A semi-analytical method for determining homogenized relaxation times and moduli in Prony series of a heterogeneous viscoelastic material

Abstract

This work introduces a semi-analytical method to obtain homogenized relaxation times and moduli in terms of Prony series of a heterogeneous viscoelastic material. This can be treated as an extension of linear homogenization theory which calculates homogenized elastic properties of a heterogeneous elastic material. The heterogeneous viscoelastic material is consist of multiple phases where each phase is assumed as a standard-solid viscoelastic material. The idea of so-called reduced-order-homogenization method is introduced to propose a set of residual-free governing equations with respect to the averaged strains and eigenstrains of all the phases. The set of governing equations can be rewritten as a set of ordinary differential equations (ODEs), which can be solved analytically to obtain relationship between the phase strains and the macroscopic strains. The solution of the ODEs stems from an eigenvalue problem, where the eigenvalues are the homogenized relaxation times. In addition, the relaxation moduli can be evaluated through the ODEs as well. Accordingly, a homogenized viscoelastic material in term of Prony series can be determined. Four sets of numerical tests are proposed to verify the semi-analytical method: unit cell tests, tension tests on a plate with a hole, pure bending tests, and torsion tests. The results from these tests demonstrate a strong agreement between the homogenized model and direct numerical simulations. Additionally, we compared our model against experimental measurements, further confirming the accuracy and reliability of our proposed approach.