Configurational-force-based finite element mesh refinement for elastic-plastic problems

Gábor Hénap

doi:10.3311/pp.me.2012-1.04

Abstract

In this paper the computation of configurational forces in case of elastic-ideally plastic material will be examined. Numerical computation of the error in configurational forces will also be introduced in elastic and plastic domain. It will be shown that the so-called r-adaptive mesh refinement procedure \cite 1 is also applicable for elastic-plastic problems as well as the configurational force driven h-adaptive scheme. In some special examples configurational forces are computable in analytic way. This is useful to compare the solution with numerical results, therefore validating the finite element procedure. Two plane problems will be considered where analytical solutions are known. The first one is the thick walled tube model loaded by internal pressure. Second one is an artificial problem where the displacement field assumed to be known in every point of the domain considered. According to the papers Krieg \cite 5 and Szabó \cite 8 analytical solution is obtainable for stress and strain distributions, if the time derivative of the strain is constant. R- and h-adaptive procedures will demonstrated on these two examples.

Highlights

Configurational mechanics presented by Eshelby in the early fifties is widely used on several areas of theoretical continuum mechanics and computational mechanics as well
In case of finite element computation this requirement won’t be fulfilled and nodal configurational forces appear on interior nodes too
Considering that configurational force vectors point to direction of increasing total potential, moving the nodes in opposite direction the optimal mesh is obtainable

Summary

Introduction

Configurational mechanics presented by Eshelby in the early fifties is widely used on several areas of theoretical continuum mechanics and computational mechanics as well. In case of finite element computation this requirement won’t be fulfilled and nodal configurational forces appear on interior nodes too. Since configurational force indicates the error on finite element mesh, computation of this quantity is suitable to drive h-adaptive mesh refinement. These methods were demonstrated in several articles for elastic case, e.g. Elastic domain Strain energy density for linear elastic, isotropic and homogeneous material could have the form (W(ε) where ε is the elastic strain. The value of c could be a small fraction of the quotient of maximal configurational force norm and minimal element size This expression is only applicable on free degrees of freedom which means that boundaries of the model have to remain intact during the mesh reconfiguration.

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Conclusions