I've recently developed a nonlinear viscoelastic constitutive equation which is supposed to model the mechanical properties of HDPE in a spectrum of elastic deformations.
The constitutive model is based on the methodology developed by Pioletti et. al:
To be specific: the total stress is the sum of the elastic response, viscous stress and so-called "long term response" modelled with a heridetary integral (see Pioletti's paper for the details). Somethin'g like Fung's quasi-linear viscoelasticity but with an extra viscous potential Wv to model strain rate dependency.
Now I'd like to conduct some one finite element tests in Abaqus. I'm not very experienced in this matter but for whole I know I should use UMAT subroutine.
I should determine the tangent moduli tensor for UMAT. And here comes my dilema becouse in this kind of constitutive relation I have to potential funtions: We (elastic) and Wv (viscous) so I can compute two various tangent moduli tensors. I have quite no idea how to deal with it.
In case of normal quasi-linear viscoelasticity I would determine the tangent moduli from the elastic potential We. Then, if I'm correct, I would get the elastic stress which then would be incorporated in numerical integrating of a heridetary integral.
But there is this viscous potential... Maybe somebody here has some clue to this (recommended reading for instance).