| Preface |
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xxiii | |
| About The Author |
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xxvii | |
| 1 Introduction |
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1 | (4) |
| 2 Concepts And Mathematical Preliminaries |
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5 | (52) |
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5 | (1) |
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5 | (1) |
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2.3 Dummy index and dummy variables |
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6 | (1) |
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7 | (1) |
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2.5 Vector and matrix notation |
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8 | (1) |
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2.6 Index notation and Kronecker delta |
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8 | (2) |
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10 | (2) |
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2.8 Basic operations using vector and matrix and Einstein's notations |
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12 | (2) |
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2.9 Change of frames, transformations, concept and representation of tensors, tensor operations and tensor calculus |
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14 | (37) |
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2.9.1 Coordinate transformation T |
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15 | (2) |
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2.9.2 Induced transformations |
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17 | (1) |
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2.9.3 Isomorphism between coordinate transformations and induced transformations |
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17 | (1) |
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2.9.4 Transformations by covariance and contravariance |
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18 | (4) |
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2.9.5 Covariant and contravariant tensors |
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22 | (5) |
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2.9.6 Cartesian frames and orthogonal transformations |
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27 | (5) |
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2.9.7 Alternate representation of tensors and tensor operations |
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32 | (5) |
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2.9.8 Transformations of tensors defined in orthogonal frames due to orthogonal transformation of coordinates |
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37 | (2) |
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2.9.9 Invariants of tensors |
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39 | (4) |
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2.9.10 Hamilton-Cayley theorem |
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43 | (1) |
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2.9.11 Differential calculus of tensors |
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44 | (7) |
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2.10 Some useful relations |
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51 | (1) |
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52 | (5) |
| 3 Kinematics Of Motion, Deformation And Their Measures |
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57 | (96) |
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3.1 Description of motion |
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57 | (2) |
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3.2 Lagrangian and Eulerian descriptions and descriptions in fluid mechanics |
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59 | (5) |
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3.2.1 Lagrangian or referential description of motion |
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60 | (1) |
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3.2.2 Eulerian or spatial description of motion |
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60 | (1) |
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3.2.3 Descriptions in fluid mechanics |
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61 | (2) |
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63 | (1) |
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3.3 Material particle displacements |
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64 | (1) |
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3.4 Continuous deformation of matter, restrictions on the description of motion |
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65 | (2) |
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67 | (2) |
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3.5.1 Material derivative in Lagrangian or referential description |
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67 | (1) |
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3.5.2 Material derivative in Eulerian or spatial description |
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67 | (2) |
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3.6 Acceleration of a material particle |
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69 | (1) |
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3.6.1 Lagrangian or referential description |
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69 | (1) |
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3.6.2 Eulerian or spatial description |
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69 | (1) |
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3.7 Coordinate systems and bases |
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70 | (1) |
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71 | (1) |
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72 | (2) |
