Numerical continuum mechanics:
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| Language: | English |
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Berlin [u.a.]
de Gruyter
2013
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| Series: | De Gruyter studies in mathematical physics
15 |
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| Physical Description: | XVIII, 429 S. Ill., graph. Darst. |
| ISBN: | 3110273225 9783110273229 |
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| 100 | 1 | |a Kukudžanov, Vladimir Nikolaevič |d 1941- |e Verfasser |0 (DE-588)1029626855 |4 aut | |
| 245 | 1 | 0 | |a Numerical continuum mechanics |c Vladimir N. Kukudzhanov |
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| 300 | |a XVIII, 429 S. |b Ill., graph. Darst. | ||
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IMAGE 1
CONTENTS
PREFACE V
I BASIC EQUATIONS OF CONTINUUM MECHANICS
1 BASIC EQUATIONS O F CONTINUOUS MEDIA 3
1.1 METHODS O F DESCRIBING MOTION O F CONTINUOUS MEDIA 3
1.1.1 COORDINATE SYSTEMS AND METHODS O F DESCRIBING MOTION OF CONTINUOUS
MEDIA 3
1.1.2 EULERIAN DESCRIPTION 4
1.1.3 LAGRANGIAN DESCRIPTION 5
1.1.4 DIFFERENTIATION O F BASES 5
1.1.5 DESCRIPTION O F DEFORMATIONS AND RATES O F DEFORMATION O F A
CONTINUOUS MEDIUM 7
1.2 CONSERVATION LAWS. INTEGRAL AND DIFFERENTIAL FORMS 9
1.2.1 INTEGRAL FORM O F CONSERVATION LAWS 9
1.2.2 DIFFERENTIAL FORM O F CONSERVATION LAWS 11
1.2.3 CONSERVATION LAWS AT SOLUTION DISCONTINUITIES 13
1.2.4 CONCLUSIONS 14
1.3 THERMODYNAMICS 15
1.3.1 FIRST LAW O F THERMODYNAMICS 15
1.3.2 SECOND LAW O F THERMODYNAMICS 16
1.3.3 CONCLUSIONS 18
1.4 CONSTITUTIVE EQUATIONS 18
1.4.1 GENERAL FORM O F CONSTITUTIVE EQUATIONS. INTERNAL VARIABLES . . .
. 18 1.4.2 EQUATIONS O F VISCOUS COMPRESSIBLE HEAT-CONDUCTING GASES . .
