Contact interaction of a circular punch with a preliminarily stressed isotropic layer

Abstract

UDC 539.3 Using the linearized elasticity theory, we obtained the solution of the contact problem of pressure of a rigid circular punch of complex geometry on a preliminarily stressed isotropic layer. A numerical example of constructing the distribution function of contact stresses is considered. The effects of residual strains in the layer and the shape of a rigid punch on the distribution of contact stresses are analyzed. The enhancement of the reliability and durability of structures and mechanisms is one of the most urgent tasks of modern building and mechanical engineering. As is known [2], the residual deformations are almost always present in elements of structures and articles of machines. The nature of their appearance can be very diverse: irreversible deformations (plasticity, creep), structural transformations in materials, change of the aggregate state in separate places of structures, mechanical, chemical, and technological processes, etc. The stresses arising in these cases, like any other ones, can cause fracture and accelerate certain phase transitions and corrosion. Consideration of residual deformations in the calculations of critical elements of structures, machines, and buildings allows one to estimate more exactly the safety factor of a material and, hence, to decrease essentially its consumption, by preserving the necessary functional characteristics of elements as a whole. For this reason, studies of the contact interaction of elastic bodies with residual deformations are extremely crucial at the present time and will be such in the future. Studies of problems of contact interaction of preliminarily stressed bodies in our country and abroad appeared in significant numbers only at the end of the last century. This is related, in the first turn, to the fact that linear elasticity theory does not consider the presence of residual stresses in bodies. In the general case, the strict statement of such problems requires the application of the apparatus of nonlinear elasticity theory. However, if the initial stresses are sufficiently high, we may restrict ourselves to its linearized version. The present level of the linearized elasticity theory and mathematical methods together with the intense development of computers allow one to efficiently form various calculation models for a wide circle of problems. For example, the apparatus of the linearized elasticity theory was successfully used in works [5, 6] for the construction of a three-dimensional model of bounded elements and for studying the effects of the interaction of fibers during microdeformations in joints strengthened with isotropic and anisotropic fibers. A fairly complete description and the classification of works devoted to the theory of contact interaction of preliminarily stressed bodies with rigid punches can be found in [1]. However, the question on the interaction of ring punches of complicated configurations with the elastic half-space or a layer with residual deformations remains insufficiently studied. Let us consider the axisymmetric problem of the pressing of a rigid ring punch on a preliminarily stressed isotropic layer of thickness h that lies on a rigid absolutely smooth base. The problem will be solved within the framework of the linearized elasticity theory with the use of the terminology and notation of [3]. We assume that the elastic potentials are continuous twice differentiable functions of algebraic invariants of the Green’s tensor of deformations [3].