Investigating the influence of interface in a three phase composite using variational asymptotic method based homogenization technique

Journal Pre-proofs Investigating the influence of interface in a three phase composite using variational asymptotic method based homogenization method Pandi Pitchai, Harald Berger, P.J. Guruprasad PII: DOI: Reference:

S0263-8223(19)33195-2 https://doi.org/10.1016/j.compstruct.2019.111562 COST 111562

To appear in:

Composite Structures

Received Date: Accepted Date:

23 August 2019 9 October 2019

Please cite this article as: Pitchai, P., Berger, H., Guruprasad, P.J., Investigating the influence of interface in a three phase composite using variational asymptotic method based homogenization method, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111562

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© 2019 Elsevier Ltd. All rights reserved.

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M M M εij εkl Cijkl

+

M 2βijM εM ij θ

*

+

CvM



dR ≡ + M

θ2 T0M

2I

dM

+

θ 1 I Cijkl εIij εIkl + 2βijI εIij θI + CvI I dI + T0 I 2 * + Z 2F θ 1 F Cijkl εFij εFkl + 2βijF εFij θF + CvF F dF + T0 F 2 Z

Rk3 RkN Rjy RjR

r?2`2- (εij ) `2 i?2 +QKTQM2Mib Q7 b2+QM/ Q`/2` HBM2` 2HbiB+Biv bi`BM i2MbQ` M/ `2 2tTM/2/ +Q``2bTQM/b iQ i?2 +QMbiBim2Mib Q7 Ki`Bt- BMi2`7+2 M/ }#2` b 7mM+iBQM Q7 ;HQ#H M/ HQ+H +QQ`/BMi2 bvbi2K Q7 bBM;H2 mMBi +2HHX h?2 /2}MBiBQM Q7 bi`BM mb2/ BM i?2 7Q`KmHiBQM Bb " # M M ∂u (x; y) 1 ∂u (x; y) j i εM + ≡ uM ij (x; y) = (i|j) 2 ∂yj ∂yi " # I I ∂u (x; y) 1 ∂u (x; y) j i εIij (x; y) = + ≡ uI(i|j) 2 ∂yj ∂yi " # F F (x; y) ∂u (x; y) 1 ∂u j i εFij (x; y) = + ≡ uF(i|j) 2 ∂yj ∂yi

Rjk Rjj Rj9

UjV

U9V

h?2 T2`BQ/B+ +QMiBMmBiv +QM/BiBQM Q7 HQ+H /BbTH+2K2Mi }2H/ (ui ) QM i?2 BMi2`7+2 #2ir22M i?2 Ki`Bt M/ }#2` BM i?2 i?`22 T?b2 +QKTQbBi2 mMBi +2HH +M #2 2tT`2bb2/ rBi? ;HQ#H M/ HQ+H +QQ`/BMi2b UtcvV b

   d1 −d1 Kj x1 , x 2 , x 3 ; , y 2 , y 3 = u i x1 + d1 , x2 , x3 ; , y2 , y3 2 2     −d2 d2 Kj Kj ui x1 , x 2 , x 3 ; y 1 , , y 3 = u i x1 , x2 + d2 , x3 ; y1 , , y3 2 2     d3 −d3 Kj Kj ui x1 , x 2 , x 3 ; y 1 , y 2 , = ui x1 , x 2 , x 3 + d 3 ; y 1 , y 2 , 2 2

uKj i

e

U8V

Rj8 Rje Rjd Rj3

r?2`2- K +M K2M 2Bi?2` M - I Q` F iQ `272` iQ Ki`Bt- BMi2`7+2 Q` }#2``2bT2+iBp2HvX h?2 2tT`2bbBQM 7Q` i?2 2M2`;v +M #2 `2r`Bii2M BM i2`Kb Q7 i?2 +QMbi`BMib mbBM; i?2 G;`M;2 KmHiBTHB2`b b+ 2K 1 θ 1 K K K K K K K uK Cijkl JK = + λK (i|j) u(k|l) + βij u(i|j) θ + Cv i (hui i − vi ) K 2 2 T K 0 ) Z Z Z K K K (u+K3 − u−K3 )ds3 dK (u+K2 − u−K2 )ds2 + γi3 (u+K1 − u−K1 )ds1 + γi2 + γi1 i i i i i i Z (*

