Variational bounds on elastic constants for the penetrable sphere model

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Abstract
Since analytical results are known for the two-point and three-point spatial correlation functions of the penetrable sphere model. Milton's geometric parameters zeta and eta may be computed numerically as accurately as desired. Once tabulated these geometric parameters may be then used to provide variational bounds on elastic constants for a wide variety of two-phase composite materials assuming that the geometrical distribution of constituents is similar to that of the penetrable sphere model. The present paper develops the required numerical methods for calculating the Milton numbers, provides a table of results, and demonstrates the application to variational bounds in a few cases. In those cases considered, the bounds of McCoy (1970) and of Milton and Phan-Thien (1982) on the shear modulus are found to be virtually indistinguishable for the penetrable sphere model.