| Preface | 5 |
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| 1 Introduction | 13 |
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| 2 Basic concepts of the linear theory of elasticity | 18 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 9.1 Projection formulation of linear elasticity problems | 230 |
| 9.2 Projections vs. variations and asymptotics | 234 |
| 10 3D static beam modeling | 239 |
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| 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 |
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| 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 |
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| B Sobolev spaces | 283 |
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| Bibliography | 286 |
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| Index | 291 |
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