Extension of Mori-Tanaka Theorem to Crack Problem: 5th Report, Macroscopic Elastic Moduli of the Material Containing Mutual Perpendicular Penny-Shaped Cracks

A partial differential equation is derived for the macroscopic total strains of a material containing mutual perpendicular penny-shaped cracks with respect to the crack densities of the cracks by using the incremental form of the Mori-Tanaka theorem. By solving the partial differential equation, the macroscopic total strain, the average interaction stress and hence the macroscopic elastic moduli are formulated as a function of the crack densities of the cracks. On the contrary to the results obtained by the ordinary Mori-Tanaka theorem, the resultant macroscopic elastic muduli asymptotically tend to zero as the crack densities of the cracks increase. The present results are in good agreement with the numerical results by means of the differential scheme when the magnitudes of the crack densities of the mutual perpendicular penny-shaped cracks are equal to each other. The volume fraction of the randomly oriented penny-shaped cracks in physical meaning is obtained by comparing the resultant interaction stress with that derived from the Mori-Tanaka theorem.