Calculation of the effective properties of thermo-viscoelastic composites using asymptotic homogenization in parametric space

In this paper we use the method of asymptotic homogenization in parametric space to determine the effective properties of thermo-viscoelastic composite materials. These materials are composed of multilayered spherical inclusions imbedded in the matrix. In comparison with the traditional method of asymptotic homogenization, our approach allows for regular non-periodic distributions of inhomogeneities as well as dependences of the material characteristics on temperature. We start with the Laplace transform of the governing equations together with their boundary and initial conditions. To do so, we treat temperature and spatial coordinates responsible for non-periodic distribution of inclusions in the material as parameters (along with the parameter of Laplace transform itself). Then we define and implement a two-level scheme of asymptotic homogenization of the resulting equations in parametric space. At the first step, we solve the problem on the microscale level (a cell problem). At the second step, for the images of Laplace transform, we derive the macroscopic equation with effective coefficients. Finally, we perform the inverse Laplace transform to compute relaxation functions and determine thermo-viscoelastic properties of the composite material. The obtained results provide an information on how the change in properties and concentration of the inclusions affect the rheological characteristics and stress relaxation patterns for the thermo-viscoelastic composites.