s1

s3

s2

UeV

RjN R9y R9R R9k R9j R99 R98 R9e

r?2`2- (J K ) Bb i?2 7Q`K Q7 biiBQM`v 7mM+iBQMH +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi T?b2b Q7 i?2 i?`22 T?b2 +QKTQbBi2 Ki2`BH M/ (λK i ) b r2HH b (γijK ) `2 i?2 G;`M;2 KmHiBTHB2`b- r?B+? `2 mb2/ iQ BMi`Q/m+2 i?2 +QMbi`BMib ;Bp2M BM 1[X URV M/ 1[X U8V- `2bT2+iBp2HvX h?2b2 `2 TTHB+#H2 iQ 2+? T?b2X 1+? +QMbiBim2Mi T?b2 bm`7+2 Bb /2MQi2/ BM i2`Kb Q7 i?2 +QKTQM2Mib si X h?2 2t+i bQHmiBQM (uK i ) i?i KBMBKBx2b i?2 7mM+iBQMH ;Bp2M BM 1[X UeV Bb +QMbB/2`2/ iQ #2 i?2 bmK Q7 pQHmK2 p2`;2 Q7 /BbTH+2K2Mi (vi ) THmb bQK2 /Bz2`2M+2 (wK ) M/ Bb ;Bp2M #v K uK i (x; y) = vi (x) + wi (x; y)

R9d R93 R9N

?2`2- (wiK ) Bb +QMbB/2`2/ iQ #2 bvKTiQiB+HHv bKHH2` i?M i?2 ;HQ#H /Bb@ TH+2K2Mib vi M/ i?2 p2`;2 Q7 hwiK i b?QmH/ #2 2[mH iQ x2`Q BM ++Q`/M+2 rBi? 1[X URVX aQ- Bi +M #2 2tT`2bb2/ 7Q` 2+? +QKTQbBi2 +QMbiBim2Mi b wiK v ηvi

R8y R8R R8k

UdV

U3V

am#biBimiBM; 1[X UdV BMiQ 1[X UeV M/ mbBM; 1[X U9V- r2 +M MQr `2T`2b2Mi i?2 H2/BM; i2`Kb Q7 i?2 7mM+iBQMH +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi T?b2 Q7 +QKTQbBi2 b-

d

J

K

=

Z (* K

R8j R89

+ K 1 K K K 1 K θ2 K K K K Cijkl w(i|j) w(k|l) + βij w(i|j) θ + Cv K + λK i hwi i+ 2 2 T0 Z ∂vi K d1 )ds1 + γi1 (wi+K1 − wi−K1 − ∂x1 s1 Z UNV ∂vi +K2 −K2 K − wi − γi2 (wi d2 )ds2 + ∂x2 s2 ) Z ∂vi +K3 −K3 K γi3 (wi − wi − d3 )ds3 dK ∂x3 s3

k K ?2`2- i?2 b2+QM/ Q`/2` bi`BM i2MbQ` +QKTQM2Mib- w(i|j) M/ w(k|l) `2 /2@ }M2/ b

" # K K (x; y) ∂w (x; y) ∂w 1 j i K w(i|j) ≡ + 2 ∂yj ∂yi  K  1 ∂wk (x; y) ∂wlK (x; y) K + w(k|l) ≡ 2 ∂yl ∂yk R88

R8e R8d R83 R8N Rey ReR

URyV

h?2 T2`BQ/B+ #QmM/`v +QM/BiBQM BM i2`Kb Q7 (wiKj ) Bb 2tT`2bb2/ b    d1 −d1 Kj Kj wi x; , y2 , y3 = wi , y2 , y 3 x; 2 2     d2 −d2 Kj Kj URRV , y3 wi x; y1 , , y3 = wi x; y1 , 2 2     −d3 d3 Kj Kj wi x; y1 , y2 , = wi x; y1 , y2 , 2 2