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3.10 Alternate way to visualize covariant and contravariant bases gi, gi and the relationships between them |
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74 | (1) |
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3.11 Jacobian of deformation |
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75 | (3) |
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3.12 Change of description, co- and contra-variant measures |
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78 | (1) |
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3.13 Notations for covariant and contravariant measures |
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79 | (1) |
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3.14 Deformation, measures of length and change in length |
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80 | (4) |
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3.14.1 Covariant measures of length and change in length |
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81 | (1) |
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3.14.2 Contravariant measures of length and change in length |
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82 | (2) |
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3.15 Covariant and contravariant measures of strain in Lagrangian and Eulerian descriptions |
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84 | (11) |
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3.15.1 Covariant measures of finite strains |
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87 | (3) |
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3.15.2 Contravariant measures of finite strains |
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90 | (5) |
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3.16 Changes in strain measures due to rigid rotation of frames |
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95 | (16) |
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3.16.1 Change in Lagrangian descriptions of strain measures due to rigid rotation of x-frame to x'-frame for the reference configuration |
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97 | (4) |
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3.16.2 Changes in Eulerian measures of strains due to rigid rotation of x-frame to x'-frame |
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101 | (3) |
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3.16.3 Changes in Lagrangian measures of strain due to change of xi coordinates in the current configuration to x'i by a rigid rotation of the x-frame to x'-frame |
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104 | (3) |
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3.16.4 Changes in Eulerian measures of strain due to change of xi coordinates to x'i by the rotation of x-frame to x'-frame |
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107 | (4) |
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3.17 Invariants of strain tensors |
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111 | (1) |
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3.18 Expanded form of strain tensors |
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112 | (3) |
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3.19 Physical meaning of strains |
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115 | (12) |
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3.19.1 Extensions and stretches parallel to ox1, ox2, ox3 axes in the x-frame: covariant measure of strain in Lagrangian description using element of [ 0], Green's strain in Lagrangian description |
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118 | (2) |
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3.19.2 Extensions and stretches parallel to oxi, ox2, ox3 axes in the x-frame: contravariant measure of strain in the Eulerian description using element of [ 0], Almansi strain in Eulerian description |
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120 | (2) |
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3.19.3 Angles between the fibers or material lines |
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122 | (5) |
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3.20 Polar decomposition: rotation and stretch tensors |
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127 | (15) |
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3.20.1 Strain measures in terms of [ Sr], [ Sl] and [ R] |
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136 | (4) |
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3.20.2 Invariants of [ C[ 0]], [ B[ 0], [ Sr] and [ Sl] in terms of principal stretches λ1, λ2 and λ3 |
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140 | (2) |
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3.21 Deformation of areas and volumes |
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142 | (3) |
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142 | (2) |
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144 | (1) |
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145 | (8) |
| 4 Definitions And Measures Of Stresses |
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153 | (24) |
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153 | (6) |
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4.2 Contravariant, covariant stress tensors, Cauchy stress tensors and Jaumann stress tensor |
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159 | (10) |
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4.2.1 Contravariant Cauchy stress tensor σ(0) in Lagrangian description and σ(0) in Eulerian description |
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161 | (2) |
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4.2.2 Covariant Cauchy stress tensor in σ(0) Eulerian description and σ(0) in Lagrangian description |
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163 | (2) |
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4.2.3 Jaumann stress tensor (0)σ-J and (0)σJ |
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165 | (1) |
