Beyond additive decomposition in Rational Extended Thermodynamics: A coupled model for one-dimensional nonlinear viscoelasticity

Abstract

We develop a one-dimensional isothermal model of nonlinear viscoelasticity within Rational Extended Thermodynamics (RET) without postulating an additive decomposition of either the stress or the internal energy. The total stress is taken as an independent field and is associated with an additional balance law whose density, flux, and production are initially unspecified. Compatibility with the energy-dissipation principle determines the thermodynamically admissible structure of this balance law and leads to a theory governed by a single, generally non-separable internal-energy potential. The total stress is thereby represented as the sum of two potential-derived contributions associated, respectively, with deformation and the internal relaxation variable; in general, both depend on the full constitutive state. The relaxation variable is constitutively centered so that the same value represents equilibrium for every admissible deformation. The production law is then chosen so that the first relaxation approximation recovers the prescribed parabolic constitutive behavior. For the Newtonian class considered here, this yields a linear dissipative relation between the production and the nonequilibrium stress contribution associated with the relaxation variable. The resulting equilibrium reduction reproduces the formal step underlying the first Maxwellian iteration for gases, whereas the same constitutive strategy also accommodates non-Newtonian parabolic responses. Under suitable convexity assumptions, the resulting system is symmetric hyperbolic. Its linearization recovers the classical Zener solid, while the previous RET model with a separable energy potential is included as a special case. A simple coupled potential is used to illustrate the effects of the new constitutive freedom on nonlinear stress relaxation and wave dispersion, and the K -condition is verified on the equilibrium manifold.