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.

在本文中，我们使用参数空间中的渐近均匀化方法来确定热粘弹性复合材料的有效特性。这些材料由嵌入基质中的多层球形夹杂物组成。与传统的渐近均质方法相比，我们的方法允许不均匀性的规则非周期性分布以及材料特性对温度的依赖性。我们从控制方程的拉普拉斯变换及其边界条件和初始条件开始。为此，我们将负责材料中夹杂物非周期性分布的温度和空间坐标作为参数（以及拉普拉斯变换本身的参数）。然后，我们定义并实现参数空间中所得方程的渐近均化的两级方案。第一步，我们在微观尺度上解决问题（细胞问题）。第二步，对于拉普拉斯变换的图像，我们导出具有有效系数的宏观方程。最后，我们执行拉普拉斯逆变换来计算松弛函数并确定复合材料的热粘弹性。获得的结果提供了有关夹杂物的性质和浓度变化如何影响热粘弹性复合材料的流变特性和应力松弛模式的信息。对于拉普拉斯变换的图像，我们推导了具有有效系数的宏观方程。最后，我们执行拉普拉斯逆变换来计算松弛函数并确定复合材料的热粘弹性。获得的结果提供了有关夹杂物的性质和浓度变化如何影响热粘弹性复合材料的流变特性和应力松弛模式的信息。对于拉普拉斯变换的图像，我们推导了具有有效系数的宏观方程。最后，我们执行拉普拉斯逆变换来计算松弛函数并确定复合材料的热粘弹性。获得的结果提供了有关夹杂物的性质和浓度变化如何影响热粘弹性复合材料的流变特性和应力松弛模式的信息。