Discrete modeling of fiber reinforced composites using the scaled boundary finite element method

Journal Pre-proofs Discrete modeling of fiber reinforced composites using the scaled boundary finite element method J. Zhang, J. Eisenträger, S. Duczek, C. Song PII: DOI: Reference:

S0263-8223(19)33593-7 https://doi.org/10.1016/j.compstruct.2019.111744 COST 111744

To appear in:

Composite Structures

Received Date: Accepted Date:

24 September 2019 26 November 2019

Please cite this article as: Zhang, J., Eisenträger, J., Duczek, S., Song, C., Discrete modeling of fiber reinforced composites using the scaled boundary finite element method, Composite Structures (2019), doi: https://doi.org/ 10.1016/j.compstruct.2019.111744

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© 2019 Published by Elsevier Ltd.

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6B;m`2 R,  TQHv?2/`H 2H2K2Mi Q#iBM2/ #v b+HBM; i?2 bm`7+2 K2b? iQr`/b i?2 b+HBM; +2Mi2` OX

9

Q7 a"61J Bb i?i QMHv i?2 #QmM/`v Q7 i?2 pQHmK2 2H2K2Mi M22/b iQ #2 /Bb+`2iBx2/ #v K2Mb Q7 bm`7+2 2H2K2MibX h?2 pQHmK2 Q7 i?2 2H2K2Mi Bb Q#iBM2/ #v b+HBM; i?2 /Bb+`2iBx2/ #QmM/`v iQr`/b i?2 b+HBM; +2Mi2`X h?2 bm`7+2 2H2K2Mib QM i?2 #QmM/`v Q7  pQHmK2 2H2K2Mi `2 +QMbi`m+i2/ b BM i?2 }MBi2 2H2K2Mi K2i?Q/X AbQT`K2i`B+ i`BM;mH` M/ [m/`BHi2`H 2H2K2Mib `2 b?QrM BM 6B;X k M/ 6B;X k#- `2bT2+iBp2HvX h?2 b?T2 7mM+iBQM Q7  bm`7+2 2H2K2Mi +M #2 2tT`2bb2/ b N (η, ζ)- r?2`2 η M/ ζ `2 i?2 HQ+H +QQ`/BMi2b Q7 i?2 2H2K2MiX b BHHmbi`i2/ BM 6B;X k+-  /BK2MbBQMH2bb `/BH +QQ`/BMi2 ξ bTMMBM; [0, 1] 7`QK i?2 b+HBM; +2Mi2` O iQ i?2 #QmM/`v Bb BMi`Q/m+2/ 7Q` i?2 Tm`TQb2 Q7 b+HBM;X AM i?2 b+H2/ #QmM/`v +QQ`/BMi2b (ξ, η, ζ)-  TQBMi (ˆ x, yˆ, zˆ) BMbB/2 i?2 /QKBM +M #2 /2}M2/ b x ˆ (ξ, η, ζ) = ξx (η, ζ) = ξN (η, ζ) x,

URV

yˆ (ξ, η, ζ) = ξy (η, ζ) = ξN (η, ζ) y,

UR#V

zˆ (ξ, η, ζ) = ξz (η, ζ) = ξN (η, ζ) z,

UR+V

r?2`2 x, y, z `2 i?2 MQ/H +QQ`/BMi2 p2+iQ`b Q7 i?2 bm`7+2 2H2K2Mi BM *`i2bBM +QQ`/BMi2bX h?2 +Q``2bTQM/BM; C+Q#BM Ki`Bt QM i?2 #QmM/`v (ξ = 1) Bb  x (η, ζ) y (η, ζ) z (η, ζ) J (η, ζ) =  x (η, ζ),η y (η, ζ),η z (η, ζ),η  . x (η, ζ),ζ y (η, ζ),ζ z (η, ζ),ζ 

UkV

h?2 mMFMQrM /BbTH+2K2Mib u (ξ) `2 BMi`Q/m+2/ QM i?2 `/BH HBM2b +QMM2+iBM; i?2 b+HBM; +2Mi2` M/ i?2 #QmM/`v MQ/2b X u (ξ) = ue (ξ) , UjV e

ζ ζ

ζ (0, +1)

(−1, +1)

(+1, +1)

η

η (0, 0)

ξ

zˆ

(+1, 0)

UV  i`BM;mH` 2H2K2Mi BM HQ+H +QQ`/BMi2b η M/ ζX

(−1, −1)

(+1, −1)

U#V  [m/`BHi2`H 2H2K2Mi BM HQ+H +QQ`/BMi2b η M/ ζX

x ˆ

6B;m`2 k, h?2 b+H2/ #QmM/`v +QQ`/BMi2b ξ- η M/ ζX

8

η

O

yˆ

U+V _/BH +QQ`/BMi2 ξ 2KMiBM; 7`QK i?2 b+HBM; +2Mi2` O iQ i?2 #QmM/`v Q7 i?2 pQHmK2 2H2K2MiX

r?2`2

P

BM/B+i2b biM/`/ }MBi2 2H2K2Mi bb2K#Hv T`Q+2bbX qBi?BM i?2 pQHmK2 2H2K2Mi- i?2

e

/BbTH+2K2Mib i  TQBMi BMbB/2  b2+iQ` 7Q`K2/ #v  bm`7+2 2H2K2Mi e `2 BMi2`TQHi2/ b U9V

u (ξ, η, ζ) = Nu (η, ζ) ue (ξ) . h?2 b?T2 7mM+iBQMb `2 +QKKQMHv r`Bii2M BM Ki`Bt 7Q`K b

U8V

Nu (η, ζ) = [N1 (η, ζ) I, N2 (η, ζ) I, . . . , Nn (η, ζ) I] ,

r?2`2 I Bb i?2 B/2MiBiv Ki`BtX AM i?2 b+H2/ #QmM/`v +QQ`/BMi2b- i?2 HBM2` /Bz2`2MiBH QT2`iQ` L 7Q` bi`BM@/BbTH+2K2Mi `2HiBQMb?BT UeV

ε (ξ, η, ζ) = Lu (ξ, η, ζ) Bb 2tT`2bb2/ b (8e) 1 ∂ + L = b1 (η, ζ) ∂ξ ξ



∂ ∂ b2 (η, ζ) + b3 (η, ζ) ∂η ∂ζ



UdV

,

r?2`2 b1 (η, ζ)- b2 (η, ζ) M/ b3 (η, ζ) `2 /2}M2/ b  y,η z ,ζ − z ,η y ,ζ 0 0  0 z x − x z 0 ,η ,ζ ,η ,ζ  1  0 0 x y − y ,η x,ζ ,η ,ζ  b1 (η, ζ) =  0 x,η y ,ζ − y ,η x,ζ z,η x,ζ − x,η z ,ζ |J|   x,η y ,ζ − y ,η x,ζ 0 y,η z ,ζ − z ,η y ,ζ z,η x,ζ − x,η z ,ζ y,η z ,ζ − z ,η y ,ζ 0   zy,ζ − yz ,ζ 0 0   0 xz − zx 0 ,ζ ,ζ    1  0 0 yx − xy ,ζ ,ζ   b2 (η, ζ) = ,  0 yx,ζ − xy ,ζ xz,ζ − zx,ζ  |J|    yx,ζ − xy ,ζ 0 zy,ζ − yz ,ζ  xz,ζ − zx,ζ zy,ζ − yz ,ζ 0   yz,η − zy ,η 0 0   0 zx,η − xz ,η 0    1  0 0 xy − yx ,η ,η  . b3 (η, ζ) =  0 xy,η − yx,η zx,η − xz ,η  |J|    xy,η − yx,η 0 yz,η − zy ,η  zx,η − xz ,η yz,η − zy ,η 0



    ,  

U3V

U3#V

U3+V

AM i?2 #Qp2 2[miBQMb- |J| Bb i?2 /2i2`KBMMi Q7 C+Q#BM Ki`Bt QM i?2 #QmM/`v Q#iBM2/ 7`QK 1[X UkVX am#biBimiBM; 1[bX U9V M/ UdV BMiQ 1[X UeV- i?2 bi`BMb `2 1 ε (ξ, η, ζ) = B1 (η, ζ) ue (ξ) ,ξ + B2 (η, ζ) ue (ξ) , ξ

UNV

rBi? B1 (η, ζ) =b1 (η, ζ) Nu (η, ζ) , e

URyV

B2 (η, ζ) =b2 (η, ζ) Nu (η, ζ) ,η +b3 (η, ζ) Nu (η, ζ) ,ζ . bbmKBM; HBM2` 2HbiB+Biv- i?2 bi`2bb2b `2 2[mH iQ   1 e e σ (ξ, η, ζ) = D B1 (η, ζ) u (ξ) ,ξ + B2 (η, ζ) u (ξ) , ξ

URy#V

URRV

r?2`2 D Bb i?2 2HbiB+Biv Ki`BtX kXkX aQHmiBQM Q7 b+H2/ #QmM/`v }MBi2 2H2K2Mi 2[miBQM 7Q` 2HbiQbiiB+b h?2 b+H2/ #QmM/`v }MBi2 2H2K2Mi 2[miBQM BM /BbTH+2K2Mib +M #2 /2`Bp2/ #b2/ QM :H2`FBMǶb r2B;?i2/ `2bB/mH K2i?Q/ Q` i?2 pB`imH rQ`F T`BM+BTH2- r?B+? `2 `2TQ`i2/ BM (8e) M/ (8d)- `2bT2+@ iBp2HvX 6Q` 2+? pQHmK2 2H2K2Mi- i?2 2[miBQM BM j. +M #2 r`Bii2M b

T URkV E0 ξ 2 u (ξ) ,ξξ + 2E0 − E1 + ET 1 ξu (ξ) ,ξ + E1 − E2 u (ξ) = 0. h?2 +Q2{+B2Mi Ki`B+2b Ee0 - Ee1 M/ Ee2 Q7  bm`7+2 2H2K2Mi e `2 Q#iBM2/ #v ˆ e E0 = BT 1 DB1 |J|dηdζ, ˆe Ee1 = BT 2 DB1 |J|dηdζ, ˆe Ee2 = BT 2 DB2 |J|dηdζ.

