!"#%$ &!'($)
*,+.-0/132547698:2;2=<?>A@
BCD#EGFIHKJMLONQPORMN(STUPVTWJYX.JZTU[\]O^ORMTW\O_[#\ONa`#NXR"b TU\cXJZ[edf\gTUh,X\O[#RMi XjWjW`)]OTUhkJZRlTUmO^VJMN]nREpoqE
dfTWJMLrNtsVPqNSuJYXvJMTU[#\:wC?TW\OSYLON(hX\O]rXnS[;N(xySTWN\=Jz[vHovXRMTUXvJZTW[#\{[H CV|}B~€S
XjWS^OjUXvJZN"JMLON
H3[#jUjW[edfTU\O_O
‚ X;ƒ „GLON…hlJYXv\O]0XRl]†]ONuo;T‡XvJMTU[#\H3[#RbˆE
‚ m‰ƒrŠ ‚ b!‹Œ#C;ƒvE
‚ S.ƒrŠ ‚ b!Ž#C;ƒvE
‚ ]‰ƒrŠ ‚‘ Cy‹rb!’“ww5ƒvE
‚ N.ƒz„GLON…PORM[#mOXmOTUjWTWJ”`QJZLOXvJab TWhfdfT•JZLOTW\{w–TW\OSYLONhGH3Rl[#i—JZLVNX
oNRZXv_#N,PORMN(STUPVTWJYX.JZTU[\AE
‚ Htƒ „GLON"ovXjU^ONc˜0™hM^VSYLšJML0XvJ9JMLON"PORM[m0XmOTWjUTWJ”`nJML0XvJ9JMLON"PORMN(STWPOTWJZXvJZTW[#\7]O^ORlTU\O_[\ON
`#NXRfNusDSN(N]Ohy˜0™QTUh›l^OhkJ"œ  [vHJMLON,PORM[#mOXmOTUjWTWJ”` \O[JGJM[yNtsDSNN(]r˜0™DE
BC#BDEGžŸhlTUiyPOjUN,hlJMRM^OSuJZ^ORlN9S[\OhMTWhlJZTW\O_z[vH X SX\=JZTUjWN(oNRm‰NXi¢¡¤£¥X\O]–X S
XvmOjUN)£9¦§TWh¨^OhlN]
JZ[†S
XRlRl`gX†jW[=X]“© ‚ hMN(NªTW_#^ORMN)vƒZE„GLONi X_#\OT•JZ^O]VNa[H«JZLVNzjU[#X]covXRlTUN(h…]OXTUj•`#~X\O]
TWJMhiy[#\=JZLOj•`i X.sDTUi9^Vi§S
X\gXPOPVRM[
sDTUi XvJZN(jW`mqNQS[#\OhlTU]ON(RMN(]­¬…Xv^OhMhlT‡X\ndfTWJML®Xi NX\
[HB#wC#C#C)¯;_V~°X\O]†XQS([DNuxySTUN(\=J¨[vHovXRlT‡XvJMTU[#\"[HŒC=±gE
ªTU_^ORMNa#²³Z´µ;¶”·”¸k¹5³Z·»º¼´µQ½¾º¼·»¿aÀ¤ÁOÂtÀeÃÄÀeµ5ÅQÀ…³YÀ
Á;ƼÂeÇ
‚ X;ƒÈ«XjUS(^Oj‡X.JZN JMLONšeC0ÉIi [\5JML®i X.sDTUi9^ViÊjU[=X] ‚ JML0XvJaTWh~X–i X.sDTUi^OiÊjU[=Xv]gdfTWJML{X
RMNuJZ^ORl\†PqNRlTU[;])[vH,eC–iy[#\=JZLOhtƒYE
B
‚ m‰ƒ G„ LONgS
XmOjWNË£9¦ X\O]ÌmqN
Xvi ¡£ XvRMNn]ONhlTU_#\VN]ˆJM[šdfT•JZLOhkJYX\V]“JMLONÍeCOɔiy[#\=JZL
i X.sDTUi9^OiÎjW[=X])dfTWJZL)H‘XS(JM[#RMh¨[HhMXvH3N(J”`†[H9|ÏB#wX\V]Ì#|  C~‰RlNhMPqNSuJZT•o#Nj•`#EÐL0XvJ
XRlN,JMLON,PORM[m0XmOTWjUTWJMTUN(hK[HÑH‘XTUjW^ORMN[HJMLON,S
XmVjUN,X\O])[HÑJZLON,mqN
XizҝÓ
‚ S.ƒzžhlhM^OiyTU\V_{hlJYX.JZTUhkJZTWS
XjTU\O]VNPqN\O]ON(\OSNnmqN(J”d€NN(\“JMLONH‘XTUjW^ORMN(hz[HÔJMLONmqN
