Effect of inclusion shape on stiffness of isotropic and transversely isotropic two-phase composites

Abstract

Constitutive relations are developed for linear and non-linear incompressible, two-phase composites. The second phase is assumed to be comprised of spheroidal inclusions, and both Isotropie composites (containing randomly oriented inclusions) and transversely isotropic composites (containing aligned inclusions) are examined. Constitutive relations are established first for linearly elastic constituents and dilute concentrations of inclusions. Corresponding (approximate) constitutive relations applicable for non-dilute concentrations of inclusions are then obtained using a differential self-consistent scheme. Next, constitutive relations for non-linear matrix materials are developed using a procedure recently introduced by Ponte Castaneda [J. Mech. Phys. Solids (1991)]. The procedure uses a variational framework to exploit the constitutive relations developed for linear matrix behavior in order to obtain approximate results for non-linear matrix behavior. The results obtained are rigorous bounds on the behavior of the composites when the concentration of inclusions is sufficiently small, and are estimates for the behavior at larger concentrations of inclusions. The constitutive relations established for linear and non-linear composites are used to examine the effect of inclusion shape on the stiffness of composites, and a wide range of inclusion shapes ranging from thin disk-shaped inclusions to slender needle-like inclusions is considered.