PSI - Issue 68

Roman Kushnir et al. / Procedia Structural Integrity 68 (2025) 32–38 R. Kushnir et al. / Structural Integrity Procedia 00 (2025) 000–000

36

5

!

(

)

( ) ( ) ! $% ! !

" ! = ! ! ! ! # " $

$ ! !

"

,

(23)

#

=

"

!"

!"

#

%

!

!

! ! ! !

! ! ! !

where is a unit vector collinear with !

; and is a unit vector normal to

.

4. Boundary integral relations According to Pasternak et al. (2016) the following extended Green’s second identity can be constructed for differentiable functions and and a symmetric 2nd order tensor ( ) ! !

! !

!" "! # # =

!" #

(

)

(

)

%%

%%%

! !" " ! " " ! $ #

! !" " ! # $ %&

! !" !" ! " " ! $ ! !" !" # #

%'

=

.

(24)

#

B

B

!"#$ % !

The same can be derived for

differentiable vector-functions

and

and symmetric 4th order tensor

! !

! !

! !

( ) !"#$ #$!" % % = ! ! ( % %% !

:

)

(

)

% !

% !

% !

%%%

(25)

"

! ! !"#$ #$ ! !"#$ #$ " ! " " ! $ ! C '(

')

=

! ! !"#$ # "$ ! !"#$ # "$ ! " " ! $ !

#

B

B

Here

is a boundary of the 3D domain

and

is a unit outward normal to the surface

.

! "

! B

! B

B

! !

Substituting

and

instead of

and

in Eq. (24) and accounting for Eqs. (2), (9) one obtains the heat

!

!

!

conduction integral formulae

(

) ( ) !

( ) " !

(

) ( ) ( ) ( ! " ! $ ! ! ! ! ! ! #

)

( ) ( ) ( ) & $' ! ! ! ! #

%%

$ # %%%

!

!

# # $%

!

=

#

,

(26)

"

"

"

!

"

"

B

B

(

)

( ) ( ! " ! ! ! #

)

where

.

# ! !

!

#

!" ! $ % = ! ! ! " ! = !

"

#

"

"

(

) ! "

Assuming that

,

in Eq. (25) and accounting for Eqs. (3), (8), and (20) one obtains

! "! # ! = ! !

(

) ( ) " ! ! ! ! ! ) ( ) ! " $ ()

( ) ! !

(

) ( ) " ! ! ! ! !

(

# %% %%% B

" % &

$ !

'

=

$

!

!"

!"

(27)

(

) ( ) ( ) * (+ ! ! ! ! ! "

( ) !

(

) ( ) " (+ !

"# ! " %%% !

" ! !

%

%

+

+

!

" !" #

!

!"

"

B

B

(

)

( )

(

)

!

where . The last integral in Eq. (27) can be converted to a surface integral as follows. According to Eqs. (13)–(15) the following dependence holds, ! !" " # $ ! = ! ! " ! " " !" "#$% # !$ % C ' ( ) = ! ! ! ! ! and

(

)

(

)

!

.

(28)

! ' ! = ! !

! ! !

% &

! !" # !"

"

! $% #$ %

"

( ) ! " ! " ! = ! !

( ) ! " = !

Accounting for Eq. (2) and assuming that

,

in Eq. (24) one can write that

( ) !

(

) ( ) (' !

(((

! ! !

& '

!

! !" # !"

"

!

(29)

(

) ( ) !

(

) ( ) $ ! ! ! " !

( ) !

(

) ( ) ( ) ! + (' ! ! ! ! !

((

(((

! ! ! !

)

'

% (* %

'

=

+

#

$ &

'

"

"

#

#

#

%

!

!

"

(

) = ! ! ! "

(

) ( ) " ' ! ! ! ! "

where

.

$

% &

" "# ! #

!