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4.2.4 Second Piola-Kirchhoff stress tensor σ[ 0] in the x-frame using contravariant Cauchy stress tensor σ(0) in the x-frame |
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165 | (2) |
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4.2.5 First Piola-Kirchhoff or Lagrange stress tensor σ in the x-frame using contravariant Cauchy stress tensor σ(0) in the x-frame |
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167 | (1) |
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4.2.6 Second Piola-Kirchhoff stress tensor cr[ 0] in the x-frame using covariant Cauchy stress tensor σ(0) in the x-frame |
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168 | (1) |
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169 | (1) |
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4.4 Summary of stress measures and general considerations in their derivations |
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169 | (3) |
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4.4.1 General considerations |
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170 | (1) |
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4.4.2 Summary of stress measures |
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171 | (1) |
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4.4.3 Conjugate strain measures |
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172 | (1) |
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4.5 Relations between different stress measures and some other useful relations |
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172 | (2) |
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174 | (3) |
| 5 Rate Of Deformation, Area, Volume, Strain Rate Tensors, Spin Tensor And Convected Time Derivatives Of Stress And Strain Tensors |
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177 | (34) |
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177 | (2) |
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5.1.1 Lagrangian description |
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178 | (1) |
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5.1.2 Eulerian description |
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179 | (1) |
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5.2 Decomposition of [ L], the spatial velocity gradient tensor |
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179 | (1) |
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5.3 Interpretation of the components of [ D] |
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180 | (5) |
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5.3.1 Diagonal components of [ D] |
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180 | (5) |
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5.3.2 Off diagonal components of [ D]: physical interpret ation |
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n182 | |
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5.4 Rate of change or material derivative of strain tensors [ C[ 0]] and [ epsilon[ 0]] |
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185 | (1) |
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5.5 Physical meaning of spin tensor [ W] |
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185 | (2) |
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5.6 Vorticity vector and vorticity |
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187 | (1) |
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5.7 Material derivative of det[ J] i.e. D|J|/Dt or |J|: rate of change of |J| at a material point |
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188 | (2) |
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5.8 Rate of change of volume, i.e. material derivative of volume |
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190 | (1) |
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5.9 Rate of change of area: material derivative of area |
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191 | (1) |
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5.10 Stress and strain measures for convected time derivatives |
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192 | (1) |
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5.11 Convected time derivatives of Cauchy stress tensors and strain tensors |
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193 | (14) |
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5.11.1 Convected time derivatives of the Cauchy stress tensors: incompressible matter |
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194 | (6) |
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5.11.2 Convected time derivatives of the Cauchy stress tensors: compressible matter |
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200 | (3) |
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5.11.3 Convected time derivatives of the strain tensors |
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203 | (4) |
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5.12 Conjugate pairs of convected time derivatives of stress and strain tensors |
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207 | (1) |
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5.13 Objectivity of the convected time derivatives (rates) |
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207 | (1) |
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208 | (3) |
| 6 Conservation And Balance Laws In Eulerian Description |
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211 | (36) |
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211 | (2) |
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213 | (1) |
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6.3 Conservation of mass: continuity equation for compressible matter |
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213 | (1) |
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214 | (4) |
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215 | (1) |
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216 | (1) |
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6.4.3 Continued development of transport theorem |
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217 | (1) |
