Numerical Continuum Mechanics 9783110273380, 9783110273229

Table of contents :
Preface
I Basic equations of continuum mechanics
1 Basic equations of continuous media
1.1 Methods of describing motion of continuous media
1.1.1 Coordinate systems and methods of describing motion of continuous media
1.1.2 Eulerian description
1.1.3 Lagrangian description
1.1.4 Differentiation of bases
1.1.5 Description of deformations and rates of deformation of a continuous medium
1.2 Conservation laws. Integral and differential forms
1.2.1 Integral form of conservation laws
1.2.2 Differential form of conservation laws
1.2.3 Conservation laws at solution discontinuities
1.2.4 Conclusions
1.3 Thermodynamics
1.3.1 First law of thermodynamics
1.3.2 Second law of thermodynamics
1.3.3 Conclusions
1.4 Constitutive equations
1.4.1 General form of constitutive equations. Internal variables
1.4.2 Equations of viscous compressible heat-conducting gases
1.4.3 Thermoelastic isotropic media
1.4.4 Combined media
1.4.5 Rigid-plastic media with translationally isotropic hardening
1.4.6 Elastoplastic model
1.5 Theory of plastic flow. Theory of internal variables
1.5.1 Statement of the problem. Equations of an elastoplastic medium
1.5.2 Equations of an elastoviscoplastic medium
1.6 Experimental determination of constitutive relations under dynamic loading
1.6.1 Experimental results and experimentally obtained constitutive equations
1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory
1.7 Principle of virtual displacements. Weak solutions to equations of motion
1.7.1 Principles of virtual displacements and velocities
1.7.2 Weak formulation of the problem of continuum mechanics
1.8 Variational principles of continuum mechanics
1.8.1 Lagrange’s variational principle
1.8.2 Hamilton’s variational principle
1.8.3 Castigliano’s variational principle
1.8.4 General variational principle for solving continuum mechanics problems
1.8.5 Estimation of solution error
1.9 Kinematics of continuous media. Finite deformations
1.9.1 Description of the motion of solids at large deformations
1.9.2 Motion: deformation and rotation
1.9.3 Strain measure. Green-Lagrange and Euler-Almansi strain tensors
1.9.4 Deformation of area and volume elements
1.9.5 Transformations: initial, reference, and intermediate configurations
1.9.6 Differentiation of tensors. Rate of deformation measures
1.10 Stress measures
1.10.1 Current configuration. Cauchy stress tensor
1.10.2 Current and initial configurations. The first and second Piola-Kirchhoff stress tensors
1.10.3 Measures of the rate of change of stress tensors
1.11 Variational principles for finite deformations
1.11.1 Principle of virtual work
1.11.2 Statement of the principle in increments
1.12 Constitutive equations of plasticity under finite deformations
1.12.1 Multiplicative decomposition. Deformation gradients
1.12.2 Material description
1.12.3 Spatial description
1.12.4 Elastic isotropic body
1.12.5 Hyperelastoplastic medium
1.12.6 The von Mises yield criterion
II Theory of finite-difference schemes
2 The basics of the theory of finite-difference schemes
2.1 Finite-difference approximations for differential operators
2.1.1 Finite-difference approximation
2.1.2 Estimation of approximation error
2.1.3 Richardson’s extrapolation formula
2.2 Stability and convergence of finite difference equations
2.2.1 Stability
2.2.2 Lax convergence theorem
2.2.3 Example of an unstable finite difference scheme
2.3 Numerical integration of the Cauchy problem for systems of equations
2.3.1 Euler schemes
2.3.2 Adams-Bashforth scheme
2.3.3 Construction of higher-order schemes by series expansion
2.3.4 Runge-Kutta schemes
2.4 Cauchy problem for stiff systems of ordinary differential equations
2.4.1 Stiff systems of ordinary differential equations
2.4.2 Numerical solution
2.4.3 Stability analysis
2.4.4 Singularly perturbed systems
2.4.5 Extension of a rod made of a nonlinear viscoplastic material
2.5 Finite difference schemes for one-dimensional partial differential equations
2.5.1 Solution of the wave equation in displacements. The cross scheme
2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations)
2.5.3 The leapfrog scheme
2.5.4 The Lax-Friedrichs scheme
2.5.5 The Lax-Wendroff Scheme
2.5.6 Scheme viscosity
2.5.7 Solution of the wave equation. Implicit scheme
2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points
2.5.9 Heat equation
2.5.10 Unsteady thermal conduction. Explicit scheme (forward Euler scheme)
2.5.11 Unsteady thermal conduction. Implicit scheme (backward Euler scheme)
2.5.12 Unsteady thermal conduction. Crank-Nicolson scheme
2.5.13 Unsteady thermal conduction. Allen-Cheng explicit scheme
2.5.14 Unsteady thermal conduction. Du Fort-Frankel explicit scheme
2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives
2.6 Stability analysis for finite difference schemes
2.6.1 Stability of a two-layer finite difference scheme
2.6.2 The von Neumann stability condition
2.6.3 Stability of the wave equation
2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition
2.6.5 Stability of schemes for the heat equation
2.6.6 The principle of frozen coefficients
2.6.7 Stability in solving boundary value problems
2.6.8 Step size selection in an implicit scheme in solving the heat equation
2.6.9 Step size selection in solving the wave equation
2.7 Exercises
3 Methods for solving systems of algebraic equations
3.1 Matrix norm and condition number of matrix
3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix
3.1.2 Relative error of solution for perturbed coefficient matrix
3.1.3 Example
3.1.4 Regularization of an ill-conditioned system of equations
3.2 Direct methods for linear system of equations
3.2.1 Gaussian elimination method. Matrix factorization
3.2.2 Gaussian elimination with partial pivoting
3.2.3 Cholesky decomposition. The square root method
3.3 Iterative methods for linear system of equations
3.3.1 Single-step iterative processes
3.3.2 Seidel and Jacobi iterative processes
3.3.3 The stabilization method
3.3.4 Optimization of the rate of convergence of a steady-state process
3.3.5 Optimization of unsteady processes
3.4 Methods for solving nonlinear equations
3.4.1 Nonlinear equations and iterative methods
3.4.2 Contractive mappings. The fixed point theorem
3.4.3 Method of simple iterations. Sufficient convergence condition
3.5 Nonlinear equations: Newton’s method and its modifications
3.5.1 Newton’s method
3.5.2 Modified Newton-Raphson method
3.5.3 The secant method
3.5.4 Two-stage iterative methods
3.5.5 Nonstationary Newton method. Optimal step selection
3.6 Methods of minimization of functions (descent methods)
3.6.1 The coordinate descent method
3.6.2 The steepest descent method
3.6.3 The conjugate gradient method
3.6.4 An iterative method using spectral-equivalent operators or reconditioning
3.7 Exercises
4 Methods for solving boundary value problems for systems of equations
4.1 Numerical solution of two-point boundary value problems
4.1.1 Stiff two-point boundary value problem
4.1.2 Method of initial parameters
4.2 General boundary value problem for systems of linear equations
4.3 General boundary value problem for systems of nonlinear equations
4.3.1 Shooting method
4.3.2 Quasi-linearization method
4.4 Solution of boundary value problems by the sweep method
4.4.1 Differential sweep
4.4.2 Solution of finite difference equation by the sweep method
4.4.3 Sweep method for the heat equation
4.5 Solution of boundary value problems for elliptic equations
4.5.1 Poisson’s equation
4.5.2 Maximum principle for second-order finite difference equations
4.5.3 Stability of a finite difference scheme for Poisson’s equation
4.5.4 Diagonal domination
4.5.5 Solution of Poisson’s equation by the matrix sweep method
4.5.6 Fourier’s method of separation of variables
4.6 Stiff boundary value problems
4.6.1 Stiff systems of differential equations
4.6.2 Generalized method of initial parameters
4.6.3 Orthogonal sweep
4.7 Exercises
III Finite-difference methods for solving nonlinear evolution equations of continuum mechanics
5 Wave propagation problems
5.1 Linear vibrations of elastic beams
5.1.1 Longitudinal vibrations
5.1.2 Explicit scheme. Sufficient stability conditions
5.1.3 Longitudinal vibrations. Implicit scheme
5.1.4 Transverse vibrations
5.1.5 Transverse vibrations. Explicit scheme
5.1.6 Transverse vibrations. Implicit scheme
5.1.7 Coupled longitudinal and transverse vibrations
5.1.8 Transverse bending of a plate with shear and rotational inertia
5.1.9 Conclusion
5.2 Solution of nonlinear wave propagation problems
5.2.1 Hyperbolic system of equations and characteristics
5.2.2 Finite difference approximation along characteristics. The direct and semi-inverse methods
5.2.3 Inverse method. The Courant-Isaacson-Rees grid-characteristic scheme
5.2.4 Wave propagation in a nonlinear elastic beam
5.2.5 Wave propagation in an elastoviscoplastic beam
5.2.6 Discontinuous solutions. Constant coefficient equation
5.2.7 Discontinuous solutions of a nonlinear equation
5.2.8 Stability of difference characteristic equations
5.2.9 Characteristic and grid-characteristic schemes for solving stiff problems
5.2.10 Stability of characteristic and grid-characteristic schemes for stiff problems
5.2.11 Characteristic schemes of higher orders of accuracy
5.3 Two- and three-dimensional characteristic schemes and their application
5.3.1 Spatial characteristics
5.3.2 Basic equations of elastoviscoplastic media
5.3.3 Spatial three-dimensional characteristics for semi-linear system
5.3.4 Characteristic equations. Spatial problem
5.3.5 Axisymmetric problem
5.3.6 Difference equations. Axisymmetric problem
5.3.7 A brief overview of the results. Further development and generalization of the method of spatial characteristics and its application to the solution of dynamic problems
5.4 Coupled thermomechanics problems
5.5 Differential approximation for difference equations
5.5.1 Hyperbolic and parabolic forms of differential approximation
5.5.2 Example
5.5.3 Stability
5.5.4 Analysis of dissipative and dispersive properties
5.5.5 Example
5.5.6 Analysis of properties of finite difference schemes for discontinuous solutions
5.5.7 Smoothing of non-physical perturbations in a calculation on a real grid
5.6 Exercises
6 Finite-difference splitting method for solving dynamic problems
6.1 General scheme of the splitting method
6.1.1 Explicit splitting scheme
6.1.2 Implicit splitting scheme
6.1.3 Stability
6.2 Splitting of 2D/3D equations into 1D equations (splitting along directions)
6.2.1 Splitting along directions of initial-boundary value problems for the heat equation
6.2.2 Splitting schemes for the wave equation
6.3 Splitting of constitutive equations for complex rheological models into simple ones. A splitting scheme for a viscous fluid
6.3.1 Divergence form of equations
6.3.2 Non-divergence form of equations
6.3.3 One-dimensional equations. Ideal gas
6.3.4 Implementation of the scheme
6.4 Splitting scheme for elastoviscoplastic dynamic problems
6.4.1 Constitutive equations of elastoplastic media
6.4.2 Some approaches to solving elastoplastic equations
6.4.3 Splitting of the constitutive equations
6.4.4 The theory of von Mises type flows. Isotropic hardening
6.4.5 Drucker-Prager plasticity theory
6.4.6 Elastoviscoplastic media
6.5 Splitting schemes for points on the axis of revolution
6.5.1 Calculation of boundary points
6.5.2 Calculation of axial points
6.6 Integration of elastoviscoplastic flow equations by variation inequality
6.6.1 Variation inequality
6.6.2 Dissipative schemes
6.7 Exercises
7 Solution of elastoplastic dynamic and quasistatic problems with finite deformations
7.1 Conservative approximations on curvilinear Lagrangian meshes
7.1.1 Formulas for natural approximation of spatial derivatives
7.1.2 Approximation of a Lagrangian mesh
7.1.3 Conservative finite difference schemes
7.2 Finite elastoplastic deformations
7.2.1 Conservative schemes in one-dimensional case
7.2.2 A conservative two-dimensional scheme for an elastoplastic medium
7.2.3 Splitting of the equations of a hypoelastic material
7.3 Propagation of coupled thermomechanical perturbations in gases
7.3.1 Basic equations
7.3.2 Conservative finite difference scheme
7.3.3 Non-divergence form of the energy equation. A completely conservative scheme
7.4 The PIC method and its modifications for solid mechanics problems
7.4.1 Disadvantages of Lagrangian and Eulerian meshes
7.4.2 The particle-in-cell (PIC) method
7.4.3 The method of coarse particles
7.4.4 Limitations of the PIC method and its modifications
7.4.5 The combined flux and particle-in-cell (FPIC) method
7.4.6 The method of markers and fluxes
7.5 Application of PIC-type methods to solving elastoviscoplastic problems
7.5.1 Hypoelastic medium
7.5.2 Hypoelastoplastic medium
7.5.3 Splitting for a hyperelastoplastic medium
7.6 Optimization of moving one-dimensional meshes
7.6.1 Optimal mesh for a given function
7.6.2 Optimal mesh for solving an initial-boundary value problem
7.6.3 Mesh optimization in several parameters
7.6.4 Heat propagation from a combustion source
7.7 Adaptive 2D/3D meshes for finite deformation problems
7.7.1 Methods for reorganization of a Lagrangian mesh
7.7.2 Description of motion in an arbitrary moving coordinate system
7.7.3 Adaptive meshes
7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes
7.8.1 Algorithms for constructing moving meshes
7.8.2 Selection of a finite difference scheme
7.8.3 A hybrid scheme of variable order of approximation at internal nodes
7.8.4 A grid-characteristic scheme at boundary nodes
7.8.5 Calculation of contact boundaries
7.8.6 Calculation of damage kinetics
7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains
7.9 Exercises
8 Modeling of damage and fracture of inelastic materials and structures
8.1 Concept of damage and the construction of models of damaged media
8.1.1 Concept of continuum fracture and damage
8.1.2 Construction of damage models
8.1.3 Constitutive equations of the GTN model
8.2 Generalized micromechanical multiscale damage model
8.2.1 Micromechanical model. The stage of plastic flow and hardening
8.2.2 Stage of void nucleation
8.2.3 Stage of the appearance of voids and damage
8.2.4 Relationship between micro and macro parameters
8.2.5 Macromodel
8.2.6 Tension of a thin rod with a constant strain rate
8.2.7 Conclusion
8.3 Numerical modeling of damaged elastoplastic materials
8.3.1 Regularization of equations for elastoplastic materials at softening
8.3.2 Solution of damage problems
8.3.3 Inverse Euler method
8.3.4 Solution of a boundary value problem. Computation of the Jacobian
8.3.5 Splitting method
8.3.6 Integration of the constitutive relations of the GTN model
8.3.7 Uniaxial tension. Computational results
8.3.8 Bending of a plate
8.3.9 Comparison with experiment
8.3.10 Modeling quasi-brittle fracture with damage
8.4 Extension of damage theory to the case of an arbitrary stress-strain state
8.4.1 Well-posedness of the problem
8.4.2 Limitations of the GTN model
8.4.3 Associated viscoplastic law
8.4.4 Constitutive relations in the absence of porosity (k < 0.4, f = 0, σr = 0)
8.4.5 Fracture model. Fracture criteria
8.5 Numerical modeling of cutting of elastoviscoplastic materials
8.5.1 Introduction
8.5.2 Statement of the problem
8.6 Conclusions. General remarks on elastoplastic equations
8.6.1 Formulations of systems of equations for elastoplastic media
8.6.2 A hardening elastoplastic medium
8.6.3 Ideal elastoplastic media: a degenerate case
8.6.4 Difficulties in solving mixed elliptic-hyperbolic problems
8.6.5 Regularization of an elastoplastic model
8.6.6 Elastoplastic shock waves
Bibliography
Index

Citation preview

De Gruyter Studies in Mathematical Physics 15 Editors Michael Efroimsky, Bethesda, USA Leonard Gamberg, Reading, USA Dmitry Gitman, São Paulo, Brasil Alexander Lazarian, Madison, USA Boris Smirnov, Moscow, Russia

Vladimir N. Kukudzhanov

Numerical Continuum Mechanics

De Gruyter

Physics and Astronomy Classification 2010: 46.15.-x, 02.70.Bf, 02.60.Lj, 46.05.+b, 46.50.+a, 46.35.+z, 46.70.De, 46.70.Hg, 83.60.Df, 62.20.-x, 61.72.Lk, 81.40.Cd

ISBN 978-3-11-027322-9 e-ISBN 978-3-11-027338-0 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen  Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The book is devoted to computational methods in continuum thermomechanics and is based on the lecture course the author delivered for 15 years to students of the Moscow Institute of Physics and Technology and the Tsiolkovskii Russian State Technological University. The book has a dual purpose: educational, to introduce the reader to advanced foundations of computational mechanics, and scientific, to acquaint the reader, by more complex examples, with the state-of-the-art directions of research and lead them to an independent research work in this area of continuum mechanics. The presentation of the material is intended for engineering physicists rather than computational mathematicians. The book does not aim to give rigorous substantiation of methods. Proofs of theorems are often avoided. Presentation of ideas and qualitative considerations, illustrated by specific examples related to important applications, is preferred to rigorous exposition of theorems with detailed statements of all conditions imposed on the functions involved. At the same time, the conditions and limitations that are essential for practical applications of methods are discussed quite thoroughly. Insufficient attention to such conditions can result in serious errors. This style of presentation is justified by the fact that the book is primarily intended for engineers and physicists, who are more interested in the essence of the problem in question rather than a formally rigorous approach to its solution. Along with the solution of typical examples, illustrating the application of methods, conditions that may lead to inefficiency of the methods are discussed and possible ways of overcoming drawbacks are suggested. The book is organized into parts, chapters and sections, which are numbered sequentially. The formulas and figures have double numbering, with the first number indicating the section and the second showing the number within the section. The facultative information – such as proofs of theorems, subtleties of the application of methods, and so on – is given in smaller type and may be skipped at first reading. Part I, consisting of Chapter 1, covers issues and statements of problems that differ from traditional approaches in the modern nonlinear continuum mechanics and are rarely included in the educational literature or differ in the formulation of equations specially adapted for efficient numerical analysis. Discussed in Chapter 1 are integral and divergence forms of conservation laws, variational principles and generalized solutions of continuum mechanics, the thermodynamic theory of continuous media with internal variables allowing the description of material structure, constitutive equations of composite media with complex rheology, elastoviscoplastic media with damage,

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Preface

and the theory of large elastoplastic deformations. Also discussed are nonclassical methods of describing the motion of continua, including mixed Lagrangian–Eulerian methods and description in arbitrary adaptable moving coordinates. The presentation is performed in such a way that the reader does not have to use additional literature to learn the material. The chapter can be useful to more advanced readers as well. Part II, comprising Chapters 2 to 4, outlines the basics of numerical methods for the solution of finite difference equations. This part is close to the content of traditional courses on numerical methods with focus on efficient methods for nonlinear problems of continuum mechanics. Solution methods for stiff and singularly perturbed boundary-value problems and nonlinear wave unsteady problems are discussed. Stability analysis methods using differential approximations are outlined. Part III (Chapters 5 to 8) gives the description and development of special numerical methods of continuum mechanics and also discusses their application to solving certain classes of one-dimensional and multidimensional unsteady dynamic problems and generalization to two- or three-dimensional problems for elastic and elastoviscoplastic media. Finite difference schemes for unsteady problems with discontinuous solutions are analyzed by the method of differential approximations. The methods of splitting in directions and physical processes for media with complex rheological relations are developed; these methods allow one to reduce complex problems to successive solution of problems for simpler media. Efficient numerical-analytical methods for elastoplastic and elastoviscoplastic problems in two or more dimensions are suggested. Special methods are considered that allow one to solve problems involving large or very large deformations of elastoplastic solids under extreme thermomechanical loads. These methods are based on nonclassical mixed Lagrangian–Eulerian approaches to the description of the motion of continuous media, adaptable moving grid techniques, and the particle-in-cell technique and its modifications. Solutions of several problems are given: penetration of a rigid indenter into an elastoplastic material with fracture, formation of a cumulative jet under the action of a detonation wave, indentation of a sine-shaped rigid stamp into an elastoplastic material, fracture of an elastic layer (glass) when impacted by a steel cylinder, impact of a deformable cylindrical projectile on a deformable slab at a supersonic speed and their fracture, and some others. Chapter 8 deals with damage and continuum fracture of elastoplastic and elastoviscoplastic media with defects, under quasistatic and dynamic thermomechanical actions. Also included in Chapters 5 to 8 are new results, which only appear in journal publications and are not included in the educational literature. The new methods can be useful to the advanced readers who specialize in numerical simulation of continuum mechanical problems as well as to postgraduate and PhD students. At the end of Chapters 2 to 7, there are numerous exercises designed to supplement the text and consolidate the concepts discussed. They serve to stimulate the reader to further study and to reinforce and develop practical skills.

Preface

vii

The book has an extended table of contents to help the reader quickly locate the desired information. The brief list of notations includes symbols and terms most frequently used in computational mathematics and mechanics. The bibliography includes references cited in the text to indicate the authors who contributed to the results and refer the interested reader to more detailed information and other educational literature. The book is intended for graduate and postgraduate students in the area of applied mathematics, mechanics, and engineering sciences, who are acquainted with the basics of mechanics of continuous media and main concepts of computational mathematics. The first two parts of the book aim at extending the reader’s knowledge in these disciplines. The book may also be helpful for a wide range of engineers, scientists, university teachers, and PhD students engaged in the fields of computational mathematics and mechanics of continuous media. I am very grateful to my colleagues and pupils Nikolai Burago, Alexander Levitin, and Sergei Lychev for their valuable comments and fruitful discussions, which helped improve the book. I would also like to thank Alexei Zhurov for translating the manuscript into English thoroughly and conscientiously. Moscow, October 2012

Vladimir Kukudzhanov

Contents

Preface

v

I Basic equations of continuum mechanics 1

Basic equations of continuous media

3

1.1 Methods of describing motion of continuous media . . . . . . . . . . . . . . . 1.1.1 Coordinate systems and methods of describing motion of continuous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Eulerian description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Lagrangian description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Differentiation of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Description of deformations and rates of deformation of a continuous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 5 5 7

1.2 Conservation laws. Integral and differential forms . . . . . . . . . . . . . . . . 9 1.2.1 Integral form of conservation laws . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Differential form of conservation laws . . . . . . . . . . . . . . . . . . . 11 1.2.3 Conservation laws at solution discontinuities . . . . . . . . . . . . . . 13 1.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 First law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 18

1.4 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 General form of constitutive equations. Internal variables . . . . 1.4.2 Equations of viscous compressible heat-conducting gases . . . . 1.4.3 Thermoelastic isotropic media . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Combined media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Rigid-plastic media with translationally isotropic hardening . . 1.4.6 Elastoplastic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18 18 21 21 22 24 25

1.5 Theory of plastic flow. Theory of internal variables . . . . . . . . . . . . . . . 26 1.5.1 Statement of the problem. Equations of an elastoplastic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5.2 Equations of an elastoviscoplastic medium . . . . . . . . . . . . . . . . 30

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1.6 Experimental determination of constitutive relations under dynamic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6.1 Experimental results and experimentally obtained constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.7 Principle of virtual displacements. Weak solutions to equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.7.1 Principles of virtual displacements and velocities . . . . . . . . . . 40 1.7.2 Weak formulation of the problem of continuum mechanics . . . 42 1.8 Variational principles of continuum mechanics . . . . . . . . . . . . . . . . . . . 1.8.1 Lagrange’s variational principle . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Hamilton’s variational principle . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Castigliano’s variational principle . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 General variational principle for solving continuum mechanics problems . . . . . . . . . . . . . . . . . . . . . . . . 1.8.5 Estimation of solution error . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 44 45

1.9 Kinematics of continuous media. Finite deformations . . . . . . . . . . . . . 1.9.1 Description of the motion of solids at large deformations . . . . 1.9.2 Motion: deformation and rotation . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Strain measures. Green–Lagrange and Euler–Almansi strain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Deformation of area and volume elements . . . . . . . . . . . . . . . . 1.9.5 Transformations: initial, reference, and intermediate configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.6 Differentiation of tensors. Rate of deformation measures . . . .

49 49 50

1.10 Stress measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Current configuration. Cauchy stress tensor . . . . . . . . . . . . . . . 1.10.2 Current and initial configurations. The first and second Piola–Kirchhoff stress tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.3 Measures of the rate of change of stress tensors . . . . . . . . . . . .

57 57

46 49

52 53 54 55

57 59

1.11 Variational principles for finite deformations . . . . . . . . . . . . . . . . . . . . 60 1.11.1 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.11.2 Statement of the principle in increments . . . . . . . . . . . . . . . . . . 60 1.12 Constitutive equations of plasticity under finite deformations . . . . . . . 1.12.1 Multiplicative decomposition. Deformation gradients . . . . . . . 1.12.2 Material description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.3 Spatial description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.4 Elastic isotropic body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 64 65

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1.12.5 Hyperelastoplastic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.12.6 The von Mises yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 66

II Theory of finite-difference schemes 2

The basics of the theory of finite-difference schemes

71

2.1 Finite-difference approximations for differential operators . . . . . . . . . . 2.1.1 Finite-difference approximation . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Estimation of approximation error . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Richardson’s extrapolation formula . . . . . . . . . . . . . . . . . . . . .

71 71 73 77

2.2 Stability and convergence of finite difference equations . . . . . . . . . . . . 2.2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Lax convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Example of an unstable finite difference scheme . . . . . . . . . . .

78 78 78 79

2.3 Numerical integration of the Cauchy problem for systems of equations 2.3.1 Euler schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Adams–Bashforth scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Construction of higher-order schemes by series expansion . . . 2.3.4 Runge–Kutta schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 82 83 85 85

2.4 Cauchy problem for stiff systems of ordinary differential equations . . 2.4.1 Stiff systems of ordinary differential equations . . . . . . . . . . . . 2.4.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Singularly perturbed systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Extension of a rod made of a nonlinear viscoplastic material . .

88 88 89 90 91 92

2.5 Finite difference schemes for one-dimensional partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Solution of the wave equation in displacements. The cross scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations) . . . . . . . . . . . . . . . . 2.5.3 The leapfrog scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 The Lax–Friedrichs scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 The Lax–Wendroff Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Scheme viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 Solution of the wave equation. Implicit scheme . . . . . . . . . . . . 2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points . . . . . . . . . . . . . . . . . . . . . . 2.5.9 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.10 Unsteady thermal conduction. Explicit scheme (forward Euler scheme) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95 95 96 97 97 98 99 100 100 101 103

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2.5.11 Unsteady thermal conduction. Implicit scheme (backward Euler scheme) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.12 Unsteady thermal conduction. Crank–Nicolson scheme . . . . . 2.5.13 Unsteady thermal conduction. Allen–Cheng explicit scheme . 2.5.14 Unsteady thermal conduction. Du Fort–Frankel explicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Stability analysis for finite difference schemes . . . . . . . . . . . . . . . . . . . 2.6.1 Stability of a two-layer finite difference scheme . . . . . . . . . . . . 2.6.2 The von Neumann stability condition . . . . . . . . . . . . . . . . . . . . 2.6.3 Stability of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition . . . . . . . . . . . . . . . . 2.6.5 Stability of schemes for the heat equation . . . . . . . . . . . . . . . . 2.6.6 The principle of frozen coefficients . . . . . . . . . . . . . . . . . . . . . . 2.6.7 Stability in solving boundary value problems . . . . . . . . . . . . . . 2.6.8 Step size selection in an implicit scheme in solving the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.9 Step size selection in solving the wave equation . . . . . . . . . . . .

103 103 103 104

104 106 107 107 108 109 112 113 115 116 117

2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 3

Methods for solving systems of algebraic equations

122

3.1 Matrix norm and condition number of matrix . . . . . . . . . . . . . . . . . . . . 3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Relative error of solution for perturbed coefficient matrix . . . . 3.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Regularization of an ill-conditioned system of equations . . . . .

122 122 123 124 125

3.2 Direct methods for linear system of equations . . . . . . . . . . . . . . . . . . . 3.2.1 Gaussian elimination method. Matrix factorization . . . . . . . . . 3.2.2 Gaussian elimination with partial pivoting . . . . . . . . . . . . . . . . 3.2.3 Cholesky decomposition. The square root method . . . . . . . . . .

126 126 127 128

3.3 Iterative methods for linear system of equations . . . . . . . . . . . . . . . . . . 3.3.1 Single-step iterative processes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Seidel and Jacobi iterative processes . . . . . . . . . . . . . . . . . . . . . 3.3.3 The stabilization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Optimization of the rate of convergence of a steady-state process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Optimization of unsteady processes . . . . . . . . . . . . . . . . . . . . .

130 130 131 133 135 137

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3.4 Methods for solving nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Nonlinear equations and iterative methods . . . . . . . . . . . . . . . . 3.4.2 Contractive mappings. The fixed point theorem . . . . . . . . . . . . 3.4.3 Method of simple iterations. Sufficient convergence condition

140 140 141 143

3.5 Nonlinear equations: Newton’s method and its modifications . . . . . . . 3.5.1 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Modified Newton–Raphson method . . . . . . . . . . . . . . . . . . . . . 3.5.3 The secant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Two-stage iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Nonstationary Newton method. Optimal step selection . . . . . .

145 145 147 147 148 149

3.6 Methods of minimization of functions (descent methods) . . . . . . . . . . 3.6.1 The coordinate descent method . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 The steepest descent method . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 The conjugate gradient method . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.4 An iterative method using spectral-equivalent operators or reconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 152 154 155 156

3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4

Methods for solving boundary value problems for systems of equations 160 4.1 Numerical solution of two-point boundary value problems . . . . . . . . . 160 4.1.1 Stiff two-point boundary value problem . . . . . . . . . . . . . . . . . . 160 4.1.2 Method of initial parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.2 General boundary value problem for systems of linear equations . . . . . 163 4.3 General boundary value problem for systems of nonlinear equations . . 164 4.3.1 Shooting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.3.2 Quasi-linearization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 4.4 Solution of boundary value problems by the sweep method . . . . . . . . . 4.4.1 Differential sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Solution of finite difference equation by the sweep method . . . 4.4.3 Sweep method for the heat equation . . . . . . . . . . . . . . . . . . . . .

166 166 170 171

4.5 Solution of boundary value problems for elliptic equations . . . . . . . . . 4.5.1 Poisson’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Maximum principle for second-order finite difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Stability of a finite difference scheme for Poisson’s equation . 4.5.4 Diagonal domination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Solution of Poisson’s equation by the matrix sweep method . . 4.5.6 Fourier’s method of separation of variables . . . . . . . . . . . . . . .

172 172 175 176 176 178 181

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4.6 Stiff boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Stiff systems of differential equations . . . . . . . . . . . . . . . . . . . . 4.6.2 Generalized method of initial parameters . . . . . . . . . . . . . . . . . 4.6.3 Orthogonal sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 185 186

4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

III Finite-difference methods for solving nonlinear evolution equations of continuum mechanics 5

Wave propagation problems

197

5.1 Linear vibrations of elastic beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Longitudinal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Explicit scheme. Sufficient stability conditions . . . . . . . . . . . . 5.1.3 Longitudinal vibrations. Implicit scheme . . . . . . . . . . . . . . . . . 5.1.4 Transverse vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Transverse vibrations. Explicit scheme . . . . . . . . . . . . . . . . . . . 5.1.6 Transverse vibrations. Implicit scheme . . . . . . . . . . . . . . . . . . . 5.1.7 Coupled longitudinal and transverse vibrations . . . . . . . . . . . . 5.1.8 Transverse bending of a plate with shear and rotational inertia 5.1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 197 199 200 202 203 204 206 209

5.2 Solution of nonlinear wave propagation problems . . . . . . . . . . . . . . . . 5.2.1 Hyperbolic system of equations and characteristics . . . . . . . . . 5.2.2 Finite difference approximation along characteristics. The direct and semi-inverse methods . . . . . . . . . . . . . . . . . . . . 5.2.3 Inverse method. The Courant–Isaacson–Rees grid-characteristic scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Wave propagation in a nonlinear elastic beam . . . . . . . . . . . . . 5.2.5 Wave propagation in an elastoviscoplastic beam . . . . . . . . . . . 5.2.6 Discontinuous solutions. Constant coefficient equation . . . . . . 5.2.7 Discontinuous solutions of a nonlinear equation . . . . . . . . . . . 5.2.8 Stability of difference characteristic equations . . . . . . . . . . . . . 5.2.9 Characteristic and grid-characteristic schemes for solving stiff problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.10 Stability of characteristic and grid-characteristic schemes for stiff problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.11 Characteristic schemes of higher orders of accuracy . . . . . . . .

209 209 211 211 212 215 219 220 222 222 224 225

5.3 Two- and three-dimensional characteristic schemes and their application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.3.1 Spatial characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 5.3.2 Basic equations of elastoviscoplastic media . . . . . . . . . . . . . . . 229

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5.3.3 5.3.4 5.3.5 5.3.6 5.3.7

Spatial three-dimensional characteristics for semi-linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic equations. Spatial problem . . . . . . . . . . . . . . . . . Axisymmetric problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference equations. Axisymmetric problem . . . . . . . . . . . . . A brief overview of the results. Further development and generalization of the method of spatial characteristics and its application to the solution of dynamic problems . . . . . . . . . . .

231 235 236 238

244

5.4 Coupled thermomechanics problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.5 Differential approximation for difference equations . . . . . . . . . . . . . . . 5.5.1 Hyperbolic and parabolic forms of differential approximation . 5.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Analysis of dissipative and dispersive properties . . . . . . . . . . . 5.5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Analysis of properties of finite difference schemes for discontinuous solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.7 Smoothing of non-physical perturbations in a calculation on a real grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

248 248 249 250 251 253 254 259

5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 6

Finite-difference splitting method for solving dynamic problems

263

6.1 General scheme of the splitting method . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Explicit splitting scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Implicit splitting scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 263 264 265

6.2 Splitting of 2D/3D equations into 1D equations (splitting along directions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.2.1 Splitting along directions of initial-boundary value problems for the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.2.2 Splitting schemes for the wave equation . . . . . . . . . . . . . . . . . . 268 6.3 Splitting of constitutive equations for complex rheological models into simple ones. A splitting scheme for a viscous fluid . . . . . . . . . . . . . . . 6.3.1 Divergence form of equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Non-divergence form of equations . . . . . . . . . . . . . . . . . . . . . . 6.3.3 One-dimensional equations. Ideal gas . . . . . . . . . . . . . . . . . . . 6.3.4 Implementation of the scheme . . . . . . . . . . . . . . . . . . . . . . . . .

270 270 272 273 275

6.4 Splitting scheme for elastoviscoplastic dynamic problems . . . . . . . . . . 6.4.1 Constitutive equations of elastoplastic media . . . . . . . . . . . . . . 6.4.2 Some approaches to solving elastoplastic equations . . . . . . . . . 6.4.3 Splitting of the constitutive equations . . . . . . . . . . . . . . . . . . . .

276 276 277 279

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6.4.4 6.4.5 6.4.6

The theory of von Mises type flows. Isotropic hardening . . . . . 281 Drucker–Prager plasticity theory . . . . . . . . . . . . . . . . . . . . . . . 283 Elastoviscoplastic media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

6.5 Splitting schemes for points on the axis of revolution . . . . . . . . . . . . . . 286 6.5.1 Calculation of boundary points . . . . . . . . . . . . . . . . . . . . . . . . . 286 6.5.2 Calculation of axial points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 6.6 Integration of elastoviscoplastic flow equations by variation inequality 290 6.6.1 Variation inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 6.6.2 Dissipative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 7

Solution of elastoplastic dynamic and quasistatic problems with finite deformations

298

7.1 Conservative approximations on curvilinear Lagrangian meshes . . . . . 7.1.1 Formulas for natural approximation of spatial derivatives . . . . 7.1.2 Approximation of a Lagrangian mesh . . . . . . . . . . . . . . . . . . . . 7.1.3 Conservative finite difference schemes . . . . . . . . . . . . . . . . . . .

298 298 299 301

7.2 Finite elastoplastic deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Conservative schemes in one-dimensional case . . . . . . . . . . . . 7.2.2 A conservative two-dimensional scheme for an elastoplastic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Splitting of the equations of a hypoelastic material . . . . . . . . .

303 303 305 306

7.3 Propagation of coupled thermomechanical perturbations in gases . . . . 7.3.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Conservative finite difference scheme . . . . . . . . . . . . . . . . . . . . 7.3.3 Non-divergence form of the energy equation. A completely conservative scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 307 307

7.4 The PIC method and its modifications for solid mechanics problems . 7.4.1 Disadvantages of Lagrangian and Eulerian meshes . . . . . . . . . 7.4.2 The particle-in-cell (PIC) method . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 The method of coarse particles . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Limitations of the PIC method and its modifications . . . . . . . . 7.4.5 The combined flux and particle-in-cell (FPIC) method . . . . . . 7.4.6 The method of markers and fluxes . . . . . . . . . . . . . . . . . . . . . .

311 311 311 314 315 316 317

7.5 Application of PIC-type methods to solving elastoviscoplastic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Hypoelastic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Hypoelastoplastic medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Splitting for a hyperelastoplastic medium . . . . . . . . . . . . . . . . .

317 318 319 321

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7.6 Optimization of moving one-dimensional meshes . . . . . . . . . . . . . . . . 7.6.1 Optimal mesh for a given function . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Optimal mesh for solving an initial-boundary value problem . . 7.6.3 Mesh optimization in several parameters . . . . . . . . . . . . . . . . . 7.6.4 Heat propagation from a combustion source . . . . . . . . . . . . . . .

324 325 326 327 328

7.7 Adaptive 2D/3D meshes for finite deformation problems . . . . . . . . . . . 7.7.1 Methods for reorganization of a Lagrangian mesh . . . . . . . . . . 7.7.2 Description of motion in an arbitrary moving coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.3 Adaptive meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

330 330

7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes . . . 7.8.1 Algorithms for constructing moving meshes . . . . . . . . . . . . . . 7.8.2 Selection of a finite difference scheme . . . . . . . . . . . . . . . . . . . 7.8.3 A hybrid scheme of variable order of approximation at internal nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.4 A grid-characteristic scheme at boundary nodes . . . . . . . . . . . . 7.8.5 Calculation of contact boundaries . . . . . . . . . . . . . . . . . . . . . . . 7.8.6 Calculation of damage kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 335 337

331 333

339 341 344 346 347

7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 8

Modeling of damage and fracture of inelastic materials and structures 354 8.1 Concept of damage and the construction of models of damaged media 8.1.1 Concept of continuum fracture and damage . . . . . . . . . . . . . . . 8.1.2 Construction of damage models . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Constitutive equations of the GTN model . . . . . . . . . . . . . . . . .

354 354 355 361

8.2 Generalized micromechanical multiscale damage model . . . . . . . . . . . 8.2.1 Micromechanical model. The stage of plastic flow and hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Stage of void nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Stage of the appearance of voids and damage . . . . . . . . . . . . . . 8.2.4 Relationship between micro and macro parameters . . . . . . . . . 8.2.5 Macromodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.6 Tension of a thin rod with a constant strain rate . . . . . . . . . . . . 8.2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

363

8.3 Numerical modeling of damaged elastoplastic materials . . . . . . . . . . . 8.3.1 Regularization of equations for elastoplastic materials at softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Solution of damage problems . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Inverse Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

375

364 365 366 367 368 373 375

375 376 377

xviii

Contents

8.3.4

Solution of a boundary value problem. Computation of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Splitting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Integration of the constitutive relations of the GTN model . . . . 8.3.7 Uniaxial tension. Computational results . . . . . . . . . . . . . . . . . . 8.3.8 Bending of a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.9 Comparison with experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.10 Modeling quasi-brittle fracture with damage . . . . . . . . . . . . . . 8.4 Extension of damage theory to the case of an arbitrary stress-strain state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Limitations of the GTN model . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Associated viscoplastic law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Constitutive relations in the absence of porosity (k < 0:4, f D 0, r D 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Fracture model. Fracture criteria . . . . . . . . . . . . . . . . . . . . . . . .

379 379 382 386 387 389 390 393 394 395 396 396 397

8.5 Numerical modeling of cutting of elastoviscoplastic materials . . . . . . . 398 8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 8.5.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 8.6 Conclusions. General remarks on elastoplastic equations . . . . . . . . . . . 8.6.1 Formulations of systems of equations for elastoplastic media . 8.6.2 A hardening elastoplastic medium . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Ideal elastoplastic media: a degenerate case . . . . . . . . . . . . . . . 8.6.4 Difficulties in solving mixed elliptic-hyperbolic problems . . . . 8.6.5 Regularization of an elastoplastic model . . . . . . . . . . . . . . . . . 8.6.6 Elastoplastic shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

406 406 406 407 408 408 409

Bibliography

411

Index

421

Part I

Basic equations of continuum mechanics

Chapter 1

Basic equations of continuous media

Chapter 1 presents the main equations of continuum mechanics as well as some models of continuous media that will be required in subsequent sections of the book. These equations and models will not be discussed in detail, since the author presumes that the reader is already familiar with them from a main university course of continuum mechanics taught to engineering students; for example, see [38, 49, 59]. The current chapter should therefore be treated as a summary of facts required for numerical modeling of problems arising in continuum mechanics rather than a guide to the systematic study of continuum mechanics. Modern computational continuum mechanics studies and solves complete systems of equations for not only the classical models of continua, such as elastic or elastoplastic media or viscous thermally conductive fluids, but also recent nonclassical models with complex rheologies and damage as well as fracture models of elastoviscoplastic materials and more. Chapter 1 gives a brief description of the methods for constructing such models. Also outlined are benefits of representing the basic equations of continuum mechanics in different forms, as integral or differential relations or variational principles; these are important for the discrete approach and will be used in subsequent sections. Let us begin with the common ways of describing motion of continuous media and different forms of representation of conservation laws.

1.1 Methods of describing motion of continuous media 1.1.1 Coordinate systems and methods of describing motion of continuous media The conventional approach to describing the motion of a continuous medium is to introduce a frame of reference, conditionally accepted to be fixed, and consider the motion relative to this frame. The simplest kind of a reference frame is a rectangular, Cartesian coordinate system with orthonormal basis vectors ei such that ei  ej D ıij ; where ıij is the Kronecker delta (ıij D 1 if i D j and ıij D 0 if i ¤ j ). The dot between vectors denotes the scalar product. Other general reference frames include curvilinear and skew coordinate systems. In this case, apart from the main, covariant basis ei , it is convenient to use the con-

4

Chapter 1 Basic equations of continuous media

travariant basis ej , which orthogonal to ei : ei  ej D ıij ;

ei  ej ¤ ıij ;

In such bases, an arbitrary vector v can be represented as v D v˛ e˛ D v ˇ eˇ ;

v˛ D v  e˛ ;

v ˇ D v  eˇ ;

(1.1)

where the v˛ are covariant coordinates of v and v ˇ are its contravariant coordinates. An arbitrary tensor  can be represented in terms of its covariant, contravariant, and mixed components as  D  ij ei ej D ˛ˇ e˛ eˇ D ˇ˛ e˛ eˇ D ˛ˇ e˛ eˇ :

(1.2)

The squared distance between two infinitely near points is ds 2 D d r  d r D e˛  eˇ dx ˛ dx ˇ D g˛ˇ dx ˛ dx ˇ ;

(1.3)

where the g˛ˇ are covariant components of the metric tensor g D g˛ˇ e˛ eˇ or g D g˛ˇ e˛ ˝ eˇ . In what follows, the dyadic (tensor) product of vectors a and b will conventionally be denoted ab or a ˝ b. Formulas for various transformations of covariant and contravariant basis vectors, including differentiation, as well as those for tensor transformations are the subject matter of vector and tensor analysis, which is widely used in continuum mechanics.

1.1.2 Eulerian description Given a Cartesian coordinate system (reference frame) xi , all changes of physical quantities (fields) that occur as the medium moves can be characterized in the coordinate space xi and time t . All physical fields are functions of xi and t : a D f .x; t /: With such a description, known as Eulerian, all quantities are assumed to be functions of a fixed point in space (cell) through which various points of the medium (particles) pass. In order to track changes in the quantities associated with a moving particle, the position vector of the particle, r D r.x; t /, must be introduced, whose components ri D xi .t / determine the path along which the particle moves in the reference frame xi . This path is determined by the velocity field as dx D v.x; t /; dt

ˇ xˇ tD0 D x0 :

Section 1.1 Methods of describing motion of continuous media

5

1.1.3 Lagrangian description An alternative approach to describing the motion of a medium is to characterize the motion of each individual particle. With this description, known as Lagrangian, particle’s characteristics – for example, its coordinates x0 at t D 0 and time t – are taken to be the independent variables. Let i denote Lagrangian coordinates and let all quantities associated with the motion of the medium be functions of i and t : ˇ a D a.; t / with  D xˇ tD0 : Any quantity a (whether it is a scalar, vector, or tensor field) that depends of the Eulerian variables, a D a.x; t /, can be treated as a composite function of the Lagrangian variables, a.x; t / D a.x.; t /; t /. Conversely, any function defined in terms of the Lagrangian variables can be treated as a composite function of the Eulerian variables, a.; t / D a..x; t /; t /. Accordingly, the partial derivatives with respect to time are taken at i D const in the Lagrangian description and xi D const in the Eulerian description. A partial derivative at i D const, called a material derivative, is a quantity that has a physical meaning; it determines the rate of change of the quantity a associated with a particle. The relationship between the partial derivatives at xi D const and i D const follows from the chain rule: ˇ ˇ ˇ @a ˇˇ @a ˇˇ @a @xi @a @a.x.; t /; t / ˇˇ D D C C vi : (1.4) ˇ ˇ ˇ @t @t x @xi @t @t x @xi  The material derivative is also known as the Lagrangian, substantial, or full derivative. The first term on the right-hand side in (1.4) is the partial derivative with respect to t , while the second term represents the convective derivative [47, 157]. Apart from the above two approaches to describing the motion of a continuous medium, it is possible to introduce infinitely many descriptions in an arbitrary moving reference frame i whose law of motion is prescribed relative to a fixed reference frame, i D i .x; t /. Such arbitrary coordinate systems will be discussed later on, in connection with the construction of optimal (in a sense) coordinate systems and associated computational grids. These kinds of characterization are intermediate between the Eulerian description in a fixed reference frame and the Lagrangian description in a frame moving together with the particle.

1.1.4 Differentiation of bases The vectors and tensors used to characterize the motion of a medium with both kinds of description can be referred to the basis of a fixed frame or any other basis unrelated to the motion. If one chooses the moving basis vectors of a Lagrangian coordinate system, then one says that a “frozen” (or convective) frame and a “frozen” basis are used. In a frozen basis, the coordinates of a moving particle remain constant, while the basis vectors change.

6

Chapter 1 Basic equations of continuous media

The vectors of a frozen basis are written as ei . j ; t / D

@r ; @ i

where r. i ; t / D  i ei is the position vector in the Lagrangian coordinate system. The differentiation of the basis vectors is performed as follows: ˇ ˇ @ ˇˇ @ ˇˇ @r.; t / @v @ ei D ˇ D i D i .v ˛ e˛ / D e˛ ri v ˛  e˛ v;i˛ ; (1.5) ˇ i @t  @t  @ @ @ where ri v ˛ is the covariant derivative of the contravariant ˛-component of the vector v; also ˇ @ ˇˇ j e D ri v j ei D e˛ r˛ v j : @t ˇ Using the above formulas for differentiating a vector field a D a˛ e˛ with respect to time, one obtains ˇ ˇ ˇ    ˛  @a .; t / ˇˇ @a˛ .; t / ˇˇ @ ˇˇ ˇ ˛ ˇ e e˛ : a D C a r v D  a r v (1.6) ˛ ˇ ˇ ˛ ˇ ˇ @t ˇ @t @t   Likewise, one can obtain the formulas for the material derivatives of a tensor written in a frozen basis [157]. The differentiation formulas for Eulerian curvilinear basis vectors eQ i with respect to t are as follows: ˇ ˇ @ ˇˇ j @ ˇˇ j ˇ ˛ eQ i D vQ ˇ i eQ ˛ ; eQ D Qei iˇ vQ ˇ (1.7) @t ˇ @t ˇ where the ˇ˛i are the triple-index Christoffel symbols of the second kind. A vector a is differentiated in an Eulerian basis using the formula ˇ ˇ   @ ˇˇ @aQ i .x; t / ˇˇ j Qi : a D  a Q v Q  j k ki e ˇ @t ˇ @t 

(1.8)

Formulas (1.5)–(1.8) will subsequently be used in deriving the objective rates of change of vectors and tensors. In a similar way, the vectors and tensors used to characterize the motion of a medium can be referred to the basis vectors of any other coordinate system i D i .x; t /. If the vector and tensor fields are referred to the basis of the same coordinate system where the motion is described, then the equations acquire their simplest form. See Section 7.7 in Chapter 7.

Section 1.1 Methods of describing motion of continuous media

7

1.1.5 Description of deformations and rates of deformation of a continuous medium The description of the kinematics of a medium as well as the selection of deformation measures and deformation rate measures depends on the approach used to characterize the motion, reference frames, and the basis to which the tensor quantities are referred. Here and henceforth, we use orthogonal Cartesian coordinates in calculations for simplicity and write out final results in invariant, indexless form. Invariant form is valid for any required coordinates and not only in continuous but also discrete representation, allowing one to use known discrete relations (see Section 7.1). This approach is perhaps the simplest and most convenient for numerical solution. The deformation of a medium is determined by the change in the distance between two infinitely near points along a selected direction by formula (1.3). Let us write out the change of the squared distance in terms of the Lagrangian variables: @x˛ @x˛ d i d j  d i d i D 2Eij d i d j ; @i @j (1.9) where the Eij are components of the Green–Lagrange finite strain tensor in the Lagrangian coordinates with    1 @x˛ @x˛ 1 (1.10) Fi˛ F˛j  ıij :  ıij D Eij D 2 @i @j 2 ds 2  ds02 D dxi dxi  d i d i D

The Green–Lagrange strain tensor E can be expressed in terms of the deformation gradient tensor F, referred to the basis ei of the coordinate system xi : ED

 1 T F FI ; 2

FD

@xi ei ej ; @j

(1.11)

where FT is the transpose of F. The components of the Green–Lagrange strain tensor Eij are easy to express in terms of the displacement vector of a particle, u D x  , as   1 @ui @uj @u˛ @u˛ C C : Eij D 2 @j @i @i @j Omitting the quadratic terms, one obtains the components of the small strain tensor "ij :   1 @ui @uj " D "ij ei ej ; ; "ij D C 2 @xj @xi since @ui @ui D @xj @x˛

  @ui @u˛ ı˛j   : @xj @j

8

Chapter 1 Basic equations of continuous media

If the change of the squared length in (1.9) is rewritten in the Eulerian coordinates and referred to a finite length, then ds 2  ds02 D

@˛ @˛ dxi dxj  dxi dxj D 2Aij dxi dxj : @xi @xj

The Euler–Almansi finite strain tensor A describes deformation near a particle in the Eulerian variables:     1 @˛ @˛ @uj @u˛ @u˛ 1 @ui ıij  C  D : (1.12) Aij D 2 @xi @xj 2 @xj @xi @xi @xi  1 I  FT F1 D FT E F1 : AD 2 In a similar way, one introduces tensors that characterize the rate of change of the length of a particle’s small directed element expressed via the velocity gradients as vi .x C d x/  vi .x/ D

@vi dxj D Lij dxj : @xj

The velocity gradient tensor Lij can be additively decomposed into the symmetric, eij , and antisymmetric, !ij , parts:   1 @vi @vj eij D C ; eij D ej i ; 2 @xj @xi   1 @vi @vj !ij D ; !ij D !j i :  2 @xj @xi It is not difficult to find the relationship between the time derivatives of the above strain tensors F and " and the velocity tensors L and e. In the index form, we have       d @xi @˛ @vi @ dxi .; t / d @xi @˛ D D ; D Lij D @xj @xj dt dt @˛ @xj dt @˛ @xj since d ˛ =dt D 0. In the indexless tensor notation, the velocity gradient tensor L is expressed in terms of the material time derivative of the deformation gradient tensor F as d F 1 F : (1.13) LD dt In this section, we have summarized the most common concepts and information from kinematics of continuous media. For more details, see texts on continuum mechanics (e.g., [38, 49, 59]). The differentiation formulas for vectors and tensors in curvilinear coordinates, fixed (Eulerian) bases, and frozen (Lagrangian) bases, the expressions of the time derivatives of tensors in different bases, the concepts of objective derivatives and other concepts from tensor analysis, and some additional information on kinematics of continuous media, required subsequently, will be given below in Sections 1.9–1.12 and later on as they are required for studying equations for specific media.

Section 1.2 Conservation laws. Integral and differential forms

9

1.2 Conservation laws. Integral and differential forms. Divergence and non-divergence forms This section focuses on different representations of conservation laws common to any continuous medium. The possibility of representing one and the same law in different forms is crucial when using discrete forms of the law. While the different representations of a conservation law are all equivalent in continuum form, there is no equivalence any more when the law is discretized and, depending on the purposes of the study, one should prefer one or another discrete form. The considerations that should taken into account in doing so depend on which properties of the original equation are required to be preserved in the discrete representation in the first place. This will become clearer subsequently, in applying these representations to specific problems.

1.2.1 Integral form of conservation laws The laws of conservation of mass, linear momentum, angular momentum, and energy must hold for any continuous medium. For a moving medium, suppose V is an arbitrary volume consisting of the same particles during motion; we will call it a fluid volume. Then the above laws can be written as follows: conservation of mass:

d dt

Z  d V D 0I

(1.14)

V

conservation of linear momentum: Z Z Z d vi d V D ij nj dS C Fi d V I dt V S V

(1.15)

conservation of energy:  Z Z Z  Z v2 d  UC qi ni dS C .Fi vi C r/ d V: d V D .ij vj /ni dS  dt V 2 S S V (1.16) Here  is the mass density, vi the particle velocity, Fi the body force per unit volume, r the energy source intensity, ij the stress, U the potential (internal) energy per unit volume, and qi the heat flux, with d=dt denoting the total (material) derivative. The law of conservation of angular momentum holds identically, provided that the stress tensor ij is symmetric. Each of the above equations (1.14)–(1.16) has the form Z Z Z d pi d V D Aij nj dS C gi d V (1.17) dt V S V

10

Chapter 1 Basic equations of continuous media

This is a general form of conservation laws, which is valid for any finite fluid volume V .xi ; t / with surface S . Here p.xi ; t / and g.xi ; t / are k-dimensional vector fields, A.xi ; t / is an m  k matrix field, n is a unit normal vector to S , and m is the dimension of the physical space. Equation (1.17) means that the change of the quantity p within the volume V , consisting of the same particles, is balanced by the flux of the quantity A through the surface S and by the action of sources g in the volume V . Equations (1.14)–(1.16) are written in the Eulerian (spatial) variables xi and t . These equations can also be rewritten in a different form if the total derivative of the integral on the right-hand side is transformed as follows: Z  Z Z d 1 p.xi ; t / d V D lim p.xi ; t C t / d V  p.xi ; t / d V dtV t!0 t V  ZV CV h i 1 p.xi ; t C t /  p.xi ; t / d V D lim t!0 t V  Z  p.xi ; t C t / d V V ˇ Z Z @p ˇˇ p vn dS: (1.18) D ˇ dV C V @t x S The volume V D vn t S is shaded in Figure 1.1 and vn D vi ni is the normal velocity component to the surface S at the time instant t . V + ∆V

V

χi

Figure 1.1. Change of a fluid volume (shaded) as the medium moves; the fluid volume consists of the same particles.

Then the general conservation law (1.17) in an Eulerian reference frame becomes ˇ Z Z Z @p ˇˇ d V D .A  p ˝ v/ n dS C g d V ˇ (1.19) V @t xi S V p vn D p.v n/ D .p ˝ v/ n;

Section 1.2 Conservation laws. Integral and differential forms

11

or, in the component notation, ˇ Z Z Z   @pi ˇˇ Aij  pi vj nj dS C gi d V: ˇ dV D V @t xi S V The symbol ˝ denotes the dyadic product of vectors.

1.2.2 Differential form of conservation laws Let B D A  p ˝ v be a continuous tensor-valued function in the domain V and let its first partial derivatives be also continuous in V . Then, by converting the surface integral in (1.19) to a volume integral with the Gauss–Ostrogradsky theorem, one can rewrite the general conservation law as  Z    @pi  rj Aij  pi vj  gi d V D 0: (1.20) @t V Taking into account that (1.20) holds for any volume V , one arrives at the following differential form of the conservation law in the Eulerian coordinate system:   @pi D rj Aij C pi vj C gi @t

(1.21)

A relation of the form (1.21) is called a divergence equation, since the right-hand side represents the divergence of a quantity in the three-dimensional space xi . In the four-dimensional space of xi and t , the four-vector Ci D .pi ; Aij  pi vj / with a fixed i and j D 1; 2; 3 can be used. By applying Ostrogradsky’s formula for the four-dimensional space of xi and t to the divergence of the vector Ci , one obtains a different integral form of conservation laws than (1.21) [45]. By introducing the material derivative of p, characterizing the change of p in a particle , with the formula ˇ ˇ @p ˇˇ dp @p ˇˇ D D C vi ri p: @t ˇi dt @t ˇxi where the i are Lagrangian (material) coordinates associated with the particle, one can convert relations (1.21) to the non-divergence form of conservation laws ˇ @pi ˇˇ D rj Aij  pi rj vj C gi : (1.22) @t ˇi In discrete representation, one should distinguish between the divergence form (1.21) of a conservation law and its non-divergence form like (1.22). Let us rewrite the law of conservation of linear momentum in the form (1.22): ˇ @vi ˇˇ D rj ij  vi  rj vj C Fi : @t ˇi

12

Chapter 1 Basic equations of continuous media

Further, we have 

ˇ ˇ @vi ˇˇ @ ˇˇ C v D rj ij  vi  rj vj C Fi i @t ˇi @t ˇi

and see that the underlined terms cancel out by virtue of the mass conservation law written in the form (1.22). So we obtain the equation ˇ @vi ˇˇ  D rj ij C Fi : (1.23) @t ˇi This is also true for any conservation law written for the mass density fi of the quantity pi D fi . Therefore, it follows from the conservation law (1.22) that ˇ @fi ˇˇ D rj Aij C gi (1.24)  @t ˇi For example, the law (1.24) for the energy density becomes  ˇ v 2 ˇˇ @ UC D rj .ij vi /  rj qj C .r C Fi vi /:  @t 2 ˇi

(1.25)

With (1.23), the conservation equation (1.25) can be converted to another form, known as an equation of heat influx. We have vi

@U @vi D vi rj ij C ij rj vi  rj qj C .r C Fi vi /: C @t @t

The underlined terms cancel out due to the equation of motion (1.23) and so ˇ @U ˇˇ  D ij rj vi  rj qj C  r D ij "Pij  rj qj C  r @t ˇi 1 dq  dq e dU D ij "Pij C C ; dt  dt dt

(1.26)

where q e and q  are specific heat fluxes per unit mass; q e is the external heat, while and q  is the heat influx due to internal sources. Equations (1.24) are easy to obtain directly from the integral law (1.17) by converting the volume integrals to integrals over mass, which does R not change during the R motion. Since mass M is time invariant and V gi d V D M gi d m, the following conversion formulas hold true: ˇ Z Z Z d d dfi ˇˇ  fi d V D fi d m D ˇ d m; dt V dt M M dt i Z Z Z 1 rj Aij d m; Aij nj dS D rj Aij d V D S V M  ˇ Z Z Z @fi ˇˇ 1 r d m  A d m  gi d m D 0: j ij ˇ M @t i M  M

Section 1.2 Conservation laws. Integral and differential forms

13

It follows that

dfi D rj Aij C gi (1.27) dt This is the main non-divergence form of conservation laws for the mass density fi of the quantity pi . 

1.2.3 Conservation laws at solution discontinuities Let Bij be a piecewise continuous function that suffers a discontinuity at a moving surface †. This is possible in continuum mechanics, since the quantities , vi , ij , and qi appearing in (1.14)–(1.16) can undergo discontinuities (it is only the displacements of the medium that must be continuous). Then the Gauss–Ostrogradsky theorem is applicable to the domains of continuity of Bij , outside †. Compatibility conditions must hold at the surface †, which relate the value BijC just upstream and value Bij just downstream of the surface of discontinuity †. Suppose † moves in the space xi with a velocity Dn, where n is the outward normal to †. Let †C and † denote two parallel surfaces to †, each spaced by a distance h=2 from it (see Figure 1.2), and let Vh denote the volume of the medium between the two surfaces. The motion will be considered in a reference frame moving with the velocity Dn. Let us take the fixed volume Vh to be the fluid volume V at a given instant of time t . By writing the conservation law (1.19) for Vh and letting h tend to zero, we obtain Z Z  @pi dV  Aij  pi .vj  Dnj / nj dS  gi d V Vh @t S Vh Z °  ±  C   D  p .v  Dn / Aij  piC .vjC  Dnj / nj  A j nj d † D 0; ij i j

Z lim

h!0

†

where vj  Dnj denote the components of the velocity vector in the moving reference frame. It follows that, by virtue of the fact that the surfaces †˙ can extend arbitrarily far, the integrand must be zero:  C    Aij  piC .vjC  Dnj / nj D A (1.28) ij  pi .vj  Dnj / nj : In (1.28), replacing Aij and pi with the respective quantities from the conservation laws (1.14)–(1.16), we arrive at the following relations at the front of the discontinuity surface †: conservation of mass: C .viC  Dni / D  .vi  Dni /I

(1.29)

14

Chapter 1 Basic equations of continuous media

n D Vh

h/2

Figure 1.2. A volume moving with a velocity Dn together with a discontinuity surface †.

conservation of momentum: ijC nj  C viC .vjC  Dnj /nj D ij nj   vi .vj  Dnj /nj I

(1.30)

conservation of energy:  .v C /2 .vjC nj  D/   U C 2   .v  /2 D ij vi nj  qj nj   U  C .vj nj  D/; 2

ijC viC nj

qjC njC

C



C

(1.31)

To summarize, for piecewise continuous Bij , the integral form of the conservation law (1.19) is equivalent to the differential form (1.21) in domains of continuity of solutions and the jump relations (1.29)–(1.31) at discontinuity surfaces. The conservation laws (1.14)–(1.16) and (1.21) are dynamic conservation laws and hold true for any continuous medium. One can easily see that these laws do not form a closed system of equations. To close the system of equations of continuum mechanics, constitutive equations of the medium are required that would link the dynamic quantities, the stress and heat flux, which appear in (1.14)–(1.16), to the kinematic and thermodynamic characteristics of the medium that determine its deformation and entropy.

1.2.4 Conclusions To summarize, the following conclusions can be drawn: 1. Conservation laws are initially formulated in the integral form (1.17) for a finite volume of the medium having a fixed composition of particles; the differential form (1.21) is a corollary of the integral form.

15

Section 1.3 Thermodynamics

2. The differential equations can be rewritten in the divergence form (1.21) or slightly simpler, non-divergence form (1.24), in the Lagrangian variables; they can also be written in the non-divergence form (1.27) in the Eulerian variables, or in the invariant form 

@f D rA C g: @t

The equations have the simplest form when rewritten in terms of the mass density fi of the quantity pi (Eq. (1.27)). 3. The integral representation of the conservation laws (1.19) is equivalent, in the Eulerian variables, to the differential equations (1.21) in domains of continuity of the motion, which have a divergence form, and relations (1.29)–(1.31) at discontinuity surfaces. In order to derive constitutive equations, closing the system of equations for a specific continuum, in a correct manner, one should consider thermodynamic laws.

1.3 Thermodynamics Apart from conservation laws, the first and second laws of thermodynamics should be considered in order to describe non-isothermal, reversible and irreversible, processes in continuous media.

1.3.1 First law of thermodynamics The first law of thermodynamics, or the law of conservation of energy, has already been presented in integral form, equations (1.14)–(1.16). It can be converted to a different form. To this end, let us integrate the equation of heat influx (1.26) with respect to mass, d m D  d V , and apply the Gauss–Ostrogradsky formula to obtain the following integral form of the equation of heat influx: Z Z Z Z d U d V D ij "Pij d V C qj nj dS C r d V: (1.32) dt V V S V This equation can be rewritten as d U D dAi C dQe C dQ :

(1.33)

The internal energyRincrement d U is equal to the work done by the power of internal stresses, dAi D  V ij "Pij d V , plus the amount of heat supplied, dQ. The latter equals the external heat influx due to heat exchange, dQe , which is determined by the second term on the right-hand side in (1.32), and the internal heat influx, dQ , due to other sources and mechanisms R (e.g., electrical). In (1.32), the internal heat influx is described by the source term V r d V , where r is the source intensity.

16

Chapter 1 Basic equations of continuous media

1.3.2 Second law of thermodynamics The second law of thermodynamics postulates the existence of a function of state S , called entropy, that satisfies the inequality dS 

dQe : T

(1.34)

Alternatively, this law can be stated in the form of an equation [157, 118, 70], dS D

dQ0 dQe C ; T T

(1.35)

for any process that happens in a homogeneous field of quantities; the quantity dQ0 >0 represents uncompensated heat. Equation (1.35) can be viewed as the definition of dQ0 . In general, dQ0 does not coincide with the heat term dQ D  dq  in the equation of heat influx (1.26). It should be emphasized that dQ0 and dQe are not total differentials, while dS is the total differential of S with respect to state variables and d U is the total differential of U . Equations (1.34) and (1.35) can be generalized to the case of an arbitrary fluid volume in which the filed of quantities is inhomogeneous by assuming that entropy is additive with respect to mass. Introducing specific entropy per unit mass, s, we can rewrite equation (1.35) in the form   1 dqe dq 0 ds D C : (1.36) dt T dt dt Integrating with respect to mass yields Z Z Z ds 1 dqe 1 dq 0 dm D dm C d m; M dt M T dt M T dt

(1.37)

with ²

q  q  grad T ³ div q dqe 1 D D  div C  ; dt T T T T2 and so Z M

ds dm D  dt

Z @V

qi n i d! C  T

Z  V

grad T T

2

Z dV C

 V

dq 0 1 d V: dt T

It has been taken into account that q D  grad T (heat equation), where  is the thermal conductivity. The symbol @V stands for the surface bounding the volume V and d! denotes the area element of @V . Also ds D dse C dsi ;

17

Section 1.3 Thermodynamics

where dse in the external entropy influx, corresponding to the surface integral, and dsi is the entropy increment due to internal processes. If dq 0 D 0, for a thermally insulated body, the first term on the right-hand side of the equation is zero, but nevertheless  Z  dsi grad T 2 D d V  0; (1.38) dt T V which means that the heat conduction process is irreversible; the reversibility criterion for a body with inhomogeneous distribution of parameters is the condition dsi > 0 rather than dq 0 > 0. Consequently, the second law of thermodynamics (1.37) for a body with inhomogeneous temperature distribution can be represented as the inequality Z Z Z Z ds qi n i r d dm D d! C d V: (1.39) s d V  dt V M dt  T V T This inequality has the structure of a conservation law, and hence all transformations valid for the conservation laws dealt with above are applicably to inequality (1.39) as well. In particular, inequality (1.39) implies a non-divergence differential inequality similar to equation (1.24), ˇ

q   r ds ˇˇ i C ; (1.40)  ˇ  ri dt T T 

which is known as the Clausius–Duhem inequality [119]. Rewriting (1.40) in divergence form, one finds that

q  r @ i .s/  ri C svi C (1.41) @t T T in the domain of smooth variation. Also one obtains an entropy inequality at a surface of discontinuity: hq n i  i i  s.vi  D/ni  0; (1.42) T where Œa D aC  a stands for the jump of the quantity a at the surface. Inequality (1.40) can be converted to a different form with the help of the equation of heat influx in the form (1.26). Eliminating the sources r, one obtains q  grad T 1  0: T sP  UP C ij "Pij   T Introducing free energy A D U  sT instead of internal energy U , one arrives at the entropy inequality in the form 1 q  grad T AP  TP s C ij "Pij   0:  T

(1.43)

18

Chapter 1 Basic equations of continuous media

1.3.3 Conclusions Any thermodynamic process occurring in a body must satisfy condition (1.43) in domains of continuity of the motion and condition (1.42) at surfaces of discontinuity. In some problems of nonlinear mechanics, this requirement allows one to select a unique solution amongst all possible discontinuous solutions.

