Numerical Continuum Mechanics (eBook)

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Vladimir N. Kukudzhanov, Ishlinsky Institute for Problems in Mechanics, Russia.

Preface 5
I Basic equations of continuum mechanics 19
1 Basic equations of continuous media 21
1.1 Methods of describing motion of continuous media 21
1.1.1 Coordinate systems and methods of describing motion of continuous media 21
1.1.2 Eulerian description 22
1.1.3 Lagrangian description 23
1.1.4 Differentiation of bases 23
1.1.5 Description of deformations and rates of deformation of a continuous medium 25
1.2 Conservation laws. Integral and differential forms 27
1.2.1 Integral form of conservation laws 27
1.2.2 Differential form of conservation laws 29
1.2.3 Conservation laws at solution discontinuities 31
1.2.4 Conclusions 32
1.3 Thermodynamics 33
1.3.1 First law of thermodynamics 33
1.3.2 Second law of thermodynamics 34
1.3.3 Conclusions 36
1.4 Constitutive equations 36
1.4.1 General form of constitutive equations. Internal variables 36
1.4.2 Equations of viscous compressible heat-conducting gases 39
1.4.3 Thermoelastic isotropic media 39
1.4.4 Combined media 40
1.4.5 Rigid-plastic media with translationally isotropic hardening 42
1.4.6 Elastoplastic model 43
1.5 Theory of plastic flow. Theory of internal variables 44
1.5.1 Statement of the problem. Equations of an elastoplastic medium 44
1.5.2 Equations of an elastoviscoplastic medium 48
1.6 Experimental determination of constitutive relations under dynamic loading 50
1.6.1 Experimental results and experimentally obtained constitutive equations 50
1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory 54
1.7 Principle of virtual displacements. Weak solutions to equations of motion 58
1.7.1 Principles of virtual displacements and velocities 58
1.7.2 Weak formulation of the problem of continuum mechanics 60
1.8 Variational principles of continuum mechanics 61
1.8.1 Lagrange’s variational principle 61
1.8.2 Hamilton’s variational principle 62
1.8.3 Castigliano’s variational principle 63
1.8.4 General variational principle for solving continuum mechanics problems 64
1.8.5 Estimation of solution error 67
1.9 Kinematics of continuous media. Finite deformations 67
1.9.1 Description of the motion of solids at large deformations 67
1.9.2 Motion: deformation and rotation 68
1.9.3 Strain measure. Green-Lagrange and Euler-Almansi strain tensors 70
1.9.4 Deformation of area and volume elements 71
1.9.5 Transformations: initial, reference, and intermediate configurations 72
1.9.6 Differentiation of tensors. Rate of deformation measures 73
1.10 Stress measures 75
1.10.1 Current configuration. Cauchy stress tensor 75
1.10.2 Current and initial configurations. The first and second Piola-Kirchhoff stress tensors 75
1.10.3 Measures of the rate of change of stress tensors 77
1.11 Variational principles for finite deformations 78
1.11.1 Principle of virtual work 78
1.11.2 Statement of the principle in increments 78
1.12 Constitutive equations of plasticity under finite deformations 79
1.12.1 Multiplicative decomposition. Deformation gradients 79
1.12.2 Material description 81
1.12.3 Spatial description 82
1.12.4 Elastic isotropic body 83
1.12.5 Hyperelastoplastic medium 84
1.12.6 The von Mises yield criterion 84
II Theory of finite-difference schemes 87
2 The basics of the theory of finite-difference schemes 89
2.1 Finite-difference approximations for differential operators 89
2.1.1 Finite-difference approximation 89
2.1.2 Estimation of approximation error 91
2.1.3 Richardson’s extrapolation formula 95
2.2 Stability and convergence of finite difference equations 96
2.2.1 Stability 96
2.2.2 Lax convergence theorem 96
2.2.3 Example of an unstable finite difference scheme 97
2.3 Numerical integration of the Cauchy problem for systems of equations 99
2.3.1 Euler schemes 100
2.3.2 Adams-Bashforth scheme 101
2.3.3 Construction of higher-order schemes by series expansion 103
2.3.4 Runge-Kutta schemes 103
2.4 Cauchy problem for stiff systems of ordinary differential equations 106
2.4.1 Stiff systems of ordinary differential equations 106
2.4.2 Numerical solution 107
2.4.3 Stability analysis 108
2.4.4 Singularly perturbed systems 109
