Finite Stretching and Shearing of an Internally Balanced Elastic Solid

Abstract

When seeking to separately describe elastic and inelastic effects in solids undergoing finite strain, it is typical to factor the deformation gradient \(\bf F\) into parts, say \(\mathbf{F}={\hat {\mathbf{F}}}{\mathbf{F}}^{*}\). Such a factorization can also be used to describe a notion of internal elastic balance. For equilibrium deformations this balance emerges naturally from an appropriately formulated energy minimization by considering the variation with respect to the multiplicative decomposition itself. The internal balance requirement is then a tensor relation between F, \({\hat {\mathbf{F}}}\) and F∗. Such theories holds promise for describing various substructural reconfigurations in solids. This includes the development of surfaces that localize slip once a threshold value of load is obtained. We consider an incompressible internally balanced elastic material with a constitutive law that is motivated by neo-Hookean behavior in the standard hyperelastic theory. The role of the internal balance relation is examined in uniaxial loading and also in simple shearing. In uniaxial loading the principle directions remain fixed. In simple shearing the principle directions change as the deformation proceeds. To address the latter, we establish a specific sense in which the internal balance relation can be solved analytically for general isochoric \(\bf F\).