2.8.2
Referential Description (Lagrangian Description) . . . . . . 42
2.8.3
Spatial Description (Eulerian Description) . . . . . . . . . . . 42
2.9
Rate of Deformation and Rate of Spin Formulation . . . . . . . . . . . 43
2.9.1
Comparison of Rate of Deformation Tensor, D,
and Time Derivative of the Strain Tensor, _
ε . . . . . . . . . . 44
2.9.2
True Strain (Natural Strain) (Logarithmic Strain) . . . . . . 45
2.10 Finite Strain and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.10.1 Green Deformation Tensor, C, and Cauchy
Deformation Tensor, B
21 . . . . . . . . . . . . . . . . . . . . . . . 49
2.10.2 Relation Between Deformation, Strain,
and Deformation Gradient Tensors . . . . . . . . . . . . . . . . 50
2.10.3 Comparing Small Strain and Large (Finite) Strain . . . . . 52
2.10.4 Strain Rate and Rate of Deformation Relation . . . . . . . . 54
2.10.5 Relation Between the Spatial Gradient of Velocity
Tensor, L,and the Deformation Gradient Tensor, F . . . . . 56
2.11 Rotation and Stretch Tensors in Finite Strain . . . . . . . . . . . . . . . 57
2.12 Compatibility Conditions in Continuum Mechanics . . . . . . . . . . . 57
2.13 Piola-Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.13.1 First Piola-Kirchhoff Stress Tensor σ
0 . . . . . . . . . . . . . . 60
2.13.2 Second Piola-Kirchhoff Stress Tensor e σ . . . . . . . . . . . . . 61
2.14 Direct Relation Between Cauchy Stress Tensor and
Piola-Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.15 Conservation of Mass Principle . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.16 The Incompressible Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.17 Conservation of Momentum Principle . . . . . . . . . . . . . . . . . . . . 65
2.18 Conservation of Moment of Momentum Principle . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1
Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2
First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1
Work Done on the System (Power Input) . . . . . . . . . . . . 76
3.2.2
Heat Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3
Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1
Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.2
Quantification of Entropy in Thermodynamics . . . . . . . . 85
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1
Literature Review of Use of Thermodynamics in Continuum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2
Laws of Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . 132
4.2.1
Second Law of Unified Mechanics Theory . . . . . . . . . . . 133
4.2.2
Third Law of Unified Mechanics Theory . . . . . . . . . . . . 134
4.3
Evolution of Thermodynamic State Index (Φ) . . . . . . . . . . . . . . . 135
x
Contents
Referential Description (Lagrangian Description) . . . . . . 42
2.8.3
Spatial Description (Eulerian Description) . . . . . . . . . . . 42
2.9
Rate of Deformation and Rate of Spin Formulation . . . . . . . . . . . 43
2.9.1
Comparison of Rate of Deformation Tensor, D,
and Time Derivative of the Strain Tensor, _
ε . . . . . . . . . . 44
2.9.2
True Strain (Natural Strain) (Logarithmic Strain) . . . . . . 45
2.10 Finite Strain and Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.10.1 Green Deformation Tensor, C, and Cauchy
Deformation Tensor, B
21 . . . . . . . . . . . . . . . . . . . . . . . 49
2.10.2 Relation Between Deformation, Strain,
and Deformation Gradient Tensors . . . . . . . . . . . . . . . . 50
2.10.3 Comparing Small Strain and Large (Finite) Strain . . . . . 52
2.10.4 Strain Rate and Rate of Deformation Relation . . . . . . . . 54
2.10.5 Relation Between the Spatial Gradient of Velocity
Tensor, L,and the Deformation Gradient Tensor, F . . . . . 56
2.11 Rotation and Stretch Tensors in Finite Strain . . . . . . . . . . . . . . . 57
2.12 Compatibility Conditions in Continuum Mechanics . . . . . . . . . . . 57
2.13 Piola-Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.13.1 First Piola-Kirchhoff Stress Tensor σ
0 . . . . . . . . . . . . . . 60
2.13.2 Second Piola-Kirchhoff Stress Tensor e σ . . . . . . . . . . . . . 61
2.14 Direct Relation Between Cauchy Stress Tensor and
Piola-Kirchhoff Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.15 Conservation of Mass Principle . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.16 The Incompressible Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.17 Conservation of Momentum Principle . . . . . . . . . . . . . . . . . . . . 65
2.18 Conservation of Moment of Momentum Principle . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1
Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2
First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1
Work Done on the System (Power Input) . . . . . . . . . . . . 76
3.2.2
Heat Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3
Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1
Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.2
Quantification of Entropy in Thermodynamics . . . . . . . . 85
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1
Literature Review of Use of Thermodynamics in Continuum
Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2
Laws of Unified Mechanics Theory . . . . . . . . . . . . . . . . . . . . . . 132
4.2.1
Second Law of Unified Mechanics Theory . . . . . . . . . . . 133
4.2.2
Third Law of Unified Mechanics Theory . . . . . . . . . . . . 134
4.3
Evolution of Thermodynamic State Index (Φ) . . . . . . . . . . . . . . . 135
x
Contents