Deformation analysis for finite elastic-plastic strains in a lagrangean-type description

Abstract

A general concept is presented to analyse the deformation of structures undergoing arbitrarily large elastic and arbitrarily large plastic strains. Based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions the kinematics of two superposed finite, non-coaxial deformations are investigated. Lagrangean-type elastic and plastic stretch tensors are introduced and multiplicative decompositions of the total stretch into these elastic and plastic stretches are derived. It is shown that the result is independent of any decomposition of the total rotation into an elastic and a plastic rotation.For the second, superposed deformation the total Lagrangean logarithmic (Hencky) strain tensor with corresponding elastic and plastic logarithmic strains is defined. If in a large deformation analysis the first deformation is updated such that the second deformation is constrained to be moderately large, then the total logarithmic strain tensor of the second deformation can be additively decomposed into purely elastic and purely plastic parts. This enables an appropriate formulation of constitutive equations for isotropic hyperelastic material behavior with associated flow rule and evolution laws for combined isotropic-kinematic hardening. Work-conjugate to the elastic logarithmic strain tensor is a “back-rotated” Kirchhoff stress tensor. The rotational change of its reference configuration during the update is given explicitly.Finally the principle of virtual work with corresponding equilibrium equations and static and geometric boundary conditions is given. The virtual work functional is transformed to deliver the consistent tangent stiffness matrix as basis for a finite element solution algorithm.