coefficients de restitution et efforts aux impacts: …issn 0249-6399 apport de recherche thème num...
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Submitted on 19 May 2006
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Coefficients de restitution et efforts aux impacts : Revueet comparaison des estimations analytiques
Vincent Acary, Bernard Brogliato
To cite this version:Vincent Acary, Bernard Brogliato. Coefficients de restitution et efforts aux impacts : Revue et com-paraison des estimations analytiques. [Rapport de recherche] RR-5401, INRIA. 2004, pp.162. inria-00070602
ISS
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249-
6399
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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Coefficients de restitution et efforts aux impacts.Revue et comparaison des estimations analytiques.
Vincent Acary — Bernard Brogliato
N° 5401
Décembre 2004
Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)
Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52
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mξ + cξ + kξ = 0»2½
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ξ(t) =v0ω0
exp(−αω0t) sin(ω0
√
1 − α2t)»Ç¼P½
y ω0 =
√
k
m
£w~^q^£wkomutyp9~^suyP~^so`-lq9kujlknmo]_=`²α =
c
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en = exp
(
− απ√
1 − α2
)
»GP½
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3kuyqJk£w¡ª®yso_=`#kq^t¢pDmu`
¦¦¦ξ +
k
cξ +
k
mξ = 0
»nQP½
¢`Q,£`QkyPpltmutypktp^tmutw£`LkL²ξ(t = 0) = 0, ξ(t = 0) = v0, ξ(t = 0) = 0
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ξ(t) = exp(−νω0t)
(
v0
ω√
1 − ν2sin(ω0
√
1 − ν2t) − 2νv0ω0
cos(ω0t)
)
+2νv0ω0
»Q½
y ν =
√km
2c=
1
4α
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c²~^~so ml`9ªÇ yPp ltSrLsu`LpPmo`^pk£`¡yH`¬t`LpPm
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α → 1tA¦ `P¦
c → cc²yp,ylmot`LpDm pk
£`Lk ^`ql,ken → 0
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tf£yskuvDq^`,£w ª®yPso`0k« ppDq£`,~Syq^s
0 ≤ α < 1
tf =1
ω0
√
1 − α2arctan
(
2α√
1 − α2
2α2 − 1
)
»Q½
´ep3y^mut`pDmc£ysk q^p^`8`Lkmut_mutypÈlqÈyD`¬t`pDml`#su`Qknmotmoqlmutyp,kuyqk£wª®yso_¡`#kuq^t¢µpDmu`
en = exp
(
− α√1 − α2
arctan
(
2α√
1 − α2
2α2 − 1
))
» ) ½
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k1
k2
c
S A ' MP= F
§9ª®yPso`#^pk£A« kok`L_9^£wP`e`Qknm£ysk su`L£tr`#q,^r~^£w`_=`pDmξ~Js £wso`£wmotyPpkuq^t¢µpDmu`
(k1 + k2)F + cF = k2k1ξ + k2cξ»D½
`#vPqt¨yPplq^tmc¥¡£Y« rLvDqmutyp,lq,_¡yPq^¢`L_=`pDm ¢µ£^£`~S`pJ^pDm £`ypDmoPm
2αη
ω0
¦¦¦ξ + ξ + 2αω0ξ + ω2
0ξ = 0»nµ½
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y £w9~^q£komotyPpω0
`Lkm £w~^q^£wkumutyppmuq^so`££`lq3kjlkmu]L_¡`P²α£`~s_=]mosu`el`e¢HtwkuyPkutmorc`m
ηq^p
s~~JyPsm l`#stwl`q^s lr¯p^twk ~s
ω0 =
√
k1k2
(k1 + k2)m
»LP½
α =k2c
2(k1 + k2)mω0
» 2½
η =k1
k1 + k2
»L¼P½
°£`Lkm¥p^ymo`sevDq^`£yPsokovDq^`k2 → ∞ ²SypÈso`musoyq¢`£`_=yll]£`kmop^sÈl` 8`£¢Dtpl¤ SRytm»
ω0 →√
k1/m²α → c/2mω0
²η → 0
½¦§,surQkyP£qlmotyPp³l`'`mmo`rLvDqmutypÀltWrso`pDmot`L££`p6« `Lkm9rLp^rs£`_=`LpPm#~Pk#~SyPkokt^£`¦6¾Opk« ~l¤
~q^jPpPmkq^se£'su`L_svPq`#~^surQrQl`pDmu`P²t£`Qknm~syPpPmosu`9~JyDkukut£`9^`lyPp^p^`seq^p^`¡kyP£qlmotyPp0s `¥,qp^`mu`Q\^p^twvDq^`l`~S`sumuq^somutypÀ~JyPq^s
η ≪ 1¦±`mumu`mo`L\^p^twvDq^`=ª®yqsup^tm9q^p``Lkmut_=mutyplq
yH`¬t`pDml`#so`LkmutmuqlmotyPp¢£w£`#¥¡£Y« yslso`1`Lp
η
en = exp
[(
− α√
1 − α2+ ηf1(α)
)(
arctan
(
2α√
1 − α2
2α2 − 1
)
+ ηf2(α)
)]
»AGP½
µ¢P`Lf1(α) = α− α3/2 + O(α5)
`mf2(α) = 2α− 3α3 + O(α5)§¨`c~so_¡]muso`c^t_=`pJktyp^p`£
η~S`so_=`m~^\HjlktwvDq^`_=`pDml`eypDmus£`s£#sorPq^£wsotkomotyPp=kuq^s £`
koq^m ^`eª®yPso`#¥£A« tpkmopDmtp^tmutw£¨^`£A« t_=~PmL¦ShHtη = 0
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ξ
8`£¢Htp^¤SRyPtmd0 `££S`p`s
η = 0.05S`p`s
η = 0.2S`p`s
η = 0.4
¨ $ %D! ! ! ! ) WRR= ! %D P $ ' ω0 = 1
α = 0.1
ν = 0.1
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en
±yH`¬t`pDml`#¢HtkoyDktmurαyq
ν
8`L£¢Htpl¤ SRytm `mcd0 `L££
8`L£¢Htpl¤ SRytm _=yH^t¯r
S`Lp^`sη = 0.05
S`Lp^`sη = 0.2
S`Lp^`sη = 0.4
W = ) R = # !$ $ '
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W! !
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p£jDmutwvDq^`LkL²Hq^pÈ\^ytkut_=~^£`l`8_=yH^]£`« _=ysumutwkuku`q^s`Qknm £`kq^t¢pDm
F = −c|ξ|ξ − kξ»YP½
¹ ¶ ¨¸¨º »nGGPP½_=ypDmuso`pDmevDq^`9`_=yll]£`tplq^tmcq^p^`¡lrLsuyPtkokop`8^qyH`¬t`pDml`so`LkmutmuqlmotyPpµ¢`L£#¢Htmu`Qkuku`ctp^tmutw£`e« t_=~PmQ¦PxO£qk ~^sorLtkur_=`pDmQ²~Syq^s l`cPsopl`Lk¢£`Lq^sok#^`¢Dtmu`Qkuku`« tpt^`p`²¨£`yH`¬t`LpDm#^`su`QknmotmoqlmutypÀ`LkmtpD¢P`sk`L_=`pDm#~^suyP~JyPsmotyPp^p^`L£¥`mumu`#¢Htmo`Lkok`
(1 − en) ∝ v−1n
»AlQ½
9$,M9 Q!*H Æ * *HLQ Qà /  / ÄoÆ0 / ,'0 * *PÃ$^ Â/µÃA0YL *H§-« qlmot£twkumutypÈ« q^pÈsu`Qkukuysump^ypl¤Z£tp^rQtsu`8~S`so_=`mcl`#so`p^su`#lrL~J`Lp^pDmk £`8mo`_=~kel`yPpPmm
tpJktRvDq^`¡£`=yH`¬t`LpPm#l`¡su`Qknmotmoqlmutyp"l`=£'¢Htmo`Lkok`¡tptmot£`=« t_=~PmL¦6§ª®yPsu_=`£~^£qk#yP_¤_q^p^rL_¡`LpDm \ytwkt`kL« tpku~^tsu`8lq3_=yH^]£`8^`8ypDmoPm l`8É`sumuÊP¦^§¨ª®ys`p^yPsu_£`e^q^`#q3su`QkukuysumkL« `l~^sot_=`#£yskl`8£=_pt]Lsu`8kqt¢pDmo`
Fn = −knξ32
»A½y
kn`Lkmqp^` sot^`q^sso`£trL`qlLsmorsotkmutwvDq^`Lkr£wkmutwvDq^`Lklq=_=t£t`q=`mOqsµjyPpl` yPq^so^q^so`
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pk £`Lkekt_9q£mutypk^`_murLsutwq^'spHq^£wtsu`Qk8» ¹¸ O²¨AG 2 GP½¦l¾O££`9yplqtme¥qp^``Lkmut_motyPp,^q yH`¬t`LpPmc^`su`Qknmotmoqlmutyp0vDq^tsoy wmcµ¢`Qe£w¢Htmu`Lkoku`#tp^tmutw£`9« t_¡~JmQ¦Jacyqk£w¡£tkoku`soypklyPp8l`#ymor#~^q^twkuvDq^`#`£w¡ypDmuso`L^tm £`Lkypkmomutypk `l~Srsot_=`LpPm£`Lk qkuq^`££`Lk
xyqs'`0vDq^t`Qknm3lq \^yt « q^p _=yPsmotkok`Lq^sp^yPp £tp^rLtso`Èy\rso`pDmµ¢`L3£`Lkypkmomutypk`l~JrLsut_=`pDmo£`Qk²£`Qk=`l`_=~^£`Lk=^pk¡£©£tmumursmoq^so`3kuypDm¡muso]Lk¡p^y_^so`ql³`m`3kuqzn`mypkmutmuq^``LpyPsu`eq^pmo\^]_=`#l`su`Q\^`s\^`cPmotªn¦l§6`Qk _¡yll]L£`QkkyPpPm `p3rLp^rs£S\^ytwktwk~JyPq^s£`Lq^s ªÇt£tmorl`sorLkuy£qlmotyPpp£jDmutwvDq^`¦l´ep~S`qlm tmu`Ls£w9£Pkuku`cl`e_¡yll]L£`lr¢P`£y~^~Sr`e~s¶ ¹ e¸ ·
»AG>2½²Dª®yplrL`kqs q^p,`Wysumpyso_=£kyPqk£ª®yPsu_=`#kqt¢pDmo`
Fn = −c|ξ|nξ − kξn »A ) ½xyPq^s`Lk_¡yll]L£`Qk²Dt£J`Htwkmu`cqp^`ekuy£qlmutypp£jDmutwvDq^`e¥#£A« rLvDqmotyPp'l`e£ljHp_=twvPq`¦°£W`Qknm£ysk~SyPkokut^£`9l`¡\^ytwkuts
n = 32
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1 = R1, R′
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1
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a
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uz =1 − ν2
E
πp0
4a(2a2 − r2), r ≤ a
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4aE∗(2a2 − r2) = δ − 1
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1 − ν21
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1 − ν22
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−(y
b
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12 »Y¼P½
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uz =1 − ν2
πE(L−Mx2 −Ny2)
»YGP½kuytm ~JyPq^s £`Lkl`Lqlyso~k5
uz1 + uz2 =1
πE∗(L−Mx2 −Ny2)
» 2P½§so`£wmotyPpPryP_¡rmusotvDq^`8q,yPpDmoml`É`LsmoÊ
uz1 + uz2 = δ −Ax2 −By2 » 2HQ½
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b
e2a2 (K(e) − E(e))»2P½
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p0
E∗
b
e2a2
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a2
b2E(e) − K(e)
) » 2 ) ½
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p0
E∗bK(e)
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∫ π/2
0
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1 − e2 sin2 φ
» 2½
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∫ π/2
0
√
1 − e2 sin2 φ dφ»2P½
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3p0πab
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pm~J`Lqlmc£ysk ¿mosu`#lrQlq^tmu`
pm =2
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kuq^t¢pDmu`Qk5
B
A=
R′
R′′=
(a
b
)2
E(e) − K(e)
K(e) − E(e)
» 2GP½
(AB)−12 =
1
2
(
1
R′R′′
)
−12
=1
2Re
»Y¼P½
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E∗
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(a
b
)2
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§¨`Lk¢£`qsok l`QktpDmors£`Lk `L££t~^mutwvPq`LkK(e)
`mE(e)
tpDmusoyH^q^tmo`Lk q 6 9¦ kuypDmmoq^£rL`LkL¦^¾O££`Qk~S`q¢`pDm¿muso`8musoyq^¢Pr`Lk pk» ^·^¸²AGPGP½¦l[¨yPqlmu`ª®ytwk²^qp^`9~~^suyµlt_mutyp~JyPq^s£`#so~^~Sysumb/a
~JyPq^scl`Qk¢£`q^sk yP_¡~sutwk`Qk `pDmuso`1`m
5~S`q^mc¿mosu`#lyPp^p^r`8~s
b
a≈ (
B
A)−
23 = (
R′
R′′)−
23 ,
b
a∈ [1, 5]
»Ç¼D½
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( )
±`mmo`~^~^soyµlt_mutyprL¢Dtmu`£yPsok« `Lkmut_=`s £`Lk tpDmurLs£`Lk`££t~lmutwvDq^`LkL²K(e)
`mE(e)
¦¾OpÈtpDmusoyllq^twkupPm _tpDmu`LppDm q^p^`#p^yPq^¢`L££`8¢sot^£`P²
c = (ab)12`m`Lp3kuq^kmutmuqJpDm
p0~s £`
\Jso`_=`LpPm muymo£Y²lyPp~S`qlmerLsotsu`
c3 = (ab)32 =
(
3PRe
4E∗
4
πe2
)
(
b
a
12
)32 [(
(a
b
)2
E(e) − K(e)
)
(K(e) − E(e))
]12 »Y¼ ) ½
kuytm
c = (ab)12 =
(
3PRe
4E∗
)13
F1(e)»Ç¼D½
±`mmu`#rQsotmoq^so`~S`so_=`mc« r¢£q^`Ls £Y« tpl`pDmmutyp3`p3ª®yPpmotyPp3lqÈ\suP`_=`pDm muymo£
δ =3P
2πabE∗bK(e)
»Ç¼D½
=
(
9p2
16E∗Re
)
13 2
π
(
b
a
)12
(F1(e))13 K(e)
»Ç¼PP½
=
(
9p2
16E∗Re
)
13
F2(e)»Ç¼>2½
¾Op^¯p6²£`8_lt_q^_1^`~su`QkukutyPp`QknmlyPp^p^r#~s
p0 =3P
2πab=
(
6PE∗
π3R2e
)13
(F1(e))−
23
»Y¼¼P½
$ ±yP_=_¡`£`_=ypDmuso`£A« rQvPqJmutyp » 2GD½²£Y« `^`LpDmusottmor
ep^`lrL~J`Lp~Jk9^q\suP`_=`pDm_twk^q
s~~JyPsm l`Lkcyqsu^qsu`Qktp^tmutw£`Lk R′
R′′
¦§-« rQvPqJmutyp » ) D½²vDq^tOsorPtm£A« tpDmu`Lsokmutw``LpPmosu`£`Lk9l`qlÀyPsu~k8pyp³\sorQk²W_=ypDmosu`vDq^`£`Lk
twkuy¢µ£`Lq^sk« rP£etpDmu`Lsokmutw` kuypDm,l`Lk`££t~ku`Lk3µjPpPm~Syqs~^soy~sutrmor b
a= (
A
B)
12 = (
R′′
R′)
12¦
±`L~J`Lp^pPmQ²£w'kuq^suªÇ`^`yPpDmomc`££t~lmotvDq^``Lkmeku`pkut£`L_¡`LpDmc~^£qkcrL£prL`~S`p^pDme£wlrª®yPs¤_mutyp6¦
$ bcpsµjyPp9rQvPqt¢£`pDmOl`£wkuq^suªÇ` l` ypDmoPmL²
c = (ab)12²tpkut^vDq6« qp^`yPq^suq^su`su`L£mut¢`P²
Re =
(R′R′′)12²¨ypDm#rmurtpDmusoyllq^tmu`Lk9qÀyqsok#l`Qk#£q^£kL¦±`Lkspl`Lq^ske~S`so_=`mumu`pDm9l`y_=~su`Ls
£`=~^so`Lkoktyp0_Ht_£`²p0`m#£Y« tpJl`pDmomutyp
δµ¢P`L£`=~^soy^£]_=`=ltkujH_=rmusotwvPq`¦¨´ep©~S`q^m9£ysk
so`_sovDq^`Ls9vDq^`£`Lk~^so`_=t`skmo`so_¡`Qk^`Lk`l~^so`Lkoktypk»Ç¼D½² »Ç¼D½`mȻǼ¼D½kuypDmtwl`LpPmotvDq^`QkqlsorLkuq^£momokOHtwkujD_=rmusotvDq^`Qk¦hH`Lq^£qpmu`Lsu_=`l`cysosu`QmotyPp3»
F1(e), (F1(e))23 , F2(e)
½`Qknmznyqlmor¯p^`emo`p^tscy_=~lmu`#l`#£A« `^`LpPmosutwtmur#l`#£A« `££t~ku`²
e¦
ÍÍ ÒT'UWVXY
) W! !
O *HN6Æ * 0* ÅÂ / Ã!^ÅÃ * / Ã* 0*H:OÅ+6Æ0 / 0*H0 / ÅPÆ0 / H I ?FH ¯Jp l`,lyp^p`svDq^`L£vDq^`yPso^su`¡l`Psopl`Lq^seqlÈmu`Lsu_=`Lk8ysosu`Qmu`Lq^sk
F1(e), (F1(e))23`mF2(e)
²lqk#¥£A« `l`pDmusotwtmur²Sypypkutwl]so`e£`8~^soy^£]_=`#l`#yPpPmm ^`8^`ql,jH£tplso`Lkl`#sµjyPp
RlyPpDm£`Lk l`LkkyPpDmtp£tp^rLk ¥
HP°¦§¨`#so~^~JyPsm`pDmosu`8£`Lkyq^so^q^so`eso`£wmot¢P`Lk `LkmrD£¥
(R′
R′′)
12 = (
B
A)
12 = 2.41
»Ç¼=GP½`m£`8sµjyPp'rLvDq^t¢£`pDm `LkmrD£¥
Re = (R′R′′)12 =
√2R.
»GPP½hHyPqkR\suP`_=`pDmL²µ£` so~^~Sysum`LpPmosu`£`LkRl`L_=t¤°l`Lk~S`q^mO¿muso` £q^£rPsP`¥¡» 2GD½^` _=p^t]so`
`^mo`a
b= 3.18
»GQ½yPq3^`#_=p^t]so`8~^~^soyl\^r`s `¥,»Ç¼D½
a
b≈ 3.25
»GD½§¨`Lk mu`Lsu_=`QkyPsuso`Lmu`qsok~^so`p^p^`LpDm £yPsok £`Lk ¢£`Lq^sokkuq^t¢µpDmu`Lk
F1(e) ≈ F2(e) = 0.95»G ) ½
(F1(e))23 ≈ 1.08
»GD½´ep ypJknmmu`"vDq^`©~Syqs,^`Lk3so~^~Sysumok'^`"l`L_=t¤°l`Lk'ªÇt^£`LkÀ»
b/a < 5½²yp ~J`LqlmÈ\^ytwkts
F1 = F2 = 1`p qlmut£twkopDmc£`Lk ª®yPsu_q^£`Lk ~Syq^seq^p^`¡kq^suªÇ`l`9ypDmoPmetsq^£wtso`#~JyPq^sq^p0sµjyPp
rQvDq^t¢£`LpPmRe¦6epk£Y« `l`L_¡~£`l`Qk^`ql©jH£tplso`LkL²`mumu`=~^~suyµlt_=mutyp yp^q^tm¥q^p^`kuq^s¤
`Qknmot_mutyp,l`#£mt££`8_¡yjP`p^p`8^q,ypDmoPmc`mcl`8£A« tpl`LpPmmotyPp
δ« `LpH¢Dtsoyp
5%`mq^p^`kuyqk
`Qknmot_mutyp,l`#£¡~^so`LkokutyPp'_lt_=£`p0q,ypDmml`
8%» ²AGP¼PP½¦
% '% ./, #(' # #5,1 .0%, % )+"1) # #" % #
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Plytm¿mosu`8`l~^sot_¡r8`Lp3qp^tmorl`8ª®ys`8~sq^p^tmurl`#£yp^Pq^`qsL¦
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2a¦
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btpl¯p^tA¦
bcp^`=£mu`sopmut¢`²S~^£qk8kt_=~^£`²¥'`mmo`¡~^~^soyl\^`9`Qknm#l`=ypJktwlrso`svDq^`£`9~suyP^£]L_=`~S`qlm8¿mosu`
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h = z1 + z2 = Ax2 =1
2
(
1
R 1+
1
R 2
)
x2 =1
2
1
Rx2 »GP½
§¡ypDmustpDmu`#l`8p^yPp3~SrprmusmotyPp`LpDmuso`£`Lk kuy£twl`Lk#»ÇPP½ l`L¢Ht`LpPmc^yp
uz1 + uz2 = δ −Ax2 = δ − 1
2
1
Rx2 »GP½
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P¥£A« tpl`LpDmomotyPp
δ²
`m`el`eªÇ yptplrL~J`Lp^pPmo`^`eyPpkutlrLsomutypk-£y£`Qk¦l±`mumu`8vPq`Lkmutyp'ku`s9ltwkoqlmor`c~£qk `Lp^rmot£6pk£` 6 ^¦¦
±`L~J`Lp^pDmQ²R£`,~^soy^£]_=`3`p lrª®yPsu_mutypk¡yq `p yPpDmustpPmo`Lk~J`Lqlm¿mosu`3sorLkuy£q6¦ -sot¢Pypk~Syqs`£w¡£`#PsoPlt`pDm ~Js s~^~Sysum¥
xl`#£A« rLvDqmotyPp"»GD½+
∂uz1
∂x+∂uz2
∂x= − 1
Rx
»G 2½ s `8¥¡£A« rQvDqmotyPp"»Y 2 ) ½²^p^yPqk~Syq^¢PypkrQsotso`vDq^`
∂uz1
∂x+∂uz2
∂x= − 2
πE∗
∫ a
−a
p(s)
x− sds
»GP¼P½
kuytm `pÈkuq^kmutmuqpPm^pk#»G>2½+∫ a
−a
p(s)
x− sds =
πE∗
2Rx
»GGP½±`mmu`rLvDqmutypÀ~J`Lqlm¿muso`sorLkuy£q^`=`p ypkut^rspDm#£È~su`QkukutyPp³y_=_=`tpyPp^pHq^`¦¾O££`ªÇtm
~^~S`£qlÀ¢£`q^sk~^sotpt~£`Lk9l`LktpDmors£`Lk9l`,± q\Hj¦c~^~S`£ypkkt_=~^£`_=`pDm£`sorLkuq^£mompkp^ymuso`#k~Jsumutwq^£t`Ls
p(x) = −πE∗
2R
x2 − a2/2
π(a2 − x2)12
+P
π(a2 − x2)12
»nQD½
±`mmo`ckyP£q^mutyplr¯Jp^tmp(x)
l`_pt]Lsu` q^ptvDq^`ckut^p^yPqkOkuy_=_=`LkOL~^£`« `l\^tS`s-q^p^`so`£wmotyPp`LpDmuso`£`¡\Jso`_=`LpPmemuymo£
P`m#£3l`L_¡t¤Z£suP`q^s8l`¡£,kuq^sªÇP`¡l`ypDmm
a¦SxyPq^s#`£w^²Wp^yPqk
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) ¼ W! !
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P ≥ πa2E∗
4R
»nQ^µ½hlt-£`'\so`L_=`pDm8muym£`^]Ql`£0_¡tp^yPsomutyp³lypp^r`~s'»nQ^µ½²¨t£-`Qknm£yPsok~JyDkukut£`l`lr¤_=yPpPmosu`LsevDq^`£`slt`pDmel`£'kq^suªÇ`l`L¢Ht`LpPmtp^¯p^t¥£w£t_=tmu`^`9£w'kuq^sªÇP`9\Jsor`P¦W±`mmu`kutmuqmutyp0rmopDm#¥`l£qsu`=^pJkc£`¡Lkl`=kq^suªÇ`LkcsurLq^£t]Lsu`Qk²S£w'ku`q^£`¡yp£qJktyp0~SyPkokut^£`9`QknmvDq^`
P =πa2E∗
4R
»nQPP½kuytm
a2 =4PR
πE∗
»nQ ) ½´ep,y^mut`pDmlyp8£A« `l~^so`Lkoktyp3l`8£=~^so`LkokutyPpl`ypDmoPm kq^t¢pDmu`
p(x) =2P
πa2 (a2 − x2)12
»nQH½
§¨`#~^su`Qkukutyp'_Ht_9q_4`Lkmclypp^r`8~s
p0 =2P
πa=
4
πpm =
(
PE∗
πR
)12 »nQPP½
y pm
`Lkm £¡~^so`LkokutyPp_=yj`Lp^p^`P¦§¨`Lk#yPpDmustpPmo`Lk^pJke£`Lk8kuy£t^`Lk~J`Lq^¢`LpDm8¿muso`¡L£wq^£r`Qks `¡ql0`l~^su`Qkukutypk¡»Ç>2½¦´ep
yPlmut`pDm ~Syq^sc`#k~Jsumutwq^£t`Ls l`#~^so`Lkoktyp`m£`#£yPp^l`8£Y« l`z(x = 0)
σx = −p0
a
(
(a2 + 2z2)(a2 + z2)−12 − 2z
) »nQD½
σz = −p0a(a2 + z2)−
12
»nQ 2½
4 #-'# ./,3/) ./% ,($% 1 1
Ë®ÌÍ6ËÏÎ
!"#$&%'! (
) G
z p0 =3P
2πa2 =
(
6PE∗2
π2R2
)13
δ =a2
R=
(
9P 2
16RE∗2
)
13
a =
(
3PR
4E∗
)13
y p0 =2P
πa=
4
πpm =
(
PE∗
πR
)12 a2 =
4PR
πE∗
p0 =3P
2πab=
(
6PE∗
π3R2e
)13
(F1(e))−
23 δ =
(
9p2
16E∗Re
)
13
F2(e) c = (ab)12 =
(
3PRe
4E∗
)13
F1(e)
1
E∗=
1 − ν21
E1+
1 − ν22
E2
1
R=
1
R1+
1
R2Re = (R′R′′)1/2
F1(e) =
(
4
πe2
)13(
1 − e2)
16
[((
1
1 − e2
)
E(e) − K(e)
)
(K(e) − E(e))
]12
F2(e) = (F1(e))13 K(e)
K(e) =
∫ π/2
0
dφ√
1 − e2 sin2 φ
E(e) =
∫ π/2
0
√
1 − e2 sin2 φ dφ
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vz1`m
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G1,m1R1
E1, ν1vz1
G2,m2
R2
E2, ν2
vz2
P
FG ! $&%'! ! P" %&%('
ypkutwl]so`9vDq^`£`Lk`LpPmosu`Qkcl`sµ¢Htmor9l`\Jqp^`9l`Qkeku~^\^]Lsu`Qkcku`9s~^~^soyl\^`pDmc« q^p^`¢£`q^sδzvDq^tysosu`Qk~Syp¥£wlrª®yso_mutyp'r£wkmutwvPq`el`Qk yso~kL¦D§9¢sotwmutyp^`8`mumu`8lrª®yPsu_motyPp~S`qlm
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vz2 − vz1 =d δzd t
»nQ¼D½±yPpktwlrLsuyPpk=_tpPmo`ppDm=£wÀsorLkuq^£mpDmu`Èl`Qkmutypk^`0ypDmm`LpPmosu`È£`Lkyso~k
P¦-acyqk
~Syq¢ypJkrQsotso`#kt_=~^£`_=`pDmcvDq^`
P = m1d vz1
d t= −m2
d vz2
d t
»nQGD½kuytmL²l`LpÈ`#vDq^t¨yPp`Lsup`e£A« tpl`LpDmomotyPp0
−m1 +m2
m1m2P =
d (vz2 − vz1)
d t=d2 δzd t2
»nPLD½
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P =4
3R
12E∗δ
32z = Kδ
32z
»nPµ½y
1
R=
1
R1+
1
R2
»nPQP½
1
E∗=
1 − ν21
E1+
1 − ν22
E2
»nP ) ½
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1
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md2 δzd t2
= −Kδ32z
»nPQP½hHt¨£Y« yp,tpPmo]Psu`8`mmu`#rQvDqmotyPp3ltWrso`pDmut`££`#~Jsso~^~Sysum
δz²yp3ylmot`LpPm
1
2
[
V 2z −
(
d δzd t
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=2
5
K
mδ
52z
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y Vz = (vz2 − vz1)|t=0
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vDq^tku`8~^soyH^q^tm~Syqs d δz
d t= 0
²^kuytm
δmz =
(
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4K
)
25
=
(
15mV 2z
16R12E∗
)
25 »nPL¼D½
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T0 =1
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z
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8R12E∗
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§-« `Qknmot_mutypÈl`δmz
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(
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(
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8R3E∗
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d δzd t
= δ′z = Vz
√
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δzδmz
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≥ 0»nµ ) ½
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t =
∫ t
0
δzδ′zdt =
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0
d δzδ′z
=
∫ δz
0
d δz
Vz
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1 −(
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kuytmL²l~Syqs £`8mo`_=~kl`#y_=~^so`Lkoktyp_=lt_q^_
tm =δmz
Vz
∫ δmz
0
1(
1 − (δz/δmz )
52
)12
d(δz/δmz )
»nµP½
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tm = 1.4716376δmz
Vz
»nµD½
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
t/tm
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sin(πt/2tm)
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(x∗, y∗)`LkmclrLkutp^r8~s £¡¢sot^£`y_=~^£`l`#kq^t¢pDmu`
z∗ = x∗ + i y∗»nL ) ½
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F∗
F∗
R
R′
R′′
x∗
y∗
z∗l
λ∗
Q ' !W ' F' %'!=" R =
ÍÍ ÒT'UWVXY
P W! !
