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Pee - e iew unde esponsibili y o P o esso Fe i Aliabadi
F ac u e, Damage and S uc u al Heal h Moni o ing
In luence o ini ial mis i s ains on small scale domain swi ching
ahead o in e ace c ack be ween piezoelec ic laye and dielec ic
iso opic subs a e
Mi osla H s kaa, Michal Ko oula,b
*
, Tomáš P o an a, Fe i Aliabadic
aIns i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y o Mechanical Enginee ing, BUT, Technická 2896/2, B no, 616 69,
Czech Republic
bFacul y o Special Technology, Alexande Dubček Uni e si y o T enčín, S uden ska 2, 911 50 T enčín, Slo ak Republic
cDepa men o Ae onau ics, Impe ial College London, Exhibi ion Rd, SW7 2AZ Sou h Kensing on, London, Uni ed Kingdom
Abs ac
This s udy de i es he e ec o discon inuous ini ial s ain dis ibu ions on he small-scale domain swi ching ahead o he bi-
ma e ial no ch o med be ween a piezoelec ic laye and dielec ic iso opic subs a e. As a piezoelec ic laye he piezoelec ic
ce amics PZT-5H g own on he elas ic subs a e o med by amo phous silicon dioxide (SiO2) is conside ed. The ene ge ic swi ching
p inciple and mic omechanical domain swi ching amewo k p oposed by Hwang e al. (1995) is applied. The ini ial he mal mis i
cons an s ain is included in he cons i u i e ela ions. The analysis o he asymp o ic in-plane ield o he bi-ma e ial no ch is
conduc ed u ilizing he ex ended Lekhni skii-Eshelby-S oh o malism. The asymp o ic in-plane ield is used o p edic he domain
swi ching zone applying he ene gy-based c i e ion. The in luence o he he mal mis i s ain on he size and shape o he swi ching
zone in he piezoelec ic laye is compu ed o a ious ini ial poling di ec ions.
Keywo ds: Small-scale domain swi ching; Piezoelec ic in e ace c ack; Expanded LES o malism; Two-s a e H-in eg al.
1. In oduc ion
This wo k ollows up ou ecen pape s H s ka e al. (2025) which in es iga es small-scale domain swi ching ahead
o he in e ace c ack in he piezoelec ic bi-ma e ial comp ising piezoelec ic ce amics PZT-5H and BaTiO3 and i s
impac on he in-plane in ensi y o singula i y a he ip o in e ace c ack is compu ed. Howe e , no ini ial he mal
mis i s ains ha e been conside ed. Ini ial s ain in piezoelec ic composi es is usually o med du ing he
* Co esponding au ho : E-mail add ess: ko oul@ me. u b .cz
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2 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
manu ac u ing p ocess, especially in piezoelec ic laye / ilm and elas ic subs a e s uc u es. These s uc u es
ine i ably p oduce c acks and o he de ec s inside he coa ing o a he in e ace junc ion du ing he manu ac u ing
p ocess, which educes he eliabili y o he de ice and sho en he se ice li e o he de ice unde he ac ion o loads.
He e, he piezoelec ic laye o PZT-5H g own on amo phous silicon dioxide (SiO2) subs a e is conside ed. The
analysis o he asymp o ic in-plane ield o in e ace c ack is conduc ed employing he ex ended Lekhni skii-Eshelby-
S oh o malism. The ini ial subs a e-induced cons an elas ic mis i s ains a e included in he analysis. The
asymp o ic in-plane ield is employed o p edic he domain swi ching zone using he ene gy-based c i e ion p oposed
by Hwang e al. (1995). To he bes o ou knowledge he e is no li e a u e whe e he e ec o he domain swi ching
including he e ec o he he mal mis i s ain pa allel o he in e ace on he change in singula i y in ensi y a he bi-
ma e ial no ch ip has been s udied. This pa icula con igu a ion is common in senso s and ac ua o s wi hin in elligen
s uc u es, and i will be he subjec o discussion in he p esen s udy.
2. P oblem o mula ion
In ou p e ious s udy o small scale swi ching ahead o bi-ma e ial no ches we conside ed bi-ma e ial con igu a ion
o med by wo piezoelec ic ce amics and he analysis o he asymp o ic in-plane ield a ound a bi-ma e ial sha p
no ch was conduc ed applying he ex ended Lekhni skii-Eshelby-S oh o malism, see e.g. Ting (1996), while no
ini ial he mal mis i s ains we e aken in o accoun . In s uc u es which employ piezoelec ic elemen s, piezoelec ic
ma e ials a e coupled o elec odes, which conduc he elec ic cha ge, o o insula o s, e.g. an unde lay o insula ing
pads be ween piezoelec ic and elec odes, o simply o he body o a cons uc ion. Sol ing p oblems o bi-ma e ials
consis ing o combina ions o piezoelec ic and non-piezoelec ic solids equi es speci ic changes in he expanded
LES o malism used o modelling o bi-ma e ial no ches. The i s s ep in he modi ica ion o he expanded LES
o malism o piezoelec ic ma e ials o pu e elas ic non-piezoelec ic ma e ials is o se he piezoelec ic cons an s o
ze o, i.e. eijk = 0 o any i,j,k. The elas ic and elec ic ields a e hen decoupled and bo h di ec and con e se
piezoelec ic e ec s anish. The second s ep is o modi y he p oblem acco ding o he case o an insula o o
conduc o . Bo h cases a e di e en om he physical poin o iew. In he amewo k o he Lekhni skii and S oh
o malism, Hwu and Kuo (2010) p oposed a me hod which ul il he condi ion o he in e ace impe meabili y by
educing he pe mi i i y o a su icien ly small alue when modelling an insula o o inc easing o a e y la ge alue
when conside ing a conduc o . I s pu pose was o be in ag eemen wi h au ho s in Xu and Rajapakse (2000), Weng
and Chue (2004), Chen e al. (2006), who modelled he insula o /piezoelec ic bi-ma e ial by p esc ibing elec ic
displacemen
( )
00
insula o
D
=
along he in e ace. Howe e , he condi ion o an insula o /piezoelec ic in e ace is
no physically exac . The assump ion o ze o elec ic displacemen exp esses an impe meable in e ace condi ion, i.e.
