The second edition of An Introduction to Nonlinear Finite Element Analysis has the same objective as the first edition, namely, to facilitate an easy and thorough understanding of the details that are involved in the theoretical formulation, finite element model development, and solutions of nonlinear problems. The book offers an easy-to-understand treatment of the subject of nonlinear finite element analysis, which includes element development from mathematical models and numerical evaluation of the underlying physics.
The new edition is extensively reorganized and contains substantial amounts of new material. Chapter 1 in the second edition contains a section on applied functional analysis. Chapter 2 on nonlinear continuum mechanics is entirely new. Chapters 3 through 8 in the new edition correspond to Chapter 2 through 8 of the first edition, but with additional explanations, examples, and exercise problems. Material on time dependent problems from Chapter 8 of the first edition is absorbed into Chapters 4 through 8 of the new edition. Chapter 9 is extensively revised and it contains up to date developments in the large deformation analysis of isotropic, composite and functionally graded shells. Chapter 10 of the first edition on material nonlinearity and coupled problems is reorganized in the second edition by moving the material on solid mechanics to Chapter 12 in the new edition and material on coupled problems to the new chapter, Chapter 10, on weak-form Galerkin finite element models of viscous incompressible fluids. Finally, Chapter 11 in the second edition is entirely new and devoted to least-squares finite element models of viscous incompressible fluids. Chapter 12 of the second edition is enlarged to contain finite element models of viscoelastic beams. In general, all of the chapters of the second edition contain additional explanations, detailed example problems, and additional exercise problems. Although all of the programming segments are in Fortran, the logic used in these Fortran programs is transparent and can be used in Matlab or C++ versions of the same. Thus the new edition more than replaces the first edition, and it is hoped that it is acquired by the library of every institution of higher learning as well as serious finite element analysts.
The book may be used as a textbook for an advanced course (after a first course) on the finite element method or the first course on nonlinear finite element analysis. A solutions manual is available on request from the publisher to instructors who adopt the book as a textbook for a course.
To request a copy of the Solutions Manual, visit: //global.oup.com/uk/academic/physics/admin/solutions
Author(s): J. N. Reddy
Edition: 2
Publisher: Oxford University Press
Year: 2015
Language: English
Pages: 768
Tags: Engineering;Aerospace;Automotive;Bioengineering;Chemical;Civil & Environmental;Computer Modelling;Construction;Design;Electrical & Electronics;Energy Production & Extraction;Industrial, Manufacturing & Operational Systems;Marine Engineering;Materials & Material Science;Mechanical;Military Technology;Reference;Telecommunications & Sensors;Engineering & Transportation;Mathematical Physics;Physics;Science & Math;Engineering;Aeronautical Engineering;Chemical Engineering;Civil Engineering;Electrical
1 General Introduction and Mathematical Preliminaries . . . 1
1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . 2
1.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . 4
1.4 The Finite Element Method . . . . . . . . . . . . . . . . . . 6
1.5 Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.2 Classi cation of Nonlinearities . . . . . . . . . . . . . . . 8
1.6 Review of Vectors and Tensors . . . . . . . . . . . . . . . . 12
1.6.1 Preliminary Comments . . . . . . . . . . . . . . . . . 12
1.6.2 De nition of a Physical Vector . . . . . . . . . . . . . . 13
1.6.2.1 Vector addition . . . . . . . . . . . . . . . . . . 13
1.6.2.2 Multiplication of a vector by a scalar . . . . . . . . . 13
