4.3.1
On the Relationship Between the Second Fundamental
Theorem of the Mechanical Theory of Heat and
Probability Calculations Regarding the Conditions for
Thermal Equilibrium, by Ludwig Boltzmann (1877) . . . . 137
4.3.2
Kinetic Energy Has Discrete Values . . . . . . . . . . . . . . . 139
4.4
Critic of Boltzmann’s Mathematical Derivation . . . . . . . . . . . . . . 180
4.5
On the Law of Distribution of Energy in the Normal Spectrum,
By Max Planck, Annalen Der Physik, Vol. 4, P. 553 Ff (1901) . . . 181
4.5.1
Calculations of the Entropy of a Resonator as
a Function of Its Energy . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5.2
Introduction of Wien’s Displacement Law . . . . . . . . . . . 186
4.5.3
Numerical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.6
Thermodynamic State Index (TSI) in Unified Mechanics
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6.1
Experimental Verification Example . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5 Unified Mechanics of Thermo-mechanical Analysis . . . . . . . . . . . . . . 203
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2
Unified Mechanics Theory-Based Constitutive Model . . . . . . . . . 203
5.2.1
Flow Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2.2
Effective Stress Concept and Strain Equivalence
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3
Return Mapping Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.3.1
Linearization (Consistent Jacobian) . . . . . . . . . . . . . . . . 211
5.4
Thermodynamic Fundamental Equation in Thermo-mechanical
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.4.1
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.5
Numerical Validation of the Thermo-mechanical Constitutive
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.5.1
Thin Layer Solder Joint-Monotonic and Fatigue
Shear Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.6
Thermo-mechanical Analysis of Cosserat Continuum:
Length-Scale Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.6.2
Cosserat Couple Stress Theory . . . . . . . . . . . . . . . . . . . 237
5.6.3
Toupin-Mindlin Higher-Order Stress Theory . . . . . . . . . 242
5.6.4
Equilibrium Equations and Problem Formulation . . . . . . 245
5.7
Finite Element Method Implementation . . . . . . . . . . . . . . . . . . . 248
5.7.1
General Couple Stress Theory: Variational
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.7.2
Reduced Couple Stress Theory: Variational
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
5.7.3
Reduced Couple Stress Theory: Mixed Variational
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Contents
xi
On the Relationship Between the Second Fundamental
Theorem of the Mechanical Theory of Heat and
Probability Calculations Regarding the Conditions for
Thermal Equilibrium, by Ludwig Boltzmann (1877) . . . . 137
4.3.2
Kinetic Energy Has Discrete Values . . . . . . . . . . . . . . . 139
4.4
Critic of Boltzmann’s Mathematical Derivation . . . . . . . . . . . . . . 180
4.5
On the Law of Distribution of Energy in the Normal Spectrum,
By Max Planck, Annalen Der Physik, Vol. 4, P. 553 Ff (1901) . . . 181
4.5.1
Calculations of the Entropy of a Resonator as
a Function of Its Energy . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5.2
Introduction of Wien’s Displacement Law . . . . . . . . . . . 186
4.5.3
Numerical Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.6
Thermodynamic State Index (TSI) in Unified Mechanics
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.6.1
Experimental Verification Example . . . . . . . . . . . . . . . . 194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5 Unified Mechanics of Thermo-mechanical Analysis . . . . . . . . . . . . . . 203
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2
Unified Mechanics Theory-Based Constitutive Model . . . . . . . . . 203
5.2.1
Flow Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2.2
Effective Stress Concept and Strain Equivalence
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3
Return Mapping Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.3.1
Linearization (Consistent Jacobian) . . . . . . . . . . . . . . . . 211
5.4
Thermodynamic Fundamental Equation in Thermo-mechanical
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.4.1
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.5
Numerical Validation of the Thermo-mechanical Constitutive
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.5.1
Thin Layer Solder Joint-Monotonic and Fatigue
Shear Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.6
Thermo-mechanical Analysis of Cosserat Continuum:
Length-Scale Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.6.2
Cosserat Couple Stress Theory . . . . . . . . . . . . . . . . . . . 237
5.6.3
Toupin-Mindlin Higher-Order Stress Theory . . . . . . . . . 242
5.6.4
Equilibrium Equations and Problem Formulation . . . . . . 245
5.7
Finite Element Method Implementation . . . . . . . . . . . . . . . . . . . 248
5.7.1
General Couple Stress Theory: Variational
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.7.2
Reduced Couple Stress Theory: Variational
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
5.7.3
Reduced Couple Stress Theory: Mixed Variational
Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Contents
xi