`Qknm#~SyPkutmotyPp^p^r¥z∗ = i R
¦We`q^"`Lso£`Qksuttwl`Lk#^`¡sµjyPpkR′
`mR′′
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x∗`my∗¦S´epÈlr¯p^tmcl`8_¡¿L_=`
w∗ = u∗ + i v∗»nLH½
´epp^ymo`e~sλ∗£A« tpl`LpDmomotyPp'so`£wmot¢P`elq3`s£`suttwl`e^`sµjyPp
R′pk £``s£`etpl`LpPmor¦
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F ∗¦^§¨¡¢£tltmur
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εlr¯p^tl`£9_p^t]so`ekuq^t¢µpDmu`
ε =l
R
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ε′ =l
R′
»nLPD½§-« rQvPqJmutyp,l`#£w=kq^suªÇ`8tplª®rLsut`qsu`8p^yPp3lrª®yso_=r`#lqÈ`Lso£`8r£wkmutwvDq^`8`Lkmlyp^pr`#~s
y∗(x∗) = R −√
R2 − x∗2,»nL>2½
vDq^t~S`q^mc¿mosu`#lrL¢`£y~~JrL`8¥¡£A« yPso^su` O(ε)^pk£A« tpDmu`Lsu¢££` |x∗| ≤ l/2
²lkuyqk£wª®yso_¡`kuq^t¢pDmu`
y∗(x∗) ≈ x∗2
2R, |x∗| ≤ l/2
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y∗(x∗) ≈ −x∗2
2R′, |x∗| ≤ l/2
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motyPp »n GP½¦ ´ep su`musoyq^¢P`ÈkyPqkq^p^`0ª®yso_¡`0kut_¡~£`0£w³yPpDmustpPmo`,ryP_=rmusotwvPq`0¥À£Y« yslso` O(ε)vDq^t~suyllq^tm£`lr~^£w`_=`pDmkuq^t¢pPm
v∗(x∗) = −1
2
(
1
R+
1
R′
)
x∗2 + const., |x∗| ≤ l/2»nµP½
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Z =z∗
l, W =
w∗
l, Λ =
λ∗
l, F =
F ∗
µl, P =
p∗
l,
»nµlµ½
y µ`Lkm £`#_=yllq^£`l`#twkot££`_=`pDmlq,_morsotqrL£PknmotvDq^`8`m
p∗£=~su`QkukutyPpl`ypDmoPmQ¦
´ep©su`L_=sovDq^`¡vDq^`£w£t_=tmo`²ε → 0
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l¯^l`=y_=_=`¡q^p^`
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bcp©ku`LyPp=zn`Lq©l`¢µsutw£`Qk`LkmetpDmusoyllq^tm8vDq^`¡k`Lso^rp^yP_¡_=rczn`q©^`¢sotw^£`Lk nyPqlmu`Ls²Jl`£w¡_p^t]so`#kq^t¢pDmu`
z =z∗
R, w =
w∗
R, λ =
λ∗
R, f =
F ∗
µR, p =
p∗
R,
»nµP½
´ep su`L_=sovDq^`¡£yskcvDq^`¡£'£t_=tmo`²ε → 0
~Syq^sz¯Hr¡_tpDmut`pDm£w'^t_=`pJktyp"smurLsutwknmotvDq^`
^qkuy£tl`R¯^lr`P¦H§¨`elrL¢`L£yP~^~J`L_=`pDmPkjH_=~lmoymutwvDq^`eysosu`Qk~SypJ^pDmku`slyPpc¢£twl`epk q^p^`
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z = εZ, w = εW, λ = εΛ, f = εF, p = P»nQ ) ½
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Vi(X) =1
2(ε+ ε′)X2 + const.
»nQD½§3kuy£qlmotyPp"lqÀ~^soy£]L_¡`l`| yPqkukutp^`LkovÈkyPq^_=tk#¥3£w,ypDmosotpDmu`¡rLy_=rmosutwvDq^`=~^sorLrQl`pDmo`
~S`q^mkL« rQsotso`#`p,mo`so_¡`Qkcl`9¢µsutw£`Qk ntp^p^`Ls9kuyqk£=ª®yPsu_=`9kuq^t¢pDmu`'»Ç¢yPts9» 6· A²G ) ½½
4πWi(Z) = C − i F[
κ [2ZQ(Z)− ln(Q(Z))] + 2ZQ(Z) − ln(Q(Z)) + 4(Z − Z)Q(Z)]
»nµP½y
C`Qknm£w³yPpknmpDmu`,tpyPp^pHq^`Èlq ~^soy^£]_=`0l`0| yqJkukutp`LkovÀ^t^t_=`pJktyp^p`£c`m
κ`Lkmq^p^`
ypkmopDmu`_¡rQp^tvDq^`vDq^t6^r~S`p,lqÈyH`¬t`LpDmcl`8xyPtkokuypν^`£w=_p^t]so`8kq^t¢pDmu`
κ = 3 − 4ν`pÈlrª®yso_=mutypk ~^£wp`LkL² »nQP½
κ =3 − ν
1 + ν
`Lp,ypDmustpDmu`Qk~£p^`LkL¦ »nµ=2½
ÍÍ ÒT'UWVXY
W! !
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Q`Qknmclr¯p^t`#~s
Q(Z) = 2Z − i√
1 − 4Z2»nµ¼D½
y √1 − 4Z2
`Lkm \^y£y_=yso~^\^`#^pk£`#~^£pZyq~Jr#~s £Y« tpPmo`so¢££`
[−1
2,1
2]l`#£Y« l`8sor`£A¦^§¨
sop\^`#\^ytwkut`#`Qknmc£w=^sp\`#~JyDktmut¢`9kuq^s£A« tpDmu`Lsu¢££`¦eypp^ypk£`QkcyP_=~JyPsmo`_=`pDmokckujD_=~l¤moymotvDq^`Qk l``mmu`8ª®yPpmotyPp0
√
1 − 4Z2 = −2i Z
(
1 − 1
4Z2
)
1
2 ≈ −2i Z
(
1 − 1
8Z2
)
,~JyPq^s |Z| → ∞ »nµ GP½
§¨`kjH_9Sy£`lnlrLp^ymo`8£¡^sp\`~sutpt~£`#lq,£yPsutmu\_¡`8kuq^t¢pPmo`
lnQ = ln |Q| + i argQ»LD½
§ kuy£qlmutyp »QP½#komutwkªÇtm£w yPpDmustpPmo`rLy_=rmosutwvDq^`'µ¢`Q£wÈsu`L£mutypkuq^t¢µpDmu`'`LpDmuso`£sorLkuq^£mopPmo`#l`Lk`SyPsmk`m£w£wso`Lq^s l`8£=Jpl`#l`ypDmoPm
F = πε+ ε′
2(κ+ 1)
»nQ^µ½
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z = 0`mz = i
¦ °£R`Lkm8~SyPkokut^£`=l`¡^r_=ypDmuso`s8vDq^`=£3kuy£qlmutyp"l``¡~^suyP^£]_=`¡ysosu`Qk~SypJqlr¢P`£y~^~S`_=`LpPm¥#£A« yPsolso` O(1)
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±`mmu`kuy£qlmotyPp~^so`pJ£wª®yso_=`8kuq^t¢pDmu`8`p,¢sotw^£`yq^mu`s
4πwo(z) = −i f[
κ ln (2z
2 + i z) + ln (
2z
2 + i z) +
z
z+
2 + i z
2 − i z+ (κ− 1)(1 + iz) − κ
]
+ C ′
»nQPP½y
C ′`Lkm£wypJknmpDmu`¥lrmu`Lsu_=tp^`s`LpÈ\^ytwktwkokupPm q^pÈ~SytpDm~sumutwq£t`sc^`#£=sorPtyPpyqlmo`s ²
y_=_=`8~s `H`L_=~^£`8£``LpPmosu`#lqÈ`s£`8kut£`Lk`Wysumok kuypDm rLvDq^t£t^sorLk
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l`Qknm-_tpDmu`LpHq=¯^l`e£yskRvDq^`
Rmu`Lp¢`LsokO£Y« tp^¯p^tA¦l´epso`mosuyPq^¢`£yPsokq^p'~^suyP^£]_=`cl`
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R`LkmyPpk`Lsu¢Pr`=£ysk8vPq`
lmu`Lp"¢`Lsok
0¦´ep
so`mosuyPq^¢`8£ysk £`8~^soy£]L_¡`#« q^pÈkuy£tl`#kuyq^_=twk ¥¡q^p^`8ª®ys`#yp`pDmusor`P¦
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lq'lrL¢`£y~~J`L_¡`LpDm nyPqlmu`Ls¥#£A« yPso^su` O(δ)lytm-¿mosu`et^`pDmutwvDq^`cq^r¢`L£yP~^~S`_=`pDm
yq^mu`s ¡¥,£A« yPso^su` O(δ)lq³lrL¢`£y~~J`L_¡`LpDm ntp^p^`Ls¥,£Y« yslso` O(∆)
¦acyqk#`l~^sut_=`soypk#`mumu`so]P£`8kuyqk£wª®yso_¡`#kuq^t¢µpDmu`
wi]o = wo]i»nQ ) ½
acyqk££ypJk¢Pyts y_=_=`pDmc`mmo`8su]L£`~S`so_=`mcl`#lrmu`Lsu_=tp^`s £`Lk ypkmopPmo`LktpyPp^pHq^`Lk ^q~suyP^£]L_=``mcl`ª®yq^sop^ts q^pÈ\_=~,l`lrL~^£P`L_¡`LpDm ¥¡£A« yPso^su` O(1)
kuq^s £`8kuy£twl`8`pDmut`sQ¦¯pl`8yPpkmusoq^tso`
wi]o²P£`Lk¢sot^£`Qk tpp^`s e^pJkc»nµP½OkuypDmso`_=~^£wr`Lk-~s£`Qk-¢sutw^£`Lk
yq^mu`s cs `e¥3»nµ ) ½¦^§6`surQkq^£mom `QknmlrL¢`L£yP~^~Jr`pª®ypJmutyp3l`ε¥9£A« yslsu` O(1)
¦^§¨`^r¢`L£yP~l¤~S`_=`LpPm yq^mu`s ¥¡£A« yPsolso` O(1)
lqÈlrL¢`£y~~J`L_¡`LpDm tp^p^`s ¥=£A« yPso^su` O(1)¢qlmclyPp
4πwi]o = εC + i f(κ+ 1) ln ε− i f
[
κ ln z + ln z +z
z+ 2(κ+ 1) ln 2 +
1
2(κ− 1)
]
, ε→ 0»nQH½
±`mmo`Osu`L£mutyp8~S`qlmmuyq^mqJkukut^t`p#¿muso`OsurL`l~^sot_=rR`Lp8mu`so_=`-l`Qk6¢sot^£`Lktpp^`s R^`O£_=p^t]so`kuq^t¢pDmu`
4πWi]o = C − i F
[
κ lnZ + ln Z +Z
Z+ 2(κ+ 1) ln 2 +
1
2(κ− 1)
]
, |Z| → ∞ »nQP½
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4πWo]i =C ′
ε− i F
[
(κ+ 1) ln ε+ κ lnZ + ln Z +Z
Z
]
, ε→ 0»nQP½
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4πwo]i = C ′ − i f[
κ ln z + ln z +z
z
]
, |z| → ∞ »nQ 2½±`mmo`¡l`Lsupt]Lsu`su`L£mutypÈ`l~^sot_¡`¡£`yP_=~JyPsmo`_=`pDmPkjH_=~lmoymutwvDq^`^q©lrL~^£P`L_¡`LpDm~^soyl\^`^q~SytpDml`ktp^Pq^£sutmur8y 3kuypDm~^~£twvDq^r`Qk£`Qk ª®ys`Lk yp`pDmusor`Qk¦
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εC − C ′ = i f
[
(κ+ 1)(2 ln 2 − ln ε) +1
2(κ− 1)
] »nQ¼D½
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2f ln
(
eε2
64
) »nQGD½
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λ =eλ∗
4R (1 +R/R′)
»n2Hµ½
f = e(κ+ 1)F ∗
32πµR (1 +R/R′)
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λ =η1
η1 + η2f ln
(
R2
R1f
)
− η2η1 + η2
f ln
(
R1
R2f
) »n2 ) ½
λ =e(λ∗1 + λ∗2)
4(R1 +R2)
»n2µH½
f =eF ∗(η1 + η2)
32π(R1 +R2)
»n2P½
ηi =κi + 1
µi
»n2D½
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4πwc = −i fκ[
2z
εQ(z
ε
)
− ln
(
1
εQ(z
ε
)
)
− ln(2 + i z)
]
−i f[
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(
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ε
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−i f[
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(
z − z
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1
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]
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]
2a
" 2∆ =4P (1 − ν2)
πE
[
1
2− ln(
a
2l)
]
l
" δ =4P (1 − ν2)
πE
(
1
3+ ln
2D
a
)
!" " " " #
$ % & '
() * + % " " , "
# - % & . ( / " δ =2P (1 − ν2)
πE
(
2
3+ ln
2D1
a+ ln
2D2
a
)
0 D1 D2
1 "
2 " % 3 ) * 4 5 " ( " 6 " 2∆ =4P (1 − ν2)
πE
[
1
2+ ln 2 + ln
(
Q
a
)]
2a
2c 7 (xr, zr)
$
Q =
√
2(x2r + z2
r)
1 +√
1 + (xr/c)2 + (zr/c)2exp
(
−z2r
2(1 − ν)(x2r + z2
r)√
1 + (xr/c)2 + (zr/c)2
)
ÍÍÒT'UWVXY
D W! !
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u(x, y), v(x, y), w(x, y)ql'~^so`LkokutyPpk~^~^£twvPqr`Lk kuq^s q^p`kq^suªÇ`S
u(x(, y) =1
2πE
∫
S
X(ξ, η)
[
1 − ν
r+ν(x− ξ)2
r3
]
dξdη»nL¼PP½
+1
2πE
∫
S
Y (ξ, η)
[
ν(x − ξ)(yη)
r3
]
dξdη
− 1 − 2ν
4πE
∫
S
Z(ξ, η)x− ν
r2dξdη
v(x(, y) =1
2πE
∫
S
X(ξ, η)
[
ν(x− ξ)(yη)
r3
]
dξdη»nL¼Q½
+1
2πE
∫
S
Y (ξ, η)
[
1 − ν
r+ν(x− ξ)2
r3
]
dξdη
− 1 − 2ν
4πE
∫
S
Z(ξ, η)x− ν
r2dξdη
w(x(, y) =1 − 2ν
4πE
∫
S
X(ξ, η)x− ν
r2dξdη
»nL¼D½
+1 − 2ν
4πE
∫
S
Y (ξ, η)x− ν
r2dξdη
− 1 − ν
2πE
∫
S
Z(ξ, η)Z(ξ, η)
rdξdη
y £`Lk ª®yPpmotyPpkX,Y, Z
kyPpPmlr¯p^t`Lk¥¡~smots^`Lk yPpDmustpPmo`Lkl`#£w¡_=p^t]so`8kuq^t¢pDmu`
X(x, y) = −τxz(x, y)»nQ¼ ) ½
Y (x, y) = −τyz(x, y)»nQ¼D½
Z(x, y) = −σz(x, y)»nQ¼P½
§spJl`q^srsu`L~^surQk`LpDmu`#£w'ltwknmp`9« q^p ~SytpDml`\so`L_¡`LpDmeqÈ~SytpDmey £Y« yp0\^`Lso\`£`
^r~^£w`_=`pDmr =
√
(x− ξ)2 + (y − η)2»nQ¼D½
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!"#$&%'! ( )
§¨`8kujlknmo]_=`8¥sorLkuyqJlsu`vDq^t6yPsuso`Lku~JyPpq'~^soy^£]_=`8l`#| yqkoktp^`Qkuv¡so`¢Ht`LpDm¥¡r¢£q^`Ls£`Qk6tpDmurLs£`Lkkqt¢pDmo`Lk~Syq^s q^p`8ª®ypmutyp,l`\so`L_=`pDm
f(ξ, η)lyp^pr`
I1(f, x, y) =
∫
S
f(ξ, η)
rdξdη
»nL¼>2½
I2(f, x, y) =
∫
S
f(ξ, η)(y − η)2
r3dξdη
»nL¼P¼P½
I3(f, x, y) =
∫
S
f(ξ, η)(x− ξ)2
r3dξdη = I1 − I2
»nQ¼GD½
I4(f, x, y) =
∫
S
f(ξ, η)(x− ξ)(yη)
r3dξdη
»nAGPP½
I5(f, x, y) =
∫
S
f(ξ, η)(x− ξ)
r2dξdη
»nAGQ½
I6(f, x, y) =
∫
S
f(ξ, η)(y − η)
r2dξdη
»nAGD½
§-« `Lpku`_9£`#l`QktpDmurLs£`Lk¥¡`Lkmut_=`s~S`qlm¿muso`8_¡twkkuyqk£wª®yso_=`
I =
∫
S
K(x, y, ξ, η)f(ξ, η) dξdη»nG ) ½
*H 0* 0 Æ *D 6ÂJ Æ * 0 *HÆYÂ'6 6 *H * / à +6N6ÃLÂJÃA0;* §,lr_s\^`^y~lmor`=~s ·^¸ »AG 2lLD½#`Lkml`ÈypJktwlrso`s¡q^p^`Èkuq^suªÇ`,l`,\so`L_=`pDm
Sy£©£suP`q^s
l`Lkm=t`Lp
tplª®rLsut`q^so`¥#ko#£yPp^q`q^sL²£` muyPqlm-rmpDmOtplª®rsot`Lq^s¥8£wlt_=`pkutypsoPmurLsutwkmutwvPq`
Rl`LkOylzn`mok
`Lp3ypDmoPmQ²^kyPtml ≪ L ≪ R
»nAGD½ ~smotsRl``mumu`\HjD~Symo\^]Lku`²t£l`W`Lmuq^` qp=^r¢`L£yP~^~S`_=`pDmRkujD_=~lmoymotvDq^` l` `QkRltSrLsu`LpPmo`Lk
tpDmurLs£`Lk~ss~^~Sysumql's~^~Sysumok `LpPmosu`£yp^Pq^`q^s`m£wso`Lq^sl`#£¡kuq^sªÇP`P¦DxyPq^s surQ£tku`s`^r¢`L£yP~^~S`_=`pDmL²l£`Lk \Jp^P`_=`pDmok ^`¢sutw^£`Lk kuq^t¢pDmok kuypDm tpDmosuyllq^tmok
ξ = na(m) + b(m), |n| ≥ 1»AGPP½
m =η
L,
»nGD½
y =y
L
»nG 2½`#vPqt¨lypp^`8~JyPq^s £=_=`Qkq^so`#l`8£Y« tpPmorPso£`
dξdη = La(m) dndm»nG¼D½
pJk `#\p`_=`LpPml`8¢sotw^£`LkL²a(m)
so`~^sorLku`pDmo`e£w=l`_=t¤A£wso`Lq^s ^`#£=kuq^sªÇP`8l`yPpDmomL²ξ = b(m)
so`~^sorLku`pDmo`£Y« rLvDqmutyp³l`'£wÈ£tPp^`_=yj`Lp^p^`l`3yPpPmm`mL£0^`_=t-£yp^Pq^`qslq
ypDmoPmQ¦J´ep3kuq^~^~SyPku`#vPq`a(m)
`mb(m)
kuypDmyPpDmutp _¡`LpDmcltWrso`pDmot^£`Qk ~JyPq^s |m| > 1¦
ÍÍ ÒT'UWVXY
W! !
§-« tpDmurPso£`=»nG ) ½-~S`q^mclyp8¿muso`8surLrLsotmu`#kyPqk £¡ª®yPsu_=`#kuq^t¢pPmo`
I =
∫ 1
−1
∫ 1
−1
LK(x, Ly, ξ(n,m), Lm)f(ξ(n,m), Lm)a(m) dmdn»AG=GP½
´ep,kuq^~^~SyPku`8^`#~^£qJkvDq^`8£ª®ypJmutypf(ξ(n,m), Lm)a(m)
~S`q^m¿muso`8_=tku`kyPqk£w9ª®yPsu_=`
f(ξ(n,m), Lm)a(m) = s(n)g(n,m)»YP½
µ¢P`Le£`Qk yPpltmutypk kuq^t¢µpDmu`Lk kuq^sg`ms
g(n,m) ∈ C2, |n| ≥ 1, |m| < 1»YQ½
|g(n,m)| < G(m), G(m) ∈ L1, |m| < 1»YP½
∫ 1
−1
|s(n)| dn < +∞ »A ) ½∫ 1
−1
|s(n) ln |x− n|| dn < +∞, ∀x ∈ IR»AH½
bcp3`l`_=~^£`l`ª®ypmutypÈl`#£w¡kuysumu`~S`qlmc¿muso`#lyp^pr`LpÈypJktwlrspDm
f(ξ, η) =
(
1 −(
ξ
l
)2
−( η
L
)2)
12 »YPP½
´ep,=£ysk
a(m) = l√
1 −m2»YD½
a(m)f(n,m) = l(1− n2)−12
»Y 2½s(n) = (1 − n2)−
12
»Y¼D½g(n,m) = l
»Y=GP½§-« tpDmurLs£`¡»AGGD½ l`¢Ht`pDm£ysk5
I =
∫ 1
−1
s(n) dn
∫ 1
−1
LK(x, Ly, ξ(n,m), Lm)g(n,m) dndm»Y^LP½
±`mmo`RtpDmurLs£`¢µ¿muso`OlrL¢`£y~~JrL`PkjH_=~lmoymutwvDq^`_=`LpPm6~SyqsL→ +∞ `p8_tpDmu`pJpDm¨yPpknmpDmu`Qk
£`Lkqlmosu`QkO¢sotw^£`LkL¦D±`e^r¢`L£yP~^~S`_=`pDmkL« µ¢P]so`lypJ tpDmorso`LkokupPm~JyPq^sl`Qk-kuq^suªÇ`Qk-l`yPpPmmrL£prL`LkL¦l§6`Lkclr¯ptmotyPpk kqt¢pDmo`Lk kuypDm tpDmusoyllq^tmu`Qk5
ξ = ξ(n,m), ξ = (n, y)»YlPQ½
ξ = ξ + (m− y)ξ′ + O((m− y)2)»Y^QP½
9.oËÏÕHØ)5 3# Þ 5 µÒµÓRÛ1&nÖ Û®Þ'# Þ$#×ÜÒµÓRÖµÛÇÓnØAØ'#×ÜÒeÓnÕ×Õ%#×ÖµØAÜ6 Õ×Ó8µÓÓnÛ®Þ
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( D
y £`kujD_JyP£`′^rLkutPp^`8£=lrLsut¢rL`e~smot`L££`8~s s~~JyPsm¥
m¦
bcp3~S`motme~s_=]mosu`8l`lrL¢`L£yP~^~J`L_=`pDmε²l`QknmtpDmusoyllq^tm ^`#£¡_p^t]so`#kqt¢pDmo`
ε =ξ′(ξ − x)
L2
»Y^ ) ½
vDq^t6yp^q^tmcvDq^tlr¢P`£y~^~S`_=`pDmkuq^t¢pPm¥¡£A« yPsolso`1
(ξ − x)2 + (η − y)2 = (ξ − x)2 + 2(ξ − ξ)(ξ − x) + (ξ − ξ)2 + L2(m− y)2»Y^H½
= (ξ − x)2 + L2(m− y + ε)2»Y^QP½
*HÆÇÂ6 6 *H * / à 0*H 0 / ÃQ ^Æ *H$0 K 0 / * / Å* ´ep,p^`9lyPp^p^`s=twtvDq^`9£`9^rmot£lq0lrL¢`¤£y~~J`L_¡`LpDmcl`8£Y« tpDmors£`
I1
I1(f, x, y) =
∫
S
f(ξ, η)√
(x− ξ)2 + (y − η)2dξdη
»YlQP½
= J1(sg, x, y)»Yl2½
=
∫ 1
−1
s(n) dn
∫ 1
−1
Lg(n,m) dm√
(x − ξ)2 + L2(y −m)2
»Y^L¼D½
±yPpkutlrLsuyPpk ~JyPq^s`L£¡£A« tpDmurLs£`kuq^t¢µpDmu`
A1 =
∫ 1
−1
s(n) dn
∫ 1
−1
Lg(n, y) dm√
(x− ξ)2 + L2(y −m− ε)2
»Y^AGD½
=
∫ 1
−1
s(n)g(n, y)
[
arcshL(y −m− ε)
|x− ξ|
]1
m=−1
dn»YPP½
§¨`#lr¢P`£y~^~S`_=`pDmkujH_=~lmuymutwvPq`8^`#£ª®ypJmutyparcsh
¢qlm~JyPq^sl`#Psopl`Lk ¢£`qsokl`z
arcsh z =ktp
(z)[
ln(2z) + O(1/z2)] »YPlµ½
´ep,=lypJ8~JyPq^s |y| < 1`mε = O(1/L2)
£`^r¢`L£yP~^~S`_=`pDmkqt¢pDm ~Syq^sA1
A1 =
∫ 1
−1
s(n)g(n, y)[
ln(
4L2(1 − y2))
− 2 ln |x− ξ|]
dn»YP½
er¯ptkokyPpk_tpDmu`pJpDmA2
l`8£w=_p^t]so`kuq^t¢pDm
A2 =
∫ 1
−1
s(n) dn
∫ 1
−1
g(n,m) − g(n, y)
|m− y| dm»YP ) ½
ÍÍ ÒT'UWVXY
P W! !
`m£Y« yp,yPpkutl]Lsu`£`#su`Qknmo`8^`#£Y« tpDmors£`8kq^t¢pDmu`
J1−A1−A2 =
∫ 1
−1
s(n) dn
∫ 1
−1
L(g(n,m)−g(n, y))
1√
(x− ξ)2 + L2(y −m− ε)2− 1
L|m− y|
dm
»YPH½hlpk6`LpDmuso`s¨^pk6~£qklrmot£wk¨£q^£wmuyPtso`LkL² ·¸»AG>2HLD½J_¡yPpDmuso`-vDq^`O£`-lr¢P`£y~^~S`_=`LpPm
^`8`#su`Qknmo`#lyp^p`J1 −A1 −A2 = O
(
lnL
L2
)
, L→ ∞ »YP½kuytm
I1 = Ff (y) ln(4(L2 − y2) − 2
∫
C(y)
f(ξ, y) ln |x− ξ| dξ »AP½
+
∫ L
L
Ff (η) − Ff (y)
|η − y| dη + O(
lnL
L2
) »YP=2½
µ¢P`Le£`Qkp^ymmotyPpk kqt¢pDmo`Lk5
Ff (y) =
∫
C(y)
f(ξ, y) dξ»Y¼P½
C(y) = x, b(y) − a(y) ≥ x ≥ b(y) + a(y) »YP GP½§¨`¡_=¿L_¡`=mnjH~S`l`lr¢P`£y~^~S`_=`pDm3rmor¡sorL£twkur~Syq^s#£`Lk#qlmuso`LktpDmurPso£`Qk8^pk ·^¸
»AG>2HS½¦´ep,p^`8su`L~JyPsmo`twtvPq`£`Lk~sutpt~ql'sorLkuq^£mmk5
I2(f, x, y) = −2Ff (y) + I1(f, x, y) + O(
lnL
L2
) »Y ) D½
I3(f, x, y) = I1(f, x, y) − I2(f, x, y) = 2Ff (y) + O(
lnL
L2
) »Y ) µ½
I4(f, x, y) = O(
lnL
L2
) »Y ) P½
I5(f, x, y) = π
∫
C(y)
f(ξ, η)ktp
(x− ξ) dξ + O(
1
L
) »Y )) ½
I6(f, x, y) = −∫ L
−L
Ff (η)dη
y − η+ O
(
1
L2
) »Y ) H½
y £`kujD_JyP£`−
∫ so`~surQk`LpPmo`e£w¡¢£`q^s ~^sotpt~£`#l`#£A« tpDmurPso£`8l`± q\Hj¦
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( >2
;lÃA0Y / 0 Å / Ã!ÅÃÆ0 / >0;* §¨`Lk rLvDqmutypk8»nQ¼D½-kuq^s £`8^r~^£w`_=`pDm`pª®ypmutyp,l`LktpDmurLs£`Lk« tpJq^`p`#~J`Lq^¢`LpDm £yPsok ¿muso`sorrQsotmo`Lk kuyqk£wª®yso_=`8kuq^t¢pDmu`
2πEu(x, y) = FX(y)(
(1 − ν) ln(4(L2 − y2)) + 2ν) »Y ) P½
− 2(1− ν)
∫
C(y)
X(ξ, y) ln |ξ − x| dξ
+ (1 − ν)
∫ L
−L
FX(η) − FX(y)
|η − y| dη
+1
2(1 − 2ν)π
∫
C(y)
Z(ξ, y)kutPp
(ξ − x) dξ
+ O(
|X | lnLL2
)
+ O(
ν|Y | lnLL
)
+ O(
(1 − 2ν)|Z|L
)
2πEv(x, y) = FY (y)(
(1 − ν) ln(4(L2 − y2)) − 2ν) »Y ) D½
− 2(1 − ν)
∫
C(y)
Y (ξ, y) ln |ξ − x| dξ
+
∫ L
−L
FY (η) − FY (y)
|η − y| dη
+1
2(1 − 2ν)π−
∫ L
−L
FZ(η)dη
η − y
+ O(
|Y | lnLL2
)
+ O(
ν|X | lnLL
)
+ O(
(1 − 2ν)|Z|L
)
2πEw(x, y) = FZ(y)(
(1 − ν) ln(4(L2 − y2))) »Y ) 2½
− 2(1 − ν)
∫
C(y)
Z(ξ, y) ln |ξ − x| dξ
+ (1 − ν)
∫ L
−L
FZ(η) − FZ(y)
|η − y| dη
− 1
2(1 − 2ν)π
∫
C(y)
X(ξ, y)kutPp
(ξ − x) dξ
− 1
2(1 − 2ν)−
∫ L
−L
FY (η)dη
η − y
+ O(
(1 − 2ν)|X |L
)
+ O(
(1 − 2ν)|Y |L
)
ÍÍ ÒT'UWVXY
P¼ W! !
