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E-raamat: Introduction To Mathematical Elasticity

(Lawrence Technological Univ, Usa), (National Univ Of Colombia, Colombia)
  • Formaat: 316 pages
  • Ilmumisaeg: 03-Sep-2009
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814467797
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Introduction To Mathematical ElasticitySuurem pilt
  • Formaat: 316 pages
  • Ilmumisaeg: 03-Sep-2009
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814467797
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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.
Foreword v
Preface vii
Some Notation xi
1. Models and Ideas of Classical Mechanics 1
1.1 Orientation
1
1.2 Some Words on the Fundamentals of Our Subject
2
1.3 Metric Spaces and Spaces of Particles
4
1.4 Vectors and Vector Spaces
8
1.5 Normed Spaces and Inner Product Spaces
11
1.6 Forces
16
1.7 Equilibrium and Motion of a Rigid Body
21
1.8 D'Alembert's Principle
23
1.9 The Motion of a System of Particles
25
1.10 The Rigid Body
31
1.11 Motion of a System of Particles; Comparison of Trajectories; Notion of Operator
33
1.12 Matrix Operators and Matrix Equations
40
1.13 Complete Spaces
44
1.14 Completion Theorem
48
1.15 Lebesgue Integration and the Lp Spaces
54
1.16 Orthogonal Decomposition of Hilbert Space
60
1.17 Work and Energy
63
1.18 Virtual Work Principle
67
1.19 Lagrange's Equations of the Second Kind
70
1.20 Problem of Minimum of a Functional
74
1.21 Hamilton's Principle
83
1.22 Energy Conservation Revisited
85
2. Simple Elastic Models 89
2.1 Introduction
89
2.2 Two Main Principles of Equilibrium and Motion for Bodies in Continuum Mechanics
89
2.3 Equilibrium of a Spring
91
2.4 Equilibrium of a String
95
2.5 Equilibrium Boundary Value Problems for a String
100
2.6 Generalized Formulation of the Equilibrium Problem for a String
105
2.7 Virtual Work Principle for a String
108
2.8 Riesz Representation Theorem
112
2.9 Generalized Setup of the Dirichlet Problem for a String
115
2.10 First Theorems of Imbedding
116
2.11 Generalized Setup of the Dirichlet Problem for a String, Continued
120
2.12 Neumann Problem for the String
122
2.13 The Generalized Solution of Linear Mechanical Problems and the Principle of Minimum Total Energy
126
2.14 Nonlinear Model of a Membrane
128
2.15 Linear Membrane Theory: Poisson's Equation
131
2.16 Generalized Setup of the Dirichlet Problem for a Linear Membrane
132
2.17 Other Membrane Equilibrium Problems
145
2.18 Banach's Contraction Mapping Principle
151
3. Theory of Elasticity: Statics and Dynamics 157
3.1 Introduction
157
3.2 An Elastic Bar Under Stretching
158
3.3 Bending of a beam
168
3.4 Generalized Solutions to the Equilibrium Problem for a Beam
175
3.5 Generalized Setup: Rough Qualitative Discussion
179
3.6 Pressure and Stresses
181
3.7 Vectors and Tensors
188
3.8 The Cauchy Stress Tensor, Continued
196
3.9 Basic Tensor Calculus in Curvilinear Coordinates
202
3.10 Euler and Lagrange Descriptions of Continua
207
3.11 Strain Tensors
208
3.12 The Virtual Work Principle
214
3.13 Hooke's Law in Three Dimensions
218
3.14 The Equilibrium Equations of Linear Elasticity in Displacements
221
3.15 Virtual Work Principle in Linear Elasticity
224
3.16 Generalized Setup of Elasticity Problems
227
3.17 Existence Theorem for an Elastic Body
231
3.18 Equilibrium of a Free Elastic Body
232
3.19 Variational Methods for Equilibrium Problems
235
3.20 A Brief but Important Remark
243
3.21 Countable Sets and Separable Spaces
243
3.22 Fourier Series
245
3.23 Problem of Vibration for Elastic Structures
249
3.24 Self-Adjointness of A and Its Consequences
252
3.25 Compactness of A
255
3.26 Riesz Fredholm Theory for a Linear, Seif-Adjoint, Compact Operator in a Hilbert Space
262
3.27 Weak Convergence in Hilbert Space
267
3.28 Completeness of the System of Eigenvectors of a Self-Adjoint, Compact, Strictly Positive Linear Operator
272
3.29 Other Standard Models of Elasticity
277
Appendix A Hints for Selected Exercises 281
Bibliography 293
Index 295
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