Contact problem for punch of complex shape and preliminary stressed semi-space

Наведено розв’язок осесиметричної контактної задачі про тиск штампа складної конфігурації на пружний ізотропний півпростір з урахуванням попередніх напружень. Побудовано функції розподілу контактних напружень для граничної площини. Проаналізовано вплив конфігурації штампа на розподіл контактних напружень.Determination of contact stresses and deformations in the interaction of rigid and elastic bodies stamps with existing permanent deformations 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. Resultant stress can cause fracture and accelerate some phase transitions, corrosion in particular. To improve the accuracy of calculations the residual must be takeninto account, which directly affect the state of bodies. Taking into account all the factors that affect the interaction of the body is one of the key problems to determine the stability, reliability and other characteristics. Therefore, consideration of residual stress in the study of contact interaction of elastic bodies is an important task. Research problems investigations 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 the residual stresses in bodies. In general, strict proper statement 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. In the article the solution of axisymmetric contact problem of pressure a punch of complex shape for an elastic isotropic half-space, taking into account preliminary stresses is described. Besides distribution function of contact stresses and displacements for the plane boundary of semi-space was created. Influence of residual stress on the distribution of contact stresses under the punch was investigated. With basic relations of the linearized elasticity theory, the problem is treated as the construction of the solution of the double integral equations. 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 punch has been analyzed