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Anisotropic elastic properties of human femoral cortical bone and relationships with composition and microstructure in elderly

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HAL Id: hal-02107625

https://hal.archives-ouvertes.fr/hal-02107625

Submitted on 23 Apr 2019

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Anisotropic elastic properties of human femoral cortical

bone and relationships with composition and

microstructure in elderly

Xiran Cai, Hélène Follet, Laura Peralta, Marc Gardegaront, Delphine Farlay,

Rémy Gauthier, Boliang Yu, Evelyne Gineyts, Cécile Olivier, Max Langer, et

al.

To cite this version:

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Accepted Manuscript

Anisotropic elastic properties of human femoral cortical bone and relationships

with composition and microstructure in elderly

Xiran Cai, Hélène Follet, Laura Peralta, Marc Gardegaront, Delphine Farlay,

Rémy Gauthier, Boliang Yu, Evelyne Gineyts, Cécile Olivier, Max Langer,

Aurelien Gourrier, David Mitton, Françoise Peyrin, Quentin Grimal, Pascal

Laugier

PII:

S1742-7061(19)30218-1

DOI:

https://doi.org/10.1016/j.actbio.2019.03.043

Reference:

ACTBIO 6025

To appear in:

Acta Biomaterialia

Received Date:

24 November 2018

Revised Date:

19 March 2019

Accepted Date:

20 March 2019

Please cite this article as: Cai, X., Follet, H., Peralta, L., Gardegaront, M., Farlay, D., Gauthier, R., Yu, B., Gineyts,

E., Olivier, C., Langer, M., Gourrier, A., Mitton, D., Peyrin, F., Grimal, Q., Laugier, P., Anisotropic elastic properties

of human femoral cortical bone and relationships with composition and microstructure in elderly, Acta

Biomaterialia (2019), doi:

https://doi.org/10.1016/j.actbio.2019.03.043

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KK@i?B+F +`Qbb b2+iBQM rb +mi T2`T2M/B+mH` iQ i?2 #QM2 tBb 7`QK i?2 KB/@/BT?vbBbX h?2 +`Qbb b2+iBQM rb b2T`i2/ iQ 7Qm` b2+iBQMb +Q``2bTQM/BM; iQ 7Qm` [m/`Mib UHi2`H- K2/BH-Mi2`BQ` M/ TQbi2`BQ`VX AM 2+? Q7 i?2 Hi2`H M/ K2/BH MiQKB+H [m/`Mib- j `2+iM@ ;mH` T`HH2H2TBT2/ b?T2/ bT2+BK2Mb Ub2i OR- Ok M/ OjV r2`2 T`2T`2/ HQM; i?2 tBH /B`2+iBQM 7Q` /Bz2`2Mi i2biBM; Tm`TQb2b U6B;X RVX h?2 2HbiB+ T`QT2`iB2b- i?2 KB+`Qbi`m+im`H +?`+i2`BbiB+b Q7 i?2 TQ`2 M2irQ`F M/ i?2 /2;`22 Q7 KBM2`HBxiBQM r2`2 Q#iBM2/ 7`QK b2i ORX aT2+BK2Mb Ok r2`2 T`2T`2/ 7Q` K2+?MB+H i2biBM; BKBM; i 2biBKiBM; iQm;?@ M2bbX h?2 `2bmHib Q7 K2+?MB+H i2biBM; ?p2 #22M `2TQ`i2/ BM  b2T`i2 Tm#HB+iBQM (j9)X h?2 `2bB/m2b `QmM/ i?2 b2i Ok 7i2` +miiBM; 7Q` Ok r2`2 mb2/ iQ bb2bb +QHH;2M T`QT2`@ iB2bX bb2bbK2Mi Q7 KBM2`H T`QT2`iB2b rb +?B2p2/ mbBM; b2i OjX h?2 MQKBMH bT2+BK2M bBx2 Q7 b2i OR- Ok M/ Oj rb j×9×8 KK3- j×9×k8 KK3 M/ j×9×yX8 KK3 BM `/BH

UtBb RV- +B`+mK72`2MiBH UtBb kV M/ tBH /B`2+iBQM UtBb jV- `2bT2+iBp2Hv- /2}M2/ #v i?2 MiQKB+ b?T2 Q7 i?2 72KQ`H /BT?vbBbX HH bT2+BK2Mb r2`2 F2Ti ?v/`i2/ /m`BM; bKTH2 T`2T`iBQMX

(7)

Lateral & Medial #1 #3 #2 RUS (stiffness) Biochemistry (collagen cross-links) FTIRM (mineral, crystal) SR- CT 6.5 m (DMB, microstructure) a d f e b c 1 2 3 3x4x5 mm3 3x4x25 mm3 3x4x0.5 mm3 residues close-by #2 spectroscopy micro-computed tomography infrared microspectroscopy RUS = resonant ultrasound

FTIRM = Fourier transform SR- CT = synchrotron radiation Mechanical testing

(data not shown in this work)

2500 2000 1500 1000 500 0.25 0.50 0.75 1.00 1.25 1.75 2.00 Wave number (cm-1) Absorbance

6B;m`2 R, aKTH2 T`2T`iBQM T`Q+2/m`2 M/ i?2 `2T`2b2MiiBQMb Q7 i?2 2tT2`BK2MiH KQ/HBiB2b BM+Hm/BM; _la- a_@µ*h- #BQ+?2KBbi`v M/ 6hA_JX UV  +`Qbb@b2+iBQM Q7 72KQ`H #QM2 i i?2 KB/@/BT?vbBb rb 2ti`+i2/X U#V h?`22 `2+iM;mH` T`HH2H2TBT2/ b?T2/ bT2+BK2Mb Ub2i OR- Ok M/ OjV r2`2 T`2T`2/ HQM; i?2 tBH /B`2+iBQM i #Qi? i?2 Hi2`H M/ K2/BH [m/`MibX U+V _la K2bm`2K2Mib 7Q` #QM2 biBzM2bbX U/V SQ`2 M2irQ`F Q#iBM2/ i`2iBM; #QM2 T?b2 b #+F;`QmM/X U2V "QM2 `2bB/m2b `QmM/ Ok 7i2` +miiBM; 7Q` Ok r2Mi iQ #BQ+?2KBbi`v 2tT2`BK2Mib iQ [mMiB7v i?2 +QHH;2M M/ +`Qbb@HBMFbX U7V  6hA_J bT2+i` +QMiBMBM; +QKTQbBiBQMH BM7Q`KiBQM Q7 i?2 #QM2 bT2+BK2MX

kXkX "QM2 biBzM2bb K2bm`2K2Mib

h?2 biBzM2bb +QMbiMib Q7 #QM2 bT2+BK2Mb Ub2i ORV r2`2 B/2MiB}2/ mbBM; _la 7QHHQrBM; i?2 T`Q+2/m`2 T`2pBQmbHv /2b+`B#2/ BM "2`M`/ 2i HX (j8)X AM _la- i?2 biBzM2bb +QMbiMib `2 2biBKi2/ #v bQHpBM; M BMp2`b2 T`Q#H2K r?B+? +QMbBbib Q7 i?2 +QKT`BbQM Q7 K2bm`2/ `2bQMMi 7`2[m2M+B2b iQ KQ/2H@T`2/B+i2/ 7`2[m2M+B2b U2[miBQM XRVX "`B2~v- i?2 2tT2`BK2Mi +QMbBbi2/ Q7 i?2 7QHHQrBM; bi2Tb, URV  #QM2 bT2+BK2M rb TH+2/ QM irQ QTTQbBi2 +Q`M2`b #2ir22M irQ mHi`bQMB+ b?2`rp2 i`Mb/m+2`b UoR89_J- SMK2i`B+b- qHi?K- JV- QM2 7Q` 2KBbbBQM M/ QM2 7Q` `2+2TiBQM- iQ +?B2p2  7`22 #QmM/`v +QM/BiBQM 7Q` pB#`iBQMX UkV h?2 7`2[m2M+v `2bTQMb2 Q7 i?2 pB#`iBQM Q7 i?2 bT2+BK2M BM i?2 Ryy @ eyy F>x 7`2[m2M+v #M/@ rB/i? rb KTHB}2/ #v  #`Q/#M/ +?`;2 KTHB}2` U>Z@R8 J@Ry h- 62KiQ J2bbi2+?MBF :K#>- "2`HBM- :2`KMvV M/ i?2M `2+Q`/2/ #v  p2+iQ` M2irQ`F MHvx2` U"Q/2 Ryy- PKB@ +`QM 1H2+i`QMB+b :K#>- EHmb- mbi`BV- 7`QK r?B+? i?2 ky iQ jy }`bi `2bQMMi 7`2[m2M+B2b r2`2 2ti`+i2/X UjV bbmKBM;  i`Mbp2`b2Hv BbQi`QTB+ bvKK2i`v (je- jd) UTHM2 R@k Bb i?2 THM2 Q7 BbQi`QTvV- i?2 biBzM2bb +QMbiMib Cij UoQB;i MQiiBQM, ij = 11; 33; 13; 44; 66V- r2`2

miQKiB+HHv +H+mHi2/ #v QTiBKBxBM; i?2 KBb}i 7mM+iBQM #2ir22M i?2 2tT2`BK2MiH M/ KQ/2H T`2/B+i2/ `2bQMMi 7`2[m2M+B2b UBMp2`b2 T`Q#H2KV- r?B+? rb 7Q`KmHi2/ BM  "v2bBM

