Numerical study of consistency of rate constitutive equations with elasticity at finite deformation

The present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional Oldroyd, Truesdell, Cotter–Rivlin, Jaumann and Green–Naghdi rates and the three novel co-rotational E- and L-based, logarithmic rates, and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated Kirchho stress and material logarithmic strain. To this end, three integration procedures for updating stress are presented. The stress responses of several typical deformation processes are simulated. According to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters. Integrating the other spatial rate equations will provide path-dependent stress response. These numerical conclusions support the arguments in H. Xiao et al. (Acta Mechanica 1999; 138:31–50). The reasons leading to elasticity inconsistency of spatial rate equations are analysed. If the material parameters are assumed to be strain-dependent, the logarithmic rate equation loses also its elasticity-consistent property. The numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other