Integrodifferential Relations in Linear Elasticity (eBook)

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Georgy V. Kostin and Vasily V. Saurin, Ishlinsky Institute for Problems in Mechanics, Russia.

Preface 5
1 Introduction 13
2 Basic concepts of the linear theory of elasticity 18
2.1 Stresses 18
2.2 Linearstrains 25
2.3 Constitutive relations 31
2.4 Boundary value problems 38
2.4.1 Static statements 38
2.4.2 Dynamic problems 42
2.5 Simplified models 43
2.5.1 Elastic rods and strings 43
2.5.2 Beam models 47
2.5.3 Membranes 50
2.5.4 Plane stress and strain states 52
3 Conventional variational principles 55
3.1 Classical variational approaches 55
3.1.1 Energy relations 55
3.1.2 Direct principles 56
3.1.3 Complementary principles 59
3.2 Variational principles in dynamics 61
3.3 Generalized variational principles 67
3.3.1 Relations among variational principles 67
3.3.2 Semi-inverse approach 72
3.4 Finite dimensional discretization 73
3.4.1 Ritz method 74
3.4.2 Galerkin method 76
3.4.3 Finite element method 78
3.4.4 Boundary element method 84
4 The method of integrodifferential relations 85
4.1 Basic ideas 85
4.1.1 Analytical solutions in linear elasticity 85
4.1.2 Integral formulation of Hooke's law 90
4.2 Family of quadratic functionals 93
4.3 Ritz method in the MIDR 95
4.3.1 Algorithm of polynomial approximations 95
4.3.2 2D clamped plate - static case 97
4.4 2D natural vibrations 102
4.4.1 Eigenvalue problem 102
4.4.2 Free vibrations of circular and elliptic membranes 105
5 Variational properties of the integrodifferential statements 113
5.1 Variational principles for quadratic functionals 113
5.2 Relations with the conventional principles 115
5.3 Bilateral energy estimates 118
5.4 Body on an elastic foundation 126
5.4.1 Variational principle for the energy error functional 126
5.4.2 Bilateral estimates 130
6 Advance finite element technique 136
6.1 Piecewise polynomial approximations 136
6.2 Smooth polynomial splains 139
6.2.1 Argyris triangle 139
6.2.2 Stiffness matrix for the Argyris triangle 144
6.2.3 C2 approximations for a triangle element 145
6.3 Finite element technique in linear elasticity problems 148
6.4 Mesh adaptation and mesh refinement 157
7 Semi-discretization and variational technique 170
7.1 Reduction of PDE system to ODEs 170
7.1.1 Beam-oriented notation 170
7.1.2 Semi-discretization in the displacements 172
7.1.3 Semi-discretization in the stresses 174
7.2 Analysis of beam stress-strain state 178
7.3 2D elastic beam vibrations 182
8 An asymptotic approach 189
8.1 Classical variational approach 189
8.2 Integrodifferential approach 194
8.2.1 Basic ideas of asymptotic approximations 194
8.2.2 Beam equations - general case of loading 199
8.3 Elastic beam vibrations 202
8.3.1 Statement of an eigenvalue problem 202
8.3.2 Longitudinal vibrations 206
8.3.3 Lateral vibrations 211
8.4 3D static problem 216
9 A projection approach 230
9.1 Projection formulation of linear elasticity problems 230
9.2 Projections vs. variations and asymptotics 234
10 3D static beam modeling 239
10.1 Projection algorithms 239
10.2 Cantilever beam with the triangular cross section 253
10.3 Projection beam model 258
10.4 Characteristics of a beam with the triangular cross section 260
11 3D beam vibrations 264
11.1 Integral projections in eigenvalue problems 264
11.2 Natural vibrations of a beam with the triangular cross section 267
11.3 Forced vibrations of a beam with the triangular cross section 278
A Vectors and tensors 281
B Sobolev spaces 283
Bibliography 286
Index 291

lt;P>"The aim of this very important book is the study of problems in solid mechanics whose variational forms are equalities, expressing the principle of virtual power in its equality form. This book is unique in that it presents a profound mathematical analysis of general elasticity of deformed bodies, including the resulting interior stresses and displacements." Mathematical Reviews

"This well-written and good-organized monograph can be recommended to highly-qualified experts in the field." Zentralblatt für Mathematik