Numerical Continuum Mechanics

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 10
1.1.1 Coordinate systems and methods of describing motion of continuous media......Page 20
1.1.2 Eulerian description......Page 21
1.1.4 Differentiation of bases......Page 22
1.1.5 Description of deformations and rates of deformation of a continuous medium......Page 24
1.2.1 Integral form of conservation laws......Page 26
1.2.2 Differential form of conservation laws......Page 28
1.2.3 Conservation laws at solution discontinuities......Page 30
1.2.4 Conclusions......Page 31
1.3.1 First law of thermodynamics......Page 32
1.3.2 Second law of thermodynamics......Page 33
1.4.1 General form of constitutive equations. Internal variables......Page 35
1.4.3 Thermoelastic isotropic media......Page 38
1.4.4 Combined media......Page 39
1.4.5 Rigid-plastic media with translationally isotropic hardening......Page 41
1.4.6 Elastoplastic model......Page 42
1.5.1 Statement of the problem. Equations of an elastoplastic medium......Page 43
1.5.2 Equations of an elastoviscoplastic medium......Page 47
1.6.1 Experimental results and experimentally obtained constitutive equations......Page 49
1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory......Page 53
1.7.1 Principles of virtual displacements and velocities......Page 57
1.7.2 Weak formulation of the problem of continuum mechanics......Page 59
1.8.1 Lagrange’s variational principle......Page 60
1.8.2 Hamilton’s variational principle......Page 61
1.8.3 Castigliano’s variational principle......Page 62
1.8.4 General variational principle for solving continuum mechanics problems......Page 63
1.9.1 Description of the motion of solids at large deformations......Page 66
1.9.2 Motion: deformation and rotation......Page 67
1.9.3 Strain measures. Green–Lagrange and Euler–Almansi strain tensors......Page 69
1.9.4 Deformation of area and volume elements......Page 70
1.9.5 Transformations: initial, reference, and intermediate configurations......Page 71
1.9.6 Differentiation of tensors. Rate of deformation measures......Page 72
1.10.2 Current and initial configurations. The first and second Piola–Kirchhoff stress tensors......Page 74
1.10.3 Measures of the rate of change of stress tensors......Page 76
1.11.2 Statement of the principle in increments......Page 77
1.12.1 Multiplicative decomposition. Deformation gradients......Page 78
1.12.2 Material description......Page 80
1.12.3 Spatial description......Page 81
1.12.4 Elastic isotropic body......Page 82
1.12.6 The von Mises yield criterion......Page 83
2.1.1 Finite-difference approximation......Page 86
2.1.2 Estimation of approximation error......Page 88
2.1.3 Richardson’s extrapolation formula......Page 92
2.2.2 Lax convergence theorem......Page 93
2.2.3 Example of an unstable finite difference scheme......Page 94
2.3 Numerical integration of the Cauchy problem for systems of first-order ordinary differential equations......Page 96
2.3.1 Euler schemes......Page 97
2.3.2 Adams–Bashforth scheme......Page 98
2.3.4 Runge–Kutta schemes......Page 100
2.4.1 Stiff systems of ordinary differential equations......Page 103
2.4.2 Numerical solution......Page 104
2.4.3 Stability analysis......Page 105
2.4.4 Singularly perturbed systems......Page 106
2.4.5 Extension of a rod made of a nonlinear viscoplastic material......Page 107
2.5.1 Solution of the wave equation in displacements. The cross scheme......Page 110
2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations)......Page 111
2.5.4 The Lax–Friedrichs scheme......Page 112
2.5.5 The Lax–Wendroff Scheme......Page 113
2.5.6 Scheme viscosity......Page 114
2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points......Page 115
2.5.9 Heat equation......Page 116
2.5.13 Unsteady thermal conduction. Allen–Cheng explicit scheme......Page 118
2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives......Page 119
2.6 Stability analysis for finite difference schemes......Page 121
2.6.2 The von Neumann stability condition......Page 122
2.6.3 Stability of the wave equation......Page 123
2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition......Page 124
2.6.5 Stability of schemes for the heat equation......Page 127
2.6.6 The principle of frozen coefficients......Page 128
2.6.7 Stability in solving boundary value problems......Page 130
2.6.8 Step size selection in an implicit scheme in solving the heat equation......Page 131
2.7 Exercises......Page 132
3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix......Page 137
3.1.2 Relative error of solution for perturbed coefficient matrix......Page 138
3.1.3 Example......Page 139