URjV URj#V URj+V

e

h?2v `2 bb2K#H2/ iQ 7Q`K i?2 +Q2{+B2Mi Ki`B+2b E0 - E1 M/ E2 Q7  pQHmK2 2H2K2Mi BM 1[X URkV ++Q`/BM; iQ 2H2K2Mi +QMM2+iBpBivX h?Bb T`Q+2/m`2 Bb B/2MiB+H iQ i?i FMQrM 7`QK i?2 61JX ++Q`/BM; iQ (dN)- i?2 BMi2`MH MQ/H 7Q`+2b q (ξ) QM  bm`7+2 rBi?  +QMbiMi ξ +M #2 7Q`KmHi2/ b
q (ξ) = ξ E0 ξu (ξ) ,ξ +ET UR9V 1 u (ξ) .

h?2 Q`/2` Q7 i?2 /Bz2`2MiBH 2[miBQMb BM 1[X URkV +M #2 `2/m+2/ #v BMi`Q/m+BM;  p`B#H2 X (ξ)   0.5 ξ u (ξ) X (ξ) = . UR8V ξ −0.5 q (ξ) lbBM; 1[bX UR9V M/ UR8V- 1[X URkV Bb i`Mb7Q`K2/ iQ ξX (ξ),ξ = −ZX (ξ) , r?2`2 Z Bb  >KBHiQMBM Ki`Bt M/ Bb /2}M2/ b   E−1 ET − 0.5I −E−1 1 0 0
. Z= T − E E−1 − 0.5I −E2 + E1 E−1 1 0 0 E1

h?2 2B;2MpHm2 /2+QKTQbBiBQM Bb TTHB2/ iQ i?2 Ki`Bt Z M/ Bi H2/b iQ    −λ   Φu1 Φu2 ξ 0 c1 0 X (ξ) = , Φq1 Φq2 0 c2 0 ξλ d

UReV

URdV

UR3V

r?2`2 −λ M/ λ `2 i?2 2B;2MpHm2b Q7 ZX LQi2 i?i i?2 `2H T`ib Q7 HH λ `2 M2;iBp2 bBM+2 r2 +QMbB/2` QMHv #QmM/2/ /QKBMbX Φu1 - Φu2 - Φq1 M/ Φq2 `2 i?2 bm#@Ki`B+2b Q7 i?2 2B;2Mp2+iQ` Ki`Bt- r?BH2 c1 M/ c2 /2MQi2 i?2 BMi2;`iBQM +QMbiMibX b M Hi2`MiBp2-  MmK2`B+HHv `Q#mbi bQHmiBQM T`Q+2/m`2 #b2/ QM  a+?m` /2+QKTQbBiBQM Bb T`QTQb2/ BM (d3- dN)X .m2 iQ i?2 `2[mB`2K2Mi i?i 7Q`  #QmM/2/ /QKBM U0 ≤ ξ ≤ 1V i?2 /BbTH+2K2Mib b?QmH/ #2 }MBi2 i ξ = 0- i?2 bQHmiBQM Q#iBM2/ 7`QK 1[bX UR8V M/ UR3V Bb ;Bp2M BM i?2 7QHHQrBM; 7Q`K u (ξ) = Φu1 ξ −λ−0.5I c1 , q (ξ) = Φq1 ξ

−λ+0.5I

c1 .

URNV URN#V

PM i?2 #QmM/`v (ξ = 1)- i?2 biBzM2bb Ki`Bt Q7 i?2 pQHmK2 2H2K2Mi Uq (ξ = 1) = Km u (ξ = 1)V Bb Q#iBM2/ #v 2HBKBMiBM; i?2 BMi2;`iBQM +QMbiMib c1 b Km = Φq1 Φ−1 u1 ,

UkyV

r?2`2 i?2 BM/2t K /2MQi2b p`B#H2b rBi? `2bT2+i iQ i?2 Ki`Bt Ki2`BHX h?2 biBzM2bb Ki`Bt 7`QK 2+? pQHmK2 2H2K2Mi Bb bm#b2[m2MiHv bb2K#H2/ BMiQ i?2 ;HQ#H biBzM2bb Ki`Bt- r?B+? Bb i?2 bK2 b i?2 bb2K#HBM; T`Q+2bb BM i?2 61JX GbiHv-  bvbi2K Q7 HBM2` H;2#`B+ 2[miBQMb rBi? #QmM/`v +QM/BiBQMb Bb bQHp2/ M/ i?2 MQ/H /BbTH+2K2Mib `2 Q#iBM2/X h?2 bi`2bb2b M/ bi`BMb `2 +H+mHi2/ mbBM; 1[bX UNV M/ URRVX jX .Bb+`2i2 KQ/2HBM; Q7 }#2`b AM i?Bb b2+iBQM- i?2 /Bb+`2i2 KQ/2HBM; Q7 }#2`b Bb BMi`Q/m+2/X AM i?2 61J- i?2 K2b? Q7 i?2 }#2`b mbmHHv M22/b iQ #2 +QM7Q`KBM; iQ i?2 K2b? Q7 i?2 Ki`Bt- BX2X 7Q` 2+? MQ/2 QM i?2 }#2`i?2`2 Kmbi #2  +QBM+B/2Mi MQ/2 BM i?2 Ki`Bt K2b?X .m2 iQ i?2 /pMi;2 Q7 TQHviQTH 2H2K2Mib r?B+? `2 bi`B;?i7Q`r`/Hv 7Q`KmHi2/ BM i?2 a"61J-  bBKTH2 H;Q`Bi?K +M #2 BKTH2K2Mi2/ iQ 2Mbm`2 i?2 +QKTiB#BHBiv Q7 i?2 }#2` M/ Ki`Bt K2b?2bX aBM+2 /BbiQ`i2/ 2H2K2Mib +M #2 +`2i2/ #v 2K#2//BM; i?2 }#2`b- M //BiBQMH b?B7iBM; T`Q+2/m`2 Bb /2bB;M2/ iQ BKT`Qp2 i?2 K2b? [mHBivX jXRX aiBzM2bb Ki`Bt Q7 }#2`b h?2 }#2`b `2 KQ/2H2/ mbBM; i`mbb 2H2K2Mib r?B+? BM bT+2X h?mb- i?2 biBzM2bb Ki`Bt Q7  }#2` b2;K2Mi  Cx Cy Cx Cz Cx2 2  Cx Cy Cy Cz C y  Ef Af  C C C C Cz2 x z y z  Kf = 2  −Cx Cy −Cx Cz Lf  −Cx  −Cx Cy −Cy Cz −Cy2 −Cx Cz −Cy Cz −Cz2

`2 i`Mb7Q`K2/ ++Q`/BM; iQ i?2B` TQbBiBQM Kf BM j. Bb 2tT`2bb2/ b  −Cx Cy −Cx Cz −Cx2 −Cy Cz  −Cx Cy −Cy2  −Cx Cz −Cy Cz −Cz2  , UkRV Cx2 Cx Cy Cx Cz   Cy Cz  Cx Cy Cy2 Cx Cz Cy Cz Cz2

r?2`2 Ef Bb i?2 uQmM;Ƕb KQ/mHmb- Af Bb i?2 +`Qbb b2+iBQM `2- M/ Lf Bb i?2 H2M;i? Q7 i?2 }#2` b2;K2MiX Cx - Cy M/ Cz `2 i?2 /B`2+iBQM +QbBM2 r?B+? `2 /2}M2/ b x2 − x1 , Lf y2 − y1 , Cy = Lf 3

Cx =

UkkV Ukk#V

Cz =

z2 − z1 . Lf

Ukk+V

h?2 /B`2+iBQM +QbBM2 `2 +H+mHi2/ #b2/ QM i?2 +QQ`/BMi2b Q7 i?2 bi`iBM; (x1 , y1 , z1 ) M/ 2M/ (x2 , y2 , z2 ) TQBMi Q7 i?2 }#2` b2;K2MiX b K2MiBQM2/ #27Q`2- i?2 bvbi2K biBzM2bb Ki`Bt Bb +QKTmi2/ #v bmT2`BKTQbBM; i?2 ;HQ#H biBzM2bb Ki`Bt Q7 i?2 Ki`Bt Ki2`BH Km rBi? i?i Q7 i?2 }#2`b Kf UkjV

K = Km + Kf . jXkX 1K#2//BM; K2i?Q/

AM i?2 T`QTQb2/ K2i?Q/- i?2 BMBiBH K2b? Q7 i?2 Ki`Bt Bb ;2M2`i2/ rBi?Qmi +QMbB/2`BM; i?2 HQ+iBQM Q7 i?2 }#2`bX h?2 }#2`b `2 2K#2//2/ BM i?2 Ki`Bt Hi2` #v BMb2`iBM; i?2 BMi2`b2+iBQM TQBMib Q7 }#2`b M/ Ki`Bt pQHmK2 2H2K2Mib b //BiBQMH MQ/2bX PMHv M2` i?2 }#2` iBTb-  pQHmK2 /Bb+`2iBxiBQM Bb `2[mB`2/X jXkXRX 1K#2//BM; }#2`b BM k. h?2 T`Q+2/m`2 Q7 2K#2//BM;  }#2` BMiQ i?2 Ki`Bt Bb BHHmbi`i2/ BM 6B;bX j M/ 9 7Q` i?2 k. +b2X  [m/`BHi2`H 2H2K2Mi ABCD BM i?2 BMBiBH K2b? Bb b?QrM BM 6B;X jX  }#2` l1 rBi? `M/QK Q`B2MiiBQM Bb TbbBM; i?`Qm;? i?2 [m/`BHi2`H 2H2K2MiX h?2 BMi2`b2+iBQM TQBMib `2 K`F2/ b E M/ F - b22 6B;X j#X h?2 irQ BMi2`b2+i2/ 2/;2b Q7 i?2 [m/`BHi2`H 2H2K2MiAB M/ BC- `2 T`iBiBQM2/ BMiQ M2r 2/;2b AE- EB M/ BF - F C- `2bT2+iBp2HvX >2M+2- i?2 [m/`BHi2`H 2H2K2Mi #2+QK2b  TQHv;QM rBi? e 2/;2b U?2t;QMVX A7 b2p2`H }#2`b BMi2`b2+i M 2H2K2Mi- HH i?2 BMi2`b2+i2/ 2/;2b `2 T`iBiBQM2/ i i?Qb2 BMi2`b2+iBQM TQBMibX h?2 T`iBiBQMBM; T`Q+2/m`2 i`Mb7Q`Kb i?2 BMBiBHHv [m/`BHi2`H 2H2K2Mi BM 6B;X j+ BMiQ  TQHv;QM rBi? 3 2/;2b UQ+i;QMVX Ai Bb MQi2/ i?i i?2 BMi2`b2+iBQM TQBMi #2ir22M i?2 irQ }#2`b l1 M/ l2 Bb MQi BMb2`i2/ BMiQ i?2 K2b?- BX2X i?2 irQ }#2`b `2 MQi +QMM2+i2/ i?2`2 M/ i?mb +M /27Q`K 7`22Hv rBi?BM i?2 2H2K2MiH /QKBMX AM KMv bBimiBQMb-  }#2` iBT rBHH #2 HQ+i2/ BMbB/2  [m/`BHi2`H 2H2K2Mi b b?QrM BM 6B;X 9BX2X i?2 }#2` #2;BMbf2M/b BM i?i 2H2K2MiX *QMb2[m2MiHv- MQ BMi2`b2+iBQM TQBMi +M #2 +QKTmi2/ M/  /Bz2`2Mi TT`Q+? ?b iQ #2 /2pBb2/X AM i?2 +m``2Mi BKTH2K2MiiBQM- i?Bb +b2 Bb ?M/H2/ #v bm#/BpB/BM; i?2 +Q``2bTQM/BM; b+H2/ #QmM/`v TQHv;QM BMiQ b2p2`H bKHH2` 2H2K2Mib mbBM;

