reduce the diameter of a given body to zero. However, we may regard any body as the union of an arbitrary number of arbitrarily small (n) (n) disjoint sub-bodies Pt(n) , i.e., κt = ∪∞ n=1 Pt , with d(Pt ) → 0. Imagine separating these sub-bodies and unloading them individually. We then have T(y, t) → O for every y ∈ Pt(n) , for every n. Because every y in κt belongs to some Pt(n) , this process results in a state in which the material is pointwise unstressed. Of course,

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each piece Pt(n) has in general experienced some distortion in this process, and so the unstressed sub-bodies cannot be made congruent to fit together into a connected region of three-dimensional space. Thus, there is no global stress-free configuration of the body, and hence no position field χ i , say, such that dχ i = Gdx (or H−1 dy), that is, there is no neighborhood in the vanishingly small unloaded sub-bodies that can be used to define a gradient of a position field. Accordingly, as we have already noted, neither G nor H is a gradient. It follows that for any closed curve Γ ⊂ κr , with image γ = χ (Γ, t) in κt under the deformation map, the vector 

 B=

Γ

G dx =

H−1 dy

(3)

γ

does not vanish. This is called the Burgers vector associated with the specified curve, induced by the plastic deformation. In view of the foregoing, we regard the vector space κi induced by the map of the translation space of κr by G (or that of κt by H−1 ) as being associated with a material point x, rather than as a configuration per se. In fact, it may be regarded as the tangent space to a certain body manifold, but this manifold is not Euclidean as it does not support a position field. This interpretation is the basis of an elegant differentialgeometric theory of plastically deformed bodies [11, 31], which, however, is not emphasized here as it is largely superfluous as far as the formulation of a predictive theory is concerned. Throughout this chapter, we use notation that is currently standard in nonlinear continuum mechanics. Boldface is used for vectors and tensors and a dot interposed between bold symbols is used to denote the standard Euclidean inner product; for example, if A1 and A2 are second-order tensors, then their inner product is A1 · A2 = T tr(A1 AT2 ), where √ the superscript is used to denote the transpose. The associated norm is |A| = A · A. The notation ⊗ identifies the standard tensor product of vectors. We use Orth+ to denote the group of rotation tensors, and Sym and Skw, respectively, to denote the linear spaces of symmetric and skew tensors. The symbol Div is used for the three-dimensional referential divergence operator. We use the notation A−1 , A∗ and J A . These are, respectively, the inverse, the cofactor and the determinant of a tensor A; they are connected by A∗ = J A A−T if A is invertible. For a fourth-order tensor A, the notation A[B] stands for the second-order tensor resulting from the linear action of A on B. Its transpose AT is defined by B · AT [A] = A · A[B], and A is said to possess major symmetry if AT = A. If A · A[B] = AT · A[B] and A · A[B] = A · A[BT ], then A is said to possess minor symmetry. Finally, the notation (•)A stands for the derivative of (•) with respect to the tensor A.

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2 Materially Uniform Elastic Bodies Noll’s framework accommodates constitutive response that is thermo-viscoelastic in the sense that the current value of the stress is sensitive to the history of temperature and also of H. In the present work, we confine attention to materials that respond elastically to H. Further, we assume the existence of a strain-energy function and confine attention to the purely mechanical theory. The first Piola–Kirchhoff stress tensor is thus given by P = Ψ F,

(4)

the derivative with respect to the deformation gradient F of the strain energy, Ψ (F; x), per unit reference volume. We assume that κr could be occupied by the body, at least in principle, and thus require that JF > 0. The first Piola–Kirchhoff stress is related to the Cauchy stress T by Ψ F = TF∗ . In the same way, (5) WH = TH∗ , where W (H) with

H = FK,

(6)

is a suitable strain-energy function that does not depend explicitly on x. We assume that JK > 0 and conclude that J H > 0. Then, with H∗ = F∗ K∗ , we derive the connection (7) WH = Ψ F K ∗ , and integration at fixed K gives Ψ (F; x) = JK−1 W (FK(x)),

(8)

apart from an unimportant function of x. The strain-energy function W thus encodes the intrinsic elastic properties of the material. The familiar strong-ellipticity condition is a ⊗ b · Ψ FF (F; x)[a ⊗ b] > 0 for all a ⊗ b = O.

(9)

To interpret this in terms of W , we differentiate (7), at fixed K, on a one-parameter family H(u) = F(u)K, reaching ∗ ˙ ˙ = (Ψ FF [F])K , WHH [FK]

(10)

where the superposed dot is the derivative with respect to u. Scalar multiplication by ˙ and use of the rule A · BC = BT A · C, with B = F˙ and C = K, yields FK

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˙ · WHH [FK] ˙ ˙ FK = JK F˙ · Ψ FF [F],

(11)

which we apply with F˙ = a ⊗ b to express (9) in the form a ⊗ m · WHH (H)[a ⊗ m] > 0 for all a ⊗ m = O, with m = KT b.

(12)

It follows that Ψ is strongly elliptic at F if and only if W is strongly elliptic at H = FK [22]. Evidently, the strain-energy function describes, for fixed K, a nonuniform but otherwise conventional elastic material. Accordingly, we impose the usual Galilean invariance restriction in the form Ψ (F; x) = Ψ (QF; x) for every rotation Q. From (6) and (8) this is equivalent to W (H) = W (QH),

(13)

which, as is well known, is satisfied if and only if W depends on H via HT H, or, equivalently, via the elastic strain E=

1 T (H H − I), 2

(14)

where I is the identity for 3-space. Thus, W (H) = U (E), say. Galilean invariance also implies, via (5), that the Cauchy stress is symmetric. To see this, we observe that (15) WH = HS, where S = UE is the symmetric second Piola–Kirchhoff stress, referred to κi . The symmetry of the Cauchy stress then follows from (5). To show the converse, i.e., that the symmetry of the Cauchy stress implies the Galilean invariance of the strain energy, we note, from (5), that (WH )HT · Ω vanishes for all skew Ω. Consider the initial-value problem ˙ = ΩQ with Q(0) = I Q (16) for a one-parameter family Q(u) of tensors and Ω a fixed skew tensor. Then Q(u) is a rotation, and ˙ = (WH )HT · Ω = 0, (17) WH · H where H(u) = Q(u)H0 . This implies that W (H0 ) = W (H(u)) and hence the Galilean invariance of the energy. Proceeding from (15), an application of the chain rule furnishes the useful connection (18) WHH (H)[B] = BS(E) + HC(E)[HT B] for any tensor B, where C(E) = UEE . This possesses the major and minor symmetries.

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We impose the normalization W (I) = 0. Galilean invariance then implies that W (H) = 0 if H ∈ Orth+ . We assume the converse to be true, and hence that W (H) = 0 if and only if H ∈ Orth+ . We further assume that S = O if and only if E = O, i.e., if and only if H ∈ Orth+ , and that C(O) is positive definite in the sense that A · C(O)[A] > 0 for all nonzero symmetric A. We have S(E) = C(O)[E] + o(|E|),

(19)

and it follows from (18) that WHH (I)[B] = C(O)[B].

(20)

Our assumptions thus imply strong ellipticity at zero elastic strain. Normally metals undergo only small elastic strains before yielding, at least if the rate of strain is sufficiently small. We simplify matters accordingly by supposing that |E| is always small enough to justify the use of the quadratic-order approximation 1 U (E) = U (O) + E · UE (O) + E · C(O)[E] + o(|E|2 ). 2

(21)

Because κi is associated with vanishing stress by assumption, the coefficient UE (O) of the linear part of the expansion vanishes. The leading-order strain energy is then purely quadratic: 1 (22) U (E) E · C(O)[E]. 2 This, of course, is just the usual elastic energy for small strains, yielding S = C(O)[E].

(23)

Because κi is free from elastic distortion, in the case of a crystalline metal the lattice is perfect and undistorted in κi . This has the consequence that W (H) = W (HR),

(24)

for all rotations R characterizing the symmetry of the lattice, and hence that U (E) = U (RT ER),

(25)

which implies that the transformation E → RT ER induces the transformation S → RT SR.

(26)

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The collection of all such rotations is a group, the symmetry group of the lattice. For crystalline solids, this group is always discrete, whereas for isotropic or transversely isotropic solids it is connected. In particular, isotropic materials satisfy (24) for all rotations.

3 Plastic Evolution 3.1 Energy and Dissipation The strain energy stored in an arbitrary part Pr ⊂ κr of the body is  U(Pr ) =

Ψ (F, K) dv,

(27)

Ψ (F, K) = JK−1 W (FK).

(28)

Pr

where, from (8),

The total mechanical energy in this part is then given by  E(Pr ) = Pr

1 Φ dv, with Φ = Ψ + ρr |˙y|2 , 2

(29)

where ρr is the mass density in κr and superposed dots are used to denote material derivatives (∂/∂t at fixed x). The power of the forces acting on Pr is  P(Pr ) =

 ∂ Pr

p · y˙ da +

ρr b · y˙ dv,

(30)

Pr

where p = Pn is the Piola traction acting on the boundary with local (referential) orientation given by the unit vector field n. The local equations of motion are Div P + ρr b = ρr y¨ , PFT = FPT in κr ,

(31)

where P = TF∗ is the first Piola–Kirchhoff stress, T is the Cauchy stress, and Div is the referential divergence (i.e., the divergence with respect to x). We are concerned mainly with the constitutive structure of the theory and therefore restrict attention to smooth processes. Using the first of (31), we obtain ρr b · y˙ =

∂ ∂t



   1 ρr |˙y|2 − Div (PT y˙ ) − P · ∇ y˙ . 2

(32)

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Following a standard procedure, we invoke conservation of mass (ρ˙r = 0), substitute into (30), apply the divergence theorem and use p · y˙ = PT y˙ · n to arrive at the mechanical energy balance P= where

d K + S, dt

1 K(Pr ) = 2

is the kinetic energy and

(33)

 ρr |˙y|2 dv

(34)

Pr



P · F˙ dv

S(Pr ) =

(35)

Pr

is the stress power. The dissipation D is defined as the difference between the power supplied and the rate of change of the total energy; thus, D=P−

d E. dt

(36)

˙ +U ˙ and combining with (33), it follows immediately Using (29) in the form E˙ = K that  D dv, (37) D(Pr ) = Pr

where

D = P · F˙ − Ψ˙ .

(38)

In the purely elastic context it is evident, from (4), that D vanishes identically. Here, we impose the requirement D ≥ 0 for all Pr ⊂ κr and conclude, from the localization theorem, that D≥0 (39) pointwise. This assumption serves as a surrogate for the second law of thermodynamics in a purely mechanical setting. To obtain a useful expression for the dissipation, we proceed from (28), obtaining Ψ˙ = JK−1 [W˙ − ( J˙K /JK )W ].

(40)

˙ together with Here we use the identity J˙K /JK = K−T · K ˙ = WH KT · F˙ + FT WH · K. ˙ W˙ = WH · H

(41)

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Recalling that WH = T(FK)∗ = PK∗ and hence that WH KT = JK P and FT WH = JK FT PK−T , (40) is reduced to ˙ −1 , Ψ˙ = P · F˙ − E · KK

(42)

E = Ψ I − FT P

(43)

where

is Eshelby’s Energy–Momentum Tensor. Accordingly, the local dissipation may be written in the form ˙ −1 . (44) D = E · KK This result, first announced in [13], highlights the role of the Eshelby tensor as the driving force for dissipation. We use it here, in conjunction with (39), to derive ˙ We note in passing, restrictions on constitutive equations for the plastic evolution K. relying on (42) and the chain rule, that P = Ψ F (F, K) and E = −Ψ K (F, K)KT .

(45)

The expression (44) for D makes clear the fact that the dissipation vanishes in the ˙ = O. On the basis of empirical obserabsence of plastic evolution, i.e., D = 0 if K vation, we introduce the hypothesis that plastic evolution is inherently dissipative; ˙ = O. In view of (39), this means that thus, we suppose that D = 0 if and only if K ˙ = O if and only if D > 0. K

(46)

It may be observed, from the definition (43), that the Eshelby tensor is purely referential, mapping the translation space of κr to itself. It proves convenient to introduce a version of the Eshelby tensor, E i , that maps κi to itself. This is given by the relation (47) E = JK−1 K−TE i KT . The derivation proceeds from the observation that if E  is the Eshelby tensor derived by taking the current configuration as reference, i.e., E  = ψI − T (with ψ = JF−1 Ψ ), then E = JF FTE  F−T . Thus, E is the pullback of E  from κt to κr . It follows that E is the pullback of E i from κi to κr , and E i is the pullback of E  from κt to κi . Further, it is straightforward to show that E i = W I − H T WH ,

(48)

which implies that E i is determined entirely by H and hence purely elastic in origin. Alternatively, we may write (49) E i = U I − CS,

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where C (= HT H) is the elastic Cauchy–Green deformation tensor, and further conclude that E i is Galilean invariant, i.e., that it is invariant under superposed rigid-body motions. In the case of small elastic strain, (49) implies that E i = −S + o(|E|),

(50)

where S is given by (23). It follows that the Eshelby tensor based on the local intermediate configuration is given, to leading order and apart from sign, by the second Piola–Kirchhoff stress referred to the same configuration. Using (44) and (47), we may also establish that ˙ JK D = E i · K−1 K

(51)

and hence that the assumption of inherent dissipativity is equivalent to the statement: ˙ > 0. ˙ = O if and only if E i · K−1 K K

(52)

It is interesting to observe that if E = O, then U = 0, S = O and hence D = 0; then (46) implies that there can be no plastic evolution. That is, without stress, there can be no change in the plastic deformation. This comports with the observed phenomenology.

3.2 Invariance We have observed that the symmetry of the Cauchy stress is equivalent to the state¯ ¯ Because the argument leading to this ment W (H) = W (QH) for all rotations Q. ¯ conclusion is purely local, the rotation Q may conceivably vary from one material point to another. This stands in contrast to the rotation Q(t) associated with a superposed rigid-body motion, which must be spatially uniform and hence the same at all material points; our notation is intended to distinguish these cases explicitly. In a superposed rigid-body motion, the deformation χ(x, t) is changed to χ + (x, t) = Q(t)χ(x, t) + c(t),

(53)

where c is a spatially uniform vector. As is well known, it follows immediately that F (= ∇χ ) goes into F+ (= ∇χ + ), with F+ = QF. The argument cannot be adapted to H, however, because it is not the gradient of any position field. This observation raises the question of how plastic deformation transforms under a superposed rigid motion when the plastic deformation is allowed to vary, as distinct from the case considered previously in which the plastic deformation was considered to be fixed. ¯ whereas H+ = F+ K+ = QFK+ , Nevertheless, we may infer that H+ = QFK, and therefore that

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¯ QFK = QFK+ .

(54)

We would like to use this to arrive at some conclusion about the relationship between K+ and K, but this requires a further hypothesis. A natural one is that the dissipation is insensitive to superposed rotations. To explore the implications we define Z = K+ K−1 and note, from (54), that J Z = 1. Suppose Z(t0 ) = I, so that the superposed rigid motion commences at time t0 . Using (51) we find that the dissipation transforms to + −1 ˙ + −1 −1 ˙ −1 ˙ −1 −1 ˙ JK D + = E + i · (K ) K = E i · (K Z ZK + K K) = J K D + E i · K Z ZK,

(55) wherein we have invoked the invariance of JK and of the Eshelby tensor E i . Accordingly, if D + = D, as assumed, then ˙ = 0, E i · K−1 Z−1 ZK

(56)

and this purports to hold for any K with JK > 0. It, therefore, holds for K = I, yielding Z = K+ . This Z is a plastic flow, and therefore subject to our strong dissipation hypothesis (52). This requires that Z˙ vanish, and hence, given the initial condition, ¯ = Q(t) and hence that K+ = K; thus, G+ = G. From (54) it then follows that Q that (57) F+ = QF, H+ = QH and K+ = K.

3.3 Yielding, the Work Inequality, and Plastic Flow ˙ = O, occurs when We assume the onset of yield, and hence the possibility that K the elastic strain is such that G(E) = 0, where G is an appropriate yield function pertaining to the material at hand. In a crystalline material, for example, this implies that yield occurs when the lattice is sufficiently distorted. Of course, we may derive this from the more basic assumption that the yield function is dependent on H, and that yield is insensitive to superposed rigid motions. Thus yield occurs when the elastic distortion lies on a certain manifold in 6-dimensional space. The material is said to respond in the elastic range if E is such that G(E) < 0, whereas, in the rate-independent theory, the region of strain space where G(E) > 0 is deemed to be inaccessible. For metals, the diameter of the elastic range is typically such as to severely limit the size of the elastic strain, so that as a practical matter we may assume the elastic strain—at least in the rate-independent theory—to be small. In these circumstances, our constitutive hypotheses imply that the relation between E and S is one-to-one, so that we may equally well characterize yield in terms of the statement F(S) = 0, where F(S) = G(E(S)) is the yield function, expressed in terms of the stress. We suppose elastic response to be operative in the elastic range defined by F(S) < 0, and, in the case of rate-independent response, that no state of

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stress existing in the material can be such that F(S) > 0. We consider rate-dependent response later. We further suppose that F(0) < 0, and hence that the stress-free state belongs to the elastic range. In this way, we partition 6-dimensional stress space into the regions defined by positive, negative, and null values of F, with the first of these being inaccessible in any physically possible situation. This appears to disallow behavior of the kind associated with the Bauschinger effect, in which yield can occur upon load reversal before the unloaded state is attained. However, empirical facts support the view that this effect is accompanied by the emergence of dislocations; these give rise to nonuniform distributions of stress and elastic strain in the material which cannot be directly correlated with the overall global response represented in the test data. From this point of view, the Bauschinger effect is thus an artifact of the test being performed, not directly connected with constitutive properties per se. Nevertheless, it is possible to model the Bauschinger effect in terms of more general constitutive functions that involve the gradients of elastic and plastic deformations explicitly [28]. Consider now a cyclic process in which the deformation and velocity fields start and end at the same values. Assuming that nonnegative work must be performed to effect such a process, it follows by a well-known argument [21] that 

t2

P · F˙ dt ≥ 0,

(58)

t1

where t1,2 , respectively, are the times when the cycle begins and ends. Suppose these times are such that the associated stresses satisfy F < 0; the cycle begins and ends in the elastic range. Suppose the cycle is such that there exists a sub-interval of time [ta , tb ] ⊂ [t1 , t2 ] during which F = 0, and that F < 0 outside this sub-interval. Then ˙ = O, during this sub-interval, while K ˙ = O outside we may have plastic flow, i.e., K it, implying that K(t1 ) = K(ta ) and K(t2 ) = K(tb ). Substituting (38) and noting that the process is cyclic in the sense that F(t2 ) = F(t1 ), we arrive at the statement 

tb

Ψ (F(t1 ), K(tb )) − Ψ (F(t1 ), K(ta )) +

D dt ≥ 0.

(59)

ta

Equivalently,



tb

˙ [Ψ K (F(t1 ), K(t)) · K(t) + D(t)] dt ≥ 0.

(60)

ta

To ensure that a cycle beginning in the elastic range (i.e., F(S(t1 )) < 0) also ends there, we pass to the limit tb − ta → 0. This yields H(t2 ) = F(t2 )K(tb ) = F(t1 )K(tb ) → F(t1 )K(ta ) = H(t1 ), implying, via the elastic stress–strain relation, that S(t2 ) → S(t1 ). Dividing (60) by tb − ta (> 0) and passing to the limit, we conclude, from the mean value theorem, that

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˙ a ) + D(ta ) ≥ 0, Ψ K (F(t1 ), K(ta )) · K(t

(61)

which may be written, using (44) and (45)2 , as ˙ a )K(ta )−1 ≥ 0. E(F(ta ), K(ta )) − E (F(t1 ), K(ta ))] · K(t [E

(62)

Using (47), this inequality may be restated in the form ˙ a ) ≥ 0, Ei (E(ta )) − E i (E(t1 ))] · K(ta )−1 K(t [E

(63)

where E i (E) is the function of elastic strain obtained by recasting (49) with the aid of (14) and (23). This means that the dissipation is maximized by states E (equivalently, S, if the elastic strain is small) that lie on the yield surface. ˙ = −GG ˙ −1 , In the case of small elastic strain, we substitute (50) with K−1 K which follows from GK = I, divide by |E|, and pass to the limit in (63) to derive the restriction ¯ ≤ 0, F(S) = 0. ¯ · GG ˙ −1 ≥ 0, F(S) (64) (S − S) ˙ −1 solving: Stated differently, we seek GG ˙ −1 ) subject to F(S) ≤ 0 and W = O, where W = Skw S, (65) max(S · GG which is a standard optimization problem involving both equality and inequality constraints. If the function F(S) is differentiable, then the Kuhn–Tucker necessary condition [32] generates the flow rule ˙ −1 = (λF + Ω¯ · W)S , GG

(66)

¯ It is straightwhere λ and Ω¯ are Lagrange multipliers, with λ ≥ 0 and Ω¯ = −Ω. forward to derive (Ω¯ · W)S = Ω¯ and thus obtain T

¯ ˙ −1 = λFS + Ω. GG

(67)

¯ The skew tensor field Ω(x, t) is called the plastic spin. In our earlier discussion of material symmetry, we imposed the invariance of the strain-energy function under the transformation H → HR, where R is any rotation of κi belonging to the material symmetry group. We saw that this yields the transformation (26) for the stress, and so it is natural to impose the symmetry requirement F(S) = F(RT SR) on the yield function. A symmetry transformation thus induces the transformation FS → RT FS R. Now, invariance of material response under the transformation H → HR is equivalent to invariance under the transformation FK → FKR, which is tantamount to invariance under K → KR at fixed F, or, equivalently, under G → RT G. Accord˙ −1 )R. This implies that Sym (GG ˙ −1 ) → RT [Sym (GG ˙ −1 )] ˙ −1 → RT (GG ingly, GG

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R and hence that the plastic multiplier λ is invariant under symmetry transformations. ˙ −1 )]R and thus conclude that ˙ −1 ) → RT [Skw (GG We also have Skw (GG ¯ Ω¯ → RT ΩR

(68)

under symmetry transformations. From (67), the dissipation is given (to leading order in the small elastic strain) by JK D = λS · FS .

(69)

Because λ ≥ 0, the dissipation is positive only if λ > 0 and hence only if S · FS > 0,

(70)

which may be imposed as an a priori restriction on the yield function. Because the ˙ is nonzero, it follows that we may dissipation is deemed to be positive if and only if G ¯ ˙ is nonzero, and thus recast always find Ω ∈ Skw such that Ω = λΩ whenever G (67) as ˙ −1 = λ(FS + Ω), F(S) = 0. (71) GG The foregoing considerations about yield and flow are quite general and apply to both crystalline and noncrystalline materials. Theory for crystalline media is still in a state of active development [16], particularly with respect to issues such as work hardening—the evolution of the yield function with plastic flow—and plastic spin. Nevertheless, sufficient progress has been made to warrant a brief description of the current state of the art.

4 Crystalline Materials The conventional theory of crystal plasticity [16, 17] rests on a kinematical interpretation of plastic deformation according to which the rate of plastic deformation in a single crystal is presumed to be expressible as a superposition ˙ −1 = GG



νi si ⊗ ni

(72)

of simple shear rates, in which G is the plastic part of the deformation gradient, νi are the slips and the si and ni are orthonormal vectors specifying the ith slip system. The sum ranges over the currently active slip systems. Here, the νi are determined by suitable flow rules, arranged to ensure that the response is dissipative, and the skew part of (72), in which the slip-system vectors are specified, furnishes the plastic spin. This decomposition, though virtually ubiquitous, has been criticized on the grounds that for finite deformations it cannot be associated with a sequence of simple shears

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unless these are suitably restricted [25]. In particular, the order of the sequence generally affects the overall plastic deformation, a fact which is not reflected in (72). In [8] conditions are derived under which (72) yields an approximation to the deformation associated with a sequence of slips. Interestingly, it is found that such deformations are well approximated by (72) in face-centered cubic crystals, but the issue remains unresolved for other crystal classes. Further, it has been reported [1] that sequential slip is typical in experiments but that simultaneous multi-slip is not usually observed. This state of affairs provides impetus for alternative phenomenological models based purely on considerations of material symmetry. Indeed, the decoupling of the ˙ −1 in the flow rule (71) affords considerably more symmetric and skew parts of GG latitude in the fitting of theory to experiment than is possible using (72).

4.1 Lattices In crystal-elasticity theory, the stress arises in response to lattice distortion. The theory is based on the idea that linearly independent, undistorted lattice vectors li , i ∈ {1, 2, 3}, are mapped to their images ti in κt in accordance with the Cauchy– Born hypothesis, that is, the li are convected as material vectors. To accommodate plasticity, this hypothesis is assumed to apply to the elastic deformation. Thus, ti = Hli , where li are the lattice vectors in κi . The lattice set {li } associated with κi is assumed to be an intrinsic property of the crystal. Accordingly, it is regarded as a uniform field (i.e., independent of x) in a single crystal, regarded as a materially uniform body. The ti are observable in principle. In practice, they are computed from their measurable duals ti [7]. Equation (6) yields ti = Fri , where ri = Kli are the lattice vectors in κr . The plastic deformation is then given by K = ri ⊗ li , where the li are the duals of the li . The elastic deformation is H = ti ⊗ li , and the deformation gradient is F = ti ⊗ ri . ˙ i + K˙li . These The material derivatives of the referential lattice vectors are r˙ i = Kl ˙ imply that if li = 0, then the lattice vectors are nonmaterial (˙ri = 0) in the absence of ˙ = O). However, we assume that plastic flow is solely responsible for plastic flow (K the non-materiality of the lattice, i.e., that plastic flow alone accounts for the evolution of material vectors relative to the lattice. Thus, we impose ˙li = 0 and regard the set {li } of lattice vectors as assigned data. This in turn yields the materiality of the set {ri } in the absence of plastic flow, in accordance with the conventional statement of the Cauchy–Born hypothesis for elastic deformations. This view comports with (72) in which the slip-system vectors are considered to be fixed; these, in turn, may be obtained from the fixed set {li }. Further, for the purpose of integrating the flow rule it is necessary to assign an initial value of plastic deformation at time t0 , say, and this is another reason why it is necessary to specify the lattice {li }. If we choose κr = κt0 , for example, then because the referential lattice

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coincides with the actual lattice at time t0 and is therefore measurable in principle, the specification of {li } provides the initial value of K, and hence that of G. The fact that the lattice {li } is fixed, independent of x and t, implies that the elements R of the (discrete) material symmetry group are also fixed, i.e., invariant in time and the same at all material points of the crystal. This has important consequences for the further development of the theory.

4.2 Work Hardening The present framework may be extended to model work hardening. In standard theories of plasticity, it is common to model hardening in terms of a measure of the history of plastic flow [16]. This feature may be incorporated by requiring the t ˙ −1 |dτ , for example. This is invariant yield function to depend on the scalar t0 |GG under superposed rigid motions and time translations, insensitive to the choice of κr , and invariant under symmetry transformations, and thus qualifies as an argument of a constitutively admissible function. Here, however, we pursue an alternative development inspired by the scale effects observed in plastically deforming solids [18, 29]. For example, motivated by G.I. Taylor’s formula [30] giving the flow stress as a function of dislocation density, we may assume that yielding occurs when

where

F(S, α) = 0,

(73)

α = JK K−1 Curl (K−1 )

(74)

is the geometrically necessary dislocation density. Here Curl is the referential curl operation, defined, in terms of the usual vector operation, by (CurlA)c = Curl (AT c)

(75)

for any fixed vector c and second-order tensor A. This implies that yield is inherently scale dependent. Further, α vanishes if and only if K−1 is a gradient, and so our model tacitly assumes that yield is sensitive to the incompatibility, or inhomogeneity, associated with the plastic deformation G. The dislocation density may also be expressed in terms of the elastic deformation by using (6) in the form H = (∇χ )K. This furnishes [15] α = J H H−1 curl (H−1 ),

(76)

where curl is the spatial curl. In the literature, it is common to identify dislocations as either geometrically necessary or statistically stored, with the classification depending on the considered

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length scale. Here we make the role of geometrically necessary dislocations explicit, whereas the role of the statistically stored dislocations is relegated to the details of the structure of the flow rule itself, in the form of the particular function used as well as the (scale-dependent) material parameters within it. ˙ = O, is possible As in the foregoing, we assume that plastic evolution, i.e., K only when (73) is satisfied, and that the stress S is always constrained to belong to the current elastic range defined by F(•, α) ≤ 0. In view of our restriction to materially uniform bodies, we require, as tacitly assumed previously, that all material points have the same yield function. Because this function does not depend on the time derivative of K, the considerations culminating in (63) are seen to carry over unchanged. Accordingly, (71) remains in effect, with the proviso that F(S) is replaced by F(S, α). To determine the plastic multiplier λ in the flow rule (71), we assume the existence of a time interval during which plastic deformation evolves, this being necessary for ˙ (equivalently, K, ˙ where the existence of a continuous, nonvanishing derivative G −1 K = G ). Our hypotheses imply that the yield function F vanishes identically in this interval, and hence the consistency condition FS · S˙ + Fα · α˙ = 0,

(77)

˙ + (GG ˙ −1 )α − tr(GG ˙ −1 )α. α˙ = JK G(Curl G)

(78)

where, from (74),

Substitution of (71) results in a linear constraint on λ and ∇λ jointly, furnishing a differential equation for λ whose solutions are subject to the restriction λ > 0. This is a significant complication relative to standard scale-independent hardening models in which the constraint on λ is algebraic. For stresses lying on the current yield surface, elastic unloading, and the consequent cessation of plastic evolution, is associated with stress increments that result in stress values lying in the interior of the current elastic range. Thus, elastic unloading is identified by the conditions: F(S, α) = 0, λ = 0, FS · S˙ < 0. The alternative, commonly referred to as plastic loading, is associated with the conditions: F(S, α) = 0, λ > 0, FS · S˙ ≥ 0. In this case, (77) is operative and implies that Fα · α˙ ≤ 0. This case subsumes neutral loading, defined by FS · S˙ = 0 and Fα · α˙ = 0. The formidable obstacle to analysis posed by the differential equation for the plastic multiplier does not arise in the case of rate-dependent viscoplastic response. For example, in the spirit of the classical over-stress theories of viscoplasticity [24], we may propose the flow rule (cf. (71)) ˙ −1 = ν −1 (FS + Ω), GG

F(S, α) ≥ 0

(79)

to model rate-dependence, where ν (> 0) is a material viscosity coefficient. This is far simpler than the rate-independent theory, for two reasons: first, there is no need to ensure that the state of stress associated with a given dislocation density remains

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confined to the elastic range; and second, there is no consistency condition and hence no multiplier field λ. This is replaced, in the viscoplastic framework, by an assigned viscosity. Remarkably, α is invariant under differentiable changes of the reference and current configurations κr and κt , respectively [2]. An explicit proof is provided in [15]. It, therefore, reflects the state of the material. For this reason, it is sometimes referred to as the true dislocation density. It is insensitive to superposed rigid motions in particular, ensuring that (73) provides a Galilean-invariant model of yield, provided that the function F is likewise invariant. Moreover, any constitutive function which is insensitive to the choice of κr and which depends on K and its (referential) gradient necessarily involves the latter in the combination α defined by (74) [2]. We remark that no constitutive function that purports to describe the local state of the material can depend explicitly on K. For example, we can exploit the arbitrariness in the choice of reference configuration to reduce K to the identity. This is accomplished by constructing a new (global) reference configuration using a map whose gradient coincides—at the particular material point in question—with K−1 [15]. This is possible despite the fact that K−1 is not itself a gradient. In fact, we have tacitly invoked this observation in the course of deriving the transformation rules (57). The discussion of material symmetry is naturally complicated somewhat by a constitutive dependence on the dislocation density. To see how, we use the fact that the transformation K → KA(x) induces the transformation JK K−1 Curl (K−1 ) → J A JK A−1 K−1 Curl (A−1 K−1 ),

(80)

provided, of course, that A is invertible. If A is also uniform, then [2] Curl (A−1 K−1 ) = Curl (K−1 )A−T .

(81)

This covers single-crystal symmetry. For, as we have seen, the relevant transformation is K → KR in which the uniform rotation R is an element of the fixed symmetry group. Accordingly, for single crystals a material symmetry transformation induces the transformation (82) α → RT αR, and the restriction on the yield function is F(S, α) = F(RT SR, RT αR).

(83)

A model of this kind, in which the plastic spin is assumed to be a function of S, has been developed in [9] for viscoplastic cubic crystals and applied to the numerical simulation of transient dislocation loops under plane-strain conditions. Reference may be made to [16] for guidance on the use of single-crystal models in the construction of models for polycrystalline aggregates.

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5 Isotropy In the case of isotropy the yield function, in the absence of work hardening, satisfies the material symmetry restriction

with (cf. (26))

¯ F(S) = F(S)

(84)

S¯ = RT SR

(85)

FS¯ = RT FS R,

(86)

for any rotation R. Accordingly,

and this has the important consequence that plastic spin may be suppressed in the flow rule. To see how this arises, recall that invariance of constitutive functions under material symmetry transformations is tantamount to their invariance under replacement ¯ = KR—equivalently, under replacement of G by G ¯ = RT G—with F of K by K remaining fixed. To see how this replacement affects plastic flow, we compute ˙¯ G ¯ −1 = RT GG ˙ −1 R + R ˙ T R, (G)

(87)

where we have allowed for the possibility that R may be time dependent. In particular, the argument leading to the conclusion that R is fixed in the case of crystalline response is not applicable here. Substituting (67) and (86), we conclude that ˙¯ G ¯ −1 = λFS¯ + RT (Ω¯ + RR ˙ T )R. (G)

(88)

Now, for any skew Ω¯ we can always find a rotation R(t) to nullify the parenthetical term in (88). Thus, suppose B(t) satisfies the initial-value problem B˙ = WB with B(0) = B0 ,

(89)

where W is skew and B0 is a rotation. Let Z = BBT . Then, Z˙ = WZ − ZW, with Z(0) = I.

(90)

Clearly Z(t) = I is a solution, and the uniqueness theorem for solutions to ordinary differential equations ensures that it is the only one. Thus B is orthogonal. Further, ˙ −1 ), J˙B = B∗ · B˙ = J B tr(BB

(91)

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and this vanishes because W is skew. Then, J B (t) = J B (0) = 1 and B is a rotation. ¯ Because the rotation in (88) is arbitrary, we are free to pick R = B (with W = Ω, of course), to conclude that ˙¯ G ¯ −1 = λF ¯ S¯ , with λ¯ = λ. (G)

(92)

Thus, by exploiting the degree of freedom afforded by the material symmetry group in the case of isotropy, we can effectively suppress plastic spin in the flow rule and thus reduce it to ˙ −1 = λFS . (93) GG Because the deformation gradient F is unaffected by symmetry transformations, this reduction has no effect on the deformation of the material. This major simplification, first announced in [11] and further developed in [16, 27], is not generally possible for crystalline materials.

5.1 Von Mises’ Yield Function Empirical facts indicate that yield in metals is insensitive to pressure over a very large range of pressures and certainly for the pressures normally encountered in practice. Thus, yield is insensitive to the value of trT, where T is the Cauchy stress. From (5), (15) and (23), we conclude that in the case of small elastic strain, trT = trS + o(|E|).

(94)

As the model we are pursuing purports to be valid to leading order in elastic strain, it follows that the yield function should be insensitive to trS. It should, therefore, depend on S entirely through its deviatoric part, Dev S; we write ˜ F(S) = F(Dev S).

(95)

With reference to (85) we have Dev S¯ = RT (Dev S)R and hence the material symmetry condition ˜ ˜ T (Dev S)R). (96) F(Dev S) = F(R Recall that in the theory for small elastic strain, the strain-energy function was expanded through quadratic order in the elastic strain. Moreover, the stress was approximated by an invertible, linear function of elastic strain. Accordingly, the strain energy may be regarded as a quadratic function of the stress S. For consistency, we also approximate the yield function by a quadratic function of S. Because Dev S is a linear function of S, this means that F˜ should be approximated by a quadratic function. The most general such function in the case of isotropy is a linear

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combination of tr(Dev S), [tr(Dev S)]2 and tr[(Dev S)2 ] = |Dev S|2 , the first two of which vanish identically. The most general yield function of the required kind such that the yield surface F = 0 partitions stress space into regions defined by F > 0 and F < 0, in which the latter contains the stress-free state, is then of the form 1 ˜ F(Dev S) = |Dev S|2 − k 2 . 2

(97)

This is the famous yield function proposed by von Mises [14]. Because the linear space of symmetric tensors may be decomposed as the direct sum of the 5-dimensional space of deviatoric tensors and the 1-dimensional space of spherical tensors, it follows that the √ yield surface defined by F = 0 is a cylinder in 6-dimensional stress space of radius 2k. Here k is the yield stress in shear. That is, if the state of stress is a pure shear of the form S = S(i ⊗ j + j ⊗ i),

(98)

with i and j orthonormal, then |Dev S|2 = 2S 2 and the onset of yield occurs when |S| = k. Here k may be a fixed constant, corresponding to perfect plasticity, or may depend on appropriate variables that characterize the manner in which the state of the material evolves with plastic flow. The latter pertains to work hardening, the understanding of which is the central open problem of the phenomenological theory of plasticity. Some tentative ideas concerning hardening in isotropic materials are discussed below. To generate the flow rule for the plastic deformation, we use (95) and (97) with the chain rule, obtaining 1 ∂ (Dev S · Dev S) FS · S˙ = F˙ = F˜˙ = 2 ∂t ∂ ˙ = Dev S · (Dev S) = Dev S · Dev S˙ = Dev S · S, ∂t

(99)

and hence FS = Dev S, which conforms to (70) whenever the yield condition is satisfied. Finally, (93) provides ˙ −1 = λDev S. GG

(100)

This implies that JG is fixed, and hence that no volume change is induced by plastic flow.

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5.2 The Classical Theory for Isotropic Rigid-Plastic Materials The elastic strain is invariably small in the case of rate-independent response or under low strain-rate conditions because it is then bounded by the diameter of the elastic range. If the overall strain is nevertheless large, then the main contribution to the strain comes from plastic deformation. In this case it is appropriate to consider the idealization of zero elastic strain, which entails the restriction HT H = I. The elastic deformation is, therefore, a rotation field, which we denote by Q. Because the elastic strain vanishes identically, the strain energy is fixed in value and the stress is arbitrary, i.e., ˙ with E˙ = O. 0 = U˙ = S · E, (101) At this level of the discussion, S is an arbitrary symmetric tensor, constitutively indeterminate, as in a rigid body, granted that it satisfies the yield criterion. Further, J H = 1 and the relation between the Cauchy stress and S becomes S = QT TQ.

(102)

Dev S = Dev (QT TQ) = QT (Dev T)Q,

(103)

˜ ˜ ˜ T (Dev T)Q) = F(Dev T), F(Dev S) = F(Q

(104)

Therefore,

implying that

the second equality being a consequence of isotropy (cf. (96)). The yield function is thus expressible in terms of the Cauchy stress alone, as in conventional expositions of the classical theory. Using (103), the flow rule (100) may be recast as ˙ −1 )QT = λDev T. Q(GG

(105)

This may be reduced to the classical form via the decomposition L=D+W

(106)

of the spatial velocity gradient L into the sum of the symmetric straining tensor D ˙ −1 together with (6), we find in the and the skew vorticity tensor W. Using L = FF present specialization to H = Q that ˙ −1 )QT ˙ T + Q(GG L = QQ

(107)

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in which the first term is skew whereas the second, according to (105), is symmetric. ˙ −1 )QT and hence the The uniqueness of the decomposition then yields D = Q(GG classical flow rule D = λDev T, (108) due to St. Venant [14]. This is the central equation of the classical theory and predates the modern theory for finite elastic–plastic deformations by at least a century. Its straightforward derivation in the framework of the modern theory, relying in particular on simple ideas about material symmetry, promotes confidence in both the classical and modern theories. Indeed, prior to the advent of the modern theory, the classical theory was regarded as a separate model, seemingly unrelated to the theory for elastic–plastic response and without a clear connection to the notion of isotropic material symmetry as espoused in nonlinear continuum mechanics.

5.3 Work Hardening in Isotropic Materials To describe scale-dependent work hardening in isotropic materials we may propose, as we have in the case of crystalline materials, that the dislocation density should be included among the arguments of the yield function. However, dislocation density is not a state variable in the case of isotropy. The proof is given in Noll’s paper [23], but it is instructive to consider an alternative line of reasoning. To set the stage for this discussion, recall that the strain-energy function satisfies the material symmetry condition W (H) = W (HR) for all R ∈ G,

(109)

the material symmetry group. This is relative to a local intermediate state wherein the solid is supposedly relaxed, so as usual we take G ⊂ Orth+ , the group of rotations. To arrive at restrictions like (109) we construct a map of a local neighborhood of a material point—-but not the entire body—that is undetectable by experiment. Here R is the gradient of such a map. If this gradient is a rotation, then, as is well known [3], it must be uniform in the considered neighborhood. But of course it may vary from one local neighborhood to another and thus from one material point to another. Therefore the symmetry group may be nonuniform in principle, even in a materially uniform body. Our earlier considerations about lattices in materially uniform crystals led us to conclude that an element R of G should nevertheless be uniformly distributed, but these considerations are not relevant to isotropic materials. To elaborate, recall that W delivers a symmetric Cauchy stress if and only if it depends on H via C = HT H, the elastic distortion. Combining this with material symmetry, we conclude that the energy is invariant under C → RT CR. In the case of isotropy, for example, the solution is well known: the energy is a function of the usual invariants, namely, I1 = trC, I2 = [I12 − tr(C2 )]/2, I3 = det C. Obviously these are

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invariant under every rotation R, uniform or not. It follows that for isotropy G includes all the nonuniform rotations automatically. That is, there can be no restriction to uniform R in this case. Notice that (6) and (109) combine to give W (FK) = W (FKR),

(110)

so that we may interpret (109) as a statement to the effect that our constitutive function must be invariant under K → KR for any R ∈ G with F fixed. We have already made use of this interpretation in our earlier discussion of plastic flow. Suppose the yield function, for example, depends on the dislocation density α(K), where α is the function of K(x) and its derivatives defined by (74). To compute this function, it is necessary to limit consideration to smooth K(x). Material symmetry demands that the yield function remain invariant when α(K) is replaced by α(KR). Here we replace K by KR and use (74) to compute the new value of α. For this we require that KR be smooth, and hence, crucially, that R(x) be smooth. For R ∈ Orth+ uniform, we recover (82), i.e., α(KR) = RT α(K)R.

(111)

However, for nonuniform R this is replaced by α(KR) = RT α(K)R + β,

(112)

where β involves the derivatives of R(x). To verify this, we use (75) to derive the Cartesian-coordinate representation [Curl (AB)]i j = eikl A jm,k Bml + A jm eikl Bml,k = eikl A jm,k Bml + (Curl B)im A jm , (113) where, as usual, commas followed by subscripts refer to partial derivatives with respect to Cartesian (referential) coordinates and eikl is the permutation symbol (e123 = +1, etc.). We note that this yields (81) if A is uniform (A jm,k = 0). Using K−1 = G, we write (74) in the form α(K) = JG−1 G Curl G.

(114)

Then, for R(x) a rotation field, α(KR) = JG−1 RT G Curl (RT G),

(115)

Curl (RT G) = (Curl G)R + eikl Rm j,k G ml ei ⊗ e j ,

(116)

in which where {ei } is the standard orthonormal basis associated with the Cartesian coordinates. This yields (112), with

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β = JG−1 RT (eikl Rm j,k G ml ei ⊗ e j ).

(117)

Suppose the symmetry is crystalline, so that G is a discrete group of rotations. Then, as observed by Noll [23], either R is uniform or it varies discontinuously (it can jump to another element of G). In the latter case β—and hence α(KR)—does not exist because KR then violates our smoothness requirement. The only remaining option is a uniform field R. In this case, as we have already seen, the yield function must be invariant under α → RT αR. In the case of isotropy, our smoothness requirement on KR allows us to impose, for the assumed-smooth K, an arbitrary smooth rotation field R(x). In this case (112) is meaningful, and the yield function must remain invariant under α → RT αR + β. In this general requirement, we are allowed to impose R(¯x) = I at a material point x¯ , say. Then, at this fixed (but arbitrary) point, we require invariance of the constitutive ¯ where β¯ is obtained by putting R = I in the expression function under α → α + β, for β, without any restriction on the derivatives of R(x) at x¯ . Because the latter can be given arbitrary values, so too the value of β¯ is arbitrary. To prove these assertions, we use (117) to derive β¯ = eikl Rl j,k ei ⊗ e j ,

(118)

where, for simplicity’s sake, we have invoked the freedom in the choice of local reference configuration to reduce K(¯x) to the identity (JG = 1 and G ml = δml , the Kronecker delta). Differentiation of RT R = I yields the conclusion that R jm,k = −Rm j,k at the material point x¯ . Let Γ be the tensor with components Γnk = 21 em jn R jm,k , where the derivatives are evaluated at x¯ . Using the well-known e − δ identity, this is inverted to give Rl j,k = en jl Γnk . Thus, the rotation gradient at x¯ may be assigned arbitrary values by choice of Γ . We obtain β¯ = eikl en jl Γnk ei ⊗ e j = (δin δk j − δi j δkn )Γnk ei ⊗ e j = (Γi j − Γkk δi j )ei ⊗ e j , (119) or

2 β¯ = Dev Γ − (trΓ )I. 3

(120)

This implies that the deviatoric and spherical parts of β¯ can be assigned arbitrarily and hence that β¯ can be given arbitrary values, as claimed. We conclude, in the case of isotropy, that the yield function (indeed, any constitutive function) is unaffected by variation of the value of the dislocation density at an arbitrary material point. Therefore, the yield function cannot depend on dislocation density in the case of isotropy. This conclusion is natural in view of the fact that dislocations are associated intimately with crystal lattices, whereas isotropic materials are not crystalline. Nevertheless, following the example of crystalline materials, we wish to contemplate a model of hardening in isotropic materials that reflects the incompatibility, or inhomogeneity, associated with the plastic deformation G. The measure of inhomo-

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geneity should qualify as a state variable; that is, it should be uniquely determined, insensitive to superposed rigid motions, and invariant under differentiable variations of the reference configuration. To explore this issue, we recall that because of material symmetry, the field K is nonunique to the extent that no distinction can be made between K and KR, where R(x) is any element of the material symmetry group. Equivalently, no distinction can be made between G and RT G. Of course, in the present context, R(x) is any rotation field whatsoever. Following a standard argument, we select R to be the rotation in the polar factorization of G to conclude that G is then mechanically indistinguishable from its right-stretch factor, which in turn is equivalent to M = GT G.

(121)

Conversely, if G and RT G yield the same value of M, then R is an arbitrary rotation and hence an element of the material symmetry group. This does not mean that M qualifies as a state variable, however. For, at a given material point, K (hence G and M) may be reduced to the identity merely by choice of the reference configuration. In this case material inhomogeneity is uniquely characterized by the Riemann– Christoffel curvature induced by a Riemannian metric derived from (121). We use the notation of tensor analysis on differentiable manifolds [20] in the manner of the original work on the subject of inhomogeneity [31]. What has been proved [10, 23, 31] is that the (local) inhomogeneity associated with M is completely characterized by the Riemann–Christoffel curvature tensor m m n m n m Rm . jkl = Γ jl,k − Γ jk,l + Γ jl Γnk − Γ jk Γnl ,

(122)

in which the Levi–Civita connection is [20, 31] Γimj =

1 lm M (M jl,i + Mli, j − Mi j,l ), 2

(123)

where (•),i = ∂(•)/∂θ i are partial derivatives with respect to (convected) coordinates θ i . Here all indices range over {1, 2, 3} in accordance with the fact that that the body is regarded as a three-dimensional manifold and the usual summation convention is in force. Further, Mi j is the smooth, positive-definite metric given by Mi j = Gi · MG j , with (M i j ) = (Mi j )−1

(124)

in which Gi = x,i is the natural basis induced by the coordinates in κr . Importantly, this metric is entirely independent of κr . To see this, we note that Mi j = mi · m j , where mi = GGi = H−1 gi , and where gi = y,i (= FGi ) is the natural basis induced by the coordinates in κt . Nevertheless, (124) is often convenient in computational work where it is customary to specify the parametrization x(θ i ) of κr . The curvature tensor defined in (122) stands in one-to-one relation to the covariant tensor

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Ri jkl = Mim Rm . jkl ,

(125)

which, therefore, fully characterizes material inhomogeneity. Because this is induced by a Levi–Civita connection, it satisfies the minor skew symmetries [20] Ri jkl = −R jikl , Ri jkl = −Ri jlk

(126)

Ri jkl = Rkli j .

(127)

and the major symmetry

Accordingly, it stands in one-to-one relation to the symmetric Einstein tensor [20, 31] (128) Π pq = ε pi j εqkl Ri jkl , where

ε pi j = M −1/2 e pi j ,

(129)

in which M = det(Mi j ) and e pi j is the permutation symbol (e123 = +1, etc.). The rules governing the variables on the right-hand side of (128) under coordinate transformations ensure that Π pq is an absolute contravariant tensor [20]. The Einstein tensor thus furnishes a complete characterization of material inhomogeneity in the case of isotropy. It may be used to generate the invariant Π = Π i j mi ⊗ m j ,

(130)

which, of course, is insensitive to superposed rigid motions and to the choice of reference configuration. Accordingly, it is appropriate to model (scale-dependent) hardening in terms of a yield function F(S, Π). Most of what we have said before about isotropic response carries over unchanged, except for restrictions imposed on the yield function by material symmetry. In particular, this requires invariance of the yield function with respect to the transformation mi → RT mi . Because the metric (121) is invariant under the rotation group, and because κr and its parametrization are unaffected by material symmetry transformations in the present theory, the metric components (124), and hence the components (128) of the Einstein tensor, are also invariant. Thus, the yield function is subject to the symmetry condition F(S, Π) = F(RT SR, RT ΠR) for all rotations R.

(131)

A simple example of such a yield function is proposed in [19] to model scaledependent hardening in isotropic materials.

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References 1. Bell JF, Green RE Jr (1967) An experimental study of the double-slip deformation hypothesis for face-centered cubic crystals. Philso Mag 15:469–476 2. Cermelli P, Gurtin ME (2001) On the characterization of geometrically necessary dislocations in finite plasticity. J Mech Phys Solids 49:1539–1568 3. Chadwick P (1976) Continuum mechanics: concise theory and problems. Dover Publications, New York 4. Cleja-Tigoiu S (2003) Consequences of the dissipative restrictions in finite anisotropic elastoplasticity. Int J Plast 19:1917–1964 5. Cleja-Tigoiu S (2003) Dissipative nature of plastic deformations in finite anisotropic elastoplasticity. Math Mech Solids 8:575–613 6. Cleja-Tigoiu S, Soós E (1990) Elastoviscoplastic models with relaxed configurations and internal variables. Appl Mech Rev 43:131–151 7. Cullity B (1978) Elements of X-ray diffraction. Addison-Wesley, Reading, MA 8. Deseri L, Owen DR (2002) Invertible structured deformations and the geometry of multiple slip in single crystals. Int J Plast 18:833–849 9. Edmiston J, Steigmann DJ, Johnson GJ, Barton N (2013) A model for elastic-viscoplastic deformations of crystalline solids based on material symmetry: theory and plane-strain simulations. Int J Eng Sci 63:10–22 10. Epstein M (2010) The geometrical language of continuum mechanics. Cambridge University Press, Cambridge 11. Epstein M, Elzanowski M (2007) Material inhomogeneities and their evolution. Springer, Berlin 12. Epstein M, de León M (2000) Homogeneity without uniformity: toward a mathematical theory of functionally graded materials. Int J Solids Struct 37:7577–7591 13. Epstein M, Maugin GA (1995) On the geometrical material structure of anelasticity. Acta Mech 115:119–131 14. Geiringer H (1973) Ideal plasticity. In: Truesdell C (ed) Mechanics of Solids, vol III. Springer, Berlin, pp 403–533 15. Gupta A, Steigmann DJ, Stölken JS (2007) On the evolution of plasticity and incompatibility. Math Mech Solids 12:583–610 16. Gurtin ME, Fried E, Anand L (2010) Mechanics and thermodynamics of continua. Cambridge University Press, Cambridge 17. Havner KS (1992) Finite plastic deformation of crystalline solids. Cambridge University Press, Cambridge 18. Hutchinson JW (2000) Plasticity at the micron scale. Int J Solids Struct 37:225–238 19. Krishnan J, Steigmann DJ (2014) A polyconvex framework for isotropic elastoplasticity theory. IMA J Appl Math 79:722–738 20. Lovelock D, Rund H (1989) Tensors, differential forms and variational principles. Dover Publications, New York 21. Lucchesi M, Šilhavý M (1991) Il’yushin’s conditions in non-isothermal plasticity. Arch Ration Mech Anal 113:121–163 22. Neff P (2003) Some results concerning the mathematical treatment of finite multiplicative elasto-plasticity. In: Hutter K, Baaser H (eds) Deformation and failure in metallic and granular structures. Lecture notes in applied and computational mechanics, vol 10. Springer, Berlin, pp 251–274 23. Noll W (1967) Materially uniform simple bodies with inhomogeneities. Arch Ration Mech Anal 27:1–32 24. Prager W (1961) Introduction to the mechanics of continua. Ginn & Co., Boston 25. Rengarajan G, Rajagopal K (2001) On the form for the plastic velocity gradient L p in crystal plasticity. Math Mech Solids 6:471–480 26. Sadik S, Yavari A (2015) On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math Mech Solids 22:771–772

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27. Steigmann DJ, Gupta A (2011) Mechanically equivalent elastic-plastic deformations and the problem of plastic spin. Theor Appl Mech 38:397–417 28. Steinmann P (1996) Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int J Eng Sci 34:1717–1735 29. Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46:5109–5115 30. Taylor GI (1934) The mechanism of plastic deformation of crystals. Part 1: theoretical. Proc R Soc A 145:326–387 31. Wang C-C (1967) On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch Ration Mech Anal 27:33–94 32. Zangwill WI (1969) Nonlinear programming. Prentice-Hall, Englewood Cliffs, NJ

Nonlinear Constitutive Modeling of Electroelastic Solids Luis Dorfmann and Raymond Ogden

Abstract In this chapter, the equations governing the mechanical behavior of electroelastic solids capable of finite deformations are summarized with particular reference to the development of constitutive equations describing the electromechanical interactions in soft dielectric materials. Following a brief summary of some background from continuum mechanics and nonlinear elasticity theory, the equations of electrostatics are given in Eulerian form and then re-cast in Lagrangian form. The electroelastic constitutive equations are based on the existence of a so-called total energy function, which may be regarded as a function of the Lagrangian form of either the electric field or electric displacement as the independent electric variable, together with the deformation gradient. For each form of the total energy function, corresponding expressions for the (total) nominal and Cauchy stress tensors are provided, both in full generality and for their isotropic specializations for unconstrained and incompressible materials. The general formulas are then applied to the basic problem of homogeneous deformation of a rectangular plate, and illustrated by the choice of a simple specific example of constitutive equation. As a further illustration, the theory is applied to the analysis of a nonhomogeneous deformation of a circular cylindrical tube subject to an axial force, a torsional moment, and a radial electric field generated by a potential difference between flexible electrodes covering its inner and outer curved surfaces.

L. Dorfmann (B) School of Engineering, Tufts University, Medford, MA 02155, USA e-mail: [email protected] R. Ogden School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_6

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1 Introduction Electroactive elastomers are dielectric materials that have unique advantages in that they are both capable of large deformations and that their mechanical properties can be changed under the action of an electric field. Consequently, these “smart” materials have attracted increasing attention for use in industrial applications in, for example, actuators, energy harvesting devices, sensors, artificial muscles, and biomedical devices [1, 2, 14, 19, 21]. The design and prediction of the mechanical behavior of these devices is in part dependent on the use of a consistent theoretical framework for describing the mechanical response of the materials in question. The main purpose of this chapter is, therefore, to present such a framework, based on the nonlinear theory of continuum electromechanics that fully integrates the theories of nonlinear elasticity and electrostatics. For a detailed discussion of the history of the development of electroactive materials capable of large deformations and a description of the various contributions to the nonlinear theory of electromechanical interactions, we refer to the review article [12], which contains extensive references to the literature. For more details of the development of the theory and its applications to particular boundary-value problems, see the monograph [10] and references therein. We begin, in Sect. 2 with a brief summary of the relevant background from the equations of continuum mechanics in chapter “Basic Equations of Continuum Mechanics” [17] and nonlinear elasticity in chapter “Finite Deformation Elasticity Theory” [18]. Section 3 summarizes the equations and boundary conditions of electrostatics in both Eulerian and Lagrangian forms, while Sect. 4 develops the full combination of electrostatic and elastic influences. In particular, the notions of the total energy function, the total nominal and Cauchy stresses are introduced and the connections between the stresses, deformation and electric and electric displacement fields are given, both in general form and for the isotropic material specialization. A simple application to the homogeneous biaxial deformation of a rectangular slab is then provided for illustration of the general theory, along with a simple prototype example of a constitutive equation. Section 5 contains a more detailed example involving the nonhomogeneous deformation of a circular cylindrical tube, which undergoes axial extension and torsion in the presence of a radial electric field that is generated by a potential difference between flexible electrodes attached to its inner and outer circular boundaries. Finally, some concluding remarks are contained in Sect. 6.

2 Essential Background from Continuum Mechanics and Nonlinear Elasticity In this section, in order to set the scene for further developments in the context of nonlinear electroelasticity, we provide a brief summary of the basic notation from continuum mechanics in chapter “Basic Equations of Continuum Mechanics” [17]

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and constitutive equations of nonlinear elasticity from chapter “Finite Deformation Elasticity Theory” [18]. The stress-free reference configuration is denoted by Br and the corresponding deformed configuration, connected to Br by the deformation gradient F, is denoted by B, with the subscript t omitted since time dependence is not considered in this chapter. Equilibrium of the configuration B is governed by either of the two equivalent equations (1) div σ + ρ f = 0, Div S + ρr f = 0, where σ and S are the Cauchy and nominal stress tensors, respectively, ρ and ρr the mass densities in B and Br , respectively, f is the body force per unit mass, and Div and div are the divergence operators with respect to X ∈ Br and x ∈ B, respectively. In the absence of couple stresses σ is symmetric. For a hyperelastic material, the deformation and stresses are related through the strain-energy function W = W (F), defined per unit reference volume, through the equations ∂W ∂W , σ = J −1 F (2) S= ∂F ∂F for an unconstrained material. For an incompressible material, the stress–deformation relations (2) are replaced by S=

∂W ∂W − p F−1 , σ = F − pI, ∂F ∂F

(3)

where p is the Lagrangian multiplier required by the incompressibility constraint det F = 1

(4)

and I is the identity tensor. It is generally assumed that in the reference configuration Br , where F = I, there is no stored energy, so that W (I) = 0. If Br is stress free then we also have ∂ W (I) = 0, ∂F

∂ W (I) − p0 I = 0, ∂F

(5)

for unconstrained and incompressible materials, respectively, where p0 is the value of p in Br , i.e. since p depends only on x, p0 = p(X) (refer to Sect. 2.2 of chapter “Finite Deformation Elasticity Theory” [18] for discussion of constraints in general). As discussed in Sect. 2.1 of chapter “Finite Deformation Elasticity Theory” [18], objectivity requires that W be invariant with respect to rotations superimposed on the deformation, a condition that is automatically satisfied when W is treated as a function of the right Cauchy–Green deformation tensor C = FT F. With W = W (C), we obtain the alternative expressions

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σ = 2J −1 F

∂ W (C) T ∂ W (C) T F , σ = 2F F − pI ∂C ∂C

(6)

for the Cauchy stress for unconstrained and incompressible materials, respectively. Details of the specializations of the above equations in the case of isotropy have been described in Sect. 3.1 of chapter “Finite Deformation Elasticity Theory” [18]. Here we just summarize results for an incompressible isotropic hyperelastic material for a strain-energy function expressed either as W¯ (I1 , I2 ) in terms of the principal invariants of C defined in (81) in chapter “Finite Deformation Elasticity Theory” [18] with I3 = 1, or as W (λ1 , λ2 , λ3 ) in terms of the principal stretches defined in Sect. 1.1.3 of chapter “Basic Equations of Continuum Mechanics” [17] subject to λ1 λ2 λ3 = 1. For the first of these, the Cauchy stress tensor can be expanded in the form  σ =2

 ∂ W¯ 2 ∂ W¯ ∂ W¯ B−2 + I1 B − p I, ∂ I1 ∂ I2 ∂ I2

(7)

where B = FFT is the left Cauchy–Green tensor, σ being coaxial with B. For the second, the principal Cauchy stress components are given by σi = λi

∂W − p, i = 1, 2, 3. ∂λi

(8)

A useful alternative representation is to make use of the incompressibility constraint λ1 λ2 λ3 = 1 to eliminate one of the principal stretches, say λ3 and p, and to consider the energy function Wˆ (λ1 , λ2 ). It then follows that the principal stress differences are given by σ1 − σ3 = λ1

∂ Wˆ ∂ Wˆ , σ2 − σ3 = λ2 , ∂λ1 ∂λ2

(9)

or, from (7), σ1 − σ3 = 2(W¯ 1 + I1 W¯ 2 )(λ21 − λ23 ) + 2 W¯ 2 (λ41 − λ43 ), σ2 − σ3 = 2(W¯ 1 + I1 W¯ 2 )(λ22 − λ23 ) + 2 W¯ 2 (λ42 − λ43 ),

(10) (11)

where the notation W¯ 1 = ∂ W¯ /∂ I1 , W¯ 2 = ∂ W¯ /∂ I2 has been adopted. These representations will be generalized for an isotropic electroelastic material in Sect. 4.2.1.

3 Electrostatics In this section, we summarize the basic equations and boundary conditions governing the electric field vector E and electric displacement vector D of the theory of electrostatics, defined in the configuration B without reference to the deformation,

Nonlinear Constitutive Modeling of Electroelastic Solids

159

i.e., the Eulerian forms of the equations. These are then cast in Lagrangian form, which involves the deformation gradient.

3.1 Eulerian Formulation The properties of an electroactive material depend on the polarization vector P, which is related to E and D through the connection P = D − ε0 E, where ε0 is the electric permittivity of free space. This means that only two of the three field vectors are needed in the subsequent analysis. We, therefore, omit P from explicit further consideration and work in terms of E and D, which satisfy the equations of electrostatics curl E = 0, div D = ρf

(12)

in B. These are the specializations of Maxwell’s equations for the static situation in which there are no magnetic fields or free currents, where ρf is the free charge density within the material, defined per unit volume in B, and curl and div are differential operators with respect to x ∈ B. Note that ρf = 0 for a dielectric material. The properties of a given material then require a relationship between E and D, i.e., a constitutive equation. This may be developed by adopting either E or D as the independent field variable. If required, the polarization is then given by P = D − ε0 E. The equations in (12) are accompanied by boundary conditions on ∂B, which are now described. Consider a surface S, which is the interface between two materials with different electrical properties. The two sides of S are identified as side S− and side S+ , and the corresponding field vectors by the superscripts − and + . The jump in E, i.e., the difference E+ − E− across S, evaluated on S is denoted [[E]], and similarly for D. The standard boundary conditions to be satisfied on S are n × [[E]] = 0, n · [[D]] = σf ,

(13)

where n is the unit normal to S pointing from S− to S+ and σf is the free charge density per unit area of S. For the purpose of developing the constitutive theory, it is more convenient to adopt a Lagrangian approach with E and D replaced by their Lagrangian counterparts, which are introduced in the next subsection.

3.2 Lagrangian Formulation Following the analysis in [5], the field equations and boundary conditions expressed in Eulerian form in (12) and (13) are now transformed into Lagrangian form, which involves the operators Div and Curl based on Br .

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First, we note that on integrating Eq. (12)1 over an open surface S with bounding closed curve ∂S and applying Stokes’ theorem, we obtain 

 curl E · n da = S

∂S

E · dx = 0,

(14)

where n is the unit normal to S defined by the right-hand screw rule with respect to traversal of ∂S. Using the connection dx = FdX from (12) in chapter “Basic Equations of Continuum Mechanics” [17] and application of Stokes’ theorem in the reference configuration, Eq. (14) can be written in the Lagrangian form 

 Curl (F E) · N d A = T

Sr

∂Sr

(FT E) · dX = 0,

(15)

where the unit normal N is the referential counterpart of n, Sr the surface in the reference configuration that deforms into S and the closed curve ∂Sr is its boundary. This prompts the introduction of the Lagrangian version of E, denoted EL and defined by EL = FT E, the pull back of E from B to Br . Thus, as (15) applies for arbitrary Sr , it may be deduced that EL satisfies the equation Curl EL = 0,

(16)

the counterpart of (12). In a similar vein, by integrating Eq. (12)2 over the volume B and applying the divergence theorem we obtain 





ρf dv = B

div D dv = B

∂B

D · n da,

(17)

n here being the unit outward normal to ∂B. Application of Nanson’s formula to Eq. (17) followed by use of the divergence theorem in the reference configuration Br yields    ∂B

D · n da =

∂Br

J (F−1 D) · N d A =

Div (J F−1 D) dV.

(18)

Br

By defining ρF = Jρf as the charge density per unit reference volume, Eq. (17) then gives   ρF dV = Div (J F−1 D) dV. (19) Br

Br

This leads to the definition DL = J F−1 D of the Lagrangian version of D, the pull back of D from B to Br , and hence, since Br can be regarded as an arbitrary subset of the reference configuration, to the equation

Nonlinear Constitutive Modeling of Electroelastic Solids

161

DivDL = ρF ,

(20)

the Lagrangian counterpart of (12)2 . In summary, the connections between the Lagrangian and Eulerian field vectors are (21) EL = FT E, DL = J F−1 D, and the Lagrangian forms of the field equations are CurlEL = 0, DivDL = ρF .

(22)

Associated with Eqs. (22) are boundary conditions analogous to (13). These are given by (23) N × [[EL ]] = 0, N · [[DL ]] = σF , where N is the unit normal to the reference boundary ∂Br and σF is the surface charge per unit reference area. To translate from (13) to (23) requires the use of standard vector identities.