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f(x) =
∫ b
a
ϕ(ξ) ln |ξ − x|, dξ »A ) ¼P½
hlt£A« yPpltWrso`pt`3`mumu`rLvDqmotyPp6²£Y« tpDmors£`sorLkuq^£mopDmo`kL« `H~sut_=``p ª®ypmutyp l`3£w0¢£`q^s~sutpt~£`#l`9± qJ\Djl`8£=_pt]Lsu`#kuq^t¢pPmo`
f ′(x) = −−∫ b
a
ϕ(ξ)
ξ − xdξ
»Y ) GP½
±`mmo`#rLvDqmotyPp~S`q^m¿muso`8tpH¢P`skrL`e`m_¡twku`kyPqk £¡ª®yPsu_=`8rLvDq^t¢£`pDmu`#kuq^t¢pPmo`
ϕ(x) =1
π2
∫ b
a
√
(ξ − a)(b− ξ)
(x− a)(b− x)
f ′(ξ)
ξ − xdξ +
C√
(x − a)(b− x)
»YPD½
y C`Lkmq^p^`#ypkmopPmo`#vPqt6~J`Lqlm¿mosu`#lrmu`Lsu_=tp^r`#`Lp'mo`ppPmcy_=~lmo`8^`#£Y« rLvDqmutypÀ»Y ) ¼P½
C =
(
π2 ln
(
b− a
4
))
−1 ∫ b
a
f(x)
(x − a)(b− x)dx =
1
π
∫ b
a
ϕ(x) dx»YQ½
±`sorLkuq^£mmR`LkmO¥so~^~^soyH\`s^q¡sorLkuq^£momp^ymur`pp^p^`l`|³yϕznyPq^`£` s£` 6« q^p` ~su`QkukutyPp
`mf« q^p'lrL~^£w`L_=`pDmL¦HepkO£`eLklq'yPpDmomO£tp^rLtvDq^`P²P£`Lk`l~^so`LkoktypJkOkqt¢pDmo`Lk-kuypDm-q^mut£`Qk
ϕ(x) =√
a2 − x2»YDP½
Fϕ =
∫ a
−a
ϕ(x) dx =1
2πa2 »Y ) ½
f(x) =
∫ a
−a
√
a2 − ξ2 ln |ξ − x| dξ »YPD½
=1
2π ln
(
a2
4
) »YDP½
pk£`Lk-^qyPpDmom« q^pjH£tplso`sotPtl`kqsq^p'~^£pr£wkmutwvDq^`²D£`elrL~^£P`L_¡`LpDmp^yPsu_£¥¡£w=kq^suªÇ`#k« `l~^sot_=`8~s £=yPpltmutyp,l`#p^yPp~Srp^rmusmotyPp0
w(x, y) = −Ax2 + δ, AyPpknmpDm »YPP½
±yP_=_¡`cyp£A« #¢Hq^pkO£A« p^p^`l`c|²£`clr~£P`_=`LpPmQ²PvDqPlsomutwvDq^` `px²Pyplqtm¥#q^p^`~su`QkukutyPp
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Z(x, y) =2AE
(1 − ν)
√
a2(y) − x2»Y>2½
FZ =πAEa2
1 − ν
»YP¼D½
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w(x, y) = −Ax2 +1
2Aa2
[
ln
(
16(L2 − y2)
a2
)
+ 1
]
+1
2
∫ L
−L
Aa2(η) −Aa2(y)
|η − y| dη»Y=GP½
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δ = w(0, y) =1
2Aa2
[
ln
(
16(L2 − y2)
a2
)
+ 1
]
+1
2
∫ L
−L
Aa2(η) −Aa2(y)
|η − y| dη»YPP½
¯p0l`lyPp^p^`scq^p^`#`Lkmut_motyPp,^`#£Y« tpDmors£`#^pJk`mmu`#rQvDqmotyPp6² ·¸ »AG 2lLD½O~^soy¤~SyPku`9« `W`Lmuq^`Lseqp p^yPq^¢`QqÈlr¢P`£y~^~S`_=`LpPm8l`9£w'^`_=t¤Z£wso`qscl`ypDmoPmeqlmuyPq^se^`
y = 0kuyqJk£ª®yso_=`#kq^t¢pDmu`
a2(y) = B2a0(y)
(
1 +1
Λa1(y) + . . .
)
, a0(0) = 1»Y^Q½
µ¢P`LΛ£`#~Js_=]mosu`8kqt¢pDm
Λ = ln
(
16L2
B2
)
+ 1»YP½
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δ =1
2AB2a0(y)
(
1 +1
Λa1(y) + . . .
) »YP ) ½
×[
Λ + ln
(
1 − y2
L2
)
− ln
(
1 +1
Λa1(y) + . . .
)]
+1
2AB2
∫ L
−L
[
a0(η)
(
1 +1
Λa1(η) + . . .
)
− a0(η)
(
1 +1
Λa1(η) + . . .
)]
dη
|η − y|
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ln
(
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δ =1
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»YD½
a1(y) = − ln
(
1 − y2
L2
) »YPP½
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a(y) = B
√
1 − 1
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(
1 − y2
L2
) »YPD½
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−L
FZ dy =πAEB2
1 − ν
∫ L
−L
1 − 1
Λln
(
1 − y2
L2
)
dy»YP=2½
=πAEB2L
1 − ν
[
2 +0.614
Λ
] »YP¼D½
`m£¡~Srp^rmusmotyPp`QknmclyPprD£`¥δ =
1
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∫ L
−L
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(
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L2
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dy = L
∫ +1
−1
ln(
1 − x2)
dx»YP½
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~q^tkoku`kL« rQsotso`kuyqk£wª®yso_=`
∫
ln(
1 − x2)
dx = x ln(1 − x2) − 2x+ 2 argth(x), −1 < x < 1»Y^Q½
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limx→−1
x ln(1−x2)−2x+2 argth(x) = limx→−1
(1+x) ln(1+x)+(x−1) ln(1−x)−2x = −2 ln(2)+2»YPP½e`#_=¿_=`#`p
1²lyPp,ylmot`LpPm
limx→1
x ln(1−x2)−2x+2 argth(x) = limx→1
(1+x) ln(1+x)+(x−1) ln(1−x)−2x = 2 ln(2)−2»Y ) ½h^\pPm vDq^`
2 ln(2) − 2 = −0.6137056»YD½
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a
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4 + . . .+Anr2n + . . .
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R1
R2
∆R
φ
ψ
Q
(r, φ)(R2, ψ)
µ ' ! W )$ ' ! P ! !
LsmursotwknmotvDq^`Qk-_morsot`L££`Lk kuypDmlrQsotmo`Lk~s
Ei =Ei
1 − ν2i
, νi =νi
1 − νi,
lrª®yPsu_motyPpk ~^£wp^`Qk »Y 2 ) ½
Ei = Ei, νi = νi,ypDmustpDmu`Qk~£p^`Lk »Y 2µH½
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η =E1
E2
»Y 2P½
λ = (1 − ν1) − η(1 − ν2)»Y 2D½
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ypkut^rsor y_=_=` ~S`motm l`¢pDm£`LksµjyPpk
R1`mR2²kyPqk£w9ª®yPsu_=`#kuq^t¢pPmo`
u1 − u2 = δ cosφ− ∆R(1 − cosφ)»Y 22½
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u1 − u2 = −∫
∞
a
∂u2
∂rdr −
∫ a
0
∂u1
∂rdr = −
∫
∞
a
ε2r dr −∫ a
0
ε1r dr»Y 2¼P½
y a ≈ R1 ≈ R2
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^yp^p^rL`Lk~sc·^¸ »nGH½+R2
Q(−σrr + σφφ) =
1
π
∫ ε
−ε
q(ψ) dψ + 2
∫ ε
−ε
q(ψ)L1(ρ, θ) dψ +3 − ν22πρ
cosφ»Y 2GP½
R2
Q(σrr) =
ρ− 1
2πρ
∫ ε
−ε
q(ψ) dψ +
∫ ε
−ε
q(ψ)L2(ρ, θ) dψ +3 − ν2
4π
ρ2 − 1
ρ3 cosφ»Y¼P½
R2
Q(σrφ) = +
∫ ε
−ε
q(ψ)L3(ρ, θ) dψ +3 − ν2
4π
ρ2 − 1
ρ3 sinφ»Y¼^µ½
y ρ = r/R2
²θ = ψ − φ
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q(φ)so`~^sorLku`pDmo`£wÀ~^so`Lkoktyp
Plt_¡`LpkutyPp^p^r`q(φ) = R2
p(y)
Q
»Y¼PP½µ¢P`L
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Li(ρ, θ)kyPpPm
^r¯p^t`Lk~Js
L1(ρ, θ) =1
2π
(1 − ρ2)
(ρ2 − 2ρ cos θ + 1)
»Y¼ ) ½
L2(ρ, θ) =1
2π
(1 − ρ2)2
ρ
(cos θ − ρ)
(ρ2 − 2ρ cos θ + 1)2»Y¼H½
L3(ρ, θ) =1
2π
(1 − ρ2)2
ρ
(sin θ)
(ρ2 − 2ρ cos θ + 1)2»Y¼PP½
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R2
Q(σφφ(φ) − σrr(φ)) =
3 − ν22π
cosψ +1
π
∫ ε
−ε
q(ψ) dψ»Y¼PP½
σrr(φ) =R2
Qq(y) = p(y)
»A¼>2½
σrφ(φ) = 0»Y¼¼P½
§¨`Lklrª®yPsu_motyPpkε1r
`mε2r
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ε1r =1
E1
Q
a
[
−(1 + ν1)
∫ ε
−ε
q(ψ)L2(ρ, θ) dψ + 2ν1
∫ ε
−ε
q(ψ)L1(ρ, θ) dψ
]
+Q
aF1(r) cosφ
»Y¼=GP½
ε2r =1
E2
Q
a
[
(1 + ν2)
∫ ε
−ε
q(ψ)(ρ2 − 1
2πρ2 + L2(ρ, θ)) dψ − 2ν1
∫ ε
−ε
q(ψ)(1
2π+ L1(ρ, θ)) dψ
]
+Q
aF2(r) cosφ
»Y GPP½µ¢P`L
F1(r) = −1 + ν14πE1
[
1 − ν1 − (1 − 3ν1)ρ2] 1
ρ
»YG^µ½
F2(r) =3 − ν24πE2
[
1 − ν2 − (1 + ν2)ρ2] 1
ρ
»YGPP½
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(1 + η)
∫ ε
−ε
q′(ψ) cotθ
2dψ + [2(1 + η) − λ]
∫ ε
−ε
q(ψ) cos θ dψ»A G ) ½
= −λπq(φ) − η
∫ ε
−ε
q(ψ) dψ − πE1∆R
Q
»YGH½
¥¡£wvDq^`L££`8yp,znyq^mu`8£Y« rLvDqmutyp,« rLvDq^t£t^so`∫ +a
−a
q(ψ) cos(θ) dψ = cosφ»YGPP½
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y = tan(φ/2), t = tan(ψ/2), b = tan(ε/2),»A GD½
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−b
q′(t)dt
t− y+
λπ
1 + η
q(y)
1 + y2 = − 4k
π(1 + y2)2+
B
1 + y2
»YG 2½
µ¢P`LB =
2k
π− 2η
1 + η
∫ b
−b
q(t)dt
1 + t2− π
1 + η
Ei∆R
Q
»YG¼D½
k =π
2
2(1 + η) − λ
1 + η
»YGGD½
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−b
q(t)1− t2
(1 + t2)2dt =
1
2
» ) D½
pk£`8kl`_morsotqltl`LpDmutwvPq`Lk#»η = 1, λ = 0
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−b
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t− y= − 4
π(1 + y2)2+
B
1 + y2 , B = 2 −∫ b
−b
q(t)dt
1 + t2− π
1 + η
Ei∆R
Q
» ) ^µ½
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∆Rp(y)
Q=
2
π√
b2 + 1
√
b2 − y2
1 + y2 +1
2πb2(1 + b2)ln
(√
b2 + 1 +√
b2 − y2
√
b2 + 1 −√
b2 − y2
)
» ) PP½¦
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Qq3^`_=t¤°p£`8^`8ypDmoPmb
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2
π
1 − b2
b2− Ib
π2b2(1 + b2)
» ) ) ½
µ¢P`LIb£A« tpDmurLs£`8kuq^t¢pDmu`
Ib =
∫ b
−b
ln
(√
b2 + 1 +√
b2 − y2
√
b2 + 1 −√
b2 − y2
)
dt
1 + t2» ) H½
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r£wkmutwvDq^` ²σ
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εvDq^`=kq^tm8£
_mursotwq'yPp,ylmot`LpPm £wl`Lkosot~lmutyp3kq^t¢pDmu`
σ = σ + σ
» ) LD½ε = ε = ε
» ) µ½§~q^tkokop`
σ ¯⊗ ε`Lkm#£yPsoke£w'~^qtkokup`¡¢yP£q_¡twvDq^`¡surL¢`skut^£`9`m
σ ¯⊗ ε
£,ltwkukut~motyPptpDmusotpJk]QvPq`¡¢Py£q^_=tvDq^`P¦¨§¨3mo\^ryPsut`=£tp^rQtsu`twkymusoy~S`l`¢HtkoyDktmurkL« yPlmut`pDm`LpÀypkutwlrspDm#£`_=¿L_¡`~Symo`pDmut`£Omu\^`Lsu_=ylljHp_=twvDq^`²
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kut~motyPp k`L_9^£w£`ϕ²R»YvPqJlsmotvDq^`Qkc`m8lr¯p^t¤A~SyPkutmutªÇko½e¯p"l`^r¯p^ts8q^p^`¡¢HtkoyDktmur£tp^rLtso`²
kuytm
Ψ =1
2ρλ [Tr(ε)]
2+ 2µTr(ε2)
» ) QP½
ϕ =1
2λθλ [Tr(ε)]
2+ 2µθµTr(ε
2)» ) ) ½» ) H½
y λ`mµkuypDme£`LkeyH`¬t`LpPm8« r£wkmutwtmur9l`¡§¨_¡rP²
θλ`mθµ£`LkyH`¬t`LpPml`Lke¢HtkoyDktmur`m
ρ£w¡_kok`8¢yP£q_¡twvDq^`¦e`'£w,_=¿_=`'_pt]Lsu`vDq6« `p³rL£Pknmottmor'knmps"`LkyH`¬t`pDmok9~J`Lq^¢`LpDm9¿mosu`so`£trLk¥Èqp
_=yllq^£` « Oyq^p^¢HtkovDq^`q^rLvDq^t¢µ£`LpDmE′`mO¥cqp¡yD`¬t`pDm^` ~SytwkokyPp9¢HtkovDq^`q^rLvDq^t¢µ£`LpDmL²
ν′yPqt`Lp¡£yPsokq^p=_=yllq^£`^`tkot££`L_¡`LpDm¢HtwkuvDq^`Lql9rQvPqt¢£`pDmG′`mqp¡_=yllq^£` l`yP_=~^su`Qkukutyp
\HjllsoyPkmomutwvPq`erQvPqt¢£`pDmK
λθλ = λ′ =ν′E′
(1 + ν′)(1 − 2ν′), µθµ = µ′ =
E′
2(1 + ν′)
» ) QP½
E′ = µ′3λ′ + 2µ′
λ′ + µ′, ν′ =
λ′
2(λ′ + µ′)
» ) LD½
G′ =E′
2(1 + ν′)= µ′, K ′ =
E′
3(1 − 2ν′)=
3λ′ + 2µ′
3
» ) 2½
§¨`Lk6£ytwk¨^`yP_=~JyPsmo`_=`pDmok¨k« rm^£tkoku`pDm¥ ~smotsl`-`Qk¨l`Lql#~Jymu`LpPmot`L£kl`-_p^t]Lsu`RqJkq^`L££`
σ = ρ∂
∂ε(Ψ) = λTr(ε)1l + 2µε
» ) L¼D½
σ =
∂
∂ε(ϕ) = λ′Tr(ε)1l + 2µ′ε
» ) AGD½¾Op"kyP_=_=pDm£`Qkel`LqlÈypDmosut^qlmotyPpÈ`p ypDmustpDmu`LkL²lyp©Syqlmotme¥'££yPtl`¡y_=~Sysumu`L_¡`LpDm
£tp^rLtso`8~JyPq^s £`#kuy£t^`8^`8`£¢Dtpl¤ SRytm
σ = λ [Tr(ε) + θλTr(ε)] 1l + 2µ [ε+ θµε]» ) D½
àÒeÛYÓ°ÞYÛYÜÓ*#%(#DÕ ÒµÜÞ'#×ÜÒ µÓ8 ÜÒÞ 3ÓRÓnÒÖ Û Õ×Õ%nÕ×Ó9µÓnØ (nÜ ÖÜØ ÒÞYØÛÇÐ&nÜÕ×Ü3#%-.QÓØ
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¼ W! !
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ε`pÈq^p^`#~Jsumut`sor¢P`skt^£`
ε `m
qp^`#~sumut`8tsosor¢`Lsokut^£`εPkukuyltr`8¥¡q^p^`yPpDmustpPmo`
σ²JkyPtm
ε = ε + ε
» ) lµ½σ = σ = σ
» ) P½§£ytR« rmm9lrLsut¢`=lq"~Jymu`LpPmot`L£mu\^`Lsu_=ylljHp_¡twvDq^`
Ψ(ε − ε )`m#££yPtOyP_¡~£rL_¡`LpDmotso`
^q3~Symo`pDmut`£l`#ltkokut~mutyp,lq£ϕ?(σ)
²Jkuytm
σ = ρ∂
∂ε (Ψ(ε− ε
))
» ) ) ½
ε =
∂
∂σ(ϕ?(σ))
» ) D½¯p"^`ª®yPsu_q^£`s`l~^£ttmo`_=`pDm#£`¡_=yll]£`l`¡y_=~Sysumu`_=`LpPmQ²Wyp©ypkutwl]so`9Prp^rLso£`L_¡`LpDm
£`8~Jymu`LpPmot`L£mu\`so_¡ylljHp_=tvDq^`#lq£Y²Ψ? kq^t¢pDm
Ψ?(σ) =1
2ρ
(
1 + ν
ETr(σ2) − ν
E[Tr(σ)]
2
) » ) P½
`#vPqt¨yPplq^tmc¥¡£¡£yt6« rmmckuq^t¢pPmo`
ε = ρ∂
∂σ(Ψ?(σ)) =
1 + ν
Eσ2 − ν
E[Tr(σ)1l
» ) P½e`_¡¿L_=`²£wc£ytly_=~^£r_=`pDmotso` `Lkm`l~^sut_=r` ~sq^p¡~Symu`LpDmut`£^l` ltwkukut~motyPpvDq^somutwvPq`
`mclr¯Jp^t6~SyPkutmotª1ϕ? =
1
2
(
1 + ν
Eτ1Tr(σ2) − ν
Eτ2[Tr(σ)]2
) » ) =2½y
τ1`mτ2kuypDml`q^p^yPq^¢`L££`LkyPpkmopDmo`Lk l`8¢HtkoyDktmurP¦
¾Op3kyP_=_=pDm-£`Qk-ypDmusotqlmutypk-`plrª®yso_mutypyPpy^mut`pDm-£w£ytJ^`eyP_=~JyPsmo`_=`pDm-£tprLtso`^q,kuy£twl`#l`#d0 `££
ε =1 + ν
E
[
σ +σ
τ1
]
− ν
E
[
Tr(σ) +Tr(σ)
τ2
] » ) ¼D½
§¨`8y_=~Sysumu`_=`LpPm lqmnjH~J`#d0 `££Wp6« `LkmPrp^rLso£`L_=`pDm~Pk qlmut£twkur8ku`q£~JyPq^s_¡yllrL£twku`s£`y_=~Sysumu`_=`LpPm8l`Lk8kuy£t^`LkL¦6±e« `Lkm8~^£q^m m#q^p`£ytRl`=yP_¡~Sysumu`L_=`pDm8^`9mnjH~S` qtl`=`Lp©`=ku`pkvDq6« t£p¨« `Htwkmu`=~k8« rmm« rLvDq^t£t^so`¡¥3yPpDmustpPmo`ypkmopPmo`pyp pHq^££`¦%°£`Lkm#£`¡~^£qk#kyPq^¢`LpDmq^mut£twkr#`Lp~Js££]L£`#µ¢`Qcqp3so`Lkokuysum ¯Jp,^`eª®yPsu_=`Ls £`Qk_¡yll]L£`Qk ~^£qkrLp^rsq^W¦Y XàÒeÛYÓ°ÞYÛYÜÓ*(nÓ°ÞYÞYÓ Ü#×Ø(#HÕ×Ó ÜÒÞ 3ÓRÓnÒeØ'&nÛ$#×Ó
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Â/"Æ lÃA0Y / Z / ÅÃA0Ç / / *HÆÇÆ * 0* Æ 0ÇQÅÂS0®ÃL bp`mosuyPtkut]_=`,¢yt`,k« µ¢]Lsu`,kyPq^¢`LpDm¡~^£qk\J^t£`=~Syq^s8ª®yPsu_q^£`s8£`QkyP_=~JyPsmo`_=`pDmok¢HtwkuvDq^`Lql¦%°£-k« Ptm#l`£wª®yso_9q^£wmotyPp0ª®ypJmutypp^`££`kuyqJk ª®yso_¡`#l`8~^soyllq^tmcl`yPpH¢y£qlmotyPpµ¢`Qe^`Lk ª®ypJmutypJk l` q`8`mcl`8so`£w^motyPp6¦
±yPpktwlrLsuyPpk ¥¡mutmuso`« `l`_=~^£`£w=surL~JyPpk`8q^ptlt_=`pkutyp^p^`L££`9« q^p0yso~k¢HtkovDq^`ql,kyPq^_=tk¥¡q^p,rQ\^`£yp,l`#yPpPmosotpDmo`Lklr¯Jp^t6~s
σ = σ0H(t− τ)» ) GP½
y H`Lkm£w9ª®yPpmotyPpÈl`#É`Lµ¢Htwktwl`8`m
τ`Lkm£A« tpkmopDm« ~^~£twmutyp,l`#£Y« rL\`£yp6¦
§sor~SypJk`8q^p^twlt_=`pkutyPp^p^`L££`#`p,lrª®yPsu_mutyp3~J`Lqlm¿mosu`#rQsotmo`8kuyqJk£ª®yso_=`#kq^t¢pDmu`
ε = J(t− τ)σ0» )=) D½
xRslr¯ptmotyPp6²R£"ª®yPpmotyPpJ`Lkm£©ª®ypJmutyp l` qJ`ÈLsmorsotkmutwvDq^`,l`,£"¢HtkoyDktmur,q^pt¤
^t_=`pJktyp^p`££`'^qyso~kL¦pk9£`k9y³£A« yPpyPpktwl]Lsu`q^p`'krLsut`'« rL\`£ypkkuyqJk£w3ª®yso_¡`kuq^t¢pDmu`
σ =n∑
j=1
σjH(t− τj),» )=) µ½
£w¡surL~JyPpku`8^q3_mursotwq'~S`q^mmuyPqlmqkoktt`Lp3¿muso`8_¡twkkuyqk£wª®yso_=`8kuq^t¢pDmu`
ε =
n∑
j=1
J(t− τj)σj ,» )=) P½
`m,6« q^p`©_pt]Lsu`©rprs£`0~JyPq^s,q^p^`ÀkyP£q^mutyp ypDmutpHq^`σ(τ)
²yp tpDmosuyllq^tm,£ª®yso_9q^£wmotyPpª®yPpmotyPp^p^`L££`#lqÈy_=~Sysumu`L_¡`LpDm kuyqJk£ª®yso_=`#kq^t¢pDmu`
ε(t) =
∫ t
0
J(t− τ)d σ(τ)
d τdτ +
n∑
j=1
J(t− τj)σj» )=)) ½
¾Op'tpDmusoyllq^twkupDm-q^p^`c^rsot¢Pr` `m q^p~^soyllq^tm l`cypH¢yP£q^mutypqku`pk-l`Lk ltwknmosut^qlmotyPpk²D`mmu`ª®yPsu_q^£wmutyp~J`Lqlmeku`8_¡`mmosu`kuyqk £w¡ª®yso_=`8£PkukutwvPq`
ε(t) = J ⊗ Dσ
Dτ
» )=) H½¾Op,yPpkutlrLsopDmL²HlrLkuyso_twk²Pq^p^`#kyP££twtmomotyPp'kuyqk-ª®yso_=`« rL\`£ypk l`#lrª®yPsu_mutypk yP_¤
tp^rL`#µ¢`Lq^p`8kuy££ttmmutyp3`pÈlrª®yso_mutypsurLq^£t]so`²lyp3~S`qlmrQsotso`#l`8_=p^t]so`8^q£`
σ(t) =
∫ t
0
R(t− τ)d ε(τ)
d τdτ +
n∑
j=1
R(t− τj)εj» )=) P½
y £weª®ypmutypR`QknmO£weª®ypmutyp=q^p^twlt_¡`LpkutyPp^p^`££`^` su`L£lmutyp¡lq=_=murLsutwq¢DtwkovPq`ql¦D±`mmu`
so`£wmotyPp'~S`qlmk`8_=`mumuso`kyPqk £¡ª®yPsu_=`#« q^p,~^suyllqtml`ypH¢yP£qlmotyPp0
σ(t) = R⊗ Dε
Dτ
» )=) D½
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ε = (J +K) ⊗ Dσ
Dτ−K ⊗ DTr(σ)
Dτ1l
» )=) 2½y
J`Lkm9£wª®ypmutypl` qJ`twl`LpPmot¯Jr``pÀmosoPmotyPp©`m
K£3ª®yPpmotyPpl` q``p³tkot££`¤
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σ = 2M ⊗ Dσ
Dτ+ L⊗ DTr(ε)
Dτ1l
» )=) ¼D½±`Qk8ª®yso_9q^£wmotyPpk8`p³ypH¢yP£qlmotyPpk8kyPpDm#kuyq¢`pDm#S`LqJyq~"~£qk#ªÇt£`Lk#¥,_p^t~^q^£`s6« q^p
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uz1`muz2
so`£trLk#¥,q^p^`~su`QkukutyPpYY Y U ÒµÜ, Ó (nÓnÞÇÞYÓ& Û1(ZеÓRÖµÛ1#×Ò(#×ÖÓ9µÓ YX
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^q^`ql'mu`so_=`Lk¢HtwkuvDq^`Lqlp
_=]p`ql'sorLkuq^£momokkqt¢pDmk5
uz1 =1 − ν′1
2
πE′
1
∫
S
pf (s, φ)
rdsdφ
» )=) GP½
¾Op©lrsot¢pPme£ypDmustpDmu`#Pry_=rmusotvDq^`»AP½ ~seso~^~JyPsmcq,mu`L_¡~JkcyPpÈylmot`LpPme£so`£wmotyPpkuq^t¢pDmu`
δ = uz1 + uz1 =1
πE′∗
∫
S
p (s, φ)
rdsdφ
» ) PP½y pJmuqsu`L££`_=`pDm
1
E′∗=
1 − ν′12
E′
1
+1 − ν′2
2
E′
2
» ) µ½±yP_¡_=`8~Syq^s £`yPpPmml`Éc`sumuÊ8r£wkmutwvDq^`²lyPp~S`q^m_¡yPpDmuso`s l`#£w_=¿L_¡`8_pt]Lsu`#vDq^`8£
~su`QkukutyPpPlrLvDqmu`8~^so`p£ª®yPsu_=`kuq^t¢pPmo`
p (x, y) =P
2πa2
(
1 −(r
a
)2)
−12 » ) DP½
`#vPqt¨lypp^`8~JyPq^s £=¢Htmu`Qkuku`#« tpl`LpDmomotyPpδ
δ =P
2E′∗(πa)2
∫
S
1
r
(
1 −( r
a
)2)
−12
dxdy» ) ) ½
¾OpÈsorL£twkopDm`mumu`8tpDmorsmotyPp6²lyp3y^mut`pDmc£ysk
P = 2aE′∗δ = 2E′∗R12 δ
12 δ
» ) H½±`mmu``Hmu`LpkutyPp8rmur-surQ£tkur`R~Syq^sl`LkyPpPmmok`LpPmosu`Oku~^\^]Lsu`Qk¦!~^sotyPsutA²`RmnjH~J`-6« ~^~^soyH\`
~S`q^mck« ~^~£twvDq^`sl`#£w¡_=¿_=`#_p^t]so`#¥9moyqlmcypDmoPm tp^tmot£`L_=`pDm~JyPpmoq^`£vPqt¨k`#lrL¢`L£yP~^~J`kuq^t¢pDm q^p^`#`££t~ku`¦
« qp^`_p^t]so`rLp^rs£`²µ£A« p£jlk` l`| yqJkukutp`Lkov8~JyPq^sOq^p=_morsotq¢HtwkuyDrL£PknmotvDq^`£tprLtso`yp^q^tmc¥¡£wª®ys`#ltwkukut~Jmut¢`
P = A∂P (δ)
∂δδ
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2x
52
» ) 2P½§¨`^t£wp¡rp`sormotvDq^` ~S`p^pDm£w~^\Pk` l` yP_=~^su`Qkukutyp9~S`q^mRku`_=`mmosu`kuyqJk£weª®yso_=` kuq^t¢µpDmu`
∫ 1
0
dE(x)
d xdx = −f(v0)
∫ 1
0
x12 x
12 dx,
» ) 2 ) ½£`8_=`_9su`#l`lsoytmyPsuso`Lku~JyPp^pPm ¥¡£Y« rp`sot`8^tkokt~Sr`8~J`Lp^pPm£`#\^yl¦
pk£`#Pk 6« q^p,\^yler£wkmutwvPq`²l£wypku`so¢mutypl`8£Y« rp^`LsuPt`~S`so_¡`ml`#lyp^p`s£¢Htmo`Lkok``Lp'ª®yPpmotyPpÈlqÈlr~^£w`_=`pDmkuyqk£wª®yso_¡`#kuq^t¢µpDmu`»®¢Pyts rLvDqmotyPp"»Q ) ½n½
x(x) =
√
1− x52
» ) 2µH½ ¸ "»nG¼>2½kq^~~JyDk`LpPm-vDq^`£` _=yq^¢P`_=`pDmR`p=~^sorLku`p` l` ¢HtkoyDktmur ªÇt^£`²
« `Qknm¥ltsu`~JyPq^s^`Lk¢µ£`Lq^sk¨lq9yH`¬t`pDm^`-so`LkmutmuqlmotyPp~^soyH\`Lk6l`e²Q`Lkm¥ ~S`q~^so]Lk6twl`LpPmotvDq^`q_=yPq^¢`L_¡`LpDm pk£`Pkr£wkmutwvPq`²^kuytm
x(x) ≈ ±√
1 − x52
» ) 2P½±`mmo`#\DjH~Symo\^]Lku`8p^yqk yp^q^tmc£yPsok¥¡£A« `Qknmot_motyPp3kuq^t¢pDmu`#lqÈyH`¬t`LpDmcl`8su`Qknmotmoqlmutyp0
e2n − 1 = −4f(v0)
∫ 1
0
x12 x
12 dx ≈ −4f(v0)
∫ 1
0
√
1 − x52x
12 dx
» ) 2P½
§¨`ªÇPmu`Lq^sl`L¢pDm £A« tpDmurLs£`~^soy¢Ht`LpPmlqªÇtm vDq^`c£`^t£wprp^`LsuPrmotvDq^`¡» ) 2 ) ½Op6« `QknmrLsutm vDq^`~Syqs£w3~^\Pk`l`yP_=~^su`Qkukutyp6¦W§¨3l`sop^t]so`=tpDmurLs£`9`Qknm#r¢£q^rL`=s `¡¥3q^p^`| `mo¤Yª®yPpmotyPpB²^yq,kut£Y« yp,~^sorª®]Lsu`8q^p^` _¡_¤Yª®yPpmotyPp
Γl`8£w_p^t]Lsu`#kuq^t¢µpDmu`
∫ 1
0
√
1 − x52x
12 dx =
√
(π)Γ
(
3
5
)
5Γ
(
21
10
) =2
5B
(
3
5,3
2
)
≈ .5044549» ) 22½
´ep,Syq^mutmc£ysk ¥¡£A« `Qknmot_mutyp,kq^t¢pDmu`^q,yH`¬t`pDml`#so`LkmutmuqlmotyPp5
en ≈ 1 − 1.009f(v0)» ) 2¼D½
ÍÍ ÒT'UWVXY
G W! !
