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 |  موضوع: كتاب Finite Element Analysis - Thermomechanics of Solids الثلاثاء 16 يوليو 2024, 3:54 pm | |
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Finite Element Analysis - Thermomechanics of Solids
David W. Nicholson
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Table of Contents
Chapter 1 Mathematical Foundations: Vectors and Matrices .1
1.1 Introduction 1
1.1.1 Range and Summation Convention 1
1.1.2 Substitution Operator 1
1.2 Vectors 2
1.2.1 Notation .2
1.2.2 Gradient, Divergence, and Curl 4
1.3 Matrices 5
1.3.1 Eigenvalues and Eigenvectors .8
1.3.2 Coordinate Transformations 9
1.3.3 Transformations of Vectors .9
1.3.4 Orthogonal Curvilinear Coordinates .11
1.3.5 Gradient Operator 16
1.3.6 Divergence and Curl of Vectors 17
Appendix I: Divergence and Curl of Vectors in Orthogonal
Curvilinear Coordinates 18
Derivatives of Base Vectors 18
Divergence .19
Curl 20
1.4 Exercises 20
Chapter 2 Mathematical Foundations: Tensors 25
2.1 Tensors .25
2.2 Divergence, Curl, and Laplacian of a Tensor .27
2.2.1 Divergence .27
2.2.2 Curl and Laplacian 28
2.3 Invariants 29
2.4 Positive Definiteness 30
2.5 Polar Decomposition Theorem 31
2.6 Kronecker Products on Tensors .32
2.6.1 VEC Operator and the Kronecker Product .32
2.6.2 Fundamental Relations for Kronecker Products .33
2.6.3 Eigenstructures of Kronecker Products 35
2.6.4 Kronecker Form of Quadratic Products 36
2.6.5 Kronecker Product Operators for Fourth-Order Tensors 36
2.6.6 Transformation Properties of VEC and TEN22 37
2.6.7 Kronecker Product Functions for Tensor Outer Products 38
 2003 by CRC CRC Press LLC2.6.8 Kronecker Expressions for Symmetry Classes
in Fourth-Order Tensors 40
2.6.9 Differentials of Tensor Invariants .41
2.7 Exercises 42
Chapter 3 Introduction to Variational and Numerical Methods .43
3.1 Introduction to Variational Methods 43
3.2 Newton Iteration and Arc-Length Methods 47
3.2.1 Newton Iteration 47
3.2.2 Critical Points and the Arc-Length Method .48
3.3 Exercises 49
Chapter 4 Kinematics of Deformation .51
4.1 Kinematics .51
4.1.1 Displacement .51
4.1.2 Displacement Vector 52
4.1.3 Deformation Gradient Tensor .52
4.2 Strain 53
4.2.1 F, E, EL and u in Orthogonal Coordinates 53
4.2.2 Velocity-Gradient Tensor, Deformation-Rate Tensor,
and Spin Tensor .56
4.3 Differential Volume Element .60
4.4 Differential Surface Element .61
4.5 Rotation Tensor 63
4.6 Compatibility Conditions For EL and D .64
4.7 Sample Problems .67
4.8 Exercises 69
Chapter 5 Mechanical Equilibrium and the Principle of Virtual Work .73
5.1 Traction and Stress 73
5.1.1 Cauchy Stress 73
5.1.2 1st Piola-Kirchhoff Stress .75
5.1.3 2nd Piola-Kirchhoff Stress 76
5.2 Stress Flux .77
5.3 Balance of Mass, Linear Momentum, and Angular Momentum 79
5.3.1 Balance of Mass 79
5.3.2 Rayleigh Transport Theorem 79
5.3.3 Balance of Linear Momentum 79
5.3.4 Balance of Angular Momentum 80
5.4 Principle of Virtual Work 82
5.5 Sample Problems .85
5.6 Exercises 89
2003 by CRC CRC Press LLCChapter 6 Stress-Strain Relation and the Tangent-Modulus Tensor 95
6.1 Stress-Strain Behavior: Classical Linear Elasticity 95
6.2 Isothermal Tangent-Modulus Tensor .97
6.2.1 Classical Elasticity 97
6.2.2 Compressible Hyperelastic Materials .97
6.3 Incompressible and Near-Incompressible Hyperelastic Materials 99
6.3.1 Incompressibility .99
6.3.2 Near-Incompressibility 102
6.4 Nonlinear Materials at Large Deformation .103
6.5 Exercises 104
Chapter 7 Thermal and Thermomechanical Response .107
