Introduction to Unified Mechanics Theory with Applications

Foreword
Preface
	References
Contents
Abbreviations
Chapter 1: Introduction
	1.1 What Is the Mechanics of Continuous Medium?
	References
Chapter 2: Stress and Strain in Continuum
	2.1 Newton´s Universal Laws of Motion
		2.1.1 First Universal Law of Motion
			2.1.1.1 Formulation of the First Law
		2.1.2 Second Universal Law of Motion
			2.1.2.1 Formulation of Newton´s Second Law
		2.1.3 Third Universal Law of Motion
		2.1.4 Range of Validity of Newton´s Universal Laws of Motion
		2.1.5 Relation to the Thermodynamics and Conservation Laws
	2.2 Stress
		2.2.1 Definitions of Stress and Traction
		2.2.2 Stress Vector on an Arbitrary Plane
		2.2.3 Symmetry of Stress Tensor
		2.2.4 Couple Stresses
		2.2.5 Principal Stresses and Principal Axes
		2.2.6 Stress Tensor Invariants
			2.2.6.1 Stress Invariants in Principal Axes
			2.2.6.2 Representation of Stress Tensor in Spherical and Deviatoric Components
			2.2.6.3 Invariants of the Deviatoric Stress Tensor
		2.2.7 Octahedral Plane and Octahedral Stresses
	2.3 Deformation and Strain
		2.3.1 Small Strain Definition
			2.3.1.1 Elementary Definition of Pure Uniaxial Strain
			2.3.1.2 Pure Shear Strain
			2.3.1.3 Pure Rigid Body Motion
		2.3.2 Small Strain and Small Rotation Formulation
			2.3.2.1 Definition of Material (Local) Coordinates
			2.3.2.2 Small Strain in Local Coordinates
			2.3.2.3 Shear Strain in Local (Material) Coordinates
		2.3.3 Small Strain and Rotation in 3-D
	2.4 Kinematics of Continuous Medium
		2.4.1 Material (Local) Description
		2.4.2 Referential Description (Lagrangian Description)
		2.4.3 Spatial Description (Eulerian Description)
		2.4.4 Material Time Derivative in Spatial Coordinates (Substantial Derivative)
	2.5 Rate of Deformation Tensor and Rate of Spin Tensor
		2.5.1 Comparison of Rate of Deformation Tensor, D, and Time Derivative of the Strain Tensor,
		2.5.2 True Strain (Natural Strain) (Logarithmic Strain)
	2.6 Finite Strain and Deformation
		2.6.1 Green Deformation Tensor, C, Cauchy Deformation Tensor, B-1
		2.6.2 Relation Between Deformation, Strain, and Deformation-Gradient Tensors
		2.6.3 Comparing Small Strain and Large (Finite) Strain
		2.6.4 Strain Rate and Rate of Deformation Tensor Relations
		2.6.5 Relation Between, the Spatial Gradient of Velocity Tensor, L and the Deformation-Gradient Tensor, F
	2.7 Rotation and Stretch Tensors in Finite Strain
	2.8 Compatibility Conditions in Continuum Mechanics
	2.9 Piola-Kirchhoff Stress Tensors
		2.9.1 First Piola-Kirchhoff Stress Tensor σ0
		2.9.2 Second Piola-Kirchhoff Stress Tensor
	2.10 Direct Relation Between Cauchy Stress Tensor and Piola-Kirchhoff Stress Tensors
	2.11 Conservation of Mass Principle
	2.12 The Incompressible Materials
	2.13 Conservation of Momentum Principle
	2.14 Conservation of Moment of Momentum Principle
	2.15 Lagrangian Mechanics
	2.16 Hamilton´s Principle: The Principle of Stationary Action
	References
Chapter 3: Thermodynamics
	3.1 Thermodynamic Equilibrium
	3.2 First Law of Thermodynamics
		3.2.1 Work Done on the System (Power Input)
		3.2.2 Heat Input
	3.3 Second Law of Thermodynamics
		3.3.1 Entropy
		3.3.2 Quantification of Entropy in Thermodynamics
		3.3.3 Gibbs-Duhem Relation
		3.3.4 Euler Equation
		3.3.5 Entropy Production in Irreversible Process
		3.3.6 Clausius-Duhem Inequality
		3.3.7 Traditional Use of Entropy as a Functional in Continuum Mechanics
		3.3.8 Entropy as a Measure of Disorder
		3.3.9 Thermodynamic Potential
		3.3.10 Time-Independent Dissipation (Instantaneous Dissipation)
		3.3.11 Dissipation Power and Onsager Reciprocal Relations
	References
Chapter 4: Unified Mechanics Theory
	4.1 Literature Review of Use of Thermodynamics in Continuum Mechanics
	4.2 Laws of Unified Mechanics Theory
		4.2.1 Second Law of Unified Mechanics Theory
		4.2.2 Third Law of Unified Mechanics Theory
	4.3 Evolution of Thermodynamic State Index (Φ)
		4.3.1 On the Relationship Between the Second Fundamental Theorem of The Mechanical Theory of Heat and Probability Calculations...