1.4 Constitutive equations 1.4.1 General form of constitutive equations. Internal variables The motion of a continuous medium must satisfy the conservation laws (1.14)–(1.16) and the second law of thermodynamics in the form (1.39) or (1.42)–(1.43). Although thermodynamic laws do not allow one to obtain a closed system of equations, these impose certain restriction on the constitutive equations of the medium. In order to close the system of equations of continuum mechanics, one requires equations that would determine thermomechanical properties of the medium and link the quantities U , s, q, and  , appearing in the conservation laws, to kinematic and thermal characteristics determining the deformation " and temperature T of the medium as well as g D grad T . Let us take the last three quantities to be independent parameters of the medium. Apart from dynamic and kinematic variables, rheologically complex media are characterized, as a rule, by additional variables, k , that describe internal processes of structural changes caused by thermomechanical actions. These additional variables do not enter the conservation laws and are called internal variables. For a general continuous medium, the quantities constituting the first group (U , s, q, and  ) can be treated as functions or functionals of the independent parameters and internal variables: U D U."ij ; T; gi ; k /;

s D s."ij ; T; gi ; k /;

qi D qi ."ij ; T; gi ; k /; ij D ij ."ij ; T; gi ; k /:

(1.44)

Apart from these equations, it is necessary to specify equations that determine the evolution of the internal variables k . In general, such equations are represented as functionals; in simples cases, these are written as differential equations: d k D ˆ."ij ; T; gi ; k /: dt

(1.45)

Broadly speaking, relations (1.44) and (1.45) must satisfy the general laws of continuum mechanics; specifically, they must be invariant to orthogonal transformations of coordinates in the current configuration (cf. Section 1.9) and compatible with the second law of thermodynamics. Otherwise, the forms of the functions in (1.44) and (1.45) are determined by experimental data or the mechanisms of the physical processes occurring at the structural level in the material. This is the general scheme

19

Section 1.4 Constitutive equations

of constructing constitutive equations in continuum mechanics. This scheme will be illustrated with specific examples later on. Let us derive the restrictions on the form of the constitutive relations (1.44)–(1.45) that follow from the second law of thermodynamics written in the form of the Clausius–Duhem inequality (1.43): 

1 q grad T @U C T sP C ij "Pij C  0: @t  T

Introducing free energy A D U  T s, one obtains 

@A 1 q grad T  TP s C ij "Pij C  0: @t  T

Taking the variables of state .T; "ij ; k ; gi ; TP ; "Pij ; P k / to be independent and taking into account that free energy A is a function of these quantities, one finds that 

@A  sC @T





 ij @A qi g i  "Pij C  @"ij T @A @A @A R @A @A

P k 

R k  "Rij  gP i  0: T   @ k @ P k @"Pij @gi @TP

TP C

(1.46)

Since R k , TR , "Rij , and gP are not variables of state, it follows from (1.44) that the dependent variables s, ij , A, and qi are independent of them. Hence, it is always possible to choose a process of changing the state of the medium so that there is only P for example, TR , and all other first and one nonzero quantity out of R k , TR , "Rij , and g, second derivatives in (1.46) are zero. Then inequality (1.46) becomes 

@A R T  0: @TP

Since the derivative @A=@TP D ATP is independent of TR and TR can change sign in the process concerned, the condition ATP D 0 must hold. With similar reasoning, it can be found that AP k D A"Pij D Agi D 0, and hence free energy A can only depend on the parameters T , ", and k : A D A.T; "; k /: Inequality (1.46), whose left-hand side represents a sum of products of “generalized forces” Fi by “generalized fluxes” Xi , is called a dissipative inequality; it can be rewritten as X Fi Xi  0; (1.47) DD i

20

Chapter 1 Basic equations of continuous media

where each term corresponds to an individual irreversible thermodynamic process. The force vector F has four components,   @A @A q @A 1 ;  ; ; ; FD sC @T  @" @ T and the flux vector is X D .TP ; "Pij ; P k ; gi /. Let us assume that entropy s and stress  are representable as the sum of two terms, dissipative and conservative: s D s d C s c ,  D  d C  c . The conservative components can be determined by considering reversible processes occurring in the medium, in which case the dissipative components are zero. With reasoning similar to that above, one can establish the relationship between the conservative components and free energy: s c D @A=@T and  c D @A=@". In modern continuum mechanics, an additional thermodynamic principle, the principle of maximum rate of dissipation, is adopted. The principle reads: the rate of dissipation D acquires its maximum value at actual “fluxes” Xi amongst all possible fluxes that correspond to arbitrary processes [187, 61, 75, 157, 59]. It follows from inequality (1.47) that F D F.X/. If, in addition, the force Fi .Xi / is assumed to be independent of the Xk with k ¤ i , then inequality (1.47) will hold for each individual dissipative process: Di D F.i/ X.i/  0;

i D 1; : : : ; 4;

where there is no summation over the repeated subscript in brackets .i /. It follows from the principle of maximum rate of dissipation that the dissipation function D is related to the generalized forces by @D.i/ : @X.i/

(1.48)

@D.i/ X.i/  0: @X.i/

(1.49)

Fi D .i/ Then, one can find from equation (1.47) that Di D .i/

From this relation it is easy to determine .i/ as a function of Xi , given the dissipation function Di .Xi /. In the case that .i/ is independent of Xi and so is constant, it follows from (1.49) that Di is a homogeneous function of degree 1 Euler’s homogeneous function theorem. Thus, based on the aforesaid, the following theorem can be stated [75, 18]. Theorem P 1.1. Two functions, free energy A D A.T; "; k / and dissipation function D D 4iD1 Di .Xi /, completely determine the general form of constitutive equations satisfying the general and additional principles of thermodynamics:

21

Section 1.4 Constitutive equations

stress equation:

1 ij D

entropy equation:

@A @D1 C .1/ D 1 .ijc C ijd /I @"ij @"Pij

sD

@D2 @A C .2/ D sc C sdI @T @TP

(1.50)

(1.51)

structural parameters equation:

k

heat equation:

@A @D3 D .3/ ; @ k @ P k

k D 1; : : : ; nI

(1.52)

qi @D4 D .4/ ; T @gP i

i D 1; 2; 3:

(1.53)

The coefficients .i/ are determined from (1.49). In the case where A and D are quadratic functions of their arguments, equations (1.50)–(1.53) are linear. By specifying A and D, one can obtain various models of continuous media. Let us consider some special cases of the constitutive equations (1.50)–(1.53).

1.4.2 Equations of viscous compressible heat-conducting gases Assuming that, in equations (1.50)–(1.53), free energy is a function of density  and temperature T and dissipation is a quadratic form of strain rates with a thermal term, A D A.; T /;

D1 D



qi gi .P"i i /2 C "Pij "Pij C ; 2  2T

i; j D 1; 2; 3;

one finds that ijc D pıij ; ijd

3p D 2

D "Pkk ıij C 2 "Pij ;

@A @A ; s D sc D  ; @ @T q D g D  grad T;

(1.54)

where and are viscosity coefficients,  is the thermal conductivity, and p is the hydrostatic pressure.

1.4.3 Thermoelastic isotropic media For a three-dimensional problem (i; j D 1; 2; 3), by setting   gi gi 1 1 2 .grad T /2 "i i C "ij "ij  .3 C 2 /˛T "i i ; D D D ; AD  2 2T 2T

22

Chapter 1 Basic equations of continuous media

with "ij being the strain tensor components, one obtains from (1.48)–(1.53) the following constitutive stress, entropy, and heat equations: ij D ijc D "Pkk ıij C 2 "Pij  .3 C 2 /˛T ıij @A q D g D  grad T: s D sc D  ; @T

(1.55)

1.4.4 Combined media Let us prove the following important property of thermodynamically well-posed models. By combining together simple models that satisfy the thermodynamic principles, one can obtain new, more complicated models that will also satisfy the thermodynamic principles. Suppose that the Clausius–Duhem inequality holds for two materials, ˛ and ˇ, individually: 1 gi  gi  0; APi C si TPi C  i "P i C  T

i D ˛; ˇ

(model index)

(1.56)

By adding together inequalities (1.56) for model ˛ and model ˇ, one obtains   gˇ  gˇ 1 g˛  g˛ P P P P .A˛ C Aˇ / C .s˛ T˛ C sˇ Tˇ / C . ˛ "P ˛ C  ˇ "P ˇ / C C  T˛ Tˇ 1 g  g   D AP C s TP C .  "P  / C  0: (1.57)  T It follows that inequality (1.56) is also valid for a combined model  with  D ˛ D ˇ D , provided that the following conditions hold: additivity of free energy (see Figure 1.3): A˛ C Aˇ D A I

(1.58)

additivity of the work done by internal stresses:  ˛ "P ˛ C  ˇ "P ˇ D   "P  I

(1.59)

additivity of internal heat sources: s˛ TP˛ C sˇ TPˇ D s TP I

(1.60)

and additivity of external heat sources: gˇ  gˇ g  g g˛  g˛ C D : T˛ Tˇ T

(1.61)

23

Section 1.4 Constitutive equations

For condition (1.59) to be satisfied, it suffices that one of the following two sets of conditions holds: A1 /  ˛ D  ˇ D   ; "P ˛ C "P ˇ D "P  I A2 / "˛ D "ˇ D "P  ;  ˛ C  ˇ D   : Condition A1 corresponds to a series connection of the models and condition A2 corresponds to their parallel connection; see Figure 1.3. σ

A1

σ

A2

Figure 1.3. Series (A1 ) and parallel (A2 ) connections of mechanical models.

For the additivity condition of internal heat sources (1.60) to be satisfied, it suffices that one of the following sets of conditions holds: B1 /

s˛ D sˇ D s ;

B2 / T˛ D Tˇ D T ;

T˛ C Tˇ D T I s˛ C sˇ D s :

Condition B1 corresponds to isentropic processes and condition B2 corresponds to isothermal processes. For condition (1.61) to be satisfied, it suffices that the conditions C/

g˛2 C gˇ2 D g2 ;

T˛ D Tˇ D T

hold. The above allows us to conclude that for condition (1.56) to hold for model  , it is sufficient that condition (1.58) is satisfied as well as any combination of three rows of conditions Ai (i D 1; 2), Bj (j D 1; 2), and C, in which case, we have D D D˛ C Dˇ  0:

24

Chapter 1 Basic equations of continuous media

It is needless to say that the above elements can again be combined together and with the original one to obtain more and more complex models for which inequality (1.56) surely holds. This method of connecting mechanical models is widely used in continuum mechanics; in particular, it is applicable to constructing an elastoplastic model that satisfies the thermodynamical requirements [41, 61].

1.4.5 Rigid-plastic media with translationally isotropic hardening First, let us consider a rigid-plastic model for which the total strain equals the plastic strain, " D "p . Then the stress equation (1.50) becomes @A @D ij D p C p :  @"ij @"Pij

(1.62)

p

The body remain rigid ("Pij D 0) if the condition .ij  ijc /.ij  ijc /  k 2 .Wp / p is satisfied. In the case of equality, the condition of neutral loading, ij "Pij D 0, must Rt p additionally be satisfied. Here Wp D 0 ij "Pij dt is the work done by plastic strain. p The body is deformed plastically, in which case "Pij ¤ 0, if .ij  ijc /.ij  ijc / D k 2 .Wp /

(1.63)

p

and the condition of active loading ij "Pij > 0 holds. Equation (1.63) is a condition of plasticity with translationally isotropic hardening. In the stress space ij , the yield surface is displaced as a rigid body relative to the origin of coordinates. This displacement characterizes translational hardening of the material and is determined by the coordinates ijc . In addition, the surface radius increases with Wp , resulting in isotropic hardening of the plastic material. The transformation of the yield surface is illustrated in Figure 1.4. By prescribing, in accordance with the theorem on page 20, free energy and dissipation function, AD

a p p " " ; 2 ij ij

DD

1 p p b.P" "P /1=2 ;  ij ij

one finds that ijc D 

@A p p D a"ij : @"ij

Then equation (1.62) becomes p

ij 

p a"ij

Db

"Pij Ip

;

p

p

where Ip D .P"ij "Pij /1=2 :

(1.64)

25

Section 1.4 Constitutive equations σij

σij K(Wp) b0 aε pij = σ cij b0

Figure 1.4. Transformation of the yield surface in translationally isotropic hardening in the space of ij .

Taking the square of the left- and right-hand sides of equation (1.64), one obtains .ij  ijc /.ij  ijc / D b 2 ; whence follows the identity b 2 D k 2 .Wp /. Finally, p

"Pij D

Ip p .ij  a"ij /: k.Wp /

1.4.6 Elastoplastic model Let us derive an elastoplastic model by connecting in series the elastic and rigidplastic models. For an elastic solid, one obtains, from (1.55), the following equations for the elastic component, labeled with superscript e: ije D "ekk ıij C 2 "eij C .3 C 2 /˛T ıij : For an elastoplastic solid, the conditions of series connection A1 must hold: p

ije D ij D ij ;

p

"eij C "ij D "ij : p

It follows that for a plastically incompressible solid ("kk D 0), p P ij D "Pkk ıij C 2 .P"ij  "Pij /  .3 C 2 /˛ TP ıij :

26

Chapter 1 Basic equations of continuous media

The plastic strain rate is determined by equation (1.64): p "Pij

IPp .ij  ijc / or D k.Wp /

p

p "Pij

D

c / "P˛ˇ .˛ˇ  ˛ˇ

k 2 .Wp /

.ij  ijc /:

If the material does not possess hardening, which means that a D 0, k.Wp / D k0 , p and "Pi i D 0, one finds that sij sij D k02 , and also the following relations must hold for p the deviatoric components Pij : .Pkl  12 sPkl = /skl

D

k02

sij D

"Pkl skl sPkl skl  2 k0 2 k02

!

1 ij D "ij  "kk ıij : 3 (1.65) In the case of von Mises ideal plasticity, the second term in (1.65) is zero, since after differentiating sij sij D k02 , one finds that sij sPij D 0. Then the final equations are the equations of the theory of Prandtl–Reuss elastoplastic flow [72]: ! skl Pkl 1 sij ; sij D ij  i i ıij ; i i D .3 C 2 /."i i C 3˛T /: sPij D 2 Pij  3 k02 (1.66) The conditions of active, passive (unloading), and neutral loading remain the same as in the case of rigid-plastic material. There is another, more common approach to deriving constitutive equations compatible with the thermodynamic laws. With this approach, the equations are obtained from theoretical hypotheses and experimental results followed by their coordination with the thermodynamic principles. The coordination involves substituting the equations, involving arbitrary functions, into the Clausius–Duhem inequality and selecting arbitrary functions and parameters so as to satisfy the inequality. For example, the same equations of plastic flow are obtained from the general postulate of plasticity [187, 72, 59] and other hypotheses generalizing experimental data; one should then verify that the particular model does not contradict thermodynamics. It is this approach, based on general postulates of plasticity (associated flow rule), that will be used below for constructing a theory of flow for finite elastoplastic strains. p Pij

sij ;

1.5 Theory of plastic flow. Theory of internal variables 1.5.1 Statement of the problem. Equations of an elastoplastic medium Let us consider the plasticity theory based on the associated flow rule, where the p plastic strain increment tensor d "ij is collinear with the normal to the loading surface F .ij ; / D 0 in the stress tensor space: p

d "ij D d

@F : @ij

(1.67)

27

Section 1.5 Theory of plastic flow. Theory of internal variables

R "p p The quantity  D 0 ij ij d "ij D W p is the hardening parameter of the material, which is taken to be equal to the work done by plastic strains; d  0 is a scalar parameter determined from the condition F .ij ; / D 0. Differentiating F .ij ; / D 0 gives @F @F p p ij d "ij D 0; dij C d  D ij d "ij : (1.68) @ij @ p

Substituting d "ij from (1.67) into (1.68), one arrives at the following expression for d : @F @F 1 @F D  kl dij H; I @ij H @ @kl @F d D 0 if ij D 0 and F D 0: @ij d D

(1.69)

Condition (1.69) corresponds to neutral loading, where the stress vector is orthogonal to the normal to the yield surface. @F If @ dij < 0, one has d < 0 and this corresponds to unloading. ij The total strain is the sum of the elastic and plastic components:   1 p 1 (1.70) d "ij D d "eij C d "ij D Dij kl C HFij Fkl dkl D Aij kl dkl ; where 1 Dij kl

    1 2 ıik ıj l  1  ıij ıkl ; D 2 3k

Fij D

@F ; @ij

1 denoting the elastic compliance tensor and A1 the elastoplastic compliwith Dij kl ij kl ance tensor. Equations (1.70) can be represented in the matrix-vector form accepted in the finite element analysis. To this end, instead of the stress and strain tensors, appropriate vectors should be used; vectors and matrices are conventionally denoted by curly braces ¹ º and square brackets Œ , respectively (see [188]). We have     ¹d "º D ŒD e 1 C H ¹F º¹F ºT ¹d º D ŒD e 1 C H ¹F º ˝ ¹F º ¹d º (1.71)

with ŒD e 1 C H ¹F º ˝ ¹F º D ŒD ep 1 ; where ˝ stands for tensor (or dyadic) multiplication of vectors, ŒD e 1 is the elastic compliance matrix, and ŒD ep 1 is the elastoplastic compliance matrix, whose entries depend on the stress-strain state level of the body. Equations of the form (1.70) are hypoelastic; such equations are independent of changes in the scale of time and describe irreversible deformation, unlike equations in hyperelastic form, which describe reversible deformation.

28

Chapter 1 Basic equations of continuous media

The deformation process is locally invertible, which means that it is invertible for an infinitesimal loading cycle; however, it depends on the loading path for a finite cycle. Hypoelastic relations differ from hyperelastic ones in that the former are nonintegrable [137, 75]. If the yield condition is taken in the form of von Mises, which means that it only depends on the second deviatoric stress invariant S and hardening parameter  so that F .S; / D 0;

S D .sij sij /1=2 ;

(1.72)

then @F sij dsij @F @S @F sij ; d D  H; D @S @ij @S S @S S @F @F 1 D F SFS ; ; FS D ; skl dskl D skl dkl ; F D H @ @S   skl dkl sij skl 1 1 d "ij D Dij d  s D D  dkl : jj kl kl ij kl F S 2 F S 2 Fij D

The last equation can be rewritten in the matrix-vector form   ¹sº ˝ ¹sº ¹d º D ŒD ep 1 ¹d º: ¹d "º D ŒD e 1  F S 2

(1.73)

Equation (1.70) can be inverted by solving it for the stress increments. To this end, let us perform the contraction of the equation with Fij :   1 2 1 Fij dij  1  di i Fkl C HFmn Fmn Fkl dkl : Fij d "ij D 2 3k 2 p

In view that "i i D Fi i D 0, we have 

1 1 Fkl dkl D C HFmn Fmn Fkl d "kl ; 2  1 1 1 C HF d "ij D Dij d C HF F Fkl d "kl ; ij mn mn kl kl 2  1 1 C HFmn Fmn Fkl d "kl Dij kl Fkl : dij D Dij kl d "kl  H 2 where Dij kl D 2 ıki ılj C ıij ıkl . Finally, we arrive at the following equation of the theory of plastic flow: i h  1 Fkl d "kl D Aij kl d "kl : dij D Dij kl  .2 /2 Fij H 1 C 2 Fmn Fmn (1.74)

Section 1.5 Theory of plastic flow. Theory of internal variables

29

In the matrix-vector notation, it has the form ¹d º D ŒD e ¹d "º 

.2 /2 ¹F º ˝ ¹F º ¹d "º F . /¹F º C 2 ¹F º¹F º

(1.75)

with ŒD ep D ŒD e 

.2 /2 ¹F º ˝ ¹F º ; F . /¹F º C 2 ¹F º¹F º

where ¹F º D ¹Fij º D ¹@F=@ij º, ŒD ep is the stiffness matrix of an elastoplastic solid, and ŒD e is that of an elastic solid. In case the yield criterion is adopted in the form (1.72), equations (1.74) and (1.75) become 

e 2 3 1 sij skl d "kl (1.76) dij D Dij kl  .2 / FS .F S C 2 FS / S2 and    .2 /2 FS3 e 1 ¹d º D ŒD  .F S C 2 FS / ¹sº ˝ ¹sº ¹d "º: S2 Thus, in the flow theory, the constitutive equations (1.72) connect the stress increments to the strain increments, and the matrix D ep is independent of the stress and plastic strain tensors. The properties of a medium are independent of the scale of time t but depend on the loading history [19]. The equations considerably simplify if the yield criterion is rewritten in the form solved for the stress intensity S : S  k0 ./ D 0: Then FS D 1, F D k00 ./, and Fij D sij =S . So we have " # .2 /2 sij skl e  d "kl dij D Dij kl   0 k0 ./S C 2 S 2 2 3 2 sij skl 5 e  d "kl : D 4Dij kl 

2 k00 ./ 1C S S

(1.77)

2

For an ideal plastic medium, k00 ./ D 0, S 2 D k02 , and equation (1.77) coincides with (1.66).

30

Chapter 1 Basic equations of continuous media

1.5.2 Equations of an elastoviscoplastic medium Most materials exhibit, in one way or another, a viscosity, or sensitivity to the rate of loading. For metals, this effect is negligible at low loading rates and moderate temperatures. However, at fast loading ("P  10 s1 ) and increased temperatures (T  100ı C), the effect already becomes noticeable and then becomes more and more significant as "P increases further. In such materials as polymers and composites with organic components, the effect of viscosity is already significant at low loading rates and room temperatures. It is assumed that the principle of additivity of elastic and viscoplastic strains holds true. This means that the total strain rate "P can be represented as the sum of the elastic strain rate "Pe and viscoplastic strain rate "Pvp : vp

"Pij D "Peij C "Pij :

(1.78)

The elastic deformation is determined by the total stress according to Hooke’s law P ij D Dij kl "Pekl

(1.79)

written in matrix form, where Dij kl is the elasticity tensor. Just as in the case of elastoplastic medium, the viscoplastic strain rate is determined from the gradientality principle. According to this principle, the viscoplastic strain rate vector is collinear with the gradient vector to the instantaneous viscoplastic loading surface, which can be defined in the form vp

F .ij ; ; T; "Pij / D k0 ;

(1.80)

where F is some function determined from experiment, k0 is the initial yield stress in uniaxial stress state, and  is the hardening parameter. The loading surface (1.80) differs from that for an elastoplastic medium (1.72) in that the function F has an additional argument, the viscoplastic strain rate "Pvp . As "Pvp ! 0, the expression of F becomes a function that holds true in the theory of elastoplastic flow: vp

lim F .ij ; ; T; "Pij / D F ep .ij ; ; T / D S0 .; T /:

"Pvp !0

Based on the gradientality principle, the equation for the viscoplastic strain rate tensor can be written as @F vp : (1.81) "Pij D @ij Equation (1.80) can be conveniently rewritten as [94, 87] O JPvp D ˆ.S  S0 .; T //;

O ˆ.0/ D 0;

(1.82)

vp vp where S D . 12 sij sij /1=2 is the shear stress intensity and JPvp D . 12 "Pij "Pij /1=2 is the viscoplastic strain intensity. The function S0 .; T / is a quasi-equilibrium dependence

Section 1.5 Theory of plastic flow. Theory of internal variables

31

of the shear stress intensity on the hardening parameter  and temperature T ; in the elastoplastic flow theory, it is adopted as the yield condition S D S0 .; T /. The O function ˆ.z/ is nonzero outside the surface S D S0 .; T / and must be identically vp zero inside, S < S0 .; T /, where the viscoplastic strain does not change and "Pij D 0: ´

ˆ.z/ O ˆ.z/ D 0

if z  0; if z < 0;

ˆ.0/ D 0I

the hypoelasticity condition "Pij D "Peij must hold inside the surface (at small strains). Then equation (1.81) becomes vp

"Pij D

O @S O sij @F @ˆ @ˆ : D D @ij @S @ij @S S

(1.83)

Taking the square of equation (1.83), performing the contraction, and extracting the square root, one finds that D JPvp

 O 1 @ˆ : @S

With this formula and in view of (1.82), one obtains the constitutive equation (1.83) in the form O ˆ.S  S0 .; T // vp sij ; "Pij D (1.84) S where  is the (constant) relaxation time of the viscoplastic material. The right-hand side of equation (1.83) only depends on the stress tensor ij , parameter , and temperature T ; it does not involve time derivatives of unknown quantities. O The form of the dimensionless function ˆ.z/ can be determined from experimental data obtained in uniaxial tension/compression tests at constant stress rates if (1.82) is rewritten in the form solved for S : O 1 . JP vp /: S D S0 .; T / C ˆ For the uniaxial case, we have O 1 . "Pvp /:  D 0 .; T / C ˆ

(1.85)

This dependence describes a family of curves obtained by translating the quasiequilibrium curve  D 0 .; T /. Ways of approximating the curves and related issues are addressed in [85, 87]. Power-law and exponential representations of ˆ.z/ are most common.

32

Chapter 1 Basic equations of continuous media

1.6 Experimental determination of constitutive relations under dynamic loading The first attempts to determine laws of deformation of metals at large rates of loading date back to the beginning of the 20th century. It already became clear at that time that the rate of deformation affects the mechanical characteristics beyond the elastic limits of materials. In the plastic state, many materials behave differently under dynamic loading than in static conditions; for example, the yield stress, the residual deformations at fracture, and other characteristics can change by a factor of several times. However, the difficulties associated, on the one hand, with the measurement of fast processes, lasting for several microseconds, and, on the other hand, with theoretical solution of the problem did not allow researchers for a long time to advance in obtaining constitutive equations of solids at large rates of loading. By now, some progress on this issue has been made due to increased capabilities of both theoretical and experimental methods. Nevertheless, we are still far from a complete solution in both theoretical and experimental aspects. The current status of the issue is outlined below. Experimental results are discussed first.

1.6.1 Experimental results and experimentally obtained constitutive equations The simplest kind of test where deviations from static response are already observed as the rate of deformation increases is a quasistatic test of cylindrical specimens subjected to tension or compression. Ludwik [110] was the first address this issue. In this kind of test, the rate of deformation does not exceed "P  1–10 s1 . The inertial forces emerging in the specimen can be neglected, and hence no complications arise due to the wave character of the stress and strain fields. For example, Figure 1.5 displays curves obtained by Campbell [21] in quasistatic compression tests for steel specimens and Figure 1.6 depicts experimental curves obtained in tension tests for low-carbon steel specimens [23]. The experiments show that materials with a pronounced yield point, such as carbon iron and steel, are most sensitive to the rate of deformation, while aluminium alloys and some other metals and alloys that do not have pronounced proportionality limit are significantly less sensitive. To obtain experimental results in the range of large strain rates, 102 s1 . "P . 104 s1 , Kolsky [73] suggested a method for dynamic tests of thin disk-shaped specimens placed between two colliding bars of the same diameter as the specimens. The bars are made of a tempered material whose yield point is significantly higher than that of the specimen, so that only longitudinal waves can propagate through the bars after the collision. It is assumed that since the specimen is short, the nonuniformity

Section 1.6 Experimental determination of constitutive relations under dynamic loading 33

σ · 102 kg/cm2 56 20 5 1 0.2 0.00017 s–1

49

42

35

28

21

14 0

1

2

3

4

5

6

7

8

9

10

ε, %

Figure 1.5. The -" diagram in compression for steel specimens at different strain rates.

of quantities caused by the waves can be neglected. By measuring the deformation in the elastic bars with electrical strain gauges, one can use the one-dimensional theory of elastic waves in bars to calculate the time dependences of the stress, strain, strain rate in the specimen and construct the  -" curves at constant strain rates. The results obtained in [108] and depicted in Figure 1.7. It is noteworthy that, although the method became widely used, the results should be interpreted with some caution, since the elementary theory of bars does not provide satisfactory predictions when sharp pulses are used to obtain large strain rates. As shown in [76], the reflection of plastic waves from the lateral surface must be taken into account if the specimen length is of the order of one centimeter. Other experimental schemes are also possible, including twisting impact tests, impact pressure tests in a cylindrical tube, and magnetopulse extension tests for annular specimens [76, 176]. Tests for studying stress waves in long thin rods and tests with colliding plates have become quite widespread. In these tests, very large strain rates can be attained, up to "P D 106 –107 s1 [176, 167, 69]. However, a priori constitutive equations must be adopted in this case; as a result, by measuring one or another quantity, one has to compare it with the value predicted by the theoretical model based on the adopted equations of state. Consequently, these experiments can only confirm or refute a certain hypothesis that is to be tested.

34

Chapter 1 Basic equations of continuous media σ kg/mm2

60

A × × C

40 C

× ×

A

×

×

×

× ×

B

× ×

B

×

×

20 –1

–2 –10 log (ε• p)average (ε• p, s– 1)

2

Figure 1.6. Dependence of  on the logarithmic plastic strain rate in tension for low-carbon steel specimens with different carbon content.

Equations that take into account the effect of strain rates can be written as [164, 114, 32, 82]: P D E "P;  < s ."/; p  D s ."/ C G. "P /'."/;   s ."/;

(1.86)

where the plastic strain rate is expressed as "Pp D "P  P =E, E is the elastic (instantaneous deformation) modulus,  D s ."/ is the stress-strain dependence obtained in static tests, '."/ is a decreasing function that characterizes the decrease of the effect of the strain rate on the amount of stress as " increases, and G.P"p / is a function that characterizes the effect of the strain rate or viscosity on the stress-strain diagram  D s ."/. Equation (1.86) can be rewritten in a more convenient form for further analysis:    1   s ."/ P C H   s ."/ ˆ ; (1.87) "P D E  '."/ where H.z/ is the Heaviside function and ˆ.z/ is the inverse of G.P"p /, which is determined from experimental data. The advantages of this approach as well as the comparison of theoretical predictions with experimental findings [115, 167, 2, 53, 10] are discussed in the papers [131, 82] and others. A number of interesting experiments

Section 1.6 Experimental determination of constitutive relations under dynamic loading 35 12 pure aluminium

10

ε = 0.08 σ, klb/inch2

8 0.06 0.04

6

0.03 0.02

4 2 0 10–3

10–2

10–1

1

10 ε,

102

103

104

s–1

Figure 1.7. Dependences  D ."/ P at constant strain, " D const, obtained by the Hopkinson– Kolsky pressure shear bar.

were carried out that employed complex programs of dynamic loading; see [107] and others. These experiments showed that the  -" dependence is influenced by the entire loading history of the specimen. To describe this influence on the current stress-strain state, one has to use functionals. For arbitrary loading history, such dependences were suggested in [142]. A large number of experimental studies, mostly in the quasistatic range of loading rates, deal with the analysis of the phenomenon of delayed yield, observed in low carbon steels. A survey of these studies can be found in [169]. To describe this phenomenon, relations were suggested similar to those used in the ageing theory [105]. Based on experimental data and the ideas of the theory of dislocations, Rabotnov [140] suggested a model of an elastoplastic medium with delayed yield where the transition from an elastic state to a plastic state occurs when the Cottrell condition is satisfied. Within the framework of this model, Burago and Kukudzhanov [15] studied the effect of the strain rate on the lower yield point. More detailed surveys of experimental and theoretical studies dealing with dynamic constitutive equations can be found in [176, 22, 69, 113, 128, 82, 119]. Let us now consider the modern theoretical basic concepts of solid state physics that enable one to substantiate, to a certain degree, the equations of elastoviscoplastic deformation of materials.

36

Chapter 1 Basic equations of continuous media

1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory The preceding section outlined the phenomenological description of the laws of deformation of solids observed in experiments, without looking into how the microstructure of the material changes in the course of deformation. Meanwhile, it is well known that crystalline materials undergo structural changes under plastic deformation. The modern theory of plastic flow, which takes into account the strain rate effects, has sufficiently sound physical foundations laid in the second half of the 20th century owing to the rapid development of the theory studying the origination and propagation of defects in crystals, especially dislocations. Taylor and Gillman [171, 44] developed the dislocation theory of deformation of crystals at the stage of hardening. According to this theory, a crystal deforms plastically due to the motion of dislocations under the action of thermomechanical loads, while the macroscopic state of the material is determined by some averaged quantities that characterize the densities and velocities of moving dislocations as well as their interaction. Later on, for elastoplastic materials, a theory of nucleation and growth of defects in the form of elliptic pores was suggested [52, 26]. This theory generalizes the Taylor– Gillman approach to the stage of softening when the material is damaged as dislocations develop until mesolevel defects (micropores and microcracks) are formed. As a result, the von Mises yield criterion is replaced with the Gurson yield criterion for a porous medium; the Gurson condition depends on the strain rate, temperature, and triaxiality factor (the first-to-second invariant ratio, which characterizes the type of the stress-strain state). By using the correlation between the micro and macro parameters, macroscopic equations for the damaged medium were further derived [95] (see also Chapter 8). On the basis of these ideas, the plastic shear strain rate P p can be expressed in the one-dimensional case as (1.88) P p D aNm bV; where Nm is the density of moving dislocations in a sliding plane, b is the Burgers vector, a is an orientation coefficient, and V is the average speed of dislocations. There are quite reliable direct methods for measuring the dislocation density and dislocation speed. For more details, see [126]. For example, it was found for iron [44] that   Nt ; Nt D N0 C ˛ p ; Nm D Nt exp  N where Nt is the total dislocation density and N , N0 , and ˛ are material constants. The most significant experimental problem is to determine the dislocation speed depending on the applied stress  (in the case of a complex stress-strain state,  should be replaced with the tangential stress intensity S ) and temperature T . It was found out that the dislocation speed is a very sensitive function of the effective shear stress

Section 1.6 Experimental determination of constitutive relations under dynamic loading 37

  D   A , where A is the inter-dislocation long-range elastic stress impeding the motion of dislocations. Figure 1.8 displays the dependences V D V . / for single crystals of lithium fluoride from [64]. Curves of this form are characteristic of many substances. 106 105 104 103

Dislocation speed V, cm/s

102 10

boundary components helical components

1.0 10–1 10–2 10–3 10–4 10–5 10–6 10–7

yield stress 0.1

0.5 1.0 5 10 50 100 shear stress τ, kgf/mm2

Figure 1.8. Dependence of the dislocation speed V on the tangential stress  for single crystals of lithium fluoride [64].

For small  , the dislocation speed is negligibly small; as  increases, the dislocation speed increases almost proportionally in logarithmic coordinates and then, starting from a certain  , drastically slows down. The first segment is fairly well approximated by a power law:

 m ; (1.89) V D V0 0 where V0 and m are constant and 0 is dependent on the temperature T and plastic shear strain  p .

38

Chapter 1 Basic equations of continuous media

For large V , close to the speed of elastic shear waves c0 , Jonhson and Gillman [64] suggested the dependence

  0 ; (1.90) V D c0  where 0 is, as before, dependent on T and  p . Relations (1.88) and (1.90) were used in [171, 127] for determining the decay of plastic waves in studying the collision of plates. The dependences (1.89) and (1.90) are purely empirical and do not suggest any interpretation in the language of the dislocation motion mechanism. However, there have been attempts to obtain a theoretical dependence V D V . / based on analyzing a certain mechanism of motion of dislocations between obstacles and overcoming the obstacles. The simplest variant of such a mechanism of overcoming energy barriers due to thermal activation and applied stress [186] is described by the following expression of the plastic shear strain rate:   U0  .  A / p ; (1.91) P D bNA!0 exp  kT where  D bxl is the activation volume, !0 is the frequency at which a dislocation overcomes the barrier when the thermal activation energy is U D 0, U0 is the energy of a local barrier, A is the area swept by a dislocation on the sliding plane after overcoming the barrier. Clifton [27] suggested the dependence V D V .; T / that results from treating the motion of dislocations as a thermally activated process of overcoming energy barriers and a process of viscous drag when dislocations move between barriers. In this case, V is expressed as 8 c0 h. / exp.U=kT / < 0 <   <  p; /; exp.U=kT /CŒ1exp.U=kT / h.  (1.92) V D : c h. /;   >  p; 0 p 0 B0 where  D v0 l=c0 , h. / D 1  2 ,  D c2b , and B0 is the viscosity coefficient as V ! c0 . The energy U is assumed to have the form [132] ´ U D U0 1 



 p

2=3 μ3=2 ;

where  p is the stress required for overcoming the barrier; if   >  p , the barriers do not affect much on the motion of dislocations. The dependence (1.92) behaves in the same way as the experimental dependence shown in Figure 1.9. However, using (1.92) for solving specific macrodynamic problems with the purpose of obtaining quantitative coincidence with experimental data does not give good results [63], since these dependences were obtained by significantly simplifying the

Section 1.6 Experimental determination of constitutive relations under dynamic loading 39 V/c0 1.0 thermally active processes long-range stress field

0.8

viscous resistance

viscous resistance and relativistic effects

0.6

0.4 h(τ) 0.2 Bcs 0

b 0

τA

τA + τP

τ

Figure 1.9. Dependence of the dimensionless dislocation speed V =c0 on the shear stress  for different resistance mechanisms to dislocation propagation; b is the magnitude of the Burgers vector, B the viscosity coefficient, and cs the shear wave speed.

actual mechanism of the phenomenon. For this reason, it is advisable to use empirical dependences for specific analyses. The comparison of the dependences (1.88)– (1.92) for "Pp with the phenomenological equation (1.87) shows that these have qualitatively similar forms with the only difference in the specific expression of the function O ˆ.; "/. Consequently, dislocation theory provides a physical substantiation for the equations of viscoplastic flow taking into account the effect of strain rate. The most common kinds of test are schemes relying on the propagation of onedimensional waves arising at the impact of long bars (uniaxial stress state) and plane collision of plates (one-dimensional strain state). These are two simplest test schemes and, at the same time, simplest problems for theoretical treatment; there are a large number of studies devoted to the numerical solution of these problems (e.g., see [131, 164, 114, 32, 82, 101, 81]); see also Chapter 5 of the present book. The obtained constitutive equations can be generalized to the case of complex stress state by following the procedure outlined Sections 1.4 and 1.5. Hypotheses that reduce obtaining multi-dimensional constitutive equations for complex loading programs to constitutive equations for the stress-strain state of simple shear and hydrostatic uniform tension-compression verified experimentally. For small deformations of elastoviscoplastic materials at fast loading without specially holding specimens for some time, it was shown by Lindholm [107] that the loading history does not affect noticeably the constitutive relations of the medium and so, in these cases, it suffices to use differential relations of the form (1.84)–(1.85) or, in the one-dimensional case, (1.86)–(1.87).

40

Chapter 1 Basic equations of continuous media

In studying specific problems in subsequent sections, we will be using relations of the form (1.87) and their generalizations to the case of complex stress state, and the function ˆ.z/ will be assumed to be known from experimental data.

1.7 Principle of virtual displacements. Weak solutions to equations of motion Consider another, energy form of the equation of motion of a continuous medium, alternative to the differential form discussed in the preceding sections. The energy formulation of the continuum mechanics problem relies on the principle of virtual displacements or principle of virtual velocities and is closely associated with the notion of a weak form of a solution to the problem.

1.7.1 Principles of virtual displacements and velocities Suppose a domain V of some continuous medium is bounded by a surface S . On part of the surface, Su , either conditions of relative displacement or velocities are prescribed, while on the remaining part of the surface, S D S n Su , a stress vector (traction vector) t is specified (Figure 1.10): u  n D un ; u   ˛ D u˛ ; ˛ D 1; 2I

(a) x 2 Su W (b)

x 2 S n Su D S W   n D t  :

(1.93)

The boundary conditions (1.93a) imposed on kinematic quantities (displacements or velocities) are called principal or kinematic, while those imposed of the stresses, (1.93b), are called natural or static.

p

×

×× ×× × Sσ ××

××

× ×

× ×

Su

V

Figure 1.10. The spatial domain V occupied by a solid under a load p applied at the surface S with a displacement u at the surface Su .

Section 1.7 Principle of virtual displacements. Weak solutions to equations of motion

41

Any continuous medium satisfies the principle of virtual displacements: the work done by all external forces over the virtual displacements ıui equals the work done by the internal stresses over the field of virtual strains ı"ij linked to the field of virtual displacements by the Cauchy relations: Z Z Z Z ij ı"ij d V D fi ıui d V C ti ıui dS  uR i ıui d V (1.94) V

V

S

V

The field of virtual displacements ıui is understood as any field of displacements ui that satisfies the kinematic boundary conditions (1.93a). The integral relation (1.94) involves the following quantities: fi is the mass force distributed over the volume of the body V and ti D ij nj is the surface force acting on the surface S . The last term on the right-hand side of equation (1.94) is the work done by the inertia force on the virtual displacements. The principle of virtual velocities for the field ıvi can be written likewise, with condition (1.93a) specified with respect to velocities and condition (1.93b) remaining the same: Z Z Z Z ij ı "Pij d V D fi ıvi d V C ti ıvi dS  vP i ıvi d V (1.95) V

V

S

V

Relations (1.94) and (1.95) are equivalent to the differential equations of motion (1.23) and can be obtained by multiplying by ıvi or ıui , summing up over i , integrating over the volume of the body Vi . For example, let us derive the equation (1.95) of the virtual velocity principle. It follows from equation (1.23) that Z Z   ij;j C fi ıvi d V: vP i ıvi d V D (1.96) V

V

Let us transform the first integral on the right-hand side by using the Gauss–Ostrogradsky theorem, taking into account the boundary conditions (1.93), which imply ıvi D 0 if

x 2 Sv ;

and employing the Cauchy relations between the virtual velocity field and the virtual strain rate ı "Pij , 1 (1.97) ı "Pij D .ıvi;j C ıvj;i /; 2 to obtain Z Z Z ij;j ıvi d V D  ij ı "Pij d V C ti ıvi dS: (1.98) V

V

S

Substituting (1.98) into (1.96) results in relation (1.95), which holds for any continuous medium.

42

Chapter 1 Basic equations of continuous media

1.7.2 Weak formulation of the problem of continuum mechanics If equation (1.95) is supplemented with constitutive equations, relating the stresses to the kinematic quantities, strains and strain rates, and the stresses in (1.95) are expressed in terms of the velocities vi , one arrives at the weak formulation of the original problem of continuum mechanics in terms of the kinematic variables: Z Z Z Z   ij "ij ; "Pij ı "Pij d V C vP i ıvi d V D fi ıvi d V C ti ıvi dS: (1.99) V

V

V

S

The weak formulation of the equations of motion differs from the differential formulation quite significantly. The former does not involve spatial derivatives of the actual velocity and stress fields; in view of (1.97), equation (1.99) contains only derivatives of the virtual velocity field, ıvi;j , which can always be chosen to be sufficiently smooth. This, therefore, reduces the requirements for the smoothness of the desired solution; it only suffices that the integrals appearing in (1.94) and (1.95) exist. This makes it possible to take into consideration discontinuous functions as well, thus avoiding the treatment of the discontinuities – the relevant equilibrium conditions at the discontinuities (see (1.29)–(1.31)) will be satisfied automatically. Another advantage of the weak formulation is that there is no need to satisfy separately the so-called natural boundary conditions (1.93b) for the stresses; these conditions enter relation (1.95) and will be satisfied whenever (1.95) is satisfied. The unknown functions must only satisfy the principal boundary conditions (1.93a) for the kinematic quantities. The above advantages of the weak formulation of problems significantly simplify the solution and make this formulation primary when applying approximate methods. For example, these advantages are effectively used in variational difference methods or the finite element method, where the stress field is, as a rule, discontinuous by construction between elements; this approach facilitates the solution and, in addition, there is confidence that the approximate solution converges (in a certain sense) to the true solution of the problem as the mesh is refined. The virtual displacement and velocity principles are not the only ones that provide a weak formulation for continuum mechanics problems. There are various modifications and generalizations [36], which, however, are not as simple and common. Mixed variational principles can also be formulated where the displacement, strain, and stress fields are varied simultaneously rather than the fields of the kinematic quantities or the stress field individually. These include the Hu–Washizu principle, Hellinger–Reissner principle, and others [180, 181]. See Section 1.8 for the general variational principle.

43

Section 1.8 Variational principles of continuum mechanics

1.8 Variational principles of continuum mechanics For certain classes of continuous media, the principles of virtual displacements and velocities outlined in Section 1.6.2 can be used to obtain complete variational principles and so reduce the problem to minimizing special functionals.

1.8.1 Lagrange’s variational principle Let us focus on continuous media whose constitutive equations can be written in a potential form, which means that the stresses can be expressed in terms of derivatives of a potential function that depends on the kinematic variables: ij D

@ˆ1 .P"ij / @"Pij

or

ij D

@ˆ2 ."ij / : @"ij

(1.100a)

Such media are called nondissipative or conservative; in this case, the work done as the body is deformed does not depend on the deformation path. All external forces are assumed to be conservative and, hence, potential: fi D

1 @‰f ;  @vi

ti D

@‰ t @vi

or

fi D

1@ f ;  @ui

ti D

@ t @ui

(1.100b)

It follows from (1.95) that if there are no inertial forces, the following relation holds: Z  Z  ı ‰ t .vi / dS D 0: ˆ1 .P"ij /  ‰f .vi / d V  V

S

Denoting the expression that is varied by L.vi /, one arrives at Lagrange’s variational principle: the true solution of the problem corresponds to an extremum point of the Lagrangian function L.vi /: Z Z  ˆ1 .P"ij /  ‰f .vi / d V  ‰ t .vi / dS: (1.101a) ıLv D 0 ; where Lv D V

S

A similar principle holds true for an elastic medium that has a potential ˆ2 ."ij / with respect to the strains: Z Z  ˆ2 ."ij /  f .ui / d V  ıLu D 0 ; where Lu D t .ui / dS; (1.101b) V

S

where Lu is the potential energy of all forces, both internal and external, that act on the body or system of bodies. Lagrange’s variational principles (1.101a)–(1.101b) hold true for quasistatic problems, where the inertial terms are zero.

44

Chapter 1 Basic equations of continuous media

1.8.2 Hamilton’s variational principle Let us derive a dynamic variational principle for an elastic medium with a strain potential ˆ2 ."ij /. Using the virtual displacement principle (1.94) and taking into account the inertial term followed by integrating with respect to time over a finite interval Œt1 ; t2 and converting the volume integral to a mass integral for the inertial term, one obtains  Z t2 Z Z Z Z t2Z uR i ıui d m dt D ij ı"ij d V C fi ıui d V C ti ıui dS dt t1 M

t1

V

V

S

(1.102) for a body of volume V and mass M . Integrating the left-hand side of equation (1.102) by parts and taking into account that ıui .t1 / D ıui .t2 / D 0, one finds that  Z t2Z Z Z t2 uR i ıui d m dt D  uP i ıui d m dt M

t1

t1 M

Z t2Z D

t1 V



.uP i /2 ı 2



Z

t2

d V dt D ı

K dt; t1

R P 2 where K D V .u/ 2 d V is the kinetic energy of the body. Substituting the resulting expression into (1.102) and taking into account (1.101b), one arrives at Hamilton’s variational principle Z t2 ı .Lu C K/ dt D ı D 0; (1.103) t1

Rt where  D t12 .Lu C K/ is the Hamilton action on a finite time interval t 2 Œt1 ; t2 . According to Hamilton’s variational principle, amongst any kinematically admissible fields on a finite time interval with fixed endpoints, the true field corresponds a stationary point of the Hamilton action. When a problem is discretized, a continuous medium is replaced with a system with finitely many degrees of freedom characterized by nodal displacements qi . In this case, the condition that the functional (1.103) is stationary is reduced to the system of Lagrange equations of the second kind   d @L @L D Fi ; i D 1; : : : ; n;  dt @qP i @qi where qi and qP i are, respectively, generalized coordinates and generalized velocities of the system, Fi are generalized forces, and L D K  U is the Lagrangian function (for an elastic medium, U D ˆ2 ."ij /).

45

Section 1.8 Variational principles of continuum mechanics

1.8.3 Castigliano’s variational principle The above variational principles are kinematic, since the quantities that are varied are kinematic and correspond to continuum mechanics problems formulated in terms of kinematic variables, displacements and velocities. Variational principles formulated in terms of stresses are also possible; these can be constructed for the same potential media with the exception that the equations of the form (1.100) that characterize such media are solved for the strains. To this end, one should make use of the Legendre transformation, according to which a function '.x1 ; : : : ; xn / such that @' D Xi .x1 ; : : : ; xn / with @xi

d' D Xi dxi

(1.104a)

dˆ D xi dXi :

(1.104b)

'.0/ D ˆ.0/ D 0:

(1.104c)

is associated with a function ˆ.X1 ; : : : ; Xn / such that @ˆ D xi .X1 ; : : : ; Xn / with @Xi The functions '.xi / and ˆ.Xi / are related by '.xi / C ˆ.Xi /  xi Xi D 0;

with

Equations (1.104b) are a solution of the system of equations (1.104a) for the unknowns xi , which are expressed as derivatives of the same function ˆ.Xi / connected to '.xi / by relation (1.104c), which is easy to verify by differentiating (1.104c) with respect to Xi .

By applying the Legendre transformation to equations (1.100), one obtains "ij D

@' : @ij

(1.105)

If, for an elastic medium, there exists a stress potential ˆ2 ."ij /, then there also exists a strain potential '.ij /. In this case, for stationary problems of elasticity, one can formulate a principle of statically admissible stress fields. A statically admissible stress field is a field ıij that satisfies the static equations of elasticity in the absence of mass forces with static (natural) boundary conditions: ıij;j D 0; xi 2 V I

(1.106)

ıij nj D 0; xi 2 S :

Then it follows from the first equation in (1.106), after multiplying it by ui and integrating over the volume V , that Z Z Z Z ıij;j ui d V D ıij nj ui dS C ıij nj ui dS  ıij "ij d V D 0; V

S

Su

V

where ui are prescribed displacements on the part Su of the body surface.

(1.107)

46

Chapter 1 Basic equations of continuous media

The first integral on the right-hand side vanishes by virtue of the second condition in (1.106); the last two integrals can be written, in view of (1.105), as the total variation of a functional K, called the Castigliano functional: Z Z ' dV  ij nj ui dS: (1.108) ıK D 0 ; where K D V

Su

Formula (1.108) expresses Castigliano’s variational principle, which can be formulates as: the true stress field delivers a stationary value to the Castigliano functional. Equation (1.107) can be rewritten as Z Z ıPi ui dS D ıij "ij d V; where ıPi D ıij nj ; (1.109) Su

V

and treated as the principle of statically admissible stress fields: the work done by all external surface statically-admissible forces equals the work done by the internal statically-admissible stresses over the true displacements. The statement (1.109) can be treated as a weak form of the strain continuity equations. Just as (1.99), this form does not involve derivatives of the true stresses and strains and, in addition, for a solution to exist it suffices that the integrals in (1.109) exist. For a hyperelastic material, for which relations (1.100) hold, the variational principle (1.108) follows from relation (1.109). However, Castigliano’s principle (1.108) has a narrower area of application than Lagrange’s principle (1.101). For a steadystate flow of a viscous fluid, governed by equations (1.100), Castigliano’s principle is not valid because of the convective transport terms in the equations of motion. The weak form of a solution in the sense of Castigliano corresponds to the differential formulation of elasticity problems in terms of the stresses.

1.8.4 General variational principle for solving continuum mechanics problems In the above variational principles, which replace the differential formulation of a problem in terms of the displacements with Lagrange’s principle and that in terms of the stresses with Castigliano’s principle, the quantities that are varied are the displacements and stresses, respectively. A general differential formulation of continuum mechanics problems is possible where the displacements, strains, and stresses are all unknowns. A general variational formulation can be associated with it, in which a functional dependent of the above unknowns is used and these unknowns are all varied simultaneously. Such a variational principle was suggested by Washizu [181]. Let

Section 1.8 Variational principles of continuum mechanics

us formulate this principle for an elastic medium with the functional  ³  Z ² 1 ij "ij  .ui;j C uj;i / C U."ij /  Fi ui d V …W D 2 V Z Z    ti ui dS  ij nj ui  ui dS; S

47

(1.110)

Su

where "ij is the total strain, U."ij / is the specific elastic strain energy, Fi D fi is the body force, ti is the surface force, and ui are displacements prescribed on the part Su of the surface, with U."ij /  ˆ2 ."ij / for an elastic medium. Equating the variation of …W with zero, one arrives at the variational equation   Z ² 1  ıij "ij  .ui;j C uj;i / ı…W D 2 V  ³  1 @U ij ı"ij  .ıui;j C ıuj;i / C ı"ij  Fi ıui d V (1.111) 2 @"ij Z Z  ıij nj .ui  ui /  ij nj ıui dS D 0: ti ıui dS   S

Su

Bearing in mind that the variations ıui , ı"ij , and ıij are independent, one can obtain the equations and boundary conditions for the medium in question. The condition that the coefficient of ıij in the volume integral is zero implies the kinematic relations  1 x 2 V: (1.112) ui;j C uj;i ; "ij D 2 The same condition in the surface integral leads to the boundary conditions for the displacements x 2 Su : (1.113) ui D ui ; The condition that the coefficient of ı"ij is zero yields the constitutive equations of an elastic material @U : (1.114) ij D @"ij Performing appropriate transformations (similar to those performed when deriving Lagrange’s and Castigliano’s principles) and equating the coefficient of ıui with zero, one arrives at the equilibrium equations and boundary conditions for the stresses ij;j C Fi D 0; x 2 S : ij nj D ti ;

(1.115a) (1.115b)

Thus, the condition that the functional …W must be stationary leads to the complete system of equations and boundary conditions (1.112)–(1.115) for the continuous medium in question.

48

Chapter 1 Basic equations of continuous media

The general variational principle can be used to obtain more particular forms of functionals and associated variational principles if some of the above differential equations are assumed to be satisfied in advance for a particular medium and so not to be subjected to varying. For example, if the constitutive equations (1.114) are assumed to hold a priori and so "ij must not be varied, one arrives at the functional suggested by Reissner [144]: Z h i 1 ij .ui;j C uj;i /  ˆ0 .ij /  Fi ui d V …R D V 2 Z Z (1.116)   ij nj .ui  ui / C ij nj ıui dS: ti ui dS  S

Su

where ˆ0 .ij / D ij "ij  U0 ."ij / is the specific additional strain energy, which is plotted in Figure 1.11 for the case of uniaxial stress state (shaded area). F

Φ0

U0

u

Figure 1.11. Specific additional strain energy.

By varying ij and ui , one obtains the kinematic equations (1.112), boundary conditions (1.113), equilibrium equations (1.115a), and boundary conditions (1.115b). If, apart from the constitutive relations (1.114), one assumes that the equations (1.112) and boundary conditions (1.113) are also satisfied a priori, and so the only quantities that are subject to varying are the kinematically admissible displacements ui , one arrives at the Lagrange functional Z Z  U0 ."ij /  Fi ui d V  ti ui dS: (1.117) …L D V

S

Castigliano’s principle can be obtained by assuming that, apart from the constitutive equations (1.114), the equations (1.115) are also satisfied in advance, and hence by varying the statically admissible stresses ij : Z Z ˆ0 .ij / d V  ij nj ui dS: (1.118) …K D V

Su

Section 1.9 Kinematics of continuous media. Finite deformations

49

Formally, the functional (1.110) can be used to obtain a number of other variational functionals by assuming some combinations of the equations (1.112)–(1.115) to hold a priori [141].

1.8.5 Estimation of solution error In using one or another variational principle, one can assess the solution accuracy. As will be shown below, Lagrange’s principle provides a lower estimate for the strain energy, since the system stiffness is higher here than in the exact solution (certain constraints are imposed on the displacements and so there are fewer degrees of freedom in the exact solution), while Castigliano’s principles provides an upper estimate. Hence, with these principles, one can determine bounds within which the exact solution resides. The application of mixed variational principles is usually due to the fact that Lagrange’s principle, or the principle of least (stationary) total potential energy of a system, although allowing one to obtain the displacement fields in a relatively easy manner, requires the differentiation of the solution in order to obtain the stresses, which reduces the accuracy of their determination. Especially large errors arise in areas of stress concentration – near inclusions, in the vicinity of inhomogeneities, and at the interfaces between inhomogeneous layers. With mixed principles, where the displacements and stresses are varied simultaneously, these quantities are determined with the same accuracy when certain approximation schemes are used [42].

1.9 Kinematics of continuous media. Finite deformations In previous sections, when constructing constitutive equations of elastoviscoplastic media, we confined ourselves to the case of small deformations. However, in many important applications, the strains can reach tens and hundreds of percent and so cannot be treated as small; for example, this is the case in metal forming, deformation under the action of powerful impact and explosive loads, stress-strain analysis of structures at near-critical states, etc. The theory of large (nonlinear) deformations is much more complex than that of small (linear) deformations and requires special consideration.

1.9.1 Description of the motion of solids at large deformations Consider a continuum body B consisting of a composition of particles. Suppose that in the process of deformation, the body occupies a sequence of regions in the three-dimensional Euclidean space. These regions will be called configurations of the body B at times t . The position of a particle of the body in the initial configuration, C0 , at t D 0 is determined by a vector  D ˛ e˛ (˛ D 1; 2; 3) in a Cartesian reference frame xi , where e˛ is an orthogonal vector basis in this reference frame.

50

Chapter 1 Basic equations of continuous media

A motion of the body is a sequence of its configurations in time: x D F .; t / D Fi .; t /ei D xi ei ;

xi D Fi .˛ ; t /;

(1.119)

where x D xi ei is the position vector of the particle  in the current configuration, C t , at a time t , with xi j tD0 D i . All configurations of the body are considered in the same reference frame. In order to distinguish between the initial and current configurations, let us use the Greek indices ˛, ˇ, and  for the initial configuration and the Latin indices i , j , and k for the current one. Let us assume that there is a one-to-one correspondence between the points xi and ˛ and so the function in (1.119) can be inverted:  D F 1 .x; t / D Ľ.x; t / D ‰˛ e˛ ;

˛ D ‰˛ .xi ; t /:

(1.120)

The function Ľ.x; t / determines the coordinates of the particle in the initial configuration C0 , at t D 0, that has the position x in the current configuration C t , at the time t . The coordinates ˛ are Lagrangian (or material) and the coordinates xi are Eulerian (or spatial). In the Lagrangian (material) description, motion is characterized relative to the initial (reference) configuration. It is clear that the coordinate lines ˛ D const refer to specific particles; in the course of the motion (1.119), these lines change and form a curvilinear grid in the space of xi at time t . The coordinates ˛ are also called convective. Since these coordinates are non-orthogonal, one should consider covariant and contravariant quantities, which are conventionally denoted using subscripts and superscripts (e.g., ˛ and  ˛ ). In the Eulerian (spatial) description, motion is characterized in the coordinates xi . The motion (1.120) is described for a fixed point in space, through which different particles ˛ pass with time. For example, the Eulerian approach is natural in describing fluid flows. It is especially suitable for describing steady-state flow, since time does not enter (1.120) in this case. The Eulerian description is widely used in characterizing the motion of bodies subject to large deformations, although the Lagrangian description is more natural in characterizing unsteady motions [49].

1.9.2 Motion: deformation and rotation As the body moves, the vector d  also moves to d x. In the course of the motion, it deforms – changes its length and rotates – in accordance with equation (1.119): @Fi .; t / d ˛ ei D F d ; @˛ @Fi F D Fi˛ ei e˛ ; Fi˛ D : @˛

d x D dxi ei D d  D d ˛ e˛ ;

(1.121)

Section 1.9 Kinematics of continuous media. Finite deformations

51

Here the following sequence of transformations has been used: @Fi @Fi d ˛ ei D ı˛ d  ei D Fi˛ e˛ e ı˛ d  ei D F d  @˛ @˛ where F is a two-point tensor, which is indicated by the Latin subscript of the current configuration C t and the Greek subscript of the initial configuration C0 . The tensor F maps a small neighborhood of the particle d  in the initial configuration C0 into a neighborhood of d x in the current configuration C t . The inverse map is performed by the tensor F1 : @‰˛ d D .x; t / dxi ei D F1 d x; (1.122) @xi @‰˛ 1 1 D ; F1 D F˛i e˛ ei : F˛i @xi Let  D d =d  denote the unit vector in the direction of d , where d  is the length of d , and let n D d x=dx denote the unit vector in the direction of d x. Then it follows from (1.121) that dx n D F; (1.123) d where D dx=d  is the stretch ratio or simply the stretch of the vector d . Formula (1.123) shows that the tensor F rotates  into n and stretches d  by a factor of . In other words, the tensor F can be represented as the product of two tensors: F D RU or

F D VR;

D R1 , det R D 1), which performs a where R is a proper orthogonal tensor rigid-body rotation of a line element d  into a line element d x, while U and V are positive definite symmetric tensors, called the right (or material) stretch tensor and the left (or spatial) stretch tensor, respectively, which characterize pure stretch of an element. Mathematically, this representation is expressed by the theorem of unique polar decomposition of the tensor F into products of two tensors: (RT

F D RU D VR:

(1.124)

The pure rotation tensor R and the pure stretch tensors U and V can be expressed via F as  1=2 ; V2 D FFT : (1.125) FT F D U2 ; R D FU1 D F FT F It follows from formulas (1.123) and (1.124) that if the principal axes and eigenvalues of U are denoted by p and p , and those of V, by np and p , respectively, then Up D p p ;

np D Rp ;

p D 1; 2; 3:

The relations for V and p are the same up to the notation. From the similarity of the tensors U and V, U D RT VR, it follows that p D p , which implies that the transformation R performs a pure rigid-body rotation [49].

52

Chapter 1 Basic equations of continuous media

1.9.3 Strain measures. Green–Lagrange and Euler–Almansi strain tensors One can see from formula (1.125) that it is more convenient to use U2 and V2 rather than U and V. In solid mechanics, the Green–Lagrange strain tensor is taken to be a measure of deformation in the Lagrangian description of motion. It is expressed as ED

 1 1 T .C  I/ D F FI ; 2 2

(1.126)

where C D U2 D FT F is the right Cauchy–Green tensor and I is the identity tensor. The Green–Lagrange strain tensor determines the difference of the squared line elements in the current and initial configurations relative to the initial configuration C0 : d x2  d  2 D 2 d   E d  D 2E˛ˇ d ˛ d ˇ :

(1.127)

From (1.123) and (1.127) one can express the stretch ˛ as ˛ D .1 C 2E˛˛ /1=2 ;

(1.128)

which means that the diagonal components of E are related to the stretches. The off-diagonal components of E can be expressed via the cosines of the angles between coordinate lines ˛ and ˇ in the deformed configuration: 2E˛ˇ cos .m  n/ D   1=2 ;  .1 C 2E˛˛ / 1 C 2Eˇˇ

(1.129)

where m D Fe˛ = ˛ and n D Feˇ = ˇ are unit vectors. It follows that the off-diagonal components of E are related to the shear deformation of an initially rectangular area element. The components of the Green–Lagrange strain tensor E are expressed as  1 Fk˛ Fkˇ  ı˛ˇ 2  1 u˛;ˇ C uˇ;˛ C uk;˛ uk;ˇ ; D 2

E˛ˇ D

uk;˛ D

@uk ; @a˛

(1.130)

where the displacement vector u is found from (1.119) as  u.; t / D x   D Fi .i ; t /  ıiˇ ˇ ei D ui ei : Another strain measure, the Euler–Almansi strain tensor A, can be obtained by taking the current configuration C t to be the reference one: AD

 1  1 I  B1 D I  FT F1 ; 2 2

(1.131)

Section 1.9 Kinematics of continuous media. Finite deformations

53

where B D V2 D FFT is the left Cauchy–Green tensor. The Euler–Almansi strain tensor also determines the difference of the squared line elements in the current and initial configurations, but unlike the Green–Lagrange tensor, the difference is expressed relative to the current configuration C t : d x2  d  2 D 2 d x  A d x:

(1.132)

The tensors A and E are related to each other by the transformations E D FTAF;

A D FT EF1 :

(1.133)

This follows from the comparison of formulas (1.127) and (1.132). The components of A can be expressed from (1.131) as  1 1 1 ıij  F1 ˛i F˛j D .ui;j C uj;i  u˛;i u˛;j /; 2 2 @u˛ u˛;i D : u D x   D Œı˛i xi  ‰˛ .x; t / e˛ D u˛ e˛ ; @xi Aij D

(1.134)

It is clear that the Euler–Almansi strain tensor corresponds to the Eulerian description of motion. From (1.133)–(1.134) one can see that the components of the Green–Lagrange tensor in the basis e˛ the components of the Euler–Almansi tensor in the contravariant basis g˛ of the convective frame are numerically equal. This becomes apparent from the definition of the covariant basis g˛ : g˛ D Fe˛ D

@‰i ˛ . ; t / ei : @ ˛

(1.135)

The contravariant basis is defined as g˛ D FT e˛ D "˛ˇ gˇ  gˇ ;

with "˛ˇ  D

˛ˇ  ; .g1  g2 /  g3

where ˛ˇ  are the components of the Levi–Civita tensor, also known as the third-rank permutation tensor (e.g., see [137, 157]).

1.9.4 Deformation of area and volume elements As the body moves, an area element in the material coordinate system, d S0 D dS0 , is deformed into an area element of the current configuration, d S D dS n, where  and n are unit normals. Volume elements are expressed using dot and cross products: d V0 D d   .ı ˝ / D ˛ˇ d ˛ ıˇ  ; d V D d x  .ıx ˝ x/ D ij k Fi˛ d ˛ Fjˇ ıˇ Fk  D J d V0 ;

(1.136)

54

Chapter 1 Basic equations of continuous media n ν dS0

dS

Figure 1.12. Deformation of area element.

where d , ı, and  are differentials along the basis lines in the Lagrangian coordinates, d x, ıx, and x are differentials along the Eulerian coordinates, and J D ij k Fi˛ Fjˇ Fk D det F is the Jacobian of the transformation F. On the other hand, the volume transformation formula (1.136) can be written in terms of the area element d S as d V D d x  d S D J d V0 D d   J d S0 : It follows that

dx d S D J d S0 ; d

F d S D J d S0 :

(1.137)

1.9.5 Transformations: initial, reference, and intermediate configurations The current configuration C t , rather than the initial one C0 , is often taken to be the reference configuration, to which all other configurations are referred. The deformation gradient in a configuration C at a time  > t referred to the current configuration C t will be denoted F t . / and called a relative deformation gradient to distinguish it from the deformation gradient F. / in the configuration C referred to the initial configuration C0 (see Fig 1.13). We have F t . / D

@xi . / ei ej ; @xj .t /

F. / D

@xi . / ei e˛ : @˛ .t /

(1.138)

F(τ)

F(t)

C0

Ft(τ) Ct

Cτ

Figure 1.13. Transformation of configuration C0 (initial) into C t (current) via C (intermediate).

55

Section 1.9 Kinematics of continuous media. Finite deformations

Let us map C0 onto C via the intermediate configuration C t , C0 ! C t ! C (Figure 1.13), and consider the tensor decomposition F. / D F t . /F.t /

or F t . / D F. /F1 .t /:

(1.139)

Deformation can also be measured relative to the configuration C t . For example, the Green–Lagrange strain tensor can be represented as 1 T F t . /F t . /  I 2 1  T F .t /FT . /F. /F1 .t /  I D 2 1  T F .t /F1 .t /  I D FT .t /E. /F1 .t / C 2 1 . /E. /F . /  A.t /: D FT t t

E t . / D

(1.140)

1.9.6 Differentiation of tensors. Rate of deformation measures Let us differentiate the relative deformation gradient F t . / with respect to  at t D  : ˇ ˇ @ @ ˇ ˇ P /F1 .t / D L.t /; F t . /ˇ F. /ˇ F1 .t / D F.t D Dt Dt @ @

(1.141)

where L.t / D Lij .t /ei ˝ ej ;

Lij .t / D

@xP i @˛ @xP i .t / D D FPi˛ .F1 /˛j : @xj .t / @˛ @xj

It follows that L.t / is the velocity gradient tensor. Differentiating the polar decomposition of F t . / at  D t gives ˇ ˇˇ @ @ ˇ F t . /ˇ R t . /U t . / ˇ : D Dt Dt @ @ Taking into account that, by virtue of (1.139), R t . /j tD D I and U t . /j tD D I, one finds that P t .t / D D.t / C .t /; P t .t / C R (1.142) L.t / D U where sym L.t / D D.t / is the rate of deformation tensor and asym L.t / D .t / is the rate of rotation tensor, also known as the spin tensor or vorticity tensor. The tensor D characterizes the rates of pure deformation of a line element along the principal axes, while  characterizes the rotation of the line element. One can easily find that the relative rate of deformation of a volume element equals @xP i .d V /P D D tr L D tr D: dV @xi

56

Chapter 1 Basic equations of continuous media

Differentiating the area element transformation formulas (1.137) yields .d S/P D JP FT d S0 C J.F T /P d S0 D .tr L/ d S  LT d S; .d S/P D .tr L/ n dS  LT n dS;

(1.143)

.dS /P D .tr L/ dS  .n  LT n/ dS: Differentiating the Green–Lagrange tensor gives 1 P D 1 .FT FT FP T F C FT FP T F1 F/ EP D .FP T F C FT F/ 2 2 1 D FT .LT C L/F D FT DF: 2

(1.144)

P is related to D is the same way as E is related to the Euler–Almansi One can see that E T tensor, E D F AF. It follows that the rate of deformation tensor D relates to the Euler–Almansi tensor A in the current configuration C t in the same way as EP relates to E in the material configuration C0 . The material derivative EP characterizes the rate of change of the tensor E, which is a measure pure deformation of a particle without its rotation as a rigid body. Hence, D characterizes the rate of change of A without rotation of a particle as a rigid body as well. This means that the rate of deformation tensor D is an objective measure of the rate of change of the Euler–Almansi tensor A in the current configuration. Using (1.144), one can write P 1 D FT .FTAF/PF1 D A P C LTA C AL D L.A/: D D FT EF

(1.145)

Formula (1.145) suggests a more general rule for determining the objective derivative of a tensor Q in a configuration C related to the material configuration C0 by a transformation P: C D PC0 . The tensor Q is transformed into a tensor Q0 by the formula Q0 D PT QP, then the material derivative is computed in the material configuration C0 and the inverse transformation P1 is performed. The resulting expression represents the objective derivative L.Q/ of the tensor Q in the configuration C : P C .PP P 1 /T Q C Q.PP P 1 /: L.Q/ D PT .PT QP/PP1 D Q

(1.146)

Formula (1.146) defines an objective measure of the rate of change of a tensor Q in time in an arbitrary configuration. In tensor calculus, L.Q/ is called the Lie time derivative. It relates the objective time derivative, L.Q/, of the tensor Q, defined P provided that the in a certain configuration C , with its material time derivative Q, transformation P is known [155].