. . 21 1.4.3 THERMOELASTIC ISOTROPIC MEDIA 21
1.4.4 COMBINED MEDIA 22
1.4.5 RIGID-PLASTIC MEDIA WITH TRANSLATIONALLY ISOTROPIC HARDENING . .
24 1.4.6 ELASTOPLASTIC MODEL 25
1.5 THEORY O F PLASTIC FLOW. THEORY O F INTERNAL VARIABLES 26
1.5.1 STATEMENT O F THE PROBLEM. EQUATIONS O F AN ELASTOPLASTIC MEDIUM V
2 6
1.5.2 EQUATIONS O F AN ELASTOVISCOPLASTIC MEDIUM 30
HTTP://D-NB.INFO/1027078249
IMAGE 2
X
CONTENTS
1.6 EXPERIMENTAL DETERMINATION O F CONSTITUTIVE RELATIONS UNDER
DYNAMIC LOADING 32
1.6.1 EXPERIMENTAL RESULTS AND EXPERIMENTALLY OBTAINED CONSTITUTIVE
EQUATIONS 32
1.6.2 SUBSTANTIATION O F ELASTOVISCOPLASTIC EQUATIONS ON THE BASIS OF
DISLOCATION THEORY 36
1.7 PRINCIPLE O F VIRTUAL DISPLACEMENTS. WEAK SOLUTIONS TO EQUATIONS O F
MOTION 40
1.7.1 PRINCIPLES O F VIRTUAL DISPLACEMENTS AND VELOCITIES 40
1.7.2 WEAK FORMULATION O F THE PROBLEM O F CONTINUUM MECHANICS . . . 42
1.8 VARIATIONAL PRINCIPLES O F CONTINUUM MECHANICS 43
1.8.1 LAGRANGE'S VARIATIONAL PRINCIPLE 43
1.8.2 HAMILTON'S VARIATIONAL PRINCIPLE 44
1.8.3 CASTIGLIANO'S VARIATIONAL PRINCIPLE 45
1.8.4 GENERAL VARIATIONAL PRINCIPLE FOR SOLVING CONTINUUM MECHANICS
PROBLEMS 46
1.8.5 ESTIMATION O F SOLUTION ERROR 49
1.9 KINEMATICS O F CONTINUOUS MEDIA. FINITE DEFORMATIONS 49
1.9.1 DESCRIPTION O F THE MOTION O F SOLIDS AT LARGE DEFORMATIONS . . .
. 49 1.9.2 MOTION: DEFORMATION AND ROTATION 50
1.9.3 STRAIN MEASURES. GREEN-LAGRANGE AND EULER-ALMANSI STRAIN TENSORS
52
1.9.4 DEFORMATION OF AREA AND VOLUME ELEMENTS 53
1.9.5 TRANSFORMATIONS: INITIAL, REFERENCE, AND INTERMEDIATE
CONFIGURATIONS 54
1.9.6 DIFFERENTIATION O F TENSORS. RATE O F DEFORMATION MEASURES . . . .
55
1.10 STRESS MEASURES 57
1.10.1 CURRENT CONFIGURATION. CAUCHY STRESS TENSOR 57
1.10.2 CURRENT AND INITIAL CONFIGURATIONS. THE FIRST AND SECOND
PIOLA-KIRCHHOFF STRESS TENSORS 57
1.10.3 MEASURES O F THE RATE O F CHANGE O F STRESS TENSORS 59
1.11 VARIATIONAL PRINCIPLES FOR FINITE DEFORMATIONS 60
1.11.1 PRINCIPLE O F VIRTUAL WORK 60
1.11.2 STATEMENT O F THE PRINCIPLE IN INCREMENTS 60
1.12 CONSTITUTIVE EQUATIONS O F PLASTICITY UNDER FINITE DEFORMATIONS 61
1.12.1 MULTIPLICATIVE DECOMPOSITION. DEFORMATION GRADIENTS 61 1.12.2
MATERIAL DESCRIPTION 63
1.12.3 SPATIAL DESCRIPTION 64
1.12.4 ELASTIC ISOTROPIC BODY 65
IMAGE 3
CONTENTS
X I
1.12.5 HYPERELASTOPLASTIC MEDIUM 66
