Probabilistic and stochastic analysis of the effective properties for the particle reinforced elastomers

► A collection of various constitutive models for the reinforced elastomers. ► Various stochastic computational techniques available for constitutive modeling of the reinforced elastomers. ► Comparative studies of fourth order statistics for the effective shear modulus of elastomers. The main issue in this paper is determination of the effective properties of the reinforced elastomers in terms of uncertainty of the reinforcing particles ratio or the applied strain rate. We discuss in this context various concepts for calculation of those properties like general, cluster-dependent as well as strain-dependent models and also different probabilistic strategies – thanks to analytical, simulation and the tenth order perturbation-based approaches. This is done to determine the most optimal strategy in terms of a time and computer power consumption and to compare an accuracy of the basic probabilistic characteristics determination, especially in the perspective of further usage in the Finite Element Method experiments with real and the homogenized elastomers. Numerical experiments provided for the elastomers reinforced with the carbon black and silica using the computer algebra system MAPLE show a perfect agreement of all those techniques for general models, a lack of analytical solution for the power-law breakdown theory and the very good agreement of the first two probabilistic moments determined according to the nonlinear concepts. Finally, we discuss also the stochastic aging of the reinforced elastomers in terms of the linear increase of the strain rate with the coefficients defined uniquely as the Gaussian random variables with the given first two probabilistic characteristics.