h?2 mMFMQrM i2`Kb- (wiK )- BM i?2 7mM+iBQMH +M #2 Q#iBM2/ #v KBMBKBx@ BM; 1[X UNVX h?Bb H2/b iQ  b2i Q7 1mH2`@G;`M;2 2[miBQMb M/ bbQ+Bi2/ #QmM/`v +QM/BiBQMb +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi T?b2 Q7 +QKTQbBi2X h?Bb Kv `2bmHi BM ?QKQ;2M2Qmb ;Qp2`MBM; /Bz2`2MiBH 2[miBQMb M/ BM?Q@ KQ;2M2Qmb #QmM/`v +QM/BiBQMbX h?Bb Bb HH2pBi2/ #v +QMbB/2`BM; +?M;2 Q7 p`B#H2 b #2HQr wiK (x; y) = yj

∂vi + χK i (x; y) ∂xj 3

URkV

Rek Rej Re9 Re8

r?2`2- i?2 i2`Kb (χK i ) `2 /2}M2/ b i?2 HQ+H }2H/b Q7 ~m+imiBM; 7mM+@ iBQMbX am#biBimiBM; 1[X URkV BMiQ 1[X UNV M/ mbBM; 1[X URyV- r2 +M `2r`Bi2 i?2 p`BiBQMH bii2K2Mi ;Bp2M BM 1[X UNV BM i2`Kb Q7 i?2 HQ+H }2H/b Q7 ~m+@ imiBM; 7mM+iBQMbX h?Bb Bb 2tT`2bb2/ b J

K

          2K 1 1 K K K K K Kθ = Cijkl ε¯ij + χ(i|j) ε¯kl + χ(k|l) + Cv K + βij ε¯ij + χ(i|j) θK 2 2 T0     Z Z −K1 +K1 −K2 +K2 K K K K +λi hχi i + γi1 χi − χi ds1 + γi2 χi − χi ds2 s1 s2   Z +K3 −K3 K + γi3 χi − χi ds3 s3

Ree

Red Re3 ReN Rdy

r?2`2- ε¯ij Bb i?2 K+`Qb+QTB+ bi`BM- r?B+? Bb ;Bp2M #v " # ∂v (x) ∂v (x) 1 j i + ε¯ij = 2 ∂xj ∂xi

Rdk

Rdj Rd9 Rd8 Rde

UR9V

Ai b?QmH/ #2 MQi2/ i?i i?2 p2`;2 Q7 ~m+imiBM; 7mM+iBQM Bb 2[mH iQ x2`Q B7 i?2 Q`B;BM Q7 i?2 HQ+H +QQ`/BMi2 bvbi2K Bb +?Qb2M iQ #2 i?2 +2Mi2` Q7 i?`22 T?b2 +QKTQbBi2X h?2 T2`BQ/B+ +QM/BiBQMb BM i2`Kb Q7 i?2 ~m+imiBQM 7mM+iBQMb `2 2tT`2bb2/ b 

   d1 −d1 Kj x; , y2 , y3 = χi x; , y 2 , y3 2 2     d2 −d2 Kj Kj χi x; y1 , , y3 = χi , y3 x; y1 , 2 2     d3 −d3 Kj Kj χi x; y1 , y2 , = χi x; y1 , y2 , 2 2

UR8V

JT = J M + J I + J F

UReV

χKj i

RdR

URjV

h?2 iQiH 7mM+iBQMH 7Q` i?2 i?`22 T?b2 +QKTQbBi2 Bb 2tT`2bb2/ b i?2 bmK Q7 i?2 2M2`;v 7mM+iBQMH Q7 i?2 +QMbiBim2Mi T?b2b br?2`2- (JT ) `2T`2b2Mi i?2 iQiH 7mM+iBQMH BM i2`Kb Q7 i?2 Ki`Bt- BMi2`7+2 M/ }#2`X h?2 7Q`KmHiBQM Bb ;2M2`H +M #2 mb2/ iQ ?QKQ;2MBx2 2Bi?2`  k. Q`  j. mMBi +2HHX 6BMBi2 1H2K2Mi U61V K2i?Q/ Bb mb2/ b i?2 MmK2`B+H iQQH iQ Q#iBM i?2 biiBQM`v pHm2 bQHmiBQMb Q7 i?2 7mM+iBQMHX N