4 Nonlinear Electromechanical Interactions In this section, we combine the theories of elasticity and electrostatics to form the fully integrated theory of nonlinear electroelasticity. We begin by giving the equations of mechanical equilibrium and the associated mechanical boundary conditions and introduce the notions of total Cauchy stress and total nominal stress, which embody the effects of electric body forces.

4.1 Equilibrium Equations and Boundary Conditions In a continuum with charge density ρf and polarization P, an electric body force is generated by an electric field E. This is denoted fe (per unit volume in B) and given by (24) fe = ρf E + (P · grad )E. If the mechanical body force is again denoted by f (per unit mass) then the total body force per unit volume in B is ρf + fe and the equilibrium equation takes the form divσ + ρ f + ρf E + (P · grad )E = 0,

(25)

which generalizes (1)1 , but now the Cauchy stress σ is different from that in (1)1 , and, in particular, it is no longer symmetric in general even in the absence of mechanical

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couple stresses. For a discussion of alternative forms of the “Cauchy stress tensors” and the associated “electric body forces”, we refer to [3, 4], but these are not needed here. From (12), it is straightforward to show that fe can be written in the form fe = div τm ,

(26)

where the second-order tensor τm is the electrostatic Maxwell stress tensor defined by 1 (27) τm = D ⊗ E − ε0 (E · E)I. 2 In general, this is not symmetric. By contrast, rotational balance of the forces acting on an arbitrary body requires that the combination σ + τm be symmetric in the absence of mechanical couple stresses, and hence Eq. (25) can be written in the compact form div τ + ρf = 0, (28) where the symmetric tensor τ is the so-called total Cauchy stress tensor defined by τ = σ + τm .

(29)

Thus, τ is the counterpart in nonlinear electromechanics of σ in the purely mechanical theory, and Eq. (28) has the same structure as (1)1 . From (2), we note that in elasticity theory (and indeed in any continuum mechanical theory) the nominal stress tensor S is given in terms of the Cauchy stress tensor σ by S = J F−1 σ . Correspondingly, in the electromechanical context, we define the total nominal stress tensor, denoted T, by T = J F−1 τ .

(30)

Then, analogously to (1)2 , the equation of equilibrium (28) translates into the equivalent form (31) Div T + ρr f = 0.

4.1.1

Exterior Fields and Boundary Conditions

We now consider the body in the deformed configuration B, with boundary ∂B, to be surrounded by free space within which the fields are denoted E and D , which are distinguished from the fields E and D within B. The standard relation D = ε0 E then holds, and the equations curl E = 0 and div D = 0 also hold. Since the exterior of B is not deformable the deformation gradient is not defined therein and the jump conditions (13), with E+ and E− replaced by E and E, respectively, and D+ and D− replaced by D and D, become

Nonlinear Constitutive Modeling of Electroelastic Solids

N × (FT E − EL ) = 0, N · (J F−1 D − DL ) = σF ,

163

(32)

N now being the unit outward normal to ∂Br and σF the surface charge per unit area of ∂Br . In the exterior region the Maxwell stress defined in (27), denoted τm , becomes 1 τm = D ⊗ E − (D · E )I, 2

(33)

and since D = ε0 E , τm is, therefore, symmetric and the equation div τm = 0 is satisfied. The exterior Maxwell stress generates a load on the boundary ∂B and it is therefore appropriate to consider the traction boundary condition on ∂B. Suppose that ta is the mechanical traction per unit area applied on the part ∂Bt of ∂B. Then the stress τ in B evaluated on ∂B satisfies  on ∂Bt , τ n = t a + tm

(34)

 = τm n is the traction vector due to the exterior Maxwell stress and n is the where tm unit outward normal to ∂B. On the complementary part, ∂Bx of ∂B (= ∂Bt ∪ ∂Bx ) it is usual to prescribe the position x, but here the main focus is on the traction boundary condition. From Nanson’s formula nda = J F−T Nd A, the traction boundary condition (34) may be expressed in Lagrangian form as  , TT N = tA + tM

(35)

where the Eulerian and Lagrangian traction vectors in (34) and (35) are connected   da = tM d A. by ta da = tA d A and tm Having established the governing equations and boundary conditions, we now go on to describe the (Lagrangian) connections between the variables F, EL , DL , and T through constitutive equations, together with the Eulerian counterparts of these connections.

4.2 Electroelastic Constitutive Equations There are many ways to construct constitutive equations in electroelasticity and we refer to [3] for discussion of a selection of these. However, perhaps the most elegant versions of constitutive equations within the framework of nonlinear electroelasticity are those based on the Lagrangian variables EL and DL , and we focus on these here. The formulations are based on the notion of a total energy (density) function, which was developed in [5, 6]. There are two versions of this, one dependent on EL as the independent electric variable, and one on DL , in addition to the deformation gradient.

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These are Ω(F, EL ) and Ω ∗ (F, DL ), each defined per unit volume of the reference configuration Br , and they are partial Legendre duals with the connection Ω ∗ (F, DL ) = Ω(F, EL ) + DL · EL .

(36)

Of course, by objectivity, each of Ω and Ω ∗ depends on F through the right Cauchy–Green tensor C, but it is convenient to retain the argument F in what follows. The deformation gradient and the nominal stress tensor are work conjugate when the electric field is fixed, i.e., tr (TdF) = dΩ = dΩ ∗ , and, for fixed F, EL , and DL are work conjugate in the sense that tr(EL dDL ) = dΩ ∗ , while tr(DL dEL ) = −dΩ, the latter following from (36). In terms of Ω, the resulting constitutive expressions are T=

∂Ω ∂Ω , , DL = − ∂F ∂EL

(37)

T=

∂Ω ∗ ∂Ω ∗ , EL = . ∂F ∂DL

(38)

and in terms of Ω ∗ we have

From (6) and (30), it follows that the corresponding expressions for the total Cauchy stress tensor τ and the Eulerian field vectors D and E are given by τ = J −1 F

∂Ω ∂Ω , D = −J −1 F , ∂F ∂EL

(39)

∂Ω ∗ ∂Ω ∗ , E = F−T . ∂F ∂DL

(40)

and τ = J −1 F Incompressible Materials

For an incompressible material, the expressions for the stress tensors given above require modification as for the case of pure elasticity in (3) with the incompressibility constraint (4). In respect of Ω and Ω ∗ , the required modifications for the total Cauchy and nominal stress tensors are ∂Ω ∂Ω − pI, T = − pF−1 , ∂F ∂F

(41)

∂Ω ∗ ∂Ω ∗ − p ∗ I, T = − p ∗ F−1 , ∂F ∂F

(42)

τ =F and τ =F

respectively, where p and p ∗ are Lagrange multipliers, which are not in general the same.

Nonlinear Constitutive Modeling of Electroelastic Solids

165

The formulas for DL and EL in (37)2 and (38)2 remain valid, as do the formulas for D and E in (39)2 and (40)2 , subject to the restriction J = 1.

4.2.1

Material Symmetry

Consider an electroactive material, which, in the absence of an electric field, exhibits isotropy in its response. Then, the application of an electric field induces a preferred direction, which is identified by the Eulerian vector E in B or, equivalently and more usefully from the point of view of constitutive equation development, the Lagrangian vector field EL . The effect of this electric field is similar from a mechanical point of view to that of a preferred direction in a transversely isotropic elastic material. In the latter, the preferred direction is identified by a unit vector, say M, in Br and the strain-energy function is formulated in terms of the right Cauchy–Green tensor C and the structure tensor M ⊗ M. The situation is analogous in the present context. From the theory of Spencer [22], the number of independent invariants for the combination of these two symmetric second-order tensors in three dimensions includes the three principal invariants I1 , I2 , I3 that depend on C, as defined in Eq. (59) in chapter “Finite Deformation Elasticity Theory” [18] and two additional invariants that also depend on the preferred direction M. This translates to an electroelastic material, which is said to be isotropic if the total energy function Ω is an isotropic function of the tensors C and EL ⊗ EL . As distinct from the case of transverse isotropy, however, EL is not a unit vector, and this introduces an additional invariant. Thus, the properties of an isotropic electroelastic material, on the basis of Ω, are described in terms of the invariants I1 = trC,

I2 =

1 [I1 (C)2 − tr(C2 )], 2

I3 = det C

(43)

and I4 = EL · EL ,

I5 = EL · (CEL ),

I6 = EL · (C2 EL ).

(44)

Note that the latter expressions are unaffected by reversal of the sign of EL , and that the notation I4 , I5 , I6 is different from that used in chapter “Finite Deformation Elasticity Theory” [18]. For an incompressible material, I3 = 1 and only five of the six invariants remain. The energy function Ω then depends on the principal invariants I1 , I2 , I3 defined in (43) (or just I1 and I2 in the incompressible case), together with the invariants I4 , I5 , I6 in (44). Note that this choice of independent invariants is not unique. Equally, in terms of Ω ∗ , the properties of an isotropic electroelastic material depend on the invariants of C and DL ⊗ DL , and therefore depend on the invariants (43), with I3 = 1 for incompressibility, together with invariants that depend on DL , which we denote here by I4∗ , I5∗ , I6∗ . We define these by I4∗ = DL · DL ,

I5∗ = DL · (CDL ),

I6∗ = DL · (C2 DL ).

(45)

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For convenience and consistency, we adopt the notation I1∗ , I2∗ , I3∗ in respect of Ω ∗ for the first three invariants. Thus, we have the representations   Ω = Ω¯ (I1 , I2 , I3 , I4 , I5 , I6 ) , Ω ∗ = Ω¯ ∗ I1∗ , I2∗ , I3∗ , I4∗ , I5∗ , I6∗ ,

(46)

with I1∗ = I1 , I2∗ = I2 , I3∗ = I3 . Direct calculations based on (37)1 , (39)1 , (38)1 , and (40)1 lead to 6 

T=

Ω¯ i

i=1,i=4

T=

6 

Ω¯ i∗

i=1,i=4

6  ∂ Ii ∂ Ii Ω¯ i , τ = J −1 F , ∂F ∂F i=1,i=4

(47)

6  ∂ Ii∗ ∂I∗ Ω¯ i∗ i , , τ = J −1 F ∂F ∂F i=1,i=4

(48)

¯ Ii and Ω¯ i∗ = ∂ Ω¯ ∗ /∂ Ii∗ , i ∈ {1, . . . , 6}. where Ω¯ i = ∂ Ω/∂ From (37)2 , (39)2 , (38)2 , and (40)2 , we obtain DL = −

6 

Ω¯ i

i=4

EL =

6 

Ω¯ i∗

i=4

6  ∂ Ii ∂ Ii , D = −J −1 F , Ω¯ i ∂EL ∂EL i=4

(49)

6  ∂ Ii∗ ∂I∗ Ω¯ i∗ i . , E = J −1 F ∂DL ∂DL i=4

(50)

To expand these expressions in full, we require the derivatives of the invariants with respect to F, EL , or DL as appropriate. First, the derivatives with respect to F are ∂ I2 ∂ I3 ∂ I1 = 2F, = 2I1 FT − 2FT FFT , = 2I3 F−1 , (51) ∂F ∂F ∂F as given in Eq. (82) of chapter “Finite Deformation Elasticity Theory” [18], in which Ii may be replaced by Ii∗ , i = 1, 2, 3, ∂ I5 = 2EL ⊗ FEL , ∂F

∂ I6 = 2 (EL ⊗ FCEL + CEL ⊗ FEL ) , ∂F

(52)

∂ I6∗ = 2 (DL ⊗ FCDL + CDL ⊗ FDL ) , ∂F

(53)

noting that ∂ I4 /∂F = O, and ∂ I5∗ = 2DL ⊗ FDL , ∂F

with ∂ I4∗ /∂F = O. The latter are analogous to the derivatives of the transversely isotropic invariants given in Eq. (125) of chapter “Finite Deformation Elasticity Theory” [18].

Nonlinear Constitutive Modeling of Electroelastic Solids

167

The only nonzero derivatives with respect to EL and DL , respectively, are ∂ I4 = 2EL , ∂EL

∂ I5 = 2CEL , ∂EL

∂ I6 = 2C2 EL , ∂EL

(54)

∂ I4∗ = 2DL , ∂DL

∂ I5∗ = 2CDL , ∂DL

∂ I6∗ = 2C2 DL . ∂DL

(55)

On substitution of the derivatives (51)–(53) into (47)2 and (48)2 , we obtain the following explicit expressions for the total Cauchy stress for an unconstrained material τ = 2J −1 Ω¯ 1 B + Ω¯ 2 (I1 B − B2 ) + I3 Ω¯ 3 I + Ω¯ 5 BE ⊗ BE

+ Ω¯ 6 (BE ⊗ B2 E + B2 E ⊗ BE) ,

(56)

 τ = 2J −1 Ω¯ 1∗ B + Ω¯ 2∗ (I1 B − B2 ) + I3 Ω¯ 3∗ I + Ω¯ 5∗ D ⊗ D + Ω¯ 6∗ (D ⊗ BD + BD ⊗ D) ,

(57) where we recall that B is the left Cauchy–Green tensor. The incompressible counterparts of (56) and (57) are τ = 2Ω¯ 1 B + 2Ω¯ 2 (I1 B − B2 ) − pI + 2Ω¯ 5 BE ⊗ BE + 2Ω¯ 6 (BE ⊗ B2 E + B2 E ⊗ BE),

(58)

τ = 2Ω¯ 1∗ B + 2Ω¯ 2∗ (I1 B − B2 ) − p ∗ I + 2Ω¯ 5∗ D ⊗ D + 2Ω¯ 6∗ (D ⊗ BD + BD ⊗ D), (59) respectively, with I3 = I3∗ = 1. The corresponding Lagrangian forms are obtained from (47)1 and (48)1 or directly from the above by using T = J F−1 τ , with J = 1 in the incompressible case. Next, from (49)2 , (50)2 , (54) and (55), we obtain D = −2J −1 (Ω¯ 4 B + Ω¯ 5 B2 + Ω¯ 6 B3 )E,

(60)

  E = 2 Ω¯ 4∗ B−1 + Ω¯ 5∗ I + Ω¯ 6∗ B D,

(61)

and the Lagrangian counterparts are obtained by using the connections EL = FT E and DL = J F−1 D, with J = 1 again in the incompressible case. Equations (60) and (61) can be written alternatively in the form D = εE, where ε = −2J −1 (Ω¯ 4 B + Ω¯ 5 B2 + Ω¯ 6 B3 ) =

−1 1  ∗ −1 Ω¯ 4 B + Ω¯ 5∗ I + Ω¯ 6∗ B , 2

(62)

is the permittivity tensor. It depends on the deformation and, depending on the specific form of Ω or Ω ∗ , on the electric field or electric displacement.

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Note that the simple classical formula D = εE, with scalar ε, can be obtained as a special case of these formulas by setting Ω¯ 4 = I2 Ω¯ 6 , Ω¯ 5 = −I1 Ω¯ 6 , −2J Ω¯ 6 = ε, Ω¯ 4∗ = Ω¯ 6∗ = 0, 2Ω¯ 5∗ = ε−1 .

4.3 Application to Pure Homogeneous Strain In order to illustrate the general theory of the preceding section, we now consider the pure homogeneous strain of an incompressible material defined in Cartesian components by x1 = λ1 X 1 , x2 = λ2 X 2 , x3 = λ3 X 3 subject to the incompressibility constraint λ1 λ2 λ3 = 1, where λ1 , λ2 , λ3 denote the principal stretches. The constitutive formulation can be cast in terms of either Ω or Ω ∗ , but here we restrict attention to that based on Ω ∗ , i.e. Ω¯ ∗ (I1∗ , I2∗ , I4∗ , I5∗ , I6∗ ). We take the electric field to be aligned with the x3 coordinate axis, with Eulerian electric displacement component D3 (with D1 = D2 = 0) and corresponding electric field component E 3 given by the specialization of (61) as ¯∗ ¯∗ 2 E 3 = 2(Ω¯ 4∗ λ−2 3 + Ω5 + Ω6 λ3 )D3 ,

(63)

with E 1 = E 2 = 0. The corresponding Lagrangian components are E L3 = λ3 E 3 and DL3 = λ−1 3 D3 . From (59), the nonzero components of τ are τ11 = 2Ω¯ 1∗ λ21 + 2Ω¯ 2∗ λ21 (λ22 + λ23 ) − p ∗ , τ22 = 2Ω¯ 1∗ λ22 + 2Ω¯ 2∗ λ22 (λ23 + λ21 ) − p ∗ , τ33 = 2Ω¯ 1∗ λ23 + 2Ω¯ 2∗ λ23 (λ21 + λ22 ) − p ∗ + 2Ω¯ 5∗ D32 + 4Ω¯ 6∗ λ23 D32 .

(64) (65) (66)

In terms of two independent stretches, say λ1 and λ3 , the invariants I1∗ , I2∗ , I4∗ , I5∗ , and I6∗ are written as −2 I1∗ = λ21 + λ23 + λ−2 1 λ3 ,

I4∗ = DL2 ,

−2 2 2 I2∗ = λ−2 1 + λ3 + λ1 λ3 ,

I5∗ = λ23 I4∗ ,

I6∗ = λ43 I4∗ .

(67) (68)

It is then clear that Ω¯ ∗ depends on just three independent variables, which we specify as λ1 , λ3 and I4∗ , and we introduce the notation Ωˆ ∗ (λ1 , λ3 , I4∗ ) defined by Ωˆ ∗ (λ1 , λ3 , I4∗ ) ≡ Ω¯ ∗ (I1∗ , I2∗ , I4∗ , I5∗ , I6∗ ) to represent this, with the invariants specialized according to (67) and (68). Then, by forming the stress differences from (64)–(66), we obtain the compact formulas ∂ Ωˆ ∗ ∂ Ωˆ ∗ τ11 − τ22 = λ1 , τ33 − τ22 = λ3 , (69) ∂λ1 ∂λ3

Nonlinear Constitutive Modeling of Electroelastic Solids

and from (63) E 3 = 2λ−2 3

∂ Ωˆ ∗ D3 . ∂ I4∗

169

(70)

Noting that τii (no sum over i) are now principal stress components and may be written τi , equations (69) generalize to the present electroelastic context the equations in (9) for the purely elastic situation (with λ1 and λ3 instead of λ1 and λ2 and τi instead of σi ). A particular example of the energy function Ω ∗ in common use has the decoupled form 1 ∗ 1 Ω¯ ∗ = W¯ (I1∗ , I2∗ ) + I5 = W¯ (I1∗ , I2∗ ) + D · D, (71) 2ε 2ε where W¯ (I1∗ , I2∗ ) is any form of incompressible purely elastic strain-energy function and ε is a constant, the permittivity of the material. For the present specialization, this becomes 1 Ωˆ ∗ = Wˆ (λ1 , λ3 ) + λ23 I4∗ , (72) 2ε and then the stress differences in (69) specialize to τ11 − τ22 = λ1

∂ Wˆ ∂ Wˆ , τ33 − τ22 = λ3 + ε−1 D32 , ∂λ1 ∂λ3

(73)

and (70) to E 3 = ε−1 D3 . We now apply the equations in this section to two simple boundary-value problems involving a rectangular plate of uniform finite thickness with its major surfaces normal to the x3 axis. The problems may be formulated either in terms of Ω or Ω ∗ , but here we focus specifically on the Ω ∗ formulation for illustration. Example 1: A Plate in an External Field We consider the plate to have reference geometry defined by − L 1 ≤ X 1 ≤ L 1 , −L 2 ≤ X 2 ≤ L 2 , 0 ≤ X 3 ≤ H,

(74)

where L 1 , L 2 , H are constants, H being the thickness of the plate, which is in general small compared with L 1 and L 2 so that end effects can be neglected. The deformed plate is defined by − l1 ≤ x1 ≤ l1 , −l2 ≤ x2 ≤ l2 , 0 ≤ x3 ≤ h,

(75)

where l1 = λ1 L 1 , l2 = λ2 L 2 and h = λ3 H . Now suppose that the plate is placed within free space in an electric field with component E 3 , and electric displacement field D3 = ε0 E 3 . By the continuity conditions (13), it follows that within the plate the only nonzero field components are E 3 and D3 , with D3 = D3 and E 3 given by (70) in respect of the energy function Ωˆ ∗ .

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From (33), the components of the Maxwell stress are then given by 1 1    = τm22 = − ε0−1 D32 , τm33 = ε0−1 D32 . τm11 2 2

(76)

with D32 = I5∗ = λ23 I4∗ . Suppose that a mechanical traction ta with a single component ta3 is applied on the  , faces of the plate, so that, from the boundary condition (34), we have τ33 = ta3 + τm33 and hence, from (69),  − λ3 τ22 = ta3 + τm33

∂ Ωˆ ∗ ∂ Ωˆ ∗ , τ11 = τ22 + λ1 , ∂λ3 ∂λ1

(77)

which give the stresses τ11 and τ22 required in the lateral directions to maintain a given deformation for a prescribed electric field and mechanical traction. For the important special case of equibiaxial deformation, with λ1 = λ2 , τ11 = τ22 ˆ∗ and λ3 = λ−2 1 , we obtain from (77) that ∂ Ω /∂λ1 = 0 and  − λ−2 τ11 = ta3 + τm33 1

∂ Ωˆ ∗ ∗ (λ1 , λ−2 1 , I4 ), ∂λ3

(78)

which provides an expression for τ11 as a function of λ1 and I4∗ = λ41 D32 .   = −τm33 The stress τ11 must balance the lateral Maxwell stress, so that τ11 = τm11 and hence (78) gives ∂ Ωˆ ∗ −1 2 ∗ (λ1 , λ−2 1 , I4 ) − ε0 D3 , ∂λ3

(79)

∂ Wˆ −1 (λ1 , λ−2 − ε0−1 )D32 . 1 ) + (ε ∂λ3

(80)

ta3 = λ−2 1 and, for the model (72), ta3 = λ−2 1

For all materials ε > ε0 so that the latter term is negative. Thus, if there is no applied mechanical traction then ∂ Wˆ /∂λ3 must be positive. To illustrate the consequences of this we now consider W¯ (I1∗ , I2∗ ) to have the neo-Hookean form 1 1 −2 W¯ (I1∗ ) = μ(I1∗ − 3), Wˆ (λ1 , λ3 ) = μ(λ21 + λ23 + λ−2 1 λ3 − 3), 2 2

(81)

where μ (> 0) is the shear modulus of the neo-Hookean material. Hence ∂ Wˆ −1 2 (λ1 , λ−2 1 ) = μ(λ3 − λ3 ), ∂λ3

(82)

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171

which is positive provided λ3 > 1, i.e., the plate thickens and contracts laterally. This   is tensile while τm11 is compressive. is not surprising since the Maxwell stress τm33 The lateral compressive stress is likely to lead to instability of the plate if the applied electric field is large enough, but analysis of such an instability is not considered here. Example 2: A Parallel Plate Actuator For the second example, we consider the same geometry as in Example 1, but in this case flexible electrodes are coated on the surfaces X 3 = 0, H and deform into the surfaces x3 = 0, h. A uniform electric field E 3 normal to the faces is generated by a potential difference, say V , between the electrodes so that V = E 3 h = λ3 E 3 H . Let the resulting surface charges on the electrodes have densities ±σf per unit deformed area, with the positive sign applying for x3 = 0. Then, with end effects neglected, Gauss’s theorem implies that there is no electric field outside the plate and hence no Maxwell stress, and from the specialization of the boundary condition (13)2 we have D3 = σf . For this problem, we consider only the case of equibiaxial deformations, again with λ1 = λ2 and λ3 = λ−2 1 . Then, (78) applies but without the Maxwell stress and with τ33 = ta3 : ∂ Ωˆ ∗ ∗ τ11 = ta3 − λ−2 (λ1 , λ−2 (83) 1 1 , I4 ), ∂λ3 and E 3 is given by (70) specialized for equibiaxial deformations. Since there are no Maxwell stresses, we may set the lateral stress τ11 to zero. If no mechanical tractions are applied on the major surfaces then ta3 = 0, and (83) reduces to ∂ Ωˆ ∗ ∗ (λ1 , λ−2 (84) 1 , I4 ) = 0, ∂λ3 which provides a connection between the lateral stretch λ1 and the electric field via I4∗ . For the model (72), this becomes ∂ Wˆ −1 2 2 (λ1 , λ−2 1 ) + ε λ1 D3 = 0. ∂λ3

(85)

Hence, the first term must be negative. For the neo-Hookean model, for example, this is given by (82) and this then requires that λ3 < 1, i.e., the plate thins and expands laterally. This is in contrast to the situation for a plate “floating” in an external field in Example 1. A plate, or rather a thin film, forms the basis for prototype actuators using thin-film elastomeric materials, and for a review of the background and constitutive analysis of such materials we refer to [12]. The stability of such configurations has been examined extensively in the literature and we refer to the recent review [13] for discussion of the literature and the various approaches that have been adopted for the analysis of stability/instability.

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Let A and a denote the reference and deformed areas of the plate. Then, incompressibility requires that ah = AH and hence a = λ21 A. Because the deformation and electric field are uniform for this problem, the potential difference V is given by V = E 3 h = λ3 E 3 H , while the total charge, say Q, on one electrode is given by Q = aσf = Aλ21 D3 . Then, since D3 = εE 3 for the considered model, we have the connection Q = ελ41 AV /H . For the neo-Hookean specialization, Eq. (85) can then be written as V2 Q2 . (86) μ(λ61 − 1) = ελ81 2 = H ε A2 Thus, λ1 increases monotonically with Q and with the magnitude of the electric field, while V has a maximum with respect to λ1 at λ31 = 2. The lack of monotonicity is associated with instability under voltage (potential difference) control (see, for example, [24] and references therein). On the other hand, although less of a practical option, charge control is more stable. The above illustration is based on the use of the neo-Hookean model for the purely elastic response, but similar results can be obtained for other models that are appropriate for describing the elastic response of elastomeric materials. Much of the motivation for the current modeling interest in dielectric elastomers stems from the practical possibilities that they offer, as mentioned briefly in the Introduction (Sect. 1). The two examples above are relatively simple in that they involve homogeneous deformations and uniform fields. For other geometries, the solution of more complex boundary-value problems involving nonhomogeneous deformations presents additional interest, both theoretical and from the point of view of applications, and for the analysis of a selection of such problems we refer to, for example, [5, 6, 10, 11]. One such problem, that of the extension and torsion of a circular cylindrical tube, is examined in the following section as a representative example by way of illustration, and we note that such a geometry is also relevant for actuator design.

5 Representative Example for a Tubular Geometry In this section, we analyze the response of an incompressible, isotropic electroelastic circular cylindrical tube subject to a nonhomogeneous deformation and a nonuniform electric field, specifically a radial electric field generated by a potential difference between electrodes coated on the curved surfaces of the tube, coupled with the axial extension and torsion of the tube that maintain its circular cylindrical shape under an appropriate axial load and torsional moment.

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5.1 Geometry and Deformation In the stress-free undeformed configuration, the geometry of the considered tube is described in terms of cylindrical polar coordinates R, Θ, Z in the form A  R  B, 0  Θ  2π, 0  Z  L ,

(87)

where A (> 0) and B denote its inner and outer radii and L its length. The associated unit basis vectors are E R , EΘ , E Z and a material particle in the tube has position vector X given by X = R E R + Z E Z relative to an origin at the center of the end Z = 0. The tube is deformed by an axial extension combined with a torsion such that the circular cylindrical shape of the tube is maintained. For a hyperelastic circular cylindrical tube subject to axial extension and torsional rotation, the theory was originally given by Rivlin [20]. Here, we summarize the equations in respect of an incompressible material. In terms of cylindrical polar coordinates r, θ, z, these are  2  R − A2 , θ = Θ + ψλz Z , z = λz Z , r 2 = a 2 + λ−1 z

(88)

where a is the deformed inner radius, ψ is the torsion per unit deformed length of the tube, and λz = l/L is the constant axial stretch, l being the deformed length of the tube. With r, θ, z are associated unit basis vectors er , eθ , ez , and the point X occupies the position x = r er + z ez in the deformed configuration. The deformed outer radius, denoted b, is then given by b=

  B 2 − A2 . a 2 + λ−1 z

(89)

The azimuthal stretch λθ = r/R is denoted by λ and its values on the inner and outer boundaries by λa and λb . Thus, λa = a/A, λb = b/B,

(90)

and the connections λa2 λz − 1 =

  R2  2 B2  2 λ λb λ z − 1 λ − 1 = z 2 2 A A

(91)

are then obtained on use of (88) and (89). The deformation gradient has the form F = λr er ⊗ E R + λeθ ⊗ EΘ + λz ez ⊗ E Z + λz γ eθ ⊗ E Z ,

(92)

where the notation γ is defined as γ = ψr and the radial stretch λr is given by the incompressibility condition det F = λr λλz = 1, which does not depend on γ . The

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corresponding left and right Cauchy–Green tensors, B = FFT and C = FT F, are given by B = λr2 er ⊗ er + (λ2 + γ 2 λ2z )eθ ⊗ eθ + λ2z ez ⊗ ez + γ λ2z (eθ ⊗ ez + ez ⊗ eθ ) , (93) C = λr2 E R ⊗ E R + λ2 EΘ ⊗ EΘ + λ2z (1 + γ 2 )E Z ⊗ E Z + γ λλz (EΘ ⊗ E Z + E Z ⊗ EΘ ) .

(94)

In what follows, it is convenient to use the incompressibility condition to write the radial stretch in the form λr = λ−1 λ−1 z and to adopt λ and λz as the independent stretches. The principal invariants I1 = I1∗ and I2 = I2∗ defined in (43) then specialize to   2 2 2 , I1∗ = λ−2 λ−2 z + λ + λz 1 + γ

  I2∗ = λ2 λ2z + λ−2 1 + γ 2 + λ−2 z ,

(95)

with I3 = I3∗ = 1. Note that both λ and γ depend on r , alternatively on R.

5.2 The Electric Field Contribution We now consider that the deformation is controlled both by mechanical loads on the tube and by a radial electric field. The latter is generated by a potential difference applied between flexible electrodes that are coated on the inner and outer curved surfaces of the tube, R = A and R = B in the reference configuration and r = a and r = b in the deformed configuration after application of the potential difference and mechanical loads. The analysis in this section follows closely that in [11], in which an electroelastic tube is subject to combined extension, inflation, and torsion, accompanied by an internal pressure, which is not included here. The inflation and extension of a tube was investigated in [6, 10] based on a constitutive equation using the energy function Ω and without flexible electrodes, while in [15] the same problem but with flexible electrodes and based on the energy function Ω ∗ was analyzed. In the present example, the material properties of the tube are described in terms of the energy function Ω ∗ with DL as the independent electric variable. It is appropriate to consider that end effects can be neglected provided the length L of the tube is significantly larger than the external radius B, and therefore that each of the electric field E and electric displacement field D has only a single nonzero component, the radial component. These are denoted Er and Dr , which are functions of r within the tube. The potential difference applied between the electrodes creates equal and opposite surface charges on the electrodes, and we denote by ±Q, with Q > 0, the total charges on the two electrodes. Then, by Gauss’s theorem, no field is generated outside the tube. The boundary condition (13)2 becomes D · n = −σf on r = a and r = b, where σf is the surface free charge density, which is different on the two electrodes. Let Q

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and −Q be the total charges on the surfaces r = a and r = b, respectively; hence σf is Q/(2πal) = Da and −Q/(2π bl) = −Db on r = a and r = b, respectively, Da and Db being the values of Dr thereon. Since Er depends only on r , the equation curl E = 0 given in (12)1 is satisfied automatically, while the equation divD = 0 from (12)2 reduces to Dr 1 d dDr + = (r Dr ) = 0, dr r r dr

(96)

so that r Dr is constant, and hence r Dr = a Da = bDb =

Q . 2πl

(97)

From (21)2 with J = 1, we have DL = F−1 D, which yields just a single component of DL , denoted by D R and given by D R = λλz Dr =

Q . 2π R L

(98)

It follows that the invariants I4∗ , I5∗ , I6∗ defined in (45) reduce to I4∗ = D 2R ,

∗ I5∗ = λ−2 λ−2 z I4 ,

∗ I6∗ = λ−4 λ−4 z I4 .

(99)

On adopting the energy function Ω ∗ , we obtain an expression for the radial component Er of the electric field by specializing Eq. (61) to   Dr . Er = 2 Ω¯ 4∗ λ2 λ2z + Ω¯ 5∗ + Ω¯ 6∗ λ−2 λ−2 z

(100)

This may also be expressed in terms of D R and the Lagrangian electric field component E R on use of (98) and from the connection EL = FT E, which gives E R = λ−1 λ−1 z Er .

5.3 Stress Components and Mechanical Equilibrium We now have all the ingredients that are necessary to determine the total Cauchy stress tensor (59). With respect to the cylindrical polar coordinate axes, its components have the explicit forms  

2 ¯ ∗ 1 + γ 2 λ−2 + λ−2 − p ∗ + 2Ω¯ 5∗ Dr2 + 4Ω¯ 6∗ λ−2 λ−2 τrr = 2Ω¯ 1∗ λ−2 λ−2 z + 2Ω2 z z Dr ,     − p∗ , τθθ = 2Ω¯ 1∗ λ2 + γ 2 λ2z + 2Ω¯ 2∗ λ2 λ2z + γ 2 λ−2 + λ−2 z   (101) τzz = 2Ω¯ 1∗ λ2z + 2Ω¯ 2∗ λ2 λ2z + λ−2 − p ∗ , τθ z = 2Ω¯ 1∗ γ λ2z + 2Ω¯ 2∗ γ λ−2 ,

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with the remaining two components τr θ = τr z = 0. The invariants in Eqs. (95) and (99) depend on the three deformation measures λ, λz , γ , together with I4∗ . This enables the forms of the constitutive equations to be simplified by adopting a reduced energy function, denoted Ω˜ ∗ and defined by     Ω˜ ∗ λ, λz , γ , I4∗ = Ω¯ ∗ I1∗ , I2∗ , I4∗ , I5∗ , I6∗ ,

(102)

with the invariants I1∗ , I2∗ , I4∗ , I5∗ , I6∗ given by (95) and (99). By evaluating the derivatives of Ω˜ ∗ with respect to the independent variables λ, λz , γ , I4∗ , it is easily shown that the stress components (101) satisfy the relations τθθ − τrr = λ

∂ Ω˜ ∗ ∂ Ω˜ ∗ ∂ Ω˜ ∗ ∂ Ω˜ ∗ ∂ Ω˜ ∗ +γ , τzz − τrr = λz , τθ z = . (103) −γ ∂λ ∂γ ∂λz ∂γ ∂γ

Similarly, in terms of Ω˜ ∗ the constitutive equation (100) takes on the compact form Er = 2λ2 λ2z

∂ Ω˜ ∗ Dr . ∂ I4∗

(104)

In the absence of mechanical body forces, the equilibrium condition (28) reduces to the radial component equation dτrr τθθ − τrr = . dr r

(105)

In this example, we assume that no mechanical traction is applied on the boundaries r = a and r = b, and, since there is no electric field outside the tube and therefore no Maxwell stress, the boundary condition (34) reduces to τ n = 0. Thus, τrr = 0 on r = a, b.

(106)

Then, integration of (105) and use of (103)1 leads to 

b



a

∂ Ω˜ ∗ ∂ Ω˜ ∗ +γ λ ∂λ ∂γ



dr = 0, r

(107)

with b depending on a through (89). The torsional moment and resultant axial load required to maintain the given deformation are denoted by M and F, respectively. On use of (103)3 , M is given by  M = 2π a

b

 τθ z r 2 dr = 2π a

b

∂ Ω˜ ∗ 2 r dr. ∂γ

Likewise, with (103)1,2 , F can be expressed in the form

(108)

Nonlinear Constitutive Modeling of Electroelastic Solids



b

F= a



∂ Ω˜ ∗ ∂ Ω˜ ∗ ∂ Ω˜ ∗ − 3γ −λ 2λz ∂λz ∂λ ∂γ

177

 r dr.

The latter follows from the basic definition  b τzz r dr F = 2π

(109)

(110)

a

and, on use of (105) and (107), the result 

b

(τrr + τθθ )r dr = 0.

(111)

a

5.4 Application to an Electroelastic Neo-Hookean Model The equations derived in the previous section are valid for any incompressible, isotropic electroelastic material. For purposes of illustration, we now specialize the form of Ω˜ ∗ to a simple model for which the purely mechanical contribution is given by the neo-Hookean model and the electric part characterized by a deformationdependent permittivity. Specifically,

∗   1  ∗ 1  ∗ μ I1 − 3 + α I1 − 3 + βλ−2 λ−2 I4 , (112) z 2 2ε0   2 2 2 with I1∗ = λ−2 λ−2 z + λ + λz 1 + γ , where μ is the elastic shear modulus of the neo-Hookean material in the reference configuration and α and β are two material constants related to the electrical properties of the material, α being a measure of the deformation dependence of the permittivity. From (104), we obtain Ω˜ ∗ =

Er =



1 2 2 ∗ αλ λz I1 − 3 + β Dr , ε0

(113)

and hence the deformation dependent permittivity ε has the form  

−1 ε = ε0 αλ2 λ2z I1∗ − 3 + β .

(114)

Note that evaluation in the reference configuration gives the permittivity ε = ε0 /β, so that β −1 is the relative permittivity of the material in the reference configuration. The derivatives of the energy function (112) that are needed to evaluate the integrals in (107)–(109) have the forms

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∂ Ω˜ ∗ = ∂λz ∂ Ω˜ ∗ = λ ∂λ ∂ Ω˜ ∗ = γ ∂γ

λz



 

∗  + ε0−1 αλ2z 1 + γ 2 − (α + β) λ−2 λ−2 I4 , μ λ2z 1 + γ 2 − λ−2 λ−2 z z  

∗ μ λ2 − λ−2 λ−2 + ε0−1 αλ2 − (α + β) λ−2 λ−2 I4 , z z   μ + ε0−1 α I4∗ λ2z γ 2 .

(115)

Note that each of the integrals in (107)–(109) can be evaluated in closed form, as given in [11], and for the case of zero torsion in [15]. We refer to these papers for the explicit formulas. In the following section, we obtain numerical results from these integrals in respect of the simple model (112) in order to illustrate the influence of the electric field along with the extension and torsion.

5.4.1

Numerical Results

For the numerical calculations, we introduce the notations q and E 0 for the square of the charge per unit reference area of the inner surface and for the mean value of the electric field, defined by  q=

Q 2π AL

2 ,

E0 =

φ(b) − φ(a) , B−A

(116)

where φ(b) − φ(a) is the potential difference between the two electrodes. The results are conveniently illustrated in terms of the dimensionless charge density, mean electric field, axial load, torsional moment, and torsional deformation defined, respectively, by q∗ =

q ε0 E 02 , , e∗ = με0 μ

F∗ =

F , μπ A2

M∗ =

M , ψ ∗ = ψ A. μπ A3

(117)

The electrostatic potential φ(r ) is related to Er by Er = −dφ/dr , integration of which between r = a and r = b with Er given by (113) and Dr by (97) yields φ(b) − φ(a) = −

√ qA s, ε0 λz

(118)

where s, as derived in [11], is given by  1 η2 − 1 + α (λ3z − 3λz + 1) log η + (λa2 λz − 1) 2 η2  1 + (λa2 λz − 1)λ2z ψ ∗ 2 log η + λ2z ψ ∗ 2 (η2 − 1) , (119) 2 

s = (α + β) log

ηλb λa



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where η = B/A. It follows that q ∗ and e∗ are connected via the geometry and the deformation according to (120) q ∗ = λ2z (η − 1)2 e∗ /s 2 , and it is, therefore, straightforward to translate between results for fixed charge and fixed potential difference. If an internal pressure P is applied then this is given by the integral in (107). The dependence of P on λa was examined in [11] for the fixed value λz = 1.2 of the axial stretch and for ψ ∗ = 0, 0.5, 1. Here we have set P = 0. Then, for given values of the parameters η, α, β and either q ∗ or e∗ , Eq. (107) provides an implicit formula relating λa , λz and ψ ∗ , and their connections are illustrated in Fig. 1. As a representative value we take η = 1.3, while λa measures the deformed internal radius of the tube relative to its undeformed value, which is conveniently taken as A = 1 for numerical purposes, so that λa = a. The parameters describing the permittivity in (114) are taken as α = 0.25 and β = 0.5, as in [11]. Then, in Fig. 1, λa is plotted against λz ; results in the left-hand column are for q ∗ = 0, 1, 3, 10, and in the right-hand column for e∗ = 0, 0.2, 1, 10, in each case depicted by continuous, dashed, dotted and dash-dot curves, respectively. These representative values are selected so that the curves for the different values are sufficiently distinguishable. Panels (a) and (b) represent the response without torsion, i.e. ψ ∗ = 0, (c) and (d) with ψ ∗ = 0.5, and (e) and (f) with ψ ∗ = 1. The continuous curves are for the purely elastic responses, q ∗ = 0 or e∗ = 0, and, for each value of ψ ∗ , these two curves coincide. Clearly, as expected, when there is no electric field, λa decreases monotonically as the axial stretch λz increases from 1. For ψ ∗ = 0 the curve starts from λa = 1 and this starting value falls below λa = 1 for ψ ∗ = 0, i.e., the torsion causes a reduction in the radius. The presence of an electric field increases the value of λa with either q ∗ or e∗ at each fixed λz , while the trend for increasing λz follows the pattern indicated for the purely elastic situation. This is the case for either fixed q ∗ (left-hand column) or fixed e∗ (right-hand column), the main difference being that the curves for fixed e∗ tend to that for the purely elastic case as λz increases, whereas for fixed q ∗ they remain separated. In Fig. 2, the dependence of λa on the torsion ψ ∗ is illustrated using the same values for the geometric and material parameters as for Fig. 1. Panel (a) represents the changes of the inner radius for λz = 1 for the purely elastic case q ∗ = 0 and for q ∗ = 1, 3, 10, illustrated, respectively, by continuous, dashed, dotted, and dashdot curves. Panel (b), again for λz = 1, includes the purely elastic case with e∗ = 0 and e∗ = 0.2, 1, 10, which are again depicted by continuous, dashed, dotted, and dash-dot curves. In each case λa , i.e., the inner radius, decreases monotonically with increasing ψ ∗ , but increases monotonically with either q ∗ or e∗ for each fixed ψ ∗ , with the curves for each given e∗ tending to the purely elastic curve as ψ ∗ increases. How the results in panels (a) and (b) change with an increase in the axial stretch λz are illustrated in the other panels of Fig. 2. For λz = 1.5 in panels (c) and (d) and λz = 3 in panels (e) and (f) the patterns of the results are similar to those in panels

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Fig. 1 Plots of λa versus λz from Eq. (107) for the neo-Hookean dielectric model (112). Panels (a) and (b) are for the case with no torsion ψ ∗ = 0; (c) and (d) for ψ ∗ = 0.5; (e) and (f) for ψ ∗ = 1.0. Plots for q ∗ = 0, 1, 3, 10 are shown in the left-hand column, and for e∗ = 0, 0.2, 1, 10 in the right-hand column by continuous, dashed, dotted and dash-dot curves, respectively, in each panel

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Fig. 2 Plots λa versus ψ ∗ from Eq. (107) for the neo-Hookean dielectric model (112). Panels (a) and (b) are for the case with no axial extension λz = 1; (c) and (d) for λz = 1.5; (e) and (f) for λz = 3.0. Plots for q ∗ = 0, 1, 3, 10 are shown in the left-hand column, and for e∗ = 0, 0.2, 1, 10 in the right-hand column by continuous, dashed, dotted and dash-dot curves, respectively, in each panel

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(a) and (b), the main feature being that λa decreases with an increase in λz , as already noted in respect of Fig. 1. In Fig. 3, the dimensionless torsional moment M ∗ is plotted against ψ ∗ . Reading from top to bottom, the three rows correspond to fixed values of λz = 1, 1.5, 3.0, respectively. In each panel of the left-hand column the continuous, dashed, dotted, and dash-dot curves depict the behavior for q ∗ = 0, 1, 3, 10, and in the right-hand column they represent the responses for e∗ = 0, 0.2, 1, 10. The results in the lefthand column show that the torsional response becomes stiffer as q ∗ increases while an increase in axial extension has the opposite effect. The axial extension generates a reduction of the cross-sectional area, and therefore less torque is required to obtain the torsional deformation. The results in the right-hand column, for fixed values of e∗ , again show stiffening with increasing electric field. Initially, M ∗ increases with increasing ψ ∗ until a maximum is reached. For larger ψ ∗ , the mechanical load reduces gradually approaching the purely elastic case for large torsional deformations. The increase in the axial extension is again accompanied by a reduction of the cross-sectional area, and therefore the required mechanical load to obtain a given ψ ∗ reduces significantly with increasing λz . For a given e∗ , the possible onset of instability as the maximum of M ∗ is approached needs to be examined, but this is not within the scope of the present analysis. In Fig. 4, the dependence of the dimensionless axial load F ∗ on ψ ∗ is illustrated for fixed values of the axial extension. The panels in the left-hand column show the dependence of F ∗ on ψ ∗ with the same geometric and material parameters and for q ∗ = 0, 1, 3, 10, arranged as in Fig. 3. The influence of the electric field via e∗ is depicted in the right-hand column for e∗ = 0, 0.2, 1, 10. Panels (a) and (b) show the results for λz = 1, panels (c) and (d) for λz = 1.5 and panels (e) and (f) for λz = 3. A negative value of F ∗ corresponds to compression, i.e., without the axial force the tube would extend in the axial direction. The continuous curves in panels (a) and (b) represent the purely elastic case and show that with increasing values of ψ ∗ more compression is needed to maintain the prescribed axial stretch. The magnitude of the force also increases with the application of electric charges or an electric field. In Panel (a), the magnitude of F ∗ increases with increasing values of q ∗ and ψ ∗ , in Panel (b) it also increases with e∗ but, for sufficiently large e∗ , reduces as ψ ∗ increases. Panels (c) and (d) show that tensile forces are required to maintain the axial stretch λz = 1.5 when ψ ∗ = 0, but as ψ ∗ increases F ∗ becomes compressive, in particular for ψ ∗  1 compression is needed to maintain the axial stretch λz = 1.5 constant. For λz = 3, panels (e) and (f), F ∗ is tensile and increases in magnitude as q ∗ increases (to a lesser extent as e∗ increases) but decreases as ψ ∗ increases. For larger values of ψ ∗ (not shown here) F ∗ again becomes negative, as in panels (c) and (d). The results for fixed e∗ tend to those for the purely elastic case for sufficiently large values of ψ ∗ .

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Fig. 3 Plots M ∗ versus ψ ∗ based on Eq. (108) with (117)4 for the neo-Hookean dielectric model (112). Panels (a) and (b) are for the case with no axial extension λz = 1; (c) and (d) for λz = 1.5; (e) and (f) for λz = 3.0. Plots for q ∗ = 0, 1, 3, 10 are shown in the left-hand column, and for e∗ = 0, 0.2, 1, 10 in the right-hand column by continuous, dashed, dotted, and dash-dot curves, respectively, in each panel

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-4 -6

-4 -6

1.5

6

-2

F∗

2

8

2

-6

1.5

(d)

6 4

F∗

1

ψ∗

-8 0

0.5

1

ψ∗

1.5

2

0

0.5

1

ψ∗

Fig. 4 Plots F ∗ versus ψ ∗ based on Eq. (109) with (117)3 for the neo-Hookean dielectric model (112). Panels (a) and (b) are for the case with no axial extension λz = 1; (c) and (d) for λz = 1.5; (e) and (f) for λz = 3.0. Plots for q ∗ = 0, 1, 3, 10 are shown in the left-hand column, and for e∗ = 0, 0.2, 1, 10 in the right-hand column by continuous, dashed, dotted, and dash-dot curves, respectively, in each panel

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6 Concluding Remarks Within the framework of the present volume, the main aim of this chapter is the development of constitutive equations that describe nonlinear electromechanical interactions, and the constitutive theory has been illustrated by application to basic problems with simple prototype forms of constitutive equations that demonstrate the influence of the electric field on the mechanical response of the material. Several other problems have been examined in, for example, [5, 6, 10]. Beyond these basic analyses, there is a wealth of interesting directions that the subject can take. For example, the equations governing linearized increments of the deformation and of the electric field superimposed on a known finitely deformed configuration in the presence of an electric field are given in [7, 10], and these have been applied to the analysis of stability of plates and tubes in [9, 16, 23]. We refer to the recent review article [13] for a detailed overview of the current status of electroelastic stability analysis and references to the literature. Due to space limitations, we do not include the incremental theories in this chapter. Electroelastic wave propagation, within the quasi-electrostatic approximation, has been examined in [8] as an extension of the static incremental framework. The number of references in the present chapter has deliberately been kept to a minimum, and we refer to the works cited herein for extensive access to the literature. Finally, we mention that although this volume does not include a chapter on magnetoelastic interactions the theory governing these runs in close parallel with that for electroelastic interactions discussed here, and we refer to [10] for extensive coverage of this topic and relevant pointers to the literature.

References 1. Bar-Cohen Y (2002) Electro-active polymers: current capabilities and challenges. In: BarCohen Y (ed) Proceedings of the 4th electroactive polymer actuators and devices (EAPAD) conference, 9th smart structures and materials symposium. San Diego. SPIE Publishers, Bellingham, WA, pp 1–7 2. Brochu P, Pei Q (2010) Advances in dielectric elastomers for actuators and artificial muscles. Macromol Rapid Commun 31:10–36 3. Bustamante R, Dorfmann A, Ogden RW (2009) Nonlinear electroelastostatics: a variational framework. Z Angew Math Phys 60:154–177 4. Bustamante R, Dorfmann A, Ogden RW (2009) On electric body forces and Maxwell stresses in nonlinearly electroelastic solids. Int J Eng Sci 47:1131–1141 5. Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mech 174:167–183 6. Dorfmann A, Ogden RW (2006) Nonlinear electroelastic deformations. J Elast 82:99–127 7. Dorfmann A, Ogden RW (2010) Nonlinear electroelasticity: incremental equations and stability. Int J Eng Sci 48:1–14 8. Dorfmann A, Ogden RW (2010) Electroelastic waves in a finitely deformed electroactive material. IMA J Appl Math 75:603–636 9. Dorfmann L, Ogden RW (2014) Instabilities of an electroelastic plate. Int J Eng Sci 77:79–101 10. Dorfmann L, Ogden RW (2014) Nonlinear theory of electroelastic and magnetoelastic interactions. Springer, New York

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11. Dorfmann L, Ogden RW (2018) The effect of deformation dependent permittivity on the elastic response of a finitely deformed dielectric tube. Mech Res Commun 93:47–57 12. Dorfmann L, Ogden RW (2017) Nonlinear electroelasticity: materials, continuum theory and applications. Proc R Soc A 473:20170311 13. Dorfmann L, Ogden RW (2019) Instabilities of soft dielectrics. Philso Trans R Soc A 377:20180077 14. Goulbourne NC (2009) A mathematical model for cylindrical, fiber reinforced electropneumatic actuators. Int J Solids Struct 46:1043–1052 15. Melnikov A, Ogden RW (2016) Finite deformations of an electroelastic circular cylindrical tube. Z Angew Math Phys 67:140 16. Melnikov A, Ogden RW (2018) Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field. Z Angew Math Phys 69:60 17. Merodio J, Ogden RW (2019) Basic equations of continuum mechanics. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). Springer 18. Merodio J, Ogden RW (2019) Finite deformation elasticity theory. In: Merodio J, Ogden RW (eds) Constitutive modeling of solid continua. Series in Solids Mechanics and its Applications (In Press). Springer 19. Pelrine R, Kornbluh R, Pei QB, Joseph J (2000) High-speed electrically actuated elastomers with strain greater than 100%. Science 287:836–839 20. Rivlin RS (1949) Large elastic deformations of isotropic materials VI. Further results in the theory of torsion, shear and flexure. Philso Trans R Soc A 242:173–195 21. Rogers JA (2013) A clear advance in soft actuators. Science 341:1243314 22. Spencer AJM (1971) Theory of Invariants. In: Eringen AC (ed) Continuum physics, vol 1. Academic, New York, pp 239–353 23. Su YP, Broderick HC, Chen WQ, Destrade M (2018) Wrinkles in soft dielectric plates. J Mech Phys Solids 119:298–318 24. Suo Z (2010) Theory of dielectric elastomers. Acta Mech Solida Sin 23:549–578

A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies Roger Bustamante and Kumbakonam Rajagopal

Abstract Implicit constitutive relations are discussed for elastic, electro- and magneto-elastic and thermo-elastic bodies. Approximations of implicit constitutive equations for electro-elastic bodies are developed when the displacement gradient is small, while in the case of the constitutive equations for thermo-elastic bodies approximations are developed when both the displacement gradient and the heat flux are small. Boundary-value problems are studied confining attention to some specific forms of implicit constitutive relations.

1 Introduction Elasticity, till very recently, has meant either Cauchy elasticity (wherein the Cauchy stress is a function of the deformation gradient), or Green elasticity1 wherein one can associate the existence of stored energy, which is a function of the deformation gradient from which the stress can be derived, see [47] for a detailed discussion of the various notions attributed to what is meant by a body being elastic. Rajagopal [44, 45] recognized that the class of elastic bodies is much larger than Cauchy elastic bodies (Green elastic bodies, if one requires the existence of a stored energy), but 1 Real

bodies exhibit different kinds of responses while undergoing different processes. A process is called elastic if during the process the body does not dissipate energy, and all the energy that is stored in the body due to working can be completely recovered by means of a purely mechanical process. A body that can only undergo elastic processes would qualify to be called an elastic body, but this is an idealization. Many bodies dissipate very little energy while undergoing a sufficiently large class of processes and qualify to be idealized as elastic bodies.

R. Bustamante (B) Department of Mechanical Engineering, Universidad de Chile, Beauchef 851, Santiago, Chile e-mail: [email protected] K. Rajagopal Department of Mechanical Engineering, University of Texas A&M, College Station, Austin, TX, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_7

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this would need one to give up the myopic view of thinking of the stress/the stored energy being a function of the deformation gradient but allows a much larger class of response relations, namely, a relation between the stress the deformation gradient and the density in the true sense of the mathematical definition. Rajagopal articulated the view that the identification of whether a body is elastic ought to be made by appealing to the notion that such bodies are incapable of converting mechanical work into heat, that is, such bodies are incapable of dissipation, and not on notions such as the body retaining its shape on the removal of the external load, etc. In view of this, Rajagopal and Srinivasa [52] developed a thermodynamic basis for a class of implicit elastic bodies and showed that the stress cannot be derived from stored energy that depends only on the deformation gradient. Later, Rajagopal and Srinivasa [53] provided a generalization of Green elastic bodies that would apply to implicit elastic bodies by extending the notion of a mechanical cycle, which requires both the stress and deformation gradient return to their same values as the starting values for a process to be considered as a cycle. Requiring that the work done in such cycles is zero leads to non-dissipative bodies, that is elastic bodies, that are more general than Green elastic bodies. The generalization of the class of elastic bodies allows one to be able to describe the response of non-dissipative bodies that was hitherto not possible. First, such bodies are completely in keeping with the requirements of causality, the basic tenet of Newtonian mechanics. The classical approach of providing constitutive expressions for the stress puts the cart before the horse in that they ignore the fact that it is stresses that cause deformation. Second, implicit constitutive relations provide a completely consistent framework for studying a variety of problems, within the context of small displacement gradients wherein one can approximate the strain to be the linearized strain, which within the context of the classical linearized elasticity leads to inconsistencies in that the strains become singular such as in the fracturing of brittle elastic bodies, the strains at the edges of notches, etc. Third, such implicit theories provide a means to describe the nonlinear behaviour in the small strain range of many metallic alloys which the classical linearized model is incapable of (see [51] for a detailed discussion of the relevant issues). This review article is devoted to a discussion of the recent results concerning the response of such implicit elastic bodies. Implicit constitutive relations for elastic bodies have the potential to describe a plethora of phenomena that have defied adequate explanation within the context of both classical linearized and nonlinear elasticity theories. They have applications that span the gamut from soft matter such as biological and polymeric material to materials at the other end of the spectrum such as rocks and metals. Implicit constitutive relations have not yet been gainfully exploited in describing several phenomena that have remained inexplicable and we hope that this review article will provide the impetus to use such models to describe the response of elastic bodies. The organization of the paper is as follows. In the next section, we provide some basic definitions and equations concerning deformations, strains and stresses. In Sect. 3 we document some classes of implicit constitutive relations for elastic, electroand magneto-elastic and thermo-elastic bodies. In Sect. 4 we study the implicit rela-

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tions presented in the previous section, within the context of small gradients of the displacement field in the case of elastic bodies, small gradients of the displacement field and small electric displacements in the case of electro-elastic bodies, and small gradients of the displacement field and small heat fluxes in the case of thermo-elastic bodies. In Sect. 5 some applications are considered, in particular for the subclass of bodies whose constitutive equations and relations are obtained assuming the gradient of the displacement field is small. In Sect. 6 some boundary-value problems are analysed, and finally in Sect. 7 a brief list of open problems is discussed.

2 Basic Equations We provide a very rudimentary and mathematically unsophisticated description of the kinematics of continua and some stress measures relevant to elastic bodies. An interested reader can refer to Truesdell and Toupin [62] and Truesdell and Noll [63] for a detailed discussion of the kinematics of continua and various stress measures. Let the particle X belong to the abstract body B and let κ r denote a one-to-one mapping referred to as the placer that maps B into a three-dimensional Euclidean space E and let X = κ r (X ). Let us refer to κ r (B) as the reference configuration of B. Let κ t , t ∈ R be a one parameter family of placers such that x = κ t (X ). Since κ r is invertible we can define a mapping χ κ r : κ r × R → E such that x = χ κ r (X, t). We will assume χ κ r to be sufficiently smooth and also for ease of notation we shall drop the suffix κ r and refer to the mapping as χ . The deformation gradient, left and right Cauchy–Green tensors, the Lagrange–Saint Venant strain tensor, displacement field and linearized strain tensor are defined, respectively, as2 ∂χ ∂x 1 = , B = FFT , C = FT F, E = (C − I), ∂X ∂X 2 1 T u = x − X, ε = (∇X u + ∇X u ), 2

F=

(1) (2)

where ∇X is the gradient operator with respect to the reference configuration and we assume that J = det F > 0. The Cauchy stress tensor, first Piola–Kirchhoff stress tensor and second Piola– Kirchhoff stress tensor are denoted T, P and S, respectively, and they are related through3 P = J TF−T , S = F−1 P = J F−1 TF−T . (3) The balance of linear momentum, balance of mass and first law of thermodynamics for elastic bodies (since there is no dissipation) read, respectively, as

2 All

the quantities in (1) should have a suffix κ r , but for ease of notation we shall ignore it. that the notation for the stress tensors differs from that used in some of the other chapters.

3 Note

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˙ = ρr U˙ , ρ x¨ = div T + ρb, ρ˙ + ρdiv x˙ = 0, tr(SE)

(4)

where b represents the body force (per unit mass), div is the divergence operator with respect to x, ρ is the density of the body in the configuration at time t and U is the internal energy of the body. The density of the body in the reference configuration is denoted ρr and ρr = Jρ.

3 Nonlinear Implicit Theories 3.1 Elastic Bodies In the classical theory of elasticity it is customary to assume a measure of stress as a function of some measure of the deformation gradient as in the case of a Cauchy elastic body (see [20, 21]), where the Cauchy stress T = f(F). If one assumes that there exists a stored energy function W associated with the elastic body that is a function of the deformation gradient, then one can show that the first Piola–Kirchhoff stress P is given by P = ∂ W/∂F, and such bodies are referred to as Green elastic bodies (see [31, 32]). An approximation that has proved very useful is that which is obtained by carrying out a linearization under the assumption that the displacement gradient is small in an appropriate norm, and the constitutive relation is referred to as the linearized elastic model. In the special case of the body being isotropic the constitutive expression for the Cauchy stress takes the form T = 2με + λtr(ε)I, where μ and λ are constants. Interestingly, the expression for the Cauchy stress of the linearized elastic body can be inverted to obtain ε= where λ=

ν (1 + ν) T − tr(T)I, E E

E Eν , μ= , (1 + ν)(1 − 2ν) 2(1 + ν)

with E and ν the Young’s modulus and Poisson’s ratio, respectively. In this last expression the material constants E, ν have a clear physical meaning, in contrast with the previous expression for the stress in terms of the linearized strain, where the Lamé constant λ is determined in a circuitous fashion by using the values for the shear modulus and the bulk modulus. In the case of the functions f (F) and W (F), in general they cannot be inverted. A more general relation between stresses and strains, from which we can obtain as special cases the previous constitutive equations is the implicit relation [52] F (S, E) = 0.

(5)

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Let us assume that the stored energy of the body is expressed as U = U (S, E), then for a body to be elastic, we need U and F to satisfy Sαβ E˙ αβ = ρr



 ∂U ˙ ∂U ˙ Sαβ + E αβ , ∂ Sαβ ∂ E αβ

∂Fγ δ ˙ ∂Fγ δ ˙ Sαβ + E αβ = 0. ∂ Sαβ ∂ E αβ

(6)

In the special case, if we assume that S = p (E) and U = U (E), then from (6) it is possible to show that pαβ (E) = ρr ∂U/∂ E αβ , i.e. the Green elastic body is just a special case of (5). A more general class of implicit constitutive relations than (5) is that wherein S and E must be found from (see Eq. (3.2) in [52]) Aαβγ δ (S, E) S˙γ δ + Bαβγ δ (S, E) E˙ γ δ = 0,

(7)

subject to the restriction (6)1 . Yet another implicit constitutive relation is (see [45]) G (T, B, ρ) = 0,

(8)

from where we can obtain as a special case the Cauchy elastic bodies T = f (ρ, B), and the new subclass of elastic bodies4 B = g (ρ, T). In the case that G is an isotropic relation, (8) becomes (see [49, 60]) α0 I + α1 T + α2 B + α3 T2 + α4 B2 + α5 (TB + BT) + α6 (T2 B + BT2 ) +α7 (B2 T + TB2 ) + α8 (T2 B2 + B2 T2 ) = 0,

(9)

where the scalar functions αi , i = 0, 1, 2, . . . , 8 depend on the invariants [60] ρ, I1 = tr T, I2 = tr (T2 ), I3 = tr (T3 ), I4 = tr B, I5 = tr (B2 ), I6 = tr (B3 ), I7 = tr (TB), I8 = tr (T2 B), I9 = tr (B2 T), I10 = tr (T2 B2 ).

(10) (11)

In the particular case that αl = 0, l = 4, 5, 6, 7, 8 and αm , m = 0, 1, 2, 3 do not depend on the invariants In , n = 4, 5, 6, 7, 8, 9, 10, from (9) we have the subclass B = g (ρ, T) whose explicit expression is [45] B = α¯ 0 I + α¯ 1 T + α¯ 2 T2 ,

(12)

last subclass of elastic bodies B = g(ρ, T) satisfies (6) if one assumes that there exists an implicit constitutive relation H (S, C, ρ) = 0 and also U = U (S, C, ρ) satisfies (6), and that for the special case in which such functions can be written as H (S, C, ρ) = H˜ (J −1 SC) − C = 0 ˜ αβ /∂G γ δ − and U (S, C, ρ) = U (J −1 SC), defining G = J −1 SC, the condition (6) becomes Sαβ ∂ H ρr ∂U/∂G γ δ = 0. If we assume there exists a function Wˆ (G) such that H˜ (G) = ∂ Wˆ /∂G, then if H˜ satisfies Eq. (8) the body is elastic, and we can find g directly in terms of the derivatives of Wˆ (see [10]). 4 This

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where the functions α¯ j , j = 0, 1, 2, depend on the invariants I1 , I2 , I3 and ρ from (10). Dependence on ρ in virtue of the balance of mass implies that the material functions α¯ j depend on the determinant of the deformation gradient and hence the model (12) is still implicit. If we make the further assumption that the material functions do not depend on the density ρ, then (12) would be an explicit relationship for B in terms of the stress T. Another type of implicit constitutive relation can be derived directly from (6)1 in the following manner (see Eq. (3.3) of [52] and Chap. 6.2 of [26]). Let us rewrite (6)1 as ∂W ∂W dSαβ + dE αβ , Sαβ dE αβ = ∂ Sαβ ∂ E αβ where we have defined W = ρr U . Let us take the derivative of the above expression with respect to S to obtain in indicial notation and Cartesian coordinates (see [26]) Aγ δαβ dSαβ + Bγ δαβ dE αβ = 0,

(13)

where we have defined   ∂2W 1 ∂2W ∂2W ∂2W , (14) + + + Aγ δαβ (S, E) = 4 ∂ Sαβ ∂ Sγ δ ∂ Sαβ ∂ Sδγ ∂ Sβα ∂ Sγ δ ∂ Sβα ∂ Sδγ   ∂2W 1 ∂2W ∂2W ∂2W Bγ δαβ (S, E) = + + + 4 ∂ E αβ ∂ Sγ δ ∂ E αβ ∂ Sδγ ∂ E βα ∂ Sγ δ ∂ E βα ∂ Sδγ 1 (15) − (δαγ δβδ + δβγ δαδ ), 2 and where we have used the fact that S is a symmetric tensor. The above implicit relation (13) is of the form (7) and its parts have been defined explicitly in terms of W (S, E). If A has an inverse, from (13) we obtain the subclass of implicit relations (see [26, 52]) −1

dSεζ = − A εζ γ δ Bγ δαβ dE αβ ,

(16)

−1

where A εζ γ δ are the components of A −1 . We end this section with a discussion of a model similar to (12) proposed by Srinivasa [61], who used as the basic variables the Cauchy stress tensor, the tensor V from the polar decomposition F = VR, and the complementary or Gibbs energy function G. If the energy U only depends on the strain, it is possible to show that S = ρr ∂U/∂ E αβ can be written alternatively as J T = ∂ W/∂ ln V. Using the above results as the starting point, Srinivasa proposed the constitutive equation based on the complementary function G such that ln V =

∂G , ∂TK

(17)

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where TK = J T is the Kirchhoff stress tensor and ln V is the Hencky strain tensor. Equation (17) is actually an implicit relation, since on the right side G depends on the Cauchy stress and also J . For isotropic bodies, Srinivasa proposed two different sets of three invariants. In one of them, the invariants are defined as a0 , a1 and a3 , where  1 a0 = tr(TK ), a1 = tr(Tdev TTdev ), a2 = det N, 3 in which Tdev = TK − a0 I, and N = Tdev /a1 . From (17), when G is an isotropic function, we have the representation tr(ln V) = ln J =

∂G ∂G , dev (ln V) = , ∂a0 ∂Tdev

(18)

where dev (ln V) is the deviatoric part of the tensor ln V. In the case of incompressible bodies, Srinivasa showed that G would only depend on T since J = 1 always. Another set of invariants for G in the case of G being an isotropic function is σKi , i = 1, 2, 3, which are the eigenvalues of TK . In such a case, if vi are the eigenvalues of ln V we have the alternative representation ∂G 1  ∂G − , i = 1, 2, 3. ∂σKi 3 j=1 ∂σK j 3

vi =

3.2 Electro- and Magneto-Elastic Bodies Described by Implicit Constitutive Relations 3.2.1

Basic Equations for Electro- and Magneto-Elastic Bodies

Simplified Forms for the Maxwell Equations Implicit constitutive theories have been proposed for electro- and magneto-elastic bodies in [8, 11]. Let us define E , D and P as the electric field, electric displacement and polarization, respectively, and H , B and M as the magnetic field, magnetic induction and magnetization, respectively. For simplicity let us consider only problems where there is no coupling or interaction between the magnetic and electric fields, and there is no distributed charges and no time dependence. In such a case for electro-elastic and magneto-elastic bodies the simplified forms of the Maxwell equations are [38] E = 0, divD D = 0, curlH H = 0, divB B = 0, curlE

(19)

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where curl is the curl operator in the current configuration, and for condensed matter we have P = D − ε0E and for vacuum D = ε0E , where ε0 is the electric permittivity H + M ) and for for free space or vacuum, and for condensed matter we have B = μ0 (H vacuum B = μ0H , where μ0 is the magnetic permeability for free space or vacuum. Boundary or Continuity Conditions In the mathematical modelling of the behaviour of electro- and magneto-elastic bodies, it is necessary to model not only the bodies but also the surrounding space, which we assume is vacuum. In such a case for electro-elastic bodies the electric E]] = field and electric displacement satisfy the continuity conditions [38] n × [[E D]] = 0, and for magneto-elastic bodies n × [[H H]] = 0, n · [[B B]] = 0, where 0, n · [[D [[a]] = a(o) − a(i) is the difference of a quantity between the outside and inside of a body at the boundary and n is the outer normal vector to the body in the current configuration. Equations of Equilibrium When we assume x¨ = 0 the equations of equilibrium for electro- and magneto-elastic bodies, when P is considered as the independent electric variable, and M is chosen as the independent magnetic variable, respectively, take the forms B)TM + ρb = 0, E)TP + ρb = 0, div T + μ−1 divT + (gradE 0 (gradB

(20)

where in both cases b represents the non-electric and non-magnetic body forces and B)TM can be E)TP and μ−1 T is in general non-symmetric. The terms (gradE 0 (gradB incorporated into the divT term, on the basis of which we can define a quantity called the total stress tensor τ , and (20) becomes [24, 25] div τ + ρb = 0.

