Quasistatic temperature stresses in a multilayer thermally sensitive cylinder
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QUASISTATIC TEMPERATURE STRESSES IN A MULTILAYER THERMALLY SENSITIVE CYLINDER R. M. Kushnir, B. V. Protsyuk, and V. M. Synyuta
UDC 539.3
An approach to the evaluation of nonstationary temperature fields and the thermoelastic state of multilayer hollow long thermally sensitive cylinders based on the use of the Kirchhoff substitution, generalized functions, and Green’s functions of the corresponding problems for a hollow cylinder with piecewise constant physicomechanical characteristics is illustrated. The problem of heat conduction is reduced to the solution of a Fredholm–Volterra nonlinear integral equation of the second kind for the Kirchhoff variable. In the problem of thermoelasticity, the coefficients of the equation (continuous inside each layer) are approximated by piecewise constant functions. The boundary conditions at the ends of the cylinder are satisfied in the integral form. Numerical analysis is performed for the case of a three-layer cylinder.
In finding approximate solutions of nonstationary problems of heat conduction and quasistatic problems of thermoelasticity with regard for the temperature dependences of physicomechanical characteristics, it is customary to combine the application of numerical methods with various numerical-analytic and analytic methods [1– 7]. For problems of this sort, the exact solutions can be found only in some special cases. Thus, the nonstationary temperature fields in contacting bodies can be determined either under certain assumptions which enable one to linearize, e.g., the conditions of heat contact [4] or for special conditions of heating when the problem becomes much simpler [2]. As a rule, the corresponding problems are solved for a small number of layers. In the present work, we illustrate an approach to the numerical-analytic determination of nonstationary temperature fields and the thermoelastic state of multilayer hollow long thermally sensitive cylinders based on the Kirchhoff substitution, generalized functions, and Green’s functions of the problems of heat conduction and statics for a hollow cylinder with piecewise constant physicomechanical characteristics. The problem of heat conduction is reduced to the solution of a Fredholm–Volterra nonlinear integral equation of the second kind for the Kirchhoff variable. In the problem of thermoelasticity, the coefficients of the equation (continuous in each layer) are approximated by piecewise constant functions. The boundary conditions imposed at the ends of the cylinder are satisfied in the integral form. Statement of the Nonstationary Problem of Heat Conduction and the Procedure of Its Solution In a cylindrical coordinate system r, ϕ, z, we consider a multilayer hollow cylinder whose components are in the state of perfect contact. The inner surface of the cylinder is heated by a time-dependent heat flow with intensity q ( τ ). The temperature of the outer surface tc( τ ) is kept constant. At the initial time, the temperature of the cylinder is a function of the radial coordinate. Our aim is to determine the non
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