Thermal Reaction During Thermal Shock of a Massive Body with an Internal Cylindrical Cavity

The paper examines mathematical models of thermal shock in terms of dynamic thermoelasticity and their application to specific cases during intense heating of a solid boundary. We introduce a stress compatibility equation for dynamical problems, generalizing the well-known Beltrami --- Mitchell relation for quasi-static cases. It is convenient to use this relation when considering numerous special cases in the theory of heat shock in Cartesian coordinates both for bounded bodies of a canonical form, i.e., an infinite plate, and for partially bounded bodies, i.e., space bounded from the inside by a flat surface. In the latter case, the obtained analytical solutions of dynamic problems of thermoelasticity lead to visual and convenient for physical analysis functional structures describing the kinetics of thermal stresses. For cylindrical and spherical coordinate systems, we propose a compatibility equation in displacements, which is convenient for studying the problem of thermal shock in bodies with a radial heat flux and under conditions of central symmetry. In the study, we singled out a class of problems in which the consideration of the geometric dimensions of a structure investigated for a thermomechanical reaction under conditions of intense heating concerns mainly the near-surface layers. According to the experimental results, it is these layers that absorb the main amount of heat during a time close to the beginning of heating, which corresponds to the time of the microsecond duration of the inertial effects. We investigated the thermal reaction of a massive body with an inner cylindrical cavity within the framework of dynamic thermoelasticity under various modes of intense heating of the cavity surface. Finally, we carried out numerical experiments and described the wave character of thermal stresses with the corresponding quasi-static values, and established the role of inertial effects in mathematical models of the theory of thermal shock

Tolerance and asymptotic modelling of dynamic thermoelasticity problems for thin micro-periodic cylindrical shells

The problem of linear dynamic thermoelasticity in Kirchhoff–Love-type circular cylindrical shells having properties periodically varying in circumferential direction (uniperiodic shells) is considered. In order to describe thermoelastic behaviour of such shells, two mathematical averaged models are proposed—the non-asymptotic tolerance and the consistent asymptotic models. Considerations are based on the known Kirchhoff–Love theory of elasticity combined with Duhamel-Neumann thermoelastic constitutive relations and on Fourier’s theory of heat conduction. The non-asymptotic tolerance model equations depending on a cell size are derived applying the tolerance averaging technique and a certain extension of the known stationary action principle. The consistent asymptotic model equations being independent on a microstructure size are obtained by means of the consistent asymptotic approach. Governing equations of both the models have constant coefficients, contrary to starting shell equations with periodic, non-continuous and oscillating coefficients. As examples, two special length-scale problems will be analysed in the framework of the proposed models. The first of them deals with investigation of the effect of a cell size on the shape of initial distributions of temperature micro-fluctuations. The second one deals with study of the effect of a microstructure size on the distribution of total temperature field approximated by sum of an averaged temperature and temperature fluctuations.

Analysis of the generalized heat equation applied to the solution of the dynamic problem of thermoelasticity

Based on the approximate solution of the dispersion equation, the paper presents an analysis of the system of dynamic thermoelasticity equations taking into account the generalized heat equation. It is noted that during the wave process of heat transfer, a sufficiently intensive process of energy exchange between thermal and elastic fields is realized, while depending on the relations of the characteristic relaxation times, the direction of energy exchange can change.