Rdd Rd3 RdN R3y R3R R3k R3j R39 R38 R3e R3d R33

kXkX LmK2`B+H BKTH2K2MiiBQM h?2 /2i2`KBMiBQM Q7 i?2 61 bQHmiBQM iQ i?2 biiBQM`v pHm2 T`Q#H2K BM 1[X UReV Bb /B{+mHi /m2 iQ i?2 H`;2 MmK#2` Q7 mMFMQrMbX h?Bb +M #2 pQB/2/ B7 i?2 bQHmiBQM Bb bQm;?i #v ++QmMiBM; 7Q` i?2 T2`BQ/B+Biv Q7 i?2 ~m+@ imiBM; 7mM+iBQM ;Bp2M BM 1[XUR8VX h?2b2 #QmM/`v +QM/BiBQMb Kv MQi z2+i i?2 bQHmiBQM #mi Bi ?2HTb iQ mMB[m2Hv /2i2`KBM2 i?2 ~m+imiBM; 7mM+iBQMbX h?2b2 T2`BQ/B+ +QMbi`BMib `2 BKTH2K2Mi2/ BM i?2 61 K2i?Q/ #v TH+BM; TQbBiBp2 #QmM/`v bm`7+2 UBX2 yi = di /2) b bHp2 iQ i?2 MQ/2b QM i?2 QT@ TQbBi2 M2;iBp2 #QmM/`v bm`7+2 UBX2 yi = −di /2)X am#b2[m2MiHv- HH i?2 BM/2T2M/2Mi +iBp2 /2;`22b Q7 7`22/QK Q7 i?2 bvbi2K `2 bb2K#H2/X h?Bb H2/b iQ  `2/m+iBQM BM i?2 iQiH MmK#2` Q7 mMFMQrMb BM i?2 HBM2` bvbi2K M/ Bb 2tT`2bb2/ b

ΠΩT

1 = 2ΩT

(Z

M



   M M Cijkl ε¯ij + χM ε¯kl + χM (i|j) (k|l) + 2βij M

θ2  dM ε¯ij + θ + T0M Z     I Cijkl ε¯ij + χI(i|j) ε¯kl + χI(k|l) + 2βijI + χM (i|j)





M

CvM

I

I   θ2  ε¯ij + χI(i|j) θI + CvI I dI T0 Z      F + Cijkl ε¯ij + χF(i|j) ε¯kl + χF(k|l) + 2βijF F ) 2F    θ ε¯ij + χF(i|j) θF + CvF F dF T0

R3N RNy

URdV

r?2`2- ΩT Bb i?2 iQiH pQHmK2 Q7 i?2 Ki`Bt- BMi2`7+2 M/ }#2` M/ χk(i|j) +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi Bb ;Bp2M #v

Ry

# " M M ∂χ 1 ∂χ j i + χM (i|j) ≡ 2 ∂yj ∂yi # " I I ∂χ 1 ∂χ j i χI(i|j) ≡ + 2 ∂yj ∂yi " # F F ∂χ 1 ∂χ j i χF(i|j) ≡ + 2 ∂yj ∂yi RNR RNk RNj

RN9

RN8

RNe RNd RN3 RNN kyy kyR kyk

UR3V

q2 `2T`2b2Mi i?2 ~m+imiBQM 7mM+iBQM χk BM i2`Kb Q7 i?2 b?T2 7mM+iBQM [N ] M/ /Bb+`2i2 MQ/H p`B#H2b dk +Q``2bTQM/BM; iQ i?2 Ki`Bt- BMi2`7+2 M/ }#2` bk



URNV

χK = [N K ].{dK }

h?2 p2`;2 bi`BM +M #2 `2T`2b2Mi2/ b

UkyV

 T ¯ = 11 22 33 212 223 213

h?2 bi`BM +Q``2bTQM/BM; iQ i?2 ~m+imiBQM +M #2   ∂χK 1   ∂     ∂y1 0 0   ∂y1   ∂χK   2 ∂   0  0    ∂y2  K    ∂y2 K   ∂χ3   0  ∂   χ1 0 ∂y3   ∂y3 = ∂  χK K K ∂ ∂χ ∂χ 1 2 0   2K    +  ∂y2 ∂y1 ∂y2 ∂y1   χ3         0 ∂y∂ 3 ∂y∂ 2  ∂χK ∂χK 2 3   + ∂y2     ∂y3 ∂   0 ∂y∂ 1   ∂χK ∂χK ∂y3 1 3 + ∂y3 ∂y1

r`Bii2M b-

 

