PSI - Issue 13

Ivan Shatskyi et al. / Procedia Structural Integrity 13 (2018) 1476–1481 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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1. Introduction

The adequate analysis of the strength and durability of thin-walled structures is impossible without taking into account the concentration of the stresses in proximity to crack-like defects. Among the problems of the mechanics of cracks in plates and shells there is a specific set of problems which are connected with the mutual interaction of the crack surfaces, which fall into the pressure zone. The contact interaction of the edges of the defects may significantly redistribute the stress field and affect the indicators of the construction strength. In case of thin plates, these problems naturally arise when the plate is under bending strain, which gives different signs of tension along the median surface. If the slit theory (cracks with stress-free surfaces) in bent plates is sufficiently developed (Williams (1961), Isida (1977), Berezhnitskii et al. (1979), Savruk (1981), Murakami (1987), Zehnder and Viz (2005)), then the phenomenon of partial closure of cracks by the bending of plates has only been studied within two-dimensional theories in the last decades. The most productive model here has been that of the contacts of edges along a line (Jones and Swedlow (1975), Heming (1980), Shatskyi (1988), Shatskii (1989), Young and Sun (1992), Khludnev (1995), Khludnev and Kovtunenko (2000), Shatskii (2001), Bozhidarnik et al. (2006), Lazarev (2011)). Certain problems for arrays of contact cracks in plates and shells have been solved in the papers by Shatskii (1990, 1991), Shats’kyi and Dalyak (2000), Perepichka and Shats’kyi (2002), Dalyak (2004), Opanasovych et al. (2008), Shats’kyi and Makoviichuk (2009), Shatskii and Makoviichuk (2011), Dovbnya and Shevtsova (2014), Dovbnya and Hryhorchuk (2016). In the article by Shats’kyi and Dalyak (2002), the elastic equilibrium of the bent plate weakened by rectilinear cracks, connected with coaxial slits, was addressed. The aim of this paper is to conduct numeric-analytical research of the problem of interaction of rectilinear contact cracks and narrow slits in case of bending of the infinite plate and on this basis to demonstrate qualitative differences in the concentration of stresses in proximity of the defects of different nature. We shall start with the terminology. A crack is understood as a mathematical cut with zero spacing between the edges, which may contact without mutual penetration. A slit is understood as a physical cut with small spacing between the edges, yet in a model performance we understand it as a mathematical cut with surfaces free from load, on which a negative displacement jump is possible. The level of bending load is considered to be such, that the edges of the slit do not contact with each other under any circumstances. Let us view the infinite isotropic plate 2 ( , , ) [ , ] x y z h h    R weakened by the system of N rectilinear arbitrary oriented cuts. The geometry of their location is described by such characteristics: n  is inclination angle of the defect to the abscissa axis, 0 n x , 0 n y are coordinates inside the segment of the cut in the system xOy and 2 n l is length of n -th cut. We place the local coordinate systems n n n x O y in the neighbourhood of the cuts, so that the point n O would lie in the center of the line segment ( , ) n n l l  , and the axis n n O x would be direct along the defect line. Let us assume that among the N cuts there are 1 N cracks, whose aggregate contours create the set L 1 . The remaining defects 1 ( ) N N  will be slits with multiple contours L 2 . Consideration of the interaction of edges on at least one cut by the bending will disrupt the antisymmetry of the fields of stress by thickness in the whole plate. In addition to the jumps of the rotation angles of the normal, discontinuities of the displacements appear on the cuts in the middle surface of the plate. The stress state of the plates outside of the defects is described through biharmonic equations of classical theories of plane stress state and plate bending: 2 1 2 0, 0, ( , ) \ ( ) w x y L L      R  , (1) where  is stress function, w is plate flexure,  is two-dimensional Laplace operator. Depending on the type of the cuts, conditions of smooth contact of the edges are set out on the edges of the cuts (Shatskyi (1988), Shatskii (1989)): 2. Model and method 2.1. Formulation of the problem