Analytical stress solutions of a closed deformation path with stretching and shearing using the hypoelastic formulations

In the present work a set of hypoelastic analytical stress solutions of a closed deformation path, consisting of four deformation phases with stretching and shearing, is developed. The rate constitutive equations, based on the Oldroyd, Truesdell, Cotter–Rivlin, Jaumann, Green–Naghdi and logarithmic stress rates, are exploited. According to a representation principle of the material spin tensors, issued in Xiao et al. (1998a, 1998b), a set of new and concise representation equations of the spin tensors for the four deformation phases is derived. The derived spin equations play a crucial role for formulating the hypoelastic closed-form corotational-rate-based stress solutions of the specific deformation path. The analytical stress solutions as functions of the deformation phases are illustrated and complete comparisons between the residual stresses at the end of the deformation path are performed. These analytical stress solutions, as a set of living examples of applying the hypoelastic equations, support the conclusion that in all spatial hypoelastic rate equations only the one based on the logarithmic stress rate is consistent with elasticity.