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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

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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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tf£yskuvDq^`,£w ª®yPso`0k« ppDq£`,~Syq^s

0 ≤ α < 1

tf =1

ω0

√

1 − α2arctan

(

2α√

1 − α2

2α2 − 1

)

»Q½

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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

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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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ÍÍ ÒT'UWVXY

Q¼ W! !

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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 = # !$ $ '

ÍÍ ÒT'UWVXY

W! !

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F = −c|ξ|ξ − kξ»YP½

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Fn = −knξ32

»A½y

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n = 32

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Fn = c|ξ|1/2ξ − kξ3/2 »AD½

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|F | ≤ Fmax,kutξ = 0

|F | = Fmax,ktp

(F ) =kutPp

(ξ),kutξ 6= 0,

»AP½

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Fn =

k1ξ`p,yP_¡~su`QkukutyPp

ξ ≥ 0,

k2(ξ − ξ0)`p3su`Qknmot­moqlmutyp

, ξ < 0.

»A=2½

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en =

√

k1

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b

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K(e) − E(e)

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c3 = (ab)32 =

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πe2

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(

b

a

12

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b

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kuytm

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2πabE∗bK(e)

»Ç¼D½

=

(

9p2

16E∗Re

)

13 2

π

(

b

a

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(F1(e))13 K(e)

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p0 =3P

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(F1(e))−

23

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a= (

A

B)

12 = (

R′′

R′)

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c = (ab)12²tpkut^vDq6« qp^`yPq^suq^su`su`L£mut¢`P²

Re =

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(R′

R′′)

12 = (

B

A)

12 = 2.41

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Re = (R′R′′)12 =

√2R.

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h = z1 + z2 = Ax2 =1

2

(

1

R 1+

1

R 2

)

x2 =1

2

1

Rx2 »GP½

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2

1

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xl`#£A« rLvDqmotyPp"»GD½+

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∂x+∂uz2

∂x= − 1

Rx

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∂uz1

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∫ a

−a

p(s)

x− sds

»GP¼P½

kuytm `pÈkuq^kmutmuqpPm^pk#»G>2½+∫ a

−a

p(s)

x− sds =

πE∗

2Rx

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p(x) = −πE∗

2R

x2 − a2/2

π(a2 − x2)12

+P

π(a2 − x2)12

»nQD½

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P ≥ πa2E∗

4R

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a2 =4PR

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a

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12

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(

9P 2

16RE∗2

)

13

a =

(

3PR

4E∗

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y p0 =2P

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4

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(

PE∗

πR

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4PR

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p0 =3P

2πab=

(

6PE∗

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(F1(e))−

23 δ =

(

9p2

16E∗Re

)

13

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(

3PRe

4E∗

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F1(e)

1

E∗=

1 − ν21

E1+

1 − ν22

E2

1

R=

1

R1+

1

R2Re = (R′R′′)1/2

F1(e) =

(

4

πe2

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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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d t

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−m1 +m2

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32z

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1

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1

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1

R2

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4K

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25

=

(

15mV 2z

16R12E∗

)

25 »nPL¼D½

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12E∗(δm

z )32

»nµlµ½

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3R2E∗

(

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8R3E∗

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R′

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x∗

y∗

z∗l

λ∗

Q ' !W ' F' %'!=" R =

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P W! !

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x∗`my∗¦S´epÈlr¯p^tmcl`8_¡¿L_=`

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ε =l

R

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ε′ =l

R′

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y∗(x∗) = R −√

R2 − x∗2,»nL>2½

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y∗(x∗) ≈ x∗2

2R, |x∗| ≤ l/2

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y∗(x∗) ≈ −x∗2

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v∗(x∗) = −1

2

(

1

R+

1

R′

)

x∗2 + const., |x∗| ≤ l/2»nµP½

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l, W =

w∗

l, Λ =

λ∗

l, F =

F ∗

µl, P =

p∗

l,

»nµlµ½

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F ∗

µR, p =

p∗

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»nµP½

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Vi(X) =1

2(ε+ ε′)X2 + const.

