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أخواني في الله أحضرت لكم كتاب Introduction to Nonlinear Finite Element Analysis Nam-Ho Kim
![كتاب Introduction to Nonlinear Finite Element Analysis I_t_n_12](https://i.servimg.com/u/f61/15/36/74/14/i_t_n_12.jpg) و المحتوى كما يلي :
Contents 1 Preliminary Concepts 1 1.1 Introduction . 1 1.2 Vector and Tensor Calculus . 3 1.2.1 Vector and Tensor . 3 1.2.2 Vector and Tensor Calculus 11 1.2.3 Integral Theorems . 12 1.3 Stress and Strain 14 1.3.1 Stress 15 1.3.2 Strain 26 1.3.3 Stress–Strain Relationship . 31 1.4 Mechanics of Continuous Bodies 36 1.4.1 Boundary-Valued Problem . 37 1.4.2 Principle of Minimum Potential Energy . 38 1.4.3 Principle of Virtual Work 46 1.5 Finite Element Method . 50 1.5.1 Finite Element Approximation 50 1.5.2 Finite Element Equations for a One-Dimensional Problem 54 1.5.3 Finite Element Equations for 3D Solid Element . 61 1.5.4 A MATLAB Code for Finite Element Analysis 67 1.6 Exercises . 73 References . 79 2 Nonlinear Finite Element Analysis Procedure . 81 2.1 Introduction to Nonlinear Systems in Solid Mechanics 81 2.1.1 Geometric Nonlinearity . 85 2.1.2 Material Nonlinearity . 87 2.1.3 Kinematic Nonlinearity . 89 2.1.4 Force Nonlinearity . 90 xi2.2 Solution Procedures for Nonlinear Algebraic Equations . 91 2.2.1 Newton–Raphson Method . 93 2.2.2 Modified Newton–Raphson Method 101 2.2.3 Incremental Secant Method 103 2.2.4 Incremental Force Method . 109 2.3 Steps in the Solution of Nonlinear Finite Element Analysis . 114 2.3.1 State Determination 114 2.3.2 Residual Calculation . 115 2.3.3 Convergence Check 116 2.3.4 Linearization . 116 2.3.5 Solution 117 2.4 MATLAB Code for a Nonlinear Finite Element Analysis Procedure 119 2.5 Nonlinear Solution Controls Using Commercial Finite Element Programs 132 2.5.1 Abaqus . 133 2.5.2 ANSYS . 134 2.5.3 NEiNastran 136 2.6 Summary . 137 2.7 Exercises . 138 References . 140 3 Finite Element Analysis for Nonlinear Elastic Systems . 141 3.1 Introduction . 141 3.2 Stress and Strain Measures in Large Deformation 142 3.2.1 Deformation Gradient 142 3.2.2 Lagrangian and Eulerian Strains . 145 3.2.3 Polar Decomposition . 150 3.2.4 Deformation of Surface and Volume 154 3.2.5 Cauchy and Piola-Kirchhoff Stresses . 158 3.3 Nonlinear Elastic Analysis 161 3.3.1 Nonlinear Static Analysis: Total Lagrangian Formulation . 162 3.3.2 Nonlinear Static Analysis: Updated Lagrangian Formulation . 174 3.4 Critical Load Analysis . 179 3.4.1 One-Point Approach . 181 3.4.2 Two-Point Approach . 181 3.4.3 Stability Equation with Actual Critical Load Factor . 182 3.5 Hyperelastic Materials . 183 3.5.1 Strain Energy Density 184 3.5.2 Nearly Incompressible Hyperelasticity 190 3.5.3 Variational Equation and Linearization 197 3.6 Finite Element Formulation for Nonlinear Elasticity 200 3.7 MATLAB Code for Hyperelastic Material Model 205 xii Contents3.8 Nonlinear Elastic Analysis Using Commercial Finite Element Programs 211 3.8.1 Usage of Commercial Programs . 211 3.8.2 Modeling Examples of Nonlinear Elastic Materials . 214 3.9 Fitting Hyperelastic Material Parameters from Test Data 221 3.9.1 Elastomer Test Procedures . 