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Evaluation of infrared thermography methods for analysing the damage behaviour of adhesively bonded repair solutions
Journal Pre-proofs Evaluation of infrared thermography methods for analysing the damage behaviour of adhesively bonded repair solutions U. Martens, K.-U. Schröder PII: DOI: Reference:
S0263-8223(19)33243-X https://doi.org/10.1016/j.compstruct.2020.111991 COST 111991
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
27 August 2019 14 January 2020 23 January 2020
Please cite this article as: Martens, U., Schröder, K.-U., Evaluation of infrared thermography methods for analysing the damage behaviour of adhesively bonded repair solutions, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111991
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© 2020 Published by Elsevier Ltd.
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T2` im`2 BM+`2 b2 Bb BM2z2+iBp2- #2+ mb2 i?2 +Q2{+B2Mi /Q2b MQi BM+Hm/2 i?2 ` i2 /2T2M/2M+vX h?2`27Q`2- 7`QK i?2 KQMBiQ`BM; TQBMi Q7 pB2r- i?2 T i+? i?B+FM2bb b?QmH/ #2 T`272` #Hv bK HH iQ K F2 bm`2 i? i 2MQm;? 2M2`;v /BbbBT i2b iQ i?2 bm`7 +2X hQ ; BM 7Q+mb2/ BK ;2 BM QTiB+ H Gh b2p2` H TB+im`2b `2 i F2M M/ bmT2`@
R9y
BKTQb2/ #v M TT`QT`B i2 bQ7ir `2X lT iQ +2`i BM TQBMi- i?2 KQ`2 TB+im`2b `2 bmT2`BKTQb2/- i?2 #2ii2` i?2 bmTT`2bbBQM Q7 MQBb2 2z2+ibX h?2`27Q`2- Gh Bb mbm HHv TTHB2/ iQ MQM@KQpBM; Q#D2+ibX 6Q` KQpBM; Q#D2+i- iBK2@/2T2M/2Mi +? M;2b ? p2 iQ #2 M2;H2+i #H2 7Q` i?2 MmK#2` Q7 TB+im`2b M/ i?2 +Q``2bTQM/@ BM; iBK2X qBi? i?2 BK Q7 KQMBiQ`BM; i?2 / K ;2 T`Q+2bb- b2p2` H TB+im`2b `2 d
R98
M22/2/X h?2`27Q`2- 7i2` 2 +? K2 bm`BM; KQ/mH2 i?2 bvbi2K ? b iQ #2 `2bi `i2/X aT2+BK2M ;2QK2i`v M/ i2bi +QM};m` iBQMb 6Q` i?2 +?QB+2 Q7 i?2 bT2+BK2M ;2QK2i`v- Bi ? b iQ #2 i F2M BMiQ ++QmMi i? i bQK2 TQbbB#H2 / K ;2b + MMQi #2 b22M 7`QK i?2 QmibB/2X 6Q` BMbi M+2- i?2 K2i HHB+ +` +F ;`Qri? mM/2` i?2 T i+?- BMi2`H KBM ` /2H KBM iBQM Q7 i?2 T i+?
R8y
H v2`b M/ /2H KBM iBQM Q7 i?2 T i+? `2 MQi pBbB#H2 b i?2v `2 bm#bm`7 +2 /2@ 72+ibX HH i?2b2 / K ;2 K2+? MBbKb + M BM~m2M+2 QM2 MQi?2`- M/ Bi KB;?i #2 /B{+mHi iQ /BbiBM;mBb? #2ir22M i?2KX h?2 /Qm#H2 H T DQBMi ;2QK2i`v mb2/ 7Q` i?2 T`2HBKBM `v /2bB;M Q7 i?2 /?2bBp2 xQM2 Bb ;QQ/ +?QB+2 iQ H2 p2 K2i HHB+ +` +F ;`Qri? bB/2X hQ ;m ` Mi22 i? i /?2bBp2 / K ;2 Bb i?2 QMHv K2+? MBbK
R88
i? i + M #2 Q#b2`p2/ QMHv QM2 T i+? H v2` Bb TTHB2/ i 2 +? bB/2X h?Bb ? b i?2 //BiBQM H /p Mi ;2 i? i i?2 K2 bm`2K2Mi /2Ti? Bb p2`v bK HH- #2M2}+B H 7Q` #Qi? T bbBp2 A`h M/ GhX AMp2biB; iBQMb `2 K /2 7Q` [m bB@bi iB+ HQ /BM; M/ 7Q` 7 iB;m2 HQ /BM;X hrQ /Bz2`2Mi T i+? K i2`B Hb `2 i2bi2/- mMB/B`2+iBQM H Ul.V T`2T`2; Q7 + `#QM }#`2 `2BM7Q`+2/ TH biB+b U*6_SV- U2TQtv # b2/ *6_S
Rey
T`2T`2; 7`QK >2t+2H- ivT2 >1sSGu JRyXRfj3fl.jyyfJ9yCfey M/ i?B+F@ M2bb Q7 yX9 KKV- M/ yfNy◦ KmHiB/B`2+iBQM H UJ.V rQp2M + `#QM 7 #`B+ U2TQtv # b2/ *6_S T`2T`2; 7`QK a:G h2+?MQHQ;B2b :K#> Q7 ivT2 * q9Ry@hqkfk@ 1jkjf9kWfeF M/ i?B+FM2bb Q7 yXe KKV `2 mb2/X h?2 K2i HHB+ T `ib `2 #mBHi 7`QK akj8C_Y_X
Re8
:Bp2M i?2 #QmM/ `B2b Q7 i?2 K2 bm`2K2Mi i2+?MB[m2b- i?2 bi M/ `/ p Hm2b 7Q` #Qi?- bT2+BK2M /BK2MbBQMb M/ i2bi p2HQ+Biv 7Q` [m bB@bi iB+ HQ /BM;- `2 MQi bmBi #H2X 6Q` Gh iQ #2 TTHB+ #H2- Bi ? b iQ #2 2Mbm`2/ i? i / K ;2 T`Q;`2bb Bb M2;H2+i #H2 /m`BM; i?2 +`2 iBQM Q7 QM2 TB+im`2 rBi? b2p2` H HQ+F@BM 2t+Bi iBQMbX >2M+2- i?2 i2bi p2HQ+Biv ? b iQ #2 / Ti2/ ++Q`/BM; iQ i?2 HQ+F@BM T ` K2i2`b
Rdy
M22/2/X S`2HBKBM `v i2bib TTHB2/ iQ bT2+BK2M MQi mM/2` HQ / H2 / iQ HQ+F@BM 7`2[m2M+v Q7 R >x 7Q` i?2 l. }#`2b M/ yXR >x 7Q` i?2 J. T`2T`2;X 6m`i?2`i?2 i2bib b?Qr2/ i? i MmK#2` Q7 R8 HQ+F@BM 2t+Bi iBQM T2`BQ/b + M #2 b22M b ;QQ/ +QKT`QKBb2 #2ir22M MQBb2 bmTT`2bbBQM M/ BK ;BM; iBK2X hQ;2i?2` 3
rBi? i?2 `2[mB`2/ HQ+F@BM 7`2[m2M+B2b i?2 i2bi p2HQ+BiB2b v `2 i?2M +?Qb2M iQ #2 Rd8
vU D = 0.5 KKfKBM 7Q` i?2 l. T i+? M/ vM D = 0.05 KKfKBM 7Q` i?2 J. T i+?X h?2 H `;2 bT2+BK2M rB/i? `2bmHib 7`QK i?2 /2K M/ 7Q` b2p2` H BK ;2b i? i + M /BbTH v i?2 / K ;2 T`Q;`2bb Qp2` iBK2X hQ i F2 KQ`2 i? M QM2 TB+im`2- i?2 /2b+`B#2/ HQ+F@BM T`Q+2bb ? b iQ #2 `2bi `i2/ 7i2` 2 +? TB+im`2 r?B+? K F2b Bi
R3y
iBK2@+QMbmKBM; T`Q+2/m`2X h?2 H `;2bi TQbbB#H2 bT2+BK2M rB/i? `2bmHib 7`QK i?2 +H KTBM; D r /BK2MbBQMb M/ Bb +?Qb2M iQ #2 3y KK ++Q`/BM;HvX aT2+BK2M /BK2MbBQMb `2 bF2i+?2/ BM 6B;X 9- p Hm2b ;Bp2M BM KKX 6Q` +QKT `BbQM Tm`TQb2b i?2 bT2+BK2M ;2QK2i`v +?Qb2M Bb mb2/ 7Q` HH i2bib T2`7Q`K2/X AM Q`/2` iQ `2@ /m+2 i2biBM; iBK2- i2bi p2HQ+Biv Q7 v = 1 KKfKBM Bb +?Qb2M 7Q` i?2 T bbBp2 A`h
R38
BMp2biB; iBQMbX 6 iB;m2 / i `2bmHi 7`QK i?2 TTHB+ iBQM M/ i?mb /Q MQi M22/ Mv 7m`i?2` 2tTH M iBQM- b22 h #H2 RX J2+? MB+ H 2biBK iBQMb }`bi 2biBK iBQM K /2 QM M M HviB+ H K2+? MB+ H KQ/2H ;Bp2b BM7Q`K iBQM QM r?2`2 T i+? /2i +?K2Mi Bb 2tT2+i2/X h?2 KQ/2H +?Qb2M Bb b2KB@+QMiBMmQmb
RNy
7Q`KmH iBQM 7Q` /Qm#H2 H T DQBMi `2bmHiBM; 7`QK b?2 ` r2# i?2Q`v ++Q`/BM; iQ (3) M/ (k8)Xh?Bb TT`Q +? K BMHv `2bmHib BM i?2 r2HH@FMQrM oQHF2`b2M 2[m @ iBQMb (ke)X hQ /2i2`KBM2 i?2 MQ`K H b r2HH b i?2 b?2 ` bi`2bb /Bbi`B#miBQM rBi?BM i?2 Qp2`H T- [m `i2` KQ/2H Bb +?Qb2M ++Q`/BM; iQ (3) b /2TB+i2/ BM 6B;X 8X
RN8
amKK `Bx2/- # bBb Q7 i?2 KQ/2H Bb i? i i?2 /?2`2M/b QMHv i` MbKBi MQ`K H 7Q`+2b r?BH2 i?2 /?2bBp2 QMHv i` MbKBib b?2 ` HQ /bX Pmi@Q7@TH M2 #2M/BM; 27@ 72+ib `2 M2;H2+i2/X h?2 +?Qb2M b2KB@+QMiBMmQmb KQ/2H MQr mb2b +QMiBMmQmb bi`2bb 7Q`KmH iBQM BM HQ / /B`2+iBQM UxVX 6Q` KQ`2 /2i BH2/ 2tTH M iBQM b22 (3)X h?2 `2bmHiBM; b?2 ` bi`2bb /Bbi`B#miBQM Bb ;Bp2M iQ #2 τ (x) ρ cosh(ρx/l) (1 − ψ) sinh(ρx/l) − = τm 2 sinh(ρ/2) (1 + ψ) cosh(ρ/2)
N
URV
kyy
rBi? Bib K tBKmK p Hm2 Q7 ρ ρ (1 − ψ) τmax ρ tanh , coth + = τm 2 2 (1 + ψ) 2
UkV
M/ ρ=
p
((1 + ψ)Gta2 /EA1 d).