D

C

D

C

D

C l2

l1

A

B

UV  [m/`BHi2`H 2H2K2Mi BM i?2 K2b? Q7 i?2 Ki`BtX

A

E

F

B

l1

A

U#V  }#2` BMi2`b2+iBM; i?2 [m/`BHi2`H 2H2K2MiX

6B;m`2 j, 1K#2//BM; Q7 }#2` b2;K2Mib BM k.X

N

E

F

B

U+V hrQ }#2`b BMi2`b2+iBM; i?2 [m/`BHi2`H 2H2K2MiX

D

C

A

B

UV  }#2` iBT HQ+i2/ BMbB/2  [m/`BHi2`H 2H2K2MiX

D

C

A

B

U#V "Bb2+iBM; i?2 2/;2b Q7 i?2 [m/`BHi2`H 2H2K2MiX

D

C

A

B

U+V h?2 M2r 2H2K2Mib ;2M2`i2/ #v +QMM2+iBM; i?2 }#2` iBT M/ i?2 KB/@2/;2 TQBMibX

6B;m`2 9, 1K#2//BM; Q7 }#2` iBTb BM k.X

TQHvi`22 H;Q`Bi?K- +7X (3y- 3R)X h?2 #QmM/`v Q7 i?2 TQHv;QM Bb /Bb+`2iBx2/ #v #Bb2+iBM; HH Q7 Bib 2/;2b U6B;X 9#VX  M2r MQ/2 Bb BMb2`i2/ i i?2 }#2` iBT M/ +QMM2+i2/ iQ HH i?2 KB/@2/;2 TQBMib iQ 7Q`K M2r TQHv;QMH 2H2K2Mib- r?B+? Bb b?QrM BM 6B;X 9+X 7i2`r`/b- i?2 }#2` b2;K2Mi Bb BKTH2K2Mi2/ mbBM; i?2 T`2pBQmbHv /2b+`B#2/ 2K#2//BM; H;Q`Bi?K- b22 6B;X jX Ai Bb BKTQ`iMi iQ MQi2 i?i BM k. TTHB+iBQMb i?2 M2rHv ;2M2`i2/ 2H2K2Mib `2 Hrvb [m/`BHi2`HbX hQ pQB/ /BbiQ`i2/ 2H2K2Mib-  b?B7iBM; T`Q+2/m`2 Bb /2bB;M2/- r?B+? rBHH #2 2tTHBM2/ BM a2+iBQM jXjXRX jXkXkX 1K#2//BM; }#2`b BM j. AM j. T`Q#H2Kb- i?2 T`Q+2/m`2 Q7 2K#2//BM; i?2 }#2`b Bb bBKBH` iQ k.X A7  }#2` BMi2`b2+ib  pQHmK2 2H2K2Mi b b?QrM BM 6B;X 8- i?2 BMi2`b2+iBQM TQBMib #2ir22M i?2 }#2` M/ i?2 7+2b Q7 i?2 pQHmK2 2H2K2Mi `2 BMb2`i2/ b M2r MQ/2bX h?2 7+2b BMi2`b2+i2/ #v i?2 }#2` `2 bm#/BpB/2/ mbBM; i?2 bK2 K2i?Q/ b /Bb+mbb2/ BM i?2 k. +b2 r?2M i?2 }#2` iBT Bb HQ+i2/ rBi?BM i?2 2H2K2Mi Ub22 6B;X 9+VX >Qr2p2`- MQ pQHmK2 /Bb+`2iBxiBQM Bb `2[mB`2/X AM 6B;X 8#- i?2 ?2t?2/`QM #2+QK2b  TQHv?2/`QM rBi? Rk 7+2b 7i2` i?2 2K#2//BM;X 6Q` i?2 bF2 Q7 bBKTHB+Biv- BM j. TTHB+iBQMb QMHv QM2 }#2` Bb HHQr2/ iQ BMi2`b2+i rBi?  bT2+B}+ 7+2 QM i?2 #QmM/`vX Pi?2`rBb2- KmHiBTH2 Tii2`Mb

UV  ?2t?2/`QM 2H2K2Mi BM i?2 K2b? Q7 i?2 Ki`BtX

U#V  }#2` BMi2`b2+iBM; i?2 2H2K2MiX

U+V hrQ }#2`b BMi2`b2+iBM; i?2 2H2K2MiX

6B;m`2 8, 1K#2//BM; Q7 }#2` b2;K2Mib BM j.X

Ry

UV  }#2` iBT HQ+i2/ BMbB/2  ?2t?2/`QM 2H2K2MiX

U#V h?2 7+2b Q7 i?2 ?2t?2/`QM 2H2K2Mi `2 /Bb+`2iBx2/X

U+V h?2 2H2K2Mi Bb /BpB/2/ U/V h?2 M2r ?2t?2/`QM BMiQ M2r 2H2K2Mib 2H2K2Mib 7i2` +QMM2+iBM; i?2 }#2` bm#/BpBbBQMX iBT M/ i?2 +2Mi2`b Q7 Q`B;BMH 7+2bX

6B;m`2 e, 1K#2//BM; Q7 }#2` iBTb BM j.X

`2 TQbbB#H2 M/ i?2 2{+B2M+v Q7 i?2 K2i?Q/ rQmH/ #2 /2+`2b2/X L2p2`i?2H2bb- i?Bb `2[mB`2K2Mi +QmH/ #2 biBb}2/ #v 7m`i?2` K2b? `2}M2K2MiX LQi2 i?i Bi Bb HHQr2/ iQ ?p2 KQ`2 i?M QM2 }#2` TbbBM; i?`Qm;? i?2 bK2 pQHmK2 2H2K2Mi U6B;X 8+V- b HQM; b i?2 7+2b +M #2 /Bb+`2iBx2/ /B`2+iHvX A7  }#2` iBT Bb HQ+i2/ BMbB/2  pQHmK2 2H2K2Mi BM j. U6B;X eV- BX2X i?2 }#2` #2;BMbf2M/b BM i?i 2H2K2Mi- MQ BMi2`b2+iBQM TQBMi +M #2 +QKTmi2/ M/  /Bz2`2Mi TT`Q+? ?b iQ #2 /2pBb2/X AM i?2 +m``2Mi BKTH2K2MiiBQM i?Bb +b2 Bb ?M/H2/ #v bm#/BpB/BM; i?2 +Q``2bTQM/BM; b+H2/ #QmM/`v TQHv?2/`QM mbBM;  TQHvi`22 `2}M2K2Mi BM j.X h?2 /2iBHb QM i?2 BKTH2K2MiiBQM Q7 i?2 `2}M2K2Mi T`Q+2/m`2 +M #2 7QmM/ BM (ed)X b BM k. TTHB+iBQMb- i?2 }#2` iBT Bb i?2M +QMM2+i2/ iQ i?2 BMb2`i2/ MQ/2 BMbB/2 i?2 pQHmK2 2H2K2Mi b b?QrM BM 6B;X e+X LQi2 i?i  b?B7iBM; T`Q+2/m`2 iQ pQB/ /BbiQ`i2/ 2H2K2Mib rBHH #2 2tTHBM2/ BM i?2 7QHHQrBM; b2+iBQMX jXjX a?B7iBM; K2i?Q/ iQ BKT`Qp2 i?2 K2b? [mHBiv h?2 K2b? [mHBiv Q7 i?2 Ki`Bt ?b  bB;MB}+Mi BM~m2M+2 QM i?2 ++m`+v Q7 i?2 MHvbBbX AM i?Bb b2+iBQM-  b?B7iBM; b+?2K2 Bb /2bB;M2/ iQ BKT`Qp2 i?2 K2b? [mHBiv #27Q`2 2K#2//BM; i?2 }#2`b BMiQ i?2 Ki`BtX h?2 i`2iK2Mib +QM+2`MBM; i?2 }#2` b2;K2Mib M/ iBTb `2 BMi`Q/m+2/`2bT2+iBp2HvX jXjXRX a?B7iBM; K2i?Q/, }#2` iBT h?2 2K#2//BM; H;Q`Bi?K /2b+`B#2/ bQ 7` KB;?i `2bmHi BM K2b?2b UQ7 i?2 Ki`Bt Ki2`BHV 2t?B#BiBM; b2p2`2Hv /BbiQ`i2/ TQHviQTH 2H2K2Mib B7 bT2+B}+ +b2b `2Hi2/ iQ i?2 TQbBiBQM Q7 i?2 }#2` iBT rBi?BM i?2 2H2K2Mi `2 MQi iF2M BMiQ ++QmMiX AM i?2 k. BKTH2K2MiiBQM r2 ?p2 iQ /BbiBM;mBb? irQ +b2b, UBV i?2 }#2` iBT Bb +HQb2 iQ  MQ/2 Q7 i?2 TQHv;QMH 2H2K2Mi Q` UBBV i?2 }#2` iBT Bb +HQb2 iQ M 2/;2 Q7 i?2 TQHv;QMH 2H2K2MiX h?2 }`bi +b2 Bb /2TB+i2/ BM 6B;X d r?2`2 i?2 }#2` iBT O Bb +HQb2 iQ MQ/2 A BM 2H2K2Mi ABCDX A7 i?2 2H2K2Mi Bb bm#/BpB/2/ mbBM; i?2 K2i?Q/ BMi`Q/m+2/ BM a2+iBQM jXk /B`2+iHv- i?2`2 rBHH #2  +QM+p2 [m/`BHi2`H AEOH r?B+? b?QmH/ #2 pQB/2/ BM M MHvbBb Ub22 6B;X dVX h?2`27Q`2- MQ/2 A Bb b?B7i2/ iQ i?2 }#2` iBT O b /2TB+i2/ BM 6B;X d#X LQ bm#/BpBbBQM Bb `2[mB`2/ BM i?Bb +b2- M/ i?2 K2b? [mHBiv Q7 i?2 2H2K2Mi Bb bB;MB}+MiHv BKT`Qp2/X h?2 b?B7iBM; iQH2`M+2 7Q`  MQ/2 Bb tol = εn × ln - r?2`2 εn Bb  T`2/2}M2/ `2HiBp2 iQH2`M+2 UivTB+HHv-  pHm2 #2ir22M 0.2 M/ 0.3V M/ ln Bb i?2 KBMBKmK H2M;i? Q7 i?2 2/;2b +QMM2+i2/ iQ i?2 MQ/2X RR