3.1.4 Regularization of an ill-conditioned system of equations......Page 140
3.2.1 Gaussian elimination method. Matrix factorization......Page 141
3.2.2 Gaussian elimination with partial pivoting......Page 142
3.2.3 Cholesky decomposition. The square root method......Page 143
3.3.1 Single-step iterative processes......Page 145
3.3.2 Seidel and Jacobi iterative processes......Page 146
3.3.3 The stabilization method......Page 148
3.3.4 Optimization of the rate of convergence of a steady-state process......Page 150
3.3.5 Optimization of unsteady processes......Page 152
3.4.1 Nonlinear equations and iterative methods......Page 155
3.4.2 Contractive mappings. The fixed point theorem......Page 156
3.4.3 Method of simple iterations. Sufficient convergence condition......Page 158
3.5.1 Newton’s method......Page 160
3.5.3 The secant method......Page 162
3.5.4 Two-stage iterative methods......Page 163
3.5.5 Nonstationary Newton method. Optimal step selection......Page 164
3.6.1 The coordinate descent method......Page 167
3.6.2 The steepest descent method......Page 169
3.6.3 The conjugate gradient method......Page 170
3.6.4 An iterative method using spectral-equivalent operators or reconditioning......Page 171
3.7 Exercises......Page 172
4.1.1 Stiff two-point boundary value problem......Page 175
4.1.2 Method of initial parameters......Page 176
4.2 General boundary value problem for systems of linear equations......Page 178
4.3 General boundary value problem for systems of nonlinear equations......Page 179
4.3.2 Quasi-linearization method......Page 180
4.4.1 Differential sweep......Page 181
4.4.2 Solution of finite difference equation by the sweep method......Page 185
4.4.3 Sweep method for the heat equation......Page 186
4.5.1 Poisson’s equation......Page 187
4.5.2 Maximum principle for second-order finite difference equations......Page 190
4.5.4 Diagonal domination......Page 191
4.5.5 Solution of Poisson’s equation by the matrix sweep method......Page 193
4.5.6 Fourier’s method of separation of variables......Page 196
4.6.1 Stiff systems of differential equations......Page 198
4.6.2 Generalized method of initial parameters......Page 200
4.6.3 Orthogonal sweep......Page 201
4.7 Exercises......Page 204
5.1.2 Explicit scheme. Sufficient stability conditions......Page 210
5.1.3 Longitudinal vibrations. Implicit scheme......Page 212
5.1.4 Transverse vibrations......Page 213
5.1.5 Transverse vibrations. Explicit scheme......Page 215
5.1.6 Transverse vibrations. Implicit scheme......Page 216
5.1.7 Coupled longitudinal and transverse vibrations......Page 217
5.1.8 Transverse bending of a plate with shear and rotational inertia......Page 219
5.2.1 Hyperbolic system of equations and characteristics......Page 222
5.2.3 Inverse method. The Courant–Isaacson–Rees grid-characteristic scheme......Page 224
5.2.4 Wave propagation in a nonlinear elastic beam......Page 225
5.2.5 Wave propagation in an elastoviscoplastic beam......Page 228
5.2.6 Discontinuous solutions. Constant coefficient equation......Page 232
5.2.7 Discontinuous solutions of a nonlinear equation......Page 233
5.2.9 Characteristic and grid-characteristic schemes for solving stiff problems......Page 235
5.2.10 Stability of characteristic and grid-characteristic schemes for stiff problems......Page 237
5.2.11 Characteristic schemes of higher orders of accuracy......Page 238
5.3.1 Spatial characteristics......Page 240
5.3.2 Basic equations of elastoviscoplastic media......Page 242
5.3.3 Spatial three-dimensional characteristics for semi-linear system......Page 244
5.3.4 Characteristic equations. Spatial problem......Page 248
5.3.5 Axisymmetric problem......Page 249
5.3.6 Difference equations. Axisymmetric problem......Page 251
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......Page 257
5.4 Coupled thermomechanics problems......Page 258
5.5.1 Hyperbolic and parabolic forms of differential approximation......Page 261
5.5.2 Example......Page 262
5.5.3 Stability......Page 263
5.5.4 Analysis of dissipative and dispersive properties......Page 264
5.5.5 Example......Page 266
5.5.6 Analysis of properties of finite difference schemes for discontinuous solutions......Page 267
5.5.7 Smoothing of non-physical perturbations in a calculation on a real grid......Page 272
5.6 Exercises......Page 273
6.1.1 Explicit splitting scheme......Page 276
6.1.2 Implicit splitting scheme......Page 277
6.2.1 Splitting along directions of initial-boundary value problems for the heat equation......Page 278
6.2.2 Splitting schemes for the wave equation......Page 281
6.3.1 Divergence form of equations......Page 283