UjV
q?2`2 τm /2MQi2b i?2 K2 M b?2 ` bi`2bb M/ ρ i?2 /?2bBQM BM/2tX h?2 K DQ` T ` K2i2` /2i2`KBMBM; i?2 b?2 ` bi`2bb /Bbi`B#miBQM Bb i?2 biBzM2bb ` iBQ Q7 i?2 /?2`2M/b ψX Ai Bb /2}M2/ b i?2 ` iBQ #2ir22M i?2 BMM2` M/ Qmi2` /?2`2M/ ky8
ψ=
EA1 EA2 X
AM i?2 ;Bp2M + b2- Bi Bb i?2 ` iBQ Q7 K2i H Qp2` T i+? biBzM2bb- +QMbB/2`@
BM; i? i 7Q` i?2 [m `i2` KQ/2H QMHv ? H7 i?2 K2i H i?B+FM2bb Bb mb2/ iQ /2i2`KBM2 EA1 X 6Q` biBzM2bb ` iBQ ψ 6= 1 i?2 b?2 ` bi`2bb /Bbi`B#miBQM HQM; i?2 Qp2`H T #2@ +QK2b bvKK2i`B+X h?2 ?B;?2bi b?2 ` bi`2bb i?2M TT2 `b rBi?BM i?2 r2 F2` kRy
/?2`2M/ i i?2 7`22 2M/ Q7 i?2 biBz2` QM2- b22 6B;X eX TTHB2/ iQ i?2 bT2+BK2M +QM};m` iBQM /2b+`B#2/ #Qp2- i?2 biBzM2bb ` iBQ ψ `2bmHib iQ #2 > 1X qBi? i?2 K2i HHB+ b?22i b i?2 biBz2` /?2`2M/- i?2 Q++m``2M+2 Q7 T i+? /2i +?K2Mi Bb 2tT2+i2/ iQ #2 i i?2 bT2+BK2M +2Mi`2 Ux = 0VX h?mb- HQ+ H i2KT2` im`2 /Bz2`@ 2M+2b `QmM/ i?2 +2Mi`2 HBM2 T`2bmK #Hv Q++m``BM; i i?2 7`22 2/;2b- r?2`2 i?2
kR8
?B;?2bi bi`2bb2b `2 mbm HHv 2tT2+i #H2- BM/B+ i2 /?2bBp2 / K ;2 QMb2iX h2bi /2b+`BTiBQM h?2 i2bi K +?BM2 mb2/ Bb b2`pQ@?v/` mHB+ i2bi K +?BM2 Q7 ivT2 a+?2M+F SPw Rey rBi? bi iB+ MQKBM H HQ / Q7 Yf@Rey FLX h?2 BM7` `2/ + K2` Bb ivT2 AK ;2A_3j3y a UAM7` h2+ :K#>- .`2b/2MV BM +QK#BM iBQM rBi? i2H2 xQQK
kky
H2Mb2 Uf = 50 KKVX h2+?MB+ H / i Bb HBbi2/ BM h #H2 kX h?2 i?2`K H biBKmH iBQM /2pB+2 Bb ? HQ;2MH KT Q7 ivT2 >k8b U>2/H2` avbi2KHB+?i :K#>- _mMF2HV rBi? TQr2` Q7 kX8 Fq mbBM; irQ B/2MiB+ H #mH#b Q7 RXk8 Fq #2 K TQr2` 2 +?X
Ry
aT2+BK2M T`2T ` iBQM aBM+2 i?2 BMp2biB; iBQM 7Q+mbb2b QM i?2 KQMBiQ`BM; K2i?Q/- MQi i?2 bT2+BK2M kk8
T`QT2`iB2b- bT2+BK2M T`2T ` iBQM Bb F2Ti bBKTH2X h?2 K2i HHB+ T `i Bb +mi BMiQ i?2 `B;?i /BK2MbBQMb M/ 7i2` b M/ #H biBM; M/ +H2 MBM; i?2 *6_S T i+?2b `2 ii +?2/ rB?iQmi Mv 2ti` /?2bBp2X h?2 2TQtv `2bBM Bb ? `/2M2/ mM/2` T`2bbm`2 BM +HBK i2 +? K#2` 7Q` j ?Qm`b mM/2` Rky◦ *X HH bT2+BK2Mb `2 +Q i2/ rBi? K ii #H +F T BMi 7Q` ?B;? 2KBbbBpBiv U2KBbbBpBiv Q7 i?2 Q`B;BM H bT2+BK2M
kjy
Bb #Qmi yXkNVX jX _2bmHib h?2 K2+? MB+ H #2? pBQm` Q7 i?2 bT2+BK2Mb b r2HH b i?2 Q#b2`p #H2 / K ;2 T`QT ; iBQM r2`2 BM ;QQ/ ;`22K2Mi 7Q` 2 +? i2bi b2`B2bX h?2`27Q`2- QM2 bT2+@ BK2M Bb 2t2KTH `BHv +?Qb2M iQ /2KQMbi` i2 i?2 i2bi `2bmHib Q7 2 +? i2bi b2`B2bX
kj8
6B;X d /2TB+ib i?2 TB+im`2 b2+iBQM +?Qb2M iQ 2p Hm i2 i?2 / K ;2 T`Q;`2bbX Zm bB@bi iB+ HQ /BM; 6B`bi i?2 i2bi `2bmHib 7Q` i?2 bT2+BK2Mb mM/2` [m bB@bi iB+ HQ /BM; `2 T`2@ b2Mi2/ 7Q` #Qi? KQMBiQ`BM; K2i?Q/b T bbBp2 A`h M/ GhX 1 +? i2bi r b + ``B2/ Qmi rBi? #Qi?- T i+?2b rBi? l. }#`2b M/ T i+?2b rBi? J. }#`2bX hQ /2KQM@
k9y
bi` i2 i?2 / K ;2 T`QT ; iBQM mM/2` [m bB@bi iB+ HQ /BM; 7Q` 2 +? bT2+BK2M MmK#2` Q7 3 TB+im`2b Bb +?Qb2MX //BiBQM HHv- i?2 iBK2 i?2 TB+im`2b `2 i F2M `2 K `F2/ BM i?2 +Q``2bTQM/BM; HQ /f/BbTH +2K2Mi +m`p2b- b22 6B;X 3 M/ RyX S bbBp2 h?2`KQ;` T?v h?2 / K ;2 T`QT ; iBQM T`Q+2bb 7Q` l. T i+? KQMBiQ`2/ rBi? T bbBp2 A`h
k98
Bb b?QrM BM 6B;X 3X PM i?2 i?2`K H BK ;2b i?2 +2Mi` H ?Q`BxQMi H HBM2 `2T`2b2Mib i?2 #mii DQBMi #2ir22M i?2 irQ K2i H b?22ibX h?2 HQr2` TBbiQM i` Mb72`b i?2 HQ /X .m2 iQ i?2 KQp2K2Mi Bi ?2 ib mT M/ i?mb HbQ i` Mb72`b i2KT2` im`2 BMiQ i?2 bT2+BK2MX h?2`27Q`2- i?2 HQr2` b?22i Bb r `K2` i? M i?2 mTT2` QM2X
RR
AM i?2 HQ /f/BbTH +2K2Mi +m`p2 i?2 QMb2i Q7 / K ;2 T`Q+2bb2b mbm HHv ;Q2b HQM; k8y
rBi? /2+`2 b2 BM i?2 TTHB2/ HQ /X >2`2- i?2 HQ / BM+`2 b2b +QMiBMmQmbHv iQ K tBKmK p Hm2 M/ /2+`2 b2b bi2TrBb2 mMiBH }M H 7 BHm`2 Q++m`bX h?2 bi2TrBb2 /2+`2 b2 BM HQ / + M #2 2tTH BM2/ #v bT HHBM; Q7 bBM;H2 }#`2b Q` }#`2 #mM/H2br?B+? + M #2 KQMBiQ`2/ /m`BM; i2biBM;X h?2 }`bi +QMbB/2` #H2 HQ+ H i2KT2` im`2 /Bz2`2M+2b Q++m` Dmbi #2?BM/ i?2 ?B;?@
k88
2bi HQ /- b22 BK ;2 MQX RX PM i?2 H27i@? M/ bB/2 Q7 i?2 BK ;2- p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2` im`2 + M #2 b22MX 6m`i?2`- i i?2 +2Mi`2 HBM2ě2bT2+B HHv i #Qi? 7`22 2/;2bě `2 b Q7 /Bz2`2Mi i2KT2` im`2 + M #2 K /2 QmiX GQQFBM; i i?2 bT2+BK2MǶb 7`22 `B;?i 2/;2 #Qp2 i?2 +2Mi`2 HBM2 7Q` BMbi M+2 BK ;2 MQX R /BbTH vb bK HH `2 rBi? BM+`2 b2/ i2KT2` im`2X :QBM; 7`QK BK ;2 MQX R iQ
key
BK ;2 MQX j i?2 BMi2MbBiv Q7 i?2 p2`iB+ H HBM2b /2+`2 b2b r?BH2 i?2 BMi2MbBiv M/ bBx2 Q7 i?2 `2 BM+`2 b2X i TQBMi 9 MQi?2`- #B;;2` bi2TrBb2 HQ / /2+`2 b2 + M #2 b22M BM i?2 HQ /f/BbTH +2K2Mi +m`p2X h?2 +Q``2bTQM/BM; i?2`K H BK ;2 ; BM b?Qrb p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2` im`2 BM i?2 H27i `2 - i?Bb iBK2 i 2p2M ?B;?2` i2KT2` im`2b M/ Qp2` rB/2` `2 X >2`2 ; BM- i?2 BMi2MbBiv Q7 i?2b2
ke8
HBM2b /2+`2 b2b Qp2` i?2 M2ti irQ BK ;2b r?BH2 i?2 `2 BM i?2 `B;?i T `i ?2 ib mT 7m`i?2` M/ 2tT M/b +`Qbb KQ`2 i? M ? H7 Q7 i?2 bT2+BK2M rB/i?X h?2 H bi irQ BK ;2b b?Qr irQ TQBMib BKK2/B i2Hv #27Q`2 i?2 }M H 7` +im`2 Q7 i?2 bT2+BK2MX o2`iB+ H HBM2b Q7 ?B;? i2KT2` im`2 /QKBM i2 i?2 irQ BK ;2bX AM TB+im`2 MQX d i?2b2 HBM2b `2 HQ+ HHv HBKBi2/ iQ i?2 b? T2 Q7 i?2 `2 i? i 2tT M/2/ 7`QK i?2
kdy
`B;?i 2/;2 iQ MQr HKQbi i?2 2MiB`2 bT2+BK2M rB/i?X AM i?2 }M H TB+im`2- i H2 bi BM i?2 mTT2` T `i- i?2 p2`iB+ H HBM2b 2t+22/ i?Bb `2 M/ `mM 7`QK i?2 iQT iQ i?2 +2Mi`2 HBM2 Q` 2p2M iQ i?2 #QiiQKX hQ;2i?2` rBi? i?2 T`2HBKBM `v K2+? MB+ H 2biBK iBQM M/ i?2 HQ /f/BbTH +2K2Mi +m`p2 i?2 HQ+ H `2 Q7 BM+`2 b2/ i2KT2` im`2 i? i 2tT M/b Qp2` i2biBM; iBK2 Bb
kd8
2biBK i2/ iQ #2 i?2 HQ+ H / K ;2 BM i?2 /?2bBp2 xQM2X 6Q` i?2 ;Bp2M biBzM2bb ` iBQ- i?2 / K ;2 QMb2i Bb 2tT2+i2/ i 7`22 2/;2 `QmM/ i?2 +2Mi`2 HBM2- r?B+? +Q``2bTQM/b iQ i?2 Q#b2`p iBQMb QM i?2 i?2`K H BK ;2bX +QMbi MiHv BM+`2 bBM; / K ;2 H2 /b iQ T2`K M2Mi ?2 i /BbbBT iBQM r?B+? ;Q2b HQM; rBi? i?2 bHB;?iHv BM+`2 bBM; i2KT2` im`2 BM i?2 /2i2+i2/ `2 Qp2` iBK2X h?2 ` TB/ HQ / /2+`2 b2 Rk
