A mathematical model for an upper convected Maxwell fluid with an elastic core: Study of a limiting case

In this paper we formulate a mathematical model for a continuum which behaves like an upper convected visco-elastic Maxwell fluid if the stress is above a certain threshold and like a neo-Hookean elastic solid if the stress is below that threshold. The constitutive equations for each phase are derived within the context of the theory of natural configurations and by means of the criterion of the maximization of the rate of dissipation [11]. We then focus on a limiting case in which the continuum becomes an elastic-rigid body. In this limiting case the constitutive relation of the material becomes implicit and, although there is no energy dissipation, it cannot be included in the class of hyperelastic (or Green) bodies.