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6.5 Conservation of mass: continuity equation |
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218 | (1) |
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6.6 Kinetics of continuous media: balance of linear momenta (momentum equations) |
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219 | (9) |
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6.6.1 Preliminary considerations |
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220 | (1) |
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6.6.2 Application of Newton's second law to the deformed volume of the matter v(t) with boundary partialdifferentialV in the current configuration: balance of linear momenta |
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221 | (3) |
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6.6.3 Momentum equations for compressible matter |
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224 | (4) |
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6.7 Kinetics of continuous media: balance of angular momenta |
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228 | (2) |
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6.8 First law of thermodynamics for compressible matter |
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230 | (7) |
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6.9 Second law of thermodynamics: Clausius-Duhem inequality or entropy inequality |
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237 | (4) |
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6.10 A summary of mathematical models |
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241 | (1) |
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242 | (5) |
| 7 Conservation And Balance Laws In Lagrangian Description |
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247 | (32) |
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247 | (1) |
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7.2 Mathematical model for deforming matter in Lagrangian description |
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247 | (1) |
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7.3 Conservation of mass: continuity equation |
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248 | (1) |
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7.4 Balance of linear momenta |
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249 | (6) |
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7.5 Balance of angular momenta |
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255 | (1) |
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7.6 First law of thermodynamics |
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255 | (7) |
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7.6.1 Conjugate stress and strain rate (or strain) measures |
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259 | (2) |
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7.6.2 Energy equation in equivalent conjugate stress and strain measures |
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261 | (1) |
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7.7 Second law of thermodynamics: entropy inequality using Helmholtz free energy density |
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262 | (3) |
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7.8 Second law of thermodynamics: entropy inequality using Gibbs potential |
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265 | (2) |
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7.8.1 Using σ[ 0] and epsilon[ 0] as conjugate pair |
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265 | (1) |
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7.8.2 Using σ[ 0] and C[ 0] as conjugate pair |
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266 | (1) |
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7.9 Summary of mathematical models |
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267 | (2) |
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7.10 First and second laws of thermodynamics for thermoelastic solids |
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269 | (6) |
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7.10.1 First law of thermodynamics (energy equation) for thermoelastic solids |
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270 | (2) |
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7.10.2 Second law of thermodynamics (entropy inequality) for thermoelastic solids |
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272 | (3) |
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275 | (4) |
| 8 General Considerations In The Constitutive Theories For Solids And Fluids |
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279 | (18) |
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279 | (3) |
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8.2 Axioms of constitutive theory |
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282 | (2) |
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284 | (2) |
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285 | (1) |
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285 | (1) |
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8.4 Preliminary considerations in the constitutive theories |
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286 | (4) |
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286 | (2) |
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288 | (1) |
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8.4.3 Choice of arguments of the dependent variables for solid matter and fluids |
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289 | (1) |
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8.5 General approach of deriving constitutive theories |
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290 | (4) |
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8.5.1 Assuming Φ as a function of the invariants of epsilon[ O] |
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291 | (1) |