7.1 Balance of Energy and Production of Entropy .107
7.1.1 Balance of Energy .107
7.1.2 Entropy Production Inequality 108
7.1.3 Thermodynamic Potentials 109
7.2 Classical Coupled Linear Thermoelasticity 110
7.3 Thermal and Thermomechanical Analogs of the Principle
of Virtual Work 113
7.3.1 Conductive Heat Transfer .113
7.3.2 Coupled Linear Isotropic Thermoelasticity 114
7.4 Exercises 116
Chapter 8 Introduction to the Finite-Element Method .117
8.1 Introduction 117
8.2 Overview of the Finite-Element Method 117
8.3 Mesh Development 118
Chapter 9 Element Fields in Linear Problems .121
9.1 Interpolation Models 121
9.1.1 One-Dimensional Members 121
9.1.2 Interpolation Models: Two Dimensions 124
9.1.3 Interpolation Models: Three Dimensions .127
9.2 Strain-Displacement Relations and Thermal Analogs 128
9.2.1 Strain-Displacement Relations: One Dimension 128
9.2.2 Strain-Displacement Relations: Two Dimensions 129
9.2.3 Axisymmetric Element on Axis of Revolution 130
9.2.4 Thermal Analog in Two Dimensions 131
9.2.5 Three-Dimensional Elements 131
9.2.6 Thermal Analog in Three Dimensions 132
9.3 Stress-Strain-Temperature Relations in Linear Thermoelasticity .132
9.3.1 Overview .132
9.3.2 One-Dimensional Members 132
9.3.3 Two-Dimensional Elements 133
2003 by CRC CRC Press LLC9.3.4 Element for Plate with Membrane and Bending Response .135
9.3.5 Axisymmetric Element 135
9.3.6 Three-Dimensional Element .136
9.3.7 Elements for Conductive Heat Transfer .137
9.4 Exercises 137
Chapter 10 Element and Global Stiffness and Mass Matrices 139
10.1 Application of the Principle of Virtual Work 139
10.2 Thermal Counterpart of the Principle of Virtual Work .141
10.3 Assemblage and Imposition of Constraints 142
10.3.1 Rods .142
10.3.2 Beams 146
10.3.3 Two-Dimensional Elements 147
10.3 Exercises 149
Chapter 11 Solution Methods for Linear Problems .153
11.1 Numerical Methods in FEA 153
11.1.1 Solving the Finite-Element Equations: Static Problems 153
11.1.2 Matrix Triangularization and Solution of Linear Systems .154
11.1.3 Triangularization of Asymmetric Matrices .155
11.2 Time Integration: Stability and Accuracy .156
11.3 Newmark’s Method .157
11.4 Integral Evaluation by Gaussian Quadrature 158
11.5 Modal Analysis by FEA 159
11.5.1 Modal Decomposition .159
11.5.2 Computation of Eigenvectors and Eigenvalues 162
11.6 Exercises 164
Chapter 12 Rotating and Unrestrained Elastic Bodies .167
12.1 Finite Elements in Rotation .167
12.2 Finite-Element Analysis for Unconstrained Elastic Bodies 169
12.3 Exercises 171
Chapter 13 Thermal, Thermoelastic, and Incompressible Media 173
13.1 Transient Conductive-Heat Transfer 173
13.1.1 Finite-Element Equation .173
13.1.2 Direct Integration by the Trapezoidal Rule 173
13.1.3 Modal Analysis 174
13.2 Coupled Linear Thermoelasticity 175
13.2.1 Finite-Element Equation .175
13.2.2 Thermoelasticity in a Rod .177
13.3 Compressible Elastic Media 177
13.4 Incompressible Elastic Media .178
13.5 Exercises 180
2003 by CRC CRC Press LLCChapter 14 Torsion and Buckling .181
14.1 Torsion of Prismatic Bars 181
14.2 Buckling of Beams and Plates 185
14.2.1 Euler Buckling of Beam Columns 185
14.2.2 Euler Buckling of Plates .190
14.3 Exercises 193
Chapter 15 Introduction to Contact Problems 195
15.1 Introduction: the Gap .195
15.2 Point-to-Point Contact .197
15.3 Point-to-Surface Contact .199
15.4 Exercises 199
Chapter 16 Introduction to Nonlinear FEA 201
16.1 Overview 201
16.2 Types of Nonlinearity 201
16.3 Combined Incremental and Iterative Methods: a Simple Example 202