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4πWi(Z) = C − i F[

κ [2ZQ(Z)− ln(Q(Z))] + 2ZQ(Z) − ln(Q(Z)) + 4(Z − Z)Q(Z)]

»nµP½y

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κ`Lkmq^p^`

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κ = 3 − 4ν`pÈlrª®yso_=mutypk ~^£wp`LkL² »nQP½

κ =3 − ν

1 + ν

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Q`Qknmclr¯p^t`#~s

Q(Z) = 2Z − i√

1 − 4Z2»nµ¼D½

y √1 − 4Z2

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2]l`#£Y« l`8sor`£A¦^§¨

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1 − 4Z2 = −2i Z

(

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4Z2

)

1

2 ≈ −2i Z

(

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8Z2

)

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lnQ = ln |Q| + i argQ»LD½

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F = πε+ ε′

2(κ+ 1)

»nQ^µ½

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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 ′

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wi]o = wo]i»nQ ) ½

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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

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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

]

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εC − C ′ = i f

[

(κ+ 1)(2 ln 2 − ln ε) +1

2(κ− 1)

] »nQ¼D½

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Z = 0l`Lk rLvDqmutypk8»QP½O`m»nLD½¦l¾-pqlmot£twkupPm»L¼D½O~Syqs rL£t_=tp`s

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4πλ = −κ+ 1

2f ln

(

eε2

64

) »nQGD½

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R2f

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ε

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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

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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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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

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I =

∫

S

K(x, y, ξ, η)f(ξ, η) dξdη»nG ) ½

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Rl`LkOylzn`mok

`Lp3ypDmoPmQ²^kyPt­ml ≪ L ≪ R

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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)

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`mb(m)

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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

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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

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ξ = ξ(n,m), ξ = (n, y)»YlPQ½

ξ = ξ + (m− y)ξ′ + O((m− y)2)»Y^QP½

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y £`kujD_JyP£`′^rLkutPp^`8£=lrLsut¢rL`e~smot`L££`8~s s~~JyPsm¥

m¦

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ε =ξ′(ξ − x)

L2

»Y^ ) ½

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(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

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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

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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

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^`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 ·^¸

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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£`−

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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

)

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f(x) =

∫ b

a

ϕ(ξ) ln |ξ − x|, dξ »A ) ¼P½

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f ′(x) = −−∫ b

a

ϕ(ξ)

ξ − xdξ

»Y ) GP½

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ϕ(x) =1

π2

∫ b

a

√

(ξ − a)(b− ξ)

(x− a)(b− x)

f ′(ξ)

ξ − xdξ +

C√

(x − a)(b− x)

»YPD½

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C =

(

π2 ln

(

b− a

4

))

−1 ∫ b

a

f(x)

(x − a)(b− x)dx =

1

π

∫ b

a

ϕ(x) dx»YQ½

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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

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w(x, y) = −Ax2 + δ, AyPpknmpDm »YPP½

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Z(x, y) =2AE

(1 − ν)

√

a2(y) − x2»Y>2½

FZ =πAEa2

1 − ν

»YP¼D½

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`m£`#lrL~^£w`L_=`pDm `LkmclrLlqt­m

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

§-« tpl`LpPmmotyPp,^q,jH£tplso`8`Lkm ^yp8lyPp^p^r`#~Js

δ = w(0, y) =1

2Aa2

[

ln

(

16(L2 − y2)

a2

)

+ 1

]

+1

2

∫ L

−L

Aa2(η) −Aa2(y)

|η − y| dη»YPP½

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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|

§¨`cmo`so_¡`8¥£Y« yslso`1ypDmot`LpPmq^pmu`Lsu_=``p

ln

(

1 − y2

L2

) ¦±`emo`so_=`cmo`p'¢`sk −∞ £yskuvDq^`y → L

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R1

R2

∆R

φ

ψ

Q

(r, φ)(R2, ψ)

µ ' ! W )$ ' ! P ! !

