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HAL Id: inria-00110560https://hal.inria.fr/inria-00110560v2

Submitted on 27 Nov 2006

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Nonsmooth fracture dynamics using a cohesive zoneapproach

Vincent Acary, Yann Monerie

To cite this version:Vincent Acary, Yann Monerie. Nonsmooth fracture dynamics using a cohesive zone approach. [Re-search Report] RR-6032, INRIA. 2006, pp.56. inria-00110560v2

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Nonsmooth fracture dynamics using a cohesive zoneapproach

Vincent Acary — Yann Monerie

N° 6032

Octobre 2006

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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K(β) = β

(CNn⊗ n+ CT

uT ⊗ uT

‖uT ‖2

),

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rl`twCT

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β

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β = 0'ILN>`\^ILNnx j4ILN_Zedp4N&¶_jl`LN'dgZaLpYOwLrlMOrllNFw7QTZvatrlpYlldp`Ojlcyw[N_c\ej-rlj4dwrs`:Y

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βrl`LdgZvILNFZ>´HJILNTaL`tx \edjl`

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βc^jlM rd`Ldp\edgrsrlpaLNY:dp`L<N \]!N>N_`1rl`tw

0\^j¶_N>c^jt

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wLj5NFZ`Ljs\-w[N>o<N>`uwN>²:otpdgx dp\eYjl`\eILNw[dgZvotr4x N>MON_`4\&k]aLMOo[u]

ta[\-jl`LYjl`β©Lj4cdp`tZv\^rl`tx N4E\eILN

j4pj'd`Lj4ceM x>rs`¢<NxyILj4ZeN>`

Rcoh = β R

cohmax

rl`twg(β, β, uN , uT

)= 0

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'ItN>c^NRcoh

max

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βZeatxyI¢rlZ

Rcoh = Υ (β)

rl`twg(β, β, uN , uT

)= 0

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g

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RN ∈ ∂uNΨ(uN , β),

«.s¬'ItN>c^N

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Ψ(uN , β) = ψ(β, uN ) + IR+(uN ) + I[0,1](β).« ,4¬

·&j4MOotrsc^dp`t\ej\^ILNN_f5atr\^dpj4`« #¬)5\eILNTN_f5atrs\edjl`tZ«.l¬irs`uw«7,5¬´Y:dpN_w\ej\^ILNnj4pj'd`L-jlc^M jlc&\eILNx>jlILNFZvdlNTjlcyx N4

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ψ(β, uN ).«7%4¬

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∂2uN

= 0&j4aLw

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'dp\eI[u]

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β = ∂χΛ(χ),« #F4¬

'ItN>c^Nχ

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χ = ∂βΨ(uN , β).« #&#F¬

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,« #_~5¬

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uN

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uT

δTF(λ),

« #:;5¬

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4

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27

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g(β, β, uN , uT

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(1 − λ)2, β

! ,

« #0.s¬'ItN>c^N

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N|

Rcoh

Ð]Ù ð)ðyÙGïFÑðyÖ1Ð]ï ülö

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N ) > 0x j4`tZ]\yrs`5\_<f5atrs`5\edp\]Y

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g(β, β, uN , uT

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β0

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T i ·vids, ∀v ∈ V.«G&,4¬

HJItNrlZ^ZvaLMOo[\^dpj4`tZ«Gl|4¬Jrs`uw «1-.s¬!N_`tZeaLceN\^Itr\φ ∈ W 1,2(0, T ;V )

i©d`trlpYlL!N-w[N>`tjs\eN-:Yγiu =

γ|Γcui \^ILN\ecyrlx>NTjl

ui jl`Γc

Ǭjlci = 1, 2

¬)? ZeZeaLMOdp`t rOc^rs\eN °ùd`tw[N>o<N>`uw[N>`5\nd`4\^ced`tZedx-x>jlILNFZvdlN¶_jl`LNMOj[w[N>1L\^ILN-pN>¨\RN_f5atrs\edjl`¢js'«¯~5¬Jx>rl`

<N-c^N>'c^d¦\e\eN>`¢rlZ

β = g?([u]),«G&%4¬

'ItN>c^Ng? ∈ H

Lrs`uw\^ILNx j4ILN_ZedlNTjlcyx NRcoh x_rs`uNw[NFZex>ced<N_wrlZ'raL`tx \edjl`

ξjl

[u]­

ξ([u]) = Rcoh([u]) = K g?([u]) · [u], «¯~44¬

'dp\eIξ : [L2(Γc)]

3 → [L2(Γc)]3

< dp\eILj4a[\rl`:Y c^dx \edjl`<rs`tw¢'dp\eId`Ldp\edgrsx>jl`twLd¦\^dpj4`tZu0 ∈ K rs`tw