1.6.3 Scalar and Vector Products . . . . . . . . . . . . . . . 14
1.6.3.1 Scalar product (or \dot" product) . . . . . . . . . . 14
1.6.3.2 Vector product . . . . . . . . . . . . . . . . . . . 14
1.6.3.3 Plane area as a vector . . . . . . . . . . . . . . . 15
1.6.3.4 Linear independence of vectors . . . . . . . . . . . . 16
1.6.3.5 Components of a vector . . . . . . . . . . . . . . . 16
1.6.4 Summation Convention and Kronecker Delta and
Permutation Symbol . . . . . . . . . . . . . . . . . . 17
1.6.4.1 Summation convention . . . . . . . . . . . . . . . 17
1.6.4.2 Kronecker delta symbol . . . . . . . . . . . . . . . 17
1.6.4.3 The permutation symbol . . . . . . . . . . . . . . 18
1.6.5 Tensors and their Matrix Representation . . . . . . . . . . 19
1.6.5.1 Concept of a second-order tensor . . . . . . . . . . . 19
1.6.5.2 Transformation laws for vectors and tensors . . . . . . 20
1.6.6 Calculus of Vectors and Tensors . . . . . . . . . . . . . 22
1.7 Concepts from Functional Analysis . . . . . . . . . . . . . . 27
1.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 27
1.7.2 Linear Vector Spaces . . . . . . . . . . . . . . . . . . 28
1.7.2.1 Vector addition . . . . . . . . . . . . . . . . . . 28
1.7.2.2 Scalar multiplication . . . . . . . . . . . . . . . . 29
1.7.2.3 Linear subspaces . . . . . . . . . . . . . . . . . . 29
1.7.2.4 Linear dependence and independence of vectors . . . . 29
1.7.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . 30
1.7.3.1 Holder inequality . . . . . . . . . . . . . . . . . . 30
1.7.3.2 Minkowski inequality . . . . . . . . . . . . . . . . 30
1.7.4 Inner Product Spaces . . . . . . . . . . . . . . . . . . 32
1.7.4.1 Orthogonality of vectors . . . . . . . . . . . . . . . 33
1.7.4.2 Cauchy{Schwartz inequality . . . . . . . . . . . . . 33
1.7.4.3 Hilbert spaces . . . . . . . . . . . . . . . . . . . 35
1.7.5 Linear Transformations . . . . . . . . . . . . . . . . . 36
1.7.6 Linear Functionals, Bilinear Forms, and Quadratic Forms . . 37
1.7.6.1 Linear functional . . . . . . . . . . . . . . . . . . 38
1.7.6.2 Bilinear forms . . . . . . . . . . . . . . . . . . . 38
1.7.6.3 Quadratic forms . . . . . . . . . . . . . . . . . . 38
1.8 The Big Picture . . . . . . . . . . . . . . . . . . . . . . 40
1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 42
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 Elements of Nonlinear Continuum Mechanics . . . . . . . 47
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Description of Motion . . . . . . . . . . . . . . . . . . . . 48
2.2.1 Con gurations of a Continuous Medium . . . . . . . . . . 48
2.2.2 Material and Spatial Descriptions . . . . . . . . . . . . . 49
2.2.3 Displacement Field . . . . . . . . . . . . . . . . . . . 53
2.3 Analysis of Deformation . . . . . . . . . . . . . . . . . . . 54
2.3.1 Deformation Gradient . . . . . . . . . . . . . . . . . . 54
2.3.2 Volume and Surface Elements in the Material and
Spatial Descriptions . . . . . . . . . . . . . . . . . . 55
2.4 Strain Measures . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Deformation Tensors . . . . . . . . . . . . . . . . . . 56
2.4.2 The Green{Lagrange Strain Tensor . . . . . . . . . . . . 57
2.4.3 The Cauchy and Euler Strain Tensors . . . . . . . . . . . 58
2.4.4 In nitesimal Strain Tensor and Rotation Tensor . . . . . . 59
2.4.4.1 In nitesimal strain tensor . . . . . . . . . . . . . . 59
2.4.4.2 In nitesimal rotation tensor . . . . . . . . . . . . . 60
2.4.5 Time Derivatives of the Deformation Tensors . . . . . . . . 60
2.5 Measures of Stress . . . . . . . . . . . . . . . . . . . . . 62
2.5.1 Stress Vector . . . . . . . . . . . . . . . . . . . . . . 62
2.5.2 Cauchy's Formula and Stress Tensor . . . . . . . . . . . . 63
2.5.3 Piola{Kirchho Stress Tensors . . . . . . . . . . . . . . 65
2.5.3.1 First Piola{Kirchho stress tensor . . . . . . . . . . . 65
2.5.3.2 Second Piola{Kirchho stress tensor . . . . . . . . . 67
2.6 Material Frame Indi erence . . . . . . . . . . . . . . . . . 67
2.6.1 The Basic Idea . . . . . . . . . . . . . . . . . . . . . 67