·^¸ 6· »nGGP¼P½~^soy~SyPku`pDm« _=r£tyPsu`Ls`mumu``Qknmot_motyPp,`p,su`L£^pDm £A« \DjD¤~Symo\^]Lku`,kq^s¡£`3_=yPq^¢`L_¡`LpDmQ»®rQvDqmotyPp » ) 2½½¦R[¨yPqlm« Sys²~Syqs¡£w©~^\ku`^`,y_=~^so`Lkoktyp¨²£A« ~^~^soyµlt_mutyp9so`Lkmu` mosu]QkOk`L_9^£w£` ¥#`L££`ªÇtmu` ~Js ¸ "»nG¼>2½¥#`Qt^~^so]LkvDq^`8£A« tpl`LpDmomotyPp,_=lt_£`8`p3~^sorLku`p`8^`#¢DtwkoyPkutmur`Lkm~^sutwku``Lp3y_=~lmo`
xm²kyPtm
x(x) ≈
√
1 −(
x
xm
)52
, x ∈ [0, xm]» ) 2GD½
§-« rQvDqmotyPp"» ) 2 ) ½O`Qknmc£yPsok tpDmurPsurL`²l`vDq^tSyqlmotm¥
1
2x
52m − 1
2= −f(vo)x
32m
∫ 1
0
√
1 − x52x
12 dx
» ) ¼P½
y yPp3ypkutwlrsore£w¡¢£`q^skuq^t¢µpDmu`#l`#£A« rLp^`sot`
E(xm) =1
2x
52m, E(xm) =
1
2x(0) =
1
2
» ) ¼^µ½xyqs£w~\ku` ^`so`LkmutmuqlmotyPp6²µt£wktpDmosuyllq^twku`pDm£wcpymutyp¡l` y££twktyp9tpH¢`Lsoku` »ÇkutwL½¦±`mumu`p^y¤
motyPpypkut^]so`Oq^p9_=yll]L£`qllt£twtso`µ¢`QRq^p¡_=ysumutwkuku`_=`LpPmp^rDmutªtpH¢`sku`-l` £Y« _=ysumutwkok`L_¡`LpDm^q3_=yll]L£`#kmopJ^s²^`#vDq^t¨yp^q^tmc¥¡£A« rQvDqmotyPp3Plt_=`Lpktypp^r`#kuq^t¢pPmo`
dE(x)
d x= f(v0)x
12 x
12
» ) ¼PP½±`mmo`'rLvDqmutypÀ`LkmtpDmurLsor`~Syq^sylmo`p^ts9q^pk`QypJ©t£wprLp^`sormutwvDq^`²¨y_=~^£r_=`pDmtsu`qt£wpÀ» ) 2 ) ½²^~J`Lp^pPm£w¡~^\Pk`#l`#so`LkmutmuqlmotyPp3µ¢P`Le£`QkyPpltmutypk ql'£t_=tmu`Qkkqt¢pDmo`Lk5
x(0) = 0, x(0) = en» ) ¼ ) ½
§kutp^t¯Jmutyp~^\HjHkutwvPq`8^`8`LkyPpltmutypk ql£t_=tmu`Lk~JyPq^s `#\^yl¢Dtsumuq^`L£~^suy¢Ht`pDm ^qªÇtmvDq^`3£A« yPpkuyq^\tmo`'yPlmu`Lp^tsy_=_=`¢Htmu`Lkoku`^t_=`pJktyp^pr`¯Jp£`²£w ¢£`qs=^qyH`¬t`LpDm=l`so`Lkmutmuq^mutyp
en¦^§¨`8_¡yPq^¢`L_=`pDm `QknmlyPp8~^~suyl\^rl`#£_=¿_=`8_p^t]so`#vPq`~S`pJ^pDm £¡~^\Jku`
^`8y_=~^so`Lkoktyp~s
x(x) ≈ en
√
1 −(
x
xm
)52
, x ∈ [xm, xf ]» ) ¼H½
§-« tpDmurLsmutyp"lq"Ptp"`p"rp^`LsuPt`=~S`p^pDm#`mumu` uy££tkutyPp tpH¢`sku` p^yPqk#_=]Lp^`¥'£A« rQvDqmotyPpkuq^t¢pDmu`#kuq^s £`yH`¬t`pDml`#so`LkmutmuqlmotyPp0
1
2x
52m − e2n
1
2= enf(vo)x
32m
∫ 1
0
√
1 − x52 x
12 dx
» ) ¼P½
y £A« yp,ypJktwlrsor8vPq`
E(xm) =1
2x
52m, E(xm) =
1
2x(0) =
1
2e2n
» ) ¼D½
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( GD
¾Op¡mo`ppPmOy_=~lmu`^`LkRl`Lql9^t£pkR« rp^`LsuPt`kuq^sR£`Qk~^\Pk`QkRl` y_=~^so`LkokutyPp» ) ¼D½6`mRkuq^sR£`Lk~\ku`Lkl`-so`LkmutmuqlmotyPp'» ) ¼D½²µyp9JyPqlmutm¥c£Y« rLvDqmutypkuq^t¢pPmo`-~Syq^s£`yH`¬t`LpDml`su`Qknmotmoqlmutyp0
en + 2f(vo)Ie35n = 1
» ) ¼ 2½µ¢P`L
I£A« tpDmurPso£`8kuq^t¢pPmo`
I =
∫ 1
0
√
1 − x52x
12 dx
» ) ¼¼P½e`ql0_=rmu\^yll`QkcyPpPmermur¡~^soy~SyPkur`Qk ~JyPq^ssurQkyPqlso``mumu`rLvDqmotyPp l`_=p^t]so`9p£jDmotvDq^`
§~^so`_=t]so`ckL« ~^~^q^t`ekuq^s q^plrL¢`L£yP~^~J`L_=`pDm`Lp'krLsut`c`LpDmut]so`» ·^¸ 6· ²JGG¼D½vDq^tJp^rL`Lkoktmu` qp£wq£pHq^_=rsotvDq^`^`Lk-yH`¬t`pDmokL²`m-£9k`QypJl`kqsOq^p~~^suyµlt_pDmOl`xRlr^ypDm £`Qk yH`¬t`pDmkkyPpDm L£wq^£rLkp£jDmutwvPq`_=`pDmL¦
*HÆÇÂ6 6 *H * / Ã * / L,0 * ¨·^¸ 6· , 8 ±`mmo`ÈkyP£qlmotyPp `QknmPkrL`kuq^sq^p,lr¢P`£y~^~S`_=`LpPm`p,kursot`~ss~^~Sysum ¥
f(v0)¦J±\JvDq^`#yH`¬t`LpPmrmpDm `pkuq^tmu`8£q^£r
pHq^_=rLsutwvDq^`_=`pDmQ¦µaemoq^su`L££`_=`LpPmQ²`sorLkuq^£mmR`Lkmª®ysum~surQtwk`mOyPpD¢P`so` µ¢`L-£Y« yslso` ^` £w8krLsut`¦ `m`W`mQ²^q^p^`#pyq^¢P`££`8¢Htmo`Lkok`#LsmorsotkmutwvDq^`
v?0
`Qknmclr¯p^t`l`8£=_p^t]Lsu`#kuq^t¢µpDmu`
f(v0) =1
2I
(
vo
v?0
)15 » ) ¼GD½
bcp^`kyP£q^mutypª®yso_¡`L££`#l`#£A« rQvPqJmutypÀ» ) ¼ 2½-~S`q^m¿muso`8_=tku`kyPqk£w9ª®yPsu_=`#kuq^t¢pPmo`
e−1n = 1 + 2f(vo)I(1 + 2f(vo)I(1 + . . .)
25 . . .)
25
» ) GD½vDq^tp`lytm~k ^t¢P`so`s ~Syq^s
vo → ∞ ¦¾Op"sor`l~^sot_=pDm`mmo`¡rQvDqmotyPp0`p ª®ypmutyp©^q©s~^~SysumPlt_¡`LpkutyPp^p^r=l`¢Htmu`Qkuku`²Wyp©yl¤
mot`LpDm
en = 1 − a1
(
vo
v?0
)15
+ a2
(
vo
v?0
)25
− a3
(
vo
v?0
)35
+ a4 . . .» ) G^µ½
´ep`QknmlypJ`p'~^surQk`Lp`e« qp'lr¢P`£y~^~S`_=`LpPm `p'kursot``pDmut]so`clyPpDm£`Lk yH`¬t`pDmkOypDm rmursorLkuy£qkepHq^_=rsotvDq^`L_¡`LpDme~s ·^¸ 6· »nGG¼D½¦WayPqkp^`£A« `l~^£ttmo`soypke~ktwtko\JpDmvDq6« t£`Qknm8~JyDkukut^£`=l`lyPp^p^`sq^p^`=`Qknmot_motyPp©ltso`Lmo`¡^q©yH`¬t`LpDm8l`=so`LkmutmuqlmotyPp `LpsorLkuy£¢pDm ltso`Lmu`L_¡`LpDm£A« rLvDqmotyPp"» ) ¼>2½» SRyPts 6 kuq^t¢µpDm½¦
6:6Â>O 0Ç / Ã'0* 4 0 4 ϸ·º ! , § ku`Lypl`3vDq^`'p^yPqk¡££yPpk9sorLkuq^_=`stwt8kL« ~^~^q^t`"kuq^s3q^p ~^~^soyµHt_pDml`"xRlrP¦ ±yP_=_¡`©£`Àkuyq^£tPp^`pDm 4 ϸ·Jº ! ¦#»nGG=GP½²`lrL¢`L£yP~^~J`L_=`pDm¡0muyq^mu`Lk¡£`Lk¡\pJ`Lk« ¿muso`^t¢P`so`pDm¡~JyPq^s
vo → ∞ ¦-±e« `Lkm~Syq^svDq^yt t£wk~suyP~JyDk`LpPm « qlmut£tku`sqp ¤Z n~~^suyµlt_pDm-l`#xRlrP²^vDq^t¨lyPp^p^`8^t`p,£`Qk so`£wmutypJk kuq^t¢pDmu`Qk5
limv0→0
en(v0) = 1, limv0→∞
en(v0) = 0» ) GP½
ÍÍ ÒT'UWVXY
GP W! !
±`mmo`~^~^soyµlt_pPml`#xRlr#kL« `l~^sut_=`#kyPqk£ª®yPsu_=`#kqt¢pDmo`
en =
1 + d1
(
vo
v?0
)15
1 + d2
(
vo
v?0
)15
+ d3
(
vo
v?0
)25
+ d4
(
vo
v?0
)35
+ d5
(
vo
v?0
)45
» ) G ) ½
§-« rL¢µ£qJmutyp3pHq^_=rsotvDq^`#l`QkyH`¬t`LpDmok`pª®ypJmutypÈl`Iyp^q^tmcql'sorLkuq^£momk kqt¢pDmk5
d1 = 2.5839
d2 = 3.5839
d3 = 3.9839
d4 = 1.1487
d5 = 0.3265
^Æ0 0lÃA0Y / / , 0;* ¯p l`_=t`ql y_=~^so`p^su`£`QkltWrso`pDmu`Qk`Qknmot_mutypk`m=\HjD~Sy¤mo\^]Lku`Lk,Pkukuyltr`Lk,vDq^t#yPpPm,rmor"ªÇtmu`LkL² yp ku`©~^soy~SyPku`À^pkÈ`mumu`À~sumut`Àl`³lyp^p`sÈvDq^`£wvDq^`Lk`l`_=~^£`LkpHq^_=rsotvDq^`Qk l`#yP_¡~JstwkyPp'sorL£twkr8kyPqkhlt£w6¦
pk£`Qk kt_9q^£wmotyPpk vDq^t¢yPpDm kuq^t¢Hsu`P²l£`8ªÇPmu`Lq^sf(v0)
¡rmur\ytwkt6l`8£w¡ª®yso_=`8kuq^t¢pDmu`
f(v0) = K v150
» ) GH½y
K`Qknmq^p,~so_=]muso`8lr~S`ppDml`Lk^yp^p^rL`Lk_morsot`L££`LkL¦
§¨`LkRsorLkuq^£momkR~Syq^sK = 0.2
`m-q^p` rL\^`L££`l`¢Htmu`Lkoku` ¢µsutwpDmOl`0.1
¥10_ k-kyPpPmOso`~SysumurQk
kuq^s £w¯q^so`= 2l¦´ep©~J`Lqlmso`_sovDq^`LsevDq^`²Wkuq^s8£¯q^so`'2l»YP½ vDq^`¡£Y« r¢Py£qlmutyp©lq©yD`¬t`pDml`¡so`LkmutmuqlmotyPp
µ¢P`L£w#¢Htmo`Lkok`c`Lkm-^t`p~^sorL^tmo`~s£Y« `pJk`L_9^£`cl`QkO`Lkmut_motyPpk¦Pxyq^s l`Lk-¢Dtmu`Qkuku`LkOªÇt^£`Lk-pkpymuso`9LkL²yq©« qp0_p^t]so`~^£qkerprs£`²^£yskuvDq^`9£`¡yH`¬t`pDme^`9so`LkmutmuqlmotyPp0`Lkme~^soyl\^`9l`P²J£A« `pku`_^£`l``Qknmot_mutypkc~J`Lsu_=`m8q^p^`JyPp^p^`¡~^~^soyµlt_mutypÈlq©yH`¬t`pDm^`9so`LkmutmuqlmotyPpµ¢P`Le£`Qk¢Htmo`Lkok`Qk¦
xyqsl`Qk¨¢£`Lq^sok¨~^£qkr£ytp^rQk¨l`en = 1
²Qku`q£`£csurQkyP£qlmotyPp#pHq^_=rsotvDq^` l`-£A« rQvDqmotyPp'» ) ¼ 2½ypDmutpHq^`-l` lypp^`sl`LksurQkq£mmokkumutwknªÇtkopDmk¦!°£D~^~swm¨vDq^`-£A« ~^~^soyµlt_mutyp#lq9_=yPq^¢`L_¡`LpDmªÇtm8~s ·^¸ ¨· »AG=G¼P½ck`L_9^£`9stwkuyp^p^£`9« q^p ~SytpDm8l`¡¢Hq^`¡rp^`LsuPrmutwvDq^`¦§¨`sorLkuq^£mmq³~^so`_=t`Ls9yPsolso`l`"» ¸ »nG¼ 2½`m¡£Y« ~^~suyµlt_=pDm9l`xRlrkuypDm£ysk q^p,~J`Lq,_¡yPtpJk~^surQtwk¦¯p"6« t££qkmuso`s8£`Lk8ltSrLsu`LpPmo`Lk\HjH~Jymu\^]Qk`QkcªÇtmo`Lk8kuq^s8£`=_=yq^¢P`_=`pDm8~s£`Lk8ltSrLsu`LpPmk#ql¤
mo`q^skL²DyPp'so`~Sysumu`8kqs £w9¯q^so`¡ 2l»ÇJ½£`QkyqsuS`Lk so`~^sorLku`pDmopDmokO£w¡¢Htmo`Lkok`8lt_=`pkutyp^p^rL`x`Lp
ª®yPpmotyPp3^`#£Y« tpJl`pDmomutyp,lt_=`pkutyp^p^rL`²x¦
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( G>2
0 1 2 3 4 5 6 7 8 9 10
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
0 1 2 3 4 5 6 7 8 9 10
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
0 1 2 3 4 5 6 7 8 9 10
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
0 1 2 3 4 5 6 7 8 9 10
0.68
0.72
0.76
0.80
0.84
0.88
0.92
0.96
1.00
en
v0(m/s)
xx
abcd
WÜÓ!()#×ÓnÒÞ QÓRÛYÓnØYÞ$# Þ$QÞ'#×ÜÒPÙ enÓnÒ ÜÒ(°Þ'#×ÜÒ0QÓRÕ .# ÞAÓnØAØYÓ8#×Ò# Þ$# Õ×ÓÙ v0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
−1.0
−0.6
−0.2
0.2
0.6
1.0
1.4
en
v0(m/s)
x
x
abcd
# ÞYÓØYØAÓRÓnÒ ÜÒ(°Þ$#×ÜÒ &nÖµÕ (nÓ) ÓnÒÞ x(x) ÖÜµÛ v0 = 10m/s
2 $ %'!! ' ! %=%R %' ! W ! ' ! %'! $# R ! $ ' = # '! » ) ¼D½ $ ! J!F! '# 4 ϸ·º !M( '! » ) G ) ½ $ ! !R! ' = ¸ '! » ) 2¼D½ '$ ' # '! » ) ¼ 2½ W%D R=!' ! W%%'$ P= ·^¸ . 6·
ÍÍ ÒT'UWVXY
GP¼ W! !
1#$ '% )+"$) ' # #5,1 !% )+ .0,11# #" . 1 % # .,
3, 9Â / Ã!ÅÃ 0* B*,QÃC 6-ÂJ' / ÅÂ 6=HÆ ^LÃLÂÄ6Æ Ã0;* 6 D0ÇÃpk-`mumu` ~smot`P²p^yqJkR~^sorLku`pDmuyPpk£`QkR~^sotpt~Jql^r¢`L£yP~^~S`_=`pDmokRvDq^trmu`Lpl`LpPmO£wmu\rysot`
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(1 − en) ∝ v−
14
0
» ) GPP½¯pl`3~^sorLku`pDmu`Ls9£`Lk¡~^sotpJt~ql³rL£rL_¡`LpDmok¡l``Lkmusµ¢ql²¨pyqk¡yP_¡_=`Lp`LsuyPpk~sso`¤
~su`Lplso`9£wmu\rysot`¡l`¡\^`LsmoÊ9`p©r£wkmutwvDq^`¯p©l`=lrQlq^tsu`¡£`QkeypDmustpDmu`Qke^pkeq^p©l`L_=t¤Z~^£wp6¦±`QksurQkq£mmokpyqk~S`so_=`mmosuyPpk l`8mosuyPq^¢`Ls £w¡~^so`Lkoktyp_=lt_£`8~JyPq^s £wvDq^`££`qp,sutmu]Lsu`#l`~£Pknmottmor~S`q^mc¿mosu`8so`_=~^£tO» 6 2H¦×l¦×½¦
acyqk9~^surQk`LpDmu`soypJk8`pkuq^tmu`'£`_=yll]£`Pkrkq^smusoytwk~^\Pk`Qkl` ! ¦ »AGH½8q6 2H¦×l¦ ) ¦^¾-pkqtmo`²Wp^yqJkclyPp^p^`soypJkq^p^``Lkmut_mutyp0^q yH`¬t`LpPm8l`9so`Lkmutmuq^mutypÈ~Syq^s£Y« t_¡~Jm`LpDmuso`¡l`Lql ku~^\^]so`Lk8lyPpDm£`y_=~Sysumu`L_¡`LpDm`Qknm#kq~^~JyDkr=¿muso`rL£Pknmoy~^£wkmutwvDq^`¦S¾Opl¯p¨²p^yqk8`Qkn¤kot`soypJk¨l`yPplª®suyPpDmu`s`QksorLkuq^£momokql#`l~Srsot`Lp`Qk¨pHq^_=rLsutwvDq^`Lk`mRl` £JyPsomuytso`-vDq^tH~J`Lq^¢`LpPm¿muso`musoyq^¢Pr`Lk^pk£w£tmmurLsomuq^so`¦
3 3 9 / ÃD0 / Ã!*H 0 / ¡Æ * 0*D 0ÏÄ.*HA6Å* ^0*HÈLÂSÆ0 0*H 0* SÂJÆY¨Ã0Ç / ^ÃLÂS'0*ÆLK DO *
zepk8£`k#^`Lk#kuy£tl`Qk#l`sor¢yP£q^¤
motyPp £`Lk¡yPpPmosotpDmo`Lk¡^pk£`3l`L_=t¤Z`Lku~`kuypDmlyp^pr`Lk~s · »AGPGP½¦1£"kuq^sªÇP`P²£`LkypDmustpDmu`QkkyPpDmelypp^r`Qkc~sc£`LkerLvDqmutypk»Y GD½¦6´epÈso`_svDq^`9vDq^`9£`LkeyPpPmosotpDmo`Lk sltw£`LkkuypDmR`p=yP_=~^su`Qkukutyppk£wÊyp`^`ypDmoPmOkoqlªWkq^sR£`¢yPtkutpJ`l`£wcª®soypDmot]Lsu`l` £weÊLyp^` l`ypDmoPmy£`Qk ypDmosotpDmu`QkkuypDm`p'musmotyPpµ¢P`Leq^p^`8¢£`q^s_lt_9q^_4l`
(1− 2ν)p0/3¦Hxyq^s
^`Lk_morsotqll`mnjH~J`8ª®sPt£`²ly_=_=`#£`Qk¢`soso`Lk yq3£`Qk rLso_¡twvDq^`LkL²l`mmu`#so`_sovDq^``l~^£tvDq^`£`Lk ªÇPt]Lk l`8ª®smuq^so`#kyPqk tpJl`pDmomutypÈl`#É`Lsmoʦ
pk£`kl`Qk8_=murLsutwqlÈ_=rm££twvDq^`LkL²kqt¢pDm8q^pÀy_=~Sysumu`L_¡`LpDmer£wkmuyP~^£wkmutwvPq`9qtlr~Jsq^psutmu]Lsu`Rl`SRyPp#dÈtku`Lk-»Hµ¤Z~^£PknmottmorQ½SyPq~Js6q^p#sutmu]Lsu`Ol`O[so`Lko^²£A« p£jlku`^`Lk¨ypDmustpDmu`Qk
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( G=G
_=yPpPmosu`=vDq^`¡£`sotmu]so`9`Qknm#so`_=~^£t`p"~^su`L_=t`Ls£t`q"£`¡£yPp^,l`=£Y« l`z» ¨·
Ï·^¸²OGPlµ½¦% ¯Jpl`~^sorLtwku`s#`mmu`so`_sovDq^`P²so~^~J`L£yPpk8q^p^`=ª®yPsu_=`kmopJ^s©^qÀsutmu]Lsu`l`SRyPp3d0tku`Lk
J2(σ) =1
6
[
(σ1 − σ2)2 + (σ2 − σ3)
2 + (σ3 − σ1)2]
= k2 =σ2
0
6
» ) GD½y
σ0`Lkm£=ypDmustpDmu`#« rLyPq^£`_=`pDm q^p^twltsu`Qmutypp^`££`8`LpmosoPmotyPp`m
k£w¡ypDmustpDmu`#« rLyql¤
£`_=`LpPm`LpÈtwkut££`L_=`pDm`mc£`LkσikuypDmpmuq^so`££`L_=`pDm£`Qk yPpPmosotpDmo`Lk~^sotpJt~£`LkL¦^§6`9sutmu]Lsu`#l`
[so`Lko~J`Lqlmck`#_=`mmuso`#kyPqk ª®yso_=`8rLvDq^t¢µ£`LpDmu`
max |σ1 − σ2|, |σ2 − σ3|, |σ3 − σ1| = 2k = σ0» ) G 2½
§¨`Lk yPpDmustpPmo`Lk £`#£yPp^l`#£Y« l`zkyPpDm ^yp^p^rL`Lk~s £`Qk rLvDqmutypk8»A GD½²kuytm
σr
p0=
σθ
p0= −(1 + ν)
(
1 − z
aarctan(
a
z))
+1
2
(
1 +z2
a2
)−1 » ) G¼D½
σz
p0= −
(
1 +z2
a2
)−1 » ) GGD½
§3spl`Lq^s#sotmotvDq^`~Syqs`Lk9sotmo]so`Lk#l`~^£wkmutwtmur\ytwktR`Qknm#£wÈypDmosotpDmu`=l`tkot££`_=`LpPmkuq^t¢pDmu`
2τ1 =σr − σz
p0=σθ − σz
p0= −(1 + ν)
(
1 − z
aarctan(
a
z))
+3
2
(
1 +z2
a2
)−1 »®DD½
´eplyp^p` kuq^s£¯q^so`cQ¼£`-~^soy¯J£Dl`QkypDmosotpDmu`Qk6`Lpª®ypJmutypl`£w~suyª®ypl`Lq^slt_=`pkutyPp^p^rL`λ = z/a
¦H§=lrsot¢Pr`#l`#£w=ypDmosotpDmu`8l`tkot££`_=`LpPm `Lp'ª®yPpmotyPp,^`λ~J`LqlmckL« `H~sut_=`s
∂
∂λ(2τ1) = −(1 + ν)
(
arctan(λ) − π
2+
λ
λ2 + 1
)
− 3λ
(λ2 + 1)2»®D^µ½
§ckuy£qlmutyp¡l``mumu`rLvDqmutyp#ª®yPq^sop^tmOlyPp£c~^suyª®yp^`q^s¥c£wvDq^`L££` £` _lt_q^_ `p¡yPpPmosotpDmo`^`twkut££`L_=`pDmR`Lkm-mmo`tpDmL¦PhHyPp`l~^su`Qkukutyp=p£jDmutwvDq^``pª®ypmutypl`
νp^`ku`_^£`~Pkmusot¢Dtw£`¦
´ep^yp^p^`kq^s£È¯q^so`0G^²6£`'sorLkuq^£mom¡l`£w0sorLkuy£qlmutypÀpHq^_=rsotvDq^`kyPqk9hlt£w tpJkt vPq`£`~suy¯£6l`Lk ypDmustpDmu`Qk`Lp3ª®ypmutyp,l`
λ¦
§¨`_lt_9q^_ ^`¡ypDmustpDmu`Lkel`twkut££`_=`pDme`Qknm#lyPpmosuyPq^¢r¡kyPqke£kq^suªÇ`¡^`¡ypDmoPmL¦¯pÈl`8¯^l`s £`LkyPso^su`Qk ^`#spl`Lq^sQ²P~Syq^s
ν = 0.3²H£=~suyª®ypl`Lq^sy
J2`Qknm_Ht_9q_4¢qlm
λmax(ν = 0.3) =z
a= 0.4808645
»ÇPPP½~Syqs q^p^`yPpDmustpPmo`Plt_¡`LpkutyPp^p^r`8¢£wpDm
τ1max
p0= 0.3100205
»ÇP ) ½
ÍÍ ÒT'UWVXY
Q W! !
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
−0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
−0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
−0.1
0.1
0.3
0.5
0.7
0.9
1.1
τ1
−σr/p0
σz/p0
λ
L¼ W ' ! !=$ '' ) R λ
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0.21
0.25
0.29
0.33
0.37
0.41
0.45
0.49
0.53
0.57
ν
λm
ax
τ1max
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0.26
0.30
0.34
0.38
0.42
0.46
0.50
0.54
νλmax
τ 1m
ax
OG !! $ !"=$$ W !' τ1max
E M!%R)R= M!' ! '
λmax ) R =
ν
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( L
±`L£#kutPp^t¯`vDq^` ~JyPq^sO£`sotmo]so` l` SRypdÈtwk`QkOyP_=_¡` ~SyqsO£`sutmu]Lsu` l`e[¨so`Lkoe£8~su`QkukutyPp_=yjP`p^p^`8¥¡~sumutsl`#£wvDq^`L££`8£`#sotmu]so`#l`#~^£wkmutwtmur8`Lkmcmumu`LtpDm¢qlm
p0 =σ0
2τ1max
»®DH½
kuytm ~JyPq^sν = 0.3
p0 = 1.6127966σ0»®DPP½
´ep,¢yPtmcvDq^`#£w=~^so`Lkoktyp3_=yj`Lp^p^`²pm²`Lkmcsu`L£tr`8~sq^p^`ypkmopDmu`8rLy_=rmosutwvDq^`#lt_=`pl¤
kutyp^p^rL` ¥e£weypDmustpDmu`-£t_¡tmu` ¥e£Y« rLyq^£`_=`LpPmQ¦µ¾Op9mo`ppDmy_=~lmu`l`£A« rQvPqJmutyp¨²H» 2¼D½²ypp^ymo`spk£w=kq^tmu``mumu`yPpkmopDmo`
CYkuytm
pY = CY σ0»®DD½
^'0*0*H OÅ+6Æ0 / 0(*H'0 K O *H 6 ^ÆYÆ7HÆ *H ´ep`pH¢Htwku`^pk9`~sPso~^\^`=£`Pk9l`^`ql'jD£tplso`Lkr£wkmutwvPq`LkL¦D§¨`LkyPpPmosotpDmo`Lk-kuypDm lypp^r`Qk-~s£`Qk-rQvPqJmutypJke»Y 2P½¦^e`e£w9_=¿L_¡`_p^t]Lsu`P²Wyp"~J`Lqlm_=ypDmuso`s#vDq^`=£`Qk#yPpPmosotpDmo`Lkl`twkut££`L_=`pDmn~^sotpt~£`LkkuypDm#_=lt_£`Lkkuq^s £A« l`
z¦^« ~^su]Qk£`QkrLvDqmutypk#»Y 2P½²l£`LkypDmustpDmu`Lkkuq^s £A« H`
zkL« rQsot¢P`pDm
σx = −p0
a
[
(a2 + 2z2)(a2 + z2)−12 − 2z
] »®D 2½
σz = −p0a(a2 + z2)−
12
»®D¼D½±`Qk yPpPmosotpDmo`LkkyPpDm ^`Lk yPpDmustpPmo`Lk ~^sotpJt~£`Lk lyPp£=ypDmustpDmu`#l`#twkut££`_=`pDmk« rLsutm
τ1 =p0
a
(
z − z2(a2 + z2)−12
) »®DGD½kuytml`#_p^t]so`8lt_=`pkutyPp^p^rL`
τ1p0
= λ− λ2(1 + λ2)−12
»®JLD½
§¡^rsot¢Pr`8~s s~~JyPsm ¥λlypp^`lypJ
1
p0
∂τ1∂λ
= 1− 2λ
(1 + λ2)12
+λ3
(1 + λ2)32
»®Jµ½
§¨`c_lt_9q^_%l`QkypDmustpDmu`QkO`Lkm ^ypemmo`tpDm tplrL~J`Lp^_=_=`pDm l`ν~JyPq^sl`Lk¢£`q^skL£wql¤
£r`QkpHq^_=rsotvDq^`L_¡`LpDm kuyqJkhlt£0
λmax = 0.7861514, τ1max = 0.3002831»ÇQP½
´ep0lyPp^p^`kuq^s£w¡¯qsu`9=£ltwkmusot^q^mutyp0l`QkyPpPmosotpDmo`Lk ^t_=`pJktyp^pr`Lk`p3ª®ypmutyp0l`λ¦Y 5 #×Ò#%(nÓ Y ØAÓÛ1& nÛÇÓ #×ÓnÕ%
ÍÍ ÒT'UWVXY
QP W! !