57

Section 1.10 Stress measures

1.10 Stress measures 1.10.1 Current configuration. Cauchy stress tensor Let us now consider the stress state. The total force acting on a body (of arbitrary volume) in the current configuration C t equals Z Z F D tn dS C f d V; (1.147) @V

V

where tn is the surface stress vector (traction vector), f is the force vector per unit mass, V is the volume of the body, and @V is the surface bounding the volume V . The law of conservation of momentum implies Z Z Z d tn dS C f d V D xR d V: (1.148) dt V @V V Considering the equilibrium of a tetrahedral element with one face being on the surface @V of the volume V , one finds that the surface stress vector and the Cauchy stress tensor T are related by tn D Tn; where n is the outward unit normal to the surface @V of V in the current configuration. Substituting tn into (1.148) and transforming the integral using the Gauss– Ostrogradsky theorem Z Z Tn dS D div T d V; @V

V

one arrives at the equations of motion div T C f D xR or

@Tij C fi D xR i : @xj

(1.149)

It follows from the torque equation that Tij D Tj i , or T D TT , which means that the Cauchy stress tensor is symmetric.

1.10.2 Current and initial configurations. The first and second Piola–Kirchhoff stress tensors The Cauchy equations of motion (1.149) are referred to the current configuration C t ; it is unknown for a moving deformable body and has to be determined. In addition, constitutive equations are usually formulated for the initial configuration C0 , which is known. Therefore, it is important to be able to write the main conservation laws

58

Chapter 1 Basic equations of continuous media

in different configurations and change from some stress measures to others, referred either of the two configurations, C t or C0 . In order to transform the surface force, let us make use of formula (1.137) for an area element: tn dS D Tn dS D J TFT  dS0 D P dS0 D t0n dS0 :

(1.150)

So t0n is the contact traction vector related to the initial area of the element. The stress tensor referred to the initial configuration, P, is called the first Piola–Kirchhoff stress tensor. It is related to the Cauchy tensor T by P D J TFT

1 with Pi˛ D J Tij F˛j I

(1.151)

so P, just as the deformation gradient F, is a two-point tensor referred to both the deformed configuration C t and the initial configuration C0 simultaneously [106]. Physically, the components Pi˛ can be interpreted as the components of the traction vector acting on the area element d S˛ , which initially was d S0˛ , in the spatial basis ei related to the unit area of the initial element dS0 . The equations of motion (1.149) can be rewritten in terms of the Lagrangian variables as @Pi˛ C 0 fi D 0 xR i ; Div P C 0 f D 0 xR or @˛ (1.152) or Pi˛ Fj˛ D Fiˇ Pjˇ : PFT D FPT It follows that the tensor P is nonsymmetric, just as F. The operator Div means that the divergence is taken in the Lagrangian variables. Another stress measure, completely referred to the initial configuration, is the second Piola–Kirchhoff stress tensor defined as S D J F1 TFT D F1 P:

(1.153)

In view of (1.150), the surface traction is expressed in terms of S as tn dS D t0n dS0 D FS dS0 : It follows that S is a symmetric tensor. Then the equations of motion (1.152) become   @ Fi˛ S˛ˇ Div .FS/ C 0 f D 0 xR or C 0 fi D 0 xR i .S D ST /: (1.154) @ˇ The Kirchhoff stress tensor K is defined as K D FSFT D J T:

(1.155)

Equation (1.155) shows that the contravariant components K ˛ˇ of the tensor K D K ˛ˇ g˛ gˇ are equal to the components S˛ˇ of the tensor S D S˛ˇ e˛ eˇ , where g˛ are the covariant basis vectors of the material reference frame in the deformed configuration.

59

Section 1.10 Stress measures

1.10.3 Measures of the rate of change of stress tensors The generalization of differential constitutive equations to the case of finite deformations will require measures of the rate of change of stress tensors. These rheological equations involve the time derivatives of stress tensors. Let us find out how the second Piola–Kirchhoff tensor S t . / changes as the configuration C t is transformed into a configuration C by the deformation gradient F t . /. It is clear that as  ! t;

F t . / ! I;

J t . / ! 1;

S t . / D T.t /:

In accordance with (1.153),  1  T S t . / D J t . / F t . / T. / F t . / :

(1.156)

The derivative of (1.156) with respect to time  taken at  D t is the Truesdell derivative of the Cauchy stress tensor [173]: ˇ ı @S t . / ˇˇ P  LT  TLT C T tr L: TD DT (1.157) @ ˇDt The Zaremba–Jaumann derivative of the Cauchy stress tensor [137] is obtained when a particle is rotated as a rigid body, so that F t . / D R t . / and U t . / D I with J t . / D 1; hence, S t . / D RT t . /T. /R t . /; ˇ @S t . / ˇˇ P  T  TT ; TD DT @ ˇDt 

P 1 :  D RR

Since the Piola–Kirchhoff tensor S t . / referred to the initial configuration C0 equals S. / D J.t /F1 .t /S t . /FT .t /;

(1.158)

its rate of change is expressed similarly: ı

P / D J.t /F1 .t /T.t /FT .t /: S.t

(1.159)

The convective derivative of the Kirchhoff tensor K referred to the current configuration is calculated as 

P  LK  KLT D T  DK  KD; P C D F.F1 KFT /PFT D K K 

(1.160)

P C D KP ˛ˇ g˛ ˝ gˇ . where T is the rate of change of T in the sense of Jaumann and K C T P is obtained as the Lie derivative (1.146) with P D F . So K

60

Chapter 1 Basic equations of continuous media

1.11 Variational principles for finite deformations 1.11.1 Principle of virtual work Let us generalize the variational principles of virtual displacements and virtual velocities to the case of finite deformations [129]. The work done by all external forces over virtual displacements equals the work done by the internal forces. In the current configuration C t , the equation of virtual displacements becomes Z Z Z T W ıD d V D ıv  tn dS C ıv  .f  x/ R d V: (1.161) V

@V

V

Inertial forces can also be included into this equation by making use of d’Alembert’s principle. The components ıvi are virtual velocities and ıui D ıvi dt are virtual displacements. The scalar product T W D is calculated as tr.TDT / D Tij Dj i , where Dij D

@vi @vj C : @xj @xi

Equations (1.161) can be rewritten in the initial configuration in terms of the Piola– Kirchhoff stress tensor components and the corresponding strain rate tensor compoP and FP using nents. The stress power can be expressed in terms of the tensors P, S, E, formulas (1.153) and (1.155): P T W D D J 1 P W FP D J 1 S W E: The stress tensor and the rate of deformation tensor, whose contraction equals the work power, are mutually conjugate tensors. In the initial configuration C0 , the virtual power is expressed as Z Z Z P W ıF d V0 D ıv  t0n dS0 C ıv  0 .f  x/ R d V0 ; V0

Z

Z

@V0

S W ıE d V0 D V0

ıv  @V0

t0n dS0

Z

V0

C

(1.162) ıv  0 .f  x/ R d V0 ;

V0

where the vector t0n is calculated from formulas (1.150) and (1.153).

1.11.2 Statement of the principle in increments In nonlinear problems, solutions are usually obtained by an incremental step-by-step method. To this end, the principle of virtual work (1.162) should be represented in terms of increments and virtual rates. It is required to find a solution in a configuration C for a given solution at the previous step at t D  in the configuration C t .

61

Section 1.12 Constitutive equations of plasticity under finite deformations

It should be noted that the virtual rates of the Green–Lagrange strain tensor in the configurations C t and C are related by 

      1 P / C 1 FT ı FP C ı FT PF ; (1.163) ıE. / P D FT . /ı FP C FT P F. / D ı E.t 2 2

where Fi˛ D

@ui ; @˛

F D Fi˛ ei ˝ e˛ ;

ui D xi . /  xi .t /:

Then equation (1.161) becomes Z Z Z 0 P P W ı F.t / d V0 D ıv  tn . / dS0 C ıv  0 Œf. /  xR . / d V0 V0 @V0 V0 Z (1.164) P  P.t / W ı F.t / d V0 ; V0

or Z

P / C .F  S.t // W ı F P d V0 D ŒS W ı E.t

Z

V0

@V0

ıv  t0n . / dS0

Z

C

ıv  0 Œf. /  xR . / d V0 (1.165) Z V0



P / d V0 ; S.t / W ı E.t

V0

where P D P. /  P.t /;

S D S. /  S.t /;

and so  D t C t:

If C t is taken to be the reference configuration, then one should set P.t / D S.t / D T.t /;

P / D ıD.t /; ı E.t

ı FP D ıL;

Fij D

@ui .t / @xj

in formulas (1.164) and (1.165) and integrate of the current volume V and current surface @V .

1.12 Constitutive equations of plasticity under finite deformations 1.12.1 Multiplicative decomposition. Deformation gradients Numerous formulations of constitutive equations of plasticity at finite deformations have been suggested over the last three or four decades and field continues to develop

62

Chapter 1 Basic equations of continuous media

at the present time. There are different points of view on the kinematics of materials and statement of plastic flow rules. An important approach to studying such phenomena is suggested by the multiplicative theory of elastoplastic flows [103], which is based on the multiplicative decomposition of the deformation gradient tensor F D Fe Fp ;

(1.166)

where Fe is the elastic deformation gradient associated with the unloaded configuration, also called an intermediate configuration, of all infinitesimal neighborhoods of points of the elastoplastic body. In order to implement an intermediate configuration in the real Euclidean space, it is generally required to violate the continuity of the material. For polycrystalline solids, such as metals, this can have a physical interpretation, based on mechanisms for the formation of dislocations, which lead to incompatibility of the plastic strain field with the strain rate field. Figure 1.14 gives a schematic representation of the kinematics of elastoplastic deformation based on considering three configurations of the body: initial C0 , current C t , and unloaded Cp . dx Ct

F

Fe

Fp

dξ C0

dx* Cp

Figure 1.14. Configurations of an elastoplastic body.

The multiplicative theory is not the only way of decomposing the elastoplastic deformation into an elastic and plastic component. The additive decomposition of the Green–Lagrange tensor (1.167) E D Ee C Ep can also be used to construct a theory based on thermodynamic considerations [50]. In plasticity theories used in computational research, preference is given, as a rule, to the additive decomposition of the spatial (Eulerian) rate of deformation D D De C Dp ; which is more convenient from the computational viewpoint.

(1.168)

Section 1.12 Constitutive equations of plasticity under finite deformations

63

Our subsequent presentation will be based on the multiplicative decomposition (1.166). However, it was shown previously [58, 161] that the additive decomposition of the rate of deformation (1.168) can be obtain from purely geometric considerations, within the framework of multiplicative kinematics, in both the material and spatial descriptions. For simplicity, let us restrict our consideration to isothermal deformation. The constitutive equations of elastoplastic flow can be most easily generalized to the case of finite deformations within the material (Lagrangian) description.

1.12.2 Material description It is assumed that the total strain is characterized by the Green–Lagrange tensor E and the plastic strain is characterized by the tensor Ep : 1 E D .FT F  I/; 2

1 Ep D .FT G Fp  I/; 2 p

(1.169)

where G is the metric tensor of the unloaded configuration. In an orthogonal reference frame, G D I. The elastic strain tensor is formally determined as the difference of the total strain tensor and the plastic strain tensor:   T (1.170) Ee D E  Ep D FT p Fe Fe  G Fp It follows that the elastic strain tensor depends not only on Fe but also on Fp :  T  T 2   T T 2 Ee D UT p Rp Ue Ue  G Rp Up D Up Rp Ue Rp Up  Up where U is the right stretch tensor and R is the rotation tensor. The relation between the stress and elastic strain is assumed to be the same as in the case of small strains and generalized by replacing the Cauchy stress tensor T with the second Piola–Kirchhoff tensor S and the small strain tensor "e with Ee : S D 0

@‰.Ee ; Ep / ; @Ee

(1.171)

where the free energy function ‰ is taken to be the potential. A plastic flow law can be derived from the associated flow rule or a general nonassociated rule in the form P /; (1.172) EP p D H.S; where the material derivative of the plastic strain tensor EP p is taken to be the measure of the plastic strain rate, P is an unknown scalar parameter, and  is a hardening tensor. In the special case of the associated flow rule, the tensor H is linked to the yield criterion ˆ.S; ; C/ D 0 (1.173)

64

Chapter 1 Basic equations of continuous media

by the relation for the plastic potential HD

@ˆ : @S

(1.174)

Finally, an evolution equation should be specified in order to determine the hardening tensor  in the general case; this equation is taken in the form P P D Q.S; C; /;

(1.175)

where C D FT gF and g is a metric tensor. If there are restrictions imposed on the structure of the material (e.g., one considers a plastic flow for an isotropic incompressible material, whose response is independent of the first invariant of the stress tensor), these are taken into account using the same methods as in the theory of small deformations.

1.12.3 Spatial description When an elastoplastic material is characterized using the spatial (Eulerian) approach, the following Euler–Almansi strain tensors are used as the measures of the total, plastic, and elastic strain [106]:  1 I  FT F1 ; 2  1  T 1 Fe GFe  FT F1 ; Ap D 2  1 1 Ae D A  Ap D : I  FT e GFe 2 AD

(1.176)

These tensors are obtained from the material Green–Lagrange tensors E, Ep , and Ee defined by equations (1.169) and (1.170) using the inverse transformation FT EF D A and so on. The objective rates of change of the tensors (1.176) are obtained by applying the push-forward/pull-back Lie transformation, denoted L.A/; see Section 1.9 [161]. In order to obtain the objective rate of change of the tensor A referred to a configuration C , one should carry out the pull-back transformation to the material configuration C0 , calculate the material derivative, and perform the push-forward transformation to the configuration C . The pull-back transformation of the tensor A is F ! FTAF D E and the inverse, push-forward transformation is expressed as F1 ! FT EF1 D A: L.A/ D FT .FTAF/PF1

P C AL D LTA C A

T T P P 1 Lp .Ap / D FT p .Fp Ap Fp / Fp D Lp Ap C Ap C Ap Lp

 1 L C LT ; DD 2

 1 Lp C LT Dp D p ; 2

D D; D Dp ;

Lp D FP p F1 p :

(1.177)

Section 1.12 Constitutive equations of plasticity under finite deformations

65

1.12.4 Elastic isotropic body Let us write out the equations for an elastic isotropic body in the spatial formulation [163]. In this case, the potential ‰ is a function of the Euler–Almansi tensor invariants k and material constants ci : ‰ D ‰.1 ; 2 ; 3 ; ci / The equation for the Cauchy stress tensor T in the index notation is written as @‰ @Aij or, in terms of the stress tensor components with mixed indices,  @‰

Tji D  ıki  2Aik ; @Aji T ij D 

Tji D g˛j T i˛ ;

(1.178)

Aij D gi˛ Aj˛ :

Let us expand the function ‰ in a power series in i up to the third-order terms in Aji . Taking the initial stresses and strains to be zero, we obtain 

0 ‰ D c1 12 C c2 2 C c3 13 C c4 1 2 C c5 3 C O .Aji /4 : Hence, the stress-strain relation (1.178) becomes  i˛ ˇ Tji D 2c1 1 C .3c3  2c1 /12 C c4 2 ıji C Œc2 C .c4  c2 /1 ıjˇ A˛  4c1 1 Aji 1 iˇ ˇ˛ C c5 ıj˛ı A˛ˇ Aı  2c2 ıj  A˛ Aiˇ : (1.179) 2 The equation involves five elastic constants and so this representation is called the five-constant theory of elasticity. In the transformation, the mass conservation law was used: p  D 0 1 C 21 C 42  83 : If the stress-strain relation is rewritten in terms of the Lamé constants and , formula (1.179) becomes  Tji D 1 C .3l C m  /12 C m2 ıji C Œ2  .m C 2 C 2 /1 Aji (1.180) 1 iˇ  4 Ai˛ Aj˛ C nıj˛ı A˛ˇ Aı ; 2 where the relations 1 c1 D . C 2 /; c2 D 2 ; c3 D l; c4 D m; c5 D n 2 have been taken into account. Neglecting the quadratic terms with respect to the strains in (1.180) results in the standard expression of Hooke’s law: Tji D 1 ıji C 2 Aji :

66

Chapter 1 Basic equations of continuous media

1.12.5 Hyperelastoplastic medium The equations for a hyperelastoplastic medium in the spatial formulation are obtained from equations (1.171)–(1.177). The results are summarized below [123]. 1. Additive decomposition of the Euler–Almansi tensor: Ae D A  Ap : 2. Stress-strain relation: T D 0

@‰.Ae ; Ap ; / : @Ae

(1.181)

(1.182)

3. Plastic flow law [125]: P g; /: Lp .Ap / D Dp D H.T; 4. Hardening law: P g; /: Lp ./ D Q.T;

(1.183)

ˆ.T; g; / D 0:

(1.184)

5. Yield criterion:

1.12.6 The von Mises yield criterion To illustrate the transition from the indexless notation to the covariant formulation of the plasticity equations, let us consider the von Mises yield criterion. In the spatial description, the von Mises yield criterion has the form  1 1 ˆ.T; g; k/ D  W Ig W   k 2 D  ij  kl gik gj l  k 2 ; 2 2 where g is the metric tensor in the current configuration, Ig is the unit tensor of rank 4 with components I ij kl D

1  1 ik 1 j l .g / .g / C .g1 /il .g1 /j k ; 2

and  is the deviator of the stress tensor T: 1  ij D T ij  .T kl gkl /.g1 /ij : 3 In the material description, the von Mises yield criterion is expressed as 1 ˆ.S; c; k/ D S ij S kl cik cj l  k 2 ; 2 1 ij ij s D S  .S kl ckl /.c1 /ij ; 3

Section 1.12 Constitutive equations of plasticity under finite deformations

67

where c D FT gF and s ij are the deviatoric components of the second Piola–Kirchhoff stress tensor S. The covariant representation of the other relations, (1.181)–(1.183), can be obtained likewise.

Part II

Theory of finite-difference schemes

Chapter 2

The basics of the theory of finite-difference schemes 2.1 Finite-difference approximations for differential operators In solving problems of solid mechanics by computational methods, one has to partition the body into a number of elements so as to reduce the problem to solving a system of algebraic equations. Historically, from its very birth, mechanics relied on the continuum method for solving problems. Bodies were treated as continuous sets of particles and problems were stated in terms of continuous functions. Differential and integral calculus was the main tool for studying these problems. Over the last few centuries, a most powerful mathematical machinery has been created for the analysis of problems arising in physics and mechanics, relying on the solution methods for differential equations. However, discrete analysis did not practically develop before the advent of computers. It was not until the 1940s, when computers came on the scene, that the situation changed and at the present time discrete methods and their applications are booming. There are two main lines of the development of discrete analysis: (i) direct physical modeling and (ii) mathematical modeling. Within the former approach, continuum bodies are treated as discrete ensembles of material particles, to which physical laws are applied directly and discrete equations are derived bypassing the mathematical formulation in terms of functions of continuous arguments. However, modern computational mechanics relies mainly on the latter approach. In this case, a continuous mathematical problem is first formulated and then its discretization is performed. This allows one to take advantage of the achievements in mathematical analysis obtained through the centuries. It is this approach that our further presentation will rely on.

2.1.1 Finite-difference approximation In order to approximate a problem stated in the form on an operator equation for a function u of a continuous argument, L.u/ D f; with discrete equations, one has to do the following.

72

Chapter 2 The basics of the theory of finite-difference schemes Γh

Γ

Ω

Figure 2.1. Grid approximation of a domain  and its boundary ; h is a broken line approximating .

1. Replace the domain ! of continuous variation of the argument with a discrete set of points !h . For example, if the operator L is defined in the domain ! shown in Figure 2.1, then ! can be replaced with a set of nodes !h of a square grid covering !. The boundary  of ! is approximated by a broken line h . 2. Introduce functions uh of a discrete argument, called grid functions, defined on the set !h . 3. Replace the differential operator L.u/ with a discrete analogue Lh .uh / defined on the discrete set !h : L.u/ ! Lh .uh /: The continuous problems is thus reduced to an algebraic system of equations for the values of the functions uh at the points of the discrete set !h . This general scheme must have a rigorous mathematical formalization. To this end, one introduces the concepts of a grid and a grid function. A grid is a set !h D ¹xi 2 !º (i D 1; : : : ; N ). A grid function associated with a continuous function u.x/ using an operator Ph is a discrete set of values uh D Ph .u/. Functions of a continuous argument u.x/ are elements of a functional space H . A set of grid functions forms a vector space Hh whose dimension coincides with the number of nodes N of the grid and the components of a vector are the values of the grid functions at the nodes xi : uh .xi /. One introduces a norm of grid functions kuh kHh in the space Hh ; it is analogous to the norm kukH in the space H , so that the compatibility condition lim kuh kHh D kukH

h!0

is satisfied. For example, (i) to the norm kukC in the space of continuous functions C there corresponds a norm kuh kCh in the space of grid functions Ch : kukC D max ju.x/j x2!

!

kuh kCh D max ju.xi /jI xi 2!h

73

Section 2.1 Finite-difference approximations for differential operators

(ii) to the norm kukL2 in the space of square-integrable functions L2 there corresponds a norm kuh kL2 in the space L2h : h

Z

u2 dx

kukL D

1=2 !

!

kuh kLh D

NX 1

u2i hi

1=2 I

iD1

(iii) to the norm kukW 2 in the Sobolev space W 2 there corresponds a norm kuh kW 2 h

in the space Wh2 : Z kukW2 D

Z

x

dx !

2

u dx 0

1=2 !

kuh kW2h D

NX 1 iD1

hk

k X

u2i hi

1=2 ;

iD1

and so on. These norms are generated by the scalar products of functions in the spaces L2 and W 2 and the scalar products of vectors in the vector spaces L2h and Wh2 .

2.1.2 Estimation of approximation error The main task of the theory of finite difference schemes is to estimate the closeness of the solution of a finite difference problem to that of the associated differential problem. However, these solutions are defined in different spaces, H and Hh , and have different norms, and hence one can only estimate the difference between the solutions in terms of a common norm. This difficulty can be overcome in two ways. First, the solution defined on a given set !h as a grid function uh can be extended to a function of a continuous argument u.x/ defined on the whole set ! by using an interpolation Q operator RŒuh ! u.x/. The objective is to recover a continuous function u.x/ Q from a given set of values of Q will certainly be different from u.x/, and the grid function uh .xi /. The function u.x/ so one should evaluate the norm ku.x/  u.x/k Q H for x 2 !. Such an extension is nonunique, which is related to the nonuniqueness of the interpolation operator R.uh /. The interpolation theory is a well-developed classical mathematical theory, which continues to evolve due to, in particular, new problems that are solved by the finite element method. A grid function can be extended in a number of different ways by using, for example, a polynomial interpolation, such as linear, quadratic and so on. In this case, suitable approximation errors can be estimated in the space H . It is exactly this approach that is used in the finite element method (FEM), where uh is defined in the entire domain ! as a piecewise continuous function. This enables one to use the power of continuous function techniques to prove convergence, stability, etc. Secondly, one can use an operator Ph .u/ D uh that projects the function u.x/ onto the grid to obtain a grid function uh .xi /. It should be noted that no inverse, extension operator R.uh / D u can be recovered from the projection operator Ph .u/,

74

Chapter 2 The basics of the theory of finite-difference schemes

since these are defined in different spaces. The operator PH .u/ acts from H into Hh , while R.uh / acts from Hh into H [153]. In numerical analysis, both approaches are employed. The former is used in the finite element method, where one deals with functions defined in H , the nodal values of uh are extended to u.x/ and so one constructs an operator that associates the vector space Hh with the continuous space H . Approximation errors are estimated and convergence is proved in the space H . To each operator Rh .u/ there corresponds a set of shape functions i .x/ for a selected set of nodes defining a finite element, which means that the functions defined on this finite element are recovered from the nodal values. The shape functions form a basis in H and are treated using continuous operators of integration, differentiation, etc. These questions have been discussed in detail in books on applying the finite element method to solving continuum mechanics problems. In the theory of finite difference equations, the opposite is done: instead of extending uh to u.x/, one projects u.x/ onto !h with an operator Ph .u/ ! uh and treats all functions in the space Hh . In the simplest case where the set of points xi of the grid satisfies !h 2 !, the projection operator is Ph .u/ D u.xi /. The operator Ph .u/ can be more complicated; for example, it can be an operator of weighted averaging over the neighboring nodes as shown in Figure 2.2, where x is the central point of the regular hexagonal mesh inside the domain of definition of u.x/: P6 u.x / D

iD1 .ui

C uiC1 / Si ; 2S

where Si is the area of the equilateral triangle with vertices at the points i , i C 1, and x . Now the question can be raised on how to define the projection operation for a differential operator Ph .L/ D Lh .uh /, or how to replace it with a finite difference operator. This can be done in infinitely many ways. For example, even in the simplest i+2

i+3

i+1

x

*

Si

i

Figure 2.2. To the definition of an operator Ph .u/ of weighted averaging over neighboring nodes.

Section 2.1 Finite-difference approximations for differential operators

75

case of approximating the first derivative on a three-point stencil, one can obtain a family of finite difference operators dependent on a parameter: L.v/ D

dv ; dx

viC1  vi D vx (forward difference); h N h .vh / D vi  vi1 D vxN (backward difference); PNh .L/ D L h .˛/ Lh D ˛vx C .1  ˛/vxN ; Ph .L/ D Lh .vh / D

where L.˛/ denotes a family of finite difference operators dependent on the parameh ter ˛ (0  ˛  1) and h is the step size of the grid. For example, second-order derivatives can be approximated as L2 .v/ D

d 2v ; dx 2

viC1  2vi C vi1 : h Approximation formulas for higher-order derivatives (see exercises at the end of Chapter 2) and, hence, any differential operator Lh can also be obtained quite easily. The question arises as to what is the approximation error of these formulas. For a given differential operator L, the norm k h kHh of the grid function Ph .L2 / D L2h .vh / D

h

D Lh .uh /  Ph .L.u//h

(2.1)

will be called the approximation error of replacing L with a finite difference operator Lh ; here uh D Ph .u/ with u.x/ being a function of a continuous argument and uh .xi / being a function of a discrete argument. This norm characterizes the approximation error across the entire domain of definition of the grid operator Lh .uh /. If k kHh D O.hk /, then Lh will be said to approximate L with order k. To sum up, global approximation is associated with the concept of norm and, hence, with the domain and its partitioning; therefore, it differs from local approximation in a neighborhood of a point. The local error of approximation .xi / at a point xi can be easily evaluated by expanding vi˙1 D v.xi ˙ h/ in a Taylor series. For example, for the forward difference, one obtains   1 h2 v.xi / C v 0 .xi / h C v 00 .xi / C O.h3 /  v.xi / D v 0 .xi / C O.h/; vx D h 2 0 h .xi / D vx  v .xi / D Lh .uh /  Ph .L.u// D O.h/: The local approximation error for any difference operator can be evaluated in a similar manner. It is important to emphasize the difference between the local approximation error in a neighborhood of a selected point and the global approximation error for the entire grid domain.

76

Chapter 2 The basics of the theory of finite-difference schemes

Let us show that the selection of the norm k kHh is rather significant and that the approximation errors evaluated in different spaces can happen to have different orders of magnitude. This is especially important when dealing with irregular grids. Consider an example. Suppose L D @2=@x 2 . Let us approximate L on an irregular grid with a varying step size hi as follows:   1 viC1  vi vi  vi1 hiC1 C hi Lh D :  ; hN D hiC1 hi 2 hN It can easily be shown that the local approximation of smallness: h .xi /

D

h .xi /

of Lh has the first order

hiC1  hi 000 v .xi / C O.hN 2 /: 3

The global approximation errors in Ch and L2h are also of the first order: k .xi /kCh D max j .xi /j D O.h/; NX 1=2 1 2 k .xi /kL2h D hi i D O.h/:

(2.2)

iD1

However, in terms of the norm in the Sobolev space Wh , the approximation has the second order: "N 1 N 1 2 #1=2 X X hk k D O.hN 2i /: k .xi /kWh D hN i iD1

kD1

Indeed, one can write N 1 X

hk

k

D

kD1

N 1 X

hN k

kD1

 h2kC1  h2k 000  1 000 : v .xk / D h2i vi000 C h2iC1 viC1 6 6hN k

The intermediate terms are canceled out to give the estimate [153] k .xi /kWh D

"N 1 X iD1

hN i

NX 1 kD1

hk

#1=2 2 #1=2 "NX 1  2  1 2 2 h v 000  h2i vi000 D hN i k 36 iC1 iC1 iD1

D

NX 1

1=2 4 N D O.h2 /; hi O.h /

iD1

where it has been taken into account that O.hi / D O.hN i /. The approximation order depends on the chosen stencil, the set of nodes involved in the approximation of the differential operator with finite differences. The approximation order can be increased by using stencils with more nodes, which enables one to

Section 2.1 Finite-difference approximations for differential operators

77

reduce the number of grid points while preserving the computational accuracy. However, this is not always favorable, since increasing the number of nodes in the stencil results in more complicated approximation formulas and an increased computation time per grid point. There is another possibility for increasing the order of approximation; this possibility was suggested by Lewis Fry Richardson in the early 20th century1 .

2.1.3 Richardson’s extrapolation formula To increase the approximation order, one can perform computations on embedded grids instead of using stencils with more nodes. According to this approach, one should perform computations on grids with decreasing step size: h, h=2, h=3, and so on. With the successively obtained solutions uh , uh=2 , uh=3 , . . . , one can construct an extrapolation formula that provides a higher order of approximation than those of the calculated solutions involved [117]. For example, with two solutions, uh and uh=2 , calculated by the same symmetric second-order scheme uh .x/ D u.x/ C h2 v.x/ C O.h4 /;  2 h v.x/ C O.h4 /; uh=2 .x/ D u.x/ C 2 one can compose a linear combination, 1 4 Uh D  uh C uh=2 D u.x/ C O.h4 /; 3 3 to obtain a solution accurate to the fourth order of smallness. By using three firstorder schemes with decreasing step sizes h, h=2, and h=3, one can obtain a solution accurate up to O.h3 /: Uh D auh C buh=2 C cuh=3 D u.x/ C O.h3 /;

(2.3)

where the weighting coefficients a, b, and c are determined from the system of equations a C b C c D 1; 1 1 a C b C c D 0; 2 3  2  2 1 1 aC bC c D 0: 2 3 1

Richardson, L. F. (1911). The approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London, Series A 210 (459-470): 307–357.

78

Chapter 2 The basics of the theory of finite-difference schemes

It follows that 1 aD ; 2

b D 4;

9 cD : 2

In many cases, this technique allows one to improve the accuracy at almost no cost, since computations on two or three embedded grids are usually performed anyway to check the convergence of the method employed. It should be noted, however, that as the number n of terms in formula (2.3) increases, the weighting coefficients, which have alternate signs, increase rapidly with decreasing grid step size h=n, which can result in the effect of rounding errors on the final result. To avoid this, one usually refines the grid in the ratio of h=2n , in which case the coefficients increase more slowly but each set of computations requires more time [117].

2.2 Stability and convergence of finite difference equations 2.2.1 Stability The question arises: Does it follow from the approximation condition k h k D O.hk / (see (2.1)) that the solution of the finite difference equation will always differ from the exact solution by O.hk /? No, it does not. The above approximation condition for a differential operator is necessary but not sufficient for the solution of the finite difference equation Lh .uh / D 0 to converge to the solution of the corresponding differential equation L.u/ D 0 as h ! 0. One more condition is required for the convergence; specifically, small errors introduced by the approximation into the finite difference equation must not result in large deviations in the solution. This important property of a finite difference scheme ensures the stability of the finite difference equation Lh .uh / D fh : This property is closely linked to the continuous dependence of the solution on the right-hand side of the equation; a small perturbation ıfh on the right-hand side of the equation results in a small perturbation ıuh in the solution of the equation. Definition of stability: a finite difference scheme is called stable if the condition kıuh k  C kıfh k holds for any ıfh 2 Hh , where ıuh D u  uh and ıfh D f  fh .

2.2.2 Lax convergence theorem The following theorem holds true (which is due to Peter Lax):

(2.4)

Section 2.2 Stability and convergence of finite difference equations

79

Theorem. If a difference operator Lh .uh / D fh approximates a differential operator L.u/ D f and the resulting finite difference scheme is stable, then the solution uh converges to u. So we have: 1)

    Lh Ph .u/  Ph L.u/ D O.hk /; H h

k > 0:

(2.5)

The grid function uh approximates the function u, Ph .L.u// is a projection of the differential operator L onto the grid space Hh , and Lh is a difference operator. 2) kıuh k  ckıfh k. The solution uh of the finite difference equation Lh .uh / D fh is stable with respect to the right-hand side. It is required to prove that kuh  Ph .u/k D O.hk / as h ! 0, where u is the solution of L.u/ D f . Proof. From the approximation condition (2.5) one finds that     Lh Ph .u/  Ph L.u/ D kLh .uh /  Ph .f /k D kfh  Ph .f /k D kıfh k D O.hk /:

(2.6)

Let ıuh denote the error of the solution uh of the finite difference equation: ıuh D Ph .u/  uh : It follows from stability condition (2.4) and relation (2.6) that kıuh k  C kıfh k D C O.hk /; kuh  Ph .u/k D O.hk /; which is what was to be proved. In other words, the error brought by the approximation into the right-hand side of the finite difference equation has the order of smallness O.hk /; then, by virtue of the stability condition (2.4), the error of the solution to the finite difference equation will have the same order of smallness.

2.2.3 Example of an unstable finite difference scheme Let us consider a simple example to study the stability of a finite difference scheme for the first-order ordinary equation y 0 C ˛y D 0 with y.0/ D y0 : It is easy to solve: y D y0 e ˛x .

(2.7)

80

Chapter 2 The basics of the theory of finite-difference schemes

Let us approximate equation (2.7), using a uniform three-point stencil with step size h, by the following family of difference operators dependent on a parameter  (0    1), represented by a linear combination of a forward and a backward finite difference: y0 D

ynC1  yn yn  yn1 dy D C .1  / : dx h h

So equation (2.7) becomes ynC1 

2  1  ˛h  1 yn C yn1 D 0:  

(2.8)

Let us search for the solution of this constant-coefficient finite difference equation in the form yn D C1 n . This leads to the following quadratic equation for : 2 C

1 1  2 C ˛h  D 0:  

(2.9)

Its solution is 1;2

  q 1 2 2  .1  2 C ˛h/ ˙ 1 C 2˛h.1  2/ C ˛  : D 2

The general solution of equation (2.8) is written in terms of two arbitrary constants, C1 and C2 : yn D C1 n1 C C2 n2 : (2.10) Let us analyze the behavior of the solution as h ! 0. We have 1 D 1  ˛h C O.˛ 2 h2 /;

2 D

1 .1 C ˛h/ C O.˛ 2 h2 /: 

Taking into account that nh D xn and limh!0 .1  ˛h/1= h D e ˛ , we find that         1 xn=h : (2.11) yn D C1 e ˛xn C O.˛ 2 h2 / C C2 e ˛xn C O.˛ 2 h2 /  There is only one condition, the initial condition of (2.7), for determining C1 and C2 . Hence, one of the constants remains arbitrary and so C2 is nonzero. This means that the particular solution corresponding to 2 is an artefact of the form of the approximation adopted; it is a parasitic solution. The appearance of the parasitic solution is due to the fact that the finite difference equation (2.8) is formally of the second order, which determines the number of arbitrary constants in the general solution (2.10). However, the original differential equation (2.7) is of the first order and its general solution depends on a single arbitrary constant, which is determined from the initial condition in (2.7). To determine

81

Section 2.3 Numerical integration of the Cauchy problem for systems of equations

the constants in (2.10), a second condition is required for the finite difference equation in order to determine the second arbitrary constant in solution (2.10). There are several ways for determining this constant. For example, this can be done by using different orders of approximation. First of all, it is clear that the second condition must refer to the point x D h rather than x D 0; one should set y.h/ D y1 with y1 being very close to y0 . If one sets y1 D y0 , this will result in an O.h/ error. If one uses the two-point scheme (2.8) with  D 1, then y1 D y0 .1  ˛h/ C O.h2 /: It is clear that the error in determining y1 must agree with the approximation order of the finite difference scheme employed in order not to lose the accuracy of the solution. This situation is typical of the schemes whose formal order is higher than that of the differential equations they are used to approximate. Such schemes are quite common when it is desired to increase the order of approximation of the solution; however, one should make sure that appropriate orders of approximation are used in the additional initial conditions. Thus, the second condition for determining C1 and C2 should be obtained using the two-point scheme (2.8) with  D 1. Finally, we have y D y0 at x D 0 and y1 D .1  ˛h/ y0 at x D h. Substituting these conditions into (2.10) yields C1 D y0 C O.˛ 2 h2 /;

C2 D O.˛ 2 h2 /:

It is clear from (2.11) that as h ! 0 and with 12    1, the resulting solution differs from the exact solution of equation (2.7) by O.˛ 2 h2 /, which means that the scheme (2.8) is stable and provides the second order of approximation for the solution of (2.7). However, if 0 <  < 12 , the second particular solution increases catastrophically as h ! 0 for fixed xn , which testifies that the scheme (2.8) is unstable. Indeed, it is apparent from (2.11) that a small error due to the approximation leads to a small deviation of C2 from zero if 1 >   12 , and hence the error in the solution tens to zero as h ! 0, whereas it catastrophically increases as  < 12 . To sum up, even though the approximation condition (2.7) is satisfied for  < 12 , the stability condition is violated resulting in a solution of equation (2.8) not converging to the solution of (2.7).

2.3 Numerical integration of the Cauchy problem for systems of first-order ordinary differential equations Consider a system of k first-order ordinary differential equations and represent it in the form of a single vector equation: du D f.u; t / dt

with u D .u1 ; : : : ; uk /;

f D .f1 ; : : : ; fk /:

(2.12)

82

Chapter 2 The basics of the theory of finite-difference schemes

It is required to determine a function u.t / that solves the system for 0  t  T and satisfies prescribed initial conditions at t D 0: u.0/ D u0 ;

(2.13)

where u0 is a given constant k-vector and T is the length of the interval where the solution is required. A wide class of problems arising in mechanics of rigid bodies is reducible to problem (2.12)–(2.13). Examples include problems arising in studying the motion of heavenly bodies, artificial satellites, and rockets as well as some problems of the dynamics of mechanical systems consisting material points, resulting from studying continuum mechanics problems, and many others. Considered below are some of the methods for solving problem (2.12)–(2.13) beginning with the simplest ones; the problems are solved on a uniform grid with a constant step size  D T =n, where T is the final time to which the computations are performed and n is the number of steps in time.

2.3.1 Euler schemes Let us approximate the differential operator using one of the schemes discussed in Section 2.1. For example, let us replace the derivative with a unilateral forward difference and take the right-had side of (2.12) at the lower i th point to obtain the explicit Euler scheme uiC1  ui D f.ui ; ti /: (2.14)  Here and henceforth the subscript denotes the number of the time step; ui is the vector of already computed (known) values corresponding to time ti and uiC1 is the vector of unknown values at tiC1 . If the right-hand side of (2.12) is taken at the .i C 1/st point, one arrives at the implicit Euler scheme uiC1  ui D f.uiC1 ; tiC1 /: (2.15)  If the derivative is approximated with a central difference, one obtains the explicit and implicit Euler schemes with central difference: uiC1  ui1 D f.ui ; ti /I 2

uiC1  ui1 D f.uiC1 ; tiC1 /: 2

(2.16)

A countless number of other schemes are possible but, for the time being, let us restrict ourselves to those listed above. The easiest way to determine the values uiC1 is to use the explicit Euler scheme; one finds that i D 0; 1; : : : ; n; (2.17) uiC1 D ui C  f.ui ; ti /; where all values on the right-hand side have already been calculated at the previous steps.

Section 2.3 Numerical integration of the Cauchy problem for systems of equations

83

With the implicit Euler scheme, one obtains uiC1 D ui C  f.uiC1 ; tiC1 /;

i D 0; 1; : : : ; n:

(2.18)

In this case, equation (2.18) is a nonlinear equation for uiC1 ; it can be solved by using one of the iterative methods such as the simple iterative method or Newton’s method (see Section 3.5). An important feature that facilitates the solution of equation (2.18) .0/ is that there is always a good initial approximation, uiC1 D ui , which is just slightly different from the exact solution uiC1 [39]. In the schemes (2.16), the formal order of the finite difference equation is again, just as in the preceding section, higher than the order of the original differential equation (2.12), and hence an additional boundary condition is required. Let us evaluate the computational efficiency of the above schemes leaving aside other characteristic properties of these schemes. The most amount of computations is due to evaluating the vector function f on the right-hand side of the equation, and hence the most efficient scheme is the one that evaluates the function the least number of times to obtain uiC1 . In the explicit scheme, the function is evaluated only once per step, while in the implicit scheme, it is evaluated as many times as it is required to calculate the solution iteratively with the desired accuracy. In this respect, the explicit scheme is more economical. However, the solution accuracy also depends on the order of approximation. Higher-order schemes can become more efficient than lower-order schemes, although requiring a minimum amount of computations for the function f on the right-hand side of the equation. For simplicity, the subsequent presentation will deal with only one equation (2.12). The generalization to the case of a system of equations is straightforward.

2.3.2 Adams–Bashforth scheme Consider an economical numerical scheme of a high order of approximation. A scheme is said to be economical if it requires only one additional evaluation of the right-hand side of equation (2.12) when the order of approximation increases by one. The calculation of the unknown uiC1 requires k C 1 values ui ; : : : ; uik , already known from the previous k steps, rather than only one value ui . Let us use these k C 1 values to construct a polynomial of degree k: Lk .ui ; : : : ; uik / D

k X

lkp .t /uip ;

(2.19)

pD0 p lk .t /

are basic interpolating polynomials. where For example, the Lagrange interpolating polynomials for the time interval ti  t  tik are calculated as lkp .t / D

.t  ti / : : : .t  tp1 /.t  tpC1 / : : : .t C tik / : .tp  ti / : : : .tp  tp1 /.tp  tpC1 / : : : .tp  tik /

(2.20)

84

Chapter 2 The basics of the theory of finite-difference schemes

Formula (2.19) can be used to extrapolate the function ui on the interval Œti ; tiC1 : Z ti C Z ti C du dt D ui C u.ti C  / D ui C f .u; / d : (2.21) dt ti ti Replacing the integrand f .u;  / with its interpolating polynomial by formula (2.19), one obtains Z ti C k X p fip lk ./ d  D ui C  .a0 fi C    ak fik /; (2.22) u.ti C  / D ui C  pD0

ti

where fp D f .u.tp /; tp /; the constants a0 ; : : : ; ak are independent of the integration p step length  and determined by integrating the basic interpolating polynomials lk .t / p satisfying the conditions lk .tm / D ımp (m; p D i  k; : : : ; i ); ımp is the Kronecker delta. With this approximation of the right-hand side, the error will be kf .u.t //  Lk .t /k D O. k /; and hence u.ti1 C  /  u.ti1 / D a1 fi1 C    C ak fik C O. k /I (2.23)  consequently the approximation order equals the number of points used in the interpolation. This scheme requires storing the k previous values of the right-hand side. There is a complication that these values are unavailable at the beginning of the computation. For the scheme to start working, one should find the first k values of f in a nonstandard way with another scheme of the same order or a scheme whose order increases consecutively from 1 to k in the first k steps or by the Runge–Kutta method, which is outlined below. The above family of finite difference schemes is called the Adams–Bashforth schemes. These are also known as linear multistep schemes and have an arbitrary order of approximation (the number of points used to construct the interpolating polynomial can be arbitrary); these are formally described by difference equations whose order matches the order of approximation and, hence, contains parasitic solutions, discussed above in Section 2.2. Let us discuss this issue using the second-order Adams–Bashforth scheme as an example: 3 1 uiC1  ui D f .ui /  f .ui1 /; u.0/ D u0 ; 0  t  T: (2.24)  2 2 By setting f .ui / D aui , one obtains a second-order finite difference scheme, which requires, just as the scheme (2.8), an additional boundary condition at t D  for consistency with the order of approximation O. 2 / of equation (2.24) in the same way as with equation (2.8).

Section 2.3 Numerical integration of the Cauchy problem for systems of equations

85

2.3.3 Construction of higher-order schemes by series expansion Finite difference schemes of higher order of accuracy O. k / for equation (2.12) can be obtained by expanding uiC1 D u.ti C  / in a Taylor series at ti , uiC1 D ui C u0i  C u00i

k1 2 .k1/  C    C ui C O. k /; 2Š .k  1/Š .k1/

and calculating the derivatives u0i , u00i , . . . , ui equation (2.12):

(2.25)

at ti by successively differentiating

du D f .u; t /; dt d 2u D fu u0 C f t D fu f C f t ; dt 2

(2.26)

d 3u D fuu .u0 /2 C fu u00 C f t t D fuu f 2 C fu .fu f C f t / C f t t : dt 3 Substituting (2.26) into (2.25) and retaining only the first three terms in the expansion, one arrives at a third-order finite difference scheme for equation (2.12): ˇ uiC1  ui  Df .ui ; ti /  .fu f C f t /ˇ tDt i 2Š  ˇ 2 C .fuu f 2 C fu .fu f C f t / C f t t /ˇ tDt C O. 3 /: i 3Š

(2.27)

It is apparent that the number of terms in the series coefficients in formula (2.27) increases rapidly, since increasing the order by one requires the repeated calculation of the right-hand side and its derivatives. This can be avoided through calculating the derivatives at additional points of the interval Œti ; tiC1 [39].

2.3.4 Runge–Kutta schemes The scheme 2.27 belongs to finite difference schemes of the form uiC1  ui D P .ti ; ui /; 

(2.28)

where the right-hand side P .ti ; ui / D P1 Œf .ui /; ti is constructed so as to be dependent, in a certain way, on the right-hand side f .ui / of the original differential equation (2.12) and approximate the equation up to O. k /. For example, the Euler predictor-corrector method also belongs to this class of finite difference schemes. The solution is calculated by the following two-step algorithm:

86

Chapter 2 The basics of the theory of finite-difference schemes

1. Predictor. Calculate uiC1=2 at half the step length of the explicit scheme; the right-hand side in (2.17) is taken in the form 12 f .ui ; ti /: 1 uiC1=2 D ui C  f .ui ; ti /: 2 2. Corrector. Calculate uiC1 at the central point i C 1=2 with the right-hand side f .uiC1=2 ; tiC1=2 /, where uiC1=2 D ui C 12 f .ui ; ti /. Finally, the right-hand side of equation (2.28) becomes

  uiC1  ui D P .ui ; ti / D f ui C f .ui ; ti /; ti C :  2 2 It is not difficult to verify that the scheme has the second order of approximation. Let us expand the function P .ui ; ti / in a Taylor series as a function of two variables ui and ti :

  P .ui ; ti / D f ui C f .ui ; ti /; ti C 2 2 h i   D f .ui ; ti / C fu .ui ; ti /f .ui ; ti / C f t .ui ; ti / C O. 2 / : 2 2 On the other hand, uiC1 D ui C  uP t .ti / C

2 uR t t .ti / C    C O. k / D ui C  P .ui ; ti /: 2

(2.29)

The derivatives with respect to t at t D ti are easy to determine from the original differential equation (2.12): uP t .ti / D f .ui ; ti /; uR t .ti / D fu .ui ; ti /uP i C f t .ui ; ti / D fu .ui ; ti /f .ui ; ti / C f t .ui ; ti /: Substituting the obtained solutions into (2.29), one obtains an estimate for the residual term determining the order of approximation of the Euler predictor-corrector method: riC1 D

1 .uiC1  ui /  P .ui ; ti / D O. 2 /: 

This method is the simplest amongst the schemes belonging to the family of the Runge–Kutta schemes of the second order of accuracy. The idea of the method is to replace the repeated differentiation of the right-hand side with its calculation at k intermediate points of the interval Œui ; uiC1 . The combination of these values can be chosen so at to be equivalent, up to a residual term of the order of O. k /, to the truncated Taylor series for uiC1 in (2.29). This method requires the evaluation of f .u; t / at only k additional points and provides the kth order of approximation. First,

Section 2.3 Numerical integration of the Cauchy problem for systems of equations

87

one calculates the auxiliary quantities u1 D k1 D f .ui ; ti /; 1 1 u2 D k2 D f .ui C k1 ; ti C  /; 2 2 1 1 u3 D k3 D f .ui C k2 ; ti C  /; 2 2 1 u4 D k4 D f .ui C k3 ; ti C  /: 2 Then, the final value of uiC1 is calculated with the fourth order of accuracy by the formula 1 uiC1 D ui C .k1 C 2k2 C 2k3 C k4 /: (2.30) 6 The coefficients of ki are selected so that the right-hand side of (2.30) coincides with the truncated Taylor series up to O. 5 / to ensure that the finite difference scheme equation (2.12) has the fourth order of approximation. Obviously, in order to be able to calculate P .u/, it is required that the right-hand side function f .u; t / is thrice differentiable in its arguments. Thus, the Runge–Kutta schemes are as economical as the Adams–Bashforth schemes. Each step in the Runge–Kutta schemes increases the order of approximation by one and requires only one evaluation of the right-hand side of equation (2.12) at an intermediate point of the interval ti  t  ti C  . The stability of the Runge–Kutta schemes is stated by the following theorem. Theorem 2.1. The system of difference equations (2.28) is stable if (i) the function P .u/ satisfies the Lipschitz condition kP .x/  P .y/k  C kx  yk and (ii) the integration step length  is sufficiently small and satisfies the condition C  1. For the proof of this theorem, see textbooks on computational mathematics, for example, [4, 39].

88

Chapter 2 The basics of the theory of finite-difference schemes

2.4 Cauchy problem for stiff systems of ordinary differential equations 2.4.1 Stiff systems of ordinary differential equations Amongst the systems of original differential equations, the class of the so-called stiff systems requires special treatment, since these systems are difficult to integrate because the rates of change of the solution in the equations are very diverse. The direction field of integral curves of such systems changes its direction almost instantaneously as certain trajectories are approached. The solutions to Cauchy or boundary value problems include domains of very rapid change on small intervals followed by domains of very slow evolution. In mechanics, regions where the solution changes very rapidly are called boundary layers or internal layers, while regions of slow variation are called quasistationary mode regions. Consider the system of equations du D f.u; t /: dt

(2.31)

System (2.31) will be called stiff in a neighborhood ˇ of a ˇsolution u D u0 if the @f ˇ condition number N of the Jacobian matrix G D @u D fu ˇu is equal to u 0

N.t / D

max j i .t /j

1; min j i .t /j

0

1  i  n;

(2.32)

where i is an eigenvalue of the matrix G. This means that individual components of the solution have very different scales of variation in t . The spectrum of eigenvalues of the matrix G can conditionally be split into a stiff part, for which Re i .u/  L;

jIm i .u/j < jRe i .u/j;

and a soft part, for which j i .u/j < l L: It is clear that i .u/ is solution dependent and so the nonlinear system (2.31) can have different stiffnesses in different regions of the phase space. The number n D L= l is called the stiffness ratio of the system. In real applied problems, the stiffness ratio n can amount to 107 and even up to 1015 . Then, the integration with ordinary accuracy by a standard method on a time interval Œ0; T will require a step size  at which the condition  kfu k 1 holds. Taking into account that kfu k  max j i j  L, we have  1=L and so the required number of steps will be m D T = T L  1015 . This is absolutely unacceptable if we are interested in a

Section 2.4 Cauchy problem for stiff systems of ordinary differential equations

89

quasistationary mode, for which T  O.1/, rather than the structure of boundary layers. For quasistationary modes, it suffices to have m  103 . So our primary objective will be constructing an algorithm that would allow us to perform computations with such a large step length  D T =m. ˇ The system stiffness is determined by the matrix fu ˇu0 , or, given u D u0 , by the linear part of fu . Therefore, in the first approximation, it suffices to investigate the linearized problem and, instead of (2.31), consider the system du D fu .u0 ; t /  u: (2.33) dt For illustration, let us perform the stability analysis for the model equation du D u; (2.34) dt where u is a scalar and is a complex number, since the Jacobian matrix can have complex eigenvalues. One looks for all points of the complex plane of D  for which the finite difference scheme for equation (2.31) is stable. For example, the explicit Euler method is stable only within the circle of unit radius centered at the point .1; 0/ and, therefore, is unsuitable for integration with a large step size. If a method is stable on the entire half-plane Re < 0, it is said to be A-stable or absolutely stable. Since the solution of equation (2.34) is stable for Re < 0, the Astability of its finite difference scheme means that the method is stable for any  > 0, since the stability of a finite difference scheme is determined by the product  D . Im μ α Re μ 0

Figure 2.3. Domains of stability in the complex plane .

2.4.2 Numerical solution Stiff problems must be solved using A-stable or A.˛/-stable algorithms. A finite difference scheme is called A.˛/-stable if the domain of stability of this scheme in the complex plane is restricted to within an angle ˛ (Figure 2.3): j arg. /j < ˛.

90

Chapter 2 The basics of the theory of finite-difference schemes

Such algorithms include Gear’s implicit schemes of high order of approximation. High-order multistep Gear implicit schemes are constructed in a similar manner as the Adams–Bashforth schemes, by using an interpolating polynomial for the righthand side function f .u; t / of equation (2.31) with the only difference that the set of nodes used to perform the interpolation also includes node n C 1, at which the solution is sought. The interpolating polynomial of degree m C 1 is determined by the nodal values f .unC1 ; t nC1 /, f .un ; t n /, . . . , f .unC1m ; t nC1m /. Then, unlike the explicit Adams–Bashforth schemes discussed in Section 2.3, one arrives at the family of implicit .m C 1/st-order schemes m

X unC1  un D ai f .unC1i ; t nC1i /: 

(2.35)

iD0

For example, with m D 3, one obtains an implicit scheme of the fourth order of accuracy:  unC1  un 1  nC1 D 9f C 19f n  5f n1 C f n2 : (2.36)  24 The explicit scheme employs the following nodal values: f .un ; t n /, f .un1 ; t n1 /, . . . , f .unm ; t nm /. The sum in (2.35) starts with i D 1. If m D 3, one obtains the third-order scheme  1 unC1  un D 23f n  16f n1 C 5f n2 :  12

(2.37)

To solve the system of implicit equations (2.36), one can use a predictor-corrector scheme. The predictor is calculated by the explicit scheme (2.37):  uQ nC1  un 1 D 23f n  16f n1 C 5f n2 :  12 Then the solution is refined using the implicit scheme (2.36)  1  nC1 nC1 nC1 unC1  un D 9f .uQ ;x / C 19f n  5f n1 C f n2 :  24

2.4.3 Stability analysis Let us carry out the stability analysis for the second-order implicit Gear scheme 4 1 2 unC2  unC1 C un D f .unC2 ; t nC2 /: 3 3 3

(2.38)

Suppose that the right-hand side of the linearized equation is f .u/ D u, where is a complex number.

Section 2.4 Cauchy problem for stiff systems of ordinary differential equations

91

Then the general solution of equation (2.38) is given by un D C1 .r1 /n C C2 .r2 /n ; where r1 and r2 are roots of the quadratic equation

2  4 1 r 2 1    r C D 0; 3 3 3 p 2 ˙ 1 C 2 r1;2 D ;  D : 3  2

(2.39)

As mentioned above, the solution of equation (2.38) is of interest for stiff systems with j j 1 for determining the structure of boundary layers. The solution of the difference equation (2.38) approximates the exact solution u D e t of equation (2.34). The domain j j 1 of the complex variable is called the “accuracy domain.” One can see that, for small magnitudes of , the first root of equation (2.39) approximates the exact solution:   r1 . / D e 1 C O. / ;   u1 D C1 .r1 /n D C1 e  n 1 C O. 2 / ; t D n: If j j 1, the second root is given by r2 D

1 C O. 2 /; 3

u2 D C2 .r2 /n ! 0:

It follows that the scheme is stable for small j j. The wider the accuracy domain j j < 0 , where the solution is approximated with the required accuracy, the better the scheme. The other domain is the domain of large magnitudes of : j j 1. It is called the domain of stability (quasistationary mode). One can see from equation (2.39) that if j j 1, the roots are small, r1  r2  .2j j/1=2 1, and hence the scheme is stable. However, some schemes may not be stable for all values of from the half-plane Re < 0. For example, they may only be stable in an angular domain jarg. /j < ˛ or a domain Re < a2 . Such schemes are also suitable for obtaining slowly varying solutions (quasistationary mode). The stability zone must contain a sufficiently wide neighborhood of the ray Im D 0, Re < 0. For more details on the solution methods for stiff systems, see, for example, [143].

2.4.4 Singularly perturbed systems In many physical problems that belong to the class of stiff problems, there is a small parameter " that appears explicitly in the system of equations. This facilitates the

92

Chapter 2 The basics of the theory of finite-difference schemes

integration of such systems. For example, consider the system of two equations "uP D f .u; v/; vP D '.u; v/;

(2.40)

which contains a small parameter " as the coefficient of the derivative uP in the first equation; equivalently, the right-hand side can be treated as containing the large parameter L D "1 1. Both functions f .u; v/ and '.u; v/ as well as their derivatives are quantities of the order of O.1/. The spectrum of the Jacobian matrix of the system (2.40) is determined by the equation   Lfu  Lfv D 0: (2.41) det 'u 'v  The stiff component corresponds to the function Lfu , while the component corresponding to ' is small. The quasistationary mode is determined by the equation f .u; v/ D 0; it splits the uv-plane into two domains. The domain f .u; v/ > 0 is stiff for 1 < 0. The theory of such systems has been well developed [143, 39]. The case of a singularly perturbed system with an explicitly occurring large parameter is similar to the general case of system (2.31) considered above. Here, the large parameter L plays the same role as j Re max j, and the qualitative behavior of the solution in this case is quite clear. Let us carry out the analysis of the system for a specific example.

2.4.5 Extension of a rod made of a nonlinear viscoplastic material Consider the problem of uniform tension of a rod made of a nonlinear viscoplastic material with a static stress-strain diagram of the general form  D s ."/ [93] shown in Figure 2.4. The problem generates a singularly perturbed system. Suppose that one end of the rod is subjected to a given varying tensile stress  D 0 .t /, while the other end is fixed. When written in terms of dimensionless variables, the problem can be reduced to a system of two equations with a large parameter ı 1:   @"N @N @"Np O jN j  s .N"/ ; D  D ı sign N ˆ N N N @t @t @t ˇ @ D P 0 .t /; t > 0;  ˇ tD0 D 0 : @t

(2.42)

The first equation of system (2.42) corresponds to the equation (1.85) solved for viscoplastic strain rate "Pp and rewritten in terms of the dimensionless variables N D

 ; 0

"N D

" ; "0

tN D

t ; t0

0 D E"0 ;

ıD

t0

1; 

Section 2.4 Cauchy problem for stiff systems of ordinary differential equations σ

93

F1 A1(σ 0, σ 0/E)

σ0

D C(σ0, εc)

F

B σ 0s

A

α 0

εs

ε

Figure 2.4. Static s ."/ (OABCDF ) and quasistatic (OA1 CDF1 ) stress-strain diagrams of a material; E D tan.˛/ is Young’s modulus.

where t0 is the characteristic time appearing in the function 0 .t /,  is the relaxation time of the viscoplastic material, and E is Young’s modulus in tension/compression. The first equation in (2.42) relates the stress  with the strain " in the elastoviscoplastic material. The second equation determines the rate of change of the applied O stress. The functions of the right-hand sides, ˆ.; "/ and P 0 .t /, as well as their derivatives are quantities of the order of O.1/. The spectrum of the Jacobian matrix (2.41) of the system is determined by the equation   ıˆ"  ıˆ D .ıˆ"  / D 0: det 0  The parameter ı D t0 = appearing on the right-hand side of the first equation in (2.42), equal to the ratio of the characteristic time t0 to the relaxation time  , is a large quantity for many materials, ı 1, and so the system of equations (2.42) is a singular system of the form (2.40). O is defined as The function ˆ ´ ˆ.z/; z > 0; O ˆ.z/ D 0; z  0: This means that the plastic strain rate is zero, "Pp D 0, for j j < s ."/, and hence the stress is related to the strain by Hooke’s law N D "N in dimensionless variables. Taking into account that ˆ" D ˆz

dz @s D ˆz ; d" @"

one can see that the system is stiff if ds =d " > 0 and non-stiff if ds =d "  0 with ˆz > 0.

94

Chapter 2 The basics of the theory of finite-difference schemes

The field of integral curves in the phase plane  -" is easy to analyze. The curve  D s ."/ divides the plane into two parts, with   s ."/ > 0 to the left of it and "Pp D 0 to the right. Beyond the small neighborhood O.ı 1 / of the curve  D s ."/, the direction field of integral curves is almost horizontal and the rate of change of " is very large (of the order of O.ı 1 /) and increases with ", so that the plastic strain increases rapidly. In a short time O.ı 1 /, the rod passes, along an almost horizontal line, from the state A1 .0 ; 0 =E/ to a state C.0 ; "C / in a neighborhood of the curve  D s ."/. In this neighborhood, P D O.1/ and "P D O.1/, since ˆ.z/ D O.ı 1 /, and so the stress-strain state changes along the raising branch CD of the curve  D s ."/ to the point D, where ds =d " D 0. The subsequent motion along the falling branch of the curve becomes unstable, the system loses stiffness, and the motion occurs rapidly along the horizontal line to the point F1 , as shown in Figure 2.4. In order to characterize the variation of " on the interval of rapid change from A1 .0 ; 0 =E/ to C.0 ; "C /, it suffices to set  D 0 in the first equation of (2.42) and integrate the resulting system to obtain Z "   d" d"  ; D ı ˆ 0  s ."/ ; tN D dt ı ˆ  "0 0  s ."/ where it has been taken into account that "0 D 0 =E at the initial time, since, as follows from (2.42), the instantaneous deformation occurs by Hooke’s law. At the point ." D "C ; 0 D s ."C //, depending on the asymptotic behavior of the function ˆ.z/ D az ˛ as z ! 0, the integral is convergent for ˛ < 1 and divergent for ˛  1. Accordingly, the time in which " ! "C is either finite or infinite on the scale O.ı 1 /. However, with ˛  1 too, " tends to "C in an exponentially fast manner, with the “effective time” of the passage being always a finite quantity on the scale O.ı 1 /. If the passage occurs from a point of instability, this indicates the existence of an internal boundary layer. The point at which it begins is determined by  D  C and " D "C at t D t C and the transition time to the stable branch is calculated from Z " d"  : t  tC D C   ."/ C ı ˆ  " s Thus, if the rod is subjected to a slow tensile stress, the quasistationary dependence  D  ."/ will be represented by the curve OA1 CDF1 in Figure 2.4. This dependence is characterized by an increase in the yield stress, as compared with the stationary dependence  D s ."/, and the appearance of a plato of ideal sliding.

95

Section 2.5 Finite difference schemes for 1D partial differential equations

2.5 Finite difference schemes for one-dimensional partial differential equations Let us consider the simplest finite difference schemes for evolution partial differential equations in one space coordinate and time – the wave equation (hyperbolic type) and unsteady heat equation (parabolic type).

2.5.1 Solution of the wave equation in displacements. The cross scheme The one-dimensional wave equation in terms of the displacement has the form 2 @2 u 2@ u  a D b: @t 2 @x 2 The initial conditions are specified using two functions: ˇ @u ˇˇ D v0 .x/: u.x; 0/ D u0 .x/; @t ˇ tD0

(2.43)

Let us make use of the simplest explicit three-layer second-order cross scheme, whose stencil consists of five nodes as shown in Figure 2.5: un  2uni C uni1  2uni C un1 unC1 i i 2 iC1 D a : t 2 x 2

(2.44)

Definition 2.1. The stencil of a finite difference scheme is the arrangement of grid points involved in a difference equation serving to obtain the solution at the point of interest on the .n C 1/st layer. The order of approximation of the difference equation (2.44) is O.t 2 C x 2 /: ˇ ˇ ˇ   2 ˇn ˇ @4 u ˇˇn t 2 @4 u ˇˇn x 2 @2 u ˇˇn 1 4 2 @ uˇ 1 4 C 4ˇ C O.t / D a C 4ˇ C O.x / @t 2 ˇi 2Š @t i 4Š @x 2 ˇi 2Š @x i 4Š To start the computation based a three-layer scheme, one should known the nodal values at the first two layers, whereas the initial conditions of the Cauchy problem are specified by two functions, u D u0 .x/ and v D v0 .x/, a layer n D 0. The second condition can be used to obtain the value of u1 at layer n D 1. In order to start the computation of the first time step, one should determine u1 i 0 1 from the condition for the initial speed u0i  u1 i =t D vi , whence follows ui accurate to the first order, O.t /. Accordingly, the solution of the Cauchy problem (2.43)–(2.44) will have the first order of approximation. For a second-order approximation, condition (2.44) must be approximated by a second-order expression. Let us make use of the expansion term ˇ ˇ   ˇ ˇ u0i  u1 t t i 0 00 ˇ 0 2 00 ˇ D vi C u t t ˇ D vi C a uxx ˇ C O.t 2 C x 2 /; Cb t 2 2 tD0 tD0

96

Chapter 2 The basics of the theory of finite-difference schemes n+1

n+1

n

n

n–1 i–1

i

i+1

n–1 i–1

(a)

i

i+1

(b)

Figure 2.5. Cross (a) and leapfrog (b) schemes. The solid line indicates the spatial derivative and the dashed line corresponds to the time derivative. The solid circles indicate known data and the shaded circles correspond to unknown data.

where the expression of u00t t had ˇ been by obtained by substituting the derivative of the second initial condition, u00xx ˇ tD0 , into the original equation (2.44). Then the solution of the original problem will have the order O.t 2 C x 2 /.

2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations) The complete system of equations describing the propagation of longitudinal waves in an elastic bar consists of the equation of motion, the compatibility equation between the strains and strain rates, and Hooke’s law. In the case of uniaxial tension/compression, the system has the form 

@ @v D C b; @t @x

@" @v D ; @t @x

 D E"

(2.45)

where  is density, b is the mass force, and E is Young’s modulus. Having eliminated the stress  , one can rewrite the system as two simultaneous wave equations for the strain rate v and strain ": @" @v D a2 Cb @t @x

.a2 D E=/;

@v @" D : @t @x

(2.46)

Equations (2.46) can be approximated on a rectangular grid in the xt -plane. Let x denote the step size in the x-direction and t denote that in time t . The subscript i will refer to grid points along the x-coordinate and n will refer to points in t (Figure 2.6).

97

Section 2.5 Finite difference schemes for 1D partial differential equations

2.5.3 The leapfrog scheme The leapfrog scheme has the form "n  "ni1 vinC1  vin1 2 iC1 Da C bin 2t 2x n v n  vi1 "nC1  "n1 i i D iC1 2t 2x Unlike the cross scheme, the stencil of the leapfrog scheme does not involve the central point .xi ; t n /. The order of approximation is O.t 2 C x 2 /. For example, the first equation of the system can be represented as ˇ ˇ ˇ  ˇn  ˇ @3 " ˇˇn t 2 @v ˇˇn @3 v ˇˇn t 2 4 2 @" ˇ 4 C O.t C O.t C / D a C / C b: @t ˇi @t 3 ˇi 3Š @x ˇi @x 3 ˇi 3Š The scheme has three layers and so the initial step should be calculated with any two-layer scheme; for example, the Lax scheme can be used. n+1

n+1 1 n+– 2

n

n i–1

i+1

i

1

i–1

1

i–– 2

i

(a)

i+– 2

i+1

(b)

Figure 2.6. Stencils of the Lax–Friedrichs (a) and Lax–Wendroff (b) scheme. The solid line indicates the spatial derivative and the dashed line shows the time derivative. The solid circles, shaded circles, and diamonds indicate known data, unknown data, and auxiliary nodes, respectively.

