Application of a constitutive equation for softening, yield and permanent deformation to finite plane simple shear

Abstract

The finite homogenous simple shear deformation of an incompressible material is considered. The response is modeled with a constitutive equation that reflects a continuous process of microstructural transformation as the deformation increases beyond a threshold value. The original and transformed portions of the material are both taken to respond as incompressible elastic solids. It is shown that the transformation can lead to softening of the response with increasing deformation and to a local maximum in the shear stress-shear strain curve. The existence of permanent deformation after release of the shearing traction is demonstrated. It is confirmed that a process of increasing deformation followed by decreasing deformation to the point of zero shear traction is a dissipative cycle. A special case is then considered in which both the original and transformed materials are assumed to respond as neo-Hookean solids. The critical volume fraction of transforming material at which the shear stress-shear strain curve loses monotonicity is found analytically. Representations are obtained for the dependence of the residual shear deformation on the fraction of transforming material; on the ratio of moduli of the original and transformed materials; and on the maximum shear reached before unloading.