he su aces a e ee o cha ge. This e ec is no iola ed, i one ma e ial has signi ican ly highe pe mi i i y han he
second one, e.g. a piezoelec ic ce amic in a con ac wi h ai (Qin (2013)), which is ac ually p esc ibed on he no ch
ace. Bu his canno be applicable o an insula o /piezoelec ic in e ace, because ela i e pe mi i i y o insula o s
a ains a wide ange on alues. Then, an insula o /piezoelec ic bi-ma e ial no ch can be modelled in he same way as
a piezoelec ic bi-ma e ial, bu by p esc ibing ze o piezoelec ic cons an s and gi en pe mi i i y, i known.
Fu he , i should be no ed ha he LES o malism is p ima ily de i ed o aniso opic ma e ials. In case o an
iso opic ma e ial he key ma ices o LES o malism A and L (see Appendix A) a e degene a e o non-semisimple
and canno be no longe in e ed. Since mos insula o s ha e iso opic p ope ies, he Muskhelish ili complex
po en ial me hod needs o be implemen ed in he amewo k o LES o malism o desc ibe he elas ic ield o he
iso opic ma e ial.
The las p oblem ela es o he combined ( he mal +elec o/mechanical) loading. The he mal loading due o cooling
down om he p ocessing empe a u e o oom empe a u e induces esidual s esses in he specimen. In case o he
elec o/mechanical loading, he esul s ob ained om 2D plane s ain calcula ions en i ely co espond o hose
ob ained om he ull 3D analysis in he cen e o he specimen. Howe e , in case o he he mal loading o lamina e,
ce ain disp opo ions a e ound be ween he esul s ob ained wi h 2D and 3D model, espec i ely. The eason is ha
he plane s ain condi ions canno be ul illed in case o he he mal loading (because o he ma e ial ex ension in all
di ec ions) and he biaxial he mal esidual s ess
110,
330,
22 = 0 is no ep oduced. To o e come his p oblem
while s ill aking an ad an age o 2D calcula ions also o he he mal p oblems, ce ain modi ica ion o he ma e ial
coe icien s o he mal expansion (CTEs), in all di ec ions, has o be pe o med.
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M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 3
The piezoelec ic laye o PZT-5H exhibi s he e agonal symme y. Howe e , because he c ys allog aphic
e e ence ame can be a bi a y o a ed abou x3 axis wi h espec o global coo dina e sys em in Fig.1, he elas ici y
and piezoelec ici y ma ices possess he s uc u e o monoclinic ma e ials in he global coo dina e sys em, hence he
p oblem is o mula ed o he monoclinic ma e ials wi h he symme y plane a x3 = 0.
Fig. 1. Geome y o a bi-ma e ial no ch cha ac e ized by wo egions I and II. No ch aces a e de ined by angles ω1 and ω2. Ma e ial in e ace is
always conside ed a θ = 0. Angle α1 deno es poling di ec ion o he ma e ials I
The cons i u i e laws o a linea elas ic piezoelec ic ma e ial which include homogeneous ini ial s ains due o
uni o m empe a u e change ΔT can be w i en in he ma ix o m as
,,
TT
ED
TT
−−
==
−−
−−
σ ε α ε α σ
C e S g
D E E D
eω g β
(1)
whe e σ is he s ess enso , ε is he s ain enso (bo h w i en in he Voig no a ion in he ec o o m), α is he he mal
expansion enso , E and D a e he ec o s o elec ic in ensi y and elec ic lux densi y, CE is he elas ic s i ness enso
a cons an elec ic ield, e is he s ess/cha ge piezoelec ic enso , and ωε is he dielec ic pe mi i i y enso a
cons an s ain, SD is he elas ic compliance enso a cons an elec ic induc ion, g is he s ain/ ol age piezoelec ic
enso , and βσ is he dielec ic non-pe mi i i y enso a cons an s ess (Hwu and Ikeda (2008), Hwu and Kuo (2009),
2010)). The ma e ial ma ices a e ela ed h ough an in e sion as
.