2.5.4 The Lax–Friedrichs scheme A stencil can involve two or more layers in time t . Let us approximate system (2.46) using a two-node stencil in x (Figure 2.6a) with nodes i C 1 and i  1 used at the nth layer: n vinC1  viC1=2

t

D a2

"niC1  "ni1 2x

;

"nC1  "niC1=2 i t

D

n n viC1  vi1

2x

;

(2.47)

98

Chapter 2 The basics of the theory of finite-difference schemes

where 1 viC1=2 D .viC1 C vi1 /; 2

1 "iC1=2 D ."iC1 C "i1 /: 2

The scheme (2.47) is known as the Lax–Friedrichs scheme. Let us determine the order of approximation of this finite difference scheme in x n , vn and t . Expanding viC1 , etc. in Taylor series at point .i; n/ gives iC1=2 ˇ ˇ ˇ   @2 v ˇˇn x 2 x 4 @v ˇˇn @2 v ˇˇn t 2 C C O t C C @t ˇi @t 2 ˇi 2 @x 2 ˇi 2t t ˇ   ˇn ˇ @3 " ˇˇn x 2 2 @" ˇ 4 Da C O.x /: C 3ˇ @x ˇi @x i 3Š It follows that the local approximation has the order o.t C x 2=t /. The term o.x 2=t / tends to zero only if the order of x 2 is less or equal to the order of t as x ! 0 and t ! 0. This kind of approximation is called conditional.

2.5.5 The Lax–Wendroff Scheme Let us use a three-point stencil in x with points i  1, i , and i C 1 to obtain another finite difference scheme: "nC1  "ni1 vinC1  vin D a2 iC1 ; t 2x

n v nC1  vi1 "nC1  "ni i D iC1 : t 2x

(2.48)

Determine the approximation order of this scheme: ˇ ˇ ˇ   ˇn ˇ @3 " ˇˇn x 2 @v ˇˇn @2 v ˇˇn t 2 2 @" ˇ 4 C O.t C O.x C / D a C / ; @t ˇi @t 2 ˇi 2 @x ˇi @x 3 ˇi 3Š ˇ ˇ ˇ ˇ @v ˇˇn @3 v ˇˇn x 2 @" ˇˇn @2 " ˇˇn t 2 C O.t / D C O.x 4 /: C 2ˇ C 3ˇ @t ˇi @t i 2 @x ˇi @x i 3Š It is clear that the expansion at point .i; n/ gives the approximation order O.t C x 2 /. In order to obtain a second-order approximation in both t and x, let us ext @2 @2 press the residual terms t 2 @t 2 " and 2 @t 2 v on the right-hand sides of the equations in (2.48) in terms of the second derivatives with respect to x using the original system of equations (2.46) and approximate the resulting expressions of the second derivatives by finite differences up to o.x 2 / to obtain t @2 " t a2 @2 " t a2 D D ."iC1  2"i C "i1 / C O.x 2 /; 2 @t 2 2 @x 2 2 x 2 t a2 @2 v t a2 t @2 v D D .viC1  2vi C vi1 / C O.x 2 /: 2 2 2 @t 2 @x 2 x 2

99

Section 2.5 Finite difference schemes for 1D partial differential equations

Finally, the finite difference scheme becomes "nC1  "ni1 vinC1  vin t a2 n n n 2 iC1  .v  2v C v / D a i i1 t 2 x 2 iC1 2x nC1 nC1 n n 2  vi1 v "i  "i t a iC1 n n n  ."  2" C " / D i i1 t 2 x 2 iC1 2x

(2.49)

It readily follows from (2.48) that the order of approximation on solutions to system (2.46) is now o.t 2 C x 2 /. Here the concept of approximation has been narrowed down to the class of exact solutions to the differential equations (2.46). In this case, the notion “consistency condition” is used instead of the notion “approximation condition” (see [148]). Just as the approximation condition, the consistency condition indicates how well the exact solution satisfies the finite difference equations. The finite difference scheme (2.49) is known as the Lax–Wendroff scheme. The same scheme can be obtained in a different way, by introducing an intermediate layer numbered n C 12 and using a two-step predictor-corrector scheme (Figure 2.6b). In the first half-step (predictor), one finds the solutions at points i C 1=2 and i  1=2 by the Lax–Friedrichs scheme (2.47) and then computes the final solution at layer n C 1 by the leapfrog scheme (corrector): nC1=2

nC1=2

"iC1=2  "i1=2 vinC1  vin D a2 I t 2x

nC1=2

nC1=2

viC1=2  vi1=2 "nC1  "ni i D t 2x

By eliminating the quantities at the points with half-integer indices, one gets  n   n n  "  "ni1  2vin C vi1 vinC1  vin t a2 viC1 D a2 iC1 C t 2x 2 x 2   n n viC1  vi1 "nC1  "ni t a2 "niC1  2"ni C "ni1 i D C t 2x 2 x 2

(2.50)

The resulting two-step scheme (2.50) coincides with second-order Lax–Wendroff scheme (2.49).

2.5.6 Scheme viscosity The Lax–Friedrichs (2.47) and Lax–Wendroff (2.49), (2.50) difference equation contain additional terms, which correspond viscosity. This means that a finite difference schemes brings a small numerical viscosity into the differential equation; this viscosity significantly affects discontinuous solutions. If a scheme is of the first order, the scheme viscosity of the first order results in monotone smoothing of solution discontinuities. The second-order viscosity of second-order schemes results in nonmonotone discontinuity profiles. For details on the effect of the scheme viscosity, see Section 5.5, “Differential approximation for difference equations.”

100

Chapter 2 The basics of the theory of finite-difference schemes

2.5.7 Solution of the wave equation. Implicit scheme A second-order scheme with accuracy O.t 2 C x 2 / can be constructed using the four-point stencil shown in Figure 2.7a, where two nodes are used on layers n and n C 1 each and the derivative with respect to x in (2.46) is approximated as v nC1  vinC1 v n  vin @v D  iC1 C .1  / iC1 ; @x x x nC1 n  viC1=2 viC1=2

t "nC1 iC1=2



"niC1=2

t

D a2 

D

nC1 "nC1 iC1  "i

x

nC1 viC1

viC1=2



vinC1

01

C .1  /

C .1  /

"niC1  "ni

n viC1

x 1 D .viC1 C vi / 2

!

x 

(2.51)

vin

x

If  D 1=2, the approximation has the second order of accuracy, which is easy to prove by expanding all quantities in Taylor series at point .i C 1=2; n C 1=2/.

i

i

1 +– 2

(a)

i+1

n+1

n+1

n

n i–1

i

i+1

(b)

Figure 2.7. Stencils of implicit schemes for the first (a) and second (b) derivatives.

2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points The scheme (2.51) is considerably different from the previous schemes (2.47)–(2.50): at the upper layer n C 1, either equation involves two variables rather than one and the complete system of equations for the .n C 1/st time layer does not split into a system of recurrence equations. Such schemes as called implicit in contract with explicit schemes where the solution is determined at each point of the .n C 1/st layer

Section 2.5 Finite difference schemes for 1D partial differential equations

101

independently of the other points of this layer, which implies that the matrix of the system of equations for the quantities with index n C 1 has a diagonal form. Explicit schemes enable one to calculate the solution at the .n C 1/st time layer one the solution at all points of the previous nth layer is known; in other words, such schemes allow one to solve difference Cauchy problems or problems with periodic boundary conditions specified at the endpoints of a segment of the x-axis. Solving an initial-boundary value problem, where boundary conditions are specified at the endpoints x D 0 and x D 1 in addition to initial conditions, requires constructing special schemes for these points. A scheme that serves to determine the solution at internal points of the segment is unsuitable for the endpoints, since one or more points of the stencil turn out to be beyond the segment. Implicit schemes involve two or more points at the .n C 1/st layer, which results in a system of algebraic equations for determining the values of quantities at these points; to close this system, boundary conditions are required. The solution can only be obtained for all points of the .n C 1/st time layer simultaneously once the system of algebraic equations has been solved. This property of implicit schemes contradicts the property of the wave equation that the solution at a point .x; t / of the bar is independent of the solution at other points at the same time instant t , since the speed of propagation of perturbations through an elastic body is finite. In what follows, this issue will be investigated in more detail; for the time being, this contradiction will be ignored. Implicit schemes involve two or more points at the .n C 1/st layer, which results in a system of algebraic equations for determining the values of quantities at these points; to close this system, boundary conditions are required. The solution can only be obtained for all points of the .n C 1/st time layer simultaneously once the system of algebraic equations has been solved. This property of implicit schemes contradicts the property of the wave equation that the solution at a point .x; t / of the bar is independent of the solution at other points at the same time instant t , since the speed of propagation of perturbations through an elastic body is finite. In what follows, this issue will be investigated in more detail; for the time being, this contradiction will be ignored. Boundary points of hyperbolic equations should be treated with the aid of relations along characteristics (see Section 5.2).

2.5.9 Heat equation Consider a family of finite difference schemes for the heat equation   @ @T @T D K C !: c @t @x @x

102

Chapter 2 The basics of the theory of finite-difference schemes

For K D const, it can be rewritten as @2 T ! @T DA 2 C @t @x c

(2.52)

where c is the linear specific heat,  is the linear density, K > 0 is the thermal conductivity, ! is the power of heat sorces/sinks, and A D K=.c/. Initial conditions: T .t0 ; x/ D T0 .x/I Mixed boundary conditions at the left, xL , and right, xR , boundaries: ˇ @T ˇˇ D L ; ˛L T .xL / C ˇL @x ˇxDxL ˇ @T ˇˇ D R : ˛R T .xR / C ˇR @x ˇxDxR

(2.53)

The second derivative will be approximated using a six-node stencil (Figure 2.7b) that uses nodes i  1, i , and i C 1 at layers n and .n C 1/ in time (we restrict ourselves to the homogeneous equation): nC1 n  2T n C T n T nC1  2TinC1 C Ti1 TiC1 TinC1  Tin i i1 D A iC1 C .1  /A : t x 2 x 2 (2.54) For 0 <   1, the scheme is implicit and has the order O.x 2 C t /. At  D 0, the scheme becomes explicit. At  D 12 , the scheme has the second order of approximation O.x 2 C t 2 /.

n+1

n+1

n+1

n

n i–1

i

(a)

i+1

n i–1

i

(b)

i+1

i–1

i

i+1

(c)

Figure 2.8. Six-node stencil for the heat equation; (a) explicit scheme, (b) implicit scheme, (c) Crank–Nicolson scheme. Solid circles correspond known data and shaded circles indicate unknown data.

103

Section 2.5 Finite difference schemes for 1D partial differential equations

2.5.10 Unsteady thermal conduction. Explicit scheme (forward Euler scheme) An explicit scheme follows from (2.54) with  D 0: n T n  2Tin C TiC1 TinC1  Tin D A i1 ; t x 2 n n TinC1 D Tin C C.Ti1  2Tin C TiC1 /;

where C D A

(2.55) t : x 2

The approximation order is O.x 2 C t /.

2.5.11 Unsteady thermal conduction. Implicit scheme (backward Euler scheme) An implicit scheme follows from (2.54) with  D 1: nC1 T nC1  2TinC1 C TiC1 TinC1  Tin D A i1 ; t x 2 nC1 nC1 C .1 C 2C /TinC1  C TiC1 D Tin ; C Ti1

(2.56)

where C D A

t : x 2

The approximation order is O.x 2 C t /.

2.5.12 Unsteady thermal conduction. Crank–Nicolson scheme The Crank–Nicolson scheme follows from (2.54) with  D 1=2 " nC1 # nC1 n n Ti1  2TinC1 C TiC1 Ti1  2Tin C TiC1 TinC1  Tin 1 D A C t 2 x 2 x 2 

(2.57)

C nC1 C nC1 C nC1 C nC1 Ti1 C .1 C C /TinC1  TiC1 D Ti1 C .1  C /TinC1 C TiC1 ; 2 2 2 2

where C D A t =x 2 . The approximation order is O.x 2 C t 2 /. There are other explicit scheme for the heat equation. These are listed below.

2.5.13 Unsteady thermal conduction. Allen–Cheng explicit scheme n T n  2TinC1 C TiC1 TinC1  Tin D A i1 ; t x 2 n n .1 C 2C /TinC1 D Tin C C.Ti1 C TiC1 /;

where C D A

(2.58) t : x 2

O.x 2 C t C t =x 2 /, which means that the scheme approximates the original equation conditionally at t  x 2 .

104

Chapter 2 The basics of the theory of finite-difference schemes

2.5.14 Unsteady thermal conduction. Du Fort–Frankel explicit scheme

TinC1

n T n  .TinC1 C Tin1 / C TiC1 TinC1  Tin1 D A i1 (2.59) 2 t x 2 1  C n1 C t n n Ti .Ti1 D C C TiC1 /; where C D 2A : 1CC 1CC x 2

The approximation order is O.x 2 Ct 2 Ct 2 =x 2 /, which means that the scheme approximates the original equation conditionally at t  x. To initiate the computation (to obtain layer n D 1), one has to use a two-layer scheme. n+1 n+1

n n–1

n i–1

i

i+1

(a)

i–1

i

i+1

(b)

Figure 2.9. Stencils of explicit schemes for the heat equation; (a) Allen–Cheng scheme, (b) Du Fort–Frankel scheme. The solid line indicates the space derivative and the dashed line shows the time derivative. Solid circles correspond known data and shaded circles indicate unknown data.

2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives One can easily see that if problem (2.52) is solved on the interval x 2 Œ0; 1 on a grind with nodes i D 1; : : : ; N , then one can only write out N  2 difference equations (2.54) with 0 <   1 for nodes i D 2; : : : ; N  1. For the system to be closed, two more equations must be added, which follows from the boundary conditions at the specified endpoints x D 0 and x D 1; then the number of equations will equal the number of unknowns. The simplest approximation of the boundary has the first order of approximation in space, O.x/: T1nC1  T0nC1 D 0nC1 I x nC1  TNnC1 1 nC1 nC1 nC1 TN nC1 at x D xR : ˛N D N TN C ˇN : x at x D xL : ˛0nC1 T0nC1 C ˇ0nC1

Section 2.5 Finite difference schemes for 1D partial differential equations

105

The approximation order of a boundary value problem is determined by the least approximation order of the equations and boundary conditions. Accordingly, if a boundary condition contains a derivative, the entire problem becomes first-order accurate in the space coordinate. Let us derive a second-order approximation of the right boundary condition. By expanding TNnC1 1 (adjacent to the right boundary node) into a Taylor series in x around the endpoint x D xR along the exact solution up to the second derivative inclusive, we obtain ˇ ˇ @T ˇˇnC1 @2 T ˇˇnC1 x 2 nC1 nC1 nC1 C O.x 3 / TN 1 D T .t ; xR  x/ D TN  x C 2 ˇ @x ˇN @x N 2 ˇ ˇ TNnC1  TNnC1 @2 T ˇˇnC1 x 2 @T ˇˇnC1 1 C 2ˇ C O.x 2 /: D ) @x ˇ x @x 2 N

N

Expressing the second spatial derivative from the differential equation,

)

ˇ @2 T ˇˇnC1 @x 2 ˇN

! 1 @T .t; x/ @2 T .t; x/  D @x 2 A @t c ˇ nC1 nC1 !N 1 TNnC1  TNn 1 @T .t; x/ ˇˇn!nC1 !N D  ; D  A @t ˇN c A t c

ˇnC1 ˇ nC1 nC1 nC1 @T ˇnC1 ˇ and substituting @T into the boundary condition ˛N TN C ˇN D @x N @x N nC1 N , we arrive at the difference equation ! ! nC1 nC1 !N  TNnC1 1 TNnC1  TNn x nC1 nC1 nC1 TN nC1 1 C  D N ˛N TN C ˇN ; x A t c 2 ! nC1 nC1 ˇ ˇ x nC1 nC1 nC1 TNnC1  ˇN C N C N TN 1 ˛N x A 2 t ! nC1 ! 1 x nC1 nC1 Tn C N : D N C ˇN 2 A t N c For the heat equation, the use of implicit finite difference schemes is physically relevant, since thermal perturbations propagate with an infinite speed and all points of the bar at time t influence one another. It is the explicit scheme, with  D 0, that is physically irrelevant for the heat equation in contrast with the wave equation. However, this property is only essential for rapidly changing or high-frequency solutions. For smooth low-frequency solutions, there is no significant difference between explicit and implicit schemes in such problems. It is noteworthy that by increasing the number of points in the stencil, one can increase the order of approximation but this will significantly complicate the system

106

Chapter 2 The basics of the theory of finite-difference schemes

of difference equations and its analysis. For this reason, increasing the order of approximation is not always beneficial (see Richardson’s extrapolation formula in Section 2.1). In what follows, the evolution equations will as a rule be approximated using two-layer schemes of the first or second order of accuracy.

2.6 Stability analysis for finite difference schemes A two-layer system of difference equations can be written in the general form B1 unC1  B0 un D 0 or, more precisely, N X

ˇ

B1 T ˇ .unC1 / 

ˇ D0

N1 X

ˇ

B0 T ˇ .un / D 0;

(2.60)

ˇ D0 ˇ

ˇ

where T ˇ .u/ D u.x C ˇh/ is a translation operator along the x-axis, B0 and B1 are square matrices having the same dimension as the vector of unknowns u, with entries being constant but, possibly, dependent on the step sizes  and h, and ˇ is an integer. For an explicit scheme, the number of points N D 1 and the matrix B1 is diagonal, implicit schemes have involve several points adjacent to xi . By applying the Fourier transform in x to equation (2.60), Z 1 1 u.x/ e ikx dx; u.k/ O D 2 1 where the hat over a symbol denotes a Fourier transform in the plane of the complex variable k, and taking into account the translation operator is transformed as Z 1 1 ˇ O u.x C ˇh/ e ikx dx D e ikˇ h u.k/; O T .u/ D 2  1 one obtains H1 uO nC1 .k/  H0 uO n .k/ D 0;

(2.61)

where H1 D

N X ˇ D0

ˇ

Bi exp.iˇhk/;

H0 D

N X

ˇ

B0 exp.iˇhk/;

ˇ D0

and k is the Fourier transform parameter. Solving (2.61) for uO nC1 gives the following system of recurrence equations for the transforms uO nC1 : uO nC1 .k/ D G.; h; k/Oun .k/; (2.62)

107

Section 2.6 Stability analysis for finite difference schemes

where G D H1 H0 is the transformation matrix from layer n to layer n C 1 in the space of Fourier transforms. By applying n times the operator G to uO 0 .k/, one obtains the solution at the .n C 1/st layer in the product form uO nC1 .k/ D Gn .; h; k/Ou0 .k/:

(2.63)

2.6.1 Stability of a two-layer finite difference scheme In this subsection, we use the notion of stability essentially equivalent to that given above in Section 2.2 but in a different formulation, more convenient for further treatment [147]. A finite difference scheme (2.60) will be called stable if there are some 1 > 0 and T > 0 such that the infinite set of the transformation operators Gn .; k/

with 0 <  < 1

and

0  n  T

is uniformly bounded, kGn .; k/k < C , where the constant C is independent of  and k. This condition is necessary and sufficient for the stability. Equation (2.61) is the analogue of (2.60) in the space of Fourier transforms and G.; k/ is a matrix dependent on the transform parameter k. The stability condition requires that the matrix operators Gn .; k/ for all n are uniformly bounded on a finite interval of t for any k. For a matrix A.k/, its norm kA.k/k is defined as kA.k/k D max V ¤0

jA.k/ vj jvj

(2.64)

where jvj D .†vi2 /1=2 is the magnitude of the vector. The spectral radius of a matrix A is the number R D max j i j, where i are eigenvalues of A. It is clear that R  kAk. The spectral radius Rn of the matrix An equals Rn D Rn . Furthermore, kA2 k D max v¤0

jA.Av/j jA.Av/j jAvj jAvj jAvj D max  max D kAk2 ; jvj jAvj jvj v¤0 v¤0 jvj jvj

since the space of vectors v is wider than Av. Hence, kGn .; k/k  kGkn and Rn  kGn k  kGkn .

2.6.2 The von Neumann stability condition A necessary stability condition is the condition of existence of a constant C that bounds the spectral radius of the matrix Gn .; k/: Rn .; k/  C; R.; k/  C 1=n ;

0n

T : 

108

Chapter 2 The basics of the theory of finite-difference schemes

In particular, the condition R  C =T must hold. On a finite interval 0 <  < 1 , the exponential function of  on the right-hand side of the inequality must be bounded by a linear function: R  C =T  1 C C1 : It follows that the necessary stability condition for the finite difference scheme holds if all eigenvalues of the transformation matrix G satisfy the condition R D max j i j  1 C O. /; i

(2.65)

which was obtained by von Neumann and is known as the von Neumann stability condition. If the complex matrix G is normal, i.e., it commutes with its conjugate transpose, GG D G G, then the spectral radius is equal to the norm of G and the von Neumann condition is not only necessary but also sufficient. Note that if one searches for a solution to the original difference equation (2.60) in the form unmCˇ D u0m .k/ n exp.iˇkh/ and substitutes this expression into the original system of difference equations to obtain Œ E  G.k/ u0m D 0; where E is the identity matrix, one immediately arrives at the characteristic equation of the matrix G.; k/, detŒ E  G.k/ D 0; which serves to determine the eigenvalues of G. This technique is practically useful in analyzing the stability of finite difference schemes. Below we analyze the stability of the schemes presented in Section 2.4.5 for the acoustics and heat equations, whose approximation was studied there.

2.6.3 Stability of the wave equation Let us apply the von Neumann spectral stability analysis to the wave equation in displacements (2.44). The solution is sought in the form n 0 ik.xm Ch/ O e : unC1 mC1 D q u

Section 2.6 Stability analysis for finite difference schemes

109

Substituting into (2.44) yields uO 0 Since

  E 1 nC1 .q  2q n C q n1 / D 2 uO 0 q n e ikh  2 C e ikh : 2  h 1 e ikh  2 C e ikh D 2 cos.kh/  2 D 4 sin2 . kh/; 2

we get

   E 2 2 1 kh C 1 D 0: 2 sin q  2q 1   h2 2 2



By Vieta’s formula, the product of the roots of this quadratic equation is q1 q2 D 1. It follows that the stability condition jqj  1 can be satisfied in the only case jq1 j D jq2 j D 1. If the equation coefficients are real, this means that the roots must form a complex conjugate pair; in this case, the discriminant must be negative: ˇ  ˇ ˇ ˇ ˇ1  2 E  sin2 kh ˇ < 0: ˇ h 2 ˇ For this inequalitypto hold for any k, it is necessary and sufficient that the Courant condition = h  =E D 1=a is satisfied, which implies that the scheme is conditionally stable.

2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition Consider the Lax scheme. Substituting unC1 D nC1 u0m exp.i khˇ/ mCˇ into (2.47) yields i sin kh D 0; h (2.66) i v0 .  cos kh/  "0 a2 sin kh D 0: h From the condition that the determinant of system (2.66) must vanish, one obtains "0 .  cos kh/  v0

a2  2 sin2 kh D 0; h2 a sin kh; D cos kh ˙ i h a  1; j j  1 if h

.  cos kh/2 C

which means that the scheme is stable.

(2.67)

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Chapter 2 The basics of the theory of finite-difference schemes

The condition a= h  1, called the Courant condition [29], is a necessary condition of stability. The Courant condition is also known as the Courant–Friedrichs– Lewy (CFL) condition. It relates the space step size to the time step and holds for any hyperbolic equation. The condition has he meaning that the time step  must be chosen so as not to reduce the domain of dependence of the solution at the point x on layer n C 1 of the difference equation as compared with the domain of dependence of the differential equation, which is determined by the slope of the characteristics issuing from the point x until they meet the nth layer (Figure 2.10). This condition admits a simple physical interpretation. If the condition is violated and the characteristics of equations (2.46) pass as shown by dashed lines in Figure 2.10, the deviation of the solution to the difference equation from that to the differential equation can be made arbitrarily large. To this end, one should apply sufficiently large perturbations at the segments Ai and .i C 1/A1 (shaded areas in Figure 2.10), which are beyond the domain of definition of the difference equation and, hence, have no effect on the solution at the point O. This means that the solution is unstable. O

1 i +– ,n+1 2

τ

α A

i

n h

1 i+– 2

i+1

A1

Figure 2.10. Domains of dependence of a solution to a differential equation (solid characteristic lines) and a difference equation (dashed lines).

Let us prove that for the wave equation (2.66), the matrix G is normal, and hence the Courant condition in (2.67) is not only necessary but also sufficient for convergence. Indeed, it is easy to verify that the matrix !  cos kh  hi sin kh GD 2  a h i sin kh  cos kh can be symmetrized with the change of variables v0 D v1 a and "0 D "1 to obtain !  cos kh  ah i sin kh G1 D :  ah i sin kh  cos kh It is obvious that G1 G1 D G1 G1 .

Section 2.6 Stability analysis for finite difference schemes

111

Now let us investigate the stability of the difference equation (2.48), which differs from (2.47) in only that the difference derivative with respect to t is calculated using the value at the middle point i on the nth layer rather than the half-sum at point i C 1 and i  1, as in (2.47). We have i "0 .  1/  v0 sin kh D 0; h  i v0 .  1/  "0 a2 sin kh D 0; h (2.68) 2 2 a  2 2 det G D .  1/ C 2 sin kh D 0; h a sin kh: D1˙i h It is apparent that the von Neumann condition (2.65) is violated, j j > 1, and the scheme is unstable. Lax–Wendroff scheme. Let us analyze the stability of the scheme (2.49). The equations involve a finite difference representation of the second derivative. Since this representation is frequently used in what follows, let us introduce a special designation for it: umC1  2um C um1 : ƒum D h2 Its Fourier transform is 2.1  cos kh/ 4  2 C e i kh kh : (2.69) D uO m D  2 uO m sin2 h2 h2 h 2 Then, for (2.49) one finds that   2 kh sin2 v0  . i sin kh/"0 D 0; .  1/   2 a2 2 2 2 D : (2.70)    h2 2 2 kh . i sin kh/v0 C .  1/  sin "0 D 0; 2 2 O m D uO m e ƒu

i kh

Equating the determinant with zero gives 1;2 D 1   2 .1  cos ˛/ ˙ i  sin ˛; ˛ D kh;   ˛ ˛ ˛ 1=2 1  sin2 ; 1;2 D 1  2 2 sin2 ˙ 2i  sin (2.71) 2 2 2 ˛ j j2 D 1  4 2 .1   2 / sin4  1; 2 where  is the Courant number. As ˛ varies in the range 0  ˛  2, the quantity j j describes, in the complex plane, an ellipse that lies within the unit circle j j D 1 if  < 1. For  D 1, the ellipse becomes the unit circle, which indicates that the scheme in nondissipative and so the amplitude of each Fourier component is preserved exactly. Here also GG D G G and the Courant condition is sufficient for stability.

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Chapter 2 The basics of the theory of finite-difference schemes

2.6.5 Stability of schemes for the heat equation Now let us investigate the stability of the finite difference scheme (2.54) for the heat equation (2.52) in dimensionless variables TmnC1  Tmn D ƒTmn C .1  /ƒTmnC1 : 

(2.72)

Searching for a solution in the form nC1 TmCˇ D Tm0 nC1 exp.i khˇ/

and using formula (2.69), one obtains i 4 h 2 ˛ 2 ˛ C .1  / sin ;  sin h2 2 2 4 ˛ Œ1 C .1  /p 2 D 1  p 2 ; where p 2 D 2 sin2 ; h 2 1  p 2 C p 2  p 2 p2 D D 1 : 1 C .1  /p 2 1 C .1  /p 2 1D

(2.73)

From (2.73) it follows that the von Neumann condition is satisfied if 0

p2  2: 1 C .1  /p 2

(2.74)

Inequalities (2.74) must hold for any p in order to avoid any restrictions on the time step  . The left inequality holds for any 0 <  < 1, while the right inequality provides a constraint on 1  : (2.75) 2p 2 .1  / C 2  p 2 : It follows that the condition   12 must hold. To summarize, the scheme is unconditionally, or absolutely stable if   12 and is only conditionally stable if  > 12 . For example, an explicit scheme with  D 1 implies D1

˛ 4 sin2 : 2 h 2

Consequently, for the von Neumann condition to be satisfied it is necessary that, in 2 dimensional variables,   h2 . This is a very strict constraint on the time step, which results in a too small step size in time, so that the finite difference scheme for the heat equation becomes inefficient. On the other hand, although the implicit scheme (2.72) with   1=2 does not lead to any restrictions on the time step, it makes it necessary to solve a system of algebraic equations at each step.

Section 2.6 Stability analysis for finite difference schemes

113

2.6.6 The principle of frozen coefficients The above spectral method was developed for studying linear constant-coefficient equations. However, it turns out to be helpful also for stability analysis of a much wider class of problems for linear and nonlinear equations. In nonlinear equations and linear variable-coefficient equations, the stability analysis should be performed using the following rule: all coefficients dependent on the variables x and t and the unknown u are assumed to be constant, or, as is often said, “frozen.” Then the equation becomes a linear constant-coefficient equation, which is further analyzed using the spectral method. In this case, the stability condition will depend on the frozen coefficients and, hence, on x, t and u. The time step  must be chosen so as to satisfy the stability condition for all values of the coefficients involved in the computation. For example, let us consider the following nonlinear heat equation with thermal conductivity dependent on the coordinates and temperature,  D .t; x; T /:   @ @T @T D .t; x; T / C q.t; x; u/: @t @x @x n n ; xm ; Tmn / in the same way as in The analysis is carried out with .t; x; T / D .tm the previous example by assuming that  is constant and dependent on t , x, and T as parameters. Let us investigate the stability of the explicit scheme. The function q.t; x; u/ on the right-hand side of the equation does not affect the stability and, hence, can be neglected. Consequently, for fixed  we have    TmnC1  Tmn n n n D 2 Tm1  2Tm C Tm1 :  h

The stability condition is satisfied if 

h2 h2  : 2.t; x; T / 2 max.x;T / .t n ; x; T /

(2.76)

This rule is known as the principle of frozen coefficients. It follows that the time step  at each layer t n can vary and is determined by max.x;T / .tn ; x; T /. This can significantly restrict  if .x; T / assumes a large value in a small region while being small in the rest of the bar, where the computation can be performed with a larger step than prescribed by condition (2.76). For this reason, it is desirable to obtain an explicit but unconditional scheme, which would be much more efficient. Is it possible to construct such a finite difference scheme? To answer this question, let us approximate the heat equation (2.52) with the Dufort–Frankel three-layer scheme (2.59) n T n  TmnC1  Tmn1 C Tm1 TmnC1  Tmn1 D mC1 : 2 h2

(2.77)

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Chapter 2 The basics of the theory of finite-difference schemes

This scheme uses a five-point cross stencil; its specific feature is that the second derivative on the right-hand side is approximated in an unusual way. In the usual representation of the second derivative, n n  2Tmn C Tm1 TmC1 ; h2

the second term, 2Tmn , is replaced with .TmnC1 C Tmn1 /. nC1 Let us investigate the stability of the scheme (2.77). Substituting TmC1 D 0 nC1 exp.i khm/ yields (˛ D kh) Tm       2 1 1 D 2 e i˛   C e i˛ ;  h 2 q D 2; 2 .1 C q/  2 q cos ˛ C .1  q/ D 0; h p 2 2 q cos ˛ ˙ 1  q sin ˛ : 1;2 D 1Cq Analyzing the resulting expression, we obtain ˇ ˇ ˇ q cos ˛ ˙ i q sin ˛ ˇ ˇ D q < 1I if q 2 sin2 ˛ > 1; j j < ˇˇ ˇ 1Cq 1Cq ˇ ˇ ˇ ˇ q q cos ˛ ˙ 1 ˇ  1: if q 2 sin2 ˛ < 1; j j < ˇˇ D 1Cq ˇ 1Cq It follows that the explicit scheme (2.77) is unconditionally stable. However, it turns out that, although the time step is not constrained by the stability, there are restrictions that arise from the approximation conditions. Indeed, let us check the approximation of the right-hand side of (2.77):   2 1 T .t; x C h/  T .t C ; x/  T .t  ; x/ C T .t; x  h/ D Txx T t t 2 CO.h2 /: h h It is apparent that the approximation must satisfy   o.h/; otherwise, if   h, the scheme (2.77) will approximate, instead of the heat equation, a telegraph equation of hyperbolic type that contains the second derivative with respect to time: Tt C Tt t

2  Txx D O. 2 C h2 /: h2

So, although absolutely stable, the finite difference scheme (2.77) approximates equation (2.52) conditionally. Therefore, just as the explicit conditionally stable scheme, it inefficient, since it requires a very small time step  . For the heat equation, no efficient explicit scheme can be constructed and so implicit schemes should be used. The most common implicit scheme is the scheme (2.72) with  D 0, it has

115

Section 2.6 Stability analysis for finite difference schemes

the first order of approximation in time and second order in space. For  D 1=2, the scheme uses a six-point stencil and, as one can easily see, has the second order of approximation in both variables. The spectral stability condition of a scheme does not generally guarantee stability in a real computation but is its necessary condition, which favors stability. Some difficulties may be caused by a nonlinearity in the frozen coefficient and, especially, by the approximation of the boundary conditions, which are not considered by the spectral method. The spectral method was designed for the stability analysis of solutions to Cauchy problems for partial differential equations. However, most problems arising in continuum mechanics are initial-boundary value problems. Therefore, the question of how the approximation of the boundary conditions affects of the solution stability is important. This question is very difficult to investigate in the general case. In what follows, we restrict our presentation to a simple practical method of assessing stability.

2.6.7 Stability in solving boundary value problems This approach suggests that, apart from the standard analysis of spectral stability in interior of the domain in question, one applies the same method to the boundary of the domain. For example, let us investigate the explicit scheme for the heat equation with the boundary condition @T  ˇT D 0 at @x

x D 0;

which is approximated as T1n  T0n  ˇT0n D 0; h

.1 C hˇ/T0n  T1n D 0:

(2.78)

A solution of (2.78) will be sought in the form T D T 0 n e ik˛ , which implies e i˛ D 1 C ˇh;

˛D

1 ln.1 C ˇh/ D iˇh C O.h2 /: i

For this ˛, let us calculate the spectral point .˛/. For the explicit scheme (2.73) with  D 1, we get     4 ˇ 2 h2 4 2 ˛ 2 D 2 C O.h / D  ˇ 2 C O.h/ ;  1 D  2 sin h 2 h 4 2 2 D 1 C ˇ C O.h /; j j  1 C O. /; which means that the scheme is stable at the left edge (x D 0).

116

Chapter 2 The basics of the theory of finite-difference schemes

The boundary condition at the right edge (x D 1) is analyzed likewise. It can be checked for the left ray (x  1) by setting k D 0; 1; 2; : : : I one finds that ˛ D iˇh

and so remains the same: D 1 C ˇ 2 C O.h2 /:

For details on the informal stability theory of boundary-value problems, which is quite sophisticated, see [147]. Solving a boundary value problem of the system of equation (2.72) is reduced, for fixed n, to solving a system of algebraic equations with a tridiagonal matrix, which is efficiently solved by the tridiagonal matrix algorithm (sweep method). This method is a simplified form of Gaussian elimination; it is heavily used and is crucial in computational mathematics. Many boundary value problems solved by finite difference methods are reduced to algebraic systems with matrices close to diagonal, which are solved by the sweep method. The main idea of the method admits various generalizations. Scalar, vector, and matrix sweeps are known. The method will be discussed in detail below (see Sections 4.4–4.5), once the general methods for the solution of difference equations have been presented.

2.6.8 Step size selection in an implicit scheme in solving the heat equation For stable computations based on explicit schemes, the Courant condition gives the following constraint on the time step:   0:5h2 . At the same time, explicit schemes do not impose any restrictions on  . The question arises: How to choose the time step in a real computation based on an implicit scheme? Here the restriction is imposed by the accuracy requirement rather than the stability condition. In order to answer the question, one should analyze the exact solution to a differential equation and the solution to the corresponding difference equation by representing them in terms of Fourier series. The exact solution of the differential problem (2.52) is given by X Ck0 e  k t sin kx; k D k 2  2 : (2.79) T D k

The solution of the difference equation (2.72) with  D 1 is X Ckn sin.k mh/: Tmn D k

Substituting it into (2.72) yields the following equation for Ckn : CknC1  Ckn kh nC1 4 D  2 sin2 C :  h 2 k

117

Section 2.7 Exercises

Its solution is

   kh n : Ckn D Ck0 1 C 4 2 sin2 h 2

(2.80)

It is apparent that Ckn is dependent on  and h and, hence, on t and x, whereas Ckn in the exact solution (2.79) is only dependent t and independent of x: Ckn D Ck0 e k

2  2t

:

It follows from formula (2.80) that the expression of Cnk is independent of h and, hence, of x only if jkhj 1 and then solution (2.80) can be rewritten as 1C4

kh  2 2 D 1 C k 2  2   e k  t : sin2 h2 2

For k 2  2  1, the solution is close to the exact one. Consequently, for real approximation, h should be chosen so as to satisfy the condition jkhj 1, or h k=. On the other hand, it follows from the solution to the difference equation that the time step  should be chosen so as to satisfy the condition  k 2  2 1 for the solution to be close to the exact one. In this case,   O.h2 /. For example, if k D 100, we get h 102 and  104 , which means that the relation between h and  must be the same as in the explicit scheme. This condition is natural for thermal conduction problems. This does not apply to slow-varying solutions, where k  1; in this case, one can take   h and it is reasonable to use an implicit scheme.

2.6.9 Step size selection in solving the wave equation In order to work out the time step for an implicit absolutely stable scheme for solving wave equations, there is no need to compare the exact solutions to the differential equation and the corresponding difference equation. It suffices to recall that the domain of dependence of the exact solution to the differential equation is determined by the characteristics and  D h=c (Figure 2.10). Therefore, the Courant stability condition for the explicit scheme is a constraint on the time step  and, simultaneously, on the approximation accuracy, provided that the accuracy is assessed in the Ch -metric. If integral characteristics of motion are only of interest, then the solution accuracy should be assessed in the L2h -metric; in this case, a larger time step can be taken,  > h=c.

2.7 Exercises 1. Obtain a finite difference representation of a third derivative on a four-point stencil of a uniform grid with nodal points i  2, i  1, i , and i C 1. Determine the order

118

Chapter 2 The basics of the theory of finite-difference schemes

of approximation using uxxx

  d 3u d d 2u D D dx 3 dx dx 2

and the finite difference formula for the second derivative. 2. Obtain a finite difference representation of a fourth derivative on a five-point stencil (i  2, i  1, i , i C 1, i C 2) using the representation of the third derivative obtained in Exercise 1. 3. Obtain a finite difference representation of a fourth derivative using the formula   d 2 d 2u IV ux D dx 2 dx 2 and a difference formula for the second derivative on a five-point stencil (i  2, i  1, i , i C 1, i C 2) and a seven-point stencil (i  2, i  2, i  1, i , i C 1, i C 2, i C 3). Compare the orders of approximation. 4. Write out a difference operator approximating the Poisson equation L.u/ D

@2 u @2 u C D f .x; y/ @x 2 @y 2

on a rectangular grid with step sizes h in x and H in y. Determine the order of approximation of the resulting scheme.   1  1 uiC1;j  2ui;j C ui1;j C 2 ui;j C1  2ui;j C ui;j 1 D fi;j 2 h H Prove that Lh .u/ D L.u/ C O.h2 / C O.H 2 / D f .x; y/: 5. How to combine the computations on embedded grids with step sizes h and h=2 based on a first-order scheme so as to increase the order of approximation to O.h2 / (Richardson’s formulas)? 6. For the equation y 0 C ˛y D 0;

y.0/ D y0 ;

analyze the stability of the two finite difference schemes   h˛ i yiC1 D y C yi C yiC1 ; 2 yiC1  yi1 C ˛yi D 0; 2h by looking for exacts solutions to the difference equations in the form yi D i (see Section 2.2).

119

Section 2.7 Exercises

7. Obtain the third-order Adams–Bashforth formula. Use the basis Lagrange interpolating polynomials p

lk .t / D

.t   /.t  2 / : : : .t  k / ; .t  p / !.p /

where !.t / is the expression obtained by dividing the numerator by .t  p /; lpk .i  / D ıip with i D 1; : : : ; k, p < k, and ıip being the Kronecker delta. 8. Obtain the difference equations of the Adams–Bashforth method of the third order of accuracy and formulas for evaluating the first three values ui (i D 0; 1; 2) required to begin the computation according to a third-order scheme. Apply the scheme to solve the equation du D u2 C t 2 : dt 9. Obtain a third-order Runge–Kutta scheme by the method of double predictorcorrector. Apply the scheme to solve the equation du D t 2 C ux C u2 : dt 10. Reduce the telegraph equation @2 u @2 u  2 C ku D 0 @t 2 @x to a system of three first-order equations and analyze the stability of the Lax scheme for this system. 11. Write out an explicit scheme for the parabolic constant-coefficient equation @T @T @2 T D 2 2 C a C bT @t @x @x and perform its stability analysis. 12. Reduce the nonlinear wave equation   2 @u 2 @u @ u Da @t 2 @x @x 2 to a system of two equations, write out the Lax–Wendroff scheme, and analyze its stability by the frozen coefficient method.

120

Chapter 2 The basics of the theory of finite-difference schemes

13. Perform the stability analysis of the five-point cross finite-difference scheme for the telegraph equation @2 u @2 u  2 C ku D 0: @t 2 @x 14. Perform the stability analysis of the explicit finite difference scheme for the equation y 0 C .ky 2 /y D bx where k and b are constants, by the frozen coefficient method. 15. For the system of wave equations @" @v D a2 ; @t @x

@" @v D ; @t @x

determine the order of approximation and analyze the stability of the scheme "n  "nk1 vknC1  vkn D a2 k ;  h

v n  vkn "nC1  "nk k D kC1 :  h

16. Represent the equation describing the dynamic behavior of an elastoviscous bar @" 1 @ DE   @t @t  as a system of two first-order equations; here, E is Young’s modulus and  is the relaxation time. Write out a second-order scheme using the Lax–Wendroff method. 17. For the system of equations for an elastoviscous bar (see Exercise 16), obtain an explicit finite difference scheme on a three-point L-shaped stencil. Determine the order of approximation. Analyze the stability. 18. For a one-dimensional flow of a viscous fluid through a plane channel without friction, governed by the constitutive equation 1 @ D ; @t  where  is the relaxation time, obtain an implicit scheme for the six-point stencil shown in Figure 2.7b. Analyze the stability. 19. Determine the order of approximation and analyze the stability of the Allen–Chen scheme (2.58) for the heat equation, TknC1  Tkn 

D

n n TkC1  2TknC1 C Tk1

.h/2

C f .Tkn /:

121

Section 2.7 Exercises

20. Perform the stability analysis of the Krankel–Nicolson scheme (2.57) for the heat equation subject to the initial and boundary conditions T D ‚.x/

at t D 0;

@T C b1 T D '1 .t / at @t @T C b2 T D '2 .t / at a2 @t a1

x D 0; x D 1:

Chapter 3

Methods for solving systems of algebraic equations

3.1 Matrix norm and condition number of matrix Prior to considering specific methods for solving systems of algebraic equations, let us look at some common issues associated with the solution of such systems. In solving a linear system Ax D b, where A is an n  n matrix of coefficients, x is the vector on unknowns, and b is a constant vector, one faces the question of stability of its solution with respect to small perturbations of the coefficient matrix, A C ıA, and/or the right-hand sides, b C ıb. The matrix A will be assumed to be positive definite, .Ax; x/ > 0, and symmetric, AAT D ATA.

3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix Let us examine the error arising in the solution, ıx, when the right-hand side is disturbed by ıb. We have A.x C ıx/ D b C ıb;

Aıx D ıb;

ıx D A1 ıb;

kıxk D kA1 ıbk  kA1 k kıbk: If the norm kA1 k is large, then kıxk will also be large. The maximum increase in the vector length will be in the direction of the eigenvector x1 corresponding to the maximum eigenvalue max , A1 x1 D max x1 , or when ıb is directed along the eigenvector x1 . In this case, ıb D ˛x1 ;

ıx D ˛A1 x1 D

ıb ; ƒ1

where 0 < ƒ1      ƒn are the eigenvalues of the n  n matrix A, while the i D 1=ƒi are eigenvalues of the inverse matrix A1 . If a ƒ1 is close to zero, the matrix A is close to a singular matrix and then kıxk is very large. Relative errors are more important than absolute errors. So it is essential as compared with kıbk . The worst case scenario is to evaluate the relative error kıxk kxk kbk when the error kıxk is maximum while the norm kxk is minimum. The latter is true

123

Section 3.1 Matrix norm and condition number of matrix

when x is parallel to the direction corresponding to the minimum eigenvalue n : x D A1 b; It follows that

x D ˛A1 xn D ˛ n xn D

b D ˛xn ;

b : ƒmax

ƒmax kıbk kıxk  ; kxk ƒmin kbk

(3.1)

which implies that the larger the number C D ƒmax =ƒmin , called the condition number of the matrix A, the larger the error. If the matrix A is not symmetric, then the norm is defined as the maximum increase of the vector length relative to its original length kxk and denoted kAk: kAk D max x

kAxk : kxk

(3.2)

For a non-symmetric matrix, this maximum may not necessarily be attained at the maximum eigenvector xn and so ƒmax ¤ kAk. Therefore, in the formula for the condition number, ƒmax and ƒmin must be replaced with kAk and kA1 k, respectively: C D kAk kA1 k:

(3.3)

For non-symmetric matrices, the system of equations Ax D b can be symmetrized, AT Ax D AT b D b1 , and so the condition number can be expressed in terms of eigenvalues. It should be noted, however, that the symmetrization stretches out the eigenvalue spectrum and increases the condition number. It follows from (3.1) that the more extended the eigenvalue spectrum, or the larger the ratio ƒmax =ƒmin D C , the less stable the solution is to roundoff errors and other perturbations.

3.1.2 Relative error of solution for perturbed coefficient matrix Suppose the coefficient matrix is perturbed to become A C ıA. Let us determine the as compared with kıAk : relative error kıxk kxk kAk .A C ıA/.x C ıx/ D b; 1

ıx D A

A ıx D ıA .x C ıx/; ıA.x C ıx/;

kA1 k kıAk kx C ıxk kA1 k kıAk kAk kıAk kıxk   DC : kxk kxk kAk kAk

(3.4)

So, in this case, the relative error in the solution is proportional to the condition number of the matrix A.

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Chapter 3 Methods for solving systems of algebraic equations

Below are examples of an ill-conditioned and a well-conditioned matrix:     1 1 0:0001 1 I II ; A D : A D 1 1:0001 1 1 Let us show that the matrix AI is ill-conditioned and the matrix AII is wellconditioned. The eigenvalues of AI are determined from the equation

1;2

2  2:0001 C 104 D 0; q D 1:00005 ˙ .1:00005/2  104 :

It follows that C I D 1024 D 2  104 . By solving the system with the coefficient matrix AI , let us verify that the system is unstable. We have     2 1 I ; xD : Ax D 2:0001 1 By perturbing the right-hand side with ıb D .0; 104 /T , we find that the solution changes by a quantity of the order of O.1/. Indeed,     2 0 I ; xD : Ax D 2:0002 2 The eigenvalues of AII satisfy the equation 2  1:0001  0:9999 D 0; p p 1 j1 C 5 j 5 II 1;2  ˙ ; C  p : 2 2 j1  5 j The same perturbation of the right-hand side, ıb D .0; 104 /T , results in the solution perturbation ıx D C II .0; 104 /T , which has the same order of magnitude.

3.1.3 Example Let us evaluate the condition number of the tridiagonal matrix arising in solving a simple two-point boundary value problem for a second-order equation of the form u00 .x/ D f .x/;

u.0/ D 0;

u0 .1/ D 0:

125

Section 3.1 Matrix norm and condition number of matrix

By partitioning the segment Œ0; 1 into n subsegments, tridiagonal n  n matrix A of the difference system: 0 2 1 0    0 0 B1 2 1    0 0 B B :: :: :: :: :: :: B : : : : : : B B :: : : : : : ::: : : : B : : : : ADB : B :: :: :: :: :: :: B : : : : : : B B0 0 0    1 2 B @0 0 0    0 1 0 0 0  0 0

one arrives at the following 0 0 :: : :: : :: :

0 0 :: : :: : :: :

1

C C C C C C C C: C C C 1 0 C C 2 1A 1 2

The right-hand side is of the order of unity, f .x/  O.1/, and so kbk  O.1/. It can be shown that the maximum and minimum eigenvalues of the matrix A are max D 4 and min D  2=n2 . Then the roundoff error of the right-hand side ıb D 109 at n D 12 will cause, by virtue of (3.1), a relative error in the solution kıxk=kxk  105 ; however, at n D 104 , we get kıxk=kxk  101 , which means that excessively fine partitioning can severely affect the computation accuracy due to roundoff errors. The matrix A of the equation uIV .x/ D f.x/ has min  1=n4 and, in this case, roundoff errors will decrease accuracy at already n D 102 . This examples demonstrates that using a very fine partitioning in the hope to “guarantee” high computational accuracy can result in an ill-conditioned matrix of the algebraic system of equation and cause the opposite result – loss of accuracy.

3.1.4 Regularization of an ill-conditioned system of equations When solving ill-conditioned systems of equations, it is reasonable to perform their regularization, which implies that the coefficient matrix is slightly perturbed, A1 D A C ˛E, where E is the identity matrix, A is a symmetric positive definite matrix, and ˛ > 0. The eigenvalues ƒi of the matrix A1 are equal to i C ˛, where i are the eigenvalues of A. The solution to the regularized system of equations can be represented as a decomposition in eigenvectors ei of the matrix A1 : xD

n X i

ci ei

with ci D

.b; ei / .b; ei / : D ƒi i C ˛

If ˛ is chosen so as to satisfy the condition max ˛ min , which is feasible for an ill-condition matrix A, the terms corresponding to small i will, unlike the original matrix, no longer cause a significant perturbation in the solution when the right-hand side is perturbed, b C ıb. At the same time, perturbations caused by adding ˛ to the terms corresponding to large i will be insignificant and so the solution can happen to

126

Chapter 3 Methods for solving systems of algebraic equations

have an acceptable accuracy. The optimal choice of ˛ depends on the specific features of the problem and should be performed by trial and comparison of the results for different ˛.

3.2 Direct methods for linear system of equations 3.2.1 Gaussian elimination method. Matrix factorization Let us consider the simplest direct methods for solving systems of algebraic equations of the form Ax D b; (3.5) where A is an n  n square matrix with real entries, det A ¤ 0, and b is a real vector. The Gaussian elimination method implies successive elimination of unknowns (forward sweep): x1 is first eliminated from n  1 out of n original equations, then x2 is eliminated from n  2 out of n  1 remaining equations, and so on. As a result, the original system (3.5) is transformed to a system whose matrix is upper triangular: 0 10 1 0 1 U11 U12    U1n x1 b1 B 0 U22    U2n C Bx2 C Bb2 C B CB C B C Ux D b or (3.6) B :: : C B : C D B : C: :: : : @ : : :: A @ :: A @ :: A : 0

0

   Unn

xn

bn

Here the vector b1 has been obtained by the above transformation from the right-hand side vector b of the original system (3.5). Then the system is solved backwards (backward sweep): xn is found from the last equation and substituted into the .n  1/st equation, then xn1 is found, and so on. The inversion of the upper triangular matrix is straightforward. The entire algorithm can be represented in terms matrix transformations. To show this, let us consider the process inverse to the elimination of unknowns in the forward sweep. This will enable us to determine the matrix A as the product of a matrix L by the matrix U and obtain an algorithm for calculating the entries Uik and Lkj . Multiply the .n  1/st equation by Ln;n1 and add to the nth equation, then multiply the .n  2/nd equation by Ln;n2 and Ln1;n2 and add to the nth and .n  1/st, respectively, multiply the .n3/rd equation by Ln;n3 , Ln1;n3 , and Ln2;n3 and add to the nth, .n  1/st, and .n  2/nd, respectively, etc. This algorithm coincides with the multiplication of the matrices L and U. As a result, we arrive at the matrix A: LU D A

(3.7)

127

Section 3.2 Direct methods for linear system of equations

or 0

1 0 0 BL21 1 0 B BL31 L32 1 B B :: :: :: @ : : : Ln1 Ln2 Ln3

10 U11 U12 U13 0 B 0 U22 U23 0C CB B 0C 0 U33 CB 0 :: :: :: C B :: : : :A @ :  1 0 0 0

   :: :

1 0    U1n a11 a12 a13 B 0 a22 a23    U2n C C B B    U3n C C D B 0 0 a33 :: : C B : :: :: : :: A @ :: : :    Unn 0 0 0

1    a1n    a2n C C    a3n C C: :: C :: : : A    ann

It is an important result that the matrix L is lower triangular with unit diagonal. The entries of L and U can be calculated using the recurrent formulas U11 D a11 ; U1j D a1j ; Ui i D ai i 

Lj1 D i1 X

aj1 ; U11

j D 2; : : : ; nI

Lik Uki ;

i D 2; : : : ; nI

Lik Ukj ;

i D 2; : : : ; nI

kD1

Uij D aij 

i1 X kD1

Lj i

  i1 X 1 D Lj k Uki ; aj i  Ui i

j D i C 1; : : : ; n:

kD1

So Ax D LUx D Lb1 D b. Consequently, the Gauss procedure is essentially the decomposition (factorization) of the matrix A into the product of a lower triangular matrix L by an upper triangular matrix U. Both matrices are easy to invert: in the forward sweep, one inverts L to obtain L1 b D b1 and in the backward sweep, one inverts U to get x D U1 b1 . The original problem is thus reduced to two simpler ones: the inversion of the upper and lower triangular matrices U and L. It is noteworthy that, in solving specific problems of continuum mechanics, what changes is just the right-hand side of the system, while the matrix A remains the same; in other words, what changes is the external load, while the equation and the domain where the solution is sought remain unchanged. Therefore, one inverts L and U only once and stores in the computer memory, thus reducing the solution of an particular problem to the multiplication of a matrix by a vector.

3.2.2 Gaussian elimination with partial pivoting For the Gauss process to be feasible, it is necessary that all j th-order minors at the top left corner of the matrix must not be zero, jAj j ¤ 0, j D 1; : : : ; n. For example,

128

Chapter 3 Methods for solving systems of algebraic equations

the Gauss process can not be realized for the matrix   0 1 ; AD 1 0 since jA1 j D 0, although the system determined by this matrix has an obvious solution; it suffices to swap the rows to get jAj j ¤ 0 (j D 1; 2). Furthermore, if jAj j D ", where " is small, then the elimination involves dividing by a small quantity; this may result in the loss of true information about the coefficients aij , which is due to a bad algorithm, even though the matrix A is well-conditioned. For example, if   " 1 ; AD 1 0 one should first swap the rows before performing the Gaussian elimination. Thus, prior to eliminating a kth unknown, one should first locate the main element, largest in absolute value, amongst all entries of the kth column and move the corresponding row to the top. Then the corresponding Ujj > 0 will be maximum and the Gauss algorithm will become as stable at it is allowed by the matrix A. This method is known as the Gaussian elimination with partial pivoting.

3.2.3 Cholesky decomposition. The square root method If the matrix A is positive definite, as in solving elasticity problems by the finite element method [188], then det A D det L det U D u11 u22 : : : ujj > 0, since all ujj > 0. Then the Gauss elimination process is always feasible and, furthermore, it does not require the permutation of rows if A is symmetric. Indeed, in this case, the decomposition A D LU can be rewritten as A D LD.D1 U/, where D is a diagonal matrix with entries u11 ; : : : ; ujj . Then 1 0 U1n 1 U12 U13    B0 1 U22    U2n C C B B :: C : :: :: :: D1 U D B ::: : : C : : C B @0 0 0    Un1;n A 0 0 0  1 Since A D AT , we have Q DU Q T DLT A D LDU

Q D D1 U: with U

Q D LT . In addition, the matrix A can also be repreBy symmetry, it follows that U sented as (3.8) A D LDLT D LD1=2 D1=2 LT D LLT :

129

Section 3.2 Direct methods for linear system of equations

This representation is called the Cholesky decomposition [168]. If the condition number of A is C , then the condition numbers in the direct and backward sweeps are the 1=2 same and equal to C 1=2 , since L D A1=2 . Then the eigenvalues of L are li D ƒi . Below is an example of the Cholesky decomposition:       " 1 1 0 " 0 1 "1 AD D 1 ; 1 0 " 1 0 "1 0 1    1=2   1=2  1 0 " " 0 0 1=2 LD D 1 D 1=2 1=2 ; " 1 0 "1=2 " " det LD1=2 D det L D 1: Although the matrix A requires choosing the main element, let us obtain the solution, after the Cholesky decomposition, without this. The entries of D and L are calculated as d1 D sign a11 ;

l11 D .a11 /1=2 ;

  i1 X 2 di D sign ai i  .lki / dk ; kD1

  i1 X 1 lki lki dk ; lij D aij  li i di kD1

a1j ; j D 2; : : : ; nI l11 1=2  i1 X 2 li i D ai i  .lki / dk I

l1j D

kD1

i D 2; : : : ; n; j D i C 1; : : : ; n:

The computational cost of the Cholesky method is approximately half that of the Gauss method where the symmetry of A is not taken into account. The Gauss method is often applied to sparse matrices that have a quasidiagonal or band-like structure, where nonzero entries are close to the main diagonal. Such matrices require fewer operations to invert. Therefore, it is reasonable to convert the system matrix to a form that has the minimum width of the diagonal band. For matrices with a narrow band, where the number of nonzero entries, k, in a row is much less than the dimension of the matrix, N , the gain in the efficiency can be substantial. In addition, if the band has a varying width, it is reasonable to store the entire band profile, i.e., the first and last nonzero entries in each row. Sometimes, when the matrix is sparse but cannot be converted to a band-like form, one can store the position of each nonzero entry and then perform the elimination using this arrangement [178, 168]. In many finite difference problems, the matrix A has a very simple, tri- or five-diagonal or block-diagonal structure. These cases can be treated using the most efficient algorithm, the sweep method. The next section outlines iterative methods – another way of solving systems of algebraic equations. The sweep method, which crucial for solving finite difference problems, will be elaborated later on.

130

Chapter 3 Methods for solving systems of algebraic equations

3.3 Iterative methods for linear system of equations In solving systems of algebraic equations resulting from the discretization of equations of solid mechanics, it is often reasonable to use iterative methods. The initial approximation is a very important issue here. In many cases, whether one succeeds in solving the problem or not depends on how good the initial approximation is. When studying evolution problems using implicit schemes, one has to solve systems of equations at each time step. In such problems, the solution obtained at the previous step provides an excellent initial approximation, which makes the application of the iterative method very efficient. When solving stationary problems, there is no such an easy way of determining the initial approximation. Physical intuition and clear understanding of the essence of the problem can help in such cases. Sometimes, it may be reasonable to introduce time in the problem artificially and then solve it as an evolution problem until the solution reaches a steady-state mode (stabilization method). An important advantage of iterative methods over direct methods is that the former do not require large computer memory – this problem is still essential today, although it is not as sharp as it was due a massive progress in computer hardware over the last few decades.

3.3.1 Single-step iterative processes Iterative methods are most crucial for solving nonlinear problems, where they have no alternative. However, studying iterative methods is reasonable to start with solving linear problems. Consider the system of linear algebraic equations Ax D f;

(3.9)

where A is a positive semi-definite matrix, .x; Ax/  0 8x ¤ 0. The matrix A can be treated as a linear operator in the Euclidean space with the scalar product .x; y/ D xi yi and associated norm kxk D .x; x/1=2 , where x is an n-dimensional vector. If each subsequent approximation xkC1 is calculated using only the previous approximation xk , then the iterative process will be called one-step or two-layer (just as the corresponding finite difference scheme). If two preceding approximations, xk and xk1 , are used, the process will be called two-step or three-layer. Canonically, a two-layer iterative process can be represented as B

xkC1  xk C Axk D f; kC1

k D 0; 1; : : : :

(3.10)

The matrix B and scalars k are parameters of the iterative process, which are selected so as to make the process most efficient. The form (3.10) corresponds to a finite

131

Section 3.3 Iterative methods for linear system of equations

difference scheme. Hence, there is a close relationship between iterative methods and explicit finite difference schemes. The convergence of an iterative process will be examined in an energy space, HC , generated by a positive definite matrix C with the scalar product .a; b/C D .Ca; b/ and associated norm kakC D .Ca; a/1=2 . The process is convergent if kzk kC ! 0 as k ! 1, where zk D xk  x with xk being the kth approximation of x. Since the exact solution x to equation (3.9) is not known, the accuracy is evaluated using, instead of kzk kC , the discrepancy norm kAxk  fk D k k k; which is easy to calculate at each iteration. The relative error of the discrepancy, ", will be taken as the measure of convergence and the accuracy will be estimated as k k k  "k 0 k: This accuracy estimation condition corresponds to convergence in the energy space with the matrix C D ATA. We have  1=2  k 1=2 k k k D Axk  f; Axk  f D Ax  Ax; Axk  Ax 1=2  T k k 1=2  D Azk ; Azk D A Az ; z D kzk kC : The iterative process determined by formula (3.10) can be optimized by selecting a suitable matrix Bk and parameter k . If these parameters change between iterations, the iterative process is called nonstationary or unsteady; if these parameters do not change, the iterative process is stationary or steady-state.

3.3.2 Seidel and Jacobi iterative processes Let us find out how the classical Seidel and Jacobi iterative processes are represented in the form (3.10). In Jacobi iterative process, the kth approximation in solving equations (3.9) involves each component of the vector xikC1 from the i th equation by the formula i1 X j D1

aij xjk

C

ai i xikC1

C

n X

aij xjk D fi ;

i D 1; : : : ; n;

(3.11)

j DiC1

where all components except xikC1 are taken from the previous approximation (one takes x k D 0 in the initial approximations if k < 0). In the Seidel method, xjk in the first sum of (3.11) is replaced with the already determined values xjkC1 (j D 1; : : : ; i  1). Let us represent the matrix A as the sum of an upper triangular matrix L, a diagonal matrix D, and a lower triangular matrix U: A D L C D C U:

132

Chapter 3 Methods for solving systems of algebraic equations

The the Jacobi iterative process can be represented in the form (3.10) with parameters B D D and  D 1, while the Seidel process can be represented in the same form with parameters B D L C D and  D 1. The Jacobi and Seidel process can be generalized to the nonstationary case with a varying parameter k to obtain D

xkC1  xk C Axk D f; kC1

xkC1  xk C Axk D f; .L C D/ kC1

k D 0; 1; : : : ; n:

Another generalization of such an iterative process is given by the formula .LkC1 C D/

xkC1  xk C Axk D f; kC1

which represents the upper relaxation method. Substituting A D L C D C U and solving for xkC1 , one obtains  xkC1 D L C

1 kC1

  1  f  UC 1

D

1 kC1

   D xk :

The only matrix here that needs to be inverted is the upper triangular matrix L. In the stationary case, these processes are convergent if the transition matrix G from layer k to layer k C 1 satisfies the von Neumann condition: xkC1 D Gxk C B1 kC1 f;

where G D E  kC1 B1 A:

In the Jacobi method, G D E   D1 A: In the Seidel method, G D E   .L C D/1 A: In both cases, the matrix G is permutable with its transpose and the von Neumann condition is no only necessary but also sufficient for convergence, provided that A D AT is a symmetric conjugate matrix. The steady-state iterative process (3.10) correspond to the simple iteration method. It is convergent if, by the von Neumann condition, all eigenvalues i of the matrix G satisfy the condition j i j < 1.

133

Section 3.3 Iterative methods for linear system of equations

3.3.3 The stabilization method It is easiest to analyze the convergence of iterative processes with B D E. In this case, equation (3.10) can be replaced, by passing to the limit as kC1 ! 0, with the differential equation d CA Df (3.12) dt subjected to the boundary condition t D 0;

D 0:

(3.12a)

Let us prove that problem (3.9) can be solved using the stabilization method by solving equation (3.12) and then letting t ! 1. Proof. Let us seek a solution to problem (3.12) as a superposition of the eigenvectors un of the matrix A: Aun D n un ; .t / D

N X

an .t /un ;

fD

nD1

N X

fn un ;

nD1

where an .t / are the Fourier coefficients of the function .t /. Substituting these representations into (3.12) and taking into account that an .t / D .  un /, we obtain N  X dan .t / nD1

dt

 C n an .t /  fn un D 0:

Since the basis vectors un are linearly independent, we arrive at N constant-coefficient ordinary differential equations for an .t /: dan C n an D fn ; dt

n D 1; : : : ; N:

From the initial condition (3.12a) it follows that an .0/ D 0: Then the solution to (3.12) becomes an D C e  n t C an D

fn ; n

where C D 

 fn  1  e  n t ; n

D

fn n

from an .0/ D 0

N X  fn  1  e  n t un : n

nD1

134

Chapter 3 Methods for solving systems of algebraic equations

By letting t ! 1 and taking into account that the eigenvalues n of the positive definite matrix A are all positive, we arrive at the following stationary solution to equation (3.12): D

lim

t!1

N X fn un : n

nD1

On the other hand, the solution to (3.9) can also be represented as a superposition of the eigenvectors un of the matrix A. By expanding x and f in terms of un and substituting in (3.9), one obtains xD

N X

xn un ;

fD

nD1 N X

xn Aun D

nD1

N X

N X

fn un ;

nD1 N X

n xn un D

nD1

fn un ;

nD1

xn D

fn : n

Thus, the solution to system (3.12) in the limit as t ! 1 is reduced to the solution of equation (3.9). This enables one to infer that the solution to the difference equation that approximates the differential equation (3.12) will also converge to the solution to system (3.9). Equation (3.12) can be represented in the finite difference form nC1



n

n

CA

n

D f n;

n D 1; 2; : : : ; N:

(3.13)

The right-hand side does not have a superscript because the vector f is independent of t . The solution to (3.13) can be represented as the recurrence relation nC1

D .E  An /

n

C n f;

(3.14)

where E is the identity matrix, n is the iteration number of the number or the integration step in the parameter t . Formula (3.14) can be treated as an iterative representation of the solution to equation (3.9). Relation (3.14) involves the undetermined parameter n , which must be selected so at to ensure the convergence of the iterative process. It is clear that n must depend on the properties of A. Let us rewrite equation (3.14) in terms of the new variable  n – the solution discrepancy of equation (3.9) at the nth iteration: n D A

n

 f:

Then equation (3.14) can be represented as a single homogeneous equation for  n : n

D A1 . n C f/;

nC1



n

D A1 . nC1   n /:

Section 3.3 Iterative methods for linear system of equations

135

Substituting this in (3.13) yields  nC1 D .E  An / n : Expanding  n into eigenvectors of the matrix A and matching the coefficients of the linearly independent vectors uk , one obtains n D

N X

kn uk ;

 nC1 D .1  k n / n :

kD1

It is clear that for the steady-state process to be convergent at k D n , it is necessary and sufficient that max j1  k  j  1;

k D 1; : : : ; N:

k

If the spectral boundaries of the matrix A are 1  n .A/  N , then the parameter  must satisfy the inequality 2 : (3.15)  N In other words, if  is chosen this way, then the norm of the transition operator satisfies kQn k D kE  An k  1 and the operator itself is compressive (see Section 3.4). However,  can still be chosen from a wide range even though condition (3.15) is satisfied. The question arises: How to choose  so that the iterative process converges at the maximum rate?

3.3.4 Optimization of the rate of convergence of a steady-state process The rate of convergence of an iterative process is determined by the largest eigenvalue and the corresponding eigenvector uN . The parameter  can be chosen so as to suppress the component of the discrepancy vector  nN that corresponds to the largest eigenvalue: 1 1  N  D 0; D : (3.16) N Then the other components of  nk will decay in the iterative process as qkn



k D 1 N

n I

the rate of convergence is determined by max qk D 1  k

1 1 D 1 ; N C

136

Chapter 3 Methods for solving systems of algebraic equations

where C is the condition number of the matrix A. It is clear that the larger C , the slower is the convergence of the process of interest. In the general case, the convergence rate of an iterative process is determined by the norm of the transition matrix Q: q D max qk D kQk D max n k



kQ n k k nC1 k D max : n k k k n k n

If Q does not change between iterations, then q is also independent of n. If the step size n is dependent on n, then q is also dependent on n and, hence, one has to calculate the average value of q. N The transformation matrix with a variable step size n is given by P n .Qn / D .E  1 A/.E  2 A/ : : : .E  n A/; qN D lim kP n .Qn /k1=n D lim k.E  1 A/ : : : .E  n A/k1=n ; n!1

n!1

which implies that one should evaluate the asymptotic rate of convergence. It is convenient to use the exponential rate of convergence as a characteristic of an iterative process: S D  ln q: N If A is ill-conditioned, the average rate of convergence qN is little different from unity, qN  1  ˛, where ˛ is a small quantity and so S  ˛. The smaller S the slower is the convergence of the process. In formula (3.16) above,  was chosen in an optimal way; the choice of  can be improved by finding the optimal rate of convergence. Indeed, where does it follow from that the component corresponding to N should be suppressed, max j1   j? For a constant step n D  , the rate of convergence is determined by qN D limn!1 kP n . /k1=n D max j1   j, where 1   N and 0 <   1= N . Consequently, with variable  , the step size n should be selected so as to minimize this maximum with respect to within the above range. This is a problem of determining the minimax of P . /. It is required to find min max P . /, where P . / is a linear function on the interval 1   N with P .0/ D 1. A geometric solution of the problem is obvious (Figure 3.1). The straight line P . / must pass through the midpoint of the interval. Then the maximum value of jP . /j, attained at one of the endpoints of the interval, is maximum. It is apparent from the figure that if the straight line meets the -axis to the left of the midpoint (dashed line), the value of P . / at the right endpoint of the segment increases. If the straight line meets the axis to the right of the midpoint, the value at the left endpoint increases. Since max P . / is always attained at an endpoint, min max corresponds to the midpoint. From this condition, one finds  and, then, maxk qk .