D

G

C D

C D

G

C D

O H

F O

A

B

UV  MQ/2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` iBTX

O

H A

E

U#V h?2 MQ/2 b?B7i2/ iQ i?2 }#2` iBTX

C

F E

B A

U+V M 2/;2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` iBTX

B A

B

U/V h?2 2/;2 T`iBiBQM2/ i i?2 }#2` iBTX

6B;m`2 d, a?B7iBM; Q7  MQ/2 M/ T`iBiBQMBM; Q7 M 2/;2 M2`  }#2` iBT BM k.X

h?2 b2+QM/ bT2+BH +b2 Bb b?QrM BM 6B;X d+- r?2`2 i?2 }#2` iBT O Bb +HQb2 iQ i?2 2/;2 AD #mi MQi iQ Mv Q7 i?2 MQ/2bX h?2 [m/`BHi2`H DHOG Q#iBM2/ #v  /B`2+i bm#/BpBbBQM Bb b2p2`2Hv /BbiQ`i2/ U+QM+p2 [m/`BHi2`HVX AM Q`/2` iQ BKT`Qp2 i?2 K2b? [mHBiv- i?2 2/;2 AD Bb T`iBiBQM2/ BMiQ irQ 2/;2b AO M/ OD U6B;X d/V M/ MQ 7m`i?2` bm#/BpBbBQM Bb `2[mB`2/ 7Q` i?Bb 2H2K2MiX h?2 iQH2`M+2 7Q` M 2/;2 Bb tol = εe × le - r?2`2 εe Bb MQi?2` `2HiBp2 iQH2`M+2 r?B+? Bb ;2M2`HHv bKHH2` i?M εn UmbmHHv #2ir22M 0.1 M/ 0.2V M/ le Bb i?2 H2M;i? Q7 i?2 2/;2X AM j. TTHB+iBQMb- //BiBQMH +b2b ?p2 iQ #2 iF2M BMiQ +QMbB/2`iBQM iQ 2Mbm`2  ;QQ/ [mHBiv K2b? ;2M2`iBQMX hQ i?Bb 2M/- i?2 b?B7iBM; T`Q+2/m`2 Bb m;K2Mi2/ BM i?2 7QHHQrBM; rv, h?2 b?B7iBM; Q7 MQ/2b U6B;X 3 M/ 6B;X 3#V M/ T`iBiBQMBM; Q7 2/;2b U6B;X 3+ M/ 6B;X 3/V `2 i?2 bK2 b i?2 T`Q+2bb BM k.X AM //BiBQM- B7 i?2`2 Bb  }#2` iBT HQ+i2/ M2`  7+2 #mi MQM2 Q7 i?2 2/;2b Q` MQ/2b- 2X;X 7+2 ABCD BM 6B;X 32- i?2 TQHv?2/`QMb 7i2` bm#/BpBbBQM rBHH #2 /BbiQ`i2/ `QmM/ i?2 7+2X h?2`27Q`2- i?2 7+2 Bb bm#/BpB/2/ BMiQ b2p2`H M2r 7+2b i i?2 }#2` iBT b b?QrM BM 6B;X 37X h?2 pQHmK2 Q7 i?2 TQHv?2/`QM Bb MQi bm#/BpB/2/ MvKQ`2- i?2`27Q`2 MQ /BbiQ`i2/ TQHv?2/`QM rBHH #2 ;2M2`i2/X h?2 b?B7iBM; iQH2`M+2 i  7+2 Bb tol = εf × lf - r?2`2 εf Bb  `2HiBp2 iQH2`M+2 ivTB+HHv #2ir22M 0.05 M/ 0.1 M/ lf Bb i?2 p2`;2 H2M;i? Q7 i?2 2/;2b BM i?2 7+2X jXjXkX a?B7iBM; K2i?Q/, }#2` b2;K2Mi h?2 b?B7iBM; b+?2K2 BM a2+iBQM jXjXR 2HBKBMi2b i?2 /BbiQ`iBQM `QmM/ i?2 }#2` iBTbX >Qr2p2`i?2 MQ/2b r?B+? `2 +HQb2 iQ i?2 }#2` b2;K2Mib #mi MQi i?2 }#2` iBTb Kv +mb2 /BbiQ`iBQM b r2HHX AM 6B;X N- MQ/2 C Bb +HQb2 iQ i?2 }#2` b2;K2MiX 7i2` T`iBiBQMBM; i?2 2/;2b- i?2`2 rBHH #2  b?Q`i 2/;2 CF BM i?2 TQHv;QM- r?B+? Kv +mb2 BHH@+QM/BiBQM2/ +Q2{+B2Mi Ki`B+2b /m`BM; i?2 bQHmiBQMX AM Q`/2` iQ pQB/ bm+? +b2b- i?2 MQ/2 Bb b?B7i2/ iQ i?2 M2`2bi TQBMi QM i?2 }#2`X h?2 bK2 b?B7iBM; iQH2`M+2 b BM a2+iBQM jXjXR tol = εn × ln +M #2 mb2/X AM j. i?2 MQ/2b +HQb2 iQ i?2 }#2`b `2 b?B7i2/ mbBM; i?2 bK2 TT`Q+? b b?QrM BM 6B;bX Ry M/ Ry#X  M2r bBimiBQM `Bb2b r?2M bQK2 2/;2b Q7 i?2 pQHmK2 2H2K2Mi `2 +HQb2 i?2 }#2` U6B;X Ry+Vi?2 [mHBiv Q7 i?2 bm`7+2 /Bb+`2iBxiBQM rBHH #2 HQr b i?2 7+2b QM i?2 #QmM/`v `2 b2p2`2Hv /BbiQ`i2/X h?mb-  TQBMi Bb HQ+i2/ QM i?2 }#2` r?B+? Bb +HQb2bi iQ i?2 2/;2- 2X;X TQBMi E QM i?2 }#2` BM 6B;X Ry/X h?2 2/;2 AB Bb T`iBiBQM2/ BMiQ AE M/ EBX h?2 7+2b +QMiBMBM; i?2 Q`B;BMH 2/;2 AB rBHH #2 mT/i2/ ++Q`/BM;HvX b  `2bmHi- i?2 7+2b QM i?2 #QmM/`v 7i2` /Bb+`2iBxiBQM rBHH ?p2 M BKT`Qp2/ [mHBivX h?2 b?B7iBM; T`Q+2/m`2 +M BKT`Qp2 i?2 K2b? [mHBiv r?2M i?2 MmK#2` Q7 }#2`b Bb HQr M/ i?2 Rk

B

O O

A

A

UV  MQ/2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` iBTX

A

U+V M 2/;2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` iBTX

U#V h?2 MQ/2 b?B7i2/ iQ i?2 }#2` iBTX

C

C

D

D O

O

O B

B

A

B

A

U/V h?2 2/;2 T`iBiBQM2/ i i?2 }#2` iBTX

A

U2V  7+2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` iBTX

U7V h?2 7+2 T`iBiBQM2/ i i?2 }#2` iBTX

6B;m`2 3, a?B7iBM; Q7  MQ/2 M/ T`iBiBQMBM; Q7 M 2/;2 M/ 7+2 M2`  }#2` iBT BM j.X

D

A

F

E

C

D

B

A

C

E

U#V h?2 MQ/2 b?B7i2/ iQ i?2 }#2` b2;K2MiX

UV  MQ/2 Q7 M 2H2K2Mi +HQb2 iQ  }#2` b2;K2MiX

6B;m`2 N, a?B7iBM; Q7  MQ/2 M2`  }#2` b2;K2Mi BM k.X

Rj

B

D

B

D

D C

C

A

A UV hrQ MQ/2b Q7 M 2H2K2Mi +HQb2 iQ  }#2` b2;K2MiX

U#V h?2 MQ/2b b?B7i2/ iQ i?2 }#2` b2;K2MiX

A

E B

D C

F

F

A

E

B

U/V h?2 2/;2b T`iBiBQM2/ U+V hrQ 2/;2b Q7 M i i?2 M2`2bi TQBMib 2H2K2Mi +HQb2 iQ  }#2` QM i?2 }#2`X b2;K2MiX

6B;m`2 Ry, a?B7iBM; Q7 MQ/2b M/ T`iBiBQMBM; Q7 2/;2b M2`  }#2` b2;K2Mi BM j.X

}#2`b `2 +H2`Hv b2T`i2/ 7`QK 2+? Qi?2`X A7 i?2 }#2` /2MbBiv Bb `2K`F#Hv ?B;? Q` i?2`2 `2  HQi Q7 Qp2`HTTBM; }#2`b- i?2 b?B7iBM; +MMQi BKT`Qp2 i?2 K2b? [mHBiv MvKQ`2X PM i?2 Qi?2` ?M/- BM i?Qb2 +b2b Bi Bb `2bQM#H2 iQ bBKTHv b?B7i i?2 ;2QK2i`v Q7 i?2 }#2`b iQ i?2 MQ/2b Q7 i?2 K2b? Q7 i?2 Ki`Bt- r?B+? rBHH MQi z2+i i?2 Qp2`HH `2bmHi Q7 i?2 MHvbBbX q?2M i?2`2 Bb KQ`2 i?M QM2 }#2` iBT HQ+i2/ BMbB/2  pQHmK2 2H2K2Mi- Bi +M #2 ?M/H2/ #v i`BKKBM; i?2 }#2`b i i?2 #QmM/`v Q7 i?2 pQHmK2 2H2K2MiX AM i?Bb +b2- ?Qr2p2`- i?2 pQHmK2 7`+iBQM Q7 i?2 }#2`b rBHH #2 KQ/B}2/ b i?2 iQiH H2M;i? Q7 i?2 }#2`b Bb `2/m+2/X A7 i?Bb KQ/B}+iBQM ?b  bB;MB}+Mi BM~m2M+2 QM i?2 MmK2`B+H `2bmHi- //BiBQMH }#2`b +M #2 //2/ KMmHHv iQ +QKT2Mbi2 i?2 `2/m+iBQM Q7 }#2` pQHmK2 7`+iBQMX h?2 T`Q+2/m`2 Q7 /Bb+`2i2 KQ/2HBM; Q7 }#2`b M/ KQ/B7vBM; i?2 K2b? Q7 i?2 Ki`Bt Bb bmKK@ `Bx2/ BM H;Q`Bi?K RX h?2 T`QTQb2/ K2i?Q/ #b2/ QM i?2 a"61J ?b  p`B2iv Q7 /pMi;2b Qp2` Qi?2` MmK2`B@ +H K2i?Q/bX *QKT`BM; rBi? 61J mbBM; +QM7Q`KBM; K2b?2b- i?2 T`QTQb2/ K2i?Q/ `2/m+2b i?2 K2b?BM; /B{+mHiv- b bBKTH2 #mi 2{+B2Mi [m/i`22fQ+i`22 H;Q`Bi?Kb +M #2 2KTHQv2/ BM i?2 K2b? H;Q`Bi?K R .Bb+`2i2 KQ/2HBM; Q7 i?2 }#2`b BM 6_*b R, AMTmi pQHmK2 K2b? k, AMTmi }#2` ;2QK2i`v j, .2}M2 T`K2i2`b, εn - εe - εf 9, J`F #QmM/`v Q7 pQHmK2 K2b? 8, a?B7i MQ/2b iQ }#2` iBTb e, a?B7i 2/;2b iQ }#2` iBTb d, a?B7i 7+2b iQ }#2` iBTb 3, 1K#2/ }#2` iBTb N, a?B7i MQ/2b iQ }#2` b2;K2Mib Ry, a?B7i 2/;2b iQ }#2` b2;K2Mib RR, 1K#2/ }#2` b2;K2Mib Rk, *?2+F pBbB#BHBiv Rj, *QMbi`m+i MQ/2 Ki+?BM; HBbi R9, PmiTmi }#2` M/ Ki`Bt R9