k3y
i TQBMib R- 9 M/ 3 Bb bbQ+B i2/ rBi? }#`2 bT HHBM;- r?B+? Bb BM #`mTi / K ;2 K2+? MBbKX h?Bb BM im`M +Q``2bTQM/b iQ i?2 p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2`@ im`2 i? i TQT mT M/ +QQH /QrM ; BMX hQ 7m`i?2` bmTTQ`i i?Bb i?2Q`v /B`2+i +QKT `BbQM #2ir22M i?2`K H BK ;2 MQX d M/ T?QiQ;` T? Q7 i?2 / K ;2/ bT2+BK2M Bb b?QrM BM 6B;X NX PM i?2 H27i@? M/ bB/2 }#`2 #mM/H2 + M #2 b22M
k38
i? i T22H2/ Qz 7`QK i?2 mTT2` T `i Q7 i?2 bT2+BK2MX h?2 TQbBiBQM }ib iQ i?2 Q#b2`p2/ p2`iB+ H HBM2bX 6m`i?2`- Bi Bb TT `2Mi i? i i?2 }#`2b BM i?2 `B;?i `2 /2i +?2/ 7`QK i?2 mTT2` K2i HHB+ T `iX G27iQp2`b Q7 i?2 /?2bBp2 QM i?2 K2i HHB+ T `i +Q``2bTQM/ iQ i?2 b? T2 Q7 i?2 `2 i? i + M #2 b22M QM i?2 i?2`K H BK ;2X h?mb- }#`2 T22H Qz M/ / K ;2 BM i?2 /?2bBp2 H v2` b22K iQ #2 Q#b2`p #H2
kNy
mbBM; T bbBp2 A`hX GQ+F@BM h?2`KQ;` T?v h?2 / K ;2 T`Q+2bb KQMBiQ`2/ rBi? Gh Bb b?QrM BM 6B;X RyX .Bz2`2M+2b BM i?2 HQ /f/BbTH +2K2Mi +m`p2b +QKT `2/ iQ 6B;X 3 `2bmHi 7`QK i?2 /Bz2`2Mi K i2`B H T`QT2`iB2b Q7 i?2 l. M/ J. T i+?2bX aBM+2 i?2 7Q+mb Q7 i?Bb BM@
kN8
p2biB; iBQM Bb b2i iQ i?2 KQMBiQ`BM; i2+?MB[m2 M/ MQi iQ i?2 [m HBiv Q7 i?2 K2+? MB+ H T`QT2`iB2b- HbQ i?2 +QM/2Mb2/ bT2+BK2M T`2T ` iBQM T`Q+2bb BM~m@ 2M+2b i?2 HQ /f/BbTH +2K2Mi #2? pBQm`X AM +QMi` bi iQ T bbBp2 A`h- i?2 /BbTH v2/ BK ;2b `2 T? b2 BK ;2bX S? b2 BK ;2b `2 dz`2H i2/ iQ i?2 (i2KT2` im`2) T`QT@ ; iBQM iBK2Ǵ (e) M/ /Q MQi b?Qr i2KT2` im`2 /Bbi`B#miBQMX h?2Q`2iB+ HHv
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i?2v `2 BM/2T2M/2Mi Q7 bm`7 +2 p `B iBQMb M/ + M i?mb 2HBKBM i2 2``Q`b `2bmHi@ BM; 7`QK MQM@mMB7Q`K ?2 iBM; Q` p `B iBQMb BM bm`7 +2 2KBbbBpBiv (e- kd)X ; BMi?2 +2Mi` H ?Q`BxQMi H HBM2 + M #2 bbB;M2/ iQ i?2 #mii DQBMi #2ir22M i?2 K2i H b?22ibX AK ;2 MQX R b?Qrb i?2 bi i2 `B;?i 7i2` 2t+22/BM; i?2 ?B;?2bi TQBMi BM i?2 HQ /f/BbTH +2K2Mi +m`p2X 6`QK TQBMi MQX k iQ TQBMi MQX 8- i?2 bHQT2 Q7 i?2
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HQ /f/BbTH +2K2Mi +m`p2 /2+`2 b2b +QMiBMmQmbHvX PM BK ;2 MQX k i #Qi? 7`22 2/;2b / `F bTQib + M #2 K /2 Qmi `QmM/ i?2 ?Q`BxQMi H HBM2 Ux = 0VX h?2 `2 Q7 i?2b2 bTQib 2tT M/b rBi? BM+`2 bBM; HQ / iQr `/b i?2 bT2+BK2M +2Mi`2X SQBMi MQX e HB2b i TQbBiBQM rBi? bi2TrBb2 HQ / /2+`2 b2X //BiBQM HHv iQ i?2 BM+`2 bBM; `2 Q7 i?2 /2b+`B#2/ bTQib QM i?2 `B;?i M/ H27i bB/2- MQr / `F2` Rj
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`2 `QmM/ i?2 +2Mi`2 HBM2 BM i?2 KB//H2 T `i + M #2 K /2 Qmi +QMM2+iBM; i?2 irQ bTQibX _B;?i #2?BM/ i?Bb TQBMi MQi?2` bm//2M HQ / /2+`2 b2 Q++m`bX AK ;2 MQX d b?Qrb i? i i?2 #B;;2` bTQi QM i?2 H27i@? M/ bB/2 MQr +Qp2`b i?2 2MiB`2 T i+? ?2B;?i M/- 2p2M KQ`2 BKTQ`i Mi- i?2 bK HH bTQi QM i?2 `B;?i@? M/ bB/2 bm//2MHv 2tT M/2/ M/ MQr +Qp2`b i?2 2MiB`2 ?2B;?i #2ir22M i?2 +2Mi`2 HBM2
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M/ i?2 mTT2` T i+? 2/;2X AK ;2 MQX 3 b?Qrb i?2 }M H 7 BHm`2 Q7 i?2 bT2+BK2MX "2ir22M TQBMib d M/ 3 H bi bm//2M HQ / /`QT ? TT2Mb +Q``2bTQM/BM; iQ i?2 }M H bT2+BK2M 7 BHm`2X PM i?2 T? b2 BK ;2 i?2 irQ `2 b `2 MQr 2MiB`2Hv +QM@ M2+i2/X b #27Q`2- i?2 K2+? MB+ H BM/B+ iQ` Q7 / K ;2 QMb2i i x = 0 TTHB2bX 6m`i?2`- i?2 /B`2+i +QKT `BbQM Q7 i?2 i?2`K H BK ;2 rBi? T?QiQ;` T? Q7 i?2
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/ K ;2/ bT2+BK2M b?Qrb i? i i?2 b? T2 Q7 i?2 /2i +?2/ `2 b +Q``2bTQM/ iQ i?2 / `F `2 QM i?2 T? b2 BK ;2- b22 6B;X RRX HbQ- i?2 HQ /f/BbTH +2K2Mi +m`p2 mM/2`HBM2b i? i i?2 / `F bTQib + M #2 +Q``2H i2/ iQ i?2 /?2bBp2 / K ;2 T`Q+2bbX *QMiQm` HBM2b Q7 i?2 / `F2` `2 b `2 bHB;?iHv #Hm``2/X _2K2K#2`BM; i?2 7mM+iBQMBM; Q7 Gh i?Bb + M #2 `2H i2/ iQ i?2 7 +i i? i 2Bi?2` R8 bmT2`BKTQb2/
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TB+im`2b `2 MQi 2MQm;? iQ 2MiB`2Hv }Hi2` MQBb2 2z2+ib Q` i? i / K ;2 T`Q;`2bb Bb 7 bi2` i? M 2tT2+i2/ M/ i?2 / K ;2 2tT M/b r?BH2 QM2 BK ;2 Bb ;2M2` i2/X 6 iB;m2 HQ /BM; hQ 2p Hm i2 i?2 TQi2MiB H Q7 i?2`KQ;` T?v QM 7 iB;m2- +v+HB+ i2bib `2 T2`@ 7Q`K2/X lM/2` +v+HB+ HQ /BM; i?2 bT2+BK2M /BbTH +2K2Mi Bb +QKT ` #Hv bK HHX
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6m`i?2`- i?2 2MiB`2 bT2+BK2M ?2 ib mT Qp2` iBK2 /m2 iQ i?2 +QMbi Mi +? M;2 #2ir22M HQ /BM; M/ mMHQ /BM;X "Qi? 2z2+ib H2 / iQ Km+? bK HH2` i2KT2` im`2 /Bz2`2M+2b +QKT `2/ iQ i?2 [m bB@bi iB+ HQ /BM; K2 MBM; i? i T i+? /2i +?K2Mi Bb TQbbB#Hv KQ`2 /B{+mHi iQ /2i2+iX .m2 iQ i?2 BM+`2 b2 Q7 # b2 i2KT2` im`2 Qp2` iBK2- HbQ i?2 i2KT2` im`2 b+ H2 /Bz2`b rBi? BM+`2 bBM; MmK#2` Q7 +v+H2b LX
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h?2`27Q`2- i2KT2` im`2 b+ H2b /Bz2` BM Q`/2` iQ K F2 i2KT2` im`2 /Bz2`2M+2b pBbB#H2X h?mb- i?2`K H BK ;2b + MMQi /B`2+iHv #2 +QKT `2/ [m MiBi iBp2HvX
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S bbBp2 h?2`KQ;` T?v h?2 / K ;2 T`Q+2bb mM/2` +v+HB+ HQ /BM; KQMBiQ`2/ rBi? T bbBp2 A`h Bb b?QrM BM 6B;X RkX 1 +? TB+im`2 Bb bbB;M2/ iQ i?2 HQ / +v+H2 MmK#2` N Bi Bb j9y
i F2MX AM +QMi` bi iQ i?2 [m bB@bi iB+ HQ /BM; i?2 ;Bp2M i2KT2` im`2 /Bbi`B#m@ iBQM K F2b i?2 / K ;2 #2 #2bi pBbB#H2 #2HQr i?2 +2Mi`2 HBM2X b 2tT2+i2/i2KT2` im`2 /Bz2`2M+2b `2 bK HH2` i? M mM/2` [m bB@bi iB+ HQ /BM;X "mi i?2 b K2 T`Q+2bb + M #2 b22M ?2`2ě i i?2 #mii DQBMi HBM2 HQ+ H `2 b r?2`2 i?2 rQ@ p2M }#`2 bi`m+im`2 Bb +H2 `Hv pBbB#H2 + M #2 bTQii2/ r?BH2 i?2 `2bi Q7 i?2 T i+?
