DOI: https://doi.org/10.20535/2305-9001.2012.66.39005

### ALGORITHMS OF THE DECISION OF PROBLEMS WITH BOUNDARY CONDITIONS, WITH APPLICATION OF THE FINITE ELEMENT METHOD AT THE LARGE ELASTOPLASTIC STRAINS AND TAKING INTO ACCOUNT DAMAGES STRUCTURE. MESSAGE 1. LOGARITHMIC STRAINS

К. М. Рудаков, І. Л. Сидоренко

#### Abstract

The cycle of messages about algorithms FEM for the decision of problems with boundary conditions taking into account the large strains and damage a material is offered. In the given message some practical questions connected with application of logarithmic deformations are considered: the left and right decomposition of a matrix of gradients of strains of Koshi-Green, definition of mainstreams and values of the strains, interfaced tensors stresses. The algorithm of a finding of own numbers and vectors of a matrix with components of strains tensors of Koshi-Green (algorithm of polar decomposition) is in detail stated. On a numerical example matrixes of the left and right decomposition are illustrated. The scheme of an initial, intermediate and current condition of elementary volume of a material is resulted at the left and right decomposition.

Conclusion. Using the Hencky logarithmic strain in the solution of boundary value problems with large deformations requires additional calculations. These calculations are the main directions of strain by determining the eigenvalues and eigenvectors of a nonsingular symmetric matrix of size 3x3. Such an approach should be used in the construction of high-precision algorithms. When using the logarithmic strain is necessary to use the proper measure of stress - the Noll's stress tensor, which is a tensor of the " Kirchhoff's stress excepted rotation." Most theory of the damage equations is constructed for the main directions of strain. Therefore, the use of logarithmic Hencky strain is easily combined with such theories

#### Keywords

large strains; logarithmic strains; polar decomposition; stress tensors

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