PSI - Issue 59

2

Ivan Shatskyi et al. / Procedia Structural Integrity 59 (2024) 246–252 Shatskyi et al. / Structural Integrity Procedia 00 (0000) 000 – 000

247

1. Introduction Researchers have made significant progress in modeling the stress-strain state of thin-walled structures with consideration of crack closure mainly by using two-dimensional classical theories of plane stress and plate bending. In particular, Jones and Swedlow (1975), Shatsky (1988), Youg and Sun (1992), Khludnev (1995), Dempsey et al. (1998), Shatskii (2001), Zehnder and Viz (2005), Bozhydarnik et al. (2006), Opanasowych et al. (2008), Syasky and Muzychuk (2012) used the classical Kirchhoff theory of plates to describe the crack surface contact, while Heming (1980), Joseph and Erdogan (1989), Lazarev (2011), Delyavskyy et al. (2021) applied the Timoshenko-Mindln Reissner theories. However, to date, the problem of using the results obtained to estimate the equilibrium limit of the cracked plates has not been completely solved. The reason for this is the uneven stress distribution at the through crack front in the bending plate. In this paper, based on a detailed analysis of the solution of the problem of bending a plate with a rectilinear crack, the main ways of applying the criteria of linear fracture mechanics to the construction of estimates of the limiting load in the case of an unevenly stressed crack front are considered. The authors' experience in developing asymptotic methods for analyzing ultimate loads for some contact crack systems in infinite and semi-infinite plates, as well as in a plate on an elastic foundation, is also described. 2. Material and methods 2.1. Formulation of problem and model of contact Let us consider an infinite isotropic plate [ , ] ( , , ) 2 h h x y z    R , containing a through rectilinear crack (cut) l 2 long, located along the abscissa axis. The crack edges and the front surfaces of the plate are free from external loads. The plate is bent by a uniform field of bending moments at infinity perpendicular to the crack line. Within the framework of the classical two-dimensional plane stress state and bending theories, we study the elastic and limit equilibrium of the plate, taking into account the contact of the crack edges caused by the bending deformation. The closure of cracks in the compressed part of the bending plate leads to a violation of the antisymmetry of the stress and displacement fields relative to the base plane, and membrane stresses arise in the vicinity of the defect. Following Kirchhoff's hypothesis of a rigid normal, we interpret the contact of the cut edges as the closure of its sharp ends along a line located on the front surface of the plate. The unknown reaction on the contact line is transferred to the middle surface of the plate with the corresponding compensating bending moment. This model of contact along the line under the conditions of symmetry of the stress-strain state with respect to the abscissa axis corresponds to the boundary value problem for a pair of biharmonic equations in the plane with interconnected boundary contact inequalities on the section (Shatsky (1988), Khludnev (1995), Shatskii (2001)):

2 R 

0,

x y 0, ( , )

\

w    

L



;

(1)

[ ] y

|[ ]| 0,  

], sgn[ 

0,   

( , ), x l l y

0

 u h

 M hN

N



;

(2)

y

y

y

y

y

 0, 0, N N N M M M m x y y xy x y xy x      , ( , )



(3)

;

Here  is a stress function, w is a plate deflection,  is a Laplace operator; [ ] y u , [ ] y  are the jumps in displacement and normal angle on the cut edges; ij N are the membrane forces, ij M are the bending moments, ( , ) l l L   is a crack contour.