137

Section 3.3 Iterative methods for linear system of equations P(λ) 1

(λ1 + λN)/2

λN λ

λ1

0

Figure 3.1. To the problem of geometrically determining the minimax of a linear function P . /.

Hence, 2 N C 1 D 0 H)  D ; 2 N C 1 2 k ; q D max jqk j; qk D 1  1 C N k

1

ˇ 1  C1 N  1 2 1 ˇˇ D D N C 1 ˇ N C 1 1 C C1       1 1 1 2 1 D 1 D1 CO : 1 CO C C C2 C C2

ˇ ˇ q D min max jP . /j D ˇˇ1  

For large C , the rate of convergence equals S D fast as with the first choice.

2 C

C O.C 2 /, which is twice as

3.3.5 Optimization of unsteady processes So far, the parameter  has been fixed between iterations. The question arises: Can we increase the rate of convergence by using a varying iteration step? An iterative process with parameter n dependent on the iteration number is called unsteady (nonstationary) as opposed to a steady-state (stationary) process for a fixed  . If n is varying, then n D

n Y

.E  i A/ 0 D P n .A/ 0 ;

iD1

where P n .A/ is a polynomial of degree n in the matrix A. Since Am un D m n un , where n and un eigenvalues and eigenvectors of A, by expanding  n and  0 in the

138

Chapter 3 Methods for solving systems of algebraic equations

eigenvectors of A, one obtains the following formulas for the Fourier coefficients:  nm D P n . m / 0m ; where n

P . m / D

n Y

.1  i m / :

(3.17)

iD1

Just as previously, one have to minimize the maxim of this polynomial to obtain the optimal rate of convergence: qN D min max jP n . /j; 

1   n :

This problem can easily be reduced to Tchebychev’s well-known problem of finding a polynomial P n of degree n with P n .0/ D 1 such that its maximum value on the interval Œ1; 1 is minimum. To reduce our problem to Tchebychev’s problem, it suffices to transform the independent variable as follows: yD

n C 1  2 : n  1

This transformation maps the interval Œ 1 ; n into ŒC1; 1 . It remains to normalize the solution so that P n .0/ D 1. Then

 1 2 Tn n C n  1

 ; (3.18) P n . / D

n C 1 Tn n  1 where Tn .y/ is Tchebychev’s polynomial of degree n: Tn .y/ D

1 2n1

cos.n arccos y/:

The roots of the polynomial Tn .y/ are given by yi D cos

.2i  1/ ; 2n

i D 1; : : : ; n:

Then the roots of P n . / are expressed as   1 .2i  1/ N C 1  . N  1 / cos : i D 2 2n

139

Section 3.3 Iterative methods for linear system of equations

Now i are easy to determine in terms of i using the representation (3.17):  n  Y 1 n 1 and hence i D ; P . / D i i iD1

i D

2 N C 1  . N  1 / cos .2i1/ 2n

;

i D 1; : : : ; n:

This solves the problem of finding a formula for calculating the step sizes of the optimal unsteady iterative process. Let us evaluate the asymptotic rate of convergence of this unsteady iterative process. The worst rate of convergence is at the value jmax P n . /j D qN n , attained by Tchebychev’s polynomials at the endpoints of the interval Œ1; C1 : Tn .˙1/ D .˙1/n . Therefore, from formula (3.18) we can obtain the maximum value P n . 1 / D Tn .1/=Tn .r/. Then qN n D

2 1 2 p p p D  ; jTn .r/j .r C r 2  1 /n C .r  r 2  1 /n .r C r 2  1 /n

where rD

n C 1 > 1: n  1

Expressing r in terms of the condition number C , we find that   1 C C1 1 2 D1C CO : rD C C2 1  C1 Then the exponential rate of convergence for an ill-conditioned matrix A will be given by the asymptotic estimate p   1 S D  ln qN D  ln 2 C ln r C r 2 C 1 n r

  2 2 2 1 1C 1 D  ln 2 C ln 1 C C n C C     1 2 2 1 D  ln 2 C ln 1 C C p Co n C C C with

    2 2 1 lim S D ln 1 C p : D p Co n!1 C C C

(3.19)

By comparing (3.19) with the formula S D C2 C O.C 2 / for the steady-state process, one can see that the convergence rate S of the unsteady iterative process is square root faster.

140

Chapter 3 Methods for solving systems of algebraic equations

Thus, the selection of i based on the roots of Tchebychev’s polynomials gives a massive acceleration of the iterative process. In practice, however, the implementation of a Tchebychev iterative process may not result in acceleration of convergence and, sometimes, can even lead to divergence. This may be caused by poor computational stability; this problem is resolved by a special selection of the parameters i . For more details and the selection algorithm, see [135]. The above results on the convergence acceleration of iterative processes where the discrepancy is calculated by the formula  nC1 D .E  k A1 / k are applicable to any simple iterative method, inclusive of the Jacobi method with A1 D D1 A and Seidel method with A1 D .L C D/1 A. In all above examples, the parameter i was independent of the previous approximations and was only dependent on the properties of the matrix A. Such iterative processes are linear. In nonlinear processes, i depends on previous approximations and is adjusted at each iteration, depending on the solution obtained. Nonlinear iterative processes can exceed the optimal linear process in the rate of convergence (e.g., see [39]). In conclusion, let us discuss the accuracy to which the iterative process should be carried out. Since it is assumed that the system of algebraic equations of interest is obtained by approximating differential equations with an error " D O.hk /, the iterations should be conducted with the same accuracy. In practice, this means that the iterative process should run until the inequality k nC1   n k  " is satisfied, where  is the discrepancy of the solution to equation (3.9). Further iterations do not make sense, since the error of approximation will prevail over the accuracy of the iterative process.

3.4 Methods for solving nonlinear equations 3.4.1 Nonlinear equations and iterative methods Consider a system of nonlinear equations written in vector form f.x1 ; : : : ; xn / D 0

or

f.x/ D 0;

(3.20)

where x D .x1 ; : : : ; xn / is the vector of unknowns and f.x/ is a given vector-valued function. In the special case where f D Ax C b, the system of equations (3.20) is linear. Nonlinear equations are solved by iterative methods that can be treated as a generalization of linear iterative methods discussed in the previous section. Formula (3.10)

141

Section 3.4 Methods for solving nonlinear equations

for a two-layer interactive process for a nonlinear vector equation becomes BkC1

xkC1  xk C f.xk / D 0; kC1

k D 0; 1; : : : ;

(3.21)

or, in the form solved for xkC1 , xkC1 D P.xk /; where P.x/ D x   B1 f.x/ is a nonlinear operator and B is an n  n invertible kC1 square matrix. For a steady-state process, B and  are independent of k. In general, the nonlinear equation (3.20) may not have a unique solution. Identifying a domain where equation (3.20) has a unique root is a separate, often difficult problem. We assume that this problems has been solved and so we aim at finding this root by an iterative method. We are also interested in determining the rate of convergence of the iterative process. An iterative method will be said to have a linear rate of convergence if n  x / or a quadratic rate of convergence if xn  x D xn x D O.x  O .xn  x /2 . Here x is the root of equation (3.20) of interest. Let us determine the rate of convergence of the iterative process. The method will be said to converge with a rate of order m if .xkC1  x /  O..xk  x /m /. The method is linearly convergent if .xkC1  x /  O..xk  x // and quadratically convergent if .xk  x / D O .xk  x /2 . Here x is a root of equation (3.20) and .x k ; x  / is the distance the kth iteration solution and exact solution.

3.4.2 Contractive mappings. The fixed point theorem So P.x/ is a nonlinear operator that maps a n-dimensional Euclidean space En into itself. Any point x 2 En satisfying the condition P.x/ D x

(3.22)

is called a stationary point of the operator P. Solution (3.22) is a solution to the problem of a stationary? point. A solution to any nonlinear equation f.x/ D 0 can be represented as a solution to the problem of a stationary point x D x C f.x/ D P.x/: To find stationary points, let us apply the method of successive approximations. Let x0 be a test solution and x1 D P.x0 /, where x1 can be taken as a refinement of x0 . The refinement process can be continued to get xkC1 D P.xk /;

k D 1; 2; : : : :

Introducing the powers of P, one can write xkC1 D PkC1 .x0 /;

  Pk .x0 / D P : : : P.: : : P.x0 / : : :/ : : : : ƒ‚ … „ k

(3.23)

142

Chapter 3 Methods for solving systems of algebraic equations

Definition 3.1. An operator P that maps space En into itself is said to be compressive in a closed ball R.x0 ; r/ D .x; kx  x0 k  r/ if for any x and y 2 R.x0 ; r/, the Lipschitz condition kP.x/  P.y/k  kx  yk

(3.24)

holds or if kP.x/k  kxk for y D 0, where < 1 is the compression coefficient. Theorem 3.1. 1) Let P.x/ be a compressive operator in R.x0 ; r/ with coefficient < 1 and 2) let x0 satisfy the condition kP.x0 /  x0 k < r0 .1  / < r: Then the sequence ¹xr º converges to x 2 R, and x is the only stationary point in R.x0 ; r/. For the zeroth approximation, condition (3.22) guarantees that all approximations stay within R.x0 ; r/: kxr0C1  x0 k D kPr C1 .x0 /  x0 k D kPr C1 .x0 /  Pr .x0 / C Pr .x0 /     C P.x0 /  x0 k  kPr C1 .x0 /  Pr .x0 /k C    C kP.x0 /  x0 k  r .1  /r0 C    C .1  /r0 D .1  r C1 /r0 < r0 ; where each difference has been transformed using the formula kPr C1 .x0 /  Pr .x0 /k D kPr .P.x0 /  x0 /k  kPr ..1  /r0 / k  r .1  /r0 : The proof of convergence is based on a convergence test for the Cauchy sequence ¹xn º. For any m and n > 0 we have kxm  xmCn k D k.xm  xmC1 C    C .xmCn1  xmCn k  kxmC1  xm k C    C kxmCn  xmCn1 k  m .1  /r0 C    C mCn1 .1  /r0  m .1  n /r0  m r0 : For any " > 0 there exists an N."/ such that for any m > N."/ and n > 0 the condition kxm  xmCn k < " holds. Indeed, let us choose N so that N < "=r0 ; then kx m  x mCn k < ", whence follows the convergence of the sequence ¹xn º to its limit x .

143

Section 3.4 Methods for solving nonlinear equations

Proof of uniqueness. Suppose the solution is no unique, so that there are two distinct solutions, x and x : x D P.x /;

x D P.x /;

kx  x k D kP.x /  P.x /k  kx  x k;

0 < < 1:

It follows that kx  x k D 0. Suppose that x D limm!1 Pm .x0 /. Then the point x will be said to be attainable from x0 and the set of all points ¹x0 º from where x is attainable will be called the domain of attainability. It follows from the theorem that x is attainable from any point of the ball R.x0 ; r/. Indeed, we have kP.x/  P.x0 /k  kx  x0 k  r0 ;

kP.x/  x1 k  r0 ;

which means that P maps R.x0 ; r0 / into R.1/ .x1 ; r0 /, P2 W R ! R.2/ ! R.2/ .x2 ; 2 r0 /, etc.; so the radius of the ball R.n/ decreases with the iteration number and ¹xn º ! x .

3.4.3 Method of simple iterations. Sufficient convergence condition Let us show that for the Lipschitz condition to be satisfied it suffices that the condition kP0x .x/k  =n < 1 holds, where n is the dimensionality of x; hence, the norm of the Jacobian matrix must be less than =n ( < 1). Here the Jacobian matrix of the function P.x/ is denoted by 0 1 @Pn @P1  B @x1 @x1 C B C @P i : 0 : B ::: C : : DB : Px D : C @xj @ @P1 @Pn A  @xn @xn and the norm is understood as the maximum norm of a vector or matrix: kxk D max jxi j; i

kAk D max jaij j: ij

The proof follows from expanding the functions appearing in the Lipschitz condition in Taylor series at the point x0 :  n  X @Pi 0 .xj  xj0 / C O.xj  xj0 /2 ; Pi .xj / D Pi .xj / C @xj x0 j D1  n  X @Pi .yj  xj0 / C O.yj  xj0 /2 ; Pi .yj / D Pi .xj0 / C @xj x0 j D1

144

Chapter 3 Methods for solving systems of algebraic equations

whence ˇ ˇ n ˇ 0 ˇ X ˇ @Pi .xj / ˇ jPi .xj /  Pi .yj /j  ˇ ˇ jxj  yj j: ˇ @xj ˇ j D1

By rewriting this inequality in terms of the maximum norm, one obtains n   X @Pi kPi .xj /  Pi .yi /k  max .xj  yj / @x j

j D1

 kx  yk

n X j D1

ˇ ˇ ˇ @Pi .i0 / ˇ ˇ  kx  yk: ˇ maxˇ @xj ˇ

Then it is clear that for the Lipschitz condition to be satisfied it suffices that ˇ ˇ ˇ @P . 0 / ˇ ˇ i i ˇ where i0 2 R.x0 ; r0 /; < 1: max ˇ ˇ ; ˇ n i;j ˇ @xj At n D 1, the condition ˇ ˇ ˇ dP ˇ ˇ ˇ ˇ dx ˇ < 1 must hold. Figures 3.2a and 3.2b illustrate the geometric interpretation (in the one-dimensional case, with n D 1) of the method of simple iterations. It is apparent from the figures that the process converges for jP 0 .x/j  1; furthermore, the convergence is sign z

z

x0 x2 x x3 x1 *

(a)

x

x x3 x2 x1 x0 x x0 *

*

(b)

Figure 3.2. Geometric interpretation for finding a fixed point of P .x/ D x.

x

Section 3.5 Nonlinear equations: Newton’s method and its modifications

145

alternate and monotonic if P 0 .x/ < 0 (Figure 3.2a). The approximations satisfy the inequalities x0  x  < 0;

x1  x  > 0;

x2  x  < 0;

etc.

The quantity xn changes its sing after each iteration. If P 0 .x/ > 0 near the root x1 in Figure 3.2b, the convergence does not change sign: xn > 0. If kP 0 .x/k > 1 near the larger root x2 , the process diverges (Figure 3.2b).

3.5 Nonlinear equations: Newton’s method and its modifications 3.5.1 Newton’s method Apart from the method of simple iterations, which is linearly convergent, a family of more accurate iterative methods can be constructed, which have a faster, quadratic rate of convergence. In formula (3.21), let us set  D 1 and take the matrix B to be the Jacobian matrix BDJD

@.f1 ; : : : ; fn / @.x1 ; : : : ; xn /

to arrive at Newton’s iterative method (also known as the Newton–Raphson method): xkC1 D xk  Œf 0 .xk / 1 f.xk /:

(3.25)

In the one-dimensional case, J D

@f D fx ; @x

J 1 D

1 : fx

Then x kC1 D x k 

f .x k / ; fx .x k /

x kC1 D 

f .x k / : fx .x k /

Equation (3.25) can be treated as x D P.x/ D x  J1 .x/f.x/: The operator P.x/ can be proven to be compressive. The following theorem holds. Theorem 3.2. If the following conditions hold: 1) the matrix .J0 /1 is nonsingular, det.J0 /1 ¤ 0, in the ball R.x0 ; r/, k.J0 /1 k  B

with B > 0;

146 2)

Chapter 3 Methods for solving systems of algebraic equations

2 @ f @x @x i

3)

j

 C;

C > 0;

kx1  x0 k < r0 ;

i; j D 1; : : : ; n; r0 > B;

then the iterative process is convergent. Suppose x0 is an initial approximation and x is a solution to the system of nonlinear equations f.x/ D 0. By expanding f.x/ in a Taylor series at x D x0 , one obtains   f.x/ D f.x0 / C J.x0 /.x  x0 / C O .x  x0 /2 D 0; whence, for small x D x  x0 , one finds, keeping only the linear term, that x D x0  .J0 /1 .x0 / f.x0 /: By applying this formula to the above xk , one arrives at the recurrence relation xkC1 D xk  J1 .xk / f.xk /: kC1

The process runs until the condition kx

(3.26)

k  " is satisfied.

Proof of the quadratic convergence of the Newton–Raphson method. Let us rewrite (3.26) in the form xkC1 D '.xk /;

where '.x/ D x  .fx /1 f.x/:

Using the formula for the differentiation of the inverse of a matrix, .A1 /x D A1 Ax A1 , we obtain '0 .x/ D .fx /1 fxx .fx /1 f.x/;

'0 .x / D 0:

The notation fx or f 0 .x/ stands for the matrix obtained by differentiating the vector function f with respect to the vector x, with fxx D Hij D

@f.x/ @xi @xj

denoting the Hessian matrix of the function f.x/ [129]. In addition, ˇ '00 .x / D .fx /1 fxx ˇ  : xDx

Then xkC1 D xkC1  x D '.xk /  x D '.x / C '0 .x /xk C '00 .x /.xk /2  x   D ' 00 .x /.xk /2 C O .xk /3 :

(3.27)

It follows that the convergence rate of the Newton–Raphson method is quadratic, provided that the conditions of the above theorem are satisfied in a small neighborhood of x .

Section 3.5 Nonlinear equations: Newton’s method and its modifications

147

3.5.2 Modified Newton–Raphson method A disadvantage of the Newton–Raphson method is that one has to calculate and invert the Jacobian matrix at each step. Furthermore, the original system of equations (3.20), which can be quite complicated, has to be differentiated to obtain the Jacobian matrix. An improvement of the Newton–Raphson method would be a minimization of the computations and an extension of the neighborhood of the root where an initial approximation can be specified. The simplest modification of the Newton–Raphson method is the case where A D J1 .x0 / is independent of k. Here the interpolation method is stationary and the rate of convergence is linear rather than quadratic, as with A D J1 .xk /. Figures 3.3 and 3.4 provide a geometric interpretation of the convergence conditions for the Newton–Raphson method. In Figure 3.3, the conditions of the theorem are satisfied, while in Figure 3.4a, the first condition is violated, f 0 .x/ ¤ 0, and hence the successive approximations are divergent. In Figure 3.4b, the first condition is satisfied, f 0 .x/ ¤ 0, by the second condition is violated: in an neighborhood of the root x , the second derivative fxx changes its sign and the iterative process diverges. It is apparent from Figure 3.3 that in the modified method, with J1 .x0 / D Œf.x0 / 1 fixed, the process converges much more slowly (dashed line) than in the method with varying tangent. f(x)

x

*

x2 x’2

x1

x0

x

Figure 3.3. Geometric interpretation of finding a root of an equation (convergent process) by the Newton–Raphson (solid line) and modified Newton–Raphson method (dashed line).

3.5.3 The secant method Another modification of the Newton–Raphson method, the secant method, is obtained by replacing the inverse of the derivative in (3.25) with a finite difference: xkC1 D xk 

xk  xk1 f.xk /; f.xk /  f.xk1 /

k D 0; 1; : : : :

(3.28)

148

Chapter 3 Methods for solving systems of algebraic equations

f(x) y

x3

x0 x

x1 x

*

x1

x0

x2

*

x2 x

x

(a)

(b)

Figure 3.4. Geometric interpretation of finding a root of an equation by the Newton–Raphson method (divergent process).

This is a two-step method. To obtain the approximation xkC1 , one must use two previous approximation, xk and xk1 . In order to start the iterative process at the first step, one should find x1 with a one-step iterative process using, for example, the modified Newton–Raphson method. Figure 3.5 illustrate the convergence of the secant iterative method. This method converges more slowly than the Newton–Raphson method but faster than the modified Newton–Raphson method.

x

*

x3 x2

x1

x0

x

Figure 3.5. Geometric interpretation of finding a root of an equation by the secant method.

3.5.4 Two-stage iterative methods The Newton–Raphson method is based on the linearization of a nonlinear system of equation. It reduces the solution of the nonlinear vector equation to a multiple solution of linear equations, or to the inversion of the matrix of the linearized equations. The linear problem can be solved, in turn, by iterative methods by using, for example, the methods of accelerated convergence discussed above in Section 3.3. Then the

Section 3.5 Nonlinear equations: Newton’s method and its modifications

149

complete iterative solution cycle for the nonlinear equation consists of two stages involving an external cycle (implies the application of the Newton–Raphson method to the original system of equations, thus reducing the solution of the nonlinear problem to the solution of a nonlinear one) and an internal cycle, where the linear problem is solved using, for example, the nonstationary Seidel method with acceleration. To solve nonlinear equations, one can directly apply a generalization of standard linear iterative methods to the nonlinear case. For example, the nonlinear Seidel method has the form k k ; : : : ; xm / D 0; fi .x1kC1 ; x2kC1 ; : : : ; xikC1 ; xiC1

i D 1; : : : ; m:

Then k / D 0; f1 .x1kC1 ; x2k ; : : : ; xm

k f2 .x1kC1 ; x2kC1 ; x3k ; : : : ; xm / D 0; : : : ;

where m is the number of equations. At each stage, one solve a nonlinear equation in one unknown (m D 1): for x1kC1 at the first state and x2kC1 at the second state. These m equations, each containing a single unknown, are solved by the Newton–Raphson method, thus reducing the problem to a linear one. In this case, one has to deal with a two-stage method as well, where the external iterations are carried out with the nonlinear Seidel method and the internal iterations are preformed using the Newton–Raphson method for a system of functions dependent on a single variable. It is clear that other two-stage methods are possible, involving different combinations of external and internal iterative processes.

3.5.5 Nonstationary Newton method. Optimal step selection Suppose one has to solve the equation f.x/ D 0:

(3.29)

Let us consider a family of Newton’s equations with parameter  and optimize the process of successive approximations in this parameter. Solving equation (3.29) by the Newton–Raphson iterative method with parameter  can be treated as solving the difference equation xkC1 D  fx1 .xk / f.xk /;

xkC1  xk D fx1 .xk / f.xk /: 

(3.30)

A continuous analogue of the Newton–Raphson method suggests that instead of (3.26), one uses equation (3.30) with  ! 0. This matrix differential equation is easy to in-

150

Chapter 3 Methods for solving systems of algebraic equations

tegrate @x D fx1 .x/ f.x/; ln f.x/ D t C C; @t 0 t D t 0; f.x/ D f.x0 / e t=t ; x D x0 ; @.f1 ; : : : ; fm / @f D : fx D J D @x @.x1 ; : : : ; xm /

(3.31)

One can readily see that solution (3.31) tens to solution (3.29) as t ! 1 for any x0 : lim f.x/ D 0;

t!1

lim x.t / D x :

t!1

This solves the convergence of the process as  ! 0 but not for a finite  . Although the solution to equation (3.29) is theoretically attained as t ! 1, the function f.x/ decreases exponentially and f.x/  0 already for a finite t . In practice, the discrete analogue of equation (3.30) is realized: xkC1  xk D  k fx1 .xk / f.xk / D  k a.xk / or xkC1 D xk   k a.xk /:

(3.32)

This formula represents the nonstationary Newton method. With  k D 1, one obtains the classical stationary Newton method. The process (3.32) was proved to converge to (3.31) as  k ! 0 on a practically finite interval if fx is invertible in the neighborhood kx  x k  kx0  x k, so that jfx j ¤ 0. For practical computations, it is necessary to complete (3.32) by an algorithm of optimal selection of  k at each iteration. From the principle of nondecreasing discrepancy in an iterative process it follows that  k must decrease proportionally to the discrepancy, and hence  k should be taken so as to satisfy the condition  k D  kC1

ık ı k1

;

(3.33)

where ı k D kf.xk /k D .f.xk /; f.xk //1=2 is the norm of the discrepancy. It follows from (3.33) that the step size  k decreases with decreasing discrepancy. In order to avoid the step size to become too small, there must be a condition restricting the step size from below: 0 < ‚   k  1. There is no rigorous proof that the algorithm (3.32)–(3.33) converges; however, numerical computations indicate the convergence is faster that of the stationary Newton method. As  k decreases, the domain of convergence expands, while the rate of convergence falls. An approximate formula can be suggested for refining  k . It is natural to consider the step  k to be optimal for which the corresponding discrepancy is minimum along

Section 3.5 Nonlinear equations: Newton’s method and its modifications

151

a selected direction. The direction of the increment vector xk at the kth step is determined by the vector a calculated by formula (3.32); the square of the discrepancy along this direction is equal to     ı kC1 . / D f.xkC1 /; f.xkC1 / D f.xk   k a.xk //; f.xk   k a.xk // : (3.34) A rigorous determination of  k minimizing ı kC1 is a very difficult problem. However, this problem can be solved approximately by replacing the actual function ı k . / with a parabola by expanding the function in a Taylor series at  D 0 and assuming that ı k .0/ is the discrepancy obtained at the previous iteration. This is achieved by using formula (3.34): ˇ ˇ 2 kˇ d ı k ˇˇ k k 2d ı ˇ C ; ı k .0/ D Œf.xk / 2 ; (3.35) ı . / D ı .0/ C  d  ˇD0 d  2 ˇD0 where d ık D 2f.xk   ak / fx .xk   ak /ak D 2f.xk   ak / f.xk /; d and hence

ˇ

 d ı k ˇˇ k k D 2 f.x /; f.x / D 2ı k .0/: d  ˇD0

(3.36)

The coefficient of  2 can be determined from the value of the discrepancy at  D 1 by formula (3.35): ı k .1/ D ı k .0/  2ı k .0/ C .ı 0 /00 : Expressing .ı 0 /00 and substituting into (3.35), ı k . / D ı k .0/  2ı k .0/ C Œı k .1/ C ı k .0/  2 ; one arrives at the value of  at which the discrepancy ı k . / is minimum: ı 0k . / D 2ı k .0/ C 2 Œı k .1/ C ı k .0/ D 0; and hence  kC1 D

ı k .0/ : ı k .0/ C ı k .1/

(3.37)

In this case, the condition 0   k  1 always holds. Thus, ı k should be evaluated at each step using formula (3.34) for  D 0 and  k D 1. This formula has the flaw that situations are possible where  k is too small far away from the root. If  k is small, the convergence rate is low far away from the root, and hence the step size should be limited from below by a quantity ‚: " # k .0/ ı :  kC1 D max ‚; k ı .0/ C ı k .1/

152

Chapter 3 Methods for solving systems of algebraic equations

Proof of convergence. In a neighborhood of a simple root x D x , choosing the step size  kC1 according to formula (3.37) results in a quadratic convergence, just as in the Newton–Raphson method. Expanding (3.37) in powers of ı k .1/=ı k .0/ yields  k 2 ! ı k .1/ ı .1/ kC1 CO D 1 k D 1  O.xk  x /2 ;  ı .0/ ı k .0/ since ı k .1/ is the square of the discrepancy at the next iteration with respect to ı k .0/ of the stationary Newton–Raphson method, where it was shown that ı k .1/  Œı k .0/ 2 . Then the rate of convergence of the unsteady Newton process (3.37) is quadratic, just as in the steady process:  O.xk  x / fx1 .xk / fx .xk /; xkC1 D xk  ı nC1 fx1 .xk / fx .xk / D xkC1 N where xkC1 is the value obtained at the .k C 1/st iteration by the classical Newton N method.

3.6 Methods of minimization of functions (descent methods) 3.6.1 The coordinate descent method The previous section considered iterative processes for solving nonlinear equations. Now let us discuss iterative methods where approximations are constructed in a somewhat arbitrary manner. These include a family of methods for minimizing functions of the form F .x/ D f T .x/ f.x/, which arise from solving many problems where one has to minimize functionals. These methods are collectively called descent methods if, at each subsequent iteration, the condition F .xkC1 /  F .xk / holds, where F .x/ is a scalar function dependent on a vector argument. The direction of the descent is defined by a vector Dk and the magnitude of the correction xk is determined by a scalar k : xkC1 D xk C k Dk :

(3.38)

The simplest way to perform the minimization is to choose the vector Dk arbitrarily; for example, it can be directed along one of the coordinates xj , Dk D ejk D ¹ 0 : : : 0 1 0 : : : 0 º; 1 ::: j ::: n

j D 1; : : : ; n;

(3.39)

Section 3.6 Methods of minimization of functions (descent methods)

153

where n is the dimensionality of the vector x or space En ; the scalar k is determined from an approximate condition of fastest descent at the point xk . This method is known as the coordinate descent method. The function F .x/ is replaced with a quadratic function in accordance with the expansion in powers of k :     1 n 2 k T @2 F k k k k k @F k F .x C D / D F .x / C D C . / ŒD Dk : (3.40) k k k 2 @x @xi @xj If we adopt the assumption that F .x/ is a convex function in a neighborhood of xk , which means that F .x / D F .˛X C .1  ˛/Y/  ˛F .X/ C .1  ˛/F .Y/; 0  ˛  1; X  x  Y; then the maximum step size will be determined from the condition ˇ   ˇ @F @F .xk C k Dk / k k k T @F ˇ Dk D 0: D D C ŒD k k @xi @xj ˇxk @ @x

(3.41)

Then, in view that Dk is defined by (3.39), we obtain the j th component of grad F ,   @F k gjk D D ; @xk j and the diagonal element Fjjk of the Hessian matrix: ˇ ˇ k T @F ˇ Dk D @F D F k : ŒD jj @xi @xj ˇx k @xj @xj Hence, k D 

gjk Fjjk

:

It is clear that the arbitrariness in choosing the direction of Dk can be used to increase the rate of descent k . To this end, one should take a j such that gjk is maximum amongst the n quantities, 1  j  n, where n is the dimensionality of x or the space En : ˇ ˇ @F ˇ @F ˇ ˇ ˇ: @x D max j ˇ @xj ˇ j This is the simplest way of taking advantage of the above arbitrariness in the coordinate descent method, also known as the coordinate relaxation method. Other modifications of the method are possible.

154

Chapter 3 Methods for solving systems of algebraic equations

3.6.2 The steepest descent method The descent can be accelerated by choosing the decent direction along the gradient @F=@xk rather than a coordinate, Dk D 

@F D Gk ; @xk

(3.42)

and by determining k as before, from condition (3.41). Then one obtains ˇ Gk Gk @F ˇˇ k D k T ; where H D is the Hessian matrix: @xi @xj ˇxk ŒG HGk

(3.43)

If F .x/ is a quadratic function of the form F .x/ D A C 2bT x C xT Hx; where b is a vector and H is a symmetric positive definite matrix (H D HT with xT Hx > 0 for any x) then minimizing F .x/ is reduced to solving a linear system of equations with the matrix H. Then it can be rigorously proven that there exist a unique solution, and hence that the iterative process is convergent. The iterative process (3.38), (3.42), (3.43) is called the steepest descent method for solving the system of linear equations Hx D b arising from taking F .x/ to be a quadratic function. Figure 3.6 illustrates successive approximations of the steepest descent method and depicts level lines of F .x/. If F .x/ is not quadratic, the process can converge under the condition that there is an initial point x0 , very close to x , such that the quadratic terms in the expansion of F significantly dominate the higher-order terms and the Hessian matrix is positive definite in the vicinity of x . x0

x1 x2 x

*

Figure 3.6. Geometric interpretation of the steepest descent method.

Section 3.6 Methods of minimization of functions (descent methods)

155

3.6.3 The conjugate gradient method The conjugate gradient method is intended for solving systems of linear algebraic equations with a symmetric positive definite matrix. This methods enables one to minimize a quadratic function F .x/ in n unknowns in k iterations, k  n. Let us represent the quadratic function in terms of xx , where x is the point at which F .x/ attains its minimum: 1 F .x/ D F0 C .x  x /T H.x  x /: 2

(3.44)

The gradient of F .x/ is expressed as G.x/ D rF D H.x  x /:

(3.45)

Suppose that the vectors D0 ; D1 ; : : : ; Dk in formula (3.38) are linearly independent and form a basis. Denote am D m Dm . Then the mth iteration can be expressed via the previous one as xm D xm1 C am1 D    D xmq C amq C    C am1 : For the kth iteration, we get k1 X

xk D xmC1 C

aq ;

k  m C 2:

qDmC1

In addition,  k1  X aq  Hx : Gk D H.xk  x / D HxmC1 C H qDmC1

Since Hx D 0 and HxmC1 D GmC1 , then the kth iteration for Gk can be written as a decomposition in the basis vectors Dq : k1 X

Gk D GmC1 C

q HDq ;

m D 1; 0; : : : ; k  2;

qDmC1

Gk D G0 C

k1 X

(3.46)

q HDq :

qD0

The contraction of (3.46) with Dq gives k T

k

ŒG G D ŒG

mC1 T

G

mC1

C

k1 X qDmC1

q ŒDq T HDq :

(3.47)

156

Chapter 3 Methods for solving systems of algebraic equations

Since the Dq are linearly independent vectors, they can be chosen so that they are H-conjugate or H-orthogonal. Then ŒDq T HDm D 0;

q ¤ m:

(3.48)

It follows from (3.47) that ŒGk T Dm D 0:

(3.49)

Since D0 ; : : : ; Dn form a basis, then relation (3.49) implies that Gk  0: Furthermore, by virtue of (3.45) and the fact that H is positive definite, we find that xk D x ; so the iterative process converges to the minimum point of F .x/ in k iterations. An H-conjugate basis that satisfies conditions (3.48) is easy to obtain from the original basis D0 ; : : : ; Dk in the following manner: D0 D G0 D rF .x0 /; DmC1 D Gm C ˇ m Dm :

(3.50)

The constants ˇ m are chosen so as to satisfy the condition (3.48) that the basis is H-conjugate. Using (3.50), we get ŒDmC1 T HDm D ŒGm T HDm C ˇ m ŒDm T HDm D 0; and hence ˇm D

ŒGm T HDm : ŒDm T HDm

In order to calculate xm , the step size m should be evaluated from (3.43). If F .x/ is not a quadratic function but the initial approximation is sufficiently good, the convergence is also achieved, which may require more than n iterations.

3.6.4 An iterative method using spectral-equivalent operators or reconditioning As pointed out in Section 3.3, one often introduces an additional matrix parameter B into an iterative process and chooses it so at to accelerate the convergence. Let us illustrate this with an example based on the simple iteration method: BxkC1 D Bxk   .Axk  b/:

(3.51)

The matrix B must be easy to invert. Then, by multiplying (3.51) by B1 , one obtains xkC1 D xk   B1 .Axk  b/ D .E   B1 A/xk   B1 b;

(3.52)

157

Section 3.7 Exercises

where E is the identity matrix. The iterative process (3.51) represents the simple iteration algorithm with the matrix E   B1 A instead of A. Suppose the eigenvalues i of the positive definite matrix A are in the range  i  M with the condition number M= 1. Then the iterative method (3.52) is known to converge slowly for B D E. Can we find a positive definite matrix B such that the process (3.52) converges faster? Denote M1 D sup x

.Ax; x/ ; .Bx; x/

1 D inf x

.Ax; x/ : .Bx; x/

If B D E, we have M1 D M and 1 D . It can be shown [4] that if B is chosen so that M M1 ; (3.53)

1

then the iterative process with the reconditioning matrix B converges significantly faster. The iteration convergence factor q2 is expressed as q2 D

M1  1 1  1 =M1 D : M1 C 1 1 C 1 =M1

(3.54)

By virtue of inequality (3.53), we have q2 < q1 , where q1 is the convergence factor in the iterative process with B D E. Consequently, the introduction of the reconditioning matrix B accelerates the simple iteration method. Reconditioning can also be used in the other iterative processes considered above in a similar manner. It is noteworthy that, in some cases, the matrix B turns out to be fairly easy to choose. For example, one can take B to be the matrix consisting of the diagonal elements of A. This often suffices for the improvement of the convergence of the iterative process.

3.7 Exercises 1. Construct examples of ill-conditioned and well-conditioned systems of linear algebraic equations in two and three unknowns. Analyze their conditionality and carry out regularization for the ill-conditioned systems. 2. Prove that the condition det A D " 1 is insufficient for the matrix A to be ill-conditioned and, conversely, the condition det A  1 is insufficient for A to be well-conditioned. Give counterexamples. 3. Prove that the condition number of a positive definite matrix A D BT B equals the square of that of the matrix B. Use the definition of the norm of A.

158

Chapter 3 Methods for solving systems of algebraic equations

4. Prove that if A D BT B is a semi-positive definite matrix, then the matrix BT AB is also semi-positive definite. 5. Analyze the convergence of the Seidel method with the matrix A having a diagonal prevalence in columns: qjaij j >

n X

aij ;

i D 1; : : : ; n;

q < 1:

i¤j D1

6. Solve the system of equations Ax D b with 0 1 1 0:5 0 A D @0:5 1 0:4A ; 0 0:4 2

0 1 2 b D @1A 1

using Cholesky decomposition. 7. A bar made of a nonlinear elastic material with the stress-strain diagram  D E." C "0 /1=2 is subjected to the tensile stress  D 10 E"0 . Can the tensile strain be determined using: 1) the simple iteration method; 2) the Newton–Raphson method? 8. By the secant method, determine the tensile strain of a bar subjected to the tensile stress  D 18 if the material stress-strain diagram in dimensionless variables is given by ´ "; "  1I D 2  1="; " > 1: Take ".0/ D 1 as the initial approximation. Compute five iterations. 9. By the simple iteration method ukC1 D uk C  .Axk  f/; solve the system of two equations (Ax D f) x1 C x2 D 1; x1 C .1 C "/x2 D 1;

" 1:

Show that the system is ill-conditioned and determine the parameter  at which the convergence rate of the iterative process is optimal. Verify this with a numerical analysis for the initial approximation x1.0/ D 0:8, x2.0/ D 0 with " D 0:1. Compare with the exact solution.

159

Section 3.7 Exercises

10. Solve the nonlinear system of equations x1 C x2 D 5; x1 C x2 C 0:01.x2 /2 D 5 by the simple iteration method. Determine the value of the parameter  at which the iterative process is convergent. Find the number of iterations required to obtain .0/ .0/ a solution accurate to 104 with the initial approximation x1 D 0:9 and x2 D 0. 11. Solve the nonlinear system of equations .x1 /2 C .x2 /2 D 1; x1 C x2 D 0:1 by the secant method and the modified Newton method in the half-plane x1 > 0. Compute the first five successive approximations with the initial approximation p x1.0/ D x2.0/ D 2=2. Evaluate the accuracy of the resulting solution. Compare the solutions obtained by the two methods with the exact solution. 12. Consider a nonlinear elastoviscous bar with the following constitutive equation in dimensionless variables: @" @ D  ˛. /n : @t @t Solve the problem of stress relaxation by the implicit Euler method on the time interval t 2 Œ0; 3 with the initial condition  j tD0 D 1. Solve the arising system of algebraic equations using the Newton–Raphson method by partitioning the time interval into l D 3; 5; 7 subintervals with ˛ D 10, n D 5, and tmax D 3.

Chapter 4

Methods for solving boundary value problems for systems of differential equations 4.1 Numerical solution of two-point boundary value problems. Stable and unstable algorithms Chapter 2 discussed the solution of initial value (Cauchy) problems for ordinary differential and partial differential equations. However, continuum mechanics mostly deals with boundary and initial-boundary value problems. This section discusses the solution of boundary value problems. First, consider the simplest problem for a second-order ordinary differential equation.

4.1.1 Stiff two-point boundary value problem Numerical solution of a two-point boundary value problem for a second-order ordinary differential equation is associated with certain difficulties, although analytical solution may be fairly straightforward. Even though the original boundary value problem is well-conditioned, not every solution algorithm has the property of being well-conditioned and stable with respect to small perturbations. This may lead to a rapid accumulation of rounding errors and inadequate solution of the problem. First, consider the solution of a boundary value problem for a constant coefficient differential equation with a large parameter a2 1. Suppose it is required to solve the following problem on the interval x 2 Œ0; 1 : y 00  a2 y D 0;

y.0/ D Y0 ;

y.1/ D Y1 :

(4.1)

The general solution to the equation is given by y D C1 cosh ax C C2 sinh ax with y 0 D a.C1 sinh ax C C2 cosh ax/: By satisfying the boundary conditions of (4.1), one obtains a system of two equations for determining the arbitrary constants C1 and C2 , thus arriving at the solution to problem (4.1) in the form y.x/ D Y0 cosh ax C

Y1  Y0 cosh a sinh ax: sinh a

161

Section 4.1 Numerical solution of two-point boundary value problems

By collecting the coefficients of Y0 and Y1 , one can rewrite this solution as y.x/ D Y0

sinhŒa.1  x/ sinh ax C Y1 D A.x/Y0 C B.x/Y1 : sinh a sinh a

(4.2)

Let us analyze its behavior at large values of the parameter a 1. Find A.x/ and B.x/ as a ! 1: sinhŒa.1  x/ D 0; a!1 sinh a sinh ax D 0; B.x/ D lim a!1 sinh a A.x/ D lim

x 2 .0; 1 ; A.0/ D 1I x 2 Œ0; 1/; B.1/ D 1:

It is apparent that A.x/ and B.x/ are discontinuous at the endpoints x D 0 and x D 1 as a ! 1. Figure 4.1a illustrates the solution to the boundary value problem (4.1) for a 1. The coefficients A.x/ and B.x/ of Y0 and Y1 in (4.2) are bounded as a ! 1 for any x 2 Œ0; 1 , and hence the solution to problem (4.1) is stable. A small perturbation in Y0 or Y1 leads to a small deviation in the solution y.x/; the error does not increase even though the solution changes rapidly near the endpoints and forms boundary layers or edge effects (Figure 4.1a). This phenomenon also arises in more complicated systems of equations with small parameters as coefficients of the highest derivatives. y

y

Y1

Y0

Y2

Y1

0

1

x

(a)

0

1

x

(b)

Figure 4.1. (a) Edge effects at the endpoints of Œ0; 1 for a 1. (b) Two linearly independent solutions to a Cauchy problem that lead to an ill-conditioned system of equations.

4.1.2 Method of initial parameters To solve the boundary value problem (4.1) numerically, one can take advantage of the method of initial parameters. The method involves searching for two linearly

162

Chapter 4 Methods for solving boundary value problems for systems of equations

independent particular solutions each satisfying two conditions at the left endpoint: 1/ y1 .0/ D 1;

y10 .0/ D 0

H)

2/ y2 .0/ D 0;

y20 .0/ D 1

H)

y1 D cosh axI 1 y2 D sinh ax: a

(4.3)

Then, by using a linear combination of the two solutions, one satisfies the boundary conditions (4.1) at both the left and right endpoints: yD

2 X iD1

Ci yi ;

y.0/ D C1 y1 .0/ C C2 y2 .0/ D Y0 ; y.1/ D C1 y1 .1/ C C2 y2 .1/ D Y1 :

(4.4)

The resulting system of equations serves to determine the arbitrary constants C1 and C2 : C1 D

Y0 y2 .1/  Y1 y2 .0/ ; y1 .0/y2 .1/  y1 .1/y2 .0/

C2 D

Y0 y1 .1/  Y1 y1 .0/ ; y1 .0/y2 .1/  y1 .1/y2 .0/

lim C1 D

1 D Y0 I 1

lim C2 D

1 D Y1 : 1

a!1

a!1

Since y1 .1/ ! 1 and y2 .1/ ! 1 as a ! 1 (Figure 4.1b), none but absolutely precise calculations can satisfy the boundary conditions as a!1: lima!1 C1 D Y0 and lima!1 C2 D Y1 . With any approximate calculations, an arbitrarily small deviation ıy1 .0/ or ıy2 .0/ can result in an arbitrarily large error, since one has to obtain finite values C1  O.1/ and C2  O.1/ using linear combinations of infinitely large numbers. A solution to system (4.4) is constructed in terms differences of the solutions y1 and y2 at x D 1. However, y1 and y2 are both large numbers as a 1. Consequently, C1 and C2 will be determined by the “tails” of y1 and y2 , which are affected by rounding errors and other small perturbations. The leading parts of the large numbers are canceled out when subtracted, and hence the solution will be highly dependent on rounding errors rather than the real physical conditions of the problem. It is clear that such an approach to calculating C1 and C2 is unsatisfactory. The algorithm of the initial parameter method relies on obtaining two linearly independent solutions to Cauchy problems, which results in an ill-conditioned problem for a 1. Indeed, the matrix A of system (4.4) is   y1 .0/ y2 .0/ : AD y1 .1/ y2 .1/ Its eigenvalues are expressed as     1=2 y1 .0/ C y2 .0/ y1 .0/ C y2 .0/ 2  ˙ 1;2 D  y1 .0/y2 .1/  y2 .0/y1 .1/ : 2 2

Section 4.2 General boundary value problem for systems of linear equations

163

Since it follows from (4.3) that y1 .0/  y0 .0/  O.1/ and y1 .1/  y0 .1/  O.e a /, we have  1=2 1 1;2 D O.e a / ˙ ŒO.e 2a / C O.e a / 1=2 D O.e a / ˙ 1 C O.e a / : 2 Hence, 1 D O.e a / and 2 D O.1/ and so the condition number of problem (4.4) is C D O.e a /. Consequently, since the Cauchy problems for the second-order equation with a

1 lead to a rapidly increasing solution of the order of O.e a /, it should not be included in the numerical algorithm for the boundary value problem, because this results in an unsatisfactory numerical solution for large a. Of course, if a  O.1/, the initial parameter method provides a reasonably accurate solution to the boundary value problem, since the matrix of system (4.4) will have a good condition number, C D O.1/, in this case. Thus, neither the initial parameter method nor any other method based on numerically solving Cauchy problems can provide a good numerical solution for secondorder differential equations of the form (4.1) with large a. This, for example, also holds true for the shooting method (see Section 4.3), where the second condition at x D 0 in (4.3) is varied until the other condition at the right endpoint x D 1 is met. The original boundary value problem (4.1) with large a2 1 belongs to the class of the so-called stiff problems, whose numerical solution is associated with certain difficulties; overcoming these difficulties calls for adequate special methods to be used.

4.2 General boundary value problem for systems of linear equations A similar situation can also arise in the general case when solving a boundary value problem for a linear system of nonhomogeneous differential equations with variable coefficients: du D A.x/ u C a.x/; 0  x  1; (4.5) dx where u.x/ and a.x/ are n-dimensional vector functions A.x/ is an n  n matrix. In general, boundary conditions can be written as B u.0/ C C u.1/ D f;

(4.6)

where B and C are n  n constant matrices and f is an n-vector. The general solution to (4.5), as follows from the theory of linear ordinary differential equations, is expressed as u.x/ D u0 .x/ C

n X iD1

Ci ui .x/;

(4.7)

164

Chapter 4 Methods for solving boundary value problems for systems of equations

where u0 .x/ a particular solution to the nonhomogeneous system and ui .x/ are n linearly independent solutions to the homogeneous system (4.5) with a.x/ D 0 and any homogeneous linearly independent boundary conditions of the form (4.6) with f D 0. Problem (4.5)–(4.6) is nondegenerate and has a unique solution, provided that the determinant of the system matrix obtained from (4.6) by substituting the solution of (4.7) for the unknowns Ci is nonzero. A solution to this general problem can be obtained numerically by the method of initial parameters, in a similar manner to the case of a second-order equation considered above, by reducing it to solving n C 1 Cauchy problems. First, one uses a numerical method for Cauchy problems, e.g., the Euler method, to find a particular solution u0 .x/ satisfying the zero initial conditions u.0/ D 0. Then, one finds n particular solutions ui to Cauchy problems for the homogeneous system of equations (4.5) with a.x/  0 that satisfy the conditions ui .0/ D li ;

i D 1; : : : ; n;

where li are n linearly independent unit vectors of an n-dimensional basis. Substituting the resulting solutions into (4.7) and then (4.6), one solves system (4.6) for the constants Ci . The condition number of the resulting matrix will be large and the problem ill-conditioned if, amongst the solutions ui .x/, there are rapidly increasing solutions. This is the case when some of the eigenvalues of the matrix A.x/ satisfy the condition Re i  C , with C 1, just as in solving the second-order equation. If there are no such i , the problem is well-conditioned and this approach is adequate.

4.3 General boundary value problem for systems of nonlinear equations A two-point boundary value problem for an nth-order system of nonlinear equations is stated in the same manner as for linear equations with the only exceptions that the linear functions in equation (4.5) are replaced with nonlinear ones. Suppose it is required to solve a system of equations of the form du D f.u; x/; dt

x0  x  x1 ;

(4.8)

with the general nonlinear boundary conditions Ĺ.u.x0 /; u.x1 // D 0; where u.x/, f.x; u/, and Ĺ are n-dimensional vector functions.

(4.9)

Section 4.3 General boundary value problem for systems of nonlinear equations

165

4.3.1 Shooting method The shooting method is based on reducing the problem to a multiple solution of a Cauchy problem. To obtain a solution u.x/ to the Cauchy problem for an nth-order system, one specifies an initial condition: u.x0 / D ˛. Let u.x; ˛/ denote the corresponding solution. To determine the parameters ˛, one uses the boundary conditions of the original problem (4.9) to obtain the following system of n nonlinear equations for ˛:     (4.10) F .˛/ D Ĺ u.x0 ; ˛/; u.x1 ; ˛/ D Ĺ ˛; u.x1 ; ˛/ D 0: Equation (4.10) is solved iteratively with the rough initial approximation ¸.0/ . The successive approximations ¸.k/ are determined by, for example, Newton’s method from  .k/   .k/  ˛ F ˛ : (4.11) ˛.kC1/ D ˛.k/  F 1 ¸ The iterations are performed until the condition k˛.kC1/  ˛.k/ k  " is satisfied to obtain ˛./ . For this value, one calculates u.x; ˛./ /, the solution to the boundary value problem (4.8)–(4.9). Needless to say, the above approach has at least the same drawbacks as those pointed out in solving stiff linear equations, which arise due to the reduction of the original boundary value problem to Cauchy problems, apart from additional difficulties associated with the nonlinearity.

4.3.2 Quasi-linearization method Another solution method is based on reducing the original boundary value problem for a system of nonlinear equations to a boundary value problem for a system of linear equations. Suppose it is required to solve the system of equations dy D f.y; x/ dx

(4.12)

defined on an interval 0  x  L and subject to the general nonlinear condition   F y.0/; y.L/ D 0; (4.13) where y.x/, f.y; x/, and F.y.0/; y.1// are n-dimensional vector functions, y.0/ D ˛ is a p-vector (p < n), and y.L/ is an .n  p/-vector. The initial approximation y0 .x/ will be chosen so that it is as close to the actual solution of the problem (4.12)–(4.13) as possible. Let us linearize equation (4.12) about y0 .x/: y00 .x/

y1 .x/ D y0 .x/ C ıy.x/; C ıy0 D f.y0 .x/; x/ C fy .y0 ; x/ıy:

(4.14)

166

Chapter 4 Methods for solving boundary value problems for systems of equations

This results in a nonhomogeneous system of linear differential equations with a nonzero right-hand side for ıy, where fy is the Jacobian matrix of the function f.y/. The boundary conditions are also linearized:   F y0 .0/ C ıy.0/; y0 .L/ C ıy.L/ D 0;       F y0 .0/; y0 .L/ C Fy0 .0/ y0 .0/; y0 .L/ ıy.0/ C Fy0 .L/ y0 .0/; y0 .L/ ıy.L/ D 0: (4.15) Eventually, we arrive at a linear nonhomogeneous system of boundary equations for ıy0 .0/ and ıy0 .L/ similar to (4.6). Problem (4.14)–(4.15) is solved in the same way as problem (4.5)–(4.6). Once ıy.x/, and hence y1 .x/, has been determined, the procedure is repeated: one looks for y2 .x/ D y1 .x/ C ıy1 .x/

and so on.

Thus, the solution is reduced to solving a sequence of nonhomogeneous linear boundary value problems (4.14)–(4.15). In the literature, this method is known as the quasilinearization method. The convergence of the method depends on how well the initial approximation y0 .x/ is chosen. To this end, the parameter continuation method together with the immersed boundary method initial approximation and used in the numerical solution of nonlinear solid mechanics problems by the parameter loading method [51].

4.4 Solution of boundary value problems for ordinary differential equations by the sweep method As follows from the preceding discussion, in order to obtain an adequate numerical solution to a stiff boundary value problem, one needs an algorithm that would not involve the integration of Cauchy problems in the direction of rapid increase of the function. This can be achieved by using the sweep method.

4.4.1 Differential sweep To illustrate the application of the sweep method, consider the general second-order equation "y 00 D p.x/y C F .x/;

" 1;

p.x/ > 0;

x 2 Œ0; 1 ;

(4.16)

subject to the general boundary conditions ˇ y 0 D ˛0 y C ˇ0 ˇxD0 ; ˇ y 0 D ˛N y C ˇN ˇxD1 ;

(4.17)

167

Section 4.4 Solution of boundary value problems by the sweep method

The first condition, at x D 0, continued to every point x of the interval Œ0; 1 determines a family of integral curves of equation (4.16). This family satisfies a linear first-order equation that can be written as y 0 .x/ D ˛.x/y C ˇ.x/;

(4.18)

where ˛.x/ and ˇ.x/ are unknown functions to be determined from equation (4.16). Relation (4.18) can be treated as the transfer of the boundary condition at x D 0 to any point of the interval 0  x  1, including the right endpoint x D 1. Substituting (4.18) into (4.16) yields equations for ˛.x/ and ˇ.x/: y 00 D ˛ 0 y C y 0 ˛ C ˇ 0 D ˛ 0 y C ˛ 2 y C ˇ˛ C ˇ 0 D

F .x/ p.x/ yC : " "

Equating the coefficients of y and the free terms with zero, one obtains two differential equations for ˛.x/ and ˇ.x/: p.x/ ; " ".ˇ 0 C ˛ˇ/ D F .x/; ˛0 C ˛2 D

˛.0/ D ˛0 I

(4.19)

ˇ.0/ D ˇ0 :

(4.20)

The initial conditions in (4.19)–(4.20) follow from condition (4.17) at x D 0. The solution algorithm for the boundary value problem (4.16)–(4.17) involves the following stages. 1. Solving the Riccati equation (4.19) gives the function ˛.x/. 2. From equation (4.20), one finds the function ˇ.x/. 3. With ˛.x/ and ˇ.x/ known, one obtains y.x/ from (4.18) to arrive at the additional boundary condition at the right endpoint x D 1 y 0 .1/ D ˛.1/y.1/ C ˇ.1/;

(4.21)

which, together with the second condition in (4.17), at x D 1, gives a well-conditioned system of two equations for y.1/ and y 0 .1/. The forward sweep suggests the integration of the equations for ˛.x/ and ˇ.x/ from left to right. 4. With y.1/ known, one integrates equation (4.18) from right to left. This is the so-called backward sweep. Thus, the boundary value problem (4.16)–(4.17) is reduced to three Cauchy problems for three first-order equations (4.19), (4.20), and (4.18). Note that the solution of the Cauchy problem for the second-order equation (4.16) with two obtained conditions at right endpoint x D 1 in the backward direction is

168

Chapter 4 Methods for solving boundary value problems for systems of equations

inadequate, since the problem contains a rapidly increasing solution in the integration direction. Let us justify the well-posedness of the solution for all three Cauchy problems; the idea is to perform the integration in the direction of decreasing solution, so that the error can only decay with x. Proof. Consider equation (4.19) ˛ 0 .x/ C ˛ 2 .x/ D

2 ; "

 2 D p.x/ > 0:

˛.0/ D ˛0 ;

For a constant p.x/ D  2 , equation (4.22) admits the exact solution      ˛.x/ D 1=2 coth 1=2 x C C1 ; j˛0 j > 1=2 ; " " "      ˛.x/ D 1=2 tanh 1=2 x C C2 ; j˛0 j < 1=2 ; " " " C1 D

 C ˛0 "1=2 1 ln > 0; 2  C ˛0 "1=2

(4.22)

(4.23)

1  C ˛0 "1=2 C2 D  ln < 0: 2   ˛0 "1=2 α

α

coth z κε– 1/2

tanh z tanh z

α0

0

x0

1

x

α0 x0

0

1

x

(a)

κε– 1/2

(b)

Figure 4.2. Proof of stability of the differential sweep.

I. a) Suppose ˛0  0. Then, depending on the sign of the inequality j˛0 j ? ="1=2 , the behavior of ˛.x/ changes: in the first case, ˛.x/ increases, while in the second case, it decreases staying nonnegative, ˛.x/  0, for all x (Figure 4.2a). Let us prove that the error ı˛.x/ decays. Indeed, ı˛.x/,which is the error of the solution

169

Section 4.4 Solution of boundary value problems by the sweep method

to equation (4.22), satisfies the equation in variations ı˛.x/0 C 2˛.x/ ı˛.x/ D 0I for ˛.x/  0, as x increases, the quantity jı˛.x/j decreases. b) The solution to the second problem, (4.20), decays with x for ˛.x/  0; hence, the error also decays. c) The third equation, (4.18), is integrated in the direction of decreasing x; for ˛.x/  0, the quantities y.x/ and ıy.x/ both decrease. II. Now suppose that ˛0  0. Then, since j˛0 j ="1=2 , the second expression in (4.23) is valid; this situation is shown in Figure 4.2b. We have      ˛.x/ D 1=2 tanh 1=2 C C2 ; where  1=2 < ˛0  0: " " " The point of intersection x0 of the curve ˛.x/ with the x-axis is to the right of zero, and hence ˛.x/ is negative on the interval 0  x  x0 ; the error ı˛.x/ will be increasing here. Let us estimate this increase: ı˛ 0 .x/ C 2˛.x/ ı˛.x/ D 0;

ı˛ D ı.0/e 2

Rx 0

˛.x/ dx

:

Since ˛.x/ is monotonic on 0  x  x0 , the maximum of ı˛ is attained at x D x0 . Let us evaluate the integral in the argument of the exponential: p ˇ    Z x0 Z x0   1  C ˛0 " ˇˇx0  ˛.x/ dx D p coth p x C C2  ln coth p x  ln p ˇ 2   ˛0 " 0 " " " 0 0  D  ln q  2  ˛02 " with    1 ı˛.x0 / 2 D exp 2 ln q D D D 2  ˛0 2  1; 2 ı˛.0/   "˛0 1  "  2  ˛2 "

" 1:

0

It follows that ı˛.x/ is little different from ı˛.0/. Problems are possible only if ˛0 1=2 ; this, however, contradicts the original assumption that all parameters of   " the problem but " are O.1/. On the remaining portion of the interval, x0  x  1, we have ˛.x/ > 0, and hence the error ı˛.x/ decays, as was shown above. It is clear that the same holds true for the errors ıˇ.x/ and ıy.x/ as well. This completes the proof of the well-posedness of the sweep method for solving the two-point boundary value problem (4.16)–(4.17).

170

Chapter 4 Methods for solving boundary value problems for systems of equations

4.4.2 Solution of finite difference equation by the sweep method Now let us consider the finite difference approximation of a boundary value problem for an arbitrary second-order equation and extend the sweep method to the solution of difference equations. A finite difference approximation for any second-order ordinary differential equation with smooth coefficients A.x/u00 C B.x/u0 C C.x/u D F .x/ subject to the boundary conditions u.0/ D ' at x D 0 and u.1/ D the general form an un1 C bn un C cn unC1 D fn ;

at x D 1 has (4.24)

which is only valid for internal points, n D 1; : : : ; N  1 of the interval x 2 Œ0; 1 , with the boundary conditions x D 0;

u0 D ';

x D 1;

uN D ;

(4.25)

prescribed at the endpoints. Equations (4.24)–(4.25) form a system of linear equations with a tridiagonal matrix; this system can be solved using the sweep method in the form of recurrence relations. A first integral of equation (4.24) (the analogue of equation (4.18)) can be written as (4.26) un D LnC 1 unC1 C KnC 1 2

2

where LnC 1 and KnC 1 are coefficients to be determined from equation (4.24). The 2 2 initial values LnC 1 and KnC 1 are found from the boundary conditions (4.25): 2

2

n D 0;

u0 D L1=2 u1 C K1=2 :

It follows from (4.25) that L1=2 D 0 and K1=2 D '. Further, n D 1;

a1 .L1=2 u1 C K1=2 / C b1 u1 C c1 u2 D f1 ; f1  K1=2 a1 c1 u 2 C ; u1 D  a1 L1=2 C b1 a1 L1=2 C b1

with L3=2 D  :: : LnC1=2 D 

c1 ; a1 L1=2 C b1

K3=2 D

f1  K1=2 a1 ; a1 L1=2 C b1

:: : fn  Kn1=2 an cn ; KnC1=2 D : an Ln1=2 C bn an Ln1=2 C bn

(4.27)

Section 4.4 Solution of boundary value problems by the sweep method

171

Formulas (4.27) for the coefficients are finite difference analogues of equations (4.19) and (4.20). The forward sweep suggests that the coefficients LnC1=2 and KnC1=2 are calculated by formulas (4.27). The backward sweep suggests that all un are determined from unC1 using formula (4.26). For n D N  1, it follows from (4.26) that uN 1 D LN 1=2 uN C KN 1=2 D LN 1=2

C KN 1=2 :

further, for n D N  2; : : : ; 1, one successively calculates uN 2 ; : : : ; u1 . One can see that the more general boundary conditions (4.17) result in a system of two equations at x D 1 for the unknowns uN and uN 1 . Once the system has been solved, the backward sweep can be started. For countable stability, it follows from the maximum principle (see Section 4.5), in the special case of a single variable, that the conditions an > 0;

bn < 0;

cn > 0;

jan C cn j  jbn j

(4.28)

must hold. In this case, the rounding error will not increase. Just as the Gauss method, the sweep method is based on factorization. The latter is essentially an implementation of the Gauss method in the form of explicit recurrence formulas resulting from a simpler, tridiagonal form of the matrix A of the system of difference equations (4.24) [46]. It may seem that the sweep method is only suitable for solving boundary value problems for second-order ordinary differential equations. However, this is not so; solving this kind of problems is an integral part of the analysis of a wide class of problems for partial differential equations. A few examples will be considered below to illustrate this [168].

4.4.3 Sweep method for the heat equation Consider the following initial-boundary value problem for the one-dimensional heat equation: @2 u @u D 2 2 ; @t @x

x D 0W u.0; t / D .t /; x D 1W u.1; t / D .t /; t D 0W u.x; 0/ D u0 .x/:

(4.29)

Approximating the function u.x; t / with discrete values un .x/ in t and replacing unC1 .x/un .x/

with  equations in x



, one arrives at the system of N second-order ordinary differential

d 2 u.x/   dx 2 2

@u @t

nC1

 unC1 .x/ D un .x/;

n D 1; : : : ; N;

172

Chapter 4 Methods for solving boundary value problems for systems of equations

subject to the boundary conditions unC1 .1; t nC1 / D

.t nC1 /;

unC1 .0; t nC1 / D '.t nC1 /;

which can be solved using the differential sweep method. t

τ

n+1

n k–1

k

k+1

0

h

1 x

Figure 4.3. Scheme of the sweep method for unsteady heat conduction equations.

By choosing the simplest implicit scheme for the equation of (4.27), one obtains  unk unC1 k

2

unC1  2unC1 C unC1 kC1 k k1

D ; h2    2  2  2  nC1 ukC1  1 C 2 2 unC1 C 2 unC1 D unk : k 2 h h h k1 

(4.30)

For a fixed n, equation (4.30) and boundary conditions (4.29) have the form (4.24)– (4.25) and can be solved successively for each time layer n C 1, with the solution for layer n known (Figure 4.3), by the difference sweep method (4.26)–(4.27). It is clear that the countable stability conditions (4.28) for the coefficients of equation (4.30) are satisfied.

4.5 Solution of boundary value problems for elliptic equations 4.5.1 Poisson’s equation We now proceed to difference equations for boundary value problems described by elliptic equations. The simplest elliptic equation is Poisson’s equation @2 u @2 u C 2 D f .x; y/; 2 @x @y

(4.31)

173

Section 4.5 Solution of boundary value problems for elliptic equations

which arises in solving numerous problems of mathematical physics. The boundary conditions can have one of the following three forms (s 2 @/: u D '.s/I @u D '.s/I @n

Dirichlet conditions: von Neumann conditions: mixed conditions:

a.s/

(4.32) (4.33)

@u C b.s/u D '.s/: @n

(4.34)

@ Here @n indicates a derivative with respect to the outward normal to the contour @ of the domain ; a.s/, b.s/, and '.s/ are functions defined on @. Furthermore, different types of boundary conditions can be set on different portions of the surface @. Perform a finite difference approximation of problem (4.31)–(4.34). For simplicity, introduce a square grid with step size h and cover the domain  with this grid. Introduce a grid function u.xk ; ym / D uk;m defined at the grid nodes .k; m/ with k D 0; 1; : : : ; K and m D 0; 1; : : : ; M .

(k, m + 1) internal nodes

1

(k, m) boundary nodes (k – 1, m)

0

(k + 1, m)

1

(a)

(k, m – 1)

(b)

Figure 4.4. (a) A rectangular grid for the solution of the discrete Poisson equation (boundary value problem, first-order approximation). (b) A five-point stencil.

With the simplest five-point stencil, the second derivatives in equation (4.31) can be approximated by second-order central finite differences (Figure 4.4) to obtain ukC1;m  2uk;m C uk1;m uk;mC1  2uk;m C uk;m1 C D fk;m : 2 h h2 (4.35) Equation (4.35) is only defined at the nodes of the stencils that lie completely within the domain . To simplify the approximation of the boundary conditions, we assume the domain to be rectangular. For example, let us approximate condition (4.34) by a left unilateral uk;m D

174

Chapter 4 Methods for solving boundary value problems for systems of equations

y

h

b

m+1

h

k

k+1

h – 2

a

x

h – 2

Figure 4.5. Arrangement of nodes on the boundary to provide a second-order accurate solution to a boundary value problem.

finite difference at x D 0 and a right unilateral finite difference at x D 1: u0;m  u1;m C b0;m u0;m D '0;m ; a0;m h m D 0; 1; : : : ; M: (4.36) uK;m  uK1;m C bK;m uK;m D 'K;m ; aK;m h The boundary conditions at y D 0 and y D l will be approximated likewise. The order of approximation of the difference equation (4.31) is O.h2 / and that of the boundary conditions is only O.h/. In order to obtain second-order approximation for the boundary conditions as well, one should extend the grid by h=2 in both x and y and introduce outer nodes along the contour @ as shown in Figure 4.5. The approximation in, for example, x should be taken in the form u1;m  u0;m u1;m C u0;m C b1=2;m D '1=2;m (4.37) a1=2;m h 2 where the quantities labeled with k D 12 coincide exactly with the corresponding values at the contour x D 0. The solution u.x; y/ is assumed to have four continuous derivatives, which requires the function f .x; y/ to have two continuous derivatives in  and the function ' to be continuous on @. It is noteworthy that general irregular finite-element meshes [168, 89] are used to solve elliptic continuum mechanics equations for complex-shaped domains.

175

Section 4.5 Solution of boundary value problems for elliptic equations

4.5.2 Maximum principle for second-order finite difference equations The stability of the finite difference scheme (4.35)–(4.37) can be proved based on the maximum principle valid for the elliptic differential equation (4.31). It follows from the maximum principle that the value of a harmonic function at a point .x; y/ is equal to its average taken along a circumference centered at .x; y/. The maximum principle is also valid for the finite difference approximation (4.35). Consider the second-order elliptic ordinary differential equation 

  2 X @u @ a.˛/ .x/ C q.x/u D f .x/; @x˛ @x˛ ˛

x 2 :

(4.38)

The finite difference approximation (4.38) on a rectangular grid with step size h˛ along the coordinate x˛ (˛ D 1; 2) is given by  ukC1;l  uk;l .1/ .1/ uk;l  uk1;l  ak;l Qh .u/ D  akC1;l h1 h1  u  u u  uk;l1 k;lC1 k;l k;l .2/ .2/ C ak;lC1  ak;l C qk;l xk;l uk;l h2 h2 (4.39) D fk;l ; where akC1;l D a.xkC1  0:5h1 ; y/; ak;lC1 D a.xk ; ylC1  0:5h2 /;

q.xk ; yl / D qk;l ; f .xk ; yl / D fk;l :

Let us rewrite the difference equation (4.39) in the conditional form Qh .uk;l / D Auk;l C

N1 X

B.xk;l ; /u./;

(4.40)

D1

where  is the node number in the stencil N1 obtained from the full stencil N by removing the central point .k; l/. The coefficients A and B satisfy the conditions A.x/ > 0;

B.x/ > 0;

C.x/ D A 

N1 X

B.x; / > 0;

x 2 :

(4.41)

D1

The maximum principle for the solution to the difference equation (4.40) reads: if the grid function uh satisfies the homogeneous boundary conditions uh .xi / D 0, xi 2 @, and f .xi /  0 or f .xi /  0), then u.xi /  0 (u.xi /  0) and maximum of juh .xi /j or, respectively, minimum of juh .xi /j is attained at the boundary (xi 2 @). The convergence of the finite difference schemes (4.39)–(4.40) can also be proved using the maximum principle.