1.12.6 THE VON MISES YIELD CRITERION 66
II THEORY OF FINITE-DIFFERENCE SCHEMES
2 THE BASICS O F THE THEORY O F FINITE-DIFFERENCE SCHEMES 71
2.1 FINITE-DIFFERENCE APPROXIMATIONS FOR DIFFERENTIAL OPERATORS 71
2.1.1 FINITE-DIFFERENCE APPROXIMATION 71
2.1.2 ESTIMATION O F APPROXIMATION ERROR 73
2.1.3 RICHARDSON'S EXTRAPOLATION FORMULA 77
2.2 STABILITY AND CONVERGENCE O F FINITE DIFFERENCE EQUATIONS 78
2.2.1 STABILITY 78
2.2.2 LAX CONVERGENCE THEOREM 78
2.2.3 EXAMPLE O F AN UNSTABLE FINITE DIFFERENCE SCHEME 79
2.3 NUMERICAL INTEGRATION O F THE CAUCHY PROBLEM FOR SYSTEMS O F
EQUATIONS 81 2.3.1 EULER SCHEMES 82
2.3.2 ADAMS-BASHFORTH SCHEME 83
2.3.3 CONSTRUCTION OF HIGHER-ORDER SCHEMES BY SERIES EXPANSION . . . 85
2.3.4 RUNGE-KUTTA SCHEMES 85
2.4 CAUCHY PROBLEM FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
. . 88 2.4.1 STIFF SYSTEMS O F ORDINARY DIFFERENTIAL EQUATIONS 88
2.4.2 NUMERICAL SOLUTION 89
2.4.3 STABILITY ANALYSIS 90
2.4.4 SINGULARLY PERTURBED SYSTEMS 91
2.4.5 EXTENSION O F A ROD MADE O F A NONLINEAR VISCOPLASTIC MATERIAL . .
92
2.5 FINITE DIFFERENCE SCHEMES FOR ONE-DIMENSIONAL PARTIAL DIFFERENTIAL
EQUATIONS 95
2.5.1 SOLUTION O F THE WAVE EQUATION IN DISPLACEMENTS. THE CROSS SCHEME
95
2.5.2 SOLUTION O F THE WAVE EQUATION AS A SYSTEM OF FIRST-ORDER
EQUATIONS (ACOUSTICS EQUATIONS) 96
2.5.3 THE LEAPFROG SCHEME 97
2.5.4 THE LAX-FRIEDRICHS SCHEME 97
2.5.5 THE LAX-WENDROFF SCHEME 98
2.5.6 SCHEME VISCOSITY 99
2.5.7 SOLUTION O F THE WAVE EQUATION. IMPLICIT SCHEME 100
2.5.8 SOLUTION O F THE WAVE EQUATION. COMPARISON O F EXPLICIT AND
IMPLICIT SCHEMES. BOUNDARY POINTS 100
2.5.9 HEAT EQUATION 101
2.5.10 UNSTEADY THERMAL CONDUCTION. EXPLICIT SCHEME (FORWARD EULER
SCHEME) 103
IMAGE 4
XII CONTENTS
2.5.11 UNSTEADY THERMAL CONDUCTION. IMPLICIT SCHEME (BACKWARD EULER
SCHEME) 103
2.5.12 UNSTEADY THERMAL CONDUCTION. CRANK-NICOLSON SCHEME 103 2.5.13
UNSTEADY THERMAL CONDUCTION. ALLEN-CHENG EXPLICIT SCHEME . 103 2.5.14
UNSTEADY THERMAL CONDUCTION. DU FORT-FRANKEL EXPLICIT SCHEME 104
2.5.15 INITIAL-BOUNDARY VALUE PROBLEM OF UNSTEADY THERMAL CONDUCTION.
APPROXIMATION O F BOUNDARY CONDITIONS INVOLVING DERIVATIVES 104
2.6 STABILITY ANALYSIS FOR FINITE DIFFERENCE SCHEMES 106
2.6.1 STABILITY O F A TWO-LAYER FINITE DIFFERENCE SCHEME 107
2.6.2 THE VON NEUMANN STABILITY CONDITION 107
