Computational Continuum Mechanics / Edition 1

by Ahmed A. Shabana

Overview

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 then a formulation that can efficiently solve small deformation problems that characterize very and moderately 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.

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

Product Details

ISBN-13:	9780521885690
Publisher:	Cambridge University Press
Publication date:	03/10/2008
Pages:	348
Product dimensions:	7.24(w) x 10.24(h) x 0.98(d)

About the Author

Ahmed A. Shabana, PhD, is University Distinguished Professor and the Richard and Loan Hill Professor of Engineering at the University of Illinois at Chicago. Professor Shabana is the author of several books and serves on the Editorial Board of several journals. He served as the Chair of the ASME Design Engineering Division, the Founding Chair of the ASME Technical Committee on Multibody Systems and Nonlinear Dynamics, and the Founding Chair of the ASME International Conference of Multibody Systems, Nonlinear Dynamics, and Control. Professor Shabana is a Fellow of the American Society of Mechanical Engineers (ASME) and a Fellow of the Society of Automotive Engineering (SAE International). He has received several awards, including the Humboldt Prize, the Fulbright Research Scholar Award, the ASME D'Alembert Award, and Honorary Doctorate, Honorary Professorship, and Best Paper Awards, as well as several teaching and research awards from the University of Illinois at Chicago.

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Table of Contents

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