Pressure with friction of a perfectly rigid die upon an elastic half space with cracks
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PRESSURE WITH FRICTION OF A PERFECTLY RIGID DIE UPON AN ELASTIC HALF SPACE WITH CRACKS M. P. Savruk1 and A. Tomczyk2
UDC 539.3
We study the interaction of a rigid die with a base of any shape and the surface of an elastic half space containing cracks in the presence of friction in the contact zone. The solution of the plane contact problem of the theory of elasticity is obtained by the method of singular integral equations. The detailed analysis of the problem is performed for the case where the base of the die is parabolic and a crack is rectilinear and appears on the surface of half space. We also investigate the effects of the friction coefficient, crack length, its orientation, and location on stress intensity factors K I and K II at the crack tip and the distribution of contact stresses under the die. Keywords: contact problem, stress intensity factor, elastic half space, crack, friction.
In the strength and durability analyses of contact couples from the positions of fracture mechanics, it is customary to assume that cracklike defects are present in one or both elements of the couple. This leads to the redistribution of stresses near defects, which may cause damage to the contact surfaces or even lead to failures of these elements. Therefore, the investigations carried out in this field are always urgent. In the mathematical modeling of contact interaction between two bodies, it is traditionally assumed that one of these bodies is an elastic half plane, weakened by a collection of cracks and the other body is a rigid die. In the major part of investigations of this sort, the action of the die is replaced by the corresponding distribution of contact pressure. Approaches of this kind can be found in the works by Hasebe, et al. [1–7], where method of conformal mappings is applied to the solution of the problem [8]. Thus, e.g., the case where one edge of a die acting upon the boundary of a half plane containing an edge crack is rounded and the other edge is sharp is considered in [6]. In addition, the influence of the distance between the crack and the die on the values of the stress intensity factors (SIF) is investigated. Panasyuk, Datsyshyn, and Marchenko [9, 10] used the method of singular integral equations for the solution of the two-dimensional problem of interaction of a rigid die (with base of any convex shape) with a half space weakened by cracks whose lips may be in contact. The problem was reduced to singular integral equations written in the operator form. The numerical results were obtained for the case of a plane die pressed into the half plane without friction. In the works by Datsyshyn, et al. [11–18], the method of singular integral equations is also applied to the solution of contact problems of this type. Much attention is given to the construction of the paths of propagation of fatigue cracks under the action of model contact loads applied to the edge of the half space (half plane). Goshima et al. [19–24] studied the problem of rolling (and rolling with sliding) of a rigid cylinder over th
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