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
τ1
−σr/p0
σz/p0
λ
W W ' ! !=$ '' ) R λ
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( L )
xyqs£` sotmo]so`l` [su`QkuL^²Qypylmot`LpPmRlyPp-qp^`~^su`Qkukutyp_=yj`Lp^p^` l` kq^suªÇ`¥e£Y« rLyq^£`_=`LpPm~£PknmotvDq^`8tplr~S`pJ^pDmcl`
νkuytm
p0 =σ0
2τ1max
= 1.6650953σ0»®J ) ½
xyPq^s£`8sotmo]so`el`ESRyp'd0tku`LkL²H£musoytwkt]_=`e^tso`Lmutyp~^sotpt~£`8l`eypDmustpDmu`Lk`pDmosu`e`pzn`Lq'`Lp^rª®yso_motyPpk ~^£p^`LkL¦J±`mumu`8mosuyPtkut]L_=`8ypDmustpDmu`~^sotpt~J£`¢qlm
σ3 = ν(σ1 + σ2)»®JH½
h^pk9`pDmuso`s¡^pk9£`Qk9lrmot£wk^q £q^£lqsotmo]so`'l`#SRyPpdÈtwk`Qk²yplyPp^p^`3kq^s£È¯q^so`^vDq^`L£vDq^`Qk¢£`q^skl`
J2/p0`p ª®ypJmutyp ^`È£À~^soyª®yPpl`Lq^s
λ¦ ´ep lyp^p`0kq^s'£"¯Pq^so` ©£
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
ν = 0
ν = 0.2
ν = 0.3
ν = 0.5
λ
J2/p0
W^ ! '! NJ2/p0
) R λ%D ' ! W
ν
~suyª®ypl`Lq^s¥£wvDq^`££`e£`e_Ht_9q_4lqsutmu]Lsu``Lkmmmo`tpPm tpkutvPq`e£w¢£`qs^``e_lt_9q^_,²J2/p
20max
`pª®ypmutyp l`ν¦Rxyq^s=l`,¢£`q^sk=l`
νtp^ª®rsot`Lq^su`Qk=¥
0.195²ypypkmomu`,vDq^`3`
_Ht_9q_`LkmsorL£twkr#¥=£wkq^suªÇ`l`yPpDmomL¦^xyPq^sl`Lkc¢£`q^sk kq~JrLsut`q^so`LkL²l£`_lt_q^_`Qknmqp^`8ª®ypmutyp3£`LpDmu`_=`LpPmc^rLsoytwkokupPmo`el`
ν¦/ mutmuso`« tplª®yPsu_mutyp6²l~Syq^s
ν = 0.3²l£`#sutmu]Lsu`8`Lkm
ÍÍ ÒT'UWVXY
Q W! !
0 0.1 0.2 0.3 0.4 0.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
λmax
J2/p20max
ν
W ! J2/p2
0max
λmax
)R =ν
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( LD
sorL£twkure~Syq^s £w=~^so`_=t]so`cª®yPtk ~Syqs
λmax = 0.7042916,J2
p20
= 0.1036081, p0 = 1.7936698σ0»®JQP½
^ 0*H6 ÆÇ S / ,^ O §¨`8Lkl`Qk~^suy¯£wkPrp^rLsoql~S`qlm¿muso`¢Hq,yP_¡_=`8q^p,PktpDmu`Ls¤_=rQlttso`c`LpDmuso`c£`Lkl`8l`ql'kuy£twl`Lk l`surL¢y£qlmotyPpqlmoyq^sl`£Y« l`
z`ml`Lql'jH£tpJlsu`Qk6« H`
~Js££]L£`P¦´epkL« mmo`p=lyPp¥8¢ytsr¢Py£q^`sR£w~^suyª®yp^`q^sO« rLyq^£`_=`LpPmR`mO£8~^so`LkokutyPp9£t_=tmu``Lpª®yPpmotyPp'l`e£Y« `^`LpDmusottmore« q^pLk-¢`skO£A« qlmosu`P¦Hhlpk`pDmuso`s ^pk~^£qk l`^rmot£wkL²P£`LksurQkq£mmokyPlmu`LpHqk~s#rL¢µ£qJmutyp©pHq^_=rsotvDq^`=l`Lk8rQvDqmotyPpk»Ç 2½» ·^¸ ²AGP¼P½ekyPpDm^yp^p^rQk ^pJk£`8m^£`Lq,^¦
b/a ¦ ^¦ ¦ ^¦ ¼ ¦
z/b^¦ 2¼D ¦ 2D ^¦ P ^¦× GP ¦ ) ^¦ P¼P
(τ1)max/p0^¦ ) P ¦ ) ^¦ ) P ¦ ) ) ^¦ ) 2 ^¦ ) Q
W0 ! ! ' # W $ ' % M! 'P )R = # ">W'
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§-« `Lpk`L_9^£`l`QksurQkq^£momok vDq^t6¢PypDmkqt¢Hso`k« ~^~£twvDq^`#q3~^suyP^£]_=`l`yPpDmom tp^tmutw£`_=`LpPm~SypJmuq`£A¦l´ep'^yp^p^`Lso8`Lp¯p3l`c~sos~^\`6« qlmuso`Lk-surQkq^£momoklqk ¥ ¹¸$·,»nAG=G¼D½yp`soppDm£`ypDmoPmr£wkmuyP~^£PknmotvDq^`8`Lp3lrª®yso_mutypJk~^£p^`LkL¦
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pY = CY σ0»®JLD½
ÍÍ ÒT'UWVXY
Q W! !
xyqs8`mumu`¡¢µ£`Lq^s#l`¡~^su`Qkukutyp6²S£Y« tpl`pDmmutypδY
`m#£w'\suP`PY
~S`q¢`pDm#k`¡_=`mumuso`¡kuyqk£wª®yso_=`8kuq^t¢pDmu`
δYR
=
(
3π
4
)2(CY σ0
E∗
)2 »®J 2½
PY
CY σ0R2 = π
(
3π
4
)2(CY σ0
E∗
)2 »®JL¼D½
¯p « `Lkmut_=`s£`'yH`¬t`LpDm9rp^`LsuPrmotvDq^`l`so`Lkmutmuq^mutyp6²t£p^yqk9k`LsoÈqkukutOp^rQ`Qkukotsu`l`yp^pJwmuso`£Y« rp`sot`9r£wkmutwvDq^`kmuyl Pr`pke£`¡_morsotq6¦W±`L££`¤°t~S`qlm#k« rLsutso``p ª®ypmutyp©^qmosoµ¢t£l`#£wª®ys`8r£wkmutwvDq^`8~J`Lp^pDmc`mmo`#~^\Pk`
WY
E∗R3 =
∫ δY /R
0
P (δ′/R)
E∗R3 d
(
δ′
R
)
=8
15
(
δYR
)52 »®JAGD½
kuytmkyPqk ª®yso_=`#lt_=`pkutyPp^p^rL`
WY
R3CY σ0
=2π
5
(
3π
4
)4(CY σ0
E∗
)4 »®HD½
4 @^ * 0* 0 ZÂ/ lÃ0Ç / HÆ ^ÃQÂ'6Æ Ã0;* ±`mumu`R~^\Jku`Ok« ~^~q^t`Okuq^s6£A« \DjH~Symo\^]Lku`OvDq^`R£Y« rLyql¤£`_=`LpPm ~^£wkmutwvPq`cso`Lkmu`cypl¯pr`epkq^p^`eÊyp`²Hkq^~~JyDkrk~^\rsotvDq^`ckuyqJkO£w9kuq^sªÇP`c^`ypDmoPmL¦´epkuq^~^~SyPku`c£yskRvDq^` £A« rmom l`LkypDmosotpDmu`QkR^pJkR£`c_=murLsutwq=~S`q^sO¿muso`ckur~sur`p=mosuyPtkOÊyp`Lk5
bp #q^seku~^\^rLsutwvDq^`l`soµjPypakuyqkc£kuq^suªÇ`l`9yPpDmomcyÈ£`LkeyPpPmosotpDmo`LkkuypDmckuq^~l¤
~JyDkrL`Lk¿muso`\HjllsoyPkmomutwvPq`Lkl`#~^so`LkoktyprD£`8¥p²
q^p^`8ÊyPp^`\^`_=t¤Zku~^\^rLsutwvDq^`²lyP_=~^sutwku`e`LpPmosu`£`sµjyPpa`m q^p3soµjPyp
cy£Y« yp,kuq^~^~SyPku`vDq^`
£`y_=~Sysumu`_=`LpPm `Qknm r£wkmuyP~^£wkmutwvPq`~JsuªÇtmL² bp^`el`sop^t]so` ÊLyp^` ~SyqsOl`LkRsµjyPpkkq~JrLsut`q^skR¥
cy=£A« yPp=kuq^~^~SyPku` vDq^` £`ey_=~Sysumu`L_¡`LpDm
su`Qknmo`8r£wkmutwvPq`¦pke£wÊyPp^`9\`_=t¤°ku~^\^rsotwvPq`²£`LkyPpDmustpPmo`LkekyPpPmkq^~~JyDkrL`LkerP£`Qk¥'`££`Qke6« q^p`~^\Jku`
~£PknmotvDq^`#^pkq^p,_=t£t`Lq,r£wkmutwvPq`tpl¯ptY²kuytm« ~^so]Lk#»A¶ ²GPP½+
σr
σ0= −2 ln
( c
r
)
− 2
3, a ≤ r ≤ c
»®D^Q½
σθ
σ0= −2 ln
( c
r
)
+1
3
»®HP½
pk£¡ÊyPp^`8r£wkmutwvDq^`8`HmurLsut`q^so`²l£`Lk ypDmustpDmu`Qk kuypDmkuq^~^~SyPkur`Lk ^`£w¡ª®yso_=`#kq^t¢pDmu`
σr
σ0= −2
3
( c
r
)3
, r ≥ c»®H ) ½
σθ
σ0=
1
3
( c
r
)3 »®HH½
Ë®ÌÍ6ËÏÎ
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£'£t_=tmo`lq #qse\HjllsoyPkmomotvDq^`P²£w'~su`QkukutyPpÈ\HjH^suyDknmmutwvDq^``Qknm8lyp^pr`¡~s£yP_¡~Sy¤kopDmo`s^t£`8l`Lk ypDmustpDmu`Qk~Syqs
r = akyPtm
p
σ0= −σr(r = a)
σ0= 2 ln
( c
a
)
+ 2/3»®DP½
¾Opkuq^~^~SyPkopDmOvDq^` £w^rª®yso_motyPp~^£wkmutwvDq^``LkmOtpJy_=~^so`Lkokut^£`²¶ »nAGDD½6yPlmut`pDmO£A« rL¢y¤£qlmotyPp,^qsµjyPpl`#£w¡Êyp`rL£Pknmoy~^£wkmutwvDq^``p3ª®ypmutyp,lq,soµjPyp3l`#ypDmm
da
dc=
σ0
E∗
[
3(1 − ν)c2
a2 − 2(1 − 2ν)a
c
] »®HD½
§9_=¿L_¡`#`l~^su`Qkukutyp~S`qlm¿muso`8ylmo`pHq^`#l`8_p^t]so`8~^£qJkrprs£`eyPp`LsupJpDm £`#lr~£P`_=`LpPms^t£Wpk£w=~sumut`8r£wkmuyP~^£PknmotvDq^`
du(r)
dc=
σ0
E∗
[
3(1 − ν)c2
r2− 2(1 − 2ν)r
c
] »®H=2½
¾Opkuq^~^~SyPkopDml`~^£qkL²¨vDq^`=£È^rª®yso_motyPp"lq³#q^s#`Lkm#tpy_=~^so`Lkokt^£`²£wÈyPpk`Lsu¢mutyp^`8kuyp3¢yP£q_¡`8p^yPqk yPplq^tmc¥
2πa2du(a) = πa2dδ = πa2 a
Rda
»®H¼P½¾-p qlmot£tkopDm8£Y« rLvDqmutyp »ÇDP½ `m£A« rLvDqmotyPp©l`¡yPpk`Lsu¢mutyp lq©¢yP£q^_=`P²JyPp0ylmot`LpDmq^p``Qknmot¤_mutyp,lq,so~^~JyPsm
c/akq^t¢pDmu`
E∗a
Rσ0= 6(1 − ν)
( c
a
)3
− 4(1 − 2ν)»ÇDGP½
hHt£A« yPp kq~^~JyDk`©l` ~^£qkvPq` £wlrª®yPsu_mutyp r£wkmutwvDq^` l`"yP_¡~su`QkukutyPp `Lkm'p^rP£t`Q^£`²£`'_mursotwq³~S`qlm£ysk#¿mosu`3yPpktwlrLsurL`'yP_¡_=`tpy_=~^so`Lkokt^£` »
ν = 0.5½¦O´ep³ylmot`LpDm=£yPsok
£A« `Lkmut_=mutyp,l`c/a
kq^t¢pDmu`c
a≈(
E∗a
3Rσ0
)13 »® ) D½
§~su`QkukutyPp3pk£`# #q^s ¢q^mckq^t¢pDm`Qk\DjH~Symo\^]Lku`Lk
p
σ0= 2/3
[
1 + ln
(
E∗a
3Rσ0
)] »® ) µ½
¾Opmu\rysot`²R`mmu`,~^so`LkokutyPp kq~^~JyDkrL`qp^tª®yPsu_=`,^pJk£`, #q^s=\HjllsuyDknmmotvDq^`3l`¢Hstm¿mosu`rLP£`¥,£w,~^su`QkukutypÀ_=yj`pp^`l`'yPpDmom
pm¦°£ku`=musoyq¢`vDq^`£A« rmom9^`Lk9ypDmustpDmu`Qk#kyPqk8£
kuq^suªÇ`,^` yPpPmm=p¨« `Qknm'~Jksor`L££`_=`pDm\HjllsuyDknmmotvDq^`,`mQ²-`Lp sorL£tmurP²-ku`,musoyq^¢P`3~suyl\^`0l`£A« rLyPq^£`_=`pDm ~^£wkmutwvPq`¦ »nG 2P½-~^suyP~JyDk`~^£qlm mcl`8su`L£t`s £w¡~^so`Lkoktyp3l`#ypDmm ¥¡£ypDmustpDmu`soPltw£`kuq^suªÇ`P²^`#vDq^t6~S`qlmckL« rLsotsu`#tpkut
pm = p+2σ0
3
»® ) P½
ÍÍ ÒT'UWVXY
Q¼ W! !
kuytmpm
σ0= 2/3
[
2 + ln
(
E∗a
3Rσ0
)] »® )) ½¾OpÀrQsot¢pDm`mmo`¡rQvDqmotyPp"q"lrL^qlml`£A« rQyPq^£`L_=`pDm~Syqs#£3~^so`Lkoktyp
pY`m8£`=soµjPyp ¥
£A« rLyPq^£`_=`pDmaY²^yPp'mosuyPq^¢`8£wso`£wmotyPp3Plt_¡`LpkutyPp^p^r`8kuq^t¢pPmo`
pm
pY= 1 +
2
3CYln
(
a
aY
) »® ) H½
±`mmo`#su`L£mutyp'~S`q^mc¿mosu`8sorrLsutmu``p3ª®yPpmotyPp,^`£w\suP`muymo£`
P
PY=
(
a
aY
)2 [
1 +2
3CYln
(
a
aY
)] »® ) P½
¯p l` su`L£t`s£\so`È~~^£twvDq^r`"¥³£A« tpl`LpPmmotyPp6²£` sµjyPp ^`©yPpDmom'`Lkm3L£wq^£r0`Lpª®yPpmotyPpÈl`#£Y« tpl`pDmmutyp3s `#¥£A« \DjH~Symo\^]Lku`« tpy_=~^so`Lkokt^t£tmur#lqÈ#q^s `p,p^rL£tP`LpDm q^p^`ª®yPtk ^`#~^£qJk£lrª®yso_mutypr£wkmutwvDq^`¦J´ep,y^mut`pDm£ysk5
δ
δY=
1
2
(
(
a
aY
)2
+ 1
)
»® ) D½
§-« rQvPqJmutypÀ»® ) H½-~S`q^mc£ysk kL« rLsotsu``p3ª®ypJmutypÈl`8£A« tpl`LpDmomotyPp0
pm
pY= 1 +
1
3CYln
(
2δ
δY− 1
) »® ) 2½
`m£Y« rLvDqmutypÀ»® ) ½
P
PY=
(
2δ
δY− 1
)[
1 +1
3CYln
(
2δ
δY− 1
)] »® ) ¼D½
¾Op^¯p,~JyPq^s`mmu`#~\ku`8t£6ypH¢Ht`LpPm l`L£wq^£`s£`emosoµ¢t£WsorL£twkr~s £w¡\so`8q,yPq^sokl`£A« tp^`pDmomutyp6¦hlP\pDmvDq^`
WY = 3/5FY δY²^`£wso`¢Ht`pDm¥=£wq£`Ls £Y« tpDmors£`8kq^t¢pDmu`
W
WY= 1 +
5
2
∫ δ/δY
1
(
2δ′
δY− 1
)[
1 +1
3CYln
(
2δ′
δY− 1
)]
d
(
δ′
δY
) »® ) GD½
kuytm
W
WY= 1 +
5
8
[
(
1 − 1
6CY
)
(
(
2δ
δY− 1
)2
− 1
)
+1
3CY
(
2δ
δY− 1
)2
ln
(
2δ
δY− 1
)
]
»®PPD½
Ë®ÌÍ6ËÏÎ
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pm`Lkm£ysk
ypkmopDmu`q¡yq^skl`£Y« tpJl`pDmomutyp6¦¯p¡^`^rmu`Lsu_=tp^`s£wc¢£`Lq^s~JyPq^s£PvPq`££``mmu`~^\Jku` l`^rª®yso_motyPp8^r^qlmo`²µ£A« \HjH~Jymu\^]Qk`vPqtP`Qknm6mosoPltmutyp^p^`L££`_=`LpPmªÇtmu`O`Qknml`ypkutwlrso`s6£w _morsotqy_=_=` suttwl`~^£wkmutwvDq^` ~suªÇtmL¦P§O« r£wkmutwtmur `Lkm-£yskp^rL£tPr` yP_¡~£]mu`_=`LpPm-pkO`mumu`~^\Jku`^`8^rª®yso_motyPp6¦
§-« p£jlk`c^`e£A« tpl`LpDmomotyPp« q^p^`8ku~^\^]so`c^pk q^p'l`L_=t¤Z~^£wp'sottwl`~^£wkmutwvDq^`ku`csor¢P]£`¿mosu`so^q^`¦H9« ~su]Qk ³»nG¼D½²HqJq^p^`8kuy£qlmutyp'p£jDmutwvDq^`cp6« rmuremusoyq¢r`c~JyPq^s£`8yPpPmmkopJkª®suymmu`L_=`pDmL²6`m9`QtRrmpDm~^sotpJt~£`_=`pDm9³q ªÇtm9vDq^`£A« qlmuy¤Zkut_¡t£wsotmor¡lq³~suyP^£]L_=``Qknm=~S`slq^`¦xyqs¡l`Qk=ypJltmotyPpk¡« Pl\^rLsu`Lp`~suªÇtmu`P²:4 ! ¦»AG>2µD½#~^suyP~JyDk`LpDmqp^`9kuy£qlmutypÈpJ£jPmotvDq^`#Jkur`#kqsc£w¡mu\^rLysot`l`9£tp^`Qkl`#P£twkuku`_=`LpPmµ»®¢Pyts»Y¶ ²6AGDD½½¦^§¨`Lk¢£`Lq^sokl`=£w'~^so`LkokutyPp _=yj`Lp^p^`9~Syqs£wvDq^`L££`¡£Y« rLyq^£`_=`pDm#~^£`tp^`_=`LpPm8~^£wkmutwvDq^`k`¡~^soyH^q^tm¢sut`l`
3.02σ0~Syq^s
a/R = 0.07¥
2.95σ0~Syq^s
a/R = 0.30¦ »AG¼D½e~^soy~SyPku`l`
so`mo`p^ts q^p^`8¢£`qsqlmoyq^sl`3¦J´eppymu`Lso=^pk£w¡kuq^tmu``mmo`#¢£`q^sl`8£w¡ªÇ yp,kq^t¢pDmu`
pp = Cpσ0»®Pµ½
y pp`Lkm £=~su`QkukutyPp'_=yj`Lp^p^`#q,lrL^qlmcl`#£w¡~^\ku`8~^£`tp^`_=`pDm~^£wkmutwvDq^`¦
pk8£w~^\ku`¡r£wkmuy~£PknmotvDq^`P²J£w'~su`QkukutyPp0_=yj`Lp^p^`¡`Lkmq^ptª®yPsu_=`¡`m#su`Qknmo`¡ypkmopDmu`¦W§¨\Jso`emuym£`#q^P_¡`LpDmu`#lypJµ¢P`Le£mt££`l`8£kuq^suªÇ`l`ypDmoPm
F
FY=Cp
CY
(
2δ
δY− 1
)
, F > Fp»®PDP½
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W
WY=Wp
WY+
5
2
∫ δ/δY
δp/δY
F
FYd
(
δ
δY
) »®P ) ½
kuytmW
WY=Wp
WY+
5
2
Cp
CY
[(
(
δ
δY
)2
− δ
δY
)
−(
(
δpδY
)2
− δpδY
)]
»®PH½
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PL W! !
) !! # M! %RWW $ R =&W' ! ) R # M! )$ ! !=$ '
E∗a/(σ0R)( $ A ' ## R= ! J %(%'! W%D R ! $ ' PR $$
! A= ( # ! %F '
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W0 !' =!D $ M% !N $ % F$ ' % ! '($ %D! ! ! W $ ' = % !W 'A( $# ! %R '
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PQ W! !
W %'! ! P! %% ) R = # F' ! F( # !W%R '
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x`pJ^pDm£lrL\suP`²P£9su`L£mutypª®yPso`µtpl`LpDmomotyPp`LkmrL£PknmotvDq^`P¦D§~^sotpJt~£`c\HjH~Jymu\^]Qk`vDq^t`Lkm#ªÇtmu``Qknm¡vPq¨« `L££`'kuq^tmq^p^`£yt-^`'É`LsmoÊr£wkmutwvDq^`~^so`ppPm`LpyP_¡~^mu`£w,_=yH^t¯JLmotyPp^`#£kuq^suªÇ`#l`ypDmmlq`9¥=£Y« `_=~^so`tpDmu`#~^£wkmutwvDq^`#£wtwkukur`8~s £y_=~^so`Lkoktyp6¦^§¨`#p^yq¢`LqsµjyPp'l`yq^so^q^so`
R`Lkm£yskpymur#~Js
R =
R1R2
R1 + R2, R1 ≥ 0, R2 ≥ 0
R1R2
R1 − R2, −R1 > R2 ≥ 0
»®PDP½
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´eppymu`-£Y« tpJl`pDmomutyp#`Lp#¯p9l`-~^\ku`-l` yP_¡~su`QkukutyPpδc`m¨£A« tpl`LpDmomotyPp9vDq^tP£t`q9qyPq^sok
^`'£wÈsu`Qknmotmoqlmutyp³rL£PknmotvDq^``Lkm9p^ymur`δr¦eqªÇtm¡^`Lk9^rª®yso_motyPpk~^£wkmutwvDq^`Lk9~J`Lsu_p^`pDmu`Qk²
£A« tp^`pDmomutyp¯p£`²δf`LkmclyPp^p^r`#l`#£w¡_p^t]so`#kq^t¢pDmu`
δf = δc − δr»®PPD½
bcp^`\HjH~Jymu\^]Qk` `QknmOPrp^rLso£`L_¡`LpDmªÇtmo`~JyPq^s-Lsmorsotku`s£A« tpl`LpPmmotyPp=^q^`¥8£8so`Lkmutmuq^mutyp6¦´ep©kq^~~JyDk`¡vPq`9£A« tpl`LpPmmotyPp0~J`Lp^pPm8£w'lrQ\so`#rL£PknmotvDq^``LkmPry_=rmusotvDq^`L_¡`LpDmekt_=t£wtso`¥=`££`#vDq^t¨¡`q,£t`Lq,~J`Lp^pDm £w\so`er£wkmutwvPq`²^kuytm
δYR
=δrR
»®P>2½§9so`£wmotyPp`pDmuso`£`8sµjyp^`ypDmoPm`m £Y« tpl`pDmmutypr£wkmutwvDq^`e`Qknm lyPp^p^r`~s£A« rQvDqmotyPp
»YD½²kuytm~SyqsδY
`mδr
δrR
=a2
r
R2 ,δYR
=a2
Y
R2
»®PP¼D½
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qsµjyPpacl`#£w¡_p^t]so`#kq^t¢pDmu`
2δcδY
=a2
c
r2+ 1
»®P=GP½`#vPqt6~J`Lqlm¿mosu`8sorrQsotm s `¥,»ÇP¼D½ kyPqk£ª®yPsu_=`#kqt¢pDmo`
2δcR
=a2
c
R2 +δYR
»®HD½¾Op^¯p6²~JyPq^s=lrLlqtso`'£`'s~^~Sysum`pDmosu`£Y« tpJl`pDmomutyp
δr`m=£A« tpl`LpPmmotyPpq_lt_9q^_ l`
y_=~^so`Lkoktypδc²ypÀkq~^~JyDk`vDq^`=£w'¢sotwmutyp"lqÀsoµjPyp ^`=ypDmoPm
ar`Qknm#rLP£`=q"soµjPyp l`
ypDmoPm `p¯pÈl`y_=~^so`Lkoktypac²`vPqt6lyp^p`
R
R=δrδc
=
(
2δcδY
− 1
)12 »®Hlµ½
°£S~S`qlm ¿mosu`etpDmorso`LkokupPm l`epymu`Ls vDq^``mmu`so`£wmutyp^`lrQ\so`cr£wkmutwvPq`cso`Lkmu`c¢£^£`ktS£A« yPp^rL\suP`¥¡~smots l`#£w¡~^\ku`#l`#~^£wkmutwtmur#yPpl¯p^rL`¦
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F =4
3E∗R
12 (δ − δf )
32
»®HP½kuq^s £A« tpDmu`Lsu¢££`8^`so`Lkmutmuq^mutyp
δf ≤ δ ≤ δc¦
±`mmu`tpDmorsmotyPp0yPplq^tm¥qpÈmusµ¢t£¨p^rLPmotªn²Wr
²WvPqt~S`qlm¿mosu``l~^sot_=r9kuyqkc£ª®yso_¡`kuq^t¢pDmu`
Wr
σ0R3 = −
∫ δr/R
0
4
3
E∗
σ0
(
R
R
)3(δ
R
)32
d
(
δ
R
)
= − 8
15
E∗
σ0
(
δrδY
)3(δYR
)52 »®H ) ½
¾Op qlmut£twkopDm£=so`£wmotyPp,ryP_=rmusotwvPq`~Syqs£^rL\suP`=»Ç2½`mc£`Lkc`l~^so`LkoktypJk ^q,musµ¢µt£`mcl`8£Y« tpl`pDmmutypÈ¥¡£A« rQyq£`L_¡`LpDm~£PknmotvDq^`P²lyp,ylmot`LpDm
δYR
=δrR
=
(
3π
4
)2(CY σ0
E∗
)2
,WY
σ0R3 =
8
15
E∗
σ0
(
δYR
)52 »®HH½
´ep©ylmot`LpDm8£A« `l~^so`Lkoktyp ^q©s~^~Sysume`LpDmuso`£`¡musµ¢µt£so`Lkmutmuq^rWr
`m8£`¡musµ¢t£¥£Y« rLyq^£`_=`LpPm~£PknmotvDq^`
WY
Wr
WY= −
(
δrδY
)3
= −(
2δcδY
− 1
)32 »®HP½
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PL W! !