=
−−
TT
ED
C e S g I
eω g β
(2)
The elas ici y and piezoelec ici y ma ices o a monoclinic ma e ial poled in he x1x2-plane ha e he ollowing
s uc u e:
11 12 13 16 111
12 22 23 26 222
13 23 33 36 3 33
423
44 45
51
45 55
6
16 26 36 66
00
00
00 ,
0 0 0 0
0 0 0 0
00
= = =
E E E E
E E E E
E E E E
EEE
EE
E E E E
C C C C
C C C C
C C C C
CC
CC
C C C C
Cσ
111
222
333
423
5
3 13
6
12 12
, 2
2
2
==
ε
(3)
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4 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
11 12 13 16
21 22 23 26
34 35
00
00
0 0 0 0
=
e e e e
e e e e
ee
e
,
11 12
12 22
33
0
0
00
=
ε
ω
,
1
2
3
=
E
E
E
E
,
1
2
3
D
D
D
=
D
The s uc u e o he ma ices SD, g and
β
is simila o he s uc u e o hei in e se coun e pa s and is no s a ed
he e o he sake o b e i y. The di ec ional p ope ies o he ma ices depend on he poling axis, which can a ain wo
limi con igu a ions, ei he i coincides wi h x1-axis o wi h x2-axis. Be ween hese s a es hei s uc u e co esponds
o he abo e men ioned monoclinic one. The ma e ial cons an s in he gene al poling di ec ion a e ob ained by using
ans o ma ion ela ions
( )
1 1 1 1
, ,
T
DD
− − − −
= = =
* * *
S K S K g Ωg K β Ωβ Ω
. (4)
whe e
D
*
S
,
*
g
and
*
σ
β
a e ma e ial coe icien ma ices wi h he poling di ec ion aligned wi h ce ain p incipal
di ec ion (e.g. in which he p ope ies we e measu ed). The o m o he ans o ma ion ma ices K and Ω a e gi en in
Appendix A.
Ex e nal loads a e assumed o be pa allel o he plane de ined by x3 = 0 and plane s ain de o ma ions cha ac e ized
by linea piezoelec ici y a e conside ed. These assump ions allow o decouple he in-plane and an i-plane ela ions
and o sol e he in-plane and an i-plane p oblem sepa a ely. In he ollowing only he in-plane p oblem is conside ed.
The g- ype cons i u i e ela ions o he in-plane p oblem can be exp essed as (H s ka (2019), H s ka e al. (2019),
Hwu and Ikeda (2008))
''
''
σ
ˆ
,
ˆ
ˆ
ˆ
T
ps D
T
−
=
−
−
Sg σ
ε
D
Egβ
(5)
whe e he componen s o he e ec i e plane s ain CTE
ps
o iso opic he mal expansion α (see Table1) a e
13 23 36
33 33 33
1 , 1 ,
T
D D D
ps ID D D
S S S
αS S S
= − − −
(6)
and
1 11 1 11
1
11
2 22 2 22
22
6 12 6 12
2
, , ,
2
x
ED
ED
x
−
= = = = = = =
−
σ ε E D
(7)
a e he s ess ec o , he s ain ec o , he elec ic ield ec o , he elec ic po en ial, and he elec ic displacemen
ec o , espec i ely,
'''
11 12 16 ' ' ' ''
11 12 16 11 12
' ' ' ' ' '
12 22 26 ' ' ' ''
21 22 26 12 22
' ' '
16 26 66
ˆˆˆ
ˆˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆˆˆ
ˆ ˆ ˆ
ˆ
, ,
ˆ
,
ˆ
DDD
D D D
D
D D D
SSS g g g
S S S g g g
S S S
= = =
Sgβ
(8)
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M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 5
whe e
' ' '
ˆˆ
ˆ, ,
D
Sgβ
a e he compliance ma ix a cons an induc ion, he piezoelec ic s ain/ ol age ma ix, and he
dielec ic impe meabili y ma ix a cons an s ess, espec i ely, e alua ed unde he assump ion o he gene alized
plane s ain a sho ci cui
30
=
and E3 = 0 as
3 3 3 3 3 3
' ' ' ' '
33 33 33
3 3 3 3 3 3
33 33 33
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
ˆˆ
ˆ ˆ ˆ
ˆ
, , ,
, , ,
,
ˆ ˆ ˆ
ˆ
i j i j i j
D D D
ij ij ji ij ij ij ij ji
D D D
i j i j i j
D D D
ij ij ji ij ij ij ij ji
D D D
g g g
S S S g g
S S g S g g
S S S g g
S S S
= + = = − = − =
+==−==−=
(9)
o
,3ij
, whe e
1 1 1 1 1 1 1
, ,
TT
D E E E E E
− − − − − − −
= − = + =S C C e ω eC ω eC e ω g ω eC
.