		4.3.2 Critic of Boltzmann´s Mathematical Derivation
		4.3.3 On the Law of Distribution of Energy in the Normal Spectrum, [Max Planck, (1901) Annalen Der Physik, Vol. 4, p. 553 Ff, ...
		4.3.4 Thermodynamic State Index (TSI) in Unified Mechanics Theory
	4.4 Experimental Verification Example
		4.4.1 Tension-Compression Cyclic Loading
		4.4.2 Monotonic Loading Test
	4.5 Dynamic Equilibrium Equations in Unified Mechanics Theory
		4.5.1 Derivation of the Dynamic Equilibrium Equation
		4.5.2 Vibration of a 1-DOF Mass-Spring System
		4.5.3 Entropy Generation in Sliding Friction Contact: Thermodynamic Fundamental Equation
		4.5.4 Comparison with Newtonian Mechanics Results
		4.5.5 Comparison with Experimental Data
	References
Chapter 5: Unified Mechanics of Thermomechanical Analysis
	5.1 Introduction
	5.2 Unified Mechanics Theory-Based Constitutive Modeling
		5.2.1 Flow Theory and Yield Criteria
		5.2.2 Effective Stress Concept and Strain Equivalence Principle
		5.2.3 Unified Mechanics Theory Implementation
	5.3 Return Mapping Algorithms
		5.3.1 Linearization (Consistent Jacobian)
	5.4 Thermodynamic Fundamental Equation in Thermomechanical Problems
	5.5 Conservation Laws
		5.5.1 Conservation of Mass
		5.5.2 Momentum Principle in Newtonian Mechanics
		5.5.3 Conservation of Energy
	5.6 Entropy Law: Second Law of Thermodynamics
	5.7 Fully Coupled Thermomechanical Equations
	5.8 Numerical Validation of the Thermomechanical Constitutive Model
		5.8.1 Thin Layer Solder Joint: Monotonic and Fatigue Shear Simulations
	5.9 Thermomechanical Analysis of Cosserat Continuum: Length Scale Effects
		5.9.1 Cosserat Couple Stress Theory
		5.9.2 Toupin-Mindlin Higher-Order Stress Theory
		5.9.3 Equilibrium Equations
		5.9.4 Finite Element Method Implementation
		5.9.5 General Couple Stress Theory: Variational Formulation
		5.9.6 Reduced Couple Stress Theory: Variational Formulation
		5.9.7 Reduced Couple Stress Theory: Mixed Variational Principle
		5.9.8 General Couple Stress Theory Implementation
	5.10 Cosserat Continuum Implementation in Unified Mechanics Theory
		5.10.1 Rate-Independent Plasticity Without Degradation
		5.10.2 Rate-Dependent Plasticity (Viscoplasticity) Without Degradation
		5.10.3 Introducing the Thermodynamic State Index
		5.10.4 Entropy Generation Rate in Cosserat Continuum
		5.10.5 Integration Algorithms
	References
Chapter 6: Thermomechanical Analysis of Particle-Filled Composites
	6.1 Introduction
	6.2 Ensemble-Volume Averaged Micromechanical Field Equations
	6.3 Noninteracting Solution for Two-Phase Composites
		6.3.1 Average Stress Norm in Matrix
		6.3.2 Average Stress in Filler Particles
	6.4 Pairwise Interacting Solution for Two-Phase Composites
		6.4.1 Approximate Solution of Two-Phase Interaction
		6.4.2 Ensemble-Average Stress Norm in the Matrix
		6.4.3 Ensemble-Average Stress in the Filler Particles
	6.5 Noninteracting Solution for Three-Phase Composites
		6.5.1 Effective Elastic Modulus of Multiphase Composites
		6.5.2 Ensemble-Average Stress Norm in the Matrix
		6.5.3 Ensemble-Average Stress in Filler Particles
