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

34

3

!"#$ % !

linear thermal expansion coefficients. Moreover, the resulting extended tensor

of phonon-phason constants is

symmetric:

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

(5)

.

and positively defined

! !"#$ ! " # $ % C C > ! ! ! !

.

(6)

"

The equilibrium equations for a quasicrystal solid can also be rewritten in the following compact form (Pasternak et al., 2024)

! # !

.

(7)

! !" " ! = "

!

!

where are the components of phonon and phason body force vectors, respectively. Substituting Eq. (3) into the equilibrium equations (7) one obtains the compact form of the governing equations of thermoelasticity of quasicrystals: ! ! " " = ! ! ! " # + = ! " ! " , and ,

!

% C ! !

!

.

(8)

'

! " # = #

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

!

In Eqs. (3), (7) and (8) the capital letter indices range from 1 to 6, while the lowercase letter indices continue to range from 1 to 3. 3. 3D fundamental solutions Consider the fundamental solutions of Eqs. (1), (2), and (8) for 3D heat conduction and thermoelasticity in quasicrystals subjected to a point heat source or a point force in an infinite anisotropic thermoelastic quasicrystal medium. Since uncoupled steady-state thermoelasticity is considered, the point force does not affect the temperature field. Therefore, there is one fundamental solution for heat conduction governed by

(

) ( ! =

)

,

(9)

" ! !

# ! !

! "

#

"

#

#

!"

!"

and two for thermoelasticity: one for the action of a unit point force, where the temperature field is absent,

(

)

(

)

C ' !

! ! ! = " " ! ! ! !

,

(10)

! !"#$ %# "$

"

"

!%

and the other for the action of a point unit heat sink:

(

)

(

)

% C !

! + " !

" ! !

" ! !

.

(11)

!

#

=

" !"#$ # "$

#

"

#

!"

"

( ) ! !

Here

is the Kronecker delta and

is the 3D Dirac delta function.

!" !

Using the Radon integral transform (Deans, 1983)

!

( ) ( ) ! "

( ) $ !

( ) ( ! "

) ! " # #$#$ #$

$$$

! "

%

! "

!

"

=

=

" #

,

(12)

! ! " = ! "

!

!

where is a unit vector, normal to a plane

, Eqs. (9)–(10) results in