Introduction to Non-linear Mechanics. A Unified Energetical Approach 9783031519192, 9783031519208

Table of contents :
Preface
Contents
List of Figures
1 Introduction
1.1 Some General Features
1.2 Description of the Motion
1.3 Homogeneous Deformations
1.4 The Mobility and the Interactions
1.4.1 On the Initial Configuration
1.5 Conservation of Energy and Entropy Production
1.6 The Linear Thermoelasticity
1.7 More General Cases
1.7.1 Generalized Standard Materials
1.7.2 Linear Visco-elastic Behaviour
1.7.3 Normality Rule
1.8 The Quasistatic Evolution
1.8.1 Dissipative Function
1.8.2 The Isothermal Boundary Value Problem
1.9 The Lagrangian and the Dynamical Case
1.10 The Hamiltonian
1.11 Some Properties
1.11.1 Expression of the Conservation of Energy
1.11.2 Conservation Law
1.11.3 Property of Stationarity
1.12 On Discontinuities
1.12.1 Change of Scale
References
2 Non-linear and Linear Elasticity
2.1 Introduction
2.2 Universal Deformation
2.3 Properties of Equilibrium Solution
2.4 Example of Non-linear Elastic Deformation
2.4.1 The Flexion of a Prismatic Bar
2.4.2 The Antiplane-Shear
2.5 Linear Elasticity: Small Perturbations
2.6 Equilibrium Solution of a Linear Elastic Body
2.7 Stability and Bifurcation in Non-linear Elasticity
2.7.1 Notion of Stability
2.7.2 The Metronome
2.7.3 The Euler Column
References
3 Elasto-plasticity
3.1 Introduction
3.2 The Domain of Reversibility
3.3 The Evolution of Internal State
3.4 A Model of Perfect Plasticity
3.5 The Rate Boundary Value Problem
3.5.1 Characterization of Equilibrium
3.5.2 The Internal State Evolution
3.5.3 Primal Formulation
3.6 On the Adjoin State of Evolution Problem
3.7 Cyclic Plasticity
3.8 Classical Solutions in Elasto-plasticity
3.8.1 A Three Bars Lattice Under Traction
3.8.2 Case of a Hollow Sphere
3.9 Finite Elasto-plasticity
3.9.1 Case of Homogeneous Polycristal
3.10 Stability and Bifurcation in Elastoplasticity
3.10.1 The Shanley Column
3.10.2 A Model of Elastoplastic Beam
References
4 Fracture Mechanics
4.1 Introduction
4.2 Case of Linear Elasticity
4.3 Crack Propagation in Plane Conditions
4.4 Energetical Interpretation
4.5 Invariance and J-integral
4.6 Dual Approach in Linear Elasticity
4.7 On the Rate Boundary Value Problem
4.8 Interaction of Cracks
4.9 Stability and Uniqueness: A Simple Example
4.10 Case of Hyperelasticity
4.11 Case of Dynamics
4.12 On Inhomogeneous Body
4.12.1 On the Rate Boundary Value Problem
4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity
4.13.1 Invariant Integrals upper JJ, upper G Subscript thetaGθ
4.13.2 Mode I
4.13.3 Mode II
4.13.4 Mode III
4.13.5 General Remark
4.14 Separation of the Modes of Rupture
4.15 For a Non Planar Crack
References
5 Moving Discontinuities
5.1 Introduction
5.2 Dissipation Analysis
5.2.1 In the Dynamical Case
5.3 General Features for Quasi-static Evolution
5.4 Moving Discontinuity
5.4.1 The Equilibrium State
5.4.2 Variations of the Potential Energy
5.4.3 Dissipation and Evolution of the Interface
5.4.4 Examples on a Bar
5.4.5 A Model with Dissipation: A Quasi-brittle Material
5.5 Problem of Evolution
5.6 The Rate Boundary Value Problem
5.6.1 Stability and Bifurcation
5.7 An Example
5.8 Connection with Fracture
5.8.1 The Quasi-Crack Problem
5.8.2 Peculiar Solutions of Equilibrium Equation
5.9 The Quasi-Crack Solution in Mode III
5.9.1 Determination of the Constants
5.9.2 Solutions for alpha greater than or equals 0αge0
5.9.3 Solution for alpha less than or equals 0αle0
5.9.4 A Particular Constitutive Law
5.9.5 The Particular Case alpha equals 0α=0
References
6 Damage Modelling and Initiation of Defect
6.1 Introduction
6.2 A Simple Local Damage Model
6.2.1 Evolution of Damage Parameter
6.2.2 Properties of Damage Field
6.2.3 Models with Local Discontinuities: An Axial Description
6.3 Models with Damage Gradient
6.3.1 The Total Potential Energy and its Variations
6.3.2 On the Bar in Extension
6.4 A Model of Graded Damage
6.4.1 The Equilibrium Problem
6.4.2 On the Regularity of the Fields
6.4.3 The Total Potential Energy
6.4.4 The Bar Under Uni-axial Extension
6.5 A Regularized Graded Damage Model
6.5.1 On the Bar in Extension
6.6 Comparison Between Graded Damage and Thick-Level Set Model
6.6.1 Model with Convex Constrains
6.7 The State of Equilibrium
6.7.1 On the Evolution of Damage
6.8 On the Rate Boundary Value Problem
6.9 On the Role of the Curvature: Example on a Sphere
6.9.1 The Inhomogeneous Sphere Under Radial Loading
6.9.2 The Sharp Interface
6.9.3 A Graded Damaged Sphere
6.10 Coupling with Plasticity
6.10.1 Sharp Interface
6.10.2 Solution with Transfer of Internal State
6.10.3 Sharp Versus Diffuse Interface
References
7 A Thermodynamical Approach to Contact Wear
7.1 Introduction
7.2 The Energetical Approach
7.3 The Dissipation
7.3.1 Interface Propagation Law
7.3.2 Description of the Interface
7.4 An Application of the Model
7.5 Global Approach of the Interface
7.6 On Change of the Contact Surface
References
8 Delamination of Laminates
8.1 Introduction
8.2 The Kinematic of the Plates
8.3 Conservation of the Momentum
8.4 Dissipation Analysis
8.5 The Rate Boundary Value Problem
8.6 Delamination of a Thin Membrane Under Pressure
References
9 On Relationships Between Micro–Macro Quantities
9.1 Introduction
9.2 Mode and Process of Localization
9.3 Potentials and General Properties
9.4 Macrohomogeneous Body and Linear Elasticity
9.5 On the Decomposition of the Macroscopic Strain
9.6 Moving Interfaces
9.7 Case of Linear Elastic Phases
9.8 More General Cases
9.9 The Composite Sphere Assemblage
9.10 Extension to Finite Deformation
9.11 From Monocrystal to Polycrystal
9.11.1 On the Elastic Behaviour
9.11.2 On Elastoplastic Behaviour
References
10 Homogenization in Linear Elasticity
10.1 The Problem of Inhomogeneous Elasticity
10.2 Introduction of a Comparison Material
10.3 Isotropic Spatial Distribution of Mechanical Phases
10.4 On Particulate Composite Material
10.5 On the Hashin's Spheres Assemblage
10.6 Extension to Imperfect Interface
10.6.1 Estimation of the Global Behaviour
10.6.2 Choice of the Reference Medium
10.6.3 Interpretation
10.6.4 Case of Conduction
10.6.5 Evaluation of upper Q left parenthesis upper K Subscript o Baseline right parenthesisQ(Ko) and upper Q asterisk left parenthesis 1 divided by upper K Subscript o Baseline right parenthesisQ*(1/Ko)
References
11 Optimal Control and Non Linear Inverse Problems
11.1 Inverse Problems in Linear Elasticity
11.1.1 The Problem Setting
11.1.2 A Well Posed Problem
11.1.3 The Idea of Control
11.1.4 The Optimization Method
11.2 Inverse Problem in Elastoplasticity
11.2.1 Inverse Problems on Three Bars Lattice
11.2.2 Inverse Problem When h Subscript o Baseline equals 0ho=0
11.3 Estimation of the Internal State in Elastoplasticity
11.3.1 The Inverse Problem on a Sphere
11.4 Boundary Control and Extension in Viscoplasticity
References
12 Conclusion
Appendix A Tensorial Analysis
A.1 Bilinear form Associated to a Linear Mapping
A.2 Euclidean Vector Space
A.3 Differential Operators
A.3.0.1 Cartesian Coordinates
A.3.0.2 On Other Basis
Appendix B General Relations
B.1 Continuous Case
B.2 Discontinuous Case
Appendix C Particular Solution in Linear Elasticity
C.1 Cylinders and Spheres Under Radial Loading
C.1.1 Case of Uniform lamdaλ
C.2 A Cylindrical or Spherical Shell Under Shear
C.3 Fundamental Linear Elastic Solution
C.3.1 Plane Isotropic Elasticity
C.3.2 3D-Elasticity
C.3.3 Case of on Half Plane in Plane Strain
C.4 Anti-plane Elasticity
C.4.1 Case of the Half-plane y greater than 0y>0
Appendix D Hodograph Transformation
Appendix E Convex Analysis
Appendix F Optimal Control
Appendix G Some Integrals
Index

Citation preview

Springer Series in Solid and Structural Mechanics 14

Claude Stolz

Introduction to Non-linear Mechanics A Unified Energetical Approach

Springer Series in Solid and Structural Mechanics Volume 14

Series Editors Michel Frémond, University of Rome Tor Vergata, Rome, Italy Giuseppe Vairo, Department of Civil Engineering and Computer Science Engineering, University of Rome Tor Vergata, Rome, Italy

The Springer Series in Solid and Structural Mechanics (SSSSM) publishes new developments and advances dealing with any aspect of mechanics of materials and structures, with a high quality. It features original works dealing with mechanical, mathematical, numerical and experimental analysis of structures and structural materials, both taken in the broadest sense. The series covers multi-scale, multi-field and multiple-media problems, including static and dynamic interaction. It also illustrates advanced and innovative applications to structural problems from science and engineering, including aerospace, civil, materials, mechanical engineering and living materials and structures. Within the scope of the series are monographs, lectures notes, references, textbooks and selected contributions from specialized conferences and workshops.

Claude Stolz

Introduction to Non-linear Mechanics A Unified Energetical Approach

Claude Stolz Institut of Mechanical Sciences and Industrial Applications, UMR CNRS 9219, ENSTA-Paris Palaiseau, France

ISSN 2195-3511 ISSN 2195-352X (electronic) Springer Series in Solid and Structural Mechanics ISBN 978-3-031-51919-2 ISBN 978-3-031-51920-8 (eBook) https://doi.org/10.1007/978-3-031-51920-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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 Paper in this product is recyclable.

Preface

An introduction to non-linear mechanics based on energetic approaches is proposed. The framework of a thermo-dynamical description of continua is introduced. Determination of the evolution of a system is studied through the definition of functionals presented in the case of non-linear dynamics. After a short description of the motion of the system and analysis of the mechanical interactions, the first part is devoted to Lagrangian and Hamiltonian functionals of the system. Quasi-static characterization is then deduced: the role of potential energy and pseudo-potential of dissipation is emphasized. Using the driving force derived from potential energy, the evolution of the internal state is a solution of variational inequalities issued from second derivatives of potential energy taking account of normality rule, existence and uniqueness of solutions are then characterized. First, finite strain in elasticity is introduced, stability and uniqueness of equilibrium positions are investigated, and variational formulations in linearised elasticity are then presented. The evolution of elastoplastic systems is investigated in small and finite strains. The internal state being governed by the normality law, formulation of the rate boundary value problem for the system evolution is given in terms of variational inequality. The conditions of uniqueness are then discussed. This framework is extended to fracture mechanics. The evolution of a system of cracks is analysed, based on an energy criterion as a generalisation of Griffith’s law. The formulation of the boundary value problem has the same form as in elastoplasticity; the internal variables for the structure are the crack lengths. Primal and dual approaches are given, with extension to thermoelasticity and elastoplasticity. Conservation laws and invariance of integrals are used to characterize the dissipation due to cracking, with an analysis of the influence of the geometry on the mechanical fields. For damage mechanics and wear, a similar energetic approach is proposed. The transition zone from sound material to damaged one can be sharp or continuous. In the first case, the dissipation is associated with a moving surface along which the mechanical fields have discontinuities. Such discontinuities produce entropy and v

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Preface

dissipation. When the moving surface is governed by an energetic criterion, the formulation of the boundary value problem of propagation of the damage front is given. Some properties are given; stability and uniqueness of the solution are also discussed. The regularity of damage models is discussed in order to emphasize the fundamental role of discontinuities. The introduction of a smooth and continuous transition from sound material to damaged one permits a unified framework to follow the nucleation and propagation of defects. The connection between damage and crack is analysed through the definition of a quasi-crack as the propagation of a damaged zone of constant thickness in an infinite medium. The framework of a moving surface associated with degradation is applied to model wear contact. A model of wear is obtained by the description of the interface behaviour between the surfaces in relative motion. The transition between two materials along a moving front is generalised to continua as beams, plates or shells. The delamination of stratified composite is investigated in this framework. The delamination separates a sound beam or a sound plate into two beams or two plates. Along the boundary of the delamination, a jump of mechanical quantities occurs. In particular, the strain energy changes and the propagation of the front induces dissipation. This energetic approach is applied to homogenization in non-linear mechanics. The general relationships between microscopic and macroscopic quantities are presented for elastoplastic behaviour and locally damaged material. The definition of irreversible strains and the analysis of dissipation emphasized the role of residual stresses on the macroscopic behaviour. For elastic brittle material, an irreversible macroscopic strain appears even with non-plasticity; this is due to the influence of the change of geometry on the equilibrium of residual stresses. In finite plastic strain, the relation between the internal state and the global residual state is studied using a generalisation of the notion of macro-homogeneity. This permits to define the global plastic gradient and to determine the global evolution of a poly-crystal knowing the behaviour of mono-crystals. Analysis of stability and bifurcation of equilibrium path during the process of loading in order to describe the global behaviour is necessary: a constitutive law must be well defined and reproducible. An example is proposed in non-linear mechanics, based on the concept of assemblage of composite spheres and local damage model. In linear heterogeneous elasticity, bounding processes are given for particular composite materials, with perfect and imperfect interfaces. Finally, inverse problems in non-linear mechanics are discussed using adjoin state. Typical problems are studied: determination of the internal state inside a body from its residual shape, estimation of plastic zone or determination of loading history compatible with the residual measured shape. I hope that this modest contribution offers the opportunity to discover different aspects of non-linear mechanics and suggests new developments in this context. Palaiseau, France July 2023

Claude Stolz

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some General Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Description of the Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Homogeneous Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Mobility and the Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 On the Initial Configuration . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conservation of Energy and Entropy Production . . . . . . . . . . . . . . 1.6 The Linear Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 More General Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Generalized Standard Materials . . . . . . . . . . . . . . . . . . . . . 1.7.2 Linear Visco-elastic Behaviour . . . . . . . . . . . . . . . . . . . . . 1.7.3 Normality Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The Quasistatic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Dissipative Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 The Isothermal Boundary Value Problem . . . . . . . . . . . . . 1.9 The Lagrangian and the Dynamical Case . . . . . . . . . . . . . . . . . . . . 1.10 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Expression of the Conservation of Energy . . . . . . . . . . . . 1.11.2 Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.3 Property of Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 On Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Change of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 6 8 9 10 11 11 12 13 13 14 15 16 17 19 19 19 20 20 22 25

2

Non-linear and Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Universal Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Properties of Equilibrium Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Example of Non-linear Elastic Deformation . . . . . . . . . . . . . . . . . . 2.4.1 The Flexion of a Prismatic Bar . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Antiplane-Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 29 30 30 30 33 vii

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2.5 2.6 2.7

Linear Elasticity: Small Perturbations . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Solution of a Linear Elastic Body . . . . . . . . . . . . . . . . Stability and Bifurcation in Non-linear Elasticity . . . . . . . . . . . . . . 2.7.1 Notion of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Metronome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 The Euler Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 36 36 37 40 43

3

Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Domain of Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Evolution of Internal State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Model of Perfect Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Rate Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Characterization of Equilibrium . . . . . . . . . . . . . . . . . . . . . 3.5.2 The Internal State Evolution . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Primal Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 On the Adjoin State of Evolution Problem . . . . . . . . . . . . . . . . . . . 3.7 Cyclic Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Classical Solutions in Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . 3.8.1 A Three Bars Lattice Under Traction . . . . . . . . . . . . . . . . 3.8.2 Case of a Hollow Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Finite Elasto-plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Case of Homogeneous Polycristal . . . . . . . . . . . . . . . . . . . 3.10 Stability and Bifurcation in Elastoplasticity . . . . . . . . . . . . . . . . . . 3.10.1 The Shanley Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 A Model of Elastoplastic Beam . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 48 50 51 52 53 54 54 57 60 60 65 69 71 74 74 76 78

4

Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Case of Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Crack Propagation in Plane Conditions . . . . . . . . . . . . . . . . . . . . . . 4.4 Energetical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Invariance and J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Dual Approach in Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 On the Rate Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 4.8 Interaction of Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Stability and Uniqueness: A Simple Example . . . . . . . . . . . . . . . . . 4.10 Case of Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Case of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 On Inhomogeneous Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.1 On the Rate Boundary Value Problem . . . . . . . . . . . . . . . . 4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity . . . . . 4.13.1 Invariant Integrals J, G θ . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.2 Mode I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 81 84 87 87 89 90 94 96 97 98 100 101 106 107 110

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4.13.3 Mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.4 Mode III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13.5 General Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Separation of the Modes of Rupture . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 For a Non Planar Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 114 114 115 119

5

Moving Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dissipation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 In the Dynamical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 General Features for Quasi-static Evolution . . . . . . . . . . . . . . . . . . 5.4 Moving Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Variations of the Potential Energy . . . . . . . . . . . . . . . . . . . 5.4.3 Dissipation and Evolution of the Interface . . . . . . . . . . . . 5.4.4 Examples on a Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 A Model with Dissipation: A Quasi-brittle Material . . . . 5.5 Problem of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Rate Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Stability and Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Connection with Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 The Quasi-Crack Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Peculiar Solutions of Equilibrium Equation . . . . . . . . . . . 5.9 The Quasi-Crack Solution in Mode III . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Determination of the Constants . . . . . . . . . . . . . . . . . . . . . 5.9.2 Solutions for α ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3 Solution for α ≤ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.4 A Particular Constitutive Law . . . . . . . . . . . . . . . . . . . . . . 5.9.5 The Particular Case α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 123 125 127 127 128 128 129 131 133 134 137 138 139 141 142 144 146 146 147 149 152 155 158

6

Damage Modelling and Initiation of Defect . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Simple Local Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Evolution of Damage Parameter . . . . . . . . . . . . . . . . . . . . 6.2.2 Properties of Damage Field . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Models with Local Discontinuities: An Axial Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Models with Damage Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Total Potential Energy and its Variations . . . . . . . . . 6.3.2 On the Bar in Extension . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 162 162 163 164 165 166 167

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Contents

6.4

A Model of Graded Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Equilibrium Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 On the Regularity of the Fields . . . . . . . . . . . . . . . . . . . . . 6.4.3 The Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 The Bar Under Uni-axial Extension . . . . . . . . . . . . . . . . . 6.5 A Regularized Graded Damage Model . . . . . . . . . . . . . . . . . . . . . . 6.5.1 On the Bar in Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Comparison Between Graded Damage and Thick-Level Set Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Model with Convex Constrains . . . . . . . . . . . . . . . . . . . . . 6.7 The State of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 On the Evolution of Damage . . . . . . . . . . . . . . . . . . . . . . . 6.8 On the Rate Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 6.9 On the Role of the Curvature: Example on a Sphere . . . . . . . . . . . 6.9.1 The Inhomogeneous Sphere Under Radial Loading . . . . 6.9.2 The Sharp Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.3 A Graded Damaged Sphere . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Coupling with Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1 Sharp Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 Solution with Transfer of Internal State . . . . . . . . . . . . . . 6.10.3 Sharp Versus Diffuse Interface . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 169 169 169 172 174 174

7

A Thermodynamical Approach to Contact Wear . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Energetical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Interface Propagation Law . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Description of the Interface . . . . . . . . . . . . . . . . . . . . . . . . 7.4 An Application of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Global Approach of the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 On Change of the Contact Surface . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 198 199 200 201 204 206 207 209

8

Delamination of Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Kinematic of the Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Conservation of the Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Dissipation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Rate Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Delamination of a Thin Membrane Under Pressure . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 211 214 216 217 221 223

176 179 179 181 181 184 184 186 186 189 189 191 193 194

Contents

9

xi

On Relationships Between Micro–Macro Quantities . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Mode and Process of Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Potentials and General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Macrohomogeneous Body and Linear Elasticity . . . . . . . . . . . . . . 9.5 On the Decomposition of the Macroscopic Strain . . . . . . . . . . . . . 9.6 Moving Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Case of Linear Elastic Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 More General Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 The Composite Sphere Assemblage . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Extension to Finite Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 From Monocrystal to Polycrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.1 On the Elastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.2 On Elastoplastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 226 228 230 232 233 234 235 238 244 248 248 249 252

10 Homogenization in Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Problem of Inhomogeneous Elasticity . . . . . . . . . . . . . . . . . . . 10.2 Introduction of a Comparison Material . . . . . . . . . . . . . . . . . . . . . . 10.3 Isotropic Spatial Distribution of Mechanical Phases . . . . . . . . . . . 10.4 On Particulate Composite Material . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 On the Hashin’s Spheres Assemblage . . . . . . . . . . . . . . . . . . . . . . . 10.6 Extension to Imperfect Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Estimation of the Global Behaviour . . . . . . . . . . . . . . . . . 10.6.2 Choice of the Reference Medium . . . . . . . . . . . . . . . . . . . 10.6.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Case of Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.5 Evaluation of Q(K o ) and Q ∗ (1/K o ) . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 257 258 262 265 267 269 270 273 273 275 276

11 Optimal Control and Non Linear Inverse Problems . . . . . . . . . . . . . . . 11.1 Inverse Problems in Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 A Well Posed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 The Idea of Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 The Optimization Method . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inverse Problem in Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Inverse Problems on Three Bars Lattice . . . . . . . . . . . . . . 11.2.2 Inverse Problem When h o = 0 . . . . . . . . . . . . . . . . . . . . . . 11.3 Estimation of the Internal State in Elastoplasticity . . . . . . . . . . . . . 11.3.1 The Inverse Problem on a Sphere . . . . . . . . . . . . . . . . . . . 11.4 Boundary Control and Extension in Viscoplasticity . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 277 277 278 278 279 280 281 284 285 287 289 292

12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

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Contents

Appendix A: Tensorial Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Appendix B: General Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Appendix C: Particular Solution in Linear Elasticity . . . . . . . . . . . . . . . . . 311 Appendix D: Hodograph Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Appendix E: Convex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Appendix F: Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Appendix G: Some Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6

Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 5.1

A simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The motion of areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenvectors, simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The metronome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case λ1 < 0 = 0, Stable (s) and unstable (u) paths, Phase diagram for λ < λc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case λ1 = 0, λ2 > 0, Stable (s) and unstable (u) paths, Phase diagram for λ > λc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case λ1 = 0, λ2 < 0, Stable (s) and unstable (u) paths, Phase diagram for λ < λc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A quasi-static tensile curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The scheme of experiments on copper . . . . . . . . . . . . . . . . . . . . . . Perfect plastic tensile curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three elastoplastic bars, initial (left) and residual configuration (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial domain of reversibility (bold line), (- -) (1-4) |n 1 | = n c ; (2-5) |n 2 | = n c ; (3-6) |n 3 | = n c ; Actual p p p p domain with δ1 , δ2 = δ3 = 0, (· · · ) |n i − n ri (δ1 )| = n c . . . . . . . The hollow sphere under radial extension . . . . . . . . . . . . . . . . . . . The global response for a hollow sphere in perfect plasticity . . . Decomposition with a stress free state . . . . . . . . . . . . . . . . . . . . . . The Shanley column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The description of the behaviour near the crack tip . . . . . . . . . . . The geometry to study the singularities in a corner . . . . . . . . . . . The boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decomposition of  in  ∪ V . . . . . . . . . . . . . . . . . . . . . . . . . . The Knowles and Sternberg behaviour class . . . . . . . . . . . . . . . . . Shock curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A section of the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A moving surface: evolution of o and 1 . . . . . . . . . . . . . . . . . .

3 6 7 37 40 40 40 46 47 50 60 60

62 65 69 71 75 82 82 84 84 97 98 107 127 xiii

xiv

Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 5.13 Fig. 5.14 Fig. 5.15 Fig. 5.16 Fig. 6.1

Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 Fig. 6.8 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 9.1

List of Figures

Reversible Phase transformation, (Left); with dissipation (Right); < − > reversible path . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation of the interface a = φ(s, t) . . . . . . . . . . . . . . . . . . . The composite sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The response of the composite sphere . . . . . . . . . . . . . . . . . . . . . . The sharp interface for a cavity: σrr (R), for R/Re = 0.2, 0.4, 1, Ri (0)/Re = 0.1 . . . . . . . . . . . . . . . . . . . . A quasicrack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical constitutive behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . The ellipticity (left) domain and the hyperbolicity domain (right) for α = −0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The shape of the shock curve L, for α = −0.5, 1 is the circle ρ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intersection of cycloid (–) for elliptic domain, and cycloid (-) in hyperbolic domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The domains and the shock curve L for α = −0.5, Rm = ∞, for ρu = 8 . . . . . . . . . . . . . . . . . . . . . . . The shock line for special law α = −0.5, ρm = 20, ρu = 8 . . . . The shock curve for α = 0, Rm = ∞, ρu == ρo = 1, a = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The shock line for α = 0, ρm = 2, ρu = ρo = 1, a = 4 . . . . . . . . The shock line for α = 0, ρm = 2, ρu == ρo = 1 . . . . . . . . . . . . Homogeneous response of the bar, left: α = 0, h o = 0 and h o = 10, β = 1(−), 2(.), 5(−−) right: α = 1, h o = 0 and h o = 10, β = 1(−), 2(−), 5(−−) . . . . . . . . . . . . . . . . . . . . . A model of continuous transition; profile of damage along a layer of finite thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . The bar in extension: G = 1 − α, h = 0, depending on L: “o”, L = 1; “x”, L = 2, “*”, L = 3 . . . . . . . . . . . . . . . . . . . . . . . . . The Lagrange multiplier γ2 , for increasing values of αm = 0.2, 0.4, 0.6, 0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The profile of damage (x, α(x)) for different αm , (xa , α(xa )): “+”, (xb , α(xb )):“o” . . . . . . . . . . . . . . . . . . . . . . . . . . The local stress σrr (r ), r/Re ∈ {0.2, 0.3, 0.5, 0.7, 0.9, 1} . . . . . Geometry of the composite sphere in elastoplasticity . . . . . . . . . The response of an elastoplastic brittle sphere . . . . . . . . . . . . . . . Macroscopic description of contact . . . . . . . . . . . . . . . . . . . . . . . . The moving boundaries and the interface medium . . . . . . . . . . . . The mesoscopic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A moving punch on an elastic half-space . . . . . . . . . . . . . . . . . . . Delamination of a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modelization with beams or plates, local geometry . . . . . . . . . . . Kinematics of plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Delamination of a thin membrane . . . . . . . . . . . . . . . . . . . . . . . . . The response of the composite sphere . . . . . . . . . . . . . . . . . . . . . .

133 136 139 140 141 141 143 150 151 152 153 154 156 157 158

164 172 173 173 176 189 189 191 198 198 202 204 212 212 212 222 240

List of Figures

Fig. 9.2 Fig. 9.3 Fig. 11.1

The global response for a hollow sphere in perfect plasticity (left), and damage (right) . . . . . . . . . . . . . . . . . . . . . . . . Global decomposition F = E.P, and corresponding local decomposition f = e.s.p. f −1 ............................ o On left: a non-well posed problem: on o , To and uo (–) are applied (Left). On right: the well posed problem: v on i (- -) and To on o are prescribed . . . . . . . . . . . . . . . . . . . . .

xv

242 246

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Chapter 1

Introduction

1.1 Some General Features In order to explain and to predict the motion and the equilibrium of bodies or structures subjected to various physical interactions, first kinematics of the motion is performed. In the case of a continuum, this description must ensure the continuity of the body during its motion. Usually one looks for the motion of a material point .X from a reference configuration .Co by describing its displacement .u(X, t). After the description of the kinematics, one has to deal with the mechanical interactions. Many statements permit the description of these interactions, for example the virtual power statement can be used. This shows the manner to describe the mechanical interaction between each material point of the body with respect to a given loading distribution. For sake of simplicity and conciseness of this presentation, thermodynamics of interaction is then chosen.

1.2 Description of the Motion Let a body .Ω submitted to external forces described by vector fields . f over .Ω and vector fields .T along the boundary .∂Ω. The external forces are generally functions of time. Under this loading the body is transformed. The actual position .x of a material point is a function .Φ of it’s initial position .X and of the time .t. The displacement .u is then defined by: .x(X, t) = Φ(X, t) = X + u(X, t). (1.1) Consider now, two material points .X and .X + dX, then we have: x(X + dx, t) = x(X, t) + dx = Φ(X + dX, t) = Φ(X, t) +

.