	6.6 Effective Thermomechanical Properties
		6.6.1 Effective Bulk Modulus
		6.6.2 Effective Coefficient of Thermal Expansion (ECTE)
		6.6.3 Effective Shear Modulus
		6.6.4 Effective Young´s Modulus and Effective Poisson´s Ratio
		6.6.5 Numerical Examples
	6.7 Micromechanical Constitutive Model of the Particulate Composite
		6.7.1 Modeling Procedures for Particulate Composites
		6.7.2 Elastic Properties of Particulate Composites
		6.7.3 A Viscoplasticity Model
		6.7.4 Thermodynamic State Index
		6.7.5 Solution Algorithm
		6.7.6 Consistent Elastic-Viscoplastic Tangent Modulus
	6.8 Verification Examples
		6.8.1 Material Properties of ATH
		6.8.2 Properties of PMMA
		6.8.3 Properties of Matrix-Filler Interphase
		6.8.4 Properties of Particulate Composites
		6.8.5 Finite Element Simulation Results
		6.8.6 Cyclic Stress-Strain Response
	References
Chapter 7: Unified Micromechanics of Finite Deformations
	7.1 Introduction to Finite Deformations
	7.2 Frame of Reference Indifference
	7.3 Unified Mechanics Theory Formulation for Finite Strain
		7.3.1 Thermodynamic Restrictions
		7.3.2 Constitutive Relations
	7.4 Thermodynamics State Index
	7.5 Definition of Material Properties
	7.6 Applications of Finite Deformation Models
		7.6.1 Material Properties
	7.7 Numerical Implementation of Dual-Mechanism Model
		7.7.1 Simulating Isothermal Stretching of PMMA
		7.7.2 Simulating Non-isothermal Stretching of PMMA
	References
Chapter 8: Unified Mechanics of Metals Under High Electrical Current Density: Electromigration and Thermomigration
	8.1 Introduction
	8.2 Physics of Electromigration Process
		8.2.1 Driving Forces of Electromigration Process
		8.2.2 Laws Governing Electromigration and Thermomigration
		8.2.3 Electromigration Electron Wind Force
		8.2.4 Temperature Gradient Diffusion Driving Force
		8.2.5 Stress Gradient Diffusion Driving Force
	8.3 Laws of Conservation
		8.3.1 Vacancy Conservation
	8.4 Newtonian Mechanics Force Equilibrium
	8.5 Heat Transfer
	8.6 Electrical Conduction Equations
	8.7 Thermodynamic Fundamental Equation for Electromigration and Thermomigration
		8.7.1 Entropy Balance Equations
	8.8 Example
	References
Chapter 9: Fatigue of Materials
	9.1 Predicting High Cycle Fatigue Life of Metals
		9.1.1 Thermodynamic Fundamental Equation
		9.1.2 Entropy Generation Mechanisms
		9.1.3 Comparison Between Different Entropy Generation Mechanisms
		9.1.4 Temperature Evolution due to Mechanical Work
		9.1.5 Entropy and TSI Calculations
	9.2 Predicting Ultrasonic Vibration Fatigue Life
		9.2.1 Thermodynamic Fundamental Equation
		9.2.2 Temperature Evolution due to Mechanical Work
		9.2.3 Comparison Between Different Entropy Generation Mechanisms
		9.2.4 Computing Thermodynamic State Index
	9.3 Predicting Low Cycle Fatigue Life
	References
Chapter 10: Corrosion-Fatigue Interaction
	10.1 Corrosion
	10.2 Thermodynamic Fundamental Equation of Corrosion-Fatigue
		10.2.1 Entropy Generation Mechanisms During Corrosion
		10.2.2 Entropy Generation Mechanisms: Fatigue
	10.3 Thermodynamic State Index (TSI)
	10.4 Comparing Simulation Results and Test Data
	References
Index