222 3.9.2 Data Preparation 224 3.9.3 Curve Fitting . 227 3.9.4 Stability of Constitutive Model 229 3.10 Summary . 231 3.11 Exercises . 232 References . 239 4 Finite Element Analysis for Elastoplastic Problems . 241 4.1 Introduction . 241 4.2 One-Dimensional Elastoplasticity . 242 4.2.1 Elastoplastic Material Behavior . 242 4.2.2 Finite Element Formulation for Elastoplasticity 247 4.2.3 Determination of Stress State . 250 4.3 Multidimensional Elastoplasticity . 265 4.3.1 Yield Functions and Yield Criteria . 266 4.3.2 Von Mises Yield Criterion . 272 4.3.3 Hardening Models . 275 4.3.4 Classical Elastoplasticity Model . 280 4.3.5 Numerical Integration 290 4.3.6 Computational Implementation of Elastoplasticity 299 4.4 Finite Rotation with Objective Integration . 308 4.4.1 Objective Tensor and Objective Rate . 309 4.4.2 Finite Rotation and Objective Rate . 314 4.4.3 Incremental Equation for Finite Rotation Elastoplasticity . 317 4.4.4 Computational Implementation of Finite Rotation 321 4.5 Finite Deformation Elastoplasticity with Hyperelasticity . 325 4.5.1 Multiplicative Decomposition 325 4.5.2 Finite Deformation Elastoplasticity . 326 4.5.3 Time Integration 330 4.5.4 Return-Mapping Algorithm 333 4.5.5 Consistent Algorithmic Tangent Operator 336 4.5.6 Variational Principles for Finite Deformation . 337 4.5.7 Computer Implementation of Finite Deformation Elastoplasticity . 338 4.6 Mathematical Formulas from Finite Elasticity 343 4.6.1 Linearization of Principal Logarithmic Stretches . 343 4.6.2 Linearization of the Eigenvector of the Elastic Trial Left Cauchy-Green Tensor . 344 Contents xiii4.7 MATLAB Code for Elastoplastic Material Model . 345 4.8 Elastoplasticity Analysis of Using Commercial Finite Element Programs 350 4.8.1 Usage of Commercial Programs . 350 4.8.2 Modeling Examples of Elastoplastic Materials 355 4.9 Summary . 359 4.10 Exercises . 360 References . 365 5 Finite Element Analysis for Contact Problems . 367 5.1 Introduction . 367 5.2 Examples of Simple One-Point Contact . 369 5.2.1 Contact of a Cantilever Beam with a Rigid Block 369 5.2.2 Contact of a Cantilever Beam with Friction . 374 5.3 General Formulation for Contact Problems 378 5.3.1 Contact Condition with Rigid Surface . 379 5.3.2 Variational Inequality in Contact Problems . 382 5.3.3 Penalty Regularization 385 5.3.4 Frictionless Contact Formulation 389 5.3.5 Frictional Contact Formulation 393 5.4 Finite Element Formulation of Contact Problems 398 5.4.1 Contact Between a Flexible Body and a Rigid Body 398 5.4.2 Contact Between Two Flexible Bodies 404 5.4.3 MATLAB Code for Contact Analysis . 406 5.5 Three-Dimensional Contact Analysis . 408 5.6 Contact Analysis Procedure and Modeling Issues 412 5.6.1 Contact Analysis Procedure 413 5.6.2 Contact Modeling Issues 417 5.7 Exercises . 423 References . 426 Index . 