Finite Element Analysis: Thermomechanics of Solids

Finite Element Analysis: Thermomechanics of Solids......Page 1
Dedication......Page 3
Preface......Page 4
Acknowledgment......Page 6
About the Author......Page 7
Table of Contents......Page 8
1.1.2 S UBSTITUTION O PERATOR......Page 14
Table of Contents......Page 0
1.2.1 N OTATION......Page 15
1.2.2 G RADIENT , D IVERGENCE , AND C URL......Page 17
1.3 MATRICES......Page 18
1.3.1 EIGENVALUES AND EIGENVECTORS......Page 21
1.3.3 TRANSFORMATIONS OF VECTORS......Page 22
1.3.4 ORTHOGONAL CURVILINEAR COORDINATES......Page 24
1.3.5 GRADIENT OPERATOR......Page 29
1.3.6 DIVERGENCE AND CURL OF VECTORS......Page 30
DERIVATIVES OF BASE VECTORS......Page 31
DIVERGENCE......Page 32
1.4 EXERCISES......Page 33
2.1 TENSORS......Page 37
2.2.1 DIVERGENCE......Page 39
2.2.2 CURL AND LAPLACIAN......Page 40
2.3 INVARIANTS......Page 41
2.4 POSITIVE DEFINITENESS......Page 42
2.5 POLAR DECOMPOSITION THEOREM......Page 43
2.6.1 VEC OPERATOR AND THE KRONECKER PRODUCT......Page 44
2.6.2 FUNDAMENTAL RELATIONS FOR KRONECKER PRODUCTS......Page 45
2.6.3 EIGENSTRUCTURES OF KRONECKER PRODUCTS......Page 47
2.6.5 KRONECKER PRODUCT OPERATORS FOR FOURTH-ORDER TENSORS......Page 48
2.6.6 TRANSFORMATION PROPERTIES OF VEC AND TEN22......Page 49
2.6.7 KRONECKER PRODUCT FUNCTIONS FOR TENSOR OUTER PRODUCTS......Page 50
2.6.8 KRONECKER EXPRESSIONS FOR SYMMETRY CLASSES IN FOURTH-ORDER TENSORS......Page 52
2.6.9 DIFFERENTIALS OF TENSOR INVARIANTS......Page 53
2.7 EXERCISES......Page 54
3.1 INTRODUCTION TO VARIATIONAL METHODS......Page 55
3.2.1 NEWTON ITERATION......Page 59
3.2.2 CRITICAL POINTS AND THE ARC-LENGTH METHOD......Page 60
3.3 EXERCISES......Page 61
4.1.1 D ISPLACEMENT......Page 62
4.1.3 D EFORMATION G RADIENT T ENSOR......Page 63
4.2.1.1 Deformation Gradient and Lagrangian Strain Tensors......Page 64
4.2.1.2 Linear-Strain Tensor in Cylindrical Coordinates......Page 66
4.2.2 VELOCITY-GRADIENT TENSOR, DEFORMATION-RATE TENSOR, AND SPIN TENSOR......Page 67
4.2.2.2 Cylindrical Coordinates......Page 68
4.2.2.3 Spherical Coordinates......Page 69
4.3 DIFFERENTIAL VOLUME ELEMENT......Page 71
4.4 DIFFERENTIAL SURFACE ELEMENT......Page 72
4.5 ROTATION TENSOR......Page 74
4.6 COMPATIBILITY CONDITIONS FOR EL AND D......Page 75
4.7 SAMPLE PROBLEMS......Page 78
4.8 EXERCISES......Page 80
5.1.1 CAUCHY STRESS......Page 83
5.1.2 1ST PIOLA -KIRCHHOFF STRESS......Page 85
5.1.3 2ND PIOLA -KIRCHHOFF STRESS......Page 86
5.2 STRESS FLUX......Page 87
5.3.3 BALANCE OF LINEAR MOMENTUM......Page 89
5.3.4 BALANCE OF ANGULAR MOMENTUM......Page 90
5.4 PRINCIPLE OF VIRTUAL WORK......Page 92
5.5 SAMPLE PROBLEMS......Page 95
5.6 EXERCISES......Page 99
6.1 STRESS-STRAIN BEHAVIOR: CLASSICAL LINEAR ELASTICITY......Page 104
