The finite deformation of nonlinear composite materials .2. Evolution of the microstructure

This work deals with the development of constitutive models for two-phase nonlinearly viscous and rigid-perfectly plastic composites with evolving microstructures. Part I of the work was concerned with the estimation of instantaneous constitutive relations for the class of particulate microstructures with aligned ellipsoidal inclusions. This second part deals with the identification of appropriate variables characterizing the state of the microstructure, and with the development of evolution equations for these variables. Under the assumption of triaxial loading conditions, it is argued that aligned ellipsoidal inclusions deform-in some average sense-into ellipsoidal inclusions with different size and shape. The appropriate state variables are thus the current values of the volume fractions of the phases and the aspect ratios of the inclusions. The pertinent evolution laws then follow from well-known kinematical relations, together with appropriate estimates for the average strain rate in the inclusion and matrix phase. The resulting constitutive models take the form of standard homogenized stress-strain rate relations, supplemented by evolution equations for the above-mentioned state variables. Although the ultimate goal of this study is to be able to model complex forming processes, illustrative results are given here only for axisymmetric and plane strain deformations of composites with rigid-perfectly plastic phases. The main conclusion is that effective behavior of these composites will not be perfectly plastic, but may exhibit hardening, or even softening, depending on the specific nature of the applied loading conditions.