2.6.2 Objectivity of Strains and Strain Rates . . . . . . . . . . 69
2.6.3 Objectivity of Stress Tensors . . . . . . . . . . . . . . . 69
2.6.3.1 Cauchy stress tensor . . . . . . . . . . . . . . . . . 69
2.6.3.2 First Piola{Kirchho stress tensor . . . . . . . . . . 70
2.6.3.3 Second Piola{Kirchho stress tensor . . . . . . . . . 70
2.7 Equations of Continuum Mechanics . . . . . . . . . . . . . . 70
2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 70
2.7.2 Conservation of Mass . . . . . . . . . . . . . . . . . . 71
2.7.2.1 Spatial form of the continuity equation . . . . . . . . 71
2.7.2.2 Material form of the continuity equation . . . . . . . 72
2.7.3 Reynolds Transport Theorem . . . . . . . . . . . . . . . 72
2.7.4 Balance of Linear Momentum . . . . . . . . . . . . . . 73
2.7.4.1 Spatial form of the equations of motion . . . . . . . . 73
2.7.4.2 Material form of the equations of motion . . . . . . . 73
2.7.5 Balance of Angular Momentum . . . . . . . . . . . . . . 74
2.7.6 Thermodynamic Principles . . . . . . . . . . . . . . . . 74
2.7.6.1 Energy equation in the spatial description . . . . . . . 75
2.7.6.2 Energy equation in the material description . . . . . . 76
2.7.6.3 Entropy inequality . . . . . . . . . . . . . . . . . 77
2.8 Constitutive Equations for Elastic Solids . . . . . . . . . . . . 78
2.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 78
2.8.2 Restrictions Placed by the Entropy Inequality . . . . . . . 79
2.8.3 Elastic Materials and the Generalized Hooke's Law . . . . . 80
2.9 Energy Principles of Solid Mechanics . . . . . . . . . . . . . 83
2.9.1 Virtual Displacements and Virtual Work . . . . . . . . . . 83
2.9.2 First Variation or G^ateaux Derivative . . . . . . . . . . . 83
2.9.3 The Principle of Virtual Displacements . . . . . . . . . . 84
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 88
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 91
3 The Finite Element Method: A Review . . . . . . . . . . 97
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 One-Dimensional Problems . . . . . . . . . . . . . . . . . . 98
3.2.1 Governing Di erential Equation . . . . . . . . . . . . . 98
3.2.2 Finite Element Approximation . . . . . . . . . . . . . . 98
3.2.3 Derivation of the Weak Form . . . . . . . . . . . . . . . 101
3.2.4 Approximation Functions . . . . . . . . . . . . . . . . 104
3.2.5 Finite Element Model . . . . . . . . . . . . . . . . . . 107
3.2.6 Natural Coordinates . . . . . . . . . . . . . . . . . . . 114
3.3 Two-Dimensional Problems . . . . . . . . . . . . . . . . . 116
3.3.1 Governing Di erential Equation . . . . . . . . . . . . . 116
3.3.2 Finite Element Approximation . . . . . . . . . . . . . . 118
3.3.3 Weak Formulation . . . . . . . . . . . . . . . . . . . 118
3.3.4 Finite Element Model . . . . . . . . . . . . . . . . . . 121
3.3.5 Approximation Functions: Element Library . . . . . . . . 122
3.3.5.1 Linear triangular element . . . . . . . . . . . . . . 122
3.3.5.2 Linear rectangular element . . . . . . . . . . . . . 125
3.3.5.3 Higher-order triangular elements . . . . . . . . . . . 126
3.3.5.4 Higher-order rectangular elements . . . . . . . . . . 128
3.3.6 Assembly of Elements . . . . . . . . . . . . . . . . . . 130
3.4 Axisymmetric Problems . . . . . . . . . . . . . . . . . . . 136
3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 136
3.4.2 One-Dimensional Problems . . . . . . . . . . . . . . . . 137
3.4.3 Two-Dimensional Problems . . . . . . . . . . . . . . . 138
3.5 The Least-Squares Method . . . . . . . . . . . . . . . . . . 139
3.5.1 Background . . . . . . . . . . . . . . . . . . . . . . 139
3.5.2 The Basic Idea . . . . . . . . . . . . . . . . . . . . . 141
3.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . 143
3.6.1 Preliminary Comments . . . . . . . . . . . . . . . . . 143
3.6.2 Coordinate Transformations . . . . . . . . . . . . . . . 143
3.6.3 Integration Over a Master Rectangular Element . . . . . . 147
3.6.4 Integration Over a Master Triangular Element . . . . . . . 148
3.7 Computer Implementation . . . . . . . . . . . . . . . . . . 149
