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P ocedia S uc u al In eg i y 42 (2022) 1584–1590
2452-3216
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Pee - e iew unde esponsibili y o he scien i ic commi ee o he 23 Eu opean Con e ence on F ac u e – ECF23
10.1016/j.p os .2022.12.200
10.1016/j.p os .2022.12.200 2452-3216
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h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0
)
Pee - e iew unde esponsibili y o he scien i ic commi ee o he 23 Eu opean Con e ence on F ac u e – ECF23
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
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Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
In luence o lexoelec ici y on an in e ace c ack be ween wo
dissimila dielec ic ma e ials
Jan Sladeka* , Vladimi Sladeka , Ma yan H y synaa , Tomas P o an a,b
a Ins i i e o Cons uc ion and A chi ec u e, Slo ak Academy o Sciences, 84503 B a isla a, Slo akia
b Ins i u e o Solid Mechanics, B no Uni e si y o Technology, Technicka 2, 61669 B no, Czech Republic
Abs ac
In he p esen pape , he in e ace c ack be ween wo dissimila dielec ic ma e ials unde a mechanical load is in es iga ed wi h
including lexoelec ici y e ec s. Flexoelec ici y is a size dependen elec o-mechanical coupling phenomenon, whe e he elec ic
pola iza ion is induced by a s ain g adien in dielec ics. The s ain g adien s may po en ially b eak he in e sion symme y in
cen osymme ic c ys als and pola iza ion is obse ed e en in all dielec ic ma e ials. The pola iza ion is p opo ional o he s ain
g adien s in he di ec lexoelec ici y. Laye ed composi e s uc u es a e equen ly u ilized in mic oelec onics. Due o a poo
adhesion o p o ec ion laye and basic ma e ial, he in e ace c ack can be c ea ed he e and o he p edic ion o ailu e o hese
s uc u es i becomes essen ial o in es iga e dis ibu ion o he in e acial s ess and s ain ields.
Go e ning equa ions in he g adien heo y con ain highe -o de de i a i es han in he s anda d con inuum mechanics. The e o e,
a eliable compu a ional ool is equi ed o sol e hese bounda y- alue p oblems. The mixed ini e elemen me hod (FEM) is
de eloped, whe e he s anda d C0 con inuous ini e elemen s a e u ilized o independen app oxima ions o displacemen s and
s ains. The cons ain s be ween he s ain g adien s and displacemen s a e sa is ied by colloca ion a Gaussian in eg a ion poin s
inside elemen s. In nume ical examples, a pa ame ic s udy is pe o med wi h espec o lexoelec ic and elas ic coe icien s o
bo h ma e ial egions. The in luence o hese pa ame e s on he c ack opening displacemen is discussed.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: Di ec lexoelec ici y, g adien heo y, a pu e mechanical load, induced elec ic po en ial
* Co esponding au ho . Tel.: +421-904885687
E-mail add ess: jan.sladek@sa ba.sk
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
In luence o lexoelec ici y on an in e ace c ack be ween wo
dissimila dielec ic ma e ials
Jan Sladeka* , Vladimi Sladeka , Ma yan H y synaa , Tomas P o an a,b
a Ins i i e o Cons uc ion and A chi ec u e, Slo ak Academy o Sciences, 84503 B a isla a, Slo akia
b Ins i u e o Solid Mechanics, B no Uni e si y o Technology, Technicka 2, 61669 B no, Czech Republic
Abs ac
In he p esen pape , he in e ace c ack be ween wo dissimila dielec ic ma e ials unde a mechanical load is in es iga ed wi h
including lexoelec ici y e ec s. Flexoelec ici y is a size dependen elec o-mechanical coupling phenomenon, whe e he elec ic
pola iza ion is induced by a s ain g adien in dielec ics. The s ain g adien s may po en ially b eak he in e sion symme y in
cen osymme ic c ys als and pola iza ion is obse ed e en in all dielec ic ma e ials. The pola iza ion is p opo ional o he s ain
g adien s in he di ec lexoelec ici y. Laye ed composi e s uc u es a e equen ly u ilized in mic oelec onics. Due o a poo
adhesion o p o ec ion laye and basic ma e ial, he in e ace c ack can be c ea ed he e and o he p edic ion o ailu e o hese