∂Φ .dX + o(dX). ∂X (1.2)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 C. Stolz, Introduction to Non-linear Mechanics, Springer Series in Solid and Structural Mechanics 14, https://doi.org/10.1007/978-3-031-51920-8_1

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2

1 Introduction

Hence, a material element .dX, defined on .Co , is convected by the motion in the actual material element .dx defined on .Ct , the corresponding transformation is the linear application associated with the gradient .F: dx =

.

∂x .dX = F.dX. ∂X

(1.3)

The actual length of the material element is given by: dx.dx = dX.FT .F.dX = dX.C.dX.

.

(1.4)

The variations of the local geometry, stretching of fibers and gliding, are determined by the Cauchy-Green tensor .C = FT .F. Then we have seen that a vector . V defined on .Co is convected as . v on .Ct as: . v = F. V. A volume element. dΩo is transformed in volume. dΩ as. dΩ = det F dΩo . An elementary volume . dΩo is defined by a vector . V and an elementary area . dSo of normal .N, then . dΩo = V.N dSo . During the motion, . V is convected to . v and . dSo becomes . dSt with normal .n, then the transformed volume is obtained as .

dΩ = v.n dS = det F V.N dSo .

(1.5)

This relation is valid for any vector . V, then an elementary area is moved according n.F dS = det F N dSo .

.

(1.6)

This emphasizes that the linear form are convected as . Ad jF.

1.3 Homogeneous Deformations General homogeneous deformation is given by the transformation x = F.X + c,

.

(1.7)

where .F is independent of .X and .c a constant vector. Consider an elementary direction . V, it becomes . v = F. V, (1.8) then its initial length .|| V|| is transformed, its actual value becomes .

v. v = V.FT .F. V = λ2 ( V)|| V||2 ,

where .λ is the stretch in direction of . V (Fig. 1.1).

(1.9)

1.3 Homogeneous Deformations

3

Fig. 1.1 A simple shear

E2

E2 m

E1

E1

The Cauchy-Green tensor.C = FT .F is a scalar product on the initial configuration 2 .Co , . C has three positive eigenvalues .μi = λi , i = 1..3, the principal stretches .λi associated with three eigenvectors . Si . s = F. Si = λi ui , || ui || = 1.

. i

(1.10)

The frame . ui associated to a direct orthogonal basis . Si is direct orthogonal too ; the orthogonal basis . Si , i = 1..3 is the unique basis which remains orthogonal by the convected motion, then 3 Σ .R = ui ⊗ Si , (1.11) i=1

is a rotation, the polar decomposition of .F is recovered: F=

3 Σ

.

λi ui ⊗ Si = R.S = U.R,

i=1

S=

3 Σ

Si ⊗ Si , U =

i=1

3 Σ

ui ⊗ ui .

(1.12)

i=1

The Cauchy-Green tensor has three principal invariants I = Tr C =

3 Σ

. 1

μi ,

i=1

) 1( 2 I1 − Tr(C.C) = μ1 μ2 + μ2 μ3 + μ3 μ1 , 2 I3 = det C = μ1 μ2 μ3 .

I2 =

(1.13)

Cayley-Hamilton: The Cauchy-Green tensor is solution of its characteristic polynomial . P(μ)

= det(C − μ I) = −μ3 + I1 μ2 − I2 μ + I3 = −(μ − μ1 )(μ − μ2 )(μ − μ3 )

Σ As . P(μi ) = 0, . i P(μi )Si ⊗ Si = P(C) = 0

(1.14)

4

1 Introduction

Simple elongation Consider the uniform extension of circular cylinder with principal axis . S1 , without no change of the orientation of principal axis and preservation of circular symmetry, then .F = λ1 S1 ⊗ S1 + λ2 ( S2 ⊗ S2 + S3 ⊗ S3 ). (1.15) Simple shear For this motion, the point .X is moving on the position .x such that x = X + m X.E2 E1 ,

.

(1.16)

where .m is the amount of shear, and assuming orthonormal coordinates: x 1 = X 1 + m X 2; x 2 = X 2, x 3 = X 3,

(1.17)

F = E1 ⊗ E1 + E2 ⊗ E2 + E3 ⊗ E3 + mE1 ⊗ E2 .

(1.18)

.

then .

As the basis is orthonormal .Ei = Ei . In this case, the Cauchy-Green tensor .C becomes 2 . C = I + m(E1 ⊗ E2 + E2 ⊗ E1 ) + m E2 ⊗ E2 . (1.19) Its characteristic equation determines the eigenvalues .μ = λ2 : .

det(C − λ2 I) = −(μ − 1)(μ2 − (2 + m 2 )μ + 1) = 0,

(1.20)

μ = λ = 1 is the eigenvalue associated to direction .E3 . Other eigenvalues are .μ and 1/μ. Considering the sum of the eigenvalues

. .

μ+

.

we obtain m =λ−

.

1 1 = λ2 + 2 = 2 + m 2 , μ λ

√ 1 = 2γ , λ± = γ ± 1 + γ 2 , λ+ λ− = 1. λ

Discussion Elongation in direction .E1 E1 .C.E1 = 1, λ(E1 ) = 1.

.

Elongation in the direction .E2

(1.21)

(1.22)

1.3 Homogeneous Deformations

5

E2 .C.E2 = 1 + m 2 , λ(E2 ) =

√

.

1 + m2.

The unit areas of initial directions .E2 , E3 are convected respectively as E2 .F ds2 = E2 dS2 , E3 .F ds3 = E3 dS3 .

.

(1.23)

The normal vector .E1 to area . dS1 becomes 1 n= √ (E1 − mE2 ), 1 + m2

(1.24)

1 E1 . n.F = √ 1 + m2

(1.25)

.

and

.

As . ds1 =

√

1 + m 2 dS1 and .det F = 1 this illustrates the relation n.F ds = det F N dS.

.

(1.26)

Vectors . AB and . Ba are initially orthogonal vectors AB = λ− E1 + E2 , Ba = λ+ E1 − E2 ,

.

(1.27)

and associated convected vectors are orthogonal too Ab = λ+ E1 + E2 , ba = λ− E1 − E2 ,

.

(1.28)

then the eigenvectors of .C are .

S1 =

Ba AB , S2 = , || AB || || Ba ||

(1.29)

ba Ab , u2 = . || Ab || || ba ||

(1.30)

these vectors are convected to .

u1 =

Evolution of the geometry The velocity of the particle initially at .X is actually at position .x(X, t) and v(X, t) =

.

the gradient of the velocity satisfies

∂x , ∂t

(1.31)

6

1 Introduction

Fig. 1.2 The motion of areas

E2

E2

m

N = E2

N = −E1

n = E2

−E1 + mE2 √ 1 + m2 E1

dv = grad v.dx, grad v = K =

.

∂v ∂X ˙ −1 . . = F.F ∂X ∂x

E1

(1.32)

We have the property .

d d d C = FT .F + FT . F = FT .(K + K T ).F = 2 FT . D.F. dt dt dt

(1.33)

The strain rate on the actual configuration is given by .

D=

1 (K + K T ) = ε(v), 2

(1.34)

and .

grad v = D + Ω,

(1.35)

where the antisymmetric part .Ω is related to the vorticity (Fig. 1.2).

1.4 The Mobility and the Interactions The body .Ω is considered as a continuous set of elements, positions of which are denoted by .x. Each element of volume . dΩ has an elementary mass .dm = ρ dΩ, where .ρ is the density (Fig. 1.3). The mobility of the body is defined by the set of virtual motions associated with any vector field . v˜ ∗ which can be interpreted as a virtual velocity field, [1–3]. Forces are defined by a linear form:. v˜ ∗ → P( v˜ ∗ ) where.P is a real number, named the virtual power of the forces developed by the virtual motion . v˜ ∗ . In classical continuum mechanics, the external forces applied on the body are given by vector field .f defined over the volume .Ω and by vector field .T defined over the boundary .∂Ω. In this case the virtual power of external forces is expressed as: Pe ( v˜ ∗ ) =

∫

.

Ω

f. v∗ dΩ +

∫ ∂Ω

T. v∗ dS.

(1.36)

1.4 The Mobility and the Interactions

7

Fig. 1.3 Eigenvectors, simple shear

B

b γ

A

1

γ

√ − 1 + γ2

γ

γ O

a √

1 + γ2

The power of internal interactions is given by a field of second order tensors .σ˜ (x) such that: ∫ .Pi ( v ˜ ∗ ) = − σ : grad v∗ dΩ, (1.37) Ω

σ is the Cauchy stress tensor. This expression is the simplest form, for a local description of internal interactions, compatible with external loading defined only in terms of vector fields. Taking account of the axiom of objectivity: the power of local interaction .Pi is equal to zero for any rigid body motion:

.

Pi (˜v∗ ) = 0, ∀ v˜ ∗ ∈ R.B.M.

.

(1.38)

The set .R.B.M is the set of rigid body motion: R.B.M = {˜v/˜v = vo + ωo .X},

.

(1.39)

with .ωoT = −ωo . By application of (Eq. 1.38) for any domain .Ω, the Cauchy stress .σ is a symmetric second order tensor. If we assume that there is no jump for the velocity, the virtual power of acceleration quantities is: ∫ A(˜v∗ ) =

.

Ω

ργ .v∗ dΩ, γ = v˙ .

(1.40)

Virtual Power: The fundamental statement of dynamics is written in terms of virtual power: the sum of the virtual power of internal interactions and of the virtual power of external forces is equal to the virtual power of acceleration quantities: Pi (˜v∗ ) + Pe (˜v∗ ) = A(˜v∗ ),

.

(1.41)

for any virtual motion .v˜ ∗ . Using this equality for any virtual motion and any subdomain .Ω∗ , the local form of the balance of momentum is obtained:

8

1 Introduction .

div σ + ρf = ργ ,

over Ω, σ T = σ , σ .n = T along ∂Ω.

(1.42)

By the same reasoning the principle of action and reaction is recovered. Along an elementary internal surface . dS with normal .n the stress vector is continuous, if there is no jump of velocity: + − .σ .n = σ .n . (1.43)

1.4.1 On the Initial Configuration The power of internal stresses can be written on the reference configuration .Co using the properties of convected frame. For a virtual motion.v∗ we can evaluate its gradient with respect to the reference configuration .

∂v∗ = grad v∗ .F = F∗ , ∂X

(1.44)

C ∗ = 2FT . D∗ .F,

(1.45)

and .

then, the power by unit of mass satisfies: .

p(v∗ ) = −

σ σ π θ : grad v∗ = − : D∗ = − : C ∗ = − : F∗ , ρ ρ 2ρo ρo

(1.46)

then, we have the definitions: .

π σ = F. .FT , ρ ρo

θ σ = F−1 . . ρo ρ

(1.47)

This defines two new stress tensors: the Piola-Kirchoff .π and the nominal stress tensor .θ . Using the definition of the convection of unit area, n.F dS = det F N dSo ,

(1.48)

n.σ dS = N.θ dSo ,

(1.49)

.

we obtain first .

and with integration by part on the initial configuration, we have .

Div θ T + ρo f = 0, over Ωo .

(1.50)

These equations of conservation are not sufficient to determine the internal state, some complementary informations are needed.

1.5 Conservation of Energy and Entropy Production

9

A thermodynamical point of view is chosen. The body is a thermodynamical system formed by a collection of small elements defined as material points. Each small element has a density .ρ and the local state is characterized by a set of state variables.

1.5 Conservation of Energy and Entropy Production The internal state is described by the actual value of a set of state variables. To described effectively the behaviour of the material, we must measure a great number of mechanical quantities, to gain in efficiency the concept of internal parameters is adopted. The choice of these parameters is governed by the observation and the ability of the modelling to describe the studied phenomenon with accuracy. The state variables are the strain .ε, the temperature .ϑ and a set of internal variables .α. Attached with these parameters an internal energy density .e and a entropy density .s are determined. Then the internal energy and the entropy of the body .Ω are given by whole integration over the body: ∫ E=

∫

.

Ω

ρ e dΩ, S =

Ω

ρ s dΩ.

(1.51)

Conservation of energy: The conservation of energy is written as: .

˙ + E˙ = Pcal + Pe , K

(1.52)

where .K is the kinetic energy ∫ K=

.

Ω

1 2 ρv dΩ, 2

(1.53)

and .Pcal is the calorific power. Assuming that the calorific power is due to conduction: ∫ Pcal = −

∫

.

∂Ω

q.n dS = −

div q dΩ,

(1.54)

Ω

the local expression of the energy conservation is deduced from (Eqs. 1.41–1.52– 1.54) applied to any volume .Ω: ρ e˙ = σ : ε(v) − div q.

.

(1.55)

Entropy production: The internal production of entropy must be positive, this is the second law of thermodynamics written here for a continuum:

10

1 Introduction

.

S˙ +

∫ ∂Ω

q.n dS ≥ 0, ϑ

(1.56)

After integration by parts, the entropy production contains two contributions: ∫ .

Ω

ρ s˙ +

1 div q dΩ − ϑ

∫ Ω

q. grad ϑ dΩ ≥ 0. ϑ2

(1.57)

The two terms have different origin, the first one is due to internal mechanical irreversibility, the second one is due to conduction. In the total dissipation, we distinguish the part due to the conduction and the part due to internal forces. We assume that the choice of state parameters is a normal set of variables. In this case the two contributions are individually positive: 1 div q ≥ 0, ϑ q. grad ϑ . Dth = − ≥ 0. ϑ2 .

Dm = ρ s˙ +

(1.58) (1.59)

By introducing the conservation of energy in the first equation, we can use the free energy .ψ instead of internal energy .e = ψ + sϑ. Then, the intrinsic dissipation . Dm is rewritten as: ˙ + s ϑ) ˙ ≥ 0. . Dm = σ : ε(v) − ρ(ψ (1.60) This inequality must be satisfied by any real evolution of the body, from the state defined by the actual values of the state variables .ε, α, ϑ.

1.6 The Linear Thermoelasticity Definition: A constitutive behaviour is elastic, if all thermodynamic quantities are functions of the actual state defined by the Cauchy-Green tensor .C and the temperature .ϑ. For linear thermo-elasticity and small perturbations around a natural state at the temperature .ϑo , the free energy has the following form (.τ = ϑ − ϑo ): ρψ=

.

1 1 ε : C : ε + k.ε τ − Cτ 2 . 2 2

(1.61)

1 where .ε(u) is the linearised part from .C : .ε(u) = (∇u + ∇ T u). The positivity of 2 entropy production is satisfied by any real variations of the state near a thermodynamical equilibrium state (i.e. mechanical equilibrium under uniform temperature), then we deduce the equations of state:

1.7 More General Cases

11

σ =ρ

.

∂ψ ∂ψ , s=− . ∂ε ∂ϑ

(1.62)

In this case the stresses .σ must satisfy the balance of momentum. Therefore the elastic behaviour is essentially reversible.

1.7 More General Cases In general, the intrinsic dissipation . Dm has a form driven by the choice of the free energy .ψ, which depends on the strain .ε, internal parameters .α and temperature .ϑ. The entropy production is rewritten as: .

Dm = (σ − ρ

∂ψ ∂ψ ∂ψ ) : ε˙ − ρ( + s)ϑ˙ − ρ α˙ ≥ 0. ∂ε ∂ϑ ∂α

(1.63)

Defining the driving forces associated with the state variables by the state equations: ∂ψ ∂ψ ∂ψ , A = −ρ , s=− , .σ r = ρ (1.64) ∂ε ∂α ∂ϑ the dissipation takes the form: .

Dm = (σ − σ r ) : ε˙ + Aα˙ = σ ir : ε˙ + A.α˙ ≥ 0.

(1.65)

Then, two sources of entropy production exist, one due to the variations of internal parameters and one due to the strain rate. The equations of state do not provide the full constitutive equations, some complementary laws are needed to describe the irreversibility. Such laws are determined by observations and experimentations. First we must define the domain of reversibility, we must discuss the influence of the strain rate and finally we must determine constitutive relations between the rates .ε˙ , α˙ and the driving forces .σ ir , A.

1.7.1 Generalized Standard Materials A powerful method is to consider the existence of potentials for the dissipation. Let us assume that the behaviour belongs to the class of generalized standard materials, [4]. Let us consider the potentials .φ ∗ and .φ as convex functions of their arguments, with minimum value at the origin. The evolution of the internal state satisfies normality law: ∂φ ∂φ ∂φ ∗ ∂φ ∗ , or A = (1.66) .α ˙ = , ε˙ = , σ ir = . ∂A ∂σ ir ∂ α˙ ∂ ε˙

12

1 Introduction

1.7.2 Linear Visco-elastic Behaviour For linear visco-elasticity the potential of dissipation is chosen as: φ(˙ε) =

.

1 ε˙ : η : ε˙ , 2

(1.67)

where .η is a positive definite operator. The complementary law gives: σ ir =

.

∂φ ˙ = η : ε. ∂ ε˙

(1.68)

The stresses .σ which satisfy the balance of momentum are decomposed in reversible part .σ r and irreversible contribution .σ ir σ = σ r + σ ir .

.

(1.69)

Let us described two particular behaviours. • The model of Kelvin-Voigt of linear viscoelasticity is defined by the potentials: ρ ψ(ε) =

.

1 1 ε : C : ε ; φ(˙ε) = ε˙ : η : ε˙ , 2 2

(1.70)

then, the constitutive behaviour satisfies: σ = C : ε + η : ε˙ .

.

(1.71)

• The Maxwell description is obtained by choosing the free energy: ρ ψ(ε, α) =

.

1 (ε − α) : C : (ε − α), 2

(1.72)

and the pseudo potential of dissipation as: φ=

.

1 α˙ : η : α. ˙ 2

(1.73)

Then, the driving forces are .σ ir = 0, σ = A = C : (ε − α) and the complementary law gives the relation: .A = η : α. ˙ (1.74)

1.8 The Quasistatic Evolution

13

1.7.3 Normality Rule In the case of a regular and differentiable function, the convexity of the potential of dissipation gives us the characterization of internal state evolution by the equalities: σ ir =

.

∂φ ∂φ , A= . ∂ ε˙ ∂ α˙

(1.75)

For non-differential function, the definition of gradient is replaced by the notion of subgradient.1 Normality rule: The internal state satisfies the evolution laws given by the normality rule .(σ ir , A) ∈ ∂φ(˙ ε, α). ˙ (1.76) The set .∂φ is the set of thermo-dynamical forces whose satisfy the inequality: φ(˙ε, α) ˙ + σ ir : (ε ∗ − ε˙ ) + A : (α ∗ − α) ˙ ≤ φ(ε∗ , α ∗ ),

.

(1.77)

for all admissible values .ε ∗ , α ∗ . The existence of a potential for the dissipation ensures the positivity of the entropy production: σ ir : ε˙ + A : α˙ =< ∂φ(˙ε, α), ˙ (˙ε , α) ˙ > ≥ φ(˙ε, α) ˙ − φ(0, 0) ≥ 0.

.

(1.78)

1.8 The Quasistatic Evolution Consider a body.Ω submitted to some prescribed boundary conditions. The boundary is decomposed into.∂Ωu where the displacement is imposed and.∂ΩT where the stress ˜ ˜ ϑ) vector is prescribed, (.∂Ω = ∂Ωu ∪ ∂ΩT and .∅ = ∂Ωu ∩ ∂ΩT ). A solution .( u˜ , α, 2 of the problem of quasi-static evolution satisfies : • the compatibility equations for strains and displacements: the strain field .ε˜ is associated with the displacement.u˜ ,.2ε( u) = ∇u + ∇ T u, the displacement satisfies the boundary condition .u = ud over .∂Ωu , • the state equations: σr = ρ

.

1

∂ψ ∂ψ ∂ψ , A = −ρ , s=− , ∂ε ∂α ∂ϑ

(1.79)

Elements of convex analysis can be founded in J.J. Moreau [5]. is prescribed or solution of a thermal problem, omitted here to simplify the presentation.

2 .ϑ ˜

14

1 Introduction

• the equations of evolution for the state variables: (σ ir , A) ∈ ∂φ(˙ε, α), ˙

(1.80)

σ = σ r + σ ir ,

(1.81)

.

• the constitutive law:

.

• the conservation of the momentum and boundary conditions: .

div σ = 0, on Ω, σ .n = Td over ∂ΩT .

(1.82)

For the overall system the rule of the free energy is replaced by the global free energy: ∫ ˜ = .Ψ(˜ ε , α, ˜ ϑ) ρ ψ(ε, α, ϑ) dΩ. (1.83) Ω

We recall the definition of the Gâteaux differential: .

F(q˜ + ηq˜ ∗ ) − F(q) ˜ ∂F · q˜ ∗ = lim , η→0 ∂ q˜ η

(1.84)

and then we get for our particular case: .

∂Ψ ∗ · q˜ = ∂ q˜

∫ Ω

ρ

∂ψ ∗ q dΩ. ∂q

(1.85)

The equations of state are now relations between fields: σ˜ r =

.

∂Ψ ˜ = − ∂Ψ , s˜ = − ∂Ψ . , A ∂ ε˜ ∂ α˜ ∂ ϑ˜

(1.86)

In a global description the equations of state have the same form as in the local one, and the state of the system is defined by fields of state variables.

1.8.1 Dissipative Function By integration of the dissipation potential over the body, we define the dissipative function: ∫ ˜˙ , α) ˜˙ = .Φ(ε φ(˙ε, α) ˙ dΩ. (1.87) Ω

By integration of the local inequality (Eq. 1.77), we have: ˜˙ − Φ(˜ε∗ , α˜ ∗ ) + σ˜ ir · (˜ε∗ − ε˜˙ ) + A ˜˙ ≤ 0, ˜ · (α˜ ∗ − α) Φ(ε˜˙ , α)

.

(1.88)

1.8 The Quasistatic Evolution

15

for all admissible fields .(˜ε ∗ , α˜ ∗ ). This shows that the evolution of internal state satisfies the normality rule, rewritten in terms of fields: ˜˙ ˜ ∈ ∂Φ(ε˜˙ , α). (σ˜ ir , A)

(1.89)

.

For example, in the case of regular function we have: .

∫ ∂Φ σ ir : ε(δu) dΩ, · ε˜ (δu) = ∂ ε˙˜ Ω ∫ ∂Φ A.δα dΩ. · δ α˜ = ∂ α˙˜ Ω

These equations can be rewritten as: σ˜ ir =

.

∂Φ ˜ = ∂Φ . , A ∂ ε˙˜ ∂ α˙˜

(1.90)

1.8.2 The Isothermal Boundary Value Problem We consider now for sake of simplicity isothermal processes. Let us consider that the external loading derives from a potential given in terms of traction .Td applied on the external surface .∂ΩT of the body. Then, the global free energy .Ψ is replaced by the potential energy .E of the system: E(˜u, α, ˜ T˜ d ) =

∫

∫

.

Ω

ρ ψ(ε(u), α) dΩ −

∂ΩT

Td .u dS.

(1.91)

By combining all the equations in terms of fields of state variables, the quasistatic evolution is described in a global manner by the variational system: ∂Φ ∂E · δ u˜ + · ε˜ (δu), ∂ u˜ ∂ ε˜˙ ∂Φ ∂E · δ α˜ + · δ α. ˜ 0= ∂ α˜ ∂ α˜˙ 0=

.

(1.92)

These equations are defined on a set of admissible fields: • The displacement is submitted to boundary conditions .u = ud over .∂Ωu . Then the fields .δ u˜ must satisfy .δu = 0 over .∂Ωu . • The .δ α˜ have some constraints depending on the nature of irreversibility. The preceding equations are general. They contain the essential structure of a quasistatic evolution. The first equation of this system explains the conservation of momentum taking into account the constitutive law:

16

1 Introduction .

div σ = 0, σ = σ r + σ ir , σ .n = Td over ∂ΩT ,

(1.93)

the second one explains the complementary law as a relation between the forces .A and the internal parameters: A · δ α˜ = −

.

∂Φ ∂E · δ α˜ = · δ α. ˜ ∂ α˜ ∂ α˙˜

(1.94)

1.9 The Lagrangian and the Dynamical Case By definition the Lagrangian is the difference between the kinetic energy and the potential of interactions applied to the system. For all kinematically admissible fields, the potential of interaction is related to the potential energy: ˜ T˜ d ) = E(˜u, α, ˜ ϑ,

∫

.

Ω

∫ ψ(ε, α, ϑ)ρ dΩ −

∂ΩT

Td .u dS.

(1.95)

The kinetic energy is defined as: ∫ K(˜v) =

.

Ω

1 2 ρv dΩ, 2

(1.96)

and the Lagrange’s functional is then: ˜ T˜ d ) = K(˜v) − E(˜u, α, ˜ T˜ d ). L(˜u, v˜ , ϑ, ˜ ϑ,

.

(1.97)

The acceleration is denoted .γ , .γ = v˙ . Let us consider variations of the Lagrangian: ∫ ∫ ∂L · δ u˜ = − σ r : δε dΩ + Td .δu dS, ∂ u˜ Ω ∂ΩT ∫ ∂L · δ v˜ = ρv.δv dΩ, ∂ v˜ Ω ∂L d ∂L d ∂L ( · δ u˜ ) = · δ v˜ + ( ) · δ u˜ dt ∂ v˜ dt ∂ v˜ ∫∂ v˜ ∫ .

=

Ω

(ρv.δv + ργ .δu) dΩ +

Γ

m[ v ]Γ .δu dS.

(1.98)

m is the mass flux through the moving surface .Γ along which the velocity .v has discontinuities .[ v ]Γ . The equations of motion are explained by the conservation of the momentum:

.

1.10 The Hamiltonian

17 .

div σ = ργ , over Ω,

[ σ ]Γ .n = m[ v ]Γ , along Γ.

(1.99)

Then, a variational form for the conservation of momentum is easily deduced: ∫

∫ .

Ω

σ : ε(δu) dΩ =

∂Ω

∫ n.σ .δu dS +

∫

Γ

n.[ σ ]Γ .δu dS +

Ω

ρ γ .δu dΩ. (1.100)

The stresses .σ are decomposed as previously as .σ = σ r + σ ir taking the constitutive law into account. .Γ is a moving surface, where the velocities have discontinuities. Taking all these relations into account, the evolution of the system is governed by the generalized Lagrange’s equations: .

−(

∂Φ d ∂L ∂L − ) · δ u˜ = · ε˜ (δu), dt ∂ v˜ ∂ u˜ ∂ ε˜˙ ∂L ∂Φ · δ α, ˜ · δ α˜ = ˜ ∂ α˜ ∫∂ α˙ ∂L ρs δϑ dΩ. · δ ϑ˜ = ∂ ϑ˜ Ω

(1.101)

These equations are a generalization to non linear dynamics ([6]) of the classical Lagrange’s formulation, they have the same form as the expression given by Biot ([7]) in visco-elasticity. In this formulation, we have defined the dissipative function as: ∫ ˜˙ , α) ˜˙ = .Φ(ε φ(˙ε, α) ˙ dΩ. (1.102) Ω

The first equation is the equation of motion, the second one the evolution law for the internal state, the last one defines the local entropy. To this set of equations the conduction law for completeness must be added.

1.10 The Hamiltonian The Hamiltonian is a Legendre transformation of the Lagrangian, with respect to velocity and to temperature [6]: H(˜u, p˜ , α, ˜ s˜ , T˜ d ) =

∫

.

then, we have:

Ω

˜ T˜ d ), (p. u + ϑ ρ s) dΩ − L(˜u, v˜ , α, ˜ ϑ,

(1.103)

18

1 Introduction

H(˜u, p˜ , α, ˜ s˜ , T˜ d ) =

∫

.

Ω

1 2 p /ρ dΩ + 2

∫

∫ ρ e(ε(u), α, ϑ) dΩ −

Ω

∂Ω

Td .u dS.,

(1.104) where .p is the momentum .ρv. In this expression appears the density of internal energy: .e = ψ + ϑ s. In a global formulation, we obtain successively: .

∂H ∗ du˜ ∗ · p˜ = v˜ · p˜ ∗ = · p˜ , ∂ p˜ dt ∫ ∫ ∂ψ ∂H · δ u˜ = : ε(δu) dΩ − ρ Td .δu dS ∂ u˜ ∂ε Ω ∂ΩT ∫ ∫ = σ r : ε(δu) dΩ − Td .δu dS. Ω

∂ΩT

Taking account of the momentum conservation, of decomposition of stress in reversible and irreversible parts, of the boundary conditions and of jump conditions, the expressions are then modified: .

∂H · δ u˜ = − ∂ u˜

∫

∫ Ω

σ ir : ε(δu) dΩ −

Ω

∫ ργ .δu dΩ +

Γ

n.[ σ ]Γ .δu dS.

Recall that .σ˜ ir = ∂Φ/∂ ε˜˙ and consider the relation: [ σ ]Γ .n = m[ v ]Γ , p = ρv, ∫ ∫ d p˜ · δ u˜ = ργ .δu dΩ + m[ v ]Γ .δu dS, dt Ω Γ .

we obtain the conservation of the momentum in the Hamiltonian’s form: ∫ ∫ ∂H ∂Φ ργ .δu dΩ + m[ v ]Γ .δu dS. . · ε˜ (δu) − · δ u˜ = − ∂ u˜ ∂ ε˜˙ Ω Γ Finally, the Hamiltonian formulation of the evolution problem is obtained: .