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, νi =νi

1 − νi,

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∫ a

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∞

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cosφ»Y 2GP½

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ρ2 − 1

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ρ2 − 1

ρ3 sinφ»Y¼^µ½

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p(y)

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ρ

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∫ ε

−ε

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∫ ε

−ε

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∫ ε

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]

+Q

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∫ ε

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2π+ L1(ρ, θ)) dψ

]

+Q

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»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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∫ ε

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Q

»YGH½

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y = tan(φ/2), t = tan(ψ/2), b = tan(ε/2),»A GD½

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1 + η

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1 + η

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1 + η

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Q

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Q=

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√

b2 − y2

1 + y2 +1

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b2 − y2

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b2 − y2

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σ = σ + σ

» ) LD½ε = ε = ε

» ) µ½§~q^tkokop`

σ ¯⊗ ε`Lkm#£yPsoke£w'~^qtkokup`¡¢yP£q_¡twvDq^`¡surL¢`skut^£`9`m

σ ¯⊗ ε

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kuytm

Ψ =1

2ρλ [Tr(ε)]

2+ 2µTr(ε2)

» ) QP½

ϕ =1

2λθλ [Tr(ε)]

2+ 2µθµTr(ε

2)» ) ) ½» ) H½

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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

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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

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ε`pÈq^p^`#~Jsumut`sor¢P`skt^£`

ε `m

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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

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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

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ε =1 + ν

E

[

σ +σ

τ1

]

− ν

E

[

Tr(σ) +Tr(σ)

τ2

] » ) ¼D½

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σ = σ0H(t− τ)» ) GP½

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ε = J(t− τ)σ0» )=) D½

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σ =n∑

j=1

σjH(t− τj),» )=) µ½

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ε =

n∑

j=1

J(t− τj)σj ,» )=) P½

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ε(t) =

∫ t

0

J(t− τ)d σ(τ)

d τdτ +

n∑

j=1

J(t− τj)σj» )=)) ½

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ε(t) = J ⊗ Dσ

Dτ

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σ(t) =

∫ t

0

R(t− τ)d ε(τ)

d τdτ +

n∑

j=1

R(t− τj)εj» )=) P½

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σ(t) = R⊗ Dε

Dτ

» )=) D½

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ε = (J +K) ⊗ Dσ

Dτ−K ⊗ DTr(σ)

Dτ1l

» )=) 2½y

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σ = 2M ⊗ Dσ

Dτ+ L⊗ DTr(ε)

Dτ1l

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uz1 =1 − ν′1

2

πE′

1

∫

S

pf (s, φ)

rdsdφ

» )=) GP½

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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

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p (x, y) =P

2πa2

(

1 −(r

a

)2)

−12 » ) DP½

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δ =P

2E′∗(πa)2

∫

S

1

r

(

1 −( r

a

)2)

−12

dxdy» ) ) ½

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P = 2aE′∗δ = 2E′∗R12 δ

12 δ

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P = A∂P (δ)

∂δδ

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» ) 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! !

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(1 − en) ∝ v−

14

0

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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

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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

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νkuytm

p0 =σ0

2τ1max

= 1.6650953σ0»®J ) ½

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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`

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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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pY = CY σ0»®JLD½

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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

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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^`

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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

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r = akyPt­m

p

σ0= −σr(r = a)

σ0= 2 ln

( c

a

)

+ 2/3»®DP½

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da

dc=

σ0

E∗

[

3(1 − ν)c2

a2 − 2(1 − 2ν)a

c

] »®HD½

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du(r)

dc=

σ0

E∗

[

3(1 − ν)c2

r2− 2(1 − 2ν)r

c

] »®H=2½

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2πa2du(a) = πa2dδ = πa2 a

Rda

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c/akq^t¢pDmu`

E∗a

Rσ0= 6(1 − ν)

( c

a

)3

− 4(1 − 2ν)»ÇDGP½

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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

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pm¦°£ku`=musoyq¢`vDq^`£A« rmom9^`Lk9ypDmustpDmu`Qk#kyPqk8£

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pm = p+2σ0

3

»® ) P½

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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 ¥

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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

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δ

δY=

1

2

(

(

a

aY

)2

+ 1

)

»® ) D½

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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

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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

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pm`Lkm£ysk

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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

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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

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− δpδY

)]