s uc u es i becomes essen ial o in es iga e dis ibu ion o he in e acial s ess and s ain ields.
Go e ning equa ions in he g adien heo y con ain highe -o de de i a i es han in he s anda d con inuum mechanics. The e o e,
a eliable compu a ional ool is equi ed o sol e hese bounda y- alue p oblems. The mixed ini e elemen me hod (FEM) is
de eloped, whe e he s anda d C0 con inuous ini e elemen s a e u ilized o independen app oxima ions o displacemen s and
s ains. The cons ain s be ween he s ain g adien s and displacemen s a e sa is ied by colloca ion a Gaussian in eg a ion poin s
inside elemen s. In nume ical examples, a pa ame ic s udy is pe o med wi h espec o lexoelec ic and elas ic coe icien s o
bo h ma e ial egions. The in luence o hese pa ame e s on he c ack opening displacemen is discussed.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: Di ec lexoelec ici y, g adien heo y, a pu e mechanical load, induced elec ic po en ial
* Co esponding au ho . Tel.: +421-904885687
E-mail add ess: jan.sladek@sa ba.sk
2 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
1. In oduc ion
The delamina ion is obse ed equen ly in laye ed s uc u es due o induced high s esses on he in e ace a a
he mal load. Then, o he p edic ion o ailu e, i is needed o analyse s ess and s ain ields nea he ip o in e ace
c acks. In li e a u e one can ind a lo o s udies on in e ace c acks analysed by he classical heo y o elas ici y, whe e
he mic os uc u e o he ma e ials is igno ed (Askes and Gi man 2009). The mic os uc u e has o be conside ed i
he cha ac e is ic leng h o he ma e ial s uc u e is compa able wi h he size o he s uc u e (Buhlmann e al. 2002;
Ca alan e al. 2011). The g adien heo y has been success ully applied o modelling mic o/nano-sized s uc u es and
c acks in homogeneous ma e ials (Al an S, Ai an is 1992; Ai an is 2003; A a as and Giannakopoulos 2009; Sladek
e al. 2017; Hu SL and Shen SP 2009). Resul s o in e ace c acks be ween wo dissimila ma e ials analysed by he
g adien heo y a e e y seldom in li e a u e (I ou 1991; Picco oaz e al. 2012; Ko oul and P o an 2018).
In he di ec lexoelec ici y he elec ic pola iza ion is induced by s ain g adien s. Due o he la ge s ain g adien s
he pola iza ion is induced e en in all dielec ic ma e ials (Yudin and Tagan se 2013). The bigges in luence o
lexoelec ici y is obse ed a he c ack ip icini y since he s ain g adien s a e he la ges he e (Huang e al. 1999;
Geo giadis 2003; Gou gio is and Geo giadis 2009). Recen expe imen al obse a ions and analy ical s udies o he
lexoelec ic e ec nea he c ack ip ha e been epo ed by Wang e al. (2020) and Tian e al. (2022) in homogeneous
ma e ials. Nume ical in es iga ion o he lexoelec ic e ec nea in e ace c ack ips be ween wo dissimila ma e ials
is s ill missing. The e o e, we sol e his p oblem in his pape .
Fo his pu pose i is needed o ha e a eliable compu a ional ool. The ini e elemen me hod (FEM) seems o be a
con enien ool o sol e gene al bounda y- alue p oblems in he g adien heo y o elas ici y wi h lexoelec ic e ec .
The s anda d C0 con inuous ini e elemen me hod canno be applied o sol e p oblems in he g adien heo y due o
highe o de de i a i es in he go e ning equa ion (Sladek e al. 2017, 2019; Tian e al. 2021). In his pape he mixed
FEM is de eloped o an in e ace c ack be ween wo dissimila dielec ics, whe e he C0 con inuous app oxima ion
is employed independen ly o displacemen and displacemen g adien s. The cons ain s be ween hem a e sa is ied
by colloca ion inside elemen s (Tian e al. 2021).
Nume ical esul s illus a e he in luence o he lexoelec ic coe icien and a io o elas ic coe icien s on he c ack
opening displacemen and induced elec ic po en ial in laye ed c acked s uc u es.
2. Bounda y alue p oblems o di ec lexoelec ici y
In he di ec lexoelec ici y elec o-mechanical ields a e coupled by s ain g adien s. The la ge s ain g adien s
can b eak he in e sion symme y in cen osymme ic c ys als. Then, he pola iza ion is obse ed also in dielec ics.
The cons i u i e equa ions o Cauchy s esses
ij