6.2.2 C OMPRESSIBLE H YPERELASTIC M ATERIALS......Page 106
6.3.1 I NCOMPRESSIBILITY......Page 108
6.3.2 N EAR -I NCOMPRESSIBILITY......Page 111
6.4 NONLINEAR MATERIALS AT LARGE DEFORMATION......Page 112
6.5 EXERCISES......Page 113
7.1.1 B ALANCE OF E NERGY......Page 116
7.1.2 E NTROPY P RODUCTION I NEQUALITY......Page 117
7.1.3 T HERMODYNAMIC P OTENTIALS......Page 118
7.2 CLASSICAL COUPLED LINEAR THERMOELASTICITY......Page 119
7.3.1 C ONDUCTIVE H EAT T RANSFER......Page 122
7.3.2 COUPLED LINEAR ISOTROPIC THERMOELASTICITY......Page 123
7.4 EXERCISES......Page 125
8.2 OVERVIEW OF THE FINITE-ELEMENT METHOD......Page 126
8.3 MESH DEVELOPMENT......Page 127
9.1.1.1 Rods......Page 130
9.1.1.2 Beams......Page 131
9.1.1.4 Temperature Model: One Dimension......Page 132
9.1.2.2 Plate with Bending Stresses......Page 133
9.1.2.3 Plate with Stretching and Bending......Page 134
9.1.2.5 Axisymmetric Elements......Page 135
9.1.3 INTERPOLATION MODELS: THREE DIMENSIONS......Page 136
9.2.1 STRAIN- DISPLACEMENT RELATIONS: ONE DIMENSION......Page 137
9.2.2 STRAIN-DISPLACEMENT RELATIONS: TWO DIMENSIONS......Page 138
9.2.3 AXISYMMETRIC ELEMENT ON AXIS OF REVOLUTION......Page 139
9.2.5 THREE-DIMENSIONAL ELEMENTS......Page 140
9.3.2 ONE-DIMENSIONAL MEMBERS......Page 141
9.3.3.1 Membrane Response......Page 142
9.3.5 AXISYMMETRIC ELEMENT......Page 144
9.3.6 THREE-DIMENSIONAL ELEMENT......Page 145
9.4 EXERCISES......Page 146
10.1 APPLICATION OF THE PRINCIPLE OF VIRTUAL WORK......Page 148
10.2 THERMAL COUNTERPART PRINCIPLE OF THE OF VIRTUAL WORK......Page 150
10.3.1 RODS......Page 151
10.3.2 BEAMS......Page 155
10.3.3 TWO-DIMENSIONAL ELEMENTS......Page 156
10.3 EXERCISES......Page 158
11.1.1 S OLVING THE F INITE -E LEMENT E QUATIONS : S TATIC P ROBLEMS......Page 161
11.1.2 M ATRIX T RIANGULARIZATION AND S OLUTION OF L INEAR S YSTEMS......Page 162
11.1.3 TRIANGULARIZATION OF ASYMMETRIC MATRICES......Page 163
11.2 TIME INTEGRATION: STABILITY AND ACCURACY......Page 164
11.3 NEWMARK’S METHOD......Page 165
11.4 INTEGRAL EVALUATION BY GAUSSIAN QUADRATURE......Page 166
11.5.1 MODAL DECOMPOSITION......Page 167
11.5.2 COMPUTATION OF EIGENVECTORS AND EIGENVALUES......Page 170
11.6 EXERCISES......Page 172
12.1 FINITE ELEMENTS IN ROTATION......Page 174
12.2 FINITE-ELEMENT ANALYSIS FOR UNCONSTRAINED ELASTIC BODIES......Page 176
12.3 EXERCISES......Page 178
13.1.2 D IRECT I NTEGRATION BY THE T RAPEZOIDAL R ULE......Page 180
13.1.3 M ODAL A NALYSIS......Page 181
13.2.1 F INITE -E LEMENT E QUATION......Page 182
13.3 COMPRESSIBLE ELASTIC MEDIA......Page 184
13.4 INCOMPRESSIBLE ELASTIC MEDIA......Page 185
13.5 EXERCISES......Page 187
14.1 TORSION OF PRISMATIC BARS......Page 188
14.2.1.1 Static Buckling......Page 192
