Nonstationary heat conduction in three-dimensional bodies with nonsmooth inclusions
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NONSTATIONARY CONTACT PROBLEM OF THERMOELASTICITY FOR BODIES HEATED TO DIFFERENT TEMPERATURES Z. Olesiak and R. Kul’chyts’kyi-Zhyhailo
UDC 539.3
The solution of a nonstationary contact problem of thermoelasticity for bodies heated to different temperatures is obtained by using the Laplace–Hankel integral transformation. The expression for contact pressure is deduced in the form of an explicit dependence on two unknown functions: the distribution of heat flow and the radius of the contact zone. An algorithm of simplified solution of the contact problem is proposed.
The first works devoted to the solution of axisymmetric contact problems of thermoelasticity were published by George and Sneddon [1] and Borodachov [2] in 1962. In these works, the method of coupled integral equations was used to solve mixed boundary-value problems of linear thermoelasticity. Thus, George and Sneddon studied the axisymmetric contact problems of thermoelasticity for two contacting elastic bodies. In particular, they investigated the case of pressing of a heated cylinder with plane base and a heated cone into an elastic half space. In his lectures [3], Sneddon analyzed some other examples of problems connected with pressing of heated elastic bodies of revolution into the elastic half space, including the case of of a heated ball. Borodachov [2, 4] solved similar problems independently of George and Sneddon. We can also mention many other works generalizing the problems described above, e.g., the problems of pressing of heated punches into an elastic layer. These problems satisfy all relations of the mathematical theory of thermoelasticity. However, at that time, the authors did not know that the direction of heat flow can also be significant. Therefore, it was quite unexpected to discover [5] that the solution of the contact problem of thermoelasticity loses its physical meaning in the case where a ball whose temperature is lower than the temperature of a half space is pressed into the latter. Thus, there is absolutely no symmetry with the solution obtained for the case of a heated ball and, moreover, we observe the so-called paradox of “cold ball,” i.e., a strange fact that the function of contact pressure obtained as a result of the solution of the problem is not sign-preserving. This means that a common contact surface may appear only in the case of existence of surface tensile forces in a part of the contact zone. Similar difficulties are encountered [6] under the conditions of perfect thermal contact between the bodies and thermal insulation of their unloaded surfaces if the heat flow is directed from the body with higher coefficient of thermal distortivity [7] to the body with lower value of this coefficient: δ =
α (1 + ν) , λ
where α is the coefficient of linear thermal expansion, λ is the heat-conduction coefficient, and ν is Poisson’s ratio. Olesiak demonstrated that the effects similar to the effects described above are also observed in the case where the condition of thermal insulation is replaced by the condition of preser
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