, highe -o de s esses
ijk

and elec ic displacemen
i
D
in
dielec ic ma e ials (non piezoelec ic) can be w i en as (Hu and Shen, 2009)
ij ijkl kl
c

=
,
jkl ijkl i jklmni mni
E g

=−+
i ij j ijkl jkl
D aE

= +
, (1)
whe e aij and cijkl a e he pe mi i i y and elas ic s i ness enso s, espec i ely. The di ec lexoelec ic coe icien s
a e deno ed by ijkl and he highe -o de elas ic coe icien s by gijklmn.
S ains
ij

, elec ic in ensi y ec o
j
E
and s ain-g adien s

ijk can be exp essed h ough displacemen s
i
u
and
elec ic po en ial

, espec i ely:
( )
,, ,
/2,
ij i j j i j j
uu E

=+=−
, (2)
( )
, ,,
/2
ijk ij k i jk j ik
uu

= = +
. (3)
In he simpli ied g adien elas ici y, he highe -o de elas ic pa ame e s
jklmni
g
a e educed o elas ic s i ness
coe icien s
klmn
c
and one addi ional in e nal leng h ma e ial pa ame e l (Gi man e al., 2010) as
2
jklmni li jkmn
g lc

=
.
Simila ly wo independen pa ame e s
1
and
2
a e used o exp ession o he di ec lexoelec ic coe icien s ijkl,
(Deng e al. 2017):
Jan Sladek e al. / P ocedia S uc u al In eg i y 42 (2022) 1584–1590 1585
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
In luence o lexoelec ici y on an in e ace c ack be ween wo
dissimila dielec ic ma e ials
Jan Sladeka* , Vladimi Sladeka , Ma yan H y synaa , Tomas P o an a,b
a Ins i i e o Cons uc ion and A chi ec u e, Slo ak Academy o Sciences, 84503 B a isla a, Slo akia
b Ins i u e o Solid Mechanics, B no Uni e si y o Technology, Technicka 2, 61669 B no, Czech Republic
Abs ac
In he p esen pape , he in e ace c ack be ween wo dissimila dielec ic ma e ials unde a mechanical load is in es iga ed wi h
including lexoelec ici y e ec s. Flexoelec ici y is a size dependen elec o-mechanical coupling phenomenon, whe e he elec ic
pola iza ion is induced by a s ain g adien in dielec ics. The s ain g adien s may po en ially b eak he in e sion symme y in
cen osymme ic c ys als and pola iza ion is obse ed e en in all dielec ic ma e ials. The pola iza ion is p opo ional o he s ain
g adien s in he di ec lexoelec ici y. Laye ed composi e s uc u es a e equen ly u ilized in mic oelec onics. Due o a poo
adhesion o p o ec ion laye and basic ma e ial, he in e ace c ack can be c ea ed he e and o he p edic ion o ailu e o hese
s uc u es i becomes essen ial o in es iga e dis ibu ion o he in e acial s ess and s ain ields.
Go e ning equa ions in he g adien heo y con ain highe -o de de i a i es han in he s anda d con inuum mechanics. The e o e,
a eliable compu a ional ool is equi ed o sol e hese bounda y- alue p oblems. The mixed ini e elemen me hod (FEM) is
de eloped, whe e he s anda d C0 con inuous ini e elemen s a e u ilized o independen app oxima ions o displacemen s and
s ains. The cons ain s be ween he s ain g adien s and displacemen s a e sa is ied by colloca ion a Gaussian in eg a ion poin s
inside elemen s. In nume ical examples, a pa ame ic s udy is pe o med wi h espec o lexoelec ic and elas ic coe icien s o
bo h ma e ial egions. The in luence o hese pa ame e s on he c ack opening displacemen is discussed.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: Di ec lexoelec ici y, g adien heo y, a pu e mechanical load, induced elec ic po en ial
* Co esponding au ho . Tel.: +421-904885687
E-mail add ess: jan.sladek@sa ba.sk
A ailable online a www.sciencedi ec .com
^ĐŝĞŶĐĞŝƌĞĐƚ
S uc u al In eg i y P ocedia 00 (2019) 000–000
www.else ie .com/loca e/p ocedia
2452-3216 © 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
23 Eu opean Con e ence on F ac u e - ECF23
In luence o lexoelec ici y on an in e ace c ack be ween wo
dissimila dielec ic ma e ials
Jan Sladeka* , Vladimi Sladeka , Ma yan H y synaa , Tomas P o an a,b
a Ins i i e o Cons uc ion and A chi ec u e, Slo ak Academy o Sciences, 84503 B a isla a, Slo akia
b Ins i u e o Solid Mechanics, B no Uni e si y o Technology, Technicka 2, 61669 B no, Czech Republic
Abs ac
In he p esen pape , he in e ace c ack be ween wo dissimila dielec ic ma e ials unde a mechanical load is in es iga ed wi h
including lexoelec ici y e ec s. Flexoelec ici y is a size dependen elec o-mechanical coupling phenomenon, whe e he elec ic
pola iza ion is induced by a s ain g adien in dielec ics. The s ain g adien s may po en ially b eak he in e sion symme y in
cen osymme ic c ys als and pola iza ion is obse ed e en in all dielec ic ma e ials. The pola iza ion is p opo ional o he s ain