β0 ∈ H ∩ [0, 1]t\eILNf4aurlZed¦°

Zv\^rs\edgxcedgx)\^dpj4`trsx jl`5\yrlx)\notcej4LpN_M jli·&j[x>aN \TrlG " #+%-%l|'#x>rs`<NN>²5\^N>`twLN_w\ejO\^ILNx>jlILNFZvdlNx_rlZeNY:dN>gw[d`L\eILNj4pj'dp`trsc^dgr\edjl`ursEoLc^jlLN>M¢­

a(u, v − u) +

∫

Γc

ξ([u])·([v] − [u]) ds− (φ, v − u) > 0, ∀v ∈ K, «¯~#F¬

'ItN>c^N

a(u, v) =

∫

Ωi

ε(ui) : Ai : ε(ui)dx.«¯~5l¬

HJILNTjlj'dp`L\eILN_jlc^N>M jsN ²[dgZ]\^N>`tx>NTrl`twaL`Ldgf5aLN>`tN_Z^Zjl#rZejla[\edjl`O\^j\eItNToLc^jltpN_M «G~6#¬´x>rl`<N-ldlN_`­

ËË ÏA )÷

#, ¾ :)º7

Â:Æ Â 6 y¼l¼<½¦º¯¾ s¿Gº s : k → ξ([u(k)])

u(k) ∈ K º ¿6 : ½ [¿1º

a(u, v − u) − (φ, v − u) +

∫

Γc

k · ([v] − [u])ds > 0 ∀v ∈ K, «G~l~4¬

-»º¨¿ º: * ¼ º ¿k

º λ2c1/m < 1

º ¼<½)4¾ >» ¿u(k)

º ¿* ¿6 : ½ [¿1º ¿6'¼ @>½) »

ÆÆ #N \

u(k1)<NJ\eILNnZejla[\^dpj4`jl«G~l~4¬j4c

k = k15rs`tw

u(k2)<NJ\eILNnZejla[\edjl`js´«¯~4~4¬#jlc

k = k2´©Ljlc

r ldlN>`ku\^ILNN>²[dZv\eN_`tx Nrs`tw¢\^ILNat`Ldf5aLN_`LN_Z^ZTjsr Zejla[\edjl`\^j \^ILNoLc^jltpN_M «G~l~4¬ndZr x>r4ZeZedx_rs

c^N_ZeaLp\d`rlcedgr\^dpj4`trsdp`LNFf5atrsd¦\^dpNFZ " 2dp¤:atxyILd!rs`twwLN>` #+%&,-,'ùm!YZvaLuZ]\^d¦\^a[\ed`Lv = u(k2)

d`«G~l~5¬ij4c

k = k1trl`tw

v = u(k1)j4c

k = k2Lj4`LNjl[\yrsd`tZ'r¨\^N>cnZeaLMOMr\edjl`#­

a(u(k1) − u(k2), u(k1) − u(k2)) 6

∫

Γc

(k2 − k1) · ([u(k1)] − [u(k2)])ds.

HJItNx j:N>cyx d:d¦\]Yjsa

«¯x>jl`tZv\^rl`4\m

2Rj4ce`#¸ Z'd`LNFf4aursd¦\]Y¢rs`twouj5Zvdp\edlNMONFrlZeaLceN-jl´\eILNotrsce\Rjl´\eILN<jlat`twLrsc^Y'ILN>c^Nw[dgZvotr4x N>MON_`4\yZTrlceN-otceNFZex>ceduNFwt¬)7rl`tw\eILNW[xyI:Jrsc^¶l¸ Z'dp`tN_f5atrsdp\]Yj4cn\eItNZ^x>rlrlcotcej[w[atx \'jl`

[L2(Γc)]3 LN_r4w \^jt­

m‖u(k1) − u(k2)‖2H1(Ω) 6 ‖k2 − k1‖L2(Γc)‖[u(k1)] − [u(k2)]‖L2(Γc).

«G~;5¬m!Y\^ILNw[N>§t`Ldp\edjl`¢js\eILNw[dgZeoLr4x N_MN_`5\´k]aLMOo[\eItN§tc^Zv\J\ec^drl`Llatrlc!d`LNFf4aursd¦\]Y 4dp4N_Z_­

‖[u(k1)] − [u(k2)]‖L2(Γc) 6 ‖γ2u(k1) − γ2u(k2)‖L2(Γc) + ‖γ1u(k1) − γ1u(k2)‖L2(Γc).«G~4l¬

HJItNx j4`4\^dp`:aLdp\]Yjl\^ILNd`k]NFx)\edjl`¢c^jlM[H1/2(∂Ωi)]3

\^j[L2(Γc)]

3 rl`tw¢js\eItN-\ecyrlx>NrsoLoLdgx>r\^dpj4`c^jlM

[H1(Ω)]3j4`

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λi «i = 1, 2

¬ZeatxyI\eItrs\_­

‖γiu(k1) − γiu(k2)‖L2(Γc) 6 λi‖u(k1) − u(k2)‖H1(Ω),j4c

i = 1, 2.«¯~4|4¬

5Ò`5\^cej[w[atx>dp`tλ = λ1 + λ2 > 0

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m‖u(k1) − u(k2)‖2H1(Ω) 6 λ‖k2 − k1‖L2(Γc)‖[u(k1)] − [u(k2)]‖H1(Ω).

