Mathematical models of dynamic thermoviscoelasticity

Аuthors

Kartashov E. M.^{1, 2*}, Krylov S. S.²

1. MIREA — Russian Technological University (Lomonosov Institute of Fine Chemical Technologies), 78, Vernadsky prospect, Moscow, 119454, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: professor.kartashov@qmail.com

Abstract

The article provides the development of theoretical foundations of a new scientific direction – dynamic thermoviscoelasticity under conditions of locally nonequilibrium heat transfer process. Dynamic models describing thermal shock have attracted the attention of researchers especially in recent decades in connection with the creation of powerful energy emitters and their use in technological operations. At the same time, most studies of the thermal reaction of solids during sudden heating (or cooling) have been performed for 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 a more or less distinct phenomenon of viscous flow. A real body begins to exhibit elastic and viscous properties and becomes viscoelastic. A rather complex problem arises: the development of dynamic (quasi-static) thermoviscoelasticity within the framework of the corresponding mathematical models of classical applied thermomechanics and mathematics. The purpose of the work is to consider an open problem of the theory of thermal shock in terms of a generalized model of thermoviscoelasticity under conditions of a locally nonequilibrium process of heat propagation in solids. Three types of intense heating are considered: temperature, thermal, heating by the environment. Intensive cooling modes can be considered equally. The task is to develop model representations of dynamic (quasi-static) thermoviscoelasticity that allow accurate analytical solutions to the corresponding boundary value problems on their basis. This direction is practically absent in the scientific literature. As a result, model representations of the thermal response of viscoelastic bodies were developed using the proposed new equation of compatibility in displacements.
The latter allowed us to propose new integro-differential relations 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 canonical shape simultaneously in three coordinate systems with a parameter defining the system, which allows us to identify the influence of the topology of the domain on the magnitude of the corresponding temperature stresses.

Keywords:

thermal shock; thermoviscoelasticity; generalized dynamic models; analytical solutions; thermal stresses

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