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TT2 `b ?QKQ;2M2QmbX h?2b2 `2 b `2 2tT M/BM; 7`QK i?2 bT2+BK2M 2/;2b iQ i?2 +2Mi`2 rBi? BM+`2 bBM; iBK2 Q` MmK#2` Q7 +v+H2bX 6B;X Rj b?Qrb i?2 /B`2+i +QKT `BbQM #2ir22M i?2 i?2`K H BK ;2 M/ i?2 7 BH2/ bT2+BK2MX b #27Q`2i?2 /2i +?2/ `2 b +Q``2bTQM/ iQ i?2 bbmKTiBQMb K /2 # b2/ QM i?2 i?2`K H BK ;2bX
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GQ+F@BM h?2`KQ;` T?v AM 6B;X R9 T? b2 BK ;2b Q7 i?2 Gh M HvbBb b?Qr i?2 +v+HB+ / K ;2 T`QT@ ; iBQM- 2 +? TB+im`2 `2H i2/ iQ i?2 +Q``2bTQM/BM; HQ / +v+H2X AM +QMi` bi iQ i?2 `2bmHib `2+2Bp2/ rBi? T bbBp2 A`h ?2`2 i?2 T i+? bi`m+im`2 Bb pBbB#H2 HH Qp2` i?2 T i+? M/ i?mb /Q2b MQi +Q``2bTQM/ iQ HQ+ HHv /2i +?2/ `2 bX GQ+ H /2@
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i +?K2Mib `2 /BbiBM;mBb? #H2 b / `F bTQib QM i?2 i?2`K H BK ;2bX >2`2 ; BM+QMiQm`b TT2 ` #Hm``2/ TQbbB#Hv /m2 iQ i?2 bT2+BK2M KQp2K2Mi /m`BM; BK ;2 +[mBbBiBQMX b #27Q`2- bi `iBM; 7`QK i?2 +2Mi`2 HBM2 / `F HBM2 + M #2 bTQii2/ r?B+? 2tT M/b rBi? BM+`2 bBM; MmK#2` Q7 +v+H2bX AM i?2 T2MmHiBK i2 BK ;2 i?Bb `2 +Qp2`b TT`QtBK i2Hv ? H7 i?2 T i+? #2HQr i?2 +2Mi`2 HBM2 M/ BM i?2 H bi
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BK ;2- r?B+? Bb i?2 HQ / +v+H2 #27Q`2 7 BHm`2- HKQbi i?2 2MiB`2 HQr2` ? H7 Q7 i?2 T i+?X h?2 7 BH2/ bT2+BK2M- /2TB+i2/ BM 6B;X R8- +Q``2bTQM/b iQ i?2 Q#b2`p iBQMb K /2X h?2 / `F `2 b ;Q HQM; rBi? `2 b Q7 T i+? /2i +?K2Mi r?BH2 i?2 HB;?i2` `2 b `2 biBHH #QM/2/ iQ i?2 K2i H b?22ibX
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.B`2+i +QKT `BbQM #2ir22M T bbBp2 M/ HQ+F@BM i?2`KQ;` T?v 6Q` /B`2+i +QKT `BbQM Q7 i?2 irQ KQMBiQ`BM; K2i?Q/b +QmTH2 Q7 bT2+B@ K2Mb r2`2 KQMBiQ`2/ rBi? #Qi? K2i?Q/bX h?2 /B`2+i +QKT `BbQM #2ir22M i?2 i?2`K H BK ;2b ; BM2/ pB T bbBp2 A`h M/ i?2 T? b2 BK ;2b `2bmHiBM; 7`QK Gh Bb BHHmbi` i2/ BM 6B;X ReX Ai +QKT `2b i?2 `2bmHib Q7 #Qi? KQMBiQ`BM; K2i?Q/b i
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i?`22 /BbiBM+i +QKT ` #H2 MmK#2` Q7 HQ / +v+H2bX b + M #2 b22M- i?2 /2i +?2/ `2 b `2 Km+? #2ii2` pBbB#H2 rBi? Gh i? M rBi? T bbBp2 A`hX h?Bb r b 2tT2+i2/ bBM+2 +iBp2 i?2`KQ;` T?v K2i?Q/b + M T?vbB+ HHv K F2 bm#bm`7 +2 /Bb+QMiB@ MmBiB2b pBbB#H2 r?BH2 T bbBp2 A`h b?Qrb bm`7 +2 /Bb+QMiBMmBiB2bX P#pBQmbHv- HbQ bm#bm`7 +2 /272+ib H2 / iQ i2KT2` im`2 +? M;2b i? i `2 /2i2+i #H2 rBi? T bbBp2
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A`h b HQM; b i?2B` BKT +i QM bm`7 +2 i2KT2` im`2 +? M;2 Bb bi`QM; 2MQm;?X 9X .Bb+mbbBQM AM ;2M2` H- #Qi? K2i?Q/b- T bbBp2 A`h M/ Gh- `2 + T #H2 iQ KQMBiQ` T i+? /2i +?K2Mi M/ iQ T`QpB/2 BK ;2b iQ #2ii2` mM/2`bi M/ i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2bX
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h?2 bT2+BK2M ;2QK2i`v +?Qb2M K22ib i?2 +QM/BiBQM i? i QMHv /?2bBp2 / K ;2 Q++m`bX AM i2`Kb Q7 KQMBiQ`BM; / K ;2 T`QT ; iBQM i?BM T i+? ;Bp2b ;QQ/ `2bmHibX 6Q` `2 H TTHB+ iBQM T i+? i?B+FM2bb ? b iQ #2 /2bB;M2/ BM i2`Kb Q7 K2+? MB+ H T`QT2`iB2b /2iQ`B iBM; i?2 KQMBiQ`BM; T`QT2`iB2bX h?2 TTHB+ iBQM Q7 T bbBp2 A`h QM /2i2+iBM; bm#bm`7 +2 /272+ib ;Bp2b T`QKBb@
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BM; `2bmHibX q?2M KQMBiQ`BM; /?2bBp2 / K ;2 T`QT ; iBQM mM/2` K2+? MB+ H HQ /BM;- #Qi? 7Q` [m bB@bi iB+ HQ /BM; b r2HH b 7Q` +v+HB+ HQ /BM;- i?2 2tT M/BM; T i+? /2i +?K2Mi +QmH/ #2 Q#b2`p2/X P#pBQmbHv- i?2 2M2`;v /BbbBT iBQM i?`Qm;? i?2 T i+?- r b ?B;? 2MQm;? iQ #2 K2 bm`2/ b i2KT2` im`2 BM+`2 b2 i i?2 Q#D2+iǶb bm`7 +2X 6Q` T i+?2b i?B+F2` i? M i?2 QM2b i2bi2/ ?2`2- i?2 2z2+i Bb
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T`Q# #Hv H2bb bi`QM; M/ i?2`27Q`2 KB;?i MQi #2 pBbB#H2X HbQ- /2H KBM iBQM rBi?BM KmHiBH v2` T i+? + M Q++m` M/ TQbbB#Hv H2 / iQ bBKBH ` `2bmHib BM i2`Kb Q7 i2KT2` im`2 BM+`2 b2 QM i?2 bm`7 +2X S`QpB/2/ i? i i?2 KQp2K2Mi /m2 iQ HQ /BM;fmMHQ /BM; Qp2` BK ;BM; iBK2 Bb Re
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Bb ;Bp2MX M HvbBM; i?2 i?2`K H BK ;2b- Bi + M #2 b22M i? i BM T`BM+BTH2 Bi Bb TQbbB#H2 iQ ; BM ;QQ/ FMQrH2/;2 #Qmi b? T2 M/ bBx2 Q7 i?2 / K ;2 2tT MbBQM Qp2` iBK2X "mi- Bi Bb BKTQ`i Mi iQ `2K2K#2` i? i i?2 TB+im`2 /Q2b MQi T`QpB/2 i?2 +im H bBx2 M/ + M QMHv #2 b22M b `272`2M+2 TQBMiX AM Q`/2` iQ K2 bm`2 i?2 bBx2 Q7 bm#bm`7 +2 / K ;2 Gh ? b iQ #2 TTHB2/ iQ MQM@KQpBM; Q#D2+iX
9yy
MQi?2` /` r# +F Bb- i? i i2bi `2 iBK2@+QMbmKBM;X 6Q` i?2 l. bT2+BK2Mb i2bib i F2b `QmM/ kkĜk8 b- r?BH2 7Q` i?2 J. bT2+BK2Mb Bi i F2b KQ`2 i? M kyy bX 8X *QM+HmbBQMb h?2 irQ i?2`KQ;` T?v K2i?Q/b Gh M/ T bbBp2 A`h r2`2 TTHB2/ iQ KQMBiQ` bm#bm`7 +2 / K ;2 T`QT ; iBQM 7Q` HQ /2/ bT2+BK2MbX >2`2- Bi r b BMp2biB; i2/
9y8
?Qr i?2b2 K2i?Q/b `2 TTHB+ #H2 iQ H #Q` iQ`v i2bib BM Q`/2` iQ ;2i BM7Q`K @ iBQM #Qmi i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2b Q7 #QM/2/ `2T B`b mM/2` K2+? MB+ H HQ /BM;X h?2 BMp2biB; iBQM 7Q+mb2/ QM i?2 K BM T ` K2i2`, i?2 / K ;2 rBi?BM i?2 /?2bBp2 H v2`X .Qm#H2 H T DQBMi bT2+BK2Mb +QMbBbiBM; Q7 K2i HHB+ M/ *6_S /?2`2M/b r2`2 i2bi2/ mM/2` [m bB@bi iB+ M/ 7 iB;m2 HQ /BM;X aT2+BK2M /BK2M@
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bBQMb r2`2 / Ti2/ iQ i?2 ;Bp2M +QM/BiBQMb iQ K22i i?2 `2[mB`2K2Mib `2bmHiBM; 7`QK i?2 KQMBiQ`BM; K2i?Q/bX Ai r b b?QrM i? i mM/2` i?2 ;Bp2M +B`+mKbi M+2b #Qi? K2i?Q/b `2 #H2 iQ ;Bp2 BM7Q`K iBQM QM i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2bX h2KT2` im`2 Q` T? b2 /Bz2`2M+2b QM i?2 i?2`K H BK ;2b +QmH/ #2 `2H i2/ iQ i?2 T`2HBKBM `v K2+? M@
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B+ H 2biBK iBQMb QM i?2 TH +2 Q7 / K ;2 QMb2iX 6Q` i?2 [m bB@bi iB+ i2bib HbQi?2 K2+? MB+ H #2? pBQm` +QmH/ T`QpB/2 bbBbi M+2 BM i?2 BK ;2 BMi2`T`2i iBQMX *? ` +i2`BbiB+ TQBMib BM i?2 HQ /f/BbTH +2K2Mi +m`p2 +Q``2H i2 rBi? i?2 / K ;2 T`Q;`2bb T`2bmK2/ 7`QK i?2 BK ;2bX h?2 BM/B+ iQ` 7Q` i?2 / K ;2 HQ+ iBQM Bb /B`2+i +QKT `BbQM Q7 i?2 7 BH2/ bT2+BK2Mb M/ i?2 +Q``2bTQM/BM; i?2`K H BK@
9ky
;2b i }M H 7 BHm`2X >2`2 ; BM- ;QQ/ +Q``2H iBQM #2ir22M i?2 BK ;2b M/ i?2 bT2+BK2Mb + M #2 b22MX 6Q` T bbBp2 A`h- HQ+ H BM+`2 b2b BM i2KT2` im`2 + M #2 K /2 Qmi QM i?2 bT2+BK2M Rd
bm`7 +2 i? i `2 2tT M/BM; rBi? BM+`2 bBM; HQ /X lM/2` [m bB@bi iB+ HQ /BM; i?2 K2 bm` #H2 i2KT2` im`2 /Bz2`2M+2 Bb ?B;?2` i? M mM/2` 7 iB;m2 HQ /BM;X h?2 9k8
Qp2` HH i2KT2` im`2 BM+`2 b2 BM i?2 bT2+BK2M mM/2` HQM;@i2`K HQ /BM; K F2b Bi #Bi KQ`2 +? HH2M;BM; iQ /Bz2`2MiB i2 #2ir22M / K ;2/ M/ mM/ K ;2/ `2;BQMbX AM7Q`K iBQM #Qmi i?2 +m``2Mi bi i2 `2 /BbTH v2/ BM HBp2 BK ;2 r?B+? K F2b Bi 2 bv iQ ? M/H2 T`QpB/BM; +QKT`2?2MbBp2 T`2b2Mi iBQM Q7 QM;QBM; T`Q+2bb2bX Gh + M T`QpB/2 M Qp2`pB2r QM i?2 / K ;2 T`Q;`2bb i /BbiBM+i TQBMib BM [m H@
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Bi iBp2 K MM2`X h?2 /Bz2`2MiB iBQM #2ir22M / K ;2/ M/ mM/ K ;2/ `2;BQMb Bb Km+? 2 bB2` i? i 7Q` T bbBp2 A`hX q?2M +QKT `BM; i?2 `2bmHib Q7 T bbBp2 A`h M/ Gh QM2 + M b v i? i mM/2` bT2@ +B}+ +B`+mKbi M+2b 2 +? K2i?Q/ + M #2 mb2/ iQ KQMBiQ` i?2 /2i +?K2Mi T`Q+2bb Q7 *6_S `2T B` T i+?2bX 6Q` bm#bm`7 +2 /272+ib Gh T`QpB/2b i?2`K H BK ;2b
9j8
rBi? ?B;?2` +QMi` bi #2ir22M / K ;2/ M/ mM/ K ;2/ `2 bX 6m`i?2`- 7Q` T b@ bBp2 A`h i?2 BMBiB H / K ;2 /2i2+iBQM Bb KQ`2 /B{+mHi i? M 7Q` Gh- #2+ mb2 i?2 bK HH2` i?2 /272+i Bb- i?2 H2bb 2M2`;v Bb +QMp2`i2/ BMiQ i?2`K H 2M2`;vX "mi- BM i2`Kb Q7 K M ;2 #BHBiv T bbBp2 A`h ? b +QMbB/2` #H2 /p Mi ;2b, Ai Bb 2 bv iQ ? M/H2 M/ HBp2 BK ;2b `2 i F2MX h?2`K H BK ;2b `2 MQi T`Q+2bb2/ M/ i?2
99y
KQpBM; HBp2 BK ;2- r?B+? mM7Q`imM i2Hv + MMQi #2 T`2b2Mi2/ ?2`2- `2p2 Hb i?2 / K ;2 T`Q;`2bb #2? pBQm` 2p2M #2ii2` i? M i?2 TB+im`2b + M b?QrX *QM+Hm/BM;- #Qi? A`h K2i?Q/b `2 ;2M2` HHv + T #H2 iQ KQMBiQ` i?2 / K ;2 T`Q;`2bb Q7 T i+? /2i +?K2Mi mM/2` [m bB@bi iB+ M/ +v+HB+ HQ /BM; 7Q` irQ /B7@ 72`2Mi 2TQtv@# b2/ l. M/ J. *6_S T i+?2b rBi? i?B+FM2bb Q7 yX9 KK M/
998
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S0263-8223(19)33243-X https://doi.org/10.1016/j.compstruct.2020.111991 COST 111991
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
27 August 2019 14 January 2020 23 January 2020
Please cite this article as: Martens, U., Schröder, K.-U., Evaluation of infrared thermography methods for analysing the damage behaviour of adhesively bonded repair solutions, Composite Structures (2020), doi: https://doi.org/ 10.1016/j.compstruct.2020.111991
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© 2020 Published by Elsevier Ltd.
