Computational Continuum Mechanics

by Ahmed A. Shabana

• ISBN13: 9780521885690
• ISBN10: 0521885698
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-03-10
• Publisher: Cambridge University Press

Summary

This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and first-year graduate students. It is also a reference for researchers, practicing engineers, and scientists working in computational mechanics, bio-mechanics, computational biology, multibody system dynamics, and other fields of science and engineering using the general continuum mechanics theory.

Author Biography

Ahmed A. Shabana is the Richard and Loan Hill Professor of Engineering at the University of Illinois, Chicago.

Table of Contents

Preface	p. ix
Introduction	p. 1
Matrices	p. 2
Vectors	p. 6
Summation Convention	p. 12
Cartesian Tensors	p. 13
Polar Decomposition Theorem	p. 25
D'Alembert's Principle	p. 27
Virtual Work Principle	p. 34
Approximation Methods	p. 37
Discrete Equations	p. 40
Momentum, Work, and Energy	p. 43
Parameter Change and Coordinate Transformation	p. 45
Problems	p. 48
Kinematics	p. 51
Motion Description	p. 52
Strain Components	p. 60
Other Deformation Measures	p. 67
Decomposition of Displacement	p. 69
Velocity and Acceleration	p. 71
Coordinate Transformation	p. 75
Objectivity	p. 82
Change of Volume and Area	p. 85
Continuity Equation	p. 89
Reynolds' Transport Theorem	p. 90
Examples of Deformation	p. 92
Problems	p. 100
Forces and Stresses	p. 103
Equilibrium of Forces	p. 103
Transformation of Stresses	p. 106
Equations of Equilibrium	p. 107
Symmetry of the Cauchy Stress Tensor	p. 109
Virtual Work of the Forces	p. 111
Deviatoric Stresses	p. 120
Stress Objectivity	p. 123
Energy Balance	p. 127
Problems	p. 129
Constitutive Equations	p. 131
Generalized Hooke's Law	p. 132
Anisotropic Linearly Elastic Materials	p. 134
Material Symmetry	p. 135
Homogeneous Isotropic Material	p. 137
Principal Strain Invariants	p. 144
Special Material Models for Large Deformations	p. 146
Linear Viscoelasticity	p. 150
Nonlinear Viscoelasticity	p. 164
A Simple Viscoelastic Model for Isotropic Materials	p. 171
Fluid Constitutive Equations	p. 173
Navier-Stokes Equations	p. 174
Problems	p. 175
Plasticity Formulations	p. 177
One-Dimensional Problem	p. 179
Loading and Unloading Conditions	p. 180
Solution of the Plasticity Equations	p. 181
Generalization of the Plasticity Theory: Small Strains	p. 190
J[subscript 2] Flow Theory with Isotropic/Kinematic Hardening	p. 197
Nonlinear Formulation for Hyperelastic-Plastic Materials	p. 214
Hyperelastic-Plastic J[subscript 2] Flow Theory	p. 225
Problems	p. 230
Finite Element Formulation: Large-Deformation, Large-Rotation Problem	p. 231
Displacement Field	p. 233
Element Connectivity	p. 240
Inertia and Elastic Forces	p. 243
Equations of Motion	p. 246
Numerical Evaluation of the Elastic Forces	p. 250
Finite Elements and Geometry	p. 256
Two-Dimensional Euler-Bernoulli Beam Element	p. 263
Two-Dimensional Shear Deformable Beam Element	p. 267
Three-Dimensional Cable Element	p. 269
Three-Dimensional Beam Element	p. 270
Thin-Plate Element	p. 272
Higher-Order Plate Element	p. 274
Element Performance	p. 275
Other Finite Element Formulations	p. 280
Updated Lagrangian and Eulerian Formulations	p. 282
Problems	p. 284
Finite Element Formulation: Small-Deformation, Large-Rotation Problem	p. 286
Background	p. 287
Rotation and Angular Velocity	p. 291
Floating Frame of Reference	p. 296
Intermediate Element Coordinate System	p. 297
Connectivity and Reference Conditions	p. 300
Kinematic Equations	p. 306
Formulation of the Inertia Forces	p. 307
Elastic Forces	p. 311
Equations of Motion	p. 313
Coordinate Reduction	p. 314
Integration of Finite Element and Multibody System Algorithms	p. 317
Problems	p. 319
References	p. 321
Index	p. 327
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