Thermoelastic state of layered thermosensitive bodies of revolution for the quadratic dependence of the heat-conduction
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THERMOELASTIC STATE OF LAYERED THERMOSENSITIVE BODIES OF REVOLUTION FOR THE QUADRATIC DEPENDENCE OF THE HEAT-CONDUCTION COEFFICIENTS R. M. Kushnir1, 2 and Yu. B. Protsyuk1
UDC 539.3
We propose an analytic-numerical method for the solution of one-dimensional static problems of thermoelasticity for layered cylinders and balls subjected to the action of the surface loads for various modes of heating with regard for the quadratic dependence of the heat-conduction coefficients and arbitrary dependences of the other physicomechanical characteristics on temperature. Independently of the number of layers, the problems of heat conduction are reduced, by using the constructed exact solutions of special problems, to the solution of a single nonlinear algebraic equation or a system of two equations of this sort. The solutions of the problems of thermoelasticity are obtained by approximating the coefficients of equations continuous inside each layer by piecewise constant functions with subsequent application of Green’s functions of the problems of statics for many-layer cylinders and balls. We perform the numerical analysis of the temperature fields and the thermoelastic state in two-layer bodies whose outer surface is heated by convective-radiation heat exchange and the inner surface is kept at a constant temperature. Keywords: layered cylinder, layered ball, modes of heating, temperature dependence of the characteristics, static temperature stresses.
It is known that the solution of uncoupled problems of thermoelasticity, including the problems posed with regard for thermal sensitivity (dependences of the physicomechanical characteristics on temperature) is preceded by the construction of solutions of the corresponding problems of heat conduction. In the course of the solution of these nonlinear problems of heat conduction [1–7], it is customary to use the Kirchhoff substitution. Moreover, it is most often assumed that the heat-conduction coefficients are linear functions of temperature. However, for numerous materials, it is necessary to consider more complicated (in particular, quadratic) temperature dependences of the heat-conduction coefficients [6, 8, 9]. As in the case of the solution of the problems of thermoelasticity for bodies whose physicomechanical characteristics depend on the coordinates, the solutions of the problems of thermoelasticity for thermosensitive bodies are constructed by using the method of successive approximations, the method of perturbations, etc. [4, 5, 7, 10, 11]. In what follows, by using the well-known approach from [12], we propose an analytic-numerical method for the solution of one-dimensional static problems of thermoelasticity for layered bodies of revolution (cylinders or balls) subjected to the action of the surface loads for various modes of heating with regard for the quadratic dependence of the heat-conduction coefficients on temperature and arbitrary temperature dependences of the other physicomechanical characteristics. By using the exact solutions of the corresponding equations
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