(21)

When we use the total stress tensor, the mechanical boundary conditions are τ n = tˆ + τ m n, where τ m is the Maxwell stress tensor, which for electro-elastic problems D(o) · E (o) )I, and for magneto-elastic problems takes the form τ m = D (o) ⊗ E (o) − 21 (D B(o) · H (o) )I, and in both cases the Maxwell stress is given as τ m = B (o) ⊗ H (o) − 21 (B tˆ represents the external non-electric and non-magnetic traction.

3.2.2

Constitutive Equations and Relations

The classical approach used to model the behaviour of electro- and magneto-elastic bodies has been to assume that the stresses and one of the electric or magnetic variables are expressed as functions of the strain (or deformation gradient) and the independent electric or magnetic variable. For example, the classical linearized electro-elastic model is based on considering P as the electric independent variable, and the constitutive equations are of the form (in indicial notation and Cartesian coordinates) τi j = Ci jkl εkl + Ki jk Pk , Ei = Ki jk ε jk + Di j P j , where the elec-

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tric displacement would be found from P = D − ε0E . In the nonlinear case, for electro-elastic bodies the usual constitutive assumptions are of the form5 τ = f (F, P ), E = q (F, P ), where q is a vector function. For nonlinear magneto-elastic bodies, assuming the magnetization as the independent magnetic variable, we have6 τ = f (F, M ), H = q (F, M ). For example, for nonlinear electro-elastic bodies Dorfmann and Ogden [25] have proposed the constitutive equations τ = J −1 F∂Ω/∂F, EL , where Ω = Ω(F, E L ) is called the total energy function, the elecD L = −∂Ω/∂E tric field has been chosen as the independent electric variable and D L = J F−1D , E L = FT E . In view of what has been presented in Sect. 3.1, a natural extension of the above constitutive equations is to assume implicit relations that for electro-elastic bodies would be of the form7 [8] G (τ , B, E , D ) = 0, l (τ , B, E , D ) = 0,

(22)

where G is a tensorial implicit relation, and l would be a vector implicit relation. They are generalizations of the functions f , q mentioned previously. In (22) we have chosen E , D as the electric variables for the problem and from P = D − ε0E we can obtain P . In the case of magneto-elastic bodies we would have G (τ , B, H , B ) = 0, l (τ , B, H , B ) = 0, H + M ). where M can be found from B = μ0 (H For electro-elastic bodies, in the case G and l are isotropic relations (22) yields8 E + ψ3 τ 2E + ψ4 BE E + ψ5 B2E + ψ6D + ψ7 τD D + ψ8 τ 2D + ψ9 BD D l = ψ1E + ψ2 τE 2 2 2 E + ψ12 (τ B + Bτ )D D + ψ13 (τ B + Bτ )E E + ψ10 B D + ψ11 (τ B + Bτ )E D + ψ15 (τ B2 + B2 τ )E E + ψ16 (τ B2 + B2 τ )D D + ψ14 (τ 2 B + Bτ 2 )D E + ψ18 (τ 2 B2 + B2 τ 2 )D D = 0, + ψ17 (τ 2 B2 + B2 τ 2 )E (23)

5 See, for example, Sect. 3(a) in [8] for a brief survey of the constitutive equations for electro-elastic

bodies. Sect. 3 of [11] for a short review of the constitutive equations in magnetoelasticity. 7 We could have both the implicit relations include the density, but by conservation of mass the density can be replaced in terms of the deformation gradient (or B) and the reference density ρr . 8 The expressions for G (τ , B, E , D ) and l (τ , B, E , D ) represented in (23), (24) do not appear explicitly in the works of Spencer [60] or Zheng [64]. If one assumes that G and l can be obtained from some scalar potential, let us say Π = Π (τ , B, E , D ), Ξ = Ξ (τ , B, E , D ), respectively, as E + ∂Ξ/∂D D, where we have used dimensionless stresses, elecG = ∂Π/∂τ + ∂Π/∂B, l = ∂Ξ/∂E tric fields and electric displacement, these scalar potentials would depend on the list of invariants (132)–(146) given in the Appendix. In such a case it is possible to recover some, but not all, of the different terms that appear in (23) and (24). 6 See

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and9 G = γ0 I + γ1 τ + γ2 B + γ3 τ 2 + γ4 B2 + γ5 (τ B + Bτ ) + γ6 (τ 2 B + Bτ 2 ) E ⊗ (τE E) + (τE E) ⊗ E ] + γ7 (τ B2 + B2 τ ) + γ8 (τ 2 B2 + B2 τ 2 ) + γ9E ⊗ E + γ10 [E E ⊗ (BE E) + (BE E) ⊗ E ] + γ12 [E E ⊗ (τ 2E ) + (τ 2E ) ⊗ E ] + γ11 [E E ⊗ (B2E ) + (B2E ) ⊗ E ] + γ14 [E E ⊗ (τ BE E) + (τ BE E) ⊗ E ] + γ13 [E E ⊗ (BτE E) + (BτE E) ⊗ E ] + γ16 [(τE E) ⊗ (BE E) + (BE E) ⊗ (τE E)] + γ15 [E D ⊗ (τD D) + (τD D) ⊗ D ] + γ19 [D D ⊗ (BD D) + (BD D) ⊗ D ] + γ17D ⊗ D + γ18 [D D ⊗ (τ 2D ) + (τ 2D ) ⊗ D ] + γ21 [D D ⊗ (B2D ) + (B2D ) ⊗ D ] + γ20 [D D ⊗ (τ BD D) + (τ BD D) ⊗ D ] + γ23 [D D ⊗ (BτD D) + (BτD D) ⊗ D ] + γ22 [D D) ⊗ (BD D) + (BD D) ⊗ (τD D)] + γ25 (E E · D )(D D ⊗ E + E ⊗ D) + γ24 [(τD E · D )[D D ⊗ (τE E) + (τE E) ⊗ D + (τD D) ⊗ E + E ⊗ (τD D)] + γ26 (E E · D )[D D ⊗ (BE E) + (BE E) ⊗ D + (BD D) ⊗ E + E ⊗ (BD D)] + γ27 (E E · D )[D D ⊗ (τ 2E ) + (τ 2E ) ⊗ D + (τ 2D ) ⊗ E + E ⊗ (τ 2D )] + γ28 (E E · D )[D D ⊗ (B2E ) + (B2E ) ⊗ D + (B2D ) ⊗ E + E ⊗ (B2D )] + γ29 (E E · D )[E E ⊗ (τ BD D) + (τ BD D) ⊗ E + D ⊗ (τ BE E) + (τ BE E) ⊗ D ] + γ30 (E E · D )[E E ⊗ (BτD D) + (BτD D) ⊗ E + D ⊗ (BτE E) + (BτE E) ⊗ D ] + γ31 (E E · D )[(τE E) ⊗ (BD D) + (BD D) ⊗ (τE E) + (τD D) ⊗ (BE E) + (BE E) ⊗ (τD D)] + γ32 (E = 0,

(24)

where the scalar functions γi , ψ j , i = 0, 1, . . . , 32, j = 1, 2, . . . , 18 depend on the invariants given in Eqs. (132)–(146) of the Appendix. It is of course impossible to devise an experimental program within which to determine the various material functions that depend on as many as 35 invariants and the density (see (132)–(146)), uniquely. The point is that one needs to simplify the constitutive relation drastically but yet maintain the ability to describe experimental observations. This is no mean task and we shall not try to develop such simplified models here.

3.3 Thermo-Elastic Bodies In this section we show how to develop implicit constitutive equations for the case of thermo-elastic bodies [17]. Let us consider as basic variables the second Piola–

(23) we have omitted terms of the form [A(a × b) + (Ab) × a], which are suggested by Zheng (Table 2 of [64]) for a vector function that depends on a tensor field A in addition to two vectors fields a, b.

9 In

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Kirchhoff stress tensor, the Lagrange–Saint Venant strain tensor and the absolute temperature θ . We propose the following implicit constitutive relation for thermoelastic bodies: F (S, E, θ ) = 0. (25) Let us study now what restrictions such an implicit relation must satisfy for the body to be thermo-elastic. The first law of thermodynamics is the statement ρr U˙ = ω + Div hr + ρr r,

(26)

˙ Div is the divergence operator with respect to where we have defined ω = tr(SE), the reference configuration, hr is the heat flux and r is the internal source of heat per unit of mass, both in the reference configuration. If we define the dissipation d as d = θ η˙ − U˙ + ω/ρr , where η is the entropy, and γ = θ ∇X (1/θ ), the dissipation must satisfy the inequality d ≥ 0, while hr must satisfy the so-called Fourier inequality −hr · γ ≥ 0. Both inequalities together lead to the Clausius–Duhem inequality10   ω − hr · γ ≥ 0. ρr θ η˙ − U˙ + ρr

(27)

Now, if the Helmholtz potential ψ is defined as ψ = U − θ η, using this and (26), it is possible to write (27) in terms of ψ. For implicit thermo-elastic bodies we assume now that ψ = ψ(S, E, θ ) and (27) becomes (in index notation and Cartesian coordinates) −

∂ψ ˙ ∂ψ ˙ 1 Sαβ − E αβ + Sαβ E˙ αβ − h rα γα ≥ 0, ∂ Sαβ ∂ E αβ ρr

(28)

and taking the derivative of (25) in time we obtain ∂Fγ ζ ˙ ∂Fγ ζ ˙ ∂Fγ ζ θ˙ = 0. Sαβ + E αβ + ∂ Sαβ ∂ E αβ ∂θ

(29)

˙ E˙ and For a body to be thermo-elastic F and ψ must satisfy (28) and (29) for any S, θ˙ . There is an additional relation that is necessary, which is a generalization of Fourier’s model for heat transfer (see [17]), namely, m (S, E, hr , h˙ r , ∇X θ, θ ) = 0,

(30)

where m is a vector implicit relation. 10 We shall not use the Clausius–Duhem inequality to obtain necessary and sufficient conditions under which the constitutive relations will satisfy the same.

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In the case of F being an isotropic relation, its representation would be exactly as (9), interchanging T by S, and B by E (the same for the invariants defined in (10), (11)), adding to the list of invariants the temperature θ . In the case of the vector implicit relation (30), if m is isotropic, it becomes β1 hr + β2 Shr + β3 S2 hr + β4 Ehr + β5 E2 hr + β6 h˙ r + β7 Sh˙ r + β8 S2 h˙ r + β9 Eh˙ r + β10 E2 h˙ r + β11 ∇X θ + β12 S∇X θ + β13 S2 ∇X θ + β14 E∇X θ + β15 E2 ∇X θ + β16 (SE + ES)hr + β17 (S2 E + ES2 )hr + β18 (SE2 + E2 S)hr + β19 (S2 E2 + E2 S2 )hr + β20 (SE + ES)h˙ r + β21 (S2 E + ES2 )h˙ r + β22 (SE2 + E2 S)h˙ r + β23 (S2 E2 + E2 S2 )h˙ r + β24 (SE + ES)∇X θ + β25 (S2 E + ES2 )∇X θ + β26 (SE2 + E2 S)∇X θ + β27 (S2 E2 + E2 S2 )∇X θ = 0, where the scalar functions βi , i = 1, 2, . . . , 27, depend on the invariants presented in (147)–(180) in the Appendix.

4 The Linearized Strain as a Nonlinear Function of the Stress 4.1 Purely Elastic Deformations In this section we study the case of (9) when the gradient of the displacement field is very small, i.e. when |∇X u| ∼ O(δ), δ  1 (see [5, 6, 46, 48]). In such a situation we have B ≈ I + 2ε, replacing this in (9), and appealing to the approximation αi = αi (T, B) ≈ αi (T, ε) ≈ αi (T, 0) +

∂αi (T, 0) · ε, i = 0, 1, 2, . . . , 8, ∂ε

using the notation αi(0) = αi (T, 0) and ζ i = ∂αi /∂ε(T, 0) and neglecting terms of order δ 2 or higher, (9) becomes (see, for example, [43])  1 (T) + 2 (T)ε + [ 3 (T) · ε]T + [ 4 (T) · ε]T2 + 5 (T)(Tε + εT) +6 (T)(T2 ε + εT2 ) = 0, (31) where  i (T), i = 1, 3, 4, and  j (T), j = 2, 5, 6, are vector and scalar functions, respectively, which only depend on the stress tensor and are defined in terms of αk(0) and ζ k . We notice from Eq. (31) that the strains always appear as linear expressions, whereas the stresses can be arbitrarily large and can appear in a nonlinear manner. Equation (31) can be rewritten as (in indicial notation and Cartesian coordinates) Ai jkl (T)εkl = Bi j (T).

(32)

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−1

If the fourth-order tensor function A has an inverse then εi j = A i jkl (T)Bkl (T), −1

where A i jkl are the i jkl components of A−1 . From (32) we obtain ε = h(T).

(33)

This subclass of constitutive equations is very useful in describing metallic alloys and even materials like concrete that have been in use for a very long time, and its applications are studied in more detail in Sect. 5.2 (see [46, 48]). Another way to obtain (33) is to consider the subclass of constitutive equation (35): B = α¯ 0 I + α¯ 1 T + α¯ 2 T2 , where the material functions α¯ i are only functions of the invariants of the stress (and do not depend on the density) and further, assuming that |∇X u| ∼ O(δ), δ  1, we once again obtain (33).

4.1.1

Isotropic Bodies

In the case h (T) is an isotropic function we have the representation [6] ε = ϑ0 I + ϑ1 T + ϑ2 T2 ,

(34)

where the scalar functions ϑi , i = 0, 1, 2, depend on the invariants I1 = trT, I2 = tr(T2 )/2 and I3 = tr(T3 )/3. Let us consider a subclass of (34), wherein we assume that there exists a scalar function Π = Π (T) such that [5] h (T) = ∂Π/∂T. Restrictions on Π such that the body is actually elastic, in the sense there is no dissipation, are studied in [10]. In the case Π (T) is an isotropic function Π (T) = Π (I1 , I2 , I3 ), where these invariants have been defined above, we obtain for ε the expression ε = Π1 I + Π2 T + Π3 T2 ,

(35)

where Πi = ∂Π/∂ Ii , i = 1, 2, 3. If instead of using the classical invariants Ii , i = 1, 2, 3, we use the principal stresses σi , i = 1, 2, 3, of T, then for Π (T) isotropic we have the alternative representation (see, for example, [18]) Π (T) = Π (σ1 , σ2 , σ3 ) that must satisfy the symmetry conditions Π (σ1 , σ2 , σ3 ) = Π (σ2 , σ1 , σ3 ) = Π (σ1 , σ3 , σ2 ), and for ε we have the expression [18] ε=

3  ∂Π i=1

∂σi

t(i) ⊗ t(i) ,

(36)

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where t(i) , i = 1, 2, 3, are the eigenvectors of T. Finally, if εi , i = 1, 2, 3, are the principal strains of ε it is possible to show that εi =

4.1.2

∂Π , i = 1, 2, 3. ∂σi

(37)

Transversely Isotropic Bodies

The case of a transversely isotropic function h (T) has been briefly considered in [12] within the context of analysing inextensible bodies, and currently is being considered for the case of transversely isotropic rocks. If one uses the classical results of Rivlin and Spencer (see, for example, [60]), in the case h (T) is a transversely isotropic function and if we assume the existence of the function Π = Π (T, a) where h (T) = ∂Π/∂T, and a is the preferred direction with respect to which the body is transversely isotropic, we have that Π (T, a) = Π (I1 , I2 , I3 , I4 , I5 ), where Ii , i = 1, 2, 3, have been defined in Sect. 4.1.1, and I4 = a · (Ta), I5 = a · (T2 a). It then follows that ε = Π1 I + Π2 T + Π3 T2 + Π4 a ⊗ a + Π5 [a ⊗ (Ta) + (Ta) ⊗ a],

(38)

where Πi = ∂Π/∂ Ii , i = 1, 2, 3, 4, 5. Shariff [56] has proposed different sets of invariants for anisotropic functions of tensors and vectors. In the case of a function of the form Π (T, a) he suggested the use of the spectral invariants σi , ζi , i = 1, 2, 3, where σi are the principal stresses of T and ζi = [a · t(i) ]2 , where t(i) , i = 1, 2, 3, are the eigenvectors of T. One of the invariants ζi , i = 1, 2, 3 is not independent since ζ1 + ζ2 + ζ3 = 1, so for the set σi , ζi , i = 1, 2, 3, we would have five independent invariants. We have Π (T) = Π (σ1 , σ2 , σ3 , ζ1 , ζ2 , ζ3 ), which must satisfy the symmetry conditions Π (σ1 , σ2 , σ3 , ζ1 , ζ2 , ζ3 ) = Π (σ2 , σ1 , σ3 , ζ2 , ζ1 , ζ3 ) = Π (σ1 , σ3 , σ2 , ζ1 , ζ3 , ζ2 ), and which must be independent of ζi if the principal values of T are all the same. For the components of the strain tensor we have (see [56]) ∂Π , i = 1, 2, 3, ∂σi   ∂Π ∂Π t(i) · (At( j) ) , i = j, i, j = 1, 2, − εi j = ∂ζi ∂ζ j (σi − σ j ) εii =

εk3 =

∂Π t(k) · (At(3) ) , k = 1, 2, ∂ζk (σk − σ3 )

(39) (40) (41)

where we have defined A = a ⊗ a, and where εi j = t(i) · (εt( j) ), and in all the above expressions there is no sum with respect to the repeated index. In the case of (40)

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and (41) we assume that as σi → σ j and σk → σ3 the function Π is regular enough such that the limits exist.

4.2 Electro-Elastic Bodies In the case of electro-elastic bodies, from (23) and (24), if we assume that |∇X u| ∼ O(δ), δ  1, following a procedure similar to that outlined in Sect. 4.1, it is possible to obtain the subclass [8] ε = h (τ , E , D ), l (τ , E , D ) = 0,

(42)

where (42)2 is an implicit vector relation that does not depend on the strains. If Do is used to denote some characteristic value for the electric displacement, for example, the magnitude of the electric displacement near the electric saturation D|/Do ∼ O(δ), δ  1, from point for some materials, then assuming further that |D (42)2 we obtain D = r (τ , E ), (43) and as a result ε = h (τ , E ). D|/Do ∼ O(δ), where Let us study in more detail the cases |∇X u| ∼ O(δ) and |D δ  1, when the functions h and r are transversely isotropic, and where a is a preferred direction for the body, i.e. ε = h (τ , E , a), D = r (τ , E , a). This case is relevant to many technical problems since many electro-elastic materials present a direction where the behaviour is different. In this case we have the representations (in terms of the classical invariants [60]) E ⊗ (τE E) + (τE E) ⊗ E ] ε = γ¯0 I + γ¯1 τ + γ¯2 τ 2 + γ¯3E ⊗ E + γ¯4 [E 2 2 E ⊗ (τ E ) + (τ E ) ⊗ E ] + γ¯6 a ⊗ a + γ¯7 [a ⊗ (τ a) + (τ a) ⊗ a] +γ¯5 [E E · a)(a ⊗ E + E ⊗ a) +γ¯8 [a ⊗ (τ 2 a) + (τ 2 a) ⊗ a] + γ¯9 (E E) + (τE E) ⊗ a + (τ a) ⊗ E + E ⊗ (τ a)] +γ¯10 [a ⊗ (τE +γ¯11 [a ⊗ (τ 2E ) + (τ 2E ) ⊗ a + (τ 2 a) ⊗ E + E ⊗ (τ 2 a)], and

E + ψ¯ 3 τ 2E + ψ¯ 4 a + ψ¯ 5 τ a + ψ¯ 6 τ 2 a, D = ψ¯ 1E + ψ¯ 2 τE

(44)

(45)

where the scalar functions γ¯i , i = 0, 1, 2, . . . , 11, and ψ¯ j , j = 1, 2, . . . , 6, depend on the invariants obtained from τ , E and a, which are not documented here but the interested reader can find in [8].

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4.3 Thermo-Elastic Bodies Here we repeat the analysis of the two previous sections in the case of thermo-elastic bodies. On assuming that |∇X u| ∼ O(δ), δ  1, we have E ≈ ε, S ≈ T, hr ≈ h, h˙ r ≈ h˙ and ∇X θ ≈ ∇θ , where h is the heat flux in the current configuration and ∇ is the gradient operator with respect to the current configuration. From (25) and (30) we obtain (see [17]) ˙ ∇θ, θ ) = 0. ε = h (T, θ ), m (T, h, h,

(46)

In the case h is an isotropic function the expression for (46)1 is the same as that in (34) and (35), but in this case the functions ϑi or Π would also depend on θ . In the case (46)2 being isotropic we have the representation β˘1 h + β˘2 Th + β˘3 T2 h + β˘4 h˙ + β˘5 Th˙ + β˘6 T2 h˙ + β˘7 ∇θ + β˘8 T∇θ + β˘9 T2 ∇θ = 0,

(47)

where the scalar functions β˘i , i = 1, 2, . . . , 9, depend on some of the invariants defined in (147)–(180). The following subclass of (46) is also interesting. Let us ˙ respectively, and let us assume define h and h d as characteristic values for h and h, ˙ that |h|/ h ∼ O(δ) and |h|/ h d ∼ O(δ), where δ  1, it is possible to show that from (47) we obtain h + α(T, θ )h˙ = −qq(T, ∇θ, θ ), (48) where the scalar function α may also depend on ∇θ . The above subclass of (46) is a generalization of models for heat transfer, which has been proposed in [34], and which does not present some of the problems that the classical Fourier model of heat transfer does, in particular the infinite speed of propagation for heat. The classical Fourier model is obtained from (48) in the case α = 0 and q = q (∇θ ) = k∇θ , where k is a constant.

5 Applications 5.1 Implicit Constitutive Relations In the case of the implicit relations of the form (5) and (7), in [26, 27] it has been shown that they can be used to model biological fibres and tissues. For example, in [27], for a 1D problem, where ε and σ denote the 1D strain and stress, respectively, the first law of thermodynamics can be written as (recalling that W = ρr U ) dW = −ηdθ + σ dε, and assuming that U = U (ε, σ, θ ) one can show that

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∂W ∂W ∂W , σ dε = dε + dσ. ∂θ ∂ε ∂σ

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(49)

For biological fibres the expression     θ W = −C θ ln − (θ − θ0 ) + E[ε − α(θ − θ0 )] − σ θ0 + βσ [ε − α(θ − θ0 )]

(50)

for W was proposed, where C, θ0 , E, α and β are constants. In [41] a similar model was proposed for applications to the modelling of biomaterials, in this case starting with an implicit relation of the form (5), rewritten in terms of the first Piola–Kirchhoff stress tensor P and the deformation gradient as F (P, F) = 0. For the particular case when F is isotropic, the following expression written in terms of the principal directions of F takes the form 

Φ Pi − λi

       Ψ 1 Φ 1 1 2 2 m λi − 2q Pi − λi − 2q − λi − n = 0, J λi J λi λi

(51)

where there is no sum over the repeated index, Pi , i = 1, 2, 3, are the principal values of P, λi , i = 1, 2, 3, are the principal stretches, J = det F = λ1 λ2 λ3 , and Φ, q, Ψ , m and n are constants.

5.2 Modelling in Fracture Mechanics: Applications to the Modelling of Metallic Alloys, Concrete and Rocks In this section, we summarize some expressions for constitutive equations obtained from the implicit relations discussed in Sect. 4.1, for the special case |∇X u| ∼ O(δ), δ  1, i.e. when we restrict ourselves to the linearized strain tensor ε and the Cauchy stress tensor T. We discuss some applications within the context of the models given by (33) and (42). The case of the constitutive equation (33): ε = h (T) has been used to study problems such as the concentration of stresses, but where the strains must remain small. The problem of the fracture of brittle materials even within the small strain regime and the nonlinear response of some metallic alloys, rocks and concrete, can be gainfully described within the context of the above class of constitutive relations. With regard to problems concerning fracture, Rajagopal and Walton [55] have studied the problem of the state of strain and stress in the problem of anti-plane stress, in a body containing a crack, within the context of strain-limiting theories that are discussed in the next section. Gou et al. [29] have studied the state of the stress and strain in a body containing a crack in the case of plane stress problems.

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Modelling Strain-Limiting Behaviour

In this subsection, we discuss constitutive relations that reflect the fact that the strains in the body are limited by a value decided a priori, irrespective of the value of the stress. Of course, in reality, once the material reaches the limiting strain, the strain might remain constant for a slight increase in the stress, but the body will ultimately get damaged and cease to behave in an elastic fashion. However, strain-limiting models have important applications, provided one uses such models with care. Such models are particularly useful in describing the response of brittle elastic bodies which respond in an elastic manner until a limiting strain is reached, and fail after that. Such models seem to have diverse applications, and seem to be useful for describing the response of biological fibres and material such as DNA and at the same time materials such as rocks. We discuss a few studies in the literature where strain-limiting behaviour for the linearized strain is considered. This discussion is by no means exhaustive and even with regard to the cases discussed we merely state the model used; we do not get into a detailed discussion of these studies. For example, in [43] Ortiz et al. proposed an expression for Π = Π (T) (see (35)), namely,   1 αγ  1 + 2ιI2 , Π (T) = −α I1 − ln(1 + β I1 ) + β ι and from (35), since ε = h (T) = ∂Π/∂T, we obtain that  ε = −α 1 −

 αγ 1 I+ √ T, (1 + β I1 ) 1 + 2ιI2

(52)

where values of the constants that appear in the expression (52) are given by α = 10−9 , β = 10−3

1 1 1 , γ = 10 , ι = 10−11 . [Pa] [Pa] [Pa2 ]

(53)

In [43] using the above constitutive equation some problems involving stress concentrations such as a plane plate with an elliptic hole and a stepped flat bar with shoulder fillets were studied. A modification of (52) was considered in [42], where it was assumed that Π (T) = − which leads to

γ α ln[cosh(β I1 )] + 1 + 2ιI2 , β ι

γ ε = −α tanh(β I1 )I + √ T. 1 + 2ιI2

(54)

This was used to study the problem of a plane plate with an elliptic hole, a plane plate with a hyperbolic boundary and the behaviour of a semi-infinite medium with

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a concentrated force at a point. Values for the constants that appear in (54) are given by 1 1 1 , γ = 4.02 × 10−9 , ι = 10−14 . [Pa] [Pa] [Pa2 ] (55) In [46, 47] Rajagopal proposed the expression (56) below in order to study strainlimiting behaviour, primarily to ascertain the qualitative response of bodies described by such models than with a view to correlating against experimentally observed facts. He used the constitutive relation ⎧ ⎡ ⎤⎫ ⎨ ⎬ −λtrT β ⎦  T, (56) ε = α 1 − exp ⎣  I+   1/2 ⎩ ⎭ [1 + γ tr(T2 )]1/2 1 + tr(T2 ) α = 0.01, β = 9.277 × 10−8

where α, β, γ and λ are constants. We notice that the above model reduces to the linearized elastic model when we require the model to be linear in the stress, or put differently, for small strains and a linear stress the model reduces to a linearized elastic model, but for large stresses a body described by the model behaves nonlinearly and exhibits limiting strain. In [2], Bridges and Rajagopal, based on the specific choice of a Gibbs potential develop a nonlinear elastic model which they then linearized with respect to the displacement gradient to obtain a constitutive relation that predicts a limiting value for the linearized strain. Their constitutive relation takes the form   μ βtr T I+  T , (57) ε=α − (1 + βtr T) 1 + γ 2 tr(T2 ) where α, β, γ and μ are constants. Using the above model Bridges and Rajagopal [2] studied the problem of stress concentration in a body described by such a constitutive relation. Though they were primarily interested in studying the state of stress and strain in an annular region, by allowing the outer radius to become infinite they can study the problem of a hole in an infinite body. In keeping with the study of Ortiz et al. [43], they find that the strain grows much more slowly than the stress and in fact the strain never exceeds the limiting value that is set a priori. As with the previous model, the above model also reduces for small strains and the requirement that the model be linear in the stress to the linearized elastic model and it exhibits strain-limiting behaviour. Finally, in [3] Buli˘cek et al. proposed the expression ε = α0 (trT, tr (T2 ))I +

1 T, μ0 (1 + |T|r )1/r

(58)

for h (T) for a theoretical analysis of strain-limiting behaviour, where μ0 is a constant and α0 = α0 (trT, tr (T2 )) is a function. The aim of this study was a purely

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mathematical analysis, namely, the determination of existence of solutions to the partial differential equations that arise from the balance laws by assuming such a constitutive relation.

5.2.2

Fracture Mechanics of Brittle Bodies

In the previous section we discussed some models that have been used to explore stress concentration in a general manner, and in this section we discuss constitutive relations that have been used specifically to study the behaviour of bodies with cracks. In [4] the following model was used to investigate the behaviour of the strains near the tip of a crack (in an anti-plane stress problem) ε = g1 (trT, tr (T2 ))I + g2 (tr(T2 ))T,

(59)

where g1 = g1 (trT, tr (T2 )) and g2 = g2 (tr(T2 )) are scalar functions of the stresses, g1 (0, ·) = 0 and g2 (tr(T2 )) = g2 (|T|2 ) = 1/(1 + |T|a )1/a , a > 0. Similar analyses were carried out in [55] for the problem of a crack in a body under anti-plane shear, considering the simpler expression ε = φ(β|T|)T,

(60)

where φ(r ) = 1/(1 + βr ), β > 0 is a constant and T is the non-dimensional stress. Finally, in [39] we see some numerical results for the problem of a crack in a body under anti-plane shear, considering    1 −λtrT I+ T, ε = β 1 − exp 1 + |T| 2μ (1 + κ|T|a )1/a

(61)

where β, λ, κ, μ and a are constants. All the models presented in this and the previous sections have not been corroborated against actual experimental data, but rather they have been proposed on the basis of their possessing the mathematical property that independently of the magnitude of the stresses, the strains remain small. In the following three sections we consider some examples of h (T) that have been corroborated against actual experimental data, for Gum metal and other metallic alloys, concrete and rock.

5.2.3

Modelling of Gum Metal and Other Metallic Alloys

In this section we shall document some of the models that have been used wherein either there is an explicit expression for the linearized strain in terms of the stress, or an implicit relationship between the strain and the stress to describe the response of metallic alloys in the small strain regime. Gum metal, an alloy developed in a laboratory belonging to Toyota, and a variety of Titanium, Tantalum and Zirconium

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based metallic alloys that exhibit a nonlinear relationship between the strain and the stress even in the strain regime which would be considered small enough for the nonlinear terms in the strain to be neglected. Such response, as we observed earlier, cannot be described by the classical linearized elastic solid or for that matter any Cauchy elastic model. On the other hand, we can describe such materials within the context of the nonlinear response between the linearized strain and the stress that comes out of the implicit theory of elasticity. It was Rajagopal [48] who first recognized that one could obtain nonlinear relationships between the linearized strain and the stress within the construct of implicit theories for elastic bodies, and he proposed models of the form (33) for such metallic alloys. He introduced the following models with a view of describing the response of such bodies   n ε = λ1 tr(T)I + 2λ2 exp[n tr (T)]T, ε = λ1 tr(T)I + λ2 1 + α tr T2 T, (62) where λ1 , λ2 and n, which in general are different for the two models, and α are constants. In [40] Kulvait et al. within the context of a Gibbs potential G (see (17)), proposed a constitutive equation of the form ε=−

1 9 Kˆ (|T|2 )

(trT)I +

1 Tdev , 2μ(|T ˆ dev |2 )

(63)

where11  Kˆ (|T| ) = K 0 2

τ2  2 0  τ0 + |T|2



(s−2)/2 , μ(|T ˆ dev | ) = μ0 2

τ02  2  τ0 + 3|Tdev |2 /2

(q−2)/2

and K 0 > 0, τ0 > 0, 1 < s < ∞, μ0 > 0 and 1 < q < ∞ are constants, and Tdev = T − (trT)I/3. The constant τ0 can be considered as a characteristic value for the stress. Numerical values for the constants are given by τ0 = 0.5[GPa], q = 2.23, s = 7.65,

K 0 = 6226[GPa], μ0 = 20.2[GPa]. (64) In [22, 23] Devendiran et al. proposed the following two alternative expressions for a constitutive relation of the form (32) to model the behaviour of titanium alloys    n/2  ε − αˆ 1 {[trT − 2tr(Tε)] I + 2tr(T)ε} − αˆ 2 + αˆ 3 exp 1 + α4 tr T2        (n/2−1)  × 1 + nα4 1 + α4 tr T2 (trε)tr T2 − 2tr εT2 T = 0, (65)

11 For

a tensor A we have |A| =

 tr (AAT ).

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where αˆ 1 , αˆ 2 , αˆ 3 , α4 and n are constants given in (67), (68) below for the particular case of the alloy TNTZ30. In [22] Devindiran et al. also proposed the alternative constitutive equation for the same material       n 2 T, 1 + β4 tr T ε = β1 tr(T)I + β2 + β3 exp

(66)

where the constants β1 , β2 , β3 , β4 and n are given in (69), (70). In the models (64) and (65) the material constants are given by 1 1 , αˆ 2 = 2.0906 × 10−5 , [MPa] [MPa] 1 1 αˆ 3 = 2.685 × 10−7 , α4 = 1.242 × 10−3 , n = 1, [MPa] [MPa]2 1 1 β1 = −5.973 × 10−6 , β2 = 2.0906 × 10−5 , [MPa] [MPa] 1 1 , β4 = 3.736 × 10−5 , n = 1. β3 = 2.685 × 10−7 [MPa] [MPa]2

αˆ 1 = −5.973 × 10−6

(67) (68) (69) (70)

Notice that if one ignores the terms in which both T and ε appear in a bilinear fashion in (65), then one obtains Eq. (66). Also, we note that while the constitutive relation (65) is truly a mathematical relation in that it is an implicit expression for the stress and the strain, Eq. (66) is an explicit expression for the linearized strain in terms of the stress.

5.2.4

Application to the Modelling of Rock

The paper by Bustamante and Rajagopal [18] is devoted to the application of a subclass of (33) to the modelling the elastic behaviour of rock. Two simplifications were required, to assume that for a range of external loads rock is approximately elastic, i.e. that there was no conversion of mechanical work into heat and there is no fracture, and that rock is isotropic. From the experimental information provided, for example, in [35] it is possible to see that the first assumption is approximately satisfied; regarding the second assumption, there are several types of rocks that do not show any preferred directional response in their mechanical behaviour. The model (36) was of the form Π (σ1 , σ2 , σ3 ) = f1 (σ1 ) + f1 (σ2 ) + f1 (σ3 ) + f2 (σ1 )(σ2 + σ3 ) + f2 (σ2 )(σ1 + σ3 )   σ1 + σ2 + σ3 , (71) + f2 (σ3 )(σ1 + σ2 ) + f3 3

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where     f1 (x) = α1 d1c1 x − c1 ln(d1 )x , f2 (x) = α2 d2c2 x − 1 ,   f3 (x) = 3α3 d3c3 x − c3 ln(d3 )x , and the constants αi , ci , di , i = 1, 2, 3, are given by α1 = 0.011[MPa], α2 = −0.0004, α3 = 0.001[MPa], 1 1 1 c1 = −0.08 , c2 = −0.05 , c3 = −0.08 , [MPa] [MPa] [MPa] d1 = 0.1, d2 = 0.2, d3 = 0.3.

5.2.5

(72) (73) (74)

Modelling of Concrete

In this section, we document an expression belonging to a subclass of (33) which has been used successfully in modelling the behaviour of concrete. Grasley et al. [30] proposed the constitutive equation12 ε = β0 I + β1 T + β2 T2 ,

(75)

  where β0 = γ2 tr(T) + sinh tr(T)γ2 /γ3 , β1 = γ4 and β2 = 0, where the constants γi , i = 1, 2, 3, 4, are given by γ1 = 2.3398

5.2.6

1 , γ2 = 1.3105, γ3 = 37.52[MPa]γ2 , γ4 = 19.8398. (76) [MPa]

Simple Boundary-Value Problems Within the Context of the Models Discussed in the Previous Subsection

We are interested in presenting the behaviour of a cylinder under uniform tension and compression and a slab under shear, within the context of the models (52), (54), (63), (66), (71) and (75). In the case of a cylinder under tension and compression, we assume that the body deforms due to the application of a stress of the form13 T = σ ez ⊗ ez , where σ does not depend on the position. In such a case, for the quasistatic problem, assuming no body forces, Eq. (4)1 is satisfied automatically. From (33) and (2)2 we can assume that the displacement field produced by such a stress

12 In

(75) the strains must be divided by a factor 106 . Sect. 6.1.1 for details about the boundary-value problems considering these new classes of constitutive theories. 13 See

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R. Bustamante and K. Rajagopal 0.01

1.04

0.008

1.03

0.006 1.02

0.004 0.002

1.01

λ

c

1

0 −0.002 −0.004

0.99

−0.006 0.98 0.97 −1

−0.008 −0.5

0

0.5

−0.01 −1

1

−0.5

0

0.5

1

0.02 0.018 0.016

A B C D

0.014 0.012 0.01

E F

0.008 0.006 0.004 0.002 0 0

0.2

0.4

0.6

0.8

1

Fig. 1 Results for the homogeneous tension and compression of a cylinder with λ and c plotted against σ¯ , and for the homogeneous shear of a slab with κ plotted against τ¯ . For each panel the results correspond to: A, strain-limiting behaviour based on (52); B, strain-limiting behaviour based on (54); C, Gum metal based on (63); D, Gum metal based on (66); E, modelling of rock based on (71) with εzz = f 1 (σ ) + f 3 (σ/3)/3, εrr = f2 (σ ) + f 2 (0)σ + f 3 (σ/3)/3 and κ = f1 (τ ) − f 1 (−τ ) + f2 (−τ ) − f2 (τ ) − τ [f 2 (τ ) + f 2 (−τ )]; F, modelling of concrete based on (75)

is of the form14 (in Cylindrical coordinates) u r = cr , u θ = 0 and u z = (λ − 1)z, where c and λ do not depend on r , thus obtaining the uniform strain εrr = εθθ = c and εzz = λ − 1. In Fig. 1 we see the variation of λ = εzz + 1 and c = εrr for some of the models presented in the previous section. Regarding the shear of a slab, we assume that such a body deforms due to the application of a stress of the form T = τ (e1 ⊗ e2 + e2 ⊗ e1 ), which if τ does not depend on the position is also a solution of (4)1 for the quasi-static case without body forces. For this simple shear we assume that the displacement field presented by the slab is of the form u 1 = (λ1 − 1)x + κ y, u 2 = (λ2 − 1)y, u 3 = (λ3 − 1)z, where λi , i = 1, 2, 3, κ do not depend on x. For the components of the strains we 14 See

Sect. 7 for a discussion concerning uniqueness and non-uniqueness for the boundary-value problems in the context of these new theories.

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obtain εii = λi − 1 (no sum in i) and ε12 = κ/2. In Fig. 1 we show the behaviour of κ for some of the models presented in the previous section. In Fig. 1 the dimensionless stresses σ¯ and τ¯ are defined as σ¯ = σ/σmax and τ¯ = τ/τmax , where σmax and τmax correspond to maximum values of such stresses, which are different for each case, and for brevity are not shown here.

5.3 Constraints In this section, we discuss kinematic constraints within the context of the new constitutive theories presented in this paper. In the case of a fully implicit relation, of the form, for example, (8), it is possible to incorporate the constraint of incompressibility det F = 1, which is equivalent to det B = 1, by decomposing the stress into a spherical plus a deviatoric part as T = (trT)I/3 + Tdev , where Tdev = T − (trT)I/3, thus requiring that G (T, B) would not be influenced by the spherical part of the stress (trT)I/3, i.e. to impose restrictions on G (T, B) such that (8) would be equivalent to G (Tdev , B) = 0. Regarding the condition det B = 1, further restrictions on G could be applied such that any deformation for the class of models defined through G (Tdev , B) = 0 would be isochoric.15 In the case of large elastic deformations, where some measure of the strain is given as a function of a stress (see (12), (17) and (33)), there are two options; for example, in the case of the subclass (12), namely B = g (T), the constraint of incompressibility can be incorporated directly into such a constitutive equation (see [15]), obtaining det(gg(T)) = 1, which is a direct restriction on g . In [15] such a restriction was studied for the particular case when g is isotropic and is obtained from some scalar potential (see Sect. 3.1). The expression for g obtained in such a way does not depend on a spherical stress, i.e. g (− pI) = I; however, it has not been possible so far to show that g ((trT)I/3 + Tdev ) = g (Tdev ). The model proposed by Srinivasa [61] (see (17) and (18)) does not present such a problem, and with the use of the Kirchhoff stress and the Hencky strain, for the particular case of incompressible bodies and using the Gibbs potential G, it is possible to obtain the explicit expression dev(ln V) = ∂G/∂Tdev , where G depends only on the deviatoric part of the stress tensor. In the rest of this section we explore kinematical constraints for the particular subclass of the constitutive equations (33), wherein the linearized strain tensor is given as a function of the stress: ε = h (T). 5.3.1

Incompressibility

In the case of small strains the incompressibility constraint reads tr ε = 0, and replacing the subclass of the model presented in (35) into that equation we obtain the linear first-order partial differential equation 15 See,

for example, [28], Sects. 6.2.1 and 6.3.1 of [26] and Sect. 3 of [61] for a treatment of incompressibility within the context of a class of isotropic implicit models.

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∂Π ∂Π ∂Π + I1 + 2I2 = 0, ∂ I1 ∂ I2 ∂ I3

(77)

whose solution is of the form (see [16]) Π = Π¯ ( I¯1 , I¯2 ), where I¯1 = I2 − I12 /6, I¯2 = I3 + 2I13 /27 − 2I1 I2 /3, from which we obtain     2   ¯ 2I1 I1 I1 ∂ Π¯ I2 2 ∂Π I− + 2 . ε = T− I − T+T 3 9 3 3 ∂ I¯1 ∂ I¯2

(78)

In [16] it was proved that if the stress is decomposed as T = (trT)I/3 + Tdev then ε = h (T) = h (Tdev ). Therefore, we have a constitutive equation that satisfies the constraint automatically, and which is not affected by the spherical part of the stress. Let us study the constraint trε = 0 for the case of (36), which is an alternative representation for h for isotropic bodies. In such a case trε = 0 becomes ε1 + ε2 + ε3 = 0, and substituting (37) we obtain the first-order linear partial differential equation ∂Π ∂Π ∂Π + + = 0, (79) ∂σ1 ∂σ2 ∂σ3 whose solution is ¯ 1 − σ2 , σ1 − σ3 ) + Π(σ ¯ 2 − σ1 , σ2 − σ3 ) + Π¯ (σ3 − σ1 , σ3 − σ2 ), Π = Π(σ ¯ ¯ where we have the additional restriction Π(x, y) = Π(y, x).

5.3.2

Inextensibility

Let us consider a transversely isotropic body, for which the direction a is inextensible, and the constraint reads a · (εa) = 0. It happens that such a constraint is used as an idealization of bodies composed of a matrix filled with a much stiffer family of fibres in the direction a, and in such a case it is expected that in compression the fibres may not present much resistance to the deformation, unlike that in tension; therefore, for such problems a better constraint is a · (εa) ≤ 0. Substituting the class of constitutive equation presented in (38) in a · (εa) = 0 we obtain the first-order linear partial differential equation ∂Π ∂Π ∂Π ∂Π ∂Π + I4 + I5 + + 2I4 = 0, ∂ I1 ∂ I2 ∂ I3 ∂ I4 ∂ I5

(80)

whose solution is (see [12]) Π = Π¯ ( I¯1 , I¯2 , I¯3 , I¯4 ), where I¯1 = I4 − I1 , I¯2 = I12 /2 + I2 − I1 I4 , I¯3 = I12 − 2I1 I4 + I3 and I¯4 = −I13 /3 + I3 + I12 I4 − I1 I5 , and the expression for ε is

A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies ∂ Π¯ ∂ Π¯ ∂ Π¯ (a ⊗ a − I) + (− I¯1 I + T − I1 a ⊗ a) + [−2 I¯1 I − 2I1 a ⊗ a ∂ I¯1 ∂ I¯2 ∂ I¯3 ∂ Π¯ + a ⊗ (Ta) + (Ta) ⊗ a] + {− I¯3 I + T2 + I12 a ⊗ a − I1 [a ⊗ (Ta) + (Ta) ⊗ a]}. ∂ I¯4

213

ε =

(81)

In [12] it was shown that a stress of the form −qa ⊗ a does not produce any deformation, i.e. ε = h (T − qa ⊗ a) = h (T). In the case of a body with a family of fibres that do not support compression a · (εa) ≤ 0, and thus we have  Π (T) =

Π (I1 , I2 , I3 , I4 , I5 ) if a · (εa) < 0 ¯ I¯1 , I¯2 , I¯3 , I¯4 ) ifa · (εa) = 0, Π(

(82)

where the expression for ε in the case a · (εa) < 0 is given in (38), while for the case corresponding to a · (εa) = 0 is given in (81). As in the case of incompressibility studied in the previous section, it is possible to consider the alternative constitutive equation (39)–(41) in terms of the spectral invariants Π = Π (σ1 , σ2 , σ3 , ζ1 , ζ! ζ3 = 1 − ζ1 − ζ2 . Considering that the strain ten2 , ζ3 ),! 3 3 (i) ⊗ t( j) , taking into account (39)–(41), sor can be expressed as ε = i=1 j=1 εi j t (i) 2 recalling the definition ζi = [a · t ] , i = 1, 2, 3, from the constraint a · (εa) = 0, we obtain the partial differential equation   ζ1 ζ2 ∂Π ∂Π ∂Π ∂Π ∂Π + + +2 − ∂σ1 ∂σ2 ∂σ3 ∂ζ1 ∂ζ2 (σ1 − σ2 ) ∂Π ζ2 ζ3 ∂Π ζ1 ζ3 +2 = 0. +2 ∂ζ1 (σ1 − σ3 ) ∂ζ2 (σ2 − σ3 ) It is not simple to try to find an exact solution for the above equation, whose structure is more complicated than (80).

5.3.3

The Rigid Body

A body said to be rigid if there is no deformation irrespective of the stresses applied to it. A possible way to impose that restriction on an isotropic body is by considering the constraints a(i) · (εa(i) ) = 0, i = 1, 2, 3, where {a(i) } is a set of orthonormal vectors. Using (36) in the above constraints, assuming that a(i) = t(i) , from a(i) · (εa(i) ) = 0, we obtain ∂Π/∂σi = 0, i = 1, 2, 3, from which we conclude that Π does not depend on the stress, i.e. ε is not affected by the stresses and moreover ε = 0.

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6 Boundary-Value Problems In this section, we consider some boundary-value problems, for the subclass of the constitutive equation (33), but first, let us discuss the boundary-value problem for the implicit relation (5), where we work with the second Piola–Kirchhoff stress tensor S = F−1 P and the Lagrange Saint–Venant strain tensor E as the main variables for the problem. In such a case, in order to find the stresses and the deformation, we need to solve in parallel the equation of motion (4)1 , rewritten in the reference configuration, and (5), i.e. (83) ρr u¨ = Div (FS) + ρr b, F (S, E) = 0, X ∈ κr (B), ˆ where pˆ is the external traction for a subwith the boundary conditions (FS)N = p, part ∂κr (B) p of the boundary, with N the unit normal vector to that surface in the ˆ where uˆ is a specification of the displacement reference configuration. Also u = u, on some part of the boundary of the body X ∈ ∂κr (B)x , with ∂κr (B) = ∂κr (B) p ∪ ∂κr (B)x and ∂κr (B) p ∩ ∂κr (B)x = Ø. We need to recall that u = x − X = χ (X, t) − X and (1)4 , and we thus have three equations (from the equation of motion (83)1 ), plus six equations from the implicit relation (83)2 , which must be solved to find the three components of x and the six independent components of S. Therefore, for this class of implicit relations, if we want to solve boundary-value problems using the semi-inverse method, we need to not only propose some simplified expressions for the displacement u, but also some simplified expression for S, which should be compatible with the above field, and to solve the resulting simplified equations from (83). In comparison, in the classical theory of nonlinear elasticity, we just need to propose a simplified expression for x and from T = f (F) we obtain the components of the stress, which are replaced in the equation of motion that is now written in terms of simplified expressions for the deformation field. Now, in the case of the subclass (33), we need to find the displacement field u and the Cauchy stress tensor T by solving ρ u¨ = div T + ρb, ε =

1 (∇u + ∇uT ) = h (T), 2

(84)

where ρ should be found from the mass balance16 (4)2 , and Tn = tˆ, for x ∈ ∂κr (B)t , n is the unit normal vector to that surface, u = uˆ for x ∈ ∂κr (B)u , where tˆ and uˆ are the external traction, and the specification of the displacement field, respectively, and ∂κr (B) = ∂κr (B)t ∪ ∂κr (B)u and ∂κr (B)t ∩ ∂κr (B)u = Ø. In this case we need to solve the three components of the equation of motion, plus the six components of (84)2 , in order to find the three components of u and the six independent components of T. 16 In the rest of this section we will assume that the mass density changes little, and that such small changes in density do not have an important impact with regard to the behaviour of the bodies; thus we do not solve (4)2 and assume that ρ is approximately constant.

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Following the approach used in the classical theory of elasticity, one may be tempted to try to invert h (T) to express the stresses in terms of the strains, and to replace such expressions in the equation of motion, to have three equations in terms of the components of the displacement field. However, as explained in [50], while it might be mathematically possible to invert such an expression, the expression so obtained could involve terms of higher order in the linearized strain, which would not have been acceptable within the context of the linearized theory. Another approach that can be used when ε = h (T) is to recall what is done in the linearized theory of elasticity, when the strains are expressed in terms of the stresses. For example, in the case of isotropic bodies we have ε = (1 + ν)T/E − νtr (T)I/E, where for the quasi-static case and plane stress problem (no body forces), the components of the stress are expressed in terms of an Airy stress potential, thus satisfying the equations of equilibrium automatically, and by invoking the compatibility equation to obtain the well-known biharmonic equation. However, it is not mandatory to consider the compatibility equations for the strains (see [54]) and second, even if we use them, in our case the resulting equation is a nonlinear fourth-order partial differential equation for the stress potential, which so far has not been amenable to the determination of exact solutions for problems (see, for example, Sect. 3.3 of [6]). Therefore, for the problems presented in the following sections, in general we solve them assuming simplified expressions for the displacement field and the stress field simultaneously.

6.1 Quasi-Static Deformations In this section we study boundary-value problems for the quasi-static case, so that (4)1 becomes div T + ρb = 0, (85) and for all the problems considered here we assume additionally that b = 0.

6.1.1

Homogeneous Distributions of Stresses

Let us consider the case of the displacement field being of the form u = A0 x, where A0 is a constant second-order tensor. Let us assume that such a displacement field is caused by the homogeneous stress distribution T0 . Since T0 does not depend on x and b = 0 the above stress field is a solution of (85). From (2)2 we have ε = (A0 + AT0 )/2 and from (35), for an isotropic body, we have 1 (A0 + AT0 ) = Π1 + Π2 T0 + Π3 T20 , 2

(86)

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which is the equation that allows us to find the symmetric part of A0 in terms of T0 , where the values of the functions Πi , i = 1, 2, 3, depend on the stress tensor T0 . Let us examine two special cases already mentioned in Sect. 5.2.6. The first case is the uniform tension/compresion of a cylinder under uniform stress of the form (in cylindrical coordinates) T0 = σ ez ⊗ ez . Assuming that such a stress tensor produces a displacement field of the form u r = cr , u θ = 0, u z = (λ − 1)z, where c and λ are constants, from (86) we obtain c = Π1 , λ − 1 = Π1 + Π 2 σ + Π 3 σ 2 ,

(87)

i.e. we obtain directly the radial and axial deformations in terms of the stress σ . The second problem that is interesting is the case of the simple shear of a slab (described in Cartesian coordinates). For simple shear, we assume that we apply a uniform stress of the form T = τ (e1 ⊗ e2 + e2 ⊗ e1 ), and we assume that such a stress produces the displacement field u 1 = (λ1 − 1)x + κ y, u 2 = (λ2 − 1)y, u 3 = (λ3 − 1)z, where λi , i = 1, 2, 3, and κ are constants. From (86) we obtain λ1 − 1 = λ2 − 1 = Π1 + Π3 τ 2 , λ3 − 1 = Π1 , κ = 2Π2 τ.

6.1.2

(88)

Non-homogeneous Distributions of Stresses: Problems with Cylindrical and Spherical Symmetry

Some problem involving non-homogeneous stresses and strains have been solved numerically by describing the bodies in terms of cylindrical and spherical coordinates; see, for example, [7, 13, 14]. Let us study the deformation of the annulus ri ≤ r ≤ ro , 0 ≤ θ ≤ 2π − α, 0 ≤ z ≤ L, which we assume is under the stress distribution Trr = Trr (r ), Tθθ = Tθθ (r ), Tzz = Tzz (r ), Tr θ = Tr θ (r ), Tr z = Tr z (r ) and Tθ z = Tθ z (r ), which produces a displacement field of the form u r = f (r ), u θ = kr θ + g(r ) + τ0 r z, u z = (λ − 1)z + h(r ),

(89)

where k, τ0 and λ are constants. The above displacement field includes as special cases several deformations already studied in the literature. For example, f (r ) can be used to study the radial inflation of such an annulus, g(r ) is connected with circumferential shear, h(r ) with telescopic shear, τ0 is connected with torsion and k can be used to study the opening and closing of an annulus in the azimuthal direction. If Trr = Trr (r ), Tθθ = Tθθ (r ), Tzz = Tzz (r ), Tr θ = Tr θ (r ), Tr z = Tr z (r ) and Tθ z = Tθ z (r ), the equations of equilibrium (85) (again in the absence of body forces) become 1 dTrr + (Trr − Tθθ ) = 0, dr r

2 dTr θ + Tr θ = 0, dr r

1 dTr z + Tr z = 0. dr r

(90)

These last two equations can be solved exactly, and if Tr θi , Tr zi denote the stresses evaluated at r = ri we have Tr θ (r ) = Tr θi (ri /r )2 and Tr z (r ) = Tr zi ri /r .

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Using (89) in (2)2 from (35) we obtain f (r ) = Π1 + Π2 Trr + Π3 (Trr2 + Tr2θ + Tr2z ), f (r ) = Π1 + Π2 Tθθ + Π3 (Tr2θ + Tθθ2 + Tθ2z ), k+ r λ − 1 = Π1 + Π2 Tzz + Π3 (Tr2z + Tθ2z + Tzz2 ),   1 g(r ) g (r ) − = Π2 Tr θ + Π3 (Trr Tr θ + Tr θ Tθθ + Tr z Tθ z ), 2 r 1 h (r ) = Π2 Tr z + Π3 (Trr Tr z + Tr θ Tθ z + Tr z Tzz ), 2 1 τ0 r = Π2 Tθ z + Π3 (Tr θ Tr z + Tθθ Tθ z + Tθ z Tzz ). 2

(91) (92) (93) (94) (95) (96)

We have seven equations in (91)–(96) plus (90)1 , which can be used to find the seven unknown Trr , Tθθ , Tθ z , Tzz , f , g and h. From (90)1 we can express Tθθ in terms of Trr as Tθθ = r dTrr /dr + Trr ; on the other hand from (92) we have   #   2  dTrr dTrr 2 2 f (r ) = r Π1 + Π2 r + Trr + Π3 Tr θ + r + Trr + Tθ z − k , dr dr "

which, after being substituted in (91), leads to d dr

  #%   2  dTrr dTrr 2 2 + Trr + Π3 Tr θ + r + Trr + Tθ z − k r Π1 + Π2 r dr dr

$ "

= Π1 + Π2 Trr + Π3 (Trr2 + Tr2θ + Tr2z ).

(97)

From (94) and (95) we obtain &

  Π2 Tr z + Π3 (Trr Tr z + Tr θ Tθ z + Tr z Tzz ) dξ + h i , (98) r     &i r   1 dTrr dξ Π2 Tr θ + Π3 Trr Tr θ + Tr θ ξ + Trr + Tr z Tθ z g(r ) = 2r ξ dr ri r + gi , (99) ri

h(r ) = 2

r

where h i and gi are the values of the functions h(r ) and g(r ) evaluated at r = ri . Therefore, we end up with three equations to be solved, (97), (93) and (96). Equation (97) is a nonlinear second-order differential equation for Trr and we can assume as boundary conditions that Trr (ri ) = Trri and Trr (ro ) = Trro , where Trri and Trro would be the external traction on the inner and outer surface of the annulus.

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Equations (93), (96) can be considered as nonlinear algebraic equations that can be used to find Tθ z (r ) and Tzz (r ). If we are studying the deformation of the spherical body ri ≤ r ≤ ro , 0 ≤ θ ≤ π , 0 ≤ φ ≤ 2π , the only possibility of reducing the original partial differential equations to a system of equations depending on one variable, is to assume that the spherical body deforms under the influence of the stress tensor T = Trr (r )er ⊗ er + Tθθ (r )eθ ⊗ eθ + Tφφ (r )eφ ⊗ eφ , which we assume produces the displacement field u r = f (r ), u θ = 0, u φ = 0. From the equilibrium equations we obtain that Tθθ = Tφφ and (85) reduces to dTrr /dr + 2(Trr − Tθθ )/r = 0, while from (35) we obtain f (r ) = Π1 + Π2 Trr + Π3 Trr2 and f (r )/r = Π1 + Π2 Tθθ + Π3 Tθθ2 . These equations have the same structure as that for the problem of the cylindrical annulus, and therefore we do not discuss this problem further.

6.1.3

Non-homogeneous Distributions of Stresses: Incompressible Bodies

Equation (97) cannot, in general, be solved exactly. One of the open problems (see Sect. 7) would be to look for particular expressions for Π such that some exact solutions could be found. One can ask if it is possible to repeat the classical technique developed originally by Rivlin (see, for example, Chapter D, Section b of [63]), of appealing to the constraint of incompressibility, in order to find exact solutions that could be valid for any Π for a given family of elastic bodies. We explore such a question in this section. In the classical theory of nonlinear elasticity, for incompressible isotropic bodies we have a constitutive equation of the form T = − pI + α1 B + α2 B2 , and the indeterminacy of p is exploited in order to simplify the equations in the following manner. Let us consider as an example the case of the problem of inflation and extension of a cylindrical annulus, where it assumed that the deformation is of the form r = f (R), θ = Θ, z = λZ , where λ is a constant. Calculating F and imposing directly the constraint det F = 1 on the above deformation we obtain the solution ' R 2 − Ri2 + ri2 , r= λ where Ri and ri are the inner radii of the annulus in the reference and current configurations. If we use the notation T˜ = α1 B + α2 B2 , in the case of the above deformation, it is easy to see that T˜ only depends on the radial position, and only has normal components. Substituting T = − pI + T˜ in the equations of equilibrium without body forces we obtain that p = p(r ) = p(R) and dp 1 ˜ d T˜rr − + (Trr − T˜θθ ) = 0, dr dr r

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which can be solved easily for p(r ), and is subjected to the boundary conditions for Trr (r ) known at ri and ro , where ro is the radius of the outer surface of the annulus in the current configuration. Thus, the solution is valid for any functions α1 , α2 , i.e. it is a universal solution (see [63]). With the above discussion in mind, let us see if something similar can be carried out for the case of the incompressible isotropic body (78). In the case of (78) we have the property that ε = h (T) = h (Tdev ), where Tdev = T − tr(T)I/3. Moreover, we can add any spherical stress of the form − pI to T and the above result will always hold, i.e. h (− pI + T) = h (Tdev ). The constitutive equation (78) can be rewritten in an alternative way as (100) ε = ϑ¯ 0 I + ϑ¯ 1 Tdev + ϑ¯ 2 T2dev , ¯ I¯1 and ϑ¯ 2 = ∂ Π/∂ ¯ I¯2 , where Idev2 = where ϑ¯ 0 = −2Idev2 /3 ∂ Π¯ /∂ I¯2 , ϑ¯ 1 = ∂ Π/∂ tr(T2dev )/2. As an illustration let us consider the problem of the inflation of a cylindrical annulus, where we assume that we have a stress of the form T = Trr (r )er ⊗ er + Tθθ (r )eθ ⊗ eθ + Tzz (r )ez ⊗ ez , which we assume produces a displacement field of the form u r = f (r ), u θ = 0, u z = (λ − 1)z, where λ is a constant. The non-zero components of the deviatoric stress are 2 1 2 1 Trr − (Tθθ + Tzz ), Tdev22 (r ) = Tθθ − (Trr + Tzz ), 3 3 3 3 2 1 Tdev33 (r ) = Tzz − (Trr + Tθθ ). 3 3 Tdev11 (r ) =

From (100) we obtain that 2 f (r ) = ϑ¯ 0 + ϑ¯ 1 Tdev11 + ϑ¯ 2 Tdev , 11 2 λ − 1 = ϑ¯ 0 + ϑ¯ 1 Tdev33 + ϑ¯ 2 Tdev . 33

f (r ) 2 = ϑ¯ 0 + ϑ¯ 1 Tdev22 + ϑ¯ 2 Tdev , 22 r

(101) (102)

Let us add to the stress T a spherical stress − p(r )I; then, from the previous comments we know that the strains are the same and given by (101), (102). On the other hand, the equation of equilibrium and the boundary conditions do change due to the addition of such a spherical stress, and we have −

1 d p dTrr + + (Trr − Tθθ ) = 0, dr dr r

(103)

where p(ri ) − Trr (ri ) = tˆi , − p(ro ) + Trr (ro ) = tˆo , with tˆi and tˆo the external tractions. Equation (103) can be solved easily and we obtain & p(r ) = Trr (r ) − Trr (ri ) + p(ri ) + ri

and from the boundary conditions we have

r

1 [Trr (ξ ) − Tθθ (ξ )] dξ, ξ

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tˆo + tˆi =

& ri

ro

1 [Tθθ (ξ ) − Trr (ξ )] dξ. ξ

(104)

Therefore, the equation of equilibrium has been solved for p(r ) in a manner similar to that in the classical theory of nonlinear elasticity, and we need to solve the three equations (101)–(102) to find f (r ), Trr , Tθθ and Tzz . From (101)2 we have   2 , f (r ) = r ϑ¯ 0 + ϑ¯ 1 Tdev22 + ϑ¯ 2 Tdev 22 and substituting this in (101)1 we obtain ) d (  2 2 r ϑ¯ 0 + ϑ¯ 1 Tdev22 + ϑ¯ 2 Tdev = ϑ¯ 0 + ϑ¯ 1 Tdev11 + ϑ¯ 2 Tdev . 22 11 dr

(105)

We finally have one algebraic equation (102) and one ordinary differential equation (105) to find three functions Trr (r ), Tθθ (r ) and Tzz (r ). However, the above Eqs. (101), (102) are not independent. From the theory described in Sect. 5.3.1, we have trε = 0, which means that f (r ) + f (r )/r + λ − 1 = 0, whose solution is f (r ) = (1 − λ)r/2 + C/r , where C is a constant, i.e. in (101) f (r ) is not an unknown variable. The final conclusion is that although the equilibrium equation (103) is solved in an easy manner when assuming the addition of a spherical stress, the equations that appear from (100) are not simplified due to that addition and the constraint of incompressibility, and in general the problem (100) along with the equilibrium equation cannot be solved exactly.