3.7.1 General Comments . . . . . . . . . . . . . . . . . . . 149
3.7.2 One-Dimensional Problems . . . . . . . . . . . . . . . . 152
3.7.3 Two-Dimensional Problems . . . . . . . . . . . . . . . 157
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 163
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 164
4 One-Dimensional Problems Involving a Single Variable . 175
4.1 Model Di erential Equation . . . . . . . . . . . . . . . . . 175
4.2 Weak Formulation . . . . . . . . . . . . . . . . . . . . . 177
4.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . 177
4.4 Solution of Nonlinear Algebraic Equations . . . . . . . . . . . 180
4.4.1 General Comments . . . . . . . . . . . . . . . . . . . 180
4.4.2 Direct Iteration Procedure . . . . . . . . . . . . . . . . 180
4.4.3 Newton's Iteration Procedure . . . . . . . . . . . . . . . 185
4.5 Computer Implementation . . . . . . . . . . . . . . . . . . 192
4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 192
4.5.2 Preprocessor Unit . . . . . . . . . . . . . . . . . . . . 193
4.5.3 Processor Unit . . . . . . . . . . . . . . . . . . . . . 194
4.5.3.1 Calculation of element coecients . . . . . . . . . . . 194
4.5.3.2 Assembly of element coecients . . . . . . . . . . . 198
4.5.3.3 Imposition of boundary conditions . . . . . . . . . . 200
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 208
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 209
5 Nonlinear Bending of Straight Beams . . . . . . . . . . 213
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 213
5.2 The Euler{Bernoulli Beam Theory . . . . . . . . . . . . . . 214
5.2.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . 214
5.2.2 Displacement and Strain Fields . . . . . . . . . . . . . . 214
5.2.3 The Principle of Virtual Displacements: Weak Form . . . . . 216
5.2.4 Finite Element Model . . . . . . . . . . . . . . . . . . 222
5.2.5 Iterative Solution Strategies . . . . . . . . . . . . . . . 224
5.2.5.1 Direct iteration procedure . . . . . . . . . . . . . . 225
5.2.5.2 Newton's iteration procedure . . . . . . . . . . . . 225
5.2.6 Load Increments . . . . . . . . . . . . . . . . . . . . 228
5.2.7 Membrane Locking . . . . . . . . . . . . . . . . . . . 228
5.2.8 Computer Implementation . . . . . . . . . . . . . . . . 230
5.2.8.1 Rearrangement of equations and computation of
element coecients . . . . . . . . . . . . . . . . . 230
5.2.8.2 Computation of strains and stresses . . . . . . . . . 235
5.2.9 Numerical Examples . . . . . . . . . . . . . . . . . . 237
5.3 The Timoshenko Beam Theory . . . . . . . . . . . . . . . . 242
5.3.1 Displacement and Strain Fields . . . . . . . . . . . . . . 242
5.3.2 Weak Forms . . . . . . . . . . . . . . . . . . . . . . 243
5.3.3 General Finite Element Model . . . . . . . . . . . . . . 245
5.3.4 Shear and Membrane Locking . . . . . . . . . . . . . . 247
5.3.5 Tangent Sti ness Matrix . . . . . . . . . . . . . . . . . 249
5.3.6 Numerical Examples . . . . . . . . . . . . . . . . . . 251
5.3.7 Functionally Graded Material Beams . . . . . . . . . . . 256
5.3.7.1 Material variation and sti ness coecients . . . . . . . 256
5.3.7.2 Equations of equilibrium . . . . . . . . . . . . . . 257
5.3.7.3 Finite element model . . . . . . . . . . . . . . . . 258
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 261
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 261
6 Two-Dimensional Problems Involving a Single Variable . 265
6.1 Model Equation . . . . . . . . . . . . . . . . . . . . . . 265
6.2 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . 266
6.3 Finite Element Model . . . . . . . . . . . . . . . . . . . . 268
6.4 Solution of Nonlinear Equations . . . . . . . . . . . . . . . 269
6.4.1 Direct Iteration Scheme . . . . . . . . . . . . . . . . . 269
6.4.2 Newton's Iteration Scheme . . . . . . . . . . . . . . . . 269
6.5 Axisymmetric Problems . . . . . . . . . . . . . . . . . . . 271
6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 271
6.5.2 Governing Equation and the Finite Element Model . . . . . 272