W W=W != )F N # R= ! $# ! %F ' ¹W¸$¨·
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3 H - N6 ^ÅÃ$* / Ã* 0*H O 6+@:7,*Hacyqk so`~su`Lp^ypk£A« `l`_=~^£`9l`#£A« t_=~mc`pDmosu`l`Lql3ku~^\^]Lsu`Qk¯Jp,^`9lypp^`s £`Lkc`Lkmut_=mutypk
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µ¢PytsrLyPq^£`_=`pDm-~^£wkmutwvDq^` ~S`qlm-¿muso`clrQlq^tmu`cl`c£Y« rLvDqmutypÈ»nPL¼P½~SyqsOq^p^`tpl`LpDmomotyPp_lt¤_£`
δm = δY
V 2Y =
16R12E∗
15mδ
52
Y
»®HD½¾-p mo`ppPmy_=~lmu`Èl`,£A« `l~^su`Qkukutyp^`
δYlyp^pr`È¥"£Y« rLvDqmutyp »Ç2½²RyPpyPlmut`pDm£©¢Htmo`Lkok`
_=tp^t_=£`8~Syq^seµ¢yts rLyPq^£`_=`pDm ~^£wkmutwvPq`
V 2Y =
16R3E∗
15m
(
3π
4
)5(CY σ0
E∗
)5 »®H=2½
xyqs¡qp^`Èk~^\]so`3l`,sµjyPpR = 10mm
`LpPt`sknmps »E = 210Gpa, ν = 0.3, ρ =
7000kg/m3, σ0 = 1000Mpa,CY = 1.1½²Syp0yPlmut`pDmq^p^`¡¢Dtmu`Qkuku`9_=tp^t_=£`¡l`
0.15m/s¦W±`£w
ysosu`Qk~SypJ¥8q^p`c\Hqlmu`kuyqkOsµ¢Dtmur l`1.1mm
¦D±`QkRsurQkq£mmok-vPqt~S`q^¢P`pDmO~swmuso` rmoyp^pJpDmokyPpDm~SyqsmpDm rmur#¢£twlrLk`l~JrLsut_=`pDmo£`L_¡`LpDm~Js ¸»AGDlQ½¦
LÃA0Ç lÃA0Y / 0 Å * Å0 * / Ã0**HLÃA0®ÃQ¨Ã0Ç / L *Q / *9 *HÆ * 6 @^ *0*0 A lÃ0Ç /6Æ *>0 / *H * / Ã6Æ Ã0;* »AG¼D½~^soy~SyPku`#q^p^``Qknmot_mutyp0lq0yD`¬t`pDmkur`kuq^sqp^`eku`q^£`c~\ku`el`lrª®yPsu_mutyp~^£`tp^`L_¡`LpDm~^£wkmutwvDq^`¦,°£kq^~~JyDk`vPq`c£A« tpl`LpPmmotyPpmuymo£`c`Qknm^yp^p^rL`8~s £w¡su`L£mutyp,kq^t¢pDmu`
δ =a2
2R
»®H¼D½`m£w~^so`LkokutyPp_=yj`pp^`~S`p^pDm£`ypDmoPm-`QknmrD£`c¥
pp~S`pJ^pDm-moyqlm£`eyPpDmomL²D~^q^twkuvDq^`e£
ypDmustpDmu`ckuq^suªÇtvDq^`e`Lkm ¥£A« rLyPq^£`_=`pDmL¦ PqkovPq¨« ¥#£A« tpkmopDml`yP_¡~su`QkukutyPp_lt_q^_,²H£Y« rp^`Ls¤Pt`#tp^rmutwvDq^`#su`L£mut¢``LkmckuysoSr`~s £wmusµ¢µt£l`#£w¡surQkq£mpDmu`#l`ypDmoPm
1
2mV 2 = Wc =
∫ δc
0
P dδ =
∫ ac
0
πa2ppa
Rda = πa4
c
pp
4R
»®H GD½
xyqs£"~^\Pk`Èl`,so`LkmutmuqlmotyPp6²R£`Èsu`LJyPp `Qknmkuq^~^~SyPkur,r£wkmutwvPq``m£`3musµ¢t£ l`È£©ª®yPso`rL£PknmotvDq^`=`Qknmso`Lkmutmuq^r`LpÀrp^`LsuPt`tp^rmutwvPq`¦hHt
Vf`Lkm£3¢Htmo`Lkok`~^so]Lk8t_¡~JmQ²6yp"y^mut`pDm£
so`£wmotyPpkuq^t¢µpDmu`#~Syq^s £`musµ¢µt£6^`#£ª®ys`8rL£PknmotvDq^`
1
2mV 2
f = −Wr =3Pc
10acE∗
=3
10π2a3
c
p2p
E∗
»®DD½
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PL¼ W! !
´ep3y^mut`pDmc£ysk £Y« `l~^so`Lkoktyp3lqÈyH`¬t`LpPml`#so`Lkmutmuq^mutyp,kq^t¢pDmu`
e2n = −Wr
Wc=
3π54 4
54
10
( pp
E∗
)
1
2mV 2
ppR3
−14
»®D^µ½
LÃA0Ç lÃA0Y / 0 Å * Å0 * / à ^Q *L / * 6 @^ *'HÆ Ã0;* *DÃ8 / * 6+@^ * 6Æ *>0 / *H * / Ã6Æ Ã0;* ¹¸$· »AGGD½O~^suyP~JyDk`8l`mu`Lp^tsy_=~lmo`#l`8£¡~^\Pk`8l`#\suP`_=`pDmr£wkmutwvPq`kuq^t¢Ht`96« q^p`9~^\Pk`9~^£`Ltp`_=`pDme~£PknmotvDq^`P¦J§O« tpl`pDmmutyp0¥£Y« rLyq^£`_=`pDm`mc£wmosoµ¢t£¨l`¡`mmu`tpl`LpDmomotyPp3kuypDmlyPp8lyp^prLk `Lp'ª®yPpmotyPp,^`£wyPpkmopDmo`
Cp»Yq3£t`qÈl`#£A« rLvDqmotyPp"»ÇDD½ ½
δYR
=
(
3π
4
)2(Cpσ0
E∗
)2
,WY
σ0R3 =
8
15
E∗
σ0
(
δYR
)52 »®DPP½
kuytmWY
Cpσ0R3 =
2π
5
(
3π
4
)4(Cpσ0
E∗
)4 »®D ) ½x`pJ^pDm#£'~^\Pk`¡~^£`tp^`_=`pDm#~^£wkmutwvDq^`²W£`Lk\HjD~Symo\^]Lku`Lkkuq^s8£Y« tpJl`pDmomutyp©`p0ª®yPpmotyPp"^q
sµjyPp ^` yPpPmm'su`Qknmo`pDmtp\pr`Qk¦´ep lypJ,muyPqznyq^sk£A« rQvDqmotyPp»® ) P½=vDq^`0£A« yp ~S`qlm_=`mmuso`#kyPqk£w9ª®yPsu_=`
δ
R=δY2R
+a2
2R2 , δ > δY»®DH½
§¡surQkq^£mopDmu`#l`#yPpPmm~S`pJ^pDmc`mmo`#~^\ku`8~J`LqlmckL« `H~sut_=`s l`#£w_p^t]Lsu`#kuq^t¢µpDmu`
P
Cpσ0R2 = π
(
2δ
R− δY
R
) »®DPP½
`m~Js#yPpkrQvDq^`pDme£`¡musµ¢t£~S`ppDm#£3yP_¡~su`QkukutyPpÈ~^£wkmutwvPq`¡kL« `l~^sut_=`¡kuyqJkc£wª®yPsu_=`=kqt¤¢pPmo`
W
Cpσ0R2 =
WY
Cpσ0R2 + π
(
δ2
R2 − δδYR2
) »®DD½cq_Ht_9q_4l`8£=yP_=~^su`Qkukutyp »
δ = δc½²^£`emusµ¢µt£W`S`Qmuqr~Js£sorLkuq^£mpDmu`p^yPsu_£`¢µqlm
Wc
WY= 1 +
5
2
(
δ2cδ2Y
− δcδY
) »®D 2½
xyqs £lrQ\so`er£wkmutwvDq^`²l£w=_=¿_=`#p£jlku`8vDq6« q 6 2H¦×l¦ =`Lkm_=`Lp^r`P¦S´epyPlmut`pDmc£yPsok
Wr
WY= −
(
δrδY
)3
= −(
2δcδY
− 1
)32 »®D¼D½
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s `0ql rQvDqmotyPpk0»®D 2½9`m©»®D¼D½²Ot£cl`¢Ht`pDm'~SyPkokt^£`Èl` lyPp^p^`sqp^`0`Lkmut_motyPp ^qyH`¬t`pDm rp^`LsuPrmotvDq^`#l`8so`LkmutmuqlmotyPp0
e2n = −Wr
Wc=WY
Wc
(
8
5
WY
Wc− 3
5
)34 »®DGD½
µ¢P`LWc
WY=
5
4π
mV 2
(3π/4)4(Cpσ0/E∗)4Cpσ0R
3
»®2D½
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WY /Wc`Lkm`Qknmot_=r=¥'~Jsumutsl`,»®PH½
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pd¦ cq £t`Lq"^`¡kuq^~^~SyPku`s£3yPpltmutyp©ryP_=rmusotwvPq`'»®H¼P½²t£
kuq^~^~SyPku`#vDq^`8£Y« rp^`LsuPt`#ltwkokt~JrL`#~s~^£wkmutwtmurWc −Wr
`Lkmclyp^pr`8~s £A« rLvDqmotyPp3kuq^t¢pDmu`
Wc −Wr = pdvr»®2P½
y vr`Lkm-£`¢yP£q_¡` sorLkutwlq^`£Jl`c£A« `L_=~^su`LtpDmo`~^so]LkRt_=~mL¦l±`mmo`c\HjH~Jymu\^]Qk` _=yllt¯`£A« rQvDqmotyPp
»ÇDGP½l`#£¡_pt]Lsu`#kuq^t¢pPmo`1
2m(V 2 − 3
8V 2
f ) = pdvr = πa2c
pd
4R
»®2 ) ½y
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sorLkuq^£mopPmo`#l`Lk`SyPsmk l`yPpPmm `QknmclyPp^p^rL`~Js
δ =P
πE
(
2 ln(2d
a) − ν
1 − ν
) »®2µH½
Ë®ÌÍ6ËÏÎ
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y d`Lkm£¡~^soyª®yPpl`qs6« q^p,~SytpDmcl`#surª®rso`p`pk£`l`_=t¤A~£p6¦¯p l`"yP_¡~su`Lplso`0£Y« tpq^`Lp`Àl`
dkuq^s£wso`£wmotyPp»®2µD½tpDmusoyllq^twkyPpkq^p ~s_=]muso`
Plt_¡`LpkutyPp^p^`£ξmo`£vPq`
d = ξR¦^¾Op0q^mut£twkupDmc£=so`£wmotyPp³»nL ) ½²Jp^yqJk ~JyPq^¢yPpk sorrQsotso`»®>2D½
kuyqJk£ª®yso_=`#kq^t¢pDmu`
δ =P
πE
[
2 ln
(
ξ
√
πER
(1 − ν2)P
)
− ν
1 − ν
]
»®2P½
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¦§9¢£`Lq^sknmps'l`
CY~Syq^s £`#ypDmm « q^pÈjH£tp^su`#kuq^s q^p3~^£pyPJrLtkokopDmqsutmu]Lsu`#l`
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PY = 3σ0aY»®2D½
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PY
σ0R=
3aY
R=
36(1− ν2)σ0
πE
»®22½
`#vPqt6~J`Lsu_=`mc« `l~^sut_=`s£A« tpl`LpPmmotyPp,Plt_¡`LpkutyPp^p^r`
δ
R=
P
PY
(
6σ0
πE
)2
(1 − ν2)
[
2 ln ξ − lnP
PY+ 2 ln
πE
6(1− ν2)σ0
− ν
1 − ν
] »®2¼D½
£A« rLyPq^£`_=`pDmL²^`mumu`l`Lsup^t]so`8su`L£mutyp3l`¢Ht`pDm
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(
6σ0
πE
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2 ln ξ + 2 lnπE
6(1 − ν2)σ0
− ν
1 − ν
] »®2GD½
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F/FY
δ
δY=
P
PY
1 −ln
(
P
PY
)
2 ln ξ + ln
(
πE
6(1 − ν2)σ0
)
− ν
1 − ν
»®D¼D½
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δY≈ δ
δY
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Ea
σ0R
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1
CY
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lna
aY
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P
PY=
a
aY
(
1 +1
CY
√3
lna
aY
) »®D¼H½±yP_¡_=`#~Syq^s £`Pk« q^pÈyPpPmmtp^tmutw£`_=`pDm~Sypmuq^`L£Y²£Y« tpJl`pDmomutyp,`Lkmso`£trL`8q,soµjPyp
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δY= 1 +
a2 − a2Y
δY R
»®D¼PP½¾Op#tpDmurLspDm6`mumu`-^`sop^t]Lsu`Rso`£wmotyPppk£A« rLvDqmotyPp»®P¼D½²yp8ylmot`LpDm¨£so`£wmotyPpcª®yPso`tp^`pDmomutyp
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δ
δY=
[
δYR
a2Y
(
δ
δY− 1
)
+ 1
]12[
1 +1
2CY
√3
ln
(
δYR
a2Y
(
δ
δY− 1
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pp = Cpσ0»®D¼ 2½
y Cp
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PY=
Cp
CY
a
aY
»®D¼¼D½hHqt¢pDm¡£`Qk9_=¿_=`Qk\HjH~Jymu\^]Qk`Qk9vDq^`'~surQrL^`_=_=`pDmL²£Y« tpJl`pDmomutyp`Qknm¡su`L£tr`q³sµjyp³l`
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δY= 1 +
1
δYR
(
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) »®D¼GD½§¡su`L£mutyp'`LpDmuso`£w¡ª®ys`8l`ypDmoPm `m£A« tpl`LpDmomotyPp~S`q^melypJkL« rLsotsu`
P
PY=
√2Cp
CY
[
δY R
a2Y
(
δ
δY− 1
)
+ 1
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δ
δR=
1
2
(
δcδY
− 1
)
, 1 ≤ p
pY≤ Cp
CY(
δcδY
− 1
)
− a2Y
2δYR,p
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CY
»® G^µ½
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σ = σ0εM
(
ε
ε0
)N »® GPP½
y M
`mNkyPpDm^`Lk=`l~SyPkopDmok=~SyPkutmutªÇk²
σ0q^p^`,~so_¡]muso`_mursot`£yPpkmutmuqlmotª8`m
ε0q^p^`
¢Htmu`Qkuku`3^`0lrª®yPsu_motyPpl`,sorª®rLsu`Lp`P¦pJkku©ª®yso_=`3y_=~^£]mo`²O`mmo`È£yPt^`0y_=~Sysumu`L_¡`LpDmso`~surQk`LpPmo`q^pyP_¡~Sysumu`L_=`pDm-¢HtkovDq^`q^yqqpy_=~Sysumu`L_¡`LpDm-l`~^£wkmutwtmurc^r~S`p^pDmlq=moql^`,^rª®yso_motyPp6¦Repkl`Lk¡ª®yPsu_=`Qk¡^rrLp^rsor`Qk²yp so`musoyq¢`^`Lk=£ytwk¡^`,y_=~Sysumu`_=`LpPm~^£qkqJkq^`L££`LkL¦xyPq^s
M = 0²£w0£ytku`'sorLlq^tm¡¥©l`3£ ¢DtwkoyPkutmur'l`3ayPsmoypl¤°Éy¦xyq^s
N = 0²yp
so`mosuyPq^¢`8l`8£w=~^£Pknmottmor#µ¢`LerLsuyPq^twkukoP`¦hlpkO`LpDmuso`spk-~^£qJk-l`lrmt£kkq^s `QkO£ytwk²P`££`Qk-ku`rLp^rs£tku`pDm-^pkOqp'Plsu`mosutwlt_=`pl¤
kutyp^p^`L£¨`Lp \^ytwkutkokupDmcl`Lke~Jymu`LpPmot`L£klqqlψ`mφ²Wsu`Qk~S`Lmut¢`_=`LpPmcª®ypmutypkc\^y_=yP]p^`Qkcl`
^`sorM + 1
`mN + 1
mu`L£k vDq^`
σij =∂ψ
∂ε, εij =
∂φ
∂σ, σ ¯⊗ ε = ψ + φ
»®=G ) ½§=l`sop^t]so`ypltmutypÈl`#lqJ£tmor#t_=~SyPku`ql'`l~JyDkupPmk £¡so`£wmutyp,kuq^t¢pPmo`
M + 1 =N + 1
N
»® GH½
ÈÂJÃA0Y / 0 K ^ÃLÂSA0YJ0ÇÆ 0®ÃQ0*H8QÂJÆY¨Ã0Ç / ¶ ! ¦»nG¼GD½6~suyPq^¢`LpPmRvDq^` £`LkRkuy£qlmutypkR^q~suyP^£]L_=`l`cÉ`LsmoÊekmop^s,»ÇypDmoPmOtp^tmot£`L_=`pDm~Sypmuq^`L£w½R~JyDkuku]Ll`LpDm-£~^suyP~^sotrmur« qlmoyPkut¤_=t£sutmur#~Syqsel`Qk£ytwk« rLsoyqtkoku``pÈ~^qtkokup`P¦J§=p^ymotyPp,6« qlmuyDkt_=t£wsotmor#`Lkmcq^p^`9p^ymotyPpy_=~^£tvDq^rL`¦H¾Op,vDq^`£wvDq^`Lk_=ymokL²H`££`~S`so_¡`ml`e_=yPpPmosu`Ls vDq^`£`e~^soy£]L_¡``LkmtpH¢sotpDm~s qp\Jp^P`_=`pDm6« rQ\^`££`¡lrQvDqmQ¦WpJkc£`~suyP^£]L_=`« tpJl`pDmomutyp"l`Lk_morsotqlÈ`p £ytwke~^q^twkn¤kopJ`LkL²`=\p`_=`LpPm#« rL\^`L££`¡~J`Lsu_=`ml`kuq^kmutmuq^`Ls#£Y« rL\`££`=l`¡mu`L_¡~Jk»ÇvDq^tOyPsuso`Lku~Syp0¥
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( QP
qp,mu`_=~k¯Jmotª-6« q^p0\Jso`_=`LpPmvDqPkt¤ZkmomutwvDq^`Q½ ~sq^p^`9rL\^`L££`9ku~mot£`P¦§6`~suyP^£]L_=`« tpl¤^`pDmomutypÀ~s#qp^`'kuq^sªÇP`lrL¢`£y~~JrL``pÀ~sSy£y Zl`=« q^pl`_=t¤A`Qk~J`=`Qknm9£yPsok8so`_=~^£wr~Js£`~^soy^£]_=`8« tp^`pDmomutyp3~s q^p3~JyPtp yp3~^£wmy3£¡lt_¡`LpkutyPplq,yPpPmm p^`8¢sot`c~Pk¦
±`QkR\p^P`_=`pDmk6« rQ\^`££`kuypDmRsu`Lplqk~SyPkokt^£`Lk~sR£` soPmu]Lsu`-\^y_=yP]p^`l`Qk~Jymu`pDmot`L£k^qql0Pyq^¢P`soppDm£`Qke£ytwk8l`yP_¡~Sysumu`L_=`pDm`m#~s8£`=Lsmu]so`9\y_=yP]p^`¡l`=£3yPpPmosotpDmo`PryP_¡rmusotvDq^`~^~^£tvDq^rL`9kuq^s£`l`L_¡t¤Z`Lku~`¦^¾Op ~sumutwq^£t`sQ²~Syq^sl`Lkekq^suªÇ`Lk ~Syq¢µpDmek`lr¤¢P`£y~^~S`s `p3~soJyP£yAl`P²£wypDmustpDmu`e`Lkm\y_=yP]p^`« yPso^su`8l¦´eqlmuso`8vPq¨« t£k~S`so_=`mumu`pDm q^p^`kut_¡~£t¯Jmutyplq~^soy£]L_¡`P²`QkO\p^`L_¡`LpDmokR« rL\^`L££`ª®ypDm~^~so mosu` l` ~^£qkl`LkOypkmopDmu`Lkqp^t¢P`sk`L££`Lk-l`c`QkR~^soy^£]_=`Qk-« tpJl`pDmomutyp6¦l´ep~J`Lqlm-~S`pku`s~s`l`_=~^£`cql¡ypkmopPmo`Lk
CY`mCp
vDq^typDm rmor#tpDmusoyllq^tmu`Qk~^surQrQl`_=_=`pDmL¦¶ ! ¦¨»AGP¼GP½-_=ypDmuso`pDm `p,~smotq^£t`LsvDq^`£A« `l~^so`Lkoktyp
C2 =a2
δ2R
»® GPP½`QknmtpH¢sotpPmo`=`mvDq^`
C2p^`lr~S`p³vDq^`^`Lk#`H~SyPkopDmk#l`£w,£yPtO~^q^twkukopJ`¦6bcp^`Syp^p`'~l¤
~suyµlt_=mutyp~JyPq^sC2
`Lkm£=kuq^t¢pPmo`
C2 = 1.45 exp(−(M +N))»® GD½
xyqs~^£qkl`lrmt£kkq^s`QkPk~S`Lmok« qlmoyPkut_=t£sutmurP²yp9su`LpD¢Pyt`Oq#mosoµ¢ql8kuq^t¢pDmok »Y¶ ! ¦²WAGP¼G ¹ ¸ ·^¸ % ϲJAGG ¹ ¸ ·¸ ²JAGGD ¹ ¸ ·¸ ! ¦²GG>2½
*HÆ lÃ0Ç / Z Å* -/ 0* / Ã!lÃ0Ç / bpl`Lk-surQkq^£momokO^`e`mmo`cp£jlku` ~s \p`_=`LpPm-6« rQ\^`££``Qknm £A« rQsotmoq^so`86« q^p`#su`L£mutypª®ys`°pl`LpPmmotyPpkuyqk £w¡ª®yso_¡`#kuq^t¢µpDmu`
P
πa2 = σ0α(N,M)[
βM (M,N)a
2R
]M[
βN (M,N)a
ε02R
]N »® G 2½y £`Lk9ª®ypmutypk
α, βM`mβN
kyPpDm¡l`Qkª®yPpmotyPpk9qp^tvDq^`L_=`pDm¡lrL~J`Lp^pPm=l`Lk`l~SyPkopDmokM`m
NvDq^tW~J`Lq^¢`LpPm ¿muso`elrmu`so_=tprLk `l~Srsot_=`pDm£`_=`pDmL¦Hxyqs^`e£w9~£Pknmottmorµ¢P`LrQsoyq^twkoku`
(N = 0)²`QkRypJknmpDmu`QkRysoso`Lku~JyPpl`pDmR¥8`££`LkRvDq^` ¸0»AGDlQ½µ¢tmR_=tk`Lp¡rL¢Htl`Lp` ^pk
`Lk`H~Srsot`p`Lk5α = 2.8
`mβM = 0.4
¦~sumutsel`¡kt_9q£mutypk pHq^_=rLsutwvDq^`LkL² ¹ ¸ ·^¸ ! ¦»nGG>2½ ~^soy~SyPku`£A« `Qknmot_motyPp0kqt¤
¢pPmo`~Syqs£`Lkeª®yPpmotyPpkα, βM
`mβN²6vDq^tRku`_^£`¡¿mosu`¡q^p^`¡mosu]QkeSypp^`~^~^soyµlt_mutypÈ~Syq^s
£`Lk¢£`Lq^sokqJkq^`L££`Lk ^`Lk`l~SyPkopDmok
P
πa2 = 3(1 + 2N)σ0
[ a
6R
]M[
a
ε06R
]N »® G¼D½±`mmu`Èp£jlk`,vDq^t rmur,_=`p^rÈ^pk¡q^p ~^su`L_=t`Lsmo`_=~k=~Syq^sl`Qk=tpJl`pDmu`Lq^sk¡ku~^\^rLsutwvDq^`Lk
sottwl`Lkkuq^s8l`Lkc~^£wpkelrª®yPsu_£`Qk~S`qlm¿muso`ªÇPt£`L_=`pDmcrLp^rs£twkrL`L¥l`Lql,yso~kc^rª®yso_^£`Lk^ypDme£`Lk£ytwkel`=yP_=~JyPsmo`_=`pDme~SyPkoku]Ll`LpPm8£`Lke_¡¿L_=`Lk`l~SyPkopDmokL¦/°£ªÇq^m#£yPsokeyP_¡~su`Lplso`£¢£`Lq^s
σ0y_=_=`qp^`#¢£`q^s rQvPqt¢£`pDmo`8kuyqJk£ª®yso_=`#kq^t¢pDmu`
1
σ1/(M+N)0
=1
σ1/(M+N)1
+1
σ1/(M+N)2
»® GGD½
ÍÍ ÒT'UWVXY
µ W! !
y σ1`mσ2soPmurLsutwku`pDm£`Lk l`Lql_mursotwql¦
M9$ 3 LÃA0Y lÃA0Y / 0 ÅP * Å0 * / Ã$0* *DLÃA0ÇÃLÃA0Y / [soµ¢ql'l`» ¹ ¸ ·^¸ ¸ ²D½¯p¡^`^yp^p^`Lsqp^``Lkmut_=mutyp¡lqyD`¬t`pDmOl`so`Lkmutmuq^mutyp6²µ£A« `H`L_=~^£` l` £A« t_=~mR^` l`Lql
ku~^\^]Lsu`Qk `Lkmcso`~^sotwk^pJkc`9~sPso~^\^`¦l§-« rQvPqJmutyp³»Ç=GGD½`LkmcsorrQsotmo`#`p,mu`Lsu_=`96« tpl`LpPmmotyPpδkyPqk£w9ª®yPsu_=`kuq^t¢µpDmu`
P = γσ0(2R)2(
δ
2R
)1+(M−N)2(
δ
ε02R
)N »YD½
µ¢P`Lγ = 31−M−N2−N (1 + 2N)πC
1+(M+N)/22
»Y^µ½§-« rQvPqJmutyp,lq,_=yq^¢P`_=`pDm sortwkokupPm£A« t_=~m~J`LqlmclypJku`ª®yso_9q^£`s^`#£=kuysumu`
δ +γ
mσ0(2R)2
(
δ
2R
)1+(M−N)2(
δ
ε02R
)N
= 0»YP½
bcp^`#~^so`_=t]Lsu`8tpDmurPsomutyp~s s~^~Sysum q3mu`L_¡~Jk lyp^p`
δ2−N +2(2 −N)γσ0(2R)1−(M+N)/2
m(4 +M −N)εN0δ2+(M−N)/2 = V 2−N »Y ) ½
e`e`mmo`~^so`_=t]so` tpDmurLsmotyPp6²yp~S`q^m lrQlq^tsu`cy_=_=`^pJkO£`cPkOr£wkmutwvDq^` £A« tpl`LpPmmotyPp_Ht_9q_4~J`Lp^pPm£A« t_=~PmL²
δm
δm = 2R
m
(
1 +M +N
2(2 −N)
)
ε20
2Rγσ0
2
4 +M −N(
V
2Rε0
)
2(2 −N)
4 +M −N »YH½
§-« rL¢y£qlmotyPpl`£w\so` `Lp=ª®ypmutyp'^`£Y« tpsor_=`pDm-~S`q^m£yPsok¿muso`elyPp^p^r`c`pso`_=~^£w pDm£w¡¢£`q^sl`
δlypp^r`#~s#»Y ) ½ ^pk£Y« rLvDqmutypÀ»YP½ kuytm
P
πa2 =γσ0
πC2
2Rγσ0
m
(
1 +M +N
2(2 −N)
)
ε20
N
2 −N(
δm
2R
)
M +N
2 −N
×(
δ
δm
)
M −N
2
1 −
(
δ
δm
)2+M −N
2
N
2 −N »YPP½
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( Q 2
§¨`8mu`_=~kcl`#ypDmm `Qknm ª®yPq^supt~skut_=~^£`8tpDmorsmotyPp3^`δ~s so~^~JyPsm
δkuytm
t = ε−10
m
(
1 +M +N
2(2−N)
)
ε20
2Rγσ0
1
2 −N(
2R
δm
)
M +N
2(2−N)
×∫ δ/δm
0
dx
(1 − x2+(M−N)/2)1/(2−N)
»YD½
`Lpq^mut£twkupDm£Y« rLvDqmutypÀ»YD½²J`mumu`#so`£wmotyPpku`kt_=~^£t¯`#l`#£w=kyPsmo`
t =δm
V×∫ δ/δm
0
dx
(1 − x2+(M−N)/2)1/(2−N)
»Y 2½
±yP_¡_=`#`LpÈr£wkmutwtmur²£`#mo`_=~k^`9yPpPmm ¥=£A« ~^~^soyl\^`#_lt_9q^_»δ/δm = 1
½ ~J`Lqlm£yPsok¿muso`#lrLlqtm`Lpª®yPpmotyPp3^`Lk ¢£`qsokl`#£wª®ypmutyp
Γ»®¢yPts» ¨¸ ²AG=G ) ½u½+
tm =δm
V× 2
4 +M −N×
Γ(2
4 +M −N)Γ(
1 −N
2 −N)
Γ(1 − M +N
(2 −N)(4 +M −N))
»Y¼D½
pkc£`ke« q^p0yPpPmm~^£PknmotvDq^`9~suªÇtm9»N = 0
`mM = 0
½²yPpÈylmot`LpPme~JyPq^sc£`#mu`L_=~k^`8ypDmoPm
t =2
πtm arcsin
(
πδ
2tmV
) »YGD½`m£`mo`_=~kl`y_=~^so`Lkoktyp0
tm =
(
πm
12C2σ02R
)12 »Y^LD½
¯p©« `Lkmut_=`s8£`¡yD`¬t`pDml`¡so`LkmutmuqlmotyPp6² ¹ ¸ ·^¸ ¸ »APP½c£wq£`LpPm£`mosoµ¢t£ltwkukut~Jr#~s £¡sorLkuq^£mopDmo`8^`8ypDmoPm^pk£`Lk~^qsu`L_¡`LpDm~£PknmotvDq^`
N = 0²^kyPtm
W =γσ0(2R)3
2 +M/2
(
δm2R
)2+M/2 »Y^µ½
§¨`_=yll]£`¡p^`~SyPkoku]Ll`~k« r£wkmutwtmur¡tpkmopDmopr`²S£`Lk8qlmo`q^ske~^suyP~JyDk`LpDmcmuyPqlm#l`_=¿L_¡`¥=~Jsumutsl`9`mrmmel`#~su`Lplso`#`pÈyP_¡~^mu`q^p`lrL\suP`8r£wkmutwvPq`8^`9É`Lsmoʦ§¨`#musµ¢t£6l`#£ª®yPso`8r£wkmutwvPq`so`Lkmutmuqr`Lp,rp^`LsuPt``Lkmc£ysk5
We =8p2
ma3
3E∗
»Y^QP½
ÍÍ ÒT'UWVXY
µ¼ W! !
