2D versus 3D probabilistic homogenization of the metallic fiber-reinforced composites by the perturbation-based stochastic Finite Element Method

The main purpose of this work is computational simulation of the expectations, standard deviations, skewness and kurtosis of the homogenized tensor for some composites with metallic components. The Representative Volume Element (RVE) of this composite contains a single cylindrical fiber and their components are treated as statistically homogeneous and isotropic media uniquely defined by the Gaussian elastic modulus. Probabilistic approach is based on the generalized stochastic perturbation technique allowing for large random dispersions of the input random variables and is implemented using the polynomial response functions recovered using the Least Squares Method. Homogenization technique employed is dual and consists of (1) stress version of the effective modules method and (2) its displacements counterpart based on the deformation energies of the real and homogenized composites. The cell problem is solved for the first case by the plane strain homogenization-oriented code MCCEFF and, in the 3D case, using the system ABAQUS® (8-node linear brick finite elements C3D8), where the uniform deformations are imposed on specific outer surfaces of the composite cell; probabilistic part is carried out in the symbolic computations package MAPLE®. We compare probabilistic coefficients of the effective elasticity tensor computed in this way with the corresponding coefficients for their upper and lower bounds and this is done for the composite with small and larger contrast between Young moduli of the fiber and the matrix. The main conclusion coming from the performed numerical analysis is a very good agreement of the probabilistic moments resulting from 2 and 3D computer models; this conclusion is totally independent from the contrast between elastic moduli of both composite components.