Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity: Multiplicative Decomposition with Subloading Surface Model

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity
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Contents
Preface
1 Mathematical fundamentals
	1.1 Matrix algebra
		1.1.1 Summation convention
		1.1.2 Kronecker’s delta and alternating symbol
		1.1.3 Matrix notation and determinant
	1.2 Vector
		1.2.1 Definition of vector
		1.2.2 Operations of vector
			1.2.2.1 Scalar product
			1.2.2.2 Vector product
			1.2.2.3 Scalar and vector triple products
			1.2.2.4 Primary and reciprocal vectors
			1.2.2.5 Tensor product
	1.3 Definition of tensor
	1.4 Tensor operations
		1.4.1 Properties of second-order tensor
		1.4.2 Tensor components
		1.4.3 Transposed tensor
		1.4.4 Inverse tensor
		1.4.5 Orthogonal tensor
		1.4.6 Tensor decompositions
			1.4.6.1 Symmetric and skew-symmetric tensors
			1.4.6.2 Spherical and deviatoric tensors
		1.4.7 Axial vector
		1.4.8 Determinant
		1.4.9 Simultaneous equation for vector components
	1.5 Representations of tensors
		1.5.1 Notations in tensor operations
		1.5.2 Operational tensors
		1.5.3 Isotropic tensors
	1.6 Eigenvalues and eigenvectors
		1.6.1 Eigenvalues and eigenvectors of second-order tensors
		1.6.2 Spectral representation and elementary tensor functions
		1.6.3 Cayley–Hamilton theorem
		1.6.4 Scalar triple products with invariants
		1.6.5 Second-order tensor functions
		1.6.6 Positive-definite tensor and polar decomposition
		1.6.7 Representation theorem of isotropic tensor-valued tensor function
	1.7 Differential formulae
		1.7.1 Partial derivatives of tensor functions
		1.7.2 Time derivatives in Lagrangian and Eulerian descriptions
		1.7.3 Derivatives of tensor field
			1.7.3.1 Gradient
			1.7.3.2 Divergence
			1.7.3.3 Rotation (or curl)
		1.7.4 Gauss’ divergence theorem
		1.7.5 Material-time derivative of volume integration
	1.8 Variations of geometrical elements
		1.8.1 Deformation gradient and variations of line, surface and volume elements
		1.8.2 Velocity gradient and rates of line, surface and volume elements
2 Curvilinear coordinate system
	2.1 Primary and reciprocal base vectors
	2.2 Metric tensor and base vector algebra
	2.3 Tensor representations
3 Tensor operations in convected coordinate system
	3.1 Advantages of description in embedded coordinate system
	3.2 Convected base vectors
	3.3 Deformation gradient tensor
	3.4 Pull-back and push-forward operations
	3.5 Convected time-derivative
		3.5.1 General convected derivative
		3.5.2 Corotational rate
		3.5.3 Objectivity of convected rate
4 Deformation/rotation (rate) tensors
	4.1 Deformation tensors
	4.2 Strain tensors
		4.2.1 Green and Almansi strain tensors
		4.2.2 General strain tensors
		4.2.3 Logarithmic strain tensor
	4.3 Volumetric and isochoric parts of deformation gradient tensor
	4.4 Strain rate and spin tensors
		4.4.1 Strain rate and spin tensors based on velocity gradient tensor
		4.4.2 Strain rate tensor based on general strain tensor
5 Conservation laws and stress tensors
	5.1 Conservation laws
		5.1.1 Conservation law of physical quantity
		5.1.2 Conservation law of mass
		5.1.3 Conservation law of linear momentum
		5.1.4 Conservation law of angular momentum
	5.2 Cauchy stress tensor
		5.2.1 Definition of Cauchy stress tensor
		5.2.2 Symmetry of Cauchy stress tensor
	5.3 Balance laws in current configuration
		5.3.1 Translational equilibrium
		5.3.2 Rotational equilibrium: symmetry of Cauchy stress tensor
		5.3.3 Virtual work principle
		5.3.4 Conservation law of energy
	5.4 Work-conjugacy
		5.4.1 Kirchhoff stress tensor and work-conjugacy
		5.4.2 Work-conjugate pairs
		5.4.3 Physical meanings of stress tensors
			5.4.3.1 Two-point contravariant pull-back: first Piola–Kirchhoff and Nominal stress tensors
			5.4.3.2 Contravariant pull-back: second Piola–Kirchhoff stress tensor
			5.4.3.3 Covariant–contravariant pull-back: Mandel stress tensor
		5.4.4 Relations of stress tensors
		5.4.5 Relations of stress tensors to traction vectors
	5.5 Balance laws in reference configuration
		5.5.1 Translational equilibrium
		5.5.2 Virtual work principle
		5.5.3 Conservation law of energy
	5.6 Simple shear
6 Hyperelastic equations
	6.1 Basic hyperelastic equations
	6.2 Hyperelastic constitutive equations of metals
		6.2.1 St. Venant–Kirchhoff elasticity
		6.2.2 Modified St. Venant–Kirchhoff elasticity
		6.2.3 Neo-Hookean elasticity
		6.2.4 Modified neo-Hookean elasticity (1)
		6.2.5 Modified neo-Hookean elasticity (2)
		6.2.6 Modified neo-Hookean elasticity (3)
		6.2.7 Modified neo-Hookean elasticity (4)
	6.3 Hyperelastic equations of rubbers
	6.4 Hyperelastic equations of soils
	6.5 Hyperelasticity in infinitesimal strain
7 Development of elastoplastic and viscoplastic constitutive equations
	7.1 Basis of elastoplastic constitutive equations
		7.1.1 Fundamental requirements for elastoplasticity
			7.1.1.1 Decomposition of deformation/rotation (rate) into elastic and plastic parts
			7.1.1.2 Incorporation of yield surface
			7.1.1.3 Stress rate versus strain rate relation
		7.1.2 Requirements for elastoplastic constitutive equation
