Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1478–1483

# Analysis of Finite Elasto-Plastic Strains: Integration Algorithm and Numerical Examples

• L. U. Sultanov
Selected Articles from the Journal Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki

## Abstract

The paper is devoted to the development of a calculation technique for elasto-plastic solids with regard to finite strains. The kinematics of elasto-plastic strains is based on the multiplicative decomposition of the total deformation gradient into elastic and inelastic (plastic) components. The stress state is described by the Cauchy stress tensor. Physical relations are obtained from the equation of the second law of thermodynamics supplemented with a free energy function. The free energy function is written in an invariant form of the left Cauchy–Green elastic strain tensor. An elasto-plasticity model with isotropic strain hardening is considered. Based on an analog of the associated rule of plastic flows and the von Mises yield criterion, we develop the method of stress projection onto the yield surface (known as the radial return method) with an iterative refinement of the current stress-strain state. The iterative procedure is based on the introduction of additional virtual stresses to the resolving power equation. The constitutive relations for the rates and increments of the true Cauchy stresses are constructed. In terms of the incremental loading method, the variational equation is obtained on the basis of the principle of possible virtual powers. Spatial discretization is based on the finite element method; an octanodal finite element is used.We present the solution to the problem of tension of a circular bar and give a comparison with results of other authors.

## Keywords and phrases

nonlinear elasticity finite strains plasticity

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