16.4 Finite Stretching of a Rubber Rod under Gravity: a Simple Example 203
16.4.1 Nonlinear Strain-Displacement Relations .203
16.4.2 Stress and Tangent Modulus Relations .204
16.4.3 Incremental Equilibrium Relation .205
16.4.4 Numerical Solution by Newton Iteration 208
16.5 Illustration of Newton Iteration .211
16.5.1 Example .212
16.6 Exercises 213
Chapter 17 Incremental Principle of Virtual Work 215
17.1 Incremental Kinematics .215
17.2 Incremental Stresses 216
17.3 Incremental Equilibrium Equation 217
17.4 Incremental Principle of Virtual Work 218
17.5 Incremental Finite-Element Equation .219
17.6 Incremental Contributions from Nonlinear Boundary Conditions .220
17.7 Effect of Variable Contact .221
17.8 Interpretation as Newton Iteration .223
17.9 Buckling .224
17.10 Exercises 226
Chapter 18 Tangent-Modulus Tensors for Thermomechanical
Response of Elastomers .227
18.1 Introduction 227
18.2 Compressible Elastomers 227
18.3 Incompressible and Near-Incompressible Elastomers 228
18.3.1 Specific Expressions for the Helmholtz Potential 230
2003 by CRC CRC Press LLC18.4 Stretch Ratio-Based Models: Isothermal Conditions 231
18.5 Extension to Thermohyperelastic Materials 233
18.6 Thermomechanics of Damped Elastomers 234
18.6.1 Balance of Energy .235
18.6.2 Entropy Production Inequality 235
18.6.3 Dissipation Potential .236
18.6.4 Thermal-Field Equation for Damped Elastomers .237
18.7 Constitutive Model: Potential Functions .238
18.7.1 Helmholtz Free-Energy Density .238
18.7.2 Specific Dissipation Potential .239
18.8 Variational Principles .240
18.8.1 Mechanical Equilibrium 240
18.8.2 Thermal Equilibrium .240
18.9 Exercises 241
Chapter 19 Inelastic and Thermoinelastic Materials 243
19.1 Plasticity .243
19.1.1 Kinematics .243
19.1.2 Plasticity 243
19.2 Thermoplasticity 246
19.2.1 Balance of Energy .246
19.2.2 Entropy-Production Inequality 247
19.2.3 Dissipation Potential .248
19.3 Thermoinelastic Tangent-Modulus Tensor 249
19.3.1 Example .250
19.4 Tangent-Modulus Tensor in Viscoplasticity 252
19.5 Continuum Damage Mechanics 254
19.6 Exercises 256
Chapter 20 Advanced Numerical Methods 257
20.1 Iterative Triangularization of Perturbed Matrices .257
20.1.1 Introduction .257
20.1.2 Notation and Background .258
20.1.3 Iteration Scheme 259
20.1.4 Heuristic Convergence Argument .259
20.1.5 Sample Problem 260
20.2 Ozawa’s Method for Incompressible Materials 262
20.3 Exercises 263
Monographs and Texts 265
Articles and Other Sources .267
Monographs and Texts
Abramawitz, M. and Stegun, I., Eds, Handbook of Mathematical Functions, with Formulas,
Graphs and Mathematical Tables, Dover Publications, Inc., New York, 1997.
Belytschko, T., Liu, W.K., and Moran, B., Nonlinear Finite Elements for Continua and
Structures, John Wiley and Sons, New York, 2000.
Bonet, J. and Wood, R., Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, Cambridge, 1997.
Brush, D. and Almroth, B., Buckling of Bars, Plates and Shells, McGraw-Hill Book Company,
New York, 1975.
Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, 2nd ed., John Wiley
and Sons, New York, 1985.
Chandrasekharaiah, D. and Debnath, L., Continuum Mechanics, Academic Press, San Diego,
1994.
Chung, T.J., Continuum Mechanics, Prentice-Halls, Englewood Cliffs, NJ, 1988.
Dahlquist, G. and Bjork, A., Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974.
Ellyin, F., Fatigue Damage, Crack Growth and Life Prediction, Chapman & Hall, FL, 1997.
Eringen, A.C., Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962.
