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

35

4

$ !

!

#

( ) $ $ !

(

)

! "

"

,

(13)

!" ! " # !!

$

=

% &

"

#

!

$

#

!

!

#

( ) # !

(

)

C "

! "

,

(14)

%# ' (

(

= $

$ %

!"#$ " $ ! !

%! " "

"

!

(

#

!

% !

!

#

#

( ) # !

( ) # ' !

% "

!" " " ! "

"

,

(15)

# C '

= $

!"#$ " $ ! !

!

'

'

#

#

since

!

( ) !

( ) ! !

"

" #

"

"

# % '

$ & (

(16)

"

=

! !

$

#

"

"

!

and

(

)

(

)

(

) ( !

)

(

)

$$$

" ! !

" ! !

! "

! "

%

!

" # $ "#"# "#

!

!

=

!

" #

=

!

" #

.

(17)

!

!

!

Since the inverse Radon transform in 3D is defined as

!

!

!

"

#

( ) ( ) ! " !

( ) !

( ! "

) ( ) #$ ! " ! ! $

" # %% ! =

,

(18)

" # $ "

!

=

= "

= $

!

!

%

"

!

the 3D fundamental solutions of heat conduction and thermoelasticity can be easily obtained from Eqs. (13)–(15) in the following form

(

)

(

)

$ % ! " "

!

$

(

)

( ) !

!

# # = '' !

,

(19)

% ! !

"

#$

&

= %

!

&

%

!" ! " ""

$

!

(

) ( ) $% !

(

)

( ) ! ! " " ( ! % #

)

= && !

$ ! !

!

#

,

(20)

#

=

$

"

"

!"

!"

#

%

"

!

( ) !

"# # ! " !

!

$

%

!

(

) ( ) () !

(

)

(

)

= '' !

!"

$ ! !

& $ ! " "

,

(21)

&

'

=

"

"

!

#

%

#

$% $ % " "

#

!

( ) !"

!

( ) ! !

( ) !

Here

is a Heaviside step function,

and

are the components of the matrix, which is

= !

!#"$ # $ % ! !

"

!" ! " !

( ) !" ! !

( ) ( ) "# ! !

inverse of the matrix

, i.e.

. The latter exists due to the property (6) of the extended

! !" " # #

=

!# !

!"#$ % !

phonon-phason elasticity tensor . Accounting for the properties of the Dirac delta function Eqs. (19) and (20) can be further reduced to

!

(

) ( ) ! # !

(

)

# ! = ! ! ! ! # %

% ! !

"

,

(22)

#

$% !!

$

= #

#

!" ! "

$

&

"