On implicit constitutive relations for materials with fading memory

Abstract

Starting with an implicit constitutive relation between the stress and relative deformation gradient histories to describe the response of a fluid, and using the assumption of fading memory, we show that both rate type and differential type fluid representations can be obtained, as approximations in retarded motions, that have the same form as the Maxwell, Oldroyd-B, Rivlin–Ericksen, and other popular fluid models. This result provides further evidence that the recently proposed implicit constitutive framework provides a very robust and general methodology to describe fluid response. Thus, fluids defined through an implicit constitutive relation between the stress and relative deformation gradient histories can be seen as an appropriate generalization of the notion of simple fluids and rate type fluids.