2.4.5 Extension of a rod made of a nonlinear viscoplastic material 110
2.5 Finite difference schemes for one-dimensional partial differential equations 113
2.5.1 Solution of the wave equation in displacements. The cross scheme 113
2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations) 114
2.5.3 The leapfrog scheme 115
2.5.4 The Lax-Friedrichs scheme 115
2.5.5 The Lax-Wendroff Scheme 116
2.5.6 Scheme viscosity 117
2.5.7 Solution of the wave equation. Implicit scheme 118
2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points 118
2.5.9 Heat equation 119
2.5.10 Unsteady thermal conduction. Explicit scheme (forward Euler scheme) 121
2.5.11 Unsteady thermal conduction. Implicit scheme (backward Euler scheme) 121
2.5.12 Unsteady thermal conduction. Crank-Nicolson scheme 121
2.5.13 Unsteady thermal conduction. Allen-Cheng explicit scheme 121
2.5.14 Unsteady thermal conduction. Du Fort-Frankel explicit scheme 122
2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives 122
2.6 Stability analysis for finite difference schemes 124
2.6.1 Stability of a two-layer finite difference scheme 125
2.6.2 The von Neumann stability condition 125
2.6.3 Stability of the wave equation 126
2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition 127
2.6.5 Stability of schemes for the heat equation 130
2.6.6 The principle of frozen coefficients 131
2.6.7 Stability in solving boundary value problems 133
2.6.8 Step size selection in an implicit scheme in solving the heat equation 134
2.6.9 Step size selection in solving the wave equation 135
2.7 Exercises 135
3 Methods for solving systems of algebraic equations 140
3.1 Matrix norm and condition number of matrix 140
3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix 140
3.1.2 Relative error of solution for perturbed coefficient matrix 141
3.1.3 Example 142
3.1.4 Regularization of an ill-conditioned system of equations 143
3.2 Direct methods for linear system of equations 144
3.2.1 Gaussian elimination method. Matrix factorization 144
3.2.2 Gaussian elimination with partial pivoting 145
3.2.3 Cholesky decomposition. The square root method 146
3.3 Iterative methods for linear system of equations 148
3.3.1 Single-step iterative processes 148
3.3.2 Seidel and Jacobi iterative processes 149
3.3.3 The stabilization method 151
3.3.4 Optimization of the rate of convergence of a steady-state process 153
3.3.5 Optimization of unsteady processes 155
3.4 Methods for solving nonlinear equations 158
3.4.1 Nonlinear equations and iterative methods 158
3.4.2 Contractive mappings. The fixed point theorem 159
3.4.3 Method of simple iterations. Sufficient convergence condition 161
3.5 Nonlinear equations: Newton’s method and its modifications 163
3.5.1 Newton’s method 163
3.5.2 Modified Newton-Raphson method 165
3.5.3 The secant method 165
3.5.4 Two-stage iterative methods 166
3.5.5 Nonstationary Newton method. Optimal step selection 167
3.6 Methods of minimization of functions (descent methods) 170
3.6.1 The coordinate descent method 170
3.6.2 The steepest descent method 172
3.6.3 The conjugate gradient method 173
3.6.4 An iterative method using spectral-equivalent operators or reconditioning 174
3.7 Exercises 175
4 Methods for solving boundary value problems for systems of equations 178
4.1 Numerical solution of two-point boundary value problems 178
4.1.1 Stiff two-point boundary value problem 178
4.1.2 Method of initial parameters 179
4.2 General boundary value problem for systems of linear equations 181
4.3 General boundary value problem for systems of nonlinear equations 182
4.3.1 Shooting method 183
4.3.2 Quasi-linearization method 183
4.4 Solution of boundary value problems by the sweep method 184
4.4.1 Differential sweep 184
4.4.2 Solution of finite difference equation by the sweep method 188
4.4.3 Sweep method for the heat equation 189
4.5 Solution of boundary value problems for elliptic equations 190
4.5.1 Poisson’s equation 190
4.5.2 Maximum principle for second-order finite difference equations 193
4.5.3 Stability of a finite difference scheme for Poisson’s equation 194
4.5.4 Diagonal domination 194
4.5.5 Solution of Poisson’s equation by the matrix sweep method 196
