Introduction to Finite Strain Theory for Continuum Elasto-Plasticity

Perspective: Distinction of finite elastoplasticity from infinitesimal elastoplasticity Chapter 1 Mathematical preliminaries 1.1 Basic symbols and convention 1.2 Definition of tensor 1.2.1 Objective tensor 1.2.2 Quotient law 1.3 Vector analysis 1.3.1 Scalar product 1.3.2 Vector product 1.3.3 Scalar triple product 1.3.4 Vector triple product 1.3.5 Reciprocal set of vectors 1.3.6 Tensor product 1.4 Tensor analysis 1.4.1 Properties of second-order tensor 1.4.2 Components of tensor 1.4.3 Transposed tensor 1.4.4 Inverse tensor 1.4.5 Orthogonal tensor 1.4.6 Decompositions of tensor (a) Symmetric and skew-symmetric tensors (b) Spherical and deviatoric tensors (c) Normal and tangential tensors: Projection tensors 1.4.7 Axial vector 1.4.8 Determinant 1.4.9 Uniqueness of solution of linear simultaneous equation 1.4.10 Scalar triple products with tensor 1.4.11 Transformation of vector set to right- and left-handed set 1.4.12 Pseudo scalar, vector and tensor 1.5 Representations of tensor 1.5.1 Notations of tensors 1.5.2 Components of tensor and transformation rule 1.5.3 Various notations 1.5.4 Fourth-order identity, permutation, symmetrizing, etc. tensors 1.5.5 Isotropic tensors 1.6 Principal values and vectors 1.6.1 Principal values and vectors of second-order tensor 1.6.2 Spectral representation and general tensor functions 1.6.3 Calculations of principal values and vectors (a) Eigenvalues (b) Eigenvectors 1.6.4 Principal values and vectors of orthogonal tensor 1.6.5 Principal values and vectors of skew-symmetric tensor and axial vector 1.6.6 Cayley-Hamilton theorem 1.7 Polar decomposition 1.8 Isotropy 1.8.1 Isotropic material 1.8.2 Representation theorem of isotropic tensor-valued tensor function 1.9 Differential formulae 1.9.1 Partial derivatives 1.9.2 Directional derivatives 1.9.3 Taylor expansion 1.9.4 Time derivatives and Lagrangian and Eulerian descriptions 1.9.5 Derivatives of tensor filed 1.9.6 Gauss' divergence theorem 1.9.7 Material-time derivative of volume integration 1.10 Variation of line, surface element, volume elements and their rates 1.10.1 Variation of line, surface and volume elements 1.10.2 Rates of line, surface and volume elements 1.11 Continuity and smoothness conditions 1.11.1 Continuity condition 1.11.2 Smoothness condition Chapter 2 General (curvilinear) coordinate system 2.1 Primary and reciprocal base vectors 2.2 Metric tensor 2.3 Representations of vector and tensor 2.4 Physical components of vector and tensor 2.5 Covariant derivative with Christoffel symbol 2.6 Covariant derivatives of scalar, vector and tensor 2.7 Riemann-Christoffel curvature tensor 2.8 Component description from symbolic notation and Cartesian coordinate system Chapter 3 Description of physical quantities in embedded coordinate system 3.1 Necessity for description in embedded coordinate system 3.2 Embedded base vectors 3.3 Deformation gradient 3.4 Pull-back and push-forward operations Chapter 4 Strain and strain 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 Hecnky strain tensor 4.3 Compatibility condition 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 tensors based on general strain tensors 4.5 Representations of strain rate and spin tensors in Lagrangian and Eulerian triads 4.6 Decomposition of deformation gradient tensor into isochoric and volumetric parts Chapter 5 Convected derivative 5.1 Convected derivative 5.2 Corotational rate 5.3 Objectivity Chapter 6 Conservation laws and stress (rate) tensors 6.1 Conservation laws 6.1.1 Basic conservation law 6.1.2 Conservation law of mass 6.1.3 Conservation law of linear momentum 6.1.4 Conservation law of angular momentum 6.2 Stress tensors 6.2.1 Cauchy stress tensor 6.2.2 Symmetry of Cauchy stress tensor 6.2.3 Various stress tensors a) Contravariant pull-back of Kirchhoff stress b) Two-point pull-back of Kirchhoff stress c) Contravariant-covariant pull-back of Kirchhoff stress d) Covariant-contravariant pull-back of Kirchhoff stress e) Covariant pull-back of Kirchhoff stress f ) Rotational pull-back of Kirchhoff stress g ) Contravariant-rotational pull-back of Kirchhoff stress 6.3 Equilibrium equation 6.4 Equilibrium equation of angular moment 6.5 Conservation law of energy 6.6 Virtual work principle 6.7 Work conjugacy 6.8 Stress rate tensors a ) Contravariant covnvected derivatives b ) Covariant-contravariant convected derivatives c ) Covariant convected derivatives d ) Corotational convected derivatives 6.9 Some basic loading behavior 6.9.1 Uniaxial loading followed by rotation 6.9.2 Simple shear 6.9.3 Combined loading of tension and distortion Chapter 7 Hyperelasticity 7.1 Hyperelastic constitutive equation and its rate form 7.2 Examples of hyperelastic constitutive equations 7.2.1 St. Venant-Kirchhoff elasticity 7.2.2 Modified St. Venant-Kirchhoff elasticity 7.2.3 Neo-Hookean