g adien s in he di ec lexoelec ici y. Laye ed composi e s uc u es a e equen ly u ilized in mic oelec onics. Due o a poo
adhesion o p o ec ion laye and basic ma e ial, he in e ace c ack can be c ea ed he e and o he p edic ion o ailu e o hese
s uc u es i becomes essen ial o in es iga e dis ibu ion o he in e acial s ess and s ain ields.
Go e ning equa ions in he g adien heo y con ain highe -o de de i a i es han in he s anda d con inuum mechanics. The e o e,
a eliable compu a ional ool is equi ed o sol e hese bounda y- alue p oblems. The mixed ini e elemen me hod (FEM) is
de eloped, whe e he s anda d C0 con inuous ini e elemen s a e u ilized o independen app oxima ions o displacemen s and
s ains. The cons ain s be ween he s ain g adien s and displacemen s a e sa is ied by colloca ion a Gaussian in eg a ion poin s
inside elemen s. In nume ical examples, a pa ame ic s udy is pe o med wi h espec o lexoelec ic and elas ic coe icien s o
bo h ma e ial egions. The in luence o hese pa ame e s on he c ack opening displacemen is discussed.
© 2020 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o 23 Eu opean Con e ence on F ac u e - ECF23
Keywo ds: Di ec lexoelec ici y, g adien heo y, a pu e mechanical load, induced elec ic po en ial
* Co esponding au ho . Tel.: +421-904885687
E-mail add ess: jan.sladek@sa ba.sk
2 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
1. In oduc ion
The delamina ion is obse ed equen ly in laye ed s uc u es due o induced high s esses on he in e ace a a
he mal load. Then, o he p edic ion o ailu e, i is needed o analyse s ess and s ain ields nea he ip o in e ace
c acks. In li e a u e one can ind a lo o s udies on in e ace c acks analysed by he classical heo y o elas ici y, whe e
he mic os uc u e o he ma e ials is igno ed (Askes and Gi man 2009). The mic os uc u e has o be conside ed i
he cha ac e is ic leng h o he ma e ial s uc u e is compa able wi h he size o he s uc u e (Buhlmann e al. 2002;
Ca alan e al. 2011). The g adien heo y has been success ully applied o modelling mic o/nano-sized s uc u es and
c acks in homogeneous ma e ials (Al an S, Ai an is 1992; Ai an is 2003; A a as and Giannakopoulos 2009; Sladek
e al. 2017; Hu SL and Shen SP 2009). Resul s o in e ace c acks be ween wo dissimila ma e ials analysed by he
g adien heo y a e e y seldom in li e a u e (I ou 1991; Picco oaz e al. 2012; Ko oul and P o an 2018).
In he di ec lexoelec ici y he elec ic pola iza ion is induced by s ain g adien s. Due o he la ge s ain g adien s
he pola iza ion is induced e en in all dielec ic ma e ials (Yudin and Tagan se 2013). The bigges in luence o
lexoelec ici y is obse ed a he c ack ip icini y since he s ain g adien s a e he la ges he e (Huang e al. 1999;
Geo giadis 2003; Gou gio is and Geo giadis 2009). Recen expe imen al obse a ions and analy ical s udies o he
lexoelec ic e ec nea he c ack ip ha e been epo ed by Wang e al. (2020) and Tian e al. (2022) in homogeneous
ma e ials. Nume ical in es iga ion o he lexoelec ic e ec nea in e ace c ack ips be ween wo dissimila ma e ials
is s ill missing. The e o e, we sol e his p oblem in his pape .
Fo his pu pose i is needed o ha e a eliable compu a ional ool. The ini e elemen me hod (FEM) seems o be a
con enien ool o sol e gene al bounda y- alue p oblems in he g adien heo y o elas ici y wi h lexoelec ic e ec .
The s anda d C0 con inuous ini e elemen me hod canno be applied o sol e p oblems in he g adien heo y due o
highe o de de i a i es in he go e ning equa ion (Sladek e al. 2017, 2019; Tian e al. 2021). In his pape he mixed
FEM is de eloped o an in e ace c ack be ween wo dissimila dielec ics, whe e he C0 con inuous app oxima ion
is employed independen ly o displacemen and displacemen g adien s. The cons ain s be ween hem a e sa is ied
by colloca ion inside elemen s (Tian e al. 2021).
Nume ical esul s illus a e he in luence o he lexoelec ic coe icien and a io o elas ic coe icien s on he c ack
opening displacemen and induced elec ic po en ial in laye ed c acked s uc u es.
2. Bounda y alue p oblems o di ec lexoelec ici y
In he di ec lexoelec ici y elec o-mechanical ields a e coupled by s ain g adien s. The la ge s ain g adien s
can b eak he in e sion symme y in cen osymme ic c ys als. Then, he pola iza ion is obse ed also in dielec ics.
The cons i u i e equa ions o Cauchy s esses
ij

