PSI - Issue 81

Ivan Shatskyi et al. / Procedia Structural Integrity 81 (2026) 129–134

130

estimates of the stress-strain state and limit load of thin-walled structural elements. The classical plate bending theory without accounting for contact gives kinematically contradictory results – mutual penetration of crack faces in the compression zone (Williams, 1961; Isida, 1977; Berezhnitskii et al., 1979; Zehnder and Viz, 2005). To eliminate this contradiction, a line contact model within the framework of classical Kirchhoff theory was proposed in works (Shatsky, 1988; Young and Sun, 1992; Shatskii, 2001). For homogeneous plates with crack systems, periodic contact problems were studied in works (Shatskii, 1990, 1991; Perepichka and Shats’kyi, 2002; Dalyak, 2004; Opanasovych et al., 2008). At the same time, the influence of multilayer structure on the behavior of contact cracks remains insufficiently studied. The problem of cracked laminated plates under bending has been addressed by several researchers (Berezhnitskii et al., 1984, 1986; Delyavskyy et al., 2021), but the combined effect of layered inhomogeneity and crack face contact interaction has not been fully explored. The aim of this work is to investigate the stress-strain state and limit equilibrium of a multilayer plate with a periodic system of collinear cracks, taking into account the contact interaction of their faces. 2. Materials and Methods 2.1. Problem Statement and Contact Model Consider an infinite plate 2 ( , , ) [ , ] x y z h h    R , made of a layered material with elastic modulus ( ) E z that varies symmetrically through the thickness, and a constant Poisson's ratio  . The plate is weakened by a periodic system of straight through cracks of length 2 l , located along a line with period d . To describe partial crack closure within the theories of plane stress state and Kirchhoff bending, we use the line contact model (Shatsky, 1988; Young and Sun, 1992; Shatskii, 2001). The essence of the model is as follows: due to the loss of antisymmetry relative to the mid-surface, the stress state of a plate with a contact crack is represented as a superposition of bending and plane stress state components. Incomplete through-thickness contact of cut edges is interpreted as closure of its edges on the front surface of the plate. The stress-strain state of the plate is described by a system of biharmonic equations:

2

 R

0,    

0, ( , ) w x y

\ , L

(1)

2 2 2 2 / / x y      is the two

where  is the Airy stress function for plane stress state, w is the plate deflection,

( /2, /2) L l kd l kd     , k  Z is the set of crack location contours.

dimensional Laplace operator,

On crack faces, smooth unilateral contact conditions are formulated:

(2)

[ ] |[ ]| 0, u h   

sgn[ ], 

0, N x L y ,

0,

M hN 

  

y

y

y

y

y

y

y u is the normal

where square brackets denote the jump of the corresponding quantity when crossing the crack line; displacement in the base plane, y  is the angle of rotation of the normal; ij N are membrane forces,

ij M are bending moments.

At infinity, uniform bending conditions are specified:

(3)

0, xy y N N N M M M m x y        0, , ( , ) . x xy y x

For a multilayer structure, the elastic modulus is given as:



   

1

1

M

m

m 





  

 

   





( )

| |

| | h H z        i

,

(4)

E z

E H z  

h







m

i

 

0

0

0

m

i

i







where ( ) H  is the Heaviside function, M is the number of layers, m E and m h are the elastic modulus and thickness of the m th layer, respectively. For a symmetrical three-layer plate, this simplifies to:

,

| | h z h  

E E

z h   

  

outer inner

( )

,

(5)

E z

, | |



[0,1]   is the relative thickness parameter of the inner layer.

where