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

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2

and phason fields, which are unique to quasicrystals and govern their deformation. In contrast, traditional crystal deformation is solely driven by phonon fields. While substantial progress has been made in solving 2D elasticity problems for anisotropic materials (e.g. see Ting, 1996), these methods remain underdeveloped for quasicrystal materials, especially in the context of 3D thermoelasticity. Most existing studies focus on plane elasticity or axisymmetric problems. For example, Fan et al. (2017) investigated cracks in one-dimensional hexagonal quasicrystals by reducing the problem to four harmonic potentials. Long and Li (2022) addressed axisymmetric thermoelastic problems for cubic quasicrystal disks. Pi et al. (2022) proposed a method for analyzing plane electroelastic problems in 1D quasicrystals. Fan and Mai (2004) developed a Lekhnitski-type approach for plane elasticity in quasicrystals. Despite these advances, a significant gap remains in the literature regarding the study of 3D thermoelasticity in quasicrystals. To date, only a few works have addressed planar cracks in 3D thermoelastic quasicrystals. For instance, Li et al. (2017) derived a 3D fundamental solution for hexagonal 2D quasicrystals with a penny-shaped crack, while Li et al. (2019) proposed a method for analyzing 2D thermoelastic quasicrystals with planar cracks. Both approaches utilize potential functions to derive the solutions. However, there is still no comprehensive approach for analyzing 3D thermoelasticity problems for quasicrystals of any structural type (1D, 2D, or 3D) that contain either planar or non-planar cracks. This limitation hampers the full understanding of quasicrystals' behavior under complex thermal and mechanical loading conditions. To address this, the present paper develops a boundary element method (BEM) for the analysis of such problems, providing a unified framework capable of handling 3D thermoelastic quasicrystals with arbitrary crack geometries. This approach aims to fill the existing gap and contribute to the advancement of quasicrystal mechanics. 2. Governing equations The law of stationary thermal conductivity (Fourier’s law) for quasicrystal solids is assumed (Pasternak et al., 2024) to be the same as for anisotropic crystalline solids, i.e.

,

(1)

! " # $ ! = " ! !"

where

is temperature change compared to the reference temperature;

are thermal conductivity coefficients,

!" #

!

!

moreover

. Heat flux vector field satisfies the following stationary heat balance equation

!" "! # # =

# ! # ! " ! = !

,

(2)

"#$

%

! " "

!

!

where is the heat volume density. According to Pasternak et al. (2024), the constitutive equations for the thermoelasticity of 3D quasicrystals can be rewritten in the following compact form: ! "

, !" !"#$ # $ !" % C ! " # = $ ! ! ! !

(3)

where

" ! ! ! = ! !

" % C C C ' = = ! ! "

#

=

( ) ! ! +

!

!"

!"

!"

!

!

!

!

+

"

!

!

" ) ( ! !"#$

" ) ( ! #$!"

(4)

" ( ( ( = !"#$ !"#$

#

*

=

=

=

(

)

( ) ! ! +

( ) ( ! !

)

!"#$

! !" # $ +

! " # $

"#$

+ +

(

)

!

" # " $ !

# $

" " " ! " # $

$"%"! &

(

)

)

*

!" "

#

#

=

+

=

+

=

( ) ! ! +

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

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

"

Here

is a phonon stress tensor (symmetric);

is a phason stress tensor;

is a phonon displacement;

is

! "

! "

!" #

!" !

phason displacement;

is a tensor of phonon elastic constants;

is the phason elastic constant tensor;

!"#$ %

, and !"#$ %

!"#$ %

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

!" ! "

and

are elastic constants of phonon-phason coupling;

are phonon and phason components of

!" !