, highe -o de s esses
ijk

and elec ic displacemen
i
D
in
dielec ic ma e ials (non piezoelec ic) can be w i en as (Hu and Shen, 2009)
ij ijkl kl
c

=
,
jkl ijkl i jklmni mni
E g

=−+
i ij j ijkl jkl
D aE

= +
, (1)
whe e aij and cijkl a e he pe mi i i y and elas ic s i ness enso s, espec i ely. The di ec lexoelec ic coe icien s
a e deno ed by ijkl and he highe -o de elas ic coe icien s by gijklmn.
S ains
ij

, elec ic in ensi y ec o
j
E
and s ain-g adien s

ijk can be exp essed h ough displacemen s
i
u
and
elec ic po en ial

, espec i ely:
( )
,, ,
/2,
ij i j j i j j
uu E

=+=−
, (2)
( )
, ,,
/2
ijk ij k i jk j ik
uu

= = +
. (3)
In he simpli ied g adien elas ici y, he highe -o de elas ic pa ame e s
jklmni
g
a e educed o elas ic s i ness
coe icien s
klmn
c
and one addi ional in e nal leng h ma e ial pa ame e l (Gi man e al., 2010) as
2
jklmni li jkmn
g lc

=
.
Simila ly wo independen pa ame e s
1
and
2
a e used o exp ession o he di ec lexoelec ic coe icien s ijkl,
(Deng e al. 2017):
1586 Jan Sladek e al. / P ocedia S uc u al In eg i y 42 (2022) 1584–1590
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 3
( )
12ijkl jk il ij kl ik jl
   
=++
. (4)
De i a ion o go e ning equa ions o 2D p oblems in he di ec lexoelec ici y can be ound in wo ks (Sladek e
al., 2017, Tian e al. 2020):
,,
() () 0
ij j ijk jk

−=xx
,
,() 0
ii
D=x
. (5
)
In bounda y condi ions he e a e occu ed no mal de i a i es o displacemen , ac ion ec o o highe -o de o
s esses, elec ic cha ge
,
:/
i i j ij
s u nu= =n
,
:
i k j ijk
R nn

=
,
:
ii
Q nD=
, (6)
and he ac ion ec o
( )
,()( )
cc
i
i j ij ijk k j i
c
j
n x

   

= −− + −

x xx
, (7)
wi h
:
i k j ijk
n
 
=
, (8)
and he jump a a co ne on he o ien ed bounda y con ou

is de ined as
( ): ( ) ( )
cc c
ii i
 
= −-0 +0xx x
,
and
i
n
and
i

a e he Ca esian componen s o he uni no mal and angen ec o on bounda y, espec i ely.
3. Fini e elemen me hods and esul s
The p inciple o i ual wo k can be applied o ge he weak- o m o go e ning equa ions (5). The a ia ion o he
de o ma ion ene gy has o equal o he a ia ion o ex e nal o ces (Sladek e al. 2017, Tian e al. 2020)
( )
, ,,ij i j ijk i jk i i i i i i
V R Q
u u D dV u d R s d Q d
       
 
+ + = + + 
  
. (9)
The mixed FEM wi h independen C0 con inuous in e pola ion o bo h elas ic displacemen s and s ains is de eloped
he e. The displacemen ec o and elec ic po en ial in each elemen a e app oxima ed by
( )
12
[ ,]
uu

=uN q
( )
12
[ ,]

 
=Nq
, (10)
whe e
u
q
and

q
a e ec o s o nodal displacemen s and elec ic po en ial, espec i ely.
Abo e exp essions can be u ilized o app oxima ion o s ains and elec ic in ensi y ec o
11 1
1
22 2 1 2
2
12 2 1
0
0 [ ( , )]
2
uu
u
u

 


  

  
===

  

  

  
ε Bq
,
11
12
22
[ ( , )]
E
E

 


−=− = =



E Bq
. (11)
In he mixed FEM, we apply also an independen app oxima ion o s ains
( )
12
ˆ() [] ,
In

=ε x αp
, (12)
whe e [] a e unknown coe icien s, and
( )
12
,

p
is he polynomial unc ion ec o
( )
 
1 2 1 2 12
,1
T
    
=p
.
Since wo independen app oxima ions o s ains gi en by (11) and (12) ha e o be equal a selec ed in e nal poin s
12
(,)
c cc

=ξ
, one can de i e he inal app oxima ion o mula
4 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
12
ˆ(, )
In
u

=ε p Lq
(13)
whe e
( ) ( )
1
12 12
,,
cc cc
u
 
−
=Lp B
.
Then, he de i a i es o s ains can be ob ained om (13)
( )
( ) ( )
1 1 12 *
12
2 12
2
ˆ,
ˆ,
,
ˆ
Ïn
In
u
Ïn
 


  
= = =







ε pα
η p Lq
pα
ε
. (14)
Subs i u ing abo e app oxima ions in o a ia ional o m (9) and aking in o accoun a bi a iness o he a ia ions
 
u

q
and
 


q
, we ob ain he sys em o algeb aic equa ions o unknown nodal quan i ies
 
[ ( )] [ ][ ( )]
T
u uu
V
dV +
Bξ CBξ q
 
 
( )
 
2
[ ( )] [ ][ ( )] [ ] [ ( )] [ ]
TT T
u us
V R
l dV d d
  

+ + = + 
 
Bξ GBξq FBξq NT BR
, (15)
( )
 
