On the dynamic dilatation of a compressible Rivlin cube beyond its elastic limit

When an isotropic hyperelastic unit cube subjected to dynamic tri-axial extension/compression dilates successfully beyond its elastic limit, namely into its work-hardening deformation regime, plastic flow transforms any kind of induced into permanent anisotropy. If, for instance, two pairs of forces are identical while the third pair is different, then the initially isotropic material properties will transform permanently into those of transverse isotropy. For this problem, a plasticity model is presented that enables the energy stored during the work-hardening deformation stage of the resulting cuboid to be influenced not only by a tensorial measure of the observed deformation, but also by a measure of the plastic flow that takes place simultaneously. The model considers that plastic flow still obeys conventional plastic yield criteria, but does not postulate a-priory a rule that splits the observed deformation into elastic and plastic parts. Derivation of constitutive equations is based instead on the postulate that the strain energy density of the material is a function of the deformation gradient tensor and either the rate-of-plastic-deformation tensor encountered during loading within the work-hardening deformation regime or the residual strain tensor encountered after unloading is completed from some relevant offset yield point. An example application presents a complete analytical solution to the deformation problem of a dynamically loaded Rivlin cube which is made initially of a compressible Rivlin-Mooney material.