A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

Abstract

Parameter estimation for generalized Maxwell models for viscoelastic materials can become ill-posed when insufficient experimental data is available. In this article, we introduce a rheological model containing Maxwell elements, 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 simulate a relaxation experiment by applying a strain to the material and measure the generated stress. Since the mechanical response varies with the number of Maxwell elements, the forward operator of the underlying inverse problem depends on parts of the solution. Thereby, the forward problem consists in computing stress responses for a given number of Maxwell elements, stiffness parameters and relaxation times. The inverse problem means to compute these parameters from given stress measurements, where an additional difficulty lies in the fact that the forward mapping changes with the number of Maxwell elements and, thus, with a quantity to be computed as part of the solution. Under the assumption that every relaxation time is located in one temporal decade we propose a clustering algorithm to resolve this problem. We provide the calculations that are necessary for the minimization process and conclude by investigating unperturbed as well as noisy data. Different reconstruction approaches for the stiffnesses and relaxation times based on minimizing a least squares functional are presented. We look at individual stress components to analyze different strain rates and displacement rates, respectively, and study how experimental duration affects the identified material parameters.