Branching during axial compression histories of solid cylinders of time-dependent materials

This work considers the uniaxial compression of a solid circular cylinder of time-dependent material. Initially, the cylinder undergoes a homogeneous compression history. The purpose is to determine a time when this deformation history can form a new inhomogenous branch. Material time dependence is described by a nonlinear single-integral constitutive equation that relates the stress in the material to its deformation history. A criterion is developed for determining branching times using a Pipkin–Rogers constitutive equation for nonlinear incompressible isotropic viscoelastic solids. For the purpose of numerical examples, material parameters are chosen so that the material acts as a neo-Hookean material in its short-time response and a softer one in its long-time equilibrium response. Examples show that characteristic relaxation times and deformation histories influence the time to branch and the corresponding compressive stretch and compressive force.