176

Chapter 4 Methods for solving boundary value problems for systems of equations

4.5.3 Stability of a finite difference scheme for Poisson’s equation Let us prove that if .u/k;m > 0 at all internal nodes, then maxk;m uk;m is attained at a boundary node. Proof. From .u/k;m > 0 it follows that uk;m
0 must be ensured by additional efforts (step size reduction, step size selection during the iterations, etc.). In this case, the scheme will be completely conservative, since it provides a sufficiently correct approximation for the total as well as internal energy. A sourceless approximation of equations (7.30) that satisfies the conditions of a completely conservative scheme can be written as [154, 39] nC1 n  emC1=2 emC1=2

t

nC1 nC1  vmC1  vm 1 n .p C q/nC1 C .p C q/ mC1=2 mC1=2 2 x nC1 nC1   v v 1 m mC1 n D 0: (7.34) C .p C q/nC1  .p C q/ mC1=2 mC1=2 2 x

C

The above implicit difference equations (7.25)–(7.34) subject to the boundary conditions (7.24) are solved by the sweep method.

Section 7.4 The PIC method and its modifications for solid mechanics problems

311

7.4 The PIC method and its modifications for solid mechanics problems 7.4.1 Disadvantages of Lagrangian and Eulerian meshes The solution of continuum mechanics problems at large deformations by using classical approaches based on Eulerian or Lagrangian mesh (as described above) can be associated with certain difficulties. With sufficiently coarse meshes, which are often used in real computations, the computation becomes unfeasible starting from a certain degree of deformation. On Lagrangian meshes with explicit schemes, this is due to severe distortions of cells; in addition, the CFL stability condition results in serious restrictions on the step size and the diameter of the base of the characteristic cone, d ! 0, inscribed into the cell in the physical space xi : t  dc ! 0. On Eulerian meshes, as the medium moves relative the fixed mesh, it fills previously empty cells and forms partially filled cells, so-called fractional cells, having most peculiar shapes; the calculations of such cells is associated with certain difficulties and loss of accuracy. For this reason, one attempts to overcome these difficulties somehow by using new, nonclassical approaches. We dwell on two main approaches developed in continuum mechanics to solve large deformation problems. The first approach is based on improved characterization of motion of media on Eulerian meshes and is known as the particle-in-cell (or PIC) method. The other approach is represented by adaptive moving mesh methods, where the advantages of the Eulerian and Lagrangian approaches are combined to minimize their disadvantages.

7.4.2 The particle-in-cell (PIC) method The main idea of this method is to combine the continuum description of a medium with its characterization as large ensemble of individual particles moving on an Eulerian spatial mesh. Unlike such media as plasma or interstellar space, where collisions between particles are rare and so collisionless models for many particles can be used [136], liquids or solids, where the density of particles is incomparably higher, such models are inapplicable. Due to continual collisions between neighboring molecules, the momentum and energy of individual particles are not conserved being constantly redistributed among the neighbors, and hence the molecules in a small volume can be treated as being in thermal equilibrium. These considerations made it possible to develop a unified model that combines the description of the motion of individual particles with that of the continuum [136]. Let us first study the example of an ideal gas to illustrate how the PIC method works. The laws of conservation of substance and momentum can be written in the

312

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

Eulerian coordinates as @.nm/ C Div.mnv/ D 0; @t @.nmv/ C Div.mnv ˝ v/ D  grad p; @t

(7.35)

where n.x; t / is the number of particles per unit volume at the point .x; y/, m is the mass of a particle in a cell, v is the average velocity of particles in a cell (see Figure 7.5), p is pressure, and v ˝ v is the dyadic product. The ideal gas equation will be taken in the form p D const or d .p / D 0; dt

(7.36)

where d=dt is the total (Lagrangian) derivative. For further convenience, let us represent equation (7.36) as a conservation law in Eulerian form to obtain @ .p / C Div.p v/ D 0: @t

(7.37)

For simplicity, the PIC method will be presented below for the two-dimensional case; the generalization to the three-dimensional case is straightforward. The spatial domain under study is covered by an Eulerian mesh. Each cell of the mesh, referred to by the indices ij , is assumed to contain sufficiently many particles, referred to by the number ˛ and coordinates .x˛n ; y˛n / at a time instant t n . As a particle moves, its coordinates are determined from its velocity components: dx˛ D u˛ dt

dy˛ D v˛ dt

(7.38)

where .u; v/ are the velocity components, v D .u; v/. Apart from coordinates, each particle is assigned with a mass m˛ and momentum .m˛ un˛ ; m˛ v˛n / as well as an adiabatic invariant e˛n D .p /n˛ , which is related to the equation of state of the particles. Thus, the vector of unknowns U that characterizes the particle is a 6-vector:   1  ˛  N: Un˛ D x˛n ; y˛n ; m˛ ; m˛ un˛ ; m˛ v˛n ; e˛n ; Since particles within a cell experience a large number of collisions, there momenta and invariants equalize in a short period of time and so can be determined as the average values over all particles in the cell. Consequently, the introduction of the particle density nij , gas density ij , momentum .v/ij , and adiabatic invariant eij in cell ij is meaningful for a time instant t n . Essentially, it is assumed that a particle reaches a thermodynamic equilibrium in time t . We will use the notation    n    n y˛ x˛ n  i ı Int j ; ˛ D ı Int h h

313

Section 7.4 The PIC method and its modifications for solid mechanics problems

where h is the Eulerian mesh step size, Int.z/ is the integer par of the argument z, ´ 1 if k D m; ı.k  m/ D 0 if k ¤ m: Then n˛ D 1 whenever the particle is inside the cell and n˛ D 0 if the particles is outside the cell. The average values in a cell can be written as nnij D

1 X n  ; h2 ˛ ˛

n ij D

1 X n  m˛ ; h2 ˛ ˛

.v/nij D

1 X n  m˛ v˛n : h2 ˛ ˛

(7.39)

n Then the cell pressure can be calculated as pij D .e /nij . The instantaneous particle velocity .u; Q v/ Q in cell ij can be determined from the equations of motion (7.35), where the divergence term is discarded and so mass transfer is not taken into account (it will be included in the next computational stage):

D unij  uQ nC1 ij

t n n n .piC1;j  pi1;j /; 2ij h

nC1 n vQ ij D vij 

t n n n .pi;j C1  pi;j 1 /; 2ij h

(7.40)

where the temporal step t is chosen so as to satisfy the CFL condition. In order to find the particle velocities vn D .un˛ ; v˛n / from .uQ n˛ ; vQ˛n /, the author suggested a spatial interpolation algorithm based on four neighboring cells adjacent to the node closest to particle ˛ (see Figure 7.5).

Vi, j + 1

Vi + 1, j + 1

ai, j + 1 ai, j Vi, j

ai + 1, j + 1 α

ai + 1, j Vi + 1, j

Figure 7.5. Algorithm for determining the particle velocity from four neighboring cells.

314

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

To determine the velocity of particle ˛, one introduces a fictitious area (volume) of the particle equal to that of the cell centered at this particle (Figure 7.5, shaded square). This allows one to evaluate the weighting factors aij in (7.41), which are taken equal to the areas associated with each of the four cells:   1 nC1 N iC1;j C1 C .au/ N i;j C1 C .au/ N i;j ; u˛ D 2 .au/ N iC1;j C .au/ (7.41) h where uN D 12 .uQ nC1 C un /. From the velocities .unC1 ; v˛nC1 / determined in (7.41) one find the new position of ˛ the particle t; x˛nC1 D x˛n C unC1 ˛

y˛nC1 D y˛n C v˛nC1 t

(7.42)

and determine the position of the cell ij to which particle ˛ moves. If the particle gets into a fictitious cell, which is behind a rigid wall of the axis of rotation, then the particle is mirror reflected. A particle moving with the velocities v˛nC1 obtained carries the momentum .mv/nC1 ˛ , proportional to its mass, and adiabatic invariant e˛nC1 , which corresponds to the integration of the transfer equations, equations (7.35)–(7.37) with zero right-hand sides. The temporal step is completed here, since the solution vector U has now been found. The PIC method can be treated as a technique for splitting the conservation laws (7.35) where the convective terms are dropped in the first stage. In the second stage, the pressure gradient is omitted and the integration of the transfer equations is replaced with displacement of individual particles, which realize transfer of mass m, momentum mv, and adiabatic invariant e. Hence, the PIC method is equivalent to a splitting method where each cell has a single particle, completely occupying the cell.

7.4.3 The method of coarse particles Under the assumption that each cell contains only one particle, the PIC method coincides with the splitting of ideal gas equations in physical processes. The transport of particles is replaced with the flow of a fluid across cell boundaries followed by the calculation of the flow densities [11]. nC1 / by formulas (7.40). In the second In the first stage, one calculates .uQ nC1 ij ; vQ ij stage, the fluid is assumed to transfer a quantity Q across cell boundaries by a mass flow according to the conservation law @Q C Div.Qv/ D 0: (7.43) @t The mass flow across a face of cell .i C1=2; j / centered at node i; j in time t equals 8 nC1 nC1 ˆ 0; 2 nC1 D MiC1=2;j (7.44) nC1 nC1 ˆ u Q ij Cu Q i C1;j : nC1 nC1 y if uQ ij C uQ iC1;j < 0: iC1;j 2

Section 7.4 The PIC method and its modifications for solid mechanics problems y

315

j+1 6 uB

5 ρB

B

νB

7 C

ρC

uC νA j

8

1

4

νC

uA ρA j–1 i–1

3

A

D 2 i

ρD 9 i+1 x

Figure 7.6. Flow of a medium between neighboring cells.

Density comes from the cell that the medium flows out from (see Figure 7.6). The nC1 due to transfer by mass flow across all faces of the cell is calculated change of Qij as nC1 nC1 n DQij  Qij Qij  nC1 t nC1 nC1 nC1 D Mi1=2;j C QQ iC1;j MiC1=2;j QQ ij xy i1;j  nC1 nC1 nC1 Q nC1  QQ i;j 1 Mi;j 1=2  Qi;j C1 Mi;j C1=2 :

(7.45)

This formula corresponds to the evaluation of the divergence in equation (7.43) by the natural approximation formula (7.3) for a fixed Eulerian cell. It follows from the above algorithm that the mass, momentum, and adiabatic invariant e are conserved for the whole body, which means that the scheme is conservative. With the PIC method, it is fairly easy to visualize the motion of body boundaries and contact interfaces between media with different densities at large displacements and strains as well as the formation of eddies and other fine structures.

7.4.4 Limitations of the PIC method and its modifications The PIC also has some disadvantages. It is quite demanding of computer power, since a cell can contain a large number of particles. It also has high memory demands, as it is required to store not only values of mesh functions but also all characteristics of the particles. Furthermore, computational time increases considerably. If the number of particles is relatively small, main physical quantities fluctuate as particles cross cell boundaries, which is due to transfer discreteness. These fluctuations can give rise to instability. A stability analysis of the PIC-method numerical scheme has shown that

316

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

the scheme is stable if the temporal step size t [90] satisfies the condition jvj t  2 ; h jvj C ac 2

(7.46)

where a is a constant determined by the properties of the medium and c is the speed of sound. Numerical experiments result in the following estimate for the minimum number of particles in a cell at which the fluctuation do not yet lead to instability: ˇ ˇ h ˇ @p ˇ N  2 ˇˇ ˇˇ (7.47) v @x It follows from conditions (7.46)–(7.47) that N 1Ca

c2 : v2

(7.48)

The estimate ˇ ˇ ˇ ˇ ˇ ˇ ˇ @p ˇ ˇ @v ˇ ˇ v ˇ ˇ ˇ ˇ D ˇ ˇ  ˇ ˇ @x ˇ ˇ @t ˇ ˇ t ˇ has been used here. It follows from (7.46)–(7.48) that as the medium velocity decreases, the step size t required for stability decreases and the number of particles in a cell increases. Therefore, if cells with small velocity arise in the solution domain, this increases demands on the number of particles and the method becomes less efficient.

7.4.5 The combined flux and particle-in-cell (FPIC) method In order to decrease the number of particles introduced and reduce the computation time, a combined method was suggested in [40], based on a joint flux and particlein-cell approach (FPIC methods). It suggests that the flux method (coarse particle method) should be used everywhere in the computational domain except for neighborhoods of contact and free surfaces, where the PIC method should used. If a medium flows into a cell containing a different material, then the two materials are represented as particles. If a particle gets into a cell that already contains a larger, coarse particle and both particles are of the same material, the smaller particle is removed and its mass and energy are added to the larger particle. If a particle gets into a cell containing a different material, the cell material is represented by particles. Although the hybrid method reduces the computation time, it weakens but does not remove the flaws of the PIC approach, since the mass flow is discrete. At low velocities, a huge number particles can be formed, with mass, momentum, and energy fluctuating at contact and free boundaries due the discreetness of flow; however, as mentioned in [40], these fluctuations are weaker than in the PIC method.

Section 7.5 Application of PIC-type methods to solving elastoviscoplastic problems

317

7.4.6 The method of markers and fluxes For problems on interaction and mixing of several different media, it was suggested in [68] to consider flows on an Eulerian mesh instead of studying the motion of Lagrangian particles, in order to completely suppress the oscillations at free surfaces or material interfaces. The problem is integrated by the flux method (7.37) and the motion of boundaries is traced with special markers. The markers are massless particles that do not possess any properties of the medium but have a velocity equal to average velocity of the medium at the spatial points where the markers are located. Different media have different kinds of markers. Theoretically, boundary markers must follow the boundary and indicate its position in space. However, this correspondence may be violated due to discreteness of the mesh. To remove the discrepancy, a periodic adjustment procedure is used. Roughly speaking, the adjustment procedure suggests that if the mass within an Eulerian cell near the boundary exceeds a certain threshold value, M > M0 , but their is no marker associated with the cell, then the nearest marker is dragged to the cell. If M < M0 and there is no marker, the mass M is returned to the previous cell and the flux is set to zero. In the other cases, it is assumed that the markers fully correspond to the motion of the medium and no adjustment procedure is undertaken. Computations based on the flux method provide sufficiently smooth flows of the medium at boundaries, thus reducing oscillations and making the solution more stable than in the particle methods. Markers allow one to trace boundaries and remove nonphysical diffusion and mixing up of the medium components. In all the methods outline above, the hydrodynamic equations are split as follows: (i) the convective terms are dropped (Eulerian stage) and (ii) particulate flows or property fluxes are considered (Lagrangian stage) that transfer mass, momentum, and energy. Thus, although the PIC method and its modifications allows the analysis of large deformations, it calls for further improvements and combination with other methods, especially for subsonic flows. The adaptable mesh method can be used as one of such methods; due to its generality, it can easily be combined with other numerical methods.

7.5 Application of PIC-type methods to solving elastoviscoplastic problems with complicated constitutive equations In order to extend the PIC method to solids, which are described by rheologic relations involving not only pressure but also the total stress tensor and its material derivative with respect to time, one has to modify the solution algorithm outlined above for ideal gases.

318

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

The equations of motion become more complicated to include additionally the stress deviator sij . Accordingly, it has an additional constitutive equation, which cannot generally be written as a conservation law, since it represents a nonintegrable differential relation between the stress rate deviator sPij , strain rate deviator Pij , and p large inelastic strain tensor ij . Two types of constitutive relations must be distinguished. One type contains a derivative of the stress deviator or stress tensor and applies to hypoelastic or more complex media, which include hypoelasticity as a constituent element. The other type represents hyperelastic relations, which a finite functional relations between the stress tensor and finite strain tensor; it also applies to hyperelastoviscoplastic media, which include hyperelasticity.

7.5.1 Hypoelastic medium Consider hypoelastic or more complex media, which include hypoelasticity. The core of the PIC method scheme can preserved in this case also. For simplicity, we consider an isothermal deformation process with pressure p D p./. The conservation laws can be written as @.nm/ C Div.mnv/ D 0; @t @sij @.nmv/ C Div.mnv ˝ v/ D  grad p C : @t @xj

(7.49) (7.50)

Just as in the case of ideal gases, the equation for the spherical part of the stress tensor can be represented in the differential form  d  p  p./ D 0: dt

(7.51)

Equation (7.51) can be rewritten in the form of a conservation law in the same manner equations (7.37). A hypoelastoplastic material can be characterized with the following constitutive equations for the stress deviator sij [137, 56]: dsij Dsij D C ik skj C ski j k Dt dt 2 dƒ Div vıij C sij ; D .vi;j C vj;i /  3 dt ˇ dsij ˇˇ @sij dsij D C vi ; ˇ dt dt xDconst @xj where the second term in the last equation in (7.52) is due to convection.

(7.52)

Section 7.5 Application of PIC-type methods to solving elastoviscoplastic problems

319

Form (7.51)–(7.52) we find the relation between the elastic constants and function p./: ˇ dp ˇˇ 2 C 3  D K; D dt ˇ D 0 3 where and are the elastic constants and K is the bulk modulus. In the predictor step, the convective terms are excluded from equations (7.49)– (7.52). To determine the intermediate velocity components .uQ nC1 ; vQ˛nC1 / and inter˛ nC1 mediate stress deviator components sQij , we can use, for example, the explicit Lax scheme to obtain D unij  uQ nC1 ij

nC1 vQ ij

t n n n h .piC1;j  pi1;j / 2ij

 C .s11 /niC1;j  .s11 /ni1;j C .s12 /ni;j C1  .s12 /ni;j 1 ;  t n n n pi;j C1  pi;j D vij  n 1 2ij h  C .s11 /ni;j C1  .s11 /ni;j 1 C .s12 /niC1;j  .s12 /ni1;j ;

(7.53)

.Qs11 /nC1 D .Qs11  1k sk1  sk1 1k /nij ij     @u n 2 @u @v n C 2  C ; @x1 ij 3 @x1 @x2 ij .Qs22 /nC1 D  ; ij .Qs12 /nC1 D  ij with the nonzero spin tensor components, describing rigid rotation of particles, .12 /nij D

  .u/ni;j C1  .u/ni;j 1 .v/niC1;j  .v/ni1;j 1 @u @v n  :  D 2 @x2 @x1 ij 2h 2h

The subsequent integration steps are the same as in Section 7.4. First, one performs the interpolation (see Figure 7.5) and determines the velocity components of particle ˛. Then one finds the new position of the particles and calculates the momentum mQv, invariant  D p  p./, and stress deviator sQij at this point. This completes the step with respect to t .

7.5.2 Hypoelastoplastic medium The integration of the constitutive equation (7.52) in the predictor step is the same as in the algorithm for hypoelastic media. The corrector step splits into two substeps. In the first substep, one solves a stress relaxation problem for sQij at zero total strain rates, "Pij D 0; the resulting stress deviator sQij is adjusted using the correction procedure

320

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations ep

described in Section 6.2.2. In the second substep, the value of sQij , together with mQv and  D p  p./, is transferred to the new position. It is convenient to single out as a separate step the calculations of the correction for sij due to rigid rotation of the particle. In the plane case, the spin tensor ij has only one nonzero component, 12 , which determines the angle of rotation ! of the particle: sin 2! D 12 Furthermore, at large deformations, it is necessary to take into account the terms that cause the stress tensor to change due to rigid rotation of the particle. The rigid rotation correction can be taken into account in two ways: (i) based on the form of covariant differentiation of the stress tensor in the “frozen” basis (see Section 1.1 and [157, 134]) to obtain derivatives like the Jaumann derivative or (ii) based on physical considerations, by analyzing rotation of a cell through an angle ! and recalculating the stress tensor in the rotated axes; in the two-dimensional case, the recalculation formulas are n n n s n  s22 s11 C s22 n C 11 cos.2!/ C s12 sin.2!/; 2 2 n n s n  s22 s n  s22 n  11 cos.2!/  s12 D 11 sin.2!/; 2 2 n s n  s22 n sin.2!/: D s12 cos.2!/  11 2

n D sN11 n sN22 n sN12

n are The rotation corrections ıij n  sn s11 n n 22 cos.2!  1/  s12 sin.2!/ D ı22 ; 2 n s n  s22 n sin.2!/: D s12 cos.2!  1/  11 2

n n n ı11 D sN11  s11 D n ı12

D

n sN12



n s12

(7.54)

The rotation angle in time t is calculated from the angular velocity determined P with the only nonzero component  P 12 D 1 .u;2  v;1 / in the two-dimensional by , 2 case:   t @u @v sin.!/ D :  2 @x2 @x1 Since the angle ! remains small over time t , we have sin.2!/  2 sin.!/ D t .u;2  v;1 /.

Section 7.5 Application of PIC-type methods to solving elastoviscoplastic problems

321

Formulas (7.54) can also be simplified by dropping the quantities of the order of O.! 2 /. Then .cos.2!/  1/  O.! 2 /, and hence  nC1=2    1 V u v nC1=2 nC1 n n D s11 C 2 "11  C s12 t  ; s11 3 V x2 x1  nC1=2    1 V v nC1=2 u nC1 n n D s12 C 2 "12   s12 t  : s12 3 V x2 x1 (7.55) n The rotation corrections ıij calculated following the second approach coincide with those obtained in Section 7.4 from the expression of the Jaumann objective derivative.

7.5.3 Splitting for a hyperelastoplastic medium A hypoelastic medium is known to be described by nonintegrable differential relations, that is, incremental relations between the objective stress rate and objective strain rate in the form (7.52), is not elastic in the strict sense, since it deforms in an irreversible manner and, the more so, the energy accumulated is not potential (see Section 1.12). For a hypoelastic solid, deformations are only reversible in the small. At large strains, deformations are irreversible and hysteresis losses arise, which means that the medium is dissipative [137]. If it is required to describe a medium with large reversible deformations (e.g., rubber-like and other organic materials), one has to use a hyperelastic model. This also holds true for complex models where a hypoelastic body is a constituent element. The equations of a hyperelastoplastic model at large deformations can be found in Section 1.12; it was mentioned there that the spatial formulation of the constitutive equations is most computationally suitable, the same that was used above when considering the hypoelastoplastic model. It therefore suffices to consider only the integration of the hyperelastoplastic constitutive equations and related changes in the algorithm. The other equations are integrated in exactly the same way as those of the hypoelastic model. For a hyperelastoplastic material, the predictor step involves the integration of equations written in the of finite functional relations between the Almansi finite strain tensor Aij and Cauchy stress tensor ij , which significantly simplifies the computation algorithm for Q ij . The corrector step also simplifies due to the fact that the integration is performed at zero total strain rate and so the objective derivatives coincide with material ones. The objectivity principle is satisfied identically by the constitutive model. There are difficulties remaining, related to the multiplicative decomposition of the displacement gradient tensor F into an elastic and a plastic component. It follows from the aforesaid that the simplifications listed above are not related to the numerical method employed (in particular, the PIC method) but stem from the

322

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

splitting of the hyperelastoplastic equations and, hence, hold for any formulation of the conservation laws, both dynamic and static. The algorithm outlined above is described below in detail with a rigorous proof of the statements used. Let us adopt the additivity hypothesis for the elastic and plastic strains: L D Le C Lp :

(7.56)

P 1 is the total strain rate tensor, F is the displacement gradient tensor, where L D FF e P and L D EE1 and Lp are the elastic and plastic strain rate tensors resulting from the hypothesis of multiplicative decomposition of the displacement gradient tensor, F D EP. Suppose Lp satisfies the associated plastic flow rule under the assumption of the von Mises yield criterion. Then D p dƒ A D s; Dt dt where Ap is the plastic part of the Almansi strain tensor (see Section 1.9). Let us take the corotational (Jaumann) derivative to be the objective derivative of the plastic strain tensor: Lp D

d p DAp D A C Ap LR C LTR Ap ; Dt dt

P 1 ; LR D RR

F D RU

P is its where U is the right stretch tensor, R is the orthogonal rotation tensor, and R material derivative. The tensor F D @x=@x0 is determined from the velocity field by integrating the equation x nC1  xin dxi nC1=2 D i D vi : dt t The Almansi tensor is then found as 1 AnC1 D .I  FT F1 /nC1 : 2 P 1 D 0, we have P P D 0, Predictor. Let us split equation (7.56). By setting Lp D PP nC1 n D P . Consequently, and hence P

.Ae /nC1

EnC1 D FnC1 .Pn /1 FnC1 D EnC1 PnC1 ; 1 1 D .I  ET E1 /nC1 D I  .Pn /T .FnC1 /T .FnC1 /.Pn / : 2 2

The tensor FnC1 is determined by solving the elastic problem by one of the available methods using an explicit scheme, while the stress Q is found from the hyperelastic law  N nC1 @ˆ nC1 D : (7.57) .Q / @Ae

Section 7.5 Application of PIC-type methods to solving elastoviscoplastic problems

323

Corrector. Let us prove that D d  (7.58) Dt dt in the corrector step. For a hyperelastic material, one should correct the stresses calculated from the hyperelastic relations (7.57). P 1 D 0. It follows that In the splitting, we have L D FF FP D 0;

P C RU P D 0; RU

P 1 C RUU P 1 R1 D 0; RR

(7.59)

where R is an orthogonal tensor and U is a symmetric tensor. Let us prove that LR D 0 and D=Dt  d=dt . P 1 D 0, since LR is skew-symmetric. We From (7.59) it follows that LR D RR have P T D R.R1 /P D RR1 RR P 1 D RR P 1 /T D RT R P 1 : LTR D .RR P 1 is symmetric, since U D UT and U1 D UT . We have The tensor UU P 1 /P D UT U P T D U1 U P D UU P 1 : .UU P and U1 commute is easy to prove by differentiating the identity The fact that U 1 .UU /P D 0: P  UU P 1 D 0 U1 U Then it follows from equation (7.59) that the symmetric tensor is equal to the skewsymmetric tensor, and hence P 1 D 0: LR D RR Further, we have d d 2 ˆ d Ae d 2ˆ d 2ˆ d 2 ˆ dƒ D e p D D D : L D  L D  Dt dt D.Ae /2 dt D.Ae /2 D.Ae /2 D.Ae /2 dt (7.60) Since relation (1.105) holds for a hyperelastic body, Ae D

@ˆ1 . / ; @

the right-hand side of equation (7.60) is a function of the stress tensor alone. Consequently, in general, the relation characterizes stress relaxation of a medium with a nonlinear elastic modulus . O /:

.Ae / D

@2 ˆ D . O /: .@Ae /2

324

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

For a uniaxial stress state, equation (7.60) can be integrated in analytical form as easily as in Section 6.4 (a hypoelastic body with constant modulus ): d D  . O /: dƒ In the general case of a complex stress-strain state, the system of relaxation equations (7.60) does not now split and one has to solve a coupled system of nonlinear tensor equations together with the equations for the internal variables i : 8 d 2ˆ ˆ dij ˆ ˆ D ij D ij kl kl ; < dƒ .d Aeij /2 (7.61) ˆ d i ˆ ˆ : D fi . i ; ij /: dƒ Even in linearized form, system (7.61) is quite cumbersome for closed-form solution in a step. Therefore, the system is easier to integrate numerically by the locally implicit scheme 8 nC1 ij  ijn ˆ ˆ nC1 ˆ D ij kl .ijn /kl ; < ƒ     ˆ

nC1  ni @fi n nC1 @fi n nC1 ˆ i ˆ D kl C

k : ƒ @kl @ k and find ƒ in each step by an iterative method from the solution of (7.61) together with the yield criterion. An alternative way of solving (7.61) suggests that ƒ should first be eliminated from the yield criterion and associated flow rule.

7.6 Optimization of moving one-dimensional meshes In many continuum mechanics problems, the use of uniform Eulerian or Lagrangian meshes turns out to be insufficient for the reasons mentioned above. It becomes necessary to use nonuniform moving meshes adaptable to the problem solution. For example, condensing meshes in the regions where solutions change rapidly near moving quasi-discontinuities can be used to obtain sufficiently accurate solutions with relatively few computation points. Such meshes are also necessary in large deformation analyses. Some restrictions can be applied to the meshes; for example, mesh deformation can be limited, since too large deformation leads to a decrease in the allowed temporal step size. Due to approximation errors, coarse meshes are susceptible to the effect of cell eversion, where the Jacobian first becomes zero and then negative. In most cases, the researcher does not have a priori information where the mesh should be changes; this depends on the solution behavior, which is unknown in advance. For this reason, constructing a mesh and finding a solution become coupled problems.

Section 7.6 Optimization of moving one-dimensional meshes

325

7.6.1 Optimal mesh for a given function To begin with, consider a simple uncoupled problem for constructing an optimal nonuniform mesh for a function T D T .x/ defined on an interval. Suppose there are no specific conditions imposed on the function and, instead, it is only known to be n C 1 times differentiable. Then, for a given number of nodes, an optimal mesh is understood as a (nonuniform) mesh for which the interpolation error, or the norm of the remainder term of the interpolating polynomial on an interval Œ0; X0 , is minimum. Constructing an optimal mesh is reduced to solving the well-known Chebyshev problem of finding the minimax ˇN ˇ³ ² ˇY ˇ ˇ min max ˇ .x  xi /ˇˇ x0 ;:::;xN

0xXN

iD0

to determine a polynomial of degree N , with the leading coefficient equal to 1, least deviating from zero on the interval Œ0; XN , with x0 D 0 and xN D X D 0. Chebyshev’s mesh is .4=e/N times more beneficial as compared with the uniform mesh. In addition, Chebyshev’s mesh has been proved to be least sensitive to perturbations of node positions . If the function T D T .x/ possesses a certain property, the problem of constructing, in a physical space, an optimal mesh can be formulated as a variational problem of finding a function x D x./ that maps the physical space into the computation one, takes the interval 0  x  1 of the physical space into itself, 0    1, and minimizes the functional Z 1

I D

W .T .//J 2 d ;

(7.62)

0

which depends on the norm of the function W .z/ characterizing the required property of the mesh, with J D @.x; t /=@.;  / denoting the Jacobian determinant of the transformation of the physical space into the computational one. For example, the minimization of the gradient of T .x/ can be written as Z 1 max kgrad T .x.//k J 2 d : (7.63) min I D min x./

x./

0

It is noteworthy that the function W .z/ must meet appropriate smoothness and differentiability conditions required for a solution to exist. For more details on the mathematical issues of optimization, see [77]. The problem can be formulated as follows: find an optimal discretization h.xi / of the interval Œ0; 1 that minimizes the error of a given (e.g., piecewise linear) approximation Th .x/ in the sense of a certain norm. In other words, it is required to find a set h.xi / that minimizes a functional dependent on the norm kT .x/  Th .x/k: min I D min kT .x/  Th .x/kHh ;

h.xi /

h.xi /

(7.64)

326

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

where the function h.xi / is defined on a discrete set of numbers xi (i D 0; 1; : : : ; N ) from the interval Œ0; 1 . The function h.xi / associates with each xi an interval of length hi D xi  xi1 . The piecewise approximation Th .x; h.xi // depends on h.xi /. This problem is more convenient to reformulate as a variational problem of minimizing a functional dependent on a continuously differentiable function x D x./ and solve using the well-known methods of variational calculus. For example, we take x D x./ to be a function that maps the interval Œ0; 1 into itself and the nonuniform discretization h.xi / into a uniform one in the space of . This property is preserved on the uniform mesh , and hence d T =d  must be close to a minimum constant value on the entire interval. Then, to condition (7.64) in the physical space x there corresponds the condition Z 1 min I D min kT .x.//kC d  (7.65) x./

x./

0

in the computational space . This means that the norm of the composite function T .x.// must be minimum with respect to  on the entire interval Œ0; 1 . The function F .x; x .// in the functional (7.65) will depend on the unknown function x./ and its derivative x ./, since T D

d T dx dT D : d dx d 

Hence, relation (7.65) represents a classical variational problem.

7.6.2 Optimal mesh for solving an initial-boundary value problem Now let us consider a more complex problem of construction an optimal mesh where the function T D T .x; t / is unknown in advance and obtained as a solution to an initial-boundary value problem. It is required to find T .x; t / and, simultaneously, an optimal mesh for integrating the function. This problem can be reduced to the previous one by using an iterative procedure. A first approximation can be obtained by solving the boundary value problem on a uniform mesh in the physical space. Using the resulting solution T D T .x; t /, one finds x D x./ by minimizing a suitable functional of the form (7.65). Once this has been done, one solves the minimization boundary value problem again in the computational space  on a uniform mesh to obtain a new T D T ./, which is equivalent to finding T D T .x/ on a nonuniform mesh, and so on until a required accuracy of the solution is reached. Since the solution to the boundary value problem it is assumed to be rapidly changing, the mesh will change fairly rapidly as well. Consequently, the minimization of the functional (7.65) in actual analyses may lead to sharp changes of the mesh in both space and time. This may result in an unstable solution. To smooth out mesh condensations, one inserts additional weighted terms in the functional and tries to achieve stability by choosing suitable weights.

327

Section 7.6 Optimization of moving one-dimensional meshes

7.6.3 Mesh optimization in several parameters Let us represent the total functional as the sum of three terms with weighting coefficients i : I D 1 I1 C 2 I2 C 3 I3 : (7.66) The first term in (7.66) equals Z

1

I1 D

W .grad T /J 2 d ;

0

where W .f / stands for a norm of the function f , for example, W .grad T / D max jgrad T j, and J is the Jacobian determinant of the transformation from the physical space .x; t / to the computational space .;  /. The Jacobian determines the cell area ratio before and after the transformation and so is a condensation characteristic of the mesh. In the case of one spatial coordinate, the transformation and its Jacobian are expressed as " @x @x # ´ @x @.x; t / x D x.;  /; @ @ D det @t @t D : (7.67) J D @.;  / @ t D ; @

@

The minimization of the first term in (7.66) is responsible for mesh condensation in the regions of rapid variation of the solution T .x/ in much the same way as the functional (7.65) was minimized in the simple problem. The second term in (7.66), Z 1 J2 d I2 D 0

determines the nonuniformity of the mesh. The minimization of I2 make the mesh uniform. Therefore, one can control the rate of mesh condensation across space through the choice of 1 and 2 . Since it is assumed that the desired function has regions of rapid change, the function x D x./ will also change sharply in time. The functional I2 is introduced in order to make the mesh adapt more smoothly in time. Through the choice of the coefficient 3 of the third term Z 1 .J nC1  J n /2 d ; I3 D 0

one can control the rate of mesh condensation in time between the nth and .n C 1/st temporal layers. The integrand in the first term of the functional (7.66) depends on both the function x D x./, which is varied, and its derivative x , while that in the second and third terms depends only on x . Hence, the minimization of the functional represents a classical problem of variational calculus.

328

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

The Euler equation for the functional with integrand F .x; x / is   @ @F @F  D 0: @x @ @x Minimizing (7.66) with the above expressions of Ii (i D 1; 2; 3) yields the equation  2  J @W nC1 nC1 n . 1 C 2 C 3 / C 1  3 x D0 (7.68) x 2W @x for x D x.; t / subject to the boundary conditions x.0/ D 0;

x.1/ D 1;

arising from the condition that the interval must map into itself. Let us write out equation (7.68) for the .n C 1/st layer. The second term in (7.68) depends on x and x, while the third term is a known function of , taken from nth temporal layer. Time appears in the equation as a parameter and there is no differentiation with respect to t . The two-point problem (7.68) on a layer will be solved using the iterative procedure kC1 D ˆ.xk ; x k ; /: x

The initial approximation x 0 ./ D x n ./ is taken from the previous layer. For each iteration, the problem is solved with the sweep method. The iterative process is stopped as soon as a required accuracy " is achieved: max jxjkC1  sjk j < ": j

In case the gradient of T .x/ is large, smoothing of the functional may not suffice for the problem to be well-conditioned and then one has to smooth the function W over three neighboring points in order to ensure that the mesh adapts more gradually: Wj 1 C 2Wj C Wj C1 : WNj D 4

(7.69)

It is clear the above mesh optimization algorithm (7.66)–(7.68), which minimizes the functional I , can also be extended to the cases of non-one-dimensional spatial meshes.

7.6.4 Heat propagation from a combustion source As an example, consider the problem on heat propagation through a bar from a combustion source   @ @T @T D k.T / C f .T /; where f .T / D ˛T ˇ ; (7.70) @t @x @x

329

Section 7.6 Optimization of moving one-dimensional meshes

with thermally insulated ends, T .0; t / D T .; t / D 0, and the initial temperature distribution T .0; x/ D sin.n.  x//. At ˇ D 5 or more, the solution changes quite rapidly and so a uniform mesh requires too many nodes. Let us apply the transformation x D x.;  / to (7.70) to obtain   @T @ k.T / @T @x @T  VJ D J C f .T /; where V D : (7.71) @ @ @ J @ @t We approximate this equation using the following implicit scheme with a four-node stencil: TjnC1  Tjn 

.Jjn /1 

TjnC1  Tjn1=2 x C1=2 



nC1 nC1 nC1 KjnC1 T nC1  TjnC1 1 Kj C1=2 Tj C1=2  Tj 1=2 j 1=2  nC1 D nC1  Jj C1=2   Jj 1=2 nC1=2

! (7.72)

nC1=2

C fj C1=2 Jj C1=2 : We solve the problem iteratively. An initial approximation T D T .x; t / can be found by solving (7.70) on a uniform mesh by the scheme (7.72) with the sweep method. Once the initial T .x; t / has been found, we calculate @T =@x to obtain W .x/ and then solve equation (7.68). A final value of T D T .; t / is found by solving (7.72). The function T D T .x/ is obtained from the parametric representation by eliminating : T D T .;  /;

x D x.;  /:

It is noteworthy that, as one can see from the above analysis, the construction of adaptive meshes is a fairly costly procedure even in a one-dimensional problem. For this reason, it is beneficial to take advantage, wherever possible, of any available a priori information on the solution behavior to construct an initial nonuniform mesh and condense it in the regions where the solution is expected to change rapidly. Such information can be available in advance for many classes of problems. For example, if there is a small parameter multiplying the highest-order operator with respect to the spatial derivatives, the region of rapid change lies near the boundary of the body and represents, for example, a boundary layer in gas dynamics or an edge effect in shell theory, plastic strain localization strips or stress concentrators in elasticity, shock waves in condensed media, and other phenomena. A mesh refinement method is often used, where the initially uniform mesh is refined, as the problem is solved in a region of rapid change (e.g., near a shock wave), and interpolation formulas are used to calculate the solution in a refined cell. This

330

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

technique does not require an optimization but calls for dynamically expanding array while coding. Although this method is less accurate than the method of adaptive meshes, it has become widespread due to its simplicity and cheapness in solving problems with finite difference and finite element methods.

7.7 Adaptive 2D/3D meshes for finite deformation problems 7.7.1 Methods for reorganization of a Lagrangian mesh The solution of large deformation problems on Lagrangian meshes may result in considerable distortions of cells, which complicate the subsequent computations or even make them impossible. To remove this drawback of Lagrangian meshes, one has to either abandon using them or reorganize them periodically during the computation to make new, more regular meshes close, as much as possible, to uniform orthogonal ones. In the latter case, computational cells will change, thus requiring a recalculation of not only the new nodes but also all computational fields in terms of the new nodes. Such recalculations should be carried out after a certain number of temporal steps whenever a cell exceeds a threshold, unallowed deformation determined from the distortion of the angle between sides of the cell. The field variables, defined on the old mesh, must then be recalculated to the new mesh using interpolation formulas and the computation based on the preceding algorithm should be continued. Figure 7.7 illustrates the distortion of a square mesh caused by the displacement of a node O to a new position O1 . A

B III O

s2 s3

s5 s4

H

s6

s1 O

IV G

C II

D

I F

E

Figure 7.7. Illustration of the reorganization algorithm for a Lagrangian mesh.

The areas of the new cells are recalculated through those of the old cells using obvious geometric constructions. For example, the area of a new cell SN1 is calculated as SN1 D S1 C S1 C S3 C S4 C S6 :

Section 7.7 Adaptive 2D/3D meshes for finite deformation problems

331

The mass MN 1 within the new cell is determined in terms of the old values Si and i as MN 1 D M1 C 2 S1 C 3 .S3 C S4 / C 4 S6 : The new value of a field variable, UN 1 , in cell I (Figure 7.7) is expressed through the old values Ui using an interpolation formula with coefficients equal to the respective mass shares ˛i of Ui : UN 1 D U1 C ˛1 U2 C .˛3 C ˛4 /U3 C ˛6 U4 with ˛i D

k Si Mi D : MN i MN i

Reorganization algorithms are fairly simple in principle. However, their computer implementation is not as simple due to the fact that in actual computations, unforeseen situations may arise, where visualization of the old mesh and potential reorganized meshes is desirable in order to choose a suitable new mesh.

7.7.2 Description of motion in an arbitrary moving coordinate system In the case of large deformations, a mesh can be reorganized, as already mentioned above, using adaptive meshes, which adjust themselves to the solution and rely on moving coordinates. To this end, one should rewrite all equations in terms of moving coordinates i (i D 1; 2; 3) defined by a law of motion with respect to the fixed reference frame x k as x k D x k .i ; t /;

i 2 VN ;

t  0;

(7.73)

where VN is the domain of variation of the moving coordinates. The motion of material particles in the moving reference frame will be described by the equations (7.74) i D i . k ; t /; where  k are Lagrangian coordinates of a particle. The motion of a particle  k in the original reference frame is then defined parametrically and obtained by solving (7.74) for  k D  k .i ; t /. It is clear that i D  i corresponds to the Lagrangian description, as follows from (7.73), and i D x i corresponds to the Eulerian description, as follows from (7.74). Conservation equations can be represented in either the original basis vectors ek or moving basis vectors eQ k . The latter turns out to be preferable in many cases, since the

332

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

equations preserve their tensorial structure, thus allowing one to perform splitting in directions or physical processes more adequately. The two sets of basis vectors are related by eQ i D

@x k ek : @i

The velocity of a material particle in the moving basis is expressed as ˇ @k @x i ˇˇ k k ; vQ k D .Fik /1 v i ; v D vQ eQ k ; vQ D @x i @t ˇi while the velocity of a node of the moving mesh is ˇ @k @x i ˇˇ k k ; wQ k D .Fik /1 w i ; w D wQ eQ k ; wQ D @x i @t ˇi

(7.75)

(7.76)

with ˇ @x i ˇˇ ; v D @t ˇi i

ˇ @x i ˇˇ w D ; @t ˇi i

where Fik D @x k=@i is the displacement gradient tensor in the moving frame; it characterizes the deformation of the mesh. The above relation allow us find the relationship between the material time derivative and the time derivative in the moving frame: ˇ ˇ i @ui ˇˇ @ui ˇˇ k k @u D C .v  w / : (7.77) @t ˇi @t ˇi @x k By changing in (7.77) to the velocity components vQ i in the basis eQ k , one obtains ˇ ˇ @ui ˇˇ @ui @ui ˇˇ D C vQ rk k ; (7.78) ˇ ˇ @t i @t i @ where vQ rk D vQ k  wQ k is the velocity of a particle in the moving frame. Formula (7.78) generalizes the relation between the material and Eulerian time derivatives. It follows from (7.78) that the derivative of an integral over a moving volume in the reference frame i is given by ˇ Z Z Z d @F ˇˇ F.; t / d V D d V C .F ˝ vr /n dS; ˇ dt V V @t  S which is a generalization of formula (1.17).

Section 7.7 Adaptive 2D/3D meshes for finite deformation problems

Conservation laws in the moving coordinates i are written as Z Z Z @ F.; t / d V C .FQ ˝ vr  A/n dS D f d V; V @t S V

333

(7.79)

where FQ is a vector and A is a tensor (see Section 1.2). This formula generalizes the representation of a conservation law (1.17) to an arbitrary moving coordinate system i . In the domains of smoothness of solutions, it follows from (7.79) that the differential conservation laws for mass, momentum, and energy are expressed as ˇ @  @./ ˇˇ C k .vQ k  wQ k / D 0; ˇ @t  k @ ˇ  @  @ ˇ .vQ k  wQ k / ˇ k C k .vQ i  wQ i /.vQ k  wQ k / C Q ik  @t @  i D kj .vQ j  wQ j /.vQ k  wQ k / C Q j k C f i ; ˇ @  @.E/ ˇˇ C k  E.vQ k  wQ k / C Q j k gQj n vQ n   gQ k n rn T D .fQk vQ k C r/; ˇ @t @ k 

(7.80) where  D det.@x k =@i /, E D U C 12 v 2 is the total energy per unit mass, gQj n is the metric tensor, rn T is the covariant derivative of temperature T , r is the bulk heat source term, and fQk is the bulk force. The constitutive equations of an elastoplastic medium in tensor form can easily be rewritten in componentwise notation in an arbitrary curvilinear coordinate system as Ds ˛ˇ D .r ˛ v ˇ C r ˇ v ˛ / C ƒs ˛ˇ : Dt

(7.81)

7.7.3 Adaptive meshes The moving coordinates are chosen so that the mesh combines the advantages of a Lagrangian mesh when describing boundaries and remains as close as possible to the original mesh inside the body to ensure that the mesh’s deformation and deviation from orthogonality are small. These requirement can be met by formulating the problem of determining the transformation xi D xi .k / as a variational problem of minimizing a functional dependent on the measures H1 D

3 X

.vi  wi /2 ;

iD1

H2 D jgrad 1 .xi /j C jgrad 2 .xi /j;

H3 D .grad 1  grad 2 /2

334

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

for a given normal component of the vector w, wn D w  n D w.; t /;

 2 ;

at the boundary of the domain . The first measure, H1 , characterizes the deviation of the mesh from a Lagrangian one, the second measure, H2 , is the dimension of the mesh, and the third measure, H3 , characterizes the mesh’s orthogonality. The last two characteristics can be replaced with similar measures for the velocity field: HQ 2 D jgrad w1 .xi /j C jgrad w2 .xi /j;

HQ 3 D .grad w1  grad w2 /2 :

A functional dependent of the listed measures and, possibly, the solution itself has the form Z ˆ.i / D '.H1 ; H2 ; H3 ; u/ dx1 dx2 dt  Z (7.82) 1 '.H1 ; H2 ; H3 ; u/ d1 d2 dt; D h

where  is the Jacobian of the transformation,  D @.x1 ; x2 /=@.1 ; 2 /. Furthermore, additional conditions can be imposed on i in the form equalities or inequalities such as, for example, convexity conditions for the polygons forming the mesh cells [77] and others. In this case, conditional minimization must be carried out using the method of Lagrange multipliers. Often, it is convenient to be able to control different properties of meshes separately by representing the functional in the form of several weighted additive terms: ˆ.H1 ; H2 ; H3 ; u/ D ˛1 ˆ1 .H1 / C ˛2 ˆ2 .H2 / C ˛3 ˆ3 .H3 /: The weighting coefficients ˛i are chosen so as to ensure the dominance of one or another property of the mesh. For example, the functional ˆ in the two-dimensional case has the form “  0 .v  w 0 /2 C .v 2  w 2 /2 1 d1 d2 ; ˆ1 D 

“  ˆ2 D 

“  ˆ3 D 

@w 1 @1



2 C

@w 1 @2



2 C

@w 1 @w 2 @w 1 @w 2 C @1 @1 @2 @2

2

@w 2 @1



2 C

@w 2 @2

2 

1 d1 d2 ;

1 d1 d2

with the condition that the normal component of the nodal velocity vr equals the velocity of particles, vr  n D 0; at the boundary  of the domain .

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

335

Determining an optimal mesh through the variational principle is a costly task commensurable in complexity with the solution of the original problem. Therefore, it should only be undertaken in full in the cases where simpler approaches fail. For example, one may want to perform the optimization in only one coordinate, 1 , in which the solution changes faster, and use a priori information about the solution behavior in the other coordinate, 2 ; see [185].

7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes The presentation of this section follows the study by Zapparov and Kukudzhanov (1986) [185].

7.8.1 Algorithms for constructing moving meshes The system of dynamic equations for continuous media in two-dimensional spatial problems (plane and axisymmetric) can be represented in a moving reference frame as ˇ @U @U @U @U ˇˇ C .v i  w i / i C A 1 C B 2 C F D 0: (7.83) ˇ @t  k @x @x @x In the axisymmetric case, we have x 1 D r.1 ; 2 ; t /;

x 2 D z.1 ; 2 ; t /

and v 1 D vr ;

v 2 D vz ;

U D .vr ; vz ; r ; z ;  ; r z /;

where A.U; x i / and B.U; x i / are square matrices, F.U; x i / is a vector, and w i are the nodal velocity components of the moving coordinate frame k . m ; t /. Differentiating x i D x i .k ; t / as a composite function with k D k . m ; t /, we get ˇ ˇ ˇ @x i ˇˇ @x i @k ˇˇ @x i ˇˇ D C : (7.84) @t ˇ m @t ˇ k @k @t ˇ m Relation (7.84) allows one to address different variants of moving coordinates.  Euler–Lagrange coordinates 1 D x 1 , 2 D  2 (the first coordinate is fixed and the second one moves together with medium particles): ˇ ˇ  ˇ ˇ  ˇ @U ˇˇ @U @x2 ˇˇ @U ˇˇ 1 @U ˇ D Cv C @t ˇ m @t ˇ k @x 2 ˇ 1 @x 2 @1 ˇ 2

336

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

with 1

2

w D 0;

2

w Dv 

ˇ

@x2 ˇ v 1 1 ˇˇ : @  2

 Moving Euler–Lagrange coordinates (the velocity component w 1 is given): ˇ @x 1 ˇˇ D w1; 1 D  2 ; @t ˇ 2 ˇ ˇ     @U ˇˇ @U @U @x2 @x 1 1 @U ˇˇ 1 1 D C .v  w / C 2 1 ; (7.85) @t ˇ k @t ˇ k @x 1 @x @ @1 ˇ 2  @x 1 1 @x 2 ˇˇ 2 1 1 @x D v  .v  w / : @t ˇ k @1 @1  Arbitrary moving coordinates (w 1 and w 2 are given): ˇ ˇ @x 2 ˇˇ @x 1 ˇˇ 1 Dw ; D w2; @t ˇ k @t ˇ k ˇ ˇ @U ˇˇ @U @U ˇˇ D C .v i  w i / i : ˇ ˇ @t  k @t  k @x

(7.86)

The special case of w 1 D v 1 and w 2 D v 2 corresponds to Lagrangian coordinates. The velocity components w 1 and w 2 are generally arbitrary and one of them can be used to control the moving mesh at the boundary of the domain. The moving Euler– Lagrange coordinates (7.85) were used in the computations discussed below. Let us consider how the mesh can be controlled in more detail. At the boundary 2 D const, the velocity component w 1 can, for example, be chosen arbitrarily, w 1 D w 1 .1 ; t /, and then w 2 is determined from the condition that the normal velocity components of the material particles and moving coordinates coincide at the boundary:   @x 2 @x 1 1 : w D v  .v  w / 1 @ @1 2

2

1

1

The function w 1 D w 1 .1 ; t / was chosen from geometric considerations, computational experience, and character of deformation of bodies so as to prevent excessive distortions of the finite difference mesh. The following algorithms, minimizing a one-dimensional analogue of the functional (7.83) under some assumptions, were found [185] to be efficient.

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

337

1. Linear interpolation on an interval Œ0;  with respect to 1 :  w 1 .1 ; t / D .1  1 / v 1 .0; t / C 1 v 1 .10 ; t / .1 /1 : 2. Piecewise-linear interpolation with respect to 1 along segments of the boundary in accordance with the motion character of the boundary. 3. Interpolation of the tangential velocity component of the moving mesh, w D w 1 n2  w 2 n1 , followed by a change of variable to w 1 and w 2 ; the ni are direction cosines. Depending on the specific features of the problems considered, one of the above algorithms was chosen. For problems of deformation under pulse-like pressure, it suffices to use the first algorithm, linear interpolation. For collision problems, the second or third algorithm should be used. At internal nodes, algorithms based on linear interpolation with respect to the coordinates 1 and 2 were also used in order to control the moving mesh. One of these algorithms is given by w 1 .1 ; 2 ; t / D Œ.1  1 / w 1 .0; 2 ; t / C 1 w 1 .1 ; 2 ; t / .1 /1 ; w 2 .1 ; 2 ; t / D Œ.2  2 / w 2 .1 ; 0; t / C 2 w 2 .1 ; 2 ; t / .2 /1 :

(7.87)

It is noteworthy that linear functions of the form (7.87) in conjunction with one of the interpolation algorithms along the boundary provide an approximate solution to the problem of minimizing the functional (7.83).

7.8.2 Selection of a finite difference scheme Section 7.2 considers the Wilkins scheme [183], which is historically one of the first (and most widespread) schemes on a Lagrangian mesh used for the analysis of elastoplastic flows. Acknowledging its undoubted merits and fairly high efficiency, one should note that the development and implementation of other finite difference schemes are also important for the following reasons. 1. The Wilkins scheme is based on the Lagrangian representation of difference equations. As already mentioned, the Lagrangian form of representation may become insufficient for studying large elastoplastic deformations. It is more advantageous to rewrite the original equations in terms of arbitrary moving coordinates and approximate the resulting equations on a moving finite difference mesh. 2. The artificial viscosity is an essential part of the Wilkins scheme. Lately, preference has been given to finite difference schemes that include an approximation viscosity, which is required for stability, and to methods of parametric control for the approximation viscosity in shock-capturing analyses of wave and impact phenomena.

338

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

3. The problem of the statement and effective realization of boundary conditions on a moving curvilinear boundary has not been properly solved. In the first place, this applies to the contact boundary conditions at the collision between bodies. Significant progress in this direction can be obtained with the use of characteristic finite difference schemes. 4. The Wilkins scheme has an inhomogeneity due to the fact that some variables refer to one kind of nodes, while the others refer to another kind of nodes. For this reason, the implementation of computation algorithms for fracture processes faces certain difficulties. The system of dynamic equations for an elastoplastic medium is hyperbolic. Therefore, it is natural to use the characteristic properties of the original system in constructing numerical algorithms (see Section 5.3). Grid-characteristic schemes, as has been repeatedly noted, have some advantages but are more difficult to implement and more time-consuming. In solving dynamic two-dimensional spatial problems for elastoplastic media, one has to determine eigenvectors of the characteristic system numerically when they cannot be obtained in explicit form. For one node of the mesh, the computations based on a grid-characteristic scheme require 5–10 times more operations than those based on the Lax–Wendroff scheme. For this reason, it seems most efficient to apply grid-characteristic schemes for implementing the boundary conditions at boundary nodes of the mesh. This essentially resolves the issue of well-posed statement of boundary conditions and improves the accuracy of computation of the boundaries (including contact boundaries) as well as that of the entire analysis. At internal node, it is desirable to apply a scheme that allows one to use, on the one hand, a shock-capturing algorithm for strong discontinuities arising under impact loading and, on the other hand, analyze the regions of smooth solution with sufficient accuracy. One of the ways to obtain schemes possessing such properties is to construct hybrid schemes that have a varying order of approximation. The key idea of constructing two-step hybrid schemes suggests the following. One chooses a first-order scheme with positive approximation (stable and monotonic) as the basic scheme for the first step. In the second step, this scheme is supplied with variable control parameters so as to achieve a higher-order (second- or third-order) approximation. The values of the control parameters for each node of the scheme are chosen based on some a priori information about the solution and its behavior near the node to ensure an efficient computation of discontinuous solutions (shock waves, contact discontinuities) by a first-order scheme. In the regions where the solution is smooth and slow-varying, the advantages of higher-order schemes can be used. This makes it possible to remove some unwanted computational effects such as significant wavefront spreading for compression waves due to first-order schemes and nonphysical oscillations of the solution near the wavefront due two higher-order schemes (see Section 5.5). To isolate strong discontinuity wavefronts, one can also use differential analyzers [77]. The paper [185] proposed, analyzed, and implemented a one-parameter hybrid scheme.

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

339

The Lax scheme was taken as the basic one, which can be parametrically extended to a second-order Lax–Wendroff type scheme.

7.8.3 A hybrid scheme of variable order of approximation at internal nodes The rectangular domain ¹0  1  1 , 0  2  2 º in the coordinates 1 and 2 , which corresponds to the domain occupied by a body in the physical space (cylindrical coordinates r; z), is covered with a fixed uniform scheme with nodes .1 ; 2 /k m D .k1 ; m2 /;

k D 0; 1; : : : ; K;

m D 0; 1; : : : ; M:

On each nth temporal layer at t D n t , the derivatives @u=@i are approximated at internal nodes by the central differences     ukC1;m  uk1;m uk;mC1  uk;m1 @u @u D ; D : 1 1 2 @ k;m 2  @ k;m 2 2 The derivatives @r=@i and @z=@i are approximated likewise. The derivatives with respect to 1 and 2 are recalculated through those with respect to r and z as follows:            @u @u @z @z @u D  .Jk;m /1 ; @r k;m @1 k;m @2 k;m @2 k;m @1 k;m            @u @u @r @r @u D  .Jk;m /1 ; @z k;m @2 k;m @1 k;m @1 k;m @2 k;m where  Jk;m D

@r @1

 k;m



@z @2



  k;m

@r @2





k;m

@z @1

 k;m

is the Jacobian determinant of the transformation between the coordinates. The time derivative is approximated as follows: 

@u @t

 D k;m

with uQ nk;m D

unC1  uQ nk;m k;m t

 or

@u @t

 D k;m

unC1  un1 k;m k;m 2t

 1 n u C unk1;m C unk;mC1 C unk;m1 : 4 kC1;m

340

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

At internal nodes of the finite difference mesh, the following hybrid explicit twolayer scheme with variable approximation order is used:   n  n @u @u nC1 n n n C .vz  wz /k;m uk;m D uQ k;m  .vr  wr /k;m @r k;m @z k;m      @u @u CAnk;m C Bnk;m C Fnk;m ; @r k;m @z k;m D ˛unk;m C .1  ˛/uQ nk;m unC2 k;m     nC1 nC1 @u nC1 @u  .1 C ˛/t .vr  wr /k;m C .vz  wz /k;m @r k;m @z k;m  nC1  nC1  nC1 @u nC1 @u nC1 CAk;m C Bk;m C Fk;m ; @r k;m @z k;m (7.88) where v is the particle velocity, w is the node velocity, Ak;m and Bk;m are the matrices from the original system (7.83), and Fk;m is the vector from (7.83). The first step in (7.88) is performed by the Lax scheme. The second step the hybridity parameter ˛. The second step coincides with the first-order accurate Lax scheme at ˛ D 0 and with the second-order accurate Richtmyer scheme [147] at ˛ D 1. For each mesh node, the parameter ˛ is chosen from the interval 0  ˛  1, depending on the solution behavior near the node. For example, in order to remove the unwanted oscillations of the solution near the front of a compression wave interacting with a free boundary, one should set ˛ D 0:5 for internal nodes near the boundary and ˛ D 0 at the boundary itself. Necessary conditions for stability of the finite difference scheme (7.88) can be obtained from the analysis of the …-form of the first differential approximation (see Section 5.5). Let us write out the first differential approximation for the system of difference equations (7.88) by dropping the free term and leaving only linear terms, thus considering the case of small strains, to obtain  2  2 h1 1˛ @u @u @u 2 @ u (7.89) C A0 C B0 D E  t A0 @t @r @z 2.1 C ˛/ 2t @r 2  2  2 h2 @ u E  t B20 C 2t @z 2  @2 u ; C .A0 B0 C B0 A0 /t @r@z where A0 and B0 are the constant coefficient matrices A.u0 ; x0 / and B.u0 ; x0 /, with h1 and h2 being the step sizes along the coordinates r and z. The system of equation (7.89) is dissipative, thus satisfying the incomplete parabolicity condition

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

341

(Section 5.5), if t  p

h1 2 cmax

;

h2 t  p : 2 cmax

(7.90)

The constant cmax is the maximum absolute eigenvalue of the matrix A0 n1 C B0 n2 , provided that n21 C n22 D 1, where ni are the direction cosines of the normal to the characteristic surface. For elastoplastic flow problems, the estimate cmax  c1 holds, where c1 is the longitudinal speed of sound. Conditions (7.90) are necessary conditions for stability of the finite difference scheme (7.88). In the numerical analysis, by the scheme (7.88), of the nonlinear problems discussed below in Section 7.8.7, conditions (7.90) were ensured for all mesh nodes with a margin by taking   h2 h1 t D  min p ;p ;  D 0:6–0:8: k;m 2 c1 2 c1 For a moving mesh we set   h2 h1 t D  min p 0 ; p 0 ; k;m 2 c1 2 c1

c10 D c1 C jv  wjk;m :

In the first step, one can use a modified scheme that differs from the Lax scheme by the method of approximation with respect to the space variables:   1 @u D .ukC1;mC1 C ukC1;m1  uk1;mC1  uk1;m1 /; 1 @ k;m 41   1 @u D .ukC1;mC1 C uk1;mC1  ukC1;m1  uk1;m1 /: 2 @ k;m 42 Once a solution to (7.88) has been obtained for the nth layer, equations (7.86) are integrated at internal nodes of the finite difference mesh by the second-order accurate scheme   nC1 n D rk;m C 0:5 t wrnC1 C wrn k;m ; rk;m   nC1 n zk;m D zk;m C 0:5 t wznC1 C wzn k;m :

7.8.4 A grid-characteristic scheme at boundary nodes At boundary nodes, the system of equations must be locally transformed to new coordinates .;  /, with  measured along the inward normal ‌ D .1 ; 2 / to the contour of the domain occupied by the body and  measured along the tangent to the contour. We have ˇ @u @u @u ˇˇ C B C F D 0; (7.91) C A ˇ @t k @ @

342

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

where A D A 1 C B2 ; v D vr  2  vz  1 ;

B D A2  B1 C .v  w /E; w D wr 2  wz 1 :

System (7.91) is reduced to the characteristic form   @ui   @ui  @ui C c˛ C Bij C Fi D 0; ˛i @t @ @ or ˛i

@ui @ui C F0˛ D 0; C B0˛i @l˛ @

˛ D 1; : : : ; 4;

i D 1; : : : ; 6;

(7.92)

where c˛ are eigenvalues of the matrix A , ˛i is a rectangular matrix of left null vectors of the matrix A , and @=@l˛ is the derivative operator along the characteristic that has a nonpositive propagation speed, c˛  0. There is no summation over ˛. System (7.92) is supplemented with the boundary conditions a1 v C b1  D f1 .t /; a2 v C b2  D f2 .t /; where v and v are the normal and tangential velocity components to the boundary contour;  and  are the normal and tangential projections of the stress vector. The spatial derivatives are approximated by first-order accurate difference relations. To be specific, consider the boundary 2 D const to obtain 

@ @1

 D k;m

unkC1;m  unk1;m 21 unk;mC1  uQ nk;m

;

 @ D @2 k;m 2 ;   unC1  uQ nk;m @ k;m : D @t k;m t



The derivatives @r=@1 , @r=@2 , @z=@1 , and @z=@2 are approximated likewise. The difference derivatives (7.8.4) with respect to 1 and 2 are recalculated in terms of

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

343

those with respect to r and z. The finite difference scheme is written as ²   n  n @u @u  n n ˛i .uQ nk;m /unC1 D  . u Q / u Q  t .v  w / C .v  w / r r z z ˛i k;m k;m k;m @r k;m @z k;m  n @u C Ank;m @r k;m  n ³ @u n n C Bk;m CF ; (7.93) @z k;m k;m a1 .u /nC1 C b1 . /nC1 D f1 .t nC1 /; k;m k;m C b2 . /nC1 D f2 .t nC1 /: a2 .u /nC1 k;m k;m It generalizes the first-order accurate grid-characteristic scheme suggested in [75] to the case of finite deformations. The system of algebraic equations (7.93) is of the sixth order (and even seventh order in the non-isothermal approximation). It is solved at each boundary node numerically by Gaussian elimination with partial pivoting. On the boundary segments where external pressure p.t / is prescribed, we have a1 D 0;

b1 D 0;

a2 D 0;

b2 D 1;

f1 D p.t /;

f2 D 0:

On the free surface, f1 .t / D 0. For the symmetry axis, we have a1 D 1;

b1 D 0;

f1 D 0:

Furthermore, at the axial nodes, 0=0 indeterminate forms have to be evaluated, which arise in the free term Fi as r ! 0 (see Section 6.5). For example, ur @ur D : r !0 r @r lim

The equations of motion of the boundaries can be written, using (7.77), in the form @x 2 @x 2 C c 1 D v 1 @t @

(7.94)

where c D .v 1  w 1 /.@x 1=@1 /1 is the transfer speed along the coordinate 1 . Equation (7.94) can be integrated with a Courant–Isaacson–Rees type grid-characteristic scheme (see Section 5.2):   1 nC1 nC1 n nC1=2 ˙ n 1 1 ˙n x2 D x2  t c  x2 . /  .v2 C v2 / 2 with 1 cnC1=2 D .cnC1 C cn /; 2

 1 cn D .v1n  w1n /21 .x1 /nkC1;m  .x1 /nk1;m ;

344

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

where ˙ x2n is the forward or backward difference, depending on the sign of c , and v2˙n is the value of the velocity component v2 on the nth at the boundary node with coordinates x2˙n D x2n  t cn

˙ x2n : 1

7.8.5 Calculation of contact boundaries1 For a contact boundary, the following conditions of normal contact are set: C D  ; C D  D 0; vC D v ;

˙  0;

(7.95)

where v is the normal velocity component, the plus and minus superscripts refer to different contacting bodies, and  is a normal vector to the contact boundary. The boundaries of contacting bodies are represented by broken lines consisting straight line segments between boundary nodes of the mesh. The mutual arrangement of nodes on contact boundaries of adjacent bodies is arbitrary and can change during the computation. The boundary conditions (7.95) allow the bodies to slip relative to each other. A boundary node of one body (to be specific, we will refer to this body as body “C”) is considered to be a contact point if, at a distance not exceeding some " from it, there is a boundary segment of body “”. So every boundary node of the contacting bodies can be checked whether it belongs to a free or contact surface. In the latter case, an adjacent counterpart on the boundary of the other body can also be found. Consider a segment of the contact boundary in the coordinates 1 ; 2 (Figure 7.8). Let .k; m/ be a contact point and let .k; m/ denote its counterpart, with the line over k; m indicating that the point does not generally coincide with a node of the finite difference mesh. A common normal vector ‌ is drawn at node .k; m/ and system (7.83) is reduced to the characteristic form (7.92): C ˛i

C @uC @uC i i C C @ui C c˛C C C  B ˛ ˛i ij @t @ @  2 1 @u @x @x C C .v 1  w 1 / C C C ˛i ˛i Fi D 0; (7.96) @ @ @

where c˛ are eigenvalues of the matrix AC , C ˛i is the matrix of left null vectors, and @u=@ D @u=@.

1

A modern survey of the methods used to solve contact problems in continuum mechanics can be found in [16].

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

345

η+2 k–1

k

k+1

+

m+1 η+2

m

(k, m) k – (s, m)

m

η2–

m+1

– s–1

s

s+1

s+2

η2–

Figure 7.8. Calculation of contacting bodies during a dynamic interaction.

The finite difference scheme is written as  ²  C n  n @ui C nC1 C C n C n C @uj .u / D  / t .c / C B . u Q C ˛ k;m ˛i i k;m ˛i i k;m ij @ k;m @1 k;m  1 1 n  C n @ui @x @x n C .v 1 w 1 /k;m 1 1 @ @ k;m @1 k;m ³ n C .FiC /k;m (7.97)  n C ˛i D ˛i .uC /;

n

.uC i /k;m D

1 C n n .ui /kC1;m C .uC i /k1;m : (7.98) 2

The spatial derivatives in (7.97) are approximated using nodes .k C 1; m/, .k  1; m/ and .k; m C 1/. The finite difference scheme for body “” is constructed likewise, using linear interpolation between nodes. The interpolation between closest neighbors turns out to violate the stability condition for the scheme. The interpolation of the difference operator (7.97) itself was found to be effective: ²   n  n  @ui  nC1   n   @u . u Q .u / D  / t .c / C B  ˛ s;m ˛i i s;m ˛i ij s;m @ s;m @1 s;m  2 1 n   n @ui @x @x n C .v 1 w 1 /s;m 1 1 @ @ s;m @1 k;m ³ C .Fi /ns;m (7.99)

346

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

with n .uQ  i /s;m D

i h i± 1° h  n n  n  n  .ui /sC2;m C .u C .1  / .u / / C .u / i s;m i sC1;m i s1;m ; 2

D

Q 1s;m 1s;m 1

;

0    1;

     uQ s;m : !˛i D !˛i

The derivatives @u=@i are approximated as follows:   i 1° h  n @u n  n  .u D /  .u / i i s;m sC2;m @1 s;m 2 i± h n  n 1 1 / C .1  / .ui /sC1;m  .u (7.100) i s1;m . / ;  n i h @u n n  n 1 1 D .u Q i /sC1;m1 C .1  /.ui /s;m1 .u i /s;m . / : 2 @ s;m Equations (7.97) and (7.99) are supplemented with the boundary conditions (7.95). The system of 12 linear equations (7.97), (7.99), and (7.95) for 12 unknowns nC1 C nC1 .ui /k;m , .u i /s;m is solved by Gaussian elimination with partial pivoting. In the nC1 solution obtained, the values .u i /s;m relating to body “” are dropped. The inequalnC1

ity .C /k;m  0 is checked and if it does not hold, the computations are repeated with the boundary conditions C D C D 0. This algorithm is applied to all contact boundary nodes of both bodies. The equations of motion of contact boundaries coincide with (7.94) and are integrated after obtaining the solution on the new temporal layer, in a similar way to the calculation of boundary nodes.