Xi X\O]
S
XvmOjUNv~DdfL0XvJ«TUh€JMLON¤PORl[#m0XmVTUjUT•J”`Q[HÕH‘XTWjU^ORlN¨[HÕJMLON¤hkJZRM^VS(JZ^VRMN ‚ JML0XvJGTWh~5JZL0X.JGTWJ€dfTUjUj
mqN…^V\0XmOjWN¤JZ[yS
XRlRl`JMLON,jU[=X]qƒkÒ
BCŒVEGžÄS[#\=JZRMXS(JM[#RTUhfhM^OmVi T•JMJZTW\O_QmOTU]VhJM[zŒ›l[#mOh(~¡)~£c~qX\O]š¦E«„GLVNPORl[#m0XmOTWjUT•JZTUN(h€JML0XvJ
LON dfTWjUjdfTW\®N
XvSYL{[H«JZLVRMNNԛl[#mOhXRMNšŠ ‚ ¡,ƒ…Ö×CV|}wÕ~Š ‚ £yƒ,ÖÎCV|ÙØ~X\O]:Š ‚ ¦9ƒ…Ö×CD|ÏBÕ~
RMN(hMPqNS(JMTWoNjW`E¨ž¨hMhl^Oi NN(oN\=JZh ¡)~¾£n~%Xv\O]¦ÊXvRMN…hlJYX.JZTUhkJZTWS
XjWjW`TW\O]ON(P‰N(\O]ON\=JE¤ÚÕN(Jyb
m‰NÔJMLONÔJZ[JZXjA\5^Oi9mqNRf[H0›l[#mOhfXQS[#\=JMRZXSuJZ[#R«dfTWjUj%dfTW\AE
‚ X;ƒyÐL0XvJKXRlNfJMLONfP‰[hMhMTWmOjUN«ovXjW^ONh [H¾bÒ–ÈK[#iyPO^VJMNXv\O] PVjU[JJMLONfPORM[#mOXmOTUjWTWJ”`izXvhMh
H3^O\OSuJZTW[#\[HJMLON,REpo‰EbˆE
‚ m‰ƒzÛ¤N(JMNRli TW\ONŠ ‚ b!’“B5ƒvE
‚ S.ƒÛ¤N(JMNRli TW\ONŠ ‚ Cy‹rb!’“B5ƒvE
BC  EfÜKN(H3[RMNÔJZLON,]ON(hMTW_#\†[vHXaJZ^O\V\ONjÕJZLVRM[#^O_L–XaRM[;SY¯=`RMN(_#TU[#\%~O_#N[jU[#_#TWS
Xj‰NusDPOjW[#RZXvJMTU[#\dKXh
S[#\V]O^OS(JMN]aJZ[¨TU\=o#N(hlJMTU_=XvJMNKJZLVN›l[#TW\5JMhX\V]9JZLON€P‰[JMN\=JZTUXj;hMjUTWPhl^ORlH‘XS(NKJZLOXvJNtsVTWhlJMhTU\JMLON
RM[;SY¯–hkJZRZX.JYXVEQª0[#R¤N(S[#\V[#i TWS9RMNXhM[\Oh~Ñ[#\Oj•`–Pq[#RlJMTU[#\VhÔ[H JZLONQhlJMRZXvJZXXRMNaNtsVPVjU[#RlN]AEaF”\
X]O]OT•JZTW[#\A~JZLONGiyN
Xhl^ORMN(i N(\=JRMN(S[#Rl]ON]Qm=`JMLONGTU\VhlJZRl^OiyN\=JZhXvRMN«\O[vJPqNRlH3N(S(JMjW`RMNjWT‡XmVjUNE
„GL5^OhJZLONG_#N([#jU[_#TUhkJS
Xv\y[\OjW`S[#\VSjU^V]ON«JZL0X.JJZLONGS([#\O]OT•JZTU[\9[HqJZLONGRl[DSY¯izX
`m‰NGN(TWJZLVNR
LOTU_LOjW`†Ý0hMhl^ORMN(] ‚ßÞ ƒY~iyN]OTW^OiàÝ0hlhM^ORlN] ‚âá ƒY~Õ[R…hljUTW_#L=JZjW`)Ý0hlhM^ORlN] ‚ ڃZ~AdfTWJMLgRlNj‡X.JZTWoN
jUTW¯NjWTULO[;[;]OhK[H€#•#ÙØVE
ªTU_#^ORlN…BD²·»¹5µ5µ=ÂtÆV·”´ÁVÂG³Z´µ5¶»·”¸k¹5³Z·”ÂtÅ0Ç
«Ü XhlN]…[#\…JMLOTUhÕTW\VH3[#RlizXvJMTU[#\%~(JZLVN¾N(\O_#TW\ONN(RÑ]VNhMTW_#\OhÕJZLVN JM^O\O\ON(j;X\O],NhkJZTUi XvJMNJZL0XvJT•HDJMLON
