Stressed State of a Plane with Periodic System of Closely Located Curvilinear Holes with Edge Cracks
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STRESSED STATE OF A PLANE WITH PERIODIC SYSTEM OF CLOSELY LOCATED CURVILINEAR HOLES WITH EDGE CRACKS V. S. Kravets’
UDC 539.3
A plane periodic problem of the theory of elasticity for an isotropic plane with infinite row of closely located curvilinear holes with edge cracks is solved by the method of singular integral equations. The stress intensity factors (SIF) at the tips of edge cracks propagating from symmetric holes of various shapes (elliptic holes, rhombic holes with rounded vertices, and narrow slots) are computed for arbitrary distances between the holes under the conditions of tension of the plane at infinity (mode I). The SIF for the edge cracks at the rounded vertices of the corresponding bilateral notches in the elastic plane are found as a result of the limit transition as the distances between the holes approach zero. Keywords: plane periodic problem of the theory of elasticity, edge crack, curvilinear holes and notches, singular integral equations, stress intensity factors.
Introduction Periodic problems of the theory of elasticity for a plane with infinite row of holes containing edge cracks were considered mainly for the case of circular holes and rectilinear cracks [1, 2]. In this case, the numerical results for the stress intensity factors (SIF) were obtained in the case where the relative distances between the tips of neighboring cracks are not small. For closely located holes, high stress concentrations are observed on their contours [3], which creates significant difficulties in the evaluation of the distribution of stresses near holes, and hence, in the evaluation of SIF at the tips of the cracks propagating from these holes. In what follows, by the method of singular integral equations (SIE), we consider a plane periodic problem of the theory of elasticity for a plane containing an infinite row of closely located curvilinear holes with two rectilinear edge cracks in the case of double symmetry of the problem. The obtained solutions continue the results of our previous investigations [4, 5]. As a result of the limit transition (as periodic holes with cracks approach each other), we obtain solutions of some new problems. In particular, we find the SIF for the edge cracks propagating from the vertices of semiinfinite bilateral curvilinear notches in the elastic plane. We use these solutions as asymptotic solutions for the corresponding bounded domains with deep lateral notches and relatively short edge cracks. On the basis of these results and known values of the stress intensity factors for a half plane with edge notches and cracks propagating from their tips [6–8] (the asymptotics obtained for the domains with the corresponding shallow cracked notches), we use a modified interpolation formula (of the Neuber type [9]) for the determination of the SIF for edge cracks of various lengths propagating from the lateral notches of any depth in the strip. Integral Equations of the Problem Consider the case of vertical tension of an elastic plane (domain S ) by stresses p applied at infinity. The pla
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