Energy of a non-linear viscoelastic model compatible with fractional relaxation

Recently, a non-linear model of viscoelasticity based on Rational Extended Thermodynamics was proposed in Ruggeri (2024). This theory extends the evolution of the viscous stress beyond the linear framework of the Maxwell model to the non-linear realm, provided that the viscous energy function is given. This work aims at establishing a possible constitutive law for the viscous energy such that the relaxation modulus of the fractional Maxwell model with order $\alpha \in (1/2,1]$ is contained within the solutions of the (non-linear) relaxation experiment. Necessary and sufficient conditions for the existence of this coincident solution are discussed, together with a numerical evaluation of the viscous energy associated with the nonlinear model.