6.1.4

Another Non-homogeneous Problem

In this section, we discuss a problem wherein the body is subject to a non-homogeneous distributions of stress, which we shall describe within the context of Cartesian coordinates (see Sect. 4.1 of [7]), for the plane slab 0 ≤ x ≤ L, 0 ≤ y ≤ H . We assume plane stresses and the existence of an Airy stress potential Υ (x, y) of the form Υ (x, y) = ϕ(x) − τ0 x y, where τ0 is a constant. We obtain T11 = 0, T22 = T22 (x) =

d2 ϕ , T12 = τ0 . dx 2

(106)

From (34) for the components of the strain tensor we have (on recognizing that plane stress does not imply plane strain and on neglecting ε33 ) 2 ), ε12 = ϑ1 τ0 + ϑ2 τ0 T22 , ε11 = ϑ0 + ϑ2 τ02 , ε22 = ϑ0 + ϑ1 T22 + ϑ2 (τ02 + T22 (107) where ε11 = ε11 (x), ε22 = ε22 (x) and ε12 = ε12 (x). The above deformation field is produced by a continuous displacement field if the compatibility equations are

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satisfied, and that means that d2 ε22 /dx 2 = 0, which implies that ε22 = c1 x + c0 , where c0 , c1 are constants. In the light of the above results, let us assume that the displacement field that is produced by the stress (106) is of the form u 1 (x, y) = f (x) + h(y), u 2 (x, y) = (c1 x + c0 )y + g(x).

(108)

Calculating ε12 from the above displacement field it is found that to have ε12 = ε12 (x) we need h(y) = −c1 y 2 /2 + c2 , where c2 is a constant. Therefore, in (107) we obtain f (x) = ϑ0 + ϑ2 τ02 , c1 x + c0 = ϑ0 + ϑ1 T22 (x) + ϑ2 [τ02 + T22 (x)2 ], (109) 1 g (x) = ϑ1 τ0 + ϑ2 τ0 T22 (x), 2

(110)

and from (109)1 , (110) we obtain &

x

f (x) = 0

& (ϑ0 + ϑ2 τ02 ) dξ + f 0 , g(x) = 2

x

(ϑ1 τ0 + ϑ2 τ0 T22 (x)) dξ + g0 ,

0

where f 0 and g0 are constants. Equation (109)2 can be used to find T22 (x), given c0 and c1 .

6.2 Wave Propagation When we turn our attention to time-dependent stresses and deformations for problems involving implicit constitutive equations, only a few problems have been studied in the literature [9, 33, 36, 37]. In the particular case of the subclass of constitutive equation (33), they can be classified into two types, problems involving the full study of the nonlinear equations (84) for some simple expressions for the stresses and displacement field, and incremental analysis.

6.2.1

Fully Nonlinear Problem

In this section, we provide some details of three problems that have been studied numerically, namely, wave propagation in a one-dimensional rod, the propagation of shear waves in a semi-infinite slab, and the propagation of circumferential shear waves in a cylindrical annulus.17

17 In a recent paper by Huang et al. [33], the equations of motion corresponding to a special subclass

of (33) is considered under the prescription of Riemann data, and all wave patterns that are possible are delineated. The specific waveform that arises depends on the initial data, and it is shown that rarefaction and shock waves are possible. As the paper is quite technical we shall not discuss it in detail here.

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In [9] the problem of propagation of longitudinal waves was studied for a onedimensional rod 0 ≤ x ≤ L, for the constitutive equation (33), (35), assuming that the stress tensor was of the form T = σ (x, t)e1 ⊗ e1 , which was assumed to produce a displacement field of the form u = u(x, t)e1 . For this one-dimensional problem, there is only one component of the strain, namely, ε11 = ε = ∂u/∂ x and from (35), (52), considering the above √ expression for the stress we obtain ε = h(σ ) = −α [1 − 1/(1 + βσ )] + αγ σ/ 1 + ισ 2 . The notation ε = h(σ ) can also be used to consider other expressions for the constitutive equation. From (84) we obtain the two partial differential equations ρ

∂σ ∂ 2u , = ∂t 2 ∂x

∂u = h(σ ), ∂x

(111)

which must be solved to obtain u(x, t) and σ (x, t). With regard to the boundary and initial conditions, we can assume, for example, u(L , t) = 0, σ (0, t) = σˆ (t), where σˆ (t) is an external known traction, and u(x, 0) = u(x, ˙ 0) = 0. It is necessary to point out that in general it is not a good idea to cross-differentiate to reduce the number of equations governing the original problem (111), because some of the solutions of the new problem may not be solutions of the original system of equations; see, for example, the discussion in [19]. In [37] a problem that is similar in structure to that of (111) was studied. The authors studied the behaviour of the semi-infinite slab −∞ ≤ x ≤ ∞, 0 ≤ y ≤ yo , −∞ ≤ z ≤ ∞ that is assumed have the shear stress distribution T = τ (y, t)(e1 ⊗ e2 + e2 ⊗ e1 ), which induces the deformation in the n u = u(y, t)e1 .  body of the form In [37] the constitutive equation h = βtr (T)I + α 1 + γ tr(T2 )/2 T was assumed, which for the problem under consideration leads to h(τ ) = α(1 + γ τ 2 )n τ , where α, γ , n are constants. In this case (84) become ρ

∂ 2u ∂τ , = 2 ∂t ∂y

∂u = h(τ ), ∂y

(112)

which has a structure very similar to (111), where we now need to solve for u(y, t) and τ (y, t). The last example to be discussed in this section is the propagation of shear waves in a cylindrical annulus defined through ri ≤ r ≤ ro , 0 ≤ z ≤ 2π , −∞ ≤ z ≤ ∞ (see [36]), wherein it is assumed that the stress and the displacement have the forms T = τ (r, t)(er ⊗ eθ + eθ ⊗ er ) and u = u(r, t)eθ , respectively. The constitutive equation considered is the same as in that [37], and (84) yields the partial differential equations ρ

2 ∂τ ∂ 2u + τ, = ∂t 2 ∂r r

∂u u − = h(τ ), ∂r r

(113)

that must be solved for u(r, t) and τ (r, t). All the above problems have been solved numerically, and up to date no known exact solution has been found for the particular expressions for h presented here.

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223

Infinitesimal Wave Analysis as a Consequence of Time-Dependent Stress Superposed on a Time-Independent Stress

In this section, we consider some results concerning the extension of the classical incremental analysis for the class of constitutive equations (33) (see [1]). We assume the presence of time-independent stress, on which we superpose a small time-dependent stress (incremental stress). There are many possible applications for such an analysis, in particular regarding the determination of mechanical properties, a case in point being non-destructive testing. Let us denote by T = T(x) and u = u(x) the initial time-independent stress and displacement field, which are solutions of the boundary-value problem (33), (85) in the case of quasi-static deformations (no body forces), i.e. div T = 0 and ∇u + ˆ on x ∈ ∂κr (B)u , ∇uT = 2hh(T), where Tn = tˆ(x) on x ∈ ∂κr (B)t and u = u(x) where κr (B) = κr (B)t ∪ κr (B)u and κr (B)t ∩ κr (B)u = Ø. Let us add to the above stress the incremental time-dependent stress ΔT(x, t), where we suppose that |ΔT|/|T| ∼ O(δ), δ  1, which we assume produces an increment in the displacement field Δu(x, t). Now the key ingredients in our analysis are the assumptions that the stress T(x) + ΔT(x, t) and the displacement field u(x) + Δu(x, t) are solutions of the equation of motion18 (84). Therefore, substituting such fields in (84), recalling that T = T(x) and u = u(x) are solutions of the equilibrium equations, we obtain ρΔu¨ = divΔT. (114) Regarding (33), using T(x) + ΔT(x, t) and u(x) + Δu(x, t) in that equation, recalling that |ΔT|/|T| ∼ O(δ), δ  1, we have the approximation ε + Δε ≈ h (T) + (∂hh/∂T) · ΔT, where ε = (∇u + ∇uT )/2, and where we have defined Δε = [∇(Δu) + ∇(Δu)T ]/2 and the fourth-order tensor ∂hh/∂T is evaluated at T, i.e. it does not depend on ΔT. Let ε = h (T), then on defining19 C = C(T) = ∂hh/∂T we obtain the incremental constitutive equation (see [1]) Δε = C · ΔT.

(115)

With regard to the boundary conditions, for the original quasi-static problem we assumed the application of a time-independent traction tˆ(x). We suppose that the increment in stress ΔT(x, t) is caused by the addition of a time-dependent increment 18 One ought to mention that in general for this problem it is also necessary to determine the density, which from (4)2 is given as ρ = ρr / det F. In the case of small gradients of the displacement we have det F ≈ 1 + tr ∇u = 1 + div u = 1 + tr ε, and therefore, ρ ≈ ρr (1 + tr ε)−1 ≈ ρr (1 − tr ε). There may be some problems for which even the small changes in the density due to the factor tr ε can have an important influence in the mechanical behaviour of the material. In this work we assume that this is not the case, and that the mechanical behaviour of the material is not influenced by small changes in the density. Therefore, we adopt the approximation ρ ≈ ρr and assume the density is constant in time. 19 If we assume that h (T) = ∂Π/∂T, then C = ∂ 2 Π/∂T∂T and this fourth-order tensor has the symmetries Ci jkl = C jikl = Ci jlk = Ckli j . We assume that h (T) = ∂Π/∂T for the rest of this section.

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in the traction Δtˆ(x, t), and therefore, for T(x) + ΔT(x, t) and u(x) + Δu(x, t), we have the boundary conditions [T(x) + ΔT(x, t)]n = tˆ(x) + Δtˆ(x, t) and u(x) + ˆ ˆ Δu(x, t) = u(x) + Δu(x, t), and we obtain ˆ t) x ∈ ∂κr (B)u . ΔT(x, t)n = Δtˆ(x, t) x ∈ ∂κr (B)t , Δu(x, t) = Δu(x, (116) If we assume that ΔT and Δu do not depend on time, the left side of (114) is zero, and we obtain the counterpart for (33) of the incremental equations that are used, for example, in the nonlinear theory of elasticity, to study among other things, stability of the solutions for boundary-value problems, especially when we assume that Δtˆ = 0. If ΔT and Δu depend on time, but Δtˆ = 0 and Δuˆ = 0 we can use (114) to explore uniqueness/non-uniqueness for the solution of a boundary-value problem, and also stability, by looking for the existence of such small wave solutions from (114). Let us study the problem of propagation of infinitesimal waves in an infinite medium, assuming that T and ε are constant tensors. There are two ways to carry out the analysis; in the first case we can assume that C(T) has an inverse, and in the second case we assume that that is not necessarily the case. If C(T) has an inverse, then from (115) we obtain ΔT = C −1 · Δε and (114) becomes ρΔu¨ = div (C −1 · Δε), where Δε =

1 [∇(Δu) + ∇(Δu)T ]. 2

(117)

We look for a solution of (117) of the form Δu(x, t) = Δu0 g(ζ ), where ζ = kp · x − λt, k and λ being constants and p being in the direction of propagation of the wave, Δu0 is a constant vector and g is a scalar function that is smooth enough for our purposes. If we substitute the above expression in (117) then after some manipulations we obtain   ρλ2 Q − 2 I Δu0 = 0, (118) k −1

ν = λ/k where we have defined20 Q i j = C ikl j pk pl and we identify  as the wave  speed. Equation (118) has nontrivial solutions if det Q − ρλ2 /k 2 I = 0, which is the equation that must be used to find ν. We notice that Q is a function of the initial time-independent stress since C = C(T). Next, we study the case when C(T) does not have inverse and look for a solution for ΔT(x, t) and Δu(x, t) of the form ΔT(x, t) = ΔT0 f (ζ ) and Δu(x, t) = Δu0 g(ζ ), where ζ = kp · x − λt, ΔT0 is a constant second-order tensor and f is a scalar function smooth enough for our purposes. Replacing these expressions in (114) and (115) we obtain ρλ2 Δu0 g

(ζ ) = ΔT0 pk f (ζ ), 1 (Δu0 ⊗ p + p ⊗ Δu0 )kg (ζ ) = C · ΔT0 f (ζ ). 2 20 Here

−1 C ikl j

corresponds to the components ikl j of the fourth-order tensor C −1 .

(119) (120)

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We can make further progress with the above solutions in the following way. Let us assume that the functions f and g are the same and are given as f (ζ ) = g(ζ ) = eıˆζ , where ıˆ is the imaginary number. Substituting this in (119), (120) we obtain the equation [M][V ] = [0], where the matrix [M] = [M]9×9 and the vector [V ] = [V ]9×1 are defined below and [0]9×1 is a column vector with all components equal to 0. The non-zero components of [M] are M11 = M22 = M33 = ρλ2 , M14 = −M41 = kˆı p1 , M25 = −M52 = kˆı p2 , (121) 1 1 M36 = −M63 = kˆı p3 , M17 = −M71 = kˆı p2 , M18 = −M81 = kˆı p3 , (122) 2 2 1 1 1 M27 = −M72 = kˆı p1 , M29 = −M92 = kˆı p3 , M38 = −M83 = kˆı p1 ,(123) 2 2 2 1 M39 = −M93 = kˆı p2 , M44 = C1111 , M45 = M54 = C1122 , (124) 2 (125) M46 = M64 = C1133 , M47 = M74 = C1112 , M48 = M84 = C1113 , M49 = M94 = C1123 , M55 = C2222 , M56 = M65 = C2233 , (126) M57 = M75 = C2212 , M58 = M85 = C2213 , M59 = M95 = C2223 , M66 = C3333 , M67 = M76 = C3312 , M68 = M86 = C3313 ,

(127) (128)

M69 = M96 = C3323 , M77 = C1212 , M78 = M87 = C1213 , M79 = M97 = C1223 , M88 = C1313 , M89 = M98 = C1323 ,

(129) (130)

M99 = C2323 ,

(131)

and [V ] = (Δu 01 , Δu 02 , Δu 03 , ΔT011 , ΔT022 , ΔT033 , 2ΔT012 , 2ΔT013 , 2ΔT023 )T . The system of equations [M][V ] = [0] has non-trivial solutions if det[M] = 0. In this case this last equation would give a relation between λ, k and p for such a wave to exist.

7 Open Problems Here we provide a brief, and by no means complete, summary of some of the open problems concerning implicit constitutive equations, especially those that have been discussed here. • To obtain exact solutions for a variety of boundary-value problems for different forms of F , G and h . In the specific case of h it would be particularly useful to obtain some exact solutions for problems with stress concentrations, such as the case of a plane plate with a hole, or an inclusion subject to a variety of boundary conditions. To consider systematically problems involving concentrated loads, domains including notches and domains containing cracks, for truly implicit constitutive relations (those that have been considered thus far are explicit expressions for the linearized strain in terms of the stress).

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• To analyse the problem of existence and uniqueness for the boundary-value problems for the equations governing the motion of bodies described by implicit constitutive relations. • In the case of the implicit relations for electro- and magneto-elastic bodies, to propose conditions similar to (6). To extend the theory including the coupling with changes in temperature, and to consider mechanical and electric and magnetic dissipation. Acknowledgements R. Bustamante would like to express his gratitude for the financial support provided by FONDECYT (Chile) under grant no. 1160030. K. R. Rajagopal thanks the National Science Foundation and the Office of Naval Research for support of this work.

Appendix In this Appendix we present the lists of invariants for the different functions that appear for electro-elastic bodies (see Sect. 3.2), and thermo-elastic bodies (see Sect. 3.3). In the case of electro-elastic bodies the functions γi and ψ j that appear in (24) and (23) depend on the following list of invariants: ρ,

I1 = trτ ,

I2 = tr(τ 2 ),

I3 = tr(τ 3 ),

I4 = trB,

I5 = tr(B2 ), (132)

I6 = tr(B3 ), I7 = tr(τ B), I8 = tr(τ 2 B), I9 = tr(B2 τ ), E), I13 = E · (τ 2E ), I10 = tr(τ 2 B2 ), I11 = E · E , I12 = E · (τE E), I15 = E · (B2E ), I16 = E · [(τ B + Bτ )E E], I14 = E · (BE E], I18 = E · [(τ B2 + B2 τ )E E], I17 = E · [(τ 2 B + Bτ 2 )E

(133) (134)

E] I20 = D · D , I21 = D · (τD D), I19 = E · [(τ 2 B2 + B2 τ 2 )E 2 2 D), I24 = D · (B D ), I22 = D · (τ D ), I23 = D · (BD D], I26 = D · [(τ 2 B + Bτ 2 )D D], I25 = D · [(τ B + Bτ )D 2 2 D], I28 = D · [(τ 2 B2 + B2 τ 2 )D D], I27 = D · [(τ B + B τ )D 2 D · E ) , I30 = (D D · E )D D · (τE E), I31 = (D D · E )D D · (τ 2E ), I29 = (D

(137) (138) (139)

(135) (136)

(140) (141)

D · E )D D · (BE E), I33 = (D D · E )D D · (B2E ), I32 = (D D · E){D D · [(τ B + Bτ )E E] + E · [(τ B + Bτ )D D]}, I34 = (D 2 2 2 D · E ){D D · [(τ B + Bτ )E E] + E · [(τ B + Bτ 2 )D D]}, I35 = (D 2 2 2 2 D · E ){D D · [(τ B + B τ )E E] + E · [(τ B + B τ )D D]}, I36 = (D

(142) (143)

D · E ){D D · [(τ 2 B2 + B2 τ 2 )E E] + E · [(τ 2 B2 + B2 τ 2 )D D]}. I35 = (D

(146)

(144) (145)

In the case of thermo-elastic bodies the scalar functions βk that appear following (30) depend in the following list of invariants:

A Review of Implicit Constitutive Theories to Describe the Response of Elastic Bodies

θ,

I1 = trS,

I2 = tr(S2 ),

I5 = tr(E2 ),

(147)

I6 = tr(E ), I7 = tr(SE), I8 = tr(S E), I9 = tr(E S), I10 = tr(S2 E2 ), I11 = hr · hr , I12 = hr · (Shr ), I13 = hr · (S2 hr ), I14 = hr · (Ehr ), I15 = hr · (E2 hr ), I16 = hr · [(SE + ES)hr ],

(148) (149) (150)

I17 = hr · [(S2 E + ES2 )hr ], I18 = hr · [(SE2 + E2 S)hr ], I19 = hr · [(S2 E2 + E2 S2 )hr ] I20 = h˙ r · h˙ r , I21 = h˙ r · (Sh˙ r ), I22 = h˙ r · (S2 h˙ r ), I23 = h˙ r · (Eh˙ r ), I24 = h˙ r · (E2 h˙ r ), I25 = h˙ r · [(SE + ES)h˙ r ], I26 = h˙ r · [(S2 E + ES2 )h˙ r ],

(151) (152)

3

I3 = tr(S3 ),

I4 = trE,

227

2

2

(153) (154)

I27 = h˙ r · [(SE2 + E2 S)h˙ r ], I28 = h˙ r · [(S2 E2 + E2 S2 )h˙ r ], I29 = ∇X θ · ∇X θ, I30 = ∇X θ · (S∇X θ ), I31 = ∇X θ · (S2 ∇X θ ),

(155) (156)

I32 = ∇X θ · (E∇X θ ), I33 = ∇X θ · (E2 ∇X θ ), I34 = ∇X θ · [(SE + ES)∇X θ ], I35 = ∇X θ · [(S2 E + ES2 )∇X θ ], I36 = ∇X θ · [(SE2 + E2 S)∇X θ ], I37 = ∇X θ · [(S2 E2 + E2 S2 )∇X θ ], I38 = (h˙ r · hr )2 , I39 = (h˙ r · hr )h˙ r · (Shr ), I40 = (h˙ r · hr )h˙ r · (S2 hr ),

(157) (158) (159)

I41 = (h˙ r · hr )h˙ r · (Ehr ), I42 = (h˙ r · hr )h˙ r · (E2 hr ), I43 = (h˙ r · hr ){h˙ r · [(SE + ES)hr ] + hr · [(SE + ES)h˙ r ]}, I44 = (h˙ r · hr ){h˙ r · [(S2 E + ES2 )hr ] + hr · [(S2 E + ES2 )h˙ r ]}, I45 = (h˙ r · hr ){h˙ r · [(SE2 + E2 S)hr ] + hr · [(SE2 + E2 S)h˙ r ]}, I46 = (h˙ r · hr ){h˙ r · [(S2 E2 + E2 S2 )hr ] + hr · [(S2 E2 + E2 S2 )h˙ r ]},

(160) (161) (162) (163) (164) (165)

I47 = (∇X θ · hr ) , I48 = (∇X θ · hr )∇X θ · (Shr ), I49 = (∇X θ · hr )∇X θ · (S2 hr ),

(166) (167) (168)

I50 = (∇X θ · hr )∇X θ · (Ehr ), I51 = (∇X θ · hr )∇X θ · (E2 hr ), I52 = (∇X θ · hr ){∇X θ · [(SE + ES)hr ] + hr · [(SE + ES)∇X θ ]},

(169) (170)

I53 = (∇X θ · hr ){∇X θ · [(S2 E + ES2 )hr ] + hr · [(S2 E + ES2 )∇X θ ]}, I54 = (∇X θ · hr ){∇X θ · [(SE2 + E2 S)hr ] + hr · [(SE2 + E2 S)∇X θ ]},

(171) (172)

I55 = (∇X θ · hr ){∇X θ · [(S2 E2 + E2 S2 )hr ] + hr · [(S2 E2 + E2 S2 )∇X θ ]}, I56 = (h˙ r · ∇X θ )2 , I57 = (h˙ r · ∇X θ )h˙ r · (S∇X θ ), I58 = (h˙ r · ∇X θ )h˙ r · (S2 ∇X θ ), I59 = (h˙ r · ∇X θ )h˙ r · (E∇X θ ), I60 = (h˙ r · ∇X θ )h˙ r · (E2 ∇X θ ), I61 = (h˙ r · ∇X θ ){h˙ r · [(SE + ES)∇r θ ] + ∇X θ · [(SE + ES)h˙ r ]},

(173) (174) (175)

2

(176) (177)

I62 = (h˙ r · ∇X θ ){h˙ r · [(S2 E + ES2 )∇X θ ] + ∇X θ · [(S2 E + ES2 )h˙ r ]}, (178) 2 2 2 2 ˙ ˙ ˙ I63 = (hr · ∇X θ ){hr · [(SE + E S)∇X θ ] + ∇X θ · [(SE + E S)hr ]}, (179) I64 = (h˙ r · ∇X θ ){h˙ r · [(S2 E2 + E2 S2 )∇X θ ] + ∇X θ · [(S2 E2 + E2 S2 )h˙ r ]}. (180)

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The lists of invariants (132)–(146) and (147)–(180) were determined by taking into consideration the work by Spencer [60]. Some of these invariants may not be independent (see the theory of spectral invariants proposed by Shariff [57–59]).

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20. Cauchy AL (1823) Recherches sur l’équilibre et le mouvement interieur des corps solides ou fluids, élastiques ou non élastiques. Bull Soc Philomath, 9–13—see also Oeuvres (complete works of Augustin Cauchy) 2, 300–304 21. Cauchy AL (1828) Sur les equations qui experiments les conditions d’équilibre ou le lois du mouvement intérieur, d’ un corps solide, élastique un non élastique. Ex de Math 3, 160–187— see also Oeuvres (complete works of Augustin Cauchy) 8, 195–226 22. Devendiran VK, Sandeep RK, Kannan K, Rajagopal KR (2017) A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem. Int J Solids Struct 108:1–10 23. Devendiran VK, Sandeep RK, Kannan K, Rajagopal KR (2017) Erratum to ‘A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem’. Int J Solids Struct 108, 1–10; Int J Solids Struct 124, 264–265 24. Dorfmann A, Ogden RW (2004) Nonlinear magnetoelastic deformations. Q J Mech Appl Math 57:599–622 25. Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mech 164:167–183 26. Freed AD (2014) Soft solids: a primer to the theoretical mechanics of materials. Birkhäusen, Heidelberg, New York, Dordrecht, London 27. Freed AD, Rajagopal KR (2016) A promising approach for modeling biological fibers. Acta Mech 227:1609–1619 28. Gokulnath C, Saravanan U, Rajagopal KR (2017) Representations for implicit constitutive relations describing non-dissipative response of isotropic materials. Z Angew Math Phys 68:129 29. Gou K, Mallikarjuna M, Rajagopal KR, Walton JR (2015) Modeling fracture in the context of a strain-limiting theory of elasticity: a single plane-strain crack. Int J Eng Sci 88:73–82 30. Grasley Z, El-Helou R, D’Amborsia M, Mokarem D, Moen C, Rajagopal KR (2015) Model of infinitesimal nonlinear elastic response of concrete subjected to uniaxial compression. J Eng Mech 141:04015008 31. Green G (1837) On the laws of reflexion and refraction of light at the common surface of two non-crystallized media. Trans Camb Phil Soc 7, 1–24 – see also Green G (1871) Mathematical papers of the late. In: Ferrers NM (ed), MacMillan and Company, London, pp 243–270 32. Green G (1841) On the propagation of light in crystallized media. Trans Camb Phil Soc 7, 121–140 – see also Green G (1871) Mathematical papers of the late. In: Ferrers NM (ed), MacMillan and Company, London, pp 293–311 33. Huang SJ, Dai HH, Rajagopal KR (2017) Wave patterns in a non-classical nonlinearly-elastic bar under Riemann data. Int J Non-Linear Mech 91:76–85 34. Ignaczak J, Ostoja-Starzewski M (2010) Thermoelasticity with finite wave speeds. Oxford mathematical monographs. Oxford University Press, Oxford 35. Johnson PA, Rasolofosaon PNJ (1996) Manifestation of nonlinear elasticity in rock: convincing evidence over large frequency and strain intervals from laboratory studies. Nonlin Proc Geophys 3:77–88 36. Kambapalli M, Kannan K, Rajagopal KR (2014) Circumferential stress waves in a non-linear cylindrical annulus in a new class of elastic materials. Q J Mech Appl Math 67:193–203 37. Kannan K, Rajagopal KR, Saccomandi G (2014) Unsteady motions of a new class of elastic solids. Wave Motion 51:833–843 38. Kovetz A (2000) Electromagnetic theory. University Press, Oxford 39. Kulvait V, Málek J, Rajagopal KR (2013) Anti-plane stress state of a plate with a V-notch for a new class of elastic solids. Int J Fract 179:59–73 40. Kulvait V, Málek J, Rajagopal KR (2017) Modelling gum metal and other newly developed titanium alloys within a new class of constitutive relations for elastic bodies. Arch Mech 69:223–241 41. Mollica F, Ventre M, Sarracino F, Ambrosio L, Nicolais L (2007) Implicit constitutive equations in the modeling of bimodular materials: an application to biomaterials. Comp Math Appl 53:209–218

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42. Montero S, Bustamante R, Ortiz-Bernardin A (2016) A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies. Acta Mech 227:601– 615 43. Ortiz-Bernardin A, Bustamante R, Rajagopal KR (2014) A numerical study of elastic bodies that are described by constitutive equations that exhibit limited strains. Int J Solids Struct 51:875–885 44. Rajagopal KR (2003) On implicit constitutive theories. Appl Math 48:279–319 45. Rajagopal KR (2007) The elasticity of elasticity. Z Angew Math Phys 58:309–317 46. Rajagopal KR (2011) Non-linear elastic bodies exhibiting limiting small strain. Math Mech Solids 16:122–139 47. Rajagopal KR (2011) Conspectus of concepts of elasticity. Math Mech Solids 16:536–562 48. Rajagopal KR (2014) On the nonlinear elastic response of bodies on the small strain range. Acta Mech 225:1545–1553 49. Rajagopal KR (2015) A note on the classification of anisotropy of bodies defined by implicit constitutive relations. Mech Res Commun 64:38–41 50. Rajagopal KR (2018) A note on the linearization of the constitutive relations of non-linear elastic bodies. Mech Res Commun 93:132–137 51. Rajagopal KR (2019) Rethinking the development of constitutive relations. In: preparation 52. Rajagopal KR, Srinivasa AR (2007) On the response of non-dissipative solids. Proc R Soc A 463:357–367 53. Rajagopal KR, Srinivasa AR (2009) On a class of non-dissipative materials that are not hyperelastic. Proc R Soc A 465:493–500 54. Rajagopal KR, Srinivasa AR (2015) On the use of compatibility conditions for the strains in linear and non-linear theories of mechanics. Math Mech Solids 20:614–618 55. Rajagopal KR, Walton JR (2011) Modeling fracture in the context of strain-limiting theory of elasticity: a single anti-plane shear crack. Int J Fract 169:39–48 56. Shariff MHBM (2008) Nonlinear transversely isotropic elastic solids: an alternative representation. Q J Mech Appl Math 61:129–149 57. Shariff MHBM (2017) The number of independent invariants for a n-preferred direction anisotropic solid. Math Mech Solids 22:1989–1996 58. Shariff MHBM (2019) The number of independent invariants for n symmetric second order tensors. J Elast 134:119–126 59. Shariff MHBM, Bustamante R (2015) On the independence of strain invariants of two preferred direction nonlinear elasticity. Int J Eng Sci 97:18–25 60. Spencer AJM (1971) Theory of invariants. In: Eringen CA (ed) Continuum physics, vol 1. Academic, New York, pp 239–353 61. Srinivasa AR (2015) On a class of Gibbs potential-based nonlinear elastic models with small strains. Acta Mech 226:571–583 62. Truesdell CA, Toupin R (1960) The classical theories. In: Flügge S (ed) Handbuch der Physik, vol III/I. Springer, Berlin, pp 226–902 63. Truesdell CA, Noll W (2004) The Non-linear field theories of mechanics, 3rd edn. Antmann SS (ed). Springer, Berlin 64. Zheng QS (1994) Theory of representations for tensor functions: a unified invariant approach to constitutive equations. Appl Mech Rev 47:545–587

Continuum Damage Mechanics—Modelling and Simulation Andreas Menzel and Leon Sprave

Abstract Continuum damage mechanics elaborates the continuum mechanics-based modelling and simulation of mechanical degradation effects. The objective of this contribution is to briefly review different aspects of continuum damage mechanics of solid continua with a focus on general modelling concepts and application to isotropic as well as anisotropic damage approaches on the one hand, and to discuss possible solution strategies in the context of finite element simulations on the other. In particular, viscous regularisation and gradient-enhanced regularisation—as a reduced form of general non-local theories—are considered. Several numerical examples including ductile damage, i.e. the coupling of damage with plasticity related phenomena, are addressed which show the applicability of the particular modelling and simulation frameworks highlighted.

1 Introduction Continuum damage mechanics deals with the continuum mechanics-based modelling and simulation of degradation processes affecting mechanical properties of materials. More specifically speaking, continuum damage mechanics deals with effective properties in between the initiation of microcracks and the evolution of macrocracks; see, e.g. [40]. The roots of continuum damage mechanics were laid by Kachanov [24], and were later further developed by Rabotnov [49]. It took until the late 70s for A. Menzel (B) · L. Sprave Department of Mechanical Engineering, Institute of Mechanics, TU Dortmund, Leonhard-Euler-Str. 5, 44227 Dortmund, Germany e-mail: [email protected] L. Sprave e-mail: [email protected] A. Menzel Division of Solid Mechanics, Department of Construction Sciences, Lund University, P.O. Box 118, 221 00 Lund, Sweden e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_8

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continuum damage mechanics to become a highly active research topic [23], and several monographs on continuum damage mechanics have been published since; see, e.g. [25, 33, 40]. The physical mechanisms and phenomenological effects related to continuum damage may, however, significantly differ depending on the particular material studied and loading conditions considered [40]. Early contributions on the modelling of damage effects considered mostly scalar damage variables and assumed isotropic damage evolution. In general, however, damage evolution is intrinsically anisotropic and may require the introduction of tensorial variables in order to model-related directional dependencies; see, e.g. [8, 12, 26, 32]. One approach to account for the anisotropy of damage effects is to combine the concepts of introducing a fictitious undamaged configuration and evolving tensorial quantities related to transformations between the fictitious and real configuration; see, e.g. [18, 36, 37, 39]. Ductile damage typically goes along with large plastic deformations. In view of metallic materials, damage effects typically stem from decohesion of the inclusion matrix interface or coalescence of microvoids. Two main approaches to the modelling of ductile damage have been established. On the one hand, Gurson [20] proposed a micromechanically motivated damage model based on void volume fraction, which was later extended by Tvergaard and Needleman [53]; see also [50] as well as [22, 58]. On the other hand, phenomenological damage models are often referred to the approach established by Lemaitre [33] and Krajcinovic [28]. As a main difference, Gurson-type damage models affect the evolution of plasticity related quantities, whereas Lemaitre-type damage models affect the elastic properties as well. Both types of modelling approaches have further been developed in recent research works; see, e.g. [11] for micromechanics-based models in the spirit of Gurson and [2, 27] for Lemaitre-type models, amongst others. For more details on the modelling of ductile damage and fracture phenomena the reader is referred to the review paper by Besson, [7], and references cited therein. Polycrystalline materials may experience creep damage effects, which originate from the nucleation of microcracks on grain boundaries under elevated temperatures and constant stress levels. The damage models of Kachanov [24] and Rabotnov [49] already included the description of such creep damage phenomena. Since the publication of these pioneering works, the theory of creep damage has been further extended; see, e.g. [9, 13], or the review contribution [21]. In the case where a material degrades under cyclic loading conditions, related damage effects are classified as fatigue damage. One may further classify fatigue damage based on the number of cycles before final failure occurs. Fatigue damage constitutes a wide field of research; for more detailed information the reader is referred to [34]. Further damage classifications have been established, such as creep fatigue damage and spall damage, which are discussed in the monographs cited above. Brittle damage commonly develops without plastic deformations and is exhibited in, e.g. rocks, concrete, ceramics and glass. Continuum damage models of Lemaitretype, without the incorporation of plasticity related effects, are capable of modelling

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effects related to brittle damage. For damage models specifically designed to capture brittle damage effects, see, e.g. [41] or the monographs [15, 28]. Moreover, damage in composite materials has intensively been studied since the early nineties; see, e.g. [30, 54]. In this context, Ladevèze [29] focused on the modelling of debonding effects of the fibre matrix interface and microcracking of the matrix itself, while Voyiadjis and Kattan proposed a generalised model incorporating damage and plasticity in composites [54]. For more detailed information on damage modelling of composites, the reader is referred to [55]. Another active field of research in continuum damage is the modelling of damage effects in biological materials. Biological materials typically possess complex (micro-) structures, such as layered vessel walls which can be considered as a composite with layers of different fibre reinforcements; see e.g. [3, 10, 47, 48]. Common to all damage models, respectively all continuum models including softening or degradation effects, is the possible loss of ellipticity of the governing equations; see, e.g. [35] for a discussion based on the investigation of the underlying acoustic tensor. In the case of related finite element simulations, this may lead to mesh-dependent results. To overcome such mesh dependencies, non-local damage theories have been established; see, e.g. [4, 5], where a non-local damage variable is introduced as an integral value of an only pointwise defined, local damage variable. An alternative approach to approximate fully non-local formulations is the introduction of additional gradient terms; see, e.g. [14, 45, 52]. Gradient damage approaches to include regularisation have been further developed in [16, 19, 43, 44, 51, 56, 57]. A different approach of regularisation was proposed in [17]. Instead of regularisation contributions in space, a regularisation in time is proposed by introducing ratedependent damage evolution. This type of regularisation is often denoted as viscous regularisation and intrinsically introduces dependency of the underlying parameters related to, say, relaxation time. The modelling approach has been further developed and used for several applications; see, e.g. [1, 6, 31, 42]. The article is organised as follows. Section 2 addresses the general continuum damage modelling framework. This includes basics of continuum mechanics, fundamental concepts related to the modelling of mechanical degradation effects in solids and aspects of regularisation approaches. Based on this, Sect. 3 focuses on the formulation of gradient-enhanced damage models, whereby isotropic and anisotropic damage as well as (isotropic) ductile damage are considered. Representative finite element-based numerical examples are highlighted for the particular models discussed. The contribution closes with a short summary in Sect. 4.

2 Continuum Modelling Framework Different basic modelling frameworks shall briefly be summarised in this section. This includes general aspects of continuum approaches and regularisation techniques in view of finite element applications.

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As this work proceeds, standard notation is used: let the scalar product between two real vectors v1 and v2 be denoted by v1 · v2 = s, whereas the standard dyadic product of v1 and v2 generates a rank-one second-order tensor, i.e. v1 ⊗ v2 = T. Higher order contractions are introduced by analogy; to give an example, [ v1 ⊗ v2 ] : [ v3 ⊗ v4 ] = [ v1 · v3 ] [ v2 · v4 ]. Furthermore, use shall be made of non-standard dyadic products defined as [ v1 ⊗ v2 ] ⊗ [ v3 ⊗ v4 ] = [ v1 ⊗ v3 ] ⊗ [ v2 ⊗ v4 ] and [ v1 ⊗ v2 ] ⊗ [ v3 ⊗ v4 ] = [ v1 ⊗ v3 ] ⊗ [ v4 ⊗ v2 ]. Moreover, the second-order identity tensor is denoted by I.

2.1 Finite Deformation Setting Let the motion of the solid body B considered be represented by the mapping x = ϕ(X, t), where x ∈ Bt are placements of particles in the deformed configuration and where X ∈ B0 corresponds to the vectors identified with their position in an initial configuration. Time shall be denoted by t ∈ R and the related material time derivative of some quantity • is introduced as •˙ = (d • /dt)|X . The related deformation gradient reads (1) F = ∇X ϕ with J = det F > 0. With theses definitions in hand, further deformation and strain measures can be introduced, such as the right Cauchy–Green tensor and the Finger tensor, C = FT · F,

b = F · FT ,

(2)

respectively their isochoric counterparts ¯ = J −2/3 C, C

b¯ = J −2/3 b with F¯ = J −1/3 F

(3)

for the three-dimensional case. Based on these deformation measures, typical strain tensors can be defined such as the Green–Lagrange strain tensor and the Almansi strain tensor, i.e. e = 21 [ I − b−1 ]. (4) E = 21 [ C − I ], As this work proceeds, the Helmholtz (free) energy function ψ0 is understood as a function of the deformation in form of C, due to objectivity, a set of referential and constant variables A, as well as a set of internal variables K, which may evolve in time. Adopting a hyperelastic form, the first Piola–Kirchhoff stress is introduced as P=

∂ψ0 with ψ0 (C, A, K). ∂F

(5)

The first Piola–Kirchhoff stress can be transformed to other stress tensors such as the Cauchy stress σ and the second Piola–Kirchhoff stress S as

Continuum Damage Mechanics—Modelling and Simulation

σ = P · cof(F−1 ),

S = F−1 · P = 2

235

∂ψ0 . ∂C

(6)

Based on theses stress formats, the mechanical contribution to the dissipation (in local form) reduces to ∂ψ0 ˙ · K ≥ 0, (7) D=− ∂K ˙ which restricts the form of evolution equations K(F, A, K). For the sake of completeness, mechanical equilibrium is represented here in referential, local and quasi-static form of the balance of linear momentum, i.e. 0 = ∇X · P + b0 ,

(8)

including mechanical volume forces b0 , whereas surface tractions t0 = P · N act on ∂B0 with N denoting the referential normal unit vector.

2.2 Continuum Damage Concepts The overall mechanical response of a specimen is influenced not only by its material behaviour and loading conditions but also by its geometrical properties. These may lead to localisation or geometric (global) softening effects such as necking even if the (local) material response remains in the hardening regime. This renders the experimental investigation of the real (local) material properties to become rather difficult, respectively the separation between integral (global) experimental data and (local) material behaviour. Continuum damage mechanics describes a (local) degradation of the material. This degradation may result from different mechanisms and phenomena as well as their evolution, typically related to the microstructure of the material and defects therein such as microcracks, cracks along grain boundaries in metals or interfaces at inclusions, growth of pores, loss of network connections in polymers, rupture and debonding of fibres embedded in a matrix, to name but a few. In this context, continuum damage mechanics describes the effects of these microstructures and mechanisms on the material response rather than the initiation and evolution of these microstructural phenomena themselves. Conceptually speaking, continuum damage mechanics is related to an a priori homogenisation concept, respectively represents an effective modelling framework and, moreover, damage is not to be mistaken for failure.

Brittle damage is typically referred to elastic states of deformation and prompt damage evolution which may result in macrocrack evolution or globally even in

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the failure of a specimen. Ductile damage, however, is identified with plastic loading states, whereby damage evolution may either interact with the evolution of plasticity related internal variables only, or additionally degrade the underlying elastic properties. Further categories of damage-related phenomena, such as creep damage as well as low- and high-cycle fatigue shall not be elaborated as this work proceeds. From a continuum modelling point of view, a (real) damaged configuration can be considered and linked to an effective (or rather fictitious) configuration. In view of related stress and strain measures, different mappings related to the transformation between quantities settled within the (fictitious) effective and (real) damaged configuration can be introduced. The most simple transformation reduces to weighting-related quantities with a scalar, respectively scalar-valued function f ∈ (0, 1],

(9)

whereby f = 1 represents the undamaged state and f  0 corresponds to the fully degraded state. Examples for f are f (d) = [ 1 − d ] and f (d) = exp(− ηd d) with the constant ηd > 0 and the internal variable d ≥ 0 typically restricted by d˙ ≥ 0. A straightforward homogenisation concept is referred to the assumption of strain states being equivalent within the real and the effective configuration (Voigt). Due to the idea that damage can be related to microcracks, respectively a reduction in cross-sectional area, the effective stresses take higher values when compared to the stresses present in the real configuration since the latter are referred to a larger, in other words not reduced, cross section. Denoting effective quantities by the notation  • the framework of equivalent strains results in . E= E,

S(E, d) = f (d)  S( E),

0 ( E) ψ0 (E, d) = f (d) ψ

(10)

0 ( E)/∂  E. with  S( E) = ∂ ψ Similarly, a state of stresses equivalent within the real and effective configuration may be assumed (Reuss). In consequence, damage evolution results in increasing strain measures within the real configuration. Moreover, let the effective free enthalpy 0 ( 0∗ ( S) = maxE {  S: E−ψ E) } so that the concept of stress function be denoted by ψ equivalence can be represented by . S = S,

E(S, d) =  E( S)/ f (d),

0∗ ( S)/ f (d) ψ0∗ (S, d) = ψ

(11)

with E(S, d) = ∂ψ0∗ (S, d)/∂S. Finally, the values of the Helmholtz (free) energy function referred to the real and effective configurations can be assumed to be equivalent (covariance). This assumptions generally causes neither the strain nor stress states within the real and effective configuration to coincide. The postulate of energy equivalence can, by analogy with (10) and (11), be summarised as .   ψ0 (E, d) = ψ 0 (E),

E= E/ f (d),

S = f (d)  S.

(12)

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With these relations at hand, the dissipation inequality in local form, cf. (7), takes the representation ∂ψ0 ˙ ∂ψ0 ∂ f ˙ ∂ψ0 ˙ f =− d=− d ≥ 0, ∂f ∂ f ∂d ∂d

D=−

(13)

with − ∂ψ0 /∂d possessing the units of energy and taking the interpretation as damage energy release rate by analogy to the fracture energy release rate established for fracture mechanics. In order to distinguish between admissible loading steps, at which the material undergoes purely elastic deformations without damage evolution, and loading steps which result in damage evolution, an admissible domain is commonly introduced. Within the framework of standard dissipative materials this convex set can be defined as a function of the driving forces either conjugate to the function f (d) or to the internal variable d. For the latter approach, use of notation q = − ∂ψ0 /∂d is made and the admissible domain is formally introduced as Ed = { q | d (q; d) ≤ 0 }

(14)

with the function d (q; d) being convex in q and with d (0; d) ≤ 0. Based on this, an associated evolution equation in rate-independent form follows as ∂d d˙ = λ˙ d ≥ 0, ∂q

(15)

for λ˙ d taking the interpretation as a Lagrange multiplier, together with an initial condition for d |t0 and the Karush–Kuhn–Tucker conditions λ˙ d ≥ 0,

d (q; d) ≤ 0,

λ˙ d d (q; d) = 0.

(16)

Remark 1 To give an example, adopting the concept of strain energy equivalence, cf. (12), in combination with St. Venant–Kirchhoff-type elasticity results in 0 ( ψ E) =

1 2

 E : E˜ :  E=

1 2

f 2 (d) E : E˜ : E =

1 2

E : E : E = ψ0 (E, d)

(17)

with E˜ being the constant effective fourth order and fully symmetric elasticity tensor, the isotropic form of which is E˜ =  λI ⊗ I + μ [ I ⊗ I + I ⊗ I ],

(18)

˜ Based on this, the effective and real stresses read and with E(d) = f 2 (d) E. 0 ( E) ∂ψ  S= = E˜ :  E,  ∂E

S=

∂ψ0 (E, d) = f 2 (d) E˜ : E = E(d) : E , ∂E

(19)

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so that E =  E/ f (d) results in S = f (d)  S, cf. (12). Remark 2 The transformation between real and effective quantities in terms of f (d) is commonly identified with isotropic damage and can be extended to anisotropic damage by means of more general transformations. To give an example within the framework of strain energy equivalence, a positive definite fourth-order transformation tensor T possessing minor symmetries can be introduced so that E, E = T−1 : 

S = T : S.

(20)

As a special case, let T be determined by a symmetric and positive definite secondorder tensor T, E = T−1 ·  E · T−1 , E = [ T ⊗ T ]−1 : 

S = [T ⊗ T ] :  S = T · S · T.

(21)

The assumption of a St. Venant–Kirchhoff form, as discussed in Remark 1, renders the relation     S = T · E˜ :  E · T = T · E˜ : [ T · E · T ] · T = E : E,

(22)

so that the real elasticity tensor takes the representation

with

E(T) = [ T ⊗ T ] : E˜ : [ T ⊗ T ],

(23)

μ [ T2 ⊗ T2 + T2 ⊗ T2 ] E(T) =  λ T2 ⊗ T2 + 

(24)

for the case in which the effective elasticity tensor represents isotropic response, cf. (18). Isotropic damage is included as a special case when assuming T ∝ I, specifically √ T = f (d) I; see Remark 1. Moreover, one may introduce H = T2 as a second-order and positive definite tensor representing an internal variable related to damage. In that case, the Helmholtz (free) energy function can be introduced as a function of the basic invariants IiEH = [ E · H ]i : I for i = 1, 2, 3. Moreover, damage evolution can be included by analogy with (14), (15) and (16) with d (Z; H) for Z = − ∂ψ0 /∂H— whereby d (Z; H) can be introduced as a function of the basic invariants IiZH = ˙ = λ˙ d ∂d /∂Z. [ Z · H ]i : I for i = 1, 2, 3—and H

2.3 Regularisation Concepts The simulation of boundary-value problems undergoing inhomogeneous states of deformation is strongly influenced by the properties of the particular damage model considered. In general, material degradation may lead to ill-posedness of

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239

the boundary-value problem, respectively loss of ellipticity of the field equations considered. Accordingly, related finite element simulations may show mesh dependencies. In order to overcome such problems, different regularisation concepts are established—from simple approaches including formulations with parameters depending on a representative finite element size to mathematically rigorous frameworks such as non-local theories. The introduction of viscous, i.e. rate-dependent damage evolution—in contrast to the rate-independent form in (15)—changes the type of the underlying partial differential equation of the boundary-value problem considered. On the one hand, this may render the overall problem to become well posed but also changes the underling physics, respectively material behaviour modelled on the other hand. An example of a damage evolution equation in rate-dependent Perzyna-type form is   1 1 − exp − ηd  g(q) − d d˙ = τ

(25)

with τ being a positive relaxation time type parameter, ηd a positive constant and with  • = max{ 0, • }. Moreover, [ g(q) − d ] can be interpreted as a particular choice for d , cf. (14). Karush–Kuhn–Tucker conditions, as summarised in (16), are not to be considered for this type of viscous damage evolution. This particular choice of evolution equation, (25), limits the damage rate to max d˙ = 1/τ for exp(•)  0 and d˙  0 for exp(•) 1. A general drawback of viscous damage formulations, however, is their dependency on the loading rate, respectively the relaxation time type parameter τ . The enhancement of local models by non-local interaction contributions may regularise the underlying governing equations. A general non-local contribution is based on integral expressions and weighting functions which, in referential form, can be introduced as w0 (), with  representing the distance to the particular referential position X considered. Different quantities of interest can be introduced in non-local form, such as energy expressions, driving forces, deformation measures or internal variables such as a damage variable, i.e.   d(X) =

P0

w0 () d(X + ) dV  w0 () dV

(26)

P0

with P0 ⊆B0 . A Taylor series expansion at X results in d(X + ) = d(X) + ∇X d(X) ·  + 21 ∇X ∇X d(X) : [  ⊗  ] + . . . ,

(27)

which, under the assumption of isotropic weighting functions w0 () and together with (26), yields

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d = d + ∇X · [ cd ∇X d ] + . . .

(28)

(see, e.g. [38, 46]). Alternatively, one may use the relation d = d − ∇X · [ cd ∇X d] − . . .

(29)

In view of inhomogeneous boundary-value problems, e.g. by means of the finite element method, (28) or (29) is solved as an additional field equation together with the balance of linear momentum (8). Remark 3 The rate-dependent Perzyna-type form (25) can also be extended to anisotropic damage in the context of Remark 2. To give an example, anisotropic damage evolution can be introduced as    ˙ = 1 1 − exp − ηd d  H τ

(30)

with  = sign(i ) Ni ⊗ Ni whereby i and Ni may be chosen as the principal values and directions of ∂Z d . Remark 4 Different non-local forms can be introduced, e.g. a spatial integration over Pt ⊆ Bt and parametrisation in x in contrast to the referential representation in (26). Moreover, the flux terms in (28) and (29) may also be introduced in different forms similar to introducing either the spatial heat flux vector proportional to the spatial temperature gradient or, alternatively, to introducing the referential heat flux vector proportional to the referential temperature gradient (independent of the Piola transformation linking both).

3 Gradient-Enhanced Damage Modelling The finite element method is well established to solve nonlinear initial boundaryvalue problems. The formulation of non-local continuum damage models based on field equations (8) and (28), respectively (29), results in a coupled multi-field problem which can be solved monolithically. Apart from possible continuity requirements, the main problem with solving the non-local damage- related field equation lies in the incorporation of the Karush–Kuhn–Tucker conditions related to damage evolution, cf. (16). In order to avoid such inequality constraints at (global) finite element level, so-called gradient-enhanced damage models have been developed. This approach is also established as the micromorphic modelling framework. The main idea is to introduce an additional field equation together with a penalty term that links this global field to a local (internal) variable. In contrast to (some) other non-local modelling and simulation approaches, this link of the local variable with the global field variable allows for a direct computation of the gradient of the global field variable rather than calculating gradients of (local) internal variables.

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241

In other words, the computation of the gradient of the local variable is not required within the discrete setting and, moreover, the modelling framework does not include any higher order gradients. For the problem at hand, this local variable coincides with the damage variable. As a key advantage, inequality constraints related to damage evolution can be considered at local level (integration point) rather than on global level (finite element). In this context, let the Helmholtz (free) energy be additively decomposed into local, non-local and penalty contributions, namely pen

ψ0 (F, φ, ∇X φ, d, A) = f (d) ψ0loc (F, A) + ψ0nloc (∇X φ; F) + ψ0 (φ, d) ,

(31)

wherein φ denotes the global, respectively non-local damage field and where d represents the local, respectively internal damage variable. In view of the total energy potential, dead external loads shall be assumed so that 

(ϕ, F, φ, ∇X φ, d, A) = +

ψ0 (F, φ, ∇X φ, d, A) B0 sur

vol ext (ϕ) + ext (ϕ)

dV (32)

with 

vol ext (ϕ) = −

B0

 b0 · ϕ dV and sur ext (ϕ) = −

∂B0

t0 · ϕ d A.

(33)

Related Euler–Lagrange equations are derived based on minϕ,φ , respectively δϕ = 0 and δφ = 0, which results in 



0=



P : ∇X δϕ dV − B0

 0=

b0 · δϕ dV − 

B0

Y · ∇X δφ dV − B0

∂B0

t0 · δϕ d A,

Y δφ dV,

(34) (35)

B0

whereby homogeneous Neumann boundary conditions are assumed for (35) and the following abbreviations are introduced: P=

∂ψ0 , ∂F

Y=

∂ψ0 , ∂∇X φ

Y =−

∂ψ0 . ∂φ

(36)

Alternatively, the variational framework can be represented in spatial form with the flux terms P and Y related to their spatial counterparts by a Piola transformation, whereas the volume terms b0 and Y are related to their spatial counterparts by means of J . The penalty contribution to the energy function can be chosen straightforwardly, i.e.

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A. Menzel and L. Sprave pen

ψ0 (φ, d) =

1 2

βd [ φ − d ]2 ,

(37)

linking the local damage variable d to the non-local damage variable φ. Other forms or Lagrange multiplier-based formulations can be chosen as well. The choice of the non-local energy contribution, however, may influence the computational properties of the model and different forms, by analogy with the heat flux vector, are established. The referential gradient of the non-local damage field can be included as ψ0nloc (∇X φ; F) =

1 2

cd ∇X φ · ∇X φ =

1 2

cd ∇x φ · b · ∇x φ,

(38)

where cd is the so-called regularisation parameter which, in addition, also affects the (constitutive) response of the model. An alternative form of the gradient contribution is given by ψ0nloc (∇X φ; F) =

1 2

cd ∇X φ · C−1 · ∇X φ =

1 2

cd ∇x φ · ∇x φ.

(39)

The choice of non-local energy contribution renders the non-local damage field equation of Laplace type. To be specific, (38) results in the flux term Y = cd ∇X φ, whereas (37) renders the volume term as Y = − βd [ φ − d ], so that (35) is specified as   0= cd ∇X φ · ∇X δφ dV + βd [ φ − d ] δφ dV. (40) B0

B0

Integration by parts and Cauchy’s theorem, together with the assumption of homogeneous Neumann boundary conditions, provide the related local form 0 = cd ∇X · [ ∇X φ ] − βd [ φ − d ],

(41)

which clearly is of Laplace type. The multi-field Bubnow–Galerkin-based finite element implementation may include identical shape functions for the approximation of the referential geometry X, the unknown placements ϕ and the non-local damage field φ. In general though, different interpolation functions can be applied to the approximation of φ in the context of a mixed method—typically one order lower than the approximation of ϕ. It turns out, however, that the formulation is rather insensitive to the application of different orders of the shape functions for the different fields, so that it is convenient to use identical shape functions for all fields. The coupled discrete system of nonlinear equations is monolithically solved within a Newton–Raphson scheme.

3.1 Isotropic Gradient-Enhanced Damage To set the stage, an isotropic gradient-enhanced damage model is discussed. This includes the specification of the elasticity related energy contributions, the local

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243

damage model and finite element simulations. The elastic energy contribution shall be of neo-Hookean type together with additional fibre-related contributions, i.e. ψ0loc (F, A0i ) =

1 2

μ [ I1 − 3 ] − μ ln(J ) + 21 λ ln2 (J )

2    1 k1  exp k2 E i 2 − 1 + 2 k2 i=1

(42)

with I1 = C : I and E i = χ I1 + [ 1 − 3 χ ] I4i − 1 for I4i = C : A0i = C : [ a0i ⊗ a0i ], whereby a0i are referential unit vectors. The value χ = 1/3 renders the model to be isotropic. The non-local contribution used is based on the spatial gradient of the non-local damage field, cf. (39), together with the penalty contribution introduced in (37). The damage function included introduces a damage threshold value dd such that the current value of the internal damage variable d must exceed dd in order to activate damage evolution. As a special case the value dd = 0 can be chosen. Moreover, the previously mentioned exponential form is adopted for the damage function, namely f (d) = exp(ηd [ dd − d ])

(43)

with ηd > 0. Next, the admissible elastic domain shall be determined by the function d = q − d ≤ 0 so that

d˙ = λ˙ d ,

(44)

cf. (15). In order to numerically solve this damage evolution equation, an implicit Euler backward scheme can be applied, which results in dn+1 = dn + Δλd with Δλd = Δt λ˙ d n+1 ,

(45)

whereby the additional index refers to the current and previous time step, i.e. tn+1 = tn + Δt. In the case of damage evolution, the discrete multiplier Δλd is computed based on d n+1 = 0, which in general is a nonlinear function in Δλd solved for by means of a Newton–Raphson scheme.

3.1.1

Numerical Example

In order to illustrate the performance of the isotropic gradient-enhanced damage framework, a finite element example is discussed in the sequel, whereby the benchmark-type setting of a plate with a hole is considered. The material parameters chosen are summarised in Table 1, whereas the geometry and loading conditions considered are provided in Fig. 1 with the specimen being fully clamped at the bottom and loaded by Dirichlet boundary conditions in the ‘longitudinal’ direction and ‘transverse’ displacements at the top being constrained.

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Table 1 Parameters used for the simulation of the isotropic plate with a hole. The value χ = 1/3 renders the elastic response to be isotropic and independent of the choice of referential direction vectors a01 and a02 Parameter Value Unit μ λ k1 k2 χ

14.5805 729.025 218.708 150 1/3

kPa kPa kPa – –

cd βd ηd dd

{0, 10, 100, 500, 1000} 1000 4 0.2

N kPa – –

For the results of the finite element simulations shown in Fig. 1, the non-local damage field variable φ is initialised to coincide with the damage threshold value dd , i.e. φ |t0 = dd . The results clearly indicate the dissipative response under cyclic loading induced by damage evolution. A main advantage of the gradient-enhanced damage formulation lies in the generation of mesh-independent simulation results upon further mesh refinement and for sufficiently large regularisation parameters cd . In order to highlight this key property, simulation results referring to three different discretisations of one-eighth of the specimen are shown in Fig. 2, whereby meshes with 400, 1600 and 3200 elements are chosen. As illustrated by the contour plots of the damage function f , the response turns out to be mesh-independent. In general, the value chosen for the regularisation parameter cd influences the simulations results. Larger values for cd increase the dissipation area, whereas small values for cd may not sufficiently regularise the boundary-value problem. The influence of the regularisation parameter cd on the force–displacement curve for the investigated problem of a plate with a hole is shown in Fig. 3. The value cd = 0 N does not sufficiently regularise the problem so that no convergence was obtained for the applied displacement exceeding u = 14.22 mm.

3.2 Anisotropic Gradient-Enhanced Damage In the following, anisotropic damage shall be related to the different degradation of matrix material and fibre contributions, not to be mistaken with the anisotropic damage framework discussed in Sect. 2.2. In this context, the Helmholtz (free) energy function is extended and shall include three damage fields related to the matrix response and the degradation of two fibre contributions, i.e.

Continuum Damage Mechanics—Modelling and Simulation

(a)

(b)

(c)

(d)

245

Fig. 1 Isotropic plate with a hole. a Dimensions: R = 100 mm, W = H = 200 mm, T = 20 mm. b Cyclic loading history. c Force–displacement response and mesh insensitivity—solid black curves represent elastic response of the isotropic neo-Hookean matrix (lower solid black curve) and elastic response additionally including (isotropic) fibre contributions (upper solid black curve). d Force– displacement response under cyclic loading for element types Q2Q1 and Q1Q1 using 400 elements for the discretisation of one-eighth of the specimen. Here, f denotes the sum of reaction forces and the regularisation parameter is cd = 500 N. Reprinted from [56, Elsevier] with permission

ψ0 (F, φ j , ∇X φ j , d j , A0i ) =

ψ0vol (J )

+

¯ f 0 (d0 ) ψ0iso (F)

+

2

¯ A0i ) f i (di ) ψ0iani (F,

i=1

+

2 j=0

ψ0nloc j (∇X φ j ; F) +

2

pen

ψ0 j (φ j , d j ) .

(46)

j=0

(46) extends the two-field formulation discussed in Sect. 3 to a four field formulation. The respective energy contributions are introduced as

246

(a)

A. Menzel and L. Sprave

(b)

(c)

Fig. 2 Isotropic plate with a hole. Contour plots of the damage function f for prescribed top displacement of u = 30 mm for three different discretisations: a 400 elements, b 1600 elements, c 3200 elements, for one-eighth of the specimen. Here, f d represents the damage function introduced in (43) and the regularisation parameter is cd = 500 N. Reprinted from [56, Elsevier] with permission

Fig. 3 Isotropic plate with a hole. Force–displacement response under monotonic loading for different values of the regularisation parameter cd using 400 elements for the discretisation of one- eighth of the specimen. Solid black curves represent the elastic response of the isotropic neoHookean matrix (lower solid black curve) and the overall elastic response composed of the isotropic neo-Hookean and the anisotropic exponential model (upper solid black curve). Here, f denotes the sum of reaction forces. Reprinted from [56, Elsevier] with permission

Continuum Damage Mechanics—Modelling and Simulation

ψ0vol = ψ0iso

=

ψ0iani =

1 2 1 2

247

λ [ J − 1 ]2 , μ [ I¯1 − 3 ],

(47) (48)

   1 k1i  exp k2i  E¯ i − 1 2 − 1 , 2 k2i

(49)

¯ : I and E¯ i = C ¯ : A0i for the generalised structural tensors being defined with I¯1 = C by a distribution parameter and referential unit vectors, i.e. A0i = χ I + [ 1 − pen 3 χ ] a0i ⊗ a0i . The remaining respective energy contributions ψ0nloc j and ψ0 j are introduced by analogy with (39) and (37). Moreover, the damage functions f j (d j ) are chosen in the form of (43). It is interesting to note, that the evolution of the three damage variables d j , by analogy with (44), is not coupled. In other words, three single surface damage models are introduced rather than one multi-surface damage model. The algorithmic treatment is based on the Euler backward scheme sketched in (45).

3.2.1

Numerical Example

The subsequently elaborated example refers to the simulation of a plate with a hole with geometry as discussed in Sect. 3.1.1; see Fig. 1. The particular referential fibre directions considered in the following are aligned with the X and Y axes, i.e. a01 = e1 = e X and a02 = e2 = eY . The parameters used for the simulations are summarised in Table 2. Boundary and loading conditions are displayed in Fig. 4. The anisotropic plate is loaded by a complex non-monotonic loading path. At first, cyclic loading is applied in the Y -direction such that damage evolves. Thereafter

Table 2 Parameters used for the simulation of the anisotropic plate with a hole Parameter Value Unit μ λ k11 = k12 k21 = k22 χ

15 150 20 2.1 0

kPa kPa kPa – –

cd0 cd1 = cd2 βd0 βd1 = βd2 ηd0 ηd1 = ηd2 dd0 dd1 = dd2

250 1000 1000 1000 2 7 2 0.5

N N kPa kPa – – – –

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A. Menzel and L. Sprave

(a)

(b)

(c)

(d)

Fig. 4 Anisotropic plate with a hole. a Loading path for displacements imposed on the top side in the Y -direction. b Loading path for displacements imposed on the right side in the X - direction. Solid black curves in c, d represent elastic response of the isotropic neo-Hookean matrix (lower curve) and the response including fibres (upper curve). Dashed lines mark the respective onset of damage 0  1 and – 1 . 2 d Load–displacement evolution. c Load–displacement response along paths – 2 . 3 Here, f x and f y denote the sum of the reaction forces in the X - and response along path – Y -directions. The discretisation consists of 7330 elements for one-eighth of the specimen and the global Newton–Raphson scheme is combined with an arc length method to solve the boundary-value problem. Reprinted from [48, Elsevier] with permission

the specimen is further monotonically loaded in the X -direction to further activate damage evolution; see Fig. 4. The different activations of the respective damage fields at different states of loading are illustrated in Fig. 5. The contour plots indicate that all damage fields are activated but differently distributed, which underlines the overall anisotropic response and capabilities of the anisotropic gradient-enhanced damage modelling framework established.

Continuum Damage Mechanics—Modelling and Simulation

249

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5 Anisotropic plate with a hole. The first row highlights contour plots of the damage functions 1 cf. Fig. 4. The second row displays contour plots of the damage functions f 0,1,2 at loading state , 3 cf. Fig. 4. a, d show contour plots of f 0 . b, e show contour plots of f 1 . c, f 0,1,2 at loading state , f show contour plots of f 2 . Reprinted from [48, Elsevier] with permission

3.3 Isotropic Ductile Gradient-Enhanced Damage Metal plasticity is often accompanied by ductile damage effects so that related models should account for plasticity and damage as well as for the interaction of both phenomena. A common approach to include such interactions is the introduction of an effective stress quantity driving the evolution of plasticity related deformation contributions. The effective stresses increase with increasing degradation of the material compared to the real stress state entering the balance of linear momentum equation, respectively, mechanical equilibrium, cf. (8). As this work proceeds a Helmholtz (free) energy function of the form p

ψ0 (F, φ, ∇X , F p , α, d) = f vol (d) ψ0vol (ε e ) + f iso (d) ψ0iso (¯ε e ) + ψ0 (α) pen

+ ψ0nloc (∇X φ; F) + ψ0 (φ, d),

(50)

is used, whereby α represents an internal variable enhancing the model with proportional hardening effects. Moreover, a multiplicative decomposition of the deformation gradient F into an elastic contribution Fe and an internal variable related to plastic deformation contributions F p is assumed, i.e.

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F = Fe · F p and ε e =

1 2

ln(be ) with be = Fe · FeT ,

(51)

as well as ε¯ e = 21 ln(b¯ e ). With theses relations at hand, the local contributions shall be further specified as ψ0vol (ε e ) =

1 2

κ [ ε e : I ]2 ,

ψ0iso (ε e ) = μ ε¯ e : ε¯ e ,

p

ψ0 (α) =

1 2

h α2 .

(52)

Formally, different damage functions f vol (d) and f iso (d) are introduced, so that the choice of f vol (d) can influence the volumetric contribution to the damage driving force (energy) by analogy with the degree of triaxiality of stress states considered to influence ductile damage evolution. The particular damage functions are chosen to be of exponential type, i.e. f • (d) = exp(− η• d) .

(53) pen

Moreover, the non-local and penalty contributions ψ0nloc and ψ0 shall take the formats according to (38) and (37). For the problem at hand the dissipation inequality in local form, see (7), can be shown to reduce to D = mT : l p + γ α˙ + q d˙ ≥ 0

(54)

with mT = 2

∂ψ0 ∂ψ0 ∂ψ0 , q=− and l p = Fe · F˙ p · F−1 · be , γ = − ∂be ∂α ∂d

(55)

and, in addition, P = mT · F−T . Next, admissible domains in view of damage and plasticity evolution are introduced. To be specific, two different domains are included which in general allows us to conveniently transform the model from ductile to brittle damage evolution by choosing appropriate material parameters. By analogy with (14) an admissible domain related to damage evolution is introduced which shall be defined by the function d (q; d) = q − qmax [ 1 − f q (d) ]m with

f q (d) = exp(− ηq d) .

