Квазістатичні напруження в півпросторі з покриттям при змішаних умовах нагрівання

Abstract

Побудовано розв’язок плоскої квазістатичної задачі термопружності для півпростору з покриттям, на границі якого в смузі певної ширини задана температура, а зовні відбувається теплообмін за законом Ньютона. Розв’язок отримано із використанням інтегральних перетворень Лагерра й Фур’є та методу рядів Фур’є при розв’язуванні послідовностей парних інтегральних рівнянь. Наведено результати числового аналізу термонапруженого стану в півпросторі та покритті залежно від відносної товщини покриття та інтенсивності охолодження.

Analysis of thermal stresses in bodies with coatings is important for many engineering researches. Taking into account the actual operating conditions of these structures frequently leads to mixed heating condition. The steady problem of thermoelasticity with mixed boundary conditions currently is sufficiently investigated. However, the corresponding transient problem, despite its relevance, is poorly understood. This is due to mathematical difficulties that arise in applying the integral Laplace transform. The authors of this paper developed a new effective method of constructing solutions of mixed boundary-value non-stationary problems. The half-space with a coating, on the surface of which on the band of 2d width the temperature distribution is given and outside of this area the heat transfer according to the Newton's law is performed, is analysed in the work. On the surface of separation of materials of half-space and coating the conditions of ideal thermomechanical contact are satisfied. The initial temperature of the coating and half-space is equal to zero. To the heat conductivity problem the Laguerre integral transformation in time variables and integral Fourier transformation in spatial variable are applied. As a result the triangular sequence of ordinary differential equations is obtained. The general solution of these sequences is obtained in the form of algebraic convolution. Taking into account the mixed boundary conditions results in dual integral equations. For solution of this problem the method of Newton's series is proposed. Taking advantage of this method the problem is reduced to the infinite system of algebraic equations, for which the convergence of reduction procedure is proved. The solution of thermoelasticity problem is built using resulting temperature field in the assumption, that the border of coating is free of load. The solution is obtained in the form of series in Laguerre polynomials. Calculations were carried for the half-space made of titanium alloy and ceramic coating.