(8)

7`K2rQ`F (j3)X h?2 T`BQ` BM7Q`KiBQM Q7 i?2 /Bbi`B#miBQM Q7 i?2 biBzM2bb +QMbiMib `2[mB`2/ 7Q` i?2 "v2bBM MHvbBb rb iF2M 7`QK  T`2pBQmb bim/v (kk)X

AM i?2 2HbiB+ i2MbQ`- C11 M/ C33 r?B+? +Q``2bTQM/ iQ Tm`2 HQM;Bim/BMH rp2b `2 /2@

MQi2/ b HQM;Bim/BMH biBzM2bb +QMbiMibc C44M/ C66r?B+? +Q``2bTQM/ iQ Tm`2 b?2` rp2b

`2 /2MQi2/ b b?2` biBzM2bb +QMbiMibc C12M/ C13`2 i?2 Qz@/B;QMH biBzM2bb +QMbiMib

+Q``2bTQM/BM; iQ i?2 KBt2/@KQ/2 rp2b (jN)X 6Q` i`Mbp2`b2 BbQi`QTv- i?2 7QHHQrBM; `2HiBQMb ?QH/- C11= C22- C44= C55- C13= C23 M/ C12 = C11− 2C66X 6Q` i`Mbp2`b2Hv BbQi`QTB+

bvKK2i`v- i?2 2M;BM22`BM; KQ/mHB bm+? b uQmM;Ƕb KQ/mHB M/ SQBbbQMǶb `iBQb +M #2 +QM@ p2`i2/ #v i?2 biBzM2bb i2MbQ` (8)X h?2 Kbb /2MbBiv Q7 2+? bT2+BK2M- r?B+? Bb M22/2/ BM biBzM2bb /2i2`KBMiBQM rBi? _la- rb /2`Bp2/ 7`QK i?2 p2`;2 pHm2b Q7 7Qm` Kbb Ua`iQ@ `Bmb *Skk9b- T`2+BbBQM, yXR K;V M/ /BK2MbBQMb K2bm`2K2Mib UJBimiQvQ *QQHMi S`QQ7 *HBT2` 8yy@eye- T`2+BbBQM, yXyR KKVX h?2 2tT2`BK2MiH 2``Q`b +mb2/ #v M B``2;mH`Biv Q7 i?2 bT2+BK2M ;2QK2i`v M/ #v mM+2`iBMiB2b Q7 i?2 2ti`+i2/ `2bQMMi 7`2[m2M+B2b- 7QHHQrBM; i?Bb T`QiQ+QH- ivTB+HHv +mb2 M 2``Q` Q7 TT`QtBKi2Hv RXdW 7Q` i?2 b?2` biBzM2bb +QMbiMib M/ TT`QtBKi2Hv jXRW 7Q` i?2 +QKT`2bbBQM M/ Qz@/B;QMH biBzM2bb +QMbiMib (9y)X

kXjX "QM2 KB+`Qbi`m+im`2

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AM i?2 `2+QMbi`m+i2/ pQHmK2 Q7 2+? bT2+BK2M-  pQHmK2 Q7 BMi2`2bi UoPA- bBx2/ TT`Qt@ BKi2Hv 2.8 × 3.9 × 4.8 KK3V rb b2H2+i2/ KMmHHv 7Q` KQ`T?QK2i`B+ MHvbBbX 6QHHQrBM;

(k9)- i?2 oPAb r2`2 #BM`Bx2/ #v bBKTH2 i?`2b?QH/BM; i`2iBM; i?2 pQB/ pQHmK2b b  bQHB/ M/ i?2 #QM2 T?b2 b  #+F;`QmM/ U6B;X kVX h?2M- i?2 KQ`T?QK2i`B+ T`K2i2`b HBbi2/ BM h#H2 R r2`2 +H+mHi2/ mbBM; i?2 bQ7ir`2 *hM Uo RXReXR- aFvb+M Lo- EQMiB+?- "2H;BmKVX

(9)

h#H2 R, JB+`Qbi`m+im`2 p`B#H2b bb2bb2/ #v *hM M/ i?2B` /2}MBiBQMbX

o`B#H2 lMBi .2}MBiBQM

JB+`Qbi`m+im`2 ;`QmT

φ % TQ`2 pQHmK2 7`+iBQM

SQafSQo KK−1 TQ`2 bm`7+2 iQ TQ`2 pQHmK2 `iBQ SQL KK−1 TQ`2 MmK#2` T2` KBHHBK2i2`

SQ.K µK p2`;2 /BK2i2` Q7 i?2 TQ`2b

SQaT µK p2`;2 b2T`iBQM #2ir22M TQ`2b SQS7 KK−1 TQ`2 Tii2`M 7+iQ`- HQr2` SQS7 BM/B+i2b ?B;?2` +QM+pBiv- BX2X-#2ii2`@+QMM2+i2/ TQ`2 M2irQ`F *QMM. KK−3 +QMM2+iBpBiv /2MbBiv-  K2bm`2 Q7 i?2 /2;`22 iQ r?B+?  TQ`2 Bb KmHiBTHv +QMM2+i2/

aJA XmX bi`m+im`2 KQ/2H BM/2t- i?2 `2HiBp2 T`2pH2M+2 Q7 `Q/b M/ THi2b BM  j. TQ`2 M2irQ`F

(a) (b)

6B;m`2 k, UV PM2 i`Mbp2`b2 ;`v@H2p2Hb +`Qbb@b2+iBQM Q7 +Q`iB+H #QM2 b+MM2/ #v a_@µ*h b?QrBM; i?2 /Bbi`B#miBQM Q7 TQ`2b M/ KBM2`HX U#V j@. `2M/2`BM; Q7 i?2 TQ`2 M2irQ`F 7i2` #BM`BxiBQM Q7 i?2 pQHmK2X

kX9X .2;`22 Q7 KBM2`HBxiBQM Q7 #QM2

 TBMF #2K rb mb2/ BM i?2 }`bi a_@µ*h 2tT2`BK2Mib QM i?2 #2KHBM2 A. RN r?B+? /B/ MQi T`QpB/2 biBb7vBM; pHm2b Q7 /2;`22 Q7 KBM2`HBxiBQM Q7 #QM2 U.J"VX >2M+2- a_@µ*h 2tT2`BK2Mib r2`2 HmM+?2/ 7Q`  b2+QM/ iBK2 QM i?2 bK2 bT2+BK2Mb Ub2i ORV QM i?2 #2KHBM2 A. Rd i 1a_6 mbBM;  KQMQ+?`QKiB+ #2KX h?2 #2K 2M2`;v rb ++m`i2Hv imM2/ iQ keX8 F2o #v mbBM;  Gm2@Gm2 KQMQ+?`QKiQ`X h?2 /2i2+iQ` rb  S*P 2/;2 *JPa +K2` +QmTH2/ iQ  u: b+BMiBHHiQ`X h?2 TBt2H bBx2 rb b2i iQ eXyj µK M/ +QmMiBM; iBK2 rb yX9 b T2` T`QD2+iBQMX  iQiH b2i Q7 9yyy T`QD2+iBQMb rb +[mB`2/ mM/2` jey◦vB2H/BM; iQ

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(10)

h?2 ;`v H2p2Hb Q7 i?2 BK;2b U6B;X kV- +Q``2bTQM/BM; iQ i?2 s@_v HBM2` #bQ`TiBQM +Q2{+B2Mib Q7 #QM2- r2`2 mb2/ iQ 2biBKi2 i?2 j. /Bbi`B#miBQM Q7 KBM2`H +QMi2Mi rBi?BM  #QM2 bT2+BK2M b /2b+`B#2/ BM (98)X b i?2 HBM2` ii2MmiBQM Q7 +QHH;2M M/ ri2` `2 p2`v +HQb2 (9e- 9d)- r2 ?p2 +QMbB/2`2/ QMHv  irQ@Ki2`BH /2+QKTQbBiBQM #bBb- K/2 Q7 ?v/`QtvTiBi2 M/ ri2`X AM //BiBQM- bBM+2 r2 mb2/  KQMQ+?`QKiB+ bvM+?`Qi`QM *h +[mBbBiBQM- i?2 2M2`;v Q7 i?2 BM+QKBM; s@`v #2K Bb T2`72+iHv FMQrMX h?2M i?Bb HHQrb i?2 2biBKiBQM Q7 i?2 KBM2`H +QM+2Mi`iBQM U;f+K3V- MK2Hv .J"- 7`QK i?2 K2bm`2/