≡ Γkh χK

UkRV



K r?2`2- ΓK Bb i?2 +QHmKM +QMiBMBM; i?`22 h Bb M QT2`iQ` Ki`Bt- M/ χ +QKTQM2Mib Q7 ~m+imiBM; 7mM+iBQMb +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi Q7 i?2 mMBi +2HHX h?2 }MH 7Q`K Q7 i?2 7mM+iBQMH +M #2 r`Bii2M 7i2` bm#biBimiBM; 1[bX URNV@UkyV M/ 1[X UkRV BMiQ 1[X URdV BM i2`Kb Q7 i?2 b?T2 7mM+iBQMb M/ MQ/H /2;`22b Q7 7`22/QK b

RR

ΠΩ K

1 = 2ΩK

K

χ

TK

E K χK + 2χ

TK

2 Kθ K K K K θ + 2¯ εT DεK θK + Dθθ ε¯ + 2¯ εT Dhθ ε¯ + ε¯T Dεε Dhε T0K UkkV

kyj ky9 ky8

K K K K K r?2`2 E K - Dhε - Dεε - Dhθ - Dεθ M/ Dθθ Bb ;Bp2M #v-

E K Dεε =

Z

k

kye kyd ky3 kyN kRy kRR kRk kRj kR9

K

=

Z

K DK dK, Dεθ =

K Z


K T ΓK h N

D

K

K ΓK h N

K β K dK, Dθθ =

Z




dK,

K Dhε

=

K CvK dK, Dhθ =

K

K

kRd

kRN


T

UkjV

Uk9V

JBMBKBxBM; 1[X Uk9V- r2 ;2i i?2 HBM2` bvbi2K Q7 2[miBQMb Uk8V

r?B+? bm;;2bi i?i i?2 χT Bb HBM2`Hv T`QTQ`iBQMH iQ ¯ M/ θT M/ ?2M+2 Bi +M #2 bvK#QHB+HHv r`Bii2M b χT = χ0T ¯ + χθT θT

kR3

K

K ΓK h N

6BMHHv- bm#biBimiBM; 1[X UkjV BMiQ 1[X UkkV- +Q``2bTQM/BM; iQ 2+? +QM@ biBim2Mi Q7 i?2 +QKTQbBi2- i?2 /Bb+`2iBx2/ p2`bBQM Q7 i?2 iQiH 7mM+iBQMH +M #2 2tT`2bb2/ b

ET χT = −DhεT ¯ − DhθT θT kRe

ZK

K T K (ΓK h N ) D dK,

K ?2`2- DK Bb i?2 e×e +QM/2Mb2/ Ki2`BH Ki`Bt Q#iBM2/ 7`QK Cijkl M/ β K Bb i?2 e×R +QHmKMb +QM/2Mb2/ p2+iQ` 7`QK βij K - αK Bb i?2 +Q2{+B2Mib Q7 i?2`KH 2tTMbBQM +M #2 2ti`+i2/ 7`QK i?2 `2HiBQM β K 4−DK αK +Q``2@ bTQM/b iQ +QMbiBim2Mib Q7 Ki`Bt- BMi2`7+2 M/ }#2`X

ΠΩT = ΠΩM + ΠΩI + ΠΩF kR8

Z

UkeV

h?2 2z2+iBp2 biQ`2/ bi`BM 2M2`;v Q7 i?2 i?`22 T?b2 +QKTQbBi2 mMBi +2HH +M MQr #2 b?QrM iQ #2

Rk

β K dK

!