2.6.3 STABILITY O F THE WAVE EQUATION 108
2.6.4 STABILITY O F THE WAVE EQUATION AS A SYSTEM OF FIRST-ORDER
EQUATIONS. THE COURANT STABILITY CONDITION 109
2.6.5 STABILITY O F SCHEMES FOR THE HEAT EQUATION 112
2.6.6 THE PRINCIPLE OF FROZEN COEFFICIENTS 113
2.6.7 STABILITY IN SOLVING BOUNDARY VALUE PROBLEMS 115
2.6.8 STEP SIZE SELECTION IN AN IMPLICIT SCHEME IN SOLVING THE HEAT
EQUATION 116
2.6.9 STEP SIZE SELECTION IN SOLVING THE WAVE EQUATION 117
2.7 EXERCISES 117
3 METHODS FOR SOLVING SYSTEMS O F ALGEBRAIC EQUATIONS 122
3.1 MATRIX NORM AND CONDITION NUMBER O F MATRIX 122
3.1.1 RELATIVE ERROR OF SOLUTION FOR PERTURBED RIGHT-HAND SIDES. THE
CONDITION NUMBER O F A MATRIX 122
3.1.2 RELATIVE ERROR O F SOLUTION FOR PERTURBED COEFFICIENT MATRIX . . .
. 123 3.1.3 EXAMPLE 124
3.1.4 REGULARIZATION O F AN ILL-CONDITIONED SYSTEM O F EQUATIONS 125
3.2 DIRECT METHODS FOR LINEAR SYSTEM O F EQUATIONS 126
3.2.1 GAUSSIAN ELIMINATION METHOD. MATRIX FACTORIZATION 126
3.2.2 GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING 127
3.2.3 CHOLESKY DECOMPOSITION. THE SQUARE ROOT METHOD 128
3.3 ITERATIVE METHODS FOR LINEAR SYSTEM O F EQUATIONS 130
3.3.1 SINGLE-STEP ITERATIVE PROCESSES 130
3.3.2 SEIDEL AND JACOBI ITERATIVE PROCESSES 131
3.3.3 THE STABILIZATION METHOD 133
3.3.4 OPTIMIZATION O F THE RATE OF CONVERGENCE O F A STEADY-STATE
PROCESS 135
3.3.5 OPTIMIZATION OF UNSTEADY PROCESSES 137
IMAGE 5
CONTENTS XIII
3.4 METHODS FOR SOLVING NONLINEAR EQUATIONS 140
3.4.1 NONLINEAR EQUATIONS AND ITERATIVE METHODS 140
3.4.2 CONTRACTIVE MAPPINGS. THE FIXED POINT THEOREM 141
3.4.3 METHOD O F SIMPLE ITERATIONS. SUFFICIENT CONVERGENCE CONDITION 143
3.5 NONLINEAR EQUATIONS: NEWTON'S METHOD AND ITS MODIFICATIONS 145 3.5.1
NEWTON'S METHOD 145
3.5.2 MODIFIED NEWTON-RAPHSON METHOD 147
3.5.3 THE SECANT METHOD 147
3.5.4 TWO-STAGE ITERATIVE METHODS 148
3.5.5 NONSTATIONARY NEWTON METHOD. OPTIMAL STEP SELECTION 149
3.6 METHODS O F MINIMIZATION O F FUNCTIONS (DESCENT METHODS) 152
3.6.1 THE COORDINATE DESCENT METHOD 152
3.6.2 THE STEEPEST DESCENT METHOD 154
3.6.3 THE CONJUGATE GRADIENT METHOD 155
3.6.4 AN ITERATIVE METHOD USING SPECTRAL-EQUIVALENT OPERATORS OR
RECONDITIONING 156
3.7 EXERCISES 157
4 METHODS FOR SOLVING BOUNDARY VALUE PROBLEMS FOR SYSTEMS O F EQUATIONS
160