RM[;SY¯S[#\V]OTWJMTU[#\nTUhã~JMLON RlNjWT‡XmOTWjUT•J”`†[vHKJZLVN PORl[#Pq[#hMN(]®]ON(hMTU_\®TUh…ä#äDE}䆱E Þ [ed€N(o#N(R~
å”æçkèéuêâçë
ìaíêâî.çïfðtñeò}ïKóïgô}éðuõ#ö0÷øâðtùúç»ê3ìGù•ð(û”êßéuüAétùqýuöÿþ ïGçMðtè.øqêâîvðYêAêâîçûkéuè.ø‘êâüßó.û”êßòÏéuèfò¼øAõeçløâòuèçMõ
êßéÕòÙêßî.ø‘êßðtèvõ¨ðKïfðYñeòÏïKóï7ô}éðuõý ÿþ tíKö
Œ
TWHTWJJZ^ORl\Oha[#^VJ9JML0XvJaJMLON"RM[;SY¯®S([#\O]OT•JZTU[\7TUh~JZLON"PVRM[#m0XvmOTUjWTWJ”`n[HH‘XTUjW^ORMNQdfTUjWjKmqN
]O[#^OmVjUN] hMTWi TWj‡XRljW`#~qTWH¾JZLVN Rl[DSY¯S([#\O]OT•JZTU[\TU
h ~ÑJMLONyPORM[#mOXmOTUjWTWJ”`†[H¾H‘XTUjW^ORMN9dfTWjUjmqN
eCaJZTUiyNh€JZL0XvJfH3[#RGS([#\O]OT•JZTW[#\7ãE
‚ X;ƒyÐL0XvJTUhJZLON«RlNjWT‡XmOTWjUT•J”`[HVJZLON«PVRM[#Pq[#hMN(]JZ^V\O\ONjD]ONhlTU_#\%~TßE NE._#T•o#N€JMLONKPVRM[#m0XvmOTUjWTWJ”`
JZL0X.JGJZLONÔJZ^V\O\ONjÕdfTWjUj‰H3^V\OS(JMTU[#\AE
‚ m‰ƒzžàiy[#RlN"RMNjWT‡XmVjUN"]ONuoDTWSN)TUhQhM^VmOhMN ;^VN\=JZj•`^OhlN]JM[gTWi PVRM[eo#N"JZLVN†PORlN]OTWS(JMTU[#\­[H
RM[;SY¯S[#\V]OTWJMTU[#\AEyFIJZh…RMN(hM^Oj•JZh…TU\O]VTUS
X.JZN9JZLOXvJX)LOTW_#LOj•`Ý0hMhl^ORMN(]7S[\O]OTWJMTU[#
\ H3[#R
JZLONRM[;SY¯aXRl[#^O\O]QJZLVNJM^O\O\ON(j0TUh PORZXvS(JZTWS
XjWjW`TUiyP‰[hMhMTWmOjUN #LO[ed€N(o#N(R~OT•J S
X\V\O[J¾_#TWoN
m‰NuJMJMNRTW\VH3[#RlizXvJMTU[#\,[#\…JZLON€RMN(j‡XvJMTWo#N¾jUTW¯N(jUTULV[D[;],m‰NuJ”dKN(N\QRM[;SY¯,S[#\O]VTWJZTW[#\Oh X\O]
ãE F”\"jUTW_#L=JK[HÕJMLOTUhK\ON(d“TU\DH3[#RMi XvJZTW[#\A~#dfL0X.JGdK[#^VjU]mqNJZLVNÔRMN(o;TWhMN]"RlNjWT‡XmOTWjUT•J”`9[H
JZLON,PORl[#Pq[#hMN(])JM^O\O\ON(jÑ]VNhMTW_#\°Ò
BC#wDEG„GLONÔLO[#^VRMjW`Qo[#jU^OiyN[vHÕJZRZX.x SH3[#R«XPORM[#Pq[#hlN]"LOTW_#L=d«X
`TWhK]OTWhlJMRMTUmV^VJZN(]"XhKTW\"ªTW_#^ORMN¨Œ
dfTWJML)JMLON,P‰]VH
˜–‹Œ#C#C ‚ ˜%ƒ
Ö ™”™”™ ™ ”™”™ CaŒ#C#‹?Ca‹r
˜–‹  CCV|
™”™ Ó ™” ™ žàJMRZXvxyS–N(\O_#TW\ONN(R TWhyJM[®]ON(hMTU_\JZLON†LOTU_L5dKX
`šS
XP0XvSTWJ”`EžhMhl^OiyN–JZL0X.J JMLONXSuJZ^0Xvj
S
XPOXST•J”`a[HqJZLONfLOTW_#L=d«X
