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The aircraft multilayer glazing is considered as a rectangular multilayer plate made up of isotropic layers with constant thickness. The temperature on the side surface of the plate is zero. Convective heat transfer occurs on outer surfaces of the plate; on layers' interfaces film heat sources are arranged. The heat conduction equation for an arbitrary plate layer is reduced to the functional equation. A solution of the functional equation we search in the form of three space functions product. We get the system of ordinary differential equations. Series expansion factors are determined from a system of linear algebraic equations. A transform of the required function is found by the second expansion theorem, and the problem solution has the form of double trigonometrical series. The comparative analysis of the results is carried out with the results of other method. The method offered can be used for designing a safe multilayer glazing under operational and emergency thermal and force loading in different vehicles. Keywords—safe multilayer glazing, nonstationary thermal conduction, film heat source. I. FORMULATION OF THE PROBLEM One of the urgent problems in modern technology is the task of reliable determination of the thermal state of structural elements. Its successful solution depends on the reliability and efficiency of the elements of different structures, which often have a heterogeneous structure. The aircraft's modern heating glass is a complex, large, multi-layered structure. Its performance depends primarily on the strength and durability of the glass elements, the optimal parameters of adhesive layers and structural solutions that would provide the necessary heat resistance and durability of the roofing layer under cyclic loading and the action of extreme operating factors. It should be noted that in the calculation of temperature fields in multilayer structures there are considerable mathematical difficulties. This is due to the presence of layers whose properties are significantly different, and the fulfillment of the conditions of coupling of the layers taking into account the internal heat sources. II. ANALYSIS OF PUBLICATIONS BY RESEARCH TOPIC Practical work on the design of protective glazing of vehicles has led to the creation of GOST R 51136-2008 [1]. This standard applies to multilayer safety glasses, which are designed to protect human life, ensure safety and reliability while storing and transporting material. A review of the literature indicates that most publications are devoted to the calculation of structures under steady-state heating [2]. The article [3] proposes the formulation of contact problems of heat transfer and thermal elasticity in irradiated layered bodies with components of different transparency, in particular in the presence of thin intermediate layers, taking into account the effects of radiation on their surfaces, in partially transparent regions and boundaries. To describe the propagation of thermal radiation, we have used the relations of phenomenological radiation theory for partially transparent and opaque bodies. In formulating the thermal boundary and contact conditions, the fluxes that are caused by the radiation and absorption of thermal energy are considered. The conditions of ideal thermal contact were obtained taking into account the effects of radiation. As an example, the problem is solved for a two-layer plate with layers of different transparency, connected through a thin intermediate layer. Various numerical and analytical methods of solution have been developed in [4, 5] and a number of fundamental results have been obtained. In the case of non-stationary thermal influences, one-dimensional thermal conductivity problems of unlimited multilayer plates are considered [6] or methods based on the discretization of an area or contour are used, for example, the finite element method [7]. Methods of reducing the three-dimensional problem of thermal conductivity to twodimensional by applying various hypotheses about the temperature distribution along the thickness of the layer package are also proposed [8]. In [9], a nonstationary thermal conduction problem for layered rectangular plates at a given temperature distribution on the outer and inner surfaces is solved analytically without involving hypotheses about the temperature distribution along the plate thickness. Modeling, Control and Information Technologies 2019

Calculation of Temperature Fields in Multilayer Plates and Shells with Distributed Heat Sources