 
 
[ ( )] [ ] [ ( )] [ ][ ( )] [ ][ ( )] [ ]
TT T
uu
VQ
dV Qd
   

+− =

Bξ P Bξ FBξ q ΠBξ q N
, (16)
whe e
C
,
G
,
Π
and
F
ep esen elas ic, highe -o de elas ic, dielec ic and lexoexoelec ic ma ix coe icien s,
espec i ely.
A s aigh in e ace c ack be ween wo dissimila dielec ic ma e ials is sol ed nume ically (Fig. 1). A s a iona y
bounda y condi ions wi h a pu e ension load
7
0
1 10 ( )p Pa= 
a e applied on he op and bo om su aces. Following
geome y is conside ed:
88
125 10 , 10 10 m
c
w ml
−−
==
,
8
20 10hm
−
= 
,
8
10 10
c
hm
−
= 
. Impe meable elec ic
bounda y condi ions a e conside ed on he c ack su aces wi h a e e ence alue o elec ic po en ial a he c ack ip,
0

=
. All o he su aces ha e anishing elec ic cha ge
0
ii
nD =
. The highe -o de ac ion
0
i
R=
is also anishing
on all su aces.
Fig. 1. A symme ic pa o he c acked s ip unde a uni o m axial ension
A s anda d dielec ic ce amics (SDC) wi h ollowing pa ame e s:
10
11
0
13.9*10cPa=
,
10
12
0
7.4*10cPa=
,
10
22
0
11.5*10cPa=
,
10
44
0
.562*10cPa=
, a1=15.1*10-9C(VM)-1, a2=13.0*10-9C(VM)-1 is conside ed o domain I
(uppe laye ).
Th ee a ious ma e ials a e conside ed in he lowe laye (domain II) o in es iga e in luence o dissimila elas ic
p ope ies on beha iou o he in e ace c ack:
Jan Sladek e al. / P ocedia S uc u al In eg i y 42 (2022) 1584–1590 1587
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 3
( )
12ijkl jk il ij kl ik jl
   
=++
. (4)
De i a ion o go e ning equa ions o 2D p oblems in he di ec lexoelec ici y can be ound in wo ks (Sladek e
al., 2017, Tian e al. 2020):
,,
() () 0
ij j ijk jk

−=xx
,
,() 0
ii
D=x
. (5)
In bounda y condi ions he e a e occu ed no mal de i a i es o displacemen , ac ion ec o o highe -o de o
s esses, elec ic cha ge
,
:/
i i j ij
s u nu= =n
,
:
i k j ijk
R nn

=
,
:
ii
Q nD=
, (6)
and he ac ion ec o
( )
,()( )
cc
i
i j ij ijk k j i
c
j
n x

   

= −− + −

x xx
, (7)
wi h
:
i k j ijk
n
 
=
, (8)
and he jump a a co ne on he o ien ed bounda y con ou

is de ined as
( ): ( ) ( )
cc c
ii i
 
= −-0 +0xx x
,
and
i
n
and
i

a e he Ca esian componen s o he uni no mal and angen ec o on bounda y, espec i ely.
3. Fini e elemen me hods and esul s
The p inciple o i ual wo k can be applied o ge he weak- o m o go e ning equa ions (5). The a ia ion o he
de o ma ion ene gy has o equal o he a ia ion o ex e nal o ces (Sladek e al. 2017, Tian e al. 2020)
( )
, ,,ij i j ijk i jk i i i i i i
V R Q
u u D dV u d R s d Q d
       
 
+ + = + + 
  
. (9)
The mixed FEM wi h independen C0 con inuous in e pola ion o bo h elas ic displacemen s and s ains is de eloped
he e. The displacemen ec o and elec ic po en ial in each elemen a e app oxima ed by
( )
12
[ ,]
uu

=uN q
( )
12
[ ,]

 
=Nq
, (10)
whe e
u
q
and

q
a e ec o s o nodal displacemen s and elec ic po en ial, espec i ely.
Abo e exp essions can be u ilized o app oxima ion o s ains and elec ic in ensi y ec o
11 1
1
22 2 1 2
2
12 2 1
0
0 [ ( , )]
2
uu
u
u

 


  

  
===

  

  

  
ε Bq
,
11
12
22
[ ( , )]
E
E

 


−=− = =



E Bq
. (11)
In he mixed FEM, we apply also an independen app oxima ion o s ains
( )
12
ˆ() [] ,
In

=ε x αp
, (12)
whe e [] a e unknown coe icien s, and
( )
12
,

p
is he polynomial unc ion ec o
( )
 
1 2 1 2 12
,1
T
    
=p
.
Since wo independen app oxima ions o s ains gi en by (11) and (12) ha e o be equal a selec ed in e nal poin s
12
(,)
c cc

=ξ
, one can de i e he inal app oxima ion o mula
4 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
12
ˆ(, )
In
u