;2M2`iBQM /m2 iQ i?2 TQbbB#BHBiv iQ +QMbi`m+i TQHviQTH 2H2K2Mib BM a"61JX JQ`2Qp2`- i?2 BKTH2@ K2MiiBQM Bb bi`B;?i7Q`r`/ M/ i?2 biBzM2bb Ki`Bt Bb TQbBiBp2@/2}MBi2 b MQ BMi2`7+2 +QMbi`BMib `2 BMi`Q/m+2/ b BM +QMp2MiBQMH 61J rBi? MQM@+QM7Q`KBM; K2b?2bX Ai Bb HbQ TQbbB#H2 iQ mb2 TT`Q+?2b #b2/ QM i?2 2ti2M/2/ }MBi2 2H2K2Mi K2i?Q/ Us61JV r?B+?- ?Qr2p2`- BMpQHp2b 2M`B+?@ K2Mi 7mM+iBQMbX h?Bb bi2T +M #2 pQB/2/ BM i?2 T`QTQb2/ K2i?Q/X AM i?2 HiiB+2 /Bb+`2i2 T`iB+H2 KQ/2H i?2 Ki`Bt Bb KQ/2H2/ rBi? `B;B/ TQHv?2/`H 2H2K2Mib- r?BH2 BM i?2 a"61J i?2 /27Q`KiBQM Q7 i?2 TQHv?2/`H 2H2K2Mib +M #2 BMp2biB;i2/X 9X LmK2`B+H 2tKTH2 AM i?Bb b2+iBQM- bBt MmK2`B+H 2tKTH2b `2 T`2b2Mi2/ iQ pHB/i2 i?2 T`QTQb2/ K2i?Q/X AM i?2 }`bi 2tKTH2- +QMbiMi bi`BM i2bib BM k. M/ j. `2 T`2b2Mi2/ iQ /2KQMbi`i2 i?2 HBM2` +QKTH2i2M2bb Q7 i?2 T`QTQb2/ K2i?Q/X h?2 MmK2`B+H `2bmHib `2 +QKT`2/ rBi? i?2 MHviB+H bQHmiBQM M/ K+?BM2 ++m`+v Bb Q#b2`p2/X AM i?2 b2+QM/ 2tKTH2-  bBM;H2 }#2` i2bi Bb mM/2`iF2M iQ bim/v i?2 bi`2bb /Bbi`B#miBQM M2` i?2 }#2`X h?2 bi`2bb THQi Bb +QKT`2/ rBi? 61J mbBM;  +QKK2`+BH bQ7ir`2 T+F;2 b?QrBM; bBKBH` /Bbi`B#miBQMbX  T`K2i`B+ bim/v Bb T2`7Q`K2/ BM i?2 i?B`/ 2tKTH2 iQ 2pHmi2 i?2 2z2+iBp2 uQmM;Ƕb KQ/mHmb Q7 6_* +QMbB/2`BM; /Bz2`2Mi }#2` pQHmK2 7`+iBQMbX h?2 `2bmHib ;`22 r2HH rBi? b2p2`H Qi?2` TT`Q+?2b 7`QK HBi2`im`2X AM i?2 7Qm`i? 2tKTH2-  bT2+BK2M Q7  }#2` `2BM7Q`+2/ TQ`Qmb Ki2`BH Bb KQ/2H2/X h?2 bi`2bb /Bbi`B#miBQMb rBi? M/ rBi?Qmi }#2`b `2 +QKTmi2/- M/ i?2 2z2+i Q7 i?2 }#2`b Bb MHvx2/X AM i?2 }7i? 2tKTH2 /B;BiH BK;2 Q7 +QM+`2i2 Q#iBM2/ 7Q`K  *h b+M- +QMbBbiBM; Q7 KQ`i` M/ ;;`2;i2b- Bb KQ/2H2/X >B;? /2MbBiv }#2`b `2 /Bbi`B#mi2/ BM i?2 KQ`i` iQ BMp2biB;i2 i?2B` 2z2+i QM i?2 Ki2`BH T`QT2`iB2bX AM i?2 Hbi 2tKTH2-  ?QKQ;2MBx2/ +QM+`2i2 +vHBM/2` `2BM7Q`+2/ #v bi22H }#2`b Bb T`2b2Mi2/X M Q+i`22 K2b? Bb ;2M2`i2/ 7Q` i?2 +vHBM/2` M/ i?2 bi`2bb /Bbi`B#miBQMb `2 THQii2/X 9XRX *QMbiMi bi`BM i2bi AM i?Bb 2tKTH2- irQ +QMbiMi bi`BM i2bib BM k. M/ j. `2 T`2b2Mi2/ iQ p2`B7v i?2 T`QTQb2/ K2i?Q/X h?2 ++m`+v Q7 i?2 `2bmHib Bb +?2+F2/ #v i?2 `2HiBp2 2``Q` MQ`K BM /BbTH+2K2Mi, eu =

kunum − uref k , kuref k

Uk9V

r?2`2 unum Bb i?2 MmK2`B+H `2bmHi +H+mHi2/ #v i?2 T`QTQb2/ K2i?Q/ M/ uref /2MQi2b i?2 M@ HviB+H bQHmiBQM r?B+? Bb mb2/ b `272`2M+2X 9XRXRX *QMbiMi bi`BM i2bi BM k. AM i?Bb 2tKTH2  b[m`2 #HQ+F `2BM7Q`+2/ rBi? irQ }#2`b BM THM2 bi`2bb +QM/BiBQMb Bb +QMbB/2`2/X h?2 /BK2MbBQM Q7 i?2 b[m`2 Bb 1 m × 1 m r?BH2 i?2 H2M;i? M/ `/Bmb Q7 i?2 }#2`b `2 1 m M/ 0.01 m- `2bT2+iBp2HvX i i?2 HQr2` H27i +Q`M2` Q7 i?2 Ki`Bt i?2 /BbTH+2K2Mib BM #Qi? /B`2+iBQMb `2 +QMbi`BM2/ r?BH2 i i?2 HQr2` `B;?i +Q`M2` QMHv i?2 /BbTH+2K2Mi BM p2`iB+H /B`2+iBQM Bb +QMbi`BM2/ U6B;X RRVX .m2 iQ i?2 }#2`b i?i `2 bmT2`BKTQb2/ iQ i?2 Ki`Bt Ki2`BH i?2 biBzM2bb Q7 i?2 +QKTQbBi2 Bb BM+`2b2/ HQ+HHvX AM Q`/2` iQ 2Mbm`2  +QMbiMi bi`2bb bii2 i?2 HQ/BM; ?b iQ #2 +?Qb2M bm+? i?i i?2 iQT M/ #QiiQK bm`7+2b biv THM2X PM i?2 iQT M/ #QiiQK bm`7+2b Q7 i?2 Ki`Bt- mMB7Q`K bm`7+2 i`+iBQMb t `2 TTHB2/ M/ TQBMi HQ/b F `2 TTHB2/ iQ i?2 2M/b Q7 i?2 }#2`bX uQmM;Ƕb KQ/mHmb M/ SQBbbQMǶb `iBQ Q7 i?2 Ki`Bt `2 Em = 103 Pa M/ ν = 0.3X h?2 uQmM;Ƕb KQ/mHmb Q7 i?2 }#2`b Bb Ef = 106 PaX h?2 MQ`KH bi`BM BM p2`iB+H /B`2+iBQM BM #Qi? i?2 R8

F

t

F 7

8

1m

99 rr = = 0.01 0.01 m m

66 33

44

55 F

1m t

F

11

17 15 15

13 13

22 00

UV :2QK2i`v M/ #QmM/`v +QM/BiBQMbX

16

14 14

12 12 10 10

11 11

10−3

U#V .Bb+`2iBxiBQM Q7 i?2 Ki`BtX

(4) (4)

(1) (1)

(3) (3)

0

uy [m]

(2) (2)

U+V .Bb+`2iBxiBQM Q7 i?2 }#2`bX

U/V o2`iB+H /BbTH+2K2Mi Q7 i?2 Ki`BtX

6B;m`2 RR, k. +QMbiMi bi`BM i2bi, ;2QK2i`v- #QmM/`v +QM/BiBQMb- /Bb+`2iBxiBQMb- M/ `2bmHibX

Ki`Bt M/ }#2`b Bb bbmK2/ b +QMbiMi εy = 0.001X h?2`27Q`2- i?2 2t+i /BbTH+2K2Mi }2H/ Bb HBM2`Hv /Bbi`B#mi2/ ux = −νεy x = −0.0003x,

Uk8V

uy = εy y = 0.001y,

Uk8#V

r?2`2 x M/ y `2 i?2 +QQ`/BMi2b Q7  TQBMiX .m2 iQ i?2 TTHB2/ .B`B+?H2i #QmM/`v +QM/BiBQMb i?2 QMHv MQM@x2`Q bi`2bb +QKTQM2Mi BM i?2 Ki`Bt Bb σmy = Em εy = 1 Pa.

UkeV

h?2`27Q`2- i?2 K;MBim/2 Q7 i?2 i`+iBQM TTHB2/ QM i?2 iQT M/ #QiiQK bm`7+2b Bb t = 1 PaX h?2 bi`2bb BM i?2 }#2`b Bb σfy =

Ef σmy = 1000 Pa. Em Re

UkdV

EMQrBM; i?2 pHm2 Q7 i?2 tBH bi`2bb BM i?2 }#2` M/ Bib +`Qbb b2+iBQM `2 U+B`+H2, A = πr2 V r2 +M +H+mHi2 i?2 `2[mB`2/ TQBMi 7Q`+2 F = πr2 σfy = 0.1π N.

