Dual probabilistic homogenization of polymers filled with rubber particles

Abstract

A comparison of two different probabilistic techniques in the context of a homogenization of the polymer filled with rubber particles is demonstrated in this work. Homogenization approach is based on deformation energy of the Representative Volume Element containing a single spherical particle, which must be equal for the heterogeneous and equivalent homogenized bodies. Both probabilistic methods are based on a series of the Finite Element Method experiments that leads to determination of effective characteristics as the polynomial forms of the input random variables and it is done by using the optimized Weighted Least Squares Method. These functions are next integrated in the semi-analytical probabilistic approach and embedded into the Taylor series expansions as the derivatives in the framework of the alternative stochastic perturbation-based approach. These two strategies are employed to study a polymeric matrix with rubber particle as the filler because such a composite has enormously large contrast in-between expectations of the two random Young's moduli of its constituents. Numerical optimization concerns the order of the approximating polynomial that needs to maximize correlation with the set of trial points and minimize at the same time both computational error with the least squares standard deviation.