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T2` im`2 BM+`2 b2 Bb BM2z2+iBp2- #2+ mb2 i?2 +Q2{+B2Mi /Q2b MQi BM+Hm/2 i?2 ` i2 /2T2M/2M+vX h?2`27Q`2- 7`QK i?2 KQMBiQ`BM; TQBMi Q7 pB2r- i?2 T i+? i?B+FM2bb b?QmH/ #2 T`272` #Hv bK HH iQ K F2 bm`2 i? i 2MQm;? 2M2`;v /BbbBT i2b iQ i?2 bm`7 +2X hQ ; BM 7Q+mb2/ BK ;2 BM QTiB+ H Gh b2p2` H TB+im`2b `2 i F2M M/ bmT2`@
R9y
BKTQb2/ #v M TT`QT`B i2 bQ7ir `2X lT iQ +2`i BM TQBMi- i?2 KQ`2 TB+im`2b `2 bmT2`BKTQb2/- i?2 #2ii2` i?2 bmTT`2bbBQM Q7 MQBb2 2z2+ibX h?2`27Q`2- Gh Bb mbm HHv TTHB2/ iQ MQM@KQpBM; Q#D2+ibX 6Q` KQpBM; Q#D2+i- iBK2@/2T2M/2Mi +? M;2b ? p2 iQ #2 M2;H2+i #H2 7Q` i?2 MmK#2` Q7 TB+im`2b M/ i?2 +Q``2bTQM/@ BM; iBK2X qBi? i?2 BK Q7 KQMBiQ`BM; i?2 / K ;2 T`Q+2bb- b2p2` H TB+im`2b `2 d
R98
M22/2/X h?2`27Q`2- 7i2` 2 +? K2 bm`BM; KQ/mH2 i?2 bvbi2K ? b iQ #2 `2bi `i2/X aT2+BK2M ;2QK2i`v M/ i2bi +QM};m` iBQMb 6Q` i?2 +?QB+2 Q7 i?2 bT2+BK2M ;2QK2i`v- Bi ? b iQ #2 i F2M BMiQ ++QmMi i? i bQK2 TQbbB#H2 / K ;2b + MMQi #2 b22M 7`QK i?2 QmibB/2X 6Q` BMbi M+2- i?2 K2i HHB+ +` +F ;`Qri? mM/2` i?2 T i+?- BMi2`H KBM ` /2H KBM iBQM Q7 i?2 T i+?
R8y
H v2`b M/ /2H KBM iBQM Q7 i?2 T i+? `2 MQi pBbB#H2 b i?2v `2 bm#bm`7 +2 /2@ 72+ibX HH i?2b2 / K ;2 K2+? MBbKb + M BM~m2M+2 QM2 MQi?2`- M/ Bi KB;?i #2 /B{+mHi iQ /BbiBM;mBb? #2ir22M i?2KX h?2 /Qm#H2 H T DQBMi ;2QK2i`v mb2/ 7Q` i?2 T`2HBKBM `v /2bB;M Q7 i?2 /?2bBp2 xQM2 Bb ;QQ/ +?QB+2 iQ H2 p2 K2i HHB+ +` +F ;`Qri? bB/2X hQ ;m ` Mi22 i? i /?2bBp2 / K ;2 Bb i?2 QMHv K2+? MBbK
R88
i? i + M #2 Q#b2`p2/ QMHv QM2 T i+? H v2` Bb TTHB2/ i 2 +? bB/2X h?Bb ? b i?2 //BiBQM H /p Mi ;2 i? i i?2 K2 bm`2K2Mi /2Ti? Bb p2`v bK HH- #2M2}+B H 7Q` #Qi? T bbBp2 A`h M/ GhX AMp2biB; iBQMb `2 K /2 7Q` [m bB@bi iB+ HQ /BM; M/ 7Q` 7 iB;m2 HQ /BM;X hrQ /Bz2`2Mi T i+? K i2`B Hb `2 i2bi2/- mMB/B`2+iBQM H Ul.V T`2T`2; Q7 + `#QM }#`2 `2BM7Q`+2/ TH biB+b U*6_SV- U2TQtv # b2/ *6_S
Rey
T`2T`2; 7`QK >2t+2H- ivT2 >1sSGu JRyXRfj3fl.jyyfJ9yCfey M/ i?B+F@ M2bb Q7 yX9 KKV- M/ yfNy◦ KmHiB/B`2+iBQM H UJ.V rQp2M + `#QM 7 #`B+ U2TQtv # b2/ *6_S T`2T`2; 7`QK a:G h2+?MQHQ;B2b :K#> Q7 ivT2 * q9Ry@hqkfk@ 1jkjf9kWfeF M/ i?B+FM2bb Q7 yXe KKV `2 mb2/X h?2 K2i HHB+ T `ib `2 #mBHi 7`QK akj8C_Y_X
Re8
:Bp2M i?2 #QmM/ `B2b Q7 i?2 K2 bm`2K2Mi i2+?MB[m2b- i?2 bi M/ `/ p Hm2b 7Q` #Qi?- bT2+BK2M /BK2MbBQMb M/ i2bi p2HQ+Biv 7Q` [m bB@bi iB+ HQ /BM;- `2 MQi bmBi #H2X 6Q` Gh iQ #2 TTHB+ #H2- Bi ? b iQ #2 2Mbm`2/ i? i / K ;2 T`Q;`2bb Bb M2;H2+i #H2 /m`BM; i?2 +`2 iBQM Q7 QM2 TB+im`2 rBi? b2p2` H HQ+F@BM 2t+Bi iBQMbX >2M+2- i?2 i2bi p2HQ+Biv ? b iQ #2 / Ti2/ ++Q`/BM; iQ i?2 HQ+F@BM T ` K2i2`b
Rdy
M22/2/X S`2HBKBM `v i2bib TTHB2/ iQ bT2+BK2M MQi mM/2` HQ / H2 / iQ HQ+F@BM 7`2[m2M+v Q7 R >x 7Q` i?2 l. }#`2b M/ yXR >x 7Q` i?2 J. T`2T`2;X 6m`i?2`i?2 i2bib b?Qr2/ i? i MmK#2` Q7 R8 HQ+F@BM 2t+Bi iBQM T2`BQ/b + M #2 b22M b ;QQ/ +QKT`QKBb2 #2ir22M MQBb2 bmTT`2bbBQM M/ BK ;BM; iBK2X hQ;2i?2` 3
rBi? i?2 `2[mB`2/ HQ+F@BM 7`2[m2M+B2b i?2 i2bi p2HQ+BiB2b v `2 i?2M +?Qb2M iQ #2 Rd8
vU D = 0.5 KKfKBM 7Q` i?2 l. T i+? M/ vM D = 0.05 KKfKBM 7Q` i?2 J. T i+?X h?2 H `;2 bT2+BK2M rB/i? `2bmHib 7`QK i?2 /2K M/ 7Q` b2p2` H BK ;2b i? i + M /BbTH v i?2 / K ;2 T`Q;`2bb Qp2` iBK2X hQ i F2 KQ`2 i? M QM2 TB+im`2- i?2 /2b+`B#2/ HQ+F@BM T`Q+2bb ? b iQ #2 `2bi `i2/ 7i2` 2 +? TB+im`2 r?B+? K F2b Bi
R3y
iBK2@+QMbmKBM; T`Q+2/m`2X h?2 H `;2bi TQbbB#H2 bT2+BK2M rB/i? `2bmHib 7`QK i?2 +H KTBM; D r /BK2MbBQMb M/ Bb +?Qb2M iQ #2 3y KK ++Q`/BM;HvX aT2+BK2M /BK2MbBQMb `2 bF2i+?2/ BM 6B;X 9- p Hm2b ;Bp2M BM KKX 6Q` +QKT `BbQM Tm`TQb2b i?2 bT2+BK2M ;2QK2i`v +?Qb2M Bb mb2/ 7Q` HH i2bib T2`7Q`K2/X AM Q`/2` iQ `2@ /m+2 i2biBM; iBK2- i2bi p2HQ+Biv Q7 v = 1 KKfKBM Bb +?Qb2M 7Q` i?2 T bbBp2 A`h
R38
BMp2biB; iBQMbX 6 iB;m2 / i `2bmHi 7`QK i?2 TTHB+ iBQM M/ i?mb /Q MQi M22/ Mv 7m`i?2` 2tTH M iBQM- b22 h #H2 RX J2+? MB+ H 2biBK iBQMb }`bi 2biBK iBQM K /2 QM M M HviB+ H K2+? MB+ H KQ/2H ;Bp2b BM7Q`K iBQM QM r?2`2 T i+? /2i +?K2Mi Bb 2tT2+i2/X h?2 KQ/2H +?Qb2M Bb b2KB@+QMiBMmQmb
RNy
7Q`KmH iBQM 7Q` /Qm#H2 H T DQBMi `2bmHiBM; 7`QK b?2 ` r2# i?2Q`v ++Q`/BM; iQ (3) M/ (k8)Xh?Bb TT`Q +? K BMHv `2bmHib BM i?2 r2HH@FMQrM oQHF2`b2M 2[m @ iBQMb (ke)X hQ /2i2`KBM2 i?2 MQ`K H b r2HH b i?2 b?2 ` bi`2bb /Bbi`B#miBQM rBi?BM i?2 Qp2`H T- [m `i2` KQ/2H Bb +?Qb2M ++Q`/BM; iQ (3) b /2TB+i2/ BM 6B;X 8X
RN8
amKK `Bx2/- # bBb Q7 i?2 KQ/2H Bb i? i i?2 /?2`2M/b QMHv i` MbKBi MQ`K H 7Q`+2b r?BH2 i?2 /?2bBp2 QMHv i` MbKBib b?2 ` HQ /bX Pmi@Q7@TH M2 #2M/BM; 27@ 72+ib `2 M2;H2+i2/X h?2 +?Qb2M b2KB@+QMiBMmQmb KQ/2H MQr mb2b +QMiBMmQmb bi`2bb 7Q`KmH iBQM BM HQ / /B`2+iBQM UxVX 6Q` KQ`2 /2i BH2/ 2tTH M iBQM b22 (3)X h?2 `2bmHiBM; b?2 ` bi`2bb /Bbi`B#miBQM Bb ;Bp2M iQ #2 τ (x) ρ cosh(ρx/l) (1 − ψ) sinh(ρx/l) − = τm 2 sinh(ρ/2) (1 + ψ) cosh(ρ/2)
N
URV
kyy
rBi? Bib K tBKmK p Hm2 Q7 ρ ρ (1 − ψ) τmax ρ tanh , coth + = τm 2 2 (1 + ψ) 2
UkV
M/ ρ=
p
((1 + ψ)Gta2 /EA1 d).
