Finite Element Analysis: Thermomechanics of Solids, Second Edition

by Nicholson; David W.

• ISBN13: 9781420050950
• ISBN10: 1420050958
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-04-18
• Publisher: CRC Press

Summary

Reorganized and updated to accommodate both the beginning and more advanced study of Finite Element Analysis, the new edition of this popular text contains much new information for both first and second level students. Written by a professor with years of practical engineering and instructional experience, the book is designed to provide a strong foundation for those requiring a sophisticated understanding of Finite Element Analysis' capabilities and limitations. It is also written for those preparing to enter careers in computer-aided engineering. It includes significant illustrative examples and applications, modeling strategies, and considers a range of computational issues.

Table of Contents

Preface to the Second Edition	p. xv
Preface to the First Edition	p. xvii
Author	p. xix
Acknowledgments	p. xxi
Introduction to the Finite Element Method	p. 1
Introduction	p. 1
Overview of the Finite Element Method	p. 2
Mesh Development	p. 3
Mathematical Foundations: Vectors and Matrices	p. 5
Introduction	p. 5
Range and Summation Convention	p. 5
Substitution Operator	p. 5
Vectors	p. 6
Notation: Scalar and Vector Products	p. 6
Gradient, Divergence, and Curl in Rectilinear Coordinates	p. 8
Matrices	p. 15
Eigenvalues and Eigenvectors	p. 21
Coordinate Transformations	p. 22
Transformations of Vectors	p. 24
Orthogonal Curvilinear Coordinates	p. 25
Gradient Operator in Orthogonal Coordinates	p. 30
Divergence and Curl of Vectors in Orthogonal Coordinates	p. 31
Appendix I: Divergence and Curl of Vectors in Orthogonal Curvilinear Coordinates	p. 32
Mathematical Foundations: Tensors	p. 37
Tensors	p. 37
Divergence of a Tensor	p. 39
Invariants	p. 40
Positive Definiteness	p. 40
Polar Decomposition Theorem	p. 41
Kronecker Products of Tensors	p. 42
VEC Operator and the Kronecker Product	p. 42
Fundamental Relations for Kronecker Products	p. 43
Eigenstructures of Kronecker Products	p. 45
Kronecker Form of Quadratic Products	p. 46
Kronecker Product Operators for Fourth-Order Tensors	p. 46
Transformation Properties of VEC, TEN22, TEN21, and TEN12	p. 46
Kronecker Expressions for Symmetry Classes in Fourth-Order Tensors	p. 47
Differentials of Tensor Invariants	p. 49
Examples	p. 49
Introduction to Variational Methods	p. 59
Introductory Notions	p. 59
Properties of the Variational Operator [delta]	p. 60
Example: Variational Equation for a Cantilevered Elastic Rod	p. 60
Higher Order Variations	p. 62
Examples	p. 63
Fundamental Notions of Linear Solid Mechanics	p. 75
Displacement Vector	p. 75
Linear Strain and Rotation Tensors	p. 76
Examples of Linear Strain and Rotation Tensors	p. 78
Traction and Stress	p. 81
Equilibrium	p. 86
Stress and Strain Transformations	p. 90
Principal Stresses and Strains	p. 97
Stress-Strain Relations	p. 100
Principle of Virtual Work in Linear Elasticity	p. 104
Thermal and Thermomechanical Response	p. 109
Balance of Energy and Production of Entropy	p. 109
Balance of Energy	p. 109
Entropy Production Inequality	p. 110
Thermodynamic Potentials in Reversible Processes	p. 111
Classical Coupled Linear Thermoelasticity	p. 112
Thermal and Thermomechanical Analogs of the Principle of Virtual Work and Associated Finite Element Equations	p. 120
Conductive Heat Transfer	p. 120
Coupled Linear Isotropic Thermoelasticity	p. 121
One-Dimensional Elastic Elements	p. 127
Interpolation Models for One-Dimensional Elements	p. 127
Rods	p. 127
Beams	p. 128
Beam-Columns	p. 129
Strain-Displacement Relations in One-Dimensional Elements	p. 129
Stress-Strain Relations in One-Dimensional Elements	p. 130
General	p. 130
One-Dimensional Members	p. 130
Element Stiffness and Mass Matrices from the Principle of Virtual Work	p. 130
Single-Element Model for Dynamic Response of a Built-in Beam	p. 133
Integral Evaluation by Gaussian Quadrature: Natural Coordinates	p. 137
Unconstrained Rod Elements	p. 141
Unconstrained Elements for Beams and Beam-Columns	p. 144
Assemblage and Imposition of Constraints	p. 145
Rods	p. 145
Beams	p. 154
Damping in Rods and Beams	p. 155
General Discussion of Assemblage	p. 156
General Discussion on the Imposition of Constraints	p. 158
Inverse Variational Method	p. 159
