Осесиметрична контактна задача термопружності про тиск пружного циліндра на пружний півпростір з урахуванням неідеального теплового контакту

Abstract

Побудовано розв’язок осесиметричної контактної задачі термопружності про тиск пружного кругового ізотропного циліндра на пружний ізотропний півпростір з урахуванням неідеального теплового контакту між циліндром і півпростором. За допомогою методу інтегрального перетворення Ганкеля розв’язано рівняння теплопровідності й термопружності для півпростору, а методом Фур’є – для циліндра. Температурне поле, переміщення й напруження в циліндрі подано через коефіцієнти, які задовольняють нескінченну систему алгебраїчних рівнянь.

Determination of contact stresses, taking into account temperature factors, is of importance while investigating machine parts and construction elements strength in the area of their interaction and designing of the elastic basis constructions for the efficient operation of its material and the basis supporting power. Solution of the axis-symmetric contact thermo-elastic problem of the pressure of elastic circular isotropic cylinder on the elastic isotropic semi-space, taking into account non-ideal thermal contact between the cylinder and semi-space, has been built. All cylinder end points, being subject to the outside loading, are displaced in equal value. The semi-space and cylinder surfaces outside the contact area are free of outside stresses, and the tangential stresses in the contact area equal zero. On the free end of the cylinder constant temperature is provided and the thermal contact between the bodies is expected to be non-ideal. Free surfaces of the cylinder and the semi- space are kept at zero temperature or thermoisolated. Under qiven assumptions the method of determination of the temperature fields and the contact stresses in the cylinder and semi-space, has been developed. Using the Hankel integral transformation method the thermal conductivity and thermal elasticity equation for the semi-space, has been solved, and the Fourier method-for the cylinder. Temperature field, displacement and stress in the isotropic semi-space are expressed by the non-personal integrals, possessing unknown functions, which are found from the boundary conditions of the problem. Temperature field, displacement and stress in the cylinder are presented as coefficients, which satisfy non-finite system of algebraic equations. Satisfaction of the problem boundary conditions results in the system of integral equations, which connect the unknown functions with the coefficients, which specify the temperature field and, as a result, the Fredholm integral equation of the 2-nd order relatively the function, due to which the normal contact stresses in the semi-space are expressed, has been obtained. The Fredholm integral equation of the 2-nd order is reduced to the system of linear algebraic equations, which is solved by the numerical method. Numerical calculations for finding temperature and the temperature component of the normal stress in the semi-space in the area of contact for different values of contact conductivity, as well as the temperature component of the normal stress for different values of the Young’s modulus of the cylinder and semi-space, have been carried out. The analysis of graphs testifies, that non ideal thermal contact, when the cylinder is pressed into the semi-space, sufficiently affects the distribution of the temperature component of the normal stress in the contact area.