14.2.1.2 Dynamic Buckling......Page 194
14.2.1.3 Sample Problem: Interpretation of Buckling Modes......Page 195
14.2.2 EULER BUCKLING OF PLATES......Page 197
14.3 EXERCISES......Page 200
15.1 INTRODUCTION: THE GAP......Page 202
15.2 POINT-TO-POINT CONTACT......Page 204
15.4 EXERCISES......Page 206
16.2 TYPES OF NONLINEARITY......Page 208
16.3 COMBINED INCREMENTAL AND ITERATIVE METHODS: A SIMPLE EXAMPLE......Page 209
16.4.1 N ONLINEAR S TRAIN -D ISPLACEMENT R ELATIONS......Page 210
16.4.2 STRESS AND TANGENT MODULUS RELATIONS......Page 211
16.4.3 INCREMENTAL EQUILIBRIUM RELATION......Page 212
16.4.4 NUMERICAL SOLUTION BY NEWTON ITERATION......Page 215
16.5 ILLUSTRATION OF NEWTON ITERATION......Page 218
16.5.1 EXAMPLE......Page 219
16.6 EXERCISES......Page 220
17.1 INCREMENTAL KINEMATICS......Page 221
17.2 INCREMENTAL STRESSES......Page 222
17.3 INCREMENTAL EQUILIBRIUM EQUATION......Page 223
17.4 INCREMENTAL PRINCIPLE OF VIRTUAL WORK......Page 224
17.5 INCREMENTAL FINITE-ELEMENT EQUATION......Page 225
17.6 INCREMENTAL CONTRIBUTIONS FROM NONLINEAR BOUNDARY CONDITIONS......Page 226
17.7 EFFECT OF VARIABLE CONTACT......Page 227
17.8 INTERPRETATION AS NEWTON ITERATION......Page 229
17.9 BUCKLING......Page 230
17.10 EXERCISES......Page 232
18.2 COMPRESSIBLE ELASTOMERS......Page 233
18.3 INCOMPRESSIBLE AND NEAR-INCOMPRESSIBLE ELASTOMERS......Page 234
18.3.1.2 Invariant-Based Models for Compressible Elastomers under Isothermal Conditions......Page 236
18.4 STRETCH RATIO-BASED MODELS: ISOTHERMAL CONDITIONS......Page 237
18.5 EXTENSION TO THERMOHYPERELASTIC MATERIALS......Page 239
18.6 THERMOMECHANICS OF DAMPED ELASTOMERS......Page 240
18.6.2 ENTROPY PRODUCTION INEQUALITY......Page 241
18.6.3 DISSIPATION POTENTIAL......Page 242
18.6.4 THERMAL-FIELD EQUATION FOR DAMPED ELASTOMERS......Page 243
18.7.1 HELMHOLTZ FREE-ENERGY DENSITY......Page 244
18.7.2 SPECIFIC DISSIPATION POTENTIAL......Page 245
18.8.2 THERMAL EQUILIBRIUM......Page 246
18.9 EXERCISES......Page 247
19.1.2 P LASTICITY......Page 248
19.2.1 B ALANCE OF E NERGY......Page 251
19.2.2 E NTROPY -P RODUCTION I NEQUALITY......Page 252
19.2.3 DISSIPATION POTENTIAL......Page 253
19.3 THERMOINELASTIC TANGENT-MODULUS TENSOR......Page 254
19.3.1 EXAMPLE......Page 255
19.4 TANGENT-MODULUS TENSOR IN VISCOPLASTICITY......Page 257
19.5 CONTINUUM DAMAGE MECHANICS......Page 259
19.6 EXERCISES......Page 261
20.1.1 I NTRODUCTION......Page 262
20.1.2 N OTATION AND B ACKGROUND......Page 263
20.1.4 HEURISTIC CONVERGENCE ARGUMENT......Page 264
20.1.5 SAMPLE PROBLEM......Page 265
20.2 OZAWA’S METHOD FOR INCOMPRESSIBLE MATERIALS......Page 267
20.3 EXERCISES......Page 268
Monographs and Texts......Page 269
Articles and Other Sources......Page 270