=ε p Lq
(13)
whe e
( ) ( )
1
12 12
,,
cc cc
u
 
−
=Lp B
.
Then, he de i a i es o s ains can be ob ained om (13)
( )
( ) ( )
1 1 12 *
12
2 12
2
ˆ,
ˆ,
,
ˆ
Ïn
In
u
Ïn
 


  
= = =







ε pα
η p Lq
pα
ε
. (14)
Subs i u ing abo e app oxima ions in o a ia ional o m (9) and aking in o accoun a bi a iness o he a ia ions
 
u

q
and
 


q
, we ob ain he sys em o algeb aic equa ions o unknown nodal quan i ies
 
[ ( )] [ ][ ( )]
T
u uu
V
dV +
Bξ CBξ q
 
 
( )
 
2
[ ( )] [ ][ ( )] [ ] [ ( )] [ ]
TT T
u us
V R
l dV d d
  

+ + = + 
 
Bξ GBξq FBξq NT BR
, (15)
( )
 
 
 
[ ( )] [ ] [ ( )] [ ][ ( )] [ ][ ( )] [ ]
TT T
uu
VQ
dV Qd
   

+− =

Bξ P Bξ FBξ q ΠBξ q N
, (16)
whe e
C
,
G
,
Π
and
F
ep esen elas ic, highe -o de elas ic, dielec ic and lexoexoelec ic ma ix coe icien s,
espec i ely.
A s aigh in e ace c ack be ween wo dissimila dielec ic ma e ials is sol ed nume ically (Fig. 1). A s a iona y
bounda y condi ions wi h a pu e ension load
7
0
1 10 ( )p Pa= 
a e applied on he op and bo om su aces. Following
geome y is conside ed:
88
125 10 , 10 10 m
c
w ml
−−
==
,
8
20 10hm
−
= 
,
8
10 10
c
hm
−
= 
. Impe meable elec ic
bounda y condi ions a e conside ed on he c ack su aces wi h a e e ence alue o elec ic po en ial a he c ack ip,
0

=
. All o he su aces ha e anishing elec ic cha ge
0
ii
nD =
. The highe -o de ac ion
0
i
R=
is also anishing
on all su aces.
Fig. 1. A symme ic pa o he c acked s ip unde a uni o m axial ension
A s anda d dielec ic ce amics (SDC) wi h ollowing pa ame e s:
10
11
0
13.9*10cPa=
,
10
12
0
7.4*10cPa=
,
10
22
0
11.5*10cPa=
,
10
44
0
.562*10cPa=
, a1=15.1*10-9C(VM)-1, a2=13.0*10-9C(VM)-1 is conside ed o domain I
(uppe laye ).
Th ee a ious ma e ials a e conside ed in he lowe laye (domain II) o in es iga e in luence o dissimila elas ic
p ope ies on beha iou o he in e ace c ack:
1588 Jan Sladek e al. / P ocedia S uc u al In eg i y 42 (2022) 1584–1590
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 5
Ma e ial1 - SDC - c11= c011; c12= c012; c22= c022; c44=c044;
Ma e ial2 - c11= 2*c011; c12= 2*c012; c22= 2*c022; c44= 2*c044;
Ma e ial3 - c11= 5*c011; c12=5*c012; c22= 5*c022; c44= 5*c044.
One can see ha Ma e ial1 co esponds o a c ack p oblem in a homogeneous body. Two di e en lexoelec ic
coe icien s a e conside ed: 1= {10-9; 10-7}C/m , 2=0 and he mic o-s i ness leng h pa ame e is conside ed as
8
10lm
−
=
.
Fig. 2 Va ia ion o he c ack displacemen s along he c ack aces o a ious elas ic a ios and lexoelec ic
coe icien s: a)
9
11 10 / Cm
−
= 
; b)
7
11 10 / Cm
−
= 
Va ia ions o he c ack displacemen s
2
u
along he c ack ace
1
x
o a ious a ios o elas ic coe icien s in domain I
and II and lexoelec ic coe icien s
9
11 10 / Cm
−
= 
and
7
11 10 / Cm
−
= 
a e p esen ed in Fig. 2a and Fig. 2b,
espec i ely. One can obse e almos he same c ack opening displacemen s o di e en alues o lexoelec ic
6 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
coe icien s. Displacemen s on he uppe c ack ace a e only sligh ly educed o la ge elas ic coe icien s in domain
II, howe e , on he lowe c ack su ace hey a e educed signi ican ly in his case.
Fig. 3 Va ia ion o he induced elec ic po en ials along he c ack aces o dissimila elas ic ma e ials: SDC &
ma e ial2
The induced elec ic po en ials on bo h c ack aces a e p esen ed in Fig. 3. The induced po en ial on he uppe c ack
ace is la ge han lowe c ack ace. Po en ials a e la ge o bigge lexoelec ic coe icien s.
4. Conclusions
The in luence o he lexoelec ici y on an in e ace c ack be ween wo dissimila dielec ic laye s is in es iga ed
nume ically. The mixed FEM equa ions o his p oblem a e de eloped as he weak o mula ion o he highe -g ade
elec o-elas ici y. The ad an age o his app oach is ha only C0 con inuous elemen s a e su icien . Displacemen s
and s ains app oxima ions a e independen ly applied and cons ain s be ween hem a e sa is ied by colloca ion inside
o elemen s. One can obse e in nume ical examples ha c ack opening displacemen s a e educed o la ge alues
o lexoelec ic coe icien s. Also he c ack opening displacemen and he induced elec ic po en ial a e educed o
la ge alues o elas ic coe icien in lowe laye . The lexoelec ic coe icien has a s ong in luence on c acked
s uc u es and mic oelec onic s uc u es a e o be analysed wi h aking in o he lexoele ic e ec .
Acknowledgemen s
The au ho s acknowledge he suppo by he Slo ak Science and Technology Assis ance Agency egis e ed unde
numbe APVV-18-0004 and VEGA-2/0061/20.
Re e ences
Ai an ic, E.C., 2003. Upda e on a class o g adien heo ies. Mechanics o Ma e ials 32, 259-280.
Al an, S., Ai an is, E.C. 1992, On he s uc u e o he mode III c ack ip in g adien elas ici y. Sc ip a Me allic Ma e ials 26, 319-324.
A a as, N., Giannakopoulos, A.E., 2009. Plane asymp o ic c ack ip solu ions in g adien elas ici y. In e na ional Jou nal o Solids S uc u es 46,
4478-4503.