The in e se o m o he cons i u i e ela ions in Eq. (5) eads
''
''
ˆˆ
ˆˆ ,
Tps
ET
−
=
−
−
Ce
σε
DE
eω
(10)
whe e he in-plane elas ici y and piezoelec ici y ma ices unde he assump ion o he gene alized plane s ain and
sho ci cui ha e he ollowing s uc u e:
' ' '
11 12 16 ' ' ' ''
11 12 16 11 12
' ' ' ' ' '
12 22 26 ' ' ' ''
21 22 26 21 22
' ' '
16 26 66
,
ˆ ˆ ˆ
ˆˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆˆ
ˆˆ
ˆ ˆ ˆ
, ,
ˆ ˆ ˆ
= = =
E E E
E E E
E
E E E
C C C e e e
C C C e e e
C C C
Ceω
(11)
wi h
3 3 3 3 3 3
''
33 33 33
3 3 3 3 3 3
'ε ε ε ε 'ε ε 'ε
33 33 33
' ' ' '
, , ,
,
, ,
ˆˆ
ˆ ˆ ˆ ˆ
ˆˆ
ˆ ˆ ˆ ˆ ,
ˆ
,
ˆˆ.,
= − = − = −
= − = + = − =
= = = =
E
i j i j i j
ij ij ij ij ij ij
E
i j i j i j
ij ij ij ij ij ij ji
E
E E E
ij ij ij ij ij ij
e e C e
e e e e e e
C
ee
C
C C C e e e
(12)
The he moelas ic cons i u i e law o he iso opic subs a e in plane s ain is
( )
( )( ) ( )( )
( )( )
( )
( )( )
( )
( )
11
22
66
10
1 1 2 1 1 2
1
10 1 ,
1 1 2 1 1 2
00
1
II
II
T
EE
E
E
E
T
−
+ − + −
+
−
=+
+ − + −
−
−
+
(13)
whe e
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6 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
( )
12 12 44 44
12 44
12 44
32
, .
2
E E E
E
EE
EE
C C C
EC
CC
CC
+
==
+
+
The in-plane p oblem o he s ess singula i y a he sha p no ch composed o he monoclinic piezoelec ic ma e ial
and he iso opic non-piezoelec ic ma e ial equi es some ca e. The monoclinic piezoelec ic ma e ial is desc ibed in
e ms o he LES o malism while he iso opic non-piezoelec ic ma e ial needs o modi y he LES o malism by
employing he Muskhelish ili complex po en ials. We s a wi h he desc ip ion o he asymp o ic ield in he
monoclinic piezoelec ic ma e ial e e ed o as he Ma e ial I. The ollowing ec o s a e in oduced bo h in he case
o aniso opic piezoelec ic ma e ial and iso opic non-piezoelec ic one
11
22
,,
D
uT
uT
T
==
uT
(14)
whe e u1, u2 a e he displacemen componen s,
is he elec ic po en ial, T1, T2 and TD a e he componen s o he
gene alized s ess unc ion ec o o he mechanical and elec ical quan i ies. The gene alized s ess unc ion
componen s T1, T2 and TD a e ela ed o he s esses and elec ic displacemen s by
i1 ,2 i2 ,1 1 ,2 2 ,1
, , 1,2, , .
i i D D
T T D T D Ti
=== − = = −
(15)
The complex po en ial o exp ession o he s ess singula i y has he o m
1 2 3 1 2
diag , , , , 1,2,3,
ii
z z z z x x i
= = + =
Z
(16)
whe e
i
a e he ma e ial eigen alues and hey a e ob ained by sol ing he cha ac e is ic equa ion Eq. (A5) (see
Appendix A). The coupled elec omechanical ield nea he no ch e ex is sough in he o m o he ansa z o
unknown singula i y exponen s and co esponding eigen ec o s as
( ) ( ) ( )
( ) ( ) ( )
δδ
δδ
,,
, ,
=+
=+
u AZ AZ w
T LZ LZ w
(17)
whe e he complex unc ion
Z
and ma ices A and L a e de ined in Appendix A, he ba abo e he symbols deno es
complex conjuga e quan i ies.
and
w
a e eigen ec o s pe inen o he singula i y exponen eigen alue p oblem o
he conside ed sha p no ch in Fig. 1. The unknown singula i y exponen s and eigen ec o s a e de e mined h ough he
sa is ac ion o he bounda y condi ions a he no ch ip. The ma e ial eigen ec o s ma ices A, L in Eq. (17), see also
Eq. (A7), a e non-degene a e when he poling di ec ion is pe pendicula o he x3 axis and semi-degene a e o
degene a e o he wise. Only non-degene a e cases a e conside ed he eina e .
In case o he asymp o ic ield in he iso opic non-piezoelec ic ma e ial, he s uc u al and elec ical cons i u i e
equa ions a e decoupled due o he assump ion o ze o piezoelec ic coe icien
'
ˆ
g
, see Eq. (5). Thus, he modi ica ion
o he o malism uni ies he ela ions o pu e iso opic elas ici y wi h equa ions desc ibing he elec os a ic ield. By
subs i u ing he ma e ial pa ame e s in o he in-plane cha ac e is ic equa ion (A5), iple complex conjuga e oo s 𝜇1,2,3
= 𝑖 a e ob ained. The complex po en ials o an iso opic media (ma ked wi h a s a ) ha e he o m
( ) ( ) ( ) ( )
*d,
d
z
z z z z z
= + −
Q
(18)
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M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 7
whe e
000
1 0 0
000
=
Q
The complex po en ials (z) a e de ined as
( ) ( )
( )
( ) ( )
, cos sin ,
z
z z z i
z
= = +
(19)
in which
( ) ( ) ( )
1 2 3
, , .z z z z z z
= = =
(20)
The displacemen s and s ess ha e he o m
( )
( )
* * * * *
* * * * *
,
,
z
z
=+
=+
u A Z A Z w
T L Z L Z w
(21)
whe e he ma ices A* and L* a e exp essed by
33
**
11
0
44 0
1 1 1
0 , 1 1 0
4 4 2 0 0 2
00
ii
Gi Gi ii
Gi Gi
a
−
−
==
−
AL
(22)
wi h
'
2233 3
ˆ
a
=
,
34
=−
o plane s ain and
44
E
GC=
and
( )
12
12 44
2
E
EE
C
CC
=+
.