`QXvH JMNRT•J TUhmO^OTWjWJTWhNT•JZLON(R¾Œ#C#C¤[#R¾Œ=wCoNLOTWSjWNh P‰N(R LO[#^ORdfT•JZL
RMN(j‡XvJMTWo#NÔPVRM[#m0XvmOTUjWTWJZTWNh¾[H€,JM[  EÐL0XvJfTWhKJZLON,PORl[#m0XmVTUjUT•J”`QJZL0XvJGJMLON,S
XPOXST•J”`zdfTUjWj%mqN
NusDSN(N]ON(]°Ò
−3
x 10
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
350
400
450
ªTU_^ORMN,ŒV! ¿=Â#"5¸»´eÁDÀ
Á;ºúÆúº¼·%$…Å=Âtµ5¶kº¼·&$('U¹;µ5³Z·»º¼´µ)'U´e¸+*-,=·»¿=µ¹5ÃÁOÂY¸´.'0/eÂt¿5ºú³YÆúÂt¶1"VÂY¸€¿=´¹=¸tÇ
BCVEGF”\–[#Rl]ONRGJM[ RlNP0XTWR«JZLON…SRMXSY¯5hJML0XvJi Xe`NusDTUhkJ¨TW\)X­
C0ÉßH3NN(JGd€NjW]A~qXy\V[#\Dɔ]VNhlJMRM^OSuJZT•o#N
JZN(hlJZTW\O_ ‚325476 ƒa]ONuoDTWSNTWh Ý0RlhlJ"^VhMN]?JM[7]ONuJZNSuJ"JZLON†jU[;S
XvJMTU[#\š[H,S(RZXSY¯5hEÄ܀NS
Xv^OhMN
SRMXSY¯5hi X
`NtsVTWhlJGTW\ovXRMTW[#^OhKhlL0XPqNhX\O]"hlT98(Nh~OJMLONÔPORl[#m0XmOTWjUT•J”`aJZL0XvJGX9SRZXvSY¯zdfTUjWj‰mqN
]ON(JMNSuJZN]m=`JZLVN 2:4;6 ]VN(o;TUS(NTUhf[#\VjW`{CD|}ØEGžhlhM^OiyNJML0XvJJZLON…N(oN\=JZh¤[vHN
XSYL–SRZXvSY¯

‰m N(TU\O_]VN(JZN(S(JMN]rXRMN"hkJYXvJMTUhlJMTUSXjUj•`gTW\O]ON(P‰N(\O]ON\=J X\O]{JML0XvJaJZLVN 25476 ]V[DN(hy\V[Ja_#T•o#N
H‘XjUhlN,Xj‡XvRMiyhE
X;ƒ FIHÑJZLON(RMN…XRMN¤J”d€[QSRMXSY¯5h¨TW\JZLON¤d€NjW]A~OdfL0XvJfTWh«JZLONÔPORl[#m0XmVTUjUT•J”`QJZL0XvJKJZLONu`dK[^OjU]
\O[JmqN,]ON(JMNS(JMN]°Ò
m‰ƒz„GLONXS(JM^0XjÕ\5^Oi9mqNRf[HSRMXSY¯5h 2 TU\"JMLONÔdKN(jU])TUhG\O[J¯5\O[edf\AE Þ [ed€N(o#N(R~°TWJMh
p (n)
N
0.6
0.3
0.1
0
1
n, number of cracks
2
ªTW_#^ORlN  1<¸»´eÁ;À
Á;ºúƕº¼·&$…Ã…Àe¶»¶!'‡¹5µ5³Z·»º¼´µ)'U´e¸ ·»¿=Âf¸uÇ=/DÇ?>@,5·»¿5Âfµv¹5èÁOÂY¸´.'Õ³Z¸kÀe³BA#¶YÇ
UT h_#TWoN\QXhTW\aªTW_#^ORMN  EÐLOXvJTUhÑJZLONKPORM[#mOXmOTUjWTWJ”`ÔJML0XvJJMLON 2:4;6 ]ONuoDTWSN
dfTUjWjA]ONuJZNSuJ¨CyS(RZXSY¯5hTU\"JZLOTWh«dKN(jU]°Ò
S.ƒzFIH JZLONa]VN(o;TUS(N9]ON(JMNSuJZh…CzSRMXSY¯5h,TU\–JMLONd€NjU]%~%dfL0XvJ,TWh¨JZLVN9PORM[m0XmOTWjUTWJ”`"JML0XvJ¤JMLON
d€NjU])TUh F0XedfjWNhlh ‚ JZL0XvJfTWh~O\V[ S(RZXSY¯XvJXvjUj‘ƒ»Ò ‚âÞ TW\5J Ü«X
`#N(h¨JZLVN[#RlNi"ƒ
C ED