PSI - Issue 81

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

105

The integral equation (3) is written in the dimensionless form and taking into account the condition of uniqueness of shear displacement while bypassing the crack contour Savruk and Kazberuk (2017)

1

l





( )  

w

( ) f t dt 





0

(9)

d

.





2

1

1

l





The numerical solution of this integral equation with an additional condition (9) is found from the system of N v linear algebraic equations in the discrete external Chebyshev nodes   cos / , 1, 1 m v v m N m N      at the integration interval [- 1;+1] with regard to N v unknown values of the function (8) in the corresponding internal nodes   cos (2 1) /(2 ) , 1, k v v k N k N      . For an anisotropic body with a rectilinear crack Mode III on the Ox axis, the SIF     III 3 0 lim , / 2 z r K c u r r     is found based on the asymptotic distribution of the longitudinal shear displacement of the crack surface as they approach its tips Sih at al. (1965):     3/2 III 3 3 ( , ) 2 / Im cos sin z u r K r c o r       , , , 0 x l r x l     m . (10)

Using the expression of shear displacement (6) and CP (5) the relative SIFs are calculated from the boundary values ( 1) w  of the unknown function (8)

/ F K q l   

(11)

( 1) / ( ) w qa w q c   m m ( 1) / ( / )

  

III

III

3

3

III ( ) F F A  

III III ( ) F F B  

respectively at the left ( III have the dependence ( 1)

) and right (

crack tips (Fig. 1). Due to the xy -symmetries of the problem we

( 1) w w    and accordingly the SIF at both crack tips is the same ( Ш Ш Ш F F F     ). CSL ( c q q  ) of the anisotropic body were also determined based on the force fracture criterion III III c K K  Sih at al. (1965) ( III c K is the crack growth resistance of the body material under longitudinal shear in the direction of crack propagation) for the partially healed crack. It was assumed that the injected filler material would not fail before the anisotropic body fracture Sylovanyuk and Ivantyshyn (2022), and the relative values of CSL were calculated using the expressions   0 3 III / / ( 1) 1/ c c q q q c w F    m , (12)

where 0 III / c c q K l   is the CSL of this anisotropic body with an unhealed crack ( a = 0). 3. Results and discussion

Numerical solutions of the constructed equations were obtained for a range of geometric and mechanical parameters of the problem. We considered various types of orthotropic materials for body and isotropic filler materials. For antiplane strain of orthotropic (along the Ox and Oy axes) body material we have 3 3 3 3 i i       ( 3 0   ) (Lekhnitskii (1963), Savruk and Kazberuk (2017)):

2

3 44 55 45 G G c c c c  / , xz yz

1/   

a G G

3  

(13)

1/

, G a yz

55 0,   a

1/

,

,

a

G



3

xz yz

44

45

xz

where , xz yz G G is the shear modulus of the orthotropic body material along the Oz axis on planes with normal along the Ox and Oy axis, respectively. Relative SIFs (9) and the corresponding CSLs (10) significantly depend on the crack filling volume (parameter / [0;1] a l  ) and the elastic characteristics of the orthotropic material of the body S – orthotropy level parameter 3  (11) and relative average shear stiffness 3 3 13 / с с G  % ( 13 max{ , } xz yz G G G  ) of the body material (Fig. 2). The contour of the crack-like defect (isotropic filler material S 0 : 0 13 / 0.05 G G  ) were taken in the form of an oblate ellipse with semiaxes , ( / 0.1) l c c l  for which 2 2 ( ) 1 , [ , ] h x c x l x l l      .