4.1 NUMERICAL SOLUTION O F TWO-POINT BOUNDARY VALUE PROBLEMS 160
4.1.1 STIFF TWO-POINT BOUNDARY VALUE PROBLEM 160
4.1.2 METHOD O F INITIAL PARAMETERS 161
4.2 GENERAL BOUNDARY VALUE PROBLEM FOR SYSTEMS O F LINEAR EQUATIONS 163
4.3 GENERAL BOUNDARY VALUE PROBLEM FOR SYSTEMS O F NONLINEAR EQUATIONS .
. 164 4.3.1 SHOOTING METHOD 165
4.3.2 QUASI-LINEARIZATION METHOD 165
4.4 SOLUTION O F BOUNDARY VALUE PROBLEMS BY THE SWEEP METHOD 166
4.4.1 DIFFERENTIAL SWEEP 166
4.4.2 SOLUTION O F FINITE DIFFERENCE EQUATION BY THE SWEEP METHOD . . .
170 4.4.3 SWEEP METHOD FOR THE HEAT EQUATION 171
4.5 SOLUTION O F BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS 172
4.5.1 POISSON'S EQUATION 172
4.5.2 MAXIMUM PRINCIPLE FOR SECOND-ORDER FINITE DIFFERENCE EQUATIONS 175
4.5.3 STABILITY O F A FINITE DIFFERENCE SCHEME FOR POISSON'S EQUATION .
176 4.5.4 DIAGONAL DOMINATION 176
4.5.5 SOLUTION O F POISSON'S EQUATION BY THE MATRIX SWEEP METHOD . . 178
4.5.6 FOURIER'S METHOD O F SEPARATION O F VARIABLES 181
IMAGE 6
X I V
CONTENTS
4.6 STIFF BOUNDARY VALUE PROBLEMS 183
4.6.1 STIFF SYSTEMS O F DIFFERENTIAL EQUATIONS 183
4.6.2 GENERALIZED METHOD O F INITIAL PARAMETERS 185
4.6.3 ORTHOGONAL SWEEP 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 O F ELASTIC BEAMS 197
5.1.1 LONGITUDINAL VIBRATIONS 197
5.1.2 EXPLICIT SCHEME. SUFFICIENT STABILITY CONDITIONS 197
5.1.3 LONGITUDINAL VIBRATIONS. IMPLICIT SCHEME 199
5.1.4 TRANSVERSE VIBRATIONS 200
5.1.5 TRANSVERSE VIBRATIONS. EXPLICIT SCHEME 202
5.1.6 TRANSVERSE VIBRATIONS. IMPLICIT SCHEME 203
5.1.7 COUPLED LONGITUDINAL AND TRANSVERSE VIBRATIONS 204
5.1.8 TRANSVERSE BENDING O F A PLATE WITH SHEAR AND ROTATIONAL INERTIA
206 5.1.9 CONCLUSION 209
5.2 SOLUTION O F NONLINEAR WAVE PROPAGATION PROBLEMS 209
5.2.1 HYPERBOLIC SYSTEM O F EQUATIONS AND CHARACTERISTICS 209
5.2.2 FINITE DIFFERENCE APPROXIMATION ALONG CHARACTERISTICS. THE DIRECT
AND SEMI-INVERSE METHODS 211
5.2.3 INVERSE METHOD. THE COURANT-ISAACSON-REES GRID-CHARACTERISTIC
SCHEME 211
5.2.4 WAVE PROPAGATION IN A NONLINEAR ELASTIC BEAM 212
5.2.5 WAVE PROPAGATION IN AN ELASTOVISCOPLASTIC BEAM 215
5.2.6 DISCONTINUOUS SOLUTIONS. CONSTANT COEFFICIENT EQUATION 219 5.2.7
DISCONTINUOUS SOLUTIONS O F A NONLINEAR EQUATION 220
5.2.8 STABILITY O F DIFFERENCE CHARACTERISTIC EQUATIONS 222
5.2.9 CHARACTERISTIC AND GRID-CHARACTERISTIC SCHEMES FOR SOLVING STIFF
