| Preface |
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xiii | |
| Foreword |
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xv | |
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1 | (90) |
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1.1 Bifurcation and instability to explain pattern formation |
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2 | (4) |
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1.2 Bifurcations in elasticity: The elastic cylinder |
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6 | (2) |
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1.3 Bifurcations in elastoplasticity: The Shanley model |
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8 | (4) |
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1.4 Shear bands and strain localization |
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12 | (5) |
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1.5 Bifurcation, softening and size effect as the response of a structure |
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17 | (5) |
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1.6 Chains with softening elements |
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22 | (9) |
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1.7 Shear band saturation and multiple shear banding |
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31 | (2) |
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1.8 Brittle and quasi-brittle materials |
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33 | (4) |
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1.9 Coulomb friction and non-associative plasticity |
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37 | (4) |
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1.10 Non-associative flow rule promotes material instabilities |
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41 | (1) |
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1.11 A perturbative approach to material instability |
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42 | (6) |
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48 | (4) |
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1.13 Exercises, details and curiosities |
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52 | (39) |
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1.13.1 Exercise: The Euler elastica and the double supported beam subject to compressive load |
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52 | (17) |
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1.13.2 Exercise: Bifurcation of a structure subject to tensile dead load |
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69 | (1) |
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1.13.3 Exercise: Degrees of freedom and number of critical loads of elastic structures |
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70 | (3) |
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1.13.4 Exercise: A structure with a trivial configuration unstable at a certain load, returning stable at higher load |
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73 | (7) |
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1.13.5 Exercise: Flutter and divergence instability in an elastic structure induced by Coulomb friction |
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80 | (11) |
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2 Elements of tensor algebra and analysis |
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91 | (34) |
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2.1 Components onto an orthonormal basis |
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92 | (1) |
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93 | (2) |
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95 | (3) |
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98 | (1) |
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2.5 Positive definite second-order tensors, eigenvalues and eigenvectors |
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99 | (2) |
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2.6 Reciprocal bases: Covariant and contravariant components |
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101 | (1) |
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2.7 Spectral representation theorem |
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102 | (1) |
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2.8 Square root of a tensor |
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103 | (1) |
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2.9 Polar decomposition theorem |
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104 | (1) |
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2.10 On coaxiality between second-order tensors |
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104 | (1) |
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2.11 Fourth-order tensors |
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105 | (1) |
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2.12 On the metric induced by semi-positive definite tensors |
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106 | (1) |
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2.13 The Macaulay bracket operator |
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107 | (1) |
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2.14 Differential calculus for tensors |
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107 | (1) |
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108 | (2) |
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110 | (1) |
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2.17 Cylindrical coordinates |
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111 | (2) |
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113 | (1) |
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2.19 Convexity and quasi-convexity |
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114 | (2) |
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2.20 Examples and details |
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116 | (9) |
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2.20.1 Example: Jordan normal form of a defective tensor with a double eigenvalue |
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116 | (1) |
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2.20.2 Example: Jordan normal form of a defective tensor with a triple eigenvalue |
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117 | (1) |
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2.20.3 Example: Inverse of the acoustic tensor of isotropic elasticity |
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117 | (1) |
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2.20.4 Example: Inverse of the acoustic tensor for a particular class of anisotropic elasticity |
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118 | (1) |
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2.20.5 Example: A representation for the square root of a tensor |
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118 | (1) |
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2.20.6 Proof of a property of the scalar product between two symmetric tensors |
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119 | (1) |
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2.20.7 Example: Inverse and positive definiteness of the fourth-order tensor defining linear isotropic elasticity |
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120 | (1) |
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2.20.8 Example: Inverse and positive definiteness of a fourth-order tensor defining a special anisotropic linear elasticity |
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121 | (1) |
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2.20.9 Example: Inverse of the elastoplastic fourth-order tangent tensor |
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121 | (1) |
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2.20.10 Example: Spectral representation of the elastoplastic fourth-order tangent tensor |
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122 | (2) |
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2.20.11 Example: Strict convexity of the strain energy defining linear isotropic elasticity |
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124 | (1) |
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3 Solid mechanics at finite strains |
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125 | (27) |
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125 | (10) |
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3.1.1 Transformation of oriented line elements |
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127 | (2) |
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3.1.2 Transformation of oriented area elements |
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129 | (1) |
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3.1.3 Transformation of volume elements |
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129 | (1) |
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130 | (1) |
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131 | (4) |
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3.2 On material and spatial strain measures |
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135 | (2) |
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3.2.1 Rigid-body rotation of the reference configuration |
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135 | (1) |
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3.2.2 Rigid-body rotation of the current configuration |
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136 | (1) |
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3.3 Motion of a deformable body |
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137 | (4) |
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141 | (1) |
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3.5 Stress, dynamic forces |
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142 | (4) |
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3.6 Power expended and work-conjugate stress/strain measures |
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146 | (4) |
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3.7 Changes of fields for a superimposed rigid-body motion |
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150 | (2) |