Jan Sladek e al. / P ocedia S uc u al In eg i y 42 (2022) 1584–1590 1589
Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000 5
Ma e ial1 - SDC - c11= c011; c12= c012; c22= c022; c44=c044;
Ma e ial2 - c11= 2*c011; c12= 2*c012; c22= 2*c022; c44= 2*c044;
Ma e ial3 - c11= 5*c011; c12=5*c012; c22= 5*c022; c44= 5*c044.
One can see ha Ma e ial1 co esponds o a c ack p oblem in a homogeneous body. Two di e en lexoelec ic
coe icien s a e conside ed: 1= {10-9; 10-7}C/m , 2=0 and he mic o-s i ness leng h pa ame e is conside ed as
8
10lm
−
=
.
Fig. 2 Va ia ion o he c ack displacemen s along he c ack aces o a ious elas ic a ios and lexoelec ic
coe icien s: a)
9
11 10 / Cm
−
= 
; b)
7
11 10 / Cm
−
= 
Va ia ions o he c ack displacemen s
2
u
along he c ack ace
1
x
o a ious a ios o elas ic coe icien s in domain I
and II and lexoelec ic coe icien s
9
11 10 / Cm
−
= 
and
7
11 10 / Cm
−
= 
a e p esen ed in Fig. 2a and Fig. 2b,
espec i ely. One can obse e almos he same c ack opening displacemen s o di e en alues o lexoelec ic
6 Au ho name / S uc u al In eg i y P ocedia 00 (2019) 000–000
coe icien s. Displacemen s on he uppe c ack ace a e only sligh ly educed o la ge elas ic coe icien s in domain
II, howe e , on he lowe c ack su ace hey a e educed signi ican ly in his case.
Fig. 3 Va ia ion o he induced elec ic po en ials along he c ack aces o dissimila elas ic ma e ials: SDC &
ma e ial2
The induced elec ic po en ials on bo h c ack aces a e p esen ed in Fig. 3. The induced po en ial on he uppe c ack
ace is la ge han lowe c ack ace. Po en ials a e la ge o bigge lexoelec ic coe icien s.
4. Conclusions
The in luence o he lexoelec ici y on an in e ace c ack be ween wo dissimila dielec ic laye s is in es iga ed
nume ically. The mixed FEM equa ions o his p oblem a e de eloped as he weak o mula ion o he highe -g ade
elec o-elas ici y. The ad an age o his app oach is ha only C0 con inuous elemen s a e su icien . Displacemen s
and s ains app oxima ions a e independen ly applied and cons ain s be ween hem a e sa is ied by colloca ion inside
o elemen s. One can obse e in nume ical examples ha c ack opening displacemen s a e educed o la ge alues
o lexoelec ic coe icien s. Also he c ack opening displacemen and he induced elec ic po en ial a e educed o
la ge alues o elas ic coe icien in lowe laye . The lexoelec ic coe icien has a s ong in luence on c acked
s uc u es and mic oelec onic s uc u es a e o be analysed wi h aking in o he lexoele ic e ec .
Acknowledgemen s
The au ho s acknowledge he suppo by he Slo ak Science and Technology Assis ance Agency egis e ed unde
numbe APVV-18-0004 and VEGA-2/0061/20.
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