∂H ∗ · p˜ ∂ p˜ ∂H ∗ · u˜ ∂ u˜ ∂H · α˜ ∗ ∂ α˜ ∂H ∗ · s˜ ∂ s˜

= v˜ · p˜ ∗ =

d u˜ · p˜ ∗ , dt

∂Φ d · ε˜ (u∗ ) − p˜ · u˜ ∗ , ∂ ε˜ (v) dt ∂Φ ∗ =− · α˜ , ∂ α˜˙

=−

= ϑ˜ · ρ˜ s˜ ∗ .

(1.105)

(1.106)

1.11 Some Properties

19

As previously a conduction law must be given and the positivity of the entropy production must be verified to determine the evolution of the system.

1.11 Some Properties The definition of the Lagrangian and of the Hamiltonian can be extended to generalized media as beams or plates. The proposed description is performed when the behaviour of the system is described by two potentials: a global free energy and a dissipative function. If some particular internal constraints exist, the preceding description must be revisited.

1.11.1 Expression of the Conservation of Energy For the real motion, the value of the Hamiltonian is the sum of the kinetic energy, of the internal energy and of the potential energy of the external loading, then the conservation of the energy of the system can be easily rewritten as:

.

d ∂H dT˜ dH − · = Pcal . dt ∂ T˜ d dt

(1.107)

d

= 0, the exchange of energy is When the external loading is time independent . dT dt only due to the heat rate supply .Pcal . Generally this quantity has the form: ∫ Pcal = −

q.n dS,

.

(1.108)

∂Ω

where .q is the heat flux. This result is used in fracture mechanics to quantify the heat generated by the propagation of the crack as presented in the following section.

1.11.2 Conservation Law In the case of conservative system, in an adiabatic evolution (.Pcal = 0), the Hamiltonian is constant: .H(t) = H(0), (1.109) this property can be rewritten in terms of the Lagrangian: L − v˜ ·

.

∂L = H(0). ∂ v˜

(1.110)

20

1 Introduction

1.11.3 Property of Stationarity The Lagrangian has properties of stationnarity in elasticity or viscoelasticity: let us consider a variation of the Lagrangian for isothermal evolution: δL =

.

∂L d ∂L ∂Φ ∂L · δ v˜ + · δ u˜ = ( · δ u˜ ) + · ε˜ (δu), ∂ v˜ ∂ u˜ dt ∂ v˜ ∂ ε˜˙ ∫

then

t2

δ

.

L dt = δΦ,

(1.111)

(1.112)

t1

where .δΦ is the total viscous dissipation during the variation. Finally let us note that the above results may be adapted in the case of other type of boundary conditions, taking care they give rise to a well posed problem.

1.12 On Discontinuities In this section, presence of discontinuities is studied. Different type of discontinuities are encountered in continuum mechanics. To described strong discontinuities, P. Germain in fluids mechanics [8] and C. Stolz [9] for solids mechanics have introduced the notion of shock generating function. For non linear and dissipation material, the conservation of energy for a strong discontinuity needs additional relations to relay the forward and backward states of internal variables. These relations can be obtained considering that the discontinuities are related to a moving layer in a steady state process, forward and backward quantities being given at infinity. The problem can be solved as a continuous process. Mandel [10] has proposed for elasto-plasticity to consider that the internal variables are submit to a radial process. A singular surface .Γ is a thin transition layer, accompanied with dissipative processes of various kinds. The normal vector to the surface .Γ is .ν. When a moving surface of discontinuities exist inside the body with celerity .c, the must taken this motion into account. Rewriting conservation of energy ∫ (Eq. 1.107) ∫ Hamiltonian .H = Ω h dΩ − ∂ΩT Td .u dS, then .

Pcal

∫

∫ ∫ ∫ T˙d .u ds, Td .v dS − h˙ dΩ + [ h ]Γ c dS − Ω Γ ∂ΩT ∂ΩT ∫ ∫ ∫ =− q.n dS = − div q + [ q ]Γ .ν dS, (1.113)

d H= dt

∂Ω

Ω

Γ

then, the conservation of energy along the surface of discontinuity satisfies:

1.12 On Discontinuities

21

1 m[ e + v2 ]Γ + ν.[ σ .v ]Γ + [ q ]Γ .ν = 0. 2

.

(1.114)

Along the singular surface, the mechanical fields satisfy conservation laws: [1, 8, 9, 11] • mass conservation:

[ ρc ]Γ = 0,

(1.115)

[ mv ]Γ + ν.[ σ ]Γ = 0,

(1.116)

1 [ m(e + v2 ) ]Γ + ν.[ σ .v ]Γ − ν.[ q ]Γ = 0. 2

(1.117)

.

• momentum conservation: .

• energy conservation: .

These conservations laws are characterized by .m, Td , Qd , named shock constant: where m = (ρc)±

.

mTd = m(v)± + ν.(σ )± 1 mQd = m(e + v2 )± + ν.(σ .v)± − ν.(q)± 2

(1.118)

The jumps of any mechanical quantities must be compatible with the positivity of entropy production q . Ds = −m[ s ]Γ + [ (1.119) ] .ν ≥ 0. ϑ Γ Introducing the generating function of discontinuities .G G=

.

) m( 1 ψ − v2 + Td .v − Qd ϑ 2

(1.120)

we have .

Ds = [ G ]Γ ≥ 0,

(1.121)

1 taking account of (Eq. 1.118) .mTd = ν.σ¯ + m v¯ , with . f¯ = ( f + + f − ) this is quite 2 different than the heat power supply along the discontinuity surface .

Dq = [ q ]Γ .ν = m[ e ]Γ + ν.σ¯ .[ v ]Γ ,

where .e is the internal energy.

(1.122)

22

1 Introduction

1.12.1 Change of Scale At a lower scale, the discontinuities can be described by continuous transition between the states .+ and .− inside a thin layer. Inside the layer viscosity and thermal conductivity can be introduced. Two hypotheses are implicitly made in such a description. • The model of local accompanying state is still valid ; in other words, each particle of the material is considered in a thermodynamic state of equilibrium and classical constitutive laws for an homogeneous material is used, even if the transition profile has a thickness smaller than the size of local heterogeneities. • This thickness is also smaller than the radius of curvature of the discontinuity surface, the discontinuity surface can be considered as a part of a plane with normal vector .ν = ex . The local speed of the surface of the discontinuity has a smooth evolution, than at each time the local transition zone is considered in a steady state. To evaluate the internal state inside the layer, the solution is based on the research of a steady state, all the equations are written in a referential moving with the celerity of the singular surface. The inner expansion of the quantities results from a continuous process in the frame moving with the shock surface assuming that quantities depend only of the normal coordinate . X = x − ct. Then any function . F( x , t) = f ( x − cν) and .

∂F .ν. F˙ = −c ∂x

(1.123)

For the inner expansion .x varies from .−∞ to .+∞. The steady state must satisfy particular boundary conditions. Far from the thin layer the states are . F + and . F − and the discontinuity is: [ F ]Γ = F − − F + .

.

(1.124)

The mechanical quantities must satisfy the jump relations, with the given constant m, Td , Qd and + + − − .v + c∇u .ν = v + c∇u .ν, (1.125)

.

therefore, the steady state condition implies v + c∇u.ν = 0.

.

(1.126)

The local constitutive laws are given in a classical manner defined by two potentials the free energy .ψ and a potential of dissipation .Φ σr = ρ

.

∂ψ ∂ψ ∂ψ , A = −ρ , s=− , ∂ε ∂α ∂ϑ

(1.127)

1.12 On Discontinuities

23

and

∂Φ ∂Φ , σ ir = . ∂ α˙ ∂ ε˙

A=

.

(1.128)

The heat flux .q is assumed to be a linear function of .∇ log ϑ, ∇ϑ . ϑ

q = −K .

.

(1.129)

The stress state inside the layer is .σ with σ = σ r + σ ir , div σ = ρc2

.

∂ 2u . ∂ x∂ x

(1.130)

The entropy production is rewritten considering the given potential of dissipation ∫

q . − m[ s ]Γ + [ ] .ν = ϑ Γ

Ω

Dm ∇ϑ − q. 2 dΩ, ϑ ϑ

(1.131)

where the local dissipation is: .

Dm = A.α˙ + σ ir : ε˙ .

(1.132)

Using the shock constants, we have as previously: .

Ds = −m[ s ]Γ + [

q ] .ν = ϑ Γ

∫ Ω

Φm ∇ϑ − q. dω = [ G ]Γ , ϑ ϑ

1 m G = (−Qd + Td .v + ψ − v2 ), ϑ 2

(1.133)

where .G is the shock generating function, according to the condition of local steady state motion. Definition: The shock generating function is defined as: G=

.

) m( 1 − Qd − cTd .∇x u + ψ(∇x u, α) − c2 (∇x u)2 . ϑ 2

(1.134)

˙ we have Introducing the potential of dissipation .Φ(˙ε , α) .

D=

1 1 Φ(−c∇x ε, −c∇x α) + K (∇x log ϑ)2 . ϑ 2

With the notation . U = −c∇x u, we have

(1.135)

24

1 Introduction

(

) 1 2 − Q + T . U + ψ(∇x u, α, ϑ) − U , 2 ( ) 1 1 2 Φ(−c∇x ε, −c∇x α) + K (∇x log ϑ) D= ϑ 2

m .G = ϑ

d

d

(1.136)

The free energy depends on . U, knowing that any material derivative . f˙ is given by ρ f˙ = m

.

∂f , ξ = x − ct. ∂ξ

(1.137)

Shock structure: The shock internal structure satisfies the canonical equations [9] .

∂G ∂D = , ∂ϑ ∂∇x ϑ

∂G ∂D = , ∂∇x u ∂∇x ε

∂G ∂D = . ∂α ∂∇x α

(1.138)

Elements of proof The variations of .G and . D with respect to .ϑ, u, α are • Variation with respect to .ϑ: ∂G δϑ ms δϑ = − G − δϑ ∂ϑ ϑ ϑ ( 1 δϑ mδϑ = − 2 − Qd + Td .v + e − v2 ) = −q.ν 2 , ϑ 2 ϑ ∂D 1 (1.139) δ∇x ϑ = 2 K .∇x ϑ δ∇x ϑ ∂∇x ϑ ϑ .

that is the relation .q = −K .∇ X log ϑ. • Variation with respect to .u: .

) ∂G m( σr δu,x = − Td .c∇x (δu) + : (∇x δu ⊗ ν) − mc2 ∇x u.∇x δu ∂u,x ϑ ρ m σ ir = ν. .∇x δu ϑ ρ ∂Φ c ∂Φ c = (1.140) = σ ir , ∂ε,x θ ∂∇x ε θ

∂Φ . we recover .σ ir = ∂ ε˙ • Variations with respect to internal state

References

25

.

that gives .A =

m ∂ψ mA ∂G .δα = δα = − δα ∂α ϑ ∂α ϑ ρ ∂D c ∂Φ c =− = − A, ∂∇x α ϑ ∂ α˙ θ

(1.141)

∂Φ . ∂ α˙

The internal description of shock structure is necessary to determine the change on internal parameters during the shock, as pointed out by Mandel [12]. Conditions of stability of such an internal structure are studied in [13] for elasto-plastic material and in a more general concept in [14].

References 1. P. Germain, Functional approach in mechanics, in Proceedings of the 2nd National Conference in Theoretical and Applied Mechanics, Athens, pp. 53–69 (1989) 2. J. Salençon, Handbook of Continuum Mechnanics: General Concepts, Thermoelasticity (Springer, Berlin and Heidelberg GmbH & Co. K, 2001) 3. C. Stolz, Energy Methods in Non-linear Mechanics. Lectures Notes 11 AMAS (Polish Academy of Sciences Warsaw, 2004) 4. B. Halphen, Q.S. Nguyen, Sur les matériaux standard généralisés. Journal de Mécanique 14(1), 254–259 (1975) 5. J.J. Moreau, Fonctionnelles convexes (Instituto Poligrafico e Zecca Dello Stato Roma, 2003) 6. C. Stolz, Sur les équations générales de la dynamique des milieux continus anélastiques. C. R. Acad. Sci. Paris, Série II -307, 1997–2000 (1988) 7. M.A. Biot, On Variational Methods in the Mechanics of Solids (IUTAM, Evanston, 1978) 8. P. Germain, Shock waves jump relations and structure. Adv. Appl. Mech. 12, 131–194 (1972) 9. C. Stolz, Sur la propagation d’une ligne de discontinuité et la fonction génératrice de choc pour un solide anélastique. C. R. Acad. Sci. Paris, Série II - 308, 1–3 (1989) 10. J. Mandel, Plasticité, et viscoplasticité, classique, Cours CISM Udine, vol. 97 (Springer, New York, 1971) 11. P. Germain, Functional concepts in continuum mechanics. Meccanica 33, 433–444 (1998) 12. J. Mandel, Ondes de choc longitudinales dans un milieu é, lastoplastique. Mech. Res. Comm. 5(6), 353–359 (1978) 13. J.L. Dequiedt, C. Stolz, Propagation of shock discontinuity in an elasto-plastic material constitutive relations. Arch. Mech. 56(5), 391–410 (2004) 14. V.N. Kukudjanov, Investigation of shock wave structure in elasto-visco-plastic bars using the asymptotic method. Arch. Mech. 33(5), 739–751 (1981)

Chapter 2

Non-linear and Linear Elasticity

2.1 Introduction When the constitutive behaviour is elastic, all thermodynamic quantities are functions only of the actual state of strain .C and temperature .ϑ, [1, 2]. As pointed out, the transformation .Φ from an initial configuration to actual one, defines how elementary material direction, elementary surface or volume are transformed or convected, using the transformation gradient .F. The Cauchy-Green strain tensor .C has three eigenvalues .μi = λi2 and three principal invariants : I1 = Tr C =

Σ

μi ,

i .

1 2 (I − Tr(C.C) = μ1 μ2 + μ2 μ3 + μ3 μ1 , 2 1 I3 = det C.

I2 =

(2.1)

The free energy of an isotropic hyper-elastic body .ψ(C) depends only of these principal invariants in a isothermal process. Consider a virtual motion .v∗ defined on the actual geometry. The symmetric part of its gradient is the virtual strain-rate tensor and the corresponding virtual CauchyGreen tensor follows ∗ T ∗ . C = 2 F . D .F, (2.2) and the virtual power of internal stresses by unit of mass becomes: .

pi (v∗ ) = −

σ π ∂ψ : C ∗ = −θ.F∗ = − : grad(v∗ ) = − : C ∗. ρ 2ρo ∂C

We have first .

σ σ π = F. .FT , θ T = .F−1 . ρ ρo ρ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 C. Stolz, Introduction to Non-linear Mechanics, Springer Series in Solid and Structural Mechanics 14, https://doi.org/10.1007/978-3-031-51920-8_2

(2.3)

(2.4) 27

28

2 Non-linear and Linear Elasticity

Denoting . Mi = 2

.

∂ψ and taking the invariant derivatives into account: ∂Ii ∂I2 = Tr(C) I − C, ∂C

∂I1 = I, ∂C

we obtain .

∂I3 = I3 C −1 , ∂C

( ) π = M1 I + M2 Tr(C) I − C + M3 I3 C −1 , ρo

(2.5)

(2.6)

and the Cauchy stress tensor is given as: .

π σ = F. .FT , σ = μ1 B + μ2 B −1 + μ3 I. ρ ρo

(2.7)

The Cauchy stress depends on three moduli, functions of the principal invariants of .C. The Eulerian strain tensor . B = F.FT has . ui as eigenvectors and the same eigenvalues as .C (Eq. 1.12) .

Σ

C=

λi2 Si ⊗ Si ,

i

B=

Σ

λi2 ui ⊗ ui .

(2.8)

i

As .Tr(C) I − C = I2 C −1 − C −2 I3 obtained using polynomial characteristics of .C, .

π = M1 I − I3 M2 C −2 + (M3 I3 + I2 M2 )C −1 , ρo

(2.9)

th for the moduli the relations: μ1 =

.

1 1 1 M1 , μ2 = − I3 M3 μ3 = (M3 I3 + I2 M2 ) J J J

(2.10)

ρo 1/2 = I3 ρ When an hyperelastic material is incompressible, any motion of the body satisfies the internal constraint . J = 1. To take account of this relation we consider Lagrange multiplier .q and the strain energy where . J = det F =

ψin (C, q) = ψ(C) − q(det F − 1),

.

and successively θ=

.

and we obtain

∂ψ ∂ψin = − q det F F−1 , ∂F ∂F

σ = μ1 B + μ2 B −1 + η I

.

(2.11)

(2.12)

(2.13)

2.2 Universal Deformation

29

where .η = q + I2 M2 , analogous to a pressure, is undetermined. It is noticed that .

Tr(σ ) = μ1 Tr B + μ2 Tr B −1 + 3η.

(2.14)

2.2 Universal Deformation Let us consider here the set of equilibrium solutions whatever is the local isotropic incompressible hyperelastic behaviour. Then, the displacement solution must satisfy the momentum conservation and the constitutive behaviour: .

div σ = 0, div(μ1 B + μ2 B −1 ) + grad η = 0.

(2.15)

To obtain a solution, the condition of integrability of the field .η(x) must be satisfied, the first part of the equation must be the gradient of a scalar field too, then the condition is ( ) −1 . curl div(μ1 B + μ2 B ) = 0, ∀(μ1 , μ2 ), (2.16) under the condition .

∂μ1 ∂μ2 = . ∂I2 ∂I1

(2.17)

Eriksen [3] has shown the existence of five sets of universal deformation, studied partially in [2]: • bending, stretching of a rectangular block: r=

√

.

2 A X , θ = B Y, z =

Z − B C Y, AB

AB /= 0,

(2.18)

AB /= 0,

(2.19)

• stretching, gliding of a partial circular sector: .

x=

1 A B2 R2, 2

y=

Z CΘ Θ , z= + , AB B AB

• inflation, extension and torsion of circular cylinder: r=

.

√

A R 2 + B, θ = C Θ + D Z , z = E Θ + F Z ,

A(C F − D E) = 1, (2.20)

• inflation, spherical return of a sphere: 1

r = (±R 3 + A) 3 , θ = ±Θ, φ = Φ,

.

• dilatation, elongation, shear of a circular cylinder:

A,

(2.21)

30

2 Non-linear and Linear Elasticity

r = A R, θ = B Log(R) + C Θ, z = D Z A2 C D = 1,

.

(2.22)

where constants . A, B, C, D, E, F are used to control boundary conditions and isochoric constraint . J = 1.

2.3 Properties of Equilibrium Solution In general, the equilibrium solution is non-unique, and few results are concerned by the existence of a solution for a non-linear elastic body. Ball [4] has given condition of existence, using the property of polyconvexity. Polyconvexity: The function .ψ(F) of the gradient .F is polyconvex if a function 1 .Φ(A, B, C) convex of its argument such that ψ(F) = Φ(F, Ad j (F), det F ). Additional conditions of regularity are needed in order to ensure the existence of a solution: • .n going to .∞: consider .Fn → F, and . H n → H, .δn → O + , then .Φ(Fn , H n , δn ) tends to infinity. • coercivity: p , r > 1, .∃α > 0, β ∈ R, p ≥ 2, q ≥ (2.23) p−1 Φ ≥ α(|| F || p + || Ad j (F) ||q + || det F ||r ) + β.

.

(2.24)

These conditions indicate nothing about uniqueness.

2.4 Example of Non-linear Elastic Deformation 2.4.1 The Flexion of a Prismatic Bar Let us consider a rectangular beam of length. L, with thickness.a and width.b. There is no boby forces and the surface . X = ±a/2 are stress free. The section .Y = 0, Y = L are submitted to a relative rotation with angle .β. The transformation is defined as r = R f (X ), θ =

.

Y , z = Z. R

The compatibility implies that . Rβ = L. Then the transformation gradient is 1 . Ad j (F)

= det F F−T .

(2.25)

2.4 Example of Non-linear Elastic Deformation

31

F = R f ' er ⊗ Ex + f eθ ⊗ E y + ez ⊗ Ez .

.

(2.26)

As the material is incompressible .det F = R f ' f = 1 and .

f (X ) =

f2 X b = + . 2 R 2

r , R

(2.27)

On internal and external radius, we have the relations f e2 =

a + b, R

f i2 = −

a + b, R

.

2b = f e2 + f i2 ,

f e2 f i2 = b2 −

a2 . R2

(2.28)

The strain tensors satisfy: C = (R f ' )2 Ex ⊗ Ex + f 2 E y ⊗ E y + Ez ⊗ Ez , 1 B = 2 er ⊗ er + f 2 eθ ⊗ eθ + ez ⊗ ez , . f 1 B −1 = f 2 er ⊗ er + 2 eθ ⊗ eθ + ez ⊗ ez . f

(2.29)

As .Ex , E y , Ez are convected in .er , eθ , ez , the polar decomposition is given as .

R = er ⊗ Ex + eθ ⊗ E y + ez ⊗ Ez , S = R f ' E x ⊗ E x + f E y ⊗ E y + Ez ⊗ Ez .

(2.30)

For a Mooney-Rivlin behaviour Let us consider a Mooney-Rivlin material: σ = μ1 B + μ2 B −1 + η I.

.

(2.31)

As . f depends only on .r , the function .η depends on .r and satisfies the equilibrium equation 1 1 (μ1 − μ2 )( 2 + f 2 ) + C. .σrr = (2.32) 2 f Using the boundary conditions in.re , ri , the stretches.( f e = f (re ), f i = f (ri )) satisfy .

as

f4 +

2C f 2 + 1 = 0, μ1 − μ2

(2.33)

32

2 Non-linear and Linear Elasticity

f 2 + f e2 = −

. i

C = b, μ1 − μ2

f i2 f e2 = b2 −

a2 = 1. R2

(2.34)

The constants are / 2 a2 a 2 .b = 1 + 1 + , C = −(μ − μ ) . 1 2 R2 R2

(2.35)

Ogden behaviour Let us consider now the strain energy ψ=

3 Σ

.

μα λαk ,

(2.36)

k=1

with λ =

. 1

1 , λ2 = f = λ, λ3 = 1. f

(2.37)

The state of stress is σ = αμα (λα1 er ⊗ er + λα2 eθ ⊗ eθ + ez ⊗ ez ).

.

(2.38)

The momentum conservation gives η = μα λα + μα (1 − α)λ−α + D,

.

(2.39)

where . D satisfies boundary conditions: .

D = −μα (λαe +

1 1 ) = −μα (λiα + α ). α λe λi

(2.40)

Using the relation Eq. 2.28 a solution of an equilibrium state is determined, .λe , λi are solutions of 2α .μα λ + Dλα + μα = 0. (2.41) with λ λi = 1 = b2 − a 2 , λαe + λiα = −

. e

D . μα

The last equation determines the value of . D with respect to . R.

(2.42)

2.5 Linear Elasticity: Small Perturbations

33

2.4.2 The Antiplane-Shear Consider an isotropic homogeneous non linear elastic body. Further we introduce rectangular coordinates .(x1 , x2 , x3 ). Assume the interior of the body occupies an open region .Ω and consider a deformation corresponding to anti-plane shear which involves displacements .w in the direction .e3 normal to the plane .(x1 , x2 ): .

y( x ) = x + w(x1 , x2 )e3 , F = I + ∇w.

(2.43)

The displacement.w is the out-of-plane displacement at point. x of coordinates.xi , i = 1, 2. The modulus of displacement gradient is denoted by . R: ∇w = w,i e3 ⊗ ei ,

R 2 = w,i w,i = ∇w T .∇w = ||∇w||2 .

.

(2.44)

The local behaviour is an incompressible, homogeneous, isotropic elastic material that possesses a strain energy density .ψ per unit volume in initial configuration. This energy is a function .ψ(I1 , I2 ) of the first two invariants .I1 , .I2 of the left (or right) Cauchy-Green tensor. The third fundamental invariant is unity since the material is incompressible. For anti-plane shear .I1 = I2 = I = 3 + R 2 and the strain energy becomes a function .Φ(I) = ψ(I, I). Then, the Cauchy stress .σ satisfies the state equation 2 Σ ∂Φ = μ(R)w,i . (2.45) .σ = τ i (e3 ⊗ ei + ei ⊗ e3 ), τi = ∂w, i i=1 The balance of linear momentum in the absence of body forces leads to the equilibrium equations: T . div σ = 2(Φ' (I )w,i ),i = 0, (2.46) where.i = 1, 2 and the summation is implied. These differential equations are elliptic under the condition: ' 2 .2Φ + 4R Φ" ≥ 0. (2.47) This conditions implies that the local stress-strain curve .τ (R) = 2RΦ' (I) = Rμ(R) is an increasing function of . R.

2.5 Linear Elasticity: Small Perturbations This hypothesis assumes that the transformation should be infinitesimal [1] F = I + ∇u, ||∇u|| ≤ 1.

.

(2.48)

34

2 Non-linear and Linear Elasticity

To ensure stability of elastic body, near its initial configuration, the strain energy .ψ is convex with respect to the linear strain ε(u) =

.

1 (∇u + ∇uT ). 2

(2.49)

The state equations define the state of internal stresses σ =

.

∂ψ ∂ψ , s=− . ∂ε ∂ϑ

(2.50)

2.6 Equilibrium Solution of a Linear Elastic Body Let us consider a homogeneous linear elastic body, submitted on traction .Td along d .∂ΩT , with imposed displacement .u = u on the complementary part .∂Ωu . The total potential energy is defined as ∫ E(u, T ) =

∫

d

.

Ω

ψ(ε) dΩ −

∂ΩT

Td .u dS,

(2.51)

on the set of kinematically admissible displacement .K.A: K.A = {u|u = ud , over ∂Ωu }.

.

(2.52)

The variation of potential energy defines the condition of equilibrium of the body .

∂E .δu = 0, with δu = 0 over ∂Ωu , ∂u

∫

that is .

Ω

∂ψ : ε(δu) dΩ − ∂ε

(2.53)

∫ ∂ΩT

Td .δu dS = 0.

(2.54)

By integration by part this condition is equivalent to the local set of equilibrium σ =

.

∂ψ , div σ = 0, n.σ = Td over ∂ΩT . ∂ε

(2.55)

As .ψ is convex with respect to .ε we have the property ψ(u) ≤ ψ(u∗ ) +

.

∂ψ : (ε(u∗ ) − ε(u)). ∂ε

(2.56)

Using the variations of the potential energy with .δu = u∗ − u where .u is the solution of the problem of equilibrium

2.6 Equilibrium Solution of a Linear Elastic Body

∫ .

Ω

then ∫ .

Ω

σ : (ε(u∗ ) − ε(u)) dΩ −

∫

ψ(ε) − ψ(ε∗ ) dΩ ≤

Ω

and finally

35

∫ ∂ΩT

Td .(u∗ − u) dS = 0,

σ : (ε ∗ − ε) dΩ =

∫ ∂ΩT

Td .(u∗ − u) dS,

E(u, Td ) ≤ E(u∗ , Td ).

(2.57)

(2.58)

(2.59)

.

As .ψ is convex, the complementary energy .ψ ∗ is the Legendre-Fenchel transformation of .ψ with respect to .ε: ψ ∗ (σ ) = min(σ : ε − ψ(ε)).

.

ε

(2.60)

The complementary energy is given by ∫

E∗ (σ , ud ) =

.

Ω

ψ ∗ (σ ) dΩ −

∫ ∂Ωu

n.σ .ud dS.

(2.61)

This energy is defined on the set of statically admissible stresses .S.A S.A = {σ / div σ = 0, over Ω, n.σ = Td along ∂ΩT }.

.

(2.62)

A family of statically admissible field is defined by .δσ = curlr curll χ where .χ is the symmetric Beltrami tensor. The variations of .E∗ gives: .

∂E∗ : δσ = ∂σ

∫

∫ ε s : δσ dΩ −

Ω

∂Ωu

n.δσ .ud dS = 0,

(2.63)

then, by integration by part, the compatibility condition on .ε s is recovered ∫ .

Ω

χ : curll curlr ε s dΩ = 0,

(2.64)

this is a necessary condition to define a displacement field .us such that ε =

. s

∂ψ ∗ = ε(us ), ∂σ

(2.65)

with this displacement and .δσ satisfying .div δσ = 0, by integration by part we have ∫ .

∂Ωu

n.δσ .(us − ud ) dS = 0,

(2.66)

36

2 Non-linear and Linear Elasticity

the boundary condition .us = ud over .∂Ωu is fulfilled. Due to convexity of complementary energy, we have E∗ (σ s , ud ) ≤ E∗ (σ ∗ , ud ),

.

(2.67)

among the set of statically admissible fields .S.A. For linear elastic isotropic body the free energy is given by ψ(ε) =

.

1 λ(Tr ε)2 + μ Tr(ε.ε). 2

(2.68)

The state equations define the Cauchy stress σ = λ Tr(ε) I + 2με,

.

(2.69)

λ, μ are Lamé’s moduli of elasticity. Given Young’s modulus . E and Poisson’s coefficient .ν we have

.

2μ =

.

ν E E , λ = 2μ , 3κ = 3λ + 2μ = . 1+ν 1 − 2ν 1 − 2ν

(2.70)

2.7 Stability and Bifurcation in Non-linear Elasticity 2.7.1 Notion of Stability For conservative system, when the loading.T depends on one parameter.λ, the dynamical system associated to the evolution of the body .Ω is defined by a functional x˙ = F(x, λ)

.