PROBLEMS 222
5.2.10 STABILITY O F CHARACTERISTIC AND GRID-CHARACTERISTIC SCHEMES FOR
STIFF PROBLEMS 224
5.2.11 CHARACTERISTIC SCHEMES OF HIGHER ORDERS O F ACCURACY 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
IMAGE 7
CONTENTS
X V
5.3.3 SPATIAL THREE-DIMENSIONAL CHARACTERISTICS FOR SEMI-LINEAR SYSTEM
231
5.3.4 CHARACTERISTIC EQUATIONS. SPATIAL PROBLEM 235
5.3.5 AXISYMMETRIC PROBLEM 236
5.3.6 DIFFERENCE EQUATIONS. AXISYMMETRIC PROBLEM 238
5.3.7 A BRIEF OVERVIEW O F THE RESULTS. FURTHER DEVELOPMENT AND
GENERALIZATION O F THE METHOD O F SPATIAL CHARACTERISTICS AND ITS
APPLICATION TO THE SOLUTION O F DYNAMIC PROBLEMS 244
5.4 COUPLED THERMOMECHANICS PROBLEMS 245
5.5 DIFFERENTIAL APPROXIMATION FOR DIFFERENCE EQUATIONS 248
5.5.1 HYPERBOLIC AND PARABOLIC FORMS OF DIFFERENTIAL APPROXIMATION . 248
5.5.2 EXAMPLE 249
5.5.3 STABILITY 250
5.5.4 ANALYSIS O F DISSIPATIVE AND DISPERSIVE PROPERTIES 251
5.5.5 EXAMPLE 253
5.5.6 ANALYSIS O F PROPERTIES O F FINITE DIFFERENCE SCHEMES FOR
DISCONTINUOUS SOLUTIONS 254
5.5.7 SMOOTHING O F NON-PHYSICAL PERTURBATIONS IN A CALCULATION ON A
REAL GRID 259
5.6 EXERCISES 260
6 FINITE-DIFFERENCE SPLITTING METHOD FOR SOLVING DYNAMIC PROBLEMS 263
6.1 GENERAL SCHEME O F THE SPLITTING METHOD 263
6.1.1 EXPLICIT SPLITTING SCHEME 263
6.1.2 IMPLICIT SPLITTING SCHEME 264
6.1.3 STABILITY 265
6.2 SPLITTING O F 2D/3D EQUATIONS INTO I D 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 270
6.3.1 DIVERGENCE FORM O F EQUATIONS 270
6.3.2 NON-DIVERGENCE FORM O F EQUATIONS 272
6.3.3 ONE-DIMENSIONAL EQUATIONS. IDEAL GAS 273
6.3.4 IMPLEMENTATION O F THE SCHEME 275
6.4 SPLITTING SCHEME FOR ELASTOVISCOPLASTIC DYNAMIC PROBLEMS 276
6.4.1 CONSTITUTIVE EQUATIONS OF ELASTOPLASTIC MEDIA 276
6.4.2 SOME APPROACHES TO SOLVING ELASTOPLASTIC EQUATIONS 277
6.4.3 SPLITTING O F THE CONSTITUTIVE EQUATIONS 279
IMAGE 8
X V I
CONTENTS
6.4.4 THE THEORY O F VON MISES TYPE FLOWS. ISOTROPIC HARDENING 281 6.4.5
DRUCKER-PRAGER PLASTICITY THEORY 283
6.4.6 ELASTOVISCOPLASTIC MEDIA 285
6.5 SPLITTING SCHEMES FOR POINTS ON THE AXIS O F REVOLUTION 286
6.5.1 CALCULATION O F BOUNDARY POINTS 286
6.5.2 CALCULATION OF AXIAL POINTS 288
6.6 INTEGRATION O F 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 O F ELASTOPLASTIC DYNAMIC AND QUASISTATIC PROBLEMS WITH
