A method of asymptotic averaging is developed for the equations of thermal viscoelasticity with rapidly oscillating coefficients. In contrast to the traditional approach, an asymptotic analysis of the equations introduces an additional parameter corresponding to the dependence of the material characteristics on temperature, and the functions of fast variables are considered in the parametric space. The averaging procedure is formulated in a corresponding manner so that nonlinear dependences that have a smoothly changing character with respect to fast variables are resolved in the asymptotic analysis parametrically. To implement this scheme, a complex analogy is used for the integro-differential equations of thermal viscoelasticity. The defining relationships between stresses and deformations are integral equations with relaxation kernels of difference type, and therefore they can be described by means of the integral Laplace or Fourier transformation through complex moduli for elasticity equations with rapidly oscillating coefficients that depend on spatial coordinates and temperature. A two-level scheme for solving auxiliary problems of the considered asymptotic method is developed, based on the analytical-numerical and finite-element approaches to determining the auxiliary functions at the micro- and macro levels, which is involved in the asymptotic representation of the solution. In particular, an algorithm for calculating the effective parameters of relaxation kernels is determined taking into account the dependence of the viscoelastic properties of the material on temperature. The developed approach allows to determine the effective properties of materials with a quasiperiodic structure, for example, the functional-gradient properties of viscoelastic composite materials with dependence of these characteristics on temperature. For solving auxiliary problems for fast variable functions at the micro level, a special block analytic-numerical method is developed for approximation of the solution with inclusions of arbitrary geometrical shape, in particular, for a spherical one with intermediate interphase layer.

effective mechanical and thermophysical characteristics of rheonomaterials, nonlinear equations, parametric method of asymptotic averaging, structurally inhomogeneous viscoelastic media, нелинейные уравнения, параметрический метод асимптотического усреднения, структурно-неоднородные вязкоупругие среды, эффективные механические и теплофизические характеристики реономных материалов

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