(56)

In view of the plasticity related evolution, the admissible domain is introduced as E p = {mT , γ |  p (mT , γ ; d) ≤ 0 }

(57)

and defined by a von Mises-type function including a linear hardening contribution, namely

Continuum Damage Mechanics—Modelling and Simulation

T 

mdev

 0  2

 p (m , γ ; d) =

f (d) − 3 σy − γ with m T

251

f m (d) = exp(− ηm d) (58)

T and mdev = mT − 13 [ mT : I ] I. The contribution mT / f m (d) represents an effective stress contribution increasing with increasing material degradation, which induces coupling between plasticity and damage evolution in addition to the coupling of plasticity and damage contributions present in the Helmholtz (free) energy. Furthermore, associated evolution equations shall be used as this work proceeds, which take the representations

l p = λ˙ p

∂ p , ∂mT

α˙ = λ˙ p

∂ p ∂d and d˙ = λ˙ d . ∂γ ∂q

(59)

Due to the presence of the two separate admissible domains Ed and E p , i.e. one damage-related domain and the other plasticity related, together with the Lagrange multipliers λ˙ d and λ˙ p , two sets of Karush–Kuhn–Tucker conditions are included, namely λ˙ d ≥ 0, λ˙ p ≥ 0,

d (q; d) ≤ 0,

λ˙ d d (q; d) = 0,

(60)

 p (m , γ ; d) ≤ 0,

λ˙ p  p (mT , γ ; d) = 0.

(61)

T

Their algorithmic treatment is based on an active set strategy. Alternative solution approaches can be based on Fischer–Burmeister type or other NCP methods. The numerical integration of the underlying evolution equations is implemented by means of an exponential scheme, which ensures plastic incompressibility, and Euler backward integration. In addition to (45) this results in F p n+1 = exp(Δt L p n+1 ) · F p n and αn+1 = αn + Δλ p ,

(62)

whereby L p = Fe−1 · l p · Fe . The resulting set of nonlinear equations is solved by means of a Newton–Raphson scheme.

3.3.1

Numerical Example

The numerical example elaborated in the following refers to a geometry of a notched plate. The plate is clamped at the bottom and loaded by Dirichlet boundary conditions, respectively, applied displacements at the top. Due to symmetry, the discretisation can be reduced to one-eighth of the specimen, cf. Fig. 6 where the dimensions of the notched plate considered are also mentioned. Linear shape functions are used to approximate both fields of degrees of freedom (Q1Q1), whereby an F-bar approach is adopted. Moreover, the parameters used are summarised in Table 3. Figure 6 shows simulation results for different discretisations and different regularisation parameters cd by means of contour plots of the damage field φ at the

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(a)

(b)

Fig. 6 Notched plate. Radius = 50 mm, Width = Length = 200 mm, Thickness = 20 mm. Discretisation of one-eighth of the specimen. Contour plots of the damage field variable φ for a 12150 (left) and 5400 (right) elements with cd = 200 N at u max = 10 mm and b cd = 50 N (left) and cd = 200 N (right) for 5400 elements at u max = 10 mm Table 3 Parameters used for the simulation of the notched plate. For completeness the relations between elasticity related material parameters are mentioned, namely κ = E/3[ 1 − 2 ν ] and μ = E/2[ 1 + ν ] Parameter Value Unit E ν

208 0.3

GPa –

σy0 h qmax m

809 6000 2 2/3

MPa MPa MPa –

cd βd ηvol = ηiso = ηm ηq

{50, 200, 500} 500 0.7 10

N MPa – –

Continuum Damage Mechanics—Modelling and Simulation

cd =200 cd =500 nel =1350 nel =5400 nel =12150

600

400

F [kN]

Fig. 7 Notched plate. Force–displacement curves for different regularisation parameters cd = {200, 500} N and discretisations of one-eighth of the specimen with {1350, 5400, 12150} elements. Here, F denotes the sum of reaction forces in the (longitudinal) loading direction

253

200

0

0

5

10

u[mm]

maximum loading state considered. To be specific, Fig. 6a compares the distribution of φ for a discretisation consisting of 12150 elements (left) with a discretisation consisting of 5400 elements (right). The distribution of φ turns out to be quasi-identical, which underlines the fundamental property of the gradient-enhanced framework to generate mesh-independent results. Moreover, Fig. 6b highlights the influence of the regularisation parameter cd . The smaller value for cd , Fig. 6b, left, shows a small damage zone together with higher values of φ and a more localised deformation (larger deformation of the ‘first element row’) than the one in the simulation results that are based on a larger value for cd , Fig. 6b, right. Overall load–displacement curves are displayed in Fig. 7 whereby, by analogy with Fig. 6, different discretisations and regularisation parameters are investigated. The results underline the mesh-independent properties of the gradient-enhanced modelling framework on the one hand, and the influence of the regularisation parameter cd on the (integral) force–displacement relation on the other.

4 Summary Different modelling approaches for isotropic and anisotropic continuum damage are discussed in this article. Special focus is set on regularisation concepts in the context of finite element simulations so that mesh-independent simulation results can be generated. The numerical examples addressed show the applicability of the different gradient-enhanced continuum damage models investigated. In view of future research work, the further elaboration of (computational) homogenisation approaches to link damage models to micromechanical phenomena and mechanisms is of key interest. Additional coupling effects such as thermomechanical coupling or diffusion processes driving damage evolution are an active field of research in addition to establishing further advanced and regularised

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damage models including anisotropic damage evolution together with anisotropic elastic degradation and microcrack opening and closure, to name but a few interesting and highly relevant phenomena. Moreover, the verification and validation of continuum damage models, and in particular of regularised continuum damage models, are of key importance. This involves advanced parameter identification approaches on the one hand, and advanced experimental investigations on the other hand, including the measurement of inhomogeneously distributed (displacement) fields. Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the collaborative research centre SFB/TRR 188—Projektnummer 278868966—under subproject C02 is gratefully acknowledged.

References 1. Allix O (2012) The bounded rate concept: a framework to deal with objective failure predictions in dynamic within a local constitutive model. Int J Damage Mech 22:808–828 2. Balieu R, Kringos N (2015) A new thermodynamical framework for finite strain multiplicative elastoplasticity coupled to anisotropic damage. Int J Plast 70:126–150 3. Balzani D, Brinkhues S, Holzapfel GA (2012) Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls. Comp Meth Appl Mech Eng 213–216:139–151 4. Bažant ZP, Belytschko TB, Chang T-P (1984) Continuum theory for strain-softening. J Eng Mech 110:1666–1692 5. Bažant ZP, Pijaudier-Cabot G (1988) Nonlocal continuum damage, localization instability and convergence. J Appl Mech 55:287–293 6. Berdin C, Besson J, Bugat S, Desmorat R, Feyel F, Forest S, Lorentz E, Maire E, Pardoen T, Pineau A, Tanguy B (2004) Local approach to fracture. Presse des Mines, Paris 7. Besson J (2010) Continuum models of ductile fracture: a review. Int J Damage Mech 19:3–52 8. Betten J (1992) Application of tensor functions in continuum damage mechanics. Int J Damage Mech 1:47–49 9. Betten J, Sklepus S, Zolochevsky A (1998) A creep damage model for initially isotropic materials with different properties in tension and compression. Eng Fract Mech 59:623–641 10. Calvo B, Peña E, Martinez MA, Doblaré M (2007) An uncoupled directional damage model for fibred biological soft tissues. Formulation and computational aspects. Int J Num Meth Eng 69, 2036–2057 11. Cao T-S, Mazière M, Danas K, Besson J (2015) A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation. Int J Solids Struct 63:240–263 12. Chaboche J-L (1993) Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage. Int J Damage Mech 2:311–329 13. Cocks ACF, Leckie FA (1987) Creep constitutive equations for damaged materials. In: Wu TY, Hutchinson JW (eds) Advances in applied mechanics, vol 25. Elsevier, Amsterdam, pp 239–294 14. De Borst R, Sluys LJ, Muhlhaus H-B, Pamin J (1993) Fundamental issues in finite element analyses of localization of deformation. Eng Comput 10:99–121 15. Desmorat R, Gatuingt F, Ragueneau F (2007) Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials. Eng Fract Mech 74:1539–1560 16. Dimitrijevic BJ, Hackl K (2008) A method for gradient enhancement of continuum damage models. Tech Mech 28:43–52

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17. Duvaut G, Lions JL, John CW, Cowin SC (1976) Inequalities in mechanics and physics. Springer, Berlin 18. Ekh M, Menzel A, Runesson K, Steinmann P (2003) Anisotropic damage with the MCR effect coupled to plasticity. Int J Eng Sci 41:1535–1551 19. Forest S (2009) Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech 135:117–131 20. Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part I–yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15 21. Hayhurst DR (2001) Computational continuum damaged mechanics: its use in the prediction of creep in structures - past, present and future. In: Murakami S, Ohno N (eds) IUTAM symposium on creep in structures, vol 86. Solid mechanics and its applications. Springer, Dordrecht, pp 175–188 22. Holopainen S (2014) Influence of damage on inhomogeneous deformation behavior of amorphous glassy polymers. Modeling and algorithmic implementation in a finite element setting. Eng Fract Mech 117, 28–50 23. Hult J (1979) Continuum damage mechanics—capabilities, limitations and promises. In: Easterling KE (ed) Mechanisms of deformation and fracture. Pergamon, Oxford, pp 233–247 24. Kachanov L (1958) Time of the rupture process under creep conditions [in Russian]. Izv Akad Nauk SSSR Otdel Technich Nauk 8, 26–31 25. Kachanov L (1986) Introduction to continuum damage mechanics. Springer, Dordrecht 26. Kattan PI, Voyiadjis GZ (1990) A coupled theory of damage mechanics and finite strain elastoplasticity—I. Damage and elastic deformations. Int J Eng Sci 28, 421–435 27. Kiefer B, Waffenschmidt T, Sprave L, Menzel A (2018) A gradient-enhanced damage model coupled to plasticity—multi-surface formulation and algorithmic concepts. Int J Damage Mech 27:253–295 28. Krajcinovic D (1996) Continuum models. In: Krajcinovic D (ed) Damage mechanics, northholland series in applied mathematics and mechanics, vol 41. Elsevier, Amsterdam, pp 415–612 29. Ladevèze P, Allix O, Deü J-F, Lévêque D (2000) A mesomodel for localisation and damage computation in laminates. Comp Meth Appl Mech Eng 183:105–122 30. Ladevèze P, LeDantec E (1992) Damage modelling of the elementary ply for laminated composites. Compos Sci Technol 43:257–267 31. Langenfeld K, Junker P, Mosler J (2018) Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization. Contin Mech Thermodyn 30:1125–1144 32. Leckie FA, Onat ET (1981) Tensorial nature of damage measuring internal variables. In: Hult J, Lemaitre J (eds) Physical non-linearities in structural analysis. Springer, Berlin, Heidelberg, pp 140–155 33. Lemaitre J (1996) A course on damage mechanics. Springer, Berlin, Heidelberg 34. Lemaitre J, Desmorat R (2005) Engineering damage mechanics. Springer, Berlin, Heidelberg 35. Liebe T, Steinmann P, Benallal A (2001) Theoretical and computational aspects of a thermodynamically constistent framework for geometrically linear gradient damage. Comp Meth Appl Mech Eng 190:6555–6576 36. Menzel A, Steinmann P (2001) A theoretical and computational setting for anisotropic continuum damage mechanics at large strains. Int J Solids Struct 38:9505–9523 37. Menzel A, Steinmann P (2003) Geometrically non-linear anisotropic inelasticity based on fictitious configurations: Application to the coupling of continuum damage and multiplicative elasto-plasticity. Int J Num Meth Eng 56:2233–2266 38. Mühlhaus H-B, Alfantis EC (1991) A variational principle for gradient plasticity. Int J Solids Struct 28:845–857 39. Murakami S (1988) Mechanical modeling of material damage. J Appl Mech 55:280–286 40. Murakami S (2012) Continuum damage mechanics: a continuum mechanics approach to the analysis of damage and fracture. Springer, Dordrecht 41. Murakami S, Kamiya K (1997) Constitutive and damage evolution equations of elastic-brittle materials based on irreversible thermodynamics. Int J Mech Sci 39:473–486

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42. Needleman A (1988) Material rate dependence and mesh sensitivity in localization problems. Comp Meth Appl Mech Eng 67:69–85 43. Nguyen V-D, Lani F, Pardoen T, Morelle XP, Noels L (2016) A large strain hyperelastic viscoelastic-viscoplastic-damage constitutive model based on a multi-mechanism non-local damage continuum for amorphous glassy polymers. Int J Solids Struct 96:192–216 44. Ostwald R, Kuhl E, Menzel A (2019) On the implementation of finite deformation gradientenhanced damage models. Comp Mech 64:847–877 45. Peerlings RHJ, de Borst R, Brekelmans WAM, de Vree JHP, Spee I (1996) Some observations on localisation in non-local and gradient damage models. Eur J Mech A/Solids 15:937–953 46. Peerlings RHJ, Geers MGD, deBorst R, Brekelmans WAM (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38:7723–7746 47. Polindara C, Waffenschmidt T, Menzel A (2016) Simulation of balloon angioplasty in residually stressed blood vessels - application of a gradient-enhanced continuum damage model. J Biomech 49:2341–2348 48. Polindara C, Waffenschmidt T, Menzel A (2017) A computational framework for modelling damage-induced softening in fibre-reinforced materials - application to balloon angioplasty. Int J Solids Struct 118–119:235–256 49. Rabotnov YN (1969) Creep problems in structural members. North-Holland, Amsterdam 50. Rousselier G (1987) Ductile fracture models and their potential in local approach of fracture. Nucl Eng Design 105:97–111 51. Steinmann P (1999) Formulation and computation of geometrically non-linear gradient damage. Int J Num Meth Eng 46:757–779 52. Triantafyllidis N, Aifantis EC (1986) A gradient approach to localization of deformation. I Hyperelastic Mater J Elast 16:225–237 53. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169 54. Voyiadjis GZ, Kattan PI (1993) Local approach to damage in elasto-plastic metal matrix composites. Int J Damage Mech 2:92–114 55. Voyiadjis GZ, Woody Ju, J-W, Chaboche, J-L (1998) Damage mechanics in engineering. Studies in applied mechanics, vol 46. Elsevier, Amsterdam 56. Waffenschmidt T, Polindara C, Menzel A, Blanco S (2014) A gradient-enhanced largedeformation continuum damage model for fibre-reinforced materials. Comp Meth Appl Mech Eng 268:801–842 57. Wcislo B, Pamin J, Kowalczyk-Gajewska K (2013) Gradient-enhanced damage model for large deformations of elastic-plastic materials. Arch Mech 65:407–428 58. Zairi F, Nait-Abdelaziz M, Gloaguen JM, Lefebvre JM (2001) A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation. Int J Plast 27:25–51

Theories of Growth Marcelo Epstein

Abstract The modelling of processes of growth and the associated phenomena of remodelling, ageing and morphogenesis requires a rethinking and reformulation of the fundamental notions of material body, balance equations and constitutive theory. Some of these ideas are presented and treated with various degrees of detail.

1 Introduction This chapter is limited to the description of some of the continuum mechanics aspects involved in the formulation of models of material growth, without regard to the biological or material science considerations underlying the physical motivations of such models. Even within this declared restricted framework, the scope of the discipline is too large to be covered in detail in the available space. To acquire a more comprehensive view, the reader is referred to the classical review article [24], which covers the state of the art as of 1995, and to the more recent and extensive treatment in [11] (see also [17]). It is generally admitted that a clear distinction can be drawn in continuum mechanics between three foundational ingredients, namely, kinematics, physical laws and constitutive theory. Perhaps the boldest assertion that can be made in regard to the modelling of growth phenomena is that all three ingredients necessitate a radical reformulation if one is to attempt to embrace as general a scope as possible within the bounds of the assumption that the material body is a continuum. Mathematically speaking, the hypothesis as to the continuous nature of matter can be expressed in differential geometric terms by stating that the body is a differentiable manifold. It is usual to further suppose [29] that it can be covered with a single coordinate chart. The points of this manifold are identified as material points that maintain their individuality throughout any process of deformation. Since biological M. Epstein (B) Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_9

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growth processes can and do involve accretion or resorption at a surface, it is easy to understand why it may be more appropriate to consider the body as a manifold with boundary, which is a somewhat more technical concept in differential geometry. Moreover, even without this additional structure, there seems to be no clear epistemological reason to cavalierly associate a point in the body manifold with an identifiable material particle. As delicate as the kinematic issues just raised maybe, one may feel inclined to disregard them as a flight of mathematical fancy. Even if that were the case, it is absolutely clear that, whatever point of view a modeller might wish to adopt, the influence of growth processes on the formulation of the balance equations of mass, momentum, energy and entropy is undeniable and must be taken into consideration with utmost rigour. Finally, from a constitutive point of view, growth processes are particular instances of processes of material evolution. Results of experiments conducted at different times on what appears to be the same material location in a materially evolving body will in general differ from each other. This difference may be accounted for by an increase or decrease of mass, but it may also be due to processes of remodelling and ageing, even without growth or resorption of mass. The problem of the unequivocal attribution of the change in material properties to each of those factors is just one of the challenges of the constitutive theory of evolving materials.

2 Body and Kinematics of Boundary Growth As already intimated, the notion of a material body undergoing growth requires a more careful consideration than its standard counterpart. Consider a process of growth of an initially annular domain in R2 into a solid circle. In this case, because of the closing of a hole, there is a drastic change in the topology of the body. A change of comparable nature can be observed in a process of coalescence of two initially disjoint entities into one, such as illustrated in the physics of black holes [18]. It would appear, therefore, that the mere consideration of the body as a differentiable manifold is placed at risk. It is possible, however, to rescue the required smoothness at the price of introducing a body-time manifold, as schematically illustrated in Fig. 1, which has the appearance of a pair of trousers or a tree trunk and its roots. The body-time coordinate has been drawn vertically. These examples, whose treatment requires further study, have been brought to bear only to illustrate, in perhaps extreme circumstances, the challenges brought about by the modelling of growth processes. Assuming from now on that the topology remains intact, we still have to deal with the fact that new material is being incorporated or extracted as time goes on. There are at least two paradigms to account for this fact; they are known as bulk or volumetric growth, and surface growth. In bulk growth, the points of the body manifold are identified as material particles. Put in mundane terms, a marker inscribed on a material point accompanies it throughout the deformation and indicates one and the same material particle. Growth

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Fig. 1 A body-time manifold illustrating the smooth coalescence of two circles

y

x

is manifested by changes in the material density. It is important, however, to bear in mind that these changes take place in the body itself (or, from a practical point of view, in its reference configuration). These are indeed genuine configurational changes which, in addition to growth, may also entail changes in material properties, such as remodelling and ageing. The representation of surface accretion, or, more generally, of singular growth on surfaces, lines, and points, requires a detailed specification of the process of entry or exit of new material particles. In an important early treatment of these matters, Skalak et al. [23] propose the introduction of a new time variable indicating for each particle the time elapsed from birth, which is akin to the material time coordinate of Fig. 1.1 A more precise geometrical treatment, proposed by Segev [22], postulates the existence of a material universe, namely, a three-dimensional manifold M of material particles devoid of any specific constitutive properties. A material body is then defined as a connected trivial three-dimensional manifold B together with an embedding μ : B → M. Without this embedding, the manifold B serves only the purpose of establishing the topology of the body. Thus, the body manifold B is a timeindependent entity. The embedding μ, called the content embedding, serves the function of capturing material particles, like a cookie cutter applied to a sheet of dough. In conventional continuum mechanics, this embedding is fixed and, therefore, can be incorporated into the definition of the material body, without any explicit mention. In the case of growth, however, we may allow the capturing embedding μ to vary smoothly in time. Proceeding along these lines, as detailed in [4], we can accommodate growth occurring on the boundary ∂B by defining the notion of configuration as an embedding (1) κ : μ(B) → E3 . 1 An

[3].

interesting different perspective on the unification of bulk and surface growth is proposed in

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Notice how this definition differs from that of a configuration as an embedding κ˜ : B → E3 that maps the points of B directly into the Euclidean space E3 , regardless of their appropriation of a material identity. The definition proposed in Eq. (1) provides us with boundary growth whenever the set μ(∂B) depends on time. As an example, one may think of the body as the part of a violin string between the musician’s finger and the bridge. As the violinist moves the finger along the fingerboard, more or fewer material particles enter the body through the boundary, thus providing a concrete example of a time-dependent content embedding μ.

3 The Balance Equations of Volumetric Growth The equations of balance applicable to bulk growth can be derived from an axiom of invariance of the energy balance under observer transformations. This approach, advocated in different ways by Noll [19] and by Green and Rivlin [12], is particularly suited to the correct derivation of the reduced forms of the equations of balance when phenomena of mass self-diffusion may take place [7], as shown in [13].2 The following axiom is adopted, whether implicitly as in [12] or explicitly as in [19], as the cornerstone of the derivation.3 Axiom 1 The energy balance equation is invariant under all possible observer transformations provided that the difference a−f between the acceleration and the body force per unit mass be a frame-indifferent vector. In other words, as far as an accelerating observer is concerned, there appear to exist additional body forces, whose dependence on the state of motion of this noninertial observer compensates exactly for the additional terms (such as Coriolis and centripetal) in the acceleration. The phenomenon of mass diffusion within one species has been discussed at length in [7], where a mass influx m and a diffusive momentum density pd are postulated. These quantities are justified in [7] a posteriori by an argument invoking the equations of a binary mixture in the limit as the two components become one and the same. The terms containing these additional quantities may be ignored altogether in the treatment that follows if self-diffusion is not of interest. The diffusive momentum pd , which is assumed to be frame indifferent, does not affect the particle velocities v of the underlying gross motion, but it does make a contribution pd · v to the total energy per unit volume of the system. Moreover, a 2 To

the best of our knowledge, the first treatment of mass flux within the context of a theory of growth is given in [8]. The treatment of the flux term therein, however, leads to a form of the balance equations that is not invariant under the action of the Galilean group. The same remark applies to the treatment presented in [14]. In [10], growth and remodelling are placed within the context of mixture theory involving at least two species and mass flux is used only in the context of diffusion of nutrients in the underlying tissue. 3 A comprehensive comparative account of these works is presented in [15].

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counterpart to the body force f is introduced in the guise of a so-called non-compliant ¯ measured in units of momentum per unit volume and per unit momentum source p, time. Just as it is the case with the body force f, the non-compliant momentum source p¯ depends on the state of motion of a non-inertial observer, as discussed in [13]. Under these conditions, an appropriately formulated energy balance equation, in combination with the fundamental Axiom 1, will be shown to deliver all the balance equations of the theory. The total content of energy e per unit spatial volume is assumed to be 1 e = ρε + ρv · v + pd · v. 2

(2)

In this equation, ρ is the current density, ε is the internal energy per unit mass and v is the velocity field. The contribution of the diffusive momentum pd per unit spatial volume can be justified on physical grounds, but is here proposed as an axiom of the theory. Adopting a fixed control volume ω in space, with boundary ∂ω and exterior unit normal n, the balance of energy (first law of thermodynamics) is written as d dt

 ω

   1 π(ε + v · v) + f · (ρv + pd ) + ρr + p¯ · v dω 2 ω    1 −(v · n) e + m(ε + v · v) + h + t · v da. + 2 ∂ω

e dω =

(3)

The rate of change of the total energy content in ω is balanced by volumetric and boundary contributions, as shown on the right-hand side of this equation. The mass growth π per unit volume and per unit time contributes directly to the total energy. For simplicity, we assume that the specific internal energy ε of the mass being created is the same as that of the underlying body point. Otherwise, we can add a noncompliant term representing any desired deviation, which would have to be prescribed ¯ constitutively. We have accounted for a possible non-compliant momentum input p, contributing to the energy rate in the amount p¯ · v. The body force f per unit mass produces mechanical power on the velocity field. In addition, however, since we have postulated the existence of a small diffusive momentum pd per unit volume, we have added the corresponding power f · pd . The heat supply per unit mass is denoted by r and the surface counterpart by h. The boundary traction is t. Using the divergence theorem, Eq. (3) can also be written in terms of the material time derivative as       De 1 π(ε + v · v) + f · (ρv + pd ) + ρr + p¯ · v dω + e ∇ · v dω = Dt 2 ω ω    1 m(ε + v · v) + h + t · v da. (4) + 2 ∂ω

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Following the idea of [12], we write the balance equation (4) as seen by an observer under translation c = c(t), such that c(0) = 0, and then collect the quadratic, linear and constant terms in c˙ (0). Each of these terms must vanish, due to the arbitrariness of c˙ (0). The quadratic terms yield the balance equation 



 ω

(ρ˙ + ρ ∇ · v) dω =

ω

π dω +

m da,

(5)

∂ω

where we have used a dot to indicate the material derivative. Assuming the equation to be valid for arbitrary domains ω, the Cauchy tetrahedron argument prescribes the existence of a vector field m of mass flux such that m = −m · n,

(6)

the minus sign being the result of having assumed the influx of mass as positive. Using the divergence theorem, and localizing, yields the following form for the local spatial equation of mass balance in the presence of mass flux: ρ˙ + ρ ∇ · v = π − ∇ · m.

(7)

The linear terms in c˙ (0) provide us with the balance equation   ω

   D ¯ dω + (mv + t) da. (ρv + pd ) + (ρv + pd ) ∇ · v dω = (π v + ρf + p) Dt ω ∂ω

(8) Using again the tetrahedron argument, we obtain the Cauchy stress tensor s such that t = sn and, applying the divergence theorem, localizing, and invoking the just obtained balance of mass (7), we arrive at the following local form of the spatial equation of linear momentum balance k i − m k v,k , ρ v˙ i + p˙ di = ρ f i + p¯ i + s,kik − pdi v,k

(9)

where we have used a component notation to avoid any ambiguities. We employ the standard convention of using lower- (upper-) case indices for spatial (referential) components. The remaining term represents the reduced form of the balance of energy as ρ ε˙ + pdi v˙ i = f i pdi + ρr − q,ii + s ik vi,k − m i ε,i .

(10)

To obtain this equation, we have used the Cauchy tetrahedron argument to introduce the heat flux vector q and we have enforced the previously deduced equations of mass and linear momentum balance. Considering next a rotating observer, we obtain two important results. The first one is that (11) pd = m.

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It establishes that the diffusive momentum is a direct result of the mass flux. One cannot exist without the other. The second result is the symmetry of the Cauchy stress, namely, (12) s i j = s ji . We have obtained the collection of all mechanical balance laws out of the first law of thermodynamics. The second law is postulated independently in the form of the Clausius–Duhem inequality ρr − h¯ − m i η,i − ρ η˙ ≥ θ



q i + qdi θ

 ,i

.

(13)

In this expression, η is the entropy content per unit mass, θ is the absolute temperature ¯ is a non-compliant volumetric entropy sink. A diffusive heat flux term, and h/θ h d = −qd · n, is also included for greater generality. In Sect. 2, we had occasion to observe how the kinematic analysis of growing bodies brings into question the very notion of material particle in a continuous body. In the present section, when dealing with the balance laws, we can observe how the dissipation inequality introduces some uncertainty due to the fact that we are considering an open thermodynamic system.4 Such fundamental questions arise too in regard to the constitutive aspects of growing bodies, as we shall presently reveal.

4 Internal State Variables The hope that the main open problem of the theory of material behaviour 5 could be solved once and for all with an enterprise consisting on placing restrictions upon a generic form of a constitutive law based on dependence on the histories of the motion and the temperature alone, has not been realized. In the opening paragraph of a remarkable article by Coleman and Gurtin [1], themselves involved in this enterprise, we read In phenomenological theories of the dynamical behaviour of continua there are several ways of accounting for the dissipative effects which, in addition to heat conduction, accompany deformation. The oldest and simplest way is to introduce a viscous stress which depends on the rate of strain,... Another description of dissipation assumes that the entire past history of the strain influences the stress in a manner compatible with a principle of fading memory. A third approach is to postulate the existence of internal state variables which influence the free energy and whose rate of change is governed by differential equations in which the strain appears.

4 It is worthwhile mentioning here that, in our formulation of the second law of thermodynamics, we

have adhered strictly to the formulation of the Clausius–Duhem inequality, considered by Truesdell’s school of thought [26] as the overarching principle of the discipline. 5 See [27], p. 47.

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They add that no one approach ‘is so general as to include completely all the others’. Indeed, internal variables have opened an otherwise closed door to largely successful models used in theories of metal plasticity, continuous damage, dislocations, soil mechanics, chemically reacting mixtures, muscle mechanics, and biological ageing, remodelling and growth. Let a = a(X, t) = (a1 , . . . , a N ) be a ‘vector’ consisting of a number N of internal state variables defined over the body in some reference configuration, whose points are denoted by X at each time t. The main idea in a theory with internal state variables consists of (i) augmenting the set of independent constitutive variables (say, the present values of the deformation gradient F and of the temperature θ ) by incorporating the vector a, so that, for example, the constitutive equation for the Cauchy stress reads s = s(F, θ, a); (14) (ii) specifying an evolution equation of the form a˙ = g(F, θ, a),

(15)

where g is a constitutive function characterizing the model at hand. Our interest is now focused on exploring the restrictions that the Clausius–Duhem inequality (13) imposes on the constitutive equations and, in particular, on the evolution function g. To this end, let us consider a case devoid of mass self-diffusion, so that the dissipation inequality is reduced to the form  i q ρr − h¯ − . ρ η˙ ≥ θ θ ,i

(16)

It is convenient to replace the internal energy ε with the free energy per unit mass, defined by ψ = ε − θ η, (17) and to rewrite (16) as ρ ψ˙ ≤ −ρηθ˙ −

qi ¯ θ,i + s ik vi,k + h, θ

(18)

where the energy balance equation has been exploited to eliminate the external heat supply term r . As pointed out in condition (14) above, the free energy is a function ψ = ψ(F, θ, a), so that the left-hand side of (17) becomes  ρ ψ˙ = ρ

 N  ∂ψ ˙ i ∂ψ ∂ψ θ˙ + a˙ α . F + ∂θ ∂aα ∂ FIi I α=1

(19)

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˙ θ˙ , and θ,i , and claiming the identical satisfaction of Collecting terms affecting F, (18) for all possible processes, we obtain the following restrictions: s=ρ

∂ψ T F , ∂F

(20)

∂ψ , ∂θ

(21)

η=−

q = 0,

(22)

and the residual dissipation inequality ρ

N  ∂ψ a˙ α − h¯ ≤ 0. ∂a α α=1

(23)

The ‘vector’ b with components bα =

∂ψ , α = 1, . . . , N , ∂aα

(24)

will suggestively be called the vector of configurational forces driving the evolution of the internal state variables. The residual dissipation inequality can be compactly written as ρb ∗ a˙ − h¯ ≤ 0, (25) where ∗ denotes the standard dot product in R N . By virtue of Eq. (15), we have obtained a severe constitutive restriction to be satisfied by the evolution function g. If we set h¯ = 0, an elementary way to satisfy this restriction would be to specify g = −Kb,

(26)

where K is a positive-semidefinite constant N × N matrix. Nevertheless, it is worthwhile remembering that, in biological contexts, the neglected extra term h¯ may represent the influence of control mechanisms that guarantee the healthy and orderly existence of a living organism.

5 A Natural Choice The choice of internal variables is clearly an integral part of a material model. There is, however, one particular choice that arises naturally from the mathematical structure inherent in continuum mechanics. Not surprisingly, this natural choice has a natural physical meaning. Whether or not this particular choice is ultimately adopted

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Fig. 2 Transition map χ between two reference configurations

is, therefore, tantamount to the adoption or rejection of its physical interpretation within the context of the phenomena being modelled, but it must be given careful consideration precisely because it is canonically suggested by the very mathematical structure of the theory. The particular functional expression of a constitutive law is affected by the reference configuration chosen to express it. A change of reference configuration affects the constitutive function in a definite way. Let κ 0 : B → R3 and κ 1 : B → R3 be two reference configurations of the same body B, and let χ = κ 1 ◦ κ −1 0 be the transition from κ 0 (B) to κ 1 (B), as schematically shown in Fig. 2. The constitutive function for the free-energy density ψ, for example, at a point X in κ 0 (B) will be of the form ψ = ψ0 (F), evaluated at every deformation gradient F measured from this reference configuration. The temperature dependence has been ignored, since it doesn’t affect the reasoning that follows. In the reference configuration κ 1 , however, the same material point Y = χ(X) will have a constitutive function ψ1 (F) for every deformation gradient F measured away from κ 1 . The relation between these two functions is given canonically by ψ1 (F) = ψ0 (F∇χ (X)) ∀F ∈ GL(3, R).

(27)

In this identity, GL(3, R) is the general linear group, consisting of all non-singular real-valued 3 × 3 matrices. The tensor ∇χ (X) ∈ GL(3, R) is the gradient of the transition function at X. This well-known fact in continuum mechanics can be expressed as follows: The general linear group GL(3; R) has a natural right action on the set of all possible constitutive equations of simple material points, as per Eq. (27). Moreover, a material response is not just one point in this space of functions (such as ψ0 (F)), but a whole orbit under the action of the general linear group. In other words, two points, such as the two functions ψ0 (F) and ψ1 (F), along an orbit represent one and the same material response.

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If in a thought experiment we were to imagine that we are looking at the molecular arrangement in two different reference configurations, we would see exactly the same topology, but with somewhat different geometrical angles and distances. The material is, chemically speaking, exactly the same. Having established this fact, let us assume that we were to adopt a non-singular matrix P = P(X, t) as a collection of 9 internal state variables. Let us assume, moreover, that these variables appear multiplicatively to the right of the deformation gradient, as is the case in Eq. (27). Put differently, in the spirit of condition (14) above, the constitutive equation (in a fixed reference configuration) is made to depend on an additional list of internal variables a (now renamed P) in the form ψ = ψ(FP, θ ), (28) or, in greater detail, ψ = ψ(F, θ ; X, t) = ψ(FP(X, t), θ ; X, t0 ),

(29)

where t0 is an initial reference time. The response at time t is essentially the same as that at time t0 , except for multiplication to the right by a time-dependent map P. The physical meaning of this choice of internal variables is clearly the following: in a fixed reference configuration, the material undergoes a remodelling of sorts, that may involve addition of mass, but this phenomenon does not entail any change in material properties. The only thing that occurs as time goes on is a mere reaccommodation. For example, we may have an orthotropic material at a point X whose only configurational change consists of a time-dependent rotation (within the body, in relation to its neighbours) of the axes of orthotropy, as suggested in Fig. 3. Or we may have that nothing but the mass density changes by addition of mass of the same material. An evolution process governed by these natural internal state variables is known as a process of anelastic evolution. In light of this natural choice of internal variables, we can look at what is usually called the multiplicative decomposition of the deformation gradient, which is essentially given by Eq. (28). This decomposition arose originally in the treatment of non-linear plasticity and was later very successfully adopted as a model for growth

Fig. 3 Anelastic evolution at a point in a fixed global reference configuration

P(X,t)

X

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and remodelling in [21]. Much has been said in engineering settings about the socalled ‘intermediate configuration’, and discussions are still held as to whether this should be a stress-free configuration or, in a biological context, a homeostatic state. Some discussions involve also speculation as to whether the multiplication should be to the right or to the left of the deformation gradient. From the formulation above we can conclude that, interesting as these speculations may be, some of these arguments are idle talk. The right multiplicative decomposition is the mathematical expression of models of material evolution that assume that the material particles preserve their ‘chemical identity’ in the process of evolution, just as the proverbial spring which keeps its stiffness constant. There is one more aspect peculiar to the choice of internal variables in accordance with the multiplicative decomposition which ties it with the seminal work of Eshelby [9] in the domain of defect mechanics. Before embarking into this important point, we will record some preliminary calculations. Let ρ R (t) denote the time-dependent mass density of the reference configuration at point X. The time dependence is governed by the determinant J P of P, according to ρ R (t) = J P−1 ρ R (t0 ), where t0 is the initial time with respect to which the energy function ψ has been given. The current spatial density is given by ρ = ρ(t) = JF−1 ρ R (t). Proceeding now to the main argument, we observe that, according to Eqs. (24) and (20), the configurational forces behind the evolution of P are given by ∂ψ ∂(FP) ∂ψ = : = ρ −1 FT s (FP)−T ∂P ∂(FP) ∂P

= ρ R−1 (t) FT T P−T = ρ R−1 (t) b˜ P−T ,

(30)

where T is the first Piola–Kirchhoff stress at time t and P−T = (P−1 )T = (PT )−1 . The tensor b˜ = FT T is known as the Mandel stress. It is a purely referential tensor, whose physical meaning is precisely the driving force behind the configurational changes P. We emphasize that, as a configurational force, the Mandel stress plays a role in the theory only if the multiplicative decomposition is used to define the internal state variables of the model. The stress produces dissipation, according to Eq.  Mandel (25), in such a way that tr b˜ P˙ T ≤ 0. Had we formulated the constitutive law in terms of the free-energy density ψˆ = ρ R ψ per unit referential volume (rather than per unit mass) we would have obtained 

 ∂ J P−1 ρ R (t0 ) ψ ∂ ψˆ ˆ − b˜ P−T , = = − ψI ∂P ∂P

(31)

where Eq. (30) has been used. The bracketed quantity ˆ − b˜ b = ψI is the Eshelby stress tensor.

(32)

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6 The Theory of Adaptive Elasticity 6.1 General Formulation We will next consider a theory of growth which is not an anelastic evolution theory. It was proposed in 1976 by Cowin and Hegedus [2] to describe processes of growth and remodelling of trabecular bone. It can be considered the first theory of its kind based on a complete and rigorous thermomechanical foundation within the context of continuum mechanics. It uses a single internal state variable representing the porosity of the material. The main assumptions of the theory can be summarized as follows: 1. Bone is considered as an elastic porous matrix (made of extracellular material), whose pores are filled with a liquid perfusant; 2. The slow chemical reactions (mediated by the bone cells) responsible for growth and remodelling are controlled by the state of strain of the matrix; 3. The addition or removal of solid mass resulting from the chemical reactions takes place exclusively at the expense of the porosity, thus causing no residual stresses; 4. The porosity is included as an internal state variable; 5. The balance equations are formulated on the basis of the solid phase alone (namely, the matrix), which is an open system immersed in an isothermal perfusant bath. Since the addition (growth) and removal (resorption) of material take place exclusively at the pores, the total volume occupied by the bone (in an assumed to exist stress-free configuration at constant temperature) remains invariable. Denoting by γ the intrinsic density of the matrix material and by φ the porosity, the effective density ρ of the porous structure is given by ρ = γ ζ = γ (1 − φ),

(33)

where ζ = 1 − φ is the solid volume fraction. The equations of physical balance have already been presented in Sect. 3 and need not be modified. In the formulation of [2], no account is taken of self-diffusion. Accordingly, the corresponding terms can be ignored. What is left to be done is a treatment of the constitutive aspects of the theory. As already pointed out, one of the main assumptions (and limitations) of the theory of adaptive elasticity is that the new material occupies only the space available due to the porosity of the matrix. In other words, if a configuration is free of stress, and if no forces or temperature changes are imposed, then the fact that growth is taking place will not produce any stresses and the configuration will remain unchanged, except for the fact that its porosity is changing. Adopting a local stress-free configuration as reference, we denote the intrinsic density of the bone material by γ0 and its solid volume fraction by ζ0 = 1 − φ0 . This quantity will undergo changes in time due to the entrant mass. Enforcing the mass balance equation (7), we obtain

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Π Dζ0 Jπ = = . Dt γ0 γ0

(34)

where J is the determinant of the deformation gradient F. Certain subtleties should be brought to attention in regard to the notion of porosity. If we assume that the deformation gradient F of the porous material applies on average also to the solid itself, as one would do in a non-diffusive mixture, the free energy density per unit mass of the porous medium is the same as the free energy density of the solid for the same F. Accordingly, the intrinsic density γ0 of the undeformed matrix material will in general change to a new value γ = J −1 γ0 in the deformed state. In reality, though, because of the presence of the pores (which means the imposition of traction-free boundary conditions over the intricate geometry of the voids), the actual local deformation gradient acting on the matrix material will be different from the effective deformation gradient applied to the averaged structure. As a result, there is no reason to believe that the densities γ0 and γ are directly related by the determinant J of the applied deformation gradient. For example, if we assume that the matrix material is incompressible, in which case γ = γ0 , it is not difficult to show that the corresponding solid volume fractions are related by ζ0 = J ζ . The usual constitutive variables (free energy, entropy, stress, heat flux), and both ¯ are all assumed to be functions of the the mass production π and the entropy term h, temperature, its gradient, the volume fraction ζ0 , and the deformation gradient. This generic assumption presupposes that the chemical reactions of growth and remodelling (in bone, at least) are triggered, from the mechanical standpoint, by the state of strain of the matrix measured with respect to a putative unstressed configuration. This configuration is assumed to be uniquely defined (modulo a rigid rotation) for each state of uniform temperature, regardless of the particular value of the porosity. From the standard thermodynamic treatment of the Clausius–Duhem inequality, it is possible to conclude that the free energy is independent of the temperature gradient, and that the free energy acts as a potential for both the entropy and the stress, which are also independent of the temperature gradient. Moreover, the constitutive functions are further restricted by the residual inequality, that is, ρ ζ˙0

∂ψ 1 + q i θ,i − h¯ ≤ 0. ∂ζ0 θ

(35)

6.2 The Isothermal Quasi-static Case To obtain some specific results, we now specialize the equations of the theory of adaptive elasticity to purely mechanical (isothermal) processes and to very slow motions, so that the velocities and accelerations can be neglected. Neglecting also the body forces, the equation of balance of momentum reduces to the following equilibrium equation in the Lagrangian formulation

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T ,Ii I = 0.

(36)

The equation of mass balance is γ0

Dζ0 = Π. Dt

(37)

The equation of balance of energy is trivially satisfied by virtue of our simplifying assumptions. The constitutive equations are completely controlled by two scalar functions, namely, the free-energy density ψ = ψ(F, ζ0 ) and the mass production Π = Π (F, ζ0 ). The quantity ψ is measured per unit mass. The stress is obtained by differentiation as ∂ψ(F, ζ0 ) . (38) T = γ 0 φ0 ∂F The referential solid volume fraction ζ0 acts as an internal state variable in this theory. The change of internal energy density brought about by a change of the internal variable, namely, the derivative A=

∂ψ(F, ζ0 ) , ∂ζ0

(39)

is the driving force or configurational force behind the time evolution of the internal variable. This aspect is emphasized by the residual Clausius–Duhem inequality (35), which can be written as ζ˙0 A ≤ 0, (40) where we have set h¯ = 0. Using Eq. (34), we obtain Π A ≤ 0.

(41)

One of the simplest ways to satisfy this thermodynamic restriction is to adopt a constitutive evolution equation of the form Π = Π (F, φ0 ) = −k A,

(42)

where k is a positive material constant. If, on the other hand, the free-energy density is assumed to be independent of the porosity, then A = 0, and the Clausius–Duhem inequality is satisfied identically in a reversible manner for any assumed evolution law.

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7 Anelasticity Theory 7.1 Symmetry and Non-uniqueness The map P introduced in Eq. (29) is called a material isomorphism, a concept and terminology introduced by Noll [20] to compare the material responses of two different points. In the context of material evolution of a fixed point, however, we apply Noll’s ideas to compare the response functionals at two different times. Thus, we may say that a point undergoes an anelastic evolution if its responses at any two times are mutually material isomorphic.6 Material responses can exhibit material symmetries such as isotropy or transverse isotropy. The symmetries manifest themselves in corresponding properties of the response functional. We say that G ∈ GL(3; R) is a symmetry of the constitutive function ψ(F; X, t0 ) if the equation ψ(F; X, t0 ) = ψ(FG; X, t0 )

(43)

is satisfied identically for all deformation gradients F ∈ GL(3; R). We have disregarded the temperature dependence, since it doesn’t play a role in our reasoning. Clearly, the trivial symmetry G = I is always available, but, if other symmetries exist, we may justifiably say that the response at time t0 is non-trivially materially isomorphic to itself. The collection of all symmetries at time t0 forms a multiplicative group G0 . This symmetry group is a subgroup of the general linear group GL(3; R). On physical grounds, it is always assumed that the symmetry group is a subgroup of the special linear group U ⊂ GL(3; R), namely, the collection of all matrices with unit determinant. Let P = P(X, t) be an evolutionary material isomorphism, namely, a map satisfying Eq. (29), and let G ∈ G0 be a symmetry of ψ(F; X,t0 ). We then have ψ(F; X, t) = ψ(FP; X, t0 ) = ψ(FPG; X, t0 ) 



= ψ FPGP−1 P; X, t0 = ψ F(PGP−1 ); X, t)

(44)

We conclude that, if G is a symmetry at time t0 , PGP−1 is a symmetry at time t. In other words, the symmetry group Gt at time t is a conjugate of the symmetry group G0 at time t0 , the conjugation between these groups being given via any material isomorphism P = P(X, t) according to Gt = P G0 P−1 .

(45)

6 In terms of the group action described in Sect. 5, we may say that, as the material response evolves,

it remains within the same orbit under the action of the general linear group GL(3; R).

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Conversely, if P = P(X, t) is a material isomorphism at time t, then so is any map P of the form P = GP, where G ∈ G0 . In fact, the collection Pt of all possible material isomorphisms at time t can be expressed in any of the equivalent forms Pt = P G0 = Gt P = Gt P G0 .

(46)

This purely algebraic exercise has served to show that the same anelastic evolution can be expressed by choosing at each time t any of the elements within Pt . According to Eq. (46), therefore, the degree of freedom in this choice is governed by the symmetry group of the constitutive law. Notice, however, that if this symmetry group is discrete (such as in all the solid crystal classes, except isotropy and transverse isotropy), and if we demand the evolution to proceed smoothly in time, then this non-uniqueness is eliminated.

7.2 Physical Restrictions on Anelastic Evolution Laws We recall that, in accordance with the general tenets of the theory of internal state variables, whether anelastic or not, we need to supplement the constitutive equations of the material with an additional evolution law, as prescribed by Eq. (15). In particular, for an anelastic theory, whereby the internal state variables are encapsulated in a matrix P, we write the evolution equation in the form P˙ = g(F, P),

(47)

where, for simplicity, we are omitting the dependence on any other variables, such as the temperature or its gradient. We claim that the evolution function g cannot be completely arbitrary, but must satisfy a number of restrictions arising from physical considerations.

7.2.1

Reduction to an Archetype

The choice of an initial time t0 has provided us with an archetype, against which we compare the response at any other time t. In the case of the constitutive law, Eq. (29) tells us that the response at time t is the same as that at the archetypal time t0 , provided that we alter the actual deformation gradient from F to FP, namely, as if it were being measured from the state of affairs at time t0 . In keeping with this physical meaning, we will apply the same criterion to the evolution law (47). At a given time t, we have a material isomorphism P(t) with respect to the time t0 . If we keep t fixed and introduce a new time variable τ , measured from t, the evolution law gives us the derivative of P(τ ) with respect to τ at τ = 0 as P˙ = g (F, P(t)).

(48)

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To apply the above-stated criterion, we move to the time t0 , at which P(t0 ) = I, and replace F with FP. The evolution law provides us with the time rate P˙  = g (FP, I) = gˆ (FP).

(49)

This time rate applies to P(t0 ) = I. To recover P˙ we apply a shift as P˙ = P(t)P˙  .

(50)

Combining the above results, we obtain L P = P−1 P˙ = gˆ (FP).

(51)

The evolution is, therefore, controlled by a function of a single tensor variable. If, as suggested by its definition as the configurational force behind the configurational change P, we adopt the Mandel stress b˜ to mediate the dependence of g on F, Eq. (51) is written as  ˜ −T . (52) L P = gˆ J P PT bP

7.2.2

Actual Evolution

Because of the degree of freedom afforded by the symmetry group, it is possible to have an evolution law which leads to solutions P(t) that, starting from P(t0 ) = I, remain within the material symmetry group, that is, P(t) = G(t), G(t) ∈ G0 ∀t > t0 ,

(53)

with G(t0 ) = I. In this case, then, we have a fictitious evolution only. We derive now a condition for this situation not to happen. Taking the time derivative of Eq. (53) at time t = t0 yields ˙ 0 ). ˙ 0 ) = G(t P(t

(54)

The right-hand side of this equation can be recognized as an element of the Lie algebra (or infinitesimal generator) of the symmetry group G0 , assumed to be a Lie group. We conclude that, for a true evolution to take place, the function g must take values outside the Lie algebra of the symmetry group of the archetype. An interesting application of the actual evolution condition is obtained when the symmetry group of the archetype is the orthogonal group, which corresponds to the case of an isotropic material. The corresponding Lie algebra is the collection of all skew-symmetric 3 × 3 matrices. The principle of actual evolution requires that the function g must consistently produce matrices with a non-vanishing symmetric part.

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8 Classification of Evolution Processes 8.1 Remodelling Definition For the purpose of fixing the terminology in a precise way, we follow [5] to designate by remodelling any process of mere material reaccommodation, possibly including growth or resorption, that does not entail any change in the material constitution. Accordingly, a process of remodelling is synonymous with anelastic evolution. Any other kind of process will fall under the realms of ageing and morphogenesis. For physical reasons, a material isomorphism is always assumed to satisfy the mass consistency condition7 stipulating that ρ R (t) = J P−1 ρ R (t0 ).

(55)

Since the determinant of a linear transformation in R3 measures the ratio between the transformed and the original volume elements, the mass consistency condition implies that the mass of the volume element is conserved by the material isomorphism P. Since the reference configuration is held fixed, it follows from this remark that if J P < 1 we have a net increase in mass, as can also be understood from the fact that density has increased. The time derivative of Eq. (55) yields the instantaneous growth condition (56) ρ˙ R = −ρ R trL P . Accordingly, instantaneous growth (resorption) takes place when the trace of L P is negative (positive). The vanishing of the trace, therefore, indicates an instantaneous process of pure remodelling without growth. A case of pure growth without remodelling can be construed when the material isomorphisms are of the form P(t) = λ(t)I, where λ is a scalar function of time.

8.1.1

An Example: Growth Induced by Exercise

It follows from the discussion in Sect. 4 that the simplest non-trivial anelastic evolution law that satisfies the Clausius–Duhem inequality is of the form L P = −k J P PT b˜ P−T ,

(57)

where k is a positive constant. Following [6], p. 169, we consider a sphere of isotropic material subjected at each point of its boundary to a normal traction of magnitude a = a(t). In particular, we will consider an oscillatory time dependence of the form 7 See

[6], p. 11.

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a(t) = a0 sin(ωt),

(58)

where a0 and ω are kept constant. The idea is to simulate the effect of exercise on bone growth. The first Piola–Kirchhoff stress (in the assumed absence of significant inertial effects) is a spatially constant field given by T = a(t)I.

(59)

By symmetry arguments, we may assume the deformation gradient to given by F = f (t)I,

(60)

and the material isomorphisms to be of the spherical form P(t) = p(t)I.

(61)

We need to solve for the time-dependent quantities f (t) and p(t). Combining the equations above, we obtain the evolution law in the form p˙ = −kp 4 f a.

(62)

The constitutive response at time t0 = 0 is assumed to be given by the compressible neo-Hookean law ψ(F; 0) =

 1

μ tr(FT F) − 2 ln JF − 3 , 2

(63)

where μ is a material constant. The first Piola–Kirchhoff stress at time t = 0 is obtained as 

∂ψ = ρ R (0) μ F − F−T . (64) T = ρ R (0) ∂F At time t we obtain



T = ρ R (0) J P−1 μ FPPT − F−T .

(65)

Combining this result with the previous equations, and denoting μ0 = μρ R (0), we may write   1 μ0 2 . (66) a = 3 fp − p f Introducing this result in Eq. (62) yields ⎞ ⎛   3 2 dp ap kap 2 ⎝ ap 3 =− ± + 4 p2 ⎠ , dt 2 μ0 μ0 which is a non-linear ODE to be solved for p(t).

(67)

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Fig. 4 Exercise and anelastic growth. Solution for the relative density ρ R (t)/ρ R (0) versus the dimensionless time τ produced with Mathematica [30]

We introduce the change of variables τ = kμ0 t, α =

a , μ0

(68)

in terms of which Eq. (67) can be recast as  dp α  = − p 2 αp 3 ± (αp 3 )2 + 4 p 2 . dτ 2

(69)

Choosing, for physical reasons, the positive sign of the square root, this equation can be solved numerically. Figure 4 shows the relative density ρ R (t)/ρ R (0) = p −3 as a function of time for α0 = a0 /μ0 = 0.1 and ω = 10kμ0 . The oscillatory nature of the load is essential. A constant tension would produce growth, while a constant compression would lead to resorption, as the system tends to compensate for the external disturbance of its natural equilibrium.

8.2 Ageing and Morphogenesis 8.2.1

Symmetry Considerations

Any material evolution process ψ = ψ(F; X, t − t0 )

(70)

which cannot be put in the form (29) is, by definition, an ageing process. The time dependence is usually mediated by internal state variables, but not of the multiplica-

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tive kind that typifies anelastic evolution. By not remaining materially isomorphic to the initial response at time t0 , of necessity some of the material properties, such as the elastic moduli, are undergoing a change in time. Not all changes in material properties are so drastic as to alter the material symmetries. Consider, for example, the degradation of mineralized bone due to hormonal deficiencies. The weakened bone mineral may still retain its isotropic property. For this reason, we will reserve the term morphogenesis to indicate processes of ageing that involve a symmetry breaking, which is essential for pattern formation in plants and animals, as contemplated already by D’Arcy Wentworth Thompson (1860–1948) [25] and modelled by Alan Mathison Turing (1912–1951) [28] as a result of reaction– diffusion equations. Recall that in the case of anelastic evolution (that is, remodelling, with or without growth) the symmetry groups G0 and Gt were conjugate and, moreover, that the conjugation was achieved by means of a time-dependent material isomorphism P(t), in accordance with Eq. (45). If, on the contrary, the evolution is not anelastic, we no longer have at our disposal the existence of a material isomorphism. Nevertheless, it may so happen that the material evolves by a process of ageing that preserves the conjugacy condition between the symmetry groups, as described above. This would be a case of ageing without morphogenesis, or of pure ageing (with or without growth). Since in the case of pure ageing we are not bound by the constitutive condition (29), the set At of all available conjugations between the conjugate symmetry groups G0 and Gt is in general much ‘larger’ than the set Pt for a corresponding anelastic evolution. We will presently determine the nature of At . Recall that, in group theory, given a subgroup G of a group H, and given an element H of H, we say that the subgroup G = H G H−1 is the conjugate of G by H. The normalizer of G in H, denoted by N(G), is defined as the collection of all H ∈ H such that G = G. In other words, the normalizer consists of all H such that H G H−1 = G. It can be verified directly that the normalizer N(G) is itself a subgroup of H, and also that G ⊂ N(G). Let a process of pure ageing be such that A = A(t) is a time-dependent conjugation between the symmetry groups G0 and Gt . Then, it is a straightforward matter to show that the collection At of all possible conjugations between G0 and Gt is given by (71) At = N(Gt ) A = A N(G0 ) = N(Gt ) A N(G0 ). It turns out, therefore, that the degree of freedom available in pure ageing processes for the conjugacy map A is larger than that available for the choice of a material isomorphism P in the case of remodelling. The map A, correspondingly, cannot be interpreted physically as some kind of pure material rearrangement that can be gathered from a given ageing process. A dramatic demonstration of the sheer size of the normalizer of a unimodular subgroup of GL(3; R) is demonstrated by the obvious fact that the normalizer of any subgroup automatically contains all uniform dilatations, that is, all scalar multiples of the unit tensor.

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Fig. 5 Exercise and growth with ageing. Solution for the relative density ρ R (t)/ρ R (0) versus the dimensionless time τ produced with Mathematica [30]

8.2.2

An Example: Growth and Ageing

Let us now modify only slightly the previous example by converting the constant μ into a time-dependent property μ(t) = μ(0)e−t/s , where s is a relaxation time constant. Physically, it represents a degradation of the material quality. It should be clear that we no longer have a case of anelasticity, which would be recovered only for s → ∞. Nevertheless, since the isotropy of the material is obviously preserved, we have a case of pure ageing, without morphogenesis. Figure 5 shows the effect of the quality degradation when s = 3/(kμ(0)), while the remaining data are the same as for the previous example. Notice that, in its effort to compensate for the loss of quality, the material density is further increased.

8.2.3

The Splitting Problem

Assume that a constitutive law of the form ψ = ψ(F; t)

(72)

has been given with the property that, for any value of t within a time interval T of interest, the symmetry group of ψ preserves its type. In other words, the symmetry groups Gt for t ∈ T are mutually conjugate. Morphogenesis is explicitly excluded within the specified time interval, but ageing, growth and remodelling can be encompassed by the given constitutive law. The splitting problem consists of the following question. Can these various phenomena be separated out from each other in a canonical way?

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To further specify this question, consider a constitutive law of the form (72) with the added restrictive property that its symmetry group is preserved in time, as opposed to changing by a time-dependent conjugation. We may say that this constitutive law represents a kind of pure ageing, without any remodelling. To emphasize this special time dependence (with a constant symmetry group), we will denote our function by ψ = ψt (F). We will say that an evolving constitutive law has been split if it has been expressed as (73) ψ = ψ(F; t) = ψt (FP(t)), for some function ψt with a constant symmetry group and some time-dependent map P(t). Notice that if ψt is time-independent, we recover the anelastic evolution. At the other extreme, if ψt does depend on time but P = I, we obtain what we have called pure ageing. The splitting problem may now be reformulated as the question: given a constitutive law of the form (72), can an archetypal function ψt (Z), with a fixed symmetry group G, be shown to exist such that Eq. (73) is satisfied? If so, is this function uniquely defined? The answer to these questions is important for the prescription of phenomenological laws of evolution that control these phenomena. For example, the material degradation may be due to a hormonal deficiency, while the material reaccommodation may be attributed to the influence of external loading, such as is the claim of Wolff’s law.8 If this is the case, it should be of paramount importance to attribute the various effects to their respective causes in the formulation of their respective evolution laws. The somewhat surprising answer to the splitting problem, as presented in [5], is that, in a solid material, a necessary and sufficient condition for the splitting to exist and to be unique is that the symmetry group in the natural configuration must coincide with its orthogonal normalizer. This property is actually enjoyed by isotropic, transversely isotropic, and orthotropic materials, but not by the triclinic and n-agonal crystal classes. As a consequence, in triclinic materials one cannot in general distinguish between ageing and remodelling. In isotropic, transversely isotropic, and orthotropic materials, however, which are among the most common classes, the separation can be effected canonically and distinct evolution laws can be proposed for each of the various phenomena involved.

8.3 Pattern Formation Turning finally to morphogenesis, our intention in this section is to demonstrate that geometrical and functional patterns can emerge in appropriate circumstances out of a 8 Formulated

by Julius Wolff (1836–1902), this law asserts that bone adapts its architecture to the applied loads. In particular, the trabeculae would tend to follow the direction of the maximum principal stress.

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Fig. 6 Orthogonal and parallel molecular arrangements

purely mechanical formulation in a material where two or more free-energy functions compete for predilection. In biological contexts, this may be the case at early stages of development, when long molecules are relatively free to move within a material matrix and to form branches and more or less stable cross-links before settling into a less permissive environment. For the sake of illustration, we will assume a thin layer of isotropic matrix material (that is, a membrane) in which identical long molecules or fibres are initially arranged in a regular orthogonal lattice, as shown on the left side of Fig. 6. The molecules are assumed to be all parallel to the plane of the membrane. We will assume, moreover, that there is a single alternative local arrangement of the molecules in which they are parallel, such as the case of two identical helices side by side, as shown on the right side of Fig. 6. This arrangement will be assumed to be permanent and irreversible, once it has taken place, on the basis that the bond is stronger than in the original orthogonal counterpart. The free-energy density of the parallel arrangement will, naturally, enjoy less symmetries than the original. In going from the orthogonal to the parallel arrangement there is, therefore, a symmetry breaking mechanism at play. Both alternative energy densities are functions of the local value of the deformation gradient (or strain). For a given state of strain, we will assume that the material will choose to transfer itself to the lower energy mode, provided it was in the orthogonal lattice. The reverse passage is forbidden, as already pointed out, due to the strength of the parallel bond. Naturally, the mechanism just described is a point-wise affair governed by the local value of the strain. The reorientation of the molecules in those parts where it has taken place will manifest itself macroscopically by a change in colour or texture of the membrane and a pattern will emerge. Energetically speaking, our point of departure will be an incompressible neoHookean material representing the matrix material. Its elastic energy per unit volume of a natural (stress-free) state is given by W (C) =

1 μ (trC − 3) , 2

(74)

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M. Epstein

where μ is a material constant and C = FT F is the right Cauchy–Green tensor corresponding to the deformation gradient F. Since the material has been assumed to be incompressible, we must commit ourselves to specifying deformations such that det C = 1. The incompressibility constraint could be removed by including in the energy a term containing this determinant, as we have done in Eq. (63). Being a function of just the invariants of C, the material represented by Eq. (74) is isotropic. An orthotropic version of this constitutive law, however, can be obtained by a slight modification involving a bias tensor M, which we assume to be symmetric and positive definite. The modified neo-Hookean constitutive law is W M (C) =

1  μ [tr (M(C − I)M + I) − 3] . 2

(75)

The diagonal form of M in an eigenvector basis is ⎡ ⎤ a00 [M] = ⎣0 b 0⎦ . 00c

(76)

The constants a, b, c determine the degree of orthotropy of the material. If a = b = c, we recover the isotropic case. For the sake of our depiction, we will assume that exactly half of the molecules are fixed in the matrix in, say, the x direction, while the other half, originally aligned with the y direction, are free to rotate around the z axis and align themselves with the x-molecules. The contribution of the x-molecules to the elastic energy will be achieved by means of a bias tensor Mx given in the x, y, z coordinate system by the matrix ⎡

⎤ 1 0 0 [M]x = G ⎣0 0.1 0 ⎦ , 0 0 0.1

(77)

where G is another material constant. Correspondingly, the energy density contribution of the y-molecules is mediated by the tensor M y with matrix representation ⎡

⎤ 0.1 0 0 [M] y = G ⎣ 0 1 0 ⎦ . 0 0 0.1

(78)

The two competing energy modes are, accordingly, given by the functions W1 (C) = W (C) + W Mx (C) + W M y (C),

(79)

W2 (C) = W (C) + 2W Mx (C).

(80)

and

Theories of Growth

283

Fig. 7 Pattern formation induced by mechanical strain. Produced with MATLAB [16]

random

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

Figure 7 was produced using the values μ = 1, μ = 0.4, G = 1, and simulating in-plane displacements over the unit square with components of the form ux =

5  n=1

An sin(2nπ x) sin(π y), u y =

5 

Bn sin(2nπ y) sin(π x),

(81)

n=1

where An and Bn are randomly generated numbers between 0 and 0.001. This small range of amplitudes is chosen to ensure the in-plane impenetrability condition. The shaded portions correspond to points in the stable configuration.

References 1. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47:597–613 2. Cowin SC, Hegedus DH (1976) Bone remodelling I: theory of adaptive elasticity. J Elast 6:313–326 3. DiCarlo A (2005) Surface and bulk growth unified. In: Steinmann P, Maugin GA (eds) Mechanics of material forces, advances in mechanics and mathematics, vol 11. Springer, US, pp 53–64 4. Epstein M (2010) Kinetics of boundary growth. Mech Res Commun 37:453–457 5. Epstein M (2015) Mathematical characterization and identification of remodeling, growth aging and morphogenesis. J Mech Phys Solids 84:72–84 6. Epstein M, El˙zanowski M (2007) Material inhomogeneities and their evolution. Springer, Berlin, Heidelberg

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7. Epstein M, Goriely A (2012) Self-diffusion in remodeling and growth. Z Angew Math Phys 63:339–355 8. Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plast 6:951–978 9. Eshelby JD (1951) The force on an elastic singularity. Phil Trans R Soc A 244:87–112 10. Garikipati K, Arruda EM, Grosh K, Narayanan H, Calve S (2004) A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics. J Mech Phys Solids 52:1595–1625 11. Goriely A (2017) The mathematics and mechanics of biological growth. Springer, New York 12. Green AE, Rivlin RS (1964) On Cauchy’s equations of motion. Z Angew Math Phys 15:291– 294 13. Javadi M, Epstei M (2019) Invariance in growth and mass transport. Math Mech Solids, in press. https://doi.org/10.1177/1081286518787845 14. Kuhl E, Steinmann P (2003) Mass- and volume-specific views on thermodynamics for open systems. Proc R Soc A 459:2547–2568 15. Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs, NJ; Dover Publications, New York (1994) 16. MATLAB version 9.3.0.713579 (R2017b). The MathWorks Inc., Natick, Massachusetts (2017) 17. Menzel A, Kuhl E (2012) Frontiers in growth and remodeling. Mech Res Commun 42:1–14 18. Misner CW, Thorne KS, Wheeler JA (1973) Gravitation. W.H. Freeman and Company, San Francisco 19. Noll W (1963) La mécanique classique basée sur un axiome d’objectivité. In: Châtelet A, Destouches J-L (eds) La méthode axiomatique dans les mécaniques classiques et nouvelles. Gauthier-Villars, Paris, pp. 47–56; Reproduced in the foundations of mechanics and thermodynamics, selected papers by Walter Noll, pp 135–144. Springer, Berlin, Heidelberg (1974) 20. Noll W (1967) Materially uniform bodies with inhomogeneities. Arch Ration Mech Anal 27:1–32 21. Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27:455–467 22. Segev R (1996) Growing bodies and the Eshelby tensor. Meccanica 31:507–518 23. Skalak R, Dasgupta G, Moss M, Otten E, Dullemeuer P, Vilmann H (1982) Analytical description of growth. J Theor Biol 94:555–577 24. Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48:487–545 25. Thompson DW (1917) On growth and form. Cambridge University Press, Cambridge 26. Truesdell C (1984) Rational thermodynamics, 2nd edn. Springer, New York 27. Truesdell CA, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der physik, Vol III/3. Springer, Berlin 28. Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc B 237:37–72 29. Wang C-C, Truesdell C (1973) Introduction to rational elasticity. Noordhoff International Publishing, Leyden 30. Wolfram Research (2017) Inc., Mathematica, Version 11, Champaign

Finite-Strain Homogenization Models for Anisotropic Dielectric Elastomer Composites Morteza H. Siboni and P. Ponte Castañeda

Abstract This chapter is concerned with a homogenization framework for electroelastic composite materials at finite strains. The framework is used to develop constitutive models for dielectric elastomer composites consisting of initially aligned, rigid dielectric inclusions that are distributed randomly in a dielectric elastomeric matrix. A strategy is proposed to partially decouple the mechanical and electrostatic effects in the composite by writing the effective electroelastic energy of the composite in terms of a purely mechanical energy term together with a purely electrostatic energy term that are linked only by the unknown particle rotations. In addition to the macroscopic constitutive relation for the composite, estimates are also generated for the evolution of the average particle orientation as a function of the applied mechanical and electric fields. The resulting estimates account for the electric torques and dipolar forces on the particles that are generated as a consequence of externally applied electric fields.

