Development of model representations of thermal reaction viscoelastic bodies on the temperature field

Objectives. In recent decades, the relevance of research into the thermal response of solids to a temperature field has increased in connection with the creation of powerful energy emitters and their use in technological operations. There is a significant number of publications describing these processes using mathematical models of dynamic or quasi-static thermoelasticity, mainly for most technically important materials that obey Hooke’s law. However, at elevated temperatures and higher stress levels, the concept of an elastic body becomes insufficient: almost all materials exhibit more or less clearly the phenomenon of viscous flow. The real body begins to exhibit elastic and viscous properties and becomes viscoelastic. A rather complex problem arises: the development of dynamic (quasistatic) thermoviscoelasticity within the framework of the corresponding mathematical models of classical applied thermomechanics and mathematics. The purpose of the work is to consider the open problem of the theory of thermal shock in terms of a generalized model of thermoviscoelasticity under the conditions of classical Fourier phenomenology on the propagation of heat in solids. Three types of intense heating are considered: temperature, thermal, and medium heating. Intensive cooling modes can be equally considered. The task is posed: to develop model representations of dynamic (quasi-static) thermoviscoelasticity that allow accurate analytical solutions of the corresponding boundary value problems on their basis. This direction is practically absent in the scientific literature.

Methods. Methods and theorems of operational calculus were used.

Results. Model representations of the thermal response of viscoelastic bodies using the proposed new compatibility equation in displacements have been developed.

Conclusions. New integro-differential relations are proposed based on linear rheological models for the Maxwell medium and the Kelvin medium, including both 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 quasi-elastic bodies of a canonical shape simultaneously in three coordinate systems with a system-defining parameter, which makes it possible to identify the influence of the topology of the region on the value of the corresponding temperature stresses.

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