			7.1.2.1 Continuity condition
			7.1.2.2 Smoothness condition
	7.2 Historical development of elastoplastic constitutive equations
		7.2.1 Infinitesimal hyperelastic-based plasticity
		7.2.2 Hypoelastic-based plasticity
		7.2.3 Multiplicative hyperelastic-based plasticity
	7.3 Subloading surface model
	7.4 Cyclic plasticity models
		7.4.1 Cyclic kinematic hardening models with yield surface
		7.4.2 Ad hoc Chaboche model and Ohno-Wang model excluding yield surface
		7.4.3 Extended subloading surface model
	7.5 Formulation of (extended) subloading surface model
		7.5.1 Normal-yield and subloading surfaces
		7.5.2 Evolution rule of elastic-core
		7.5.3 Plastic strain rate
		7.5.4 Strain rate versus stress rate relations
		7.5.5 Calculation of normal-yield ratio
		7.5.6 Improvement of inverse and reloading responses
		7.5.7 Cyclic stagnation of isotropic hardening
	7.6 Implicit time-integration: return-mapping
		7.6.1 Return-mapping formulation
		7.6.2 Loading criterion
		7.6.3 Initial value of normal-yield ratio in plastic corrector step
		7.6.4 Consistent tangent modulus tensor
	7.7 Subloading-overstress model
		7.7.1 Constitutive equation
		7.7.2 Defects of past overstress model
		7.7.3 On irrationality of creep model
		7.7.4 Implicit stress integration
		7.7.5 Temperature dependence of isotropic hardening function
	7.8 Fundamental characteristics of subloading surface model
		7.8.1 Distinguished abilities of subloading surface model
		7.8.2 Bounding surface model with radial-mapping: Misuse of subloading surface model
8 Multiplicative decomposition of deformation gradient tensor
	8.1 Elastic-plastic decomposition of deformation measure
		8.1.1 Necessity of multiplicative decomposition of deformation gradient tensor
		8.1.2 Isoclinic concept
		8.1.3 Uniqueness of multiplicative decomposition
		8.1.4 Embedded base vectors in intermediate configuration
	8.2 Deformation tensors
		8.2.1 Elastic and plastic right Cauchy−Green deformation tensor
		8.2.2 Strain rate and spin tensors
			8.2.2.1 Strain rate and spin tensors in current configuration
			8.2.2.2 Strain rate and spin tensors in intermediate configuration
			8.2.2.3 Substructure spin
	8.3 On limitation of hypoelastic-based plasticity
	8.4 Multiplicative decomposition for kinematic hardening
9 Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations
	9.1 Stress measures
	9.2 Hyperelastic constitutive equations
	9.3 Conventional elastoplastic model
		9.3.1 Flow rules for plastic strain rate and plastic spin
		9.3.2 Confirmation for uniqueness of multiplicative decomposition
		9.3.3 Plastic strain rate
	9.4 Continuity and smoothness conditions
	9.5 Initial subloading surface model
	9.6 Multiplicative extended subloading surface model
		9.6.1 Multiplicative decomposition of plastic deformation gradient for elastic-core
		9.6.2 Normal-yield, subloading, and elastic-core surfaces
		9.6.3 Plastic flow rules
		9.6.4 Plastic strain rate
	9.7 Material functions of metals and soils
		9.7.1 Metals
			9.7.1.1 Hyperelastic equation
			9.7.1.2 Hyperelastic equation for kinematic hardening variable
			9.7.1.3 Hyperelastic equation for elastic-core
			9.7.1.4 Yield function
		9.7.2 Soils
			9.7.2.1 Hyperelastic equation
			9.7.2.2 Yield function
	9.8 Calculation procedure
	9.9 Implicit calculation by return-mapping
		9.9.1 Return-mapping
		9.9.2 Loading criterion
		9.9.3 Initial value of normal-yield ratio in plastic corrector step
	9.10 Cyclic stagnation of isotropic hardening
	9.11 Multiplicative subloading-overstress model
		9.11.1 Constitutive equation
		9.11.2 Calculation procedure
		9.11.3 Implicit calculation by return-mapping
	9.12 On multiplicative hyperelastic-based plastic equation in current configuration
10 Subloading-friction model: finite sliding theory
	10.1 History of friction models
	10.2 Sliding displacement and contact traction vectors
	10.3 Hyperelastic sliding displacement
	10.4 Normal-sliding and subloading-sliding surfaces
	10.5 Evolution rule of friction coefficient
	10.6 Evolution rule of sliding normal-yield ratio
	10.7 Plastic sliding velocity
	10.8 Calculation procedure
	10.9 Return-mapping
		10.9.1 Return-mapping formulation
		10.9.2 Loading criterion
	10.10 Subloading-overstress friction model
	10.11 Implicit stress integration for subloading-overstress friction model
	10.12 On crucially important applications of subloading-friction model
		10.12.1 Loosening of screw
		10.12.2 Deterministic prediction of earthquake occurrence
11 Comments on formulations for irreversible mechanical phenomena
	11.1 Utilization of subloading surface model
		11.1.1 Mechanical phenomena described by subloading surface model
		11.1.2 Standard installation to commercial software
	11.2 Disuses of rate-independent elastoplastic constitutive equations
	11.3 Impertinence of formulation of plastic flow rule based on second law of thermodynamics
Appendix 1 Proofs for formula of scalar triple products with invariants
Appendix 2 Convective stress rate tensors
Appendix 3 Cauchy elastic and hypoelastic equations
	A3.1 Cauchy elastic equation
	A3.2 Hypoelastic equation
Bibliography
Index