xyqs^`e£w9~£Pknmottmore~sªÇtmo`²H£w9~^so`LkokutyPppm
`Lkm kq^~~JyDkrL`e¿muso`erLP£`e¥3σ0
`m £`8soµjPyp`Lp¯Jpl`cyP_=~^su`Qkukutyp=¥
a2 = 3Rδm¦D§6`musµ¢t£l`so`Lkmutmuq^mutypr£wkmutwvDq^` ~S`qlm£yskRk`_=`mmuso`ckuyqk
£wª®yso_=`We =
72√
3σ20(Rδm)
32
E∗
»Y^ ) ½`m£Y« rp`sot`ltwkokt~JrL`
W =9
2πσ0Rδ
2m
»Y^H½~Syqs
C2 = 32
¦§¨`yH`¬t`pDml`#so`LkmutmuqlmotyPp3^`¢Ht`LpDmc£ysk5
e2 =16
√3σ0
πE∗
(
R
δm
)12 »Y^QP½
kuytm¨`p#so`_=~^£w LpDmδm
~skyPp`l~^su`Qkukutyp8lyp^pr`O~s6£A« rQvPqJmutyp»AH½W^pJk6£w k6~^£PknmotvDq^`~JsuªÇtm
e2 =16
√3σ0
πE∗
(
9πσ0R3
mV 2
)
14 »Y^LD½
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P¥£A« tp^`pDmomutyp
δpkcq^p0Pkkok`LÊ8rprs£A¦c`9_¡¿L_=`²£A« rmuqJl`9l`£A« t_=~Pmc`Qknm
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∂z=
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σx =2νG
1 − 2ν
[
∂ux
∂x+∂uy
∂y+∂uz
∂z
]
+ 2G∂ux
∂x
»Y ) µ½
σy =2νG
1 − 2ν
[
∂ux
∂x+∂uy
∂y+∂uz
∂z
]
+ 2G∂uy
∂y
»Y ) P½
σz =2νG
1 − 2ν
[
∂ux
∂x+∂uy
∂y+∂uz
∂z
]
+ 2G∂uz
∂z
»Y )) ½
τxy = G
[
∂ux
∂y+∂uy
∂x
] »Y ) H½
τyz = G
[
∂uy
∂z+∂uz
∂y
] »Y ) P½
τzx = G
[
∂uz
∂x+∂ux
∂z
] »Y ) D½
^ 6 µÃA0YÅÆ0 * 0 K / * A0YN6Æ * 6*HL0Ç / /  ^Æ * pJk£`,k¡y ku`q^£`qp^`~su`QkukutyPppyso_=£``Lkmc~~^£twvDq^r`P²lyp,¡£`Qk ~Jymu`LpPmot`L£k kuq^t¢pDmok
F = F1 = G = G1 = 0»Y ) 2½
6« y
ψ1 =∂H1
∂z= H =
∫
S
p(ξ, η) ln(ρ+ z) dξdη,»Y ) ¼D½
ψ =∂H
∂z=∂ψ1
∂z=
∫
S
p(ξ, η)1
ρdξdη,
»Y ) GP½
ux = − 1
4πG
[
(1 − 2ν)∂ψ1
∂x+ z
∂ψ
∂x)
] »YPD½
uy = − 1
4πG
[
(1 − 2ν)∂ψ1
∂y+ z
∂ψ
∂y)
] »Yµ½
uz =1
4πG
[
2(1 − ν)ψ − z∂ψ
∂x)
] »YDP½
¾Op0ku`8so~^~S`£wpDmvPq`£`Lk~Symo`pDmut`£wkψ`mψ1
¢Prsot¯J`pDm £A« rLvDqmotyPp3^`§¨~£P`²ltA¦ `¦²
∇2ψ = 0, ∇2ψ1 = 0»Y ) ½
`mcvDq^`8£=lt¢`LsuP`p`e`QknmclyPp^p^rL`~Js
∆ ≡ ∂ux
∂x+∂uy
∂y+∂uz
∂z=
1− 2ν
2πG
∂ψ
∂z,
»YH½
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yPpyPlmut`pDm£`Lk ypDmustpDmu`Lk `Lp,\JvDq^`~JyPtpDmclqÈkuy£tl`8`Lpª®yPpmotyPp3^`Lk~Symu`LpDmut`£wk5
σx =1
2π
[
2ν∂ψ
∂z− z
∂2ψ
∂x2− (1 − 2ν)
∂2ψ1
∂x2
] »YDP½
σy =1
2π
[
2ν∂ψ
∂z− z
∂2ψ
∂y2− (1 − 2ν)
∂2ψ1
∂y2
] »YPD½
σz =1
2π
[
∂ψ
∂z− z
∂2ψ
∂z2
] »Y>2½
τxy = − 1
2π
[
(1 − 2ν)∂2ψ1
∂x∂y+ 2
∂2ψ
∂x∂y
] »YP¼D½
τyz = − 1
2πz∂2ψ
∂y∂z
»Y=GD½
τzx = − 1
2πz∂2ψ
∂x∂z
»YPD½xyqsmu`so_=tp`sQ²ltpDmusoyllq^twkyPpk£`Lk yPpltmutypk ql'£t_=tmu`Qk `p3ª®ys`#kuq^s £w¡kuq^suªÇ`
S
σz =1
2π
[
∂ψ
∂z
]
z=0
=
−p(ξ, η) kuq^sch,
0`Lp3l`L\^ysk
.
»YPlµ½
§¨`Lk lrL~^£w`L_=`pDmok ¥¡£w¡kuq^suªÇ`~S`q¢`pDm ¿mosu`#lyPp^p^rQk`p,r¢£qpPm £`LkrQvPqJmutypJk»ADP½Z¤»YDP½O`Lpz = 0
ux = −1− 2ν
4πG
[
∂ψ1
∂x
]
z=0
»YPP½
uy = −1− 2ν
4πG
[
∂ψ1
∂y
]
z=0
»YP ) ½
uz =1 − ν
2πG[ψ]z=0
»YPH½
§¨`Lk rQvPqJmutypJk»A^Q½`m9»AH½_¡yPpDmuso`pDmvPq`#£¡~^so`Lkoktypp^yso_£`#tpkt6vDq^`#£`#lr~£P`_=`LpPmpyso_=£p^`0lr~S`pJl`pDm'vDq^`Èl`
ψ¦ ¯p « ylmo`p^ts£`0^r~^£w`_=`pDmp^yso_£A²tY¦ `¦²-kuq^t¢pPm
z`p
ª®yPpmotyPp3^`#£¡~^so`Lkoktyp¨²lt£6kL« tmlyp8« r¢£q^`Lsψ`p
z = 0¦
" §-« `Lpku`_9£`Ol`QkrLvDqmutypkW~surQrL^`pDmu`QkWª®yq^sop^twkuku`pDmq^p^`kyP£qlmotyPpª®yso_¡`L££`OyPp`LsupJpDmS£`LkyPpPmosotpDmo`Lk`m£`Lklrª®yPsu_motyPpkR`Lpª®ypmutyp'^`LkO~su`QkukutyPpkRt_=~JyDkrL`LkL¦DhHtW`QkO~^so`LkoktypJk-kuypDmyPp^pHq^`LkO`l~^£tt¤mo`_=`pDmQ²yp¡~J`LqlmOyppwmuso`-moyqlmOq_=ytpkª®yPsu_=`££`_=`pDm£Y« `pku`_^£`^`LkspJl`q^sk_¡rQp^tvDq^`Qk¦¾-p~^somutwvDq^`²£`LkO^r¢`L£yP~^~S`_=`pDmok-p£jDmutwvDq^`Lkso`Lkmu`LpDm-lt¬t£`LkL¦e` p^y_^su`Lql¡^r¢`L£yP~^~S`_=`pDmokyPpDmprLp_¡yPtpJkrmur#sorL£twkurLk^pJk l`Lk Lk~smotq^£t`sklypDmp^yPqkp^`8ªÇtkuypk ~PkrmmtwtA¦
ÍÍ ÒT'UWVXY
LP W! !
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´ep'kuq^~^~SyPku`c^pk `mumu`e~sumut`cvDq^`c£`LkyPpPmosotpDmo`LkOkuypDm~suyllq^tmu`QkO~s-q^p`ª®ys`eypJ`pDmosurL`¥e£A« yPsuttp^`¦§^tkmop` q~JyPtpDm-l` \suP`_=`pDm~J`Lqlm-£yskkL« `l~^sot_=`sRl`£w_=p^t]so` kuq^t¢µpDmu`
ρ = (x2 + y2 + z2)12
»YPP½`m£`#\suP`_=`pDm~s
P =
∫
S
p(ξ, η) dξdη»AP½
§¨`Lk~Symu`LpDmut`£wkl`| yqkokutp^`Qkuv=`m±`sosuq^mut6ku`sorL^q^tku`pDm£ysk¥
ψ1 =∂H1
∂z= H = P ln(ρ+ z)
»A 2½
ψ =∂H
∂z=P
ρ
»YP¼D½
¾OpÈq^mut£twkupDm£`LkrQvDqmotyPpk#»YP¤nH½²PyPpyPlmut`pDm£`Lk lrL~^£P`L_¡`LpDmok ^pk£`^`_=t¤Z`Lku~P`
ux = − P
4πG
[
xz
ρ3 − (1 − 2ν)x
ρ(ρ+ z))
] »YP GD½
uy = − P
4πG
[
yz
ρ3 − (1 − 2ν)y
ρ(ρ+ z)
] »YD½
uz =P
4πG
[
z2
ρ3 +2(1− ν)
ρ
] »Y^µ½
§¨`Lk yPpDmustpPmo`LkkuypDmlyp^pr`Lk~Js £`Lk rQvPqJmutypJk»AHµ¤nD½²HkyPtm
σx =P
2π
[
1 − 2ν
r2
[(
1 − z
ρ
)
x2 − y2
r2+zy2
ρ3
]
− 3zx2
ρ5
] »YPP½
σy =P
2π
[
1 − 2ν
r2
[(
1 − z
ρ
)
y2 − x2
r2+zx2
ρ3
]
− 3zy2
ρ5
] »Y ) ½
σz = −3P
2π
z3
ρ5
»YH½
τxy =P
2π
[
1 − 2ν
r2
[(
1 − z
ρ
)
xy
r2+xyz
ρ3
]
− 3xyz
ρ5
] »YPP½
τxz = −3P
2π
xz2
ρ5
»YD½
τyz = −3P
2π
yz2
ρ5
»Y 2½
y r2 = x2 + y2 ¦
Ë®ÌÍ6ËÏÎ
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pk£`'Pk9« q^p^`3\so`yp`pDmusor`P²6t£ `Lkm£ts¡vPq`£`~^soy^£]_=``QknmltkujH_¡rmusotvDq^`P¦°£~S`q^mclyp¿muso`µ¢µpDmoP`qku`_=`LpPmª®yso_9q^£r8`pÈyHyPsolyPp^p^rL`Lk jH£tplsotwvPq`LkL²^kyPtm
σr =P
2π
[
(1 − 2ν)
(
1
r2− z
ρr2
)
− 3zr2
ρ5
] »Y¼D½
σθ = − P
2π(1 − 2ν)
(
1
r2− z
ρr2− z
ρ3
) »YGD½
σz = −3P
2π
z3
ρ5
»Y 2D½
τrz = −3P
2π
rz2
ρ5
»Y 2Hµ½
´ep3y^mut`pDmcl`8£=_=¿L_¡`8_pt]Lsu`8£`LklrL~^£w`L_=`pDmok`Lp3yDyPso^yp^p^rL`LkjH£tpJlsutwvDq^`Lk
ur =P
4πG
[
rz
ρ3 − (1 − 2ν)ρ− z
ρr
] »Y 2P½
uz =P
4πG
[
z2
ρ3 +2(1 − ν)
ρ
]
,»Y 2 ) ½
`#vPqt¨lypp^`#¥£wkuq^sªÇP`8lqÈl`_=t¤A`Qk~J`
ur =1 − 2ν
4πG
P
r
»Y 2µH½
uz =1 − ν
2πG
P
r
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`L££`Lk kuypDm ¥¡£ku`#l`8£¡sorLkuy£qlmutyplq,~^suyP^£]_=`e~Syqs q^p^`#\so`ltwknmosut^q^rL`²l`p,yPpkutlrLsopDm`mumu`\Jso`ltwknmosut^q^rL`y_=_=`q^p^`¡tpDmors£`kuq^sc£w'kuq^suªÇ`l`\suP`_=`pDmcl`9ª®ys`rL£rL_=`pl¤mtsu`3yPp`LpPmosurL`¦1~Jsumutsl`QkrQvDqmotyPpk»A=2D½ »Y 2½²6t£^`¢Ht`LpDm¡~SyPkokt^£`l`,yPpkutlrLsu`Ls9£`Lk^r~^£w`_=`pDmok« q^p~JyPtpDmyqsopPm
B(x, y)l`e£kuq^suªÇ` `m£`LkyPpPmosotpDmo`LkR`Lpq^p~SytpPm yqsopPm
A(x, y, z)^q_=Pkukutªn²~JyPq^s-l`QkR~^so`LkoktypJkRl`yPpPmmOsurL~sumut`Lk
p(x, y)¦xyqs-`L£²yp=`l~^sot_=` £w
~su`QkukutyPp`p=yHyPsolyPp^p^rL`Lk¨jH£tplsotvDq^`Qkp(s, φ)
µ¢`QO~Syq^syPsuttp^` £` ~JyPtpDmBl`-mo`££` ªÇ yp¡vDq^`£w
~su`QkukutyPpvDq^ttmOkq^s-q^p=r£r_=`pDmdsdφ
l`kuq^suªÇ` `LpCkuytmrQvDq^t¢£`LpPmo`¥q^p^`ª®ys` yp`pDmusor`
ps dsdφ¦^§6`8lr~£P`_=`LpPm lq~JyPtpDm
B~S`qlm ¿muso`erQsotm ¥9~Jsumuts l`Qk rLvDqmotyPpk»Y=2D½ W»Y 2P½R^pJk
£`LkovDq^`££`Lkr = BC = s
¦J§¨`lr~^£w`_=`pDmc`LpÈ| lq`9¥¡muyPqlmu`£w=~^so`Lkoktyp,sor~smot`#kuq^sc£wkuq^sªÇP`h`QknmlyPp
uz =1 − ν2
πE
∫
S
p(s, φ) dsdφ»A=2P½
§¨`Lk yPpDmustpPmo`Lk `pAkuypDm `pkuq^tmu`lrQlq^tmu`Qkl`8£Y« tpPmorPsomutyp,l`LkrQvPqJmutypJk#»YD½ W»Y 2½¦
YLVàÒeÛ ÖµÖÓnÕ×Õ×Ó8-µÓRÕ×Ó8 Ü.µÕ×Ó µÓ8(#×Ø #×Õ×Õ×Ó ÓnÒÞ G ÓnØYÞ¨ÛÇÓÕ%#%& Ü.µÕ×Ó 5 JܵÒ3 E Ö Û G = E/(2(1 + ν))
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p(s, φ)²`m£`Lk kuq^suªÇ`Lkl`\suP`_=`pDmL²
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p = p0
(
1 − r2
a2
)n »Y 22½etWrso`pDmkPk~sumutwq^£t`sk ku`#ltwknmotp^Pq^`pDmc^pk£`\^ytl`8£A« `H~SyPkopDm
n
• n = 0²^tkmusotqlmutyp,l`#~^so`Lkoktypq^p^tª®yso_=`²
• n = −1/2²Oltkmusot^qlmutyp l`È~su`QkukutyPp vDq^teysoso`Lku~JyPp ¥Àq^p`0lr~^£w`_=`pDmq^ptª®yPsu_=`0l`È£w
kqsªÇP`8`pÈyPpDmomL²• n = 1/2
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²yPp0so`pH¢yPt`9¥ »nG¼PP½ `m8¥ ·»nAGD GD½¦W§6`=k
n = −1/2vDq^t~J`Lqlm#¿muso`9so`pJypDmosur
pk£`Lk~^soy£]L_¡`Qk « tp^`pDmomutyp3~s q^p,~Sytp yPp~£m`Lkm mustmur8~s 6· »nGPD½¦
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pkq^s¡q^pr£r_=`pDml`3kuq^suªÇ`
s dsdφ
uz =1 − ν2
πE
∫
S
p(s, φ) dsdφ»A=2¼P½
xyqsyPlmu`Lp^ts3£`"lrL~^£w`L_=`pDm`Lp q^p ~SytpDmB(r, 0)
l`©£kqsªÇP`©tsq£tso`²t£#yPpH¢Dt`pDm6« tpDmurPsu`Ls#£0~^su`QkukutypÀkq^s9muyPqlmu`£wÈkuq^sªÇP`l`\so`L_¡`LpDmL¦±`mumu`tpDmurLsmotyPp©`Qknm9t££qkmusor`¥£w¯qsu` G¦% ¯pl`=sorL£twk`Ls#`£q^£Y²¨yp"tpDmusoyllq^tm#q^pÀ~s_=rmoso`¡lrQvPqJml`£w3kuq^suªÇ`rL£rL_=`pDmotso`ktmuqr``Lp
C(s, φ)¦W§6`soµjPyp
OC²Sl`p^yso_=`
t`Lkmcso`£tr9qlÈqlmosu`Qk ¢sot^£`Qkl`£w
_p^t]Lsu`#kuq^t¢µpDmu`t2 = r2 + s2 + 2rs cos(φ)
»A=2 GP½§¨`#~^su`Qkukutypq~SytpDm
C~S`q^mc£ysk¿muso`#`l~^sot_=rL`8^`£w¡_p^t]so`#kq^t¢pDmu`
p(s, φ) =p0
a(α2 − 2βs− s2)
12
»Y¼D½y
α2 = a2 − r2, β = r cos(φ).»Y¼Q½
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a
C(s, φ)
B(r, 0)
t s
rO
s1
s2
θ φ
φ1
−φ1
λ
a
C(s, φ)
B(r, 0)
t
s
r O
s1
s2
θ
φφ1
−φ1
λ
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uz =1 − ν2
πE
p0
a
∫ 2π
0
dφ
∫ s1
0
(α2 − 2βs− s2)12 ds
»Y¼P½y
s1`Qknm £w=soPtp^`e~SyPkutmut¢`#l`
α2 − 2βs− s2 = 0»Y¼ ) ½
§-« tpDmurPso£``ps~J`Lqlmck« `l~^sot_¡`Ls l`#£¡_pt]Lsu`#kuq^t¢pPmo`
∫ s1
0
(α2 − 2βs− s2)12 ds =
1
2αβ +
1
2(α2 + β2)
[
π
2− tan−1(
β
α)
] »Y¼H½
§¨`Lk mo`so_¡`Qkβα
`mtan−1(
β
α)k« p^pHq^£wpDm £ysk ^`#£Y« tpDmorsmotyPpkuq^s
φ²^t£6su`Qknmo`
uz(r) =1 − ν2
πE
p0
a
∫ 2π
0
π
4(a2 − r2 + r2 cos2 φ)
12 dφ
=1 − ν2
πE
πp0
4a(2a2 − r2), r ≤ a
»Y¼D½¾Op3`8vDq^tSyp`sop^`c£`e^r~^£w`_=`pDm-mp^P`pDmL²DypkutmvPq`e£w9kujH_¡rmusot`l`csor¢Py£qlmutypt_=~JyDk`
vDq^`¡£`¡lrL~^£w`L_=`pDmemp^P`pDm#kyPtmsoPlt£Y¦S¾Op"qlmut£tkopDm8£Y« rLvDqmutyp»A=2D½²S£`=^r~^£w`_=`pDme`LpB^q^`¥¡q^p^`#~^so`Lkoktyp
pkq^s £wkuq^sªÇP`r£r_=`pDmtsu`
dsdφ`p
C¢µqlm
(1 − 2ν)(1 + ν)
2πEp dsdφ
»Y¼PP½kuytm `p,p^`#yPpku`so¢µpDmvDq^`#£w¡~sumut`8soPlt£``m`Lp3mu`pJpDmy_=~lmo`lqÈktp`
− (1 − 2ν)(1 + ν)
2πEp cosφ dsdφ
»A¼>2½¾Op0~^~^£tvDqpDm_tpDmu`LppDm £w~su`QkukutyPpmoymo£`P²lyp,y^mut`pDm
ur(r) =(1 − 2ν)(1 + ν)
2πE
p0
a
∫ 2π
0
cosφ dφ
∫ s1
0
(α2 − 2βs− s2)12 ds
»Y¼¼P½¾OpÈtpDmurLspDm^`#£¡_=¿_=`#_p^t]so`8vPq`#~^surQrQl`_=_=`pDm yp3ylmot`LpDm
ur(r) = − (1 − 2ν)(1 + ν)
3E
a2
rp0
[
1 − (1 − r2
a2 )32
]
, r ≤ a»Y¼GP½
hHt£A« yPp©ypkut^]so`9q^p©~JyPtpDmB¥'£Y« `HmurLsut`qsl`¡£w'kuq^sªÇP`¡\sorL`²yPp yPlmut`pDm^`£w'_=¿L_¡`
_p^t]Lsu`
uz(r) =1 − ν2
E
p0
2a
[
(2a2 − r2) sin−1(a
r) + r2
a
r(1 − a2
r2)
12
]
, r ≥ a»YGP½
ur(r) = − (1 − 2ν)(1 + ν)
3E
a2
rp0, r ≥ a
»YG^Q½
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( Q^
 / ÃD0 / Ã!*H0 / 'Æ * 0*HJ0®Ä.*HA6^Å * §6`LkeyPpPmosotpDmo`Lk¥£wkuq^sªÇP`~S`q¢`pDmªÇPt£`L_=`pDmc¿mosu`^rLlq^tmu`Qk `p,£wq£pPm £`Qklrª®yPsu_motyPpk ¥9~smots l`Qklr~^£w`_=`pDmk ¥9£¡kuq^sªÇP`8lypp^rLk ~s£`LkrQvDqmotyPpk8»Y¼D½A¤»Y¼GD½-`m9»YGP½Z¤»A G^µ½¦J´ep,ylmot`LpDm~Syqs q^p,~JyPtpDmcl`#£w=kq^suªÇ`#\sorL`
σr
p0=
1 − 2ν
3
a2
r2
(
1 −(
1 − r2
a2
)3/2)
− (1 − r2
a2 )1/2 »YGPP½
σθ
p0= −1− 2ν
3
a2
r2
(
1 −(
1 − r2
a2
)3/2)
− 2ν(1 − r2
a2 )1/2 »YG ) ½
σz
p0= −
(
1 − r2
a2
)1/2 »YGH½
`m~JyPq^s q^p,~SytpPm¥=£A« `Hmursot`q^s l`#£w=kq^suªÇ`#\sorL`
σr
p0=σθ
p0= (1 − 2ν)
a2
3r2»YGPP½
§¡ypDmustpDmu`esoPlt£`e`Lkmlypeq^p^`#ypDmustpDmu`8l`musmotyPp`p,l`L\^ysklq3`s£`vDq^t6mmo`tpPmko¡¢£`q^s _lt_9q^_1q3soµjPyp
r = a¦
§¨`LkOypDmustpDmu`LkRpkmoyqlm-£`l`L_¡t¤Z`Lku~` ~S`q¢`pDmO¿muso`musoyq^¢Pr`Lk`Lp=tpDmurLspDmR£`QkRrLvDqmutypk»AP¼P½A¤»Y 2HQ½¡l`È£w"_=¿_=`È_p^t]so`ÈvPq`È~JyPq^s£`LklrL~^£P`L_¡`LpDmokL¦-cyPp^p^yPpkkt_=~^£`_=`pDmtwt£`LkypDmustpDmu`Qk£`8£yPp^l`#£A« H`
z²^vDq^tp^yqJkk`LsuyPpDm qlmot£`Lk~JyPq^s r¢£q^`Ls £`Qk sotmo]so`Lk l`8~^£wkmutwtmur
σr
p0=
σθ
p0= −(1 + ν)
(
1 − z
atan−1(
a
z))
+1
2
(
1 +z2
a2
)−1 »YGD½
σz
p0= −
(
1 +z2
a2
)−1 »YG 2½
- ,#$$ , # .. &% 5#- %, %/ # $% , )+# #$ . % #¾Op ypkut^rspDm¡vDq^`,£`,`s£`,`Lkm=qp Lk¡~sumutwq£t`s« `££t~ku`²RyPp~S`qlm'ypkutwlrso`s=l`,£
_=¿L_¡`#_p^t]Lsu`q^p^`8ª®yPsu_=`8rLp^rsotwvPq`~Syqs £¡~^so`Lkoktyp3kyPqk£w9ª®yPsu_=`#kuq^t¢pPmo`
p(x, y) = p0
(
1 −(x
a
)2
−(y
b
)2)n »YG¼D½
~^~^£tvDq^rL`#kq^s qp^`kqsªÇP``££t~lmutwvDq^`l`#£wª®yso_¡`#kuq^t¢µpDmu`(x
a
)2
+(y
b
)2
− 1 = 0»Y G=GP½
§=_¡¿L_=`lr_s\^``Lkmc~^~£twvDq^r`vDq^`8~^sorLrLl`L_¡_=`LpPm~Syqs £`#~Symo`pDmut`£¨^rLlq^tmel`#£A« rLvDq¤motyPp"»A ) GP½
ψ(x, y) =
∫
S
p0
(
1 −(x
a
)2
−(y
b
)2)n
ρ−1 dξdη»ÇPP½
ÍÍ ÒT'UWVXY
µ W! !
µ¢P`Lρ = [(ξ − x)2 + (η − y)2 + z2]
12
»ÇP^µ½§¨`lr~^£w`_=`pDm p^yPsu_£6¥¡£w=kq^suªÇ`#\sorL`c`LkmclyPp^p^r8~s £A« rLvDqmotyPp"»AD½ kyPtm
uz =1 − ν
2πG[ψ]z=0
»ÇPPP½
§¡kuy£qlmotyPpPrp^rLso£`8l`#£A« tpDmurPso£`=»ÇPD½-~S`q^m¿muso`#rLsutmkyPqk£w9ª®yPsu_=`#kuq^t¢pPmo`
ψ(x, y, z) =Γ(n+ 1)Γ(
1
2)
Γ(n+3
2)
p0ab
∫
∞
λ1
(
1 − x2
a2 + w− y2
b2 + w− z2
w
)n+ 12 dw(
(a2 + w)(b2 + w)w)
12
»ÇP ) ½y
ΓlrLkutPp^` q^p^`ª®yPpmotyPp=P_=_`m
λ1`QknmO£w8soPtp^`~JyDktmut¢`^` £Y« rLvDqmutypq^tvDq^`kuq^t¢µpDmu`
1 − x2
a2 + λ− y2
b2 + λ− z2
λ= 0
»ÇPD½
±`8~so_¡]muso`λ1~J`Lqlm ¿muso`tpDmu`Lsu~surmurryP_=rmusotwvPq`_=`pDm y_=_=`e£`~s_=]mosu`« q^p^`8`££t~l¤
kuy Zl` l` _=¿_=`ª®yl£` vDq^` £w#kqsªÇP``££t~lmotvDq^` `m-vPqt^~koku` ~JsR£` ~JyPtpDm(x, y, z)
¦xyPq^sRmusoyq¢`s£w¡¢£`q^sl`
Ψkqs £=kuq^sªÇP`P²l£Y« tpDmors£`8lytmc£ysk ¿muso`8`l~^sot_=rL`~Syqs
λ1 = 0¦
?, $0 K / 0=6Æ Å*H * / á / 0ZÂ/ * n = −1
2
§¨`#~Jymu`pDmot`L£Ψ¢q^m~JyPq^s
n = −1
2
ψ(x, y, z) = πp0ab
∫
∞
λ1
dw(
(a2 + w)(b2 + w)w)
12
»ÇPPP½
kuytmkqs £=kuq^sªÇP`8`pz = 0
ψ(x, y, 0) = πp0ab
∫
∞
0
dw(
(a2 + w)(b2 + w)w)
12
»ÇPD½
§-« tpDmurPso£`3vDq^tlytm¿mosu`3`Lkmut_=r`,`Lkmtp^r~S`p^pDmu`Èl`x`my¦-hl©¢£`q^s=`Qknmku`q^£`_=`pDm
ª®yPpmotyPp l`È£Y« `^`pDmusottmor0l`0£A« `L££t~k`e = (1 − (b/a)2)
12¦-¾Op `S`QmoqpDm£` \p`_=`LpPml`
¢sutw^£`kuq^t¢pPmw = b2.t2
²dw/w
12 = 2bdt
²£Y« tpPmorPso£`~S`q^m¿mosu`surL`l~^sot_=rL`l`'£w,ªÇ ypkuq^t¢pDmu`
∫
∞
0
dw(
(a2 + w)(b2 + w)w)
12
=2
a
∫
∞
0
dw(
(1 + k2t2)(1 + t2))
12
»ÇP 2½
Ë®ÌÍ6ËÏÎ
!"#$&%'! ( Q )
µ¢P`Lk = b/a
£`¡so~^~JyPsml`Lkl`_=t¤Zl`LkL¦J¾Op©`W`LmuqpPmq^p pyq^¢P`Lq0\Jp^P`_=`pDmel`¡¢sot^£`²t = tanφ
kuytmdt = dφ/ cos2 φ
²P£w9ku`Lyp^`tpDmors£`~S`qlm kL« `H~sut_=`s-`Lpª®ypJmutypl`c£A« tpDmurLs£``L££t~lmutwvDq^`yP_¡~£]mu`#lq,~^so`_=t`Ls yslso`∫
∞
0
dw(
(1 + k2t2)(1 + t2))
12
=
∫ π/2
0
dφ√
1 − (1 − k2) sin2 φ=
∫ π/2
0
dφ√
1 − e2 sin2 φ
= K(e)
»ÇP¼D½§¨`lr~^£w`_=`pDm rLp^rsor8¥=£=kuq^sªÇP`8`LkmlypJ^rLlq^tmcPsP`8¥£A« rLvDqmotyPp"»YD½
uz =1 − ν2
E2p0bK(e)
»ÇPGP½
? 3 $0*ÈÆ 6*DQ0Ç / 0* B*,µÃAC n =
1
2 6Æ ^Å*H * / à I Æ ©Q ^Å * 0 0*D 0ÏÄ.*HA6Å* epk`k£`#~Symu`LpDmut`£¨kL« `l~^sut_=`#kyPqk£ª®yPsu_=`#kuq^t¢pPmo`
ψ(x, y, z) =1
2πp0ab
∫
∞
λ1
(
1 − x2
a2 + w− y2
b2 + w− z2
w
)
dw(
(a2 + w)(b2 + w)w)
12
»ÇLD½
`m~JyPq^s q^p,~SytpPml`#£w=kqsªÇP`
ψ(x, y, 0) =1
2πp0ab
∫
∞
0
(
1 − x2
a2 + w− y2
b2 + w
)
dw(
(a2 + w)(b2 + w)w)
12
»Çµ½
§¨`lr~^£w`_=`pDm¥¡£=kuq^suªÇ`8~S`qlmc^yp8¿muso`rQsotmkuyqk£wª®yso_=`8kuq^t¢pDmu`
uz =1 − ν2
πE(L−Mx2 −Ny2)
»ÇQP½µ¢P`L
M =πp0ab
2
∫
∞
0
dw(
(a2 + w)3(b2 + w)w)
12
=πp0b
e2a2 (K(e) − E(e))»Y^ ) ½
N =πp0ab
2
∫
∞
0
dw(
(a2 + w)(b2 + w)3w)
12
=πp0b
e2a2
(
a2
b2E(e) − K(e)
) »ÇH½
L =πp0ab
2
∫
∞
0
dw(
(a2 + w)(b2 + w)w)
12
= πp0bK(e)»ÇQ½
§-« tpDmurPso£`E(e)
`Lkm £A« tpDmurLs£`c`L££t~^mutwvPq`y_=~^£]mu`8lq,k`QyPpyslso`elr¯p^t`^`e£w9_p^t]Lsu`kuq^t¢pDmu`
E(e) =
∫ π/2
0
√
1 − e2 sin2 φ dφ»ÇLP½
ÍÍ ÒT'UWVXY
µ W! !