6.6 Computer Implementation . . . . . . . . . . . . . . . . . . 273
6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 273
6.6.2 Numerical Integration . . . . . . . . . . . . . . . . . . 273
6.6.3 Element Calculations . . . . . . . . . . . . . . . . . . 275
6.7 Time-Dependent Problems . . . . . . . . . . . . . . . . . . 282
6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 282
6.7.2 Semidiscretization . . . . . . . . . . . . . . . . . . . . 283
6.7.3 Full Discretization of Parabolic Equations . . . . . . . . . 284
6.7.3.1 Eigenvalue problem . . . . . . . . . . . . . . . . . 284
6.7.3.2 Time ( -family of) approximations . . . . . . . . . . 284
6.7.3.3 Fully discretized equations . . . . . . . . . . . . . . 286
6.7.3.4 Direct iteration scheme . . . . . . . . . . . . . . . 286
6.7.3.5 Newton's iteration scheme . . . . . . . . . . . . . . 287
6.7.3.6 Explicit and implicit formulations and mass lumping . . 287
6.7.4 Full Discretization of Hyperbolic Equations . . . . . . . . 289
6.7.4.1 Newmark's scheme . . . . . . . . . . . . . . . . . 289
6.7.4.2 Fully discretized equations . . . . . . . . . . . . . . 289
6.7.5 Stability and Accuracy . . . . . . . . . . . . . . . . . 292
6.7.5.1 Preliminary comments . . . . . . . . . . . . . . . . 292
6.7.5.2 Stability criteria . . . . . . . . . . . . . . . . . . 293
6.7.6 Computer Implementation . . . . . . . . . . . . . . . . 294
6.7.7 Numerical Examples . . . . . . . . . . . . . . . . . . 299
6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 306
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 306
7 Nonlinear Bending of Elastic Plates . . . . . . . . . . . 311
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 311
7.2 The Classical Plate Theory . . . . . . . . . . . . . . . . . 312
7.2.1 Assumptions of the Kinematics . . . . . . . . . . . . . . 312
7.2.2 Displacement and Strain Fields . . . . . . . . . . . . . . 312
7.3 Weak Formulation of the CPT . . . . . . . . . . . . . . . . 315
7.3.1 Virtual Work Statement . . . . . . . . . . . . . . . . . 315
7.3.2 Weak Forms . . . . . . . . . . . . . . . . . . . . . . 318
7.3.3 Equilibrium Equations . . . . . . . . . . . . . . . . . . 318
7.3.4 Boundary Conditions . . . . . . . . . . . . . . . . . . 319
7.3.4.1 The Kirchho free-edge condition . . . . . . . . . . . 320
7.3.4.2 Typical edge conditions . . . . . . . . . . . . . . . 321
7.3.5 Stress Resultant{De
ection Relations . . . . . . . . . . . 322
7.4 Finite Element Models of the CPT . . . . . . . . . . . . . . 324
7.4.1 General Formulation . . . . . . . . . . . . . . . . . . 324
7.4.2 Tangent Sti ness Coecients . . . . . . . . . . . . . . . 327
7.4.3 Non-Conforming and Conforming Plate Elements . . . . . . 331
7.5 Computer Implementation of the CPT Elements . . . . . . . . 333
7.5.1 General Remarks . . . . . . . . . . . . . . . . . . . . 333
7.5.2 Programming Aspects . . . . . . . . . . . . . . . . . . 335
7.5.3 Post-Computation of Stresses . . . . . . . . . . . . . . . 339
7.6 Numerical Examples using the CPT Elements . . . . . . . . . 340
7.6.1 Preliminary Comments . . . . . . . . . . . . . . . . . 340
7.6.2 Results of Linear Analysis . . . . . . . . . . . . . . . . 340
7.6.3 Results of Nonlinear Analysis . . . . . . . . . . . . . . . 344
7.7 The First-Order Shear Deformation Plate Theory . . . . . . . . 348
7.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 348
7.7.2 Displacement Field . . . . . . . . . . . . . . . . . . . 348
7.7.3 Weak Forms using the Principle of Virtual Displacements . . 349
7.7.4 Governing Equations . . . . . . . . . . . . . . . . . . 350
7.8 Finite Element Models of the FSDT . . . . . . . . . . . . . . 352
7.8.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . 352
7.8.2 The Finite Element Model . . . . . . . . . . . . . . . . 354
7.8.3 Tangent Sti ness Coecients . . . . . . . . . . . . . . . 356
7.8.4 Shear and Membrane Locking . . . . . . . . . . . . . . 358
7.9 Computer Implementation and Numerical Results of
the FSDT Elements . . . . . . . . . . . . . . . . . . . . . 359