(2.71)

Positions of equilibrium are given by . F(x, λ) = 0, uniqueness of .x(λ) for given .λ is not ensured. But near a known position.x(λ) under small perturbation.dλ it is possible to determine the corresponding variation .dx. Secondly, given some perturbation of equilibrium at fixed .λ, if the response remains closed to that position, the equilibrium is say stable. The stability of the position of equilibrium .xo (λ) is then determined with respect to Lyapounov definition : Definition: A position of equilibrium .xo (λ) is stable if ∀ε ∃α (||x(0, λ) − xo (λ)|| + ||˙x(0, λ)||) ≤ α → ||x(t, λ) − xo (λ)|| ≤ ε, (2.72)

.

where .x(t, λ) is solution of (Eq. 2.71) with initial conditions near the equilibrium state.

2.7 Stability and Bifurcation in Non-linear Elasticity

37

The notion of stability of an equilibrium position is essentially a dynamical notion. The evolution of the system is governed by the total potential energy, which is composed by the free energy of the material, and by the potential energy of the loading.

2.7.2 The Metronome A generic point of view is given by the study of the metronome (Fig. 2.1) and introduction of asymptotic expansion to characterized the equilibrium solution [5–7]. A vertical rigid bar is clamped by a spiral spring with free energy .Ψ(α). We study the motion of the system under a vertical loading .λ. A mass . M is attached at the top of the bar. The total potential energy .E and the kinetic energy satisfy: E(α, λ) = Ψ(α) − λL(1 − cos α), K =

.

1 M α˙ 2 L 2 . 2

(2.73)

Near the position .α = 0 the energy is developed as Ψ=

.

1 1 1 C1 α 2 + C2 α 3 + C3 α 4 + ... 2 3 4

(2.74)

The dynamical system to study becomes .

M L 2 α¨ +

∂E = 0. ∂α

(2.75)

First we characterize equilibrium position .αo , λo and the research of equilibrium position near this point is determined by asymptotic expansion as proposed by [5, 6], linking loading .λ to the position .α.

λ M

Fig. 2.1 The metronome

L

λ α

38

2 Non-linear and Linear Elasticity

Static equilibrium path An equilibrium state (.λ, α) satisfies .

( ∂Ψ α2 ) − λL sin α = α (C1 + C2 α + C3 α 2 + .) − λL(1 − .. = 0. ∂α 6

(2.76)

then two equilibrium paths exists α(λ) = 0, ∀λ, λL =

.

1 1 ∂Ψ = C1 + (C2 + C1 )α + ... sin(α) ∂α 6

(2.77)

The common point of the two equilibrium paths is the bifurcation point. λ =

. c

C1 , α = 0. L

(2.78)

Stability analysis near the fundamental path .α = 0 The dynamical behaviour around a position is a weakly non linear dynamical system, taking account of a new asymptotic expansion λ = λo + λ1 ξ + λ2 ξ 2 + · · · .

α = αo + α1 (τ )ξ + α2 (τ )ξ 2 + · · · τ = t ξ m Ω(ξ )

(2.79)

Ω = Ωo + ξ Ω1 + ξ 2 Ω2 + · · · The characteristic time .τ is chosen to satisfy the dependency of the pulsation of the system with respect to the loading. .

∗∗

M L 2 ξ 2m Ω2 α +

∂Ψ − λL sin α = 0, ∂α

(2.80)

∗∗

where . f denotes the second derivative with respect to time .τ . The motion near .α = 0 is then governed by ∗∗

∗∗

M L 2 ξ 2m (Ωo + ξ Ω1 + ξ 2 Ω2 + · · · )2 ( α 1 ξ + α 2 ξ 2 + ..) = .

(Lλo − C1 )ξ α1 + ξ 2 (L(λo α2 − λ1 α1 ) − (C1 α2 + C2 α12 )) + ···

Discussion • If .λo /= λc then .m = 0 and we have

(2.81)

2.7 Stability and Bifurcation in Non-linear Elasticity .

39

∗∗

M L 2 Ω2o α 1 = (λo − λc )Lα1 ,

(2.82)

λ c − λo then, when.λo ≤ λc the position.α = 0 is stable,.Ω2o = > 0 ; when.λ > λc , ML the position .α = 0 is unstable. 1 • If .λ = λc and .λ1 /= 0, then .m = this implies that .ξ ≥ 0 and 2 .

∗∗

M L 2 Ω2o α 1 = (Lλ1 − C2 α1 )α1 .

(2.83)

L . A position of C2 the fundamental path .(λ = λc + λ1 ξ, 0) with .λ1 < 0 is stable, unstable otherwise. See Fig. 2.2. The position .αe is stable if .λ1 > 0. Finally, the position .(λc , 0) is unstable. • .C2 = 0. It is necessary to consider .λ1 = 0, .m = 1, and the motion is governed by If.C2 /= 0, two positions of equilibrium exist:.αe = 0 and.αe = λ1

.

∗∗

M L 2 Ω2o α 1 = (λ2 L − (C3 +

Lλc 2 )α1 )θ1 . 6

(2.84)

Then we have three positions of equilibrium, one along the fundamental path (λ = λc + λ2 ξ 2 , 0), and two others

.

[ I I .λ = λc + λ2 ξ , α = ±ξ √ 2

λ2 L . λc C3 + 6

(2.85)

The fundamental path .α = 0 is stable if .λ < λc and stability of position along the secondary path if and only if .λ2 > 0, in this case the bifurcation point is a stable point of equilibrium. See Fig. 2.3. If the secondary path is defined with .λ2 < 0, see Fig. 2.4, the secondary path and the point of bifurcation are unstable positions of equilibrium. The results are resumed on the pictures (Figs. 2.2, 2.3 and 2.4), with fundamental path (.α = 0, λ), and particular phase diagram. This description of conservative systems is well known, the systematic proposed expansion can be used for study stability of beams, plates, …, the displacement .θ is replaced by a vector displacement. The second derivative of the potential energy plays a fundamental rule, when this quadratic form is positive definite then uniqueness is ensured, that is Lejeune-Dirichlet theorem. For non-conservative system, the proposed asymptotic expansion should be used, static-uniqueness doesn’t ensure Lyapounov stability, as illustrated with a bi-pendulum under following load [8].

40

2 Non-linear and Linear Elasticity

Fig. 2.2 Case .λ1 < 0 /= 0, Stable (s) and unstable (u) paths, Phase diagram for .λ < λc

λ

α˙

s

u λc α

u

s

α

Fig. 2.3 Case .λ1 = 0, > 0, Stable (s) and unstable (u) paths, Phase diagram for .λ > λc

α˙

λ

.λ2

u s λc

α

s

α

Fig. 2.4 Case .λ1 = 0, < 0, Stable (s) and unstable (u) paths, Phase diagram for .λ < λc

λ

.λ2

α˙

u λc

u

α

s α

2.7.3 The Euler Column Let us consider now an inextensible beam, with length . L, with axis .e y . The external loading is .−λe y . A point of the beam is referred by the curvilinear coordinate .s. Under the loading, the actual position of the beam is described by .

M = x(s, t) ex + y(s, t) e y ,

dx = ds sin α, dy = ds cos α.

(2.86)

The tangent vector to this curve is .

T=

dα dM = sin α ex + cos α e y , κ = = α'. ds ds

(2.87)

2.7 Stability and Bifurcation in Non-linear Elasticity

41

The local strain energy depends on .κ . The total potential energy is given by ∫

L

E(α, λ) =

.

Ψ(α ' ) + λ(y(L) − L),

(2.88)

0

where

∫ .

L

y(L) − y(0) =

cos α(s) ds

(2.89)

0

and finally

∫

L

E(α, λ) =

.

Ψ(α ' ) + λ cos α ds − λL

(2.90)

0

The potential energy is defined on the set of admissible fields K.A = {α|α(0) = 0}.

.

(2.91)

The equilibrium is given by ∂E . δα = ∂α

∫

L 0

∂Ψ dδα − λ sin α δα ds = 0, ∂α ' ds

(2.92)

which is quite equivalent to d ∂Ψ ( ) − λ sin α, ds ∂α ' . ∂Ψ 0= (L). ∂α ' 0=−

(2.93)

The Euler equation takes the final form .

∂ 2Ψ α" + λ sin α = 0. ∂α ' ∂α '

(2.94)

The fundamental path .(λ, α = 0) is solution of the system. 1 Consider that .Ψ = C1 (α ' )2 and using the proposed expansion other solutions 2 of equilibrium are researching as function of .ξ : λ = λo + ξ λ1 + λ2 ξ 2 + · · · .

α(s) = α1 (s)ξ + α2 (s)ξ 2 .

(2.95)

Introducing these developments in the equation (Eq. 2.92), we obtain at each order in .ξ , a system of the form

42

2 Non-linear and Linear Elasticity

∫ .

0

L

(C1 αn' δα '

∫

L

+ λo αn δα) ds +

Tn (αi , λi , i < n)δα ds = 0.

(2.96)

0

The same operator appears at each order ∫ .

< A(λo )[αn ], δα >=

L

0

αn' C1 δα ' + αn λo δα ds,

(2.97)

where .αn ∈ K.A. The loss of uniqueness is given by the values .λc such that .

0 = C1 ψ ' + λc ψ = 0, ω2 C1 = λc , ψ(s) = a sin ωs + b cos ωs, b = 0, a cos(ωL) = 0.

(2.98)

π When .ωL = (2k + 1) , λc = C1 ω2 , the function 2 ψ(s) = sin ωs,

.

(2.99)

π2 is a eigen-solution of the operator . A(λc ). The lowest critical value for .λc is . 2 C1 . 4L This critical value is determined with the study at order one, and we found .α1 (s) = a ψ(s). The amplitude .a is determined by analysis of further order, the second member of the Euler equation .Tn must be in the image of the operator, then because the operator is auto-adjoin this condition is ∫ .

L

Tn (s)ψ(s) ds = 0,

(2.100)

0

then we find successively at order two ∫

L

.

∫ α1 θ1 ψ ds =

0

L

λ1 aψ 2 (s) ds = 0,

(2.101)

0

then .λ1 = 0 to assume .a /= 0, and at order three ∫ .

0

L

1 (λ2 α1 − λc α13 )ψ(s) ds = 0, α1 = ae ψ(s), 6

that gives the relation λ = ae2

. 2

λc . 8

(2.102)

(2.103)

To study the stability of these three positions of equilibrium at point .λc + λ2 ξ 2 , α = 0, α(s) = ±aψ(s) we introduce the linearised kinetic energy

References

43

∫

1 (( ρ 2

L

K(α, ˙ α) =

.

0

∫

s

αdt ˙

)2

+

o

(

∫

s

α αdt ˙

)2 )

ds,

(2.104)

o

which gives the local inertia terms ∫

L

.

ρ

0

(∫

s

∫ α(σ, ¨ τ )dσ

o

s

) δα(l) dl ds.

(2.105)

o

For the mode of bifurcation .ψ(s), the solution is founded as α (s, τ ) = a(τ )ψ(s), τ = ξ m t,

. 1

(2.106)

and the reduced equation of motion (projection of the equation on the mode of bifurcation) at the order two gives .λ1 = 0 and stability condition on the fundamental path. The order three gives .m = 1, and ∫

L

a¨

.

∫

s

ρ(

0

o

∫ ψ 2 (σ )dσ )2 ds + 0

L

1 λ2 ψ 2 a(t) − λc ψ 4 a 3 (t) ds = 0. 6

(2.107)

The three positions of equilibrium are available for .λ2 > 0. The fundamental path is stable for .λ ≤ λc , and the bifurcated path is stable, the conclusion results from the linearisation of the motion near a position of equilibrium .a = ae + β(t). This example shows that the systematic approach based on the asymptotic expansion, with study of the weakly non linear dynamic analysis is powerful. The results on beams and shells given by Koîter [9] are systematically recovered.

References 1. J. Salençon, Handbook of Continuum Mechnanics: General Concepts, Thermoelasticity (Berlin and Heidelberg GmbH & Co. K, Springer, 2001) 2. Ogden, Non-linear Elastic Deformations. Dover Civil and Mechanical Engineering (1997) 3. J.L. Eriksen, Deformations possible in every isotropic incompressible perfectly elastic body. Z. Angew. Ath. Phys. 5, 466–489 (1954) 4. J.M. Ball, Convexity and existence theorems in non-linear elasticity. Arch. Rat. Mech. Anal 63, 337–403 (1977) 5. B. Budianshy, Theory of buckling and post-buckling behaviour. Advanced in Applied Mechanics, vol. 8 (1974) 6. M.J. Sewell, On the branching of equilibrium paths. Proc. Roy. Soc. London A375, 499–518 (1970) 7. Q.S. Nguyen, Stabilité et mécanique non-linéaire (Hermès, 2000) 8. H. Ziegler, Linear elastic stability, a critical analysis of methods. Zeitschrift für angewandte Mathematik und Physik ZAMP 4(2), 89–121 (1953) 9. W.T. Koiter, General Theory of Shell Stability. Springer, CISM Courses, vol. 240 (1980)

Chapter 3

Elasto-plasticity

3.1 Introduction For example, consider the tension curve on a test specimen. The behaviour is linear under a limit value .σo , upper this value some permanent strains .ε p are present after a total unloading. The reversible part of the strain .ε − ε p is the elastic strain. And the form of linear elastic behaviour is conserved: σ = C : (ε − ε p )

.

(3.1)

At microscale metals are crystalline solids: this means that they consist of atoms arranged in a pattern which is periodically repeated. The whole system is build with unit cells or lattice. At each lattice we can associated a triad of vectors (Fig. 3.1). The plastic transformation of mono-crystal is generally described by slip lines contains in crystallographic planes: the slip-plane of normal vector .n; the two parts of the crystal, which are both sides of this plane, were submitted to a small relative displacement along a crystallographic direction, the slip direction .m. This mode of deformation is the slip process. The set of slip plane and slip direction characterizes a slip system .(n, m). The activation of this transformation is reached when the resolved shear stress on the slip system yields a limit value, that is the Schmid’s law: F(σ ) = n.σ .m − τo ≤ 0.

.

(3.2)

These sets are dependent upon the crystallographic class of the crystal. The domain of reversibility is the intersection of hyperplane half-space: F α (σ ) = nα .σ .mα − τoα ≤ 0

.

(3.3)

Then the domain of reversibility is convex. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 C. Stolz, Introduction to Non-linear Mechanics, Springer Series in Solid and Structural Mechanics 14, https://doi.org/10.1007/978-3-031-51920-8_3

45

46

3 Elasto-plasticity

Fig. 3.1 A quasi-static tensile curve

Metals are polycrystalline media, composed by an aggregate of different shape, orientation. The modes of deformation of poly-crystal stay the same as mono-crystal but the grain boundary introduced supplementary modes of transformation. It is rather difficult to model directly the elasto-plastic behaviour of poly-crystal, some simplified model of such a complex reality are introduced. Classical test (tension, torsion) have been performed on cylindrical samples. Experimentally these tests show that the pressure .Tr σ does not influence the plastic behaviour of most metals. Some other results are dedicated to determine the domain of reversibility in stress space. If initially, the shape of this domain is nearly an ellipse, this shapes becomes complicated after loading history. There are three effects: an expansion, a global translation in the direction of the loading point and a local deformation near the stress point which can be very pronounced. It was pointed also out, that the projection of the plastic strain increment is almost normal to the boundary of the reversible domain defined in the stress space (Fig. 3.2) [1]. These tests show that the domain of reversibility .C is convex in the stress space. This is the principal argument to admit the principle of maximal plastic work. Maximal plastic work, normality rule: The principle of maximal plastic work defines the evolution of the plastic strain

3.2 The Domain of Reversibility

47

σxy

Fig. 3.2 The scheme of experiments on copper

Δσ Δεp

σxx

Let σ ∈ C, for all σ ∗ ∈ C, (σ − σ ∗ ) : ε˙ p ≥ 0.

.

The plastic strain rate is normal to the convex .C.

3.2 The Domain of Reversibility The domain of elasticity is modelled by a convex function .F defined on the stress space which depends of internal parameter .α, these parameters are often chosen in order to represent as well as possible the change of shape of the domain in the stress space: .C = {σ /F(σ , α) ≤ 0}. (3.4) Taking the principle of maximal plastic work into account we deduce the property: ⎧ ⎨F(σ ) < 0, ε˙ p = 0, . ∂F ⎩F(σ ) = 0, ε˙ p = μ , μ ≥ 0. ∂σ

(3.5)

Generally, the function .F is taken invariant by translation of .Tr σ then .F depends upon the deviatoric part of .σ : . s = σ − Tr3σ I. The normality rule induces that the plastic strain is isochoric. The function .F must be invariant under the group of symmetry of the material. Consider now an isotropic material then the function .F depends only upon the invariant of the stress: I = Tr σ , I2 =

. 1

1 1 Tr(σ .σ ), I3 = Tr(σ .σ .σ ), 2 3

(3.6)

σ , σ2 , σ3 .

(3.7)

or of the principal stresses: . 1

48

3 Elasto-plasticity

Generally the function .F has not a polynomial form of the invariants. Some typical yielding functions are the Tresca criterion and the Hubert-Von Misès function. The Tresca criterion:

The domain of elasticity is defined by the function F(σ ) =

.

sup {σi − σ j − σo } ≤ 0,

(3.8)

i, j∈{1,2,3}

where the value .σo has the interpretation: • .σo is the limit of elasticity in tension, • . 21 σo is the limit of elasticity in shear. This function depends only on the deviatoric part . s of the stress .σ . The Hubert-von Misès criterion: tion

The domain of elasticity is defined by the func-

F(σ ) =

√

.

where . J2 =

1 2

J2 − ko ≤ 0,

(3.9)

Tr s.s, and .ko is interpreted:

• .ko √ is the limit in shear, • .ko 3 is the limit in tension. To model the expansion of the domain of elasticity, .ko or .σo are functions of some hardening parameters .β and the translation in the stress space is obtained by replacing .σ by . A = σ − α.

3.3 The Evolution of Internal State The domain of elasticity being defined by such a function .F, the evolution of the internal parameters .hcr = (ε p , α, β) satisfies the normality law: F ≤ 0, μ ≥ 0, μ F = 0.

.

(3.10)

By derivation with respect to time of the last equation, we can deduced that: for a state of stress on the boundary of the domain of elasticity .F = 0, we have obviously ˙ ≤ 0. Then a necessary condition to obtain .μ ≥ 0 is the consistency condition .F ˙ = 0, whose expression is: .F .

F˙ =

∂F T ∂F T ˙ hcr = 0. : σ˙ + ∂σ ∂hcr

(3.11)

3.3 The Evolution of Internal State

49

To determine .μ effectively, one must give complementary laws for the hardening: h˙ = μ g(σ , hcr ),

(3.12)

. cr

where .g is a given function of the local state. Considering now the constitutive law for the stress: .σ = C : (ε − ε p ), (3.13) we determine the value of .μ1 ∂F T : C : ε˙ >+ ∂σ .μ = , T ∂F ∂F T ∂F :C: − g ∂σ ∂σ ∂hcr
+ ∂F ∂σ ˙ = C : ε˙ − , .σ C: T T ∂σ ∂F ∂F ∂F :C: − g ∂σ ∂σ ∂hcr
2+ 1 1 ∂σ , U (˙ε ) = ε˙ : C : ε˙ − 2 2 ∂F T ∂F ∂F T . :C: − g ∂σ ∂σ ∂hcr ∂U . σ˙ = ∂ ε˙
+ = 21 ( f + | f |).

50

3 Elasto-plasticity

σ

Fig. 3.3 Perfect plastic tensile curve

p

ε

ε

Elements of Proof Consider the variation of the functional .U ∫ ∫ ∂U ∂U ∗ ˜˙ d ) · δ v˜ ∗ = (ε(v)) : ε(δv . (˜ v , ) dΩ − T˙ d .δv∗ d A = 0, T ˙ ∂ v˜ ∗ ∂ ε Ω ∂ΩT

(3.18)

among the set of kinematically admissible fields.δv∗ satisfying.δv∗ = 0 on.∂Ωu . After ∂U integration by part, it is obvious that .σ˙ = satisfies the equilibrium equation and ∂ ε˙ the boundary condition over .∂ΩT .

3.4 A Model of Perfect Plasticity The model of perfect of plastic is illustrated by the tensile curve (Fig. 3.3). In this case the domain of reversibility is constant. The convex function .F depends only upon the stress .σ . The relation between stress and strain is given by: .

σ = C : (ε − ε p ).

(3.19)

1 (ε − ε p ) : C : (ε − ε p ). 2

(3.20)

The free energy is reduced to: ρψ =

.

The increment of plastic strain is normal to the domain of reversibility: at a point .σ such .F = 0, ∂F ˙ p = μ , μ ≥ 0. (3.21) .ε ∂σ Obviously, for all state of stress, we have μF = 0, μ ≥ 0, F ≤ 0.

.

(3.22)

3.5 The Rate Boundary Value Problem

51

The evolution satisfies a consistency condition, which is obtained by derivation of the equation (Eq. 3.22) with respect to time: μF ˙ + μF˙ = 0.

.

(3.23)

The only way to have .μ ≥ 0 implies that .F˙ = 0. This is the consistency condition, which defines the value of .μ: .

∂F T : C : (˙ε − ε˙ p ) = 0. ∂σ

(3.24)

∂F T : C : ε˙ >+ ∂σ . ∂F ∂F T :C: ∂σ ∂σ

(3.25)

then we deduce: μ=


0, then Let us consider a state .δ 1p , and a variation of the external loading .Σ 1 ˙ .δ p ≥ 0 and the solution satisfies the normality rule. This solution is valid if and only if .|n 2 | − n c ≤ 0, implying .Σ1 ≤ Σ2c with .Σ2c : Σ2c = (1 +

.

√ 2)n c +

Kc nc . 1 + Kc

(3.104) p

For .c = 0 the domain of reversibility changes with the value of .δ1 , see (Fig. 3.6). • Plasticity of the whole system. When .Σ1 > Σ2c the three bars are deformed plastically. The system is rewritten (Eqs. 3.82–3.84) with .Σ2 = 0 and .δ 2p = δ 3p and the condition of plasticity c n = n c + cδ 1p , n 2 = n c + √ δ 2p . 2

. 1

(3.105)

Using (Eqs. 3.83, 3.84) E1 = K n c + (1 + K c)δ 1p = 2K n c +

√

2(1 + K c)δ 2p , (3.106) √ √ 1+ 2 1 c cK nc + .Σ1 = n c (1 + 2) + √ (E − 2K n c ). (3.107) 1 + Kc 1 + Kc 2 .

3.8 Classical Solutions in Elasto-plasticity

65

Re

Fig. 3.7 The hollow sphere under radial extension

Rp u = Ed Ri p=0

The residual displacement is obtained as √ 1+ 2 1 . √ E p = δ 1p + δ 2p . 2

(3.108)

When there is no hardening, that is the case of perfect plasticity, the global stress Σ1 is constant and .E1 is not defined exactly, the uniqueness is not ensured, the limit loading is reached.

.

3.8.2 Case of a Hollow Sphere The solution of a hollow sphere under radial tension is well known for an elasto perfectly-plastic material. The external radius of the sphere is . Re , the radius of the R3 void is . Ri and the porosity is .c = i3 , (see Fig. 3.7). For an increasing loading .Ed (t) Re on the external boundary, the evolution of the internal state is determined.

3.8.2.1

Solution of the Problem of Evolution

The plastic zone .Ω p is the spherical domain .r ∈ [Ri , R p ], . p = purely radial .u = u(r )er . The Cauchy stress is given by

R 3p Re3

. The solution is

σ = σrr (r )er ⊗ er + σθθ (r )(eθ ⊗ eθ + eφ ⊗ eφ ),

.

(3.109)

The deviatoric part of the stress tensor is then s =

. r

1 1 (2σrr − σθθ − σφφ ), sθ = (2σθθ − σrr − σφφ ) = sφ 3 3

(3.110)

66

3 Elasto-plasticity

For a Hubert-Von Misès material, as (.σφφ = σθθ ) the domain of reversibility is determined by .F = 21 (sr2 + sθ2 + sφ2 ) − σo ≤ 0 where .σo is a constant. The plastic strain has the form deduced from the normality law ε p = ε p (r ) (2er ⊗ er − eθ ⊗ eθ − eφ ⊗ eφ ).

.

(3.111)

For an increasing imposed .Ed the global response of the sphere depends on the maximum value of .Emd . The solution evolves during two phases. • Elastic phase. In the beginning the behaviour of the hollow sphere is purely elastic then B E Ri3 , u(Re ) = Ed Re = A E Re + Re2 (3.112) . σrr (Ri ) = 0 = 3κ A E − 4μB E , then Ed = A E + B E c, 0 = 3κ A E − 4μB E , ⇒ 4μEd = (4μ + 3κc)A E . (3.113)

.

With notation .Σ = σrr (Re ), the solution is σ (r ) =

. rr

R3 R3 1 1 (1 − 3i )Σ, σθθ = σφφ = (1 + 3i )Σ. 1−c r 1−c r

(3.114)

This defines the stress concentration tensor σ = AΣ, A =

.

) R3 1 ( I + 3i (−er ⊗ er + eθ ⊗ eθ + eφ ⊗ eφ ) . 1−c r

(3.115)

The global bulk modulus is determined by the relation between .Ed and .Σ. Σ = 3K (c)Ed ,

.

K (c) = (1 − c)

4μ . 3κc + 4μ

(3.116)

This solution is the unique solution if and only if F(σ ) = |σθθ − σrr | − σo ≤ 0.

.

(3.117)

F(σ ) is maximum along .r = Ri . The critical value is reached when

.

6μ

.

BE − 9κ A E = σo . Ri3

(3.118)

• Plastic phase. When the critical value is reached, for increasing external loading, a plastic zone appears . Ri ≤ r ≤ Re , where .F(σ ) = 0 and assuming that .σθθ > σrr we have

3.8 Classical Solutions in Elasto-plasticity

σ

. θθ

67

σo dσrr =2 . dr r

− σrr = σo ,

(3.119)

Taking account of the boundary condition (.σrr (Ri ) = 0), the stress field is determined in the plastic domain: σ = 2σo ln

. rr

r , σθθ = σrr + σo . Ri

(3.120)

In the elastic domain . R p ≤ r ≤ Re , the displacement has the form u = Ar +

.

B . r2

(3.121)

On the boundary .r = R p , displacement and stress-vector are continuous: σrr (R p ) = 3κ A − 4μ .

σθθ (R p ) − σrr (R p ) = 6μ

Rp B = 2σo ln , 3 Rp Ri

B = σo . R 3p

(3.122)

This gives two relations 6μB = σo R 3p , 3κ A =

.

Rp 2 σo (1 + ln( )3 ). 3 Ri

(3.123)

Denoting .(E I , Σ I ) the value of .εrr , σrr at point .r = R p we have 1 1 − 3 ), r3 Rp . 1 1 σrr (r ) − Σ I = −4μB( 3 − 3 ), r Rp εrr (r ) − E I = B(

(3.124)

that corresponds to the relation of Eshelby for an inclusion σ (r ) + 4μεrr (r ) = Σ I + 4μE I .

. rr

(3.125)

Under the prescribed loading .Ed we have Ed = A +

.

B , Re3

the value of . R p with (Eq. 3.123) is determined, and the solution is

(3.126)

68

3 Elasto-plasticity

κσo p 2 p + σo (1 + ln ), 2μ 3 c 2 p Σ = σo (1 − p − ln ). 3 c

3κEd = .

(3.127)

Displacement in the plastic zone From the stress field and the elasto-plastic behaviour Tr(σ ) = 3κ Tr(ε) = 3κ(

.

then .

R 2p u(R p ) − r 2 u(r ) =

du u + 2 ), dr r

2σo ( 3 R p r ) − r 3 ln R p ln . 3κ Ri Ri

(3.128)

(3.129)

And with the continuity of the displacement at .r = R p we have .

( ) R3 u σo 2σo ( 3 R p r )1 p − r 3 ln , = Ed + (1 − p) 3 − R p ln r 6μ r 3κ Ri Ri r 3

(3.130)

using the constitutive law σ

. θθ

= λ Tr(ε) + 2μ(εθθ + ε p ), 3λ = 3κ − 2μ,

(3.131)

one obtain finally ε p (r ) =

.

R 3p 3κ + 4μ σo (1 − 3 ). 18κμ r

(3.132)

The global relation between radial extension .Ed and traction is given by: 4κμ , 3κc + 4μ . u 3κ + 4μ σo ( p) Ed = (Re ), E p = p − c − c ln . r 18μκ 1 − c c Σ = 3K (c)(Ed − E p ) = σrr (Re ),

K (c) = (1 − c)

(3.133)

As for the three bars problem, it is easy shown that: ∫ Ep =

Rp

.

A : ε p r 2 dr.

(3.134)

Ri

During the loading process, for value .Ed greater than the global elastic limit .Ecd given at . p = c, the response of the sphere satisfies:

3.9 Finite Elasto-plasticity

69 Σ

Fig. 3.8 The global response for a hollow sphere in perfect plasticity

E

1 p ) 2σo ( p + (1 + ln ) , 3 4μ 3κ c p) 2 ( . Σ = σo 1 − p + ln 3 c

Ed = .