HBM2` ii2MmiBQM +Q2{+B2Mi b /2b+`B#2/ BM (98)X h?2 .J" rb p2`;2/ QM i?2 r?QH2 j@. #QM2 T?b2 7i2` iFBM; +`2 Q7 T`iBH pQHmK2 2z2+ibX

kX8X "BQ+?2KB+H K2bm`2K2Mib

h?2 bT`2/ Ki2`BHb 7`QK i?2 T`2T`iBQM Q7 bT2+BK2Mb Ok r2`2 +mi BMiQ bKHH TB2+2b-TQr/2`2/- /2KBM2`HBx2/- 2ti2MbBp2Hv rb?2/ M/ ?v/`QHvx2/X h?2 KQmMi Q7 2MxvKiB+ +`Qbb@HBMFb U.>GLG- >GLG- Su. M/ .S.V- MQM@2MxvKiB+ +`Qbb@HBMF T2MiQbB/BM2 US1LV Uh#H2 kV M/ +QHH;2M r2`2 [mMiB}2/ mbBM; i?2 ?v/`QHvbi2bX .2iBH2/ T`Q+2bb Q7 i?2 #BQ+?2KB+H 2tT2`BK2Mib ?b #22M `2TQ`i2/ BM :mi?B2` 2i HX (93)X h?2 K2bm`2K2Mi 2``Q` Q7 i?2 +`Qbb@HBMFb rb 2pHmi2/ T`2pBQmbHv BM #QpBM2 #QM2 r?B+? rb #2ir22M 8Xk@RyXdW (9N- 8y)X h?2 p`B#H2b 7`QK #BQ+?2KB+H K2bm`2K2Mib `2 bmKK`Bx2/ BM h#H2 k ;`QmT2/ b +QHH;2M p`B#H2bX

kXeX 6Qm`B2` i`Mb7Q`K BM7``2/ KB+`QbT2+i`Qb+QTv

6hA_J MHvbBb Bb /QM2 QM k µK i?BM b2+iBQMbX hQ /Q bQ- 7i2` /Bbb2+iBQM Q7 #QM2- bKTH2b Ub2i OjV r2`2 }t2/ BM dyW 2i?MQH 7Q` irQ r22Fb- /2?v/`i2/ 7Q` 93? BM RyyW 2i?MQH- bm#@ biBimi2/ 7Q` 93? BM K2i?vH+v+HQ?2tM2 M/ i?2M 2K#2//2/ BM K2i?vHK2i?+`vHi2 UJJVX h?2 b2+iBQMb Q7 k µK i?B+FM2bb r2`2 Q#iBM2/ rBi?  KB+`QiQK2 SQHv+mi 1 U_2B+?2`i@CmM;-G2B+- :2`KMvV M/ biQ`2/ #2ir22M irQ bHB/2b i `QQK i2KT2`im`2X h?2 `r bT2+i` r2`2 +QHH2+i2/ i R +K−1 `2bQHmiBQM M/ p2`;2/ #v 9y b+Mb T2` b+MM2/ `2;BQM BM i`MbKBbbBQM

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mb2/  *bb2;`BM Q#D2+iBp2 Q7 MmK2`B+H T2`im`2 yXeX h?2 bvbi2K ?b  bTiBH `2bQHmiBQM Q7 Ry µK i ivTB+H KB/BM7``2/ rp2H2M;i?bX *QMi`B#miBQM 7`QK B` rb bm#i`+i2/ 7`QK Q`B;BMH bT2+i`mKX h?2 T`2T`Q+2bbBM; UJJ `2KQpH- #b2HBM2 +Q``2+iBQMV M/ i?2 MHvbBb #v T2F }iiBM; r2`2 2t2+mi2/ mbBM; i?2 GuPa aT2+i`mK MHvbBb bQ7ir`2 (8R)X

6`QK i?2 bT2+i`mK- i?2 7QHHQrBM; p`B#H2b Uh#H2 kV r2`2 `2i`B2p2/, KBM2`H iQ Ki`Bt `iBQ UJBMP`;V r?B+? `2T`2b2Mib i?2 `iBQ Q7 i?2 KBM2`H T?b2 Qp2` i?2 Q`;MB+ T?b2 U+QH@ H;2MB+ M/ MQM@+QHH;2MB+ T`Qi2BMbV- KBM2`H Kim`Biv UJBMJiV r?B+? `2~2+ib i?2 i`Mb7Q`@ KiBQM Q7 MQM TiBiB+ T?QbT?i2b Q7 i?2 ?v/`i2/ Hv2` BMiQ TiBiB+ T?QbT?i2b +QMiBM2/

(11)

BM i?2 +`vbiH +Q`2- +`#QMiBQM U*`#QMV r?B+? `2T`2b2Mib i?2 [mMiBiv Q7 +`#QMi2b BM@ +Q`TQ`i2/ BMiQ i?2 #QM2 KBM2`H i  H#BH2 bBi2 M/ bm#biBimi2/ BM i?2 TiBi2 HiiB+2 /m2 2Bi?2` iQ SP4 Q` P>- +`vbiHHBMBiv BM/2t U*`vAM/V r?B+? +Q``2bTQM/b iQ #Qi? +`vbiHHBM2

/QKBM bBx2 M/ ?Qr r2HH i?2 BQMb Q7 i?2 +`vbiH `2 Q`/2`2/ BM i?2 mMBi +2HHb M/ +QHH;2M Kim`Biv U*QHHJiVX h?2 `2T`Q/m+B#BHBiv Q7 2+? 6hA_J p`B#H2 rb +H+mHi2/ QM ?m@ KM +Q`iB+H #QM2 b2+iBQM- rBi?  `2;BQM Q7 BMi2`2bi Ryy µK × Ryy µK UbTiBH `2bQHmiBQMV-i k +K−1 bT2+i`H `2bQHmiBQMX Ai rb +H+mHi2/ b i?2 biM/`/ /2pBiBQM Qp2` K2M Q7

Ry K2bm`2K2Mib M/ rb- 7Q` JBMP`; UyX9WV- JBMJi UyX3WV- *`#QM UyXjWV- *`vAM/ UyXdWV M/ *QHHJi U8XeWVX JQ`2 /2iBHb Q7 i?2 /2}MBiBQM Q7 i?2b2 p`B#H2b +M #2 7QmM/ BM h#H2 kX h?2 7Q`K2` 7Qm` p`B#H2b r2`2 ;`QmT2/ BM i?2 KBM2`H p`B#H2 ;`QmT M/ i?2 Hbi QM2 *QHHJi rb ;`QmT2/ BM i?2 +QHH;2M p`B#H2 ;`QmTX

h#H2 k, *QKTQbBiBQMH p`B#H2b bb2bb2/ #v 6hA_J- a_@µ*h M/ #BQ+?2KBbi`v 2tT2`BK2Mib M/ i?2B` /2}MBiBQMbX

o`B#H2 lMBi .2}MBiBQM JQ/HBiv

JBM2`H ;`QmT

.J" ;f+K3 .2;`22 Q7 KBM2`HBxiBQM Q7 #QM2 a_@µ*h

JBMP`; MQ mMBi JBM2`H iQ Ki`Bt `iBQ- i?2 `iBQ Q7 i?2 ν1ν3SP4 `2 U910 − 1184 +K−1V Qp2` i?2 KB/2 A `2 U1592 − 1730 +K−1V

6hA_J

JBMJi MQ mMBi JBM2`H Kim`Biv- i?2 `iBQ Q7 i?2 T@ iBiB+ U∼Ryjy +K−1 T2FV Qp2` MQM TiBiB+ U∼RRRy +K−1T2FV

6hA_J

*`#QM MQ mMBi *`#QMiBQM- i?2 `iBQ Q7 i?2 ν2*P3 `2 U862 − 894 +K−1V Qp2` i?2 ν1ν3SP4 `2

6hA_J *`vAM/ +K *`vbiHHBMBiv BM/2t- i?2 BMp2`b2 Q7 i?2 7mHH

rB/i? 6hA_J i ?H7 KtBKmK Q7 i?2 ∼ey9 +K−1T2F

6hA_J

*QHH;2M ;`QmT

*QHHJi MQ mMBi *QHH;2M Kim`Biv- ∼Reey +K−1 T2F Qp2` ∼ReNy +K−1 T2F 6hA_J .>GLG KKQHfKQH +QHH;2M .B/?v/`QtvHvbBMQMQ`H2m+BM2- BKKim`2 2M@ xvKiB+ +`Qbb@HBMFb "BQ+?2KBbi`v >GLG KKQHfKQH +QHH;2M >v/`QtvHvbBMQMQ`H2m+BM2- BKKim`2 2Mxv@ KiB+ +`Qbb@HBMFb "BQ+?2KBbi`v Su. KKQHfKQH +QHH;2M

Sv`B/BMQHBM2- Kim`2 2MxvKiB+ +`Qbb@HBMFb "BQ+?2KBbi`v

.S. KKQHfKQH +QHH;2M .2QtvTv`B/BMQHBM2- Kim`2 2MxvKiB+ +`Qbb@HBMFb "BQ+?2KBbi`v S1L KKQHfKQH +QHH;2M

