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 [T. Ruggeri, Int. J. Non-Linear Mech. 160, 104658 (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 α∈(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.