(3.135)

The first equation shows that the radius . R p of the plastic zone is increasing with Ed .

.

3.9 Finite Elasto-plasticity Consider a body .Ω in its initial state. The body is transformed under a given loading. The material point initially in . X moves to actual position . x and the gradient of transformation is .F: ∂x . (3.136) .F = ∂X Due to irreversible strain, after unloading path, the initial configuration of each material elementary volume is not recovered, this local stress free state is deformed, and this deformation is characterised by a mapping.P. This corresponds to the description of the motion of a single crystal. A single crystal deforming by dislocation glide can be formulated according to the kinematic scheme based largely on the analysis of Mandel [6], Hill [2], among many others (Fig. 3.8). The total deformation .F can be decomposed as follows: a plastic deformation .P given by the set of successively simple shear on active slip systems which are referred to a fixed lattice, followed by an elastic lattice deformation .E which deforms and rotates the material and the lattice together so that: F = E.P.

.

(3.137)

70

3 Elasto-plasticity

This decomposition is unique, the lattice orientation is putting back at its initial position during the unloading process and a unique stress-free state .Cκ is defined by .P [7–10]. Relatively to this stress-free state, the reversible strain is well defined: Δκ =

.

1 T (E .E − I). 2

(3.138)

Assuming that the hardening has non influence on elasticity, the free energy density takes the form: 1 (3.139) .ψ(Δκ , γr ) = Δκ : Λ : Δκ + h(γr ), 2 where .γr are internal variables, which are related to the amount of slip on each slip system (slip in direction .mr in the plane of normal .nr ). The local constitutive law is given by the equations of state: .

πκ ∂ψ = , ρκ ∂Δκ

Ar ∂ψ = . ρκ ∂γr

(3.140)

By transport on the actual configuration .Ct the Cauchy stress tensor is given by .

πκ σ = E. .ET . ρ ρκ

(3.141)

They are the stresses in equilibrium on the actual configuration. The dissipation rate is also Φ=

.

ψ σ ˙ −1 + Ar γ˙r ≥ 0 : grad v − ψ˙ = κ : P.P ρ ρκ ρκ

(3.142)

As the plastic strain on .Cκ is the sum of simple shear rates on the active slip systems, [6, 11, 12] then ∑ ˙ −1 = .P.P γ˙r mn ⊗ nr . (3.143) r

With this definition we obtain Φ=

.

∑ ( ψκ Ar ) γ˙r ≥ 0. nr . .mr + ρκ ρκ r

(3.144)

To determine the evolution of the amount of shear we introduce a yielding function F defined for each slip system as

. r

F = nr .

. r

Ar ψκ .mr + − τo ≥ 0. ρκ ρκ

The evolution of .P system satisfies the normality law

(3.145)

3.9 Finite Elasto-plasticity

71 Co

Fig. 3.9 Decomposition with a stress free state

F

P

Ct

E Ck

∑

∑

∂Fr , ψ r r ∂ κ ρκ γ˙r ≥ 0, Fr ≤ 0, Fr γ˙r = 0.

˙ −1 = P.P .

γ˙r mr ⊗ nr =

γ˙r

(3.146)

The domain of reversibility is defined by intersection of half space .Fr ≤ 0 is convex (Fig. 3.9). The objective stress-strain rate relation is defined by the convective derivative of Cauchy stress due to the reversible motion ( π˙κ ) T σ .E = DE ( ) = Λa : De , E. ρκ ρ

.

with ˙ κ = ET .De .E , De = Δ

1 ˙ −1 (E.E + E−T .E˙ T ), 2

δ , Λiajkl = Eiα Eβ Eγk Elδ Λαβγ κ ( σ˙ ) σ ˙ −1 . σ − σ .E−T .E˙ T . DE ( ) = − E.E ρ ρ ρ ρ j

.

(3.147)

(3.148)

Combining, the evolution of internal state, conservation of the momentum and boundaries conditions, the rate boundary value problem is formulated.

3.9.1 Case of Homogeneous Polycristal For a polycristal, the domain of reversibility is convex and we introduce a convex ψ A yielding function .F( κ , ). As previously the free energy by unit of mass is given ρκ ρκ by 1 .ψ(Δκ , α) = (3.149) Δκ : Λk : Δκ + H (α), 2

72

3 Elasto-plasticity

where .α are internal variables related to the hardening. The state equations are given as πκ ∂ψ A ∂ψ . . = , = (3.150) ρκ ∂Δκ ρκ ∂α When plasticity occurs, the normality rule implies the consistency condition

.

˙ ) ∂F ( ψ˙κ ) ∂F ( A = 0. + A ρκ ψ κ ρκ ∂ ∂ ρκ ρκ

F˙ =

(3.151)

Denoting the normal to the domain of reversibility N=

.

∂F ∂F , a= , A ψκ ∂ ∂ ρκ ρκ

(3.152)

the evolution of internal state is determined .

˙ −1 = μN, α˙ = μa. G p = P.P

(3.153)

The consistency condition can be written on the actual configuration taking account of convected relation .

and

σ σ d ψκ ( ) = E−1 .(DE + 2 .De ).E, dt ρκ ρ ρ

˙ −1 = E.E ˙ −1 + E.G p .E−1 , grad v = F.F .

D = De + D p , D p = (E.G p .E−1 )s .

(3.154)

(3.155)

Finally, .n = E.N.E−1 we have ˙ σ A n : (Λa : ( D − D p ) + 2 .( D − D p )) + a : = 0. ρ ρκ

.

(3.156)

Consider .H the second derivative of .ψ with respect to .α, .ns = 21 (n + nT ), the value of .μ is obtained σ < n : Λa : D + 2n : ( . D) >+ ρ .μ = (3.157) . σ n : Λa : ns + 2n : ( .ns ) + a.H.a ρ

3.9 Finite Elasto-plasticity

73

According to the conservation of momentum on the initial configuration .Co σ T Div θ˙ o = 0, θ o = ρo F−1 . , ρ

(3.158)

˙ −1 . σ + d σ , θ˙ = ρo F−1 .θ˙ , θ˙ = −F.F ρ dt ρ

(3.159)

.

we have

. o

as .

d σ σ ˙ −1 . σ − σ .(F.F ˙ −1 )T = DE ( σ ) − g p . σ − σ .g Tp , (3.160) − F.F DC ( ) = ρ dt ρ ρ ρ ρ ρ ρ

and

σ σ σ σ θ˙ = DE ( ) + . gradT v − g p . − .g Tp . ρ ρ ρ ρ

.

(3.161)

Using the constitutive law and the normality rule we have σ σ σ θ˙ = Λa : ( D − μns ) + . gradT v − μ(n. + .nT ) ρ ρ ρ . σ σ σ T = Λa : D + . grad v − μ(Λa : ns + n. + .nT ), ρ ρ ρ

(3.162)

the last term is symmetric and we have the property θ˙ =

.

with

∂U , ∂ grad v

σ 1 1 D : Λa : D + grad v. . gradT v 2 2 ρ σ < n : Λa : D + 2n : ( . D) >2+ 1 ρ − 2 n : Λ : n + 2n : ( σ .n ) + a.H.a a s s ρ

(3.163)

U= .

(3.164)

this shows that the local behaviour is hypo-elastic in sense of Hill [10, 13]. Remark: The conservation of the momentum can be rewritten as ∫ ∫ ˙ ∗ dS, 0= θ˙ : grad v∗ ρ dΩ − T.v .

Ω

∂ωT

σ σ σ θ˙ = Λa : ( D − μns ) + . gradT v − μ n. − .nT μ. ρ ρ ρ

(3.165)

74

3 Elasto-plasticity

The normality law implies the inequality: ∫ .

( ) (σ ) (μ − μ∗ ) n : Λa : ( D − ns μ) + 2n : .( D − μ ns ) + a : H : aμ ρ dΩ ≥ 0. ρ Ω (3.166) The potential: Let introduce the potential .F ∫ ( ) 1 σ 1 F (v, μ) = D : Λa : D + grad v. . gradT v ρ dΩ 2 2 ρ ∫Ω σ μ(ns : Λa : D + 2(n. ) : D)ρ dΩ − ρ Ω ∫ . 1 2 μ (a : H : aρ + ρns : Λa : ns + 2(n.σ ) : ns ) dΩ Ω 2 ∫ ˙ dS. − T.v

(3.167)

∂ωT

A solution of the rate boundary value problem satisfies the variational inequality: The boundary value problem: The solution of the rate boundary value problem satisfies the variational inequality .

∂F ∂F (v − v∗ ) + (μ − μ∗ ) ≥ 0, ∂v ∂μ

(3.168)

on the set of admissible fields .v, μ v = vd , over ∂Ωu ,

.

and

(3.169)

{ .

μ ≥ 0 F = 0, μ = 0 otherwise .

(3.170)

3.10 Stability and Bifurcation in Elastoplasticity 3.10.1 The Shanley Column This model has been used by many authors [14], especially to study plastic buckling as discussed in [15, 16]. The rigid rod model has two degrees of freedom: the downward vertical displacement .u and the rotation .θ (Fig. 3.10). The column is supported by an uniform distribution of vertical springs along the segment .[−l, l]. The behaviour of each spring is elasto-plastic with linear hardening:

3.10 Stability and Bifurcation in Elastoplasticity

75

Fig. 3.10 The Shanley column

λ θ

u

ψ(ε, α) =

.

1 1 E(ε − α)2 + H α 2 . 2 2

(3.171)

Let us consider a state for which the plastic domain . I+τ is .[d, l]. The value of .d is determined by the condition of neutral loading .[α](d, ˙ t) = 0. The equations of equilibrium are deduced from the potential energy ∫ E(u(x), α(x), T ) =

l

d

.

−l

ψ(ε, α)d x + T d (u + L(1 − θ 2 /2)) ε(x) = u − xθ, (3.172)

then the equilibrium state obeys to ∫ 0=T +

.

d

+l

−l

∫ E(ε − α)d x, 0 = −T Lθ + d

+l

−l

x E(ε − α)d x.

(3.173)

Along the loading process , the local behaviour is purely elastic with modulus . E or elastoplastic with tangent modulus . E T = E H/(E + H ). These equations are valid during the loading process, taking account of the determination of .d(t). Then, we obtain ∫ d ∫ l 0 = T˙ d + E ε˙ d x + E T ε˙ d x, −l d . (3.174) ∫ +d ∫ l ˙ ˙ x+ ˙ x. 0 = −L T d θ + E x εd E T x εd −l

d

The local constitutive behaviour follows the requirement to have uniqueness (positive hardening) when the external loading does not depend on the geometry. The introduction of non-linear geometric effects on the loading implies that the previous requirement is not sufficient to ensure uniqueness, the second derivative must be now understand to be applied on the total potential energy. A non trivial solution in .θ is obtained by introducing the time-scale .τ such that the velocity .x1 = d˙ of propagation of the unloading domain is finite. The domain τ . I+ = [x τ , l] is defined with a asymptotic expansion

76

3 Elasto-plasticity

d = xτ =

∑

.

xi τ i .

(3.175)

i

At point .xτ , the condition .[α(x ˙ τ , τ )] = 0 where .α(x, t) = tions on the asymptotic expansion:

.

∑ i

αi (x)τi gives condi-

0 = [α1 (xo )], 0 = [α2 (xo )] + x1 [α1' (xo )], 0 = [α3 (xo )] +

2x1 [α2' (xo )]

(3.176) +

(2x12

+

x2 )[α1' (xo )].

A non trivial solution is then obtained as [16, 17]: α ∗ (x, τ ) = α(τ ) + m(τ )x.

.

(3.177)

For the time-derivative of the equilibrium equations with respect of the position of x and of discontinuities (Eq. 3.176) of the mechanical quantities on this boundary, we have: ∫ l ∫ d l x+ . f (x, τ )d x = (3.178) f˙(x, τ )d x + [ f (x, τ )]xττ− x˙τ . dt −l −l

. τ

We find .Tc = TT =

H 2l 3 E = TE , .m 1 = 0 and .m 2 = −T2 /2Hl 2 , 3L E+H 4l 2 T 2 , 3 TE − TT τ2 T = TT + T2 , 2 τ2 ElT2 + ... θ= 3L H (TE − TT ) 2

x12 = .

(3.179)

This is a bifurcated path. The condition of stability of the fundamental path.(θ = 0, T ) is preserved for loading near .T = Tc but for .T ≥ Tc another path exists which is also a stable path. This is quite different of conservative system, for which the bifurcation point correspond to a loss of stability of the fundamental path. More applications can be found in many papers for elasto-plasticity [18] with implications on the constitutive laws.

3.10.2 A Model of Elastoplastic Beam In plane motion, a beam of length. L with thickness.2l is described by the displacement of its mean line. The free energy of element of the beam is given as

3.10 Stability and Bifurcation in Elastoplasticity

ψ(ε, α) =

.

where

77

1 1 E(ε − α)2 + hα 2 , 2 2

1 ε(x, z) = ε x x = u ' − zv '' + (v ' )2 , 2

.

(3.180)

(3.181)

and .α(x, z) a field of scalar which defines the internal state. Then the total potential energy under compressive axial loading .λ is ∫ E(u, v, α) =

L

∫

l

.

0

−l

ψ dx + λ u(L).

(3.182)

The displacement is in the set of admissible displacement K.A. = {(u, v)|u(0) = v(0) = 0, v(L) = 0}.

.

(3.183)

The equilibrium state satisfies

.

∂E δu = ∂u ∂E δv = ∂v

∫ ∫

L 0

∫

l

−l L∫ l

0

−l

(ε − α)Eδu ' dx − λδu(L) = 0, (3.184) ''

'

'

(ε − α)E(zδv + v δv ) dx = 0.

And the driving force associated to .α A = E(ε − α) − hα ∈ C = {A|F(A) = ||A|| − k ≤ 0}.

.

(3.185)

The fundamental path is given as for increasing .λ: 1 − < λ − 2kl >+ , 2hl v(x) = 0, λ u(x) = (αo − )x. 2El

αo (x, z) = .

The critical state is given at .λc = λT = bifurcation is given by α I = z sin

.

(3.186)

h 2l 3 E. At this point the mode of E + h 3L

πx πx , u I = 0, v I = sin , L L

(3.187)

78

and

3 Elasto-plasticity

ξ5 ... 5! πx ξ5 . α = α +α z sin + ... o 5 5! L λ5 m5 = 2 . 2l h λ = λc + λ5

(3.188)

The parameter .ξ is related to the propagation of the unloading plastic zone defined by the line ∑ ξi l 2/5 2 .z = ci (x) = (3.189) π x (1 − Bλ ξ + .. i! sin i L The constant . B can be evaluated explicitly by the condition α(x, ˙

∑

.

i

ci

ξi ) = 0. i!

(3.190)

Developing the solution up to order ten in .ξ , a second mode of bifurcation appears as defined by Hutchinson [19], corresponding to a discontinuous solution in .u, v at this order. The effective calculation demand to derive integral on a domain which varies during the loading . Iξ = {(x, z)|F(A) ≤ 0}, (3.191) taking account of possible discontinuities on the moving interface .∂ Iξ defined by Eq. 3.189. This example shows the applicability of the proposed systematic development.

References 1. H.D. Bui, Evolution de la frontiére du domaine élastique des métaux avec l’écrouissage plastique et le comportement élastoplastique d’un agrégat de cristaux cubiques, vol. 1. Mémorial de l’Artillerie Française (1970) 2. R. Hill, A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phys. Solids 6, 336–349 (1958) 3. B. Halphen, Q.S. Nguyen, Sur les matériaux standard généralisés. Journal de Mécanique 14(1), 254–259 (1975) 4. M. Peigney, C. Stolz, Approche par contrôle optimal des structures élastoviscoplastiques sous chargement cyclique. C. R. Mécanique 329, 643–648 (2001) 5. M. Peigney, C. Stolz, An optimal control approach to the analysis of inelastic structures under cyclic loading. J. Mech. Phys. Solids 51, 575–605 (2003) 6. J. Mandel, Plasticité, et viscoplasticité, classique, Cours CISM Udine, vol. 97 (Springer, New York, 1971)

References

79

7. E.H. Lee, Elastic plastic deformation at finite strain. J. Appl. Mech. 36, 276–279 (1971) 8. J. Mandel, Sur la définition de la vitesse de déformation élastique et sa relation avec la vitesse de contrainte. Int. J. Solids Struct. 17, 873–878 (1981) 9. F. Sidoroff, The geometrical concept of intermediate configuration and elastic plastic finite strain. Arch. Mech. 25(2), 299–308 (1973) 10. C. Stolz, Large plastic deformation of polycrystals, in Large Plastic Deformation of Crystalline aggregates, CISM Courses and Lectures, vol. 376, pp. 81–108 (Springer, 1997) 11. R. Asaro, Micro and macromechanics of crystalline plasticity, in Plasticity of Metals at Finite Strain (Lee - Mallet, 1983) 12. R. Hill, J.R. Rice, Constitution analysis of elastic plastic crystals at finite strain. J. Mech. Phys. Solids 120, 401–413 (1972) 13. B. Halphen, Sur le champ des vitesses en thermoplasticité finie. Int. J. Solids Struct. 11, 947–960 (1975) 14. M.J. Sewell, On the branching of equilibrium paths. Proc. Roy. Soc. London A375, 499–518 (1970) 15. J.W. Hutchinson, Singular behaviour at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids 16, 13–31 (1968) 16. Q.S. Nguyen. Stabilité et mécanique non-linéaire (Hermès, 2000) 17. C. Stolz, Analysis of stability and bifurcation in nonlinear mechanics with dissipation. EntropyMDPI 13(2), 332–366 (2011) 18. H. Petryck, Theory of bifurcation and instability in time-independent plasticity, in Bifurcation and Stability of Dissipative Systems, CISM Courses, pp. 95–152 (Springer Verlag Wien, 1991) 19. J. Hutchinson, Plastic buckling. Adv. Appl. Mech. 14, 67–114 (1974)

Chapter 4

Fracture Mechanics

4.1 Introduction Consider a body .Ω with a crack, represented by a straight line. Around the crack tip we distinguish three domains determined by the distance from the tip (Fig. 4.1). • In zone I, the nearest zone is the domain where all physical processes of rupture occur, that is the process zone. • In zone II, the mechanical fields are represented by singularities outside the process zone. • In zone III, the mechanical fields satisfy all matching conditions with given conditions at infinity. The crack is represented at macroscopic scale by a line oriented as .ex . The normal to the line is .e y in the plane and .ez is normal to the plane. If the singularities of mechanical fields govern the propagation of the crack, it is not necessary to take into account the process of rupture in consideration. This is an approximation which leads to a global approach of rupture, this description is powerful and constitutes the key point for describing classical fracture mechanics. In this case, the singularities characterize the loading applied on the process zone.

4.2 Case of Linear Elasticity The global approach of rupture is based on the study of singularities of the mechanical fields (Fig. 4.2). The asymptotic stress fields near a crack tip in a linearly elastic material are given by solving a classical homogeneous boundary value problem with vanishing loading condition at infinity. The local displacement .u is the solution of the set of local equations given by: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 C. Stolz, Introduction to Non-linear Mechanics, Springer Series in Solid and Structural Mechanics 14, https://doi.org/10.1007/978-3-031-51920-8_4

81

82

4 Fracture Mechanics

Fig. 4.1 The description of the behaviour near the crack tip

ey III

II M(r,θ ) I

θ ex

Fig. 4.2 The geometry to study the singularities in a corner

φ r

θ

• • • •

the constitutive law: .σ = C : ε over .Ω, the compatibility: .2ε = grad u + gradT u over .Ω, the equilibrium, with null body forces: .div σ = 0 over .Ω the boundary conditions: fixed displacement .u = 0 or traction-free conditions on crack faces .σ .n = 0, or mixed conditions, fixed displacement on one face and free-traction on the other face. • conditions at infinity: .σ → 0 as r → ∞. The solution of this problem is given by a displacement of the form: u = r α f(θ ).

.

(4.1)

In general, the power .α is a complex function which depends on the local behaviour. For isotropic linear elastic homogeneous body, the value depends on the Poisson’s ratio .ν, on the angle .φ and of the boundary conditions (B-C) along the crack surface: α = α(ν, B − C, φ),

.

(4.2)

4.2 Case of Linear Elasticity

83

but for traction-free boundary conditions along a line crack, .φ = π , we obtain: α=

.

1 . 2

(4.3)

Associated with this power, we can distinguish three eigenfunctions for the problem of singularities, which corresponds to three modes for opening the crack: • mode I: • mode II:

.

[ u ]Γ .e y ≥ 0,

(4.4)

[ u ]Γ .ex /= 0,

(4.5)

[ u ]Γ .ez /= 0.

(4.6)

.

• mode III:

.

then with these properties the stresses have singularities: σ = K r − 2 f (θ ), 1

(4.7)

.

where . K denotes the stress intensity factors (. K I , K I I , K I I I ) and the eigenfunctions f are known functions for mode I and II:

.

σ

. 11

σ12 σ22

θ 3θ KI I θ 3θ KI θ θ = √ cos (1 − sin sin ) − √ sin (2 + cos cos ), 2 2 2 2 2 2 2πr 2πr θ 3θ KI I θ 3θ KI θ θ −√ = √ cos sin cos cos (1 − sin sin ), 2 2 2 2 2 2 2πr 2πr θ 3θ KI I θ 3θ KI θ θ = √ cos (1 + sin sin ) − √ sin cos cos . 2 2 2 2 2 2 2πr 2πr [ u ]Γ .e y =

.

K I (η + 1) μ

/

r K I I (η + 1) , [ u ]Γ .ex = 2π μ

/

r , 2π

where .η = 3 − 4ν in plane strain. The antiplane solution (mode III) satisfies: KI I I θ = −√ sin , 2 2πr KI I I θ σ23 = √ cos , 2 2πr / r 4K I I I [ u ]Γ .ez = . μ 2π σ

. 13

The mechanical fields being determined, we study now the propagation of the crack. We assume that the propagation is rectilinear (Fig. 4.3).

84

4 Fracture Mechanics

4.3 Crack Propagation in Plane Conditions The essential difficulty of the problem of propagation is the dependence of .Ω on the crack length and the presence of moving singularities accompanying the crack. One possibility has been investigated by [1]; by introducing a geometrical lagrangean description. We propose to based our description upon the concept of singularity transport [2–4] (Fig. 4.4). Inside a moving frame following the motion of the crack tip, the nature of the singularity is preserved. The crack singularity is surrounded by a curve .Γ delimiting a domain .VΓ . This domain moves with the tip of the crack, position of which is given by function .l(t). All mechanical quantities are expressed in terms of the classical fixed coordinates outside .VΓ and in terms of moving coordinates inside .VΓ . .

x = X − l(t),

y = Y.

(4.8) ◦

Any mechanical quantity . F possesses time derivative given by . f , which represents the variation of . f in the moving frame: .

◦ ∂f ∂F = −a + f. F(X, Y, t) = f (x, y, t), a = l˙ f˙ = ∂t ∂X

Fig. 4.3 The boundary value problem

(4.9)

d

T

l

u Fig. 4.4 Decomposition of in .ΩΓ ∪ VΓ

.Ω

d

y

x Γ

4.3 Crack Propagation in Plane Conditions

85

Applications of these definitions to average quantities on the whole domain are used to separate the contribution of the crack tip in the expression of the dissipation: ∫

∫ ∫ fρ dΩ = fρ dΩ + fρ dΩ, Ω ΩΓ VΓ ∫ ◦ ∫ d d fρ dΩ) + f ρ dΩ. F = ( dt dt ΩΓ VΓ ∫ ∫ fρ dΩ) = fρa n x dS. f˙ρ dΩ − F =

.

d ( dt

∫ ΩΓ

ΩΓ

(4.10)

Γ

Let us introduce the notations : . f x = f.ex , ∇x f = ∇ f.ex . Dissipation The dissipation of the whole system can be rewritten as d . D m = Pe − dt

∫ Ω

ψρ dΩ ≥ 0,

(4.11)

where the power of external forces is given in term of local stresses, taking into account of the conservation of the momentum: ∫ .Pe = n.σ .v dS. (4.12) ∂Ω

This quantity is decomposed in two terms using of divergence theorem: ∫

∫ .

∂Ω

n.σ .v dS =

∫ ΩΓ

σ : ε(v) dΩ +

Γ

n.σ .v dS.

(4.13)

Using now the decomposition of the overall volume (.Ω = ΩΓ ∪ VΓ ) we obtain first .

∫

d dt

Ω

ψρ dΩ =

d ( dt

∫

∫

ΩΓ

◦

ψρ dΩ) +

ψρ dΩ,

(4.14)

ρψn x dS a ,

(4.15)

VΓ

and by application of general relations: ∫

∫

d . dt

ΩΓ

∫

ψρ dΩ = ◦

ψρ dΩ =

.

VΓ

∫

ΩΓ

˙ dΩ − ψρ ◦

Γ

n.σ .v dS,

∫ Γ

(4.16)

where we have taken account of the traction-free boundary condition (.σ .n = 0) along the crack lips. The dissipation is rewritten finally as

86

4 Fracture Mechanics

∫ .

Dm =

ΩΓ

˙ dΩ (σ : ε(v) − ρ ψ)

∫ +

◦

Γ

(n.σ .(v − v) + ρψn.ex a) dS ≥ 0.

(4.17)

The displacement .u is continuous along the curve .Γ. The condition of compatibility implies Hadamard relations on the rates: ◦

[ u ]Γ = 0 ⇒ v = v − ∇ x u a .

.

(4.18)

Then the dissipation is decomposed in two terms : a volume part due to irreversibility of the body and a surface term associated with the propagation of the crack: ∫ .

Dm =

ΩΓ

˙ dΩ (σ : ε(v) − ρ ψ)

∫ +

Γ

(−n.σ .∇x u + ρψn x ) dS a ≥ 0.

(4.19)

When the constitutive law is linearly elastic, the local behaviour is reversible and ∂ψ .σ = ρ ; there is no dissipation in the volume: ∂ε ∫ ˙ dΩ = 0. . (σ : ε(v) − ρ ψ) (4.20) ΩΓ

When .Γ is reduced to the crack tip, the result is conserved: ∫ .

lim

Γ→0 Ω Γ

˙ dΩ = 0. (σ : ε(v) − ρ ψ)

(4.21)

The global dissipation contains only the contribution of the crack: ∫ .

Dm = lim

Γ→0 Γ

(−n.σ .∇x u + ρψn x ) dS a .

(4.22)

The driving force associated with the propagation is the free energy release rate .G defined by: ∫ G = lim

.

Γ→0 Γ

(−n.σ .∇x u + ρψn x ) dS.

(4.23)

This is the driving force associated to a singularity as proposed by Eshelby [5] and Rice [6].

4.5 Invariance and J-integral

87

4.4 Energetical Interpretation The total potential energy for the system is: ∫ E(u, l, Td ) =

∫

.

Ω

ρ ψ(ε(u)) dΩ −

∂ΩT

Td .u dS,

(4.24)

and the dissipation is rewritten as d (E + . D m = Pe − dt

∫ ∂ΩT

Td .u dS) ≥ 0.

(4.25)

Taking into account of the derivation of the potential energy with respect to it’s arguments, we obtain: .

As .

∂E ∂E ˙ ∂E ˜˙ d d E= .˜v + .T . l+ dt ∂ u˜ ∂l ∂ T˜ d

∂E .˜v = ∂ u˜

∫

∫ ∂Ω

n.σ .v −

then .

(4.26)

Dm = −

∂ΩT

Td .v dS,

∂E a ≥ 0. ∂l

(4.27)

(4.28)

Release rate of energy: For the overall system, the only internal parameter is the length of the crack. The driving force associated with the propagation is the release rate of energy .G obtained by the global state equation: G=−

.

∂E . ∂l

(4.29)

Remark: We must emphasise that this result is due to: • the homogeneity of the constitutive material • the absence of discontinuity of velocity inside .VΓ • some hypothesis due to steady-state and auto-similarity of the processes which are ◦

related to the regularity of . f .

4.5 Invariance and J-integral Consider now . S a closed loop inside a domain .Ω, over which the body forces are null. For an homogeneous linearly elastic material, the density .ρ is uniform. The stresses satisfy both the equation of state and the conservation of the momentum:

88

4 Fracture Mechanics

σ =ρ

.

∂ψ , div σ = 0 over Ω. ∂ε

(4.30)

ρψ n k − σi j u i,k n j dS,

(4.31)

Consider the integral .C: ∫ C=

.

S

then by divergence theorem the integral is equal to ∫ .

ΩS

∫

∂(ρψ) ∂(σi j u i,k ) − dΩ = ∂ xk ∂x j

ΩS

(σi j εi j,k − σ i j u i, jk −

∂σi j u i,k ) dΩ. ∂x j

(4.32)

Using now the conservation of momentum, the integral .C is null. This is the expression of the law of conservation of energy. From this result, it follows that the integral . JΓ : ∫ J =

. Γ

Γ

ρψ n k − σi j u i,k n j dS,

(4.33)

is independent of the choice of the loop .Γ, taking into account of the free-traction condition along the crack lips. Therefore J = lim JΓ = G.