427 xiv Contents Index A Assembly, 57 Associative plasticity, 291 B Back stress, 282, 334 Backward Euler method, 291 Balance of momentum, 37 Basis vectors, 4 Baushinger effect, 278 Broyden, Fletcher, Goldfarb, and Shanno (BFGS) method, 107 Bisection, 116 Boundary condition, 38, 54 essential, 54 natural, 54 Boundary valued problem, 38, 54 Bulk modulus, 192 C Cauchy–Green tensor, 145, 147, 176, 191, 327, 331, 343 left, 147, 327, 331, 343 right, 145, 327, 343 Cauchy’s Lemma, 20 Consistency condition, 372, 380 contact, 380, 409 Constitutive relation, 31 Constrained optimization, 384 contact, 384 Contact force, 372, 410, 417 normal, 410 Contact form, 387 normal, 387 tangential, 387 Contact pair, 413 Contact problem, 367 Contact search, 414 Contact stiffness, 410, 416 Contact tolerance,415 Contraction, 8 Convergence, 94, 421 Convex set, 382 Coulomb friction, 375, 393 Critical displacement, 180 Critical load, 179, 181, 183 actual load factor, 183 load factor, 181 one-point, 181 two-point, 181 Cross product. See Vector, product D Deformation field, 27 Deformation gradient, 144, 330 relative, 330 Deviator, 274 Directional derivative, 385 Displacement field, 27 Displacement gradient, 144 Dissipation function, 327, 328 Dissipation inequality, 328 Distortion energy theory, 268 Divergence, 11 Divergence theorem, 12 Dual vector, 10 Dyadic product, 5
Springer Science+Business Media New York 2015 N.-H. Kim, Introduction to Nonlinear Finite Element Analysis, DOI 10.1007/978-1-4419-1746-1 427E Effective plastic strain, 282 Eigenvalue, 23, 182 Eigenvector, 23 Elastic domain, 282, 326 Elasticity matrix, 34 Elasticity tensor, 32 Elastic limit, 32 Elastic modulus, 243 Elastic predictor, 291 Elastoplasticity, 241, 273, 308, 325, 360 finite deformation, 360 finite rotation, 308 infinitesimal, 273 multiplicative plasticity, 325 Euclidean norm, 157 F Failure envelope, 267 Finite element, 50, 51, 62 shape function, 62 Flow potential, 283 Form, 44 energy bilinear, 44 load linear, 44 Frame indifference, 21 Fre´chet differentiable, 43 Free energy, 327, 332 Friction, 374 G Gap, 370, 390 Gap function, 410 Gauss integration, 65 Gauss’ theorem, 47 Generalized Hooke’s law, 31, 32 Generalized solution, 40 Gradient, 11 Green’s identity, 14 H Hooke’s law, 15 generalized, 15 Hydrostatic pressure, 192 Hyper-elastic material, 184 I Impenetrability, 372 Impenetrability condition, 379, 380 Incremental force method, 109 Initial stiffness, 170, 298 Inner product, 4 Integration-by-parts, 13 Interpolation function, 53 Invariant, 185 Isoparametric mapping, 62 Isotropic hardening, 282 J Jacobian, 94 Jacobian matrix, 116 K Kinematically admissible displacement, 40 Kinematic hardening, 282, 283 Kronecker delta symbol, 4, 164 Kuhn–Tucker condition, 284, 329 L Lagrange multiplier, 284, 368, 372, 376 Lagrangian strain, 167 Lame’s constants, 33, 163, 281 Laplace operator, 11 Lie derivative, 327 Load step, 110 Lower and upper (LU) decomposition, 101 M Master, 371 Master element, 408 Material description, 168 Matrix, 5, 23, 34 determinant, 23 elasticity, 34 Modified Newton–Raphson method, 101–103 Mooney–Rivlin material, 186–187 N Natural coordinate, 379 contact problem, 379 Necking, 32 Neo–Hookean material, 186 Newton–Raphson method, 93, 168 Nonlinear elastic problem, 162 Nonlinearity, 162, 241, 367 boundary, 367 force, 