Ewing, G.M., Calculus of Variations with Applications, Dover Publications, Mineola, N.Y.,
1985.
Gent, A.N., Ed, Engineering with Rubber: How to Design Rubber Components, Hanser, New
York, 1992.
Golub, G. and Van Loan, C., Matrix Computations, Johns Hopkins University Press, Baltimore,
MD, 1996.
Graham, A., Kronecker Products and Matrix Calculus with Applications, Ellis Horwood, Ltd.,
Chichester, 1981.
Hughes, T., The Finite Element Method: Linear Static and Dynamic Analysis, Dover Publishers Inc., New York, 2000.
Kleiber, M., Incremental Finite Element Modeling in Non-Linear Solid Mechanics, Ellis
Horwood, Ltd., Chichester, 1989.
Oden, J.T., Finite Elements of Nonlinear Continua, McGraw-Hill Book Company, New York,
1972.
Schey, H.M., DIV, GRAD, CURL and All That, Norton, New York, 1973.
Thomason, P.F., Ductile Fracture of Metals, Pergamon Press, Oxford, 1990.
Wang, C-T., Applied Elasticity, The Maple Press Company, York, PA, 1953.
Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method. Volume 2, 4th ed, McGrawHill Book Company, London, 1989.
2003 by CRC CRC Press LLC267
Articles and Other Sources
ANSYS User Manual, ver 6.0, Swanson Analysis Systems, 2000.
Blatz, P.J. and Ko, W.L., Application of finite elastic theory to the deformation of rubbery
materials, Trans. Soc. Rheol., 6, 223, 1962.
Bonora, N., A nonlinear CDM model for ductile failure, Engineering Fracture Mechanics,
58, 1/2, 11, 1997.
Chen, J., Wan, W., Wu, C.T., and Duan, W., On the perturbed Lagrangian formulation for
nearly incompressible and incompressible hyperelasticity, Comp. Methods Appl. Mech.
Eng., 142, 335, 1997.
Dillon, O.W., A nonlinear thermoelasticity theory, J. Mech. Phys. Solids, 10, 123, 1962.
Green, A. and Naghdi, P., A general theory of an elastic-plastic continuum, Arch. Rat. Mech.
Anal., 18/19, 1965.
Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth, part 1: yield
criteria and flow rules for porous ductile media, J. Eng. Mat’s Technol., 99, 2, 1977.
Holzappel, G., On large strain viscoelasticity: continuum applications and finite element
applications to elastomeric structures, Int. J. Numer. Meth. Eng., 39, 3903, 1996.
Holzappel, G. and Simo, J., Entropy elasticity of isotropic rubber-like solids at finite strain,
Comp. Methods Appl. Mech. Eng., 132, 17, 1996.
LS-DYNA, ver. 95, Livermore Software Technology Corporation, Livermore, CA, 2000.
Moraes, R. and Nicholson D.W., Local damage criterion for ductile fracture with application to welds under dynamic loads, in Advances in Fracture and Damage Mechanics, Guagliano, M. and Aliabadi, M.H., Eds., 2nd ed. Hoggar Press, Geneva, 277,
2002.
Nicholson, D.W. and Nelson, N., Finite element analysis in design with rubber (Rubber
Reviews), Rubber Chem. Tech., 63, 638, 1990.
Nicholson, D.W., Tangent modulus matrix for finite element analysis of hyperelastic materials,
Acta Mech., 112, 187, 1995.
Nicholson, D.W. and Lin, B., Theory of thermohyperelasticity for near-incompressible elastomers, Acta Mech., 116, 15, 1996.
Nicholson, D.W. and Lin, B., Finite element method for thermomechanical response of nearincompressible elastomers, Acta Mech., 124, 181, 1997a.
Nicholson, D.W. and Lin, B., Incremental finite element equations for thermomechanical of
elastomers: effect of boundary conditions including contact, Acta Mech., 128, 1–2,
81, 1997b.
Nicholson, D.W. and Lin, B., On the tangent modulus tensor in hyperelasticty, Acta Mech.,
131, 1997c.
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components, Appl. Mech. Rev., 51, 5, 303, 1999.
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Chem. Tech., 59, 361, 1986.
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2003 by CRC CRC Press LLC268 Finite Element Analysis: Thermomechanics of Solids
Tvergaard, V., Influence of voids on shear band instabilities under plane strain conditions,
Int. J. Fracture, 17, 389–407, 1981.
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