FINITE DEFORMATIONS 298
7.1 CONSERVATIVE APPROXIMATIONS ON CURVILINEAR LAGRANGIAN MESHES 298
7.1.1 FORMULAS FOR NATURAL APPROXIMATION O F SPATIAL DERIVATIVES . . . .
298 7.1.2 APPROXIMATION O F A LAGRANGIAN MESH 299
7.1.3 CONSERVATIVE FINITE DIFFERENCE SCHEMES 301
7.2 FINITE ELASTOPLASTIC DEFORMATIONS 303
7.2.1 CONSERVATIVE SCHEMES IN ONE-DIMENSIONAL CASE 303
7.2.2 A CONSERVATIVE TWO-DIMENSIONAL SCHEME FOR AN ELASTOPLASTIC MEDIUM
305
7.2.3 SPLITTING O F THE EQUATIONS O F A HYPOELASTIC MATERIAL 306
7.3 PROPAGATION O F COUPLED THERMOMECHANICAL PERTURBATIONS IN GASES . .
. . 307 7.3.1 BASIC EQUATIONS 307
7.3.2 CONSERVATIVE FINITE DIFFERENCE SCHEME 307
7.3.3 NON-DIVERGENCE FORM O F THE ENERGY EQUATION. A COMPLETELY
CONSERVATIVE SCHEME 309
7.4 THE PIC METHOD AND ITS MODIFICATIONS FOR SOLID MECHANICS PROBLEMS .
3 1 1 7.4.1 DISADVANTAGES O F LAGRANGIAN AND EULERIAN MESHES 311
7.4.2 THE PARTICLE-IN-CELL (PIC) METHOD 311
7.4.3 THE METHOD O F COARSE PARTICLES 314
7.4.4 LIMITATIONS O F THE PIC METHOD AND ITS MODIFICATIONS 315
7.4.5 THE COMBINED FLUX AND PARTICLE-IN-CELL (FPIC) METHOD 316 7.4.6 THE
METHOD O F MARKERS AND FLUXES 317
7.5 APPLICATION O F PIC-TYPE METHODS TO SOLVING ELASTOVISCOPLASTIC
PROBLEMS 317
7.5.1 HYPOELASTIC MEDIUM 318
7.5.2 HYPOELASTOPLASTIC MEDIUM 319
7.5.3 SPLITTING FOR A HYPERELASTOPLASTIC MEDIUM 321
IMAGE 9
CONTENTS XVII
7.6 OPTIMIZATION OF MOVING ONE-DIMENSIONAL MESHES 324
7.6.1 OPTIMAL MESH FOR A GIVEN FUNCTION 325
7.6.2 OPTIMAL MESH FOR SOLVING AN INITIAL-BOUNDARY VALUE PROBLEM . . 326
7.6.3 MESH OPTIMIZATION IN SEVERAL PARAMETERS 327
7.6.4 HEAT PROPAGATION FROM A COMBUSTION SOURCE 328
7.7 ADAPTIVE 2D/3D MESHES FOR FINITE DEFORMATION PROBLEMS 330
7.7.1 METHODS FOR REORGANIZATION O F A LAGRANGIAN MESH 330
7.7.2 DESCRIPTION O F MOTION IN AN ARBITRARY MOVING COORDINATE SYSTEM
331
7.7.3 ADAPTIVE MESHES 333
7.8 UNSTEADY ELASTOVISCOPLASTIC PROBLEMS ON MOVING ADAPTIVE MESHES . . .
335 7.8.1 ALGORITHMS FOR CONSTRUCTING MOVING MESHES 335
7.8.2 SELECTION O F A FINITE DIFFERENCE SCHEME 337
7.8.3 A HYBRID SCHEME O F VARIABLE ORDER O F APPROXIMATION AT INTERNAL
NODES 339
7.8.4 A GRID-CHARACTERISTIC SCHEME AT BOUNDARY NODES 341
7.8.5 CALCULATION OF CONTACT BOUNDARIES 344
7.8.6 CALCULATION O F DAMAGE KINETICS 346
7.8.7 NUMERICAL RESULTS FOR SOME APPLIED PROBLEMS WITH FINITE
ELASTOVISCOPLASTIC STRAINS 347
7.9 EXERCISES 352
8 MODELING O F DAMAGE AND FRACTURE O F INELASTIC MATERIALS AND
STRUCTURES 354
8.1 CONCEPT O F DAMAGE AND THE CONSTRUCTION O F MODELS O F DAMAGED MEDIA
354 8.1.1 CONCEPT O F CONTINUUM FRACTURE AND DAMAGE 354
8.1.2 CONSTRUCTION OF DAMAGE MODELS 355
8.1.3 CONSTITUTIVE EQUATIONS O F THE GTN MODEL 361
8.2 GENERALIZED MICROMECHANICAL MULTISCALE DAMAGE MODEL 363
8.2.1 MICROMECHANICAL MODEL. THE STAGE O F PLASTIC FLOW AND HARDENING
364
8.2.2 STAGE O F VOID NUCLEATION 365
8.2.3 STAGE O F THE APPEARANCE O F VOIDS AND DAMAGE 366
8.2.4 RELATIONSHIP BETWEEN MICRO AND MACRO PARAMETERS 367
8.2.5 MACROMODEL 368
8.2.6 TENSION O F A THIN ROD WITH A CONSTANT STRAIN RATE 373
8.2.7 CONCLUSION 375
8.3 NUMERICAL MODELING O F DAMAGED ELASTOPLASTIC MATERIALS 375
8.3.1 REGULARIZATION O F EQUATIONS FOR ELASTOPLASTIC MATERIALS AT
SOFTENING 375
8.3.2 SOLUTION O F DAMAGE PROBLEMS 376
8.3.3 INVERSE EULER METHOD 377
IMAGE 10
XV111
CONTENTS
8.3.4 SOLUTION O F A BOUNDARY VALUE PROBLEM. COMPUTATION O F THE
JACOBIAN 379
8.3.5 SPLITTING METHOD 379
8.3.6 INTEGRATION O F THE CONSTITUTIVE RELATIONS O F THE GTN MODEL . . .
. 382 8.3.7 UNIAXIAL TENSION. COMPUTATIONAL RESULTS 386
8.3.8 BENDING OF A PLATE 387
8.3.9 COMPARISON WITH EXPERIMENT 389
8.3.10 MODELING QUASI-BRITTLE FRACTURE WITH DAMAGE 390
8.4 EXTENSION O F DAMAGE THEORY TO THE CASE O F AN ARBITRARY
STRESS-STRAIN STATE 393
8.4.1 WELL-POSEDNESS O F THE PROBLEM 394