S2MiQbB/BM2- MQM@2MxvKiB+ +`Qbb@HBMFb "BQ+?2KBbi`v

*QHH W *QHH;2M T2`+2Mi;2 #v r2B;?i "BQ+?2KBbi`v

(12)

kXdX aiiBbiB+b

7i2` +H+mHiBM; i?2 /2b+`BTiBp2 biiBbiB+b-  biiBbiB+H MHvbBb rb +QM/m+i2/ iQ 2t@ ?B#Bi 2KTB`B+H `2HiBQMb?BTb #2ir22M 2HbiB+Biv- QM i?2 QM2 ?M/- M/ i?2 p`B#H2b BM i?2 KB+`Qbi`m+im`H- KBM2`H M/ +QHH;2M ;`QmTb- QM i?2 Qi?2` ?M/X h?2 K2M pHm2- biM@ /`/ /2pBiBQM Ua.V M/ +Q2{+B2Mi Q7 p`BiBQM U*oV- 2tT`2bb2/ b i?2 `iBQ a.fJ2M `2 `2TQ`i2/ 7Q` 2+? p`B#H2X LQ`KHBiv Q7 i?2 /Bbi`B#miBQM Q7 i?2 p`B#H2b rb 2pHmi2/ mb@ BM; i?2 a?TB`Q@qBHF i2biX PM2@rv MHvbBb Q7 p`BM+2 ULPoV M/ qBH+QtQM i2bi U7Q` i?2 p`B#H2b 7BHBM; i?2 MQ`KHBiv i2biV r2`2 T2`7Q`K2/ iQ 2pHmi2 i?2 /Bz2`2M+2b Q7 i?2 /i b2ib 7`QK Hi2`H M/ K2/BH MiQKB+H [m/`MibX b bQK2 p`B#H2b r2`2 MQi MQ`@ KHHv /Bbi`B#mi2/- r2 mb2/ aT2`KM `MF +Q2{+B2Mib BM i?2 +Q``2HiBQM MHvb2b #2ir22M biBzM2bb +QMbiMib M/ 2+? Q7 i?2 KB+`Qbi`m+im`H M/ +QKTQbBiBQMH p`B#H2bX h?Qb2 p`B@ #H2b bB;MB}+MiHv +Q``2Hi2/ rBi? i H2bi QM2 biBzM2bb +QMbiMi r2`2 F2Ti 7Q`  b2`B2b Q7 bi2TrBb2 KmHiBTH2 HBM2` `2;`2bbBQM MHvb2bX 6Q` 2+? biBzM2bb +QMbiMi- bi2TrBb2 MHvb2b r2`2 }`biHv +``B2/ Qmi BM 2+? ;`QmT Q7 p`B#H2b UKB+`Qbi`m+im`2- KBM2`H M/ +QHH;2MVX h?2M- i?2 KQbi bB;MB}+Mi p`B#H2b Q7 2+? ;`QmT r2`2 `2iBM2/ iQ }M/ i?2 QTiBKH KmHiB@ TH2 HBM2` `2;`2bbBQM KQ/2HX 6Q` i?Qb2 p`B#H2b 7Q` r?B+? r2 Q#b2`p2/  /Bz2`2M+2 #2ir22M Hi2`H M/ K2/BH [m/`Mib- i?2 KmHiBTH2 HBM2` `2;`2bbBQM MHvb2b r2`2 +``B2/ Qmi QM Hi2`H M/ K2/BH /ib2ib b2T`i2HvX A7 i?2b2 p`B#H2b r2`2 MQi `2iBM2/ b 2tTHMiQ`v p`B#H2b BM i?2 `2;`2bbBQM KQ/2H- MHvb2b r2`2 `mM ;BM TQQHBM; i?2 Hi2`H M/ K2/BH /ib2ibX h?2 `2HiBp2 +QMi`B#miBQM Q7 i?2 p`B#H2b r2`2 [mMiB}2/ #v /Dmbi2/@_2U/D@R2V

M/ `QQi@K2M@b[m`2@2``Q` URMSEVX hQ ?B;?HB;?i i?2 `2HiBp2 BKTQ`iM+2 Q7 i?2 2tTHM@ iQ`v p`B#H2b BM i?2 KQ/2Hb- HH i?2 2tTHMiQ`v p`B#H2b r2`2 MQ`KHBx2/ #2ir22M −1 M/ 1X .i r2`2 +QMbB/2`2/ biiBbiB+HHv bB;MB}+Mi 7Q` p < 0.05X aiiBbiB+H MHvb2b r2`2 K/2 mbBM; i?2 JiH# kyRd aiiBbiB+b hQQH#Qt UJi?rQ`Fb AM+X- LiB+F- J- laVX

kX3X JB+`QK2+?MB+b KQ/2HBM;

h?2 `2bmHib Q7 i?2 biiBbiB+H MHvbBb T`2b2Mi2/ BM a2+X j bm;;2bi i?i i?2 p`B#BHBiv Q7 HH i?2 biBzM2bb +QMbiMib +M #2 2tTHBM2/ #v  bKHH MmK#2` Q7 /2i2`KBMMib- MK2Hv- i?2 TQ`QbBiv UφV M/ KBM2`H +QMi2Mi U.J"VX h?Bb BM im`M bm;;2bib i?i i?2 Q`;MBxiBQM Q7 i?2 #QM2 Ki2`BH 7QHHQrb  +2`iBM Tii2`M- r?B+? r2 ii2KTi iQ KQ/2H 7QHHQrBM; :`BKH 2i HX (Re)X q2 bbmK2 URV i?i i?2 pb+mH` TQ`2 M2irQ`F +M #2 KQ/2H2/ b  b2`B2b Q7 BM}MBi2 +vHBM/`B+H TQ`2b T2`BQ/B+HHv /Bbi`B#mi2/ BM #QM2 Ki`Bt QM  ?2t;QMH HiiB+2 bbQ+Bi2/ iQ TQ`QbBiv φ U6B;X R U?VVc M/ UkV i?i i?2 /Bz2`2Mi 2HbiB+ +Q2{+B2Mib UbiBzM2bb i2MbQ`V Q7 i?2 #QM2 Ki`Bt /2T2M/ HBM2`Hv QM .J" U2X;X- i?2 KQmMi Q7 KBM2`H b+H2b i?2 Ki`Bt 2HbiB+ T`QT2`iB2bVX h?2 2KTB`B+H `2HiBQMb?BTb `2HiBM; 2+? 2HbiB+ +Q2{+B2Mi iQ .J" +M #2 Q#iBM2/ bi`B;?i7Q`r`/Hv iFBM; φ = 0 BM 2[miBQMb R@ 8 Uh#H2 8VX

(13)

aT2+B}+HHv- i?2 KQ/2H2/ 2z2+iBp2 biBzM2bb +QMbiMib Ccyl

ij `2 +H+mHi2/ mbBM; bvKTiQiB+

?QKQ;2MBxiBQM (8k- 8j) UM BKTH2K2MiiBQM Bb pBH#H2 ?2`2 (89)VX h?2 BMTmib Q7 i?2 KQ/2H `2 φ- i?2 biBzM2bb Q7 i?2 Ki2`BH }HHBM; i?2 TQ`2b Uri2` Q7 #mHF KQ/mHmb = 2.2 :S M/  MmHH b?2` KQ/mHmbV- M/ i?2 biBzM2bb i2MbQ` Q7 i?2 Ki`Bt Q#iBM2/ b  7mM+iBQM Q7 .J" b 2tTHBM2/ #Qp2X

h?2 [mHBiv Q7 i?2 KQ/2H Bb 2pHmi2/ #v +QKT`BM; Ccyl

ij iQ biBzM2bb 2tT2`BK2MiH /i

mbBM; HBM2` `2;`2bbBQM M/ "HM/@HiKM THQiX jX _2bmHib

h?2 bT2+BK2Mb Q7 QM2 bm#D2+i rBi?  TQ`QbBiv ?B;?2` i?M jyW rb MQi BM+Hm/2/ BM i?2 MHvbBbX hrQ Qi?2` bm#D2+ib- 7Q` r?B+? T`i Q7 i?2 2tT2`BK2Mib 7BH2/ /m2 iQ /K;2 iQ i?2 bT2+BK2Mb- r2`2 HbQ 2t+Hm/2/- r?B+? }MHHv H2/ iQ ke bm#D2+ib 7Q` 2+? b2i UOR- Ok M/ OjV Q7 bT2+BK2MbX