Uk3V

h?2 b[m`2 Bb /Bb+`2iBx2/ BMiQ 9 TQHv;QMH 2H2K2Mib U6B;X RR#V- r?BH2 2+? TQHv;QM +QMiBMb e 2/;2b M/ 2+? }#2` Bb /Bb+`2iBx2/ BMiQ k b2;K2Mib U6B;X RR+VX h?2 +QMiQm` THQi Q7 i?2 p2`iB+H /BbTH+2K2Mi Q7 i?2 Ki`Bt Bb b?QrM BM 6B;X RR/- BM r?B+? HBM2` /BbTH+2K2Mi /Bbi`B#miBQM Bb Q#b2`p2/X h?2 2``Q` MQ`K +H+mHi2/ mbBM; 1[X Uk9V Bb 1.6588 × 10−14 r?B+? +QM}`Kb i?i i?2 b?T2 7mM+iBQMb `2 HBM2`Hv +QKTH2i2X LQi2 i?i 7Q` i?2 /Bb+`2iBxiBQM Q7 i?2 #QmM/`v M/ i?2 i`mbb 2H2K2Mib }`bi Q`/2` 2H2K2Mib `2 2KTHQv2/ r?B+? `2 +T#H2 Q7 2t+iHv `2T`2b2MiBM;  HBM2` /BbTH+2K2Mi }2H/X 9XRXkX *QMbiMi bi`BM i2bi BM j.  j. p2`bBQM Q7 i?2 +QMbiMi bi`BM i2bi Bb HbQ +QMbB/2`2/X h?2 ;2QK2i`v Q7 i?Bb 2tKTH2 Bb  +m#2 `2BM7Q`+2/ rBi? 9 }#2`b- r?B+? Bb b?QrM BM 6B;X RkX h?2 /BK2MbBQM Q7 i?2 +m#2 Bb 1 m × 1 m × 1 m r?BH2 i?2 H2M;i? Q7 i?2 }#2`b Bb 1 m- M/ i?2 `/Bmb ++QmMib 7Q` 0.01 mX h?2 `B;B/ #Q/v KQiBQM Bb +QMbi`BM2/ #v bT2+B7vBM; i?i uAx = uAy = uAz = 0- uCy = uCz = 0- uBz = 0X h?2 biBzM2bb Q7 i?2 +QKTQbBi2 Bb BM+`2b2/ HQ+HHv /m2 iQ i?2 }#2` `2BM7Q`+2K2MiX AM Q`/2` iQ 2Mbm`2  +QMbiMi bi`2bb bii2 i?2 HQ/BM; ?b iQ #2 +?Qb2M bm+? i?i i?2 iQT M/ #QiiQK bm`7+2b biv THM2X PM i?2 iQT M/ #QiiQK bm`7+2b Q7 i?2 +m#2- mMB7Q`K i`+iBQMb Q7 K;MBim/2 t `2 TTHB2/BM //BiBQM- TQBMi HQ/b F `2 TTHB2/ iQ i?2 2M/b Q7 i?2 }#2`b U6B;X Rk#VX uQmM;Ƕb KQ/mHmb M/ SQBbbQMǶb `iBQ Q7 i?2 Ki`Bt `2 Em = 103 Pa M/ ν = 0.3X h?2 uQmM;Ƕb KQ/mHmb Q7 i?2 }#2`b Bb Ef = 106 PaX h?2 bi`BM BM p2`iB+H /B`2+iBQM BM #Qi? i?2 Ki`Bt M/ i?2 }#2`b Bb T`2b+`B#2/ b  +QMbiMi εz = 0.001X h?2 /BbTH+2K2Mi }2H/ Bb 2tT`2bb2/ b ux = −νεz x = −0.0003x,

UkNV

uy = −νεz y = −0.0003y,

UkN#V

uz = εz z = 0.001z,

UkN+V

r?2`2 x- y M/ z `2 i?2 +QQ`/BMi2b Q7  TQBMiX .m2 iQ i?2 TTHB2/ .B`B+?H2i #QmM/`v +QM/BiBQMb i?2 QMHv MQM@x2`Q bi`2bb +QKTQM2Mi Bb i?2 MQ`KH bi`2bb BM z /B`2+iBQM σmz = Em εz = 1 Pa.

UjyV

h?2 K;MBim/2 Q7 i?2 i`+iBQM QM i?2 iQT M/ #QiiQK bm`7+2b Bb t = 1 PaX h?2 bi`2bb BM i?2 }#2`b Bb σfz =

Ef σmy = 1000 Pa. Em

UjRV

*QMbB/2`BM; i?2 +`Qbb b2+iBQM `2 Q7 i?2 }#2`b- i?2 tBH 7Q`+2 F BM i?2 }#2`b Bb F = πr2 σfz = 0.1π N.

UjkV

h?2 +m#2 Bb /Bb+`2iBx2/ BMiQ 3 ?2t?2/`H 2H2K2Mib- r?2`2b 2+? Q7 i?2 }#2`b Bb /Bb+`2iBx2/ BMiQ k b2;K2Mib M/ 2+? }#2` BMi2`b2+ib j 7+2b BM i?2 K2b? Q7 i?2 Ki`Bt- r?B+? `2 /BpB/2/ BMiQ M2r 7+2b M/ i?2 }#2`b `2 2K#2//2/ BMiQ i?2 Ki`BtX h?2 /Bb+`2iBxiBQMb Q7 i?2 Ki`Bt M/ i?2 Rd

F F F F

F F

F F

tt

.

1m

*



"

r=0.01 m tt

1m

F F

1m

F F

F F F F

U#V "QmM/`v +QM/BiBQMbX

0

uy [m]

10−3

UV :2QK2i`vX

U+V .Bb+`2iBxiBQMb Q7 i?2 Ki`Bt M/ i?2 }#2`bX

U/V o2`iB+H /BbTH+2K2Mi Q7 i?2 Ki`BtX

6B;m`2 Rk, j. +QMbiMi bi`BM i2bi, ;2QK2i`v- #QmM/`v +QM/BiBQMb- /Bb+`2iBxiBQMb- M/ `2bmHibX

}#2`b `2 b?QrM BM 6B;X Rk+- rBi? QMHv i?2 HBM2b QM i?2 pBbB#H2 #QmM/`v M/ i?2 QmiHBM2 Q7 i?2 +m#2 b?QrMX h?2 p2`iB+H /BbTH+2K2Mi Q7 i?2 Ki`Bt Bb b?QrM BM 6B;X Rk/ M/ Bi Bb Q#b2`p2/ i?i i?2 /Bb@ TH+2K2Mi Bb HBM2`Hv /Bbi`B#mi2/ BM i?2 KQ/2HX h?2 `2HiBp2 2``Q` MQ`K Bb 7.4534 × 10−14 - r?B+? K2Mb K+?BM2 ++m`+v Bb Q#iBM2/ mbBM; HBM2` 2H2K2MibX 9XkX aBM;H2 }#2` i2bi AM i?Bb 2tKTH2-  `2+iM;mH` #HQ+F `2BM7Q`+2/ #v  bBM;H2 }#2` Bb T`2b2Mi2/ iQ b?Qr i?2 bi`2bb /Bbi`B#miBQM M2`  }#2`X h?2 /BK2MbBQM Q7 i?2 #HQ+F Bb 4 m × 2 m M/ i?2 }#2` Bb HQ+i2/ i i?2 +2Mi2` Q7 i?2 #HQ+F Ub22 6B;X RjVX h?2 H2M;i? Q7 i?2 }#2` Bb 2 m M/ i?2 +`Qbb b2+iBQM `2 Bb 0.1 m2 X  mMB7Q`K i`+iBQM t = 1 MPa Bb TTHB2/ QM i?2 `B;?i 2/;2 Q7 i?2 Ki`BtX uQmM;Ƕb KQ/mHmb M/ R3

σyy [MPa]

2.0

SQBbbQMǶb `iBQ Q7 i?2 Ki`Bt `2 Em = 103 Pa M/ ν = 0.2- r?BH2 i?2 uQmM;Ƕb KQ/mHmb Q7 i?2 }#2` Bb Ef = 104 PaX  [m/i`22 K2b? Bb ;2M2`i2/ 7Q` i?2 Ki`Bt rBi? i?2 `2;BQM M2` i?2 }#2` iBTb `2}M2/ BM Q`/2` iQ `2T`2b2Mi i?2 2z2+i Q7 i?2 bi`2bb bBM;mH`BivX .m2 iQ i?2 /Bb+`2i2 KQ/2HBM; TT`Q+? 7Q` i?2 }#2`b miBHBxBM; i`mbb 2H2K2Mib-  bBM;mH` TQBMi i i?2 iBT Bb BM2pBi#H2X h?2`27Q`2- Bib 2z2+i QM i?2 `2bmHib Bb b?QrM #v i?2 HQ+H K2b? `2}M2K2Mi T`Q+2/m`2 r?B+? Bb QM2 Q7 i?2 bi`2M;i?b Q7 a"61JX h?2 KtBKmK 2H2K2Mi bBx2 Bb 0.25 m- i?2 KBMBKmK 2H2K2Mi bBx2 M2` i?2 }#2` iBTb Bb 7.8125 × 10−3 m M/ i?2 2H2K2Mi bBx2 BM i?2 KB//H2 Q7 i?2 }#2` Bb 0.0625 mX h?2 K2b? +QKT`Bb2b  iQiH Q7 e8e 2H2K2Mib M/ dej MQ/2bX AM i?Bb +b2- i?2 }#2` Bb HB;M2/ rBi? i?2 Ki`Bt r?B+? K2Mb i?2 biBzM2bb Ki`B+2b +M #2 bmT2`BKTQb2/ /B`2+iHv rBi?Qmi i?2 M22/ 7Q` i?2 2K#2//BM; M/ b?B7iBM; T`Q+2/m`2b BMi`Q/m+2/ BM a2+iBQM jX h?2 MQ`KH bi`2bb BM i?2 ?Q`BxQMiH /B`2+iBQM BM i?2 Ki`Bt Bb THQii2/ BM 6B;X Rj#X  }MBi2 2H2K2Mi K2b? Bb ;2M2`i2/ BM "Zla 7Q` +QKT`BbQM b b?QrM BM 6B;X Rj/X h?2 i`BM;mH` 2H2K2Mi *Saj U+QMiBMmmK 2H2K2Mi- THM2 bi`2bb +QM/BiBQMb- i?`22 MQ/2b- +QMbiMi bi`BM i`BM;H2V Bb mb2/ rBi? i?2 bK2 2H2K2Mi bBx2 bT2+B}2/ BM i?2 [m/i`22 K2b? r?B+? `2bmHib BM  iQiH Q7 Rdy9 2H2K2Mib M/ 3dd MQ/2bX LQi2 i?i BM "Zla i?2 i`BM;mH` K2b? 7Q` i?2 Ki`Bt M/ i?2 i`mbb K2b? 7Q` i?2 }#2` `2 HbQ +QKTiB#H2X h?2 bi`2bb HQM; i?2 Ti? Ĝ U`2/ HBM2 BM 6B;X RjV Bb THQii2/ BM 6B;X Rj+X h?2 pHm2 BM i?2 `2;BQMb M2` i?2 }#2` iBT Ub?/2/ `2V Bb MQi K2MBM;7mH #2+mb2 Q7 i?2 bi`2bb bBM;mH`Biv i i?2 }#2` iBTX Ai +M #2 Q#b2`p2/ i?i i?2 +m`p2b Q7 a"61J M/ 61J `2 BM 2t+2HH2Mi ;`22K2Mi QmibB/2 i?2 b?/2/ `2X h?2 `2bmHib Q#iBM2/ 7`QK a"61J M/ 61J `2 +QKT`2/ mbBM; HBM2` BMi2`TQHiBQM M/ i?2 2``Q` MQ`K Bb 0.84%X

x y

0

Si? Ĝ

U#V ai`2bb σyy BM Ki`Bt Ua"61JVX

UV :2QK2i`v M/ #QmM/`v +QM/BiBQMbX

σyy [MPa]