Two- and Three-Dimensional Elements in Linear Elasticity and Linear Conductive Heat Transfer	p. 163
Interpolation Models in Two Dimensions	p. 163
Membrane Plate	p. 163
Plate with Bending Stresses Only	p. 163
Plate with Stretching and Bending	p. 165
Temperature Field in Two Dimensions	p. 165
Axisymmetric Elements	p. 166
Interpolation Models in Three Dimensions	p. 167
Strain-Displacement Relations and Thermal Analogs	p. 168
Strain-Displacement Relations: Two Dimensions	p. 168
Axisymmetric Element	p. 169
Thermal Analog for Two-Dimensional and Axisymmetric Elements	p. 170
Three-Dimensional Elements	p. 170
Thermal Analog in Three Dimensions	p. 171
Stress-Strain Relations	p. 171
Two-Dimensional Elements	p. 171
Membrane Response	p. 171
Two-Dimensional Members: Bending Response of Thin Plates	p. 172
Element for Plate with Membrane and Bending Response	p. 173
Axisymmetric Element	p. 173
Three-Dimensional Element	p. 173
Elements for Conductive Heat Transfer	p. 174
Stiffness and Mass Matrices and Their Thermal Analogs	p. 175
Thermal Counterpart of the Principle of Virtual Work	p. 176
Conversion to Natural Coordinates in Two and Three Dimensions	p. 177
Assembly of Two- and Three-Dimensional Elements	p. 183
Solution Methods for Linear Problems: I	p. 187
Numerical Methods in FEA	p. 187
Solving the Finite Element Equations: Static Problems	p. 187
Matrix Triangularization and Solution of Linear Systems	p. 188
Triangularization of Asymmetric Matrices	p. 193
Time Integration: Stability and Accuracy	p. 194
Properties of the Trapezoidal Rule	p. 196
Integral Evaluation by Gaussian Quadrature	p. 200
Modal Analysis by FEA	p. 202
Modal Decomposition	p. 202
Comments on Eigenstructure Computation in Large Finite Element Systems	p. 214
Solution Methods for Linear Problems: II	p. 217
Introduction	p. 217
Solution Method for an Inverse Problem	p. 217
Inverse Problem in Elasticity	p. 217
Existence of a Unique Solution	p. 218
Nonsingularity Test	p. 221
Accelerated Eigenstructure Computation in FEA	p. 224
Introduction	p. 224
Problem Statement	p. 224
Hypersphere Path of Steepest Descent	p. 225
Hypercircle Search	p. 226
Eigenvalue Replacement Procedure	p. 228
Example: Minimum Eigenvalue of the 3 x 3 Hilbert Matrix	p. 230
Fourth-Order Time Integration	p. 231
Introduction	p. 231
Error Growth in the Newmark Method	p. 232
Undamped Free Vibration	p. 232
Adams-Moulton Formula	p. 233
Stepwise and Cumulative Error in the Adams-Moulton Method	p. 234
Stability Limit on Time Step in the Adams-Moulton Method	p. 235
Introducing Numerical Damping into the Adams-Moulton Method	p. 237
AMX: Adams-Moulton Method Applied to Systems with Acceleration	p. 239
Comments on Filtering to Remove High-Order Modes	p. 240
Additional Topics in Linear Thermoelastic Systems	p. 243
Transient Conductive Heat Transfer in Linear Media	p. 243
Finite Element Equation	p. 243
Direct Integration by the Trapezoidal Rule	p. 244
Modal Analysis in Linear Thermoelasticity	p. 244
Coupled Linear Thermoelasticity	p. 245
Finite Element Equation	p. 245
Thermoelasticity in an Elastic Conductor	p. 249
Incompressible Elastic Media	p. 250
Torsion of Prismatic Bars	p. 256
Basic Relations	p. 256
Buckling of Elastic Beams and Plates	p. 263
Euler Buckling of Beam Columns	p. 263
Static Buckling	p. 263
Dynamic Buckling	p. 264
Euler Buckling of Plates	p. 272
Introduction to Contact Problems	p. 274
Gap	p. 274
Point-to-Point Contact	p. 276
Point-to-Surface Contact	p. 278
Rotating and Unrestrained Elastic Bodies	p. 283
Finite Elements in Rotation	p. 283
Angular Velocity and Angular Acceleration Vectors	p. 283
Velocity and Acceleration in Rotating Coordinates	p. 286
Critical Speeds in Shaft Rotor Systems	p. 295
Finite Element Analysis for Unconstrained Elastic Bodies	p. 299
Body Axes	p. 299
Euler Equations of a Rigid Body	p. 300
Variational Equations of an Unconstrained Elastic Body	p. 302
Principle of Virtual Work in Body Coordinates	p. 305
Numerical Determination of the Current Position of the Body Axes	p. 305
Appendix: Angular Velocity Vector in Spherical Coordinates	p. 306
Aspects of Nonlinear Continuum Thermomechanics	p. 309
Introduction	p. 309
Nonlinear Kinematics of Deformation	p. 309
Deformation Gradient Tensor	p. 309
Lagrangian Strain Tensor	p. 311
Velocity Gradient Tensor, Deformation Rate Tensor, and Spin Tensor	p. 313
Differential Volume Element	p. 314