UjV
q?2`2 τm /2MQi2b i?2 K2 M b?2 ` bi`2bb M/ ρ i?2 /?2bBQM BM/2tX h?2 K DQ` T ` K2i2` /2i2`KBMBM; i?2 b?2 ` bi`2bb /Bbi`B#miBQM Bb i?2 biBzM2bb ` iBQ Q7 i?2 /?2`2M/b ψX Ai Bb /2}M2/ b i?2 ` iBQ #2ir22M i?2 BMM2` M/ Qmi2` /?2`2M/ ky8
ψ=
EA1 EA2 X
AM i?2 ;Bp2M + b2- Bi Bb i?2 ` iBQ Q7 K2i H Qp2` T i+? biBzM2bb- +QMbB/2`@
BM; i? i 7Q` i?2 [m `i2` KQ/2H QMHv ? H7 i?2 K2i H i?B+FM2bb Bb mb2/ iQ /2i2`KBM2 EA1 X 6Q` biBzM2bb ` iBQ ψ 6= 1 i?2 b?2 ` bi`2bb /Bbi`B#miBQM HQM; i?2 Qp2`H T #2@ +QK2b bvKK2i`B+X h?2 ?B;?2bi b?2 ` bi`2bb i?2M TT2 `b rBi?BM i?2 r2 F2` kRy
/?2`2M/ i i?2 7`22 2M/ Q7 i?2 biBz2` QM2- b22 6B;X eX TTHB2/ iQ i?2 bT2+BK2M +QM};m` iBQM /2b+`B#2/ #Qp2- i?2 biBzM2bb ` iBQ ψ `2bmHib iQ #2 > 1X qBi? i?2 K2i HHB+ b?22i b i?2 biBz2` /?2`2M/- i?2 Q++m``2M+2 Q7 T i+? /2i +?K2Mi Bb 2tT2+i2/ iQ #2 i i?2 bT2+BK2M +2Mi`2 Ux = 0VX h?mb- HQ+ H i2KT2` im`2 /Bz2`@ 2M+2b `QmM/ i?2 +2Mi`2 HBM2 T`2bmK #Hv Q++m``BM; i i?2 7`22 2/;2b- r?2`2 i?2
kR8
?B;?2bi bi`2bb2b `2 mbm HHv 2tT2+i #H2- BM/B+ i2 /?2bBp2 / K ;2 QMb2iX h2bi /2b+`BTiBQM h?2 i2bi K +?BM2 mb2/ Bb b2`pQ@?v/` mHB+ i2bi K +?BM2 Q7 ivT2 a+?2M+F SPw Rey rBi? bi iB+ MQKBM H HQ / Q7 Yf@Rey FLX h?2 BM7` `2/ + K2` Bb ivT2 AK ;2A_3j3y a UAM7` h2+ :K#>- .`2b/2MV BM +QK#BM iBQM rBi? i2H2 xQQK
kky
H2Mb2 Uf = 50 KKVX h2+?MB+ H / i Bb HBbi2/ BM h #H2 kX h?2 i?2`K H biBKmH iBQM /2pB+2 Bb ? HQ;2MH KT Q7 ivT2 >k8b U>2/H2` avbi2KHB+?i :K#>- _mMF2HV rBi? TQr2` Q7 kX8 Fq mbBM; irQ B/2MiB+ H #mH#b Q7 RXk8 Fq #2 K TQr2` 2 +?X
Ry
aT2+BK2M T`2T ` iBQM aBM+2 i?2 BMp2biB; iBQM 7Q+mbb2b QM i?2 KQMBiQ`BM; K2i?Q/- MQi i?2 bT2+BK2M kk8
T`QT2`iB2b- bT2+BK2M T`2T ` iBQM Bb F2Ti bBKTH2X h?2 K2i HHB+ T `i Bb +mi BMiQ i?2 `B;?i /BK2MbBQMb M/ 7i2` b M/ #H biBM; M/ +H2 MBM; i?2 *6_S T i+?2b `2 ii +?2/ rB?iQmi Mv 2ti` /?2bBp2X h?2 2TQtv `2bBM Bb ? `/2M2/ mM/2` T`2bbm`2 BM +HBK i2 +? K#2` 7Q` j ?Qm`b mM/2` Rky◦ *X HH bT2+BK2Mb `2 +Q i2/ rBi? K ii #H +F T BMi 7Q` ?B;? 2KBbbBpBiv U2KBbbBpBiv Q7 i?2 Q`B;BM H bT2+BK2M
kjy
Bb #Qmi yXkNVX jX _2bmHib h?2 K2+? MB+ H #2? pBQm` Q7 i?2 bT2+BK2Mb b r2HH b i?2 Q#b2`p #H2 / K ;2 T`QT ; iBQM r2`2 BM ;QQ/ ;`22K2Mi 7Q` 2 +? i2bi b2`B2bX h?2`27Q`2- QM2 bT2+@ BK2M Bb 2t2KTH `BHv +?Qb2M iQ /2KQMbi` i2 i?2 i2bi `2bmHib Q7 2 +? i2bi b2`B2bX
kj8
6B;X d /2TB+ib i?2 TB+im`2 b2+iBQM +?Qb2M iQ 2p Hm i2 i?2 / K ;2 T`Q;`2bbX Zm bB@bi iB+ HQ /BM; 6B`bi i?2 i2bi `2bmHib 7Q` i?2 bT2+BK2Mb mM/2` [m bB@bi iB+ HQ /BM; `2 T`2@ b2Mi2/ 7Q` #Qi? KQMBiQ`BM; K2i?Q/b T bbBp2 A`h M/ GhX 1 +? i2bi r b + ``B2/ Qmi rBi? #Qi?- T i+?2b rBi? l. }#`2b M/ T i+?2b rBi? J. }#`2bX hQ /2KQM@
k9y
bi` i2 i?2 / K ;2 T`QT ; iBQM mM/2` [m bB@bi iB+ HQ /BM; 7Q` 2 +? bT2+BK2M MmK#2` Q7 3 TB+im`2b Bb +?Qb2MX //BiBQM HHv- i?2 iBK2 i?2 TB+im`2b `2 i F2M `2 K `F2/ BM i?2 +Q``2bTQM/BM; HQ /f/BbTH +2K2Mi +m`p2b- b22 6B;X 3 M/ RyX S bbBp2 h?2`KQ;` T?v h?2 / K ;2 T`QT ; iBQM T`Q+2bb 7Q` l. T i+? KQMBiQ`2/ rBi? T bbBp2 A`h
k98
Bb b?QrM BM 6B;X 3X PM i?2 i?2`K H BK ;2b i?2 +2Mi` H ?Q`BxQMi H HBM2 `2T`2b2Mib i?2 #mii DQBMi #2ir22M i?2 irQ K2i H b?22ibX h?2 HQr2` TBbiQM i` Mb72`b i?2 HQ /X .m2 iQ i?2 KQp2K2Mi Bi ?2 ib mT M/ i?mb HbQ i` Mb72`b i2KT2` im`2 BMiQ i?2 bT2+BK2MX h?2`27Q`2- i?2 HQr2` b?22i Bb r `K2` i? M i?2 mTT2` QM2X
RR
AM i?2 HQ /f/BbTH +2K2Mi +m`p2 i?2 QMb2i Q7 / K ;2 T`Q+2bb2b mbm HHv ;Q2b HQM; k8y
rBi? /2+`2 b2 BM i?2 TTHB2/ HQ /X >2`2- i?2 HQ / BM+`2 b2b +QMiBMmQmbHv iQ K tBKmK p Hm2 M/ /2+`2 b2b bi2TrBb2 mMiBH }M H 7 BHm`2 Q++m`bX h?2 bi2TrBb2 /2+`2 b2 BM HQ / + M #2 2tTH BM2/ #v bT HHBM; Q7 bBM;H2 }#`2b Q` }#`2 #mM/H2br?B+? + M #2 KQMBiQ`2/ /m`BM; i2biBM;X h?2 }`bi +QMbB/2` #H2 HQ+ H i2KT2` im`2 /Bz2`2M+2b Q++m` Dmbi #2?BM/ i?2 ?B;?@
k88
2bi HQ /- b22 BK ;2 MQX RX PM i?2 H27i@? M/ bB/2 Q7 i?2 BK ;2- p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2` im`2 + M #2 b22MX 6m`i?2`- i i?2 +2Mi`2 HBM2ě2bT2+B HHv i #Qi? 7`22 2/;2bě `2 b Q7 /Bz2`2Mi i2KT2` im`2 + M #2 K /2 QmiX GQQFBM; i i?2 bT2+BK2MǶb 7`22 `B;?i 2/;2 #Qp2 i?2 +2Mi`2 HBM2 7Q` BMbi M+2 BK ;2 MQX R /BbTH vb bK HH `2 rBi? BM+`2 b2/ i2KT2` im`2X :QBM; 7`QK BK ;2 MQX R iQ
key
BK ;2 MQX j i?2 BMi2MbBiv Q7 i?2 p2`iB+ H HBM2b /2+`2 b2b r?BH2 i?2 BMi2MbBiv M/ bBx2 Q7 i?2 `2 BM+`2 b2X i TQBMi 9 MQi?2`- #B;;2` bi2TrBb2 HQ / /2+`2 b2 + M #2 b22M BM i?2 HQ /f/BbTH +2K2Mi +m`p2X h?2 +Q``2bTQM/BM; i?2`K H BK ;2 ; BM b?Qrb p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2` im`2 BM i?2 H27i `2 - i?Bb iBK2 i 2p2M ?B;?2` i2KT2` im`2b M/ Qp2` rB/2` `2 X >2`2 ; BM- i?2 BMi2MbBiv Q7 i?2b2
ke8
HBM2b /2+`2 b2b Qp2` i?2 M2ti irQ BK ;2b r?BH2 i?2 `2 BM i?2 `B;?i T `i ?2 ib mT 7m`i?2` M/ 2tT M/b +`Qbb KQ`2 i? M ? H7 Q7 i?2 bT2+BK2M rB/i?X h?2 H bi irQ BK ;2b b?Qr irQ TQBMib BKK2/B i2Hv #27Q`2 i?2 }M H 7` +im`2 Q7 i?2 bT2+BK2MX o2`iB+ H HBM2b Q7 ?B;? i2KT2` im`2 /QKBM i2 i?2 irQ BK ;2bX AM TB+im`2 MQX d i?2b2 HBM2b `2 HQ+ HHv HBKBi2/ iQ i?2 b? T2 Q7 i?2 `2 i? i 2tT M/2/ 7`QK i?2
kdy
`B;?i 2/;2 iQ MQr HKQbi i?2 2MiB`2 bT2+BK2M rB/i?X AM i?2 }M H TB+im`2- i H2 bi BM i?2 mTT2` T `i- i?2 p2`iB+ H HBM2b 2t+22/ i?Bb `2 M/ `mM 7`QK i?2 iQT iQ i?2 +2Mi`2 HBM2 Q` 2p2M iQ i?2 #QiiQKX hQ;2i?2` rBi? i?2 T`2HBKBM `v K2+? MB+ H 2biBK iBQM M/ i?2 HQ /f/BbTH +2K2Mi +m`p2 i?2 HQ+ H `2 Q7 BM+`2 b2/ i2KT2` im`2 i? i 2tT M/b Qp2` i2biBM; iBK2 Bb
kd8
2biBK i2/ iQ #2 i?2 HQ+ H / K ;2 BM i?2 /?2bBp2 xQM2X 6Q` i?2 ;Bp2M biBzM2bb ` iBQ- i?2 / K ;2 QMb2i Bb 2tT2+i2/ i 7`22 2/;2 `QmM/ i?2 +2Mi`2 HBM2- r?B+? +Q``2bTQM/b iQ i?2 Q#b2`p iBQMb QM i?2 i?2`K H BK ;2bX +QMbi MiHv BM+`2 bBM; / K ;2 H2 /b iQ T2`K M2Mi ?2 i /BbbBT iBQM r?B+? ;Q2b HQM; rBi? i?2 bHB;?iHv BM+`2 bBM; i2KT2` im`2 BM i?2 /2i2+i2/ `2 Qp2` iBK2X h?2 ` TB/ HQ / /2+`2 b2 Rk
