Тиск циліндричного кругового штампа на трансверсально-ізотропний шар при неідеальному тепловому контакті

Abstract

Побудовано розв’язок осесиметричної контактної задачі термопружності про тиск циліндричного кругового трансверсально-ізотропного штампа на пружний трансверсально-ізотропний шар з урахуванням неідеального теплового контакту між штампом і шаром. Отримано формули для визначення температурного поля і нормального напруження. Досліджено вплив контактної провідності й коефіцієнтів анізотропії матеріалів на розподіл температурних полів і нормального напруження у зоні контакту двох тіл.

Determination of the contact stresses taking into account the temperature fields are of importance for the investigation of the machine parts and construction elements strength in the area of their interrelation and while calculating elastic base constructions in order to use properly the construction material and the supporting power of the base. Solution of the axis symmetric contact task of thermo – elasticity of the cylinder circular transversal – isotropic punch stress on the elastic transversal- isotropic layer, which is free of friction, on the rigid base taking into account non-ideal thermal contact between the punch and the layer was built. The layer surface outside the contact area is free of the outside forces. Stress tangents on the contact area equal zero. In the lower part of the layer surface the vertical displacements and tangent stresses equal zero. In the free side of the cylinder the constant temperature is kept and the side surface of the cylinder is heat-insulated. The heat contact between the punch and the layer is expected to be non-ideal. Heat exchange with the outside medium occurs on the free surfaces of the layer according to the Newton’s law. Under given assumptions the method of determination the temperature fields in the cylinder and in the layer as well as the normal contact stresses has been developed. Equation of heat conductivity and thermo- elasticity for the layer was solved taking advantage of the Hankel integral transformation method. The temperature field in the cylinder was found by the Fourier method. Thus, displacement and stress in the transversal –isotropic layer is presented as the Hankel integrals, including some unknown functions, which are found from the boundary conditions of the task. The temperature field in the cylinder is expressed by means of the unlimited system of coefficients. Providing the boundary conditions of the task results in the system of the integral equations, which connect these unknown functions with the coefficients, which characterize the temperature field, and as a result, the Fredgolm integral equation of the second kind relatively the function is obtained, due to which normal contact stresses under the punch are expressed. The Fredgolm integral equation of the second kind was solved by the numerical method using the system of the linear algebraic equations. The numerical example for the transversal isotropic magnesium was analysed. The effect of the contact conductivity and isotropic coefficients of materials on the distribution of the temperature and the temperature component of the normal stress in the area of contact of the body system cylinder layer has been analysed.