A symmetric hyperbolic non-isothermal model for viscoelastic solids and non-Newtonian fluids

Viscoelastic materials and non-Newtonian fluids often exhibit a pronounced sensitivity to temperature, which significantly influences their mechanical response. Recently, Ruggeri proposed a nonlinear viscoelastic model within the framework of Rational Extended Thermodynamics and showed that, by a suitable modification of the production term, the same structure can also describe non-Newtonian fluids with a finite relaxation time in an isothermal setting. In this paper, we extend these models to non-isothermal processes in one spatial dimension in the absence of heat flux. By coupling the balance laws of momentum and energy with a balance law for an additional nonequilibrium stress variable and enforcing the entropy principle together with convexity, we derive a thermodynamically admissible system of nonlinear evolution equations. The resulting model is symmetric hyperbolic, which guarantees local well-posedness of the Cauchy problem and admits weak solutions, including shocks. For Newtonian production terms, we further verify the Shizuta–Kawashima condition, yielding global-in-time smooth solutions for sufficiently small initial data. A unified framework is thus obtained, capable of describing either nonlinear thermo-viscoelastic behavior or temperature-dependent non-Newtonian rheology in the parabolic (zero-relaxation) limit. The principal-subsystem viewpoint clarifies the nesting of reduced theories: in particular, the previously proposed isothermal model is recovered as a principal subsystem, and classical hyperelastic dynamics emerges as a common principal subsystem of the isothermal and Euler-type limits.