7.8.6 Calculation of damage kinetics In the moving coordinates, the simplest equations of the kinetics of growth of microdamage (microdefects, micropores) [87] become pp0 @N @N C .v i  w i / i D NP 0 e p1 H.p  p0 /; @t @x @R @R p  p  C .v i  w i / i D R H.p  p /; @t @x 4

(7.101)

where N is the number of microdefects per unit volume, p D .r C  C z /=3 is pressure, R is the average size of microdefects, N0 , p0 , p , , and p1 are material constants, and H.x/ is the Heaviside step function. Macrofracture is assumed to begin when the total volume of microdefects per unit volume of the material reaches a certain critical value, #0 . When # D 8NR3 reaches #0 at a computational point in the medium, a free surface (cut, crack) is formed [156]. The orientation of this free surface coincides with

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

347

the orientation of the plane of maximum principal stress. A Lagrangian finite difference mesh is introduced in a neighborhood of the affected point. One sets w 1 D v 1 and w 2 D v 2 and stores the orientation of the fracture plane in the local Lagrangian basis. In the subsequent analysis, the point is treated as double, belonging to two faces of the crack formed. It is calculated twice, once for each face, as a boundary point lying on a stress free surface. In the computer implementation of this algorithm, the only additional information that was stored for each affected point was the jumps Œv i and Œx i at the crack faces (cut edges), while the jump in the stress tensor was assumed to be zero, Œij D 0. Thus, the above algorithm does take into account the discontinuity in the tangential stress at the crack faces. Equation (7.101) is integrated along the characteristic that has a zero velocity of propagation by an explicit second-order accurate scheme in time. The equation for R , the second equation in (7.101), has an exponentially growing solution. The stability of the scheme was ensured by using an implicit approximation of the right-hand side. In the case w i D v i , the scheme has the form (Lagrangian coordinates)  RnC1  Rn 0:5.p nC1 C p n /  p0  RnC1  Rn D  H 0:5.p nC1 C p n /  p0 : t 2 4 For computations on a moving mesh, the convective terms on the left-hand sides of (7.101) must be transformed as follows: @N @N D ci i ; @x i @ @R @R .v i  w i / i D ci i ; @x @ @i ; ci D .v k  w k / @xk .v i  w i /

where ci is the speed of convective transfer along the coordinate i . With the approximation (7.8.6), forward and backward finite differences must be used depending on the signs of ci .

7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains This section presents a few solutions to some applied problems obtained by the methods described above. Figure 7.9a depicts a solution to the problem on the penetration of a rigid body into an elastoplastic slab [40]. The solution was obtained by the method of reorganization of a Lagrangian mesh (see Section 7.6). The fracture of an elastoplastic slab hit by a deformable cylinder is illustrated in Figure 7.9b. The planes along which the fracture occurs are indicated in the affected cells. The solution was obtained by the method described in Section 7.1.

348

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

– – X1

–– X1

(a)

(b)

Figure 7.9. (a) Penetration of a rigid body into an elastoplastic slab; the deformation of the Lagrangian mesh is obtained by the method of reorganization. (b) Fracture of an elastoplastic slab hit by a steel striker; the planes along which the fracture occurs are indicated in the affected cells.

2d

Z

3 R 1 2

Figure 7.10. Diagram of action of a cumulative charge on a metal cladding panel; 1, cylindrical explosive charge; 2, detonation wave; 3, cladding panel.

Figures 7.10–7.12 depict computational results illustrating the action of an explosive cumulative charge on a metal cladding panel [185]. A diagram showing how the charge acts is displayed in Figure 7.10. After the detonation of the charge, a detonation wave (2) begins to propagate through the explosive with a constant speed D. At some time, the wave reaches the conical-shaped metal cladding (3) causing it to deform. Depending on the parameters of the panel and explosive device, the following main cumulation modes: (a) eversion of the panel without a pronounced cumulation effect (Figure 7.11); (b) intensive flow of the panel material toward the axis of symmetry with the formation of a compact hight-speed element (Figure 7.12a); (c) collapse of the panel with the formation of a high-speed cumulative jet (Figure 7.12b).

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

349

Z

4

1.0

3 0.5 2

R

1 0

0.5

1.0

Figure 7.11. Eversion of a panel; no pronounced cumulation effect in the initial stage of the process.

Z Z C1

2.0 2.0 4 1.5 1.5 C1

1.0

1.0 3 2

0.5

0.5 1 R

0

0.5

(a)

1.0

R 0

(b)

Figure 7.12. (a) Intensive flow of a conical panel’s material to the axis of symmetry; formation of a high-speed cumulative jet to pierce a barrier plate. (b) Collapse of the panel with the formation of a high-speed cumulative jet. The velocities of particles are indicated by arrows.

350

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

Axis of symmetry

Rigid Punch

Workpiece

Rigid foundation

Figure 7.13. Indentation of a rigid punch with a sinusoidal profile into an elastoplastic workpiece lying on a rigid foundation.

The solutions were obtained with the method of moving adaptive meshes (see Section 7.7 and [185]). Figures 7.13a, b, and c depict computational results for the indentation of an perfectly rigid punch with a sinusoidal profile into a rectangular workpiece made of an ideal elastoplastic material, lying on a rigid foundation. Figure 7.13b shows the final penetration obtained on a Lagrangian mesh. Further penetration was impossible because of a strong distortion of the mesh. Using an adaptive mesh made it possible to perform the computation until complete penetration (Figure 7.13c). Figures 7.14 illustrate the destruction of an elastic barrier (glass) caused by the impact of a steel cylinder at a speed v0 D 0:12 c1 , where c1 is the speed of sound in glass. The regions of fracture are shaded. The problem was solved on a Lagrangian mesh [185] using the fracture kinetics as described by equations (7.101).

351

Section 7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes

0.4

Z

Z 0.5

R 0

0.5

R

1.0 0

0.5

(a)

1.0

(b)

Figure 7.14. Destruction of an elastic plate (glass) caused by the impact of a steel cylinder; the fracture regions are shaded.

t = 0.29 µs V0 = 50 km/s

0.030

+ projectile particles × slab particles 0.025

0.020

0.015

0.010

0.005

0.005

0.010

0.015

0.020

0.025

0.030

Figure 7.15. Fracture of an aluminium slab hit by a projectile (same material) at a high speed.

Figure 7.15 displays the picture of fracture of an aluminium slab hit by a projectile, made of the same material, at a high speed. The solution was obtained by the PIC method described in Sections 7.4 7.5 [68].

352

Chapter 7 Solution of elastoplastic dynamic problems with finite deformations

7.9 Exercises 1. Show that if a conservation law is written in the non-divergence form (1.22), then the application of the formulas of natural approximation (7.3) will result in a nonconservative finite difference scheme. 2. Write the system of equations of motion in two spatial dimensions for a hyperelastic material in a conservative form. Approximate it with the formulas of natural approximation (7.3). 3. Generalize the formulas of natural approximation (7.3) to the three-dimensional case. 4. Compare the natural approximation formulas for derivatives with respect to two variables on a regular quadrangular mesh with the approximation formulas for a bilinear finite element (see [188, 89] for these formulas). 5. Analyze the system of difference wave equations for the Lax scheme for conservativity. 6. Write the system of equations describing the propagation of one-dimensional waves in Eulerian variables in the divergence form for a nonlinear elastic medium at large displacements. Obtain a conservative finite difference scheme for this system. 7. Prove that the total time derivative of a function f .x1 ; x2 ; t / can be written in moving coordinates i , whose law of motion relative to the fixed reference frame is xk D xk .i ; t /, as follows: ˇ @f ˇˇ @f d D C .vQ i  wQ i / ; ˇ dt @t  @i where vQ i are the particle velocity components, wQ i are the nodal velocity components of the mesh i in the basis associated with moving mesh i . 8. Obtain the expression of differential conservation laws in the moving coordinate system i in terms of vector components referred to the moving basis eQi . 9. Write the equation of motion in a mixed Lagrangian–Eulerian reference frame with one coordinate, 1 , being Lagrangian and the other, 2 , Eulerian. 10. Derive a splitting scheme for the constitutive equations of an isotropic incompressible hyperelastoviscoplastic medium with an elastic potential dependent on p the second invariant of the plastic strain tensor, ˆ.I2 /, and a viscous potential p dependent on the second invariant of the plastic strain rate tensor ‰.IP2 /.

Section 7.9 Exercises

353

11. Solve the problem on the propagation of plastic waves in a semiinfinite bar made of an elastoviscoplastic material with the constitutive equation ´   2  1 @ 1  @" E z ˛ ; z  0; D C ' E ; where '.z/ D @t E @t  s "s 0; z < 0: Use an adaptive mesh that minimizes the solution gradient. Take the zero initial conditions and the boundary condition  .x D 0; t / D 0 H.t / with 0 D 1:5s ,  D 105 s, s D 103 MPa, E D 2  106 MPa, and H.t / being the Heaviside step function. 12. Construct a nonuniform mesh on the interval 1  x  1 that minimizes the 2 gradient of the function y D C e ˛x with C D 102 , ˛ D 103 , and discretization number n D 102 . 13. Solve the axisymmetric problem of unsteady heat conduction with a point heat source on the cylinder axis. The source intensity is q D q0 .T =T /˛ . The boundary condition is T .r D 1/ D T . Use a nonuniform mesh that minimizes the temperature gradient. The initial temperature is uniform across the body: T .t D 0/ D T0 . Set T0 D T D 20 ı C and ˛ D 5. 14. An ideal plastic material is compressed by a system of rigid plastic punches without friction. The upper punch, having a sinusoidal profile, moves vertically with a constant speed v D v0 . The lower punch is fixed. The lateral punches can only move in the horizontal direction (see Figure 7.13a). Introduce an Euler–Lagrange coordinate system. Write the system of equations for the plane problem of plasticity in these coordinates.

Chapter 8

Modeling of damage and fracture of inelastic materials and structures 8.1 Concept of damage and the construction of models of damaged media 8.1.1 Concept of continuum fracture and damage The description of fracture processes relies on viewing fracture as the cause that makes a material lose its strength, i.e., the ability to resist deformation, due breaking of internal links in the material structure. A material can lose its strength as a result of thermomechanical loading when certain thresholds, determining a fracture criterion, are reached or under non-thermomechanical actions, such as chemical reactions, irradiation, and others. Deformation and fracture experiments for standard specimens in tension and/or shear (torsion of hollow cylinders) exhibit softening segments in the stress-strain ( -") diagrams, where the stress decreases with increasing strain. When researchers interpret  -" diagrams, they should bear in mind that the stress and strain are not the only parameters of state of the material, and therefore such diagrams only show a slice of the process. Damage can be divided into two main types: quasi-brittle damage of elastic materials and ductile (viscous) damage of elastoplastic materials beyond the elastic limit. Typical pictures of fracture for ductile and brittle materials are illustrated in Figures 8.1–8.2. These pictures indicate significant differences in material structure: ductile fracture occurs predominantly through the formation of voids, while quasibrittle fracture is chiefly caused by microcracks. The basic types of fracture test include (i) shear (tension) at constant strain rate in quasistatics and (ii) collision of planes (bars) in dynamics. Shear leads to intensive local heating and thermal softening, which cause a phase transition in a shear band; this is a local phenomenon where microcracks and microvoids merge into a single adiabatic-shear band at which material fracture will further occur. Strain localization bands are observed in not only viscoplastic but also quasi-brittle materials, which is possible due to intensive local heating and softening. Figure 8.2 illustrates the mechanism of nucleation of voids and microcracks as well as the mechanism of neck formation in a specimen in tension [138]. The other type of fracture test represents, as already mentioned in Chapter 1, spall fraction during the collision of plates. The fracture is caused by the interaction of tension waves reflected from a free surface. The experiments by Curran, Seaman,

Section 8.1 Concept of damage and the construction of models of damaged media

(a)

2 μm (b)

355

2 μm

Figure 8.1. Character of fracture on microlevel for ductile (a) and brittle (b) materials.

Figure 8.2. Mechanism of formation of a strain localization band during void nucleation (top left); strain localization during the formation of large voids (bottom left); formation of voids (characteristic size 0.001–0.1 mm) in an experimental specimen subject to tension (right).

and [33, 34], Kanel, Razorenov, Utkin, and Fortov [69], and other authors have repeatedly confirmed this fact. Plastic materials exhibit ductile spall fracture with the formation of spherical voids due to hydrostatic tension. For fracture to takes place, a sufficiently high collision speed and a long enough tensile pulse are required so that the mechanism of nucleation, growth, and coalescence of microvoids could result in the formation of a crack. Figure 8.3 illustrates the formation of a macrocrack due to coalescence of microvoids.

8.1.2 Construction of damage models Historically, first attempts to describe continuum fracture of a material relied on strength criteria or strength theories, which treated fracture as a stress-strain state of

356

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

Figure 8.3. Formation of a macrocrack (left) during the coalescence of microvoids (right) in plate collision tests.

the material at which fracture onsets rather than a process developing in the course of loading. This approach contradicted in many respects the experimental data and general postulates of continuum mechanics (such as fracture criteria) and was not linked to the process of damage development. In early papers on damage theory, fracture was treated as the development of microdefects such as microcracks and microvoids. Damage was associated with the formation of voids (porosity), resulting in a decrease in the area where stress is applied and, hence, in a decrease in effective material moduli, which is easy to establish experimentally. In the simplest elastic model of a bar subjected to tension, the influence of microdefects on the load carrying capacity appears as a decrease in the Q where EQ D E.1  D/ is the effective modulus effective Young’s modulus:  D E", decreasing as damage (porosity) D increases. In modern solid mechanics, damage is understood as violation of continuity of a material due to an external action; this violation is nevertheless treated within the framework of a continuous medium containing defects. The notion of damage was first suggested by L. M. Kachanov in 1958 [65, 66] and Yu. N. Rabotnov in 1959 [139]. In the 1960s, a new approach took shape in the works by Soviet scientists in continuum mechanics where damage was treated as a process associated with loading. This point of view gained recognition very quickly and formed, within a short time, a new branch of continuum mechanics known today as damage mechanics or continuum damage mechanics. The scientists who contributed most to this area include A. A. Ilyushin and B. E. Pobedrya [60], V. N. Kukudzhanov [88], D. R. Curran, L.Seaman, and D. A. Shockey [33], A. L. Gurson [52], V. Tvergaard [174], C. C. Chu and A. Nedelman [26], and many others. The continuum approach suggests the construction of theoretical continuum models that characterizes fracture, based on deformation equations for undamaged and damaged materials, as a process. This approach describes the appearance of fracture

Section 8.1 Concept of damage and the construction of models of damaged media

357

surfaces and zones without specifying their details; for details, see [24, 43, 9, 119, 48, 79, 12, 8, 25]. The stress can drop as the material loses its strength for non-thermomechanical reasons at constant strain. This indicates that, with the continuum description, the deformation and fracture processes can and should be treated as independent (which does not rule out their mutual influence), while the development of fracture should be characterized with a separate parameter of state, damage. The description of softening (decrease in the yield stress as the strain increases) on the basis of Prandtl–Reuss type incremental yield theories results in ill-posed boundary value problems. This is due to the violation of Drucker’s postulate or its mathematical analogue Hadamard’s criterion, according to which the elastoplastic modulus changes sign and, consequently, the static equations change their type from elliptic to hyperbolic while the dynamic equations conversely change their type from hyperbolic to elliptic. For this reason, describing fracture processes directly within the framework of standard elastoplastic models turns out to inadequate. Let us illustrate the meaning of the damage parameter by the simple example of a bar subject to one-dimensional dynamic tension. Consider the simplified system of equations of the Prandtl–Reuss nonlinear elastoplastic model 

@ @v D ; @t @x @u ; "D @x

 D  ."  "p /; vD

@u ; @t

@"p D H.ˆp / p : @t

ˆp ."; "p / D 0;

The last two equations are the yield criterion and associated flow law. In terms of increments on a temporal step, the stress-strain relation is @" @ D Et ; @t @t

Et D

@ ; @"

where E t is the current elastic modulus dependent on the total and plastic strains as well as on the loading mode (active or unloading). Hence, we arrive at the system of equations 

@ @" @v D ; @t @" @x

@" @v D : @t @x

The well-posedness of this system, which is hyperbolic in dynamics and elliptic in statics (provided that inertia is neglected), is determined by Hadamard’s condition: the speed of propagation of a disturbance in a medium must be a real quantity. In softening, Hadamard’s condition results in the inequality Et D

@ < 0: @"

358

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

Drucker’s condition is an alternative form of Hadamard’s condition: d d " > 0: Fracture (segment 2–3 in the  -" diagram in Figure 8.4a) is accompanied with softening, E t < 0, and results, within the classical theory, in an ill-posed problem. ε

U

σ 2 1

x

3 ε

0

(a)

(b)

X

(c)

Figure 8.4. A typical schematic stress-strain diagram of a material (a): segment 0–1 corresponds to elastic deformation, 1–2, plastic deformation, and 2–3, softening (fracture). Qualitative graphs of the exact solution for a bar in tension with a weakened central part: strain (b) and displacement (c).

On the other hand, fracture processes can be modeled with an elastic problem for an inhomogeneous in tension with a weakened (fractured) middle segment, where E1 E. The exact solution reveals a strain burst in the fracture zone (Figure 8.4b), with the displacement changing as shown in Figure 8.4c. The question arises as to whether it is possible to construct a physically and mathematically well-posed elastoplastic model that would predict the appearance of lowstrength zones similar to the fracture zone in the above model problem. Damage theories provide a positive answer. Indeed, the system of equations describing a damaged elastoplastic medium has the form @ @v D ;  D  ."  "p ; f /, @t @x @u @u ; D v, "D @x @t @"p D H.ˆp / p ; ˆp ."; "p / D 0; @t @f D H.ˆf / f ."; "p ; f /  0; ˆf ."; "p ; f / D 0; @t where f is an internal variable that characterizes the damage of the material as it loaded. It follows that @ @ @."  "p / @f @ D Et C Ef ; Et D ; Ef D : @t @t @t @" @f 

Section 8.1 Concept of damage and the construction of models of damaged media

359

In damage theory, softening is now determined by the damage parameter rather than the plastic strain or active loading condition. Softening or, more generally, fracture is treated as the loss of strength exhibiting itself as a decrease in the resistance moduli due broken links or, in other words, the appearance of microcracks. Fracture is described as a process independent of the strain. The damage parameter, which characterizes fracture, is related to the material structure (its ability to resist loading) rather than the strain. A decrease in the stress regardless of changing strain is accounted for by the second term in the above expression of the stress rate. One can see that Ef < 0 for df > 0, which reflects the fact that the stress decreases as damage increases with E t .f / > 0. At the same time, the positivity of the elastic moduli ensures that the necessary conditions for well-posedness of initial-boundary value problems in the sense of Hadamard and Drucker, since the total differential d > 0. In the above presentation, the theory formulation and reasoning were given in a simplified manner. For example, temperature was not included, hardening parameters were not considered, and strain rate hardening was not involved. This was done on purpose to convey the main idea of damage theories more clearly. These simplifications will not be used in the sequel. It should be emphasized that prior to the appearance of damage theories, strength theories typically did not relate the fracture criterion to the deformation laws and treated it independently. In most damage models, deformation and continuum fracture are described as a single process, suggesting that a body undergoes fracture as a result of deformation when a critical value of damage is reached. Damage is modeled by nucleation and growth of microdefects, such as dislocations, microcracks, microvoids, etc., up until macrocracks are formed. This approach suggests that an averaging or homogenization of material properties over a microvolume must be performed at some stage (microdefects are typically 105 –103 cm in size) resulting in the determination of some effective properties, which are already treated within the continuum context (the mesoscale of damage carriers is typically 103 –101 cm). This is a physical approach, where damaged materials are modeled based on a certain mechanism of microstructure formation, inelastic deformation development, and continuum fracture. Alternatively, a thermodynamic, or phenomenological approach can be used to describe continuum fracture or damage of materials. With this approach, a damaged material is assumed to be initially continuous with a certain internal structure characterized by a set of internal variables (in particular, damage, irreversible plastic strain, hardening, etc.) associated with the stress-strain state of the material. To determine the internal variables, one postulates kinetic equations that agree with the basic principles of thermodynamics and the theory of constitutive relations [161, 118]. The system of constitutive equations in conjunction with the conservation laws forms a system of thermomechanical equations characterizing the material response up until failure.

360

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

This approach has been detailed by Kondaurov and Fortov [74], Maugin [119], Krajcinovic [78], Lemaitre [104, 80], and others. Intermediate approaches make use of various objects to describe the material structure on the mesoscale, which include grains, polycrystals, molecular chains, multiphase mixtures, etc. One postulates a certain mechanism of formation of reversible and irreversible deformations. The sections of mechanics that use such approaches to construct constitutive equations are known as mesomechanics, molecular mechanics, crack mechanics, etc. Any continuum fracture process occurring in tension consists of three consecutive stages. In the first stage, pre-existing voids increase in size due to the action of the tensile stress applied. Then, when a critical strain has been reached, new voids begin to nucleate and grow. In the third stage, when the yield stress has been attained in the intervoid space, large voids begin to merge. An alternative mechanism is possible, where microvoids merge into long chains connecting larger voids [13]. This results in the formation of macrocracks, whose further development and propagation eventually leads to failure of the structure and the loss of load-carrying ability. The whole process is described based on micromechanical ideas and equations, taking into account microdefects [171, 44], or phenomenological ideas and thermodynamic relations of continuum mechanics with internal variables of state characterizing damage [118]. Also a combination of the two approaches can be used to describe fracture processes [95]. Damage can be present at the initial time or can only arise due to loading (nucleation of voids) and have a threshold condition similar a yield criterion. The Gurson–Tvergaard–Needlman (GTN) model [52, 26, 174] deals with twophase materials represented by a viscoplastic matrix, described by the Prandtl–Reuss model with the von Mises yield criterion, and variable tiny defects, spherical voids, which can exist initially or nucleate under loading. As mentioned above, there are two basic approaches to describing damage, physical and phenomenological. The physical mechanism of nonlinear deformation is nucleation and development of defects in the crystalline lattice under the action of thermomechanical loads. There are numerous mechanisms of development of defects, with great diversity in their nature; these and related issues are studied by solid state physics. Today, the nature of dislocations on the microlevel as well as its connection with plastic deformation are well understood. Gurson and Tvergaard [52, 174] suggested a yield criterion for media with periodically arranged spherical voids; the criterion was obtained using a theoretical rigid plastic solution. Gurson, Tvergaard, and Needlman suggested a damage model (known as the GTN model) for an effective elastoplastic material; the model is independent of the strain rate and has porosity as the scalar measure of damage. The model includes the nucleation and growth of voids in the course of plastic deformation and describes plastic compressibility of the material and the effect of dilatancy. A critical value of porosity is taken to be the fracture criterion.

Section 8.1 Concept of damage and the construction of models of damaged media

361

The GTN model is fairly widespread; it has been used for solving many specific problems. The model well describes the effect of the first stress invariant on the plastic properties and compressibility of a softening plastic material.

8.1.3 Constitutive equations of the GTN model Consider the constitutive equations of the coupled damage elastoplastic model known in the literature as the GTN model [52, 26, 174]. In the elastic domain, the material obeys Hooke’s law:  D D W "el :

(8.1)

The elastic and plastic strain rates are additive: " D "el C "pl :

(8.2)

For a porous material, Gurson obtained a yield criterion [52] from solving a spherically symmetric deformation problem for a spherical void in an ideal plastic material. This criterion is written as    2   3 q2 p S (8.3) C 2q1 f cosh   1  .q1 f /2 D 0; ˆD Y 2 Y 1=2  is the tangential where s D pI C  is the Cauchy stress deviator, S D 32 s W s  pl  1 stress intensity, p D  3  W I is the hydrostatic pressure, Y "Nm is the yield stress of the continuous material (the matrix), dependent on the plastic strain intensity, f is the porosity (volume fraction of voids), p=Y is the stress-strain state triaxiality, and q1 and q2 are some constants. Tvergaard [174] introduced the constants q1 and q2 (as correcting factors for porosity and pressure) to ensure that the Gurson model is in agreement with the numerical analysis of the model problem on the extension of a specimen, made of the same material, with a periodic porous structure in the case of plane strain. Tvergaard found that q1 D 1:5 and q2 D 1:0. By changing these parameters, one can improve agreement between the numerical analysis and experimental data. Figure 8.5 depicts the  ."/ relationship for a material with initial porosity f0 in uniaxial tension or compression. In compression, the material hardens, since the porosity decreases, while in tension, the material softens due to the nucleation and growth of voids. The above model describes the response of materials with not-too-large void fraction ( 10–15%). Although the matrix material is assumed to be plastically incompressible, the response of the effective material (which contains voids) will depend on pressure and void fraction.

362

Chapter 8 Modeling of damage and fracture of inelastic materials and structures σ

Matrix material

σy σy 0 Effective porous material

ε

– σy 0

Figure 8.5. Typical -" diagrams for the GTN model (ideal plasticity).

The yield criterion (8.3) is taken to be the plastic potential. The plastic strain obeys the associated flow rule   1 @ˆ 3 @ˆ @ˆ pl P P D  IC s ; (8.4) "P D @ 3 @p 2S @S where P is a nonnegative scalar factor. Evolution of plastic strain intensity and porosity. Hardening and softening of ma pl  terials is characterized by the relationship Y "Nm . From the fact that the work corresponding to the plastic strain is only done by the matrix, one obtains an equation pl characterizing the evolution of "Nm : r 2 pl pl pl pl pl "m W "m ; .1  f /Y "NPm D  W "P ; "Nm D (8.5) 3 pl

where  and "P pl are the stress and strain rate tensors for the effective material, "m is the plastic strain of the matrix, and Y is the yields stress of the matrix material. The material porosity changes due to the growth of existing voids, fgr , and nucleation of new voids, fnucl : fP D fPgr C fPnucl : From the continuity equation it follows, under the assumption that the matrix material is incompressible, that the growth of voids is characterized by the equation (8.6) fPgr D .1  f /P"pl W I: Voids nucleate due to relative movement of grains and the nucleation rate depends on the plastic strain intensity. Chu and Needleman [26] suggested the relation   pl  pl  fN 1 "Nm  "N pl P P P A "N D p exp  : (8.7) fnucl D A "N ; 2 sN sN 2

Section 8.2 Generalized micromechanical multiscale damage model

363

The strain intensity at which voids begin to nucleate was assumed to obey a normal distribution with mean "N and standard deviation sN . The volume fraction of nucleating voids equals fN . Voids only nucleate in tension (to be precise, when the volume pl plastic strain is positive, "i i > 0).

8.2 Generalized micromechanical multiscale damage model for an elastoplastic material in tension This section outlines a generalized micromechanical multiscale damage model suggested and developed by the author in [88, 92, 95]. An analysis of the GTN model’s equations reveals that, apart from its considerable merits, the model has a number of drawbacks. First of all, it lacks residual stresses, which lead to kinematic hardening. Furthermore, if the material is treated as ideal plastic with f D 0, there is no strain hardening or any other kind of hardening. The hypothesis that porosity arises simultaneously with the onset of plastic deformation contradicts experimental data, since the hypothesis suggests that the elastic stage is immediately followed by a softening stage as the specimen is stretched. In problems where there is no initial geometric constraining of plastic stains, the Drucker postulate may be violated, leading to ill-posedness of the problem and an ill-conditioned matrix of the algebraic system of finite difference equations. Another drawback of the GTN model is that the material matrix is characterized by equations independent of time scale change and, hence, strain rate. This is unacceptable for most materials in velocity loading. Damage is equivalent to porosity and, hence, is a scalar quantity. Consequently, damage only applies to mesoscale defects, while microscale defects (e.g., dislocations) are not taken into account; however, as is well known, it is dislocations that the physical mechanism of plastic deformation is closely connected with. The above drawbacks can be removed by using a physical dislocation-based micromechanical approach, just as in [95], rather than the phenomenological approach underlying the GTN model. This would preserve the advantages of the GTN model, provided that void-like mesodefects arise when a critical dislocation density at dislocation grain boundaries has been reached. The physical micromechanical approach has the advantage that it enables one to use the accumulated knowledge and experimental data obtained not only on the macroscale level but also on the microlevel. This is a huge advantage even when one can only use qualitative relations and micromechanisms of the development of plastic deformation and extend these to macroprocesses using correlations between micro and macro parameters. This allows one to choose a qualitatively correct approach in theoretically and hypothetically constructing continuum models at the meso and macro levels.

364

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

The next few subsections outline the generalized dislocation theory of plastic deformation developed by Taylor and Gilman [64, 171, 44], which extends the stage of hardening to the stage of damage.

8.2.1 Micromechanical model. The stage of plastic flow and hardening The initial stage of plastic flow is described at the microlevel by the motion of dislocations. The viscoplastic strain rate P p is proportional to the dislocation flux [186, 44] (one-dimensional case): (8.8) P p D abNm V; where a is an orientation coefficient, b is the Burgers vector, Nm is the number of mobile dislocations, and V is the average dislocation velocity. The quantity V is a D sij  determined by the thermal fluctuation motion and active stresses applied, sij r sij [28, 44]:   U0  .S  S r / ; S  S r; (8.9) V D V0 exp k where U0 is the activation energy, k is the Boltzmann constant,  is absolute temperature, S is the shear stress intensity, and S r is the residual stress intensity. The number of mobile dislocations increases proportionally to the amount of plastic strain  p and decreases with increasing the total number of dislocations, N , due to their lock-up at the grain boundaries [170]: Nm D .N0 C ˛ p /n exp.N=N  /;

(8.10)

where N0 , N  , ˛, and n are material constants. Equations (8.8)–(8.10) determine the dislocation equations of plastic deformation for one-dimensional shear, relate the plastic strain rate to the active stress deviator, and represent one-dimensional elastoviscoplastic microscopic equations with kinematic hardening. Adopting the standard hypotheses of the plastic flow theory, let us extend them to the case of three-dimensional stress-strain state. First, we rewrite equations (8.8)–(8.10) in terms of dimensionless variables, express the microquantities, the dislocation velocity V and number of mobile dislocations Nm , via the plastic strain and plastic strain rate. We have   1 U0 ; (8.11) P p D .N0 C ˛ p / .SN  SN r / exp d k  r where the generalized dependence .SN  SN r / has been substituted for exp SS . k The dimensionless variables (with an overbar) are introduced as S S D ; SN D U0 Y

1 @ D NP t0 ; t0 @tN

t D tN; t0

ıD

t0

1; d

(8.12)

Section 8.2 Generalized micromechanical multiscale damage model

where

1 sij D ij  ij ıij ; 3

1 Pij D "Pij  "Pij ıij : 3

365

(8.13)

Then PN p D ı.N0 C ˛ p/ .SN  SN r /:

(8.14)

Introducing  3 a a 1=2 SN a D sNij sNij ; 2

P p D

2 3

Pij Pij

1=2

;

we obtain p NPij D ı.N0 C ˛ p/ .SN a  SN0a /

a r sNij D sij  sij ;

(8.15)

a sNij : SN a

(8.16)

0

In what follows, the bars over P p , S , and sij will be omitted for brevity. The parameter ı D t0 =d is a large dimensionless number expressing the ratio of the characteristic time t0 to the relaxation time d (related to the size of defects). In dynamic problems, ı  102 –105 . Formula (8.16) represents an elastoviscoplastic equation with the yield criterion S a  S0a or S  S r  S0  S0r with .0/ D 0 for S a < S0a . The function ´ if z < 0; O .z/ D 0 .z/ if z  0 can be introduced. For S a < S0a . p /, we have a law of elastic deformation, P p  0; for S a > S0a . p /, we have a viscoplastic flow theory with translational strain hardening. In order to describe the next stage of elastoviscoplastic deformation, the process of nucleation and development of microdefects, it is necessary to look at the balance of dislocation fluxes in the material.

8.2.2 Stage of void nucleation In accordance with (8.8)–(8.10), the total dislocation flux, pPij , at the first stage of p plastic deformation consists of (i) the flux of mobile dislocations Pij , creating the plastic deformation as such, and (ii) the flux of dislocations, !P ij , accumulated near isolated obstacles and grain boundaries, which causes hardening. Let  denote the part of pPij related to the mobile dislocation. Then 1   is the flux accumulated at grain boundaries. Hence, we have p

Pij D pPij ; p

.1  /pPij D !P ij ; p

0 <  < 1:

(8.17)

It follows that !P ij D .1  /Pij =. Further, let us take advantage of the flux balance relation and equations (8.17), which follow from it, and analyze the implications.

366

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

It is worthwhile to mention one important fact from the dislocation theory of plastic deformation. The point is that both micro and macro parameters appear in the equations of the Taylor–Gilman dislocation theory. The theory relates the dislocation fluxes to the plastic strain and stress rate macro tensors. The theory is not constructed for micro parameters separately; instead, it immediately establishes correspondence between macro and micro parameters. This allows one to use the correspondence equations for determining constitutive macro equations at different stages of deformation. This mixed approach enables one to combine theory with experimental data obtained at both the macro and micro levels. Ilyushin [59] put forward the postulate that the continuum mechanics equations must be macro determinate. Developing this idea, one can formulate a postulate of micro-determinacy of the continuum mechanics equations. This suggests that there should be a correlation between the macro and micro parameters in the Taylor–Gilman dislocation model at the damage and fracture stage as well.

8.2.3 Stage of the appearance of voids and damage When a sufficiently large amount of dislocations have concentrated at grain boundaries, the dislocations are partially annihilated and grains move relative one another; as a result, disclinations, microvoids, and microcracks can arise [170]. This, second stage of deformation is characterized by a gradual fracture of the material, accompanied by additional deformation. Plastic deformation concentrates in the areas of maximum microfracture, leading to the formation of sliding bands. At the second stage, the total dislocation flux consists of three terms and so the balance equation becomes .1  /pPij D !P ij C bPij ; where bij is the tensor of the flux of annihilating dislocations. The formation of voids is often associated with the formation of point defects such as vacancies at nodes of a crystal lattice and with the movement of these defects to boundaries of crystals, where the vacancies coagulate to form voids [170, 57]. Note that at the first stage of plastic deformation, the material is plastically incompressible, which means that the spherical part of the plastic strain tensor is zero, and so i i D 0; hence, !i i D 0. At the second stage, the bulk plastic compressibility is nonzero, pi i ¤ 0 with bi i ¤ 0. It is natural to assume that the annihilating dislocation flux bPij is proportional to the flux of dislocations accumulated at obstacles, !ij : P ij ; bPij D !

(8.18)

where P is an unknown scalar coefficient to be determined during the solution. Dislocations do not begin to discharge until after the intensity of the obstacle dis1=2  , has attained a critical value 0 . The discharge location tensor, II D 32 !ij !ij

Section 8.2 Generalized micromechanical multiscale damage model

367

O II  0 /: intensity BP II will be a monotonic function of the excessive intensity, Q. O II  0 / 1=2 3 Q. D BP II D bPij bPij 2 p with

´ Q.z/ O Q.z/ D 0

(8.19)

if z  0; if z < 0;

where Q.z/ is a dimensionless function of its argument, while p is a time parameter measured in seconds associated with the size of voids. Condition (8.19) can be used P to determine : O II  0 / Q. : (8.20) P D p II Finally, the equation for determining the flux !ij can be written, in view of (8.17)– (8.20), as p O II  0 / 1   dij Q. d!ij D  !ij : (8.21) dt  dt p II Let us analyze this equation. It follows from (8.21) that a change in the flux !ij has two components: (i) growth due to the flux of mobile dislocations, which causes an increase in the material’s resistance to plastic shear deformation, and (ii) decrease due to discharge of annihilating dislocations. The term responsible for hardening, which was present in the first stage before the appearance of mesodefects, also remains unchanged in the second stage, softening due to the appearance of the tensor bPij . This is a very important factor, which delivers stability to the relaxation of !ij and ensures that the softening process is correct, in very much the same way as in the model problem considered in Section 8.1. For further analysis, it is necessary to generalize the conditions of matching between the micro and macro parameters in the Taylor–Gilman dislocation theory of plastic deformation.

8.2.4 Relationship between micro and macro parameters The macro-physical meaning of the tensor !ij is associated with the residual stress tensor and follows from the well-known experimental fact that the amount of residual stresses is proportional to the density of dislocations at grain boundaries [170, 44]. The critical dislocation density 0 , at which microcracks begin to arise at grain interfaces and other obstacles, and the corresponding critical intensity of residual stresses S0r can be determined at the microlevel on the basis of dislocation models of fracture [145, p. 67–68]. In the space of the stress tensor components, the condition II D 0 has the meaning of a fracture onset surface, SIIr D S0r .

368

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

The annihilation tensor bij correlates to a certain degree with the damage ten0 0 of bij is related to the damage deviator Dij , which sor Dij . The deviatoric part bij is responsible for relaxation of the residual stresses (the second term on the righthand side in equation (8.21)). The spherical part bPi i correlates with the bulk damage strain, bPi i $ DP i i , and is linked to the porosity increase rate fPgr , determined by the continuity equation. The correlations between macro and micro quantities are generalized and summarized below in Table 8.1. Table 8.1. Table of micro-parameters and corresponding macro-parameters.

Micro quantities

Correlation

Total dislocation flux Mobile dislocation flux Accumulated dislocation flux Annihilating dislocation discharge Spherical part of pij Intensity of the deviator of !ij Critical value of dislocation intensity at grain boundaries

Macro quantities

pij $ sij pij p $ Pij .1  /= 2G r .1  /pij D !ijr $ sij

Stress deviator Plastic strain rate tensor Residual stress deviator

bij $ Dij

Relaxing stress deviator

pi i $ fgr II $ SIIr

Porosity Residual stress intensity

0 $ S0r

Limit value II at which residual stress relaxation begins

Using Table 8.1, one can obtain from equation (8.21) the evolution equation for the r : residual stress tensor sij r dsij

dr

p

D

dij Q.SIIr /  S0r r 1 2G  sij :  dt SIIr

(8.22)

The hardening term on the right-hand side of (8.22) ensures that the Drucker postulate of stability and well-posedness is satisfied for quasistatic problems as well as the Hadamard condition for dynamic problems.

8.2.5 Macromodel By changing from the micro to macro parameters, we rewrite equations (8.8)–(8.10) for a general three-dimensional stress-strain state given by a relation between the sec-

Section 8.2 Generalized micromechanical multiscale damage model

369

ond invariants of the plastic strain rate tensor, NP p , and active stress tensor, S a : .S a  S0a / ; NP p D .N0 C ˛ p /n d  3 a a 1=2  2 p p 1=2 a r sij ; sij D sij C sij ; NP p D Pij Pij ; S a D sij 2 3 1 1 p p p ij D "ij  ıij "kk ; sij D ij  ıij kk ; 3 3

(8.23)

r is the residual where d is the active stress relaxation time due to dislocations, sij p a stress deviator, sij is the active stress deviator, and ij is the plastic strain deviator. Adopting the hypotheses of the flow theory [67], from equations (8.8)–(8.10) one can obtain the following elastoviscoplastic equations for the general three-dimensional stress-strain state at the first stage of deformation for II < 0 :

.S a  S0a / a 1 sPij C N. p / sij ; 2G d T 1 ij D "ij  ıij "kk : i i D 3K"i i ; 3 Pij D

(8.24)

where G is the instantaneous elastic modulus between the shear strain and shear stress deviators and K is the bulk modulus. Assuming, for simplicity, the material to be p plastically incompressible, "kk D 0, one arrives at an elastic law for the spherical part i i of the stress tensor. Assuming that the residual stress deviator and accumulated dislocation flux deviator satisfy the relation r D 2  !ij0 ; (8.25) sij r corresponds the one finds that the condition II < 0 for the residual stress sij r r  2 macro-condition S  S0 , where is a constant measured in N=m . The evolution r in the first stage of plastic deformation is equation (8.17) for sij

1 p 1 r P : sPij D  2  ij

(8.26)

Integrating (8.26) with respect to t at constant  , one arrives at the well-known law of kinematic hardening p

r D 2˛ij ; sij

˛ D 

1 : 

(8.27)

The hardening modulus ˛ is determined from experimental data on the Bauschinger effect obtained in tension-compression tests, allowing one to find the parameter  from .1  /.1 C ˛=  / D 0 to obtain  D 1=.1 C ˛=  /.

370

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

Thus, when S r < S0r , i.e., before the formation of microvoids, the matrix material is described, in accordance with dislocation theory, by the equations of an elastoviscoplastic medium with kinematic hardening (8.24)–(8.27). When S r  S0r , the r tensor sij obeys the relaxation equation r C sPij

2  Q.S r  S0r .f // r p sij D 2˛ Pij : p Sr

(8.28)

Equations (8.26)–(8.28) can be rewritten as a single equation by introducing the O function Q.z/ in accordance with formula (8.19). This equation characterizes the relaxation of the residual stresses after microvoids have begun to form and, hence, softening has started. Prior to this moment, the material experienced hardening in accordance with equation (8.27). Equation (8.28) shows that the relaxation takes place until a certain nonzero stationary value S0r .f /. Further, at t p , the residual stresses will obey the plastic flow law associated with a yield surface corresponding to a porosity dependent yield stress, S r D S0r .f /; the yield stress can be obtained from (8.28) by letting p ! 0: r d "p  r mn r p s p r C H smn d "mn mnr s D 2˛ dij ; dsij ŒS0 .f / 2 ij

(8.29)

where H.z/ is the Heaviside step function. The situation changes significantly as voids appear. The material consists of a matrix and voids and so is a biphasic medium. Given the properties of the matrix, which are characterized by equations (8.24) and (8.27), one can find the effective characteristics of the medium. The yield criterion changes and, in addition, equations describing the nucleation and evolution of defects are needed. Assuming that the voids are nearly spherical in shape, we adopt the Gurson yield criterion for an ideal elastoplastic porous material. Extending this criterion to a porous material whose matrix is characterized by elastoviscoplastic equations with kinematic hardening, we obtain  a a a  3 sij sij 3q2 kk  Œ1 C .q1 f /2 D 0; (8.30) C 2q f cosh ˆ.ija ; f; Y / D 1 2 .Y /2 2 Y a where sij is the active stress deviator and Y is the yield stress of the matrix, determined from the condition that the plastic works for the matrix and effective material must be the same: p

ija "Pij D .1  f /P p ŒY . p / C Y D Y . p / C

1

. P p/:

1

. P p / ;

(8.31)

One finds  p from the first equation and then determines Y from the second. The p stress ija and dissipative strain rate "Pij in the effective material are related by the

Section 8.2 Generalized micromechanical multiscale damage model

371

associated plastic flow law @… p Pij D P a ; @sij

(8.32)

provided that the matrix material obeys the analogous associated flow law [55]. The parameter P is determined from the second equation: (8.31):    @ˆ a 1 Y  ‰ Y  Y . p /  : P D p @ija ij  a a 1=2 sij C Y . p / C 1 .p P p / with …  ˆ.f D 0/. where … D 32 sij From the continuity equation, one finds the void growth equation   3f .1  f / 3q2 kk p q1 q2 sinh : fPgr D .1  f /P"kk D P Y 2Y

(8.33)

The equations of nucleation and development of voids remain the same as in the GTN model and are given by (8.5)–(8.7); these equations close the system of constitutive equations (8.30)–(8.33) and serve to determine the stress and internal parameters for a given velocity field, which is found from the law of conservation of momentum. The condition that a critical amount of voids has accumulated is adopted as the fracture condition for the model suggested. Since crystallites are in a constrained state and so cannot deform freely, inter-crystallite voids are formed at crystallite interfaces. These voids accumulate until a certain critical porosity, f D fcr , is reached followed by a catastrophic propagation of the voids leading a complete inter-crystallite fracture of the material. The critical porosity fcr depends on many external factors such as the temperature, loading velocity, etc. as well as on the material structure; according to experimental data [170], it ranges from 0.05 to 0.50. Let us introduce additional effects in the model that account for some specific features of the material’s response in the damaged state. These include the effects of porosity on the elastic moduli and the damage surface S r D S0r .f /. Taking into account that the internal elastic energy of the damaged material is porosity dependent and independent of the plastic strain, U el .f; "el ij /, one finds that the effective elastic moduli EN and N vary in accordance with the formulas suggested by L. M. Kachanov [66]: EN D E.1  f /;

N D .1  f /;

(8.34)

where f is the porosity appearing in (8.33). Assuming that the influence of porosity on the damage surface is similar to that on the yield surface for the active stress, we adopt the approximate relation S0r .f / D

S0a .f / S r ; Sa

where S0a .f / D Y .1 C q12 f 2 /1=2 :

372

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

The inclusion of both effects results in a more intensive softening and, hence, a sharper localization. In [71], as follows from the computations based on the initial version of the model (without the above additions), these effects were not pronounced enough due to high mesoviscosity of the material: p d . It should be emphasized that, unlike the GTN model, the damage model suggested here has multiple scales. It involves three time scales: t0 , the characteristic time, p , the stress relaxation time in the damaged material, and d , the relaxation time in the original, undamaged viscoplastic material. The three times correspond to three different spatial scales – macro, meso, and micro scales – which represent the sizes of the macro-object, void, and dislocation, respectively, and satisfy the inequalities t0 p d . The scale effect is determined by the small parameter ıd D d =t0 in the hardening stage and parameter ıp D p =t0 ıd in the softening stage. As ıp and ıd tend to zero, the model tends to the GTN model with kinematic hardening, which is independent of the time scale. The yield and damage surfaces appearing in the model admit a geometric interpretation which is common in describing plastic flow theories. The model has two critical surfaces dependent on the plastic strain rate intensity "NPp and porosity f , which determine the material’s response. The two surfaces are shown in Figure 8.6 by dashed lines. The active stress yield surface S a D Y .f / determines the transition from the elastic to plastic state; the surface S r D S0r .f / determines the formation of porosity.

σ

σ

σa

a

S 0 = σy(f)

σr 0

σ r

Sr = S0(f)

Figure 8.6. Yield and damage loading surfaces.

Section 8.2 Generalized micromechanical multiscale damage model

373

The stationary positions of these surfaces are shown by solid lines, initial positions, by dashed lines, and current positions, by dot-and-dash lines. Temperature effects are easy to include in the model by adding to the strain " the temperature component "ij D ˛ıij and determining the temperature  from the equation @ p D ija "Pij ; (8.35) cp @t which follows from the assumption that the process of plastic deformation is adiabatic. In the equation, cp is heat capacitance at constant stress, D 0:8–0:9 is the thermal convertibility coefficient, ˛ is the coefficient of thermal expansion, and  is the density of the effective material. In conclusion, it should be noted that a numerical simulation of softening and strain localization processes based on the equations (8.30)–(8.35) of the micro-model suggested was performed by Kibardin and Kukudzhanov [71]; the authors solved a fracture problem for an axisymmetric cylinder in quasistatic tension with a constant velocity. A dynamic fracture problem for a bar at high loading velocities was solved in [92]. In the literature, there are other coupled meso-models of damaged viscoplastic media based on a phenomenological approach with more specific assumptions than in the above model. These models apply to both ductile continuum fracture [175] and brittle fracture under cyclic loading [35, 179] in modeling fatigue fracture.

8.2.6 Tension of a thin rod with a constant strain rate Consider the most common type of test, the uniaxial tensile-compressive test of a thin rod with a constant rate of deformation. The solution of the problem allows one to assess the model to see how well it can predict the stress-strain diagrams of materials with softening. It must be emphasized that no  -" curve is introduced in the model explicitly (as is the case in the classical theory of plasticity); it is a consequence of the multifactorial damage model employed. At the hardening stage, we adopt the simplest hypothesis of linear hardening. The integration results are shown as  -" curves at a constant strain rate, "P D "P0 . Figure 8.7 displays four characteristic curves for different types of material. The first three curves were obtained for S0 D 1:5 and different values of the hardening parameter ˛ (see Table 8.2), q1 D 1:0, q2 D 1:0, d =t0 D 0:01, and p =t0 D 1:0, with N and SN0 being dimensionless quantities referred to the initial yield stress 0Y . Table 8.2. Values of parameters for the curves displayed in Figure 8.7.

No.

1

2

3

4

S0 ˛

1:5 0:01

1:5 0:1

1:5 0:9

0:2 0:1

374

Chapter 8 Modeling of damage and fracture of inelastic materials and structures – σ (3) 2.0 (2)

(1) 1.0 (4) 0

10

20

30 –ε

Figure 8.7. Predicted -" diagrams in dimensionless variables for constant-strain-rate tension at different values of the model parameters (Table 8.2).

It is apparent from Figure 8.7 that the curves have a kink at the beginning of the second stage of deformation, which corresponds to the appearance of voids, followed by a softening segment. For small hardening parameter, ˛ D 0:01, curve 1 shows large plastic stains in the hardening stage and a rapid drop in the stress as the material softens followed by a nearly flat segment. Curve 2 was obtained for ˛N D 0:1 and SN0 D 1:5; the diagram has a sharp yield point followed by partial softening and then hardening. This behavior is characteristic of many soft steels, iron, and some other materials undergoing phase transitions. (To obtain a yield drop, a negative discontinuity is artificially introduced in the diagram after the elastic segment and on the bilinear curve). Curve 3 corresponds to a large value of the kinematic hardening parameter, ˛N D 0:9, with the other parameters specified in Table 8.2. One can see that the softening segment is very short here – the critical value f is attained at relatively small strains. This type of curve is characteristic of high-strength materials. Curve 4 corresponds to S0 < 1 (˛N D 0:1 and SN D 0:2). In this case, the  -" is smooth, does not have a kink, characteristic of the cases with S0 > 1, and has a long softening segment preceding failure at f D 0:5. Such diagrams are characteristic of some soils and clays. Thus, with only a few (four) parameters, the model allows one to describe, at least qualitatively, a wide spectrum of properties of various materials. In phenomenological quantitative description of real materials, one has to postulate the functions p Y ."m / (yield stress of the matrix, a function of continuous hardening asymptotically approaching an ideal plastic diagram), Q.z/, ‰ 1 .z/, S0 .f /, ˛."p /, and E.f / to achieve agreement with experimental data. These functions all appear in the model suggested and were taken to be linear or constant in the computations described.

Section 8.3 Numerical modeling of damaged elastoplastic materials

375

8.2.7 Conclusion The multiscale model suggested is qualitatively different from the GTN model. As one can see from the above, the model is based on dislocation mechanisms of plastic deformation occurring in thermomechanical loading of polycrystalline materials at moderate plastic strains. It is also based on the concepts of nucleation and growth of mesodefects, such as voids or cracks, at large strains. At the final stage, the fracture occurs due to the merging of voids into a macro-crack followed by its propagation until structural failure. Postulating a correlation between the micro and macro parameters allows one to connect the micro-equations with constitutive continuum macroequations and obtain kinetic equations of the coupled elastoviscoplastic model with damage. As a result, damage is described by a tensor whose spherical part is porosity and deviatoric part is connected with the relaxation of the residual stress deviator. The effective elastic moduli are assumed to change by Kachanov’s formulas (8.34). The model of the matrix material takes into account kinematic hardening and, hence, residual stresses and also describes the Bauschinger effect.

8.3 Numerical modeling of damaged elastoplastic materials (ductile and quasi-brittle fracture) 8.3.1 Regularization of equations for elastoplastic materials at softening Apart from constitutive equations, special numerical methods are required for solving these equations in modeling damage processes. Damage is related to softening and plastic strain localization, which creates certain difficulties in integrating the constitutive equations and solving initial-boundary value problems. The models of damaged media that ignore the effect of multiple scales in fracture require regularization. As noted above, the first attempts to describe damage within the framework of a flow theory with a decreasing deformation diagram were unsuccessful, since they resulted in the violation of the Drucker rheological stability condition (for quasistatic loading) or the Hadamard condition (for dynamic loading) [19, 184]: det.Cij kl ni nj / > 0 for any ni ; where Cij kl is the elastic modulus tensor of the material, dij D Cij kl d "kl . The regularization of a model can be performed by using regularizing operators that account for additional physical processes (disregarded by the initial model) or correspond to purely mathematical considerations. In the latter case, artificial higherorder terms with small parameters are introduced to regularize lower-order equations. The higher-order operators can involve a temporal scaling factor or spatial scaling factors.

376

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

For example, a flow theory that has a decreasing material diagram  D Y ."/, with dY =d " < 0, and whose yield criterion is insensitive to changing the scale of time results in an ill-posed problem. The problem can be regularized by changing to an elastoviscoplastic model:   1 @ @"   Y ."/  ; (8.36)  D @t E @t Y .0/ where  is the relaxation time and E is the elastic modulus. The operator on the lefthand side, which is responsible for the plastic strain rate, increases the order of the constitutive equation with respect to time t . Another regularization technique involves using gradient or nonlocal stress-strain relations (increasing the order of spatial derivatives) [91]. For example, with "D

@3 u @u C a 3; @x @x

(8.37)

the equation of motion acquires the form b2

@2 u @4 u @2 u C c2 2 D 2 : 4 @x @x @t

(8.38)

However, adding to the constitutive equations higher-order operators with a small parameter,  in (8.36) and b in (8.38), is not the best or unique way of regularizing the problem. This approach will be called mathematical. A simpler and more natural, physical approach suggests that more determining parameters should be included, as, for example, was the case in the above models, where a damage parameter f was introduced. With the mathematical approach, the order of equations is increased, with the limit problem as ı D =t0 ! 0 being unsolvable in the classical sense. A stiff problem can only have a solution in the asymptotic sense if at all. By contrast, with the physical approach, the limit problem remains well-posed and admits a solution in the classical sense. Furthermore, numerical solution at ı 1 is simpler than with mathematical regularization. The issue of adequacy of a numerical solution to a discrete problem cannot be removed by mathematical regularization completely – special methods for solving stiff problems are required and various artificial numerical techniques may be needed to suppress solution sensitivity to problem discretization [19].

8.3.2 Solution of damage problems At present, the finite element method is commonly used for solving a broad spectrum of nonlinear problems, in particular, elastoplastic and elastoviscoplastic ones. In most studies, the constitutive equations are numerically integrated by step-by-step methods with respect to a loading parameter. Moreover, one has to integrate a complete system of equations for a boundary value problem. The choice of an integration scheme is extremely important for the solution, especially at large deformations exceeding

Section 8.3 Numerical modeling of damaged elastoplastic materials

377

elastic deformations by an order of magnitude. A number of methods for integrating constitutive equations have been suggested in the literature [146, 130, 162]. Explicit methods lead to conditionally stable schemes [124]. At large deformations and in solving multiscale stiff problems, one has to use implicit unconditionally stable schemes [94, 124]. As shown by Ortiz and Popov [130], the inverse Euler method proves to be effective when the von Mises yield criterion is used. Aravas [3] extended this method to the case of a yield criterion dependent on the first two stress invariants and internal variables of the model; the method was then applied to solving damage and continuum fracture problems for elastoplastic materials within the GTN model.

8.3.3 Inverse Euler method Let us integrate the constitutive equations of an elastoplastic medium with a general yield criterion dependent on the first two stress invariants and structural variables. The material properties are assumed to be independent of time scale changes within the framework of the GTN model. The GTN equations form a time scale invariant system of differential equations, which can be written in terms of increments. The integration is performed step by step with respect to a loading parameter, t . The state of the material at time t is assumed to be known. In addition, the total strain increment " is prescribed at time t C t . It is required to determine the stress and internal variables that satisfy all equations of the system, inclusive of the yield criterion at t C t . To this end, let us write Hooke’s law at t C t as follows: ˇ   ˇ (8.39)  D Del W "el ˇ tCt D Del W "el ˇ t C "el D  el  Del W "pl ; where

 ˇ   el D Del W "el ˇ t C " :

The symbol  el can be treated as the elastic predictor, with   2 el D D 2G I C K  G I ˝ I 3 being the linear isotropic elasticity tensor, where G is the shear modulus, K is the bulk compression modulus, I is the second-order identity tensor, and I the symmetric fourth-order identity tensor. The total strain is assumed to be representable as the sum of two components, elastic and plastic. The yield criterion, flow law, and evolution equation for the internal variables are, respectively, ˆ. ; "Npl ; H ˛ / D 0; 1 "pl D "p I C "q n; 3 H ˛ D HN ˛ ."pl ;  ; H ˇ /

(8.40) (8.41)

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Chapter 8 Modeling of damage and fracture of inelastic materials and structures

with nD

3 s; 2q

"p D 

@ˆ ; @p

"q D 

@ˆ ; @q

q where p is pressure, s is the stress deviator, q D 32 s W s is the von Mises stress, and H ˛ (˛ D 1; 2) represents the internal variables. For the GTN model, ˛ D 2, with pl H 1 D "Nm being the strain intensity in the plastic matrix and H 2 D f the porosity. The parameter increment  can be eliminated from the relations for "p and "q to obtain "p

@ˆ @ˆ C "q D 0: @q @p

Substituting equation (8.40) into (8.39) yields  D  el  K"p I  2G"q n:

(8.42)

The tensors sel and n are coaxial (see Figure 6.2 on page 283); hence, nD

3 el s : 2q el

(8.43)

Since n is known, determining the scalar quantities "p and "q together closes the solution of the problem. Consequently, the problem of integrating elastoplastic constitutive equations dependent on pressure is reduced to solving the following two nonlinear equations for "p and "q : ˆ.p; q; H ˛ / D 0; @ˆ @ˆ C "q D 0: "p @q @p

(8.44) (8.45)

The quantities p, q, and H ˛ in (8.44)–(8.45) are defined as p D p el C K"p ; el

q D q  3G"q ; H ˛ D h˛ ."p ; "q ; p; q; H ˇ /:

(8.46) (8.47) (8.48)

Equations (8.46) and (8.47) have been obtained by projecting equation (8.42) onto I and n, respectively. Equation (8.48) is an alternative representation of equation (8.41). By solving the above system of equations for the unknowns p, q, "p , "q , and H ˛ , one can close the integration algorithm for a porous plastic material. Equations (8.44) and (8.45) are solved for "p and "q by Newton’s method. After this, one can determine p, q, and H ˛ on the next layer from (8.46)–(8.48).

379

Section 8.3 Numerical modeling of damaged elastoplastic materials

8.3.4 Solution of a boundary value problem. Computation of the Jacobian When an implicit scheme is used for solving nonlinear problems, the resulting discrete equations forms a nonlinear system for nodal variables; the system is solved with Newton’s method, which has a quadratic rate of convergence. The method suggests that the linearized modulus   @ @ D JD @" @" tCt must be calculated. The Jacobian J is calculated with formula (8.42) rewritten as    ˇ  ˇ ˇ  D 2G el ˇ t C   "q n C K "el kk t C "kk  "p I; where D "  13 "kk I is the deviatoric component of the strain tensor ". After differentiating, one obtains @ D 2G.@  @"q n  "q @n/ C K.@"kk  @"p /I:

(8.49)

The variations @"q and @"p are calculated using (8.44) and (8.45). After fairly algebraic rearrangements, one arrives at a system of linear equations for determining @"q and @"p . Substituting the resulting quantities into (8.49), one find the linearized modulus. In the general case, the linearized modulus turns out to be slightly asymmetric. However, this fact can usually be neglected in solving the system of equations.

8.3.5 Splitting method The general procedure of splitting the constitutive equations of an elastoplastic material is outlined in Section 6.4 for solving a dynamic problem by an explicit scheme. This section addresses the application of the splitting method to solving boundary value problems by an implicit scheme. As noted above, the splitting procedure here will remain the same. What will change is only the solution of the equations of the boundary value problem by the implicit scheme. Moreover, one will have to calculate the Jacobian of the complete system of equations. What will have to be split is only the equation d D D W .P"  "P pl /; (8.50) dt where D is the material’s elastic modulus tensor and d  =dt is the objective derivative of the Cauchy stress tensor. One takes the predictor at "P pl D 0, thus considering the medium elastic: d D D W ": P dt

(8.51)

380

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

In conjunction with the equations of motion for the temporal step t , one then has to solve the elastic problem with the initial conditions obtained in the previous step for the total elastoplastic problem. Then, in the corrector step, one solves the stress relaxation equations (8.50) at "P D 0 in conjunction with the equations characterizing the internal structure of the material (hardening, damage, etc.) with the initial conditions obtained in the predictor step. Using the associated flow law, one arrives at the stress relaxation equation d @ˆ d D D ; dt dt @ d Fi . ; Hj /; HP i D dt

(8.52) (8.53)

where Hi stands for the parameters characterizing the internal structure of the material. Integrating the constitutive equations to determine the stress and internal parameters at fixed strain represents a stress relaxation problem, which is important by itself. For a classical (equilibrium) elastoplastic medium, whose properties are time-scale invariant, one can eliminate time t from equations (8.52) and (8.53) change to the differentiation with respect to : @ˆ d D D ; d @ dHi D Fi . ; Hj /: d

(8.54) (8.55)

Solving equations (8.54) and (8.55) with the initial conditions  . 0 / D  n and Hi . 0 / D Hiel , obtained from the solution of the elastic problem in the predictor step, one finds a solution as functions of ,  el , and Hin :  D  . ;  el ; Hin /;

Hi D Hi . ;  el ; Hin /:

(8.56)

Substituting the resulting solution into the yield criterion, one arrives at an equation for determining the parameter D  corresponding to the correction factor:   (8.57) ˆ ; p. /; S. /; Hi . / D 0: Solving this equation gives D  . n ; Hiel /. Further, substituting  into (8.56), one obtains the desired solution to the constitutive equations for the loading step in question. Finally, one arrives at a solution to total boundary value problem for the given step in the loading parameter. Solution of the boundary value problem. Calculation of the Jacobian. The solution of boundary value problems based on the principle of virtual displacements by

Section 8.3 Numerical modeling of damaged elastoplastic materials

381

the finite element method is reduced to solving a system of nonlinear algebraic equations for nodal displacements. To solve this system by Newton’s method, one has to compute the Jacobian @ =@", where " is the total strain. The computation of the Jacobian depends on the integration algorithm for the constitutive equations. With the standard approach, the integration can only be performed numerically. With the splitting method, this problem can be significantly simplified and solved analytically. Let us exemplify this by considering an elastoplastic medium with linear hardening. In the predictor step, the total strain " is treated as elastic and the stress solving the total problem is expressed in terms of "el alone. Hence, @N @N D el : @" @"

(8.58)

Differentiating the equation for stress (8.52) with respect to " and taking into account (8.58), one obtains  el   el    @ @p @s @x el D ˝IC xCs ˝ ; (8.59) @" @" @" @" @sel 2 @p el D KI; D 2G I  G I ˝ I; (8.60) @" @" 3 where K is the bulk compression modulus, G D is the shear modulus, I is the identity tensor of rank 2, and I is the symmetric identity tensor of rank 4. Since the expression of the correction factor x is found in explicit form [94], differentiating this expression with respect to " yields   2 3G @x @ D 3GxI C K  Gx I ˝ I C el el sel ˝ sel ; (8.61) @" 3 S @S q where it has been taken into account that S el D 32 sel W sel . Computation results. As an example, let us consider the problem of shearing a bar in three dimensions. A 5  2  1 mm steel bar is clamped at the top and bottom end faces. The top face is displaced rigidly as a whole in the direction shown in Figure 8.8a by the arrow. The shearing displacement is equal to 2 mm, the bar thickness. The dimensionless elastic modulus referred to the yield stress is E D 500, Poisson’s ratio is  D 0:3, and the plastic hardening modulus equals 1 D 0:1. Figure 8.8 illustrates the deformations of the bar and the Lagrangian grid as well as isolines of the shear stress intensity. The problem was solved in two different ways, by the splitting method and the second-order accurate inverse Euler method suggested by Aravas [3]. Figure 8.9 displays the shear stress intensity S and pressure p versus the loading parameter in the element indicated by the arrow in Figure 8.8b. The solid line corresponds to the method of [3] and the x’s indicate the results obtained by the splitting method.

382

Chapter 8 Modeling of damage and fracture of inelastic materials and structures S, von Mises 1.490 1.335 1.180 1.025 0.871 0.716 0.561 0.406

(a)

(b)

Figure 8.8. (a) Initial geometry and applied load; (b) isolines of the shear stress intensity S.

The following results were obtained: (i) the temporal step size was the same in both approaches, (ii) the number of iterations in Newton’s method was the same in both cases, (iii) the computation took less time with the splitting method (12 min 18 s) than the method of [3] (13 min 26 s), and (iv) the values of the shear stress intensity and pressure were in very good agreement between the two methods. The relative error did not exceed 105 .

8.3.6 Integration of the constitutive relations of the GTN model Let us apply the numerical-analytical method to more complicated equations of a porous elastoplastic medium. For simplicity, it will be assumed that there is no generation of voids and the matrix is ideally plastic. Then the void growth equation is fP D fPgr . Transformation of the system of constitutive relations. The stress equation can be written in the form d  D D W .d "  d "el /; 3 d @ˆ d @ˆ DWI D W s; d D D W d" C 3 @p 2S @S

(8.62)

where D W s D 2Gs;

D W I D 3KI:

Pressure is governed by the equation dp D K.d " W I/  d K

@ˆ : @p

(8.63)

383

Section 8.3 Numerical modeling of damaged elastoplastic materials q

p

1.2

1.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.2

0.4

0.6

0.8

t 1.0

0.0

t 0.0

0.2

(a)

0.4

0.6

0.8

1.0

(b)

Figure 8.9. Comparison of numerical results obtained by the method of [3] (solid line) and the splitting methods (x’s): (a) shear stress intensity q and (b) pressure p in the element indicated by the arrow in Figure 8.8b versus the loading parameter t .

The stress deviator equation is d s D 2G d  6Gs d :

(8.64)

The equation for the porosity becomes df D d .1  f /

dˆ : dp

For comparison, listed below are the system of constitutive equations for the complete problem within the method of [3] and the equations of the splitting method in the corrector step: Complete problem dp D K.d " W I/  d K d s D 2G d  6G s d

Splitting method @ˆ @p

@ˆ @p d s D 6G s d

dp D d K

dˆ dˆ df D d .1  f / dp dp the initial conditions are taken the elastic solution is taken as from the preceding step the initial conditions  2   S 3 q2 p ˆD C 2q1 f cosh   .1 C q3 f 2 / D 0 Y 2 Y df D d .1  f /

384

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

Corrector. One solves the stress relaxation problem. The system of equations in dimensionless variables referred to Y is   @ˆ 3 d pN D 3Kq1 q2 f sinh q2 pN ; D K (8.65) @pN 2 d N N d sN D 6G sN d ; (8.66) dˆ df ; (8.67) D .1  f / N d pN d   3 ˆ D SN 2 C 2q1 f cosh  q2 pN  .1 C q3 f 2 / D 0: (8.68) 2 The initial conditions are taken from the solution to the equation problem obtained at D n and labeled by the zero subscript: pN0 , sN0 , SN0 , and f0 . The system of equations cannot fully be integrated analytically, unlike the case of the von Mises yield criterion. What can be integrated exactly are the stress deviator equation (8.66) and pressure equation (8.67). For the shear stress intensity, we have d SN N D 6GN d : SN

(8.69)

Integrating the equation for the dimensionless pressure pN subject to the initial condition pN D p0 , one obtains 1f : (8.70) pN D pN0  KN ln 1  f0 From (8.67) one finds the porosity in terms of pressure:   pN  pN0 : (8.71) f D 1  .1  f0 / exp  KN Eliminating the porosity f from (8.65), one arrives at a differential equation for p: N     d pN N 1 q2 sinh 3 q2 pN .1  f0 / exp pN  pN0 : D 3Kq (8.72) N 2 KN d This equation is easy to integrate on the interval Œ ; C using an implicit scheme and the pressure can be evaluated with a required accuracy. After substituting the resulting p. / and f . / into Gurson’s yield criterion (8.68), one determines the value D  corresponding to the correction factor. All subsequent solution of the boundary value problem will also have to be performed numerically. For a small step size  , it is reasonable to linearize equations (8.70)–(8.72) to obtain simple approximate analytical expressions of p. / and f . / with sufficient accuracy. This approach, based on the linearization of equations, is used below. The function ˆ.X/ is determined analytically and all subsequent mathematics is also performed analytically, which facilitates the computation of the Jacobian and further integration of the boundary value problem.

