Constitutive relation, limited stretchability, and stability of residually stressed Gent materials

Abstract

We exactly determine the constitutive relation and the strain energy function for an initially stressed material that completely follows the characteristics of the Gent model in the virtual stress-free reference. The parameters of the deduced models are expressed as functions of initial stress. Quite interestingly, we observe that when the stiffening parameter approaches infinity, the present model assumes the form of the (existing) initially stressed neo-Hookean model. We apply the developed model to perform the inflation test for a residually stressed cylinder when the axial stretch is constrained or unconstrained. The residually stressed Gent model does exhibit limited stretchability and rapid strain-stiffening in large strain. Additionally, when the axial stretch unconstrained, the inflation can exhibit either a stable characteristics, or a snap-through instability. However, the maximum allowable stretch, which is generally a function of stiffening parameter only, depends considerably on the residual stress. Furthermore, the stability of inflating the cylinder with unconstrained axial stretch also depends strongly on residual stress. Residual stress influences the critical stiffening parameter, which completely determines the stability and which is a constant for the Gent model. We observe that residual stress improves the stability and pressure-bearing capacity of a pressure vessel. Since this is an exact model, the present results should hold accurately whenever the material has the Gent model characteristics in a stress-free state.