Dual probabilistic homogenization of the rubber-based composite with random carbon black particle reinforcement

This study concerns numerical determination of the basic statistics of the effective elasticity tensor for the rubber reinforced with the carbon black particles. This goal is achieved by an application of the iterative and generalized stochastic perturbation technique implemented as the Stochastic Finite Element Method and applied to the homogenization problem of such a composite. A fundamental difference of this approach to the traditional Taylor expansion is in development of the basic equations for higher order probabilistic characteristics, where traditional linearization procedure (expectation is approximated by the zeroth order term) has been replaced by sequential (iterative) symbolic calculation of these characteristics. The radius of the carbon black particle has been chosen as the input Gaussian random parameter and it affects both FEM-based and also analytical method of the effective tensor components determination. Sensitivity analysis in addition to this radius together with the FEM computational error for the homogenization problem are carried out here prior to the principal stochastic analysis. We contrast the iterative SFEM with two other probabilistic numerical methods, namely the classical Monte-Carlo scheme and also semi-analytical probabilistic FEM strategy. Both stochastic perturbation and semi-analytical method are related to the same polynomial response functions of the input random particle radius, but the first employs Taylor series expansion while the second - symbolic integration with Gaussian PDF to calculate the final probabilistic characteristics of the effective tensor. This study shows some remarkable differences in-between numerical and analytical homogenization methods in the context of geometrical uncertainty in the RVE of such a composite.