σyy [MPa] 0

1

2 y [m]

3

4

0

2 1 0

61J

a"61J

2.0

8 7

U+V ai`2bb σyy pbX +QQ`/BMi2 y HQM; Si? Ĝ UBM+Hm/BM; }#2`VX

U/V ai`2bb σyy BM Ki`Bt U61JVX

6B;m`2 Rj, aBM;H2 }#2` i2bi, ;2QK2i`v M/ #QmM/`v +QM/BiBQMb- +QMiQm` THQib Q7 bi`2bb σyy BM Ki`Bt- M/ THQi Qp2` Si? Ĝ HQM; }#2`X

RN

9XjX S`K2i`B+ bim/v Q7 2z2+iBp2 uQmM;Ƕb KQ/mHmb AM i?Bb 2tKTH2-  T`K2i`B+ bim/v Bb T2`7Q`K2/ iQ BMp2biB;i2 i?2 BM~m2M+2 Q7 i?2 }#2` pQHmK2 7`+iBQM QM i?2 2z2+iBp2 uQmM;Ƕb KQ/mHmbX h?2 bBx2 Q7 i?2 bKTH2 Bb 200 mm × 200 mmX h?2 H2M;i? Lf Q7 i?2 }#2`b ++QmMib 7Q` 20 mm M/ i?2 /BK2i2` df Bb 0.2 mmX h?2`27Q`2- i?2 bT2+i `iBQ Bb Lf/df = 100X h?2 }#2`b `2 `M/QKHv /Bbi`B#mi2/ #mi MQM@Qp2`HTTBM;X h?2 ;2QK2i`v Q7 i?2 }#2`b Bb T2`BQ/B+- BX2X }#2`b QM i?2 QTTQbBi2 #QmM/`v 2/;2b `2 +QMiBMmQmbX h?2 /2iBHb Q7 i?2 BKTH2K2MiiBQM Q7 T2`BQ/B+ ;2QK2i`v +M #2 7QmM/ BM (3k)X h?2 MmK#2` Q7 }#2`b N Bb 0, 10, 20, . . . , 250 i?2`27Q`2 i?2 pQHmK2 7`+iBQM `M;2b 7`QK y iQ 2.31%X 6Q` 2+? N - ky /Bz2`2Mi }#2` /Bbi`B#miBQMb `2 ;2M2`i2/ M/ i?2 `2bmHib `2 p2`;2/ iQ Q#iBM  biiBbiB+HHv bB;MB}+Mi pHm2X ;BM  [m/i`22 K2b? Bb ;2M2`i2/ 7Q` i?2 Ki`Bt- BM r?B+? i?2 KtBKmK 2H2K2Mi bBx2 Bb 8 mm M/ i?2 KBMBKmK Bb 2 mmX h?2 K2b? Bb `2}M2/ HQM; i?2 }#2`b b r2HH b i?2 #QmM/`v- i?2`27Q`2 i?2 K2b? Bb HbQ T2`BQ/B+X hrQ Q7 i?2 bKTH2 K2b?2b `2 b?QrM BM 6B;bX R9 M/ R9#X S2`BQ/B+ #QmM/`v +QM/BiBQMb `2 TTHB2/ BM i?Bb 2tKTH2X h?2 #QmM/`v Q7 i?2 Ki`Bt +M #2 /BpB/2/ BMiQ irQ ;`QmTb- i?2 TQbBiBp2 #QmM/`v UiQT M/ `B;?iV M/ i?2 M2;iBp2 #QmM/`v U#QiiQK M/ H27iVX h?2 /BbTH+2K2Mi QM i?2 #QmM/`v +M #2 2tT`2bb2/ b (3j) u+ − u− = g,

UjjV

r?2`2 u+ Bb i?2 /BbTH+2K2Mi QM i?2 TQbBiBp2 #QmM/`v M/ u− `2T`2b2Mib i?2 /BbTH+2K2Mi QM i?2 M2;iBp2 #QmM/`v i i?2 +Q``2bTQM/BM; MQ/2bX g Bb  p2+iQ` #b2/ QM i?2 T`2b+`B#2/ bi`BM }2H/ εX AM i?Bb +b2- i?2 QMHv MQM@x2`Q bi`BM +QKTQM2Mi Bb εy = 0.001X h?2`27Q`2- i?2 /Bz2`2M+2 BM i?2 p2`iB+H /BbTH+2K2Mi pHm2b #2ir22M iQT M/ #QiiQK bm`7+2 Bb 0.2 mm- r?BH2 HH i?2 Qi?2` 2Mi`B2b BM g `2 x2`QX h?2 /BbTH+2K2Mi i i?2 iQT- #QiiQK- H27i M/ `B;?i #QmM/`B2b Bb 2tT`2bb2/ #v i?2 p2+iQ`b ut - ub - ul M/ ur `2bT2+iBp2HvX LQi2 i?i i?2 /BbTH+2K2Mib i i?2 H27i M/ `B;?i@?M/ bB/2 `2 2[mH Uur = ul V- r?BH2 i?2 7QHHQrBM; `2HiBQM ?QH/b 7Q` i?2 /BbTH+2K2Mib i i?2 iQT M/ #QiiQK,   0       0.2           0     0.2 ut − ub = . Uj9V   XX     X       0         0.2

h?2 ?Q`BxQMiH M/ p2`iB+H /BbTH+2K2Mib i i?2 HQr2` H27i +Q`M2` `2 +QMbi`BM2/ iQ T`2p2Mi `B;B/ #Q/v KQiBQMbX h?2 uQmM;Ƕb KQ/mHmb Q7 i?2 Ki`Bt Bb Em = 20.8 GPa M/ i?2 SQBbbQMǶb `iBQ ++QmMib 7Q` νm = 0.208- r?2`2b i?2 uQmM;Ƕb KQ/mHmb Q7 i?2 }#2`b Bb Ef = 200 GPaX h?2 MQ`KHBx2/ 2z2+iBp2 uQmM;Ƕb KQ/mHmb Bb THQii2/ ;BMbi i?2 }#2` pQHmK2 7`+iBQM BM 6B;X R9+X LQi2 i?i i?2 `M;2 Q7 i?2 MQ`KHBx2/ 2z2+iBp2 uQmM;Ƕb KQ/mHmb Bb #2ir22M 1.0 . . . 1.06X aQK2 Qi?2` `2bmHib iF2M 7`QK HBi2`im`2- BM+Hm/BM; 2tT2`BK2MiH /i #v qBHHBKbQM (39)- MHviB@ +H TT`Q+?2b #v z/H  E`/Qb (RR) M/ ?K/  G;Qm/b (38) b r2HH b  MmK2`B+H bQHmiBQM UbiQ+?biB+ KmHiBb+H2 KQ/2H #v :mM 2i HX (3e)V `2 HbQ THQii2/ 7Q` +QKT`BbQMX Ai Bb Q#b2`p2/ i?i i?2 T`QTQb2/ K2i?Q/ Ki+?2b i?Qb2 `2bmHib r2HHX

ky

UV J2b? rBi? 8y }#2`b Uνf = 0.46%VX

1.10

U#V J2b? rBi? k8y }#2`b Uνf = 2.31%VX

qBHHBKbQM URNd9V :mM 2i HX UkyR8V ?K/ 2i HX URNNRV z/H 2i HX URNdeV a"61J

E/E0 [−]

1.08

1.04 1.02 1.00

Lf/df

0

0.5

1.0

νf [%]

1.5

= 100

2.0

U+V LQ`KHBx2/ uQmM;Ƕb KQ/mHmb E/E0 pbX pQHmK2 7`+iBQM Q7 }#2`b νf X 6B;m`2 R9, S`K2i`B+ bim/v Q7 2z2+iBp2 uQmM;Ƕb KQ/mHmbX

9X9X 6B#2` `2BM7Q`+2/ TQ`Qmb Ki2`BH AM i?Bb 2tKTH2-  bT2+BK2M Q7 TQ`Qmb 2TQtv `2BM7Q`+2/ #v ;Hbb }#2`b BM THM2 bi`2bb +QM/BiBQMb Bb +QMbB/2`2/ Ub22 6B;X R8V- i?2 /BK2MbBQM Q7 r?B+? Bb 60 mm × 60 mmX h?2`2 `2 jR TQ`2b BM i?2 Ki`Bt rBi?  /BK2i2` Q7 4 mm- bQ i?2 TQ`QbBiv Q7 i?2 Ki2`BH Bb φ = 10.8%X 9d }#2`b `2 `M/QKHv /Bbi`B#mi2/ BM i?2 TQ`Qmb Ki`Bt rBi?Qmi Qp2`HTTBM; Q` TbbBM; i?`Qm;? i?2 ?QH2bX h?2 H2M;i? Q7 i?2 }#2`b Bb Lf = 10 mm M/ i?2 /BK2i2` ++QmMib 7Q` df = 0.05 mm- i?2`27Q`2 i?2 pQHmK2 7`+iBQM Bb 0.65%X  [m/i`22 K2b? Bb ;2M2`i2/ 7Q` i?2 Ki`Bt- r?2`2#v i?2 KtBKmK 2H2K2Mi bBx2 Bb 0.375 mmX h?2 K2b? M2` i?2 }#2` Bb `2}M2/ iQ  KBMBKmK 2/;2 H2M;i? Q7 0.047 mmr?BH2 i?2 `2;BQM M2` i?2 TQ`2b `2}M2/ iQ 0.023 mm U6B;X R8#V- r?B+? `2bmHib BM  iQiH Q7 Rk8de MQ/2b M/ NjRk 2H2K2MibX  mMB7Q`K p2`iB+H i`+iBQM Q7 10 MPa Bb TTHB2/ QM i?2 iQT bm`7+2 Q7 i?2 Ki`BtX h?2 uQmM;Ƕb KQ/mHmb M/ SQBbbQMǶb `iBQ Q7 i?2 2TQtv `2 Em = 3.5 GPa M/ ν = 0.3kR

U#V J2b?X

0

0

σ [MPa]

uy [mm]

400

0.021

UV :2QK2i`vX

U/V tBH bi`2bb σ BM }#2`bX

0

0

σyy [MPa]

σyy [MPa]

