  Mathematical models describing the behavior of viscoelastic materials are
often based on evolution equations that measure the change in stress depending
on its material parameters such as stiffness, viscosity or relaxation time. In
this article, we introduce a Maxwell-based rheological model, define the
associated forward operator and the inverse problem in order to determine the
number of Maxwell elements and the material parameters of the underlying
viscoelastic material. We perform a relaxation experiment by applying a strain
to the material and measure the generated stress. Since the measured data
varies with the number of Maxwell elements, the forward operator of the
underlying inverse problem depends on parts of the solution. By introducing
assumptions on the relaxation times, we propose a clustering algorithm to
resolve this problem. We provide the calculations that are necessary for the
minimization process and conclude with numerical results by investigating
unperturbed as well as noisy data. We present different reconstruction
approaches based on minimizing a least squares functional. Furthermore, we look
at individual stress components to analyze different displacement rates.
Finally, we study reconstructions with shortened data sets to obtain assertions
on how long experiments have to be performed to identify conclusive material
parameters.