The complex unc ions
*
Z
a e
( )
( )
1
*1
e 0 0
00
0 2 e sin e 0 .
0 0 0 0 e
i
ii
i
z
z z z z i
z
−
−
= − = −
Z
(23)
No e ha he uppe le 2×2 ma ix desc ibes pu e iso opic elas ici y, while he hi d diagonal elemen desc ibes he
elec ic ield. Complex conjuga ion o he unc ion (23) leads o
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8 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
( )
1
*
e 0 0
2 e sin e 0
0 0 e
i
ii
i
i
−
−− −
−
=
Z
(24)
Conside ing ac ion and cha ge ee no ch aces he ollowing bounda y condi ions a e imposed by:
( ) ( )
I II
12
0, 0.
==TT
(25)
The displacemen and ac ion con inui y condi ions a e p esc ibed along he in e ace
0
=
as
( ) ( ) ( ) ( )
0 0 , 0 0 ,
I II I II
u u T T==
(26)
The eigen alue p oblem o a bi-ma e ial no ch composed o a piezoelec ic ma e ial and an insula o is de ined in
e ms o he equa ions (17) and (21). A bi-ma e ial no ch wi h he geome y in Fig. 1 is conside ed, whe e ma e ial I
is he piezoelec ic one and ma e ial II non-piezoelec ic one de ined by elas ic cons an s
E
ij
C
and pe mi i i ies
ij
.
Le us de ine he ollowing iden i ies:
**
**
*
,
,
, , ,
.
I I II II
I I II II
I II
= = = =
= = = =
==
A A, L L, A A L L
u u T T, u u T T
Z Z Z,Z
(27)
The eigen alue p oblem is in oduced by he bounda y condi ions (25) and (26) and can be w i en in he ma ix o m
as
II
II
11
II
II II
22
II II
I I II II
0 0 0 0
II II
00
00 . 0,
L
XX
Lw
XX
L
B B B B
I I I I Lw
=
−
−−
(28)
whe e
( )
( )
( )
( )
( )
1
1
11
00
, , 1,2
i , i
−
−
−−
= = =
= = −
j j j j j j jX LZ L X LZ L
B AL B AL
(29)
and 0 deno es 3×3 ze o ma ix on he le -hand side and 12×1 ze o ec o on he igh -hand side o Eq.(28). A e
some algeb aic manipula ions one ecei es he cha ac e is ic equa ion o he singula i y exponen alues. De ini ions
o he asymp o ic s esses and elec ic displacemen s o he piezoelec ic monoclinic ma e ial and o he iso opic
non-piezoelec ic ma e ial, espec i ely, ha e he o m:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
3
12
2 2 2
3
12
1 1 1
1
11
1
1 1, 2 2, 3 3,
1
11
2
1 1, 2 2, 3 3,
, ,
,,
−
−−
−
−−
= − − −
= + +
x x x
x x x
H H H
H H H
σ λ λ λ
σ λ λ λ
(30)
PREPRINT
M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 9
o piezoelec ic ma e ial I, and
( ) ( ) ( )
( ) ( ) ( )
3
12
2 2 2
3
12
1 1 1
1
11
*1 * * *
1 1, 2 2, 3 3,
1
11
*2 * * *
1 1, 2 2, 3 3,
,
,
−
−−
−
−−
= + +
= + +
x x x
x x x
H H H
H H H
σ λ λ λ
σ λ λ λ
(31)
o iso opic non-piezoelec ic ma e ial II, whe e Hi a e gene alized s ess in ensi y ac o s and
11 21
12
12 22
12
, .
==
DD
σσ
(32)
The de i a i es o he shape unc ions (subsc ip s ,
𝑥
1 and ,
𝑥
2 deno e di e en ia ion wi h espec o x1 and x2) in Eqs.(30)
and (31) a e gi en by
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1
2
2
11
,
11
,
1
*1
* * * *
,
1
*1
* * *
,
, 1,2,3
, 1,2,3,
, 1,2,3,
, 1,2,3,
ii
ii
i
i
i
i
i
i x i i i i
i x i i i i
i x i i i i
i x i i i i
i
i
i
i
*
λ L Z L Z w
λ L Z μ L Z μw
λ B Z B Z w
λ L Z L Z w
−−
−−
−
−
−
−
= + =
= + =
= + =
= + =
(33)
whe e
1
1
22
33
00
00
0 0 , 0 0
00 00
==
μμ
,
*
3 1 0
10
20 0 2
ii
i
−
=−
B
and
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
1 2 3
1 2 3
i i i
1 2 3
i i i
1 2 3
diag e , e , e ,
diag e , e , e ,
i i i
i i i
i
i i i
i i i
i
R R R
R R R
− − −
=
=
Z
Z
(34)
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( )
1
21
*1
1
1
121
*
1
e 0 0
2 1 e sin e 0 ,
0 0 e
e 0 0
2 1 e sin e 0 .
0 0 e
i
ii
i
i
ii
i
i
i
−
−−
−
−
−−
−− − − −
−−
= − −
=−
Z
Z
(35)
PREPRINT
16 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
Fig. 5. The s ess componen s, displacemen s, elec ic displacemen componen s and elec ic po en ial o a PZT-5H/SiO2 in e ace c ack on he
ci cula pa h = 0.001mm, ω1 = 180°, ω2 = –180°, poling di ec ion α1 = 90°, loaded by σ2appl = 20 MPa and ΔT = –160 °C.
