Duality relations, correspondences and numerical results for planar elastic composites

This paper addresses three topics relating to planar elastic composites with anisotropic phases. First, the duality relations of Berdichevski are generalized to a wider class of planar elastic media. These yield phase interchange relations for the e ective compliance tensors of certain two phase media. Second, a simple derivation is given of the correspondence between a speci c class of planar elastic problems, and the associated pairs of antiplane elastic problems. This correspondence allows one to determine the e ective compliance tensor from the e ective shear matrices of the associated antiplane problems. Third, a numerical algorithm is developed and implemented for computing the e ective moduli of two phase composites with orthotropic phases. The numerical algorithm is based on an integral equation and can, more generally, be applied to solving for the elds in elastic bodies comprised of several phases di ering in their anisotropies. A series of examples demonstrates the accuracy of the numerical method and the agreement with the theoretical ndings.