Elasticity with mathematica : an introduction to continuum mechanics and linear elasticity

Title:

Personal Author:

Constantinescu, Andrei

Publication Information:

Cambridge, UK : Cambridge University Press, 2007

Physical Description:

ix, 255 p. : ill. ; 26 cm.

ISBN:

9780521842013

Subject Term:

Elasticity -- Mathematical models

Added Author:

Korsunsky, Alexander

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010209505	TA418 C66 2007	Open Access Book	Book	Searching... Unknown

This book introduces key ideas and principles in the theory of elasticity with the help of symbolic computation. Differential and integral operators on vector and tensor fields of displacements, strains and stresses are considered on a consistent and rigorous basis with respect to curvilinear orthogonal coordinate systems. As a consequence, vector and tensor objects can be manipulated readily, and fundamental concepts can be illustrated and problems solved with ease. The method is illustrated using a variety of plane and three-dimensional elastic problems. General theorems, fundamental solutions, displacements and stress potentials are presented and discussed. The Rayleigh-Ritz method for obtaining approximate solutions is introduced for elastostatic and spectral analysis problems. Containing more than 60 exercises and solutions in the form of Mathematica notebooks that accompany every chapter, the reader can learn and master the techniques while applying them to a large range of practical and fundamental problems.

Author Notes

Andrei Constantinescu is currently Directeur de Recherche at CNRS: the French National Center for Scientific Research in the Laboratoire de Mecanique des Sol ides, and Associated Professor at Ecole Polytechnique, Palaiseau, near Paris
Alexander Korsunsky is currently Professor in the Department of Engineering Science, University of Oxford

Acknowledgments	p. ix
Introduction	p. 1
Motivation	p. 1
What will and will not be found in this book	p. 4
1 Kinematics: displacements and strains	p. 8
Outline	p. 8
1.1 Particle motion: trajectories and streamlines	p. 8
1.2 Strain	p. 19
1.3 Small strain tensor	p. 28
1.4 Compatibility equations and integration of small strains	p. 29
Summary	p. 35
Exercises	p. 35
2 Dynamics and statics: stresses and equilibrium	p. 41
Outline	p. 41
2.1 Forces and momenta	p. 41
2.2 Virtual power and the concept of stress	p. 42
2.3 The stress tensor according to Cauchy	p. 46
2.4 Potential representations of self-equilibrated stress tensors	p. 48
Summary	p. 50
Exercises	p. 50
3 Linear elasticity	p. 56
Outline	p. 56
3.1 Linear elasticity	p. 56
3.2 Matrix representation of elastic coefficients	p. 58
3.3 Material symmetry	p. 65
3.4 The extension experiment	p. 72
3.5 Further properties of isotropic elasticity	p. 75
3.6 Limits of linear elasticity	p. 78
Summary	p. 80
Exercises	p. 80
4 General principles in problems of elasticity	p. 86
Outline	p. 86
4.1 The complete elasticity problem	p. 86
4.2 Displacement formulation	p. 88
4.3 Stress formulation	p. 89
4.4 Example: spherical shell under pressure	p. 91
4.5 Superposition principle	p. 94
4.6 Quasistatic deformation and the virtual work theorem	p. 95
4.7 Uniqueness of solution	p. 95
4.8 Energy potentials	p. 96
4.9 Reciprocity theorems	p. 99
4.10 The Saint Venant principle	p. 101
Summary	p. 109
Exercises	p. 109
5 Stress functions	p. 116
Outline	p. 116
5.1 Plane stress	p. 116
5.2 Airy stress function of the form A[subscript 0](x, y)	p. 119
5.3 Airy stress function with a corrective term: A[subscript 0](x, y) - z[superscript 2]A[subscript 1](x, y)	p. 122
5.4 Plane strain	p. 124
5.5 Airy stress function of the form A[subscript 0]([gamma], [theta])	p. 126
5.6 Biharmonic functions	p. 126
5.7 The disclination, dislocations, and associated solutions	p. 130
5.8 A wedge loaded by a concentrated force applied at the apex	p. 133
5.9 The Kelvin problem	p. 137
5.10 The Williams eigenfunction analysis	p. 139
5.11 The Kirsch problem: stress concentration around a circular hole	p. 145
5.12 The Inglis problem: stress concentration around an elliptical hole	p. 147
Summary	p. 152
Exercises	p. 152
6 Displacement potentials	p. 157
Outline	p. 157
6.1 Papkovich-Neuber potentials	p. 158
6.2 Galerkin vector	p. 182
6.3 Love strain function	p. 183
Summary	p. 186
Exercises	p. 187
7 Energy principles and variational formulations	p. 189
Outline	p. 189
7.1 Strain energy and complementary energy	p. 189
7.2 Extremum theorems	p. 192
7.3 Approximate solutions for problems of elasticity	p. 196
7.4 The Rayleigh-Ritz method	p. 197
7.5 Extremal properties of free vibrations	p. 204
Summary	p. 212
Exercises	p. 212
Appendix 1 Differential operators	p. 219
Appendix 2 Mathematica tricks	p. 235
Appendix 3 Plotting parametric meshes	p. 243
Bibliography	p. 249
Index	p. 251

Available:*

On Order

Summary

Summary

Author Notes

Table of Contents