1 Introduction Electro-active polymers (EAPs) are a class of materials capable of responding to external electric stimuli by spontaneously changing their shape. This special property, known as electrostriction, makes these materials close analogues of biological muscles, and as a consequence they are also known as artificial muscles [2]. Dielectric elastomers (DEs) are an important type of EAPs with many potential applications [5]. As reported in [36], the electrostrictive strains for elastomers can be fairly significant (in the order of 10% or even larger for special configurations), but require very large operating electric fields (in the order of 107 V/m). Furthermore, the perM. H. Siboni · P. Ponte Castañeda (B) Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA e-mail: [email protected] M. H. Siboni e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_10

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formance of dielectric actuators operating at high voltages is severely restricted by dielectric breakdown, or by an electromechanical (pull-in) instability followed by dielectric breakdown [35, 39]. One possible way to improve the performance of dielectric elastomers is to make dielectric elastomer composites (DECs) consisting of a soft elastomeric host with distributions of one or more filler phases with different elastic and dielectric properties [14]. Such heterogeneities can generate large electric polarizations, leading to dipolar forces and electrostatic torques, which, if harnessed by appropriate design of the microstructure, can be used to enhance the electromechanical properties of the material. Making use of the classical formulation of finite-strain electroelasticity of Toupin [38], analytical estimates for the effective response and stability of DECs with layered microstructures have been obtained by Bertoldi and Gei [6], deBotton et al. [3] and Rudykh and deBotton [29]. On the other hand, Ponte Castañeda and Siboni [24] proposed a finite-strain variational homogenization framework for DECs with more general microstructures and generated simplified estimates for DECs with periodic distributions of rigid particles accounting, in particular, for the effect of electric torques and particle rotations, which, unlike dipolar interactions, can have non-negligible effects even at dilute particle concentrations. For this purpose, they introduced a partial decoupling strategy consisting of the breakup of the total energy into a purely mechanical and a purely electrostatic contribution that are coupled by the average particle rotations. Moreover, they demonstrated that the effective electrostrictive stress for DECs undergoing finite strains can be directly related to the first derivative of the effective deformation-dependent permittivity of the composite with respect to the macroscopic strain. It should also be noted that, in the context of small strains, Tian et al. [37] developed a rigorous homogenization analysis to compute the effective electroelastic properties of composites in terms of coupled moments of the electrostatic and elastic fields in the composite, thus highlighting the strong influence of field fluctuations. They also provided results for sequentially laminated composites, where such coupled moments could be computed explicitly. Still in the context of small strains, Siboni and Ponte Castañeda [31] proposed an alternative homogenization framework for the effective electroelastic response of DECs with more general distributions of rigid particles. Their results suggest that particulate DECs consisting of slightly elongated (in the direction of the applied electric field) spheroidal conducting particles can achieve large electrostriction near the percolation limit. By means of the general homogenization framework of [24] and earlier results of Lopez-Pamies and Ponte Castañeda for purely mechanical response [17, 18], Siboni and Ponte Castañeda [32] generated analytical estimates for a special class of DECs consisting of aligned, long, rigid, and high dielectric fibers that are embedded in a soft ideal dielectric matrix. These estimates account for the contribution of the fibers to the overall properties (i.e., the stiffness, the permittivity, and the electromechanical coupling) of the composites, but were restricted to electromechanical loadings that were perfectly aligned with the fiber direction—and therefore involved no electric torques. Furthermore, making use of these results, they attempted an optimal

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design of the microstructure of the fiber-constrained DECs with the objective of achieving large electrostriction, while avoiding possible electromechanical instabilities and dielectric breakdown. These authors have shown that—due to the dipolar interactions—increasing the volume fraction or the cross section aspect ratio (in the direction of the applied field) of the fibers can lead to significant enhancements in the electromechanical coupling of the DECs. Thus, DECs with high concentrations of fibers or large aspect ratios for the fibers, in general, require smaller voltages to achieve a given deformation state. However, high fiber concentrations and/or large fiber aspect ratios can also lead to a dramatic reduction in the overall breakdown field that the DEC can withstand, due to the field magnification effect of the fibers. Several authors have also begun to study instabilities in heterogeneous active materials. Building on earlier work for the purely mechanical problem Geymonat et al. [12] and Bertoldi and Gei [3] investigated loss of positive definiteness for DECs with layered microstructures. The loss of strong ellipticity for such composites was examined by Rudykh and deBotton [29], while Destrade and Ogden [7] studied loss of strong ellipticity in the magnetoelastic context. More recently, Siboni et al. [30] have investigated the possible development of various types of instabilities for the class of (2D) fiber-constrained composites considered in [32]. In this work, we extend the partial decoupling strategy (PDS) of [24] for DECs with periodic microstructures to DECs consisting of aligned rigid fibers of ellipsoidal shape that are randomly distributed in an ideal dielectric elastomer matrix and subjected to general loadings, leading to the presence of electric torques on the fibers. In addition, we compare the resulting model with the corresponding model obtained in [32] by making use of the partial decoupling approximation (PDA), in which the simplification is made that the inclusion rotations are determined from the solution of the corresponding purely mechanical problem. The rest of the paper is organized as follows. Section 2 gives a brief overview of the fundamentals of electroelasticity, while Sect. 3 presents the general variational homogenization framework [23, 24] to be used in this work. Section 4 provides a detailed description of the class of particle-filled DECs of interest and describes the implementation of the PDS for the macroscopic response of the composite under general loading conditions, in terms of the equilibrium orientation of the particles. Finally, Sect. 5 provides a summary of the main findings of this work, as well as some directions for possible future research. In this paper, scalars are denoted by italic Roman, a, or Greek letters, α; vectors by boldface Roman letters, b; second-order tensors by boldface Roman letters, C, or boldface Greek letters, σ ; and fourth-order tensors by blackboard bold letters, P . Where necessary, index notation is adopted—e.g., bi , Ci j and Pi jkl are respectively the Cartesian components of the vector b, second-order tensor C and fourth-order tensor P .

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2 Background on Electroelasticity 2.1 Governing Equations The response of a deformable electro-sensitive material can be described by the theory of electroelastostatics; see, e.g., [9, 16, 38]. Consider a homogeneous electroelastic material occupying, in the absence of electric fields and mechanical loadings, a volume Ω0 in the reference configuration. Under the application of electric fields and mechanical loadings, a material point X in the reference configuration moves to a new x in the deformed configuration of the specimen, denoted by Ω. For simplicity, we exclude the possibility of gaps and/or interpenetration regions in the material throughout this chapter. This assumption can be enforced by taking the map x(X), which takes the material forms from the reference configuration to the deformed one, to be continuous and one to one. Then, the deformation gradient tensor F = Grad x (with Cartesian components Fi j = ∂ xi /∂ X j ) characterizes the deformation of the material, and it is such that J = det F > 0. Finally, the material satisfies the conservation of mass equation, such that the material density in the deformed configuration becomes (in local form) ρ = ρ0 / det F, where ρ0 denotes the material density in the reference configuration. The equilibrium equations in Eulerian and Lagrangian forms are given by div T + ρ f = 0, and DivS + ρ0 f0 = 0,

(1)

respectively, with div and Div being the divergence operators in the deformed (i.e., with respect to x) and reference (i.e., with respect to X) configurations. In expressions (1), T is the total Cauchy stress tensor, S = J TF−T is the (first) Piola–Kirchhoff stress tensor, and f and f0 are the given mechanical body force distributions in the deformed and reference configurations, respectively. The conservation of angular momentum requires symmetry of the Cauchy stress, i.e., T = TT , or equivalently SFT = FST .

(2)

It is important to emphasize that—unlike the mechanical body forces (couples), which are externally prescribed—the electric body forces (couples) are manifestations of the electric fields that develop in the material and, therefore, need to be determined from the solution of the coupled electroelastic problem. As a consequence, for the purposes of the present investigation, we include the effects of electric body forces (couples) in the total stress. For this reason, the stresses T and S defined above also include the effects of the electric fields, as will become clearer when we discuss the constitutive relations. The deformation gradient F and stress tensor S (or T) may be discontinuous across material interfaces, but satisfy the jump conditions

Finite-Strain Homogenization Models for Anisotropic …

[[F]] = a ⊗ N, and [[S]] N = 0 (or [[T]] n = 0),

289

(3)

where a is a vector which can be determined from the solution of the problem, and N (n) is the normal to the interface in the reference (deformed) configuration. The true (or Eulerian) electric field e and electric displacement field d must satisfy Maxwell’s equations, which, for quasi-static conditions, and in the absence of magnetic effects, are given by curl e = 0, and div d = q,

(4)

where q is the prescribed charge density (per unit volume in Ω), and curl and div are the usual differential operators (with respect to x). The Lagrangian forms of the above equations are given by [8] Curl E = 0, and Div D = Q,

(5)

where E = FT e, D = J F−1 d and Q = J q are the “pull-back” versions of the true electric field e, electric displacement field d and charge density q. The jump conditions for the electric fields are [[E]] × N = 0 (or [[e]] × n = 0), and [[D]] · N = Σ (or [[d]] · n = σ ),

(6)

where Σ (σ ) is the prescribed charge per unit area in the reference (deformed) configuration.

2.2 Constitutive Equations The constitutive behavior of a homogeneous electroelastic material has been described in many different ways (see, e.g., [15, 16]). However, the form developed by Dorfmann and Ogden [8] is most convenient for our purposes here. Thus, we introduce an energy–density or stored–energy function W (F, D), such that the first Piola–Kirchhoff stress and the Lagrangian electric field is given by S=

∂W ∂W (F, D), and E = (F, D). ∂F ∂D

(7)

The energy–density function W satisfies objectivity such that W (QF, D) = W (F, D), for all proper orthogonal tensors Q, which implies that W (F, D) = W (U, D), with F = RU being the polar decomposition of F. It is also useful to introduce the Eulerian energy–density function w(F, d) = W (F, D = J F−1 d)/J,

(8)

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such that the Cauchy stress T and the true electric field e are given by T=

∂w ∂w T F + (w − e · d)I + e ⊗ d, and e = (F, d). ∂F ∂d

(9)

Objectivity in this context implies that w(QF, Qd) = w(F, d), for all proper orthogonal tensors Q, or that w(F, d) = u(C = FT F, FT d), for some appropriately chosen function u. Using this it can be shown (for details, see [16]) that T = 2F

∂u T F + (u − e · d)I + 2e ⊗s d, ∂C

(10)

where the symbol ⊗s is used to denote the symmetric dyadic product. Note that (10) makes it evident that the total Cauchy stress, as given by (9), is symmetric. For materials with an incompressibility constraint (i.e., γ (F) = det F − 1 = 0) a hydrostatic pressure term is introduced in the expression for the Piola–Kirchhoff stress [21], while the expression for the Lagrangian electric field remains unchanged. Thus, for such materials, Eqs. (7) are written as S=

∂W ∂W (F, D) − pF−T , and E = (F, D). ∂F ∂D

(11)

The corresponding Eulerian equations (9) become T=

∂w ∂w T F − pI + (w − e · d)I + e ⊗ d, and e = (F, d), ∂F ∂d

(12)

where w is defined as before with the replacement J = 1. In this work, we are mainly concerned with deriving macroscopic forms for the potentials of heterogeneous electroelastic materials starting from the constitutive behavior of the phases. Thus, we provide next specific forms of the functions W (or w) for the matrix and inclusion phases. In this work, it will be assumed that the matrix, labeled with the superscript “1,” is made of a dielectric elastomer, while the inclusions, labeled with the superscript “2,” are made of much stiffer materials. In fact, it is desirable that the inclusions have high dielectric coefficients in order to generate stronger electroelastic couplings. However, naturally appearing materials (e.g., ceramics) with high dielectric coefficients also tend to be very stiff mechanically. As a consequence, the inclusions will be assumed to be perfectly rigid in this work. 2.2.1

Dielectric Elastomer Matrix Phase

For simplicity, the matrix phase is an “ideal dielectric elastomer” [19] with a linear dielectric response described by the isotropic permittivity ε(1) that is taken to be independent of the deformation. Thus, the matrix material will be described here by an energy–density function (in the reference configuration) W (1) of the form

Finite-Strain Homogenization Models for Anisotropic … (1) W (1) (F, D) = Wme (F) + Wel(1) (F, D),

291

(13)

(1) (F) is the usual (purely) mechanical stored–energy function of the elaswhere Wme tomer and Wel(1) (F, D) is the electrostatic part of the stored–energy function. For the purely mechanical stored energy we adopt the (incompressible) Gent model [11], as specified by

    μ(1) Jm(1) I −d ln 1 − (1) , with I = tr FT F , J = det F = 1, 2 Jm (14) where d specifies the dimension of the problem (i.e., d = 2 for 2D plane-strain problems and d = 3 for 3D problems), μ(1) is the shear modulus of the elastomer and Jm(1) , which specifies the limiting value for I − d, is the lock-up parameter. Note that the above purely mechanical energy density reduces to the (incompressible) neo-Hookean model as Jm(1) → ∞. On the other hand, the electrostatic stored energy of the ideal dielectrics can be written as 1 (15) Wel(1) (F, D) = (1) (FD) · (FD), 2ε J (1) Wme (F) = −

in the reference configuration. The Eulerian form of the above electrostatic energy  function, as defined by wel(1) (d) = Wel(1) F, J F−1 d /J , is wel(1) (d) =

1 −1 d · d, and therefore e = ε(1) d, 2ε(1)

(16)

which is consistent with the assumption that the dielectric response of the material is linear and independent of the deformation in the current configuration (i.e., ε(1) is a constant). It should be noted from the recent work of Šilhavý [34] that the material model described by (13), together with (14) and (15), for the matrix phase is polyconvex. The total Cauchy stress, as defined by (9), for the above special form of the electroelastic potential can be rewritten in the following more compact form T = Tme + Tel , where me

T

  (1) 1 ∂ Wme 1 T el F − pI and T = (1) d ⊗ d − (d · d)I = ∂F ε 2

(17)

(18)

are, respectively, the “purely mechanical” and “electrostatic” stresses in the material. Note that when the material is vacuum with permittivity ε0 , the electrostatic stress Tel reduces to the Maxwell stress   1 TM = ε0 e ⊗ e − (e · e)I . (19) 2

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For this reason, the electrostatic stress Tel is sometimes referred to as the Maxwell stress in the material; note that it is divergence free, just like the Maxwell stress TM . Finally, note that the decomposition provided in connection with expression (17) may not hold for all materials. For example, it will turn out that a composite made of a matrix material with constitutive behavior of the type (17) and rigid dielectric inclusions will not possess a constitutive response of this form (and will, in fact, include additional terms coupling the deformation and electric fields in a nontrivial manner).

2.2.2

Rigid, Polarizable Particles

The behavior of the material in this case can be described by the energy function (2) (F) + Wel(2) (D). W (2) (F, D) = Wme

(20)

(2) to be zero when F is a pure The rigidity constraint is enforced by requiring Wme (2) rotation R , and infinity otherwise. The electrostatic part of the energy for materials with linear dielectric behavior is taken to be of the standard form

Wel(2) (D) =

1 −1 D · E (2) D, 2

(21)

where E (2) is a constant, second-order tensor defining the anisotropic permittivity (or dielectric constant) of the material. Note that, because of objectivity, the tensor E (2) has to be independent of the deformation (or rotations for the special case of rigid particles), and is therefore a constant in the reference configuration. The corresponding energy function in the current configuration takes the form   1 −1 wel(2) R(2) , d = d · ε (2) d 2

(22)

where we have used objectivity, as well as the fact that for rigid materials F = R(2) . T In the above equation ε(2) = R(2) E (2) R(2) denotes the permittivity in the deformed configuration and depends on the rotation of the particle R(2) . Using (9), the Eulerian electric field e inside the rigid phase becomes −1

e = ε(2) d,

(23)

while the total stress becomes indeterminate. It is also important to mention that the second-order tensors E (2) and ε (2) are positive definite, which is consistent with the assumed convexity of the energy functions W (2) (F, D) and w(2) (F, d) in D and d, respectively. To obtain appropriate boundary conditions for the above-described electroelasticity problem one may use the jump conditions (3) and (6), taking into account

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293

the fact that neither the electric fields nor the stresses are zero outside the electroelastic specimen, even if the specimen is surrounded by empty space (or vacuum). This is due to the fact that vacuum can support electric fields, and therefore also the (self-equilibrated) Maxwell stress, as pointed out by [4, 8].

3 Homogenization Framework We consider a specimen Ω0 (in the reference configuration) made of the electroelastic composite, which consists of N homogeneous phases, occupying sub-domains Ω0(r ) in Ω0 . The distribution of the phases is described by the characteristic functions Θ0(r ) (r = 1, . . . , N ), such that Θ0(r ) (X) is equal to 1 for X ∈ Ω0(r ) and zero otherwise. Similarly, the specimen in its deformed configuration can be described by the characteristic functions Θ (r ) (r = 1, . . . , N ), such that Θ (r ) (x) = 1 for x ∈ Ω (r ) and zero otherwise, where Ω (r ) is the sub-domain of Ω (the deformed configuration of the specimen) that is occupied by phase r . Throughout this work, the microstructures of the electroelastic composites are assumed to be statistically homogeneous and to satisfy the separation of length scales hypothesis. In other words, it is assumed that the length scale at which the indicator functions Θ0(r ) vary (also referred to as the microscopic scale) is very small compared to the size of the specimen Ω0 (or the macroscopic scale), so that the specimen can be thought of as a representative volume element. In this section, we recall [24] a finite-strain homogenization framework for the above-described electroelastic composites with general microstructures in the quasistatic regime (see also [23] for similar results in the magnetoelastic context). Toward this goal, boundary conditions are prescribed that are consistent with “macroscopically uniform” fields in the composite. Here we enforce the conditions ¯ ¯ · N, on ∂Ω0 , x = FX, and D · N = D

(24)

¯ are a prescribed constant tensor and vector, respectively, and N is the where F¯ and D outward unit normal to the boundary of the composite specimen ∂Ω0 . It then follows that the macroscopic averages (over Ω0 ) for the deformation gradient and electric displacement fields are given by ¯ and D0 = D, ¯ F0 = F,

(25)

where ·0 has been used to denote a volume average in the reference configuration. ¯ can be interpreted as the macroscopic, or average, deforThis shows that F¯ and D mation gradient and electric displacement field in the composite Ω0 . Note that it is also possible to specify the electric field, or the traction on the boundary of the specimen. However, the boundary conditions (24) are preferred here since they lead to minimum-type variational formulations for the homogenization problem.

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Given the boundary conditions (24) and the assumed separation of length scales, it is expected on physical grounds that the composite material will behave like a homogeneous medium with effective, or homogenized energy function W˜ . Following [13] for purely elastic composites and [24] for electroelastic composites, we define the homogenized potential W˜ for the electroelastic composite as the volume average of the energy stored in the composite under application of the boundary conditions (24), so that ¯ D) ¯ = min min W (X, F, D)0 , (26) W˜ (F, ¯ F∈K(F)

¯ D∈G0 (D)

where W (X, F, D) is defined in terms of the uniform phase potentials W (r ) (F, D) via N  Θ0(r ) (X) W (r ) (F, D), (27) W (X, F, D) = r =1

and where

¯ on ∂Ω0 , ¯ = F | ∃ x = x(X) with F = Grad x in Ω0 , x = FX K(F) and

¯ = D | DivD = 0 in Ω0 , D · N = D ¯ · N on ∂Ω0 G0 (D)

(28)

(29)

are, respectively, sets of admissible deformation gradients and electric displacement fields that are compatible with the boundary conditions (24). It can be readily shown (see [4] for a more general version of this variational principle including contributions from the surrounding vacuum) that the Euler–Lagrange equations associated with the variational problem (26) are precisely the equilibrium equation (1)2 (with f0 = 0) and the Maxwell’s equations (5). Note that the energy contributions of the inhomogeneous terms, f0 , Q, and Σ, are ignored since they have been assumed to vary on the macroscopic length scale, and have no effect on the homogenization problem. Therefore, the minimizers of the above problem— assuming that they exist—are also solutions of the electroelastic problem described in the previous section with boundary conditions (24). To the best of our knowledge, there exist no rigorous mathematical results for the existence of the minimizers for the above variational problem. Here, it will be assumed that the minimizers of the variational problem (26) exist at least for small enough (but not necessarily infinitesimal) deformations and electric displacement fields. In the purely mechanical context, it is known that macroscopic instabilities may arise in these composites via loss of ellipticity [12, 18], leading to micro-domain formation beyond these instabilities [1, 10]. Similar phenomena are to be expected for electroelastic composites, but in this work the focus will be on the “principal” solution prior to the possible development of any instabilities. ¯ D) ¯ of the composite, it can Having defined the effective electroelastic energy W˜ (F, be shown by means of appropriate generalization of Hill’s lemma (see, for example, [26]) that the average stress and average electric field, determined by S¯ = S0 and

Finite-Strain Homogenization Models for Anisotropic …

E¯ = E0 , are given by

∂ W˜ ∂ W˜ , and E¯ = , S¯ = ¯ ¯ ∂F ∂D

295

(30)

¯ correspond to the average (or macrorespectively. As mentioned earlier, F¯ and D scopic) deformation gradient and electric displacement fields in the composite. Therefore, expression (30) provides the macroscopic, or homogenized constitutive relations for the composite. In other words, similar to the local energy functions W (r ) , which characterize the response of the constituent phases, the effective energy function W˜ , as defined by (26), completely describes the macroscopic response of the electroelastic composite. Note that although, in general, energy will be stored (via the electric field) in the free space surrounding the specimen, as has been shown above, only the energy stored inside the specimen (i.e., W˜ ) needs to be considered in the homogenization problem. In addition, it is noted that W˜ is objective, which can be easily verified by making use of the objectivity of the phase potentials. The Eulerian counterparts of the above effective constitutive equations for electroelastic composites can be obtained in terms of the volume averages, denoted by ·, of the true mechanical and electrical fields over the deformed configuration Ω of the composite. Thus, we define T¯ = T, d¯ = d and e¯ = e, which can be shown to satisfy the following relations: ¯ and d¯ = J¯−1 F¯ D. ¯ T¯ = J¯−1 S¯ F¯ T , e¯ = F¯ −T E,

(31)

¯ d) ¯ = Furthermore, we define the effective Eulerian energy–density function w( ˜ F, ¯ J¯. It then follows from Eq. (31)1 that the average Cauchy stress is ¯ J¯ F¯ −1 d)/ W˜ (F, given by ∂ w˜ T ¯ + e¯ ⊗ d. ¯ F + (w˜ − e¯ · d)I (32) T¯ = ∂F Note that T¯ is symmetric, which can be shown from objectivity using arguments completely analogous to those used in connection with Eq. (10) to show the symmetry of the local stress tensor T. Also, it is easy to show, by means of (31)2 , that e¯ =

∂ w˜ . ∂ d¯

(33)

In the next section, building on earlier works [23, 24], we propose a “partial decoupling strategy” in order to decouple the mechanical and electrostatic effects for a certain class of DECs with randomly distributed, but aligned ellipsoidal particles. This will allow us to express the solution of the variational problem (26) for the effective stored–energy function of these DECs in terms of the solutions of “purely mechanical” and “electrostatic” problems, coupled only through the average particle orientation in the deformed configuration, which, in turn, may be obtained by means of a finite-dimensional optimization process.

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4 Homogenization Estimates for Particle-Filled DECs with Random Microstructures We consider a composite consisting of a random distribution of rigid, dielectric inclusions firmly embedded in an ideal dielectric elastomer matrix. The matrix is assumed to be isotropic (both mechanically and electrically) and is capable of undergoing finite strains, as described by relations (13) and (15). For simplicity and ease of exposition, all the inclusions are assumed to be aligned ellipsoids of identical (but general) shape and general anisotropic dielectric properties, as characterized by relations (20) and (21). It is further assumed that the above-described electroelastic composite has a stress-free configuration in the absence of deformation and electric fields (i.e., when F = I and D = 0), and that its mechanical behavior for small deformations is characterized by the conventional theory of linear elasticity. Under these conditions, we may expect a unique solution to the Euler–Lagrange equations associated with the variational problem (26), at least for sufficiently small deformations and electric fields (i.e., in the neighborhood of F = I and D = 0). In the following subsections, we will describe the initial microstructure of the composite in the reference configuration, the corresponding evolution of the microstructure, and finally a “partial decoupling strategy” to simplify the calculation of the effective stored–energy function of the composite, as defined by Eq. (26).

4.1 Initial Microstructure As depicted in Fig. 1 (left), the initial microstructure of the composite is given by a random distribution of aligned ellipsoids Ω0I of identical shape, as defined by reference

deformed

¯,D ¯ F

(1)

(1)

(2)

(2) D0

¯ (2) R

I0

D

I

Fig. 1 Two-phase particulate DECs with randomly distributed but aligned inclusions in the reference (left) and deformed (right) configurations

Finite-Strain Homogenization Models for Anisotropic …

 −2 I0 = X | X · ZI0 X≤1 .

297

(34)

Here, the symmetric second-order tensor ZI0 characterizes the shape and orientation of the inclusions in the reference configuration. Then, letting Θ0I (X) denote the characteristic function of a single ellipsoidal inclusion, and the random set {Xα } denote the random positions of the inclusion centers, the characteristic function of the inclusion phase may be written as Θ0(2) (X)

=

 α

Θ0I (X

− Xα ) =

Ω0

Θ0I (X − Z)ψ0 (Z) dZ,

(35)

 where ψ0 (Z) := α δ(Z − Xα ) is the random density generated by the set of random points {Xα }. The probability density functions (PDFs) for the particle locations in the composite can then be obtained in terms of ensemble averages of the random density ψ0 (Z). Thus, letting ·0 denote ensemble averages in the reference configuration, the PDF of finding an inclusion located at Z is defined by p0I (Z) := ψ0 (Z)0 ,

(36)

while the joint PDF of finding a pair of inclusions located at Z and Z (Z = Z ) is defined by   (37) p0II (Z, Z ) := ψ0 (Z)ψ0 (Z ) 0 − ψ0 (Z)0 δ(Z − Z ). It is remarked here that, in general, higher order PDFs can also be defined in order to describe the microstructure of the composite more precisely (for more details on higher order statistics, see [20]). However, in this work, we will make use of Hashin–Shtrikman–Willis homogenization estimates that are capable of accounting for microstructural details of the composite up to the two-point probabilities. Thus, to avoid the complications associated with higher order statistics, in this study we will only consider the effects of the first- and second-order PDFs defined above. Given the assumption that the composite is statistically homogeneous, it follows [20] that p0I (Z) is independent of location, i.e., p0I (Z) = p0I ,

(38)

where p0I denotes the number density of the inclusions in the reference configuration. Therefore, the volume fraction of the inclusion phase in the reference configuration is also uniform and is given by   c0 = p0I Ω0I  .

(39)

In addition, under the assumption of statistical homogeneity, the joint probability p0II (Z, Z ) is invariant under arbitrary translations, i.e.,

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M. H. Siboni and P. Ponte Castañeda

p0II (Z, Z ) = p0II (Z − Z ),

(40)

and encompasses “ellipsoidal symmetry” [27].  Accordingly,  it willDbe assumed that

 , where Z0 is a symmetric (Z − Z ) p0II depends on Z − Z via the combination ZD 0 second-order tensor characterizing the shape of the “distributional ellipsoids”   −2 X≤1 . D0 = X | X · ZD 0

(41)

Note that in this context, statistical isotropy corresponds to the special case where ZD 0 = I. It is worthwhile to mention that in practice, the distribution of the inclusions can exhibit independent angular and radial behaviors. However, for cases where the initial distribution can be approximated as being ellipsoidal, simple analytical estimates may be obtained for the effective response of the composite, as will be seen later on, and hence the reason for the assumption of ellipsoidal symmetry.

4.2 Evolution of the Microstructure As depicted in Fig. 1, under the application of deformation and electric displacement fields, as described by the boundary conditions (24), the microstructure is expected to evolve in such a way that the particle positions and orientations, as well as the shape and size of the distribution, change with the applied deformation and electric field. Note that the particles will not change their size and shape, as they are rigid, but that their volume fraction can change, if the matrix material is compressible. Assuming that the matrix is capable of undergoing non-isochoric deformations (i.e., J¯ = det F¯ = 1), the volume fraction c of the inclusion phase in the deformed configuration is given by   (42) c = Ω (2)  / |Ω| = c0 / J¯,      where we have used the fact that Ω (2)  = Ω0(2)  and |Ω| = J¯ |Ω0 |. Under the application of the macroscopic mechanical and electrostatic loadings ¯ the rigid inclusions are expected to change position and orientation, but F¯ and D, their shape remains unchanged. Therefore, the microstructure of the composite in the deformed configuration is described by the deformed characteristic function Θ

(2)

(x) =

Ω

Θ I (x − z)ψ(z) dz,

(43)

where Θ I (x) is the characteristic function of a single rotated inclusion, i.e.,   −2 T x ≤ 1 , with ZI = RI ZI0 RI . I = x | x · ZI

(44)

Finite-Strain Homogenization Models for Anisotropic …

299

Here RI denotes the rigid rotation of the inclusions and ψ(z) denotes the random density associated with the set of random points {xα }, which characterize the locations of the particle centers in the deformed configuration. It is emphasized here that both RI and {xα } are expected to depend in a complicated manner on the local deformation ¯ Dealing with the evolution of gradient F, which in turn is a function of F¯ and D. the microstructure in its full complexity proves to be a very difficult task. For this reason and in the spirit of developing simple homogenization estimates, the following simplifying assumptions will be made. First, we assume that all the inclusions will rotate by the average rotation of the ¯ (2) , i.e., inclusion phase, denoted here by R ¯ (2) , for all inclusions. RI = R

(45)

Note that for random distributions, individual inclusions will experience slightly different local fields and are expected to undergo slightly different rotations. Such ¯ (2) may be incorporated into the slight misorientations from the average rotation R homogenization estimates by means of an orientation distribution function, as was done by [28]. However, such generalizations will not be considered in this work for the sake of simplicity. Second, as was the case for periodic microstructures, we assume that the evolution ¯ and of the distribution is solely determined by the macroscopic mechanical loading, F, is independent of the macroscopic electrostatic loading. Thus, it will be assumed here ellipthat the deformed joint probability density function p II (z − z ) encompasses    DT 

soidal symmetry, and therefore depends on z − z via the combination Z (z − z ). Here ZD is a second-order tensor describing the deformed ellipsoidal shape of the distribution, i.e.,    T −1 D D ¯ D D= x|x· Z Z x ≤ 1 , with ZD = FZ (46) 0. Similar to the inclusions orientations, the evolution of the distribution of particle centers is expected to depend on both the mechanical and electrostatic loadings, via the local deformation gradient. Therefore, in the context of random composites, the ¯ especially for large deformed joint PDF is expected to depend on both F¯ and D, volume fractions of the inclusions. In addition, the evolution of the joint PDF may also depend on higher point statistics. As was mentioned earlier, such higher point statistics are ignored here since the estimates that are used in this work only account for two-point statistics, and hence the reason for the simplifying ad hoc assumptions for the evolution of the distribution of the inclusion centers. This assumption would be expected to be reasonably accurate up to the possible development of an instability [1, 30].

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4.3 The Partial Decoupling Strategy and Approximation In this section, we describe a partial decoupling strategy to find the effective electroelastic response of the above-described DECs with random distributions of aligned particles. As we have seen in Sect. 2.2, the potential energy for the ideal dielectric (r ) (F) matrix and rigid inclusions can be split into a purely mechanical contribution Wme (r ) and an electrostatic contribution Wel (F, D). By making use of the corresponding decompositions for W (1) and W (2) in the expression (27) for the local energy of the composite, we obtain W (X, F, D) = Wme (X, F) + Wel (X, F, D),

(47)

where Wme (X, F) =

2 

(r ) Θ0(r ) (X)Wme (F) and Wel (X, F, D) =

r =1

2 

Θ0(r ) (X)Wel(r ) (F, D).

r =1

(48) Substituting the decomposition (47) into expression (26) for the homogenized potential, we obtain ¯ D) ¯ = min W˜ (F,

min {Wme (X, F)0 + Wel (X, F, D)0 } ,

¯ D∈G0 (D) ¯ F∈K(F)

(49)

¯ and G0 (D) ¯ are defined as before, and ·0 denotes the where the admissible sets K(F) volume average in the reference configuration. It is observed that the first term on the right side of (49) is independent of the electric displacement field D. Therefore, we can rewrite the variational problem (49) as ¯ D) ¯ = min W˜ (F,

¯ F∈K(F)

where

 ¯ F) , Wme (X, F)0 + W˜ el (D;

¯ F) = min Wel (X, F, D)0 W˜ el (D; ¯ D∈G0 (D)

(50)

(51)

is the homogenized electrostatic energy for a given (fixed) deformation field F. It is important to emphasize that both terms on the right side of (50) depend on the trial deformation field F, and therefore the mechanical and electrostatic energies are coupled together and cannot be separated, in general. Thus, to make the depen¯ F) on the deformation more transparent, it is useful to rewrite the dence of W˜ el (D; homogenized electrostatic energy in the current configuration, i.e., ¯ F) = J¯w˜ el (d; ¯ F), W˜ el (D;

(52)

Finite-Strain Homogenization Models for Anisotropic …

301

where J¯ = det F¯ and

  ¯ F) = min Θ (1) (x)wel(1) (d) + Θ (2) (x)wel(2) (R ¯ (2) , d) . w˜ el (d; ¯ d∈G(d)

(53)

In the above expression, wel(1) and wel(2) are given by (16) and (22), respectively, · is used to denote the volume average in the current configuration, and

¯ = d | div d = 0 in Ω, d · n = d¯ · n on ∂Ω G(d)

(54)

is the admissible set for the Eulerian electric displacement field d. Given the assumptions of Sect. 4.2 for the evolution of the microstructure, it can be seen that the deformed characteristic functions Θ (1) (x) and Θ (2) (x) only depend ¯ (2) . on the macroscopic deformation F¯ and the average rotation of the inclusions R Therefore, it is observed that, by writing the homogenized electrostatic energy of the composite in its “more natural” Eulerian form, the explicit dependence of w˜ el on the local trial field F disappears. In other words, the homogenized Eulerian electrostatic ¯ energy is seen to depend on the deformation only via the macroscopic deformation F, which determines the shape of the distribution ellipsoids in the current configuration, ¯ (2) , which determines the orientations of and the average rotation of the inclusions R the inclusions in the current configuration, so that   ¯ F; ¯ R ¯ (2) . ¯ F) = w˜ el d, w˜ el (d;

(55)

Now, using the results (50), (52) and (55), the variational problem (49) can be rewritten as      ¯ R ¯ (2) + J¯w˜ el d, ¯ F; ¯ R ¯ (2) , ¯ D) ¯ = min W˜ me F; (56) W˜ (F, ¯ (2) R

where

  ¯ R ¯ (2) = W˜ me F;

min

¯ R ¯ (2) ) F∈K (F,

Wme (X, F)0 .

(57)

  ¯ R ¯ (2) denotes the set of admissible deformations inside In this last expression, K F, the matrix phase that satisfy the affine condition on the boundary of the specimen, as given by (24)1 , and the prescribed rotations for the rigid inclusions, as determined ¯ (2) . It is seen that the variational problems (53) and (57) become by the tensor R ¯ (2) of the inclusions. In turn, the decoupled from each other for a given rotation R (2) ¯ eq is obtained form the outer minimization in (56), so equilibrium rotation tensor R that      (2) ¯ R ¯ (2) + J¯w˜ el d, ¯ F; ¯ R ¯ (2) . ¯ eq (58) = arg minR¯ (2) W˜ me F; R For this reason, we refer to (56) as the “partial decoupling strategy.” ¯ D) ¯ along with expresIn summary, expression (56) for the effective energy W˜ (F, ¯ (2) , the inner ˜ sions (53) and (57) for w˜ el and Wme , show that, for a given rotation R

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M. H. Siboni and P. Ponte Castañeda

elastic and electrostatic problems can be solved independently of each other. Having solved these two decoupled variational problems, the outer minimization in (58) can (2) ¯ eq as a function of the macroscopic be performed to obtain the equilibrium rotation R ¯ ¯ ¯ D) ¯ for the DEC. ˜ loading F and D, as well as the effective potential W (F, As is evident from Eq. (56), performing the outer minimization requires the knowl¯ (2) for both W˜ me and edge of explicit expressions in terms of the prescribed rotation R w˜ el . While this is relatively simple for the electrostatic part of the effective energy w˜ el , obtaining such explicit expressions for the effective mechanical energy, is more difficult (see [33]). For this reason, Ponte Castañeda and Galipeau [23] proposed the “partial decoupling approximation,” which makes use of the solution of the purely mechanical problem to obtain approximate estimates for the above general problem. (2) ¯ me denote the minimizer of the purely mechanical problem, such that Thus, letting R   (2) ¯ me ¯ R ¯ (2) , R = arg minR¯ (2) W˜ me F,

(59)

    (2) (2) ¯ R ¯ me ¯ F; ¯ R ¯ me ¯ D) ¯ ≤ W˜ me F; + J¯w˜ el d, . W˜ (F,

(60)

it follows from (56) that

Note that the right side of the inequality (60) can be treated as an estimate for the effec¯ D), ¯ which can be shown to become more accurate as the magnitude tive energy W˜ (F, of the elastic interactions becomes large compared to the electrostatic interactions. Such conditions are expected to be met for small electric fields or for (mechanically) stiff matrix materials. Finally, it is worthwhile to mention that the inequality (60) becomes an equality for the special case of DECs with aligned microstructure and under aligned mechanical and electrostatic loading conditions (at least up to the possible development of an instability). This is because of the fact that under aligned loading conditions and when the microstructure of the composite is also aligned with ¯ (2) = I). This feathe loading, the average rotation of the inclusions vanishes (i.e., R ture was exploited in [32] in their use of the approximation (60) for fiber-constrained DECs subjected to aligned loadings.

4.4 Estimates for the Effective Electrostatic Energy of Two-Phase DECs with Prescribed Particle Rotations In this subsection we provide Hashin–Shtrikman–Willis estimates for the effective electrostatic energy of DECs with random microstructures. For the electric behavior of the phases given by (16) and (22), the purely electrostatic homogenization problem (53) associated with w˜ el reduces to w˜ el



 ¯ F; ¯ R ¯ (2) = min d,

¯ d∈G(d)



 1 1 −1 ¯ d · ε (x)d = d¯ · ε˜ −1 d, 2 2

(61)

Finite-Strain Homogenization Models for Anisotropic …

303

where ε˜ is the homogenized permittivity of the composite in the deformed configuration. In (61), ε(x) is the local permittivity and is given by   ε(x) = ε(1) I + Θ (2) (x) ε (2) − ε(1) I ,

(62)

where it is recalled that ¯ (2) T , ¯ (2) E (2) R ε(1) = ε(1) I and ε (2) = R

(63)

denote the permittivity of the matrix and inclusion phases in the deformed configuration. Note here that, as described in Sect. 2.2, ε (1) and E (2) are fixed (independent of the deformation) tensors corresponding to permittivities of the matrix and inclusion ¯ (2) is treated as a given (fixed) phases, respectively. It is also emphasized that here R rotation tensor. Estimates for the effective permittivity of two-phase composites, defined by (61) with the constitutive behavior (62), and random microstructures with ellipsoidal symmetry, as described in Sect. 4.2, can be obtained [27] as ε˜ = ε(1) I + c

 −1 −1 ε (2) − ε(1) I +P .

(64)

In this expression, c is the (current) volume fraction of the particles, and the microstructural tensor P is given [27] by P = PI − cPD ,

(65)

where PI and PD are Eshelby-type microstructural tensors that encode the effect of the shape and distribution of the inclusions. For inclusions with general ellipsoidal shape, these microstructural tensors are given by PI =

det ZI 4π ε(1)

 −3 det ZD ξ ⊗ ξ ZI ξ  dS, PD = 4π ε(1) |ξ |=1

|ξ |=1

 −3 ξ ⊗ ξ ZD ξ  dS.

(66) It is emphasized that expression (64) for ε˜ depends on the deformation via the dependence of the shape tensors ZI and ZD , and the permittivity ε (2) , on the deformation, as is evident from Eqs. (44), (46), and (63). Therefore, it is useful to define ¯ := R p Pˆ 0I R pT − c0 Pˆ 0D (U), ¯ Pˆ 0 (U) such that P=

1 ¯ ˆ ¯ ¯T RP0 (U)R , ε(1)

(67)

(68)

in order to make the dependence of the P tensor on the deformation more transparent. ¯ are defined as In (67) the modified Eshelby tensors Pˆ 0I and Pˆ 0D (U)

304

det ZI0 Pˆ 0I = 4π

M. H. Siboni and P. Ponte Castañeda

 −3 det ZD 0 ξ ⊗ ξ  Z I0 ξ  dS, Pˆ 0D = 4π |ξ |=1

|ξ |=1

  ¯ −3 dS, ξ ⊗ ξ  ZD 0 Uξ

(69) ¯ (2) on the right side of (67) characterizes the ¯ TR and the rotation tensor R p := R ¯ relative rotation of the fibers with respect to the macroscopic rotation R. Using (68) and (63), the effective permittivity ε˜ , as given by (64), can be rewritten as    T  ¯ E˜ U; ¯ R ¯ (2) R ¯ , ¯ R ¯ (2) = R (70) ε˜ F; where        −1  ¯ Rp ¯ R ¯ (2) = ε(1) I + c0 R p E (2) − ε(1) I −1 R p T + 1 Pˆ 0 U, . (71) E˜ U; ε(1) J¯   ¯ R ¯ (2) , depends As is clear from (71), the deformation-dependent permittivity E˜ U; p ¯ on the deformation  viap J , the relative particle rotation R , and finally, the microstruc¯ ˆ tural tensor P0 U, R . In conclusion, we have obtained the following estimate for the effective Lagrangian ¯ (2) , namely, electrostatic potential of the DEC with prescribed particle rotation R       ¯ · U ¯ E˜ −1 U; ¯ D; ¯ R ¯ (2) = 1 D ¯ R ¯ (2) U ¯ D. ¯ W˜ el F, 2 J¯

(72)

4.5 Estimates for the Effective Mechanical Energy of Two-Phase Composites with Prescribed Particle Rotations In this section we provide an estimate for the effective elastic energy of the two-phase DECs with random microstructures and prescribed rotations on the inclusions. For this purpose, we use the which the  newly proposed variational framework of [33], in  M ¯ ¯ R ¯ (2) is obtained in term of a dual energy W˜ me F; M(2) effective energy W˜ me F; via the expression      M ¯ ¯ R ¯ (2) = stat c0 M(2) · R ¯ (2) − W˜ me F; M(2) . W˜ me F; M(2)

(73)

M is the In this equation, M(2) is a prescribed eigenstress on the inclusions and W˜ me effective energy of the composite in the presence of this externally applied eigenstress M , Siboni and Ponte Castañeda [33] presented a generalization M(2) . To obtain W˜ me of the homogenization procedure in which use was made of the fact that the effects of the applied eigenstress can be accounted for by the addition of a suitably chosen linear term (in the deformation) to the phase energy function of the inclusion phase. More precisely, this new procedure involves the variational formulation

Finite-Strain Homogenization Models for Anisotropic …

 M    M ¯ F; M(2) = min Wme W˜ me (X, F) 0 , F∈K(F¯ )

305

(74)

where the “modified” local energy W M (X, F) is defined as  (2)  M (1) (X, F) := Θ0(1) (X)Wme (F) + Θ0(2) (X) Wme (F) + M(2) · F . Wme

(75)

As shown in [33] estimates for the modified variational problem (74) can be obtained by means of the second-order homogenization method of [22]. In particular, [33] makes use of the generalized secant, second-order (GSO) method of [17] for the modified variational problem with eigenstress. The GSO method involves the use of an appropriately selected linear comparison composite (LCC) to obtain an estimate for the effective energy of the nonlinear composite. The GSO method utilizes the second moments of the deformation gradient in the phases of the LCC to define the constitutive response of the phases in the LCC, and is known to provide fairly accurate results even for high-contrast composites. The details for the computation of such estimates can be found in the recent work [33]; here, we will only provide a summary of the final results. Thus, estimates of the type given in [27] are used to estimate the effective properties of the LCC. On account of the fact that the particles have been taken to be rigid, the effective elasticity tensor is given by L˜ = L (1) +

c0 P −1 , 1 − c0 0

(76)

where c0 is the volume fraction of the inclusions in the reference configuration, L (1) is the modulus tensor of the matrix phase in the LCC, and, similar to the corresponding expression (64) for the effective permittivity, P 0 = P I0 − c0P D 0.

(77)

Here P I0 and P D 0 are Eshelby-type microstructural tensors encoding the effects of the shape and distribution of the inclusions in the reference configuration. In particular, P I0

det ZI0 := 4π

|ξ |=1

 −3 H (ξ ) (ZI0 )T ξ  dS,

(78)

where the fourth-order tensor H is defined in terms of the acoustic tensor K ik = −1 I Li(1) jkl ξ j ξl via Hi jkl (ξ ) := K ik ξ j ξl and it is recalled that Z0 is the shape tensor for the inclusions in the reference configuration—and similarly for P D 0. M Then, the corresponding second-order estimate for the effective (dual) energy W˜ me (2) for the two-phase rigidly reinforced elastomers in the presence of the eigenstress M in the inclusions is given by

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M. H. Siboni and P. Ponte Castañeda

  M ¯ ¯ (2) F; M(2) = c0 M(2) · R W˜ me     (1) ˆ (1) ¯ (2) − (1 − c0 )Fˆ (1) , ¯ · F¯ − c0 R F + S (1) (F) + (1 − c0 )Wme

(79)

(1) ¯ (2) is determined where S (1) := ∂ Wme /∂F and the average rotation of the particles R by the expression

   (2)  (1)  T (2)  (2) T (1)    ¯ + S ¯ − R ¯ (2) T M(2) − M(2) T R ¯ F¯ F¯ R S (1 − c0 ) R      (2) T   ¯ (2) − D 0 F¯ − R ¯ (2) T R(2) . ¯ D 0 F¯ − R = R (80) In addition, the variable Fˆ (1) is obtained from the covariance of the fluctuations of the deformation gradient in the matrix phase by means of (appropriate traces of) the expression     Fˆ (1) − F¯ ⊗ Fˆ (1) − F¯ =

   D  c0 ¯ (2) · ∂D 0 F¯ − R ¯ (2) . F¯ − R 2 (1) L (1 − c0 ) ∂L

(81)

In these expressions, the elasticity of the matrix L (1) in the LCC is obtained from the generalized secant condition       S (1) Fˆ (1) − S (1) F¯ = L (1) Fˆ (1) − F¯ ,

(82)

while the fourth-order tensor D 0 , which follows from the use of expression (76) for ˜ is defined as L L, L(1) . (83) D 0 := P −1 0 − (1 − c0 )L   M ¯ F; M(2) from Having computed the estimate for the effective (dual) energy W˜ me   ¯ R ¯ (2) is obtained by means expression (79), the corresponding estimate for W˜ me F; of (73).

4.6 Estimates for the Effective Electroelastic Energy of Particle-Enhanced DECs     ¯ D; ¯ R ¯ (2) and the estimate for W˜ me F; ¯ R ¯ (2) in Given the estimate (72) for W˜ el F, (79), the effective electroelastic energy of the composite is obtained from the expression        ¯ R ¯ (2) + W˜ el F, ¯ D; ¯ R ¯ (2) . ¯ D ¯ = min W˜ me F; (84) W˜ F, ¯ (2) R

The stationary condition associated with the above minimization problem results in the equation

Finite-Strain Homogenization Models for Anisotropic …

307

∂ W˜ me ∂ W˜ el + = 0, (2) ¯ ¯ (2) ∂R ∂R

(85)

which can be solved for the equilibrium rotation of the inclusions as a function of ¯ i.e., the macroscopic deformation F¯ and macroscopic electric displacement field D,   (2) (2) ¯ ¯ ¯ eq ¯ eq F, D . =R R

(86)

Finally, using (30), the macroscopic (Lagrangian) electric field E¯ and Piola– Kirchhoff stress S¯ corresponding to effective potential (84) can be obtained as  ˜ ˜ el   ∂ W ∂ W ∂ W˜ (2) ¯ D; ¯ R ¯ E¯ = = F, = S¯ me + S¯ el . , S¯ =  ¯ ¯ ¯  ¯ (2) ¯ (2) ¯ ¯ ∂D ∂D ∂ F R =R (F,D)

(87)

eq

In the latter equation  ˜   ¯ R ¯ (2)  ¯Sme = ∂ Wme F;  ¯ (2) ¯ (2) ¯ ¯ ∂ F¯ R =Req (F,D)  ˜   ¯ D; ¯ R ¯ (2)  ¯Sel = ∂ Wel F,  ¯ (2) ¯ (2) ¯ ¯ ∂ F¯

(88)

R =Req (F,D)

are, respectively, the purely mechanical (i.e., in the absence of electric fields) and electrostatic contributions of the (total) macroscopic Piola–Kirchhoff stress. Note that—as a consequence of the stationarity condition (85)—the derivatives in Eqs. ¯ (2) fixed. (87) and (88) are taken while holding R

5 Concluding Remarks In this work, we have discussed a homogenization framework for the finite-strain macroscopic response of electroelastic composite materials with random microstructures. In addition, the framework has been used to generate constitutive models for DECs consisting of initially aligned, rigid dielectric particles distributed with random positions in a dielectric elastomeric matrix. Assuming that the dielectric properties of the elastomer phase are isotropic and independent of the deformation, a decoupling strategy has been proposed. The strategy consists in writing the electroelastic homogenization problem in terms of a “purely mechanical” homogenization problems in the reference configuration and a “purely electrostatic” homogenization problem in the deformed configuration—with the two problems coupled only through the average particle orientations in the deformed configuration. Before the possible onset of any instabilities, the effective behavior of the DECs is determined by expression (56) for

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the effective electroelastic energy of the composite, in terms of the mechanical energy function (57) of a composite with prescribed particle rotations and given deformation on its boundary, and the corresponding electrostatic energy (53) of the composite in its deformed configuration. Explicit estimates of the Hashin–Shtrikman–Willis type for the electrostatic energy of DECs with aligned ellipsoidal particles have been given by expression (72), together with (71). On the other hand, the corresponding expressions for the mechanical energy function with prescribed particle rotations and given deformation on its boundary have been derived using a recently proposed generalization [33] of the “second-order linear comparison” method [17, 22] for composites with prescribed deformation only. Detailed expressions have been given in Sect. 4.5. In fact, completely explicit expressions have been given in [33] for the mechanical energy of DECs constrained by aligned, rigid dielectric fibers of elliptical cross section, and subjected to prescribed rotations of the fibers and given deformation on the boundary. In future work, use will be made of these estimates for the mechanical energy in combination with the corresponding estimates for the electrostatic energy to derive estimates for the macroscopic homogenized response of these fiber-constrained DECs [25]. Acknowledgements This work was begun with the support of the Applied Computational Analysis Program of the Office of Naval Research under Grant N000141110708 and completed with the support of the Applied Math Program of the National Science Foundation under Grant No. DMS1613926.

References 1. Avazmohammadi R, Ponte Castañeda P (2016) Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response. J Mech Phys Solids 97:37–67 2. Bar-Cohen Y (2004) Electroactive polymer (EAP) actuators as artificial muscles: reality, potential, and challenges, 2nd edn. SPIE, Bellingham, WA 3. Bertoldi K, Gei M (2011) Instabilities in multilayered soft dielectrics. J Mech Phys Solids 59:18–42 4. Bustamante R, Dorfmann A, Ogden RW (2009) Nonlinear electroelastostatics: a variational framework. Z Angew Math Phys 60:154–177 5. Cheng Z, Zhang Q (2008) Field-activated electroactive polymers. MRS Bull 33:183–187 6. deBotton G, Tevet-Deree L, Socolsky EA (2007) Electroactive heterogeneous polymers: analysis and applications to laminated composites. Mech Adv Mater Struct 14:13–22 7. Destrade M, Ogden RW (2011) On magneto-acoustic waves in finitely deformed elastic solids. Math Mech Solids 16:594–604 8. Dorfmann A, Ogden RW (2005) Nonlinear electroelasticity. Acta Mech 174:167–183 9. Eringen AC, Maugin GA (1990) Electrodynamics of continua, vol 1 – Foundations and solid media vol 2 – Fluids and complex media. Springer, New York 10. Furer J, Ponte Castañeda P (2018) Macroscopic instabilities and domain formation in neoHookean laminates. J Mech Phys Solids 118:98–114 11. Gent AN (1996) A new constitutive relation for rubber. Rubber Chem Technol 69:59–61 12. Geymonat G, Müller S, Triantafyllidis N (1993) Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch Ration Mech Anal 122:231–290

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13. Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc A 326:131–147 14. Huang C, Zhang QM, deBotton G, Bhattacharya K (2004) All-organic dielectric-percolative three-component composite materials with high electromechanical response. Appl Phys Lett 84:4391–4393 15. Hutter K, Ursescu A, van de Ven AAF (2006) Electromagnetic field matter interactions in thermoelastic solids and viscous fluids. Springer, Berlin, New York 16. Kovetz A (2000) Electromagnetic theory with 225 solved problems. Oxford University Press, Oxford 17. Lopez-Pamies O, Ponte Castañeda P (2006) On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I-theory. J Mech Phys Solids 54:807–830 18. Lopez-Pamies O, Ponte Castañeda P (2006) On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II-application to cylindrical fibers. J Mech Phys Solids 54:831–863 19. McMeeking RM, Landis CM (2005) Electrostatic forces and stored energy for deformable dielectric materials. J Appl Mech 72:581–590 20. Milton GW (2001) The theory of composites. Cambridge University Press, Cambridge 21. Ogden RW (1997) Non-linear elastic deformations. Dover Publications, New York 22. Ponte Castañeda P Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I–theory. J Mech Phys Solids 50 23. Ponte Castañeda P, Galipeau E (2011) Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J Mech Phys Solids 59:194–215 24. Ponte Castañeda P, Siboni MH (2012) A finite-strain constitutive theory for electro-active polymer composites via homogenization. Int. J. Non-Linear Mech. 47:293–306 25. Ponte Castañeda P, Siboni MH (2019) Constitutive models for anisotropic dielectric elastomer composites: finite deformation response and instabilities. Mech Res Commun 96:75–86 26. Ponte Castañeda P, Suquet P (1998) Nonlinear composites. In: van der Giessen E, Wu TY (eds) Advances in applied mechanics 34, pp 171–302 27. Ponte Castañeda P, Willis JR (1995) The effect of spatial distribution on the effective behavior of composite materials and cracked media. J Mech Phys Solids 43:1919–1951 28. Racherla V, Lopez-Pamies O, Ponte Castañeda P (2010) Macroscopic response and stability in lamellar nanostructured elastomers with “oriented” and “unoriented” polydomain microstructures. Mech Mater 42:451–468 29. Rudykh S, deBotton G (2011) Stability of anisotropic electroactive polymers with application to layered media. Z Angew Math Phys 62:1131–1142 30. Siboni MH, Avazmohammadi R, Ponte Castañeda P (2015) Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading. Math Mech Solids 23:907–928 31. Siboni MH, Ponte Castañeda P (2013) Dielectric elastomer composites: small-deformation theory and applications. Phil Mag 93:2769–2801 32. Siboni MH, Ponte Castañeda P (2014) Fiber-constrained, dielectric-elastomer composites: finite-strain response and stability analysis. J Mech Phys Solids 68:211–238 33. Siboni MH, Ponte Castañeda P (2016) Macroscopic response of particle-reinforced elastomers subjected to prescribed torques or rotations on the particles. J Mech Phys Solids 91:240–264 34. Šilhavý M (2018) A variational approach to nonlinear electro-magneto-elasticity: convexity conditions and existence theorems. Math Mech Solids 20:729–759 35. Stark KH, Garton CG (1955) Electric strength of irradiated polythene. Nature 176:1225–1226 36. Sundar V, Newnham RE (1992) Electrostriction and polarization. Ferroelectrics 135:431–446 37. Tian L, Tevet-Deree L, deBotton G, Bhattacharya K (2012) Dielectric elastomer composites. J Mech Phys Solids 60:181–198 38. Toupin RA (1956) The elastic dielectric. J Ration Mech Anal 5:849–915 39. Zhao X, Suo Z (2008) Electrostriction in elastic dielectrics undergoing large deformation. J Appl Phys 104:123530

Porosity and Diffusion in Biological Tissues. Recent Advances and Further Perspectives Raimondo Penta, Laura Miller, Alfio Grillo, Ariel Ramírez-Torres, Pietro Mascheroni and Reinaldo Rodríguez-Ramos

Abstract We present a review of porosity and diffusion in biological tissues from different perspectives. We first introduce the topic by illustrating experimental evidence related to diffusion in porous media and review a number of state of the art experimental techniques. We then proceed by providing a revisited derivation of the equations of poroelasticity from the microstructure (via asymptotic homogenization), which is especially aimed at giving a first insight on the topic to both students and scientists who are not familiar with the subject. Results based on this kind of models have only recently been presented in the literature and could possibly complement the experiments by getting a more thorough understanding on the complex interplay between porosity and diffusion. We investigate further the matter by exploring the role of diffusion in driving growth and stresses in the context of linear elastic modeling for tumors and cellular automata. We finally conclude the chapter by (a) R. Penta (B) · L. Miller School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow G12 8QQ, UK e-mail: [email protected] L. Miller e-mail: [email protected] A. Grillo · A. Ramírez-Torres Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, 10129 Turin, Italy e-mail: [email protected] A. Ramírez-Torres e-mail: [email protected] P. Mascheroni Braunschweig Integrated Centre of Systems Biology (BRICS), Helmholtz Centre for Infection Research (HZI), Rebenring 56, 38106 Braunschweig, Germany e-mail: [email protected] R. Rodríguez-Ramos Departamento de Matemáticas, Facultad de Matemática y Computación, Universidad de La Habana, 10400 Havana, CP, Cuba e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_11

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discussing diffusion in nonlinear, “active” materials, i.e., those which are possibly characterized by growth and remodeling, and (b) offering an overview on cutting edge research problems on diffusion for this class of complex materials.

1 An Introduction on Diffusion in Porous Media—Challenges and Experiments Porous materials are widespread throughout the nature and technology, from cracked rocks in the underground to synthetic membranes engineered to separate chemicals. Regardless of its field of application, the solid matrix of a porous medium imposes significant barriers to the diffusion of solutes within the pore space. This can be an advantage in some situations, for example, to filter multicomponent solutions in a reliable and affordable way. On the other hand, reduced diffusion may cause problems for applications in which sustained influx of chemicals to specific regions of the porous material is sought. This usually occurs in porous materials of biomedical interest, where there is a need for optimizing the delivery of nutrients or therapeutic agents to cells colonizing the solid matrix. In this section, we will discuss the latter issue, and focus our analysis on the diffusion of molecules in porous media in the context of tissue engineering and drug delivery. In tissue engineering, a set of techniques from engineering and life sciences is applied to develop biological constructs to restore, maintain or improve the function of tissues or whole organs [85]. Three-dimensional scaffolds are generally employed to provide suitable support for cellular proliferation and differentiation. Cells and growth factors are incorporated in natural or synthetic matrices, designed to mimic the mechanical and chemical cues characterizing the in vivo microenvironment. Generally, these scaffolds feature high porosities and interconnected pore networks, which facilitate the transport of nutrients and the removal of waste products. Due to the absence of a natural vascular network, the movement of these chemicals is primarily subject to diffusion across the scaffold pores. A better understanding of the factors that affect diffusion in the pore network is thus crucial for the design of these platforms for tissue engineering. The proper delivery of therapeutic agents depends on the transport mechanisms taking place in the tissue microenvironment [78]. The latter comprises an entangled network of cells and supporting structures, such as blood and lymphatic vessels and tissue interstitium. The interstitial space consists of two main components: a fluid phase made of water and dissolved biomolecules known as the interstitial fluid (IF), and a solid extracellular matrix (ECM). The latter is composed predominantly of a collagen and elastic fiber network, with interspersed macromolecular constituents (polysaccharides). Collagen and elastin impart structural integrity, whereas the polysaccharides, e.g., glycosaminoglycans (GAGs) and proteoglycans, influence the resistance to fluid and macromolecular motion in the interstitium [78]. In general, both convection and diffusion play a role in the distribution of therapeutics in the

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tissue. However, in some situations convective transfer of solutes becomes negligible and diffusion dominates transport dynamics. This occurs in cartilage, an elastic tissue that protects joints and is incorporated in many parts of the body when it is not dynamically loaded. Cartilage consists mainly of water and two solid components, namely collagen and aggrecan, which contribute to a heterogeneous and dense tissue. In addition, cartilage is mostly acellular and avascular, two conditions that make diffusion the preferred transport route for small solutes. Therefore, understanding how the porous structure of the solid matrix influences diffusion in cartilage is critical for the development of therapeutic strategies that necessitate the transport of molecules into this tissue (such as in the treatment of osteoarthritis or rheumatoid arthritis) [43]. Convection is generally poor in solid tumors, due to alterations in the microenvironment that emerges as the tumor progresses. Indeed, because of uniformly elevated interstitial fluid pressure in the tumor interstitium, convective transport is negligible and delivery of therapeutic agents relies on passive diffusion through the ECM. As novel cancer therapies rely on large molecules, for which passive delivery is often inefficient, it becomes crucial to identify the ECM constituents and geometric features that hinder diffusive transport in tumors [79]. The transport of molecules in tissue engineering and drug delivery is thus subject to several different factors, and it is of paramount importance to understand and characterize these dependencies. In this section, we describe some popular experimental techniques that have been employed to analyze molecular diffusion in scaffolds and tissues. Then, we report on experimental findings that quantify diffusive transport as a function of the characteristics of these biomedical porous materials.

1.1 Investigating Diffusion in Biomedical Porous Materials Several experimental techniques are employed to study diffusive transport in scaffolds and tissues. Usually, such analyses are performed in vitro and allow determination of diffusion coefficients of given solutes with different degrees of accuracy and spatial resolution. Figure 1 shows four common methods for measuring solute diffusivity in biomedical samples. Fluorescence recovery after photobleaching. A schematic representation of the fluorescence recovery after photobleaching (FRAP) technique is given in Fig. 1a. FRAP allows the measurement of the diffusion coefficient of solutes localized in specific subregions of the sample. After the specimen has reached equilibrium with a fluorescently tagged solute, the light from a high-intensity laser inactivates the fluorescence of a predefined area of the sample (photobleaching), causing a local decrease of fluorescence intensity. As time goes by, the fluorescence of the photobleached area gradually recovers as a consequence of diffusion of the fluorescent solute. By measuring the time needed to fully recover the original fluorescence intensity, it is possible to calculate the diffusivity of the solute of interest in specific sample regions [97].

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time

(b) Di Di using molecule

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Fig. 1 Experimental techniques used to identify the diffusion coefficient of solutes in biomedical porous materials: a in fluorescence recovery after photobleaching (FRAP), redistribution of fluorescent particles in bleached sample regions provide the local diffusion coefficient of the solute; b bulk solute diffusivities are calculated in diffusion cell experiments by measuring the transfer of solutes between two baths connected through a scaffold or tissue sample; c in solute absorption/desorption experiments, the solute exchange between a sample and a bath is monitored over time to allow the calculation of bulk diffusivities; d fluorescence correlation spectroscopy (FCS) provides the local diffusion coefficient of a solute by analyzing the correlations in fluorescence intensity resulting from solute particles microscopic movements

Diffusion cells. In a diffusion cell experiment, an upstream and downstream bath are connected through a permeable scaffold or tissue specimen (see Fig. 1b). The solute of interest is labeled with a fluorescent or radioactive tag and then dissolved in the upstream chamber. Over time, the labeled molecules diffuse from one bath to the other and their concentration is measured at regular intervals. From the values of concentration versus time, it is possible to measure the diffusion coefficient of the solute through the whole specimen [127]. Solute absorption or desorption. Solutes considered in this technique are generally labeled with fluorophores or radiolabels to allow the measurement of their concentration (Fig. 1c). In a typical solute absorption experiment, the specimen is left in the solute bath until equilibrium is reached. From the initial concentration and exposure time, one can calculate the amount of solute that has infiltrated the specimen and its bulk diffusivity. After the absorption stage, it is possible to perform a second step (desorption), in which the specimen is placed in another bath deprived of the solute of interest. In this way, another measurement of diffusion could be taken and

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compared to the one obtained in the previous stage. Both the measurements reflect the bulk properties of the sample, and their comparison provides information about the binding of the solute to the specimen matrix [43]. Fluorescence correlation spectroscopy. In fluorescence correlation spectroscopy (FCS), an optical system defines a subspace of the sample through the focusing of a laser source (see Fig. 1d). The system monitors the changes in fluorescence intensity exhibited by the labeled solute, originating from Brownian motion of the particles. Then, a correlation analysis of the fluctuations of fluorescence intensity is performed, and the calculation of the autocorrelation function is used to determine the diffusion coefficient of the solute under study. Molecules can be imaged in a small observation volume, which allows the hindrance to diffusion by obstacles on the molecules path to be captured [48].