4.5.6 Fourier’s method of separation of variables 199
4.6 Stiff boundary value problems 201
4.6.1 Stiff systems of differential equations 201
4.6.2 Generalized method of initial parameters 203
4.6.3 Orthogonal sweep 204
4.7 Exercises 207
III Finite-difference methods for solving nonlinear evolution equations of continuum mechanics 213
5 Wave propagation problems 215
5.1 Linear vibrations of elastic beams 215
5.1.1 Longitudinal vibrations 215
5.1.2 Explicit scheme. Sufficient stability conditions 215
5.1.3 Longitudinal vibrations. Implicit scheme 217
5.1.4 Transverse vibrations 218
5.1.5 Transverse vibrations. Explicit scheme 220
5.1.6 Transverse vibrations. Implicit scheme 221
5.1.7 Coupled longitudinal and transverse vibrations 222
5.1.8 Transverse bending of a plate with shear and rotational inertia 224
5.1.9 Conclusion 227
5.2 Solution of nonlinear wave propagation problems 227
5.2.1 Hyperbolic system of equations and characteristics 227
5.2.2 Finite difference approximation along characteristics. The direct and semi-inverse methods 229
5.2.3 Inverse method. The Courant-Isaacson-Rees grid-characteristic scheme 229
5.2.4 Wave propagation in a nonlinear elastic beam 230
5.2.5 Wave propagation in an elastoviscoplastic beam 233
5.2.6 Discontinuous solutions. Constant coefficient equation 237
5.2.7 Discontinuous solutions of a nonlinear equation 238
5.2.8 Stability of difference characteristic equations 240
5.2.9 Characteristic and grid-characteristic schemes for solving stiff problems 240
5.2.10 Stability of characteristic and grid-characteristic schemes for stiff problems 242
5.2.11 Characteristic schemes of higher orders of accuracy 243
5.3 Two- and three-dimensional characteristic schemes and their application 245
5.3.1 Spatial characteristics 245
5.3.2 Basic equations of elastoviscoplastic media 247
5.3.3 Spatial three-dimensional characteristics for semi-linear system 249
5.3.4 Characteristic equations. Spatial problem 253
5.3.5 Axisymmetric problem 254
5.3.6 Difference equations. Axisymmetric problem 256
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 262
5.4 Coupled thermomechanics problems 263
5.5 Differential approximation for difference equations 266
5.5.1 Hyperbolic and parabolic forms of differential approximation 266
5.5.2 Example 267
5.5.3 Stability 268
5.5.4 Analysis of dissipative and dispersive properties 269
5.5.5 Example 271
5.5.6 Analysis of properties of finite difference schemes for discontinuous solutions 272
5.5.7 Smoothing of non-physical perturbations in a calculation on a real grid 277
5.6 Exercises 278
6 Finite-difference splitting method for solving dynamic problems 281
6.1 General scheme of the splitting method 281
6.1.1 Explicit splitting scheme 281
6.1.2 Implicit splitting scheme 282
6.1.3 Stability 283
6.2 Splitting of 2D/3D equations into 1D equations (splitting along directions) 283
6.2.1 Splitting along directions of initial-boundary value problems for the heat equation 283
6.2.2 Splitting schemes for the wave equation 286
6.3 Splitting of constitutive equations for complex rheological models into simple ones. A splitting scheme for a viscous fluid 288
6.3.1 Divergence form of equations 288
6.3.2 Non-divergence form of equations 290
6.3.3 One-dimensional equations. Ideal gas 291
6.3.4 Implementation of the scheme 293
6.4 Splitting scheme for elastoviscoplastic dynamic problems 294
6.4.1 Constitutive equations of elastoplastic media 294
6.4.2 Some approaches to solving elastoplastic equations 295
6.4.3 Splitting of the constitutive equations 297
6.4.4 The theory of von Mises type flows. Isotropic hardening 299
6.4.5 Drucker-Prager plasticity theory 301
6.4.6 Elastoviscoplastic media 303
6.5 Splitting schemes for points on the axis of revolution 304
6.5.1 Calculation of boundary points 304
6.5.2 Calculation of axial points 306
6.6 Integration of elastoviscoplastic flow equations by variation inequality 308
6.6.1 Variation inequality 308
6.6.2 Dissipative schemes 310
6.7 Exercises 313
7 Solution of elastoplastic dynamic and quasistatic problems with finite deformations 316
7.1 Conservative approximations on curvilinear Lagrangian meshes 316
7.1.1 Formulas for natural approximation of spatial derivatives 316
