Determination of elastic properties of very heterogeneous media with cellular automata

Abstract

A cellular automaton is used to calculate the elastic properties of very heterogeneous media. The lattice gas method (designed to model viscous flow) is applied to elastic problems through the correspondence between the steady state velocity field in an incompressible Newtonian fluid at low Reynolds number and the displacement field in an incompressible elastic solid. The cellular automaton is applied to determine the elastic properties of a matrix containing randomly distributed hard inclusions. Because the equations of equilibrium are locally solved for any distribution of the inclusion phase, the method provides a means to estimate the influence of stress field perturbations on the macroscopic properties of heterogeneous systems. The effective elastic shear modulus is calculated as a function of the concentration C of the inclusion phase (0≤ C ≤1) for different values of R , the rheological contrast between the inclusion and the matrix. The numerical results are compared with those obtained by various micromechanical models. Estimates given by the cellular automaton agree with the analytical solution for a double periodic triangular system of discs. However, the effective shear modulus for the cellular automaton fluid does not fall within the limits of Hashin and Shtrikman. It follows the same trend as that of the lower bound but stays below the lower bound for all concentrations. Likewise, the shear modulus for a cellular automaton fluid is also systematically lower than that predicted by the differential method, although the trends are similar.