The finite deformation of nonlinear composite materials—I. Instantaneous constitutive relations

This work deals with the development of effective constitutive models for two-phase nonlinearly viscous and rigid-perfectly plastic composites subjected to finite deformations. The main new feature of the model is the ability to account approximately for the evolution of the microstructure resulting from the finite changes in geometry during a given deformation process. The model is formulated in terms of instantaneous constitutive relations for the composites, which depend on appropriate variables characterizing the state of the microstructure, together with evolution equations for these state variables. This first part of the work is concerned with the development of the instantaneous constitutive relations for classes of microstructures that are sufficiently broad to capture the changes in microstructure associated with general triaxial finite-deformation histories. Simple formulae are given for the effective potentials and stress-strain rate relations of power-law viscous composites made of aligned ellipsoidal inclusions of one phase dispersed in a matrix of a second phase. In addition, effective yield surfaces are computed for the special case of two-phase rigid-perfectly plastic composites with aligned spheroidal inclusions. The effects of the phase volume fractions, inclusion shape and yield strength ratio on these effective yield surfaces are considered. These effects will turn out to be important in the understanding of the effective response of the composites with evolving microstructures, which is considered in Part II of this work.