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8.5.2 Theory of generators and invariants |
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292 | (1) |
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8.5.3 Taylor series expansion of Φ |
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293 | (1) |
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8.5.4 Material coefficients |
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293 | (1) |
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8.5.5 Constitutive theory for heat vector q |
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293 | (1) |
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294 | (3) |
| 9 Ordered Rate Constitutive Theories For Thermoelastic Solids |
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297 | (66) |
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297 | (3) |
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9.2 Entropy inequality in Φ: Lagrangian description |
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300 | (1) |
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9.3 Constitutive theories for thermoelastic solids in Lagrangian description |
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300 | (4) |
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9.4 Constitutive theories based on σ[ 0] = σ[ 0](epsilon[ 0],θ) using theory of generators and invariants: Approach I |
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304 | (6) |
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9.4.1 Determination of material coefficients in the constitutive theory for σ[ 0] |
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305 | (2) |
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9.4.2 Further assumptions and simplifications: a constitutive theory for σ[ 0] that is linear in epsilon[ 0] and θ |
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307 | (3) |
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9.5 Rate of strain energy and strain energy density function π: Lagrangian description |
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310 | (3) |
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9.6 Constitutive theories for σ[ 0] in terms of epsilon[ 0] based on π: Lagrangian description |
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313 | (15) |
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9.6.1 Approach II: considering π = π(Iepsilon[ 0],IIepsilon[ 0],IIIepsilon[ 0],θ) and using (9.53) |
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313 | (3) |
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9.6.2 Approach III: expanding π=π(epsilon[ 0],θ in Taylor series about a known configuration and using (9.53) |
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316 | (12) |
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9.7 Constitutive theories for σ[ 0] in terms of C[ 10] based on π: Lagrangian description |
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328 | (12) |
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328 | (1) |
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9.7.2 Approach II: considering π=π(IC[ 0],IIC[ 0],IIIC[ 0],θ and using (9.57) |
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328 | (1) |
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9.7.3 Approach II, alternate derivation: considering π=π(IC[ 0],IIC[ 0],IIIC[ 0],θ and using (9.64) |
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329 | (4) |
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9.7.4 Material coefficients in the constitutive theory for σ[ 0] in terms of C[ 0] |
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333 | (3) |
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9.7.5 Further assumptions and simplifications: a constitutive theory for [ σ[ 0]] that is linear in the strain tensor [ C[ 0] and temperature θ |
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336 | (4) |
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9.8 Constitutive theories for the heat vector q: Lagrangian description |
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340 | (6) |
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9.8.1 Constitutive theory for q using entropy inequality |
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340 | (1) |
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9.8.2 Constitutive theories for q using theory of generators and invariants |
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341 | (5) |
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9.9 Alternate derivations of the constitutive theories for thermoelastic solids: constitutive theories for epsilon[ 0] in terms of σ[ 0] in Lagrangian description |
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346 | (8) |
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9.9.1 Constitutive theories based on epsilon[ 0]=epsilon[ ](σ[ 0],θ |
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347 | (4) |
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9.9.2 Constitutive theories for epsilon[ 0] in terms of σ[ 0] based on complementary strain energy density function πc |
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351 | (3) |
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9.10 Alternate derivations of the constitutive theories for thermoelastic solids: constitutive theories for q in terms of &sigam;[ 0] in Lagrangian description |
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354 | (5) |
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9.10.1 Constitutive theory for q using the entropy inequality |
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355 | (1) |
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9.10.2 Constitutive theories for q using theory of generators and invariants |
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355 | (4) |
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359 | (4) |
| 10 Ordered Rate Constitutive Theories For Thermoviscoelastic Solids Without Memory |
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363 | (70) |
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363 | (1) |
