A Study on a Grade-One Type of Hypo-Elastic Models

In Hypo-elastic constitutive models an objective rate of the Cauchy stress tensor is expressed in terms of the current state of the stress and the deformation rate tensor D in a way that the dependency on the latter is a homogeneously linear one. In this work, a type of grade-one hypo-elastic models (i.e. models with linear dependency of the hypo-elasticity tensor on the stress) is considered for isotropic materials based on the objective corotational rates of stress. A positive real parameter denoted by n is involved in the considered type. Different values can be selected for this parameter, each selection leads to a specific model within the class of grade-one hypo-elasticity. The spin of the associated corotational rate is also dependent on the parameter n. In the special case of n=0, the corresponding hypo-elastic model reduces to a grade-zero one with the logarithmic rate of stress; noting that this rate is a corotational rate associated with the logarithmic spin tensor. Moreover, by choosing n=2, the model reduces to a grade-one hypo-elastic model with the Jaumann rate, i.e. the corotational rate associated with the vorticity spin tensor. As case studies, the simple shear problem is investigated with utilizing the considered type of hypo-elastic models with various values for parameter n, and the curves for the stress-shear response are depicted.