PSI - Issue 68

13 5

Ivan Shatskyi et al. / Procedia Structural Integrity 68 (2025) 9–15 I. Shatskyi et al. / Structural Integrity Procedia 00 (2025) 000–000

! ! !

! " !

Comparing the expressions

and

in Eqs. (6) and (7), we determine the efficiency of the renovation:

& = = & & " " #

# #" !! # ! % " ! " % !

#

# $ $ ! $! # "

(8)

.

.

!

"

%

!

%

If a periodic series of cracks is healed near the surface of the half-plane, then the problem of stretching the half plane with a periodic system of parallel internal cracks will be informative. In this case, the limit equilibrium of the system is determined by the function tabulated in Nisitani et al. (1973) from the dimensionless parameters and , namely: # " ! $ " ! ± ! = = #" ! "

! !

+ " ! !

+ #

! !

$ # " ! $ = = ! "

!% ! & '

. !" ! # $

(

&

#

#

! $ "# # " ! # " ! $ % ± ± =

&

)

)

%

%

(9)

$%&

+ , + * , + * # " ! " # " ! ! " !

,

.

$

=

)

(

"

'

'

Finally, basing on Eqs. (6) and (7) we have an expression for the treatment efficiency:

! ! " ! ! # $

! ! % ! ! & '

(

"

1

(10)

# $ $ ! 2/ 1 0

# !%&' 0

$

.

.

!

=

(

"

/

.

" "

" "

(( ) *

(( ) *

++ , -

++ , -

/ /

/ /

+ .

+ .

.

+

. $ #" ! 0 /

. $ #" ! 0 /

!

!

)

)

3. Results and discussion First of all, we note that results (3) and (5) for isolated cracks are a partial case of results (8) and (10) for infinitely distant cracks ( ). Indeed , . Using the method of singular integral equations (Savruk, 1981) for periodic problems, we listed the functions , for a uniform grid of . The results of calculating of the healing efficiency according to formulas (8) and (10) for different values of parameters and are presented in Figs. 3 and 4. Only values for which a positive reinforcement effect is achieved should be taken into account. ! = ! '&% !$!"!# " = ! " ! " $#! % & ! ! ± ± = ! ! ( ) #" ! $ " ! # ! ( ) #" !%#" !$ #" ! & ! ! " ! # + # ± ! ! ! ! ! > 

Fig. 3. The efficiency of cracks healing near the tips.