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We give a mathematical statement and initial relations for the problem of determining the temperature fields and stresses in nonmetallic electrical conductors under induction heating to high temperatures. We study the influence of the temperature dependence of the characteristics of the material and the radiant heat transfer with the surrounding medium on the thermomeehanical behavior of a cylinder.In recent years the induction thermal treatment of products made of nonmetallic conducting materials has found ever wider application [8,9]. In their mechanical properties these materials are fragile. For that reason the procedures for induction heating of bodies made of nonmetallic materials are completely determined by their strength properties (the magnitude of the admissible stresses). To design rational procedures for induction thermal treatment of various products made of such materials it is important to have a mathematical model of the electromagnetic, thermal and mechanical processes in them under the action of external electromagnetic fields.We shall begin with the models known in the literature for determining the temperature fields [7] and the thermostressed state [5] of metal electrical conductors under induction heating by quasistationary electromagnetic fields in which the influence of the electromagnetic field and the stresses is taken account of by the Joule heat emission (averaged over a period of oscillation of the electromagnetic wave). In this formulation the system of original relations of the mathematical model consists of the uncoupled equations of electrodynamics, heat conduction, and quasistatic thermoelasticity [5].A characteristic property of thermal treatment of nonmetallic bodies is that it is done at relatively high temperatures [8]. For that reason it is necessary in the model to take account of the temperature dependence of the electrophysical, thermophysical, and physico-mechanical characteristics of the material and the radiant heat transfer with the surrounding medium. Here the solution of the problem of thermomechanical behavior of a nonmetallic body reduces to determining the electromagnetic field and the temperature from the simultaneous system of equations of electrodynamics and heat conduction and subsequently finding the stress state from the equations of quasistatic thermoelasticity with prescribed initial and boundary conditions on the magnetic field intensity, temperature, and components of the stress [1,5].For the external electromagnetic field defined by the magnitude of the magnetic field intensity H (~ = H0(ro, t) on the surface S of the body the system of initial equations of the problem is written as follows Here H is the magnetic field intensity, T is the temperature u is the displacement vector, 5 and ~ are the stress and strain tensors, ae is the coefficient of electrical conductivity, # is the magnetic permeability,/~ is the coefficient of thermal conductivity, cv is the three-dimensional heat capacity, Q. = a U a (curl H) 2 is the f specific power of Joule heat so...

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