Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

Abstract

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J 2-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.