7.9.1 Computer Implementation . . . . . . . . . . . . . . . . 359
7.9.2 Results of Linear Analysis . . . . . . . . . . . . . . . . 359
7.9.3 Results of Nonlinear Analysis . . . . . . . . . . . . . . . 363
7.10 Transient Analysis of the FSDT . . . . . . . . . . . . . . . 370
7.10.1 Equations of Motion . . . . . . . . . . . . . . . . . . . 370
7.10.2 The Finite Element Model . . . . . . . . . . . . . . . . 371
7.10.3 Time Approximation . . . . . . . . . . . . . . . . . . 374
7.10.4 Numerical Examples . . . . . . . . . . . . . . . . . . 375
7.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 378
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 379
8 Nonlinear Bending of Elastic Shells . . . . . . . . . . . 385
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 385
8.2 Governing Equations . . . . . . . . . . . . . . . . . . . . 387
8.2.1 Geometric Description . . . . . . . . . . . . . . . . . . 387
8.2.2 General Strain{Displacement Relations . . . . . . . . . . 392
8.2.3 Stress Resultants . . . . . . . . . . . . . . . . . . . . 394
8.2.4 Displacement and Strain Fields . . . . . . . . . . . . . . 395
8.2.5 Equations of Equilibrium . . . . . . . . . . . . . . . . 397
8.2.6 Shell Constitutive Relations . . . . . . . . . . . . . . . 399
8.3 Finite Element Formulation . . . . . . . . . . . . . . . . . 399
8.3.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . 399
8.3.2 Finite Element Model . . . . . . . . . . . . . . . . . . 400
8.3.3 Linear Analysis . . . . . . . . . . . . . . . . . . . . . 402
8.3.4 Nonlinear Analysis . . . . . . . . . . . . . . . . . . . 410
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 413
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 415
9 Finite Element Formulations of Solid Continua . . . . . . 417
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 417
9.1.1 Background . . . . . . . . . . . . . . . . . . . . . . 417
9.1.2 Summary of De nitions and Concepts from
Continuum Mechanics . . . . . . . . . . . . . . . . . . 418
9.1.3 Energetically-Conjugate Stresses and Strains . . . . . . . . 419
9.2 Various Strain and Stress Measures . . . . . . . . . . . . . . 421
9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 421
9.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . 422
9.2.3 Conservation of Mass . . . . . . . . . . . . . . . . . . 423
9.2.4 Green{Lagrange Strain Tensors . . . . . . . . . . . . . . 423
9.2.4.1 Green{Lagrange strain increment tensor . . . . . . . . 424
9.2.4.2 Updated Green{Lagrange strain tensor . . . . . . . . 424
9.2.5 Euler{Almansi Strain Tensor . . . . . . . . . . . . . . . 425
9.2.6 Relationships Between Various Stress Tensors . . . . . . . 426
9.2.7 Constitutive Equations . . . . . . . . . . . . . . . . . 426
9.3 Total Lagrangian and Updated Lagrangian Formulations . . . . . 429
9.3.1 Principle of Virtual Displacements . . . . . . . . . . . . 429
9.3.2 Total Lagrangian Formulation . . . . . . . . . . . . . . 430
9.3.2.1 Weak form . . . . . . . . . . . . . . . . . . . . 430
9.3.2.2 Incremental decompositions . . . . . . . . . . . . . 431
9.3.2.3 Linearization . . . . . . . . . . . . . . . . . . . . 432
9.3.3 Updated Lagrangian Formulation . . . . . . . . . . . . . 433
9.3.3.1 Weak form . . . . . . . . . . . . . . . . . . . . 433
9.3.3.2 Incremental decompositions . . . . . . . . . . . . . 435
9.3.3.3 Linearization . . . . . . . . . . . . . . . . . . . . 436
9.3.4 Some Remarks on the Formulations . . . . . . . . . . . . 437
9.4 Finite Element Models of 2-D Continua . . . . . . . . . . . . 439
9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 439
9.4.2 Total Lagrangian Formulation . . . . . . . . . . . . . . 439
9.4.3 Updated Lagrangian Formulation . . . . . . . . . . . . . 444
9.4.4 Computer Implementation . . . . . . . . . . . . . . . . 445
9.4.5 A Numerical Example . . . . . . . . . . . . . . . . . . 450
9.5 Conventional Continuum Shell Finite Element . . . . . . . . . 458
9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 458
9.5.2 Incremental Equations of Motion . . . . . . . . . . . . . 459
