A BEM formulation for homogenization of composites with randomly distributed fibers

Abstract In recent papers by the authors, deterministic models of distribution of fibers in composite structures have been studied. For example, problems related to optimization, homogenization, localization, etc., have been solved. The extended Hashin–Shtrikman (H–S) variational principles served as a starting point (eigenparameters were involved in the formulations), and the comparative medium was introduced. The BEM formulations were then admissible and efficient. The formulations of the above-mentioned problems require the restriction of geometry of the fibers to certain ‘locally reasonable’ structures, e.g. to periodic or pseudo-periodic cells. Since the condition of regular distribution of fibers is violated in applications, and a random distribution is more probable, another extension of the H–S principles is needed. In this paper, the problem is extended to the case of statistically distributed fibers. H–S variational principles are formulated in terms of statistical characteristics in the domain and the eigenparameters are also involved, affected by the statistical values. Following the H–S principles, an integral formulation is stated (again, thanks to the use of the comparative medium such a formulation is admissible) in a representative volume, which contains no longer regular geometry of the fibers. The boundary element method has then a special form, which is advantageous particularly for two-phase media. The above-mentioned formulation of H–S variational principles with randomly distributed fields of fibers can be extended to non-linear problems (plasticity, debonding) by introducing transformation fields (eigenstresses or eigenstrains, which are involved in the formulations for completeness). The results form the research presented in this paper basically apply to homogenization of diaphysal implants. But, there is a wide range of applications of the theory introduced in this paper. Due to results from tests on the bearing composite frame of a bicycle, which has a similar structure for certain types of composites of the diaphysal implants, a typical cross-section of the bearing frame of a bicycle is studied as an example. The frame is built of a graphite-epoxy composite.