Constitutive relations and strain energy in finite strain viscoelastic solid mechanics

I’m an applied math graduate student, and my research is straying into hyperviscoelastic models of materials. I’ve had trouble finding an answer to this question I have about the mathematical theory of finite strain viscoelasticity, so I’m hoping asking around here might yield some answers/directions.

I’ve seen a few review papers and some numerical studies that posit certain constitutive relations between the 2nd Piola-Kirchhoff stress tensor $mathbf{S}$ and the Lagrange strain tensor $mathbf{E}=frac{1}{2}(mathbf{F}^Tmathbf{F}-1)$ that mimic the integral formulations of 1D viscoelasticity, i.e.

$$mathbf{S}=int_0^tG(mathbf{E},t-tau)dot{boldsymbol{f}}(mathbf{E},tau);dtau$$

Where $G$ is some nonlinear memory kernel/relaxation function and $boldsymbol{f}$ is some nonlinear function of the strain.

The above equation is fairly general. There are of course many different classes of nonlinear constitutive laws constructed phenomenologically to model particular materials, but I’m interested in the analytical derivation of a stress-strain constitutive law from a strain energy function analogous to hyperelastic models:

Is it possible to arrive at an integral form for a nonlinear viscoelastic constitutive law by taking the gradient of some strain energy function $Psi$?
$$mathbf{S}=frac{partialPsi}{partial mathbf{E}}=int_0^tG(mathbf{E},t-tau)dot{boldsymbol{f}}(mathbf{E},tau);dtau$$

What would that be? Would such a strain energy function take on the form of a nonlinear functional? E.g.,
$$Psi(mathbf{E},t)=Lambdaleft[mathbf{E},tright]$$
where $Lambda$ contains a double integral? Would he gradient now be a Frechet derivative of some some sort? Are there any examples of such theoretical derivations of viscoelastic stress-strain relations?