PREPRINT
M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 17
Fig. 6. The s ess componen s, displacemen s, elec ic displacemen componen s and elec ic po en ial o a PZT-5H/SiO2 in e ace c ack on he
ci cula pa h = 0.001mm, ω1 = 180°, ω2 = –180°, poling di ec ion α1 = 0°, loaded by σ2appl = 20 MPa and ΔT = –160 °C.
Table 2. The con en ional s ess in ensi y ac o s KI, KII, KIV calcula ed o bo h ini ial poling di ec ions
S ess in ensi y ac o
α1 = 90°
ΔT = 0 °C
α1 = 90°
ΔT = –160 °C
α1 = 0°
ΔT = 0 °C
α1 = 0°
ΔT = –160 °C
KI [MPa m1/2]
3.44
4.34
3.49
4.32
KII [MPa m1/2]
-0.27
1.18
-0.73
0.83
KIV [μC m1/2]
0.40
1.18
-1.25
-0.55
No ice, ha o bo h conside ed ini ial poling di ec ions α1 =0°, 90° is he change o spon aneous s ain Δε12 = 0.
The swi ching zone p edic ed om he c i e ion (42) by employing he linea asymp o ic ield in Eq. (17) wi h
e alua ed GSIFs Hi is shown o wo selec ed ini ial poling di ec ions in Fig. 7 and Fig. 8. Fo compa ison, he
swi ching zone is displayed o bo h he empe a u e change ΔT = –160 °C and ΔT = 0 °C. The o ien a ion o he
swi ching angle i.e. 90°o –90°, depends on he c i e ion (42), p io i y is gi en o he o ien a ion o which he c i e ion
(42) is me i s . Mos su p ising is he ex ao dina y inc ease in he size o he swi ching zone by mo e han one o de
o magni ude o he ini ial poling di ec ion 0° when he mal mis i s ains a e included, c . Fig. 7.
Fig. 7. Domain swi ching zone o an in e ace c ack loaded by σ2appl = 20 MPa o he ini ial poling di ec ion 0° wi h espec o he in e ace. Fo
compa ison, he swi ching zone is displayed bo h o empe a u e change ΔT = –160 °C and ΔT = 0 °C; on he igh -hand side he de ailed iew is
shown
PREPRINT
18 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
Fig.8. Domain swi ching zone o an in e ace c ack loaded by σ2appl = 20 MPa o he ini ial poling di ec ion 90° wi h espec o he in e ace. Fo
compa ison, he swi ching zone is displayed bo h o empe a u e change ΔT = –160 °C and ΔT = 0 °C; on he igh -hand side he de ailed iew is
shown.
To unde s and his peculia beha io , i is necessa y o analyze indi idual e ms in he c i e ion (42). Fi s o all, no ice
ha o α1 =0° he change in spon aneous s ain componen s a e Δε11 < 0, Δε22 > 0 and he change in spon aneous
pola iza ion componen s ΔP1 < 0, ΔP2 >0 o < 0, he la e depending on he o ien a ion o he swi ching angle. When
he he mal mis i s ains a e included, he s ess componen σ11 >0 o 0 < θ < π/2, hus making he con ibu ion o
he swi ching c i e ion nega i e in his ange, and σ11 < 0 o π/2 < θ < π, hus making he con ibu ion o he swi ching
Fig. 9. Compa ison o s ess componen s σ11, σ22 in case o he ini ial poling di ec ion 0° o he mal mis i s ains included (ΔT = –160 °C ) and
he mal mis i s ains no included (ΔT = 0 °C )
Fig. 10. Compa ison o elec ic ield componen s in case o he ini ial poling di ec ion 0° o he mal mis i s ains included (ΔT = –160 °C ) and
he mal mis i s ains no included (ΔT = 0 °C )
PREPRINT
M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 19
c i e ion posi i e in his ange, c . Fig. 9. σ22 is g ea e han 0 in he whole ange hus making he con ibu ion o he
swi ching c i e ion posi i e, c . Fig. 9. When he he mal mis i s ains a e no included, he s ess componen σ11 >0
in he whole in e al o θ and i s con ibu ion o he swi ching c i e ion is always nega i e. σ22 is also g ea e han 0,
bu abou o 15% smalle . The elec ic ield plays a signi ican ole in he swi ching c i e ion. The E1 componen does
no di e much whe he o no mis i s ains a e conside ed, howe e he E2 componen di e s signi ican ly be ween
he wo cases, see Fig. 10. While wi h he he mal mis i s ains included is he E2 componen nega i e in he whole
ange, hus p o iding a posi i e con ibu ion o he swi ching c i e ion, in he case when he he mal mis i s ains a e
no included is he E2 componen nega i e only in he ange 0 < θ < π/2 and ha ing he absolu e alue mo e as 3 imes
lowe . The elec ical pa o he swi ching c i e ion is hus he main sou ce o he la ge inc ease in he swi ching zone
o he ini ial poling di ec ion 0° unde condi ions o he mal mis i s ains. Howe e , i should be emphasized ha
despi e he la ge inc ease in he swi ching zone in he p esence o he mal mis i s ains o an ini ial pola iza ion o
0°, he size o he swi ching zone is s ill an o de o magni ude smalle han he size o he swi ching zone o an ini ial
pola iza ion o 90°.