jXRX .2b+`BTiBp2 biiBbiB+b

h?2 `2bmHib Q7 i?2 2HbiB+- KB+`Qbi`m+im`H M/ +QKTQbBiBQMH UKBM2`H M/ +QHH;2MV p`B#H2b `2 bmKK`Bx2/ #v i?2 K2M pHm2 M/ a. 7Q` i?2 Hi2`H M/ K2/BH bT2+BK2Mb-`2bT2+iBp2Hv- BM h#H2 jX h?2 KB+`Qbi`m+im`H p`B#H2b U*o 7`QK dXRW iQ 83XyWV M/ +QHH;2M U*o 7`QK jXNW iQ jeX9WV b?Qr2/  ;`2i2` `M;2 Q7 `2HiBp2 p`B#BHBiv +QKT`2/ iQ i?2 biBzM2bb +QMbiMib U*o 7`QK eX9W iQ R9XkWV M/ KBM2`H p`B#H2b U*o 7`QK RX9W iQ 8XeWVX aB;MB}+Mi /Bz2`2M+2b r2`2 7QmM/ #2ir22M Hi2`H M/ K2/BH [m/`Mib Uh#H2 jV 7Q` irQ KB+`Qbi`m+im`2 p`B#H2b U*QMM.- aJAV M/ i?2 p`B#H2b K2bm`2/ #v 6hA_J UJBMP`;-JBMJi- *`#QM- *`vAM/ M/ *QHHJiVX

jXkX lMBp`Bi2 +Q``2HiBQM MHvbBb

aT2`KM `MF +Q``2HiBQM +Q2{+B2Mib URV #2ir22M Cij M/ i?2 Qi?2` p`B#H2b `2 bmK@

K`Bx2/ BM h#H2 9X 6Q` i?2 p`B#H2b /BbTHvBM;  bB;MB}+Mi /Bz2`2M+2 #2ir22M i?2 Hi2`H M/ K2/BH [m/`Mib- BX2X- *QMM.- aJA- JBMP`;- JBMJi- *`#QM- *`vAM/ M/ *QHHJi-Rrb +H+mHi2/ 7Q` Hi2`H M/ K2/BH ;`QmT b2T`i2HvX Pp2`HH- i?2 bi`QM;2bi +Q``2HiBQMb r2`2 7QmM/ 7Q` i?2 KB+`Qbi`m+m`H p`B#H2b M/ .J"X KQM; i?2 KB+`Qbi`m+im`2 p`B#H2b-φ- SQL- M/ SQ.K r2`2 M2;iBp2Hv +Q``2Hi2/ rBi? HH i?2 Cij UR 7`QK @yXe9 iQ @yX3NVX

SQbBiBp2 +Q``2HiBQMb r2`2 7QmM/ #2ir22M SQafSQo- SQaT- SQS7 M/ HH i?2 Cij UR 7`QK

yX8y iQ yX3eVX *QMM. 7`QK Hi2`H [m/`Mi bB;MB}+MiHv +Q``2Hi2/ rBi? bQK2 Cij UR 7`QK

@yX9R iQ @yX88VX KQM; i?2 KBM2`H p`B#H2b- .J" rb bB;MB}+MiHv +Q``2Hi2/ rBi? HH i?2 Cij UR 7`QK yXdd iQ yX3RVX JBMP`; 7`QK Hi2`H [m/`Mi rb bB;MB}+MiHv +Q``2Hi2/ rBi?

HH i?2 Cij UR 7`QK yX9k iQ yX8RVX JQ/2`i2 #mi bB;MB}+Mi +Q``2HiBQM rb 7QmM/ #2ir22M

(14)

h#H2 j,  bmKK`v Q7 i?2 `2bmHib UJ2MUa.VV Q7 biBzM2bb +QMbiMib- KB+`Qbi`m+im`2 M/ +QKTQbBiBQMH p`B#H2b 7`QK i?2 Hi2`H UGV M/ K2/BH UJV bT2+BK2MbX∗o`B#H2b b?QrBM; bB;MB}+Mi /Bz2`2M+2 #2ir22M Hi2`H M/ K2/BH bT2+BK2M /i M/ i?2 +Q``2bTQM/BM; T@pHm2b `2 HbQ ;Bp2M #2HQrX

aiBzM2bb +QMbiMib

*11U:SV *33U:SV *13U:SV *44U:SV *66U:SV

G RNXeUkXyV kNXyUkXjV RRXNUyXNV 8X3UyXeV 9XjUyXeV

J kyXRURX3V k3XNURX3V RkXRUyXNV 8XNUyXeV 9X8UyXeV

JB+`Qbi`m+im`2 p`B#H2b

φUWV SQafSQo UKK−1V SQL UKK−1V SQ.K UµKV SQaT UµKV

G dX9U9XjV 8NXkURdXRV yXd3UyXk8V NyUjRV jkjUj8V

J dXeUjXNV eRX3URNXdV yX3jUyXkyV 3NUj9V jR3UkdV

SQS7 UKK−1V *QMM.∗UKK−3V aJA∗UXmXV ρU;f+K3V

p < 1−7 p = 0.01

G jRXj9U3XkeV RyXjU8XyV jXkUyXjV RXNyUyXyeV

J jyXjeUNXj8V k9X3U8X3V jXRUyXkV RXNyUyXyeV

JBM2`H p`B#H2b

.J" U;f+K3V JBMP`;UMXmXV JBMJiUMXmXV *`#QMUMXmXV *`vAM/U+KV

p = 0.01 p < 1−4 p < 1−6 p < 1−4

G RXyRUyXykV 8XkeUyXjyV RX39UyXRyV yXyydRUyXyyyjV yXyj39UyXyyRRV

J RXykUyXykV 8X88UyXkeV RXdkUyXydV yXyyeeUyXyyykV yXyjNeUyXyyyeV

*QHH;2M p`B#H2b

*QHHJi∗ .>GLG >GLG Su. .S.

UMXmXV UKKQHfKQH UKKQHfKQH UKKQHfKQH UKKQHfKQH

p = 0.005 +QHH;2MV +QHH;2MV +QHH;2MV +QHH;2MV

G 9X89UyXjdV 8ddXdUkRyXkV ke8XRUd3XeV j8kXeU8jX8V N3XjURdX8V

J 9XjjUyXkNV 88dXNUR3jX8V k88XdUeNXyV j8jX8Uj9XjV Ry3XyUR3XyV

S1L *QHH UKKQHfKQH UWV +QHH;2MV G RyXjUkXNV RjXeUyX3V J NX8UkXeV RjX9UyX8V JBMP`; M/ *`#QM 7`QK K2/BH [m/`Mi M/ C13UR = 0.47V M/ C44 UR = −0.45V- `2@

bT2+iBp2HvX KQM; i?2 +QHH;2M p`B#H2b- .>GLG M/ >GLG r2`2 bB;MB}+MiHv +Q``2Hi2/ rBi? HH i?2 Cij UR 7`QK @yXjj iQ @yX93V- h?2 p`B#H2b i?i r2`2 MQi bB;MB}+MiHv +Q``2Hi2/

rBi? Mv Q7 i?2 Cij- BX2X aJA- JBMJi- *`vAM/- *QHHJi- Su.- .S.- S1L M/ *QHH r2`2

`2KQp2/ BM i?2 7QHHQrBM; `2;`2bbBQM MHvb2b BM a2+iBQM jXjX //BiBQMH BM7Q`KiBQM #Qmi i?2 `2HiBQMb?BTb #2ir22M i?2 p`B#H2b rBi?BM 2+? ;`QmT UCij- KB+`Qbi`m+im`2- KBM2`H M/

(15)

+QHH;2MV M/ KQM; i?2 ;`QmTb +M #2 7QmM/ BM TT2M/Bt "X

h#H2 9, aT2`KM +Q``2HiBQM +Q2{+B2Mi R #2ir22M CijM/ KB+`Qbi`m+im`H- KBM2`H M/ +QHH;2M p`B#H2bX h?2 p`B#H2b rBi? #QH/ 7QMi r2`2 F2Ti 7Q` i?2 7QHHQrBM; MHvbBbX .i 7`QK Hi2`H M/ K2/BH [m/`Mib r2`2 MHvx2/ b2T`i2Hv 7Q` i?2 p`B#H2b r?Q b?Qr bB;MB}+Mi /Bz2`2M+2b #2ir22M i?2 irQ [m/`Mib U//BiBQMH H#2Hb ǶGǶ M/ ǶJǶ `2 //2/ BM i?2 i#H2VX _2bmHib `2 MQi /Bz2`2MiBi2/ mbBM; H#2Hb ǶGǶ M/ ǶJǶ r?2M MQ bB;MB}+Mi `2HiBQMb?BT rb 7QmM/ #Qi? BM i?2 Hi2`H M/ K2/BH /iX 1p < 0.05-2p < 0.001-3p < 0.0001- MXbX MQi bB;MB}+MiX

