Nonstationary thermal fields in inhomogeneous materials with nonlinear behavior of the components
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NONSTATIONARY THERMAL FIELDS IN INHOMOGENEOUS MATERIALS WITH NONLINEAR BEHAVIOR OF THE COMPONENTS L. M. Zhuravchak1, 2 and N. V. Zabrods’ka1
UDC 517.958:536.12
We consider a two- or three-dimensional piecewise homogeneous body of any shape with temperaturedependent thermal characteristics. On the boundary of the body, we impose conditions of the first, second, and third kinds. The components of the body are in perfect thermal contact. We propose a procedure based on the joint application of the Kirchhoff transformation, the methods of near-boundary and boundary elements, and the stepwise time scheme of unique initial condition to the construction of the integral representations of temperature and heat flow at arbitrary space-and-time points. The numerical experiments reveal the efficiency of combining the indirect method of near-boundary elements with the Kirchhoff transformation in the case where the temperature dependence of the heat-conduction coefficient of the material is taken into account. Keywords: materials with temperature-dependent characteristics, nonstationary thermal field, indirect method of near-boundary elements, Kirchhoff transformation.
The problems of decreasing the consumption of materials in inhomogeneous structural elements operating under the conditions of ultimate thermal loads (or in the presence of sharp drops of temperature) and evaluation of their strength and reliability are urgent for various branches of industry and engineering, in particular, in the fields of machine-building, instrument making, thermal power industry, and creation of new of heat-resistant materials. The solution of these problems is based on the determination of thermal fields in piecewise homogeneous objects of any shape, i.e., on the construction of solutions of two- or three-dimensional nonstationary problems of heat conduction. The linear mathematical models constructed under the assumption of piecewise constant dependence of the thermal characteristics of materials on coordinates sometimes inadequately describe the actual processes. More reliable models taking into account the temperature dependences of the heat-conduction coefficients and thermal diffusivities of materials of the components lead to nonlinear boundary-value problems of mathematical physics solved, as a rule, by the numerical methods. Thus, in [1], the solutions of stationary problems of heat conduction with regard for the temperature dependence of the heat-conduction coefficient are obtained by combining the Kirchhoff transformation with the direct method of boundary elements. In [2], the nonstationary temperature fields are found by using the approach based on the joint application of the methods of decomposition, imbedding, near-boundary elements, nonclassical finite-difference relations, and the method of continuation of solutions in the parameter. In what follows, for the mathematical description of the nonstationary process of heat conduction in piecewise homogeneous materials with nonlinear behavior of the components, we propose to us
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