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10.2 Constitutive theories using Helmholtz free energy density |
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363 | (42) |
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10.2.1 Second law of thermodynamics using Φ and conjugate pair (σ, J), dependent variables in the constitutive theories, and their arguments |
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364 | (6) |
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10.2.2 Second law of thermodynamics using Φ and conjugate pair σ[ 0],epsilon[ 0]), dependent variables in the constitutive theories and their arguments |
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370 | (3) |
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10.2.3 Decomposition of σ[ 0] |
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373 | (1) |
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10.2.4 Constitutive theory for equilibrium stresseσ[ 0] |
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374 | (6) |
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10.2.5 Constitutive theory for deviatoric second Piola-Kirchhoff stress tensor dσ[ 0]and heat vector q |
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380 | (16) |
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396 | (8) |
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404 | (1) |
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10.3 Constitutive theories using Gibbs potential |
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405 | (17) |
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10.3.1 Entropy inequality expressed in terms of Gibbs potential |
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406 | (1) |
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10.3.2 Dependent variables in the constitutive theories, their argument tensors and the entropy inequality inΨ |
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407 | (5) |
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10.3.3 Constitutive theory for equilibrium stress tensor ecσ[ 0] |
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412 | (1) |
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10.3.4 Constitutive theories for strain tensor and heat vector |
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413 | (9) |
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10.4 Comparisons of the rate constitutive theories resulting from the entropy inequality expressed in terms of Helmholtz free energy density Φ and Gibbs potential Ψ |
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422 | (7) |
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10.4.1 Rate constitutive theories for σ[ 0] |
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422 | (4) |
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10.4.2 Rate constitutive theories for the heat vector q |
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426 | (3) |
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429 | (4) |
| 11 Ordered Rate Constitutive Theories For Thermoviscoelastic Solids With Memory |
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433 | (54) |
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433 | (1) |
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11.2 Constitutive theories using Helmholtz free energy density |
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433 | (27) |
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11.2.1 Second law of thermodynamics, dependent variables in the rate constitutive theories, and their arguments |
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434 | (3) |
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11.2.2 Stress decomposition |
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437 | (2) |
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11.2.3 Further considerations on the dependent variables and their arguments: axiom of frame invariance |
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439 | (1) |
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11.2.4 Constitutive theory for equilibrium stress eσ[ 0] |
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440 | (1) |
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11.2.5 Compressible matter: equilibrium stress tensor eσ[ 0] |
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440 | (1) |
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11.2.6 Incompressible matter: equilibrium stress tensor eσ[ 0] |
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441 | (1) |
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11.2.7 Constitutive theory for the deviatoric second PiolaKirchhoff stress tensor and heat vector |
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442 | (9) |
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451 | (8) |
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459 | (1) |
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11.3 Constitutive theories using Gibbs potential |
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460 | (17) |
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11.3.1 Entropy inequality expressed in terms of Gibbs potential Ψdependent variables in the constitutive theories and their argument tensors |
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461 | (2) |
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11.3.2 Conditions resulting from entropy inequality in Ψ and decomposition of the stress tensor |
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463 | (2) |
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11.3.3 Constitutive theory for equilibrium stress tensor eσ[ 0] |
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465 | (1) |
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11.3.4 Compressible matter: equilibrium stress tensor eσ[ 0] |
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466 | (1) |
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11.3.5 Incompressible matter: equilibrium stress tensor eσ[ 0] |
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467 | (1) |
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11.3.6 Constitutive theories for strain tensor and heat vector |