90–91 428 Indexgeometric, 85–87, 164 kinematic, 89–90 material, 87–89, 241 Nonlinear solution procedure, 91 Norm, 5, 8 Normal gap, 380 O Objective rate, 360 Operator, linear, 81 P Penalty, 368, 372, 377 Penalty method, 386 Penalty parameter, 373, 386 Penetration, 372 Permutation, 10, 156 Plane strain, 34 Plane stress, 34 Plastic consistency parameter, 283 Plastic corrector, 291 Plastic modulus, 246, 284, 334 Poisson’s ratio, 33 Polar decomposition, 150 Potential energy, 166, 384, 386 Principal stress direction, 22, 24 Principal stretch, 332 Principle of minimum potential energy, 39 Principle of virtual work, 46 Projection, 4, 290, 380 Proportional limit, 32 R Reference element, 65 Residual, 94, 116 Residual load, 170, 299 Return mapping, 292, 333, 360 Reynolds transport theorem, 13 Rigid-body motion, 421 Rigid body rotation, 315 Rotation tensor, 150 S Secant method, 104 Secant stiffness matrix, 107 Shape function, 53 Shear modulus, 33 Slave, 371 Slave-master, 368 Slave node, 399 Slip, 375, 393 Slip condition, 394 Sobolev space, 40 Solution, 44, 52 generalized, 44 trial, 52 Spatial description, 174 Spatial velocity gradient, 327 Spin tensor, 315 Stick, 375 Stick condition, 395 Stiffness matrix, 57, 64, 296 consistent, 296 solid, 64 Strain, 7, 26, 28–30, 145, 147, 167, 170, 174, 268, 332, 334 deviatoric, 30, 268 effective plastic, 334 elastic principal stretch, 332 engineering, 174 engineering shear, 28 Eulerian, 147 infinitesimal, 145, 172 Lagrangian, 145, 167, 170 normal, 28 shear, 28 symmetric, 29 tensorial shear, 28 volumetric, 30 Strain energy, 39, 163, 281 elastic, 281 Strain energy density, 268 distortion, 268 Strain hardening, 32 Stress, 17, 18, 20–22, 24, 31, 32, 159, 160, 174, 268, 291, 314, 316, 326 Cauchy, 159, 174, 314 deviatoric, 21, 268 first Piola–Kirchhoff, 159, 318 invariant, 24 Kirchhoff, 160, 326 mean, 21 normal, 20 principal, 22 second Piola–Kirchhoff, 159 shear, 20 symmetry, 18 tensor, 17 trial, 291 ultimate, 32 Index 429Stress (cont.) uniaxial, 31 yield, 32 Stress rate, 315 Jaumann, 315 Stress vector, 15 Stretch tensor, 150 Strong form, 38 Structural energy form, 168, 185, 298, 337 elastic, 168 elastoplasticity, 298 finite deformation, 337 nonlinear, 175 St. Venant–Kirchhoff material, 163 Surface traction, 15 T Tangential slip, 379, 380 Tangential traction force, 387 Tangent modulus, 243 Tangent operator, 297, 336, 337 consistent, 297, 337 material, 336 spatial, 336 Tangent stiffness matrix, 94 Tensor, 5–7, 9, 10, 17, 32 Cartesian, 5 elasticity, 32 identity, 5 orthogonal, 9 skew, 6, 10 spin, 7 stress, 17 symmetric, 6 Tensor product, 269 Time step, 110 Total Lagrangian formulation, 168 Trace, 8, 268 Transpose, 3 Trial function, 50 U Updated Lagrangian formulation, 174 V Variational equation, 43, 167 Variational inequality, 383 Vector, 3, 10 dual, 10 product, 10 Virtual displacement, 42 Virtual work, 391 contact, 391 W Weak form, 44, 115, 166, 249, 298, 383 Work, 39 Y Yield criterion, 282 von Mises, 282 Yield function, 282, 327 Yield surface, 282 Young’s modulus, 33, 83
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