»®PH½

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R =

R1R2

R1 + R2, R1 ≥ 0, R2 ≥ 0

R1R2

R1 − R2, −R1 > R2 ≥ 0

»®PDP½

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δf = δc − δr»®PPD½

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=δrR

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»YD½²kuytm~SyqsδY

`mδr

δrR

=a2

r

R2 ,δYR

=a2

Y

R2

»®PP¼D½

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=a2

c

r2+ 1

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=a2

c

R2 +δYR

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R

R=δrδc

=

(

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− 1

)12 »®Hlµ½

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F =4

3E∗R

12 (δ − δf )

32

»®HP½kuq^s £A« tpDmu`Lsu¢££`8^`so`Lkmutmuq^mutyp

δf ≤ δ ≤ δc¦

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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

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=δrR

=

(

3π

4

)2(CY σ0

E∗

)2

,WY

σ0R3 =

8

15

E∗

σ0

(

δYR

)52 »®HH½

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− 1

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V 2Y =

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15mδ

52

Y

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15m

(

3π

4

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E∗

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δ =a2

2R

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1

2mV 2 = Wc =

∫ δc

0

P dδ =

∫ ac

0

πa2ppa

Rda = πa4

c

pp

4R

»®H GD½

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1

2mV 2

f = −Wr =3Pc

10acE∗

=3

10π2a3

c

p2p

E∗

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Wc=

3π54 4

54

10

( pp

E∗

)

1

2mV 2

ppR3

−14

»®D^µ½

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=

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3π

4

)2(Cpσ0

E∗

)2

,WY

σ0R3 =

8

15

E∗

σ0

(

δYR

)52 »®DPP½

kuytmWY

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2π

5

(

3π

4

)4(Cpσ0

E∗

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δ

R=δY2R

+a2

2R2 , δ > δY»®DH½

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P

Cpσ0R2 = π

(

2δ

R− δY

R

) »®DPP½

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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

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Wr

WY= −

(

δrδY

)3

= −(

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− 1

)32 »®D¼D½

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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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ε0q^p^`

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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

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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

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*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

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¢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¦

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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

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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

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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

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N = 0²^kyPt­m

W =γσ0(2R)3

2 +M/2

(

δm2R

)2+M/2 »Y^µ½

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We =8p2

ma3

3E∗

»Y^QP½

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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

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e2 =16

√3σ0

πE∗

(

9πσ0R3

mV 2

)

14 »Y^LD½

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H(x, y, z) =∂H1(x, y, z)

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] »YP GD½»Y ) D½

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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

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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

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¢Prsot­¯J`pDm £A« rLvDqmotyPp3^`§¨~£P`²ltA¦ `¦²

∇2ψ = 0, ∇2ψ1 = 0»Y ) ½

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∆ ≡ ∂ux

∂x+∂uy

∂y+∂uz

∂z=

1− 2ν

2πG

∂ψ

∂z,

»YH½

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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

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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¦

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ρ = (x2 + y2 + z2)12

»YPP½`m£`#\suP`_=`pDm~s

P =

∫

S

p(ξ, η) dξdη»AP½

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ψ1 =∂H1

∂z= H = P ln(ρ+ z)

»A 2½

ψ =∂H

∂z=P

ρ

»YP¼D½

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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

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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 ¦

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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

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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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uz =1 − ν2

πE

∫

S

p(s, φ) dsdφ»A=2P½

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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

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kqsªÇP`8`pÈyPpDmomL²• n = 1/2

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n = −1/2vDq^t~J`Lqlm#¿muso`9so`pJypDmosur

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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

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C(s, φ)¦W§6`soµjPyp

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_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

SG WN! %=% R )!W ! A( ! ' ! % != $

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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

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pkq^s £wkuq^sªÇP`r£r_=`pDmtsu`

dsdφ`p

C¢µqlm

(1 − 2ν)(1 + ν)

2πEp dsdφ

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− (1 − 2ν)(1 + ν)

2πEp cosφ dsdφ

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ur(r) =(1 − 2ν)(1 + ν)

2πE

p0

a

∫ 2π

0

cosφ dφ

∫ s1

0

(α2 − 2βs− s2)12 ds

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ur(r) = − (1 − 2ν)(1 + ν)

3E

a2

rp0

[

1 − (1 − r2

a2 )32

]

