PSI - Issue 82

Victor Rizov et al. / Procedia Structural Integrity 82 (2026) 253–259 V. Rizov/ Structural Integrity Procedia 00 (2026) 000–000

256 4

layers. By analyzing the column compliance, we obtain formula (8) for the SERR for the two symmetric delamination cracks.

!

% "

!" $ !# ' " ! ! " #

$ % &

C

=

,

(8)

!

! ! "

!

where is the column width,

is the support reaction in the lower end of the column, is the delamination

!

length,

is the compliance. The latter is given by .

! ! " ! ! " #

$ =

(9)

The support reaction is determined with the help of the following equations. The subsidence is expressed as a function of strains by (10). , (10) " + + = " ! " " ! ! "

! #

! "

&

$%

$%

#

#

!

! "

" $

#

# #

# #

" ! !

" ! ! ! !

" ! ! ! !

where

and

are the strains in the corresponding parts of the column. These strains are presented as

" ! !!

" ! ! !

$ "#"! ! ! !

" $#"! ! ! !

functions of the stresses,

and

, in parts,

and

, of layer 2 via (7), i.e.

!

$ !#!"

% ! " " % ! " "

=

"

,

(11)

$ !

" "

!

!

! $#!"

=

"

.

(12)

! $

" "

!

Formulas (11) and (12) are inserted in (10), i.e. . " + + = " ! " " # # # # $% $% ' ! ! " " &%"$ ! "%"$ # #

! #

! "

(13)

& '

& '

!

#

"

"

" ! !

! !

The equation of equilibrium of cross-section,

, of the column is

% ! " = = & " "

# $

#$

=

!

,

(14)

& "# ! " ! ! "

"

" $#"!

! !

! !

! !

where the thicknesses,

and , are shown in Fig. 1. In view of the symmetry and by using the relaxation moduli,

formula (14) is transformed as

[

'

'

" " #$

&

# $

"#

"

+

"

=

!

.

(15)

% % % "#%! ! !

" " % "#"! ! !

" " $#"! ! !

" ! !!

Since the strains in layers in part,

, of the column are equal, (15) is re-written as

!

= " # $

% & '

&

% "#"!

& ! ! $

% % " #$ "

#

"#

!

!

+

.

(16)

" % "#"! ! !

" " $#"! ! !

"

$ "#"! ! ! !

" $#"! ! ! !

Equations (13) and (16) are used for determining of

and

.

! ! "

The support reaction,

, is found by (17), i.e. .

#

!"

! =

(17) This support reaction is inserted in (8) for obtaining the SERR for the two symmetric delamination cracks. A check-up of the SERR for the two symmetric delamination cracks is carried-out by (18). , (18) ! $#!" ! % $ $ $

!"# "$

% ! =