1.2 Factors Affecting Solute Diffusion in Biomedical Porous Media The interactions between solute particles and solid matrix determine diffusive transport in porous materials. Some of these interactions are related to geometrical features of the solute (e.g., its size or shape compared with matrix pore size), whereas others are dictated by the composition of the matrix, such as matrix fiber concentration and spatial organization. In addition, modifications of the pore network due to mechanical loading or remodeling from cells have been shown to significantly impact molecule transport in these systems. By employing the techniques discussed in the previous section, researchers are able to characterize these interactions and provide insights into the transport process. Geometrical effects. Porous materials of biomedical interest often exhibit complex pore networks, deriving from dense meshworks of fibers or interconnected cavities. As a reference, Fig. 2 shows collagen fibrillar organization in a gel (a) and in a tumor ECM (b). In both cases, an intricate network of collagen fibers is clearly visible. Geometrical factors altering diffusion in porous materials emerge from the steric interactions between the solute and the small and tortuous pores in the scaffold or tissue. A clear inverse relationship exists between diffusivity and solute size, confirmed by all the measurement techniques cited above in both scaffolds and tissues [43, 89, 118, 125]. The increase of diffusion hindrance has been tested over a broad range of solute radii, ranging from small tracers (hydrodynamic radius Rh ∼ 0.1 nm) to large molecules (Rh ∼ 10 nm). Jain reports that the diffusivity of macromolecules in tissues can be described by a power law expression in which the diffusion coefficient varies as a function of solute molecular weight [77]. Molecule shape has also a significant impact on solute mobility in porous media. Glucose and antibodies generally display spherical shapes, whereas, some Dextrans (polysaccharides of different molecular weights) and DNA molecules have a more linear (and flexible) molecular structure. Such flexibility allows the molecules to

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Fig. 2 Comparison of the fibrillar collagen network in a 2 mg/ml gel (a) and in the ECM of a tumor grown in vivo (b). Green and red colors show collagen and blood vessels, respectively. Scalebars are 10 μm (a) and 50 μm (b). Figures reproduced with permission from [68] and [100]

adjust their conformation as they pass through the pores of the network, increasing their diffusivities with respect to more rigid particles with similar radii [43, 113]. Matrix pore size is another important variable concerning the geometrical limitations of solute transport. Researchers have shown that solute diffusivity increases for increasing pore diameters, as less hindrance is experienced by the molecules along their path [4, 43, 126]. Both scaffolds and tissues show a distribution of pore sizes more or less scattered around the mean value, which ranges from a few nanometers in cartilage to hundreds of micrometers in scaffolds [86]. Pore size variability in the same specimen is a consequence of the fabrication process or emerges from a hierarchical system of pores. The latter phenomenon implies that the movement of large molecules (compared to the average pore size) is still possible throughout the porous medium, but might be restricted to some tissue regions. Porosity is another metric used to characterize the impact of geometrical effects on solute diffusivity, defined as the ratio of void volume to total volume in a reference element of the porous material. Scaffolds for tissue engineering usually display high porosities, from 30 to 99% [86]. Solutes diffuse faster in matrices with high porosities, as more space is available for the movement of the molecules. This allows tissue engineering scaffolds to be a suitable substrate for cell culture, as nutrients and growth factor can easily permeate the material. As a general trend, researchers report larger interstitial spaces in tumors than in normal tissues. In the case of tumors, porosities from 20 to 50% are reported [88]. As a consequence of a large interstitial space, diffusion coefficients of various macromolecules are an order of magnitude higher in tumors compared to several normal tissues [78]. However, due to the suppression of convection in tumors, transport of large molecules in the interstitium remains still an issue for cancer therapies.

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Effects of matrix composition and organization. Increase in matrix concentration has been shown to reduce diffusion in agarose and collagen scaffolds, due to reduced fluid content and shorter mean free path lengths [4, 51, 118, 125]. Regarding cartilage, both the local and bulk composition of the matrix are known to affect solute transport [43, 125]. In superficial layers, where matrix density and fixed charge density are lower, diffusion is faster with respect to the bulk. For larger molecules, however, this no longer holds as collagen fibers change spatial alignment from the surface to the bulk of the tissue. In tumors, collagen concentration and its combination with proteoglycans contributes significantly to diffusion hindrance [103, 112, 118]. Tumors with high collagen content and exhibiting a well-defined collagen network are more resistant to penetration by macromolecules compared with tumors that show a loose collagen assembly. The effects of matrix composition on diffusion in tumor interstitium have also been investigated in [1], in which a modification of FCS was implemented to increase its resolution in vivo. Measurements from the new technique allow identification of two solute subpopulations with different diffusion coefficients, the slow and fast diffusive components exhibiting a difference of two orders of magnitude. This twofold transport behavior reflects the complex nature of the tumor ECM network, in which collagen and different GAGs affect the transport of molecules in distinct ways. The ratio of fast to slow diffusing molecules decreases with solute size, implying an increase in transport hindrance for large molecules. As a whole, these results suggest that therapies targeting collagen synthesis in tumors not only would decompress the interstitium and decrease interstitial fluid pressure, but they would also increase the diffusion of therapeutic agents. Effects of matrix rearrangement. The solid matrix of a porous material can deform under the action of applied external loads. Usually, dynamic loading is performed in experiments studying transport in cartilage, as a way to investigate the importance of joint movement and exercise in the delivery of therapeutics and nutrients. The loading amplitude and frequency influence the movement of fluid in the tissue, determining the convective contribution to solute transport. Results collected in a recent review [43] show that convection is responsible for a significant increase in the transport of large solutes (with molecular weight larger than 1 kDa). Convective contributions to the movement of smaller solutes have instead limited effects, as diffusion occurs faster for small molecule sizes. Static compression, on the other hand, significantly decreases diffusivities of both small and large solutes. Results for scaffolds and cartilage show that the diffusion coefficients of macromolecules are inversely related to the amount of static compression [43, 53, 105, 117, 125]. Indeed, compressive stresses reduce the effective pore size and fluid content (hydration) of the porous material, leading to slow transport dynamics. In addition to externally imposed matrix rearrangements, matrix remodeling by the tissue cellular component plays a crucial role in the diffusion process. Tissue engineering scaffolds can be used to study this phenomenon by culturing fibroblasts in collagen gels. Indeed, fibroblasts grown in such in vitro environments adhere to

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collagen fibers and contract them, modifying their original alignment [15]. The diffusion coefficients of small solutes (1 to 10 nm Stokes radii) in contracted collagen gels largely decrease compared with those in solution [82]. Interestingly, diffusivities increase with increasing distance from the fibroblasts, highlighting the local nature of the cellular interactions. The effects of matrix rearrangement are even more significant in tumors, as their matrix is continuously remodeled by host cells during disease progression. It has been shown that tumors grown in mice recruit stromal cells in different amounts, depending on the tumor implantation site [112]. For tumors in which stromal cell density is higher, greater accumulation of collagen and proteoglycans occurs. As a result, a greater hindrance to diffusion emerges in these tumors, especially for larger particles.

1.3 Concluding Remarks We discussed solute diffusion in three porous media of biomedical interest, namely tissue engineering scaffolds, cartilage, and tumor tissues. In each case, diffusive transport is inversely correlated with the complexity of the solid matrix microstructure. We described four common experimental techniques used to analyze diffusion in porous media and discussed how geometric features, matrix content, and rearrangement impact the mobility of molecules across the pore network. From the analysis, it emerges that a large number of factors contributes to diffusion hindrance in the materials. However, the relative importance of single factors remains elusive. Mathematical models could be used to address the transport phenomena, elucidating the roles of solute and matrix features in molecule movement across the interconnected pores. As a matter of fact, a better understanding of solute dynamics in porous media could improve the design of scaffolds for tissue engineering and contribute to the development of successful therapeutic strategies in tissues. The next section is devoted to a pedagogical revisited derivation of Biot’s linear theory of poroelasticity via asymptotic homogenization. The latter provides a precise prescription of the relevant mechanical and hydraulic properties of poroelastic materials on the basis of the properties of the porous microstructure.

2 The Theory of Poroelasticity The mechanical behavior of a solid elastic structure interplaying with fluid percolating, its pores can be studied via the theory of poroelasticity [19–22]. There exists a large variety of scenarios of interest that can be treated by means of a poroelastic modeling approach, including soil mechanics [133], (bio) artificial constructs [31, 80, 81], and biological tissues, such as bone [34], organs, healthy, and malignant (tumorous) cell aggregates [23].

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Materials characterized by a poroelastic mechanical response exhibit an intrinsically multiscale structure. In particular, the average pore radius, and in turn, the distance between them for an approximately uniform pore distribution (the pore-scale), is typically much smaller than the average size of the medium (the macroscale) which is effectively behaving like a poroelastic material. The upscaling process that translates a pore-scale fluid-structure interaction problem into a macroscale problem governed by the equations of poroelasticity can be carried out by means of either average field techniques or asymptotic homogenization; see, e.g., [39, 75] for a comparison between these two alternative approaches in the context of fluid and solid mechanics, respectively. The former approach is focused on the derivation of the macroscopic model as such, and relies on suitable relationships between the microscale and macroscopic energy of the system at hand. Models deduced this way can be readily extended to a nonlinear constitutive behavior of the individual phases; however, the coefficients are typically not entirely related to the underlying microstructure (see, e.g., the analytic formulas relating drained and undrained coefficients for interconnected pores reported in [93, 133]), and are usually to be determined also by exploiting experimental measurements. There also, exist simplified micromechanical approaches that provide the poroelastic coefficients for specific geometries, for example, when spherical, ellipsoidal, or “penny-shaped” diluted pores are considered; see, e.g., [33]. The asymptotic homogenization technique (see, e.g., [11, 13, 17, 74, 95, 109, 110, 123]) exploits the sharp length scale separation between a fine and a coarse scale to represent the fields in terms of power series of the ratio between them. The latter approach entails, in general, a higher degree of algebraic complexity and cannot be trivially generalized to nonlinear balance equations; however, it provides a precise prescription of the coefficients of the model. These encode information concerning the microstructure as they are provided in terms of pore-scale averages which involve both the properties of the individual phases, and auxiliary variables which are to be computed by solving differential problems on the pore-scale geometry. The latter is often assumed to be periodic to allow for actual computations of the coefficients on a small and definite portion of the microstructure. In this section, we focus on the derivation of the equations of poroelasticity for a material characterized by a low Reynolds number Newtonian incompressible fluid flowing in the pores (which is often the case for biological tissues), and we embrace the asymptotic homogenization technique to derive the macroscale system of partial differential equations (PDEs). In [24] the authors derive the standard Biot’s system of PDEs via asymptotic homogenization and provide closed formulas for the coefficients of the model, as well as their associated pore-scale differential problems. In order to illustrate a formulation which can provide computationally tractable problems, we also assume pore-scale periodicity, thus embracing an approach and notations that resemble the no-growth limit of [108], which has been recently applied in the context of tumor modeling and biomimetic materials in [40].

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2.1 Governing Equations for a Poroelastic Material We begin by considering a set Ω ∈ R3 , where Ω is the union of a porous solid compartment Ωs and a fluid compartment Ωf satisfying Ω¯ = Ω¯ s ∪ Ω¯ f , the overbar indicating closure of a set. We assume a structure where the typical length scale of the pores, denoted by d, is small compared to the size of the domain, which we denote by L. This means that we have the ratio d = ε  1. L

(1)

We make the assumption that the porous solid compartment is an anisotropic linear elastic solid which can be described by ∇ · σ = 0 in Ωs ,

(2)

where σ is the solid stress tensor. Then, the anisotropic linear elastic constitutive equation can be written σ = C ∇u, (3) where u is the elastic displacement in the porous solid and C is the fourth-order elasticity tensor equipped with major and minor symmetries and with components denoted Ci jkl . In the fluid compartment we consider a low Reynolds number Newtonian fluid, so that (4) ∇ · T = 0 in Ωf , where T is the fluid stress tensor which is defined by T = − pI + 2μD(v) with D(v) =

1 [∇v + (∇v)T ], 2

(5)

I is the identity tensor, and v, p and μ are the fluid velocity, pressure, and viscosity, respectively. The incompressibility constraint reads ∇ · v = 0 in Ωf .

(6)

We note here that computing the divergence of T in (4) using (5) gives μ∇ 2 v = ∇ p,

(7)

which represents, together with the constraint (6), the Stokes problem for an incompressible Newtonian fluid. We now need to set up an appropriate fluid-structure interaction problem between the fluid and solid phases. We, therefore, require conditions across the interface between Ωs and Ωf . We define the boundary between the

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Fig. 3 A cross section of a periodic cell at the pore-scale. The fluid phase is shown in white, the elastic matrix in red, and their interface  is highlighted in blue

phases as  := ∂Ωs ∩ ∂Ωf , as depicted in Fig. 3, and assume continuity of velocities and stresses across the interface, namely u˙ = v,

Tn = σ n

on ,

(8)

where u˙ is the solid velocity.

2.2 Nondimensionalization It is important to formulate the model in non-dimensional form in order to understand the proper asymptotic behavior of the model with respect to the scale separation parameter ε. We rescale using 1  Cd 2  C , u = Lu , v = ∇ , C = C LC v, L μ σ = C Lσ  , T = C LT , p = C L p  ,

x = Lx , ∇ =

(9)

where C is the characteristic pressure gradient. Then using (2.2) and dropping the primes for simplicity of notation, Eqs. (2)–(8) can be rewritten as C∇u) = 0 ∇ · (C

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The ε2 scaling, which appears in (12) and (14) and leads to macroscale poroelasticity, arises as a direct consequence of the reference parabolic fluid profile Cd 2 /μ. Next, we introduce the asymptotic homogenization technique.

2.3 Asymptotic Homogenization Within this section we will use a two-scale asymptotic expansion to derive a macroscale model for Eqs. (10)–(18). Since ε  1, we can enforce a sharp length scale separation between the microscale d and the macroscale L with y=

x . ε

(19)

We now assume that x and y are independent variables, representing the macroscale and the microscale, respectively. Also, the elasticity tensor C = C (x, y) is a function of both variables and the gradient operator becomes 1 ∇ → ∇x + ∇ y , ε

(20)

where ∇x and ∇ y are the gradient operators with respect to x and y, respectively. We now can perform the multiple scales expansion in power series of ε for every field ψ. In particular, we assume that the latter, which collectively denotes each field and material property appearing in (10)–(18), is given by ψ ε (x, y, t) =

∞ 

ψ (l) (x, y, t)εl .

(21)

l=0

We also assume regularity of the microstructure, so that ψ (l) and C are y-periodic. Applying the asymptotic homogenization technique the Eqs. (10)–(18) then become C∇ y uε ) + ε∇ y · (C C∇x uε ) + ε∇x · (C C∇ y uε ) + ε2 ∇x · (C C∇x uε ) = 0, ∇ y · (C

(22)

∇ y · Tε + ε∇x · Tε = 0,

(23)

Tε = − p ε I + ε[∇ y vε + (∇ y vε )T ] + ε2 [∇x vε + (∇x vε )T ],

(24)

u˙ ε = vε ,

(25)

ε3 ∇x2 vε + ε2 ∇x · (∇ y vε ) + ε2 ∇ y · (∇x vε ) + ε∇ y2 vε = ∇ y p ε + ε∇x p ε , Tε n = σ ε n,

(26) (27)

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C ∇ x uε , εσ ε = C ∇ y uε + εC

(28)

∇ y · vε + ε∇x · vε = 0,

(29)

∇ y · σ ε + ε∇x · σ ε = 0.

(30)

Remark 1 We make the assumption of the regularity of the microstructure for the sake of convenience. In order to solve the problem, we need to restrict our analysis to a finite subset of the given microstructure. By assuming y-periodicity, we are achieving the desired restriction, although the equations of poroelasticity can be derived by assuming local boundedness of the fields only; see [24].

2.4 Macroscopic Model We substitute power series of the type (21) into the relevant fields in (22)–(30). Then, by equating the coefficients of εl for l = 0, 1, . . ., this allows us to derive the macroscale model for the poroelastic material in terms of the relevant leading order fields. Whenever a component in the asymptotic expansion retains a dependence on the microscale, we can take the integral average, which we define as ψ i =



1 |Ω|

Ωi

ψ(x, y, t) dy, i = f, s,

(31)

where the integral average can be performed over one representative cell due to the assumption of y-periodicity, dy is the volume element, and |Ω| is the volume of the periodic cell, with solid and fluid portions still denoted by Ωs and Ωf , respectively, and their volumes by |Ωs | and |Ωf |. Equating coefficients of ε0 in (22)–(30) gives C∇ y u(0) ) = 0 ∇ y · (C

in Ωs ,

(32)

∇ y · T(0) = 0

in Ωf ,

(33)

= −p I

in Ωf ,

(34)

(0)

on ,

(35)

=0

in Ωf ,

(36)

(0)

(0)

T

u˙

(0)

∇y p

=v (0)

(0)

T n=σ

(0)

n

on ,

(37)

C∇y u

(0)

=0

in Ωs ,

(38)

∇y · v

(0)

=0

in Ωf ,

(39)

∇y · σ

(0)

=0

in Ωs .

(40)

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Similarly, we now wish to equate the coefficients of ε1 in Eqs. (22)–(30), which gives C∇ y u(1) ) + ∇ y · (C C∇x u(0) ) + ∇x · (C C∇ y u(0) ) = 0 ∇ y · (C (1)

∇y · T (1)

T

(1)

= − p I + [∇ y v

(0)

+ ∇x · T (0)

+ (∇ y v ) ]

= ∇y p

(1)

=v

on ,

(44)

+ ∇x p

(0)

in Ωf ,

(45)

∇y · v ∇y · σ

(1)

(1)

n

on ,

(46)

(0)

in Ωs ,

(47)

+ ∇x · v

(0)

=0

in Ωf ,

(48)

+ ∇x · σ

(0)

=0

in Ωs .

(49)

= C∇y u (1)

(42) (43)

T n=σ σ

in Ωf , in Ωf ,

(1)

(1)

(0)

(41)

(1)

u˙ ∇ y2 v(0)

=0

(0) T

in Ωs ,

(1)

+ C ∇x u

From Eqs. (33) and (34) we can see that p (0) does not depend on the microscale y, so (50) p (0) = p (0) (x, t). We also have from Eq. (38) that u(0) is a rigid body motion and therefore, by yperiodicity, (51) u(0) = u(0) (x, t) does not depend on the microscale y.

2.5 Fluid Flow on the Macroscale We now wish to investigate the leading order of the velocity which we denoted v(0) . We can define (52) w(x, y, t) = v(0) (x, y, t) − u˙ (0) (x, y, t), where w is the relative fluid-solid velocity. Using Eqs. (34), (35), (42) and (43) we have a Stokes-type periodic boundary-value problem, which is given by ∇ y2 w − ∇ y p (1) − ∇x p (0) = 0

in Ωf ,

(53)

∇y · w = 0

in Ωf ,

(54)

on ,

(55)

w=0

equipped with periodicity conditions on ∂Ωf \ . Exploiting linearity and using (50), we state the Ansatz

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w = −W∇x p (0) ,

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p (1) = −p · ∇x p (0) + c(x),

(56)

where p (1) is defined up to an arbitrary y-constant function. Equation (56)2 is the solution of the problem (53)–(55) provided that the auxiliary second-order tensor W and vector p satisfy the cell problem ⎧ 2 T ⎪ ⎨ ∇ y W − ∇ y p + I = 0 in Ωf ∇ y · WT = 0 in Ωf ⎪ ⎩ W = 0 on ,

(57)

where once again periodic conditions apply to the boundary ∂Ωf \  and a further condition is to be imposed on p for the solution to be unique (e.g., zero average on periodic cell). Taking the integral average of (52) over the fluid domain and using (56)1 leads to (58) w f = −W f ∇x p (0) , i.e., the average relative leading order fluid velocity is described by Darcy’s law.

2.6 Poroelasticity on the Macroscale We now require the macroscale equations to close the system for the solid elastic displacement u(0) and the fluid pressure p (0) . Summing up the cell averages of Eqs. (42) and (49) over Ωs and Ωf , respectively, we have  Ωs

∇y · σ

(1)

 dy +

(1)

Ωf

∇y · T

 dy +

Ωs

∇x · σ

(0)

 dy +

Ωf

∇x · T(0) dy = 0.

(59) Applying the divergence theorem with respect to y to the first two integrals and rearranging the last two terms we obtain  ∂Ωs / 



+



σ (1) nΩs dS −



T(1) n dS + ∇x ·



σ (1) n dS +



Ωs

 ∂Ωf / 

σ (0) dy + ∇x ·

T(1) nΩf dS



Ωf

T(0) dy = 0,

(60)

where we recall that n is the unit outward normal to  with respect to the fluid region Ωf , which is, therefore, pointing into the solid region, i.e., the corresponding unit outward normal to  with respect to the solid region Ωs is given by −n. The unit vectors nΩs and nΩf are the outward normals corresponding to ∂Ωs /  and ∂Ωf / , respectively. Remark 2 In Eq. (60) we have assumed that the microstructure is macroscopically uniform, i.e., the periodic cell portions Ωs and Ωf do not retain any parametric depen-

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dence on the macroscale variable x. This assumption can in principle be relaxed by assuming that the medium is not macroscopically uniform. This way, only local periodicity is assumed, whereas, the periodic representative cell is parametrically varying with respect to the macroscale coordinate, leading to macroscopically heterogeneous coefficients and additional terms in (60) due to proper application of the generalized Reynolds transport theorem. This approach [107, 108] requires the solution of a periodic cell problem (of the type solved in the present manuscript) for each point of the macroscale domain, thus leading to an increase in the computational cost, although alternative strategies to reduce it are rapidly emerging in the literature; see, e.g., [38]. Since the contributions over the external boundaries of Ωs and Ωf cancel out due to y-periodicity (60) becomes  −



σ (1) n dS +

 

T(1) n dS + ∇x ·

 Ωs

σ (0) dy + ∇x ·

 Ωf

T(0) dy = 0.

(61)

Since Eq. (46) holds, the first two terms in (61) disappear and the final two terms become (62) ∇x · σ (0) s − φ∇x p (0) = 0, where φ := |Ωf |/|Ω| is the porosity of the material. Exploiting (50) and (51), together with (34) and (47), Eqs. (41) and (37) can be rewritten in terms of the cell problem C∇ y u(1) ) + ∇ y · (C C∇x u(0) ) = 0 ∇ y · (C C∇ y u (C

(1)

(0)

(0)

+ C ∇x u )n = − p n

in Ωs ,

(63)

on ,

(64)

for u(1) , equipped with periodic conditions on ∂Ωs \ Γ . The solution to the problem given by Eqs. (63) and (64), exploiting linearity, is given as u(1) = A∇x u(0) + a p (0) ,

(65)

where A is a tensor of order 3 and a is a vector. The auxiliary fields A and a solve the cell problems 

and

C∇ y A) + ∇ y · C = 0 in Ωs ∇ y · (C C∇ y A)n + C n = 0 on , (C 

C∇ y a) = 0 in Ωs ∇ y · (C C∇ y a + I)n = 0 on , (C

(66)

(67)

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where the problems (66) and (67) are to be supplemented by periodic conditions on ∂Ωs \ Γ . One further condition on both A and a is required to ensure uniqueness, e.g., prescribing their cell average equal to a chosen constant, such as zero. Since u(1) is related to σ (0) in Eq. (47), then (65) shows that σ (0) is a function of the gradient of u(0) and p (0) . Substituting u(1) into Eq. (47) gives CM∇x u(0) + C Q p (0) + C ∇x u(0) , σ (0) = CM

(68)

where M is a fourth-order tensor and Q is a second-order tensor defined by M = ∇ y A,

Q = ∇ y a.

(69)

Then, taking the integral average of (68) over the solid domain, we obtain CM + C s ∇x u(0) + C CQ s p (0) . σ (0) s = CM

(70)

∇x · σ Eff = 0,

(71)

From (62) we have where we can define σ Eff as CM + C s ∇x u(0) + (C CQ s − φI) p (0) , σ Eff := σ (0) s − φp (0) I = CM

(72)

as the effective stress. We can describe (71) and (72) as the average force balance equations for the poroelastic material. We now return to the incompressibility constraint (48) and integrate. Applying the divergence theorem to the first integral, then using (44) and applying the divergence theorem again we obtain  0=− and hence

Ωs

tr(∇ y u˙ (1) ) dy + ∇x · v(0) f ,

∇x · v(0) f = tr (∇ y u˙ (1) ) s .

(73)

(74)

Using (65) and (69) we obtain ∇ y u(1) = M ∇x u(0) + Q p (0) ,

(75)

and taking the time derivative we obtain ∇ y u˙ (1) = M ∇x u˙ (0) + Q p˙ (0) .

(76)

Therefore, we can rewrite (74) by taking the integral average over the solid domain of the trace of (76) to arrive at

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M s : ∇x u˙ (0) + trQ s p˙ (0) , ∇x · v(0) f = trM

(77)

M)iikl for i, k, l ∈ {1, 2, 3}, M is defined, in index notation, by (trM M)kl = (M where trM with repetition of i implying summation from 1 to 3. We now take the cell average of relationship (52), restated here for convenience as (78) w(x, y, t) = v(0) (x, y, t) − u˙ (0) (x, y, t), obtaining

w f = v(0) f − φ u˙ (0) ,

(79)

where we define the porosity as φ := |Ωf |/|Ω|. Rearranging (79) gives v(0) f = w f + φ u˙ (0) .

(80)

Then using (80), we can rewrite (77) as M s : ∇x u˙ (0) + trQ s p˙ (0) , ∇x · (w f + φ u˙ (0) ) = trM

(81)

where we can expand the left-hand side to obtain M s : ∇x u˙ (0) + tr Q s p˙ (0) . ∇x · w f + φ∇x · u˙ (0) = trM

(82)

We should note that we can write φ∇x · u˙ (0) as φI : ∇x u˙ (0) in Eq. (82) and we also note that φ = constant due to the assumed geometric uniformity. We can now rearrange (82) to determine an expression for p˙ (0) , i.e. p˙ (0) =

1 M s ) : ∇x u˙ (0) , ∇x · w f + (φI − trM trQ s

(83)

−1 M s . and  α := φI − trM tr Q s

(84)

where we can define M :=

In the case of isotropy,  α in (84) will reduce to  α = αI, so that we can rewrite (83) using (84) as p˙ (0) = −M(∇x · w f +  α : ∇x u˙ (0) ). (85) We now have derived all the equations required to be able to state our macroscale model. The equations in the macroscale model describe the effective poroelastic behavior of the material relating to the pressure, the average fluid velocity, and the elastic displacement. Therefore the macroscale model to be solved is given by

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⎧ w f = −W f ∇x p (0) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∇x · σ Eff = 0,

CM + C s ∇x u(0) + (C CQ s − φI) p (0) , σ Eff = CM ⎪ ⎪ ⎪ (0) ⎪ ⎪ ⎩ p˙ = −∇x · w f −  α : ∇x u˙ (0) , M

(86)

where p (0) is the macroscale pressure, u(0) is the macroscale solid displacement, u˙ (0) is the solid velocity and w is the average fluid velocity. The first equation of the macroscale model represents Darcy’s law for w. The second of the macroscale PDEs is the stress balance equation for the poroelastic material where the material has the constitutive law given by the third equation in our macroscale model. The final equation describes the conservation of mass for a poroelastic material. It also relates changes in the fluid pressure to changes in the fluid and solid volumes. Therefore, the mechanical behavior of the poroelastic material can be fully described by the effective CM + C s , the hydraulic conductivity W f , Biot’s coefficient  α elasticity tensor CM and Biot’s modulus M. We now aim to provide a clear step-by-step guide to solving the macroscale model. When we are considering a case where macroscopic uniformity applies then we can propose the following steps as a method for solving the macroscale model. The process is as follows: 1. We begin by fixing the original material properties of the poroelastic material including fixing the elasticity tensor C and the fluid viscosity μ. 2. Then, we must fix the microscale structure by defining the cell geometry of a unit cell in the material. 3. Then the cell problems can be solved. In this case we can now solve the cell problems (57), (66) and (67). 4. Then we can substitute the cell problem solution that we obtained in step 3 into the macroscale equations in order to calculate the coefficients in the corresponding macroscale model by using (69) and the integral average (31). 5. The geometry of the macroscale must also be prescribed. The boundary conditions for the homogenized cell boundary, denoted ∂ΩH , must be given, and the system is to be supplemented by appropriate initial conditions for u(0) and p (0) . 6. Finally, the macroscale model (86) for the material can be solved. The latter algorithm has been recently enforced in [40] to investigate variations of the effective poroelastic parameters against porosity and compressibility of the solid matrix in the context of tumor modeling. The next natural step is the application of relevant multiscale models for diffusion of solutes [129]; see, e.g., [90] for an example of an application for a rigid, porous tumor mass. Extension of such approaches to deformable structures via poroelasticity, such as those studied in [111], will pave the way to gain a thorough understanding of the relative importance of the various factors which affect diffusion in porous media.

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3 Diffusion-Driven Growth Inducing Stresses in Tumors. Mathematical Modeling and Applications to Cellular Automata The model described in the previous section is applicable when a cell aggregate is considered as a porous continuum medium (where the porosity effectively accounts for the intercellular distance), whereas, in this section, the idea of “cellular aggregate” is used in a different context to take into account growth as a function of cell proliferation and death. This is the reason why we use the cellular automata for studying tumor growth. This work is focused on the mathematical model developed in [120, 132], where the authors made use of linear elasticity to generalize the Ngwa and Agyngi model [104], which describes the evolution of growth-induced stresses in a spherical and isotropic growing tumor surrounded by an external medium. In [132] the generalization of the model considering real cases of stresses are reported, and the dependence of tumor growth with the stresses is analytically derived. This particular model explores the avascular stage of a solid tumor. Next, we briefly describe the mathematical model. We aim at obtaining from the latter (which is not derived here) a cellular automaton (CA) that represents tumor growth.

3.1 Kinematics and Equilibrium Equations The tumor is modeled as a solid in the three-dimensional space and the forces on it are considered acting per volume unit. As a result of the spherical symmetry hypothesis, the problem is treated in one dimension with respect to the variable r . Because of radial symmetry (the tumor has a spherical shape and its symmetry is maintained at all times), the surface S of the tumor is S = r − R(t) = 0 where r is the radial coordinate, R(t) is the radius of the tumor at time t and the velocity field has the form v = (vr , 0, 0), leading to dR = vr (R, t). dt

(87)

This equation represents the growth rate of the tumor. Removing the inertial factors and from the hypothesis that the tumor cells form a homogeneous population, which is considered as a continuum, the equilibrium equation is given by ∇ · σ + F = 0, where F is the vector of body forces, which is considered null.

(88)

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3.2 Constitutive Equation The constitutive relation, which associates the stress σi j with the material strain ei j , represents a material with a linear elastic response subject to anisotropic growth, and is given by 1+ν ν σi j − δi j σkk , with i, j, k ∈ {r, θ, φ}, E E (89) where g is the growth factor, δi j is the Kronecker delta, γr , γθ ∈ R+ and γr + 2γθ = 1. The parameters γr and γθ represent the proportions of the tumor growth in the radial and transverse directions, respectively.

Now, assuming small deformations, e = ∇u + (∇u)T /2, using the material incompressibility (i.e. ν = 1/2) from hypothesis that the tumor is in a state of diffusion equilibrium and applying a Jaumann derivative in (89), we have the relation between the rates of deformation and stress, specifically ei j = g[δ1i γr + (δ2i + δ3i )γθ ]δi j +

1 1 D ∇v + (∇v)T i j = (∇ · v)[δ1i γr + (δ2i + δ3i )γθ ]δi j + (3σ − I trσ )i j , 2 2E Dt (90) where D/Dt is the material time derivative.

3.3 External Medium Starting from the previously introduced assumption (i.e., the existence of an external medium, which is supposed to be elastic, isotropic and incompressible), the external medium satisfies the generalized Hooke’s law σiej =

Eν E e e δi j ekk e + (1 + ν)(1 − 2ν) 1 + ν ij

(91)

and since the material is incompressible, ν = 1/2, and Eq. (91) becomes σiej = − pδi j +

2E e e , 3 ij

(92)

where p is the isotropic pressure.

3.4 Growth Equation In the context of living tissues, growth can be seen as a phenomenon arising from the net difference between the production of new cells and the death of existing ones.

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We then start from the balance of mass in the tissue and assume that the nutrient consumption rate is proportional to the nutrient concentration and to the tumor cell density. Finally, in the absence of stresses, it is reasonable to assume that the cellular proliferation rate is proportional to the nutrient concentration and tumor cell density only, such that, given a cell death which is solely proportional to the cellular density, we have ∂ρ + ∇ · (vρ) = αcρ(1 + η1 trσ ) − kρ(1 − η2 trσ ), (93)       ∂t   cellular proliferation cellular death growth

where η1 , η2 ∈ R+ are constants representing the dependence of cellular proliferation and death on stress. Now, as a consequence of tumor incompressibility (assumption that the tumor material is assumed to be incompressible and responds to stress in a purely elastic and isotropic form), we obtain from (93) ∇ · v = αc(1 + η1 trσ ) − k(1 − η2 trσ ).

(94)

3.5 Nutrient Concentration The nutrient concentration variation is determined by nutrient diffusion through the boundary of the tumor and its consumption by tumor cells in the interior. Besides, we consider the assumptions that the tumor is in a state of diffusion equilibrium and that the nutrient consumption rate is proportional to the nutrient concentration and to the tumor cell density. Without stresses, the cellular proliferation rate is proportional to the nutrient concentration and to the tumor cell density, while cell death is proportional to the cellular density. Therefore, it is noticed that ∂c + v · ∇c = Dc ∇ 2 c − Ac cρ,       ∂t diffusion

(95)

consumption

where c represents the nutrient concentration, Dc is the diffusion rate (which is assumed constant) and Ac is the nutrient consumption rate. Moreover, assuming that the nutrient concentration variation is much smaller than its diffusion and consumption, Eq. (95) can be written as Dc ∇ 2 c = Ac cρ.

(96)

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3.6 Nondimensionalization of the Model An important step in modeling is to work with dimensionless variables. The dimensionless parameters have been defined in [76] and the model is given by the system of first-order partial differential equations1 dR (t) = vr (R, t), for t ∈ R∗+ , (97) dt R sinh(r ) ∂vr (r, t) = {1 + η1 E[3σr (r, t) − 2β(r, t)]} ∂r r sinh(R) vr (r, t) , for t ∈ R∗+ , r ∈ (0, R], − ε{1 − η2 E[3σr (r, t) − 2β(r, t)]} − 2 r (98) 2β ∂σr ∗ (r, t) = − , for t ∈ R+ and r ∈ [0, R), (99) ∂r   r ∂ ∂ + vr (r, t) β(r, t), for t ∈ R∗+ , r ∈ (0, R),  (r, t) = (100) ∂t ∂r with   ∂vr vr − γr  (r, t) = 2 γθ ∂r r  R sinh(r ) {1 + η1 E[3σr (r, t) − 2β(r, t)]} = 2γθ r sinh(R)  vr (r, t) , − ε{1 − η2 E[3σr (r, t) − 2β(r, t)]} − 2 r

(101)

subject to the initial and boundary conditions R(0) = R0 , σr (r, 0) = 0, β(r, 0) = 0, v(0, t) = 0,

(102)

where β is the difference between σr and the transverse component σθ of σ , i.e. β = σr − σθ . We still need two conditions for β and σr , as follows. 1. Condition for β at r = 0. As vr = 0 at r = 0 and the derivatives of β are bounded (because β is assumed C1 in [0, R] with respect to r ), then ∂β (0, t) =  (0, t). ∂t

1 Note

that here R∗+ stands for R+ excluding 0.

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Therefore, the first boundary condition for β is  R ∂β (0, t) = (2γθ − 1) {1 − η1 E[3σr (0, t) − 2β(0, t)]} ∂t sinh(R)  ∂vr − ε{1 + η2 E[3σr (0, t) − 2β(0, t)]} + (0, t). (103) ∂r 2. Condition for σr . This condition may be determined if the constitutive equation (92) of the external medium is used, assuming the continuity of the stresses at the tumor boundary. Because of the spherical tumor symmetry from the hypothesis that the tumor has a spherical shape and its symmetry is maintained at all times, and the incompressibility of the external medium, we have tre =

1 ∂ 2 (r u r ) = 0. r 2 ∂r

Substituting the solution of Eq. (92), we obtain σr |r =R = −

4(R − R0 ) . 3R

(104)

3.7 Cellular Automata Model Definition A cellular automaton based on a linear elasticity tumor model [132] is defined. The model described in [132] and summarized in the previous section is taken as an example of a system of differential equations (DEs) that represents tissue growth, tumor growth in this case. The CA solution presented in this work is a summary of previous research reported in [76]. The CA model can be extended to other similar continuous DE models. The deterministic DE linear elasticity tumor model transformation in a cellular automaton is required. Neighborhood structure and rules for a probabilistic CA are deduced from continuous differential equations.

3.7.1

General Model Definition. Neighborhood Structure Selection

In the present study, a simple potentially infinite two-state CA is considered, where a 0-state represents a normal cell and a 1-state exemplifies a tumoral cell. The neighborhood structure and a rule set that describes the continuous model behavior are explained as follows: In the continuous model described in [132], there is an equivalence between the tumor radius growth speed and the tumor cell displacement speed vr (R, t), namely

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dR = vr (R, t), dt

335

(105)

as in (97). The speed vr could be used in the definition of the transition rules of the CA, because of the “displacement speed” oriented outside the tumor is closely related to the tumor cell propagation. Moreover, we know that vr does not depend explicitly on the spatial coordinates of a point, but instead on the radial one. This fact implies that the “influence” of a tumor is equally distributed in all directions from the center of the model. In order to select the appropriate neighborhood structure, we should take into account this property of the tumor growth speed. Then, since all the discrete time steps are equal, the increments of the tumor cell coordinates should also be equal, for the same moment of time, in all directions. In the CA, the increment of the tumor cell coordinate is described as the propagation of some tumor cell into another normal cell. Therefore, the influence zone (neighbor) of any cell c must have cells with the same distance to c. That means that we can choose some constant distance q, and, for any cell c0 , the neighborhood is defined as {c ∈ C | d(c0 , c) = q}, where d(c1 , c2 ) is the common Euclidean distance in the unit square lattice. We have chosen q = 1, which generates a well-known von Neumann neighbor√ hood [41], although it is possible to choose other values to q, such as q = 2 or q = 2, which generate other unexplored neighborhoods.

3.7.2

Inference Rule

In the area of rule inference from continuous models, the main idea of some researchers [67] has been to create a stochastic rule with a structure similar to  sc (t + Δt) =

sc (t) + 1, X ≤ g(sc (t), N (c0 ), Δt) X > g(sc (t), N (c0 ), Δt), sc (t),

(106)

where X is a random variable with uniform distribution in (0, 1), and Δt is the length of the time step. The hardest part of rule inference is precisely the correct and meaningful definition of g. Actually, g express the idea of “speed” of the growing in some place of the lattice where conditions around the cell are expressed by sc (t), N (c0 ). From the continuous model, the differential equation (105) shows the relationship between the tumor radius growth speed and the radius itself, and initially, we assume that vr (R, t) ≥ 0, i.e., the tumor is nondecreasing.

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We interpret d R/dt as the average number of new tumor cells created from one tumor cell per unit of time. If d R/dt < 1, this speed is seen as the probability of the appearance of a new tumor cell during the unit time step, i.e., the g function. In the case when η1 = η2 = 0 (stress dependence is not considered), a closed form expression for vr can be obtained from (97)–(100) as vr =

cosh R 1 εR dR = − − , dt sinh R R 3

(107)

where ε is a constant. The fact that vr (R, t) = vr (R), depending only on the maximum tumor radius, is particularly important for the rule definition process, because a cellular automaton inherently does not have any notion of time. The next state depends only on the previous state. However, if η1 = 0 or η2 = 0, then stress influence is taken into account and no closed form expression can be obtained for vr . In this case, vr (R, t) is approximated using numerical methods, such as Euler’s method or the Lax–Wendroff scheme from finite differences.

3.7.3

Case Without Stress Dependence. Rule Definition Summary

When η1 = η2 = 0 the closed form expression (107) is obtained. The properties of this equation are analyzed in order to appropriately define g. Summarizing, this provides a new way for representing a model of differential equations for tumor growth in a stochastic way. For this purpose, a random variable X with uniform distribution in (0, 1) is considered. If the growth speed at the tumor border is v then a closed form expression can be found. The rule  sc (t) + 1, X ≤ v(r ) sc (t + Δt) = X > v(r ), sc (t), is now defined, where v(r  ) has a closed form expression (specifically cosh r/ sinh r − 1/r − εr/3), and r = i 2 + j 2 is the radial coordinate of the cell c in the lattice, where (i, j) is the position of the cell in the square lattice. If a closed form expression cannot be found for v, a guide set is defined as a collection of (Ri , v(Ri )) pairs ordered by the time moment ti in which the approximation is made. If v ≥ 0, the first CA rule  sc (t) + 1, X ≤ v(Ri ) sc (t + Δt) = X > v(Ri ), sc (t), can be applied to the normal cells with at least one tumoral cell in its neighborhood, where v(Ri ) is the v evaluation for the radius Ri from the guide set that is closest to the radial coordinate of the cell r . The index i on ti indicates the time instants and

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Ri the radius of each cell at the time instant ti . We remark that this rule is applied to the cells with tumoral neighbors, in particular to cells in the border of the tumor, where r ≈ R. If v < 0, the CA rule type  sc (t + Δt) =

sc (t) − 1, X ≤ |v(Ri )| X > |v(Ri )|, sc (t),

is applicable to the tumoral cells with normal neighbors, where v(Ri ) is again the v evaluation from the guide set at the closest point to the radial coordinate of the cell r .

3.7.4

Results. Tumor Growth Visualization in Time

The CA model described above was implemented using the C# programming language in an Intel Core i7 machine, with 2.0 GHz processor and 16 GB of random access memory. Common values of the parameters used in all the computations were fixed at ε = 0.1, R0 = 1, E = 64.0439. An important clarification is that the tumor radius value in the cellular automaton is computed as the average between the tumoral border cells’ distances to the center of the tumor. An example of the tumor growth visualizations is shown in Fig. 4. It corresponds to the case without stress dependence, as described above in the previous section.

3.7.5

Concluding Remarks

In the present work, a new stochastic cellular automata model of a tumor growth process is proposed. From a linear elasticity deterministic continuous tumor model described in [132], a neighborhood structure and stochastic automata rules are deduced. The results allow visualization of the growth process described in the continuous model in a more realistic manner since tumors are neither regular nor perfectly circular. Moreover, stress influence in the growing of the tumor is taken into account when deducing CA rules. Validation tests confirm that the CA captures accurately the hypothesis of the described phenomena. The methodology exposed in this work can be applied to other continuous DE models in order to represent the growth process in a nonidealized and nondeterministic way.

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Fig. 4 Visualizations of the CA tumor growth for time moments: a 0; b 20; c 40; d 70

4 Growth, Remodeling and Diffusion in Biological Tissues In this section, we start by reviewing some previously published models of growth and remodeling of biological tissues, and we propose some plans for future research in the field. We hope that these may serve as a basis for further advances in the study of diffusion in media with evolving internal structure. Among the phenomena that characterize the evolution of biological tissues, growth, and remodeling certainly play fundamental roles. Growth manifests itself through the variation of the mass of a tissue and, to occur, a complex family of intermingled interactions has to take place [128]. Remodeling, on the other hand is usually understood as the evolution of the physical, mechanical, and transport properties of a tissue [128]. It may originate from interactions internal to the tissue, or from interactions of the tissue with its environment. In both cases, remodeling may lead to a rearrangement of the tissue’s internal structure or to changes observable from outside [35]. Such types of evolution can contribute to modify the tissue’s capability

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of bearing loads and conducting fluids as well as its sensitivity to external stimuli. A review of the most acknowledged interactions, and of the theoretical tools for taking these into account can be found in [2, 3, 8, 9, 25, 26, 35, 36, 42, 45, 50, 60, 61, 63, 65, 66, 87, 96, 115, 119, 121, 128], to mention just a few.2 In the literature, the noun “growth” may describe markedly different processes. These can be physiological, as in the case of embryonic development, an increase of muscular mass, and healing of bone fractures [35, 128], or they may refer to pathological facts, like an aortic aneurysm, hypertrophic cardiomyopathy, hyperplasia and, more specifically, formation of tumors (see, e.g., [37] for a comprehensive review). Even though, in principle, all of these events are accompanied by structural adaptations, “remodeling” may also stand on its own and, in fact, it has been extensively investigated also alone, i.e., with or without growth (see, e.g., [2, 12, 18, 47, 70, 106, 134]). Since the growth of a tissue is subordinated to the availability of nourishment, an adequate amount of nutrient substances, for instance oxygen and sugars has to be supplied to the tissue cells [29, 101, 102]. Hence, to understand the processes underlying the activation, progression, and regulation of growth, it is of fundamental importance to describe the mechanisms by which the nutrient substances reach the cells. These mechanisms, in fact, become even more intriguing when remodeling is accounted for, as it contributes to vary the environment in which the nutrients are transported. The growth and remodeling of biological tissue are often studied from a mechanical point of view and, in such a context, a tissue is modeled as a deformable porous medium hosting, in its pore space, an interstitial fluid. Within this approach, the porous medium is usually taken as the representation of a system comprising one or more cell populations, and a fiber network constituting the tissue’s extracellular matrix (ECM). Although this description could be sometimes too simple, in several cases of biomechanical interest it suffices to give an idea of the environment in which the interstitial fluid flows. Recently, the picture described above has been used in [45, 91]. The role of the interstitial fluid is to bring the nutrients to the cells, to take away the byproducts of their metabolic activity, and to remove dead cells from the tissue. From this description, it is clear that the interstitial fluid is far from being “pure water”. Rather, it is a mixture of water and other substances of various nature, which is commonly denominated fluid constituents. The evolution of these substances follows a dynamics that is governed both by their reciprocal interactions and by the interactions of the fluid with the porous medium. The latter, in turn varies its shape and internal structure in response to growth and remodeling, thereby changing the flow domain of the fluid. Following [116], we report on some aspects of the dynamics of the biphasic system “porous medium–interstitial fluid”. To accomplish this task, we invoke the 2 The

literature on growth and remodeling—especially on the mechanical aspects of these phenomena—has been proliferating in the last few years, and even attempting to provide an adequate list of authors is not an easy task.

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theory of mixtures [16, 71], which provides a well-established modeling approach for framing our study. Coherently, the porous medium is described with the classical tools of continuum mechanics, appropriately adapted and reformulated under the light shed by the theory of multiphasic materials. In addition, the interstitial fluid and its interactions with the solid phase are taken care of by having recourse to the standard laws of fluid mechanics in porous media. In doing this, we regard the fluid constituents as continua, and we focus our study on the relation between remodeling and the diffusion process by which the nutrients are transported throughout the tissue. We remark that, although the evolution of the nutrients is important for understanding how growth is initiated, we consider here remodeling alone. We do this for the following two reasons. First, the mathematical formulation of remodeling is simpler than that needed for growth. Indeed, it does not require to introduce sources/sinks of mass, nor does it call for the mass balance laws that describe the dynamics of the tissue’s constituents. Second, we are interested here in drawing attention to a possible way in which the structural transformation of a tissue influences the evolution of the nutrients in the interstitial fluid. For our purposes, we consider a transversely isotropic tissue that undergoes remodeling and, by highlighting how the latter contributes to changing the tissue’s anisotropy, we discuss the influence of remodeling on the nutrients’ diffusivity tensor. In addition, we propose some future developments of the research in this field and, in particular, we point out the extension of the current models to include fractional diffusion in anisotropic media. We emphasize that the above simplifications notwithstanding, our model can be generalized to include growth and, in fact, this is one of the topics of our current research. In Sect. 4.1, we review the setup necessary for the formulation of a problem of diffusion in a tissue that undergoes remodeling. Furthermore, in Sect. 4.2, we review the steps leading to the definition of the diffusivity tensor in the case of transverse isotropy. Finally, in Sect. 4.3, we consider two already existing models of anomalous diffusion [52, 94, 124], and we adapt them to our framework with the purpose of suggesting further investigations, possibly of interest for biomechanical problems.

4.1 Mass Balance Laws and Dynamics In our framework, a tissue is viewed as a biphasic medium comprising a solid and a fluid phase. The solid phase consists of cells and collagen fibers, with the latter ones being arranged in a way that renders the tissue transversely isotropic with respect to a given spatial direction. The interstitial fluid is a mixture of chemical substances of various types, among which the most relevant ones for our study is represented by nutrient agents. To focus on the anisotropy of the considered tissue, while keeping our mathematical formulation as simple as possible, in this work we regard the solid phase as a homogenized medium, in which no distinction is made between the dynamics of the cells and that of the fibers. These, in fact, are included with the sole scope of

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describing the tissue’s anisotropy and its evolution in response to deformation and remodeling. Clearly, more general models are possible, as is the case in [45, 62, 91], in which, however, growth is considered and the tissue is regarded as isotropic. We group the mass balance laws characterizing the system under investigation in two sets. The first one refers to the solid phase and can be written as ∂t (φs ρs ) + div (φs ρs vs ) = 0,

(108)

where ∂t is defined as ∂/∂t, φs and ρs are the volumetric fraction and the mass density of the solid phase, and vs is its velocity. The second group of mass balance laws pertains to the fluid phase and to the nutrients dissolved in it (see, e.g., [91]), i.e. ∂t (φf ρf ) + div (φf ρf vf ) = 0, ∂t (φf ρf cnf ) + div (φf ρf cnf vf ) + div ynf = 0.

(109) (110)

In (109) and (110), φf and ρf are the volumetric fraction and the mass density of the fluid phase, respectively, vf is the fluid velocity, cnf is the mass fraction of the nutrients in the fluid phase, and ynf is the mass flux vector of the nutrients, i.e. ynf = φf ρf cnf unf , with unf being the velocity of the nutrients relative to the center of mass of the fluid phase. Note that, by enforcing the saturation condition, φf must comply with the equality φf = 1 − φs . Furthermore, we assume in the sequel that the mass densities ρs and ρf are constant, thereby implying that both the solid and the fluid phase are incompressible. By adhering to the picture put forward in [42, 106], the dynamics of the system discussed so far should be studied at two different, virtually independent levels. One level is associated with the “visible” degrees of freedom of the system [42], which correspond to the deformation of the solid phase and to the flow of the interstitial fluid. The other level, instead, is related to the structural transformations undergone by the tissue, and is accounted for by allotting structural degrees of freedom with which suitable kinematic descriptors are associated. A similar framework has been adopted in the majority of previous works of some of us (see, e.g., [36, 65] and references therein). In the limit in which the inertial forces and all the long-range forces (e.g., gravity) are negligible, the “visible” dynamics are represented by the set of momentum balance laws [66, 71] div(σ s + σ f ) = 0,

(111)

divσ f + p g−1 grad φf + π f = 0, π˜ nf − φf ρf cnf g−1 grad μ˜ nf = 0.

(112) (113)

Equation (111) is the momentum balance law of the biphasic medium as a whole, and involves the sum of the Cauchy stress tensors σ s and σ f associated with the solid and with the fluid phase, respectively. Equation (112) is the momentum balance law

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of the fluid: it features the sum of the terms p g−1 grad φf and π f , which represent, respectively, the non-dissipative and the dissipative parts of the linear momentum density exchange rate between the solid and the fluid phase. Here and in the following, g is the metric tensor field associated with the three-dimensional Euclidean space. We notice that the non-dissipative force density, p g−1 grad φf , features the pressure p. Equation (113) is the momentum balance law of the nutrient substances dissolved in the fluid (details about the procedure for obtaining (113) can be found in [66, 71]): it reduces to the balance between the dissipative force density π˜ nf and the generalized force φf ρf cnf g−1 grad μ˜ nf . Here, π˜ nf describes the dissipative interactions among the nutrients and the fluid itself, while μ˜ nf is the chemical potential μnf of the nutrients relative to the chemical potential μwf of water, i.e. μ˜ nf := μnf − μwf . One can prove that σ s splits additively as σ s = −φs p g−1 + σ sc , and that, under the hypothesis of a hyperelastic solid phase, σ sc can be obtained constitutively from a free energy density. Moreover, we assume that σ f reduces to the hydrostatic Cauchy stress tensor σ f = −φf p g−1 . Before going further, we remark that a similar setting, and identical expressions for σ s and σ f have been used in many previous works of some of us [36, 45, 64, 66, 131], and can be found in many renowned publications of porous media (see, e.g., [16, 71]), with or without the hypothesis of incompressibility of the solid and the fluid phase. By following a standard praxis in porous media mechanics (see, e.g., [16, 71]), we express π f and π˜ nf constitutively as linear functions of the fluid filtration velocity q = φf (vf − vs ), and of the mass flux vector of the nutrients ynf , respectively, i.e. π f = −φf g−1 k−1 q, π˜ nf = −φf cnf g−1 λ−1 nf ynf ,

(114) (115)

where k and λnf represent, respectively, the permeability tensor of the system and the mobility tensor of the nutrients. We remark that, in the present setting, both tensors are assumed to be invertible, symmetric, and positive definite from the outset. By plugging the relationships σ s = −φs pg−1 + σ sc and σ f = −φf p g−1 into (111) and (112), and using the results (114) and (115) in (112) and (113), one obtains div (− p g−1 + σ sc ) = 0, q = −k grad p, ynf = −ρf λnf grad μ˜ nf .

(116) (117) (118)

We recognize Darcy’s law of filtration and Fick’s law of diffusion in (117) and (118), respectively. Following [42], we choose a second-order tensor field, denoted by Fg and referred to as a remodeling tensor, as a kinematic descriptor of the structural changes of the tissue. The remodeling tensor is introduced by having recourse to the Bilby–Kröner– Lee (BKL) decomposition of the deformation gradient tensor of the solid phase, F (see, e.g., [98, 122] for a review). Accordingly, F is written as F = Fe Fg , where Fe is said to be the tensor of elastic distortions. We speak of “distortions” in the

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sense of Kröner [83], since in general neither Fe nor Fg are integrable (this means that there exists no deformation whose gradient is Fe or Fg ). Moreover, we notice that the subscript “g”, which usually stands for “growth”, is kept here to identify the remodeling tensor. We make this choice to emphasize that, even though no mass sources/sinks are considered here, the remodeling of the considered tissue might be induced by growth. Given Fg , we introduce the virtual velocity V g associated with it, i.e., a secondorder tensor field representing the virtual rate of change of the remodeling distortions. Then, within a “theory of grade zero” [42],3 we introduce generalized forces expending virtual power on V g . Such forces may be distinguished in an internal one Zint , and in an external one Zext , and the principle of virtual powers leads to the local force balance [30, 42] Zint = Zext ,

(119)

holding in the internal points of the reference configuration of the considered tissue. We look for constitutive expressions for Zint by exploiting the dissipation inequality for the system under study. Expressed per unit volume of the current configuration of the tissue, the dissipation D of the system reads (see, e.g., [66]) D = −π f · q − π˜ nf · unf + J −1 (Σ sc + Zint ) : L¯ g = −π f · q − π˜ nf · unf + J −1 Δ : L¯ g = −π f · q − π˜ nf · unf + J −1 (dev Δ) : L¯ g ≥ 0,

(120)

where L¯ g ≡ Fg−1 F˙ g is the rate of distortions due to remodeling, pulled back to the reference configuration, Σ sc = J FT σ sc F−T is said to be the constitutive part of the solid phase Mandel stress tensor, and Δ ≡ Σ sc + Zint is the dissipative contribution of Zint . Note that L¯ g is deviatoric. The Mandel stress tensor can also be written as Σ sc = CSsc , where Ssc is the constitutive part of the second Piola–Kirchhoff stress tensor of the solid phase and C = FT .F = FT gF is the right Cauchy–Green deformation tensor. By its own definition, Σ sc is equipped with the symmetry property Σ sc C = (Σ sc C)T = CSsc C [49, 92]. Since the definitions given in (114) and (115) satisfy the dissipation inequality (120), we focus on the remodeling part of D, i.e. Dg ≡ J −1 (dev Δ) : L¯ g ≥ 0, in order to extract information on Δ. For this purpose, we make here the simplifying assumption of setting Zext equal to the null tensor (see [30] for a discussion on this issue). Accordingly, the force balance (119) implies that Zint is null too, and the dissipative force Δ coincides with Σ sc , thereby inheriting the same symmetry properties as the Mandel stress tensor, i.e., ΔC = (ΔC)T [49, 92]. Therefore, Dg becomes 3 This means

that Grad Fg is not rated among the variables determining the kinematic picture of the theory. Of course, it can be computed a posteriori.

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Dg = J −1 (dev Δ) : L¯ g = J −1 [(dev Δ)C] : sym(L¯ g C−1 ) ≥ 0,

(121)

where sym(•) is the symmetric part of (•), and (119) can be reformulated as (dev Δ)C = (dev Σ sc )C.

(122)

If we admit a remodeling of rate-dependent type, we may suggest expressing (dev Δ)C as a linear constitutive function of sym(L¯ g C−1 ), i.e. (dev Δ)C = K : sym(L¯ g C−1 ),

(123)

where K is a positive-definite fourth-order tensor endowed with both major and minor symmetries. Hence, by plugging (123) into (121), we end up with the following evolution law for Fg : K : sym(Fg−1 F˙ g C−1 ) = (dev Σ sc )C.

(124)

Similar laws can also be found, e.g., in [46, 66], with the rate of anelastic distortions expressed as a function of the corresponding measure of stress. A review of the evolution laws for Fg is given, e.g., in [2], whereas some differential geometry aspects connected with such laws have been provided recently in [99]. Equations (108)–(110), (116)–(118) and (124) characterize our mathematical model, which has to be completed by assigning constitutive laws for k, λnf , K , and σ sc . We do not focus here on the constitutive representations of K or σ sc , as these are out of the scope of this chapter. However, we do discuss constitutive laws for k and λnf . This is, indeed, the subject of the next section.

4.2 Fick’s Law and Diffusion in Anisotropic Growing Media When the mass fraction of the nutrient substances dissolved in the interstitial fluid is sufficiently low, the mass flux vector ynf can be expressed in terms of Fick’s law: ynf = −ρf λnf grad μ˜ nf .

(125)

Equation (125) assumes that ynf is due to diffusion only, since any dispersive effect of the flow (see, e.g., [14, 59] for a review on this issue) is typically neglected for the types of tissues under study. In general, μ˜ nf is expressed as a constitutive function of a list of variables that, beyond cnf , may also contain the deformation. In this work, however, we restrict our study to the case in which μ˜ nf is a constitutive function of the sole mass fraction, cnf . Thus, with a slight abuse of notation, we set μ˜ nf = μ˜ nf (cnf ), and we rewrite (125) as

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ynf = −ρf λnf grad μ˜ nf = −ρf λnf

∂ μ˜ nf grad cnf . ∂cnf

(126)

Finally, upon introducing the diffusivity tensor [71, 72] dnf := λnf

∂ μ˜ nf , ∂cnf

(127)

we end up with the well-known expression ynf = −ρf dnf gradcnf .

(128)

If, for example, we prescribe μ˜ nf (cf. [84], Chap. 6, p. 234) as μ˜ nf (cnf ) =

RT log Mn



Mw cnf Mn [1 − cnf ] + Mw cnf

 −

RT log Mw



Mn [1 − cnf ] Mn [1 − cnf ] + Mw cnf

 ,

(129) where Mn and Mw are the molar masses of the nutrients and water, respectively, R is the gas constant, and T is the absolute temperature (regarded as a constant in the present framework), we obtain dnf = λnf

∂ μ˜ nf RT /[cnf (1 − cnf )] = λnf . ∂cnf Mn [1 − cnf ] + Mw cnf

(130)

Since the mobility tensor has to vanish for cnf = 0 and cnf = 1, i.e., in the absence of nutrients and when the nutrients are the only fluid constituent, respectively, one can choose λnf = cnf (1 − cnf )λ0nf , which enables (130) to be recast in the form dnf = λ0nf

RT . Mn [1 − cnf ] + Mw cnf

(131)

If the mass fraction cnf is so low that dnf can be taken to be independent of cnf , one can replace (131) with dnf = λ0nf

RT . Mn

(132)

This amounts to approximating (131) with its zeroth-order approximation, obtained for cnf = 0. Since, in the present framework, the term RT /Mn features only constants, it can be absorbed in the coefficients defining λ0nf . Hence, the mass flux vector ynf is entirely determined by dnf , which should be supplied experimentally, and consistently with the theorem of representation for tensor-valued functions (see [10] and references therein). By adhering to the classification done in [10], which addresses the permeability tensor in fiber-reinforced media undergoing finite deformations, and adapting it to

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our study for the case of a transversely isotropic material, we represent dnf as (cf. Eq. (30) of [10]) dnf = d0 g−1 + d1t be + 2d2t be .be + (d1a − d1t ) m ⊗ m + (d2a − d2t ) [(m ⊗ m).be + be .(m ⊗ m)] .

(133)

Equation (133) is the most general representation of a function valued in the space of second-order tensors with transverse isotropy with respect to the direction identified by the spatial vector m. In (133), be := Fe .FeT is the left Cauchy–Green tensor generated by the elastic distortions, and the dot “.” is an abbreviation for the metric tensor, g, or for its inverse, g−1 , e.g., be .be ≡ be g be , and (m ⊗ m).be ≡ (m ⊗ m)gbe . Moreover, the coefficients d0 , d1t , d1a , d2a , and d2t are scalar functions of the invariants I1e , I2e , I3e , I4e , and I5e , defined as I1e = trbe , [b−1 e

1 2



 2 I1e − tr(be .be ) ,

I3e = det be ,

−1

: (m ⊗ m)] = Ce : (νν ⊗ ν ), be : (g ⊗ g) : m ⊗ m = C2e : (νν ⊗ ν ). = b−1 : (m ⊗ m) e

I4e = I5e

I2e =

(134) (135) (136)

In (135) and (136), ν is the unit vector specifying the direction of the fiber in the natural state, and is related to m through the normalized pullback and push forward operations ν=

Fe ν Fe−1 m , m= . −1 Fe m Feν 

(137)

Moreover, in (136), the fourth-order tensor g ⊗ g is defined as g ⊗ g := 21 [g ⊗ g + g ⊗ g],

(138)

and maps symmetric second-order tensors with contravariant components into symmetric second-order tensors with covariant components (see [54]). Going back to the scalar coefficients of dnf , we notice that, while d0 accounts for the purely spherical part of the diffusivity tensor (in the jargon of [10], the term d0 g−1 is said to be “unconditionally isotropic”), the sets of coefficients {d1a , d2a } and {d1t , d2t } determine the axial and the transverse diffusivities of dnf , respectively. A final remark about (133) concerns the fact that the elastic Cauchy–Green tensor be , rather than b, is employed to construct dnf : one reason for doing so is that the use of be clearly identifies how the structural reorganization of the tissue, described by the remodeling tensor Fg , influences the evolution of dnf . This becomes evident by performing the backward Piola transformation of dnf . Indeed, by virtue of the identity be = FBg FT , with Bg := Fg−1 .Fg−T , we obtain

Porosity and Diffusion in Biological Tissues Dnf = J F−1 dnf F−T = J d0 C−1 + J d1t Bg + 2J d2t Bg CBg d1a − d1t d2a − d2t + J M ⊗ M + 2J sym[Bg C(M ⊗ M)], I4 I4

347

(139)

where I4 := C : (M ⊗ M) is the fourth invariant of the Cauchy–Green tensor C, and M = Fg−1ν /Fg−1ν  is the normalized pullback of ν to the tangent space associated with the medium’s reference configuration. From (139) it descends that remodeling has a direct impact on the evolution of both the isotropic and the anisotropic part of the diffusion tensor. To complete the picture, we need to prescribe constitutive laws for the diffusivities {d0 , d1a , d1t , d2a , d2t }. For this purpose, we follow the suggestions given in [10] for the permeability coefficients, and we adapt them to our framework in order to include remodeling. Hence, we set 



Je − Φsν κ0 d0 = d0ν exp 21 m 0 [Je2 − 1] , 1 − Φsν  

d1aν Je − Φsν κ1a d1a = 2 exp 21 m 1a [Je2 − 1] , Je 1 − Φsν  

d1tν Je − Φsν κ1t d1t = 2 exp 21 m 1t [Je2 − 1] , Je 1 − Φsν  

d2aν Je − Φsν κ2a d2a = exp 21 m 2a [Je2 − 1] , 4 2Je 1 − Φsν  

d2tν Je − Φsν κ2t d2t = exp 21 m 2t [Je2 − 1] , 4 2Je 1 − Φsν

(140) (141) (142) (143) (144)

where Φsν is the volumetric fraction of the solid phase in the natural state, and the contribution of remodeling is accounted for by the determinant Je = J/Jg , even though we have set Jg = 1 in the present study. According to the definitions (140)–(144), fifteen parameters have to be assigned. These are given by the reference values d0ν , d1aν , d1tν , d2aν , and d2tν ; the exponents κ0 , κ1a , κ1t , κ2a , and κ2t ; and the factors m 0 , m 1a , m 1t , m 2a , and m 2t . For ease of notation, in the sequel these three sets of parameters are referred to as d-coefficients, κ-exponents, and m-factors, respectively. We notice that the d-coefficients must be all nonnegative, as they represent the values of the diffusivities in the natural state, i.e., when the condition Je = 1 applies identically. This condition, in fact, does not amount, here, to invoking the constraint of isochoric elastic distortions, although such a constraint would actually compel the scalar diffusivities (140)–(144) to be equal to their corresponding reference values, for all admissible Fe . Since the κ-exponents are generally taken as positive real-valued functions (see, e.g., the experimental values presented in [73] for permeability), the fraction Je − Φsν 1 − Φsν

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has to be nonnegative in order for the scalar diffusivities to be well-defined. If Φsν is assumed to be always strictly positive (indeed, the case Φsν = 0 means that the solid phase is locally absent in the tissue), this condition is met for Φsν < 1 and, simultaneously, for Φsν ≤ Je . The first restriction is a natural consequence of the saturation constraint, whereas, the second restriction places a lower bound on the elastic distortions: under compression, Je cannot be made arbitrarily small [56]. The permeability tensor, k, is defined analogously to dnf , and can thus be obtained from (133) by replacing the scalar diffusivities d0 , d1a , d1t , d2a , and d2t with the corresponding scalar permeabilities k0 , k1a , k1t , k2a , and k2t (see [10]). These, in turn, have the same functional forms as the diffusivities given in (140)–(144), and only require the assignment of suitable model parameters of the same type as the d-coefficients, κ-exponents, and m-factors introduced above [10]. The material form of the permeability tensor is determined via the backward Piola transformation of k, i.e. K = J F−1 kF−T , which yields an expression similar to (139) for Dnf . In [57, 58], the permeability tensor of a transversely isotropic fiber-reinforced tissue is reconstructed by upscaling the tissue’s microstructural flow properties. Under the hypotheses of impermeable fibers and resistance to the flow across the fibers much larger than the axial one, the permeability tensor obtained in [57, 58] has zero transverse coefficients and the only nonzero axial coefficient is that multiplying m ⊗ m. By adapting the results reported in [57, 58] to the diffusivity tensor in (139), one may infer that the transverse diffusivities are smaller than the axial ones. This assumption leads to significant simplifications of the expression for Dnf , and, therefore, can be very helpful to reduce the overall complexity of the mathematical model. However, it leads unavoidably to a weakening of the coupling between diffusion and remodeling, and reduces the role played by remodeling on the evolution of the tissue’s anisotropy.

