Temperature Field in a Thermosensitive Plate with Local Heating

A nonlinear mathematical model for the analysis of temperature regimes in a thermosensitive isotropic plate, which is heated by a locally concentrated heat flux on the boundary surface, has been developed. To do this, the heat-active zone of the plate is described using the theory of generalized functions. In view of this, the boundary conditions contain a discontinuous right-hand side. The original nonlinear heat equation and nonlinear boundary conditions are linearized using the Kirchhoff transformation. To solve the obtained boundary value problem, the integral Fourier transform was used, and as a result, an analytical solution in the images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain an analytical expression for determining the Kirchhoff variable. As an example, a linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, an analytical ratio was obtained to determine the temperature in the heat-sensitive plate. Therefore, it becomes possible to increase the heat resistance of the plate and protect it from overheating, which can cause the destruction of not only individual elements but also the entire structure.