Introduction to Non-linear Mechanics. A Unified Energetical Approach

Preface
Contents
List of Figures
1 Introduction
	1.1 Some General Features
	1.2 Description of the Motion
	1.3 Homogeneous Deformations
	1.4 The Mobility and the Interactions
		1.4.1 On the Initial Configuration
	1.5 Conservation of Energy and Entropy Production
	1.6 The Linear Thermoelasticity
	1.7 More General Cases
		1.7.1 Generalized Standard Materials
		1.7.2 Linear Visco-elastic Behaviour
		1.7.3 Normality Rule
	1.8 The Quasistatic Evolution
		1.8.1 Dissipative Function
		1.8.2 The Isothermal Boundary Value Problem
	1.9 The Lagrangian and the Dynamical Case
	1.10 The Hamiltonian
	1.11 Some Properties
		1.11.1 Expression of the Conservation of Energy
		1.11.2 Conservation Law
		1.11.3 Property of Stationarity
	1.12 On Discontinuities
		1.12.1 Change of Scale
	References
2 Non-linear and Linear Elasticity
	2.1 Introduction
	2.2 Universal Deformation
	2.3 Properties of Equilibrium Solution
	2.4 Example of Non-linear Elastic Deformation
		2.4.1 The Flexion of a Prismatic Bar
		2.4.2 The Antiplane-Shear
	2.5 Linear Elasticity: Small Perturbations
	2.6 Equilibrium Solution of a Linear Elastic Body
	2.7 Stability and Bifurcation in Non-linear Elasticity
		2.7.1 Notion of Stability
		2.7.2 The Metronome
		2.7.3 The Euler Column
	References
3 Elasto-plasticity
	3.1 Introduction
	3.2 The Domain of Reversibility
	3.3 The Evolution of Internal State
	3.4 A Model of Perfect Plasticity
	3.5 The Rate Boundary Value Problem
		3.5.1 Characterization of Equilibrium
		3.5.2 The Internal State Evolution
		3.5.3 Primal Formulation
	3.6 On the Adjoin State of Evolution Problem
	3.7 Cyclic Plasticity
	3.8 Classical Solutions in Elasto-plasticity
		3.8.1 A Three Bars Lattice Under Traction
		3.8.2 Case of a Hollow Sphere
	3.9 Finite Elasto-plasticity
		3.9.1 Case of Homogeneous Polycristal
	3.10 Stability and Bifurcation in Elastoplasticity
		3.10.1 The Shanley Column
		3.10.2 A Model of Elastoplastic Beam
	References
4 Fracture Mechanics
	4.1 Introduction
	4.2 Case of Linear Elasticity
	4.3 Crack Propagation in Plane Conditions
	4.4 Energetical Interpretation
	4.5 Invariance and J-integral
	4.6 Dual Approach in Linear Elasticity
	4.7 On the Rate Boundary Value Problem
	4.8 Interaction of Cracks
	4.9 Stability and Uniqueness: A Simple Example
	4.10 Case of Hyperelasticity
	4.11 Case of Dynamics
	4.12 On Inhomogeneous Body
		4.12.1 On the Rate Boundary Value Problem
	4.13 Asymptotic Fields Near a Planar Crack in Linear Elasticity
		4.13.1 Invariant Integrals upper JJ, upper G Subscript thetaGθ
		4.13.2 Mode I
		4.13.3 Mode II
		4.13.4 Mode III
		4.13.5 General Remark
	4.14 Separation of the Modes of Rupture
	4.15 For a Non Planar Crack
	References
5 Moving Discontinuities
	5.1 Introduction
	5.2 Dissipation Analysis
		5.2.1 In the Dynamical Case
	5.3 General Features for Quasi-static Evolution
	5.4 Moving Discontinuity
		5.4.1 The Equilibrium State
		5.4.2 Variations of the Potential Energy
		5.4.3 Dissipation and Evolution of the Interface
		5.4.4 Examples on a Bar
		5.4.5 A Model with Dissipation: A Quasi-brittle Material
	5.5 Problem of Evolution
	5.6 The Rate Boundary Value Problem
		5.6.1 Stability and Bifurcation
	5.7 An Example
	5.8 Connection with Fracture
		5.8.1 The Quasi-Crack Problem
		5.8.2 Peculiar Solutions of Equilibrium Equation
	5.9 The Quasi-Crack Solution in Mode III
		5.9.1 Determination of the Constants
		5.9.2 Solutions for alpha greater than or equals 0αge0
		5.9.3 Solution for alpha less than or equals 0αle0
		5.9.4 A Particular Constitutive Law
		5.9.5 The Particular Case alpha equals 0α=0
	References
6 Damage Modelling and Initiation  of Defect
	6.1 Introduction
	6.2 A Simple Local Damage Model
		6.2.1 Evolution of Damage Parameter
		6.2.2 Properties of Damage Field
		6.2.3 Models with Local Discontinuities: An Axial Description
	6.3 Models with Damage Gradient