9.5.3 Finite Element Model of a Continuum . . . . . . . . . . . 460
9.5.4 Shell Finite Element . . . . . . . . . . . . . . . . . . . 462
9.5.5 Numerical Examples . . . . . . . . . . . . . . . . . . 468
9.5.5.1 Simply-supported orthotropic plate under uniform load . 468
9.5.5.2 Four-layer (0/90/90/0) clamped plate under
uniform load . . . . . . . . . . . . . . . . . . . . 469
9.5.5.3 Cylindrical shell roof under self-weight . . . . . . . . . 470
9.5.5.4 Simply-supported spherical shell under point load . . . . 471
9.5.5.5 Shallow cylindrical shell under point load . . . . . . . 472
9.6 A Re ned Continuum Shell Finite Element . . . . . . . . . . . 473
9.6.1 Backgound . . . . . . . . . . . . . . . . . . . . . . . 473
9.6.2 Representation of Shell Mid-Surface . . . . . . . . . . . . 474
9.6.3 Displacement and Strain Fields . . . . . . . . . . . . . . 478
9.6.4 Constitutive Relations . . . . . . . . . . . . . . . . . . 480
9.6.4.1 Isotropic and functionally graded shells . . . . . . . . 481
9.6.4.2 Laminated composite shells . . . . . . . . . . . . . 483
9.6.5 The Principle of Virtual Displacements and its Discretization . 486
9.6.6 The Spectral/hp Basis Functions . . . . . . . . . . . . . 488
9.6.7 Finite Element Model and Solution of Nonlinear Equations . . 491
9.6.7.1 The Newton procedure . . . . . . . . . . . . . . . 491
9.6.7.2 The cylindrical arc-length procedure . . . . . . . . . . 493
9.6.7.3 Element-level static condensation and
assembly of elements . . . . . . . . . . . . . . . . 495
9.6.8 Numerical Examples . . . . . . . . . . . . . . . . . . 497
9.6.8.1 A cantilevered plate strip under an end transverse load . 498
9.6.8.2 Post-buckling of a plate strip under axial compressive load 500
9.6.8.3 An annular plate with a slit under an end transverse load 501
9.6.8.4 A cylindrical panel subjected to a point load . . . . . . 504
9.6.8.5 Pull-out of an open-ended cylindrical shell . . . . . . . 509
9.6.8.6 A pinched half-cylindrical shell . . . . . . . . . . . . 512
9.6.8.7 A pinched cylinder with rigid diaphragms . . . . . . . 513
9.6.8.8 A pinched hemisphere with an 18 hole . . . . . . . . 515
9.6.8.9 A pinched composite hyperboloidal shell . . . . . . . . 517
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 521
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 522
10 Weak-Form Finite Element Models of Flows of
Viscous Incompressible Fluids . . . . . . . . . . . . . . 523
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 523
10.2 Governing Equations . . . . . . . . . . . . . . . . . . . . 524
10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 524
10.2.2 Equation of Mass Continuity . . . . . . . . . . . . . . . 525
10.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . 525
10.2.4 Energy Equation . . . . . . . . . . . . . . . . . . . . 526
10.2.5 Constitutive Equations . . . . . . . . . . . . . . . . . 526
10.2.6 Boundary Conditions . . . . . . . . . . . . . . . . . . 527
10.3 Summary of Governing Equations . . . . . . . . . . . . . . . 529
10.3.1 Vector Form . . . . . . . . . . . . . . . . . . . . . . 529
10.3.2 Cartesian Component Form . . . . . . . . . . . . . . . 529
10.4 Velocity{Pressure Finite Element Model . . . . . . . . . . . . 530
10.4.1 Weak Forms . . . . . . . . . . . . . . . . . . . . . . 530
10.4.2 Semidiscrete Finite Element Model . . . . . . . . . . . . 532
10.4.3 Fully Discretized Finite Element Model . . . . . . . . . . 534
10.5 Penalty Finite Element Models . . . . . . . . . . . . . . . . 535
10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 535
10.5.2 Penalty Function Method . . . . . . . . . . . . . . . . 536
10.5.3 Reduced Integration Penalty Model . . . . . . . . . . . . 538
10.5.4 Consistent Penalty Model . . . . . . . . . . . . . . . . 539
10.6 Computational Aspects . . . . . . . . . . . . . . . . . . . 540
10.6.1 Properties of the Finite Element Equations . . . . . . . . . 540
10.6.2 Choice of Elements . . . . . . . . . . . . . . . . . . . 541
10.6.3 Evaluation of Element Matrices in Penalty Models . . . . . 543
