PSI - Issue 72

Taras Dalyak et al. / Procedia Structural Integrity 72 (2025) 13–19

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The well-known solutions of plate bending problems with cracks systems that were obtained using applied theories (Williams (1961), Isida (1977), Berezhnitskii et al. (1979), Savruk (1981a, 1981b), Murakami (1987), Opanasovych and Seliverstov (2001), Zehnder and Viz (2005), Delyavskyy et al. (2021)) do not take into account crack closure possibility during bending deflection. The partial thickness contact surfaces of a single crack was studied using the two-dimensional plane stress state and bending theories by Jones and Swedlow (1975), Heming (1980), Shatsky (1988), Shatskii (1989), Joseph and Erdogan (1989), Young and Sun (1992), Khludnev (1995), Dempsey et al. (1998), Khludnev and Kovtunenko (2000), Shatskii (2001), Bozhidarnik et al. (2006), Lazarev (2011). A significant number of problems on the interference of closable cracks in homogeneous, multiply connected and piecewise-uniform bending plates were solved in the works of Shatskii (1990, 1991), Perepichka and Shats’kyi (2002), Shats’kyi and Dalyak (2002), Dalyak (2004, 2019), Opanasovych et al. (2008), Muzychuk (2013), Opanasovych and Slobodyan (2018), Shatskyi and Dalyak (2018), Sulym et al. (2021), Slobodian and Shainoga (2024). Almost all of these publications consider loads of the same sign on the all sections. Having the closure of collinear cracks under bending, Shatskii (1990) first showed a fundamental difference between the cases of loading defects with moments of the same sign and different signs. The aim of this paper is to study the influence of the parallel cracks closure on the stress-strain state of the infinite plate under the antisymmetric bending. A system of one-dimensional integral equations is formulated to find the displacement and the normal rotation angles jumps, based on the boundary conditions of smooth contact, written in the non-strain configuration, taking into account the conditions of the load antisymmetry. The influence of the relative location of cracks on the stress-strain state of the loaded plate was investigated based on the asymptotic and numerical results of the formulated problem solution.

2. Materials and methods 2.1. Problem formulation

Let us consider the infinite isotropic plate in thickness h 2 , which is with two straight-line cross-cutting parallel cracks in length l 2 , located at a distance d (Fig.1). The crack edges are loaded by uniformly distributed moments of different signs const 2 1    m m m ; the plate is free from stress at infinity. The influence of the contact interaction of crack edges on the stress-strain state of the plate is investigated.

Fig. 1. Diagram of the load of the cracked plate

Since the crack edges contact under bending leads to antisymmetry breaking of the stress fields and displacements along the thickness of the plate, the stress state of the plate outside the crack is described by a pair of biharmonic equations of the plane stress state and the classical theory of plate bending:

L x y ( , ) R \ 2  ,

0   w ,

(1)

0,   

here  is the stress function, w is the plate deflection,  is Laplace operator, L is set of segments of crack paths.