Gaussian uncertainty in homogenization of rubber–carbon black nanocomposites

The objective of this work is Gaussian uncertainty in material parameters of the rubber–carbon black particle reinforced nano-composite in the framework of determination of its effective elastic parameters. The analytical 3D micro-mechanical model is contrasted here with the Finite Element Method one realized with the use of the system ABAQUS® and its tetrahedral finite elements C3D4. They discretize cubic Representative Volume Element (RVE) with a centrally located spherical carbon black particle, whose deformation energies accumulated during the uniaxial and biaxial uniform tension are computed. The polynomial-based Response Function Method allows to determine analytical functions relating effective elasticity tensor with Young moduli and Poisson ratios of this composite original components via the Weighted Least Squares Method; this is done in the symbolic environment of MAPLE. These functions serve for the initial sensitivity analysis, where we detect Poisson ratio of the rubber matrix as the crucial design parameter and its Young modulus as having secondary importance, while particle Young modulus and Poisson ratio are totally irrelevant. Next, the most influential parameters are randomized according to the Gaussian distribution and are employed in the dual probabilistic scheme – by using the stochastic perturbation and, independently, semi-analytical techniques to determine up to the fourth order probabilistic characteristics of the effective tensor components. Deterministic and probabilistic hypersensitivity of the effective constitutive tensor, expected according to the rubber matrix presence, is confirmed in all computational experiments. Such a probabilistic energy-related FEM approach will allow for future applications of more advanced constitutive models for the carbon black nano-composites, for the RVEs of larger sizes – containing large agglomerations of these particles and for the imperfect interfaces simulation.