k3y
i TQBMib R- 9 M/ 3 Bb bbQ+B i2/ rBi? }#`2 bT HHBM;- r?B+? Bb BM #`mTi / K ;2 K2+? MBbKX h?Bb BM im`M +Q``2bTQM/b iQ i?2 p2`iB+ H HBM2b Q7 BM+`2 b2/ i2KT2`@ im`2 i? i TQT mT M/ +QQH /QrM ; BMX hQ 7m`i?2` bmTTQ`i i?Bb i?2Q`v /B`2+i +QKT `BbQM #2ir22M i?2`K H BK ;2 MQX d M/ T?QiQ;` T? Q7 i?2 / K ;2/ bT2+BK2M Bb b?QrM BM 6B;X NX PM i?2 H27i@? M/ bB/2 }#`2 #mM/H2 + M #2 b22M
k38
i? i T22H2/ Qz 7`QK i?2 mTT2` T `i Q7 i?2 bT2+BK2MX h?2 TQbBiBQM }ib iQ i?2 Q#b2`p2/ p2`iB+ H HBM2bX 6m`i?2`- Bi Bb TT `2Mi i? i i?2 }#`2b BM i?2 `B;?i `2 /2i +?2/ 7`QK i?2 mTT2` K2i HHB+ T `iX G27iQp2`b Q7 i?2 /?2bBp2 QM i?2 K2i HHB+ T `i +Q``2bTQM/ iQ i?2 b? T2 Q7 i?2 `2 i? i + M #2 b22M QM i?2 i?2`K H BK ;2X h?mb- }#`2 T22H Qz M/ / K ;2 BM i?2 /?2bBp2 H v2` b22K iQ #2 Q#b2`p #H2
kNy
mbBM; T bbBp2 A`hX GQ+F@BM h?2`KQ;` T?v h?2 / K ;2 T`Q+2bb KQMBiQ`2/ rBi? Gh Bb b?QrM BM 6B;X RyX .Bz2`2M+2b BM i?2 HQ /f/BbTH +2K2Mi +m`p2b +QKT `2/ iQ 6B;X 3 `2bmHi 7`QK i?2 /Bz2`2Mi K i2`B H T`QT2`iB2b Q7 i?2 l. M/ J. T i+?2bX aBM+2 i?2 7Q+mb Q7 i?Bb BM@
kN8
p2biB; iBQM Bb b2i iQ i?2 KQMBiQ`BM; i2+?MB[m2 M/ MQi iQ i?2 [m HBiv Q7 i?2 K2+? MB+ H T`QT2`iB2b- HbQ i?2 +QM/2Mb2/ bT2+BK2M T`2T ` iBQM T`Q+2bb BM~m@ 2M+2b i?2 HQ /f/BbTH +2K2Mi #2? pBQm`X AM +QMi` bi iQ T bbBp2 A`h- i?2 /BbTH v2/ BK ;2b `2 T? b2 BK ;2bX S? b2 BK ;2b `2 dz`2H i2/ iQ i?2 (i2KT2` im`2) T`QT@ ; iBQM iBK2Ǵ (e) M/ /Q MQi b?Qr i2KT2` im`2 /Bbi`B#miBQMX h?2Q`2iB+ HHv
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i?2v `2 BM/2T2M/2Mi Q7 bm`7 +2 p `B iBQMb M/ + M i?mb 2HBKBM i2 2``Q`b `2bmHi@ BM; 7`QK MQM@mMB7Q`K ?2 iBM; Q` p `B iBQMb BM bm`7 +2 2KBbbBpBiv (e- kd)X ; BMi?2 +2Mi` H ?Q`BxQMi H HBM2 + M #2 bbB;M2/ iQ i?2 #mii DQBMi #2ir22M i?2 K2i H b?22ibX AK ;2 MQX R b?Qrb i?2 bi i2 `B;?i 7i2` 2t+22/BM; i?2 ?B;?2bi TQBMi BM i?2 HQ /f/BbTH +2K2Mi +m`p2X 6`QK TQBMi MQX k iQ TQBMi MQX 8- i?2 bHQT2 Q7 i?2
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HQ /f/BbTH +2K2Mi +m`p2 /2+`2 b2b +QMiBMmQmbHvX PM BK ;2 MQX k i #Qi? 7`22 2/;2b / `F bTQib + M #2 K /2 Qmi `QmM/ i?2 ?Q`BxQMi H HBM2 Ux = 0VX h?2 `2 Q7 i?2b2 bTQib 2tT M/b rBi? BM+`2 bBM; HQ / iQr `/b i?2 bT2+BK2M +2Mi`2X SQBMi MQX e HB2b i TQbBiBQM rBi? bi2TrBb2 HQ / /2+`2 b2X //BiBQM HHv iQ i?2 BM+`2 bBM; `2 Q7 i?2 /2b+`B#2/ bTQib QM i?2 `B;?i M/ H27i bB/2- MQr / `F2` Rj
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`2 `QmM/ i?2 +2Mi`2 HBM2 BM i?2 KB//H2 T `i + M #2 K /2 Qmi +QMM2+iBM; i?2 irQ bTQibX _B;?i #2?BM/ i?Bb TQBMi MQi?2` bm//2M HQ / /2+`2 b2 Q++m`bX AK ;2 MQX d b?Qrb i? i i?2 #B;;2` bTQi QM i?2 H27i@? M/ bB/2 MQr +Qp2`b i?2 2MiB`2 T i+? ?2B;?i M/- 2p2M KQ`2 BKTQ`i Mi- i?2 bK HH bTQi QM i?2 `B;?i@? M/ bB/2 bm//2MHv 2tT M/2/ M/ MQr +Qp2`b i?2 2MiB`2 ?2B;?i #2ir22M i?2 +2Mi`2 HBM2
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M/ i?2 mTT2` T i+? 2/;2X AK ;2 MQX 3 b?Qrb i?2 }M H 7 BHm`2 Q7 i?2 bT2+BK2MX "2ir22M TQBMib d M/ 3 H bi bm//2M HQ / /`QT ? TT2Mb +Q``2bTQM/BM; iQ i?2 }M H bT2+BK2M 7 BHm`2X PM i?2 T? b2 BK ;2 i?2 irQ `2 b `2 MQr 2MiB`2Hv +QM@ M2+i2/X b #27Q`2- i?2 K2+? MB+ H BM/B+ iQ` Q7 / K ;2 QMb2i i x = 0 TTHB2bX 6m`i?2`- i?2 /B`2+i +QKT `BbQM Q7 i?2 i?2`K H BK ;2 rBi? T?QiQ;` T? Q7 i?2
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/ K ;2/ bT2+BK2M b?Qrb i? i i?2 b? T2 Q7 i?2 /2i +?2/ `2 b +Q``2bTQM/ iQ i?2 / `F `2 QM i?2 T? b2 BK ;2- b22 6B;X RRX HbQ- i?2 HQ /f/BbTH +2K2Mi +m`p2 mM/2`HBM2b i? i i?2 / `F bTQib + M #2 +Q``2H i2/ iQ i?2 /?2bBp2 / K ;2 T`Q+2bbX *QMiQm` HBM2b Q7 i?2 / `F2` `2 b `2 bHB;?iHv #Hm``2/X _2K2K#2`BM; i?2 7mM+iBQMBM; Q7 Gh i?Bb + M #2 `2H i2/ iQ i?2 7 +i i? i 2Bi?2` R8 bmT2`BKTQb2/
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TB+im`2b `2 MQi 2MQm;? iQ 2MiB`2Hv }Hi2` MQBb2 2z2+ib Q` i? i / K ;2 T`Q;`2bb Bb 7 bi2` i? M 2tT2+i2/ M/ i?2 / K ;2 2tT M/b r?BH2 QM2 BK ;2 Bb ;2M2` i2/X 6 iB;m2 HQ /BM; hQ 2p Hm i2 i?2 TQi2MiB H Q7 i?2`KQ;` T?v QM 7 iB;m2- +v+HB+ i2bib `2 T2`@ 7Q`K2/X lM/2` +v+HB+ HQ /BM; i?2 bT2+BK2M /BbTH +2K2Mi Bb +QKT ` #Hv bK HHX
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6m`i?2`- i?2 2MiB`2 bT2+BK2M ?2 ib mT Qp2` iBK2 /m2 iQ i?2 +QMbi Mi +? M;2 #2ir22M HQ /BM; M/ mMHQ /BM;X "Qi? 2z2+ib H2 / iQ Km+? bK HH2` i2KT2` im`2 /Bz2`2M+2b +QKT `2/ iQ i?2 [m bB@bi iB+ HQ /BM; K2 MBM; i? i T i+? /2i +?K2Mi Bb TQbbB#Hv KQ`2 /B{+mHi iQ /2i2+iX .m2 iQ i?2 BM+`2 b2 Q7 # b2 i2KT2` im`2 Qp2` iBK2- HbQ i?2 i2KT2` im`2 b+ H2 /Bz2`b rBi? BM+`2 bBM; MmK#2` Q7 +v+H2b LX
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h?2`27Q`2- i2KT2` im`2 b+ H2b /Bz2` BM Q`/2` iQ K F2 i2KT2` im`2 /Bz2`2M+2b pBbB#H2X h?mb- i?2`K H BK ;2b + MMQi /B`2+iHv #2 +QKT `2/ [m MiBi iBp2HvX
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S bbBp2 h?2`KQ;` T?v h?2 / K ;2 T`Q+2bb mM/2` +v+HB+ HQ /BM; KQMBiQ`2/ rBi? T bbBp2 A`h Bb b?QrM BM 6B;X RkX 1 +? TB+im`2 Bb bbB;M2/ iQ i?2 HQ / +v+H2 MmK#2` N Bi Bb j9y
i F2MX AM +QMi` bi iQ i?2 [m bB@bi iB+ HQ /BM; i?2 ;Bp2M i2KT2` im`2 /Bbi`B#m@ iBQM K F2b i?2 / K ;2 #2 #2bi pBbB#H2 #2HQr i?2 +2Mi`2 HBM2X b 2tT2+i2/i2KT2` im`2 /Bz2`2M+2b `2 bK HH2` i? M mM/2` [m bB@bi iB+ HQ /BM;X "mi i?2 b K2 T`Q+2bb + M #2 b22M ?2`2ě i i?2 #mii DQBMi HBM2 HQ+ H `2 b r?2`2 i?2 rQ@ p2M }#`2 bi`m+im`2 Bb +H2 `Hv pBbB#H2 + M #2 bTQii2/ r?BH2 i?2 `2bi Q7 i?2 T i+?