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4 Isotropic non-linear hyperelasticity |
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152 | (10) |
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4.1 Isotropic compressible hyperelastic material |
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153 | (2) |
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4.1.1 Kirchhoff-Saint Venant material |
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154 | (1) |
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4.2 Incompressible isotropic elasticity |
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155 | (7) |
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4.2.1 Mooney-Rivlin elasticity |
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156 | (2) |
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4.2.2 Neo-Hookean elasticity |
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158 | (1) |
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4.2.3 J2-Deformation theory of plasticity |
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158 | (1) |
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159 | (3) |
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5 Solutions of simple problems in finitely deformed non-linear elastic solids |
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162 | (26) |
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5.1 Uniaxial plane strain tension and compression of an incompressible elastic block |
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162 | (6) |
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5.2 Uniaxial plane strain tension and compression of Kirchhoff-Saint Venant material |
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168 | (2) |
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5.3 Uniaxial tension and compression of an incompressible elastic cylinder |
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170 | (3) |
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5.4 Simple shear of an elastic block |
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173 | (6) |
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5.5 Finite bending of an incompressible elastic block |
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179 | (9) |
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6 Constitutive equations and anisotropic elasticity |
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188 | (35) |
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6.1 Constitutive equations: General concepts |
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188 | (19) |
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6.1.1 Change in observer and related principle of invariance of material response |
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189 | (3) |
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6.1.2 Indifference with respect to rigid-body rotation of the reference configuration |
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192 | (3) |
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6.1.3 Material symmetries |
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195 | (3) |
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198 | (3) |
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6.1.5 Green elastic or hyperelastic materials |
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201 | (2) |
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6.1.6 Incompressible hyperelasticity and constrained materials |
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203 | (4) |
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6.2 Rate and incremental elastic constitutive equations |
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207 | (16) |
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6.2.1 Elastic laws in incremental and rate form |
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207 | (3) |
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6.2.2 Relative Lagrangean description |
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210 | (10) |
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220 | (3) |
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7 Yield functions with emphasis on pressure sensitivity |
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223 | (28) |
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7.1 The Haigh-Westergaard representation |
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225 | (4) |
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7.2 The BP yield function |
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229 | (5) |
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7.2.1 Smoothness of the BP yield surface |
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233 | (1) |
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7.3 Reduction of the BP yield criterion to known cases |
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234 | (7) |
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7.3.1 Drucker-Prager and von Mises yield criteria |
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236 | (3) |
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7.3.2 A comparison of the BP yield criterion with experimental results |
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239 | (2) |
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7.4 Convexity of yield function and yield surface |
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241 | (10) |
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7.4.1 A general convexity result for a class of yield functions |
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242 | (4) |
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7.4.2 Convexity of the BP yield function |
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246 | (1) |
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7.4.3 Generating convex yield functions |
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247 | (4) |
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8 Elastoplastic constitutive equations |
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251 | (24) |
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8.1 The theory of elastoplasticity at small strain |
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251 | (6) |
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8.2 The essential structure of rate elastoplastic constitutive equations at large strain |
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257 | (16) |
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8.2.1 The small strain theory recovered |
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264 | (1) |
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8.2.2 A theory of elastoplasticity based on multiplicative decomposition of the deformation gradient |
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265 | (2) |
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8.2.3 A simple constitutive model for granular materials evidencing flutter instability |
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267 | (1) |
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8.2.4 Elastoplastic coupling in the modelling of granular materials and geomaterials |
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268 | (5) |
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8.3 A summary on rate constitutive equations |
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273 | (2) |
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9 Moving discontinuities and boundary value problems |
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275 | (18) |
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9.1 Moving discontinuities in solids |
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275 | (10) |
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9.1.1 Local jump conditions for propagating discontinuity surfaces |
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276 | (4) |
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9.1.2 Balance equations for regions containing a moving discontinuity surface |
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280 | (5) |
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9.2 Boundary value problems in finite, rate and incremental forms |
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285 | (8) |
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9.2.1 Quasi-static first-order rate problems |
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287 | (2) |
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9.2.2 Incremental non-linear elasticity |
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289 | (4) |
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10 Global conditions of uniqueness and stability |
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293 | (17) |
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10.1 Uniqueness of the rate problem |
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298 | (5) |
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10.1.1 Raniecki comparison solids |
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299 | (1) |
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10.1.2 Associative elastoplasticity |
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300 | (2) |
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10.1.3 `In-loading comparison solid' |
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302 | (1) |
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10.2 Stability in the Hill sense |
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303 | (7) |
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10.2.1 Associative elastoplasticity |
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304 | (1) |
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10.2.2 Stability of a quasi-static deformation process |
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305 | (1) |
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10.2.3 An example: Elastoplastic column buckling |
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306 | (4) |
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11 Local conditions for uniqueness and stability |
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310 | (28) |
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11.1 A local sufficient condition for uniqueness: Positive definiteness of the constitutive operator |
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311 | (6) |
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315 | (1) |