. Γ

Γ→0

(4.34)

This is the interpretation and the invariance condition as presented by Rice [7]. Invariance in non linear elasticity Consider a closed loop . S inside a body .Ω, and a non linear elastic behaviour defined by a free energy function of the tensor .C. The nominal stresses .θ satisfy the equation of state and the conservation of momentum: θ =ρ

.

∂ψ , Div θ T = 0, over Ω. ∂F

(4.35)

By the same reasoning we deduce that: ∫ 0=

ρψn k − θi j u i,k n j dS, ∫S

.

0=

ΩS

∂ ∂ (ρψ) − (θi j u i,k ) dΩ = ∂ Xk ∂Xj

∫

(4.36) ΩS

(θi j Fi j,k − θi j u i, jk ) dΩ.

The invariance is obtained with the same properties than in small perturbations. Other integrals are also invariants with energetic interpretation [8].

4.6 Dual Approach in Linear Elasticity

89

4.6 Dual Approach in Linear Elasticity In the dual approach, the complementary energy is function of stresses E∗ (σ , l) = −

∫

.

Ω(l)

ψ ∗ (σ ) dΩ +

∫ ∂Ωu

n.σ .ud dS,

(4.37)

where .σ is statically admissible: σ ∈ S.A = {σ ∗ | div σ ∗ = 0, n.σ ∗ = Td over ∂ΩT }.

.

(4.38)

Introduce .ψ ∗ the dual Eshelby tensor of .ψ, these tensors are divergence free ψ ∗ = −ψ ∗ I + ∇σ .u, ∫ .

∫

div ψ T = 0, div ψ ∗T = 0,

∫

div ψ T dΩ = n.ψ dS = 0, C ∫ div ψ ∗T dΩ = n.ψ ∗ dS = 0.

(4.39)

ΩC

ΩC

C

For the state of equilibrium, the strain energy and the complementary energy are equal and 1 ψ(ε) = ψ ∗ (σ ) = σ : ε, 2 . (4.40) ψ − ψ ∗ = σ : ε I − (σ .∇u + ∇σ .u), ψ + ψ ∗ = ∇σ .u − σ .∇u. Lemma: For any stresses .σ˜ ∈ S.A and any fields .ε(˜u) where .u˜ ∈ K.A and for all closed loop .C ∫ σ˜ : ε(˜u)n i − n.∇i σ˜ .u − n.σ˜ .∇i u˜ dS = 0.

.

(4.41)

C

On the cracks lips co-linear to .e1 , .n.ψ.e1 = n.ψ ∗ .e1 . Then we know that . JΓ is invariant with respect to the contour .Γ ∫ .

J = JΓ =

Γ

n.ψ.e1 dS.

(4.42)

Taking the properties of Eshelby’s tensors and the lemma into account ∫ I+J= .

I−J=

∫Γ Γ

n.∇1 σ .u − n.σ .∇1 u dS, (4.43) ∗

(ψ + ψ )n 1 − n.∇1 σ .u − n.σ .∇1 u dS = 0,

90

4 Fracture Mechanics

Then we recovered the dual expression of dual integral of Bui H.D. [9] and the particular expression .

1 2

I =J=

∫ Γ

n.∇1 σ .u − n.σ .∇1 u dS.

(4.44)

4.7 On the Rate Boundary Value Problem To solve the problem of evolution of a crack inside a body .Ω a law of propagation is needed. Griffith’s law We consider a propagation law of Griffith’s form. Denoting .a = l˙ the rate of the crack length { G < G c , a = 0, . (4.45) G = G c , a ≥ 0. The evolution satisfies the property: G ≤ G c , a ≥ 0, a (G − G c ) = 0,

.

(4.46)

and by derivation with respect to time we have: .

d ˙ = 0. (a (G − G c )) = 0 ⇒ a G dt

(4.47)

At an equilibrium state such that .G = G c , the propagation satisfies the inequality: ˙ ≥ 0. ∀β ≥ 0, (a − β)G

.

(4.48)

This formulation characterizes the evolution of the crack. The equations of the rate boundary value problem The solution of the rate boundary value problem in terms of rate of displacement and rate of propagation satisfies the local set of equations: • conservation of momentum: .

0 = div σ˙ over Ω, σ˙ .n = T˙ d along ∂ΩT

• compatibility conditions: .ε˙ = ε(v) over Ω, v = vd along ∂Ωu ,

4.7 On the Rate Boundary Value Problem

• the constitutive law: .σ˙ = • the propagation law.

91

∂ 2ψ : ε˙ = C : ε˙ , over .Ω ∂ε∂ε

Choice of a representation Introducing now the decomposition of the domain.Ω = ΩΓ ∪ VΓ to take the presence of the singularity into account and the rate . fˆ of any field . f : { x ∈ ΩΓ , fˆ = f˙,

(4.49)

◦

.

x ∈ VΓ , fˆ = f . ◦

The domain .VΓ is moving with the velocity .a and . f is the rate of . f in this moving frame ◦ . f = Da f = f˙ + a ∇ f.e1 = f˙ + a ∇1 f (4.50) The field . fˆ is then discontinuous along .Γ: 0 = [ fˆ ]Γ + a ∇1 f

(4.51)

.

◦

where .[ fˆ ]Γ = f˙ − f . Therefore for the mechanical fields we obtain: .

{ x ∈ ΩΓ ,

uˆ = v, σˆ = σ˙ ,

x ∈ VΓ ,

uˆ = u; σˆ = σ .

and along .Γ .

◦

◦

0 = [ uˆ ]Γ + a ∇1 u, 0 = n.[ σˆ ]Γ + a n.∇1 σ .

(4.52)

(4.53)

˙ of the energy release rate To take account of the propagation law, the evolution .G must be evaluated. ˙ Expression of .G ˙ is given by path independent integrals The .G .

˙ = G

∫

◦

Γ

◦

n.∇1 σ .u − n.σ .∇1 u dS.

(4.54)

The proof is obtained by time differentiation of .G in the moving frame which first gives the expression:

92

4 Fracture Mechanics

∫

˙ = G

◦

.

Γ

◦

◦

σ : εn 1 − n.σ .∇1 v − n.σ .∇1 u dS.

(4.55)

and leads immediately to the result, taking (Eq. 4.41) into account: ∫

◦

.

Γ

◦

◦

(σ : ε) n 1 − n.σ .∇1 u − n.∇1 σ .u dS = 0.

(4.56)

This path invariance is naturally deduced from the definition of .G. A direct proof is given using the Green’s formula on the fields: ∇1 σ = C : ∇1 ε, div ∇1 σ = 0, ◦

.

◦

◦

σ = C : ε, div σ = 0,

(4.57)

then ∫

◦

◦

◦

◦

div(∇1 σ .u) + ∇1 σ : ε − σ : ∇1 ε − div(σ .∇1 u) dS = 0,

.

(4.58)

VS

which ensures that for a closed loop . S: ∫

◦

◦

n.σ .∇1 u − n.σ .∇1 u dS = 0.

.

(4.59)

S

Global formulation of the rate boundary value problem The necessity to take into account the singularities leads to the formulation of the rate boundary value problem in terms of rate fields .vˆ , σˆ . which satisfy the set of equations: • the conservation of the momentum: .

div σˆ = 0 over Ω, σˆ .n = T˙ d along ∂ΩT , [ σˆ ]Γ .n + a ∇1 σ .n = 0, along Γ,

(4.60)

• the compatibility: .

εˆ = ε(ˆu) over Ω, uˆ = vd along ∂Ωu , 0 = [ uˆ ]Γ + a ∇1 u, along Γ,

(4.61)

• the constitutive law: .σˆ = C : εˆ , over Ω, • the propagation law for the crack: .

K = {α|α = 0, G ≤ G c , α ≥ 0, G = G c } , ˙ ≥ 0. a ∈ K, ∀β ∈ K, (a − β)G

(4.62)

4.7 On the Rate Boundary Value Problem

93

If the propagation is known, the velocity .v = u˙ is solution of a non classical problem of linear elasticity. Indeed, .vˆ is not continuous on .Γ and surface densities (.a ∇i σ .n) are applied along .Γ. The rate boundary value problem: The solution .(ˆu, a) satisfies .

∂F ∂F · (ˆu − v˜ ) + (a − β) ≥ 0, ∂ uˆ ∂a

(4.63)

among the set of kinematically admissible fields { } K.A = (ˆu, β)/ˆu = vd over ∂Ωu , β ∈ K, [ uˆ ]Γ + β∇1 u = 0, along Γ . (4.64)

.

Where .F is the functional ∫

∫ 1 ◦ .F (ˆ u, a) = εˆ : C : εˆ dΩ − a n.∇1 σ .v dS 2 Ω Γ ∫ ∫ 1 2 T˙ d .v dS. + a n.∇1 σ .∇1 u dS − 2 Γ ∂ΩT

(4.65)

Elements of Proof The proof follows immediately from the properties of the fields. The variations of the functional .F are given by ∫

∫ ∫ 1 ◦ ◦ δF = δ εˆ : C : εˆ dΩ − δa n.∇1 σ .u dS − a n.∇1 σ .δ u dS 2 Ω Γ Γ ∫ ∫ . d T˙ .δv dS, n.∇1 σ .∇1 u dS − + a δa Γ

∂ΩT

where ∫ δ

.

Ω

1 εˆ : C : εˆ dΩ = δ 2

∫ ΩΓ

1 ε˙ : C : ε˙ dΩ + δ 2

∫ VΓ

1◦ ◦ ε : C : ε dΩ, 2

and then ∫ ∫ ( ∫ ) 1 ◦ ◦ .δ εˆ : C : εˆ dΩ − − n.σ˙ .δv + n.σ .δ u dS. T˙ d .δv dS = Ω 2 ∂ΩT Γ Taking the compatibility of the fields into account ◦

δv = δ u − δa ∇1 u,

.

(4.66)

94

4 Fracture Mechanics

the last term is rewritten as ∫ ( ) ◦ ◦ − n.(σ˙ − σ ).δ u + δa n.σ˙ .∇1 u dS. .

(4.67)

Γ

Introducing all the terms in the functional, we obtain: ∫

◦

∫

◦

◦

n.(−σ˙ + σ ).δ u dS −

δF =

.

a n.∇1 σ .δ u dS + Γ ∫Γ ∫ ◦ − δa n.∇1 σ .u dS + a δa n.∇1 σ .∇1 u dS, Γ

∫ Γ

δa n.σ˙ .∇1 u dS

Γ

and for any variation we have finally: ∫ ( ) ◦ ◦ n.(−σ˙ + σ ) − a n.∇1 σ .δ u dS ∫Γ ( ) ◦ + δa (n.σ˙ + a n.∇1 σ ).∇1 u − n.∇1 σ .u dS.

δF = .

Γ

◦

As .(n.σ˙ + a n.∇x σ ).∇1 u = n.σ .∇1 u, these terms correspond respectively to the jump of the stress vector and to the propagation law.

4.8 Interaction of Cracks For a body with many crack, we can apply the same reasoning for each tip of crack, (i = 1, . . . , n), with direction .ei .

.

The rate boundary value problem for many cracks: The solution .(ˆu, a i ) satisfies .

∑ ∂F ∂F · (ˆu − v˜ ) + (a i − βi ) ≥ 0, ∂ uˆ ∂a i i

among the set of admissible fields .K.A and .F is the functional ∫

∑∫ 1 ◦ εˆ : C : εˆ dΩ − a i n.∇i σ .u dS Ω 2 Γ i ∫ ∫ ∑1 2 ai + n.∇i σ .∇i u dS − T˙ d .v dS 2 Γ ∂Ω T i

F (ˆu, a i ) =

.

} { K.A = (ˆu, β)/ˆu = vd over ∂Ωu , β ∈ K, [ uˆ ]Γ + βi ∇i u = 0 along Γi

.

{ } K = β/βi ≥ 0, if Gi = G c βi = 0 otherwise

.

4.8 Interaction of Cracks

95

It is easy to prove in a analogous manner then previously that a solution of the inequality satisfies the set of the classical local equations given below: • the compatibility of the velocity: 1 (∇ uˆ + ∇ T uˆ ), over Ω 2 uˆ = vd , along ∂Ωu ,

ε(ˆu) = .

0 = [ uˆ ]Γ + a i ∇i u, on each Γi • the conservation of the momentum: div σˆ = 0, over Ω, . σ ˆ .n = T˙ d , along ∂ΩT , 0 =n.[ σˆ ]Γ + a i n.∇i σ , on each Γi • the constitutive law: .σˆ = C : ε(ˆu) over .Ω, • the propagation law: K = {β/βi ≥ 0, if Gi = G c , βi = 0 otherwise.} ,

.

∀β ∈ K,

.

∑ ˙ i ≥ 0. (a i − βi ) G

(4.68)

(4.69)

i

These equations are classical equations of an elasticity problem with non classical boundary conditions along each .Γi . The solution .uˆ is a linear function of the given value prescribed over .∂Ωu and of each .a i . ∫ uˆ (x) =

∫ A(x, S).v (S) dS + d

.

∂Ωu

L(x, S).T (S) dS + d

∂ΩT

∑∫ i

Ω

V i (x)a i .

(4.70) By substitution of .uˆ in terms of the propagation of cracks we obtain a reduced functional .F ∗ (a i ). 1 ∗ ¯ . a .B.a − Q.a (4.71) .F (a i ) = 2 The existence of a solution is ensured by the positivity condition: β.B.β > 0, ∀β /= 0 ∈ K,

.

where K = {β/βi = 0, Gi < G c , βi ≥ 0, if Gi = G c }.

.

(4.72)

96

4 Fracture Mechanics

and the condition of uniqueness is given by .

β.B.β ≥ 0, ∀β ∈ {β/βi = 0,Gi < G c , and βi anything otherwise},

The global formulation of the rate boundary value problem give us criterion for study the stability and the uniqueness of crack growth.

4.9 Stability and Uniqueness: A Simple Example Let us consider a straight beam under bending with fixed extremities .l1 , l2 , where the beam is clamped. The length of the beam is .l1 + l2 , the displacement is .v(x) defined on segment.[−l1 , l2 ]. The strain is defined by the vertical displacement.v :.ε = v"(x)y. We applied a load at the origin, and we study the possibility of decohesion at points .l 1 , l 2 . We study two cases, first the applied load is a vertical displacement .v(o) = V ; second the load is controlled at the origin .T (o) = F. For the first case the potential energy at the equilibrium is .

We define . Ji = −

W (l1 , l2 , V ) =

(l1 + l2 )3 1 3 = kV 2 EIV2 3 3 2 2 l1 l2

∂W ∂li J =3

. 1

(4.73)

(l1 + l2 )2 l2 l13l23 l1

(4.74)

and . J2 is obtained permuting indices. The evolution of the delamination is given by the normality law .ai ≥ 0, Ji ≤ G c , (Ji − G c )ai = 0 (4.75) and existence an uniqueness is given with respect to positivity or not of . Q such that .

Q i j = −Ji j =

l1 + l2 ∂2W = 18E I V 2 5 5 ∂li ∂l j l1 l2

[

l12 l22 l23 (l1 + 2l2 ) 2 2 3 l1 l2 l1 (2l1 + l2 )

] (4.76)

For .l1 = l2 , . Q is always positive definite, the position is then always stable and we have no bifurcation. When the force is controlled .

W (l1 , l2 , F) = −

F2 (l1 + l2 )3 , k = 3E I k l13l23

(4.77)

4.10 Case of Hyperelasticity

97

The associated . Q matrix becomes 2F 2 l1l2 .Q = − E I (l1 + l2 )2

[

l23 (l2 − l1 ) 2l12 l22 2l12 l22 l13 (l1 − l2 )

] (4.78)

is always negative definite. The symmetric equilibrium is always an unstable state with possible bifurcation, the eigenvalue of . Q having opposite signs. For multi-cracking of a body, the rate boundary value problem is formulated and condition of existence and uniqueness have been deduced [10, 11].

4.10 Case of Hyperelasticity The case of propagation of cracks in non linear mechanics is more complex and only few results exists for specific classes of behaviour. The case of propagation of crack under anti-plane shear have been studied by some authors for the class of Knowles and Sternberg materials [12, 13] ψ(I1 ) =

.

) b μ( (1 + (I1 − 3))n − 1 , 2b n

I1 = Tr(FT .F).

(4.79)

The shear curve is plotted on (Fig. 4.5). Due to the presence of non linearities the equations of motion have particular properties, which are summarized as follow: • .n > 21 : Equations of motion are always elliptic, this ensures the presence of singularities near the crack tip. • .n < 21 : Equations of motion are elliptic before the maximum and are hyperbolic after. Near the crack tip this induces jumps of gradient of displacement. In the first case, the previous analysis is conserved, the existence of .G is related to the singularities. In the other cases the existence of jump of the gradient of displacement determines a shock curve along which the dissipation is distributed, (Fig. 4.6).

Fig. 4.5 The Knowles and Sternberg behaviour class

τ n=1 n=.5

02+ ∂U ∗ 1 1 , U ∗ (σ˙ ) = σ˙ .C−1 : σ˙ − ∂ σ˙ 2 2 N T .L T .C−1 .L .N − N T .Z .N

where .< f >+ = 21 ( f + | f |). Remark We take the derivatives . fˆ as variables. This field presents jumps along .Γ [ εˆ ]Γ + a [ u,1 ]Γ = 0, [ σˆ ]Γ .n + a n.[ σ ,1 ]Γ = 0.

.

(4.112)

In .ΩΓ , the local behaviour is given by ∂U ∗ (σˆ − a σ ,1 ), ∂ σ˙ ∂U (ˆε − a ε,1 ). = ∂ ε˙

εˆ − a ε ,1 = .

σˆ − a σ ,1 The primal functional

We introduce now the overall functional .F associated to potential .U ∫ F = ∫

.

∫ Ω/ΩΓ

U(ˆε ) dΩ +

ΩΓ

U(ˆε − aε ,1 ) dΩ

1 (a σ ,1 : εˆ − a 2 σ ,1 : ε ,1 ) dΩ 2 Ω ∫ Γ ∫ 1 + (−a n.σ ,1 .ˆu− + a 2 n.σ ,1 .u,1 ) dS − T˙ d .ˆu dS. 2 ∂ΩT Γ +

(4.113)

104

4 Fracture Mechanics

The rate boundary value problem: The solution of the evolution problem .(ˆu, a) satisfies the variational inequalities .

∂F ∂F ˜ ≤ 0. (ˆu − v˜ ) + (a − l) ∂ uˆ ∂a

(4.114)

among the set of admissible fields

.

˜ | [ v˜ ]Γ + l.u ˜ ,1 = 0, along Γ, l˜ ∈ K, v˜ = u˙ d , over ∂Ωu }, KA = {(˜v, l) K = {β|β = 0, G − G c < 0, β ≥ 0, G − G c = 0}.

(4.115)

Elements of proof The variations of .F are given by: ∫

∫ δF = ∫

Ω/ΩΓ

∫

ΩΓ

∫

Γ

+ .

+ −

σˆ : δ εˆ dΩ +

ΩΓ

(σˆ − a σ ,1 ) : (δ εˆ − δlε ,1 ) dΩ

(δa σ ,1 : εˆ + a σ ,1 δ εˆ − a δa σ ,1 : ε ,1 dΩ

(−δa n.σ ,1 .ˆu− − a n.σ ,1 .δ uˆ − + a δa n.σ ,1 .u,1 dS

∂ΩT

T˙ d .δ uˆ dS

After integration by part, we obtain ∫ δF = ∫ .

+

Ω

ΩΓ

∫ +

∫

Γ

− div σˆ .δ εˆ dΩ +

∫ Γ

n.[ σˆ ]Γ .δ uˆ dS +

∂ΩT

(σ˙ .n − T˙ d ).δv dS

δa (σ ,1 εˆ − σˆ : ε ,1 ) dΩ

(4.116)

(−δa n.σ ,1 .ˆu− − a n.σ ,1 .δ uˆ − + a δa n.σ ,1 .u,1 ) dS.

The conservation of momentum is recovered : .div σˆ = 0 over .Ω and .σ˙ .n = T˙ d over .∂ΩT . Using the jump conditions .[ δ uˆ ]Γ + δa u,1 = 0, finally the variations are reduced to ∫ δF = n.([ σˆ ]Γ + a σ ,1 ).δ uˆ − dS Γ ] [∫ ∫ . − − σ ,1 : εˆ − σˆ : ε ,1 dΩ + n.σˆ .u,1 − n.σ ,1 .ˆu dS , + δa ΩΓ

Γ

4.12 On Inhomogeneous Body

105

and the jump conditions for the stresses and the law of propagation by application of the inequality are recovered. Dual functional We introduce the functional .F ∗ associated with the potential .U ∗ : F∗ =

∫ ∫

.

Ω/ΩΓ

−U ∗ (σˆ ) dΩ −

∫ ΩΓ

U ∗ (σˆ − a σ ,1 ) dΩ

l˙2 (−a σˆ : ε ,1 + σ ,1 : ε ,1 ) dΩ + 2 ΩΓ ∫ ∫ 2 ˙ l + (a n.σˆ − .u,1 − n.σ ,1 .u,1 ) ds + n.σˆ .˙ud ds 2 Γ ∂Ωu

F ∗ is defined among the set of admissible fields .K ∗ :

.

˜ div σ˜ = 0, over Ω, σ˜ .n = T˙ d along ∂ΩT , K ∗ = {(σ˜ , l)/ ˜ ,1 = 0, along Γ l˜ ≥ 0} n.[ σ˜ ]Γ + ln.σ (4.117)

.

Dual formulation: A solution of the evolution problem satisfies the variational inequalities ∂F ∗ ∂F ∗ ˜ ≤ 0, .(σˆ − σ˜ ) + (a − l) (4.118) . ∂ σˆ ∂a ˜ among the set .K ∗ of admissible fields .(σ˜ , l). Elements of proof The variations of .F ∗ satisfy: .

∗

∫

δF = − δσ : εˆ dΩ Ω/ΩΓ ∫ (δσ − δa σ ,1 ) : (ˆε − a ε ,1 ) dΩ − ΩΓ ∫ − (δa ε ,1 : σˆ + a ε ,1 : δσ − a δa σ ,1 : ε ,1 ) dΩ ΩΓ ∫ + (δa n.σˆ − .u,1 + a n.δσ − .u,1 − a δa n.σ ,1 .u,1 ) dS Γ

After integration by part and reduction, using first .δσ as Beltrami tensors associated with .χ which satisfies the conservation of momentum the strain rate .εˆ derives from displacement .uˆ and ˆ = grad uˆ t + grad uˆ . .2ε (4.119)

106

4 Fracture Mechanics

Using this property, the variations of .F ∗ are now ∫ ∫ δF ∗ = n.[ δσ .ˆu ]Γ dS − δa (σ ,1 : εˆ − σˆ : ε ,1 ) dΩ Γ ΩΓ . ∫ + δa (n.σˆ − .u,1 − a n.σ ,1 .u,1 ) + a n.δσ − .u,1 dS

(4.120)

Γ

Considering the jump condition n.δσ + − n.δσ − + δa n.σ ,1 = 0

.

we obtain: δF ∗ = .

= δa

∫ Γ

(∫

n.δσ − .([ uˆ ]Γ + a u,1 ) dS

Γ

−

−

n.σˆ .u,1 − σ ,1 : uˆ dS +

∫ ΩΓ

σˆ : ε ,1 − σ ,1

) : εˆ dΩ ,

(4.121)

then, the jump conditions for the rate of the displacement and the law of propagation for the crack, using of the variational inequality are recovered.

4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity The displacement asymptotic expansion has been presented in the two dimensional case. The study must be extended to three dimensions. It is well known that for the fundamental solution, the most singular term is reduced to superposition of plane strain and anti-plane solutions [14, 15]. But, to study the possible path of propagation it is necessary to know additional higher order terms as proposed by [16] in order to ensure the condition of equilibrium in a finite domain around the crack tip. This will be useful to make application with numerical analysis. Then the first analysis is concerned with the analysis of invariant integrals, using energy momentum tensor, taking account of the curvature of the crack front. Additional terms appear by the way of a surface integral. This term is necessary to ensure the invariance with respect to the path of integration. The second analysis proposes to define corrections of the William’s solution to take account of the front curvature. The corrections are given in terms of displacement, the resulting stresses are compared to stress development proposed in [17].

4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity

107 ez

Fig. 4.7 A section of the torus Rs

r Ri

θ

a

−N Γi Γs

4.13.1 Invariant Integrals . J, . G θ It is well known that the Eshelby’s momentum tensor .ψ satisfies a conservation law in the case of homogeneous material: .

div ψ T = 0, ψ = ψ(ε) I − σ .∇u

(4.122)

∂ψ is divergence free, where . I is the identity, .W (ε) is the strain energy and .σ = ∂ε then ∫ . div ψ T .Θ dΩ = 0, ∀Θ (4.123) Ω

Consider .Ω a section of the torus between .s, s + ds, with external radius . R S and internal radius . R I , as presented in Fig. 4.7. Let . Ai , (reps. . As ) be the disk with center . Mo and radius . R I (resp. . R S ). Chose a particular .Θ : .Θ = −Θ(r, θ )N ∫ 0=

div ψ T .NΘ dΩ ∫ ( ) div ψ T Θ(r, θ ) γ d A .N(s) ds

.

=

Ω ∫ s+ds s

As \Ai

therefore we obtain (.d A = r dr dθ ) ∫ .0 = div ψ T .N(s) Θ(r, θ ) γ d A As \Ai

(4.124)

108

4 Fracture Mechanics

By integration by part with respect to .(r, θ ) we have ∫ 0= ∫

.

− Define . JΓr = − is exactly:

∫ Γr

∫ Γ RS

n.ψ.N Θ γ R S dθ −

As \Ai

∂ψ .e S Θ Γd A + N. ∂φ

ΓRI

n.ψ.N Θ γ R I dθ

∫

As \Ai

(4.125) ψ : ∇Θ d A

n.ψ.N Θ γ r dθ , where .Γr is the circle of radius .r , the . J -integral .

J = lim JΓr

(4.126)

r →0

we have with the help of (4.125) the property ∫ .

J = JΓ RS + lim

R I →0

As \Ai

∫

then .

J = JΓ RS +

∂ψ .e S Γ Θ) d A ∂φ

(4.127)

∂ψ .e S Γ Θ) d A ∂φ

(4.128)

(ψ : ∇Θ − N.

(ψ : ∇Θ − N. As

Integral . J For the particular choice .Θ(r, θ ) = −N we obtain ∫ .

J =−

∫ Γ RS

n.ψ.N γ R S dθ −

(e S .ψ + N. As

∂ψ ).e S Γ d A ∂φ

(4.129)

Integral .G Θ For .Θ = −Θ(r, θ )N with { Θ(r, θ ) =

.

(R S − r )/(R S − R I ) , R I ≤ r ≤ R S 1 , 0 ≤ r ≤ RI

∫ .

GΘ =

(ψ : ∇Θ − ΘN. As

∂ψ .e S Γ)d A ∂φ

(4.130)

(4.131)

This expression is used essentially for computational estimation of characteristic of fracture.

4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity

109

Evaluation of . J -integral on a torus Consider the theoretical value . Jth given by the Irwin’s formula: J =

. th

1 2 1 − ν2 2 (KI + KII2 ) + K E 2μ III

(4.132)

this value is compared with the approximation of . J or .G Θ when the displacement is the first singular terms of Williams expansion u =

. o

√ ( KI I KII II KIII III ) uo (θ ) + uo (θ ) + u (θ ) r 2μ 2μ 2μ o

(4.133)

3θ θ 1 (− cos + (5 − 8ν) cos )er 2 2 2 3θ θ 1 + (8ν − 7) sin )eθ + (sin 2 2 2

(4.134)

3θ θ 1 (3 sin + (8ν − 5) sin )er 2 2 2 1 3θ θ + (3 cos + (8ν − 7) cos )eθ 2 2 2

(4.135)

where

uoI (θ ) = .

uoII (θ ) = .

θ uIII (θ ) = sin ez 2

. o

(4.136)

The . J -integral is evaluated for each mode of fracture, on a torus of external radius RS

.

• Mode I: The obtained value of . J differs from the theoretical value as I I 2 I 3 J I = Jth (1 + Jo1 η S + Jo2 η S + Jo3 ηS + · · · )

(4.137)

G Θ = Jth (1 + G 01 ηi + G 02 ηi2 + G 03 ηi3 )

(4.138)

. o

.

with .k =

RS , ηi = Γ R I , η S = Γ R S and RI JI =

. o1

k+1 I 32ν 2 − 32ν + 5 , G 01 = Jo1 8(2ν − 1) 2

The term of greater order are given in [18].

(4.139)

110

4 Fracture Mechanics

• Mode II In the same spirit for mode II, .

