LIMITING EQUILIBRIUM ANALYSIS OF CURVILINEAR BOUNDARY CRACKS IN AN ELASTIC HALF-PLANE, WITH THE STRESS ASYMPTOTICS TAKEN INTO ACCOUNT

摘要

It is frequently the case for fracture processes that the stress state near a crack is equivalent to a field of compression and shear. In such situations, the problems of fracture mechanics can be stated as a combination of a contact problem and a problem of mechanics of a deformable solid. These problems require special methods for their solution, because the distribution of the contact regions is unknown. Contact between the crack edges can be caused not only by external loads, but may be due to the shape of the crack [1, 2], or the interaction between cracks and other defects [3, 4], or their interaction with the boundary of the elastic body [1, 5], Thus, an investigation of the limiting equilibrium of boundary cracks should be performed with the possibility of contact between their surfaces taken into account. In this work, we consider a two-dimensional problem for the stress-strain state of an elastic half-plane weakened by a curvilinear surface crack whose surfaces are interacting with friction. A method for the calculation of the limiting equilibrium of such cracks is developed on the basis of the solution of a system of singular integral equations corresponding to a mixed (contact) boundary-value problem of elasticity. In this investigation, we take into consideration the behavior of the solution near the vertex on the boundary, and for this purpose an approach utilizing the results of previous asymptotic analysis [5] is proposed. In the present paper, we focus on taking into account, in a correct manner, the behavior of the solution near the surface crack. It is shown that its asymptotic behavior near the boundary substantially affects the limiting equilibrium of cracks. Solutions are obtained for problems of equilibrium of cracks with no interaction between their surfaces, as well as for cracks whose surfaces are in partial contact. Surface cracks in an elastic half-plane with no contact between their surfaces were considered, for example, in [1, 6-8]. Quasistatic growth of an arbitrary curvilinear surface crack in an elastic half-plane was studied in [9], with possible contact between its surfaces being neglected. In [10], a similar problem was formulated with possible surface contact taken into account. Two-dimensional (three-dimensional) problems for rectilinear (plane) surface cracks whose surfaces are in contact with friction were studied in [8], and some kinked cracks in an elastic half-plane with surfaces interacting with friction were considered in [11]. In [12], a problem of a rigid punch acting on an elastic half-plane with curvilinear cracks is studied with possible contact of their surfaces taken into account (without friction or with complete adhesion in the case of contact). In [5], a method is proposed for the investigation of surface cracks in two-dimensional domains, with friction between their interacting surfaces taken into account. Here, this method is refined to take into account quantitative characteristics of the asymptotic behavior of the solution near the boundary.