The explana ion o he inc ease in he size o he swi ching zone o pola iza ion 90° due o he mal mis i s ain
can be done in a simila ein as o he ini ial pola iza ion 0°. Howe e , since he di e ences be ween he mechanical
and elec ical quan i ies in his case a e no as signi ican as in he p e ious case, see Fig. 11 and 12, he inc ease in
he swi ching zone due o he mal mis i s ain is less p onounced.
Fig. 11. Compa ison o s ess componen s σ11, σ22 in case o he ini ial poling di ec ion 90° o he mal mis i s ains included (ΔT = –160 °C )
and he mal mis i s ains no included (ΔT = 0 °C )
Fig. 12. Compa ison o elec ic ield componen s in case o he ini ial poling di ec ion 90° o he mal mis i s ains included (ΔT = –160 °C ) and
he mal mis i s ains no included (ΔT = 0 °C )
PREPRINT
20 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
4. Concluding ema ks
The p esen ed wo k s udied e ec o ini ial mis i s ains eme ging om empe a u e loading o a bi-ma e ial no ch
composed o piezoelec ic laye PZT-5H and non-piezoelec ic subs a e SiO2. The wo k ollowed up au ho 's p e ious
esea ch whe e obus ness and se ings o he algo i hm we e es ed. The adius o he in eg a ion pa h enci cling he
c ack ip was se o =1 mm, so as he adius o he ci cle o he domain in eg al con aining empe a u e s ains. As
he g adien o complex po en ial o he auxilia y s ains is la ge in he close icini y o he c ack ip, he in eg a ion
o e ini e elemen s was ealized by inco po a ing 7-poin Gauss quad a u e o achie e accep able accu acy. The mesh
sensi i i y s udy was no pe o med as hese aspec s has been es ed in he p e ious au ho 's wo ks. I has been shown
ha he empe a u e change gene a ing ini ial he mal s ains has signi ican impac o he domain swi ching zone
shapes. The e ec is s onge o he piezoelec ic ma e ial poled in he x1 axis, e en hough he domain swi ching
zone was smalle han o poling in x2 di ec ion. Since he o ma ion o he swi ching zone signi ican ly a ec s he
shielding o an i-shielding o he c ack ip, leading o an inc ease o dec ease in he appa en ac u e oughness, i is
ob ious ha he he mal mis i s ain a ising om he cooling om he poling empe a u e o he piezoce amic laye
on he subs a e will ha e a signi ican indi ec impac on he esul ing appa en oughness o he in es iga ed
bima e ial s uc u e. The e o e, he nex s ep is o sol e he bounda y alue p oblem wi h he p esc ibed spon aneous
s ain and pola iza ion wi hin he swi ching domain using FEM and o employ he compu ed elec oelas ic ield in he
Be i’s ecip ocal p inciple wi h he aim o calcula e he new local GSIF
ip
i
H
acco ding o Eq. (44). The esul s o he
calcula ions will be p esen ed a he con e ence.
Da a a ailabili y
Rega ding he compu a ional p ocedu es see H s ka, M. (2025). Da a o “In luence o ini ial mis i s ains on small
scale domain swi ching ahead o in e ace c ack be ween piezoelec ic laye and dielec ic iso opic subs a e” (1.0.0).
Zenodo. h ps://doi.o g/10.5281/zenodo.17229890.
Acknowledgemen s
The au ho s acknowledge he suppo s by he p ojec BAANG – "Building Ac ions in Sma A ia ion wi h
En i onmen al Gains" unded by he Eu opean Union P og amme Ho izon Eu ope unde g an ag eemen no.
101079091 and by he p ojec "Mechanical Enginee ing o Biological and Bio-inspi ed Sys ems", unded as p ojec
No. CZ.02.01.01/00/22_008/0004634 by P og amme Johannes Amos Commenius, call Excellen Resea ch.