32ν 2 − 32ν + 9 64ν 2 − 128ν + 63 JoI I + η2S ... = 1 + ηS Jth 8(2ν − 1) 32(2ν − 1)

(4.140)

• Mode III The evaluation of integral . J for .wo (r, θ ) is then J I I I = Jth (1 − η S − η2S −

. o

η3S + ···) 3

(4.141)

Comments on the results of order .0 The difference with the theoretical value is essentially due to the fact that the plane and anti-plane fields are not statically admissible in the torus of radius . R S . When .η S or .ηi tends to zero, the theoretical value is recovered, that is conformed to the fact that the singular part of the displacement is the plane-strain or anti-plane strain solution. The fact that each term of the expansion of .G Θ depends only on the corresponding term of . J multiply by a function of .k = R S /R I is due to the choice of .Θ as a function of .r , then the integration on the domain can be decomposed into separate integration with respect to .r and to .θ . To have a best approximation of the. J -integral, we propose to define displacement fields which satisfy the equilibrium equation as the best possible. The correction is issued from the plane-strain and anti-plane-strain solutions and is obtained as an asymptotic expansion with respect to the local curvature .Γ of the crack front. To converge to the exact solution, asymptotic field is now build with respect to the equilibrium equations and boundary conditions for higher order in .Γ. Consider the expansion of the displacement, where the generalized stress intensity factors .Kα are considered as uniform: ∑ √ ∑ r Kα (r Γ) j uαj (θ ) (4.142) .u = α

j

For mode .α and terms of order . j in .Γ, the equilibrium and the boundary conditions up to order . j must be satisfied that is .

div σ = o(Γ j ), σ .n = o(Γ j )

(4.143)

We study successively each mode of rupture [18].

4.13.2 Mode I Order one As we have seen previously, the equilibrium with .uo is not satisfied on the torus. In order to cancel the linear term .Γ in equilibrium equations, we consider a correction

4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity

of order one :

111

√ KI I r (u + r Γu1I ). 2μ o

(4.144)

uI = U1I (θ )er + V1I (θ )eθ .

(4.145)

u =

. 1

where . 1

depends only on .θ , [18] U1I = UctI cos

.

3θ θ 3θ θ I + Ucu + VsuI sin . cos , V1I = VstI sin 2 2 2 2

(4.146)

With respect to equilibrium equations and boundary conditions on the lips, the constants are determined as 8ν − 3 128ν 2 − 96ν + 13 I , Ucu , = 8 24 8ν − 5 128ν 2 − 192ν + 55 VstI = − , VsuI = . 8 24

UctI = .

(4.147)

This solution is not unique, a additional term issued from the Williams solution [19] can be considered. The local stresses at order one is .σ (u1 ) = σ 0 + Γσ 1 with 3θ 13 θ 9 ) cos − ( − ν) cos , 16 2 16 2 θ 3θ 7 − 16ν ( sin + sin ), = 16 2 2 θ 3θ 9 − 16ν ( cos + 3 cos ), = 16 2 2 3θ 1( θ) + (16ν 2 − 2ν − 5) cos . = − (1 + ν) cos 2 2 2

σ rr = (ν −

. 1

σ1r θ σ1θθ σ1ss

The development of [17] is recovered. At order 1, we obtain an approximation of . J -integral at order two: .

J1I I 2 I 3 = 1 − J12 η S − J13 ηS + · · · Jth

192ν 2 − 72ν + 3 , 64(1 − 2ν) 4096ν 4 − 4608ν 3 + 1072ν 2 + 168ν − 41 , = 192(2ν − 1)

(4.148)

JI =

(4.149)

JI

(4.150)

. 12

. 13

112

4 Fracture Mechanics

and for .G θ .

Gθ k3 − 1 I 2 k4 − 1 I 3 =1+ J12 ηi + J η . Jth 3(k − 1) 4(k − 1) 13 i

(4.151)

At order two A correction of order 2 is obtained by the same way, balancing the .Γ 2 terms of the equilibrium equations with a displacement .u I2 = U2I (θ )er + V2I (θ )eθ where 5θ 3θ θ I + UctI cos + Ucu cos , 2 2 2 5θ 3θ θ I + VstI sin + Usu sin sin . 2 2 2

I U2I = Ucq cos

.

V2I = VsqI

The constants are given by the static conditions 3 − 24ν 24ν − 9 , VsqI = , 64 64 −512ν 3 + 64ν 2 + 80ν + 53 −512ν 3 + 704ν 2 + 80ν − 137 I . U , VstI = , ct = 360 360 67 − 256ν 2 256ν 2 − 192ν − 37 I , VsuI = , Ucu = 192 192 (4.152) For this new field, I Ucq =

√ u = KI r (uoI + r Γu1I + r 2 Γ 2 u2I ),

. 2

(4.153)

an evaluation of . J -integral up to order 3 is obtained [18]: I J I = Jth (1 + η3S J23 + ···)

(4.154)

6144ν 3 − 256ν 2 − 4224ν + 871 , 1536(2ν − 1)

(4.155)

. 2

with JI =

. 23

and in a similar way .

I G = Jth (1 + ηi3 J23

k4 − 1 + · · · ). 4(k − 1)

(4.156)

For a practical view point, taking account of order .n correction for the displacement make a correction of order .n + 1 for . J -integral.

4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity

113

4.13.3 Mode II The displacement .uo for mode II corresponds to the plane strain singular field uII =

. o

( ) √ r KII u IIo (θ )er + voII (θ )eθ ,

(4.157)

with θ 3θ + (8ν − 5) sin , 2 2 3θ θ II vo = 3 cos + (8ν − 7) sin . 2 2

u II = 3 sin

. o

Order one The correction of equilibrium at the order 1 gives u II = UstII sin

. 1

with .

3θ θ 3θ θ II sin , v1II = VctII cos + Usu + VsuII sin , 2 2 2 2

II UstII = (3 − 8ν)/4, Usu = (128ν 2 − 96ν − 107)/60,

VctII = (5 − 8ν)/4, VcuII = −(128ν 2 − 192ν + 79)/60.

(4.158)

The local stresses at order one is .σ = σ 0 + Γσ 1 with 3θ 107 7 θ 9 ) sin −( − ν) sin , 16 2 80 5 2 3θ θ) 1( (80ν − 35) cos + (31 − 16ν) cos , =− 80 2 2 θ 3θ 9 − 16ν ( sin + sin ), =− 80 2 2 ( 3θ θ) 1 5(1 + ν) sin + (16ν 2 + 14ν − 45) sin . =− 10 2 2

σ1rr = (ν − σ1r θ .

σ1θθ σ1ss

The development of [17] is recovered for mode II at order 1. Order two The correction at order two is given in [18].

4.13.4 Mode III √ θ The field .uo = KIII r sin ez gives the singular anti-plane shear strain. 2

(4.159)

114

4 Fracture Mechanics

JIII =

. o

1 2 K (1 − η S − η2S + · · · ). 2μ I I I

(4.160)

Order one

√ At the order one .u1 = KIII r (Wo + r ΓW1 )ez : .

1 θ sin , 4 2

W1 =

3 J1I I I = Jth (1 − η2S + · · · ). 2

(4.161)

The local stress .σ 1 is σ rs =

. 1

5 7 θ θ sin , σ1sθ = cos . 8 2 8 2

(4.162)

Order two The displacement is now √ u = KIII r (Wo + r ΓW1 + r 2 Γ 2 W2 ),

. 2

(4.163)

with: .

W2 =

3 1 sin(θ/2) − sin(3θ/2), 6 32

5 J2I I I = Jth (1 − η3S + · · · ). 4

(4.164)

4.13.5 General Remark The corrected fields are not unique. As the asymptotic expansion is defined as pro√ posed in (4.142), we know that for each term in . r r j the corresponding terms .uwj of the Williams expansion satisfies √the equilibrium, and a correction of this field with respect to .Γ is needed at order . r r j+1 . The obtained fields can be compared with analytical solutions [20, 21], they are identical as it has be proved in [18].

4.14 Separation of the Modes of Rupture To separate the three modes of rupture, the bilinear form .J associated to . J or .G θ in terms of displacement is now introduced. Consider two displacements . U, V which are admissible for the local problem of elasticity inside the torus almost locally, then .

J ( U + V) = J ( U) + J ( V) + 2J( U, V)

(4.165)

4.15 For a Non Planar Crack .

115

J ( U − V) = J ( U) + J ( V) − 2J( U, V)

(4.166)

where .J is the bilinear form associated to the quadratic form . J . It is easy to show now that ) 1( J ( U + V) − J ( U − V) .J( U, V) = (4.167) 4 this provides an extension of the bilinear form proposed by Chen and Shield [22]. An analogous relation is true for .G θ Gθ ( U, V) =

.

) 1( G θ ( U + V) − G θ ( U − V) 4

(4.168)

To extract, the local .Kα it is possible to use a test field

.

V=

∑ α

⎛ ⎞ j=n ∑ √ Kα∗ r ⎝ (r Γ) j uαj ⎠

(4.169)

j=0

where .uαj are given as in preceding sections. It can be noticed that for .n = 1 the bilinear form takes the value 2 I II J(u, V) = (1 − ν)(KI KI∗ (1 + η2S J12 ) + KII KII∗ (1 + η2S J12 ) μ . ∗ III (1 + η2S J12 ) + ··· + KIII KIII

(4.170)

and it is used to extract the .Kα along the crack front.

4.15 For a Non Planar Crack Consider now that the lips of the crack are non-plane. The front of the crack is described by a curvilinear coordinate .s, with tangent vector . T, this vector is in the tangent plane of the lips. Consider the normal vector .B to . T in this plane, and .N the normal vector to the lips. The frame .T, N, B is direct. Following the curvilinear coordinate, we introduce three curvatures along the front of the crack such as: .

dN = K B B + K T T, ds

dT = −K T N − ΓB, ds

dB = −K B N + Γ T. ds

(4.171)

To describe a point in a torus center on the crack front, we consider a point on the crack front is . Mo (s) and a point in the normal plane to . T, with local coordinates .(r, θ): .M

θ θ = Mo (s) + r (cos B + sin N). 2 2

(4.172)

116

4 Fracture Mechanics

The local basis .dM

= As ds + dr er + r dθeθ ,

(4.173)

where

θ θ = γ T − r K B eθ , γ = 1 + r cos Γ + K T r sin . 2 2 The reference frame is chosen as .er , T, eθ ; in this frame .As

.

dT = −K T N − ΓB = −er kr − kθ eθ , ds

(4.174)

(4.175)

with

θ θ θ θ + K T sin , kθ = −Γ sin + K T cos . 2 2 2 2 Consider the displacement .u r .u = u er + veθ + w T. .kr

= Γ cos

(4.176) (4.177)

Then the strain becomes ∂u 1 ∂v K B ∂w u 1 ∂w kr kθ , εθ θ = + εss = + + u + v, ∂r r ∂θ r γ ∂θ γ ∂s γ γ 1 ∂u v ∂v = − + , r ∂θ r ∂r K B ∂u 1 ∂u kr ∂w = + − w+ , γ ∂θ γ ∂s γ ∂r K B ∂v 1 ∂v kθ 1 ∂w = + − w+ . γ ∂θ γ ∂s γ r ∂θ

εrr = 2εr θ .

2εr s 2εsθ

(4.178)

In the same spirit, the divergence of a second order tensor is given by 0= .0

=

0=

kθ r θ ∂ P rr 1 ∂ Prθ K B ∂ Prs 1 ∂ Prs P rr − P θ θ kr + + + + + (P rr − P ss ) + P ∂r r ∂θ γ ∂θ γ ∂s r γ γ ∂ P θr 1 ∂ Pθs kθ 1 ∂ Pθθ K B θs P r θ + P θr kr + + P + + + P θr + (P θ θ − P ss ) ∂r r ∂θ γ γ ∂s r γ γ ∂ P sr kr kθ 1 ∂ P sθ K B ∂ P ss 1 ∂ P ss P sr r+ + + + + (P sr + P r s ) + (P sθ + P θ s ) ∂r r ∂θ γ ∂θ γ ∂s r γ γ

Corrected terms As for the plane case, the solution of plane strain, is not in equilibrium in a finite torus, around the crack tip, corrected fields at first order with respect to the different curvatures . K T , K B , Γ, Cu are build. The curvature .Cu is the curvature of the lips of the cracks in the plane normal to . T. The correction is needed to take into account of stress free conditions along the lips.

4.15 For a Non Planar Crack

117

For each curvature the corrected at order one is defined as: 3θ θ 3θ θ + Ust sin + Ucu cos + Uct cos , 2 2 2 2 θ 3θ θ 3θ . v = Vsu sin + Vst sin + Vcu cos + Vct cos , 2 2 2 2 θ 3θ θ 3θ w = Wsu sin + Wst sin + Wcu cos + Wct cos . 2 2 2 2 u = Usu sin

(4.179)

For each mode and each curvature, the non zero terms are presented. The corrected fields for the curvature .Γ have been presented in the preceding section, and we consider now the other curvatures. • Curvature . K T Mode I : Displacement 3 − 8ν 128ν 2 − 96ν − 7 , Ust = , 40 8 5 − 8ν 128ν 2 − 192ν + 59 , Vct = . = 40 8

Usu = − .

Vcu

(4.180)

Associated stresses 3θ ν 21 θ 9 ) sin − ( − ) sin , 10 2 5 80 2 3θ θ 16ν − 9 (sin + sin ), = 16 2 2 3θ 3ν 33 θ 7 + ( − ) cos , = (ν − ) cos 16 2 5 80 2 3θ 24ν 2 + ν 7 θ ν+1 sin +( − ) sin . = 2 2 5 2 2

σrr = −(ν − σθθ .

σr θ σ ss

Mode II: Displacement 128ν 2 − 96ν + 13 3 − 8ν , Uct = − , 24 8 5 − 8ν 128ν 2 − 192ν + 55 , Vst = . = 24 8

Usu = .

Vcu Associated stresses

(4.181)

118

4 Fracture Mechanics

3θ 13 − 16ν θ 16ν − 9 cos + cos , 16 2 16 2 3θ 27 − 112ν θ −16ν + 9 cos + cos ), = 16 2 16 2 3θ θ 7 − 16ν (sin + sin ), = 16 2 2 3θ 16ν 2 − 10ν + 3 θ ν+1 cos − cos . =− 2 2 2 2

σrr = σθθ .

σr θ σ ss

Mode III: Displacement Wcu = −1, Wct = −3

(4.182)

3θ 7 θ 5 − cos σ r s = − cos 4 2 4 2 . 5 3θ θ θs σ = (sin + sin ) 4 2 2

(4.183)

u=v=w=0

(4.184)

5 5 Ucu = − (8ν − 3), Vsu = − (8ν − 9). 3 3

(4.185)

.

Associated stresses

• Curvature .Cu Mode I: Displacement .

Mode II: Displacement .

Associated stresses θ θ 15 9 cos , σθθ = cos , 2 2 2 2 θ 3 θ = sin , σ ss = 12ν cos . 2 2 2

σrr = .

σr θ Mode III : Displacement

.

Wct = −1.

(4.186)

Associated stresses 3θ 3θ 3 3 σ r s = − cos , σ θs = sin . 4 2 4 2

.

• Curvature . K B Mode I: Displacement .

Wsu =

1 . 2

(4.187)

References

119

Associated stresses σ rs =

.

3θ θ 3θ θ 1 1 (3 sin + (8ν − 2) sin ), σ θs = (3 cos + (8ν − 6) cos ). 8 2 2 8 2 2

Mode II: Displacement .

Wcu =

1 8ν − 5 , Wst = . 2 2

(4.188)

Associated stresses .σ

rs

=

4ν − 1 1 3θ θ 3θ θ (3 cos + cos ), σ θ s = (3(1 − 4ν) sin + (3 − 4ν) sin ). 4 2 2 4 2 2

References 1. P. Destuynder, M. Djaoua, Sur une interprétation mathématique de l’intégrale de rice en théorie de la rupture fragile. Math. Appl. Sci. 3, 70–77 (1981) 2. Q.S. Nguyen, Méthodes énergétiques en mécanique de la rupture. Journal de Mécanique 19(2), 363–386 (1980) 3. C. Stolz, Energy Methods in Non-linear Mechanics. Lectures Notes 11 AMAS (Polish Academy of Sciences Warsaw, 2004) 4. C. Stolz, Dual approach in non-linear fracture mechanics. Int. J. Fract. 166, 135–143 (2010) 5. J.D. Eshelby, The force on an elastic singularity. Phil. Trans. Roy. Soc. London, A 244, 376–396 (1951) 6. J. Rice, G.F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material. J. Mech. Phys. Solids 16, 1–12 (1968) 7. J. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35 (1968) 8. J.K. Knowles, E. Sternberg, A class of conservation law in linearized and finite elastostatics. Arch. Rat. Mech. Anal. 44, 187–211 (1972) 9. H.D. Bui, Dualité entre les intégrales indépendantes du contour dans la théorie des solides fissurés. C. R. Acad. Sci. Paris, Série A 376, 1425–1428 (1973) 10. Q.S. Nguyen, C. Stolz, Sur le problème en vitesse de propagation de fissure et de déplacement en rupture fragile ou ductile. C. R. Acad. Sci. Paris 301II (10), 661–664 (1985) 11. Q.S. Nguyen, C. Stolz, G. Debruyne, Energy methods in fracture mechanics stability bifurcation and second variations. Eur. J. Mech. 9(2), 157–173 (1990) 12. J.K. Knowles, The finite anti-plane shear field near a tip of a crack for a class of incompressible elastic solids. Int. J. Fract. 13, 5 (1977) 13. J.K. Knowles, E. Sternberg, Antiplane-shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics. J. Elast. 11(2), 129–164 (1981) 14. H.D. Bui, Application des potentiels élastiques à l’étude des fissures de forme arbitraire en milieu tridimensionel. C.R. Acad. Sci. Paris 280, 1157–1161 (1975) 15. H.D. Bui, An integral equations method for solving the problem of a plane crack of arbitrary shape. J. Mech. Phys. Solids 23, 29–39 (1977) 16. J.B. Leblond, Crack path in plane situation 1 general form of the expansion of the stress intensity factor. Int. J. Solids Struct. 25, 1131–1325 (1989) 17. J.B. Leblond, O. Torlai, The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body. J. Elast. 29, 97–131 (1992)

120

4 Fracture Mechanics

18. C. Stolz, Invariant integrals and asymptotic fields near the front of a curved planar crack. Int. J. Fract. 230, 3–17 (2021) 19. M.L. Williams, On the stress distribution at the base of a stationary crack. ASME J. Appl. Mech. 24, 109–114 (1957) 20. V.I. Fabrikant, Green’s functions for a penny-shaped crack under normal loading. Eng. Fract. Mech. 30(1), 87–104 (1988) 21. V.I. Fabrikant, Flat crack under shear loading. Acta Mechanica 78, 1–31 (1989) 22. F.H.K. Chen, R.R. Shield, Conservation laws in elasticity of the J-integral type. J. Appl. Math. Phys. 28, 1–22 (1977)

Chapter 5

Moving Discontinuities

5.1 Introduction They exist many situations which are concerned with change of material characteristics, phase changing, local damage. A material one is changed into a material two along a surface .Γ, .Γ becomes a moving interface separating two phases or two materials. This motion can be reversible or not, depending on the discontinuities of mechanical quantities along the surface, considering a perfect bounding on the interface. In the framework of thermo-mechanical coupling as in fracture mechanics the analysis defines two different energy release rates associated with heat production and entropy production [1, 2]. Variational formulations were performed to describe the motion of the surface separating sound and damaged material [3, 4] that corresponds to the description of damage associated with a moving interface along which mechanical transformation occurs. Some connections can be made with the notion of configurational forces, [5–8]. Let.Ω be a domain composed of two distinct sub-domains .Ω1 , Ω2 of two materials with different mechanical characteristics. The bounding between the two phases is perfect and the interface is denoted by .Γ, (.Γ = ∂Ω1 ∩ ∂Ω2 ). The external boundary .∂Ω of the domain .Ω is decomposed in two parts .∂Ωu and .∂ΩT on which the displacement .ud and the loading .Td are prescribed respectively. The material 1 changes into material 2 along the interface .Γ by an irreversible process. Hence .Γ moves with the normal velocity .c = aν in the reference state, .ν is the outward .Ω2 normal, then .a is positive. When the surface .Γ is moving, all the mechanical quantities . f can have a jump denoted by .[ f ]Γ = f 1 − f 2 , and any volume average has a rate defined by d . dt

∫ Ω(Γ)

∫ f dΩ =

Ω(Γ)

f˙ dΩ −

∫ Γ

[ f ]Γ a dS

(5.1)

At a given state of equilibrium for a given value of the prescribed loading ( .ud , Td ), the position of the interface .Γ is known. At this time a variation of the loading is imposed, the mechanical quantities evolve and propagation of the interface can occur according to a given evolution law. For a prescribed history of the loading, we © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 C. Stolz, Introduction to Non-linear Mechanics, Springer Series in Solid and Structural Mechanics 14, https://doi.org/10.1007/978-3-031-51920-8_5

121

122

5 Moving Discontinuities

must determine the rate of all mechanical fields and the normal propagation .a(s) to characterize the position of the interface .Γ at each time. Along interface .Γ perfect bounding is preserved at each time. Let us introduce the notion of convected derivative. Convected Derivation. The convected derivative .Da of any function . f ( XΓ , t) is Da f = lim

.

τ →0

f ( XΓ + a(s)ντ, t + τ ) − f ( XΓ , t) . τ

(5.2)

With this definition, we can expressed the transport of the normal vector at point .x Da ν = −∇a.eα eα ,

.

(5.3)

where .e1 , e2 is a basis of the plane tangent to .Γ at point .x. The equation of the surface Γ, . S( X, t) = 0 satisfies:

.

Da S =

.

∂S ˙ ∂S .X + = 0, ∂X ∂t

(5.4)

and then the normal velocity .c of .Γ is given as: c = aν, ν =

.

∂S ∂S / || || . ∂X ∂X

(5.5)

Finally for any differentiable fields . f the convected derivative is obvious Da f =

.

∂f + a ∇ f.ν. ∂t

(5.6)

The bounding being perfect between the phases, displacement and stress vector are continuous along .Γ. Their rates have discontinuities according to the general compatibility conditions of Hadamard, rewritten with the convected derivative. Hadamard’s relations on discontinuities: • continuity of displacement [ u ]Γ = 0 ⇒ Da ([ u ]Γ ) = [ v ]Γ + a[ ∇u ]Γ .ν = 0,

.

(5.7)

• continuity of the stress vector [ σ ]Γ .ν = 0 ⇒ Da ([ σ ]Γ .ν) = [ σ˙ ]Γ .ν − divΓ ([ σ ]Γ a) = 0.

.

(5.8)

The last equation is obtained taking the equilibrium equation into account. As we have .Da ([ σ ]Γ .ν) = Da [ σ ]Γ .ν + [ σ ]Γ .Da ν, (5.9)

5.2 Dissipation Analysis

123

where Da [ σ ]Γ = [ σ˙ ]Γ + a ν.[ ∇σ ]Γ ,

.

(5.10)

Using the momentum conservation .

div σ = eα .∇σ .eα + ν.∇σ .ν = 0.

(5.11)

The above result is obtained with the expression of the surface divergence given by .

divΓ F = div F − ν.∇ F.ν,

(5.12)

Orthogonality for discontinuities: Since the displacement is continuous along the interface, we have: [ u ]Γ = 0, ⇒ [ ∇u ]Γ .eα = 0,

.

(5.13)

the discontinuities of the gradient must satisfy: [ ∇u ]Γ = U(x) ⊗ ν.

.

(5.14)

Since the stress vector is continuous on .Γ, [ σ ]Γ .ν = 0,

.

(5.15)

the discontinuities of .σ and those of .∇u have the property of orthogonality as pointed in [9]: .[ σ ]Γ : [ ∇u ]Γ = 0. (5.16)

5.2 Dissipation Analysis The mass conservation leads to the continuity of the mass flux .m = ρa. The first and the second law of thermodynamics give rise to local equations inside the volume and along the moving boundary .Γ: ρ e˙ = σ : ε˙ − div q, over Ω, .0 = m[ e ]Γ − ν.σ .[ v ]Γ + ν.[ q ]Γ , on Γ,

.

(5.17) (5.18)

e is the internal energy density (.e = ψ + ϑ s), and .q is the heat flux associated to the heat conduction. Thanks to Hadamard compatibility equations, the heat power supply is given in terms of a release rate of internal energy .Gth as an objective quantity defined along .Γ

.

124

5 Moving Discontinuities .

− ν.[ q ]Γ = Gth a, with Gth = ρ[ e ]Γ − σ : [ ε ]Γ ,

(5.19)

taking account of ν.σ .[ v ]Γ = a ν.σ .[ ∇u ]Γ .ν = a σ : [ ε ]Γ .

(5.20)

.

The value of .Gth is obtained considering the orthogonality condition on the discontinuities. When .a = 0 in the reference state the interface .Γ does not move, and the normal heat flux is continuous. When the transformation occurs, the moving interface is a surface of heat sources intensities of which are given by .Gth a. The total internal energy of the structure is ∫ E(˜u, Γ, ϑ, T d ) =

∫

.

Ω(Γ)

ρ e dΩ −

∫

∂ΩT

Td .u dS = E +

Ω(Γ)

ρ sϑ dΩ.

(5.21)

For quasi-static evolution, the first law of thermodynamics is written as follows: ∂E ˙ d d E− .T = − . dt ∂Td

∫ q.n dS,

(5.22)

∂Ω

and taking into account of the momentum conservation, we have .

∂E .Γ˙ = ∂Γ

∫ Γ

∫ [ q ]Γ .ν dS = −

Γ

Gth a(s) dS,

(5.23)

then the derivative of the total energy relatively to the position of the interface determined the source of heat due to the irreversible process, intensity of which is governed by the internal energy release rate: Gth = −

.

∂E . ∂Γ

(5.24)

The entropy production is given by ∫ .

Ω

(ρ s˙ +

∇ϑ div q − q. 2 ) dΩ + ϑ ϑ

∫ Γ

(−m[ s ]Γ − ν.[

q ] ) dS ≥ 0. ϑ Γ

(5.25)

Under the assumption of separability of the two dissipations, the term inside the volume is reduced to the conduction, and the term along the surface is then .

DΓ =

Gs ρ[ ψ ]Γ − σ : [ ε ]Γ a= a ≥ 0, ϑ ϑ

where .Gs is the release rate of free energy.

(5.26)

5.2 Dissipation Analysis

125

This quantity has an analogous form to the driving traction force acting on a surface of strain discontinuity proposed by [10]. The evolution of the interface may be written as a function of this quantity. In a thermomechanical coupling, two different release rates must be distinguished. One defined in terms of variation of the total internal energy gives rise to the heat source associated with the moving surface; the second one gives rise to the production of entropy. In the case of isothermal evolution the total dissipation is given in terms of the derivative of the potential energy relatively to the position of the interface .

∂E .Γ˙ = − ∂Γ

∫ Γ

Gs a dS, or Gs (x) = −

∂E (x). ∂Γ

(5.27)

with .Gs = ρ[ ψ ]Γ − σ : [ ε ]Γ . In this case, there is only one energy release rate to characterize the propagation, it gives the sources of entropy production and the dissipation. These relations can be generalized in the dynamical case, by replacing the internal energy of the system by its Hamiltonian, and can be extended to the case of running cracks, and to more general behaviour and structures [1, 2, 11].

5.2.1 In the Dynamical Case The first law and the second law of thermodynamics give rise to local equations inside the volume and along the moving surface .Γ: ρ e˙ = σ : ε˙ − div q, in Ω, .

0 = m[ e +

v2 ] − ν.[ σ .v ]Γ + ν.[ q ]Γ , on Γ. 2 Γ

(5.28)

Taking the momentum conservation and the continuity of the displacement into account: .[ u ]Γ = 0, [ σ ]Γ .ν = m[ v ]Γ . (5.29) The heat power supply is defined by the internal energy release rate .Gth Using the notation (.σ¯ = 21 (σ 1 + σ 2 )) and the property .

1 m[ v2 ]Γ = m v¯ .[ v ]Γ ; ν.[ σ .v ]Γ == ν.σ¯ .[ v ]Γ + ν.[ σ ]Γ .¯v, 2

(5.30)

the heat power supply is defined by the internal energy release rate .Gth .

− ν.[ q ]Γ = Gth a, Gth = ρ[ e ]Γ − ν.σ¯ .[ ∇u ]Γ .ν.

(5.31)

126

5 Moving Discontinuities

The Hamiltonian of the structure is the sum of the kinetic energy and the total internal energy, the potential energy is defined as above: ∫ H=

.

Ω

1 2 ρv dΩ + E + 2

∫ ρsϑ dΩ

Ω

(5.32)

The momentum conservation is then defined by the set of equations ∫ ∂H • δp = v.δp dΩ, ∂p Ω . ∫ ∂H d p.δ u dΩ, ◦δu = − ∂u dt Ω

(5.33)

where .p is the momentum, these equations leads to the classical equation of motion. The first law of thermodynamics is rewritten as follows: .

∂H ˙ d dH − .T = dt ∂Td

∫ ∂Ω

−q.n dS,

(5.34)

and taking into account of the momentum conservation, we have ∂H . .Γ˙ = ∂Γ

∫

∫ Γ

[ q ]Γ .ν dS = −

Γ

Gth a dS,

(5.35)

the second law has the same form as previously. The interface is perfect at each time, under the assumption of separability of the two dissipations, the term inside the volume is reduced to the conduction, and the term along the surface is then: Gs a where .Gs has also the form of a release rate of energy. . DΓ = ϑ Gs = ρ[ ψ ]Γ − ν.σ¯ .[ ∇u ]Γ .ν.

.