C11 C33 C13 C44 C66

@yX3d3 @yX3y3 @yXde3 @yX3N3 @yX393

SQafSQo yX3j3 yXd83 yXd93 yX3e3 yX3R3

SQL @yXd93 @yXe93 @yXee3 @yXdR3 @yXeN3

SQ.K @yXdd3 @yXdj3 @yXed3 @yX3j3 @yXdd3

SQaT yX893 yX8k3 yX8y2 yX893 yX8k3

SQS7 yX3R3 yXdk3 yXdk3 yX3k3 yXd33

*QMM. G @yX9R1 @yX881 MXbX @yX8R1 MXbX

J MXbX MXbX MXbX MXbX MXbX

aJA MXbX MXbX MXbX MXbX MXbX

.J" yXdN3 yXdd3 yXd33 yX3R3 yXdd3

JBMP`; G yX9d1 yX9j1 yX8R1 yX9k1 yX981

J MXbX MXbX yX9d1 MXbX MXbX JBMJi MXbX MXbX MXbX MXbX MXbX *`#QM G MXbX MXbX MXbX MXbX MXbX J MXbX MXbX MXbX @yX981 MXbX *`vAM/ MXbX MXbX MXbX MXbX MXbX *QHHJi MXbX MXbX MXbX MXbX MXbX

.>GLG @yX982 @yX992 @yX932 @yX991 @yX9y1

>GLG @yX9R1 @yX9R1 @yX9R1 @yX9y1 @yXjj1

Su. MXbX MXbX MXbX MXbX MXbX

.S. MXbX MXbX MXbX MXbX MXbX

S1L MXbX MXbX MXbX MXbX MXbX

*QHH MXbX MXbX MXbX MXbX MXbX

jXjX JmHiBp`Bi2 `2;`2bbBQM KQ/2H

AM i?2 KmHiBp`Bi2 `2;`2bbBQM KQ/2Hb- #QM2 biBzM2bb +QMbiMib Cij `2 i?2 /2T2M/2Mi

p`B#H2b M/ i?2 KB+`Qbi`m+im`2- KBM2`H M/ +QHH;2M p`B#H2b `2 i?2 BM/2T2M/2Mi p`B@ #H2bX LQM2 Q7 i?2 p`B#H2b b?QrBM;  bB;MB}+Mi /Bz2`2M+2 #2ir22M Hi2`H M/ K2/BH /i M/ F2Ti 7Q` KmHiBp`Bi2 `2;`2bbBQM MHvb2b- BX2X- *QMM.- JBMP`; M/ *`#QM Uh#H2 9V-r2`2 `2iBM2/ 7i2` MHvb2bX h?2`27Q`2- i?2 7QHHQrBM; `2bmHib Uh#H2 8V 9V-r2`2 Q#iBM2/ mbBM; i?2 TQQH2/ /i 7`QK Hi2`H M/ K2/BH [m/`MibX

(16)

i?2 CijX AM i?2 KmHiBTH2 `2;`2bbBQM KQ/2Hb mbBM; KB+`Qbi`m+im`2 p`B#H2b- φ 2tTHBM2/ KQbi

Q7 i?2 p`B#BHBiv Q7 i?2 Cij U/D@R27`QK 83X3W iQ 39X9WV- b22 h#H2 8X KQM; i?2 KBM2`H

p`B#H2b- .J" rb i?2 KQbi bB;MB}+Mi 7+iQ` 2tTHBMBM; i?2 p`B#BHBiv Q7 i?2 Cij U/D@R2

7`QK 89XeW iQ e8XNWVX b 7Q` i?2 +QHH;2M p`B#H2b- i?2 QMHv bB;MB}+Mi p`B#H2 rb .>GLG r?B+? QMHv ++QmMi2/ 7Q`  KBMQ` T`i Q7 i?2 p`B#BHBiv Q7 i?2 Cij U/D@R

2 7`QK NXNW iQ

kRX3WVX bB/2 7`QK φ- .J" M/ .>GLG- SQS7 BM KmHiBTH2 `2;`2bbBQM KQ/2H +QK#BMBM; i?2 KB+`Qbi`m+im`H p`B#H2b rb i?2 QMHv +QMi`B#miBM; p`B#H2b iQ C33X h?2 `2;`2bbBQM

+Q2{+B2Mib Q7 i?2 p`B#H2b BM i?2 HBM2` KQ/2H i2HH i?i i?2 `2HiBp2 +QMi`B#miBQM Q7 φ rb ;`2i2` i?M i?i Q7 i?2 Qi?2` 2tTHMiQ`v p`B#H2bX AM //BiBQM- SQS7 M/ φ #2BM; bi`QM;Hv +Q``2Hi2/ UR = 0.96- b22 TT2M/Bt "V- φ rb i?2 QMHv KB+`Qbi`m+im`H p`B#H2 mb2/ BM i?2 bm#b2[m2Mi KmHiBTH2 HBM2` `2;`2bbBQM MHvb2bX SQQHBM; iQ;2i?2` i?2 2tTHMiQ`v p`B#H2b 7`QK 2+? ;`QmT- φ- .J" M/ .>GLG- QMHv irQ p`B#H2b φ M/ .J" r2`2 `2iBM2/ b i?2 #2bi +QK#BMiBQM BM i?2 `2;`2bbBQM KQ/2Hb r?B+? 2tTHBM2/ deXR iQ NyXNW Q7 i?2 p`B#BHBiv Q7 i?2 Cij Uh#H2 8VX h?2 `2HiBp2 M/ +QK#BM2/ +QMi`B#miBQM Q7 φ M/ .J" iQ Cij `2

BHHmbi`i2/ BM 6B;m`2 jX

jX9X *QKT`BbQM #2ir22M KB+`QK2+?MB+b KQ/2H M/ 2tT2`BK2MiH /i

HH i?2 KQ/2H2/ biBzM2bb +QMbiMib Ccyl

ij r2`2 ?B;?Hv +Q``2Hi2/ iQ i?2B` 2tT2`BK2MiH

+QmMi2`T`i Cij U75.1% < R 2

< 91.5%VX h?2 RMSE U0.18 < RMSE < 0.85V M/ R2

U75.1% < R2

< 91.5%V #2ir22M Cijcyl M/ Cij T`2b2Mi bBKBH` `M;2 b i?2 `2bmHib Q7 i?2

irQ@p`B#H2 Uφ M/ .J"V HBM2` `2;`2bbBQM KQ/2H U0.18 < RMSE < 0.84 M/ 76.4% < R2

<90.9%V b?QrM BM h#H2 8X  #Bb Q7 #Qmi RXj :S rb 7QmM/ #2ir22M C11M/ C11cyl

M/ #2ir22M C13 M/ C13cyl #v i?2 "HM/@HiKMǶb THQib UMQi b?QrMV- r?2`2b i?2 #Bb QM

C33cyl- C cyl 44 M/ C cyl 66 Ud < 0.2 :SV `2 M2;HB;B#H2X 9X .Bb+mbbBQM

AM i?Bb rQ`F- i?2 `2HiBp2 +QMi`B#miBQMb Q7 i?2 KB+`Qbi`m+im`2 M/ +QKTQbBiBQM iQ i?2 K2bQb+H2 MBbQi`QTB+ 2HbiB+ T`QT2`iB2b Q7 ?mKM +Q`iB+H #QM2 BM i?2 2H/2`Hv `2 BMp2biB@ ;i2/X hQ Qm` FMQrH2/;2- Qm` bim/v Bb i?2 }`bi QM2 iQ T`QpB/2 /i +Qp2`BM; i?2 +QKTH2i2 MBbQi`QTB+ 2HbiB+ i2MbQ`- i?2 KB+`Qbi`m+im`2 Q7 +Q`iB+H pb+mH` TQ`QbBiv- KBM2`H M/ +QH@ H;2M +?`+i2`BbiB+bX PM2 bi`2M;i? Q7 i?2 bim/v Bb i?i HH i?2 T?vbB+H [mMiBiB2b r2`2 Q#iBM2/ 7`QK i?2 bK2 bT2+BK2Mb BM 2+? /QMQ`- Q` 7`QK BKK2/Bi2Hv /D+2Mi bKTH2b-r?B+? HBKBib i?2 +QM7QmM/BM; 2z2+i Q7 Ki2`BH ?2i2`Q;2M2Biv `QmM/ i?2 #QM2 +B`+mK72`2M+2 M/ HQM; i?2 /BT?vbBbX h?2 MBbQi`QTB+ 2HbiB+ T`QT2`iB2b ?p2 #22M K2bm`2/ #v _la- i2+?MB[m2 M2rHv BMi`Q/m+2/ iQ K2_la-bm`2 #QM2 2H_la-biB+ T`QT2`iB2bX h?2 T`QT2`iB2b Q7 i?2

(17)

h#H2 8, JmHiBTH2 HBM2` `2;`2bbBQM KQ/2Hb Q7 #QM2 biBzM2bb CijX AM i?2 irQ@p`B#H2 KQ/2Hb- QMHv φ M/ .J" `2 BM+Hm/2/X LQi2 i?i i?2 2tTHMiQ`v p`B#H2b rBi? M Qp2`#` bvK#QH ?p2 #22M MQ`KHBx2/X 3p < 0.0001X aiBzM2bb 1tTHMiQ`v p`B#H2b GBM2` KQ/2H /D@R 2 RM SE UWV U:SV C11 KB+`Qbi`m+im`2 18.14 − 4.06 × φ d3XN3 yX3d KBM2`H 19.38 + 3.63 ×.J" 8NX33 RXky +QHH;2M 19.48 − 1.34 ×.>GLG RjXy1 RXdd

