Stress/strain computation in heterogeneous bodies with discrete Fourier transforms––different approaches

In recent years the study of heterogeneous bodies, i.e., the computation of local stress–strain fields which may arise due to elastic or thermal mismatches of the constituents or the computation of effective properties (homogenisation), based on the discrete Fourier transformation (DFT) becomes more and more attractive. In fact, several contributions in the field of composite analyses are dedicated to the application of DFT (see [Int. J. Solid Struct. 3 (28) (1999) 3941; C. R. Acad. Sci. Paris 318 (II) (1994) 1417; Proceedings of the IUTAM Symposium on Transformation Problems in Composite and Active Materials, Kluwer Academic Publishers, The Netherlands, 1998, p. 61]). This paper investigates two different approaches to use DFT in order to predict the local stress/strain distribution in externally loaded two-dimensional representative volume elements (RVEs) made out of heterogeneous material. Therefore, the properties of DFT are firstly surveyed and then applied to the solution of a linear elastic material response. By the application of the equivalent inclusion method a functional equation is derived which permits the numerical computation of stresses and strains within an RVE filled with heterogeneities of arbitrary shape and stiffness. Two types of difference schemes which have to be used for the numerical solution of these functional equations are provided. To discuss these different approaches some inhomogeneity problems are numerically solved, and the results are compared to the corresponding analytical results.