ΠΩT kky kkR

kkk kkj

kk9 kk8

1 T¯ 1 ¯ θT2 T ¯ = ε¯ DT ε¯ + ε¯ βT θT + CvT 2 2 T0T

¯ T Bb i?2 ?QKQ;2MBx2/ biBzM2bb Q7 i?2 i?`22 T?b2 +QKTQbBi2 M/ r?2`2- D Bb /2i2`KBM2/ iQ #2 ¯ T = 1 (χ0 Dhε + Dεε ) D Uk3V T T T ΩT M/ β¯T Bb i?2 ?QKQ;2MBx2/ i?2`KH bi`2bb +Q2{+B2Mib- r?B+? Bb /2i2`KBM2/ iQ #2 " # 1 1 (DT χθ + χT0T DhθT ) + DεθT β¯T = UkNV ΩT 2 hεT T h?2 2z2+iBp2 +Q2{+B2Mib Q7 i?2`KH 2tTMbBQM 7Q` i?2 i?`22 T?b2 +QK@ TQbBi2 +M #2 +QKTmi2/ mbBM; UjyV

¯ −1 βT α ¯ T = −D T kke kkd

UkdV

aBKBH`Hv- 7`QK 1[X UkNV- i?2 ?QKQ;2MBx2/ bT2+B}+ ?2i Q7 i?2 i?`22 T?b2 +QKTQbBi2 Bb /2i2`KBM2/ iQ #2 i 1 h χθT DhθT T0T + DθθT UjRV ΩT >Qr2p2`- BM i?2 T`2b2Mi bim/v- Bi Bb bbmK2/ iQ #2 x2`Q (cv = 0) BM 2+? Q7 i?2 +QMbiBim2Mi T?b2b Q7 i?2 +QKTQbBi2X 7i2` ?pBM; /2i2`KBM2/ mMB[m2Hv i?2 mMFMQrM }2H/ Q7 ~m+imiBQM 7mM+@ iBQMb (χT ) Q7 i?2 i?`22 T?b2 +QKTQbBi2- i?2M r2 +M #H2 iQ `2+Qp2` i?2 HQ+H bi`BM }2H/ BM/BpB/mHHv +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi bc¯vT =

kk3 kkN kjy kjR kjk

K εK = ε¯ + ΓK χK ¯ + χK h N 0  θ θT kjj kj9 kj8 kje kjd

kjN

UjkV

r?2`2- ε¯ Bb i?2 TTHB2/ ;HQ#H bi`BM M/ θT Bb i?2 TTHB2/ ;HQ#H i?2`KH HQ/ QM i?2 i?`22 T?b2 +QKTQbBi2b Q7 mMBi +2HHX 6m`i?2`- i?2 HQ+H i?2`KH@2HbiB+ bi`2bb +QKTQM2Mi +M #2 BM/BpB/mHHv +QKTmi2/ +Q``2bTQM/BM; iQ 2+? +QMbiBim2Mi T?b2 Q7 i?2 +QKTQbBi2 mbBM; j. +QMbiBimiBp2 `2HiBQM bσ K = D K εK + β K θ K

kj3




UjjV

h?2 +QKTmiiBQMH 7`K2rQ`F Bb BKTH2K2Mi2/ BM i?2 mb2` r`Bii2M +QK@ Tmi2` Suh>PL T`Q;`KX AM i?2 M2ti b2+iBQM bBKmHiBQM `2bmHib 7`QK i?2 Rj

h#H2 R, J2+?MB+H T`QT2`iB2b Q7 i?2 MHvx2/ i?`22 T?b2 JJ*bX

Ji2`BHb

uQmM;Ƕb KQ/mHmb U:SV

SQBbbQMǶb `iBQ U@V

α (10−6 /K)

Ji`Bt ULB+F2HV 6B#2`fS`iB+H2 UhmM;bi2MV AMi2`7+2 U*`#QM +QiBM;V

kR9Xy j98Xy j9X3

yXjR yXk3 yXk

RjXjy 8Xy jXjy

k9k

+b2 bim/B2b rBHH #2 T`2b2Mi2/ iQ /2KQMbi`i2 i?2 p2`+Biv Q7 i?2 KQ/2H M/ ;BM BMbB;?i BMiQ i?2 2z2+i Q7 BMi2`7+2 +?`+i2`BbiB+b QM i?2 ?QKQ;2MBx2/ T`QT2`iB2b Q7  i?`22 T?b2 +QKTQbBi2X

k9j

jX _2bmHib M/ /Bb+mbbBQM

k9y k9R

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j9

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j8

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