RADIOELECTRONIC AND COMPUTER SYSTEMS

Today, technical systems with internal heat generation are developed and used in various areas of industry, energy, construction, as well as in microelectronics, nanotechnology, biomedicine, technical chemistry, ecology, etc. The creation of such systems is usually preceded by their mathematical and computer modeling, in which issues of strength often play a primary role. The use of purely numerical methods for this purpose does not always give results of the required accuracy, and the impossibility of parametric description of the model does not allow solving optimization problems with them. Therefore, the creation of new methods for modeling such systems and their models is an actual scientific and practical task. The subject of the research of the present article is mathematical models of the thermoelastic state of a ball with a spherical inclusion having a heat release region, as well as methods for obtaining them. The goal of the study is to create a number of parametric models for studying temperature fields and stress fields in a piecewise homogeneous sphere with an internal heat-active region under different thermomechanical conditions on its boundary. To achieve this goal, it is necessary to solve a number of problems: to perform further development of the generalized Fourier method for the class of axisymmetric stationary thermoelastic problems for a ball with an eccentric heat-active spherical inclusion, using the developed method, obtain a number of models of the thermoelastic state of a ball in cases of eccentric and concentric inclusions with a certain area of heat generation in them, perform a rigorous justification of the proposed approach, conduct a wide computer experiment with the constructed models and an analysis of the thermomechanical characteristics obtained with its help, and draw conclusions based on the results of the study. The modeling methods used in this work are the generalized and ordinary Fourier methods. The following results were obtained in the work. A number of mathematical models of the axisymmetric stationary thermoelastic state of a ball with a spherical inclusion were constructed under the condition that the entire inclusion or part of it releases heat according to a harmonic law. The cases of a stationary surface of a ball and a constant temperature on its boundary, as well as a ball whose surface is loaded with normal pressure and is in conditions of heat exchange with the environment are considered. Modeling is carried out using the generalized Fourier method, which was further developed in the work. Determination of the parameters of the models is reduced to solving infinite systems of linear algebraic equations. For the two specified types of models, the case of concentric inclusion is considered separately. Linear algebraic systems of the fourth order are obtained to determine the parameters of the models. Using a new classical inequality, lower bounds for the moduli of the determinants of systems and the existence classes of the resulting models were obtained for the first time. Separately, a thermoelastic model of a ball with a spherical thermally active layer in the inclusion was obtained in a closed form. Different types of ball and inclusion materials were used in the numerical simulation. Calculations were performed for temperature fields and stress fields with changes in geometric parameters, values and density functions of heat sources. Rich graphic material was obtained and analyzed in the work. Scientific novelty: all the above results are new. Conclusions were made based on the results of the studies.

mathematical model; sphere with heat-active inclusion; generalized Fourier method; temperature and stress fields; density of heat sources; thermomechanical characteristics; eccentric inclusion; computer simulations; lower bound of the determinant module

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