6.3.2 Non-divergence form of equations......Page 285
6.3.3 One-dimensional equations. Ideal gas......Page 286
6.3.4 Implementation of the scheme......Page 288
6.4.1 Constitutive equations of elastoplastic media......Page 289
6.4.2 Some approaches to solving elastoplastic equations......Page 290
6.4.3 Splitting of the constitutive equations......Page 292
6.4.4 The theory of vonMises type flows. Isotropic hardening......Page 294
6.4.5 Drucker–Prager plasticity theory......Page 296
6.4.6 Elastoviscoplastic media......Page 298
6.5.1 Calculation of boundary points......Page 299
6.5.2 Calculation of axial points......Page 301
6.6.1 Variation inequality......Page 303
6.6.2 Dissipative schemes......Page 305
6.7 Exercises......Page 308
7.1.1 Formulas for natural approximation of spatial derivatives......Page 311
7.1.2 Approximation of a Lagrangian mesh......Page 312
7.1.3 Conservative finite difference schemes......Page 314
7.2.1 Conservative schemes in one-dimensional case......Page 316
7.2.2 A conservative two-dimensional scheme for an elastoplastic medium......Page 318
7.2.3 Splitting of the equations of a hypoelastic material......Page 319
7.3.2 Conservative finite difference scheme......Page 320
7.3.3 Non-divergence form of the energy equation. A completely conservative scheme......Page 322
7.4.2 The particle-in-cell (PIC) method......Page 324
7.4.3 The method of coarse particles......Page 327
7.4.4 Limitations of the PIC method and its modifications......Page 328
7.4.5 The combined flux and particle-in-cell (FPIC) method......Page 329
7.5 Application of PIC-type methods to solving elastoviscoplastic problems with complicated constitutive equations......Page 330
7.5.1 Hypoelastic medium......Page 331
7.5.2 Hypoelastoplastic medium......Page 332
7.5.3 Splitting for a hyperelastoplastic medium......Page 334
7.6 Optimization of moving one-dimensional meshes......Page 337
7.6.1 Optimal mesh for a given function......Page 338
7.6.2 Optimal mesh for solving an initial-boundary value problem......Page 339
7.6.3 Mesh optimization in several parameters......Page 340
7.6.4 Heat propagation from a combustion source......Page 341
7.7.1 Methods for reorganization of a Lagrangian mesh......Page 343
7.7.2 Description of motion in an arbitrary moving coordinate system......Page 344
7.7.3 Adaptive meshes......Page 346
7.8.1 Algorithms for constructing moving meshes......Page 348
7.8.2 Selection of a finite difference scheme......Page 350
7.8.3 A hybrid scheme of variable order of approximation at internal nodes......Page 352
7.8.4 A grid-characteristic scheme at boundary nodes......Page 354
7.8.5 Calculation of contact boundaries......Page 357
7.8.6 Calculation of damage kinetics......Page 359
7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains......Page 360
7.9 Exercises......Page 365
8.1.1 Concept of continuum fracture and damage......Page 367
8.1.2 Construction of damage models......Page 368
8.1.3 Constitutive equations of the GTN model......Page 374
8.2 Generalized micromechanical multiscale damage model for an elastoplastic material in tension......Page 376
8.2.1 Micromechanical model. The stage of plastic flow and hardening......Page 377
8.2.2 Stage of void nucleation......Page 378
8.2.3 Stage of the appearance of voids and damage......Page 379
8.2.4 Relationship between micro and macro parameters......Page 380
8.2.5 Macromodel......Page 381
8.2.6 Tension of a thin rod with a constant strain rate......Page 386
8.3.1 Regularization of equations for elastoplastic materials at softening......Page 388
8.3.2 Solution of damage problems......Page 389
8.3.3 Inverse Euler method......Page 390
8.3.5 Splitting method......Page 392
8.3.6 Integration of the constitutive relations of the GTN model......Page 395
8.3.7 Uniaxial tension. Computational results......Page 399
8.3.8 Bending of a plate......Page 400
8.3.9 Comparison with experiment......Page 402
8.3.10 Modeling quasi-brittle fracture with damage......Page 403
8.4 Extension of damage theory to the case of an arbitrary stress-strain state......Page 406
8.4.1 Well-posedness of the problem......Page 407
8.4.2 Limitations of the GTN model......Page 408
8.4.4 Constitutive relations in the absence of porosity......Page 409
8.4.5 Fracture model. Fracture criteria......Page 410
8.5.1 Introduction......Page 411
8.5.2 Statement of the problem......Page 412
8.6.2 A hardening elastoplastic medium......Page 419
8.6.3 Ideal elastoplastic media: a degenerate case......Page 420
8.6.5 Regularization of an elastoplastic model......Page 421
8.6.6 Elastoplastic shock waves......Page 422
Bibliography......Page 424
Index......Page 435