Model representations of the theory of thermal shock of viscoelastic bodies

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Model representations of the theory of thermal shock of viscoelastic bodies based on two different approaches are considered. In the first approach, based on the introduction of stress and strain deviators using linear rheological models of Maxwell and Kelvin, new integral and differential relations are proposed, including simultaneously dynamic and quasi-static models for viscoelastic and elastic media, generalizing the results of previous studies. The proposed constitutive relations of the new form are applicable to describe the thermal response of bodies of canonical shape, limited by the boundaries of a rectilinear shape in Cartesian coordinates and are extended to the case of curvilinear boundaries in cylindrical and spherical coordinates. The second approach describes an elastic-viscoelastic analogy, which consists in the fact that the original problem of temperature stresses of a viscoelastic body can be reduced to the equivalent problem of thermoelasticity by replacing the shear modulus and Poisson’s ratio in the operational (according to Laplace) solution of the thermoelastic problem with their images as in the model Maxwell and in the Kelvin model. It is shown that after performing the inverse transformation, an analytical solution to the problem for a thermoviscoelastic medium is found. An illustrative example is given and the differences in the thermal response to sudden heating of an elastic and viscoelastic medium are analyzed.