Fourier transforms - An alternative to finite elements for elastic-plasticstress-strain analyses of heterogeneous materials

The intent of this paper is to apply the technique of discrete Fourier tran sforms (DFT) to the computation of the stress and strain fields around hole s in an externally loaded two-dimensional representative volume clement (RV E). This is done to show that DFT is capable to handle geometries with rath er 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 e quivalent inclusion technique is used to derive a functional equation which , in principle, allows to compute numerically the stresses and strains with in an RVE that contains heterogeneities of arbitrary shape and arbitrary st iffness (in comparison to the surrounding matrix). This functional equation is finally specialized to the case of circular and elliptical holes of var ious slenderness which degenerate into Griffith cracks in the limit of a va nishing minor axis. The numerically predicted stresses and strains are comp ared 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 clement studies (for the case of time-independent elastic/plastic material behavior ).