, r ≤ a»Y¼GP½

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_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

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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

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σr

p0=σθ

p0= (1 − 2ν)

a2

3r2»YGPP½

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r = a¦

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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

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p(x, y) = p0

(

1 −(x

a

)2

−(y

b

)2)n »YG¼D½

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a

)2

+(y

b

)2

− 1 = 0»Y G=GP½

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ψ(x, y) =

∫

S

p0

(

1 −(x

a

)2

−(y

b

)2)n

ρ−1 dξdη»ÇPP½

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12

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uz =1 − ν

2πG[ψ]z=0

»ÇPPP½

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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

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1 − x2

a2 + λ− y2

b2 + λ− z2

λ= 0

»ÇPD½

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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

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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

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∫

∞

0

dw(

(a2 + w)(b2 + w)w)

12

=2

a

∫

∞

0

dw(

(1 + k2t2)(1 + t2))

12

»ÇP 2½

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µ¢P`Lk = b/a

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kuytmdt = dφ/ cos2 φ

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∞

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)

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uz =1 − ν2

E2p0bK(e)

»ÇPGP½

? 3 $0*ÈÆ 6*DQ0Ç / 0* B*,µÃAC n =

1

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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

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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

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E(e) =

∫ π/2

0

√

1 − e2 sin2 φ dφ»ÇLP½

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£`8~Jymu`LpPmot`L£Ψ1 =

∫

∞

zΨ dz

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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

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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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σx

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τxz

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σr

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r

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∂σx

∂x+∂τxz

∂z= 0

»ÇD=2½∂σz

∂z+∂τxz

∂x= 0

»ÇD¼P½

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∂2εx∂z2

+∂2εz∂x2

=∂2γxz

∂x∂z

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εx =∂ux

∂x, εz =

∂uz

∂z, γxz =

∂ux

∂z+∂uz

∂x

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εy = 0»Ç ) Q½

σy = ν(σx + σz)»Ç ) P½

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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

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[

∂

∂x+

∂

∂z

] [

∂φ

∂x+∂φ

∂z

]

= 0»Ç ) 2½

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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¦

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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

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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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φ(r, θ) = Arθ sin θ»YP¼D½

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ÍÍ ÒT'UWVXY

µ¼ W! !

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σr = 2Acos θ

r

»Ç=GD½

σθ = τrθ = 0»ÇDP½

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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

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^t_¡]muso`d~kokopDm `Lp

O¦Jcq3ªÇtmcvDq^`

τrθ = 0²σr`mσθkuypDm £`QkyPpDmustpPmo`Lk ~^sotpJt~£`LkL¦

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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

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∂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

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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`

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r0yP_=_¡`

sorª®rLsu`Lp`P²lkyPt­muθ|θ=π/2 = −uθ|θ=−π/2 = − (1 − ν2)

πE2P ln(r0/r)

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uθµ¢P`L

ln r`_=~S¿L\^`,l`,\^ytwkts9qp^`¢£`Lq^s¡¥

£A« tpl¯Jp^tY¦

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Q~s

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σr = −2Q

π

cos θ

r

»ÇPH½

σθ = τrθ = 0»ÇPP½

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Q W! !

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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

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− 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

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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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~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

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∂x

`m ∂uz

∂x

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£`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

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∂x

ysoso`Lku~JyPp¥£lrª®yPsu_motyPpεx¥£kqsªÇP`²JvPqJpDme¥ ∂uz

∂x

²St£su`L~^sorLku`pDmu`#£w~S`pDmu`^`£wkuq^sªÇP`8lrª®yPsu_=rL`¦

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B^` £#kuq^suªÇ`kuq^sOqpr£r_=`pDm-l`ckuq^suªÇ`ds~S`q¢`pDmO¿muso`ypJktwlrsor`QkRyP_¡_=` q^p` ª®ys`yp`pl¤

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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

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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

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σx = σz = −p(x) »Ç¼PP½´ep©su`L_=sovDq^`¡lypJ9vDq^`¡£`Qk^tkmusotqlmutypk8l`=yPpDmustpPmo`Lkekq^suªÇtvDq^`Qkcpyso_=£`Qkc`memp^P`pl¤

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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