Термопружна поведінка безмежного термочутливого трискладового тіла за дії об’ємного джерела тепла

Abstract

Запропоновано чисельно-аналітичний підхід до визначення термопружного стану безмежного трискладового тіла за дії об’ємного джерела тепла з урахуванням залежності термомеханічних характеристик від температури. Він базується на використанні перетворення Кірхгофа, узагальнених функцій, функцій Гріна нестаціонарної задачі теплопровідності для трискладового простору зі сталими характеристиками та лінійних сплайнів. Наведено результати числових досліджень температури, напружень та переміщень.

The numerical-analytical approach of determining a thermoelastic state of infinite thermosensitive (thermomechanical properties depend on temperature) threelayer body with thermal insulated and smoothly fixed cylindrical surface under the action of the dependent on axial coordinate and time heat source is presented. To find the temperature field to the initial system of three nonlinear one-dimensional thermal conductivity equations and contact conditions, Kirchhoff transformation was applied. Having assumed that coefficients of thermal conductivity are in linear dependence on temperature and coefficients of thermal diffusivity are constant within each component, the obtained problem was reduced to a single equation with generalized on the coordinate derivatives relative to the function, that within each region coincides with the corresponding Kirchhoff variable. By using Green's function of the linear non-stationary termal conductivity problem for a threelayer space, integral representation, in which the subintegral functions include the time-dependent unknown Kirchhoff variable values at the surfaces of division on the right was obtained. The approach of finding them for a case where the heat source is uniformly distributed over the volume of intermediate layer and has pulsing character change of intensity over time is illustrated. Approximating the subintegral functions with these unknown values by linear splines from the integral representation, the recurrent system of two non-linear algebraic equations for the nodal values of the Kirchhoff variable was obtained. Having solved it, expressions for the entered variables and, correspondingly, the searched temperature field were found. Since it changes only in thickness, in the considered body only radial and circular stress and axial displacement will arise. The numerical investigations were carried out for the case, where the number of pulses of heat source equals two and the materials of the half infinite components and intermediate layer are niobium and platinum, for which thermal diffusivity within the range of the studied temperatures almost does not change. The influence of thermosensitivity on distribution of temperature, stresses and displacements was illustrated and analyzed.