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TT2 `b ?QKQ;2M2QmbX h?2b2 `2 b `2 2tT M/BM; 7`QK i?2 bT2+BK2M 2/;2b iQ i?2 +2Mi`2 rBi? BM+`2 bBM; iBK2 Q` MmK#2` Q7 +v+H2bX 6B;X Rj b?Qrb i?2 /B`2+i +QKT `BbQM #2ir22M i?2 i?2`K H BK ;2 M/ i?2 7 BH2/ bT2+BK2MX b #27Q`2i?2 /2i +?2/ `2 b +Q``2bTQM/ iQ i?2 bbmKTiBQMb K /2 # b2/ QM i?2 i?2`K H BK ;2bX
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GQ+F@BM h?2`KQ;` T?v AM 6B;X R9 T? b2 BK ;2b Q7 i?2 Gh M HvbBb b?Qr i?2 +v+HB+ / K ;2 T`QT@ ; iBQM- 2 +? TB+im`2 `2H i2/ iQ i?2 +Q``2bTQM/BM; HQ / +v+H2X AM +QMi` bi iQ i?2 `2bmHib `2+2Bp2/ rBi? T bbBp2 A`h ?2`2 i?2 T i+? bi`m+im`2 Bb pBbB#H2 HH Qp2` i?2 T i+? M/ i?mb /Q2b MQi +Q``2bTQM/ iQ HQ+ HHv /2i +?2/ `2 bX GQ+ H /2@
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i +?K2Mib `2 /BbiBM;mBb? #H2 b / `F bTQib QM i?2 i?2`K H BK ;2bX >2`2 ; BM+QMiQm`b TT2 ` #Hm``2/ TQbbB#Hv /m2 iQ i?2 bT2+BK2M KQp2K2Mi /m`BM; BK ;2 +[mBbBiBQMX b #27Q`2- bi `iBM; 7`QK i?2 +2Mi`2 HBM2 / `F HBM2 + M #2 bTQii2/ r?B+? 2tT M/b rBi? BM+`2 bBM; MmK#2` Q7 +v+H2bX AM i?2 T2MmHiBK i2 BK ;2 i?Bb `2 +Qp2`b TT`QtBK i2Hv ? H7 i?2 T i+? #2HQr i?2 +2Mi`2 HBM2 M/ BM i?2 H bi
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BK ;2- r?B+? Bb i?2 HQ / +v+H2 #27Q`2 7 BHm`2- HKQbi i?2 2MiB`2 HQr2` ? H7 Q7 i?2 T i+?X h?2 7 BH2/ bT2+BK2M- /2TB+i2/ BM 6B;X R8- +Q``2bTQM/b iQ i?2 Q#b2`p iBQMb K /2X h?2 / `F `2 b ;Q HQM; rBi? `2 b Q7 T i+? /2i +?K2Mi r?BH2 i?2 HB;?i2` `2 b `2 biBHH #QM/2/ iQ i?2 K2i H b?22ibX
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.B`2+i +QKT `BbQM #2ir22M T bbBp2 M/ HQ+F@BM i?2`KQ;` T?v 6Q` /B`2+i +QKT `BbQM Q7 i?2 irQ KQMBiQ`BM; K2i?Q/b +QmTH2 Q7 bT2+B@ K2Mb r2`2 KQMBiQ`2/ rBi? #Qi? K2i?Q/bX h?2 /B`2+i +QKT `BbQM #2ir22M i?2 i?2`K H BK ;2b ; BM2/ pB T bbBp2 A`h M/ i?2 T? b2 BK ;2b `2bmHiBM; 7`QK Gh Bb BHHmbi` i2/ BM 6B;X ReX Ai +QKT `2b i?2 `2bmHib Q7 #Qi? KQMBiQ`BM; K2i?Q/b i
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i?`22 /BbiBM+i +QKT ` #H2 MmK#2` Q7 HQ / +v+H2bX b + M #2 b22M- i?2 /2i +?2/ `2 b `2 Km+? #2ii2` pBbB#H2 rBi? Gh i? M rBi? T bbBp2 A`hX h?Bb r b 2tT2+i2/ bBM+2 +iBp2 i?2`KQ;` T?v K2i?Q/b + M T?vbB+ HHv K F2 bm#bm`7 +2 /Bb+QMiB@ MmBiB2b pBbB#H2 r?BH2 T bbBp2 A`h b?Qrb bm`7 +2 /Bb+QMiBMmBiB2bX P#pBQmbHv- HbQ bm#bm`7 +2 /272+ib H2 / iQ i2KT2` im`2 +? M;2b i? i `2 /2i2+i #H2 rBi? T bbBp2
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A`h b HQM; b i?2B` BKT +i QM bm`7 +2 i2KT2` im`2 +? M;2 Bb bi`QM; 2MQm;?X 9X .Bb+mbbBQM AM ;2M2` H- #Qi? K2i?Q/b- T bbBp2 A`h M/ Gh- `2 + T #H2 iQ KQMBiQ` T i+? /2i +?K2Mi M/ iQ T`QpB/2 BK ;2b iQ #2ii2` mM/2`bi M/ i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2bX
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h?2 bT2+BK2M ;2QK2i`v +?Qb2M K22ib i?2 +QM/BiBQM i? i QMHv /?2bBp2 / K ;2 Q++m`bX AM i2`Kb Q7 KQMBiQ`BM; / K ;2 T`QT ; iBQM i?BM T i+? ;Bp2b ;QQ/ `2bmHibX 6Q` `2 H TTHB+ iBQM T i+? i?B+FM2bb ? b iQ #2 /2bB;M2/ BM i2`Kb Q7 K2+? MB+ H T`QT2`iB2b /2iQ`B iBM; i?2 KQMBiQ`BM; T`QT2`iB2bX h?2 TTHB+ iBQM Q7 T bbBp2 A`h QM /2i2+iBM; bm#bm`7 +2 /272+ib ;Bp2b T`QKBb@
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BM; `2bmHibX q?2M KQMBiQ`BM; /?2bBp2 / K ;2 T`QT ; iBQM mM/2` K2+? MB+ H HQ /BM;- #Qi? 7Q` [m bB@bi iB+ HQ /BM; b r2HH b 7Q` +v+HB+ HQ /BM;- i?2 2tT M/BM; T i+? /2i +?K2Mi +QmH/ #2 Q#b2`p2/X P#pBQmbHv- i?2 2M2`;v /BbbBT iBQM i?`Qm;? i?2 T i+?- r b ?B;? 2MQm;? iQ #2 K2 bm`2/ b i2KT2` im`2 BM+`2 b2 i i?2 Q#D2+iǶb bm`7 +2X 6Q` T i+?2b i?B+F2` i? M i?2 QM2b i2bi2/ ?2`2- i?2 2z2+i Bb
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T`Q# #Hv H2bb bi`QM; M/ i?2`27Q`2 KB;?i MQi #2 pBbB#H2X HbQ- /2H KBM iBQM rBi?BM KmHiBH v2` T i+? + M Q++m` M/ TQbbB#Hv H2 / iQ bBKBH ` `2bmHib BM i2`Kb Q7 i2KT2` im`2 BM+`2 b2 QM i?2 bm`7 +2X S`QpB/2/ i? i i?2 KQp2K2Mi /m2 iQ HQ /BM;fmMHQ /BM; Qp2` BK ;BM; iBK2 Bb Re
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Bb ;Bp2MX M HvbBM; i?2 i?2`K H BK ;2b- Bi + M #2 b22M i? i BM T`BM+BTH2 Bi Bb TQbbB#H2 iQ ; BM ;QQ/ FMQrH2/;2 #Qmi b? T2 M/ bBx2 Q7 i?2 / K ;2 2tT MbBQM Qp2` iBK2X "mi- Bi Bb BKTQ`i Mi iQ `2K2K#2` i? i i?2 TB+im`2 /Q2b MQi T`QpB/2 i?2 +im H bBx2 M/ + M QMHv #2 b22M b `272`2M+2 TQBMiX AM Q`/2` iQ K2 bm`2 i?2 bBx2 Q7 bm#bm`7 +2 / K ;2 Gh ? b iQ #2 TTHB2/ iQ MQM@KQpBM; Q#D2+iX
9yy
MQi?2` /` r# +F Bb- i? i i2bi `2 iBK2@+QMbmKBM;X 6Q` i?2 l. bT2+BK2Mb i2bib i F2b `QmM/ kkĜk8 b- r?BH2 7Q` i?2 J. bT2+BK2Mb Bi i F2b KQ`2 i? M kyy bX 8X *QM+HmbBQMb h?2 irQ i?2`KQ;` T?v K2i?Q/b Gh M/ T bbBp2 A`h r2`2 TTHB2/ iQ KQMBiQ` bm#bm`7 +2 / K ;2 T`QT ; iBQM 7Q` HQ /2/ bT2+BK2MbX >2`2- Bi r b BMp2biB; i2/
9y8
?Qr i?2b2 K2i?Q/b `2 TTHB+ #H2 iQ H #Q` iQ`v i2bib BM Q`/2` iQ ;2i BM7Q`K @ iBQM #Qmi i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2b Q7 #QM/2/ `2T B`b mM/2` K2+? MB+ H HQ /BM;X h?2 BMp2biB; iBQM 7Q+mb2/ QM i?2 K BM T ` K2i2`, i?2 / K ;2 rBi?BM i?2 /?2bBp2 H v2`X .Qm#H2 H T DQBMi bT2+BK2Mb +QMbBbiBM; Q7 K2i HHB+ M/ *6_S /?2`2M/b r2`2 i2bi2/ mM/2` [m bB@bi iB+ M/ 7 iB;m2 HQ /BM;X aT2+BK2M /BK2M@
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bBQMb r2`2 / Ti2/ iQ i?2 ;Bp2M +QM/BiBQMb iQ K22i i?2 `2[mB`2K2Mib `2bmHiBM; 7`QK i?2 KQMBiQ`BM; K2i?Q/bX Ai r b b?QrM i? i mM/2` i?2 ;Bp2M +B`+mKbi M+2b #Qi? K2i?Q/b `2 #H2 iQ ;Bp2 BM7Q`K iBQM QM i?2 bm#bm`7 +2 / K ;2 T`Q+2bb2bX h2KT2` im`2 Q` T? b2 /Bz2`2M+2b QM i?2 i?2`K H BK ;2b +QmH/ #2 `2H i2/ iQ i?2 T`2HBKBM `v K2+? M@
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B+ H 2biBK iBQMb QM i?2 TH +2 Q7 / K ;2 QMb2iX 6Q` i?2 [m bB@bi iB+ i2bib HbQi?2 K2+? MB+ H #2? pBQm` +QmH/ T`QpB/2 bbBbi M+2 BM i?2 BK ;2 BMi2`T`2i iBQMX *? ` +i2`BbiB+ TQBMib BM i?2 HQ /f/BbTH +2K2Mi +m`p2 +Q``2H i2 rBi? i?2 / K ;2 T`Q;`2bb T`2bmK2/ 7`QK i?2 BK ;2bX h?2 BM/B+ iQ` 7Q` i?2 / K ;2 HQ+ iBQM Bb /B`2+i +QKT `BbQM Q7 i?2 7 BH2/ bT2+BK2Mb M/ i?2 +Q``2bTQM/BM; i?2`K H BK@
9ky
;2b i }M H 7 BHm`2X >2`2 ; BM- ;QQ/ +Q``2H iBQM #2ir22M i?2 BK ;2b M/ i?2 bT2+BK2Mb + M #2 b22MX 6Q` T bbBp2 A`h- HQ+ H BM+`2 b2b BM i2KT2` im`2 + M #2 K /2 Qmi QM i?2 bT2+BK2M Rd
bm`7 +2 i? i `2 2tT M/BM; rBi? BM+`2 bBM; HQ /X lM/2` [m bB@bi iB+ HQ /BM; i?2 K2 bm` #H2 i2KT2` im`2 /Bz2`2M+2 Bb ?B;?2` i? M mM/2` 7 iB;m2 HQ /BM;X h?2 9k8
Qp2` HH i2KT2` im`2 BM+`2 b2 BM i?2 bT2+BK2M mM/2` HQM;@i2`K HQ /BM; K F2b Bi #Bi KQ`2 +? HH2M;BM; iQ /Bz2`2MiB i2 #2ir22M / K ;2/ M/ mM/ K ;2/ `2;BQMbX AM7Q`K iBQM #Qmi i?2 +m``2Mi bi i2 `2 /BbTH v2/ BM HBp2 BK ;2 r?B+? K F2b Bi 2 bv iQ ? M/H2 T`QpB/BM; +QKT`2?2MbBp2 T`2b2Mi iBQM Q7 QM;QBM; T`Q+2bb2bX Gh + M T`QpB/2 M Qp2`pB2r QM i?2 / K ;2 T`Q;`2bb i /BbiBM+i TQBMib BM [m H@
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Bi iBp2 K MM2`X h?2 /Bz2`2MiB iBQM #2ir22M / K ;2/ M/ mM/ K ;2/ `2;BQMb Bb Km+? 2 bB2` i? i 7Q` T bbBp2 A`hX q?2M +QKT `BM; i?2 `2bmHib Q7 T bbBp2 A`h M/ Gh QM2 + M b v i? i mM/2` bT2@ +B}+ +B`+mKbi M+2b 2 +? K2i?Q/ + M #2 mb2/ iQ KQMBiQ` i?2 /2i +?K2Mi T`Q+2bb Q7 *6_S `2T B` T i+?2bX 6Q` bm#bm`7 +2 /272+ib Gh T`QpB/2b i?2`K H BK ;2b
9j8
rBi? ?B;?2` +QMi` bi #2ir22M / K ;2/ M/ mM/ K ;2/ `2 bX 6m`i?2`- 7Q` T b@ bBp2 A`h i?2 BMBiB H / K ;2 /2i2+iBQM Bb KQ`2 /B{+mHi i? M 7Q` Gh- #2+ mb2 i?2 bK HH2` i?2 /272+i Bb- i?2 H2bb 2M2`;v Bb +QMp2`i2/ BMiQ i?2`K H 2M2`;vX "mi- BM i2`Kb Q7 K M ;2 #BHBiv T bbBp2 A`h ? b +QMbB/2` #H2 /p Mi ;2b, Ai Bb 2 bv iQ ? M/H2 M/ HBp2 BK ;2b `2 i F2MX h?2`K H BK ;2b `2 MQi T`Q+2bb2/ M/ i?2
99y
KQpBM; HBp2 BK ;2- r?B+? mM7Q`imM i2Hv + MMQi #2 T`2b2Mi2/ ?2`2- `2p2 Hb i?2 / K ;2 T`Q;`2bb #2? pBQm` 2p2M #2ii2` i? M i?2 TB+im`2b + M b?QrX *QM+Hm/BM;- #Qi? A`h K2i?Q/b `2 ;2M2` HHv + T #H2 iQ KQMBiQ` i?2 / K ;2 T`Q;`2bb Q7 T i+? /2i +?K2Mi mM/2` [m bB@bi iB+ M/ +v+HB+ HQ /BM; 7Q` irQ /B7@ 72`2Mi 2TQtv@# b2/ l. M/ J. *6_S T i+?2b rBi? i?B+FM2bb Q7 yX9 KK M/
998
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