Deterministic and probabilistic homogenization limits for particulate composites with nearly incompressible components

Conventional expressions for the homogenized tensor components relevant to the particulate composites consisting of two linear elastic constituents are modified here towards application of the incompressible component. This analysis is subdivided into two important engineering situations - a combination of the incompressible rubber particle with polymeric matrix and of the incompressible rubber matrix with carbon particles. This modification is done by limit transition with Poisson ratio to its upper physical limit and, further, by assuming that contrast parameter in-between Young moduli of both components tends to 0. Such an approach enables also for an analytical calculation of the sensitivity coefficients of effective elastic characteristics with respect to material or geometrical parameters of both composites, which can be a starting point for widely available gradient optimization techniques. Analytical formulas obtained in deterministic case are then used in uncertainty analysis where analytical formulas for the basic probabilistic moments and coefficients of the effective tensor are obtained while randomizing some material or geometrical parameters of the polymer matrix or carbon particles. Fundamental value of this approach is that the first two probabilistic moments, cross-correlations as well as probabilistic entropies are given by the exact analytical equations, so that they are not affected by statistical and non-statistical computer methods errors.