On the overall properties of composites reinforced by fibers with prescribed orientations

This study is concerned with development of bounds on the elastic properties of fiber reinforced composites with arbitrary orientational distribution of fibers. Generalization of the Mori-Tanaka model [1] and Hashin-Schtrikman variational bounds [2] to the cases of non-aligned composite phases are examined. Orientation distribution functions (ODF) are used to describe orientation probability density. It is shown that the Mori-Tanaka scheme applied to the non-aligned fiber reinforced composites violates symmetry of the effective elastic moduli tensor. The study of the literature also reveals that there are no known bounds derived for the composites with orientational distribution (except for the random uniform distribution) of phases. To overcome this issue we propose to formulate a problem of finding tightest bounds for the composites with non-aligned phases as a nonlinear semidefinite optimization problem, i.e., an optimization problem where the optimization variables are represented by symmetric positive semidefinite matrices. Such a formulation guarantees that any solution of the optimization problem represents a valid tensor of elastic material properties. The optimization problem then is solved by an interior point method to produce optimal bounds for the overall elastic properties of two-phase composite with uniform distribution of carbon nanotubes in a polymer matrix.