Effects of the orientation distribution of thin soft inclusions on the effective elastic moduli of microheterogeneous material

Abstract

Many natural composite materials such as carbonate rocks contain systems of oriented or partially oriented thin inclusions (microcracks) filled with a soft elastic cement. In this paper we have studied the influence of the inclusion orientation, shape, and elastic properties on the effective elastic properties of micro-inhomogeneous materials. We have calculated the components of the compliance tensor of such materials as a function of crack density. The results were obtained for thin ellipsoidal inclusions with elastic compliance much larger than the elastic compliance of the matrix. To calculate the effective compliance tensor, we have used the “non-interaction approximation” (NIA). The application of the NIA allows us to evaluate the influence of peculiarities in the spatial distribution of inclusions on the effective properties of the medium. To simplify the calculations, we have used the special tensorial basis (T-basis). We have obtained the explicit expressions for the effective elastic compliance tensor of inhomogeneous materials. To verify the analytical expressions, we have compared our results for elastic moduli to the results obtained by numerical modeling of a material containing crack-like inclusions. The results obtained by using the two methods are in very good agreement, thus indicating that our analytic method can be applied even at sufficiently large values of crack density.