PSI - Issue 59

3

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

248

2.2. Stress-strain state near the contact crack

Based on the analytical solution of problems (1) – (3), constructed by the method of singular integral equations, we found discontinuities in the kinematic field on the crack and an expression for the contact reaction between its edges (Shatsky (1988), Young and Sun (1992), Shatskii (2001)):

h m | |



| 4 | m 

(1 ) 4 m  

2 2

2 2

( )

N x y

(4)



[ ]( ) u x

, [ ]( ) x 

;

.

l

x

l

x









y

y

1

(1 )  





Bh

Da

3

2

(3 )(1 )    a ,

3(1 ) (3 )     ,

2 /(3(1

))

 D Eh

 

 

 

Here B Eh 2  ,

,

 , E are the Young's modulus

and Poisson's ratio of the plate material. In the small neighborhood of the crack tip, the forces and moments have a root singularity, so that on the crack extension ( 0   ):

K

K

( ) 0 r

( ) 0 r

M   

( ,0)

N   

,

,

M r y

( ,0)

N r y

2

r

2

r

 )

 

and on the crack contour (

B

Da

( ) u r K r O r   2 [ ]( )

[ ]( ) r 

2

( )

K

r O r 



,

.

y

N

y

M

4

4

Let us restore the asymptotic distribution of stresses and displacements along the thickness of the plate:

1

3 h

1

2 z

B

z

  

N      

  

2    

( ) 0

( ,0, ) r z

[ ]( , ) U r z K

( )

K

K

O r

K

r O r

(5)









,

.

y

N

M

y

M

2

2

4

h

h

2

r

We can talk about the intensity coefficient of normal stresses depending on z

1

h 3

z 2

  

  

.

( )

k z 1

K

K





N

M

2

h

For a rectilinear crack with contacting edges under conditions of symmetric bending at infinity, based on solution (4) for the force and moment intensity factors and the stress intensity factor, we have:

h z

m 1 | | 3   

m



h 1

 1 | |

m l

m l



(6)

( )

;

.

l

k z 1



,

K

K





N

M

2

1







2

N M K ,K on the load, as well as their dependence on the

We note the nonlinear dependence of the coefficients

Poisson's ratio of the plate material, as a consequence of the one-sided contact of the edges. By formal substitution 0   from expressions (4), we can obtain solutions that correspond to the classical formulation without taking into account the contact (Williams (1961), Berezhnitskii et al. (1979)). In particular,

4 Da m

2 2

[ ]() 0, [ ]() x u x  

l

x





.

y

y