Fourier transforms — An alternative to finite elements for elastic-plastic stress-strain analyses of heterogeneous materials

Summary

The intent of this paper is to apply the technique of discrete Fourier transforms (DFT) to the computation of the stress and strain fields around holes in an externally loaded two-dimensional representative volume element (RVE). This is done to show that DFT is capable to handle geometries with rather sharp corners as well as steep gradients in material properties which is of importance for modeling changes in micro-morphology. To this end DFT is first briefly reviewed. In a second step it is applied to the appropriate equations which characterize a linear-elastic as well as a time-independent elastic-plastic, heterogeneous material subjected to external loads. The equivalent inclusion technique is used to derive a functional equation which, in principle, allows to compute numerically the stresses and strains within an RVE that contains heterogeneities of arbitrary shape and arbitrary stiffness (in comparison to the surrounding matrix). This functional equation is finally specialized to the case of circular and elliptical holes of various slenderness which degenerate into Griffith cracks in the limit of a vanishing minor axis. The numerically predicted stresses and strains are compared to analytical solutions for problems of the Kirsch type (a hole in an large plate subjected to tension at infinity) as well as to finite element studies (for the case of time-independent elastic/plastic material behavior).