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467 | (8) |
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475 | (2) |
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11.4 Comparisons of the rate constitutive theories resulting from the entropy inequality expressed in terms of Φ and Ψ |
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477 | (5) |
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11.4.1 Rate constitutive theories for σ[ 0] or epsilon[ 0] |
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477 | (2) |
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11.4.2 Rate constitutive theories for the heat vector q |
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479 | (3) |
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482 | (5) |
| 12 Ordered Rate Constitutive Theories For Thermofluids |
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487 | (50) |
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487 | (1) |
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12.2 Second law of thermodynamics: Entropy inequality |
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488 | (1) |
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12.3 Dependent variables in the constitutive theories and their arguments |
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489 | (3) |
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12.3.1 Choice of dependent variables in the constitutive theory |
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489 | (1) |
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12.3.2 Choice of arguments of the dependent variables in the constitutive theories |
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490 | (2) |
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12.4 Development of ordered rate constitutive theory of order n for ordered thermofluids |
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492 | (8) |
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12.4.1 Entropy inequality |
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492 | (2) |
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12.4.2 Stress decomposition |
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494 | (3) |
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12.4.3 Final choice of dependent variables and their argument tensors |
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497 | (1) |
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12.4.4 Constitutive equations for deviatoric Cauchy stress tensor and heat vector: compressible thermofluids |
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498 | (2) |
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12.5 Rate constitutive theory of order n: compressible thermofluids |
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500 | (5) |
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12.5.1 Constitutive theory for the deviatoric Cauchy stress tensor (0)dσ |
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500 | (1) |
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12.5.2 Constitutive theory for the heat vector (0)q |
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501 | (4) |
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12.6 Rate constitutive theory of order two (n=2): compressible thermofluids |
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505 | (5) |
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12.6.1 Constitutive theory for the deviatoric Cauchy stress tensor (0)dσ |
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505 | (3) |
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12.6.2 Constitutive theory for the heat vector (0)q |
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508 | (2) |
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12.7 Rate constitutive theory of order one (n=1): compressible thermofluids |
|
|
510 | (2) |
|
12.7.1 Constitutive theory for the deviatoric Cauchy stress tensor (0)dσ |
|
|
510 | (1) |
|
12.7.2 Constitutive theory for the heat vector (0)q |
|
|
511 | (1) |
|
12.8 Constitutive theory for compressible generalized Newtonian and Newtonian thermoviscous fluids |
|
|
512 | (9) |
|
12.8.1 Constitutive theory for the deviatoric Cauchy stress tensor (0)dσ |
|
|
512 | (7) |
|
12.8.2 Constitutive theory for the heat vector (0)q |
|
|
519 | (2) |
|
12.9 Incompressible ordered thermofluids of orders n, 2 and 1 |
|
|
521 | (1) |
|
12.10 Constitutive theories for incompressible generalized Newtonian and Newtonian fluids |
|
|
522 | (6) |
|
12.10.1 Constitutive theory for the deviatoric Cauchy stress tensor (0)dσ |
|
|
523 | (4) |
|
12.10.2 Constitutive theory for the heat vector (0)q |
|
|
527 | (1) |
|
12.11 Conjugate stress-strain measures and validity of rate constitutive theories in different bases |
|
|
528 | (1) |
|
|
|
529 | (8) |
| 13 Ordered Rate Constitutive Theories For Thermoviscoelastic Fluids |
|
537 | (66) |
|
|
|
537 | (2) |
|
13.2 Second law of thermodynamics: entropy inequality |
|
|
539 | (1) |
|
13.3 Dependent variables in the constitutive theories and their arguments |
|
|
540 | (3) |
|
13.3.1 Choice of dependent variables in the constitutive theory |
|
|
540 | (1) |
|
13.3.2 Choice of arguments of the dependent variables in the constitutive theories |
|
|
540 | (3) |
|
13.4 Development of ordered rate constitutive theories of orders (m,n) for ordered thermoviscoelastic fluids |
|
|
543 | (9) |
|
13.4.1 Entropy inequality |
|
|
543 | (3) |
|
13.4.2 Stress decomposition |
|
|
546 | (4) |
|
13.4.3 Final choice of dependent variables and their argument tensors |
|
|
550 | (2) |
|
13.5 Rate constitutive theory of orders m and n for the deviatoric Cauchy stress tensor and the heat vector: compressible thermoviscoelastic fluids |
|
|
552 | (6) |
|
13.5.1 Constitutive theory of orders m and n for (m)dσ |
|
|
553 | (1) |
|
13.5.2 Constitutive theory of orders m and n for (0)q |
|
|
554 | (3) |
|
13.5.3 Special forms of rate constitutive theories for compressible thermoviscoelastic fluids |
|
|
557 | (1) |
|
13.6 Rate constitutive theory of orders m=1 and n=1 for the deviatoric stress tensor and heat vector: compressible thermoviscoelastic fluids |
