Nonlinear Finite Elements for Continua and Structures

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Format: Paperback
Pub. Date: 2000-10-03
Publisher(s): WILEY
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Summary

Nonlinear Finite Elements for Continua and Structures Ted Belytschko, Wing Kam Liu, Brian Moran Northwestern University, Evanston, Illinois This book provides a comprehensive description of the major methodologies of nonlinear finite element analysis for solid mechanics, as applied to continua and structures. Treatment of the subject is integrated in such a way that the reader can gain an understanding of the fundamental methods, a feeling for the comparative usefulness of different approaches and an appreciation of the difficulties inherent in nonlinear analysis. In-depth coverage of the following is given: * Lagrangian and arbitrary Lagrangian Eulerian treatments of continua * many of the material laws used in today's software and research * solution methods, including explicit and implicit time integration methods and methods for equilibrium problems * basic concepts such as stability and smoothness, and techniques such as linearization and regularization * methods for shells and structures * contact-impact problems * element technology, including multi-field elements Ideal for self-study, no other book provides such a comprehensive description of nonlinear finite element analysis for solid mechanics. This is an invaluable reference not only for final year undergraduates, postgraduates, academics and engineers working on sophisticated finite element software and in the field of solid mechanics, but also for all users of nonlinear finite element programs.

Table of Contents

Preface xi
List of Boxes
xv
Introduction
1(18)
Nonlinear finite elements in design
1(3)
Related books and a brief history of nonlinear finite elements
4(3)
Notation
7(2)
Mesh descriptions
9(4)
Classification of partial differential equations
13(5)
Exercises
18(1)
Lagrangian and Eulerian finite elements in one dimension
19(56)
Introduction
19(1)
Governing equations for total Lagrangian formulation
20(7)
Weak form for total Lagrangian formulation
27(6)
Finite element discretization in total Lagrangian formulation
33(5)
Element and global matrices
38(10)
Governing equations for updated Lagrangian formulation
48(3)
Weak form for updated Lagrangian formulation
51(1)
Element equations for updated Lagrangian formulation
52(12)
Governing equations for Eulerian formulation
64(1)
Weak forms for Eulerian mesh equations
65(1)
Finite element equations
66(4)
Solution methods
70(2)
Summary
72(1)
Exercises
72(3)
Continuum mechanics
75(66)
Introduction
75(1)
Deformation and motion
76(16)
Strain measures
92(9)
Stress measures
101(7)
Conservation equations
108(11)
Lagrangian conservation equations
119(6)
Polar decomposition and frame-invariance
125(12)
Exercises
137(4)
Lagrangian meshes
141(74)
Introduction
141(1)
Governing equations
142(3)
Weak form: principle of virtual power
145(7)
Updated Lagrangian finite element discretization
152(10)
Implementation
162(23)
Corotational formulations
185(8)
Total Lagrangian formulation
193(3)
Total Lagrangian weak form
196(2)
Finite element semidiscretization
198(15)
Exercise
213(2)
Constitutive models
215(94)
Introduction
215(1)
The stress-strain curve
216(5)
One-dimensional elasticity
221(4)
Nonlinear elasticity
225(15)
One-dimensional plasticity
240(7)
Multiaxial plasticity
247(17)
Hyperelastic-plastic models
264(10)
Viscoelasticity
274(3)
Stress update algorithms
277(17)
Continuum mechanics and constitutive models
294(14)
Exercises
308(1)
Solution methods and stability
309(84)
Introduction
309(1)
Explicit methods
310(7)
Equilibrium solutions and implicit time integration
317(20)
Linearization
337(16)
Stability and continuation methods
353(16)
Numerical stability
369(15)
Material stability
384(8)
Exercises
392(1)
Arbitrary Lagrangian Eulerian formulations
393(58)
Introduction
393(2)
ALE continuum mechanics
395(7)
Conservation laws in ALE description
402(1)
ALE governing equations
403(1)
Weak forms
404(4)
Introduction to the Petrov-Galerkin method
408(9)
Petrov-Galerkin formulation of momentum equation
417(3)
Path-dependent materials
420(12)
Linearization of the discrete equations
432(3)
Mesh update equations
435(7)
Numerical example: an elastic-plastic wave propagation problem
442(1)
Total ALE formulations
443(8)
Element technology
451(58)
Introduction
451(2)
Element performance
453(8)
Element properties and patch tests
461(8)
Q4 and volumetric locking
469(5)
Multi-field weak forms and elements
474(13)
Multi-field quadrilaterals
487(4)
One-point quadrature elements
491(9)
Examples
500(4)
Stability
504(3)
Exercises
507(2)
Beams and shells
509(60)
Introduction
509(2)
Beam theories
511(3)
Continuum-based beam
514(10)
Analysis of CB beam
524(12)
Continuum-based shell implementation
536(14)
CB shell theory
550(5)
Shear and membrane locking
555(5)
Assumed strain elements
560(3)
One-point quadrature elements
563(3)
Exercises
566(3)
Contact-impact
569(46)
Introduction
569(1)
Contact interface equations
570(10)
Friction models
580(5)
Weak forms
585(10)
Finite element discretization
595(14)
On explicit methods
609(6)
Appendix 1 Voigt notation 615(4)
Appendix 2 Norms 619(3)
Appendix 3 Element shape functions 622(5)
Glossary 627(4)
References 631(10)
Index 641

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