20

20

U+V .BbTH+2K2Mi uy BM p2`iB+H /B`2+iBQMX

U2V LQ`KH bi`2bb σyy BM p2`iB+H /B`2+iBQM BM Ki`Bt UBM+Hm/BM; }#2`bVX

U7V LQ`KH bi`2bb σyy BM p2`iB+H /B`2+iBQM BM Ki`Bt U2t+Hm/BM; }#2`bVX

6B;m`2 R8, SQ`Qmb Ki2`BH rBi? }#2`b, ;2QK2i`v- /BbTH+2K2Mi M/ bi`2bb2b BM Ki`Bt M/ }#2`bX

kk

r?BH2 i?2 ;Hbb }#2`b 72im`2  bB;MB}+MiHv ?B;?2` uQmM;Ƕb KQ/mHmb Ef = 70 GPaX h?2 p2`iB+H /BbTH+2K2Mi Q7 i?2 TQ`Qmb Ki2`BH Bb THQii2/ BM 6B;X R8+ M/ i?2 tBH bi`2bb BM }#2`b Bb b?QrM BM 6B;X R8/X h?2 MQ`KH bi`2bb BM i?2 Ki`Bt BM p2`iB+H /B`2+iBQM Bb /BbTHv2/ BM 6B;X R82- r?BH2  THBM 2TQtv rBi?Qmi }#2` `2BM7Q`+2K2Mi Bb MHvx2/ mbBM; i?2 bK2 K2b? BM 6B;X R87X Ai +M #2 Q#b2`p2/ i?i rBi? }#2` `2BM7Q`+2K2Mi- i?2 bi`2bb +QM+2Mi`iBQM M2` i?2 TQ`2b Bb HH2pBi2/ M/ i?2 Qp2`HH bi`2bb /Bbi`B#miBQM Bb KQ`2 mMB7Q`KX h?2 p2`;2 pQM JBb2b bi`2bb `QmM/ 2+? TQ`2 Bb +H+mHi2/ M/ +QKT`2/- r?B+? b?Qrb i?i i?2 bi`2bb2b M2` kN TQ`2b `2 `2/m+2/ #v i?2 }#2` `2BM7Q`+2K2MiX :2QK2i`B+H /i 7Q` i?Bb 2tKTH2- BM+Hm/BM; i?2 HQ+iBQM Q7 i?2 TQ`2b M/ }#2`b +M #2 7QmM/ BM h#H2 XR BM i?2 TT2M/BtX 9X8X 6B#2` `2BM7Q`+2/ +QM+`2i2 BM k. h?2 +m``2Mi M/ i?2 7QHHQrBM; 2tKTH2b BHHmbi`i2 MQi?2` 72im`2 Q7 i?2 T`2b2Mi2/ H;Q`Bi?KBX2X i?2 bi`B;?i7Q`r`/ +QK#BMiBQM rBi? i2+?MB[m2b 7Q` BK;2@#b2/ MHvbBbX AM i?2 +m``2Mi b2+iBQM-  }#2` `2BM7Q`+2/ +QM+`2i2 bT2+BK2M Bb MHvx2/X h?2 KQ/2H Bb `2T`2b2Mi2/ #v  /B;BiH BK;2 Q7 i?2 +QM+`2i2 b /BbTHv2/ BM 6B;X Re- i?2 /BK2MbBQM Q7 r?B+? Bb 256 mm × 256 mmX h?2 +QM+`2i2 +QKT`Bb2b irQ +QKTQM2Mib- BX2X i?2 KQ`i` U;`2vV M/ i?2 ;;`2;i2 U#H+FVX 6B#2`b rBi? H2M;i? Q7 Lf = 40 mm M/ /BK2i2` Q7 df = 0.1 mm `2 `M/QKHv /Bbi`B#mi2/ BM i?2 KQ`i`- +7X 6B;X Re#X hrQ [m/i`22 K2b?2b `2 ;2M2`i2/ #b2/ QM i?2 BK;2- b22 6B;X ReX h?2 }`bi K2b? Bb `2}M2/ QMHv QM i?2 BMi2`7+2 #2ir22M i?2 KQ`i` M/ i?2 ;;`2;i2 b b?QrM BM 6B;X Re+X h?Bb +Q`b2 K2b? +QKT`Bb2b  iQiH MmK#2` Q7 RdNy3 MQ/2b M/ NR9e 2H2K2MibX AM  M2ti bi2T- i?2 K2b? BM i?2 r?QH2 KQ`i` Bb `2}M2/ #2+mb2 Q7 i?2 ?B;? }#2` /2MbBiv BM i?Bb 2tKTH2- +7X 6B;X Re7X h?Bb `2}M2K2Mi `2bmHib BM  K2b? +QKT`BbBM; 38Rje MQ/2b M/ 88yjR 2H2K2Mib BM iQiHX  mMB7Q`K i2MbBQM Q7 1 MPa Bb TTHB2/ QM i?2 iQT bm`7+2 Q7 i?2 Ki`BtX hQ /2b+`B#2 i?2 HBM2` 2HbiB+ #2?pBQm` Q7 i?2 Ki2`BHi?2 uQmM;Ƕb KQ/mHmb M/ SQBbbQMǶb `iBQ Q7 i?2 ;;`2;i2 `2 Ea = 55 GPa M/ νa = 0.25- r?BH2 i?2 +Q``2bTQM/BM; Ki2`BH T`K2i2`b Q7 i?2 KQ`i` UKi`BtV M/ }#2`b ++QmMi 7Q` Em = 10 GPaνm = 0.2- M/ Ef = 200 GPa- `2bT2+iBp2HvX h?2 +QMiQm` THQib Q7 i?2 MQ`KH bi`2bb BM p2`iB+H /B`2+iBQM BM i?2 +QM+`2i2 `2 b?QrM BM 6B;bX Re2 M/ Re?X 6m`i?2`KQ`2- 6B;bX Re/ M/ Re; BHHmbi`i2 i?2 tBH bi`2bb BM i?2 }#2`bX Ai +M #2 Q#b2`p2/ i?i /2bTBi2 i?2 /Bz2`2Mi K2b? bBx2b- i?2 `2bmHib Q#iBM2/ 7`QK i?2 irQ K2b?2b `2 bBKBH`X h?Bb H2/b iQ i?2 +QM+HmbBQM i?i i?2 +Q`b2` K2b? Bb HbQ #H2 iQ +Tim`2 i?2 KBM 2z2+ib 2t2`i2/ #v i?2 }#2`bX 9XeX 6B#2` `2BM7Q`+2/ +QM+`2i2 BM j. AM i?Bb 2tKTH2-  +vHBM/`B+H bT2+BK2M K/2 Q7 bi22H }#2` `2BM7Q`+2/ +QM+`2i2 Bb MHvx2/X h?2 6_* +QMbBbib Q7 b2H7@+QKT+iBM; +QM+`2i2 M/ ?QQF2/@2M/ bi22H }#2`b UH2M;i? Lf = 50 mm- /BK2i2` df = 1 mmV rBi?  }#2` +QMi2Mi Q7 80 kg m−3 - r?B+? +Q``2bTQM/b iQ TT`QtBKi2Hv 2.5 × 105 }#2`b T2` m3 X h?2 +vHBM/2` mM/2` +QMbB/2`iBQM- rBi?  ?2B;?i Q7 250 mm M/  /BK2i2` Q7 100 mmrb /`BHH2/ 7`QK  H`;2@b+H2 ~QQ` bH#X LQi2 i?i i?2 +m``2Mi bKTH2 `272`b iQ aT2+BK2M kb /2b+`B#2/ BM (k3)- r?2`2 7m`i?2` /2iBH2/ BM7Q`KiBQM QM i?2 2tT2`BK2MiH T`Q+2/m`2 M/ i?2 bKTH2 /2b+`BTiBQM +M HbQ #2 7QmM/X 7i2`r`/b- KB+`Q@+QKTmi2/ iQKQ;`T?v Uƒ*hV b+Mb r2`2 iF2M i >2HbBMFB lMBp2`bBiv rBi? +mbiQK@#mBHi ƒ*h 2[mBTK2Mi- LMQiQK R3y L6- i  `2bQHmiBQM Q7 0.123 mmX h?2 bT2+BK2M rb b+MM2/ BM b2+iBQMb bm+? i?i i?2 +m``2Mi /i `272`b iQ i?2 KB/@ b2+iBQM Q7 i?2 +vHBM/2` rBi?  ?2B;?i Q7 71 mmX "b2/ QM i?2 ƒ*h /i- TQbBiBQMb M/ Q`B2MiiBQMb Q7 i?2 }#2`b r2`2 Q#iBM2/ pB  7mHHv@miQKiB+ >2bbBM Ki`Bt@#b2/ MHvbBb b /2b+`B#2/ BM (33- 3N)X  pQHmK2i`B+ BK;2 `2bmHiBM; 7`QK i?Bb MHvbBb Bb b?QrM BM 6B;X RdX kj

UV AK;2- #b2/ QM (3d)X

0

U+V *Q`b2 K2b?X

σ [MPa]

U/V tBH bi`2bb σ BM }#2`b U+Q`b2 K2b?VX

0

U7V 6BM2 K2b?X

U#V :2QK2i`vX

σ [MPa]

U;V tBH bi`2bb σ BM }#2`b U}M2 K2b?VX

20 0

σyy [MPa]

U2V ai`2bb σyy BM p2`iB+H /B`2+iBQM BM Ki`Bt U+Q`b2 K2b?VX

20 0

σyy [MPa]

2

U?V ai`2bb σyy BM p2`iB+H /B`2+iBQM BM Ki`Bt U}M2 K2b?VX

6B;m`2 Re, 6B#2` `2BM7Q`+2/ +QM+`2i2 BM k., ;2QK2i`v- /BbTH+2K2Mi M/ bi`2bb2b BM Ki`Bt M/ }#2`bX

k9

2

U#V :2QK2i`vX

0.8

σzz [MPa]

1.2

UV oQHmK2i`B+ BK;2X

U/V o2`iB+H bi`2bb σzz U+HBT pB2r i x = 25 mmVX

0.8

0.8

σzz [MPa]

σzz [MPa]

1.2

1.2

U+V J2b?X

U2V o2`iB+H bi`2bb σzz U+HBT pB2r i x = 50 mmVX

U7V o2`iB+H bi`2bb σzz U+HBT pB2r i x = 75 mmVX

6B;m`2 Rd, 6B#2` `2BM7Q`+2/ +QM+`2i2 BM j., BK;2- ;2QK2i`v- K2b?- M/ p2`iB+H bi`2bb BM Ki`Bt M/ }#2`bX

k8

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j9

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A.

x [mm]

y [mm]

z [mm]

nx [−]

ny [−]

nz [−]

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j8

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A.

x [mm]

y [mm]

z [mm]

nx [−]

ny [−]

nz [−]

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je

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A.

x [mm]

y [mm]

z [mm]

nx [−]

ny [−]

nz [−]

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