4.3 An Outlook on Some Possible Research Problems Pulled back to the reference configuration of the tissue, Eq. (109) becomes (J − Φsν )ρf c˙nf + ρf (Grad cnf )Q + Div Ynf = 0,

(145)

where c˙nf is the material derivative of cnf , evaluated with respect to the solid phase velocity, Grad and Div are the “material” gradient and divergence operators, while Q = J F−1 q and Ynf = J F−1 ynf are the Piola transformed filtration velocity and nutrient mass flux vector, respectively. In terms of the standard Darcy and Fick laws, these two quantities read Q = −KGrad p,

Ynf = −ρf Dnf Grad cnf ,

where K and Dnf are defined in Sect. 4.2 (see, in particular, (139) for Dnf ).

(146)

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In this section, we would like to report some generalizations of (146)2 to the case of non-Fickean diffusion. Our purpose is to draw attention to diffusion processes that involve the nonlocal response of Ynf to the gradient of the mass fraction Grad cnf , as well as the nonlocality of Grad cnf in terms of orientations. To accomplish this task, we use nonlocal approaches of fractional type. Before going into the proposed generalizations, we ought to say that, although the literature on fractional calculus is very vast and keeps growing (see, e.g., [5–7, 32, 44, 114, 135], and the references therein, to mention just a few), we are not aware, to date, of stringent evidence that calls for the necessary fractionalization of the diffusion processes at the level of the transport of nutrients in remodeling tissues. Yet, we feel that it could be important to start paving the way towards the inclusion of fractional models into the standard framework of growth and remodeling. Indeed, beyond mere scientific curiosity, there is interest in understanding how nonlocal effects influence the overall response of tissues that grow in pathological conditions or for improving our comprehension of the interplay between diffusion and the reorientation of the fibers. We report a possible reframing of (146)2 in the context of nonlocal theories of fractional type, thereby suggesting expressing Ynf as the pullback of a fractional integral of order β = 2 − α ∈ ]0, 1[ [5–7, 28, 124], i.e. ynf (x, t) = −

ρf 2Γ (2 − α)

 Bt

dˇ nf (x  , t) gradcnf (x  , t) dv(x  ). x − x  1−(2−α)

(147)

In (147), Γ ( · ) is the Euler Gamma function, and dˇ nf is referred to as the fractional diffusivity tensor: it has the same structure as dnf in (133), but its physical units have to be L−2+α T−1 (where L and T stand for “length” and “time”, respectively). Note that, since vector fields cannot be integrated on manifolds, the integral in (147) can be done “as is” only if the integrand is written as the linear combination of constant, Cartesian basis vectors [130]. An alternative fractionalization of diffusion can be obtained by assuming that the mass flux vector of the nutrients is related to the fractional gradient of the mass fraction cnf . Note that, to lighten the notation, in the sequel we drop the subscripts “nf”, as it is clear that we are referring to the nutrients in the fluid phase. Hence, by defining the fractional gradient of order γ ∈ R, 0 < γ < 1, of the field c as [94] grad γμ c(x, t) ≡

 S2x S

γ m[Dm c(x, t)] dμ(m),

(148)

we require that the mass flux vector of the nutrients in the fluid phase is given by yμγ = −ρf d¯ grad γμ c.

(149)

Analogously to what has been said above, d¯ is formally equal to the diffusivity tensor introduced in (133), but its physical units have to be adjusted in order to account for the fractional gradient of c.

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In (148), m is a unit vector attached to the spatial point x ∈ S , with S being the three-dimensional Euclidean space, S2x S is the set of all the unit vectors emanating from x (it is a vector manifold contained in the tangent space Tx S ), μ( · ) is a γ “positive finite measure” on S2x S [94], and Dm c(x, t) is the fractional directional derivative of order γ of the field c along the direction m, evaluated at x ∈ S and γ time t. The definition of Dm c(x, t) is obtained in two steps. First, we introduce the Fourier transform of c, suitably extendeded over the whole three-dimensional Euclidean space S , i.e.  c(ξ ˆ , t) =

e−iξ [x−x0 ] c(x, t) dv(x),

(150)

S

where x0 is a point of S chosen as origin and ξ is the variable (a covector) in Fourier space. Its product ξ [x − x0 ] with the vector x − x0 reads ξa (x − x0 )a in index notation. Then, for a given m ∈ S2x S , we consider the quantity [iξ m]γ c(ξ ˆ , t), γ ˆ , t), i.e. and we identify Dm c(x, t) with the inverse Fourier transform of [iξ m]γ c(ξ γ c(x, t) Dm

1 = (2π )3

 K

eiξ [x−x0 ] [iξ m]γ c(ξ ˆ , t) dv(ξ ),

(151)

where K is the Fourier space (in fact, isomorphic to R3 ). For our purposes, it suffices to take the measure μ(m) in such a way that the integral on the right-hand-side of (148) can be rewritten as a surface integral, evaluated on the spherical surface enveloping S2x S . Hence, Eq. (148) becomes grad γμ c(x, t) ≡



2π π 0

0

γ

ˆ c(x, t)]| sin(ϑ)| dϑ dϕ, m(ϑ, ϕ)[Dm(ϑ,ϕ) ˆ

(152)

ˆ where (ϑ, ϕ) ∈ [0, π ] × [0, 2π [ is a system of spherical coordinates, and m(ϑ, ϕ) is the parametric representation of m. We notice that the evaluation of the integrals on the right-hand-side of (152) can be demanding in general, and that suitable numerical algorithms might be needed. A possible approach is the Spherical Design Algorithm [69], which is largely used in determining the overall elastic or flow properties of fiber-reinforced media with a statistical distribution of fibers [27, 55]. The parameters α and γ featuring in the fractional formulae (147) and (149) can be naturally associated with characteristic length scales that cannot be resolved when standard diffusion is considered. In this respect, they represent additional items of information on the tissue’s material behavior. A question that, at this stage, could arise is whether these parameters can be related to the growth and remodeling of the tissue to which they are referred. Assume, indeed, that α or γ are influenced for instance by the accumulated inelastic strain εg (X, t) =



 (2/3) 0

t

¯ g (X, τ ) dτ, D

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¯ g = sym(GL¯ g ) is the symmetric part of GL¯ g , and G is the material metric where D tensor field. Then, a feedback mechanism could be established that connects growth and remodeling with fractional diffusion. We believe that this topic could be of interest for a deeper understanding of these biological phenomena. Acknowledgements LM is funded by EPSRC with project number EP/N509668/1. ART and AG acknowledge the Dipartimento di Scienze Matematiche (DISMA) “G.L. Lagrange” of the Politecnico di Torino, Dipartimento di Eccellenza 2018–2022 (Department of Excellence 2018–2022, Project code: E11G18000350001). PM is supported by MicMode-I2T (01ZX1710B) from the German Federal Ministry of Education and Research (BMBF).

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122. Sadik S, Yavari A (2017) On the origins of the idea of the multiplicative decomposition of the deformation gradient. Math Mech Solids 22:771–772 123. Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, Switzerland 124. Sapora A, Cornetti P, Chiaia B, Lenzi EK, Evangelista LR (2017) Nonlocal diffusion in porous media: a spatial fractional approach. J Eng Mech 143, D4016007-1—D4016007-7 125. Shoga JS, Graham BT, Wang L, Price C (2017) Direct quantification of solute diffusivity in agarose and articular cartilage using correlation spectroscopy. Ann Biomed Eng 45:2461– 2474 126. Suhaimi H, Wang S, Thornton T, Das B (2015) On glucose diffusivity of tissue engineering membranes and scaffolds. Chem Eng Sci 126:244–256 127. Suhaimi H, Das DB (2016) Glucose diffusion in tissue engineering membranes and scaffolds. Rev Chem Eng 32:629–650 128. Taber LA (1995) Biomechanics of growth, remodeling, and morphogenesis. Appl Mech Rev 48(8):487–595 129. Taffetani M, de Falco C, Penta R, Ambrosi D, Ciarletta P (2014) Biomechanical modelling in nanomedicine: multiscale approaches and future challenges. Arch. Appl. Mech. 84:1627– 1645 130. Tarasov VE (2008) Fractional vector calculus and fractional Maxwell’s equations. Ann Phys 323:2756–2778 131. Tomic A, Grillo A, Federico S (2014) Poroelastic materials reinforced by statistically oriented fibers–numerical implementation and application to articular cartilage. IMA J Appl Math 79:1027–1059 132. Valdés-Ravelo F, Ramírez-Torres A, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Merodio J, Penta R, Conci A, Sabina FJ, García-Reimbert C (2018) Mathematical modeling of the interplay between stress and anisotropic growth of avascular tumors. J Mech Med Biol 18, 1850006–1–1850006–28 133. Wang HF (2017) Theory of linear poroelasticity with applications to geomechanics and hydrogeology. University Press, Princeton 134. Wilson W, Driessen N, van Donkelaar CC, Ito K (2006) Prediction of collagen orientation in articular cartilage by a collagen remodeling algorithm. Osteoarthr Cartil 14:1196–1202 135. Zingales M (2014) Fractional order theory of heat transport in rigid bodies. Commun Nonlinear Sci Numer Simul 19:3938–3953

Multiscale Homogenization for Linear Mechanics Reinaldo Rodríguez-Ramos, Ariel Ramírez-Torres, Julián Bravo-Castillero, Raúl Guinovart-Díaz, David Guinovart-Sanjuán, Oscar L. Cruz-González, Federico J. Sabina, José Merodio and Raimondo Penta

Abstract In this work, some results related to multiscale heterogeneous media under the asymptotic homogenization framework are collected. A multiscale asymptotic expansion is proposed and local problems and analytical effective coefficients are R. Rodríguez-Ramos (B) · R. Guinovart-Díaz Departamento de Matemáticas, Facultad de Matemática y Computación, Universidad de La Habana, 10400 La Habana, CP, Cuba e-mail: [email protected] R. Guinovart-Díaz e-mail: [email protected] A. Ramírez-Torres Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, 10129 Torino, Italy e-mail: [email protected] J. Bravo-Castillero · F. J. Sabina Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Apartado Postal 20-126, Alcaldía Álvaro Obregón, 01000 CDMX, Mexico e-mail: [email protected] F. J. Sabina e-mail: [email protected] D. Guinovart-Sanjuán Department of Mathematics, University of Central Florida, 4393 Andromeda Loop N, Orlando, FL 32816, USA e-mail: [email protected] O. L. Cruz-González Aix-Marseille University, CNRS, Centrale Marseille, LMA,4 Impasse Nikola Tesla, CS 40006, 13453 Marseille Cedex 13, France e-mail: [email protected] J. Merodio Departamento de Mecánica de los Medios Continuos y T. Estructuras, E.T.S.I de Caminos, Canales y Puertos, Universidad Politécnica de Madrid, 28040 Madrid, CP, Spain e-mail: [email protected] R. Penta School of Mathematics and Statistics, Mathematics and Statistics Building, University of Glasgow, University Place, Glasgow G128QQ, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 J. Merodio and R. Ogden (eds.), Constitutive Modelling of Solid Continua, Solid Mechanics and Its Applications 262, https://doi.org/10.1007/978-3-030-31547-4_12

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derived for fibrous and wavy laminated composites. The solution of the local problem is based on the application of Muskhelishvili’s complex potentials in the form of Taylor and Laurent series. Numerical implementation is done to compute the effective coefficients for elastic and viscoelastic composites. Comparisons with other theoretical approaches are shown.

1 Introduction The majority of existing materials around us can be considered heterogeneous structures or composite materials, since they are composed of several phases or components at certain spatial scales. Prediction of the physical behavior of such materials is a difficult objective. Their properties, also called effective or homogenized properties, entirely depend on the internal microstructure, which can be different from one to another composite in morphology, volume fraction, and, in properties of constituents. The interaction between components, such as damage because of fracture of the constituents must be also considered. Therefore, obtaining a good characterization of composite material behavior is, in general, a complex task and requires refined methods. The asymptotic homogenization theory has been important for modeling multiphase materials. The method is based on asymptotic expansions of displacement, strain and stress fields. Fundamental contributions to this modeling methodology are the original works of Bensoussan et al. [2] and Sanchez-Palencia [42]. The asymptotic homogenization method was continuously developed, and nowadays it is a very interesting research topic [5, 34]. In the last decade, several multiscale approaches have been developed and have become essential techniques for modeling composites materials. This is because performing a full direct numerical simulation including all the heterogeneities leads to a huge problem, which is expensive and unworkable because its computational cost. The links between different scales in the asymptotic homogenization process are accounted for, and therefore, this framework considers a strong coupling between the scales. Furthermore, the multiscale homogenization method is based on the principle of separation of scales, and then the microscale length is assumed much smaller than the macroscale length [11]. Consequently, in this approach, the length scales of microand macro-problems must be sufficiently separated. The multiscale homogenization method which uses the concept of representative volume element (RVE) [10, 46, 49] together with suitable computational approaches has emerged as one of the most promising formulations for addressing the response of composites structures. The RVE is employed to determine the homogenized properties and behavior of a material point at the macroscale level. It is defined as a microstructural subregion and must be large enough to be statistically representative of the composite material including all its microscopic heterogeneities [21, 24, 26]. The average field approach is based on an elastic-type constitutive behavior (assumed a priori) for the whole composite, where the effective elasticity tensor

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linearly relates fine-scale stresses and strains averaged on a representative volume. This volume should be large enough to statistically represent the whole structure and, at the same time, sufficiently small to enable a computationally feasible approach for the actual computation of the effective elasticity tensor. This can be achieved by performing elastic-type computations on a representative volume element (RVE). The asymptotic homogenization technique aims to find the effective governing equations for composite materials by enforcing the length scale separation between the fine and coarse scales, which are considered as independent spatial variables; multiple scale expansions of the fields are performed to obtain differential conditions which are used, under the assumption of fine-scale periodicity, to derive an elastic-type coarsescale problem for the whole composite. The fine-scale information is encoded in the homogenized moduli, which are to be computed solving elastic-type partial differential equations on the single periodic cell only once, thus reducing computational complexity. In this chapter, we have compiled some results related to the multiscale method applied to heterogeneous media working in two- and three-scale asymptotic homogenization framework. This work is based on previous ones, such as [12, 15, 37, 38, 40].

2 Multiscale Asymptotic Homogenization of Hierarchical Linear Elastic Heterogeneous Bodies 2.1 Formulation of the Problem Let us denote by Ω ⊂ R3 a periodic structure with hierarchical structure (see Fig. 1) and whose constituents behave as linear elastic materials. That is, the constitutive relationship of the stress tensor is, σ = C : E(u) in Ω,

(1)

where C is the positive definite fourth-order elasticity tensor, which satisfies the standard major and minor symmetries, i.e., Cijkl = Cjikl = Cijlk = Cklij . Moreover, E(u) = (∇u + ∇uT )/2 is the elastic strain tensor, with u being the elastic displacement. We consider that Ω possesses two hierarchical levels of organization. In this sense, we denote with 1 , 2 and L three well-separated length scales, and we introduce the small parameters ε1 and ε2 , given by ε1 =

1 2  1 and ε2 =  ε1 . L L

(2)

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Fig. 1 Top left: periodic hierarchical composite. Top right: composite material at the ε1 -structural level. Middle right: periodic cell at the ε1 -structural level. Middle left: composite material at the ε2 -structural level. Bottom: periodic cell at the ε2 -structural level

At the first structural level, referred to as the ε1 -structural level, we assume that Ω = ε1 ε1 Ω 1 ∪ Ω 2 , where the overbar indicates the closure of the set and Ω2ε1 = ∪Ni=1 i Ω2ε1 . At the second structural level, referred to as the ε2 -structural level, each domain i Ω2ε1 is also supposed to be composed of two different constituents, denoted by Ω1ε2 and ε2 ε2 ε2 Ω2ε2 , so that i Ω2ε1 = Ω 1 ∪ Ω 2 , where Ω2ε2 = ∪M j=1 j Ω2 . Ignoring inertial terms and volumetric forces, the problem in Ω can be formulated as ⎧ ε ε ε1 ε2 ⎪ ⎨∇ · [C : E (u )] = 0 in Ω\(Γ ∪ Γ ) (3) (Pε ) uε = u∗ on ∂d Ω ⎪ ⎩ ε ε ∗ [C : E (u )] · n = t on ∂n Ω, where n is the outward unit vector normal to the surface ∂Ω and u∗ and t∗ are the prescribed displacement and traction on ∂Ω = ∂d Ω ∪ ∂n Ω (with ∂d Ω ∩ ∂n Ω = ∅), respectively. Furthermore, we consider that the contact interface between the constituents at every scale is ideal. This means that the displacements and tractions are continuous across the interfaces Γ ε1 and Γ ε2 , that is, [[uε ]] = 0, [[[C ε : E (uε )] · nj ]] = 0,

(4)

for j = η, ζ . In (4), nη and nζ represent the outward unit vectors to the surfaces Γ ε1 and Γ ε2 , respectively. The operator [[•]] denotes the jump across the interface between the constituents.

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2.2 Three-Scale Asymptotic Homogenization Procedure Before addressing the asymptotic homogenization procedure, we introduce two formally independent variables [47], η=

x x and ζ = . ε1 ε2

(5)

We suppose that all fields and material properties are periodic in η and ζ , and introduce the notation Φ ε (x) = Φ(x, η, ζ ), where x ∈ R3 . The spatial decoupling introduced with (5), together with the chain rule, imply that ∇ = ∇x + ε1−1 ∇η + ε2−1 ∇ζ ,

(6)

where ∇j (j = x, η, ζ ) indicates that the derivative is performed with respect to j. Remark In the present work, the homogenization method is applied in regions far enough from the boundary of Ω. So, we do not deal with boundary layer effects, for which, to account for them properly, we refer to [1, 2]. In this sense, the homogenization process is independent of the choice of the boundary conditions imposed on ∂Ω.

2.2.1

Homogenization Procedure

Here we summarize the asymptotic homogenization procedure already depicted in [37, 38] by considering the most important results. First, we perform a formal asymptotic expansion for the elastic displacement in terms of the small parameters ε1 and ε2 as uε (x, η, ζ ) = u(0) (x, η, ζ ) +

∞  i=1

u(i) (x, η, ζ )ε1i +

∞ 

u˜ (i) (x, η, ζ )ε2i ,

(7)

i=1

 i (i) where we define u˜ (0) (x, η, ζ ) := u(0) (x, η, ζ ) + ∞ i=1 u (x, η, ζ )ε1 . Substituting expansion (7) into the original problem (3), using the spatial decoupling of the nabla operator (6) and considering only the coefficients accompanying the powers of ε2 , we find that u(0) = u(0) (x), u(i) = u(i) (x, η), for all i ∈ N,

(8)

i.e., u(0) only depends on the macroscopic variable and u(i) are periodic functions in η and independent of the variable ζ . The cell problems are obtained by substituting expansion (7) into the original problem (3) with interface conditions (4). Specifically, at the ε2 -structural level u˜ (1)

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can be written as

u˜ i(1) (x, η, ζ ) = χ˜ ikl (x, η, ζ )U˜ kl(0) (x, η),

(9)

 η  where U˜ kl(0) = ξklx u˜ (0) (x, η) + ε1−1 ξkl u˜ (0) (x, η) and ξklh (g) =

1 2



∂gk ∂gl + ∂hl ∂hk

 ,

with h = x, η, ζ . The third rank tensor χ˜ is η- and ζ -periodic and satisfies the ε2 -cell problem given by ⎧

∂ ε ζ ε ⎪ ⎪ ˜ C + C ξ ( χ) = 0 in Z \ Γζ , ⎪ ijpq pqkl ijkl ⎨ ∂ζj   ζ (P ) χ˜ ikl = 0 on Γζ , ⎪ 

 ⎪ ⎪ ⎩ Cε + Cε ξ ζ (χ˜ ) nζ = 0 on Γ , ζ j ijpq pqkl ijkl

(10)

where for a third-order tensor Π, h (Π) ξpqkl

1 = 2



∂Πqkl ∂Πpkl + ∂hq ∂hp

 ,

with h = ζ, η. The condition χ˜ ζ = O is imposed in order to guarantee uniqueness in the problem (Pζ ), where • ζ is the cell average operator defined as 1 • ζ = |Z|

 • dζ, Z

with |Z| representing the volume of the periodic cell Z. At the ε1 -structural level u(1) can be written as ui(1) (x, η) = χikl (x, η)ξklx (u(0) (x)),

(11)

where the third rank tensor χ is η-periodic and the solution of the ε1 -cell problem

η

⎧

∂ ˇε η ⎪ ⎪ Cijkl + Cˇ εijpq ξpqkl (χ ) = 0 ⎪ ⎨ ∂η j

in Y \ Γη ,

(P ) [[χikl ]] = 0 on Γη , ⎪ 

 ⎪ ⎪ ⎩ Cˇ ε + Cˇ ε ξ η (χ ) nη = 0 on Γ . η ijpq pqkl j ijkl The condition χ η = O is imposed to guarantee uniqueness, where 1 • η = |Y |

 (•) dη, Y

(12)

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with |Y | being the volume of the periodic cell Y . Finally, the homogenized problem is

(Ph )

⎧ ∂

x (0) h ˆ ⎪ ⎪ ⎨ ∂xj Cijkl ξkl (u ) = 0 in Ω , u(0) = u∗ ⎪ ⎪ ⎩ ˆ i x i (0) Cijkl ξkl (u )nj = Si∗

on ∂Ωdh , on ∂Ωnh ,

where Ω h denotes the homogeneous macroscale domain and η Cˆ ijkl = Cˇ ijkl + Cˇ ijpq ξpqkl (χ ) η ,

(13)

Cˇ ijkl =

(14)

Cεijkl

+

ζ Cεijpq ξpqkl (χ˜ ) ζ ,

are the effective stiffness tensor and the effective stiffness tensor at the ε1 -structural level, respectively. The fourth rank tensor arising from the application of the asymptotic homogenization technique is a genuine elasticity tensor for composites (see [31]).

3 Results and Discussion In the present section, we provide some numerical results for specific hierarchical composite materials and discuss their utility in biological scenarios of interest.

3.1 Hierarchical Layered Composite Materials Let Ω be a composite material representing a hierarchical layered structure. Specifically, assume that the laminates are orthogonal to the e3 direction, where {ei }3i=1 represents an orthonormal basis of Cartesian coordinates {xi }3i=1 (see Fig. 2). Therefore, in this particular case, the material properties of the composite only change along the e3 direction. Furthermore, we consider that, at the ε1 -structural level, the composite material is composed of two different constituents with elasticity tensors C γ ,η with γ = 1, 2. In addition, we consider that each of these constituents has characteristics of a composite material and that each one is composed of two constituents. So, at the ε2 -structural level we have to define four elasticity tensors C 1,γ ,ζ and C 2,γ ,ζ , with γ = 1, 2. We further assume that C 1,γ ,ζ and C 2,γ ,ζ are piecewise constant, so that the parametric dependence of the ε2 -cell problem on the variables η and x is lost and χ˜ depends only on ζ . As a consequence, Cˇε is also piecewise constant (as it is averaged on ζ ), and therefore χ depends only on η, so that also Cˆ is piecewise constant.

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Fig. 2 Schematic of a hierarchical layered composite where the laminates are oriented normal to the x3 axis

The above considerations reduce the cell problems to ordinary differential equations in the variable ζ3 and η3 , that is d dζ d dη

 d Cεi3kl (ζ ) + Cεi3p3 (ζ ) dζ  d Cˇ i3kl (η) + Cˇ i3p3 (η) dη



 kl χ˜ p (ζ ) = 0 in Z \ Γζ ,

 kl χp (η) = 0 in Y \ Γη ,

(15) (16)

where the notations ζ3 := ζ and η3 := η have been adopted. Using the results given in [38], the effective coefficients at the ε1 -structural level are ε −1 ε Cˇ ijkl = Cεijkl ζ + Cεijp3 (Cεp3s3 )−1 ζ (Cεs3t3 )−1 −1 ζ (Ct3m3 ) Cm3kl ζ

− Cεijp3 (Cεp3s3 )−1 Cεs3kl ζ ,

(17)

and the effective coefficients of the hierarchical layered composite are −1 ˇ ˇ Cˆ ijkl = Cˇ ijkl η + Cˇ ijp3 (Cˇ p3s3 )−1 η (Cˇ s3t3 )−1 −1 η (Ct3m3 ) Cm3kl η

− Cˇ ijp3 (Cˇ p3s3 )−1 Cˇ s3kl η .

(18)

Formulas (17) and (18) can be useful in the study of the mechanical properties of biological composites with plywood-like structures, e.g., collagen networks in compact bone. In [38], the effect of the laminate orientation on the effective mechanical properties of the osteon were investigated.

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Fig. 3 Schematic of a fiber-reinforced hierarchical composite material

3.2 Fiber-Reinforced Hierarchical Composite Materials Now, we consider the case of a fiber-reinforced hierarchical composite material. In particular, we consider a square-symmetric arrangement of uniaxially aligned and cylindrical fibers (see Fig. 3). Then, the cell problems given by (10) and (12) can be rewritten over the cross section of the cells (see [37]). In particular, at the ε1 -structural level the composite material is composed by two different constituents with elasticity tensors C γ ,η and γ = 1, 2. We further suppose that one of the materials at the ε1 -structural level is heterogeneous whereas the other is homogeneous. Therefore, at the ε2 -structural level we have to take into account only two elasticity tensors C γ ,ζ with γ = 1, 2. The fourth-order tensors C γ ,ζ are considered to be piecewise constant and isotropic. The established geometrical framework has as a consequence that each cell problem uncouples in sets of equations for the in-plane (four equations) and out-of-plane (two equations) stresses. The isotropic and the square-symmetric arrangement of the cylindrical fibers implies that the fiber phase at the ε1 -structural level (after it has been homogenized) can have at most tetragonal symmetry (i.e., 6 independent elastic coefficients) (see, e.g., [28, 29]. Therefore, the anti-plane cell problems (10) and (12) γ can be expressed in terms of the doubly periodic functions χ˜ 33l (ζ ) for ζ ∈ Zγ and γ χ33l (η) for η ∈ Yγ as ⎧ ⎪ ⎪ ⎪Δζ (χ˜ 33l ) = 0 ⎨ ζ (P3l ) [[χ˜ 33l ]] = 0    ⎪ ∂ χ˜ 33l ζ ζ ζ ζ ⎪ ⎪ nj = − C3131 nl ⎩ C3131 ∂ζj ⎧ ⎪ Δη (χ33l ) = 0 ⎪ ⎪ ⎨ η (P3l ) [[χ   33l ]] = 0  η ⎪ ∂χ33l η η η ⎪ ⎪ = − C3131 nl C n ⎩ 3131 j ∂ηj

in Z \ Γζ , on Γζ ,

(19)

on Γζ , in Y \ Γη , on Γη ,

(20)

on Γη ,

with j = 1, 2. The theory of analytical functions in [25] can be applied to solve the cell problems (19) and (20), where doubly periodic harmonic functions need to be found (see [37, 41] for more details). As a consequence of the constituent’s isotropic

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behavior, Cˇ 1313 = Cˇ 2323 , Cˆ 1313 = Cˆ 2323 and Cˇ 2313 = Cˇ 1323 = Cˆ 2313 = Cˆ 1323 = 0. In particular, the non-zero effective coefficient at the ε1 -structural level is given by   ˇC1313 = Cε + Cε ∂ χ˜ 33p , 1313 13pq ∂ζq ζ

(21)

and the effective coefficient of the macroscopic composite material is   ˆC1313 = Cˇ ε + Cˇ ε ∂χ33p . 1313 13pq ∂ηq η

(22)

In the context of elasticity the coefficients Cˇ 1313 and Cˆ 1313 refer to the shear modulus. In order to compute the effective coefficients (21) and (22), we fix the following 1,η 1,ζ 2,ζ material properties C1313 = 2, C1313 = 0.5 and C1313 = 0.7, and we perform a parametric study by varying the volume fraction of the fiber at the ε2 -structural level, denoted by V2,ζ and fix the volume fraction of the fiber phase at the ε1 -structural level at V2,η = 0.7. We note that in this particular situation there is no need to fix the shear properties for the fiber phase at the ε1 -structural level since it is equal to Cˇ 1313 . In Fig. 4 we show the effective coefficients Cˇ 1313 and Cˆ 1313 as functions of the fiber volume fraction. The interesting feature of the results is that the variation of the volume fraction of the fiber phase at the smaller scale influences the behavior of the homogenized macroscopic properties. This outcome is of importance in studies concerning the effect of the distribution and density of a certain constituent in a multiscale composite material. For instance, in the case of musculoskeletal mineralized tissues (such as bone and tendons), the degree of mineralization of the tissue is an important factor in characterizing its properties [32, 38]. Musculoskeletal mineralized tissues are 2 1.8 1.6 1.4 1.2 1 0.8 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 4 Effective out-of-plane shear moduli Cˇ 1313 and Cˆ 1313 plotted against the fiber volume fraction

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organized across several spatial scales, and so the present framework can be exploited to model their effective properties [39]. In this context, the scales would represent the arrangement of the basic constituents (i.e., collagen and mineral crystals), the resulting mineralized collagen fibril, and finally their packing into the mineralized collagen fibril bundle. Since the model is developed for fiber-reinforced composites, it can then serve as an approximation for both fibers embedded in a matrix, and for using inclusions of reinforcing material, such as, for example, the hydroxyapatite mineral crystals which are found in aged bone tissue.

4 Two-Scale Asymptotic Homogenization for Wavy Laminated Elastic Composites Materials and structural components with periodic wavy architectures span several scales. Most recently, multilayers with wavy lamellar architectures have been identified in the rapidly developing nanotechnology areas. For instance, wavy interfacial morphologies occur during the manufacturing process in thin metallic and ceramic multilayers that are being developed for magnetic, optoelectronic and high-speed electronic applications, or are deliberately induced to enhance certain properties. These microstructural morphologies can also be manufactured in corrugated patterns. In certain composites, the periodic distribution of the layers is parallel to a plane; for example, in Fig. 2 the layers are parallel to the x1 x2 −plane. But, for many structures the distribution of the layers is parallel to a surface (x) (see Fig. 5). This geometric change affects the properties of the microstructure and therefore the expression of the local variables η, ζ defined in Eq. (5). In order to illustrate the influence of a function (x) on the effective properties, a wavy laminated shell composite with one hierarchical level of organization is considered, i.e., the variable ζ from (5) vanishes. The geometry  of the structure is described by a function  : R3 → R, [15, 16], where  ∈ C2 R3 . In order to derive

x3

x3

x2

H1

x1

H2

L1

x2

H1

L2

x1

(a) Wavy structure along x1and x2 Fig. 5 Heterogeneous curvilinear wavy laminated structures

L1 (b) Wavy structure along x1

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the effective properties of the structure, the two-scale asymptotic homogenization method is used. Therefore, for this case the independent variable is η=

(x) . ε1

(23)

As a consequence of (23) and considering the chain rule, Eq. (6) takes the expression ∇ → ∇x + ε1−1 Dρ∇η ,

(24)

where D represents the gradient of the function . Taking into account the operator (24), the equations in (12) take the forms ∂ ∂η



∂ ε ∂ ε ∂ ∂χpkl Cijkl + C ∂xj ∂x ijpq ∂xj ∂η

 = 0 in Y \ Γη

[[χikl ]] = 0 

  ∂ ε ∂χpkl η ε Cijkl + nj = 0 C ∂x ijpq ∂η

on Γη on Γη .

(25) (26) (27)

So, for a wavy laminated shell structure the effective stiffness tensor (13) becomes   ˆCijkl = Cε + ∂ Cε ∂χpkl , ijkl ∂x ijpq ∂η η

(28)

where the average operator is the same as that mentioned in Sect. 2.2.1 and defined as • η = Y (•) dη/ |Y |, with |Y | being the periodic cell volume. For the particular case of a laminated composite with N layers, that average operator is • η = V1 (•)(1) + · · · + VN (•)(N ) , where Vj and the superscript (•)(j) denote the volume fraction and the value of the property for the layer j, respectively, for j = 1, . . . , N .

4.1 Effective Coefficients for a Structure with Isotropic Components Consider wavy laminate shell composites where the components of the structure are isotropic, i.e., the stiffness tensor has the expression  Cεijkl = λδij δkl + μ δik δjl + δil δjk ,

(29)

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where λ, μ are the Lamé parameters for the components and δij is the Kronecker delta. In order to find the effective stiffness tensor the Voigt notation is used, so Eq. (28) becomes  

ˆCab = Cε + Cε ∂ + Cε ∂ + Cε ∂ ∂χ1b ab a1 a6 a5 ∂x1 ∂x2 ∂x3 ∂η 

∂ ∂ ∂ ∂χ2b + Cεa6 + Cεa2 + Cεa4 ∂x1 ∂x2 ∂x3 ∂η  

∂χ3b ε ∂ ε ∂ ε ∂ + Ca5 + Ca4 + Ca3 , ∂x1 ∂x2 ∂x3 ∂η η

(30)

where a, b = 1, . . . , 6 and the local problem (26) is formed of the three equations ∂ ∂η



∂ ε ∂ ε ∂ ε C + C + C ∂x1 1a ∂x2 6a ∂x3 5a 





  ∂ 2 ∂ 2 ∂ 2 ∂χ1a ε ε ε + C11 + C66 + C55 ∂x1 ∂x2 ∂x3 ∂η  ∂ ∂ ∂χ2a  ε ∂ ∂ ∂χ3a + Cε66 + Cε12 + C55 + Cε13 ∂x1 ∂x2 ∂η ∂x1 ∂x3 ∂η ∂ ∂η

∂ ∂η

 = 0,

(31)



∂ ∂ ∂χ1a  ∂ ε ∂ ε ∂ ε C6a + C2a + C4a + Cε66 + Cε12 ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂η 

2

2

2  ∂χ2a ∂ ∂ ∂ + Cε66 + Cε22 + Cε44 ∂x1 ∂x2 ∂x3 ∂η   ∂ ∂ ∂χ3a = 0, + Cε23 + Cε44 ∂x2 ∂x3 ∂η

(32)



∂ ∂ ∂χ1a  ∂ ε ∂ ε ∂ ε C5a + C4a + C3a + Cε13 + Cε55 ∂x1 ∂x2 ∂x3 ∂x1 ∂x3 ∂η  ε ∂ ∂χ ∂ 2a + C23 + Cε44 ∂x2 ∂x3 ∂η  

2



  ∂ ∂ 2 ∂ 2 ∂χ3a ε ε ε + C33 + C55 + C44 = 0. ∂x2 ∂x3 ∂x1 ∂η

(33)

By integrating Eqs. (31)–(33) with respect to η and solving for the local functions ∂χja /∂η, the following system of equations is obtained:

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  ∂ ∂ ∂χ2a ∂ 2 ∂χ1a  ε + C66 + Cε12 + + ∂x3 ∂η ∂x1 ∂x2 ∂η 

 ε ∂ ∂ ∂χ3a ∂ ε ∂ ε ∂ ε ε a , (34) = θ1 − + C55 + C13 C + C + C ∂x1 ∂x3 ∂η ∂x1 1a ∂x2 6a ∂x3 5a Cε11



∂ ∂x1

2

Cε66



∂ ∂x2

2

Cε55





2

2

2  ∂χ2a ∂ ∂ ∂ ∂ ∂ Cε66 + Cε12 + Cε66 + Cε22 + Cε44 ∂x1 ∂x2 ∂x1 ∂x2 ∂x3 ∂η 

 ∂ ∂ ∂χ3a ∂ ε ∂ ε ∂ ε = θ2a − , (35) + Cε23 + Cε44 C + C + C ∂x2 ∂x3 ∂η ∂x1 6a ∂x2 2a ∂x3 4a





  ε ∂ ∂ ∂χ1a  ε ∂ ∂ ∂χ2a ∂ 2 ε ε C13 + C55 + C23 + C44 + Cε44 ∂x1 ∂x3 ∂η ∂x2 ∂x3 ∂η ∂x2

2

2  

∂ ∂ ∂ ε ∂ ε ∂ ε ∂χ3a ε ε a , = θ3 − +C33 + C55 C + C + C ∂x3 ∂x1 ∂η ∂x1 5a ∂x2 4a ∂x3 3a (36) where θia , i = 1, 2, 3, denotes an integration constant. The system of linear equations (34)–(36) can be written in matrix form as a = θ a3×1 − ba3×1 , [Dij ]3×3 N3×1

(37)

T  a = [∂χ1a /∂η, ∂χ2a /∂η, ∂χ3a /∂η]T , θ a3×1 = θ1a , θ2a , θ3a , where N3×1 D11 =

Cε11

D22 = Cε66 D33 =

Cε44

D12 = D21 D13 = D31 D23 = D32 and, finally





∂ ∂x1 ∂ ∂x1

2 + 2

Cε66

+ Cε22



∂ ∂x2 ∂ ∂x2

2 + 2

Cε55

+ Cε44



∂ ∂x3 ∂ ∂x3

2 , 2 ,





 ∂ 2 ∂ 2 ∂ 2 ε ε + C33 + C55 , ∂x2 ∂x3 ∂x1 ∂ ∂  = Cε66 + Cε12 , ∂x1 ∂x2  ∂ ∂ = Cε55 + Cε13 , ∂x1 ∂x3 ∂ ∂  = Cε23 + Cε44 , ∂x2 ∂x3

Multiscale Homogenization for Linear Mechanics

∂ ε C + ∂x1 1a ∂ ε ba3 = C + ∂x1 5a

ba1 =

∂ ε C + ∂x2 6a ∂ ε C + ∂x2 4a

371

∂ ε ∂ ε ∂ ε ∂ ε C5a , ba2 = C6a + C2a + C , ∂x3 ∂x1 ∂x2 ∂x3 4a ∂ ε C . ∂x3 3a

The local functions ∂χja /∂η are obtained analytically as solutions of the system (37). It can be seen that these functions depend on the unknown constants θia . Taking into account the periodicity of the local functions and their derivatives, i.e., ∂χja /∂η = 0 a system of linear equations for the θia was derived in [16], and it is written as (38) A3×3 θ a3×1 = Ba3×1 , where ⎡

A3×3

 D22 D33 − D23 D32 ⎢ D ⎢ ⎢ ⎢  ⎢ D23 D31 − D21 D33 =⎢ ⎢ D ⎢ ⎢ ⎢  ⎣ D21 D32 − D22 D31 D



⎤ D12 D23 − D13 D22 ⎥ D ⎥ ⎥   ⎥ D33 D11 − D13 D31 D13 D21 − D11 D23 ⎥ ⎥, ⎥ D D ⎥ ⎥   ⎥ D31 D12 − D32 D11 D11 D22 − D12 D21 ⎦ D D D32 D13 − D12 D33 D



and B3×1 has the form ⎡

    ⎤ a D22 D33 − D23 D32 a D32 D13 − D12 D33 a D12 D23 − D13 D22 b + b + b 2 3 ⎢ 1 ⎥ D D D ⎢ ⎥ ⎢ ⎥ ⎢     ⎥ ⎢ a D23 D31 − D21 D33 D D − D D D D − D D 33 11 13 31 13 21 11 23 ⎥ ⎢ b ⎥, + ba2 + ba3 ⎢ 1 ⎥ D D D ⎢ ⎥ ⎢ ⎥ ⎢     ⎥ ⎣ a D21 D32 − D22 D31 D D − D D D D − D D 31 12 32 11 11 22 12 21 ⎦ b1 + ba2 + ba3 D D D where D = det[Dij ]3×3 . Finally, by solving (38), the expressions for the integration constants θia are obtained. Substituting the results for θia into (37), the values for the local functions ∂χja /∂η are obtained [48].

4.2 Example of Wavy Structures As an example, a three-dimensional wavy structure is considered, where  : R3 → R is given by

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(x1 , x2 , x3 ) = x3 − H1 sin

 

2π 2π x1 − H2 sin x2 , L1 L2

(39)

where the constants H1 , H2 indicate the heights of the oscillations and L1 , L2 denote the period lengths of the structure waviness for the directions x1 and x2 , respectively; see Fig. 5a. Figure 5b is a particular case of Fig. 5a for which H2 = 0 in Eq. (39). To illustrate the influence of the geometry of the structure on the effective coefficients, two wavy bimetallic layered composites (N = 2) are studied (Fig. 5). Different numerical experiments are considered where the isotropic material constituents used are stainless steel (with Young’s modulus, E1 = 206.74 GPa, Poisson ratio, ν1 = 0.3) with thickness V1 = 0.2 (volume fraction) and aluminum (with Young’s modulus E2 = 72.04 GPa, Poisson ratio ν2 = 0.35) with thickness V2 = 0.8. The geometries of the composites are described by (39) with parameters, heights H1 = H2 = 0.5 and lengths of the periodicities L1 = L2 = 1 for the structure in Fig. 5a and heights H1 = 0.5, H2 = 0 and length of the periodicity L1 = 1 for the structure in Fig. 5b (this example was studied in [15]). In Fig. 6 are shown some of the components of the effective coefficient, specifically ˆC11 , Cˆ 12 , Cˆ 55 , Cˆ 66 , Cˆ 34 , Cˆ 56 , derived from (30), where the aforementioned average operator for the two-layered laminate composite is • = V1 (•) + V2 (•). In order to illustrate the influence of the wavy behavior along the x2 axis, the values of x2 are fixed at 0, 0.25 and 0.5 and the effective coefficients are compared with the results for H2 = 0. As a validation of the present model, the same effective coefficients Cˆ 11 , Cˆ 12 , Cˆ 55 , Cˆ 66 reported in [15] (H2 = 0) are obtained, as illustrated by the curves for the case d), whereas the components Cˆ 34 and Cˆ 56 are null, as was reported in the

(a)

(b)

(c)

(d)

Fig. 6 Effective coefficients Cˆ 11 , Cˆ 12 , Cˆ 55 , Cˆ 66 , Cˆ 34 , Cˆ 56 for two wavy bimetalic structures using (39), where L1 is the length of the cell along x1 ; see Fig. 5a

Multiscale Homogenization for Linear Mechanics

373

aforementioned work. The behavior of the effective properties is strongly affected by the presence of the waviness along the x2 axis, i.e., H2 = 0 in equation (39). The non-zero heights H1 , H2 in (39) generate oscillations with respect to x1 and x2 , respectively, in the wavy shell. This increases the anisotropic character of the elastic properties in the composite, making possible the appearance of new effective properties, such as Cˆ 34 , Cˆ 56 among others.

5 Modeling Elastic Transversely Isotropic Fibrous Composites Using the Two-Scale Asymptotic Homogenization Method The asymptotic homogenization method (AHM) developed by [1, 2, 42] is a mathematically rigorous technique for predicting both the local and global properties of these kinds of inhomogeneous media. The main problem of the AHM is that averaged coefficients depend on the solutions of the so-called local problems in the periodic cell [41]. These problems are given by a set of partial differential equations with periodic boundary conditions and their solution, in general, requires numerical methods [23]. In the present work, analytical expressions and exact formulas for all effective elastic coefficients are obtained using the AHM for a unidirectional reinforced twophase composite with transversely isotropic cylindrical fibers periodically distributed in a matrix [13, 13, 35]. In the analysis, the periodicity of the structure is assumed to be much smaller than the elastic wavelength. A comparison with different models [3, 19], and some experimental results [8, 9, 22] is presented. The reader can refer to [30] for a computational approach, which has been validated against the proposed analytical approach for cylindrical uniaxially aligned fibers in prismatic cells.

5.1 Formulation and Statement of the Local Problems The constitutive relations of the linear elasticity theory for an heterogeneous and periodic medium, Ω, is characterized by the Y -periodic function C. While Y denotes the periodic cell, C is the fourth-order elasticity tensor. By mean of AHM, the initial constitutive relations with rapidly oscillating material coefficients are transformed ˆ which represent the elasinto new physical relations with constant coefficients C, tic properties of an equivalent homogeneous medium and are called the effective coefficients of Ω. The main problem for obtaining the effective coefficients is to find the Y -periodic η solutions ξpqkl (χ ) of the local problems on Y in terms of the fast variable y [13]. Once the local problems are solved, the homogenized moduli Cˆ ijpq may be determined by using the formulas (13).

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x2 y2

0.578

0.5

-0.5

y1 -0.578

x1

Fig. 7 Geometric distribution of the reinforcements in the composite and its periodic hexagonal cell

The unit cell Y of the body is chosen with a side parallel to the x3 axis, with unit length in the x3 direction. The transversal sections of the periodic cell are: (A) unit squares (Fig. 3); (B) regular hexagon (Fig. 7), both with the radius of the fibers denoted by r. Due to the periodic distributions of the fibers in the isotropy plane Ox1 x2 , it is possible to reduce the general problem to the solution of the local problems over the unit cell. In this case the elasticity tensor components Cijkl take different values in the regions occupied by these two different materials, such that # Cijkl (y1 , y2 ) =

C(1) ijkl if (y1 , y2 ) ∈ Y1 (matrix) C(2) ijkl if (y1 , y2 ) ∈ Y2 (fibers).

The local problems can be written as in (12). The solution of the problem (12) must consist of doubly periodic functions in y1 and y2 subject to the perfect bonding conditions at the interface Γ . The potential method of complex variables and the properties of doubly periodic Weierstrass and related functions are used for the solution of the local problems (12). In that way, we obtain for the average coefficients of the composite given in Fig. 3 and Fig. 7 the analytic and closed-form formulas using the two-indices abbreviate notation as Cˆ 11 = Cˆ 22 = C11 + λ(Δ22 α1 /C(1) 66 − Δ1 α2 ), ˆC12 = C12 + λ(Δ2 α1 /C(1) + Δ1 α2 ), 2 66 ˆC13 = Cˆ 23 = C13 + λΔ2 Δ3 α1 /C(1) , 66

Cˆ 33 = C13 + λΔ23 α1 /C(1) 66 , Cˆ 66 = C66 + λΔ1 α3 , if μ = π/2, Cˆ 11 − Cˆ 12 , if μ = π/3, Cˆ 66 = 2 Cˆ 44 = Cˆ 55 = C55 + λΔ4 α4 ,

(40)

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375

where (2) (1) (2) (1) (2) Δ1 = C(1) 66 − C66 , Δ2 = C11 − C11 + C12 − C12 , (2) (1) (2) Δ3 = C(1) 13 − C13 , Δ4 = C55 − C55 ,

 κ2 − 1 κ2 − 1 −1 α1 = N1 Z N2 , λ − 1 − B(κ1 + 1) 2αo 2αo κ1 + 1 − 1,  α2 = (1 + χ ∗ κ1 ) λ2 − B2 U1 U −1 U2

α3 = 1 −

(41)

(1) (κ1 + 1)BC66 , λ3 − B2 V1 V −1 V2

2C(1) 55 ,  (1 + χ ∗ ) 1 + χ λ − B2 N1 Y −1 N2 π π or μ = . μ= λ = π r 2 / sin μ, 2 3

α4 = 1 −

The symbol notation “ •ˆ ” denotes the effective coefficients of the composite and λ is the volume fraction of the fiber. The magnitudes involved in the expressions of α1 , α2 , α3 , α4 in (41) can be found in detail in the works [13, 35]. The most difficult aspect in (41) is the calculation of infinite numerical matrices N1 , Z, N2 , U1 , U , U2 , V1 , V , V2 , and Y , which can be computed according to the proposed algorithm given in [41].

5.2 Comparisons with Different Models The overall properties calculated in (40) for the hexagonal and square cells of both types of composites (see Figs. 3 and 7) are now compared with theoretical and experimental results reported in the above mentioned papers. (1) In [3], theoretical expressions were obtained by means of the classical theory of elasticity for determining the composite elastic constants for fiber-reinforced plastics in terms of the elastic moduli and the geometrical parameters of the constituents. These investigations were made for a hexagonal array and the material properties used were for E-glass fiber and epoxy resin. The fiber volume content was approximately 63%. A good concordance between both models was obtained and is shown in Table 1. (2) The Hill relations [20] are satisfied identically by the effective coefficients (40). In this work it was proved that for the calculated effective coefficients, these universal relations are constant and invariant in relation with the volume fraction kˆ − k Cˆ 13 − C13 Δ2 = , = Δ3 Cˆ 13 − C13 Cˆ 33 − C33 where k = 0.5(C11 + C12 ).

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Table 1 Comparisons between AHM for both type of arrays (hexagonal and square) and the model reported in [3] Cˆ 11 Cˆ 12 Cˆ 13 Cˆ 33 Cˆ 44 Cˆ 66 Models used [3] (40) hexagonal (40) square

2.2352 2.2369 2.5749

0.8933 0.8919 0.6855

0.7625 0.7626 0.7872

6.8621 6.8631 6.8723

0.6926 0.6934 0.7467

0.6727 0.6725 0.5245

(3) The elastic constants Cˆ ij were measured and calculated for a laminated uniaxially fiber-reinforced boron–aluminum composite in [8]. In this paper, three theoretical models were considered: square array, hexagonal array, and random distribution, and relationships for predicting the full set of elastic constants for this model were derived. A comparison of AHM with the experimental and theoretical results reported in [8] at a volume fraction of 48% is shown in the Fig. 8. The hexagonal configuration by AHM agrees best with random distribution model and the experimental data. Considering all six elastic constants, the experimental data and AHM differ on the average by 6% for hexagonal, and 15% for square.

Fig. 8 Comparison of AHM (40) with the experimental and theoretical results reported in [8] for both type of arrays

Multiscale Homogenization for Linear Mechanics

377

Fig. 9 Comparison between predicted and the measured [8] values of the effective coefficients Cˆ 33 , Cˆ 11 for transversely isotropic Modmor type 1 carbon/Ciba LY558 epoxy resin versus the volume fraction of the fibers

(4) A comparison between the theoretical results derived here and experimental data [8], which were obtained for transversely isotropic Modmor type 1 carbon fibers in isotropic Ciba LY558 epoxy resin by [9] now follows. Figure 9 plots the effective stiffness Cˆ 33 and Cˆ 11 versus the volume fraction λ of the fibers. The solid line gives the results using the analytical formulas (40). The empty circles are the experimental values. It can be seen that the experimental data agree quite well with the prediction of the asymptotic homogenization method. Besides, in the case of a multiphase fiber-reinforced composite material with square periodic cell shown in Fig. 3, we can obtain the same curve as Fig. 4 (right) using the recursive homogenization scheme reported in [14], which involves two applications of (40). Figure 10 shows the behavior of the axial shear coefficient Cˆ 44 using the hierarchical (22) and the aforementioned reiteration scheme. In addition, Cˆ 44 for the two-phase composite with (40) is displayed, where we can observe the influence of the inclusions within the fiber on the effective properties.

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Fig. 10 Comparison between recursive homogenization scheme (AHM 3fR ), the hierarchical approach (22) and the two-phase model (40) for the effective coefficient Cˆ 44 . The material parameters are the same as in Fig. 4

6 The Two-Scale Asymptotic Homogenization Method: One-Dimensional Viscoelastic Heterogeneous Media 6.1 Mathematical Model A heterogeneous bar of length L and periodic structure is considered (see Fig. 11). It exhibits a non-aging linear viscoelastic behavior and is subjected to the action of a volumetric force f (x). The statement of the equilibrium equation (see [6]) is written as

 x ∂u(x, t) ∂ R ,t ◦ = f (x), (42) ∂x ε ∂x u(0, t) = u0 , ∀t ∈ R,

(43)

u(L, t) = u , ∀t ∈ R, u(x, 0) = 0, ∀x ∈ [0, L],

(44) (45)

L

Multiscale Homogenization for Linear Mechanics

379

Fig. 11 Viscoelastic heterogeneous bar

where ◦ indicates the Stieltjes convolution integral (see [50]) x ∂u(x, t)  t x

∂ ∂u(x, τ )  R ,t ◦ = dτ. R ,t − τ ε ∂x ε ∂τ ∂x 0

(46)

The constitutive law (46) is linear and the viscoelastic properties of the material are defined by the relaxation function R (see [4]). Following the scheme used in [7], the viscoelastic problem (42)–(45) can be transformed into an elastic problem by applying the Laplace–Carson transform. This is known as the correspondence principle. The Laplace–Carson transform is defined by  ∞

LC [g(x, t)] = g˚ (x, p) = p

e−pt g(x, t) dt.

0

From now on, the functions with the symbol (˚) depending on the parameter p denotes the Laplace–Carson space. Applying the Laplace–Carson transform on (42)– (45) and considering the convolution theorem (see [4]), the equilibrium viscoelastic heterogeneous problem becomes

 ∂ ˚ x ∂ u˚ (x, p) R ,p = f (x), ∂x ε ∂x u˚ (0, p) = u0 , ∀p ∈ [0, ∞], u˚ (L, p) = uL , ∀p ∈ [0, ∞], u˚ (x, 0) = 0, ∀x ∈ [0, L].

(47) (48) (49) (50)

Below are incorporated some additional conditions in order to ensure the existence of a unique weak solution to the problem: 1. x denotes the global coordinate. Also, the local scale coordinate y is introduced, where y = x/ε. The parameter ε is the small size of the periodic cell in relation to the global coordinate x (ε  L); 2. it is assumed that the relaxation modulus R(y, t) ∈ L∞ (R × R) (see [1, 33]); moreover, R(y, t) is 1-periodic in y; 3. ∃ α, β, t0 such that 0 < α ≤ R(y, t0 ) ≤ β < ∞ ∀y ∈ R (ε → 0) (see [1]); 4. f (x) ∈ L2 ([0, L]) (see [1, 33]).

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6.2 Solution Using the Two-Scale Asymptotic Homogenization Method As we are in the presence of a strongly one-dimensional heterogeneous structure when ε → 0, the two-scale asymptotic homogenization method is proposed to find the solution of the model (47)–(50) and for calculating the effective properties of the material. The asymptotic expansion is given as (see [1, 6]) u˚ (x, ε, p) =

∞ 

εi u˚ i (x, x/ε, p),

(51)

i=0

where u˚ i is 1-periodic related to the variable y = x/ε, ∀i, ∀x ∈ [0, L], ∀p ∈ [0, ∞) and u˚ i (x, x/ε, p) ∈ C ∞ ([0, L] × R × [0, ∞)). According to the chain rule, the derivative with respect to the global coordinate x applied on each term u˚ i (x, x/ε, p) in (51) yields the transformation ∂(•) ∂(•) 1 ∂(•) ≡ + . ∂x ∂x ε ∂y The operator

∂ Lαβ (•) := ∂α

 ˚R(y, p) ∂(•) ∂β

(52)

(53)

is now defined for α, β = x, y, indistinctly. Consequently, taking into account (52) and (53), the expression (51) is substituted into (47). After some simplifications and grouping in powers of ε, the following sequence of problems is obtained: ε−2 : Lyy (˚u0 ) = 0, ε

(54)

−1

: Lxy (˚u0 ) + Lyx (˚u0 ) + Lyy (˚u1 ) = 0, ε0 : Lxx (˚u0 ) + Lxy (˚u1 ) + Lyx (˚u1 ) + Lyy (˚u2 ) − f (x) = 0.

(55) (56)

The formal asymptotic solution was built such that the order of the approximation is O(ε). This truncation is enough for verifying that the solution of the homogenized problem converges to the solution of the heterogeneous problem when ε → 0 (see [1, 33] and [35]). In order to solve (54)–(56) the following lemma is required (for the proof, see Chap. 4, pp. 37–40 of [33]). Lemma 1 Consider the differential equation ∂ ∂yi

 ε ∂φ Kij = F, ∂yj

in a Y -cell, where φ is Y -periodic and F ∈ L2 (Y ). The following conditions hold:

Multiscale Homogenization for Linear Mechanics

381

(i) there exists a Y -periodic solution for φ if and only if F = 0; (ii) if the Y -periodic solution φ exists, then it is unique up to an additive constant. Problem for ε−2 Equation (54) has the trivial solution u˚ 0 (x, y, p) ≡ 0. Lemma 1 ensures that u˚ 0 (x, y, p) is a solution of (54) if and only if it is constant in relation to the variable y. Then, u˚ 0 (x, y, p) = v˚ (x, p),

(57)

where v˚ (x, t) is a infinitely differentiable function. Problem for ε−1 The solution of the problem (55) is proposed as (see [6, 7]) ∂ v˚ (x, p) u˚ 1 (x, y, p) = N˚ (y, p) , ∂x

(58)

where N (y, t) is the so-called local function. Placing (57) and (58) into (55), and after some simplifications, the local problem

 ∂ ∂ ˚ R(y, p) N˚ (y, p) + 1 = 0, ∂y ∂y ˚ N (0, p) = N˚ (1, p) = 0, ∀p ∈ [0, ∞], N˚ (y, 0) = 0, ∀y ∈ [0, 1]

(59)

is found. The analytical solution to the local problem (59), in time space, is N (y, t) =

L−1 C

$

R˚ −1 (y, p)

%−1  0

y

1 ˚ , p) R(γ

 dγ − y .

(60)

Problem for ε0 Finally, working on (56) and using Lemma 1 (see [6, 7]), the homogenized problem in Laplace–Carson space is obtained and it can be written in the form ∂ 2 v˚ (x, p) ˆ˚ R(p) ∂x2 v˚ (0, p) v˚ (L, p) v˚ (x, 0)

= f (x), = u0 , ∀p ∈ [0, ∞], = uL , ∀p ∈ [0, ∞], = 0, ∀x ∈ [0, L],

where the effective coefficient is $ %−1 ˆ˚ R(p) = R˚ −1 (y, p) .

(61)

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Relation Between the Effective Relaxation Modulus and the Effective Creep Compliance



ˆ The mathematical relation between the effective relaxation modulus R(t) and the

effective creep compliance Jˆ (t) , in Laplace–Carson space, is now presented (see [18]): ˆ˚ Jˆ˚ (p) = 1. R(p) (62) Applying the inverse Laplace–Carson transform on (62) leads to the Stieltjes convolution integral (see [17]) 

t

ˆ )Jˆ (t − τ ) dτ = t, t > 0. R(τ

(63)

0

The integral equation (63) can be solved numerically using an appropriate numerical integration scheme (see [27]). Dividing the range of integration [0, t] into many subintervals, a numerical evaluation of the integral (63) is obtained: n   i=1

ti

ˆ )Jˆ (tn − τ ) dτ = tn , t0 = 0, tn = t > 0, R(τ

n  ˆ i−1 ) Jˆ (tn − ti ) − Jˆ (tn − ti−1 ) ˆ i ) − R(t R(t i=1

(64)

ti−1

2

2

(ti − ti−1 ) = tn .

(65)

Therefore, if the effective relaxation moduli are known the effective creep compliances can be found by solution of (62) or (63).

6.3 Effective Coefficients for One-Dimensional Two-Phase Viscoelastic Composites. Example A two-phase composite material with viscoelastic constituents and perfect contact is now considered. In Fig. 12 is displayed the periodic cell in relation to the local coordinate y. The volume fractions of the constituents are γ and 1 − γ .

Fig. 12 Cell of the two-phase medium

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383

The relaxation modulus is taken as  (1) R (t) if y ∈ Ω (1) R(y, t) = R(2) (t) if y ∈ Ω (2) , where superscripts (1) and (2) indicate the corresponding material. According to the structure of the bar and based on the fact that the relaxation modulus is homogeneous in Ω (δ) , with δ = 1, 2, the effective coefficient (61) becomes (see [6]) R˚ (1) (p)R˚ (2) (p) ˆ˚ R(p) = . (66) (1 − γ )R˚ (1) (p) + γ R˚ (2) (p) Thus, an expression for the effective coefficient, in time space, is obtained  ˆ = R(t)

L−1 C

 R˚ (1) (p)R˚ (2) (p) . (1 − γ )R˚ (1) (p) + γ R˚ (2) (p)

(67)

Now, as an illustration of the above procedure, we consider as an example, a fraction exponential function or Scott Blair–Rabotnov (SBR) kernel given as α (β, t) = t −α

∞  n=0

(−β)n t n(1−α) , Γ [(n + 1)(1 − α)]

(68)

with 0 ≤ α < 1 and 0 < β (see [36, 43]). The relaxation modulus is taken as R(t) = λμ0 α (β, t), where β=

1 τ 1−α

, λ = β(1 − εmax ) =

(69)

μ∞ − μ0 β, μ0

τ is the relaxation time, μ∞ is the shear modulus at t → ∞, μ0 is instantaneous shear modulus and εmax is the maximal shear strain [44, 45]. The analytical formula for its Laplace transform is  L [α (β, t)] ≡

∞

α (β, t)e−pt =

0

1 . +β

p1−α

(70)

Applying the Laplace–Carson transform on (69) and taking into account (70) we obtain (see [6]) λ1 μ01 p λ2 μ02 p R˚ (1) (p) = , R˚ (2) (p) = , (71) x + β1 x + β2 where the variable x is defined by x ≡ p1−α .

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Table 2 Parameters for the SBR kernel β1 β2 λ1 0.98

0.45

49.6

λ2 29.6

α

μ01

μ02

0.47

1.76·106

0.98·106

Placing the expressions (71) into (66) and after some simplifications we obtain an expression for the effective coefficient, in Laplace–Carson space, ˆ˚ R(p) =

Mp , x + x0

(72)

where M =

λ1 μ01 λ2 μ02 (1 − γ )λ1 μ01 β2 + γ λ2 μ02 β1 , x0 = . (1 − γ )λ1 μ01 + γ λ2 μ02 (1 − γ )λ1 μ01 + γ λ2 μ02

Using (70), the inverse Laplace–Carson transform of (72) is obtained as (see [6]) ˆ = M α (x0 , t). R(t)

(73)

For purposes of illustration, Table 2 lists the values of the parameters in this model on which the plots in Fig. 13 are based. Specifically, the plots in Fig. 13a ˆ show, for different values of γ , comparisons of R(t) calculated from the analytical expression (73) with those obtained from the numerical inversion of (72) using INVLAP. In Fig. 13b comparisons are provided between of the plots of Jˆ (t) based on the numerical integration scheme (65) applied to (63) and on the numerical inversion of the Laplace–Carson transform in (62), again based on the use of INVLAP and for different values of γ . In each case, it is is clear that there is essentially no distinction between the results for the different approaches.

7 Conclusions In the present work, a multiple-scale asymptotic homogenization approach is presented for computing the effective properties of heterogeneous structures. The approach permits the study of problems where several length scales are present (the two-scale asymptotic homogenization only deals with two different length scales). A novel power series expansion is proposed which allows determination of the homogenized properties of multiscale periodic composites at each structural level, and we recover known results from reiterated and two-scale asymptotic homogenization techniques as particular cases of the proposed method. The novelty of this work relies on the solution, for the first time, of the effective elastic shear stiffness by considering

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(a)

(b)

ˆ obtained from the analytical expression (73) compared with those from the Fig. 13 a Plots of R(t) numerical inversion of (72) using INVLAP, for values of the volume fraction γ = 0.2, 0.5, 0.8. b Comparison of the results for Jˆ (t) based on the numerical integration scheme (65) applied to (63) and the numerical inversion of the Laplace–Carson transform of (62), using INVALP, again for values of the volume fraction γ = 0.2, 0.5, 0.8. Values of the model parameters used are given in Table 2

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a nested arrangement of cylindrical and uniaxially aligned fibers. The corresponding cell problems are solved analytically using complex variables. Furthermore, analytical formulas for the effective coefficients of a hierarchical layered composite are obtained for the first time. In fact, since analytical formulas are found, the computational cost to calculate the homogenized properties is very low at both steps. In addition, the analytic homogenization of elastic composites with generalized periodicity is presented. The two-scale asymptotic homogenization method is used to obtain the local problem and the analytical expressions for the effective coefficients. The general expression of the stress for a curvilinear structure is derived and the homogenized problem is stated. An engineering generalized periodic microstructure, such as a wavy multilayered, is analyzed. A methodology for solving the local problem of a laminated shell composite with generalized periodicity, perfect contact at the interface and anisotropic elements is given. The analytical expressions for the local functions are obtained. Analytical formulas for the effective coefficients are proposed and the effective coefficients for wavy laminated composite are computed. Besides, the general expressions for the local problems and effective coefficients for elastic heterogeneous media with a periodic structure are obtained using the AHM for a unidirectional reinforced twophase composite with transversely isotropic cylindrical fibers periodically distributed in a matrix with two different shapes: square and hexagonal arrangement. In the analysis, the periodicity of the structure is assumed to be much smaller than the elastic wavelength. A comparison with other models and some experimental results is presented. Finally, the behavior of one-dimensional viscoelastic composites for arbitrary kernel functions is studied. A mathematical justification is given and shows the effectiveness of AHM for solving one-dimensional viscoelastic heterogeneous problems. The use of fractional exponential functions reveals a great benefit for the computation of the effective properties. Besides, numerical results are shown using real parameters taken from the literature. Acknowledgements The funding of Proyecto Nacional de Ciencias Básicas 2016–2018 (Project No. 7515) is gratefully acknowledged. Thanks to the Departamento de Matemáticas y Mecánica IIMAS-UNAM, COIC-STIA-8649, COC-STIA-8651 and PREI-DGAPA, UNAM for their support and Ramiro Chávez Tovar and Ana Pérez Arteaga for computational assistance. The authors would like to thank the project PHC Carlos J. Finlay 2018 PROJECT N39142TA and to the editors for their effort in this publication.

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