PSI - Issue 81

V.S. Kravets et al. / Procedia Structural Integrity 81 (2026) 102–108

104

where * [ ( )] z u x is the jump of longitudinal shear displacements of the points at the crack-like defect surfaces ( 44 ( ) z u a qh x  is the homogeneous component of the z -component of displacements in the anisotropic body at the upper face of crack-like defect for a given body load; G 0 is the shear modulus of the injection isotropic material S 0 after its hardens; 44 a is one of the elastic constants of the anisotropic material of the body Lekhnitskii (1963). The problem formulated by the superposition method is reduced to determining the perturbed stress state of an anisotropic body with mathematically rectilinear crack [- l ;+ l ] from the following boundary value problem at the Ox axis * * 0 x y u u   ); * z 0 z z u u u   , 0

W

 

( ), x

( ) x

( ) 0, x      x a a [ ; ] ,

q q





yz

xy

y

( ) x



 

yz

(2)

,  

( ) x      U U ( ) 0, x        [ ; ] [ ; ], x l a a l ( ) 0, x [ ; ] [ ; ], l l y x

  

xy

( ) 0,

( ) x

z u x

xy

y

( ) W yz x  is given by the expression (1).

where

From boundary conditions (2) based on the dependencies for the displacements of the crack surface points along the Ox axis Lekhnitskii (1963) (for an antiplane strain of anisotropic body) and the corresponding complex potential (CP) of stresses for an anisotropic body with a rectilinear crack (Fig. 1), the singular integro-differential equation is obtained Sylovanyuk and Ivantyshyn (2022)

l

( ) f t dt f x G  ( )

c













3

0

  

1 ,    x l

44 0 H a x q a G H a x  



(3)

2

( )

t x

h x

l



relative with the unknown function of longitudinal shear displacement f ( x ) of the upper crack face on the interval [ – l , + l ] and its derivative ( ) f x    3 3 ( ) / 2Re ( ) / , z f x u x x ic x l           , (4)

2

2 2 44 44 345 45 355 55 3 / , / , / c a a c a a c a a    ; 2

3 44 55 45 c c c c   ,

where Н ( х ) is the Heaviside function,

/ z u x x l     ; 0,

2

3 44 55 45 a a a a   , ij a – elastic constants of the anisotropic material of the body Lekhnitskii (1963). Here 3 ( ) x  is the CP of stresses Sylovanyuk and Ivantyshyn (2022)

l

3 ( )

( ) f x dx 

c

d z 

3

3 3

  

3 3   ( ) z

ic



(5)

,

3

2

x z

dz

3

l



at the Ox axis, where is the complex root of the algebraic characteristic equation for longitudinal shear of an anisotropic body Lekhnitskii (1963). The shear stress components and shear displacement in anisotropic body are determined through CP (5) by the expressions 3 3 3 ( , ) 2Re[ ( )] xz x y z     , 3 3 (, ) 2Re[ ( )] yz x y z    , 3 3 (, ) 2Re[ ( )] z u x y z   (6) The numerical solution of the singular integro-differential equation (3) for the partially filled crack (a < l ) was obtained by the quadrature method Savruk and Kazberuk (2017), Kravets' (2016), Savruk at al. (2025). Using the dependencies for the unknown function f ( x ) 3 3 3 3 3 i c c ic c     45 44 3 44 ; z x y

x 

l



( ) f t dt 

f t dt 

x 

( )

( ) ,

[ ; ],

( ) 0  

f x

x l l   

f l





(7)

l



and taking into account the root singularity of the integrand at the ends of the integration interval Kravets' (2016), Savruk at al. (2025), we will proceed to the calculation of the dimensionless function

2

( ) w f t   

( ) 1 ,

, [ 1; 1]       t l

(8)

.