7.1.2 Approximation of a Lagrangian mesh 317
7.1.3 Conservative finite difference schemes 319
7.2 Finite elastoplastic deformations 321
7.2.1 Conservative schemes in one-dimensional case 321
7.2.2 A conservative two-dimensional scheme for an elastoplastic medium 323
7.2.3 Splitting of the equations of a hypoelastic material 324
7.3 Propagation of coupled thermomechanical perturbations in gases 325
7.3.1 Basic equations 325
7.3.2 Conservative finite difference scheme 325
7.3.3 Non-divergence form of the energy equation. A completely conservative scheme 327
7.4 The PIC method and its modifications for solid mechanics problems 329
7.4.1 Disadvantages of Lagrangian and Eulerian meshes 329
7.4.2 The particle-in-cell (PIC) method 329
7.4.3 The method of coarse particles 332
7.4.4 Limitations of the PIC method and its modifications 333
7.4.5 The combined flux and particle-in-cell (FPIC) method 334
7.4.6 The method of markers and fluxes 335
7.5 Application of PIC-type methods to solving elastoviscoplastic problems 335
7.5.1 Hypoelastic medium 336
7.5.2 Hypoelastoplastic medium 337
7.5.3 Splitting for a hyperelastoplastic medium 339
7.6 Optimization of moving one-dimensional meshes 342
7.6.1 Optimal mesh for a given function 343
7.6.2 Optimal mesh for solving an initial-boundary value problem 344
7.6.3 Mesh optimization in several parameters 345
7.6.4 Heat propagation from a combustion source 346
7.7 Adaptive 2D/3D meshes for finite deformation problems 348
7.7.1 Methods for reorganization of a Lagrangian mesh 348
7.7.2 Description of motion in an arbitrary moving coordinate system 349
7.7.3 Adaptive meshes 351
7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes 353
7.8.1 Algorithms for constructing moving meshes 353
7.8.2 Selection of a finite difference scheme 355
7.8.3 A hybrid scheme of variable order of approximation at internal nodes 357
7.8.4 A grid-characteristic scheme at boundary nodes 359
7.8.5 Calculation of contact boundaries 362
7.8.6 Calculation of damage kinetics 364
7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains 365
7.9 Exercises 370
8 Modeling of damage and fracture of inelastic materials and structures 372
8.1 Concept of damage and the construction of models of damaged media 372
8.1.1 Concept of continuum fracture and damage 372
8.1.2 Construction of damage models 373
8.1.3 Constitutive equations of the GTN model 379
8.2 Generalized micromechanical multiscale damage model 381
8.2.1 Micromechanical model. The stage of plastic flow and hardening 382
8.2.2 Stage of void nucleation 383
8.2.3 Stage of the appearance of voids and damage 384
8.2.4 Relationship between micro and macro parameters 385
8.2.5 Macromodel 386
8.2.6 Tension of a thin rod with a constant strain rate 391
8.2.7 Conclusion 393
8.3 Numerical modeling of damaged elastoplastic materials 393
8.3.1 Regularization of equations for elastoplastic materials at softening 393
8.3.2 Solution of damage problems 394
8.3.3 Inverse Euler method 395
8.3.4 Solution of a boundary value problem. Computation of the Jacobian 397
8.3.5 Splitting method 397
8.3.6 Integration of the constitutive relations of the GTN model 400
8.3.7 Uniaxial tension. Computational results 404
8.3.8 Bending of a plate 405
8.3.9 Comparison with experiment 407
8.3.10 Modeling quasi-brittle fracture with damage 408
8.4 Extension of damage theory to the case of an arbitrary stress-strain state 411
8.4.1 Well-posedness of the problem 412
8.4.2 Limitations of the GTN model 413
8.4.3 Associated viscoplastic law 414
8.4.4 Constitutive relations in the absence of porosity (k < 0.4, f = 0, sr = 0)
8.4.5 Fracture model. Fracture criteria 415
8.5 Numerical modeling of cutting of elastoviscoplastic materials 416
8.5.1 Introduction 416
8.5.2 Statement of the problem 417
8.6 Conclusions. General remarks on elastoplastic equations 424
8.6.1 Formulations of systems of equations for elastoplastic media 424
8.6.2 A hardening elastoplastic medium 424
8.6.3 Ideal elastoplastic media: a degenerate case 425
8.6.4 Difficulties in solving mixed elliptic-hyperbolic problems 426
8.6.5 Regularization of an elastoplastic model 426
8.6.6 Elastoplastic shock waves 427
Bibliography 429
Index 440