A two material constant, nonlinear elastic stress constitutive equation including the effect of compressibility

Abstract

A new constitutive equation for compressible, nonlinear elasticity is derived. The stress-deformation relation takes the final form σ ij = 3k 2J (J 2 3 - 1) δ ij + μ J (B ij - j 2 3 δ ij) where B ij is the left Cauchy-Green deformation tensor. J is the Jacobean of the deformation, and k and μ are the two governing mechanical properties having the same interpretation as in linear, isotropic elasticity theory. In this nonlinear theory, k and μ are separately determinable from large deformation conditions of pure dilatation and simple shear, respectively. Under isochoric conditions, the distortional part of the theory takes the form of the kinetic theory of rubber elasticity. Under conditions of small volume change, the theory coincides with the molecular theory of Flory derived for elastomers, but given explicit from only in the one dimensional case.