 / ÃD0 / Ã!*H0 / #Æ * 0*D 0ÏÄ.*HA6Å* [suyPq^¢`LsR£`Qk-yPpDmustpPmo`LkR^pkO£`l`L_¡t¤Z`Lku~` p6« `Qknm ~kqp^` \yPku`RªÇt£`¦µ[¨yPqlmR« JyPso6²Lt£DªÇqlm¿muso` `p_=`Lkuq^so`l`lrmu`so_=tp`s
λ1vDq^tH`Lkmq^p^` stp^`-6« q^p
~Sy£jHp _=` « yslso`3¦c`c~^£qJk²~SyqsO`sumotp`LkRyP_=~JyDkupPmo`Lkl` £w#yPpPmosotpDmo`²µt£^ªÇqlm-lrmu`Lsu_=tp^`s
£`8~Jymu`LpPmot`L£Ψ1 =
∫
∞
zΨ dz
¦±`Qk lt¬q^£morLk kuypDm _=ytplso`Lkkt£A« yp,\^`s\^`£`Qk ypDmosotpDmu`Qkkuq^s £A« l`
z¦l¾Op3`W`mL²
λ1 = z2`me£A« tpDmurLsmutyp0^`Ψ~scs~^~Sysume¥
z`Lkme~^£qkkt_=~^£`P¦Wh^pklypp^`se£`Qkelrmot£wkeL£wq^£wmoytsu`Qk²
yPpyPlmut`pDmckuq^s £A« H`z£`Lk ypDmustpDmu`Lkkuq^t¢µpDmu`Lk
σx
p0=
2b
e2a(Ωx + νΩ′
x)»Ç 2½
σy
p0=
2b
e2a(Ωy + νΩ′
y)»ÇL¼D½
σz
p0= − b
e2a
1 − T 2
T
»ÇAGD½µ¢P`L
Ωx =1
2(1 − T ) + ζ(F (φ, e) − E(φ, e))
»ÇDD½
Ω′
x = 1 − a2T
b2+ ζ(
a2
b2E(φ, e) − F (φ, e))
»YP^Q½
Ωy =1
2+
1
2T− Ta2
b2+ ζ(
a2
b2E(φ, e) − F (φ, e))
»YPP½
Ω′
y = −1 + T + ζ(F (φ, e) − E(φ, e))»ÇD ) ½
T =
(
b2 + z2
a2 + z2
)
12
, ζ =z
a= cotφ
»ÇPD½
§¨`Lk¢£`Lq^sok9l`Lk9tpDmurLs£`Lk8`L££t~^mutwvPq`LkF (φ, e)
`mE(φ, e)
kyPpPm9moq^£rL`Lk#`m`Lp\^ytwktwkokupPm`sumotp^`Qk#l`=£`Lq^sk#¢£`q^sk8~sumutwq£t]so`LkL²Wt£O`Qknm~SyPkokut^£`l`lyPp^p^`s^`Lk#`H~su`QkukutyPpk`l~^£ttmo`Lk^`LkyPpPmosotpDmo`LkO~Syqs^`Lk ~JyPtpDmk~surQtwk¦´ep's~~J`L££`etwtW£w^r¯p^tmutyp3l`Lk tpDmurPso£`Qk-`L££t~lmutwvDq^`Lk^`~su`L_¡t`s `mcl`ku`LyPpyslsu`
F (φ, e) =
∫ φ
0
dφ√
1 − e2 sin2 φ
»ÇDP½
E(φ, e) =
∫ φ
0
√
1 − e2 sin2 φ dφ»ÇDP½
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(x, y)`m8£'ltso`Lmutyp
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^tsotPr`O^pk6£`^`_=t¤Z`Lku~P`Okuy£tl`P¦µ´epypkut^rso`s vDq^`-£Y« l`O_=yj`Lpl`O£w pl`-l` \so`L_=`pDm`Qknm yPsut`pDmur#kuq^t¢µpDm
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x
z
b a
q(x)
p(x)
O
σx
σz
τxz
σθ
σr
τrθ
θ
r
) '$ >WR &% !
bcp^`#yq~J`8lq^`_=t¤Z`Lku~P`c`Qknmt££qkmusor`8¥£w¯Pq^so` ) ¦D§¨`\Jso`_=`LpPm `Lkmypkmutmuqr« q^p^`~su`QkukutyPp8surL~sumut`p^yPsu_£`
p(x)`m« q^p^`~^so`Lkoktyp8mp^P`pDmu`
q(x)¦µ§6`~^soy^£]_=`-ypkutkmu`O¥musoyq^¢P`s
£`Lk¡lrL~^£P`L_¡`LpDmok(ux, uz)
²`p~sumutwq^£t`s¡`m=£`Qk¡ypDmustpDmu`Qkσx, σz , τxz
pk£`,l`L_¡t¤Z`Lku~`rL£PknmotvDq^`P¦bcp^`©mmu`LpDmutyp ~smotq^£t]Lsu`0ku`sLyPsolrL`0q lrL~^£w`L_=`pDmp^yPsu_£el`0£w kuq^suªÇ`\Jsor`P²
uz¦
acyqk=p^`0^yp^p^`LsuyPpk¡ttvDq^`,£`Qk~^sotpt~J£`Lk¡rm~S`Lkl`È£w"surQkyP£qlmotyPpl`È`3~suyP^£]L_=`¦O§6`£`Lmu`Lq^stpDmurLsu`Qkukur~s~^£qk l`#lrmot£wk~JyPq^sus¡k`su`L~JyPsmo`s ¥¡l`LkyPq^¢Hso`Lk £wkokutvDq^`Qk« r£wkmutwtmury_=_=`8~s `H`L_=~^£`» 6· $Ï·¸²AGP^ ·W²AGD R²SGP¼P½¦
ÍÍ ÒT'UWVXY
µ W! !
% 1 '% ., # 5, ¾-sut¢yPpkmoyqlm« Sys8£`Qk6rLvDqmutypk« rLvDq^t£t^so`kuq^s¨£`LkypDmosotpDmu`Qk
(σx, σz , τxz)vDq^tHlyt¢`LpDm
¿muso`8¢rsot¯r`Qk`Lp3muyq^m~JyPtpDmlqÈl`_=t¤A`Qk~J`
∂σx
∂x+∂τxz
∂z= 0
»ÇD=2½∂σz
∂z+∂τxz
∂x= 0
»ÇD¼P½
§¨`Lk¨^rª®yso_motyPpk6yPsuso`Lku~Syp^pDmu`LkL²(εx, εz, γxz)
lyt¢`LpPm6¢Prsot¯J`s£A« rQvPqJmutyp^`-yP_¡~Jmut^t£tmurkuq^t¢pDmu`
∂2εx∂z2
+∂2εz∂x2
=∂2γxz
∂x∂z
»ÇD GD½y £`Lk ^rª®yso_motyPpk lypDm so`£trL`Lk q,lrL~^£P`L_¡`LpDm ^`#£¡_p^t]so`#kqt¢pDmo`
εx =∂ux
∂x, εz =
∂uz
∂z, γxz =
∂ux
∂z+∂uz
∂x
»Ç ) D½hHyPqk£A« \HjH~Jymu\^]Qk`^`Lk lrª®yso_=mutypk ~^£wp`LkL²
εy = 0»Ç ) Q½
σy = ν(σx + σz)»Ç ) P½
£`#yP_¡~Sysumu`L_=`pDm r£wkmutwvPq`£tp^rLtso`#lqÈl`_=t¤A~^£wp3~S`qlmekL« rLsotsu`
εx =1
E
[
(1 − ν2)σx − ν(1 + ν)σz
] »Ç )) ½
εz =1
E
[
(1 − ν2)σz − ν(1 + ν)σx
] »Ç ) H½
γxz =1
Gτxz =
2(1 + ν)
Eτxz
»Ç ) P½
er¯ptkokyPpk_tpDmu`pJpDm q^p^`8ª®ypJmutypφ(x, z)
mu`££`vPq`
σx =∂2φ
∂z2, σz =
∂φ
∂x, τxz =
∂2φ
∂x∂z
»Ç ) D½´ep3~S`qlmc£yPsok so`_svDq^`s vDq^`#kt
φ`Lkmq^p^`ª®yPpmotyPp3t\su_=yPp^tvDq^`P²^« `Lkm¥¡^tso`
[
∂
∂x+
∂
∂z
] [
∂φ
∂x+∂φ
∂z
]
= 0»Ç ) 2½
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e`~^£qk²t£OyPpH¢Dt`pDm#vDq^`=£`Qk#ypJltmotyPpk8ql©£t_=tmo`Lk8`pÀyPpDmustpPmo`Lk8kuyt`pDm¢Prsot¯Jr`Lk8kuq^s8£kuq^suªÇ`e£t_=tmu`P²Jkuytm
σz = τxz = 0, x < −b, x > a»Y ) ¼P½
σz = −p(x) »Ç ) GP½τxz = −q(x) »ÇDP½
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(x, z) → 0 =⇒ (σx, σz , τxz) → 0¦
¯pl`esu`ª®yso_9q^£`s-£`e~^suyP^£]_=``pyHyslypp^r`QkOjH£tp^sutwvDq^`LkL²yPp'yPpktwl]Lsu` _tpDmu`LppDmqpª®yPpmotyPpt\su_=yPp^tvDq^`
φ(r, θ)
[
∂2
∂r2+
1
r
∂
∂r+
1
r
2 ∂2
∂θ2
][
∂2φ
∂r2+
1
r
∂φ
∂r+
1
r
2 ∂2φ
∂θ2
]
= 0»ÇQ½
y
σr =1
r
∂φ
∂r+
1
r
2 ∂2φ
∂θ22
»ÇDP½
σθ =∂2φ
∂r2»Ç ) ½
τrθ = − ∂
∂r
(
1
r
∂φ
∂θ
) »ÇH½
§¨`Lk lrª®yso_mutypJk kuypDm so`£trL`Lk q^lr~£P`_=`LpPmk
εr =∂ur
∂r
»ÇDP½
εθ =ur
r+
1
r
∂uθ
∂θ
»ÇPD½
γrθ =1
r
∂ur
∂θ+∂uθ
∂r− uθ
r
»Ç>2½
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P~sq^p^tmur^`-£yp^Pq^`q^s
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^`_=t¤Z`Lku~P`=r£wkmutwvDq^`¦±`'~^suyP^£]_=``Lkm9kyPq^¢`LpDm9lrLkutPp^ryP_=_¡`£`~^soy£]L_¡`^` ¹»L¼=GP½¦
§3kuy£qlmotyPp©l``=~^soy£]L_¡`¡`Lkmlypp^r`=`pÀ\^yPtkutkokopDmy_=_=`ª®yPpmotyPp ^t\su_=yptvDq^`¡£ª®yPsu_=`#kuq^t¢pPmo`
φ(r, θ) = Arθ sin θ»YP¼D½
Y UËÏÕ7(ÜÒ .#×ÓÒÞ µÓ µÜÒµÒQÓÛ¨Ó2QÖQÕ%#%()# ÞYÓ ÓnÒÞ¨Õ×ÓnØ(ÜÒ# Þ'#×ÜÒQØ #×Ò# Þ$# Õ×ÓnØØ'µÛ6Õ×Ó ÜÛ1ØYÜ# Þ6ÓÒcÖµÛÇÓnØAØ'#×ÜÒeØAÜ# Þ6ÓnÒ &ÖQÕ (Ó°Ô ÓÒÞÙµØAÜ# Þ6ØYÜµØ ÜÛ1 Ó ,# 2ÞAÓÚÕ%µØ'#×ÓµÛÇØ(ZеÜ# 2cØAÖ &(# -µÓnØØAÓnÛÇÜÒÞ¨ÞYÛ # Þ$&nØ ÒµØ¨Õ×ÓnØ6Ö Û 3Û ÖµÐµÓnØRØ$# ÒÞAØ
ÍÍ ÒT'UWVXY
µ¼ W! !
§¨`Lk yPpDmustpPmo`LkkL« rQsot¢P`pDmlyp
σr = 2Acos θ
r
»Ç=GD½
σθ = τrθ = 0»ÇDP½
´ep"~S`qlmso`_svDq^`s8vDq^`=£,ypDmustpDmu`¡soPlt£`=lrQsoy wm8`Lp1/r
¥3£A« tpl¯p^tA²6`m#vDq^`=£w,ypDmosotpDmu``Qknmmo\^ryPsutwvDq^`_=`pDm tpl¯p^t`q,~SytpPm« ~~^£twmutyp,l`#£w¡\Jso`¦
§'yPpkmopDmo`A`Lkmcmusoyq¢r`Qknm8rLsut¢pDmevDq^`¡£\so`#`Qknm8`p0rQvDq^t£t^su`¡µ¢`Q#£yPpPmosotpDmo`
s^t£`8PtkokupDmkuq^s q^pÈl`L_¡t`s£`8l`#soµjPypr²JkyPtmQ²
−P =
∫ π2
−π2
σr cos θ rdθ =
∫ π2
0
2A cos2 θ dθ = Aπ»ÇP^Q½
´ep,y^mut`pDmlyp8~Syqs £=yPpDmustpPmo`es^t£`
σr = −2P
π
cos θ
r
»ÇDP½
´ep©~J`Lqlmso`_svDq^`svPq`σr
'q^p`=_=~^£tmuqJl`ypJknmpDmu`rP£`=¥ −2P/πd²Wkuq^smuyPqlm`Lso£`=l`
^t_¡]muso`d~kokopDm `Lp
O¦Jcq3ªÇtmcvDq^`
τrθ = 0²σr`mσθkuypDm £`QkyPpDmustpPmo`Lk ~^sotpJt~£`LkL¦
±`mmu`kuy£qlmotyPp ~Syq^s8q^p^`¡ª®yPso`¡yp`pDmusor`ª®yq^sop^tm#q^p^`kyP£q^mutyp rL£rL_=`pDmotso`¡q©~suyP^£]L_=`6« q^p`~su`QkukutyPp'ltwknmosut^q^rL`¦ ¯Jp3l`8£A« q^mut£twk`Ls ^pk £wkuq^tmu`8~Syq^sp6« t_=~Sysumu`#vDq^`£SmnjH~S`8^`ltwkmusot¤qlmutypÈl`8~^so`Lkoktyp¨²lp^yqkso`ª®yPsu_q^£ypk twt6`#sorLkuq^£mom`Lp3yHyslyp^pr`LkLsumurLkut`p^p^`Qk5
σx = σr sin2 θ = −2P
π
x2z
(x2 + z2)2»ÇD ) ½
σz = σr cos2 θ = −2P
π
z3
(x2 + z2)2»ÇDH½
τxz = σr sin θ cos θ = −2P
π
xz2
(x2 + z2)2»ÇDP½
¯p¡^`yp^pJwmuso`-£`Lklr~£P`_=`LpPmk^pk£` ^`_=t¤Z~^£wp6²yPp9qlmot£twk` ££ytl« r£wkmutwtmur £tp^rQtsu`
∂ur
∂r= εr = −1− ν2
E
2P
π
cos θ
r
»ÇDD½
ur
r+
1
r
∂uθ
∂θ= εθ =
ν(1 + ν)
E
2P
π
cos θ
r
»ÇD=2½1
r
∂ur
∂θ+∂uθ
∂r− uθ
r= γrθ =
τrθ
G
»ÇD¼D½
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c~^su]QktpDmorsmotyPp6²lyp3y^mut`pDm£`Qk lr~£P`_=`LpPmk kqt¢pDmk5
ur =1 − ν2
πE2P cos θ ln r − (1 − 2ν)(1 + ν)
πEPθ sin θ + C1 sin θ + C2 cos θ
»ÇD GP½
uθ =1 − ν2
πE2P sin θ ln r +
ν(1 + ν)
πE2P sin θ
»ÇPP½
− (1 − 2ν)(1 + ν)
πEPθ cos θ +
(1 − 2ν)(1 + ν)
πEP sin θ + C1 cos θ − C2 sin θ + C3r
§¨`Lk9^tWrso`pDmu`QkyPpkmopDmo`Lk#kuypDm9lrmo`so_=tp^rL`Lk9`p³yPpktwlrLsopPm9vDq^`£`'l`L_=t¤Z`Lku~`=p^`~S`qlmkuq^^tsl`e_=yPq^¢`L_¡`LpDm l`8kyP£twl`suttwl`P¦Deyp^pypk kt_=~^£`_=`pDm£`Qk ^r~^£w`_=`pDmok ¥£w¡kq^suªÇ`evDq^tysosu`Qk~SypJl`pDm ¥
θ = ±π/2
ur|θ=π/2 = ur|θ=−π/2 = − (1 − 2ν)(1 + ν)P
2E
»ÇP^µ½
uθ|θ=π/2 = −uθ|θ=−π/2 = − (1 − ν2)
πE2P ln r + C
»ÇD½y £w8yPpknmpDmu`
C`Lkm-lrmo`so_¡tp^r`p\^yPtkutwkukopDmqp¡~SytpDm-¥£8kuq^sªÇP`¥q^p^`ltwknmp`
r0yP_=_¡`
sorª®rLsu`Lp`P²lkyPtmuθ|θ=π/2 = −uθ|θ=−π/2 = − (1 − ν2)
πE2P ln(r0/r)
»ÇP ) ½ °£R`Qknm8t_¡~SysumopDm#^`¡pymu`Ls#vPq`£A« yPp©p`¡~S`q^m8~Jk8sorLkuyqlso`9£`=~^soy^£]_=`¡`pÀlr~^£w`_=`pDm8tpJlr¤~S`pJ^_=_=`pDml`8£¡kmusoqmoq^su`£yJ£`el`Qkylzn`mok `ml`8£w9_pt]Lsu`8lypDmt£wk kuypDmkuq^~^~SysumurQk¦H¾Op`S`mL²yp~J`LqlmypJknmmu`Ls¡vDq^`£ ¢sotmutyp l`
uθµ¢P`L
ln r`_=~S¿L\^`,l`,\^ytwkts9qp^`¢£`Lq^s¡¥
£A« tpl¯Jp^tY¦
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Q~s
qp^tmor#l`8£yPp^q^`Lq^sQ¦l±`#\suP`_=`pDm~suyllq^tm q^p^`8ltwknmosut^qlmotyPpsltw£`e^`8ypDmustpDmu`vPqt6ysoso`Lk¤~SypJ3q3Lk~^sorLrLl`LpDm moyq^sop^r#l` GHZ²kuytm
σr = −2Q
π
cos θ
r
»ÇPH½
σθ = τrθ = 0»ÇPP½
ÍÍ ÒT'UWVXY
Q W! !
§¨`Lktwky¢£`Lq^sokl`ypDmustpDmu`QkRkyPpPmR_tpDmu`LppDm-l`l`L_=t¤°`Lso£`Qk~kokupDm~s0¦¾-pyHyslypp^r`Qk
LsumurQkt`p^p`LkL²H`LkrQvPqJmutypJk~J`Lq^¢`LpPmcku`8surLrLsutsu`
σx = −2Q
π
x3
(x2 + z2)2»ÇPD½
σz = −2Q
π
xz2
(x2 + z2)2»ÇP 2½
τxz = −2Q
π
x2
(x2 + z2)2»ÇP¼D½
§¨`Lklr~^£w`_=`pDmkc¥£wkuq^suªÇ`9~J`Lq^¢`LpPm¿mosu``Lkmut_¡rQke^`9£w_=¿_=`9_p^t]Lsu`vPq`9~^sorLrLl`L_¤_=`LpPm`Lp'mo`ppDmcy_=~lmo`#l`#£¡soymmutyp,l` GH
− ur|θ=π = ur|θ=0 = − (1− ν2)
πE2Q ln r + C
»Ç=GP½
uθ|θ=π = uθ|θ=0 = − (1 − 2ν)(1 + ν)Q
πE
»Ç>2D½
,#$$ ,(5./ ,1#*, # # 1 #"##´epÈkuq^~^~SyPku`#_=tpDmo`ppPmcvDq^`#£`l`L_¡t¤Z`Lku~``Qknmc\Jsor~sqp^`#~^so`Lkoktyp3p^yso_£`
p(x)`m
qp^`#~^so`Lkoktyp'mop`pDmot`L££`q(x)
kuq^s q^p`Jpl`#tpl¯pt`l`8£wso`Lq^sa+ b
¦
x
z
b a
q(x)
p(x)
O
r
A(x, z)
B C(x, 0)
ds
s
) /"! ! %RWW D
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H, 6Æ ^Å *H * / ÃL *PÃ$0 A lÃ0Ç / I Æ Q ^Å *§¨`Lk¡lr~^£w`_=`pDmk9kuypDmlrQlq^tmok=l`'£wÈ_=¿_=`_=p^t]so``p kuy_=_pDm¡£`Qk9yPpDmusot^q^mutypkl`Lk
~su`QkukutyPpkkqs\vDq^`8r£r_=`LpPml`kuq^suªÇ`
ux = − (1 − 2ν)(1 + ν)
2E
[∫ x
−b
p(s) ds−∫ a
x
p(s) ds
] »Ç>2Hµ½
−2(1 − ν2)
πE
∫ a
−b
q(s) ln |x− s| ds+ C1
uz = −2(1 − ν2)
πE
∫ a
−b
p(s) ln |x− s| ds »Y 2½
− (1 − 2ν)(1 + ν)
2E
[∫ x
−b
q(s) ds−∫ a
x
q(s) ds
]
+ C2
§¨`-kuqlmL²L^pk¨£`Qk`l~^so`LkokutyPpk²»YQ½S`m »Ç=GP½²Ly^£t`pDm¨¥ku`P_=`pDmu`Ls£Y« tpDmo`so¢µ££`O« tpDmorsmotyPp`Lpl`Lql~smot`Qk¦xyq^s¡r¢Htmu`s=`3~^soy^£]_=`P²t£~J`Lqlm¿muso`'tpDmurLsu`QkukopDm¡^`\^yPtkuts=« r¢£q^`Ls£`LkPsoPlt`pDmok^`Lklr~^£w`_=`pDmkc¥£w'kuq^suªÇ` ∂ux
∂x
`m ∂uz
∂x
qÈ£t`Lq l`QkelrL~^£w`L_=`pDmokL¦W±`Qt~J`Lsu_=`m^`9~^£qk8l`9£`¢P`s£Y« _^tqYmur9kuq^s£`=\^yPtÈl`Lk8ypkmopPmo`LkL²
C1`mC2²lypDmeyPp '^rnz¥3kyPq^£tpr
£`LktpypH¢rLp^t`LpDmok¥=£¡so`_svDq^`8|e¦
∂ux
∂x= − (1 − 2ν)(1 + ν)
2Ep(x) − 2(1 − ν2)
πE
∫ a
−b
q(s)
x− sds
»Ç>2 ) ½
∂uz
∂x= −2(1 − ν2)
πE
∫ a
−b
p(s)
x− sds− (1 − 2ν)(1 + ν)
2Eq(x)
»Ç>2µD½
±`Qks^t`LpPmk~J`Lq^¢`LpDmc¿mosu`tpDmu`Lsu~^sormorLkªÇPt£`L_=`pDmc« q^p0~SytpPml`¢Hq^`#_=rLLp^twvDq^`¦S§6`Pso¤^t`LpPm ∂uz
∂x
ysoso`Lku~JyPp¥£lrª®yPsu_motyPpεx¥£kqsªÇP`²JvPqJpDme¥ ∂uz
∂x
²St£su`L~^sorLku`pDmu`#£w~S`pDmu`^`£wkuq^sªÇP`8lrª®yPsu_=rL`¦
H 3 Â / Ã!('0 / Ã!*H$0 / Æ *0*D 0ÏÄ.*HA6Å*±yP_¡_=` ^pk£`Pk¨mosutwlt_=`pkutyPp^p^`L£Y²µyp¡yPpktwl]Lsu` vPq` £`Lk~^so`LkokutyPpktwkukopDm`p9qp~SytpDm
B^` £#kuq^suªÇ`kuq^sOqpr£r_=`pDm-l`ckuq^suªÇ`ds~S`q¢`pDmO¿muso`ypJktwlrsor`QkRyP_¡_=` q^p` ª®ys`yp`pl¤
mosurL`¦H¾Op3tpDmorspDmkq^s£wkuq^suªÇ`c£w¡kuy£qlmotyPpr£r_=`pDmtsu`emusoyq^¢Pr`q'~Jss~\^`~^sorLrLl`LpPmQ²Ht£^`¢Ht`LpDm#£yPsok~SyPkokt^£`l`lyp^p`s#£`Lk#yPpDmustpPmo`Lk`p"q^pÀ~SytpPm9yq^spDm
A(x, z)lq³l`_=t¤A~^£wp
ÍÍ ÒT'UWVXY
QP W! !
»Yªn¦/tq^so` ) µ½¦J´ep3ylmot`LpPmc£yPsok
σx = −2z
π
∫ a
−b
p(s)(x − s)2
((x − s)2 + z2)2ds+ − 2
π
∫ a
−b
q(s)(x− s)3
((x− s)2 + z2)2ds
»Ç>2½
σz = −2z3
π
∫ a
−b
p(s)
((x− s)2 + z2)2ds+ −2z2
π
∫ a
−b
q(s)(x − s)
((x− s)2 + z2)2ds
»Ç>2P½
τxz = −2z2
π
∫ a
−b
p(s)(x− s)
((x− s)2 + z2)2ds+ −2z
π
∫ a
−b
q(s)(x − s)2
((x− s)2 + z2)2ds
»Ç>22½
hHt£`Lk ~^so`LkokutyPpk~^~^£twvPqr`Lk kuypDmyp^pHq^`Qk²Ht£6l`¢Ht`pDmlypJc~SyPkokt^£`^`8yp^pJwmuso`c`p,\PvPq`~SytpDm9£A« rmom¡l`LkypDmosotpDmu`Qk¦%°£-ªÇqlm¡`~S`pJ^pDm9so`_svDq^`svDq^`'~JyPq^s9l`Qk9ltwknmosut^qlmotyPpkvDq^`L£¤ypvDq^`Qk²^`LktpDmurPso£`Qk p^`#kyPpPm ~Pk ª®ysrL_¡`LpDm kut_¡~£`Qk¥=£wq£`LsL¦ bcp Pk~smotq^£t`Ls9t_¡~SysumopDm`Lkm¡`£q^t yku`q^£`'q^p`'~^so`LkoktypÀp^yPsu_£``Qknm=~^~£twvDq^r`3kq^s£`^`_=t¤Z`Lku~P`¦´ep,y^mut`pDm£ysk5
εx = − (1− 2ν)(1 + ν)
2Ep(x)
»Ç>2¼P½hHtH£A« yPpypkut^]so`O£e£ytHl` y_=~Sysumu`_=`LpPmr£wkmutwvDq^`-£tprLtso`²Qyp9yPlmut`pDm£esu`L£mutypkuq^t¢µpDmu`
εx =1
E
[
(1 − ν2)σx − ν(1 − ν)σz
] »Ç>2GD½
¾-p9yP_¡~JspDm6£`Qk^`ql#rLvDqmutypk¨`mtp£qJpDm¨£wcyPpltmutypql#£t_=tmo`LkL²σz = −p(x) ²QyPp8musoyq^¢P`
σx = σz = −p(x) »Ç¼PP½´ep©su`L_=sovDq^`¡lypJ9vDq^`¡£`Qk^tkmusotqlmutypk8l`=yPpDmustpPmo`Lkekq^suªÇtvDq^`Qkcpyso_=£`Qkc`memp^P`pl¤
mot`L££`LkkuypDmerLP£`LkL¦W±`mmo`¡ypltmutyp p^yqke~S`so_¡`mmoso6« `H~£twvDq^`s~JyPq^svPqyt£Y« rLyq^£`_=`pDm~^£Pkn¤motvDq^`mo`p^p`#¥=ku`lr¢P`£y~^~S`s kuyqk£w=kq^suªÇ`8~^£q^m mcvDq^`#ltso`Lmo`_=`pDmc¥¡£w¡kuq^suªÇ`¦
¾Op^¯p6²6yPp©~S`qlmso`_sovDq^`LsvDq^`=£`~^suyP^£]_=``Lp"^rª®yso_motyPpk`m`pÀyPpDmustpPmo`Lke~J`Lqlm¿mosu`sorLkuy£q kopk=_9^tqYmur,yPp`LsupJpDm£w©^rmu`Lsu_=tpmotyPpl`Qk=ypJknmpDmu`Qk3»Yªn¦O `L_svPq`|¦µ½¦#°£p¨« `Lp3`Qknm ~^£qk^`#_¡¿L_=`~Syqs£`Qk lr~£P`_=`LpPmk vDq^t6r¢Py£q^`pDm`p
ln(r)¥¡£Y« tpl¯p^tA¦
Ë®ÌÍ6ËÏÎ
Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)
Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)
Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)
Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)
ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)
ISSN 0249-6399