|
|
558 | (18) |
|
13.6.1 Further assumptions and simplifications |
|
|
559 | (5) |
|
13.6.2 Maxwell constitutive model for deviatoric Cauchy stress tensor |
|
|
564 | (5) |
|
13.6.3 Giesekus constitutive model for deviatoric Cauchy stress tensor |
|
|
569 | (7) |
|
13.7 Rate constitutive theory of orders m=1 and n=2 for the deviatoric Cauchy stress tensor and heat vector: compressible thermoviscoelastic fluids |
|
|
576 | (6) |
|
13.7.1 Further assumptions and simplifications |
|
|
577 | (1) |
|
13.7.2 Oldroyd-B constitutive model for deviatoric Cauchy stress tensor |
|
|
577 | (5) |
|
13.8 Rate constitutive theories for incompressible thermoviscoelastic fluids, i.e. polymeric liquids |
|
|
582 | (1) |
|
13.9 Numerical studies using Giesekus constitutive model |
|
|
583 | (13) |
|
13.9.1 Model Problem 1: fully developed flow between parallel plates |
|
|
584 | (7) |
|
13.9.2 Model Problem 2: fully developed flow between parallel plates using 2D formulation |
|
|
591 | (5) |
|
|
|
596 | (7) |
| 14 Ordered Rate Constitutive Theories For Thermo Hypo-Elastic Solids |
|
603 | (44) |
|
|
|
603 | (1) |
|
14.2 Second law of thermodynamics: Entropy inequality |
|
|
604 | (1) |
|
14.3 Dependent variables in the constitutive theories and their arguments |
|
|
604 | (4) |
|
14.3.1 Choice of dependent variables in the constitutive theory |
|
|
604 | (1) |
|
14.3.2 Choice of arguments of the dependent variables in the constitutive theories |
|
|
605 | (3) |
|
14.4 Development of ordered rate constitutive theory of order n for ordered thermo hypo-elastic solids |
|
|
608 | (8) |
|
14.4.1 Entropy inequality |
|
|
608 | (2) |
|
14.4.2 Stress decomposition |
|
|
610 | (4) |
|
14.4.3 Final choice of dependent variables and their argument tensors |
|
|
614 | (2) |
|
14.5 Rate constitutive theory of order n: compressible thermo hypo-elastic solids |
|
|
616 | (5) |
|
14.5.1 Constitutive theory for the deviatoric Cauchy stress tensor |
|
|
616 | (1) |
|
14.5.2 Constitutive theory for the heat vector |
|
|
617 | (4) |
|
14.6 Rate constitutive theory of order two (n=2): compressible thermo hypo-elastic solids |
|
|
621 | (4) |
|
14.6.1 Constitutive theory for the deviatoric Cauchy stress tensor |
|
|
621 | (4) |
|
14.6.2 Constitutive theory for the heat vector |
|
|
625 | (1) |
|
14.7 Rate constitutive theory of order one (n=1): compressible thermo hypo-elastic solids |
|
|
625 | (2) |
|
14.7.1 Constitutive theory for the deviatoric Cauchy stress tensor |
|
|
626 | (1) |
|
14.7.2 Constitutive theory for the heat vector |
|
|
627 | (1) |
|
14.8 Constitutive theories for compressible generalized thermo hypoelastic solids of order one (n=1) |
|
|
627 | (9) |
|
14.8.1 Constitutive theory for the deviatoric Cauchy stress tensor |
|
|
628 | (5) |
|
14.8.2 Constitutive theory for the heat vector |
|
|
633 | (3) |
|
14.9 Incompressible ordered thermo hypo-elastic solids: of orders n, 2 and 1 |
|
|
636 | (1) |
|
14.10 Constitutive theories for incompressible generalized thermo hypo-elastic solids of order one (n=1) |
|
|
637 | (5) |
|
14.10.1 Constitutive theory for the deviatoric Cauchy stress tensor |
|
|
637 | (5) |
|
14.10.2 Constitutive theory for the heat vector |
|
|
642 | (1) |
|
|
|
642 | (5) |
| 15 Mathematical Models With Thermodynamic Relations |
|
647 | (40) |
|
|
|
647 | (1) |
|
15.2 Thermodynamic pressure: equation of state |
|
|
647 | (5) |
|
15.2.1 Perfect or ideal gas law |
|
|
648 | (1) |
|
|
|
648 | (2) |
|
15.2.3 Compressible solids |
|
|
650 | (2) |
|
15.3 Mechanical pressure: incompressible matter |
|
|
652 | (3) |
|
15.3.1 Incompressible fluids or liquids |
|
|
652 | (1) |
|
15.3.2 Incompressible solids |
|
|
652 | (3) |
|
15.4 Specific internal energy |
|
|
655 | (1) |
|
15.4.1 Compressible matter |
|
|
655 | (1) |
|
15.4.2 Incompressible matter |
|
|
656 | (1) |
|
15.5 Variable transport properties or material coefficients |
|
|
656 | (1) |
|
15.6 Final form of the mathematical models |
|
|
656 | (27) |
|
15.6.1 Thermoelastic solid matter (Lagrangian description) |
|
|
657 | (6) |
|
15.6.2 Thermoviscoelastic solid matter without memory (Lagrangian description) |
|
|
663 | (4) |
|
15.6.3 Thermoviscoelastic solid matter with memory (Lagrangian description) |
|
|
667 | (4) |
|
15.6.4 Thermofluids (Eulerian description) |
|
|
671 | (4) |
|
15.6.5 Thermoviscoelastic fluids: polymers (Eulerian description) |
|
|
675 | (5) |
|
15.6.6 Thermo hypo-elastic solids (Eulerian description) |
|
|
680 | (3) |
|
|
|
683 | (4) |
| 16 Principle Of Virtual Work |
|
687 | (10) |
|
|
|
687 | (1) |
|
16.2 Hamilton's principle in continuum mechanics |
|
|
688 | (4) |
|
16.3 Euler-Lagrange equation: Lagrangian description |
|
|
692 | (1) |
|
16.4 Euler-Lagrange equation: Eulerian description |
|
|
693 | (2) |
|
|
|
695 | (2) |
| Appendix A: Combined generators and invariants |
|
697 | (8) |
| Appendix B: Transformations and operations in Cartesian, cylindrical and spherical coordinate systems |
|
705 | (16) |
|
B.1 Cartesian frame x1, x2, x3 and cylindrical frame r, θ, z |
|
|
706 | (5) |
|
B.1.1 Relationship between coordinates of a point in x1, x2, x3 and r, θ, z-frames |
|
|
706 | (1) |
|
B.1.2 Converting derivatives of a scalar with respect to xi , x2, x3 into its derivatives with respect to r, θ, z |
|
|
707 | (2) |
|
B.1.3 Relationship between bases in xi, x2, x3- and r, 9θ, zframes |
|
|
709 | (2) |
|
B.2 Cartesian frame x1, x2, x3 and spherical frame r, θ, φ |
|
|
711 | (3) |
|
B.2.1 Relationship between coordinates of a point in x1, x2, x3 and r, θ, φ-frames |
|
|
711 | (1) |
|
B.2.2 Converting derivatives of a scalar with respect to x1, x2, x3 into derivatives with respect to r, θ,φ |
|
|
711 | (2) |
|
B.2.3 Relationship between bases, i.e. unit vectors in x1, x2, x3 and r, θ, φ-frames |
|
|
713 | (1) |
|
B.3 Differential operations in r, θ, z- and r, θ, φ-frames |
|
|
714 | (3) |
|
|
|
714 | (2) |
|
|
|
716 | (1) |
|
B.4 Some examples: r, θ, z-frame |
|
|
717 | (2) |
|
B.4.1 Symmetric part of the velocity gradient tensor D |
|
|
718 | (1) |
|
B.4.2 Skew-symmetric part of the velocity gradient tensor W |
|
|
719 | (1) |
|
|
|
719 | (2) |
| Bibliography |
|
721 | (10) |
| Index |
|
731 | |