elasticity 7.2.4 Modified Neo-Hookean elasticity (1) 7.2.5 Modified Neo-Hookean elasticity (2) 7.2.5 Modified Neo-Hookean elasticity (3) 7.2.6 Modified Neo-Hookean elasticity (4) Chapter 8 Constitutive equation of finite elastoplasticity 8.1 Multiplicative decomposition 8.2 Stress and deformation tensors for intermediate configuration 8.3 Incorporation of nonlinear-kinematic hardening 8.3.1 Rheological model for nonlinear-kinematic hardening 8.3.2 Multiplicative decomposition for kinematic hardening 8.4 Strain tensors 8.5 Strain rate and spin tensors 8.5.1 Strain rate and spin tensors in current configuration 8.5.2 Contravariant-covariant pull-backed strain rate and spin tensors in intermediate configuration 8.5.3 Covariant pull-backed strain rate and spin tensors in intermediate configuration 8.5.4 Strain rate tensor for kinematic hardening 8.6 Stress tensors 8.7 Influences of superposed rotations: objectivity 8.8 Hyperelastic equations for elastic deformation and kinematic hardening 8.8.1 Hyperelastic constitutive equation 8.8.2 Hyperelastic type constitutive equation for kinematic hardening 8.9 Plastic constitutive relations 8.9.1 Normal-yield and subloading surfaces 8.9.2 Consistency condition 8.9.3 Plastic and kinematic hardening flow rules 8.9.4 Plastic strain rate 8.10 Stress rate-strain rate relation 8.10.1 Stress rate-strain rate relation in intermediate configuration 8.10.2 Stress rate-strain rate relation in reference configuration 8.10.3 Stress rate-strain rate relation in current configuration 8.11 Material functions of metals 8.11.1 Strain energy function of elastic deformation 8.11.2 Strain energy function for kinematic hardening 8.11.3 Yield function 8.11.4 Plastic strain rate and kinematic hardening strain rate 8.12 On the constitutive equation in current configuration by spectral representation 8.13 On the Clausius-Duhem inequality and the principle of maximum dissipation Chapter 9 Computational methods for finite elastoplasticity 9.1 A brief review of numerical methods for finite strain elastoplasticity 9.2 Brief summary of model formulation 9.2.1 Constitutive equations 9.2.2 Normal-yield and subloading functions 9.2.3 Plastic evolution rules 9.2.4 Evolution rule for subloading surface 9.3 Transformation to reference configuration 9.3.1 Constitutive equations 9.3.2 Normal-yield and subloading functions 9.3.3 Plastic evolution rules 9.3.4 Evolution rule for subloading surface 9.4 Time-integration of plastic evolution rules 9.5 Update of deformation gradient tensor 9.6 Elastic predictor step and loading criterion 9.7 Plastic corrector step by return-mapping 9.8 Derivation of Jacobian matrix for return-mapping 9.8.1 Components of Jacobian matrix 9.8.2 Derivatives of tensor exponentials 9.8.3 Derivatives of stresses 9.9 Consistent (algorithmic) tangent moduli tensor 9.9.1 Analytical derivation of consistent tangent moduli tensor 9.9.2 Numerical computation of consistent tangent moduli tensor 9.10 Numerical examples 9.10.1 Example 1: Strain-controlled cyclic simple shear analysis 9.10.2 Example 2: Effect of subloading surface 9.10.3 Example 3: Large monotonic simple shear analysis with kinematic hardening model 9.10.4 Example 4: Accuracy and convergence assessment of stress-update algorithm 9.10.5 Example 5: Finite element simulation of large deflection of cantilever 9.10.6 Example 6: Finite element simulation of combined tensile, compressive, and shear deformation for cubic specimen 10 Computer programs 55 10.1 User instruction and input file description 10.2 Output file description 10.3 Computer programs 10.3.1 Structure of Fortran program returnmap 10.3.2 Main routine of program returnmap 10.3.3 Subroutine to define common variables: comvar 10.3.4 Subroutine for return-mapping: retmap 10.3.5 Subroutine for isotropic hardening rule: plhiso 10.3.6 Subroutine for numerical computation of consistent tangent moduli tensor: tgnum0 Chapter 10 Computer programs 10.1 User instruction and input file description 10.2 Output file descriptions 10.3 Computer programs 10.3.1 Structure of Fortran program returnmap 10.3.2 Main routine of program returnmap 10.3.3 Subroutine for definition of common variables: comvar 10.3.4 Subroutine for return-mapping: retmap 10.3.5 Subroutine for isotropic hardening law: plhiso 10.3.6 Subroutine for numerical computation of consistent tangent moduli tensor: tgnum0 Appendices A Projection of area B Geometrical interpretation of strain rate and spin tensors C Proof for derivative of second invariant of logarithmic-deviatoric deformation tensor D Tensor exponential function and its derivative D.1 Numerical implementation of tensor exponential function D.2 Fortran subroutine for tensor exponential function: matexp D.3 Numerical implementation of derivative of tensor exponential function D.4 Fortran subroutine for derivative of tensor exponential function: matdex References Index