Stability, bifurcation and post-critical behavior of a homogeneously deformed incompressible isotropic elastic parallelepiped subject to dead-load surface tractions

• We study the equilibrium of an incompressible, isotropic elastic body.
• We consider particular dead-load surface tractions.
• We analyze stability and bifurcation issues.
• We study the post-bifurcation response for a Mooney–Rivlin material.

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.