Nonlinear Finite Elements for Continua and Structures

by Belytschko, Ted; Liu, Wing Kam; Moran, Brian

• ISBN13: 9780471987741
• ISBN10: 0471987743
• Format: Paperback
• Copyright: 2000-10-03
• Publisher: WILEY

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