Introduction to Mathematical Elasticity

by Lebedev, Leonid P.; Cloud, Michael J.

• ISBN13: 9789814273725
• ISBN10: 9814273724
• Format: Hardcover
• Copyright: 2009-09-03
• Publisher: World Scientific Pub Co Inc

Summary

This book provides the general reader with an introduction to mathematical elasticity, by means of general concepts in classic mechanics, and models for elastic springs, strings, rods, beams and membranes. Functional analysis is also used to explore more general boundary value problems for three-dimensional elastic bodies, where the reader is provided, For each problem considered, a description of the deformation; the equilibrium in terms of stresses; the constitutive equation; the equilibrium equation in terms of displacements; formulation of boundary value problems; and variational principles, generalized solutions and conditions for solvability. Introduction to Mathematical Elasticity will also be of essential reference to engineers specializing in elasticity, and to mathematicians working on abstract formulations of the related boundary value problems.

Table of Contents

Foreword	p. v
Preface	p. vii
Some Notation	p. xi
Models and Ideas of Classical Mechanics	p. 1
Orientation	p. 1
Some Words on the Fundamentals of Our Subject	p. 2
Metric Spaces and Spaces of Particles	p. 4
Vectors and Vector Spaces	p. 8
Normed Spaces and Inner Product Spaces	p. 11
Forces	p. 16
Equilibrium and Motion of a Rigid Body	p. 21
D' Alembert's Principle	p. 23
The Motion of a System of Particles	p. 25
The Rigid Body	p. 31
Motion of a System of Particles; Comparison of Trajectories; Notion of Operator	p. 33
Matrix Operators and Matrix Equations	p. 40
Complete Spaces	p. 44
Completion Theorem	p. 48
Lebesgue Integration and the Lp Spaces	p. 54
Orthogonal Decomposition of Hilbert Space	p. 60
Work and Energy	p. 63
Virtual Work Principle	p. 67
Lagrange's Equations of the Second Kind	p. 70
Problem of Minimum of a Functional	p. 74
Hamilton's Principle	p. 83
Energy Conservation Revisited	p. 85
Simple Elastic Models	p. 89
Introduction	p. 89
Two Main Principles of Equilibrium and Motion for Bodies in Continuum Mechanics	p. 89
Equilibrium of a Spring	p. 91
Equilibrium of a String	p. 95
Equilibrium Boundary Value Problems for a String	p. 100
Generalized Formulation of the Equilibrium Problem for a String	p. 105
Virtual Work Principle for a String	p. 108
Riesz Representation Theorem	p. 112
Generalized Setup of the Dirichlet Problem for a String	p. 115
First Theorems of Imbedding	p. 116
Generalized Setup of the Dirichlet Problem for a String, Continued	p. 120
Neumann Problem for the String	p. 122
The Generalized Solution of Linear Mechanical Problems and the Principle of Minimum Total Energy	p. 126
Nonlinear Model of a Membrane	p. 128
Linear Membrane Theory: Poisson's Equation	p. 131
Generalized Setup of the Dirichlet Problem for a Linear Membrane	p. 132
Other Membrane Equilibrium Problems	p. 145
Banach's Contraction Mapping Principle	p. 151
Theory of Elasticity: Statics and Dynamics	p. 157
Introduction	p. 157
An Elastic Bar Under Stretching	p. 158
Bending of a beam	p. 168
Generalized Solutions to the Equilibrium Problem for a Beam	p. 175
Generalized Setup: Rough Qualitative Discussion	p. 179
Pressure and Stresses	p. 181
Vectors and Tensors	p. 188
The Cauchy Stress Tensor, Continued	p. 196
Basic Tensor Calculus in Curvilinear Coordinates	p. 202
Euler and Lagrange Descriptions of Continua	p. 207
Strain Tensors	p. 208
The Virtual Work Principle	p. 214
Hooke's Law in Three Dimensions	p. 218
The Equilibrium Equations of Linear Elasticity in Displacements	p. 221
Virtual Work Principle in Linear Elasticity	p. 224
Generalized Setup of Elasticity Problems	p. 227
Existence Theorem for an Elastic Body	p. 231
Equilibrium of a Free Elastic Body	p. 232
Variational Methods for Equilibrium Problems	p. 235
A Brief but Important Remark	p. 243
Countable Sets and Separable Spaces	p. 243
Fourier Series	p. 245
Problem of Vibration for Elastic Structures	p. 249
Self-Adjointness of A and Its Consequences	p. 252
Compactness of A	p. 255
Riesz-Fredholm Theory for a Linear, Self-Adjoint, Compact Operator in a Hilbert Space	p. 262
Weak Convergence in Hilbert Space	p. 267
Completeness of the System of Eigenvectors of a Self-Adjoint, Compact, Strictly Positive Linear Operator	p. 272
Other Standard Models of Elasticity	p. 277
Hints for Selected Exercises	p. 281
Bibliography	p. 293
Index	p. 295
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