Appendix A
The ans o ma ion ma ices in Eq. (4) a e de ined as:
22
22
22
cos sin 0 0 0 2cos sin
sin cos 0 0 0 2cos sin
0 0 1 0 0 0
0 0 0 cos sin 0
0 0 0 sin cos 0
cos sin cos sin 0 0 0 cos sin
−
=
−
−−
K
(A1)
PREPRINT
M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000 21
cos sin 0
sin cos 0
0 0 1
=−
Ω
The complex unc ion
i
Z
in Eq.(16) is de ined as ollows:
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
1 2 3
1 2 3
i i i
1 2 3
i i i
1 2 3
diag e , e , e ,
diag e , e , e ,
i i i
i i i
i
i i i
i i i
i
R R R
R R R
− − −
=
=
Z
Z
(A2)
whe e
( )
( ) ( )
( )
22
2 ' ''
cos sin sin , 1,2,3 ,
k k k
Rk
= + + =
(A3)
( ) ( )
''
'
sin
a c an o , 1,2,3
cos sin
o
k
kkk
−
= =
+
− = −
(A4)
The symbols
'
k
,
''
k
deno e eal and imagina y pa o he ma e ial eigen alue
k
, which is he oo o he ollowing
cha ac e is ic equa ion
( ) ( ) ( ) ( )
2
2 4 2 3 0,l l m
−=
(A5)
( )
( )
( )
( )
( ) ( )
( )
' 2 ' '
2 55 45 44
' 4 ' 3 ' ' 2 ' '
4 11 16 12 66 26 22
' 3 ' ' 2 ' ' '
3 11 21 16 12 26 22
' 2 ' '
2 11 12 22
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
2
ˆˆ
,
2
ˆ ˆ ˆ ,
2 2 ,
,
2
D D D
D D D D D D
S S S
l S S S S S S
m g g g g
l
gg
= − +
= − + + − +
= − + + + −
= − + −
(A6)
and ma ices A and L in Eq.(17) a e
11 12 13 1 2
21 22 23
31 32 1 2
33
3
33
, 1 1 ,
1
a a a
a a a
a a a
− − −
==
− − −
AL
(A7)
whe e
PREPRINT
22 M. H s ka e al./ S uc u al In eg i y P ocedia 00 (2025) 000–000
( )
( )
( )
( )
( )
( )
( )
2 ' ' ' ' '
1 11 12 16 11 21
2 ' ' ' ' '
2 12 22 26 12 22
2 ' ' ' ' '
3 21 22 2
3
6 12 22
2 ' ' '
13 11 12 163
, 1, 2
/
ˆ ˆ ˆ ˆˆ
ˆ ˆ ˆ ˆˆ
ˆˆ
, 1,2 ,
/ , 1,2
ˆ ˆ ˆ
ˆ ˆ ˆ
D D D
k k k k k
D D D
k k k k k k
k k k k k k
D D D
a S S S g g k
a S S S g g k
a g g g k
a S S S
= + − + − =
= + − + − =
= + − + − + =
= + −
( )
( )
33
3 3 3 3 3
33 3 3 3 3 3
''
11 21
2 ' ' ' ' '
23 12 22 26 12 22
2 ' ' ' ' '
21 22 26 12 22
ˆˆ
ˆ ˆ ˆ ˆˆ
ˆˆ
ˆ,
ˆ
,
/
/
ˆ
,
D D D
gg
a S S S g g
a g g g
+−
= + − + −
= + − − +
(A8)
and
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
( )
23
22
23
24
, 1,2 ,
. 3 .
kk
k
kk
kk
k
kk
lm k
l
lm k
ll
= − =
= − =
(A9)
Re e ences
Chen, M.C; Zhu, J.J.; Sze, K.Y.,2006. Elec oelas ic singula i ies in piezoelec ic-elas ic wedges and junc ions. Enginee ing F ac u e Mechanics,
73, pp. 855–868.
Hwang, S. C., Lynch, C. S., McMeeking, R. M., 1995. Fe oelec ic/ e oelas ic in e ac ions and a pola iza ion swi ching model. Ac a
Me allu gica e Ma e ialia, 43, pp.2073–2084.
H s ka, M., 2019. E alua ion o F ac u e Mechanical Pa ame e s o Bi-Piezo-Ma e ial No ch [Disse a ion Thesis, B no Uni e si y o
Technology],h ps://www. u .cz/en/s uden s/ inal- hesis?zp_id=113774.
H s ka, M., P o an , T., Ko oul, M., 2019. Elec o-mechanical singula i ies o piezoelec ic bi-ma e ial no ches and c acks. Enginee ing F ac u e
Mechanics, 216,106484.
H s ka, M., Ko oul, M., P o an , T., Kianico a, M., 2025. Small-scale domain swi ching nea sha p piezoelec ic bima e ial no ches. In e na ional
Jou nal o F ac u e 249, pp. 1-30.
Hwu, C., Ikeda, T., 2008. Elec omechanical ac u e analysis o co ne s and c acks in piezoelec ic ma e ials. In e na ional Jou nal o Solids and
S uc u es, 45, pp. 5744–5764.
Hwu, C., Kuo, T. L., 2009. In e ace C acks/Co ne s in Aniso opic/Piezoelec ic ma e ials. 3 d In e na ional Con e ence on In eg i y, Reliabili y
and Failu e, Po o/Po ugal, 20–24.
Hwu, C., Kuo, T. L., 2010. In e ace co ne s in piezoelec ic ma e ials. Ac a Mechanica, 214, pp. 95–110.
Ou, Z.-C., Chen, Y.-H., 2004. In e ace c ack p oblem in elas ic dielec ic/piezoelec ic bima e ials. In e na ional Jou nal o F ac u e, 130, pp. 427–
454.
Qin, Q.-H., 2013. Ad anced Mechanics o Piezoelec ici y. Beijing: Highe Educa ion P ess.
Ting, T. T. C., 1996. Aniso opic Elas ici y. Ox o d Uni e si y P ess.
Weng, S.-M., Chue, C.-H., 2004. The s ess singula i ies a he apex o composi e piezoelec ic junc ions. A chi e o Applied Mechanics
(Ingenieu A chi ),73, pp. 638–649.
Xu, X.-L., Rajapakse, R.K.N.D., 2000. On singula i ies in composi e piezoelec ic wedges and junc ions. In e na ional Jou nal o Solids and
S uc u es, 37, pp.3253–3275.
Zeng, X., Rajapakse, R. K. N. D., 2001. Domain swi ching induced ac u e oughness a ia ion in e oelec ics. Sma Ma e ials and S uc u es,
10, pp.203–211.
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