10.6.4 Post-Computation of Pressure and Stresses . . . . . . . . . 544
10.7 Computer Implementation . . . . . . . . . . . . . . . . . . 545
10.7.1 Mixed Model . . . . . . . . . . . . . . . . . . . . . . 545
10.7.2 Penalty Model . . . . . . . . . . . . . . . . . . . . . 549
10.7.3 Transient Analysis . . . . . . . . . . . . . . . . . . . 552
10.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . 552
10.8.1 Preliminary Comments . . . . . . . . . . . . . . . . . 552
10.8.2 Linear Problems . . . . . . . . . . . . . . . . . . . . 552
10.8.3 Nonlinear Problems . . . . . . . . . . . . . . . . . . . 561
10.8.4 Transient Analysis . . . . . . . . . . . . . . . . . . . 567
10.9 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . 570
10.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 570
10.9.2 Governing Equations in Cylindrical Coordinates . . . . . . 570
10.9.3 Power-Law Fluids . . . . . . . . . . . . . . . . . . . . 572
10.9.4 White{Metzner Fluids . . . . . . . . . . . . . . . . . . 574
10.9.5 Numerical Examples . . . . . . . . . . . . . . . . . . 578
10.10 Coupled Fluid Flow and Heat Transfer . . . . . . . . . . . . 582
10.10.1 Finite Element Models . . . . . . . . . . . . . . . . . . 582
10.10.2 Numerical Examples . . . . . . . . . . . . . . . . . . 583
10.10.2.1 Heated cavity . . . . . . . . . . . . . . . . . . . 583
10.10.2.2 Solar receiver . . . . . . . . . . . . . . . . . . . . 584
10.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 587
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 587
11 Least-Squares Finite Element Models of Flows of
Viscous Incompressible Fluids . . . . . . . . . . . . . . 589
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 589
11.2 Least-Squares Finite Element Formulation . . . . . . . . . . . 593
11.2.1 The Navier{Stokes Equations of Incompressible Fluids . . . . 593
11.2.2 Numerical Examples . . . . . . . . . . . . . . . . . . 595
11.2.2.1 Low Reynolds number
ow past a circular cylinder . . . 595
11.2.2.2 Steady
ow over a backward facing step . . . . . . . . 600
11.2.2.3 Lid-driven cavity
ow . . . . . . . . . . . . . . . . 604
11.3 A Least-Squares Finite Element Model with Enhanced
Element-Level Mass Conservation . . . . . . . . . . . . . . . 607
11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 607
11.3.2 Unsteady Flows . . . . . . . . . . . . . . . . . . . . . 608
11.3.2.1 The velocity{pressure{vorticity rst-order system . . . . 609
11.3.2.2 Temporal discretization . . . . . . . . . . . . . . . 609
11.3.2.3 The standard L2-norm based least-squares model . . . . 610
11.3.2.4 A modi ed L2-norm based least-squares model with
improved element-level mass conservation . . . . . . . 611
11.3.3 Numerical Examples: Veri cation Problems . . . . . . . . 613
11.3.3.1 Steady Kovasznay
ow . . . . . . . . . . . . . . . 613
11.3.3.2 Steady
ow in a 1 2 rectangular cavity . . . . . . . 616
11.3.3.3 Steady
ow past a large cylinder in a narrow channel . . 619
11.3.3.4 Unsteady
ow past a circular cylinder . . . . . . . . . 621
11.3.3.5 Unsteady
ow past a large cylinder in a narrow channel . 627
11.4 Summary and Future Direction . . . . . . . . . . . . . . . . 632
Problems . . . . . . . . . . . . . . . . . . . . . . . . . 635
Appendix 1: Solution Procedures for Linear Equations . . . . 637
A1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 637
A1.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . 639
A1.2.1 Preliminary Comments . . . . . . . . . . . . . . . . . 639
A1.2.2 Symmetric Solver . . . . . . . . . . . . . . . . . . . . 640
A1.2.3 Unsymmetric Solver . . . . . . . . . . . . . . . . . . . 642
A1.3 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . 644
Appendix 2: Solution Procedures for Nonlinear Equations . . 645
A2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 645
A2.2 The Picard Iteration Method . . . . . . . . . . . . . . . . . 646
A2.3 The Newton Iteration Method . . . . . . . . . . . . . . . . 650
A2.4 The Riks and Modi ed Riks Methods . . . . . . . . . . . . . 654
References . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
