Probabilistic homogenization of hyper-elastic particulate composites with random interface

Abstract

A principal aim of this study is determination of deterministic and basic probabilistic characteristics of the effective material properties and stresses in hyper-elastic particulate composites with spherical reinforcement and uncertain volume fraction of the interface defects. This is done on the basis of laboratory tests of Laripur LPR 5020 High Density Polyurethane (HDPU) using the FEM homogenization method applied on the cubic multi-particle Representative Volume Element (RVE). This 3D homogenization scheme is based upon numerical determination of strain energy density in the RVE under uniaxial stretch. Each particle is spherical, linear elastic and embedded in the hyper-elastic matrix. The interphase is a thin layer around particle and contains all the particle-matrix interface defects; these defects are of the semi-spherical shape lying with the diameters on the interface. A constitutive relation of the matrix is experimentally recovered through the uniaxial stretch test and computationally approximated with three best fitting available hyper-elastic material potentials. The coincidence of these results serves for verification of the numerical results. Uncertainty of volume fraction of interface defects is assumed to be Gaussian, whose coefficient of variation does not excess 0.25. A relation of the effective stress to this uncertain parameter is recovered as some polynomial basis via the Least Squares Method (LSM) with two different approaches. The first one introduces a functional based directly on the potentials with parameters related to the strain level and to the volume fraction of interface defects, while the alternative approach is based upon a bivariate polynomial. This polynomial order is optimized via simultaneous maximization of correlation and minimization of LSM error and variance and its order is optimized for minimum LSM error. The RVE meshing is completed with the use of 20-noded brick or 10-noded tetrahedral elements. Probabilistic computations have been carried out with three independent approaches, namely the Stochastic Finite Element Method (SFEM), the crude Monte-Carlo simulation and probabilistic semi-analytical method. Probabilistic characteristics of the effective constitutive tensor include expected values, coefficients of variation, skewness and kurtosis of the effective stress. It is investigated numerically (1) if the resulting homogenized characteristics are also Gaussian, (2) how fluctuation of the interface defects volume fraction affects an effective stress in context of the deterministic and probabilistic analysis and (3) how these characteristics vary together with an increase of the applied strain.