The BEM formulation of statistically distributed fibers in composites

摘要

In recent papers of the authors, deterministic models of distribution of fibers in composite structures have been studied. For example, optimization, homogenization, localization, etc., have been solved. The extended Hashin-Shtrikman variational principles served as the starting point (eigenparameters were involved in the formulations), and the comparative medium was introduced. The REM formulations were then admissible and efficient. The formulations of the above-said 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 statistical (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. Hashin-Shtrikman variational principles are formulated in terms of statistical characteristics in the domain and the eigenparameters are also involved, burdened by the statistical values. Following the H-S principles, integral formulation is stated (again, thanks to the use of the comparative medium such a formulation is admissible) in a representative volume, which has 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 Hashin-Shtrikman variational principles with randomly distributed fields of fibers can be extended to a nonlinear problems (plasticity, debonding) by introducing transformation fields.