8.4.2 LIMITATIONS O F THE GTN MODEL 395
8.4.3 ASSOCIATED VISCOPLASTIC LAW 396
8.4.4 CONSTITUTIVE RELATIONS IN THE ABSENCE O F POROSITY (K 0.4, / =
0, A R = 0) 396
8.4.5 FRACTURE MODEL. FRACTURE CRITERIA 397
8.5 NUMERICAL MODELING O F CUTTING O F ELASTOVISCOPLASTIC MATERIALS 398
8.5.1 INTRODUCTION 398
8.5.2 STATEMENT O F THE PROBLEM 399
8.6 CONCLUSIONS. GENERAL REMARKS ON ELASTOPLASTIC EQUATIONS 406
8.6.1 FORMULATIONS O F SYSTEMS O F EQUATIONS FOR ELASTOPLASTIC MEDIA . 4
0 6 8.6.2 A HARDENING ELASTOPLASTIC MEDIUM 406
8.6.3 IDEAL ELASTOPLASTIC MEDIA: A DEGENERATE CASE 407
8.6.4 DIFFICULTIES IN SOLVING MIXED ELLIPTIC-HYPERBOLIC PROBLEMS . . . .
408 8.6.5 REGULARIZATION O F AN ELASTOPLASTIC MODEL 408
8.6.6 ELASTOPLASTIC SHOCK WAVES 409
BIBLIOGRAPHY 411
INDEX 421 |
| any_adam_object | 1 |
| author | Kukudžanov, Vladimir Nikolaevič 1941- |
| author_GND | (DE-588)1029626855 |
| author_facet | Kukudžanov, Vladimir Nikolaevič 1941- |
| author_role | aut |
| author_sort | Kukudžanov, Vladimir Nikolaevič 1941- |
| author_variant | v n k vn vnk |
| building | Verbundindex |
| bvnumber | BV040671107 |
| classification_rvk | SK 950 UF 2000 |
| ctrlnum | (OCoLC)823936221 (DE-599)DNB1027078249 |
| dewey-full | 531 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
| dewey-raw | 531 |
| dewey-search | 531 |
| dewey-sort | 3531 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV040671107 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:27:36Z |
| institution | BVB |
| isbn | 3110273225 9783110273229 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025497670 |
| oclc_num | 823936221 |
| open_access_boolean | |
| owner | DE-11 DE-19 DE-BY-UBM DE-703 DE-355 DE-BY-UBR DE-83 DE-29T |
| owner_facet | DE-11 DE-19 DE-BY-UBM DE-703 DE-355 DE-BY-UBR DE-83 DE-29T |
| physical | XVIII, 429 S. Ill., graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | de Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematical physics |
| series2 | De Gruyter studies in mathematical physics |
| spelling | Kukudžanov, Vladimir Nikolaevič 1941- Verfasser (DE-588)1029626855 aut Numerical continuum mechanics Vladimir N. Kukudzhanov Berlin [u.a.] de Gruyter 2013 XVIII, 429 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematical physics 15 Kontinuumsmechanik (DE-588)4032296-8 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Kontinuumsmechanik (DE-588)4032296-8 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Erscheint auch als Online-Ausgabe 978-3-11-027338-0 De Gruyter studies in mathematical physics 15 (DE-604)BV040141722 15 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4165192&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kukudžanov, Vladimir Nikolaevič 1941- Numerical continuum mechanics De Gruyter studies in mathematical physics Kontinuumsmechanik (DE-588)4032296-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4032296-8 (DE-588)4128130-5 |
| title | Numerical continuum mechanics |
| title_auth | Numerical continuum mechanics |
| title_exact_search | Numerical continuum mechanics |
| title_full | Numerical continuum mechanics Vladimir N. Kukudzhanov |
| title_fullStr | Numerical continuum mechanics Vladimir N. Kukudzhanov |
| title_full_unstemmed | Numerical continuum mechanics Vladimir N. Kukudzhanov |
| title_short | Numerical continuum mechanics |
| title_sort | numerical continuum mechanics |
| topic | Kontinuumsmechanik (DE-588)4032296-8 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Kontinuumsmechanik Numerisches Verfahren |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4165192&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025497670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV040141722 |
| work_keys_str_mv | AT kukudžanovvladimirnikolaevic numericalcontinuummechanics |