(5.36)

In a thermomechanical coupling, two different release rates must be distinguish, one defined in term of variation of the Hamiltonian give rise the heat source associated with the moving surface, the second one describe the production of entropy. In the case of isothermal evolution, we can define another Hamiltonian ∫ 1 2 ρv dΩ + E, .H = (5.37) 2 Ω and the total dissipation is then given by: .

∂H ˜˙ d d ∂H H− .Γ˙ = − .T = d dt ∂Γ ∂ T˜

where .Gdyn = ρ[ ψ ]Γ − ν.σ¯ .[ ∇u ]Γ .ν.

∫ Γ

Gdyn a dS,

(5.38)

5.4 Moving Discontinuity

127

5.3 General Features for Quasi-static Evolution Consider a body .Ω decomposed into two sub-domains .Ωo , .Ω1 with common boundary .Γ. At a point . X Γ , the normal vector to .Γ is .ν outward .Ω1 . The motion of .Γ is described by the normal velocity .vn (X Γ ) = aν, (see Fig. 5.1). The material of .Ωo is transformed into the material of .Ω1 when the interface .Γ is moving with .a ≥ 0. On the surface .Γ any mechanical quantity . f can experience a jump denoted by [ f ]Γ = f 0 − f 1

(5.39)

.

and any volume average has a rate defined by .

∫

∫

d dt

Ω(Γ)

f dΩ =

Ω(Γ)

f˙ dΩ −

∫ Γ

[ f ]Γ a dS

(5.40)

Time derivative of integrals defined on varying volume and surface domains has been investigated in [12].

5.4 Moving Discontinuity The domain .Ω is decomposed as a two-phases composite, sound material in domain Ωo with stiffness .Co and transformed material in domain .Ω1 with stiffness .C1 . The local free energy is defined on each phase as

.

{ .

ψ0 (ε) = 21 ε : Co : ε, over Ωo ψ1 (ε) = 21 ε : C1 : ε + H, over Ω1

(5.41)

where .ε(u) is the strain. The constant . H ≥ 0 is a stored energy due to the transformation from material (0) to material (1). The internal parameter .α presents a strong discontinuity: .α0 = 0, . over Ωo and .α1 = 1, . over Ω1 , then .[ α ]Γ = −1.

Fig. 5.1 A moving surface: evolution of .Ωo and .Ω1

vn nγ Ω1

Γt

Ω0 Γt+dt

128

5 Moving Discontinuities

At every time, bonding is perfect along the interface .Γ, therefore displacement and stress vector are continuous.

5.4.1 The Equilibrium State We study now the equilibrium of the body.Ω submitted to given boundary conditions. The boundary.∂Ω is decomposed into two complementary parts.(∂Ω = ∂Ωu ∪ ∂ΩT ): on .∂Ωu the displacement is imposed .(u(x, t) = ud (x, t), .x ∈ ∂Ωu ) and on .∂ΩT , the traction .Td (x, t) is prescribed. The unknowns of the equilibrium problem defined on .Ω are displacements .u and position of the internal boundary .Γo . The potential energy: The total potential energy of the body is: ∫ E(u, Γ) =

∫

.

Ωo (Γ)

ρψ0 (ε(u)) dΩ +

Ω1 (Γ)

∫ ρψ1 (ε(u)) dΩ −

∂ΩT

Td .u dS

(5.42)

The position of the interface .Γ plays the role of an internal parameter.1 Displacements are continuous along .Γ, then velocities are satisfying Hadamard’s relations for discontinuities (5.1): [ u ]Γ = 0, [ u˙ ]Γ + a[ ∇u ]Γ .ν = 0,

.

(5.43)

where .vn = −a(s) ν is the normal velocity associated to the motion of .Γ. As .Γ can move, variations of displacement should be compatible with possible motion of .Γ defined by .δa .[ δu ]Γ + δa[ ∇u ]Γ .ν = 0. (5.44) Any variations .δu of the displacement must be also compatible with the boundary conditions: .δu(x) = 0, x ∈ ∂Ωu .

5.4.2 Variations of the Potential Energy The variations of .E becomes: ∫ ∫ ∑∫ ∂E ∂E .δu + δΓ = . σ i : ∇δu dΩ − Td .δu dS − [ ρψ ]Γ δa dS, ∂u ∂Γ ∂ΩT Γ i=0,1 Ωi after integration by parts, we obtain two contributions: one inside the domain and another over the surface .Γ 1

During the motion of the interface the mass density is conserved: .ρo = ρ1 = ρ.

5.4 Moving Discontinuity

δE = .

∑∫ i=0,1 Ωi

129

∫ − div σ i .δu dΩ −

∫

−

Γ

∫

ν.[ σ .δu ]Γ dS −

Γ

∫

∂Ω

n.σ .δu dS −

∂ΩT

Td .δu dΩ (5.45)

[ ψ ]Γ δa dS.

Taking account of the compatibility of .δu (5.44), two independent contributions are obtained: ∫ ∫ .− ν.[ σ ]Γ .δuo dS − ([ ρψ ]Γ − ν.σ 1 .[ ∇u ]Γ .ν) δa dS. (5.46) Γ

Γ

• Variations with respect to displacement. For given.Γ, the potential energy is those of a two-phase linear composite, then for an equilibrium state, the energy is minimum among the set of admissible displacements: u ∈ K.A = {u∗ |[ u∗ ]Γ = 0, u∗ = ud over ∂Ωu }.

.

The solution .u for this minimum satisfies .

.

(5.47)

∂E δu = 0, which is exactly: ∂u

∂ψi = Ci : ε(u), div σ i = 0, over Ωi ∂ε (5.48) [ σ ]Γ .ν = 0, over Γ, n.σ = Td , over ∂ΩT . 2ε(u) = ∇u + ∇ t u, σ i = ρ

• Variations with respect to .Γ. The variations of the potential energy with respect to .Γ are described by motion with normal velocity .δa(s): .

∂E δΓ = − ∂Γ

∫ Γ

G(s)δa(s) dS.

(5.49)

The driving force associated to the interface motion is the local energy release rate .G: .G = [ ρψ ]Γ − σ : [ ε ]Γ , (5.50) as proposed in [4, 5, 10, 13].

5.4.3 Dissipation and Evolution of the Interface When interface is moving, the properties of sound material change to those of transformed material and a dissipation can occur. The dissipation due to the motion of .Γ is ∫ .

Dm =

Γ

G(s)a(s) dS.

(5.51)

130

5 Moving Discontinuities

The velocity .a(s) can be defined by a kinetic relation [5, 10] given as a function dφ or by non regular kinetic relation. .φ(G) such that .a(s) = dG Generalized Griffith’s law Let us consider that the motion is governed by a normality law based on the driving force .G .s ∈ Γ, a(s) ≥ 0, G(s) ≤ G c , (G − G c )a(s) = 0, (5.52) then .a(s) is defined on the set .KΓ KΓ = {a ∗ (s), s ∈ Γo |a ∗ (s) ≥ 0 if G(s) = G c , a ∗ (s) = 0 otherwise}.

.

(5.53)

In these framework, two families of models are now considered: models without or with dissipation. • Reversible behaviour. When no dissipation occurs whatever is loading history, contribution over .Γ vanishes, this is true for two cases – .G c = ∞, then .a = 0, the surface .Γ is fixed. There is no transformation. – .G c = 0, this implies that.G = 0 and.a can be positive. Two situations are possible depending on the constant . H . H = 0, .a can be positive and the transformation is uncontrolled under the loading; · . H > 0, the change of energy between the two phases controls the transformation as in phase transformation occurring in memory alloy or in pseudoelasticity [14–16]. ·

.

• Model with dissipation. The critical value .G c is finite. The surface .Γ is a moving surface whose velocity .a is governed by the normality law. Then the surface is defined by the implicit equation G(X Γ , t) ≤ G c ,

.

X Γ ∈ Γ.

(5.54)

For this particular behaviour, the rate boundary value problem has been presented in [4, 17] and criterion for stability and bifurcation of the evolution has been proposed based on discussion of existence and uniqueness. Remark: When∫ the dissipated energy depends only on the actual state, we can introduce .Ψd = Ω1 G c dΩ and add this term to the potential energy Ed = E + Ψd .

.

(5.55)

5.4 Moving Discontinuity

131

The problem of evolution is governed by .

∫

∂Ed ∗ (Γ − Γ) = ∂Γ

Γ

([ ρψ ]Γ − σ : [ ε ]Γ − G c )(a − a ∗ ) dS ≥ 0, ∀a ∗ ∈ KΓ .

(5.56) This point of view can∫ be employed, but the separation between dissipated energy .Ψd and stored energy . Ω1 H dΩ must be specified.

5.4.4 Examples on a Bar A bar is decomposed into two domains, one .[0, Γ] with Young’s modulus . E 1 , and the complementary part .[Γ, L] made with sound material .(E o ). The bar is loaded by applying displacement: .u(0) = 0, u(L) = ΔL, .Δ is the global strain. The local stress .σ (x) is uniform. We discuss models without or with dissipation.

5.4.4.1

Reversible Behaviour with . H > 0

The stress is uniform, then the strain is piecewise uniform. Consider .Δ the total strain, .ε 0 the strain in initial phase with modulus . E o , .ε 1 the strain in the transformed phase, with modulus . E 1 , . E o > E 1 . The proportion of transformed phase is denoted by .c. The bar is a composite, where the two phases are separated by a boundary .Γo with position . Lc. The total strain satisfies Δ = cε 1 + (1 − c)ε 0 .

.

(5.57)

The condition .0 ≤ c ≤ 1 is taken into account by a Lagrange multiplier .γ : c(c − 1) ≤ 0, γ ≥ 0, γ c(c − 1) = 0.

.

(5.58)

The total strain is imposed by a Lagrange’s multiplier .γΔ associated to the relation (5.57). The potential energy of the bar is given by unit length ([14]): .

E(c, ε 0 , ε 1 , γΔ , γ ; Δ) = cψ1 (ε 1 ) + (1 − c)ψ0 (ε 0 ) + γΔ (Δ − cε 1 − (1 − c)ε 0 ) + γ c(c − 1).

This energy is stationary with respect to all unknowns (.c, ε 0 , ε 1 , γ , γΔ )

(5.59)

132

5 Moving Discontinuities

∂E ∂E δγ = c(c − 1)δγ = 0, δγΔ = (Δ − cε 1 − (1 − c)ε 0 )δγΔ = 0, ∂γ ∂γΔ ∂E ∂E . = c(E 1 ε 1 − γΔ ) = 0, = (1 − c)(E o ε 0 − γΔ ) = 0, ∂ε1 ∂ε 0 ∂E 1 1 G=− = − E 1 ε 21 − H + E o ε 20 + γΔ (ε 1 − ε 0 ) − γ (2c − 1) = 0. ∂c 2 2 (5.60) Taking account of (5.58): .γ ≥ 0 and .δγ /= 0 if and only if .c = 0 or .c = 1; otherwise .γ = 0 and .δγ = 0. This defines the potential value of .c. The reaction associated to .Δ is given by .

∂E = γΔ = ∑. ∂Δ

(5.61)

The Lagrange’s multiplier .γΔ is the global stress. Consider now an increasing strain .Δ from initial stress free state, the response undergoes three main phases. Phase I-c = 0.

At the beginning only phase (0) exists, then .γ ≥ 0 and .G = 0.

.

1 1 E 1 ε 21 + H + Σε 0 − Σε 1 − γ = 0. 2 2

c = 0,

.

(5.62)

For .c = 0, .ε 1 is not determined, this phase does not exist yet. Hence .γ must be positive for all .ε 1 , the minimum value for .ε 1 determines the minimum value for .γ , then .

1 1 1 1 − ) ≥ 0. E 1 ε 1 = Σ, γ = H + Σ(ε 0 − ε 1 ) = H + Σ 2 ( 2 2 Eo E1

(5.63)

The transformation begins when .Σ reaches critical value .Σc defined by .γ = 0. Phase II-Transformation,0 ≤ c ≤ 1. When .Δ is increasing from the state (.Σ = Σc ), the two phases coexist and .γ = 0. The stress being uniform, we have

.

0=

.

1 Σ(ε0 − ε 1 ) + H = 0, Σ = E o ε 0 = E 1 ε 1 . 2

(5.64)

During the transformation we have .

1 2 1 1 Σ ( − ) + H = 0, 2 Eo E1

and therefore Σ = Σc , Δ = Σc (

.

1−c c + ). E1 Eo

(5.65)

(5.66)

The last equation defines the state .c: when .Δ is increasing, .c grows to one.

5.4 Moving Discontinuity

133

Phase III-Total transformation,c = 1. Now, the phase .0 has disappeared, .γ ≥ 0, the condition of no dissipation is now

.

1 1 0 = − Σε 21 + H − E o ε20 + Σε 0 + γ , 2 2

(5.67)

.

ε is now not defined, .γ ≥ 0 implies

. 0

1 1 1 γ = −H + Σ 2 ( − ) 2 E1 Eo

(5.68)

.

Then, the relation is true when .Σ > Σc , and .Δ = is increasing from the end of the previous state.

Σ so this is satisfied when .Δ E1

Unloading path from the end of phase III. From such a final point, assume now that .Δ is decreasing, then .γ ≥ 0 until .Σ equal to .Σc , at this stage .γ = 0; from that state, the equations are those of step II, then the relations between .Δ(c) is recovered, .c is decreasing as .Δ did, until .c = 0. From that point, the relations of step one are recovered, until .Δ = 0. The response is plotted on Fig. 5.2 (left). Some more general models corresponding to this framework can be found in [14–16, 18].

5.4.5 A Model with Dissipation: A Quasi-brittle Material Assume . H = 0 and consider now that .Γ is a moving surface. As previously, the bar is loaded by applying displacement: (.u(0) = 0, u(L) = ΔL). The local stress .σ is uniform with value .Σ. Under increasing global strain .Δ the response of the bar is decomposed in three phases.

Σ

Σ

Σc

Σc

Δ

a

Δ

Fig. 5.2 Reversible Phase transformation, (Left); with dissipation (Right); .< − > reversible path

134

5 Moving Discontinuities

Phase I- Elastic behaviour. The position .Γ = Lc being known, the bar is an uniaxial composite with a a global stiffness defined by

.

1 c 1−c = + . E hom E1 Eo

(5.69)

Σ2 1 1 ( − ) ≤ G c , Σ ≤ Σc , 2 E1 Eo

(5.70)

Σc2 1 1 ( − ) = Gc. 2 E1 Eo

(5.71)

.

There is no propagation if .

with .

Phase II- Propagation. When .G = G c , .Σ = Σc and propagation is possible. With respect to the normality law, the interface is moving if and only if .Σ = Σc , and the definition of the global strain imposed

.

˙ = Δ

.

1 a 1 − )Σc , ( L E1 Eo

(5.72)

where .a is the normal velocity of the moving surface. This relation is due to the continuity of displacement at the interface .Γo . .Phase III–Total transformation. When the bar is totally transformed, the material is now homogeneous with modulus .E1 . Note the main differences between the two examples. For phase transformation, with no dissipation, paths for loading and unloading are always the same. For transformation with dissipation, the unloading path depends on the state of transformation and the modulus of elasticity is always decreasing with the irreversibly of the transformation, that is a damage model, (see Fig. 5.2).

5.5 Problem of Evolution We consider isothermal processes to simplify the analysis. The state of the system is then characterized by the displacement field.u, from which the strain field.ε is derived. The spatial distribution of the two phases given by the position of the boundary.Γ. We analyse quasi-static evolution of .Γ under given loading prescribed on the boundary .∂Ω. The behaviour of the phase .i is defined by the free energy density .ψi , function of the strain .ε. The mass density .ρ of the two phases is the same. In the following the two phases are linear elastic materials.

5.5 Problem of Evolution

135

The total potential energy .E of the domain .Ω (.Ω1 ∪ Ω2 ) is defined as a two phases composite material: ∑∫

E(˜u, Γ, T˜ d ) =

∫

.

i=1,2 Ωi

ρ ψi (ε(u)) dΩ −

∂ΩT

Td .u dS.

(5.73)

The position of .Γ is an internal parameter for the global system. An equilibrium state is characterized by the minimum of the potential energy .

∫ ∑∫ ∂E ∂ψi ρ Td .δu dS = 0, · δu = : ε(δu) dΩ − ∂u ∂ε Ω ∂Ω i T i=1,2

(5.74)

among the set .K.A of fields .δu satisfying .δu = 0 over .∂Ωu . This formulation is equivalent to the set of local equations: • local constitutive relations: σ =ρ

.

∂ψi = Ci : ε, on Ωi , ∂ε

(5.75)

• momentum conservation: .

div σ = 0, on Ω, [ σ ]Γ .ν = 0 over Γ, σ .n = Td over ∂ΩT ,

(5.76)

• compatibility relations: 2ε = ∇u + ∇ t u, [ u ]Γ = 0 over Γ, u = ud over ∂Ωu .

.

(5.77)

This problem is a problem of heterogeneous elasticity. The solution is denoted by usol , this field depends upon the quantities .(u d , Td , Γ). For an equilibrium state and given position of .Γ, the total energy is

.

E(usol , Td , Γ) = W (ud , Td , Γ).

.

(5.78)

This equation emphasises the fact that position of the interface .Γ is an internal parameter. Now complementary relations are given to describe the irreversibility. An energy criterion is chosen as a generalized form of the well known theory of Griffith; we assume the normality rule a ≥ 0, if Gs = G c on Γ, a = 0, otherwise.

.

(5.79)

This is a local energy criterion. At each equilibrium state, the interface .Γ is decomposed into two subsets where the propagation is either possible or not. Let denote by .Γ + , the subset of .Γ where the critical value .G c is reached, .a ≥ 0 over this subset. The evolution of the interface is

136

5 Moving Discontinuities

Fig. 5.3 Propagation of the interface .a = φ(s, t)

X+ φ n dt

φ X S(X,t)=0

S(X,t+dt)=0

governed by the consistency condition. If at the geometrical point . x Γ (t) the criterion is reached .Gs ( x Γ (t), t) = G c , (5.80) then the derivative of .Gs following the moving surface vanishes .Da Gs = 0. This leads to the consistency condition written for all point belonging to .Γ + (a − a ∗ )Da Gs ≥ 0, ∀a ∗ ≥ 0, over Γ + .

.

(5.81)

Evaluation of .Da Gs Along the interface, displacement is continuous then velocities satisfy the Hadamard’s relation: .v2 + a∇u2 .ν = v1 + a∇u1 .ν. (5.82) To evaluate .Da Gs , the derivation is made term by term, the first one is the jump of free energy Da (ρ[ ψ ]Γ ) = −σ 2 : (∇v2 + a∇∇u2 .ν) + σ 1 : (∇v1 + a∇∇u1 .ν),

.

(5.83)

then Da G = Da [ ρψ ]Γ − Da σ 2 : [ ∇u ]Γ − σ 2 : [ Da ∇u ]Γ = [ σ ]Γ : (∇v1 + a∇∇u.ν) − (σ˙ 2 + a∇σ 2 .ν) : [ ∇u ]Γ ,

.

hence after regrouping of terms .

Da G = [ σ ]Γ : ∇v1 − σ˙ 2 : [ ∇u ]Γ − aG n ,

(5.84)

G n = −[ σ ]Γ : (∇∇u1 .ν) + ∇σ 2 .ν : [ ∇u ]Γ .

(5.85)

where .

5.6 The Rate Boundary Value Problem

137

5.6 The Rate Boundary Value Problem The solution (.v˜ , a) ˜ must satisfy: • the constitutive law: .σ˙ = Ci : ε˙ , over .Ω • the compatibility for strain and displacement: .ε˙ =

1 (∇v + ∇ T v) over .Ω, and the 2

boundary conditions . v = vd along .∂Ωu , • the conservation of the momentum: .div σ˙ = 0 over .Ω, and .σ˙ .n = T˙ d along .∂ΩT , • the compatibility conditions along the moving perfect interface: .[ Da v ]Γ = 0, .[ Da (σ .ν) ]Γ = 0, • the propagation law: .∀β ∈ K, .(β − a)Da G ≥ 0. This system is written in a global formulation. The rate boundary value problem: The evolution is determined by the functional ∫ 1 ε(v) : C : ε(v) dΩ − T˙ d .v dS 2 Ω ∂ωT ∫ ∫ 1 2 a G n dS. − a[ σ ]Γ : ∇v1 dS + Γ Γ 2

F (˜v, a, ˜ T˙ d ) =

.

∫

The solution satisfies the inequality 0≤

.

∂F ∂F (v − v∗ ) + (β − a), ∂v ∂a

(5.86)

among the set .K.A of admissible fields .(v∗ , a ∗ ): } { K.A = ( v, a)| v = vd over ∂Ωu , [ v ]Γ + a[ ∇u ]Γ = 0, a ∈ KΓ ,

.

KΓ = {β|β ≥ 0 on Γ + , β = 0 otherwise}.

.

Elements of Proof. The variations of the functional are given by ∫

ε(v) : C : ε(δv) dΩ − T˙ d .δv dS ∂ΩT ∫ ∫ ∫ − δa[ σ ]Γ : ∇v1 dS + aδaG n dS − a[ σ ]Γ : ∇δv1 dS,

δF =

.

∫

Ω

Γ

after integration by part we obtain:

Γ

Γ

(5.87) (5.88)

138

5 Moving Discontinuities

∫

∫

(σ˙ .n − T˙ d ).δv dS − Γ ∂ΩT ∫ ∫ − δa[ σ ]Γ : ∇v1 dS + aδaG n dS.

∫

n.[ σ˙ .δv ]Γ dS +

δF =

.

Γ

Γ

a[ σ ]Γ : ∇δv1 dS

Γ

Using now the compatibility conditions for the variations: δv = 0, over ∂Ωu , δv2 + δa∇u2 .ν = δv1 + δa∇u1 .ν, over Γ,

.

(5.89)

we obtain finally: ∫ δF =

.

Γ

( ) ν.[ σ˙ ]Γ − divΓ (a[ σ ]Γ ) .δv1 dS +

∫ −

Γ

∫ ∂ΩT

(n.σ˙ − T˙ d ).δv dS

δa([ σ ]Γ : ∇v1 − σ˙ 2 .[ ∇u ]Γ − a G n ) dS.

Hence, we recover the conservation of the momentum and the propagation law.

5.6.1 Stability and Bifurcation The discussion of the stability and bifurcation along an evolution process can be investigate as presented in [4]. Consider the velocity .v solution of the rate boundary value problem for any given velocity .a. The field .v satisfies: ∂ 2ψ : ε(v), over Ω, . ∂ε∂ε v = vd along ∂Ωu , σ˙ .n = T˙ d along ∂ΩT , 0 = div σ˙ , σ˙ = ρ

(5.90)

and non classical boundary conditions on .Γ: Da ([ σ ]Γ .ν) = 0, Da [ u ]Γ = 0.

.

(5.91)

Consider the value .W of .F for this solution . v(a, vd , T˙ d ): .

W (a, vd , T˙ d ) = F ( v(a, vd , T˙ d ), a, T˙ d ).

(5.92)

The stability of the actual state is determined by the condition of the existence of a solution ∂2W δa ≥ 0, δa ≥ 0 on Γ + , δa /= 0, (5.93) .δa ∂a∂a and the uniqueness and non bifurcation is characterized by

5.7 An Example

139

δa

.

∂2W δa ≥ 0, δa /= 0 on Γ + . ∂a∂a

(5.94)

The functional .W has the general form ∫ ∫ .

W =

Γ

1 a(s).B(s, s ' )a(s ' ) dS dS ' − 2

Γ

∫

¯ Q.a(s) dS,

(5.95)

Γ

where .B(s, s ' ) is an integral operator.

5.7 An Example Consider a composite sphere, whose kernel and shell are linear isotropic elastic material with different moduli. The sphere is submitted to an isotropic loading, the radial displacement is prescribed on it’s external boundary (.r = Re ). The solution of the equilibrium problem is a radial displacement (Fig. 5.4) u = u i (r )er , u i (r ) = Ai r +

.

Bi . r2

(5.96)

The boundary conditions imposed: u (Re ) = Δ Re , u 2 (0) = 0.

(5.97)

. 1

For a given history of .Δ, the reaction over the external surface is radial: σ 1 (Re ).er = T er .

(5.98)

.

Along the interface .Γ the energy release rate has the value G(Ri , Δ) =

.

9Δ2 (κ1 − κ2 )(3κ2 + 6μ1 )(3κ1 + 4μ1 ), D 2 (c)

Fig. 5.4 The composite sphere

1

Ri

2

(5.99)

Re

140

5 Moving Discontinuities

Fig. 5.5 The response of the composite sphere

Σ

Σ = 3κ1 E

Σ = 3κ2 E

Eco Ec (co )

where .

D(c) = 3κ2 + 4μ1 + 3c(κ1 − κ2 ), c =

ET

E

Ri3 . Re3

The loading parameter .Δ is increasing. Initially, the kernel does not evolve, the critical value .G c is not reached. At one time the critical value is reached and the radius of the kernel increases. The actual value of . Ri is determined by the implicit equation .G(Ri (t), Δ(t)) = G c , (5.100) this is the consistency condition. At time .to , the kernel has a radius . Ri (to ), for a decreasing loading .Δ(t) < Δ(to ), t > to , .G(Ri (to ), Δ(t)) < G c , then the composite sphere has the answer of an elastic heterogeneous medium with volume fraction .c = Ri (to )3 /Re3 . The global bulk . K (c) modulus decreases with the transformation. ∑ = 3K (c)Δ, . (K 2 + 4μ1 )K 1 − 4μ1 c(K 1 − K 2 ) . K (c) = D(c)

(5.101)

Under increasing loading and the given propagation law for the interface we have, successively ⎧ Δ < Δc , G(Ri , Δ) < G c , ⎪ ⎪ ⎪ ⎨Δ > Δ , G(R (t), Δ(t)) = G , c i c . ⎪ , G(R , Δ ) = G Δ T e T c ⎪ ⎪ ⎩ Δ > ΔT , and the answer is plotted as in Fig. 5.5.

Ri (t) = Ri (0) ⇒ Ri (t) < R Ri (T ) = Re Ri (t) = Re

(5.102)

5.8 Connection with Fracture

141 Σ Σc 1.0

Fig. 5.6 The sharp interface for a cavity: .σrr (R), for . R/Re = 0.2, 0.4, 1, . Ri (0)/Re = 0.1

0.8 0.6 0.4 0.2 0

0

Fig. 5.7 A quasicrack

2

4

6

8

Δ Δc

10

y

φ (s) Γt x

For a cavity, the evolution of the radial stress .σrr (r ) is shown on Fig. 5.6, for different values of .r . The propagation of damage is viewed regarding the value . Ri (t), σrr (Ri (t)) = 0 as function of .Δ (Fig. 5.6).

5.8 Connection with Fracture Consider a crack in mode III in an infinite medium, the displacement has the form u = w(x, y)ez ,

.

and the stress field is 1 KI I I 1 (− sin( θ ), cos( θ )). σ i3 → √ 2 2 2πr

.

142

5 Moving Discontinuities

At micro-scale that the crack can be viewed as a layer of thickness .h with a continuous boundary along which the criterion .G ≤ G c is satisfied, see Fig. 5.7. The applied loading are the stresses obtained by asymptotic expansion in mode III, at infinity. That is a matching condition between the classical crack and the quasicrack. The solution exists and we found that .Γ is a cycloid. [19, 20]. If we compare the dissipation for a steel obtained by classical fracture mechanics and the dissipation obtained by damaged modelling the thickness of the damaged zone is evaluated: 2 .G c h = K c → h : 50μ − 100μ. (5.103) This value is obtained by considering that the critical value .G c is the elastic energy at a strain of order one percent and . K c by the magnitude value for classical steel. More general results are given by [21] and more recently in [22, 23]. In fracture mechanics it is usually assumed that far from the crack tip the material is undamaged. The concentration of strain near the crack tip initiates voids and micro-cracks, the behaviour near the tip is rather quite different from the behaviour in the bulk. Then modelling of damage can be a good approximation to describe such situation.

5.8.1 The Quasi-Crack Problem The classical asymptotic expansion in linear elasticity of the field near the crack tip is considered as an outer expansion of the displacement according to the distance of the crack tip. For the inner expansion, we assume a non-linear elastic behaviour. For finite anti-plane shear, the displacement, near the tip, can exhibit singularities or discontinuities of its spatial gradient depending on the strain-stress curve as shown in [24, 25]. In these studies, the constitutive behaviour is defined for any amount of shear, which can be unbounded. For elastic power law hardening material in mode III, the HRR field can be recovered [26–28]. In 1968, Neuber [20] investigates the stress-concentration under mode III for a notch. For a non linear stress-strain laws monotonically increasing with shear, and a quasi-crack which consists of two parallel straight lines ended by a cycloid, the stress is uniform along the cycloid. Continuous damage has been used to investigate the asymptotic solution of mode III in damaged softening material [29, 30]. For elastic-brittle material, with rupture governs by a critical stretch, a closed form solution for the dynamic propagation of mode-III has been obtained [31], Under quasi-static conditions, the shape of the damaged zone is a quasi-crack [31], thickness of which is determined by the value of the stress intensity factor applied at infinity and the critical value of shear amount.

5.8 Connection with Fracture

143

τ

Fig. 5.8 Typical constitutive behaviour

α>0

τo τ1

α=0

μo

α