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

37

6

Combining Eqs. (27), (28), and (29) one obtains the following integral formula for thermoelastic anisotropic quasicrystals

(

) ( ) " ! ! ! ! ! ) ( ) ! " % )*

( ) ! !

(

) ( ) " ! ! ! ! !

(

" (( (( ((( ! ! "

" & '

% !

(

=

#

!

!"

!"

(

) ( ) !

(

) ( ) # ! ! ! ! "

( ) !

(30)

" ! ! !

+

I

$ )* %

+

+

$ &

'

!

!

!

(

) ( ) ( ) J )I ! ! ! ! ! "

(

) ( ) ( ) # J )I ! ! ! ! "

(((

&

I

+

#

!

!

!"

"

!

$

!

!

! " !

In most practical applications, both the volume extended force are zero. Therefore, Eq. (30) simplifies to a purely boundary integral formula for anisotropic thermoelastic quasicrystals with a general 3D internal structure. Moreover, if the volume heat or extended body force are non-zero, their influence can be calculated in advance using the provided relations, which is advantageous for numerical boundary element analysis. 5. Boundary integral equations and boundary element analysis of cracked quasicrystals Boundary integral equations for heat conduction and thermoelasticity in quasicrystals can be derived from Eq. (26) and Eq. (30) by approaching the internal point to some boundary point . Nevertheless, the temperature and displacement boundary integral equations degenerate in the case of a crack, as its faces are very close to each other. This problem can be addressed using the heat flux and stress integral equations. Differentiating Eqs. (26) and (30), applying the constitutive relations, and approaching the internal point to some boundary point results in ! " ! ! ! ! ! !" ! ! and the volume heat

# % &

" '

( ) !

( ) " !

(

) ( ) ( ) B &' ! ! ! ! " !

++

!! #$% C $

B

!

=

! "

#

$

"

B

(31)

' ) *

(

) ( ) ( ) &' ! ! ! ! ! C

(

) ( ) ( ) " B ) &* ! ! ! ! C

++

( $ +++

!!

!!

(F$

C

(

(

$

"

$

"

B

B

# % &

! (

( ) " !

( ) " !

(

) ( ) ( ) D )* ! ! ! ! ! % " $

(

) ( ) ( ) + )* ! ! !

**

**

!

% ! !

"#$

C'#

D

B

(

*

=

$

!

#

!#$

!#$

" $

!

!

"

"

(

) ( ) ( ) )* ! ! ! ! !

(

) ( ) ( ) ! ! ! ! " B

**

**

(32)

"#$ % I

% - & )*

+

+

!#

"

!#

!

!

"

"

' ( )

(

) ( ) ( ) " $ K )/ ! ! !

(

) ( ) ( ) " & )/ ! ! ! !

***

***

% ! !

% - K

%

(

+

$

!#$

!#

!

!

where

! !"# !"$% $# % C D ( = ! !

! !"# !"$% #C'( C $' %( ) * * +I = ! ! !

,

;

,

,

!! # ! = !

! "

!! ! !" #$ # $" % & & ' = ! " ! "

!

!"

"

!

! + % !

!

! + % !

% C ' " = $

# '

' ( ) " = $

# '

!

!

* +

,

,

(33)

&

&

" !"#$ # $

" !"#$ # %$

"

!"

!"

!"

!"

%

C% C

CPV stands for the Cauchy Principal Value and HFP for the Hadamard Finite Part of an integral. Eqs. (31) and (32) can be effectively applied to solve crack problems. For their numerical solution, the boundary element method proposed by Pasternak et al. (2017) for thermomagnetoelectroelastic materials can be adapted to quasicrystals. This requires accounting for the 6-dimensional extended vectors within the proposed 3D formalism for quasicrystals. The method involves several key components: special quadrilateral discontinuous boundary elements,