		6.3.1 The Total Potential Energy and its Variations
		6.3.2 On the Bar in Extension
	6.4 A Model of Graded Damage
		6.4.1 The Equilibrium Problem
		6.4.2 On the Regularity of the Fields
		6.4.3 The Total Potential Energy
		6.4.4 The Bar Under Uni-axial Extension
	6.5 A Regularized Graded Damage Model
		6.5.1 On the Bar in Extension
	6.6 Comparison Between Graded Damage and Thick-Level Set Model
		6.6.1 Model with Convex Constrains
	6.7 The State of Equilibrium
		6.7.1 On the Evolution of Damage
	6.8 On the Rate Boundary Value Problem
	6.9 On the Role of the Curvature: Example on a Sphere
		6.9.1 The Inhomogeneous Sphere Under Radial Loading
		6.9.2 The Sharp Interface
		6.9.3 A Graded Damaged Sphere
	6.10 Coupling with Plasticity
		6.10.1 Sharp Interface
		6.10.2 Solution with Transfer of Internal State
		6.10.3 Sharp Versus Diffuse Interface
	References
7 A Thermodynamical Approach  to Contact Wear
	7.1 Introduction
	7.2 The Energetical Approach
	7.3 The Dissipation
		7.3.1 Interface Propagation Law
		7.3.2 Description of the Interface
	7.4 An Application of the Model
	7.5 Global Approach of the Interface
	7.6 On Change of the Contact Surface
	References
8 Delamination of Laminates
	8.1 Introduction
	8.2 The Kinematic of the Plates
	8.3 Conservation of the Momentum
	8.4 Dissipation Analysis
	8.5 The Rate Boundary Value Problem
	8.6 Delamination of a Thin Membrane Under Pressure
	References
9 On Relationships Between Micro–Macro Quantities
	9.1 Introduction
	9.2 Mode and Process of Localization
	9.3 Potentials and General Properties
	9.4 Macrohomogeneous Body and Linear Elasticity
	9.5 On the Decomposition of the Macroscopic Strain
	9.6 Moving Interfaces
	9.7 Case of Linear Elastic Phases
	9.8 More General Cases
	9.9 The Composite Sphere Assemblage
	9.10 Extension to Finite Deformation
	9.11 From Monocrystal to Polycrystal
		9.11.1 On the Elastic Behaviour
		9.11.2 On Elastoplastic Behaviour
	References
10 Homogenization in Linear Elasticity
	10.1 The Problem of Inhomogeneous Elasticity
	10.2 Introduction of a Comparison Material
	10.3 Isotropic Spatial Distribution of Mechanical Phases
	10.4 On Particulate Composite Material
	10.5 On the Hashin\'s Spheres Assemblage
	10.6 Extension to Imperfect Interface
		10.6.1 Estimation of the Global Behaviour
		10.6.2 Choice of the Reference Medium
		10.6.3 Interpretation
		10.6.4 Case of Conduction
		10.6.5 Evaluation of upper Q left parenthesis upper K Subscript o Baseline right parenthesisQ(Ko) and upper Q asterisk left parenthesis 1 divided by upper K Subscript o Baseline right parenthesisQ*(1/Ko)
	References
11 Optimal Control and Non Linear Inverse Problems
	11.1 Inverse Problems in Linear Elasticity
		11.1.1 The Problem Setting
		11.1.2 A Well Posed Problem
		11.1.3 The Idea of Control
		11.1.4 The Optimization Method
	11.2 Inverse Problem in Elastoplasticity
		11.2.1 Inverse Problems on Three Bars Lattice
		11.2.2 Inverse Problem When h Subscript o Baseline equals 0ho=0
	11.3 Estimation of the Internal State in Elastoplasticity
		11.3.1 The Inverse Problem on a Sphere
	11.4 Boundary Control and Extension in Viscoplasticity
	References
12 Conclusion
Appendix A Tensorial Analysis
A.1 Bilinear form Associated to a Linear Mapping
A.2 Euclidean Vector Space
A.3 Differential Operators
A.3.0.1 Cartesian Coordinates
A.3.0.2 On Other Basis
Appendix B General Relations
B.1 Continuous Case
B.2 Discontinuous Case
Appendix C Particular Solution in Linear Elasticity
C.1 Cylinders and Spheres Under Radial Loading
C.1.1 Case of Uniform lamdaλ
C.2 A Cylindrical or Spherical Shell Under Shear
C.3 Fundamental Linear Elastic Solution
C.3.1 Plane Isotropic Elasticity
C.3.2 3D-Elasticity
C.3.3 Case of on Half Plane in Plane Strain
C.4 Anti-plane Elasticity
C.4.1 Case of the Half-plane y greater than 0y>0
Appendix D Hodograph Transformation
Appendix E Convex Analysis
Appendix F Optimal Control
Appendix G Some Integrals
Index