φY .J" 18.36 − 3.03 × φ + 1.66 ×.J" URV 3eX93 yXdy

C33 KB+`Qbi`m+im`2 26.23 − 5.97 × φ − 1.52 ×SQS7 3kXy3 yX3d

KBM2`H 28.50 + 3.61 ×.J" 8yXy3 RX98 +QHH;2M 28.57 − 1.47×.>GLG RjXk1 RXNR φY .J" 27.27 − 3.67 × φ + 1.22 ×.J" UkV 3jXk3 yX39 C13 KB+`Qbi`m+im`2 11.31 − 1.67 × φ 83X33 yX83 KBM2`H 11.78 + 1.81 ×.J" e8XN3 yX8j +QHH;2M 11.79 − 0.80 ×.>GLG kRX31 yX3y

φY .J" 11.47 − 0.92 × φ + 1.21 ×.J" UjV deX93 yX99

C44 KB+`Qbi`m+im`2 5.28 − 1.33 × φ 39X93 yXk9

KBM2`H 5.69 + 1.16 ×.J" eyXe3 yXj3

+QHH;2M 5.73 − 0.43 ×.>GLG RjXj1 yX8e

φY .J" 5.35 − 1.02 × φ + 0.49 ×.J" U9V NyXN3 yXR3

C66 KB+`Qbi`m+im`2 3.86 − 1.29 × φ deXR3 yXjy

KBM2`H 4.25 + 1.16 ×.J" 83X33 yXjN

+QHH;2M 4.30 − 0.39 ×.>GLG NXN1 yX83

φY .J" 3.93 − 0.95 × φ + 0.55 ×.J" U8V 3jXd3 yXk8

pb+mH` TQ`2 M2irQ`F M/ KBM2`H +QMi2Mi ?p2 #22M bb2bb2/ mbBM; a_@µ*h-  `272`2M+2 K2i?Q/ 7Q` i?2b2 Tm`TQb2b (88)X

h?2 /i rb Q#iBM2/ QM +/p2`B+ #QM2b 7`QK 2H/2`Hv /QMQ`b rBi?Qmi /Q+mK2MiiBQM QM i?2 2tBbi2M+2 Q7 #QM2 Ti?QHQ;B2bX Ai 7QHHQrb i?i i?2 /ib2i Q7 i?Bb bim/v T`QpB/2b `272`2M+2 MQKBMH pHm2b M/ `M;2b Q7 p`BiBQMb 7Q` i?Bb TQTmHiBQM- r?B+? BM7Q`KiBQM Bb +`BiB+H iQ BMi2`T`2i /i 7`QK /QMQ`b rBi? Ti?QHQ;B2bX

_2+2MiHv- _la rb bm++2bb7mHHv /2p2HQT2/ iQ +?`+i2`Bx2 i?2 MBbQi`QTB+ biBzM2bb Q7 ?m@ KM +Q`iB+H #QM2 (j8- j3) M/ iQ BMp2biB;i2 i?2 2HbiB+Biv@/2MbBiv `2HiBQMb?BT Q7 ?mKM iB#BH +Q`iB+H #QM2 bT2+BK2Mb (8e)X "b2/ QM Bib ;QQ/ ++m`+v M/ 2b2 Q7 mb2- _la Bb 2K2`;BM; b i?2 i2+?MB[m2 Q7 +?QB+2 7Q` BMp2biB;iBM; i?2 biBzM2bb Q7 bKHH@bBx2/ #QM2 bT2+B@ K2MbX "2+mb2 i?2 7mHH b2i Q7 i?2 2HbiB+ i2MbQ` +M #2 B/2MiB}2/ mbBM; _la- i?2 2M;BM22`BM; KQ/mHB r?B+? `2 Q7i2M mb2/ BM 2pHmiBM; #QM2 K2+?MB+H T`QT2`iB2b bm+? b bi`2bb M/ bi`BM- +M HbQ #2 /2`Bp2/X

(18)

R2= 79.3% Adj-R2= 86.4% 14 1.05 0 16 18 5 DMB (g/cm3) 20 C1 1 (G P a) 1.00 10 22 φ(%) 24 R 2= 60.6% 15 26 0.95 20 25 14 16 18 20 22 24 26 R2= 80.4% 22 Adj-R2= 83.2% 1.05 0 24 26 5 DMB (g/cm3) 28 C3 3 (G P a) 1.00 10 φ(%) 30 32 R 2= 51.0% 15 0.95 20 25 22 23 24 25 26 27 28 29 30 31 32 33 R2= 59.6% Adj-R2= 76.4% 9 1.05 0 10 11 5 DMB (g/cm3) C1 3 (G P a) 1.00 12 10 φ(%) 13 15 R2= 66.5% 14 0.95 20 25 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 R2= 84.7% Adj-R2= 90.9% 4 1.05 0 5 5 DMB (g/cm3) C4 4 (G P a) 1.00 6 10 φ(%) R2= 61.4% 15 7 0.95 20 25 4 4.5 5 5.5 6 6.5 7 R2= 76.5% Adj-R2= 83.7% 1.05 3 0 4 5 DMB (g/cm3) C6 6 (G P a) 1.00 5 10 φ(%) 6 R 2= 59.6% 15 0.95 20 25 2.5 3 3.5 4 4.5 5 5.5 6 6.5

6B;m`2 j, j. #` ;`T?b `2T`2b2MiBM; i?2 Cijb  7mM+iBQM Q7 φ M/ .J"X GBM2` `2;`2bbBQMb `2 THQii2/ BM i?2 #+F;`QmM/ 7Q` 2+? p`B#H2 rBi? i?2 +Q2{+B2Mib Q7 /2i2`KBMiBQM UR2V M/ i?2 /D@R2 pHm2b Q7 i?2 KQ/2H mbBM; φ M/ .J" b i?2 2tTHMiQ`v p`B#H2b `2 /2TB+i2/ b r2HHX

h?2 2tT2`BK2MiH 2``Q`b mbBM; _la `2 r2HH /Q+mK2Mi2/ (9y- 8d) M/ `2 ivTB+HHv HQr UTT`QtBKi2Hv RXdW 7Q` i?2 b?2` biBzM2bb +QMbiMib M/ TT`QtBKi2Hv jXRW 7Q` i?2 HQM@ ;Bim/BMH M/ Qz@/B;QMH biBzM2bb +QMbiMibV +QKT`2/ iQ Qi?2` i2+?MB[m2b pBH#H2 7Q` ǶbKHHǶ bT2+BK2Mb U 72r KK3VX 6m`i?2`KQ`2- i?2 K2bm`2K2Mi Bb [mB+F U 72r KBMmi2b T2`

bT2+BK2MV bQ i?i i?2 i2+?MB[m2 +M #2 mb2/ `QmiBM2HvX

*QKT`BM; i?2 /i 7`QK Hi2`H M/ K2/BH [m/`Mib- MQ bB;MB}+Mi /Bz2`2M+2 rb Q#b2`p2/ BM biBzM2bb- .J"- +QHH;2M p`B#H2b M/ KQbi Q7 i?2 KB+`Qbi`m+im`H p`B#H2bX h?Qm;?  bB;MB}+Mi /Bz2`2M+2 rb Q#b2`p2/ 7Q` i?2 KBM2`H p`B#H2b K2bm`2/ #v 6hA_J #2ir22M i?2 irQ bBi2b- i?Bb /Bz2`2M+2 rb MQi 7QmM/ BM .J" M/ rb MQi TT`2+Bi2/ #v i?2 biBzM2bb b i?2`2 rb MQ bB;MB}+Mi /Bz2`2M+2 7Q` biBzM2bb 2Bi?2`X

(19)

14 16 18 20 22 24 C11cyl(GPa) 14 16 18 20 22 24 C1 1 (G P a) R2= 87.4% RM SE = 0.68 GPa 22 24 26 28 30 32 C33cyl(GPa) 22 24 26 28 30 32 C3 3 (G P a) R2= 83.2% RM SE = 0.85 GPa 8 9 10 11 12 13 14 Ccyl 13 (GPa) 8 9 10 11 12 13 14 C13 (G P a) R2= 75.1% RM SE = 0.45 GPa 4 4.5 5 5.5 6 6.5 7 Ccyl 44 (GPa) 4 4.5 5 5.5 6 6.5 7 C4 4 (G P a) R2= 91.5% RM SE = 0.18 GPa 2.5 3 3.5 4 4.5 5 5.5 6 C66cyl(GPa) 2.5 3 3.5 4 4.5 5 5.5 6 C6 6 (G P a) R2= 85.6% RM SE = 0.23 GPa

6B;m`2 9, GBM2` `2;`2bbBQM #2ir22M i?2 Cij M/ CijcylX aQHB/ HBM2 Bb i?2 `2;`2bbBQM KQ/2H M/ /b? HBM2 Bb y = xHBM2X