385

Section 8.3 Numerical modeling of damaged elastoplastic materials

The linearization yields d pN p p D C0 C Cp p; N N d p

(8.73)

p

where C0 and Cp are constants arising from the expansion of the right-hand side of equation (8.72). N in N / Integrating (8.73) and taking into account that p. N N 0 / D pN0 , one finds p. explicit form:  p  Cp  p C (8.74) pN D pN0 C 0p exp Cp .  0 /  0p : Cp Cp Substituting (8.74) into (8.71) yields the explicit expression of porosity f p: N f D C0f C Cp

(8.75)

Introduce the correction factor X D expŒ103 . N  N 0 / . The factor 103 is used to avoid high degrees in the expressions of pressure and stress intensity. The linearized expressions can be rewritten as SN D SN0 X ˛ ;

pN D a C b X ˛ ;

f D c C d X˛;

(8.76)

p N a D C p =Cpp , b D pN0 a, c D C f C f a, where ˛ D 103 Cp , ˇ D 6103 G, 0 0 p f and d D Cp b. Substituting (8.76) into the yield criterion (8.68), one arrives at an algebraic nonlinear equation ˆ.X/ D 0, which is solved by Newton’s method for the correction factor X: ˆ.X n / (8.77) X nC1 D X n  0 n ; ˆ .X /

where 0

ˆ .X/ D

2ˇq02

X

2ˇ 1

C ˛X

˛1





 3 q2 pN 2q1 d cosh 2    3 C 3q1 q2 bf sinh q2 pN  2q3 f d : 2

We take X0 D 1 as the initial approximation. The iterative processes (8.77) is performed until the relative error " in determining X becomes less than "0 D 107 . The Jacobian @ =@" is computed in the same way as previously for the von Mises yield criterion. Finally, the Jacobian is expressed as 1 d D CI I C Cn I ˝ I C .CnI C CIn /.n ˝ I C I ˝ n/ C Cnn n ˝ n: d" 2 The expressions of the coefficients CI ; : : : ; Cnn can be found in [100].

(8.78)

386

Chapter 8 Modeling of damage and fracture of inelastic materials and structures u

u

1 2

1

Figure 8.10. Uniaxial tension of a square element.

8.3.7 Uniaxial tension. Computational results As an example, consider a uniaxial tension problem for a square element (see Figure 8.10) in the plane strain case. The logarithmic strain measure was used:   u.t / ; u.t / D 0:5 t: (8.79) " D ln 1 C l The initial porosity of the material was f D 0:001. This test problem was solved by (i) the method suggested by Aravas [3], which is implemented in the ABAQUS software, and (ii) the splitting method. Figure 8.11 displays pressure p and porosity f versus the loading parameter; the solid line corresponds to the precise Aravas solution [3] (obtained for a very small step size) and the x’s correspond to the approximate solution obtained by the splitting method. –d

f

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 – 0.5

t 0.0 0.2

0.4 0.6 0.8 1.0

(a)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 – 0.05

t 0.0 0.2 0.4

0.6 0.8 1.0

(b)

Figure 8.11. Comparison of the computational results obtained by the Aravas method [3] (solid line) and splitting method (x’s): (a) pressure and (b) porosity versus the loading parameter t .

Section 8.3 Numerical modeling of damaged elastoplastic materials

387

It is apparent from Figure 8.11b that the material softens and porosity increases. The results obtained by the two methods are in very good agreement. Although the matrix material is plastically incompressible, the compressibility of the effective material is due to a change in porosity; when the body increases by 50% in volume, the corresponding elastic strain is only "el D 0:002. Boundary value problem. Consider the problem of shearing a bar in three dimensions (Figure 8.8). This problem was solved above for an elastoplastic material with the von Mises yield criterion. Computations with a fixed step size have shown that the splitting method requires 2.4 times more iterations for the solution than the Aravas method [3], whereas the total computation time is nearly the same: 1 hour 31 min with the splitting method and 1 hour 27 min with ABAQUS. Thus, in integrating constitutive equations, the splitting method turned out to be about 2.5 times faster per iteration than the standard method. The comparison of the results obtained using the von Mises yield criterion and Gurson yield criterion in the above examples reveals a significant effect of porosity on the stress-strain state; an increase in porosity leads to softening of the material and, ultimately, to failure of the structure.

8.3.8 Bending of a plate To illustrate a significant difference in fracture character with and without taking into account material damage, let us consider an example problem of bending of a plate. The plate is simply supported at the ends and loaded in the middle with a rigid cylindrical punch (plane strain). By virtue of symmetry, the right half of the plane will only be considered. Figure 8.12a displays the geometry of the problem, with h D 0:25 mm. The material constants are E D 1  105 MPa,  D 0:3, Y D 400 MPa, q1 D 1:5, q2 D 1:0, f0 D 0:05, fN D 0:04, "N D 0:3, and sN D 0:1. There is no friction between the plate and punch or between the plate and supports. Since the largest deformation is in the area under the punch, we use a finer grid in this area. At large strains, the cells of the Lagrangian grid get severely distorted, thus resulting to a decrease in the computation accuracy. For this reason, the half-width of the plate is divided into four domains (Figure 8.12b). To connect a domain with coarser cells and a domain with finer cells an intermediate layer of triangles is used as shown in Figure 8.13b; the discretization step decreases by half. Domain 1 (under the punch) is covered with an adaptive mesh, which is rearranged at each temporal step (time is used as the loading parameter). Domains 2–4 use a regular Lagrangian mesh. The discretization of the computation domain is shown in Figure 8.12b. The total number of elements is 28,977 with 29,380 nodes. Figure 8.13 depicts the reaction force of the plate on the punch. It is apparent from the graph (horizontal segment of a curve) that a porous plate loses stability (it is also said that a plastic hinge is formed) at a lower load than the corresponding ideal plastic

388

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

4h 12 h 1

2

3

4

h 10 h 1.6 h

(a)

adaptive mesh

(b)

Figure 8.12. Bending of a plate simply supported at the ends and loaded in the middle by a rigid punch. (a) Geometric dimensions of the plate and supporting cylinders. (b) Connecting layer between finer and coarser meshes at the domain interfaces.

plate. This is due to the fact that the yield stress of a damaged material decreases when voids nucleate and grow and is lower than that of the matrix material, Y 0 . The lower yield stress is due to the initial porosity, f0 D 5%. This value of f0 has been chosen so as to trace the formation of sliding bands in more detail. When f0 D 0, there is no initial porosity and voids can only appear through nucleation, the formation of shear bands is observed; however, this process arises at a later stage. The jump in the contact force (Figure 8.13) is due to the formation of a gap between the punch and plate when the plastic hinge is formed (see Figure 8.14). It is apparent from the graph (Figure 8.13) that the model with damage (porosity) predicts a sharp drop in the contact force, which is absent from the ideal plastic model. This drop indicates softening of the material, preceded by localization of plastic strain (see Figure 8.15). Figure 8.14a illustrates the appearance of strain localization bands once a plastic hinge has been formed. Figure 8.14b shows the formation of a gap between the punch and plate in the region where the plate has the maximum curvature (plane strain). Figure 8.15 illustrates bending of the plate as the punch moves, where d D 2:2 h, " is the logarithmic strain, with " D ln.L= l/ D ln.1 C u= l/ in the one-dimensional case and u D L  l. For the GTN model (Figure 8.15b), one observes the formation of a pronounced band-like structure where large plastic strain zones alternate with zones of unloading. The strain localization bands have an increased concentration of voids. The predicted alternating pattern of plastic strain localization bands is similar to that observed in experiments. In the case of ideal plasticity, no localization bandlike structures are observed, with monotonous transition between strain level lines (Figure 8.15a).

389

Section 8.3 Numerical modeling of damaged elastoplastic materials porous (GTN) model ideal plastic model

3.5

1

Punch reaction force

3.0 2.5 2

2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0 1.5 2.0 2.5 Punch displacement, h

3.0

3.5

Figure 8.13. Plate contact force versus punch displacement. Comparison between the ideal plastic (curve 1) and porous (curve 2) model.

displacement of contact surface gap

(a)

(b)

Figure 8.14. Formation of a gap between the punch and the plate: (a) appearance of strain localization bands when a plastic hinge has been formed; (b) the gap arises where the plate has the maximum curvature.

8.3.9 Comparison with experiment A comparison of simulation results with experimental data for the fracture of a clamped plate is given in [166] The calibration constants of the GTN model, q1 , q2 , q3 , fN , "N , and sN , were determined from independent tension tests. The results are compared in Figure 8.16. Unlike the GTN model, an elastoplastic model without damage cannot describe the fracture of the plate.

390

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

(a)

(b)

ε = 0.15

ε = 0.24

Figure 8.15. Plate bending: (a) no localization bands arise in the ideal plastic model; (b) formation of a band-like pattern with voids in the GTN model.

The figure depicts the graph of the punch force versus the punch displacement. The experimental data and simulation results are in good agreement for both the total punch force and its axial component.

8.3.10 Modeling quasi-brittle fracture with damage A modified model of a porous elastoplastic material can be used to solve quasi-brittle fracture problems for continuous media and structures. The model is based on the Gurson yield criterion and a critical porosity criterion – a crack arises and fracture begins when a critical porosity is attained. The formation of a crack is modeled as the appearance of a stress-free surface in an element. Since quasi-brittle fracture occurs at fairly low plastic strains and the porosity is related to the hydrostatic tensile strain (see (8.6) and (8.7)), the critical porosity for a quasi-brittle material is an order of magnitude less than that in ductile fracture. Large plastic strains do not have enough time to develop and so the material begins to fracture before that, due to low critical porosity in tension. The nucleation of new defects can be neglected. The external energy supplied is spent on the formation of a free surface and fracture rather than viscoplastic deformation. This energy consumption is substantially less than in viscoplastic fracture. The quasi-brittle fracture mechanism radically different from the viscoplastic mechanism. The mathematical system of equations for deformation before cracking is formally different from the viscoplastic equations in only the critical

Section 8.3 Numerical modeling of damaged elastoplastic materials 600 F [kN]

391

Experiment Simulation

400

punch force

2 3

1

200

axial force

0 0

50

100

150 u3 [mm]

Figure 8.16. Figure from [166].

porosity constant. The criterion for the formation of a free surface in a continuous medium subject to quasi-brittle fracture is different from that adopted in the brittle fracture mechanism. However, deformation criteria are obviously superior to force (energy) criteria in quasi-brittle and ductile fracture. Quasi-brittle fracture should be analyzed using a porous elastoplastic model and, when the critical porosity is attained, an additional continuum fracture model. The simplest fracture model is the Maenchen–Sack model [111], which was outlined above. According to this model, fracture occurs abruptly and the stress drops to zero in the affected particle. A more complicated model should allow for a gradual transition to a damaged state with stress relaxation to zero and formation of a stress free surface. Below, the transition from quasi-brittle to ductile fracture is studied with the GTN model. The bodies analyzed include a plane specimen with two symmetric V-shaped notches and an axisymmetric specimen with a circular V-shaped notch. The quasi-brittle problems are solved in a dynamic formulation (taking into account inertia forces) by the stabilization method with an explicit integration scheme. The plane specimen (plane deformation) with two symmetric V-shaped notches is shown in Figure 8.17a. Such specimens are the objects of numerous experimental and theoretical studies where the transition mechanism from quasi-brittle to ductile fracture is investigated. Figure 8.17a shows a typical pattern of quasi-brittle fracture. Plastic strains only arise near the notch tips, with a crack propagating from the tips and connecting the notches. The fracture of an elastoplastic specimen is illustrated in Figure 8.17b. An elastoplastic material can withstand much higher strains without failure; its critical porosity

392

Chapter 8 Modeling of damage and fracture of inelastic materials and structures +4.614e – 01 +1.000e – 03 +9.167e – 04 +8.333e – 04 +7.500e – 04 +6.667e – 04 +5.833e – 04 +5.000e – 04 +4.167e – 04 +3.333e – 04 +2.500e – 04 +1.667e – 04 +8.333e – 05 +0.000e +00

+1.000e – 04 +9.167e – 05 +8.333e – 05 +7.500e – 05 +6.667e – 05 +5.833e – 05 +5.000e – 05 +4.167e – 05 +3.333e – 05 +2.500e – 05 +1.667e – 05 +8.333e – 06 +0.000e +00

(a)

(b)

Figure 8.17. Material damage (porosity distribution) and crack propagation for two different fracture processes: (a) brittle fracture with critical porosity fc D 1  104 , loading velocity v D 1 m/s, and loading time t D 1:35  104 s; (b) ductile fracture with loading velocity v D 2 m/s and loading time t D 3:00  104 s.

is several orders of magnitude higher than that of a brittle material. Therefore, at moderate plastic strains, the porosity accumulated near a notch tip is insufficient for a free surface to be formed and so no crack arises. Instead of a horizontal crack, sliding bands are formed at an angle to the symmetry axis, with practically all plastic strain localized within these bands and unloading occurring at the notch tips. The typical fracture pattern (Figure 8.17b) is radically different from that of quasi-brittle fracture (Figure 8.17a). A crack propagates at an angle to the direction of tension, along the lines of maximum porosity localization (Figure 8.17b). For a quasi-brittle material, the critical porosity is relatively low and the fracture occurs along the crack connecting the notch tips. For an elastoplastic material, whose critical porosity is several orders of magnitude higher, the fracture occurs due to shear along strain localization bands at an angle to the line of tension (as shown in Figure 8.17b). This confirms that a model taking into account nucleation and growth of mesodefects shaped as spherical voids is adequate in describing both types of fracture as well as the transition from quasi-brittle to viscoplastic (ductile) fracture due to different material resistance and different fracture mechanisms. The model allows one to explain the experimentally observed change in the fracture pattern in specimens with V-shaped notches.

Section 8.4 Extension of damage theory to the case of an arbitrary stress-strain state

393

8.4 Extension of damage theory to the case of an arbitrary stress-strain state So far, we have considered damage and fracture of elastic solids where damage was associated with the appearance of spherical microvoids and microcracks and so was essentially identified with porosity. Obviously, whether voids are formed in the material or not depends primarily of the type of loading and, hence, on the type of the stress-strain state of the material. If the stress-strain state changes, defects other than spherical voids may arise, which may have a different mechanism of fracture. Stressstrain states should be classified in accordance with the values of the stress invariants, especially in accordance with the first-to-second invariant ratio, known as the coefficient of triaxiality, k D 13 i i =Y , where the second stress invariant N has been replaced with the yield stress Y by virtue of the von Mises yield criterion. In [7], the coefficient k was determined for most essential, experimentally studied types of the stress state and fracture patterns for each stress state. The critical strain intensity at the onset of fracture, "Nf , was obtained experimentally as a function of the triaxiality factor k (Figure 8.19). One can see that the curve with k D 0:4 has a cusp; the investigation of specimens after failure at k  0:4 has shown that the fracture mechanism and fracture type change qualitatively. For k < 0:4, fracture occurs due to sliding in shear along bands of maximum shear at high temperatures close to the melting temperature (adiabatic shear bands), whereas for k > 0:4, mesovoids and mesocracks are formed in the material and the fracture occurs due to their coalescence into macrocracks. The values of the triaxiality factor k for most common experimental kinds of fracture test are listed below: Type of stress-strain state

k

Shear, 12 ¤ 0, with all other ij D 0, i ¤ j Uniaxial tension with 11 ¤ 0 and all other ij D 0

0 1/3 p 3=3 0:6–2:5 k3 k5 k!1

Plane strain with no compressibility Tension or bars with notches Near the tip of a blunted crack (no hardening) Near the tip of a blunted crack (with hardening) Uniaxial strain with "11 ¤ 0 and all other "ij D 0

Note that k changes together with the stress-strain state; therefore, the values corresponding to inhomogeneous stress-strain state refer to areas in the body near a stress concentrator. It will be assumed that the material is general elastoplastic with the yield stress being a multi-parameter function: Y D Y .N"; f; ; N r ; k; "PNp /, where f is damage,  is temperature, k D 13 i i =N is the stress-strain state triaxiality factor,

394

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

1=2  N D 32 sij sij is the shear stress intensity, N r is the residual stress intensity, and "PNp is the plastic strain rate intensity. In order to obtain Y D Y .N"; f; ; N r ; k; "PNp /, one has to use information on fracture from macro and micro mechanics. The coefficient k determines the stress-strain state of the material and play an important role in establishing the mechanism and pattern of fracture; however, other parameters can also be used. For example, it is clear that the critical damage also affects the fracture pattern, but its influence is less significant; therefore, we can restrict ourselves to only k in the first approximation.

8.4.1 Well-posedness of the problem The well-posedness of the initial-boundary value problem follows from the Drucker postulate for quasistatic loading or its analogue, the Hadamard condition, for dynamic loading. The simplest one-to-one relation  D Y ."/ in the softening segment cd00 leads to an ill-posed formulation of initial-boundary value problems if d 00 is in the plane f D 0 (Figure 8.18). 

For a well-posed description of the softening process, one has to assume the yield stress to be a multi-parameter function, Y D Y ."; f; ;  r ; k/, where @Y =@"  0 and dY < 0 is the total differential (Figure 8.18). Then, as follows from experimental data [62], softening must occur because @Y =@f < 0 at k > 0:4 and dY =d < 0 at k < 0:4 (Figure 8.16). 

σy

c b

d'

dσy σy = σy(σ, f) d'' × ε

a d ×

f

Figure 8.18. Yield surface. Stress relaxation process and fracture in the three-dimensional space -"-f .

Section 8.4 Extension of damage theory to the case of an arbitrary stress-strain state

395

To obtain Y D Y ."; f; ;  r ; k/, one has to use information about fracture from micro-mechanics. The coefficient k determines the stress-strain state of the material and plays the most important role in establishing the mechanism and pattern of fracture. To obtain softening, partial derivatives of the function Y must meet some restrictions for different values of k, which follow from the conditions 

for k  0:4, the influence of the temperature is week and so can be neglected: dY D



@Y @Y @Y p @Y df C dk C d" C d r < 0I p @" @f @k @ r

for k < 0:4, the porosity and residual stresses do not affect Y : f D 0;

 r D 0;

dY D

@Y @Y p @Y d C d k < 0I d" C @"p @ @k

one also has to take into account that @Y  0; @"p

@Y < 0; @

@Y > 0: @"Pp

8.4.2 Limitations of the GTN model The Gurson model [52] and its modification, the GTN model [26, 19], are applicable for k  0:4 (Figure 8.19).

0.6

ε¯f

0.4

× × ×

×

0.2 1 –– 3

×

0

0.4

k

Figure 8.19. Dependence of the fracture type "Nf .k/ and change in the fracture type at the triaxiality factor k  0:4 [7].

The curve corresponds to fracture in the plane of equivalent strain and coefficient of triaxiality. For k > 0:4, fracture is mainly due to the formation of voids, while for k < 0:4, fracture occurs chiefly along adiabatic shear bands (Figure 8.19) [96].

396

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

The experimental data of [7] were obtained for quasistatic loading; "Nf is the critical strain intensity at which fracture begins.

8.4.3 Associated viscoplastic law A Gurson type yield criterion for porous materials with an elastoviscoplastic matrix and kinematic hardening has the form  a a a  3 sij sij 3q2 kk a C 2 f q1 cosh  1  q12 f 2 D 0; F .sij ; f; SY / D 2 SY2 2 SY a where sij is the deviator of the active stress ija , SY is the yield stress of the porous elastoplastic material, determined from the condition that the plastic works for the effective material and the matrix must be the same:  p ija "Pij D .1  f /P p SYst. p / C ‰. P p/ ; SY . p ; P p / D SYst . p / C ‰ 1 . P p/; (8.80) q p 2 p p where "ij D 3 ij ij is the plastic strain intensity,  is the relaxation time, SYst is the static yield stress, and ‰ is a function characterizing the effect of the strain rate. To obtain the constitutive equations (8.80), we use the following associated flow law with additive strain and strain rate hardening:    @F a @F SY  p st p ‰ SY  SY . / (8.81) ƒD "Pij D ƒ a ; a sij : @sij  @sij

The constitutive relations of the GTN model described in detail in Section 8.1; these certainly remain valid for the generalized model for k > 0:4 as well.

8.4.4 Constitutive relations in the absence of porosity (k < 0:4, f D 0, r D 0) The Gurson yield criterion [52] becomes the von Mises yield criterion for an elastoviscoplastic material with strain and strain rate hardening and thermal softening. With the multiplicative hardening law in the form of Johnson–Cook [62], the yield stress is written as   Pp     "N 1 C O m Y D Y0 .N"p ; O / R."PNp / D A C B.N"p /n 1 C C ln "P0 with the dimensionless temperature 8 ˆ if  < transition; melt ; where transition is the temperature at which the second stage begins and melt is the melting temperature.

Section 8.4 Extension of damage theory to the case of an arbitrary stress-strain state

397

8.4.5 Fracture model. Fracture criteria f D fcr (with respect to porosity), where tensile stresses dominate, k  0:4;  p p 1=2 p  " Np D 23 ij ij D "cr (with respect to plastic strain intensity), where compressive stresses dominate, k < 0:4. 

In numerical simulation, one adopts a discrete, finite element model of fracture (Maenchen–Sack approach [111], fracture of individual elements begins when the local criterion is satisfied). If the fracture criterion is satisfied in a Lagrangian cell, the internodal links in this cell are released and the stresses either relax to zero or the cell exhibits resistance in compression only. When fracture occurs, the Lagrangian nodal masses become separate particles, which carry away mass, momentum, and energy; these masses move by inertia as rigid particles that do not interact with unaffected particles [19, 95]. For the general stress-strain state, the proposed model of damage is the simplest – it depends on only one scalar parameter, the triaxiality factor k D 13 i i =N , where   N D 32 sij sij 1=2 . The critical strain intensity at fracture is then a function of this coefficient, "Nf D "Nf .k/. The function "Nf shows that the shear mechanism of fracture changes to the tensile mechanism at k  0:4 in a medium weakened by spherical voids. Experiments for stress-strain states corresponding to k < 0:4 reveal the appearance of macrobands of adiabatic shear. Experiments corresponding to k > 0:4 reveal the formation of large spherical voids, which merge together when they touch each other directly or through a chain of smaller voids remaining spherical. The assumption that the voids preserve their spherical shape is a significant limitation of Gurson’s damage theory. As shown by experimental evidence, the void shape undergoes considerable changes as the triaxiality parameter k varies. For k  1:5, the void shape is indeed very close to spherical. For k < 0:4, voids become ellipsoidal, being elongated in the direction of maximum tensile stress [9]. As k decreases further, the coalescence of voids changes to the formation of shear bands; the energy dissipation in the shear bands leads to an increase in the temperature and, hence, softening due to temperature effects on the plastic constants. The misalignment of the principal stress axes and void axes [8] at large plastic strains leads to the necessity of taking into account the anisotropy [12]. The effect of the strain rate increases as well as other effects, which have been being intensively studied for the last 10–15 years (see the review by Besson [13]). Due to these studies, significant progress has been made in developing models of damaged media for an arbitrary stress-strain state.

398

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

8.5 Numerical modeling of cutting of elastoviscoplastic materials in three dimensions To illustrate the potentials of the above methods for modeling three-dimensional fracture problems for elastoviscoplastic materials, let us consider the unsteady process of cutting of an elastoviscoplastic layer (workpiece) with a rigid cutter. The cutter moves with a constant horizontal speed V and its front cutting edge makes an angle ˛ with the vertical Figure 8.20). The simulation was performed using a coupled thermomechanical model of an elastoviscoplastic material [99, 98]. Adiabatic cutting was compared with a cutting mode where the thermal conductivity of the workpiece material was included. A parametric analysis of the cutting process was performed for different geometries of the workpiece and cutter, different cutting speeds and depths, and different properties of the workpiece material. The workpiece thickness, the dimension in the z-direction was varied. The stress-strain state was changed from plane stress, with HN D H=L 1 (thin plate), to plane strain, HN 1 (wide plane), where H is the width L the length of the workpiece. The problem was solved on a LagrangianEulerian mesh by the finite element method with splitting and using explicit-implicit integration schemes, discussed in Section 6.4, for the constitutive equations. The numerical simulation of the problem in three dimensions was shown to have the potential of analyzing cutting processes with continuous and fragmented shavings. The mechanism of this phenomenon Cutting with fragmented shavings? can be explained in the case of orthogonal cutting (˛ D 0) by thermal softening with the formation of adiabatic shear bands with triaxiality factor k < 0:4, without invoking a model with porosity. For cutting at an acute but large angle ˛, one has to employ a coupled model of thermal and structural softening with damage. Dependences of the cutter force for different geometric and physical parameters of the problem were obtained. It was shown that quasi-monotonic and oscillating modes are possible (for both orthogonal and acute-angle cutting) and a physical interpretation of the phenomenon was suggested.

8.5.1 Introduction Cutting processes play an important role in the treatment of hard-to-deform materials on turning and milling machines. Machining is a major pricing operation in the manufacture of complex-profile parts of hard-to-deform materials such as titaniumaluminium or molybdenum alloys. The shavings arising in cutting of such materials can be fragmented (chippings), resulting in a nonsmooth treated surface and highly varying pressure on the cutter. An experimental determination of the parameters of the temperature and stress-strain states of a material at high-speed cutting is extremely difficult. Numerical simulation of the process can be a preferable alternative; it can enable one to explain the main features of the processes and look into the mechanism of cutting. Understanding the mechanism of the formation and fracture of shavings is

Section 8.5 Numerical modeling of cutting of elastoviscoplastic materials

399

of great importance for efficient cutting. Mathematical modeling of the cutting process requires taking into account large strains and strain rates as well as heating due to plastic strain energy dissipation resulting in temperature softening and fracture of the material. So far, these processes have not received due investigation, although considerable effort has been put into since the middle of the 20th century. Early studies relied on the simplest rigid plastic scheme [120, 121, 54, 158, 159]. However, the results obtained with the rigid plastic analysis could not satisfy materials engineer or theoreticians, since the simplified model failed to answer the questions raised. In the literature available, there is no solution to the three-dimensional problem that would take into account the nonlinear effects of the formation, fracture, and fragmentation of the shavings due to the three-dimensional thermomechanical response of the material. It was not until a few years ago when some progress was made in studying these processes owing to the development of efficient computational methods and numerical simulation. The effects of the cutting angle, thermomechanical properties of the workpiece and cutter, and fracture mechanism on the formation and fragmentation of shavings have been investigated [109, 122, 6]. However, most studies treated the cutting processes with significant simplifications: the problem was solved in two dimensions (plain strain), the effect of initial transient processes on the cutter force was not considered, and fracture was assumed along certain preset surfaces. All these restrictions did not allow one to give the cutting process a full enough assessment and, in some cases, led to a wrong understanding of the cutting mechanism.

8.5.2 Statement of the problem Geometry The problem is treated in three-dimensional statement. Figure 8.20 displays the domain and boundary conditions in the plane of cutting. In the direction perpendicular to the plane cutting, the workpiece has a finite thickness HN D H=L (L is the workpiece length), which was varied within a wide range. The spatial statement suggests the freedom of motion of the workpiece in the plane of cutting and a smoother emergence of the shavings, which ensures favorable conditions for cutting. Governing equations The complete coupled system of equations of thermo-elastoviscoplasticity consists of the law of conservation of linear momentum, dv D ij;j ; dt

(8.82)

dij p D Dij kl .P"kl  "Pkl  ˛ıkl P /; dt

(8.83)

 Hooke’s law with thermal stresses,

and heat influx equation, Ce

d p D K;i i  .3 C 2 /˛0 "Pi i C ij "Pkl ; dt

(8.84)

400

Chapter 8 Modeling of damage and fracture of inelastic materials and structures 0.4

V

a

1.0 0.2 u1 = 0

0.01 0.1

7°

u2 = 0

Figure 8.20. Geometry of the workpiece and cutter. Boundary conditions.

where Ce is specific heat capacity, K is the thermal conductivity, and  is the Taylor– Quinney coefficient, taking into account the material heating due to plastic dissipation. The strain of the workpiece is governed by the associated flow law dF p "Pkl D P dij

(8.85)

and yield criterion p

p

F .Ji ; Ei ; i ;  / D J2  Y .J1 ; Ei ; i ;  /  0;

(8.86)

where Ji stands for the stress invariants and Ei denotes the plastic strain tensors (i D 1; 2). The evolution equations for the internal variables are d i D fi .Jk ; i ;  /: (8.87) dt Material model A von Mises-type thermo-elasto-viscoplastic model is adopted; the yield stress is taken in the multiplicative form (8.88) with strain viscoplastic hardening and thermal softening [62]:   P p  E p Pp p n .1  O m /; (8.88) Y .E ; E ;  / D ŒA C B.E / 1 C C ln "P0 where Y is the yield stress, "Np is the plastic strain intensity, and O is the relative temperature referred to the melting temperature m : 8 ˆ if  <  ; m : The workpiece material is assumed to be homogeneous. The computations were performed for a relatively soft material Al2024-T3 (with elastic constants E D 73 GPa

Section 8.5 Numerical modeling of cutting of elastoviscoplastic materials

401

and  D 0:33 and plastic constants A D 369 MPa, B D 684 MPa, n D 0:73, "P0 D 5:77  104 , C D 0:0083, m D 1:7,  D 300 K, m D 775 K, and ˇ D 0:9) and a harder material 42CrMo4 (E D 202 GPa,  D 0:3, A D 612 MPa, B D 436 MPa, n D 0:15, "P0 D 5:77  104 , C D 0:008, m D 1:46,  D 300 K, m D 600 K, and ˇ D 0:9). An adiabatic cutting processes was compared with the solution to the complete thermomechanical problem with heat conduction. Fracture The model of fracture relies on the continuum approach of Maenchen and Sack [111], which suggests modeling of fracture zones by discrete particles. A critical p plastic strain intensity "f is chosen to determine the fracture criterion:     p  "PN J1 p O 1 C d4 ln .1 C d5 /; (8.90) "f D d1 C d2 exp d3 J2 "P0 where di stands for material constants, which are determined experimentally. If the fracture criterion is satisfied in a Lagrangian cell, internode links get broken and either the stresses relax to zero or resistance remains only in tension. As fracture progresses, Lagrangian nodal masses separate into individual particles, carrying away mass, momentum, and energy, which move by inertia as rigid bodies that do not interact with unaffected particles. A detailed survey of these models and fracture algorithms can be found in [40, 19]. The current section defines the onset of fracture as the moment p when a critical plastic strain intensity "f is reached. No fracture surface is defined in p advance – it is determined during the solution. In the analysis discussed, "f D 1:0 with the cutter speed equal to 2 m=s or 20 m=s. Method of integration For integrating the above coupled thermoplastic system of equations (8.82)–(8.90), it is reasonable to use the splitting method developed by the author [94]. The algorithm consists of splitting the whole process into (i) a predictor, a thermoelastic process with "Pp  0 and all operators related to plastic strain being zero, and (ii) a corrector, in which the total strain rate is zero, "P  0. In the predictor step, system (8.82)–(8.87), where the unknowns are marked with a tilde, becomes d vQ dt d Q ij dt d Q Ce dt p "Pij 

D Q ij;j ; D Dij kl ."PQkl  ˛ıkl PQ /;

(8.91)

D K Q;i i  .3 C 2 /˛0 "Pi i ; D 0;

P i D 0:

The last term in the heat influx equation, related to elastic dissipation, is a small quantity even at high temperatures [14] and so can be neglected. Then the heat equation can be integrated separately using an explicit conditionally stable or implicit absolutely stable scheme with splitting in directions; see Section 6.1.3. Then, one solves a dynamic problem with a given right-hand side dependent on Q by an explicit central

402

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

finite difference scheme with respect to vQ and Q ij . The resulting values are then used as the initial values in the corrector step for the following system of equations, where the unknowns are marked with a hat: 

d vO D 0; dt

p "POij D "POeij C "POij D 0:

(8.92)

Hooke’s law (8.83) and the associated flow law (8.85) lead to the stress relaxation equation (6.35) with (8.85) resulting in @F d d O ij D  Dij kl ; dt dt @O ij

(8.93)

subject to the initial values O ij j tD0 D Q ij j tCt . The heat influx equation becomes Ce

d O p D ij "POkl dt

(8.94)

Q tCt . with O j tD0 D j Equations (8.85)–(8.87), with the unknowns marked with a hat, remain unchanged. Computational results First, we discuss the computational results of soving a plane strain problem for a thick plate, HN 1. In this case, the material is constrained in thickness so that "zz D 0 and, therefore, will be pushed toward the free surface (Fig. 8.21a, b). Figure 8.21 shows the formation of shear localization bands and development of fracture surfaces in cutting a thick 42CrMo4 alloy plate: (a) formation of an initial fragment, (b) formation of a second fragment, and (c) separation of a series of fragments (fragmentation); the process is adiabatic with ˛ D 0ı and V D 2 m=s.

(a)

(b)

(c)

Figure 8.21. Formation of viscous shear strain localization bands and fracture in cutting a thick 42CrMo4 plate. The process is assumed adiabatic with orthogonal cutting, ˛ D 0ı , at a speed of V D 2 m=s. (a) Separation of an initial fragment. (b) Separation of a second fragment. (c) Separation of a series of fragments.

In a shear band, fracture occurs as follows. Some material is pushed out to form a prominence, which grows as the cutter advances. In orthogonal cutting, two shear

403

Section 8.5 Numerical modeling of cutting of elastoviscoplastic materials

bands arise, one near the cutter tip and the other in front of the prominence (due to buckling of the prominence surface). As the cutter advances, the two bands move toward each other, resulting in fracture as they meet (Fig. 8.21b). Part of material is chipped off from the workpiece. Thereafter, a new prominence is formed as well as a new shear band in the already heated and softened material. The process becomes nonmonotonic and quasiperiodic and is dependent on the cutter speed. In orthogonal cutting, fracture occurs along shear localization bands and is due to thermal softening caused by plastic strain energy dissipation. After chipping off the first fragment, the process becomes quasi-stationary leading to a continuous or fragmented shaving (Fig. 8.21c), which depends of the ratio between the strength and thermomechanical properties of the workpiece material. Smoother surfaces and continuous shavings are formed at low speeds and small angles of cutting. Taking into account heat conduction also favors smoother shavings. Under plane strain conditions (HN 1), the material is constrained laterally so that the shaving goes up in the cutter plane. In the spatial statement, for thin plates (HN 1), the shaving comes out sideways (Fig. 8.22a), with continuous shaving arising in wider ranges of cutting speeds and thermal properties of the material. Figure 8.22a displays the distribution of the plastic strain intensity and illustrates the formation of a continuous spiraling shaving. Figure 8.22b shows the evolution of the total cutter force. The workpiece material is 42CrMo4, the angle and speed of cutting are ˛ D 0ı and V D 2 m=s, and processes of cutting is analyzed taking into account heat conduction. F, MPa 1.0 0.8 0.6 0.4 0.2 0 0

(a)

20

40

60

80

t, μs

(b)

Figure 8.22. Thermomechanical cutting of a thin plate workpiece (42CrMo4). (a) Formation of a shaving and distribution of the plastic strain intensity. (b) Total reaction cutter force F . The angle of cutting is ˛ D 0ı and the cutting speed is V D 2 m=s.

The formation of a continuous shaving produces a monotonic cutter force. For a fragmented shaving, the reaction force has a sawtooth form (Fig. 8.24), which adversely affects the process of cutting.

404

Chapter 8 Modeling of damage and fracture of inelastic materials and structures

In cutting a thin plate, the first shear band is formed before the material is heated up; it is along this band that the initial fracture occurs leading to the separation of the first chip. The reaction force attains its maximum magnitude before the formation of adiabatic shear bands, arising when the material becomes sufficiently warm. Then the cutter moves along the hot and softened workpiece by flattening out and shearing off the material causing it to flow out laterally as a continuously spiraling shaving (Fig. 8.22a). After the initial transient segment, the cutter force decreases and reaches a steady-state value resulting in a quasi-monotonic process of cutting. The high-frequency oscillations arising about this value are due to material fracture at the contact surface and fragmentation of small particles (Fig. 8.22b).

(a)

(b)

Figure 8.23. Thermomechanical cutting and formation of a shaving: (a) a 42CrMo4 workpiece with ˛ D 0ı ; (b) an Al2024-T3 workpiece with ˛ D 30ı .

Figure 8.23a illustrates the shaving fragmentation in orthogonal cutting of a hardto-deform material (42CrMo4), while Fig. 8.23b shows the formation of a continuous shaving in nonorthogonal cutting of a soft material (Al2024-T3) with the cutting angle ˛ D 30ı . No strain localization bands are formed at low cutting speeds with heat conduction taken into account or at sufficiently large cutting angles and/or high plasticity of the workpiece material (e.g., duralumin); the shaving emerges continuously, with no chips (Fig. 8.23b). At low speeds and orthogonal or nearly orthogonal cutting (˛  0ı –10ı ), no fracture occurs other than in the first shear band where a single fragment is chipped off (Fig. 8.23a). Figures 8.24a and b display the evolution of the cutter reaction force for thick plates. It is apparent from Fig. 8.24a that there is a transient region related to the formation of the primary shear band, where the reaction force attains its maximum. In orthogonal cutting of a thin plate (Fig. 8.22b), the reaction force drops abruptly when the first fragment has been chipped off, the amount of drop dependent on the cutting speed, and then it gradually stabilizes reaching a quasi-stationary value. Similar drops are observed when further fragments are chipped off (curve 1 in Fig. 8.24a). The force drop is due to the fact that, after chipping off the first fragment, the cutter acts on the heated and softened material in the shear band, whose fracture strength is much lower.

405

Section 8.5 Numerical modeling of cutting of elastoviscoplastic materials F, MPa

F, MPa

1.2

1.0

2

0.8

0.8

10º

0.6

30º

0.4

1

0.4

0º

0.2

0 0

1

2

(a)

3 t, μs

0 0

40

80

t, μs

(b)

Figure 8.24. Cutter reaction force F arising in cutting a thick plate: (a) plate material 42CrMo4, cutting angle ˛ D 0ı (1, adiabatic model, 2, thermal model); (b) plate material Al2024-T3, different cutting angles.

Taking into account heat conduction of the material results in weaker softening and the absence of fracture in the secondary shear bands (curve 2 in Fig. 8.24a). For larger angles of cutting and softer materials, such as Al2024-T3, there is no fragmentation of the shaving in the primary shear band (Fig. 8.24b) and, as a consequence, the reaction force does not drop. The shaving is continuous and the reaction force gradually increases tending to a quasi-stationary value. When the shaving is fragmented, the reaction force has a sawtooth form (curve 1, Fig. 8.24a), which adversely affects the technological process of cutting. It is apparent from Fig. 8.24b that as the angle of cutting increases, the cutter force decreases while changing quasi-monotonically and tending gradually to a quasistationary value. The mechanisms of cutting and shaving fracture also change. There is no fracture along adiabatic shear bands. The fracture depends on the type of wedging. A shear zone with hydrostatic tension arises in front of the cutter, where not only thermal but also structural softening should be taken into account. This zone is related to the nucleation and growth of microdefects. Conclusion To summarize, it has been shown that the process of cutting with a constant speed has an initial transient stage, where the cutter reaction force attains a maximum value. After a drop, the reaction force changes in an oscillatory or monotonic fashion, depending on whether the shaving becomes fragmented or remains continuous. The simulation of this process requires taking into account thermal softening, fracture, and fragmentation of the workpiece material. Depending on the cutter geometry and the stress-strain state near the cutter edge, the process can cause the formation of void-like defects or can occur without them. In orthogonal or nearly orthogonal cutting, no voids are formed; what causes fragmentation of the shaving is thermal softening.

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Chapter 8 Modeling of damage and fracture of inelastic materials and structures

8.6 Conclusions. General remarks on elastoplastic equations and their numerical solution 8.6.1 Formulations of systems of equations for elastoplastic media As already noted in Section 6.4, the system of equations for an elastoplastic medium consists of conservation laws for mass, momentum, and energy as well as constitutive relations. The constitutive relations can generally include relations in different forms such as partial differential equations, finite relations, and equality or inequality constraints for the stresses, strains, and internal variables of state characterizing the internal structure of the medium. The total system of equations should be reduced to an already known mathematical problem or its generalization. A number of possible formulations were discussed in Chapter 6. To choose an efficient numerical method for solving a well-posed mathematical problem, one has to investigate the characteristic features of the governing system of equations as well as difficulties that may arise in solving the problem. Once the problem has been reduced to a system of partial differential equations, one has to determine the pure type of the system (hyperbolic, elliptic, or parabolic) or its mixed type (hyperbolic-elliptic, elliptic-parabolic, etc.). Basic formulations of mathematical problems and numerical methods for their solution relevant to studying elastoplastic problems are outlined below. Most of the formulations have already been discussed throughout the book. Here we give a summary of these results.

8.6.2 A hardening elastoplastic medium Deformation equations for hardening elastoplastic solids (deformation theory of plasticity) are, by their nature, very close to the equations for elastic solids. At the loading stage, these simply represent the equations of nonlinear elasticity. Different material response to loading and unloading can be treated as a special case of the incremental nonlinear elastic theory F .sign S; S; i / D 0, where S is the stress intensity, which only depends on the sign of the loading increment, sign S . In general, the material response predicted by the incremental equations is different from that predicted by the deformation theory in that it depends on the loading history. Accordingly, the constitutive equations contain derivatives with respect to the loading parameter , which can be interpreted as conditional time. It follows from the aforesaid that, in the case of a deformation theory, the boundary value problem for active loading of a hardening elastoplastic body is qualitatively no different from that for a nonlinear elastic body. These are elliptic equations and the methods for their solution are the same as in the case of a linear elastic body with iterations. In the case of incremental equations, one has to take into account the process history and the problem must be solved by the method of stepwise loading in the loading

Section 8.6 Conclusions. General remarks on elastoplastic equations

407

parameter . In each small step, the boundary value problem is essentially no different from the nonlinear elastic problem (deformational plastic, to be more precise) and is also an elliptic problem. In general, the solution consists of consecutively solving elliptic problems for each step in the conditional time and can be treated as the solution of a parabolic system of equations. The choice of the loading parameter may not be easy. It must ensure that the stress intensity changes monotonically during the loading process and only then is similar to the time parameter; in this case, the total solution is similar to the solution of a parabolic problem. The solution of dynamic problems for the class of hardening elastoplastic media is closely connected with that of nonlinear elastic problems, just as in statics. The only difference is that the dynamic equations contain physical time t as well as conditional time. Therefore, dynamic processes are studied in real time t and so the constitutive equations of the plastic media must include the derivative d =dt . This does not introduce any substantial additional complications. The system of equations is hyperbolic in the variables t , xi and, hence, can be solved with the methods developed for nonlinear hyperbolic equations. In dynamic problems, the plasticity theory does not present any simplifying advantages as compared with the incremental theory. In both cases, the solution is carried out with stepwise methods in real time (the loading history in time is equally important for both models).

8.6.3 Ideal elastoplastic media: a degenerate case The situation changes when an ideal (without hardening) elastoplastic medium is considered; the yield criterion will only depend on the stresses (and the more so for the case of a softening material). The model of an ideal plastic medium is so widespread because its yield criterion does not contain, by definition, any kinematic or energy parameters and, in simple cases (e.g., plane deformation), the problem becomes statically determinate. The complete system of equations splits into a static and a kinematic subsystem. The static problem can only be linked to the kinematic problem via boundary conditions set in terms of displacement rates. Furthermore, many problems deal with small elastic deformations, which can be neglected, thus leading to the rigid ideal plastic model. This model can be studied using the simplified limit equilibrium theory of [67]. From the common ground of modern solution methods, an ideal elastoplastic medium represents a degenerate case in three dimensions; specifically, in the plastic region, the equations change their type (from elliptic, in an elastic domain, to hyperbolic, in a plastic domain, in statics and from hyperbolic to elliptic in dynamics). This kind of idealization does not simplify the solution of elastoplastic boundary value problems; conversely, it complicates the solution, which is due to the introduction of an unknown boundary between the elastic and plastic zones. In the case of softening of a deformed state, the problem becomes ill-posed.

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Chapter 8 Modeling of damage and fracture of inelastic materials and structures

8.6.4 Difficulties in solving mixed elliptic-hyperbolic problems Under inhomogeneous deformation, the boundary value problem becomes mixed, elliptic hyperbolic; the equations are elliptic is the region of elastic deformation and hyperbolic in the plastic region. The solution of boundary value problems for such equations is a poorly studied area of mathematical physics. There is no general theory of these equations, with only relatively few specific problems solved. Therefore, one has to rely on experimental investigation of such problems by numerical methods. Numerical experiments have shown that the degeneration may not occur in some problems where plasticity is reached, provided that the material is in a constrained state and its “geometric hardening” takes place. In problems where there are conditions for unconstrained flow, slip bands are formed and unlimited deformations can occur (in much the same way as some solutions in the limit equilibrium theory), in which case the stiffness matrix becomes ill-conditioned. As this state is approached, the condition number of the system of equations increases and the solution quality rapidly decreases; furthermore, the convergence of iterative processes deteriorates, the solution becomes highly dependent on the discretization of the domain, and so on. These effects become even more pronounced in strain softening, when the strain hardening modulus becomes negative. This indicates the ill-posedness of the classical ideal plastic model and the need to consider additional effects, which play an important role when the material approaches its limit state and include, for example, the formation of bands with sharp localization of plastic strains, leading to material fracture and failure of the structure as a whole. In other words, the model requires a regularization that would allow the determination of the stress-strain state when the elastoplastic material is close to fracture.

8.6.5 Regularization of an elastoplastic model One of the simplest and most efficient ways of performing regularization is to consider the effect of the strain rate on the yield criterion. In doing so, one has to adopt an elastoviscoplastic material model, which takes into account the dependence of the material properties on the time scale [87]. Elastoviscoplastic equations reflect the physical behavior of the material at elastoplastic softening, since the appearance of strain localization bands is accompanied by a sharp increase in the strain and plastic strain rate. One cannot ignore this effect. The elastoviscoplastic equations have been thoroughly studied mathematically. They do not change their type in the case of ideal plasticity or at softening. Furthermore, it has been shown that the limit solution to these equations as the viscosity vanishes,  ! 0, coincides with the elastoplastic solution wherever it changes smoothly, without discontinuities [82, 93]. In this case, we have a perfectly prosperous parabolic system (in quasistatics) or hyperbolic system (in dynamics).

Section 8.6 Conclusions. General remarks on elastoplastic equations

409

Quasi-fronts are formed near discontinuities of solutions to the elastoviscoplastic equations ; the solutions change rapidly within these quasi-fronts. The viscous terms are of the same order of magnitude as the elastoplastic terms within the quasi-fronts; as ı ! 1, the elastoviscoplastic equations become elastoplastic (in this case, the solution in the transient layer is fully determinable analytically) [82, 91]. If the sequence of viscous solutions converges as ı ! 1 to some function (possibly discontinuous), this function is taken to be the generalized solution to the viscoplastic problem. A common mathematical theory has been developed, with some constraints (similar to yield criteria), for hyperbolic systems with a small parameter multiplying the highest differential operator. The questions of well-posedness of determining the generalized solution in this manner have been studies by . The problems based on this model can be solved with the same methods as those for hardening elastoplastic media. Certain difficulties, associated with the high stiffness of the elastoviscoplastic equations and multiscale effect, can arise in problems with softening, where strains can increase rapidly in very narrow zones due to the appearance of strain localization bands. However, these difficulties can be coped with by using special methods developed for solving stiff problems [93], in particular, the method of splitting in physical processes, where dissipative finite difference schemes are obtained. An alternative, physical regularization approach is to introduce in the model (a) porosity, since for k  0:4 (Figure 8.19), there is predominant tension leading to the nucleation and growth of voids, or (b) temperature, since for k > 0:4, there are predominant adiabatic shear bands, indicating significant plastic strain energy dissipation and temperature change. The two regularization approaches are formally different, since the regularization through the introduction of viscosity increases the order of the constitutive equations. The introduction of physical parameters (void-like defects, adiabatic shear) does not change the order of the constitutive equations, but increases the number of internal parameters of the system and, hence, the number of equations in the system, which is essentially equivalent to increasing the system order. The physical approach was used in this chapter. The mathematical approach (introduction of viscosity), was used in Chapters 2–6.

8.6.6 Elastoplastic shock waves Discontinuous solutions to elastoplastic initial-boundary value problems in the Prandtl–Reuss flow theory are much more complicated as well as their generalizations to the case of hardening and softening media. Within the classical statement, such problems do not have a unique solution if there is a discontinuity. In order to obtain a unique solution, one has to put forward additional hypotheses on the constitutive equations. A quasi-thermodynamic hypothesis was adopted in [20] suggesting that the energy dissipation is maximum at discontinuities. This allowed the authors

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Chapter 8 Modeling of damage and fracture of inelastic materials and structures

to close the system of equations on strong discontinuities and determine the speeds of propagation of elastoplastic waves in isotropic media under the assumption of von Mises and Tresca–Saint-Venant ideal plasticity. The difficulty is that the standard approach to solving continuum problems with discontinuities suggests replacing the system of differential equations with an equivalent system of integral conservation laws and obtaining conservation laws at the discontinuities by passing to the limit (see Section 1.2). For this approach to be feasible, the original system must be reducible to a divergence form. Unfortunately, all systems of constitutive differential equations with complex rheology cannot be reduced to a divergence form, including the equations of the Prandtl–Reuss flow theory and its generalization to hardening and, the more so, softening elastoplastic media. It becomes necessary to change the statement of the problem and give a new, more suitable generalization of the classical solution. This can be done with a variational approach similar to that based on the classical variational principles (see Sections 1.6.2 and 1.8) used in mathematical physics to introduce a generalized solution. The only difference from the classical variational principles is the usage of variational inequalities in quasistatic problems . The method of variational inequalities was used in to solve dynamic problems (see also Section 6.6), where the concept of a generalized solution was formulated within the framework of the Prandtl–Reuss flow theory and used to obtain a complete system of relations for determining strong discontinuities (shock waves) in elastoplastic media.

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Index

acoustics equation, 96 analysis discrete, 71 approach conventional, 3 phenomenological, 359, 373 physical, 359 approximation adiabatic, 230 conditional, 103, 114, 207 differential, 244, 248–259 hyperbolic () form, 249 parabolic (…) form, 249 finite-difference, 71–78 unconditional, 208 approximation error, 73–76, 293, 324 global, 75, 76 local, 75 associated flow rule, 26, 63, 277, 281, 284, 292, 357, 362, 396, 400 basis contravariant, 4, 53 covariant, 3, 4, 53, 58 frozen, 5, 6, 8, 320 bending of plate, 206, 387, 390 with shear and rotational inertia, 206 bicharacteristic, 228, 235, 239, 287 boundary condition Dirichlet, 173 kinematic, 40 left, 185, 188 mixed, 102, 173, 287 natural, 40, 42, 45 principal, 40, 42 static, 40, 45 von Neumann, 173 boundary layer, 88, 161

boundary value problem, 88, 124, 160, 165, 387 dynamic, 263, 403 elliptic hyperbolic, 404 elliptic equations, 172, 403 general, 163 linear equations, 163 nonlinear equations, 164 parabolic equations, 403 quasistatic, 263 solution methods, 160–193, 379 stability, 115 stiff, 166, 183 stiff two-point, 160 two-point, 124, 160, 169 Cauchy (initial value) problem, 81–94, 163, 165, 263 cell irregular, 300 characteristic, 110, 209, 210, 213, 217, 232, 347 spatial, 227, 239, 242 characteristic cone, 232, 238, 287, 311 characteristic equation, 108, 205, 242 characteristic form, 287, 304, 342 characteristic grid, 214, 217, 218 characteristic plane, 239, 243 characteristic scheme, 222, 224, 225, 227, 238, 244 characteristic surface, 228, 232, 341 characteristic time, 189, 215, 244, 365, 372 Cholesky decomposition, 128–129, 158 Christoffel symbols, 6 compatibility condition, 13, 72, 219 condition approximation, 78, 79, 99, 249 boundary, see boundary condition consistency, 99

Index convergence, 147 Cottrell, 35 Courant, 109, 110, 116, 198 Courant–Friedrichs–Lewy (CFL), 110, 238, 270, 313 Dirichlet, 173 dissipation, 293 Drucker, 358, 375 Gurson (criterion), 36 Hadamard, 357, 368, 375, 394 initial, see initial condition Lipschitz, 87, 142 normal contact, 344 practical stability, 206, 247 stability, 79, 107, 113, 197, 198, 220, 246, 345 von Neumann, 108, 132, 173, 199, 202 condition number of matrix, 88, 122, 123, 136, 157 configuration, 49 current (deformed) C t , 50–62 initial (material) C0 , 49–62 intermediate C , 54, 62 unloaded Cp , 62 conservation law, 9–15, 301, 333 angular momentum, 9 at discontinuities, 13–14, 219, 406 differential form, 11, 14 divergence form, 11, 15 energy, 9, 279, 333, 402 general form, 10, 11 integral form, 9, 14, 302 mass, 9, 279, 312, 333, 402 momentum, 9, 11, 57, 279, 312, 333, 399, 402 non-divergence form, 11, 13, 15 consistency condition, 99 constitutive equations, 14, 18–26, 32, 42, 47, 49, 270, 278 damaged media, 375–381, 396 elastic media, 47 elasto(visco)plastic media, 276, 279, 317, 333, 353, 409 GTN model, 361, 371 hyperelastic media, 66 hyperelasto(visco)plastic media, 321, 352

423 hypoelastic media, 279, 306, 318 hypoelastoplastic media, 297, 318, 319 plastic media, 61–67, 407 potential form, 43 viscoelastic media, 49, 396 wave propagation, 215, 353 contact, 219, 307, 338, 344–346, 388 convergence, 73, 78–80, 110, 131, 133, 135–153, 157, 175, 379, 404 Lax theorem, 78 convergence rate, 135, 136, 139, 146 asymptotic, 136, 139 average, 136 exponential, 136, 139 linear, 141, 147 Newton method, 145–147, 379 optimal, 136, 138 quadratic, 141, 145, 379 corrector, 86, 220, 255, 280, 287, 295, 319, 323, 380, 384 coupled longitudinal and transverse vibrations, 204–206 coupled thermomechanical perturbations, 307 coupled thermomechanical problems, 245–248, 296, 373, 398, 400 coupled elastoviscoplastic model with damage, 373, 375, 398 damage, 277, 346, 354–361, 366, 375, 393 ductile, 354 GTN model, 361, 363 quasi-brittle, 354, 390 viscous, see ductile damage kinetics, 346 damage surface, 371 decomposition additive, of tensors, 62, 66, 322 Cholesky, 128 multiplicative, of deformation gradient, 61, 322 polar, 51 deformation, 7, 8, 50 elastic, 30, 358, 365 finite, 49, 60, 303, 330 irreversible, 27

424 plastic, 276, 358, 363, 365 reversible, 27, 321 deformation gradient, 7, 8, 61 relative, 54 deformation rate, see rate of deformation derivative convective, 5, 59 corotational, 322 covariant, 6, 333 in Eulerian basis, 10 Jaumann, 279, 305, 322 Lagrangian, 5, 312 Lie, 56, 59 material, 5, 11, 56, 322 material time, 8, 56, 332 objective, 56, 306, 379 substantial, 5 Truesdell, 59 Zaremba–Jaumann, 59 description Eulerian (spatial), 4, 50, 64, 331 Lagrangian (material), 5, 50, 63, 66, 331 diagonal domination, 176, 177, 190 difference backward, 75, 212, 274, 344 forward, 75, 82, 212, 274, 344 discontinuous solution, 219, 220, 254, 338, 409 discrete set of points, 72 edge effect, 161, see boundary layer elastoplastic flow, 30, 63, 341 Prandtl–Reuss, 26 elastoviscoplastic dynamic problems, 276–285 entropy, 16, 20–22 equations acoustics, 96 characteristic, 108, 205, 242 constitutive, see constitutive equations difference, 74, 78, 170, 175 elliptic, 172, 175, 310, 406 finite difference, 74, 78, 170, 175 hyperbolic, 95, 110, 114, 211, 245, 251, 268, 310, 357 parabolic, 95, 119, 245, 251 parabolic-hyperbolic, 209

Index Poisson, 118, 172, 178 strictly hyperbolic, 227 estimation of solution accuracy, 49, 117 Euler predictor-corrector method, 85 Eulerian description, 4, 50, 64, 331 finite element method, 42, 73, 128, 183, 376, 398 finite-difference approximation, 71–78 five-constant theory of elasticity, 65 fracture, 354, 356, 359, 393, 397 continuum, 354, 359 ductile, 355, 373 quasi-brittle, 355, 373, 390, 392 frame of reference, 3–5, 10, 13, 49, 58, 235, 331, 335, 352 frozen coefficients, 113, 119 Gauss procedure, 127 Gaussian elimination method, 126 Gaussian elimination with partial pivoting, 127, 343, 346 Gear implicit scheme, 90 general postulate of plasticity, 26 grid characteristic, 214, 217, 218 irregular, 76 regular, 212, 218 space-like layer, 214 grid function, 72, 73, 79, 173, 176, 293 hardening, 24, 66, 362, 406 isotropic, 230, 281 kinematic, 297, 363, 369, 396 linear, 283, 373 strain, 281, 396 strain rate, 396 translationally isotropic, 24 viscoplastic, 244 hardening parameter, 27, 28, 30, 277, 281, 305, 374 hardening tensor, 63, 64 heat equation, 16, 21, 101, 112, 113, 116, 120, 171, 247, 265, 401

Index ideal gas, 273, 311 inequality Clausius–Duhem, 17, 19, 22 dissipative, 19 variation, 290, 291 initial condition, 95, 102, 165, 185, 197, 269, 380 intensity plastic strain, 361, 362 residual stress, 364 shear stress, 29, 30, 364 viscoplastic strain, 30 interpolating polynomials, 83, 90, 119, 226 iterative method, 83, 130, 140, 148, 152, 156, 282, 324 Newton, 145 nonstationary, 149 Newton–Raphson, 145 modified, 146 secant, 147, 148 simple, 143–145 two-stage, 148 iterative process, 130, 154, 157, 241, 328, 385 convergence rate, 136, 141 convergent, 131, 132, 146, 154, 159 divergent, 140, 145, 147, 148 external, 149 internal, 149 Jacobi, 131, 132 nonlinear, 139 nonstationary, 131 Seidel, 131 single-step, 130, 148 stationary, 131 steady-state, 131 steepest descent, 154 Tchebyshev, 140 three-layer, 130 two-layer, 130 unsteady, 131, 139 Lagrange equations of second kind, 44 law associated flow, 357, 371, 380, 400 associated viscoplastic flow, 396 conservation, see conservation law

425 hardening, 66, 396 Hooke, 30, 65, 93, 96, 276, 290, 361, 377 with thermal stresses, 399 plastic flow, 63, 66, 370, 377 thermodynamic first, 15 second, 16, 18 Lax convergence theorem, 78, 79 Lax scheme, 97, 109, 254, 255, 319, 340 Lax–Friedrichs scheme, 97–99 Lax–Wendroff scheme, 97–99, 111, 254, 338 leapfrog scheme, 97 loading active, 24, 26, 277, 359, 406 cyclic, 373 dynamic, 32, 35, 375, 394 impact, 338 neutral, 24, 26, 27, 277 passive, 26 quasistatic, 375, 394 thermomechanical, 354, 375 velocity, 363 matrix acoustic, 271 characteristic, 238 conservative, 271 convective, 271 decomposition, 127 dissipative, 271 elastic compliance, 27 elastoplastic compliance, 27 elastoplastic stiffness, 29 elastoviscoplastic, 245, 396 Hessian, 146, 153, 154 ill-conditioned, 124, 139 Jacobian, 88, 92, 143, 145, 166 lower triangular, 127, 131 positive definite, 154 positive semi-definite, 130 sparse, 129 stiff, 184 transformation (transition), 107, 136, 198, 200, 202, 248

426 tridiagonal, 116, 124, 180 upper triangular, 127, 131 well-conditioned, 124 maximum principle, 175, 290 media combined, 22 conservative, 43 continuous, 3 elastoplastic, 26, 244, 276, 279, 305, 333, 358, 380, 406 classical, 380 damaged, 358 hardening, 406, 410 ideal, 407 porous, 382 Prandtl–Reuss, 357 softening, 410 with linear hardening, 381 elastoviscoplastic, 30, 49, 229, 244, 285, 289, 318, 370 hyperelastic, 46, 323 hyperelastoplastic, 66, 321 hypoelastic, 279, 306, 318, 321 hypoelastoplastic, 279, 319 nondissipative, 43 rheologically complex, 18 rigid-plastic, 24, 260 thermoelastic isotropic, 21 method adaptive moving mesh, 311 coarse particles, 314, 316 conjugate gradient, 155 descent, 152 conjugate gradient, 155 coordinate, 152 steepest, 154 Euler predictor-corrector, 85 flux, 316 Fourier separation of variables, 181 FPIC, 316 Gaussian elimination, 126–129 initial parameter, 161–163, 187 generalized, 185 inverse Euler, 377 iterative, 130 Jacobi, 131 Siedel, 131

Index simple, 143 single-step, 130 two-step, 148 markers and fluxes, 317 Newton, 145 modified, 147 nonstationary, 149 PIC (particle-in-cell), 311 limitations, 315 quasi-linearization, 165 secant, 147 shooting, 165, 190, 193 splitting, 263, 286, 379, 383, 386, 401 stabilization, 133, 391 sweep, 116, 129, 166, 170, 180, 183, 200, 270 differential, 166, 172 for heat equation, 171 matrix, 178, 181, 203, 268 orthogonal, 186, 187 scalar, 183, 266, 268 upper relaxation, 132 model, see media collisionless, 22–24 combined, 22–24 damage, 354–410 generalized micromechanical multiscale, 363–375 dislocation, 367 elastoplastic, 25, 222, 361 regularization, 231, 375, 408 elastoviscoplastic, 222, 230, 376 advantages, 230, 231 fracture, 397 Gurson–Tvergaard–Needlman (GTN), 360–363, 372, 375, 389–391 constitutive equations, 361 integration, 382 limitations, 395 Maenchen–Sack, 391 Prandtl–Reuss, 357 rigid-plastic, 24 termo-elasto-viscoplastic, 400 motion description Eulerian (spatial), 4, 64 in arbitrary moving coordinate system, 331

427

Index Lagrangian (material), 5, 63 large deformations, 49–56 natural boundary conditions, 40, 42, 45 Navier–Stokes equations divergence form, 270 non-divergence form, 272 neutral loading, 24, 26, 27, 277 Newton method, 145 modified, 147 nonstationary, 149 nonlinear iterative process, 139 norm discrepancy, 131, 150 of grid function, 72, 73, 76 of matrix, 107, 122, 143 Poisson equation, 118, 172 solution by matrix sweep method, 176 stability of finite difference scheme, 178 polar decomposition, 51 porosity, 356, 361, 363, 368, 383–388, 395 critical, 371, 390, 392 predictor, 86, 90, 99, 218, 279, 294, 296, 319, 322, 377, 379, 401 predictor-corrector method, 85 predictor-corrector scheme, 90, 99 three-layer, 255 principle d’Alembert, 60 frozen coefficients, 113 gradientality, 30 Hellinger–Reissner, 42 Hu–Washizu, 42 maximum, for second-order difference equations, 175 maximum, for strain rate dissipation, 290 maximum rate of dissipation, 20 nondecreasing discrepancy, 150 statically admissible stress fields, 45, 46 thermodynamic additional, 21 general, 21

variational, 43, 60 Castigliano, 45, 46, 49 complete, 43 general, 46, 48 Hamilton, 44 Lagrange, 43, 49 mixed, 42, 49 virtual displacements, 40, 41, 60 virtual velocities, 40, 41, 60 virtual work, 60 process iterative, see iterative process thermodynamic irreversible, 23 product dyadic (tensor), 4, 11, 312 scalar, 3, 60 projection operator, 73 rate of deformation, 32, 55, 56, 60 additive decomposition, 62 rate of rotation, 55 regular grid, 212, 218 relative deformation gradient, 54 Richardson approximation formula, 77 rigid rotation, 279, 306, 319 scheme Adams–Bashforth, 83, 84, 119 averaging, 208 characteristic direct, 244, 245, 260 higher orders, 225 inverse, 212 two- and three-dimensional, 227–245 CIR, 212, 244 comparison of explicit and implicit, 100 conservative, 303, 297 completely, 309 implicit, 308 dissipative, 292 dissipative and dispersive properties, 251 Euler, 82 backward, 82, 103 central difference, 84 explicit, 82

428 forward, 82, 103 implicit, 82 predictor-corrector, 85, 90 explicit-implicit, 222, 224 Gear, 90 grid-characteristic, 211, 222, 341, 343 heat equation Crank–Nicolson, 103 explicit, 102 explicit Allen–Cheng, 103 explicit Du Fort–Frankel, 104 implicit, 102 higher-order, 83, 85, 225 hybrid, 339 Runge–Kutta, 85–87, 119, 225 shock-capturing, 245, 307 splitting, see splitting method stability analysis, 90, 106, 119, 296 viscosity, 99 wave equation, 100 cross, 96 explicit, 101 implicit, 100 Lax–Friedrichs, 97 Lax–Wendroff, 97 leapfrog, 96, 97 splitting, 268 solution boundary value problems, 166–182 discontinuous, 219, 254 dynamic problems, 244 heat equation, 101–105 nonlinear wave propagation problems, 209–226 numerical, 89, 160 parasitic, 80, 84 Poisson equation, 172 wave equation, 95–100 weak, 40 specific additional strain energy, 48 splitting method, 263–295 elastoviscoplastic dynamic problems, 276 explicit scheme, 264 finite deformation, 306 general scheme, 263, 264

Index implicit scheme, 264 viscous fluid, 270 stability, 78, 108 absolute, 89, 112, 203, 268, 401 boundary value problems, 115 conditional, 109, 112 scheme heat equation, 112 Lax–Wendroff, 112 stability condition, 78 countable, 171 Courant–Friedrichs–Lewy (CFL), 110 Drucker rheological, 375 Hadamard, 375 necessary, 107, 198 practical, 206, 247 shock, 220 spectral, 115 sufficient, 197, 202, 265 von Neumann, 107, 202 stabilization method, 133, 391 steady-state flow, 50 stencil, 75–77, 95, 97, 100, 102, 104, 173, 177, 191, 255, 267, 296 step size selection heat equation, 116 wave equation, 117 stiff systems, 88, 91, 183 stiffness ratio, 88 stretch ratio, 51 stretch tensor, 51, 63, 322 sweep, 116, 129, 166, 170, 180, 183, 200, 270 differential, 166, 172 five-point, 275 for heat equation, 171 matrix, 178, 181, 203, 268 orthogonal, 186, 187 scalar, 183, 266, 268 three-point, 275 Tchebychev polynomial, 138 tensor compliance, 27, 276 deformation gradient, 7, 54, 58, 62 metric, 4, 63, 66, 333 rate of deformation, 55, 60 rate of rotation, 55, 279

429

Index skew-symmetric, 305, 323 spin, 55 strain Euler–Almansi, 8, 52, 56, 64, 66 Green–Lagrange, 7, 52, 55, 62–64 small, 7, 63 stress Cauchy, 57, 59, 63, 65, 321, 379 first Piola–Kirchhoff, 57, 58 Kirchhoff, 58, 59 second Piola–Kirchhoff, 58, 63 velocity gradient, 8, 55 vorticity, 55 theory Drucker–Prager, 283 Prandtl–Reuss, 290, 357, 409 von Mises, 276 triaxiality factor, 36, 361, 393, 398 unconditional approximation, 208 unloaded configuration, 62 unstable scheme, 79, 111, 199, 253, 258 upper relaxation method, 132 variables Eulerian, 5, 8, 15, 279 internal, 18, 26, 286, 324, 358, 359, 377, 400 kinematic, 18, 42, 43 Lagrangian, 5, 7, 15, 58 nodal, 379 state, 16, 19, 406 structural, 377 thermodynamic, 308 vector basis, 3, 4 Burgers, 36, 39 displacement, 52

force, 20 strain rate, 30 stress, 40 surface, 57 traction, 40, 57 vibration bending, 208 coupled, 204 linear, 197 longitudinal, 197 transverse, 200–206 viscosity, 30 approximation, 222, 337 artificial, 307, 337 scheme, 99 von Neumann, 309 wave equation, 95, 100, 197 as system of two equations, 96 discontinuity contact, 219 moving, 219 discontinuous solutions, 219 in displacements, 95, 108 nonlinear, 119 weighted averaging, 74 Wilkins correction factor, 286 Wilkins correction rule, 283 yield criterion, 29, 66 differential, 281 Drucker–Prager, 283 Gurson, 36, 361, 370, 384, 390 plastic, 287 Tresca–Saint Venant, 290 von Mises, 36, 66, 277, 285, 322, 377, 384, 393