Differential Surface Element	p. 315
Mechanical Equilibrium and the Principle of Virtual Work	p. 319
Traction Vector and Stress Tensors	p. 319
Stress Flux	p. 324
Balance of Mass, Linear Momentum, and Angular Momentum	p. 326
Principle of Virtual Work under Large Deformation	p. 328
Nonlinear Stress-Strain-Temperature Relations: The Isothermal Tangent Modulus Tensor	p. 333
Classical Elasticity	p. 333
Compressible Hyperelastic Materials	p. 333
Incompressible and Near-Incompressible Hyperelastic Materials	p. 334
Incompressibility	p. 335
Near-Incompressibility	p. 338
Nonlinear Materials at Large Deformation	p. 339
Introduction to Nonlinear FEA	p. 341
Introduction	p. 341
Types of Nonlinearity	p. 341
Newton Iteration	p. 342
Combined Incremental and Iterative Methods: A Simple Example	p. 344
Finite Stretching of a Rubber Rod under Gravity	p. 345
Model Problem	p. 345
Nonlinear Strain-Displacement Relations	p. 346
Stress and Tangent Modulus Relations	p. 346
Incremental Equilibrium Relation	p. 348
Single Element Built-in at One End	p. 349
On Numerical Solution by Newton Iteration	p. 350
Assembled Stiffness Matrix for a Two-Element Model of the Rubber Rod under Gravity	p. 351
Newton Iteration near a Critical Point	p. 353
Introduction to the Arc Length Method	p. 355
Incremental Principle of Virtual Work	p. 359
Incremental Kinematics	p. 359
Stress Increments	p. 365
Incremental Equation of Balance of Linear Momentum	p. 366
Incremental Principle of Virtual Work	p. 366
Incremental Finite Element Equation	p. 368
Contributions from Nonlinear Boundary Conditions	p. 368
Effect of Variable Contact	p. 370
Interpretation as Newton Iteration	p. 372
Buckling	p. 373
Tangent Modulus Tensors for Thermomechanical Response of Elastomers	p. 375
Introduction	p. 375
Compressible Elastomers	p. 376
Incompressible and Near-Incompressible Elastomers	p. 376
Examples of Expressions for the Helmholtz Potential	p. 378
Invariant-Based Incompressible Models: Isothermal Problems	p. 378
Invariant-Based Models for Compressible Elastomers under Isothermal Conditions	p. 379
Thermomechanical Behavior under Non-Isothermal Conditions	p. 379
Stretch-Ratio-Based Models: Isothermal Conditions	p. 380
Extension to Thermohyperelastic Materials	p. 381
Thermomechanics of Damped Elastomers	p. 384
Balance of Energy	p. 385
Entropy Production Inequality	p. 385
Dissipation Potential	p. 386
Thermal Field Equation for Damped Elastomers	p. 387
Constitutive Model in Thermoviscohyperelasticity	p. 388
Helmholtz Free Energy Density	p. 388
Dissipation Potential	p. 389
Variational Principles and Finite Element Equations for Thermoviscohyperelastic Materials	p. 390
Mechanical Equilibrium	p. 390
Thermal Equilibrium Equation	p. 391
Tangent Modulus Tensors for Inelastic and Thermoinelastic Materials	p. 393
Plasticity	p. 393
Tangent Modulus Tensor in Small Strain Isothermal Plasticity	p. 393
Plasticity under Finite Strain	p. 400
Kinematics	p. 400
Plasticity	p. 400
Thermoplasticity	p. 402
Balance of Energy	p. 403
Entropy Production Inequality	p. 404
Dissipation Potential	p. 404
Thermoinelastic Tangent Modulus Tensor	p. 405
Tangent Modulus Tensor in Viscoplasticity	p. 408
Mechanical Field	p. 408
Thermoinelasticity: Thermal Field	p. 412
Continuum Damage Mechanics	p. 414
Selected Advanced Numerical Methods in FEA	p. 419
Iterative Triangularization of Perturbed Matrices	p. 419
Introduction	p. 419
Incremental Finite Element Equation	p. 420
Iterative Triangularization Procedure	p. 421
Stiff Arc Length Constraint in Nonlinear FEA	p. 427
Introduction	p. 427
Newton Iteration for Nonlinear Finite Element Equations	p. 429
Newton Iteration without Arc Length Constraint	p. 430
Arc Length Method	p. 431
Stiff Arc Length Constraint	p. 432
Stiffness of K*	p. 432
Arc Length Vector Which Maximizes Stiffness: Examples	p. 434
Arc Length Vector Which Maximizes Stiffness: General Argument	p. 436
Numerical Determination of the Optimal Arc Length Vector	p. 437
Solution Procedure	p. 438
Block Triangularization	p. 439
Solution of the Outer Problem	p. 439
Solution of the Inner Problem	p. 440
Non-Iterative Solution of Finite Element Equations in Incompressible Solids	p. 443
Introduction	p. 443
Finite Element Equation for an Incompressible Medium	p. 444
Uzawa's Method	p. 445
Modification to Avoid Iteration	p. 447
References	p. 449
Index	p. 453
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