GBM2` `2HiBQMb?BTb r2`2 bbmK2/ #2ir22M #QM2 T`QT2`iB2b BM i?2 biiBbiB+H MHvb2bX S`2pBQmb rQ`F b?Qr2/ i?i TQr2` 7mM+iBQMb Kv #2 KQ`2 TT`QT`Bi2 iQ ++QmMi 7Q` KB@ +`QK2+?MB+b `2HiBQMb #2ir22M TQ`QbBiv M/ 2HbiB+Biv 7Q` TQ`Qmb Ki2`BH BM  rB/2 `M;2 Q7 TQ`QbBiv pHm2b Uy@RyyWV (83)X >Qr2p2`- i?2 `M;2 Q7 +Q`iB+H TQ`QbBiv pHm2b Bb HBKBi2/ BM i?Bb rQ`F UH2bb i?M kkWV M/  HBM2` `2HiBQMb?BT +M #2 +QMbB/2`2/ BM i?Bb HBKBi2/ `2;BQM 7Q` biiBbiB+H MHvb2b- b Q#b2`p2/ BM Qm` /i b r2HH b BM Qi?2`bǶ (kR)X GBM2` +Q``2H@ iBQMb +?Qb2M 7Q` i?Bb rQ`F T`QpB/2 KQ/2Hb rBi? R2 U7Q` i?2 #2bi KQ/2HV ivTB+HHv ?B;?2`

i?M yX3 M/ RMSE #2ir22M yXR3 M/ yX39 :S r?B+? BM/B+i2b i?i i?2 HBM2` KQ/2Hb `2 TT`QT`Bi2X

KQM; i?2 p`B#H2b BM i?2 KBM2`H ;`QmT- bB;MB}+Mi +Q``2HiBQMb rBi? Cijr2`2 Q#b2`p2/

QMHv 7Q` .J"- JBMP`; M/ *`#QMX q2 7QmM/ i?i .J" 2t?B#Bib  HBKBi2/ `M;2 Q7 p`BiBQMb #2ir22M BM/BpB/mHb BM i?2 /mHib- r?B+? Bb +QMbBbi2Mi rBi? HBi2`im`2 /i (kd- 8N)X .2bTBi2 i?Bb bKHH `M;2 Q7 p`BiBQM- r2 +QmH/ Q#b2`p2- 7Q` i?2 }`bi iBK2- i?i i?2 pQHmK2i`B+

(20)

.J" K2bm`2/ #v a_@µ*h QM ?mKM 72Km` +Q``2Hi2b rBi? HH i?2 K2bQb+QTB+ biBzM2bb +QMbiMibX h?Bb Bb +QMbBbi2Mi rBi? i?2 bim/B2b /B`2+iHv `2HiBM; #QM2 2HbiB+ T`QT2`iB2b rBi? b? /2MbBiv (k8- ke)X Hi?Qm;? T`2pBQmb bim/B2b b?Qr2/ i?i JBMP`;- *QHHJi M/ *v`biAM/ K2bm`2/ #v 6hA_J Kv 2tTHBM T`i Q7 i?2 p`B#BHBiv Q7 i?2 KB+`QK2+?MB+H T`QT2`iB2b Q7 #QM2 iBbbm2 (jk- ey)- bm+? M 2z2+i rb MQi Q#b2`p2/ 7Q` #QM2 K2bQb+QTB+ 2HbiB+ T`QT2`iB2b BM Qm` /iX h?2 BM~m2M+2 Q7 TQ`QbBiv Bb bQ T`2/QKBMMi BM /2i2`KBMBM; K2bQb+QTB+ biBzM2bb Q7 Qm` bKTH2 b2i i?i  +Q``2HiBQM #2ir22M Qi?2` p`B#H2b M/ K2bQb+QTB+ biBzM2bb +M #2 /B{+mHi iQ Q#b2`p2- 2bT2+BHHv B7 i?2v `2 MQi K2bm`2/ 2t+iHv BM i?2 bK2 TH+2- b Bi Bb i?2 +b2 7Q` biBzM2bb M/ KBM2`H T`QT2`iB2bX

6Q` i?2 +QHH;2M p`B#H2b- r2 /B/ MQi }M/ +Q``2HiBQM #2ir22M Kim`2 2MxvKiB+ +`Qbb@ HBMFb Su.- .S.- MQM@2MxvKiB+ S1L- +QHH;2M KQmMi U*QHHV M/ Mv Q7 i?2 Cij r?B+? Bb

+QMbBbi2Mi rBi? T`2pBQmb Q#b2`piBQMb (kd- eR)X >Qr2p2`- r2 Q#b2`p2/ bB;MB}+Mi +Q``2HiBQMb #2ir22M BKKim`2 2MxvKiB+ +`Qbb@HBMFb .>GLG- >GLG M/ HH i?2 Cij UR #2ir22M −0.33

M/ −0.48- h#H2 9V r?B+? ?b MQi #22M Q#b2`p2/ #27Q`2X PM2 Kv HbQ Q#b2`p2 i?i- r?BH2 #Qi? .>GLG M/ >GLG `2 +Q``2Hi2/ rBi? i?2 Cij- MQM2 Q7 i?2b2 p`B#H2b rb `2iBM2/

BM i?2 #2bi KmHiBTH2 `2;`2bbBQM KQ/2HX .>GLG M/ >GLG `2 ?B;?Hv +Q``2Hi2/ iQ;2i?2` UR = 0.94- h#H2 "XdV M/ rBi? .J" UR = 0.55- h#H2 "XdV- ?2M+2 i?2v Kv #2 `2TH+2/ #v i?2 KQ`2 T`2/QKBMMi 2tTHMiQ`v p`B#H2 .J"X AM/22/- 7i2` i?2 /DmbiK2Mi 7Q` .J"-.>GLG M/ >GLG /Q MQi +Q``2Hi2/ rBi? Mv Q7 i?2 CijX MQi?2` BMi2`2biBM; Q#b2`piBQM

Bb i?i r2 7QmM/  bB;MB}+Mi +Q``2HiBQM #2ir22M φ M/ .J" UR = −0.59- h#H2 "XeV r?B+? rb MQi Q#b2`p2/ #27Q`2X

h?2 KB+`Qbi`m+im`2 p`B#H2b 2tTHBM  KDQ` T`i Q7 i?2 p`B#BHBiv Q7 CijUh#H2 9VX LQi2

i?i i?2 p`B#BHBiv BM KB+`Qbi`m+im`2 `2 KQ`2 BKTQ`iMi i?M i?2 p`B#BHBiv BM KBM2`H T`QT2`iB2b BM i?2 2H/2`HvX _2;`/BM; i?Bb TQBMi- iQ i?2 #2bi Q7 Qm` FMQrH2/;2- i?2 QMHv bim/v i?i +QKT`2b iQ Qm`b Bb i?i Q7 "H 2i HX (k9) BMp2biB;iBM; i?2 BM~m2M+2 Q7 i?2 TQ`2 M2irQ`F `+?Bi2+im`2 QM i?2 ?mKM +Q`iB+H #QM2 2HbiB+Biv K2bm`2/ i i?2 }#mHX Pp2`HH-Qm` `2bmHib `2 +QMbBbi2Mi rBi? i?Qb2 `2TQ`i2/ BM "HǶb bim/v- /2bTBi2 bQK2 /Bz2`2M+2b BM i?2 #bQHmi2 pHm2 Q7 i?2 K2bm`2/ p`B#H2b M/ +Q``2HiBQMb- r?B+? +M #2 2tTHBM2/ #v /Bz2`2M+2b BM bF2H2iH bBi2 M/ K2bm`2K2Mi K2i?Q/bX

M BKTQ`iMi }M/BM; Q7 i?2 bim/v Bb i?i- KQM; HH i?2 p`B#H2b BMp2biB;i2/ r?B+? `2~2+i i?2 KB+`Qbi`m+im`2 M/ +?2KB+H +QKTQbBiBQM Q7 #QM2- QMHv irQ Uφ M/ .J"V BM@ /2T2M/2MiHv +QMi`B#mi2 iQ i?2 p`B#BHBiv Q7 i?2 biBzM2bb +QMbiMib Uh#H2 8VX φ M/ .J" HQM2 2tTHBM 59.6 − 84.7% M/ 51.0 − 66.5% Q7 i?2 p`B#BHBiv Q7 Cij- `2bT2+iBp2Hv U6B;X jVX

PM+2 i?2 KQbi BKTQ`iMi /2i2`KBMMib Q7 #QM2 biBzM2bb `2 /2i2`KBM2/- QM2 b?QmH/ #2 #H2 iQ /2bB;M  bBKTHB}2/ KQ/2H #b2/ QM i?2b2 /2i2`KBMMib iQ T`2/B+i #QM2 biBzM2bbX PM2 TT`Q+? 7Q` ?2i2`Q;2M2Qmb K2/B HBF2 #QM2 Bb i?`Qm;? ?QKQ;2MBxiBQMX a2p2`H ?Q@

(21)

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(22)

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(23)

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(24)

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(25)

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(26)

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(28)

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(29)

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(30)

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