Probabilistic homogenization of polymers filled with rubber particles

The main purpose of this work is computational simulation of the basic probabilistic characteristics of the homogenized tensor for polymer filled with rubber particles. The Representative Volume Element (RVE) of this composite contains a single spherical particle and composite components are treated as statistically homogeneous and isotropic uniquely defined by the Gaussian elastic moduli. Probabilistic approach is based upon the generalized stochastic perturbation technique allowing for large random dispersions for the input random variables and is implemented using the polynomial response functions recovered via the Least Squares Method. Homogenization technique consists in equating of deformation energies for the real composite and artificial isotropic material characterized by the effective elasticity tensor. The cell problem is solved using ABAQUS® by an application of uniform deformations on specific outer surfaces of the composite cell and using tetrahedral finite elements C3D4, while probabilistic part is carried out in the symbolic computations package MAPLE®. The main conclusion coming from the performed numerical analysis is almost perfect agreement of our mean values with the analytical calculations, dominating role of the Young modulus of the polymeric matrix and negligible role of the elastic modulus for rubber. We also notice that the Gaussian character is transferred from the first modulus on the homogenized tensor, while randomization of the rubber Young's modulus is negligible. The energy approach will allow for future applications of more realistic constitutive models of rubber-filled polymers as well as for the RVEs of larger size - containing an agglomeration of the rubber particles, for instance.