Pressing by pressure of two hard parabolic punches of the isotropic layer with the residual deformations

Наведено розв’язання контактної задачі про стиснення попередньо напруженого ізотропного шару двома співвісними параболоїдальними штампами. Розглянуто числовий приклад та побудовано функції розподілу контактних напружень і переміщень для обох граничних площин шару. Проаналізовано вплив залишкових деформацій на розподіл контактних напружень під штампами.Increasing reliability and durability of structures and mechanisms is one of the most actual tasks of modern construction and engineering. As it is known [2], residual deformation is almost always available in the structural elements and machine parts. The nature of their appearance can be very different: irreversible deformation (plasticity, creep), structural changes in the material, changes of the aggregate state in some areas, mechanical, chemical and technological processes, etc. Resulted stress, can cause fracture and accelerate some phase transitions, corrosion. Consideration of the residual strains under development of the important structural elements of machines and installations can estimate more accurately the material strength life and significantly reduce its costs, while maintaining the necessary functional characteristics of the elements in general. That is why the study of the contact interaction of the elastic bodies with residual deformations is up to date and will remain so in the future. Research problems of the contact interaction of the preliminary stressed bodies in our country and abroad had appeared in the sufficient quantity only by the end of the last century. First of all it is due to the fact that the linear elasticity theory does not consider of residual stresses in bodies. In general, strict formulation of such problems requires the use of system of the nonlinear elasticity theory, however, for the sufficiently large values of the initial stresses its linearized version can be referred to. Current level of linearized elasticity theory and mathematical techniques, combined with the rapid development of computer technology makes it possible to form effectively a variety of computational models in a wide range of tasks. Complete enough description and classification of works devoted to the theory of contact interaction of the preliminary stressed bodies with rigid punches can be found in [1]. But the interaction of the complex configuration circular punches with the residual deformation elastic half-space and layer stays not studied enough. In the article the solution of the contact problem of compression of the preliminary stressed isotropic layer with two parabolic punches using the linearized theory of elasticity was built. All calculations are performed in the location of strain yi , which are associated with the coordinates of the initial state relations yi = λi xi ( i = 1, 2,3) , where λi – coefficient of linear elongation element directed along the Cartesian axis xi . Using the basic relations of the linearized elasticity theory, the problem is treated as the construction of the solution of the double integral equations with kernels involving the Bessel functions. The authors have developed a method of approximate solutions of this type. The main idea of this method is to represent the unknown distribution function of the contact stresses in the form of the Fourier series with unknown coefficients and to construct a system of the linear equations for finding them. To demonstrate the proposed methods the numerical example of the construction of the contact stresses distribution function is presented in the paper. Besides, the residual deformation field characteristics effect on the amount and nature of the contact stresses under punches has been analyzed