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11.1.2 The small strain theory |
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316 | (1) |
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11.2 Singularity of the constitutive operator |
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317 | (2) |
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318 | (1) |
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11.2.2 The small strain theory |
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319 | (1) |
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319 | (4) |
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11.3.1 The small strain theory |
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323 | (1) |
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11.4 Ellipticity, strain localisation and shear bands |
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323 | (8) |
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11.4.1 The small strain theory |
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326 | (5) |
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331 | (4) |
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331 | (1) |
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11.5.2 Flutter instability for small strain elastoplasticity with isotropic elasticity |
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332 | (3) |
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11.5.3 Physical meaning and consequences of flutter |
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335 | (1) |
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11.6 Other types of local criteria and instabilities |
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335 | (1) |
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11.7 A summary on local and global uniqueness and stability criteria |
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336 | (2) |
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12 Incremental bifurcation of elastic solids |
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338 | (47) |
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12.1 The bifurcation problem |
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339 | (1) |
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12.2 Bifurcations of incompressible elastic solids deformed in plane strain |
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340 | (25) |
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12.2.1 Local uniqueness and stability criteria for Biot plane strain and incompressible elasticity |
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340 | (11) |
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12.2.2 Bifurcations of layered structures: General solution |
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351 | (2) |
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12.2.3 Surface bifurcation |
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353 | (2) |
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12.2.4 Interfacial bifurcations |
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355 | (3) |
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12.2.5 Bifurcations of an elastic incompressible block |
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358 | (3) |
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12.2.6 Incompressible elastic block on a `spring foundation' |
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361 | (2) |
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12.2.7 Multi-layered elastic structures |
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363 | (2) |
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12.3 Bifurcations of an incompressible elastic cylinder |
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365 | (10) |
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12.3.1 Numerical results for bifurcations of an elastic cylinder subject to axial compression |
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370 | (5) |
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12.4 Bifurcation under plane strain bending |
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375 | (10) |
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13 Applications of local and global uniqueness and stability criteria to non-associative elastoplasticity |
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385 | (18) |
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13.1 Local uniqueness and stability criteria for non-associative elastoplasticity at small strain |
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385 | (3) |
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13.2 Axi-symmetric bifurcations of an elastoplastic cylinder under uniaxial stress |
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388 | (8) |
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13.2.1 Results for the axi-symmetric bifurcations of a cylinder |
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391 | (5) |
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13.3 Flutter instability for a finite-strain plasticity model with anisotropic elasticity |
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396 | (7) |
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13.3.1 Examples of flutter instability for plane problems |
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396 | (4) |
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13.3.2 Spectral analysis of the acoustic tensor |
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400 | (3) |
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14 Wave propagation, stability and bifurcation |
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403 | (24) |
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14.1 Incremental waves and bifurcation |
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405 | (2) |
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14.2 Incremental plane waves |
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407 | (2) |
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14.2.1 Non-linear elastic materials |
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407 | (2) |
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14.3 Waves and material instabilities in elastoplasticity |
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409 | (11) |
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14.3.1 Instability of uniform flow |
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413 | (6) |
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14.3.2 A discussion on waves and instability in elastoplasticity |
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419 | (1) |
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420 | (7) |
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14.4.1 Non-linear elastic material deformed incrementally |
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420 | (1) |
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14.4.2 Elastoplastic materials |
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420 | (7) |
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15 Post-critical behaviour and multiple shear band formation |
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427 | (17) |
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15.1 One-dimensional elastic models with non-convex energy |
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428 | (6) |
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15.2 Two-dimensional elastoplastic modelling of post-shear banding |
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434 | (10) |
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15.2.1 Post-shear banding analysis |
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436 | (3) |
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15.2.2 Sharp shear banding versus saturation |
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439 | (1) |
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15.2.3 Post-band saturation analysis |
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439 | (5) |
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16 A perturbative approach to material instability |
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444 | (63) |
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16.1 Infinite-body Green's function for a pre-stressed material |
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447 | (22) |
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16.1.1 Quasi-static Green's function |
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447 | (10) |
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16.1.2 The dynamic time-harmonic Green's function for general non-symmetric constitutive equations |
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457 | (7) |
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16.1.3 Effects of flutter instability revealed by a pulsating perturbing dipole |
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464 | (5) |
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16.2 Finite-length crack in a pre-stressed material |
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469 | (20) |
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16.2.1 Finite-length crack parallel to an orthotropy axis |
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471 | (9) |
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16.2.2 The inclined crack |
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480 | (2) |
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16.2.3 Shear bands interacting with a finite-length crack |
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482 | (4) |
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16.2.4 Incremental energy release rate for crack growth |
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486 | (3) |
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16.3 Mode I perturbation of a stiffener in an infinite non-linear elastic material subjected to finite simple shear deformation |
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489 | (9) |
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16.4 The stress state near a shear band and its propagation |
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498 | (9) |
| References |
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507 | (20) |
| Index |
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527 | |
Lisainfo e-raamatute kohta