«G~.s¬Sjlc^N>jlN_c_:rlZ^ZeaLMOdp`L\eIur\

ξdZ #dpotZ^xyILdp\e¶x>jl`5\ed`:aLjlatZ«Gx jl`uZ]\yrs`5\

c1¬ L&N-j4[\^rldp`#­

‖s(k1) − s(k2)‖L2(Γc) 6 c1‖[u(k1)] − [u(k2)]‖L2(Γc),

6 λc1‖u(k1) − u(k2)‖H1(Ω).

©d`trsYl:\eItN-N_f5atr\^dpj4`«G~;:¬4dp4N_Z_­

‖s(k1) − s(k2)‖L2(Γc) 6λ2c1m

‖k2 − k1‖L2(Γc).

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» ¿ 4¾ ¿ s»º¯¾ º ¾ º ^¼l¼ l¾ #%

D u

U

O

−T

u0 δu a

−E/L

u b

EL

A

B C −T

−Ta

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3 \ej[L2(Γc)]

3 Ldp\^ILNx j4`tw[dp\edjl`

λ2c1/m < 1«¯~-,4¬

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LijlaL`L-MOj[w[aLatZ

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4:YT

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U = u − L

ET.

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T = −ξ(u).«;54¬

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−T =E

L(U − u)

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«T = 0

j4cu> δ

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l ¾ :)º7

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k = E/LdgZ!\ej:jZeMrsGLrZejla[\edjl`k]aLMOotZ!cej4M

ua

\ejub > ua

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krs`uw\^ILN-grsc^lN>c'Zepj4ouNjl\^ILNx j4ILN_ZedlNlcyrsoLIdp`\^ILNZvjl¨\eN_`Ldp`totrsce\_[d1 N4´\ej

\^ILNcyr\^dpj |k|/(supu>upeak|ξ′(u)|) JHJILNx>jl`tw[dp\edjl`¢jlaL`tdf5aLN_`LN_Z^ZnjlrOj[x>rs#N_f5aLdpdLc^dpaLM ouj4dp`5\RdgZ

\^I:atZ>­

supu>upeak

|∂ξ/∂u|

|E/L| < 1.«;:l¬

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rl`tw

sup |ξ′(u)| \^j\eItNdotZ^xyILd¦\^¶l¸ Znx j4`tZv\^rs`5\c1

jsξ(u)

5ù\^ILNx>jlILNFZvdlNlcyrsotIItr4Znrs`¢d`>tN>²:djl`o<jld`5\d`¥dp\^ZZvjl¨\eN>`tdp`L otrsce\_\^ILN

sup |ξ′(u)| dgZc^N_r4xyILN_w j4cu > ua

«\eItrs\dgZ>j4cN>²[rlMOoLpN4\eILNx_rlZeNO'd¦\^I j4cH#4N>c^4r4rscyw7¸ ZRMOj:wLN>¨¬)js\eItN>c^'dZeNdp\dgZ-ceNFrlxyILNFwj4c

u = ua

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G = limΓ→0

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l

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Γ

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dZ\eItNZv\ec^N_Z^Z!jlc^¤ w[N_`tZvdp\]YltdgZ-\^ILN¤5d`LN>\edgx N>`tN>c^lY wLN>`tZed¦\]Y4irs`tw

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5Ò`j4c^w[N_cJ\ejrl`tZe&N_cJ\ejO\eItdZTf4atN_Zv\edjl`L!N-d`54N_Zv\ed4rs\eNrOoLaLc^N>YN_r4Z]\^dxMOj[w[N_'d¦\^ILjlaL\TwLrsM°rllN4­!Nw[N \^N>c^MOdp`LN4'd¦\^I \eILNILN>ojs'MOdx>cej4MON_xyItrl`Ldx_rsrlce4aLMON>`5\^Z_\^ILNN>¹<NFx)\jl'\eILdgZr4wLw[NFwx>jlMOoLdrl`tx Njl`®\eItNjlN>cyrs!N>grlZv\edgxMOj[w[aLdG HJILNx_rlZeNjlr MON_ZeI 'dp\eI ILj4Mj4lN_`LN>j4atZN_r4Z]\^dxotcej4ouN_cv\^dpNFZdgZx j4`tZedw[N_ceNFw7HJILNZ]\^atw[dN_wMOdgx c^j4Zv\ec^atx)\^aLc^N x jl`uZvdgZ]\yZd`¥o<N>`L`:YZvIurso<N_wN_r4Z]\^dxd`[°x>pauZvdjl`tZcyrs`twLjlMOpYw[dgZv\ec^dpLaL\eN_wd`Zeotr4x NOrs`twdp`jlc^dpN_`5\^r\^dpj4`N_M<N_wLw[NFwd`5\ej¢rs`ILjlMOj4lN>`tN>jlauZN_r4Z]\^dxMr\^cedp²7HJILNj4aLce\eI[°ùjlcyw[N>cZ]\^d¦¹E`LNFZeZ\^N>`tZejlcjsi\^ILNOMr\ec^dp²E7c^N_Zeo<N_x)\^dp4N>Y\eILNdp`ux atZvdjl`uZ>dgZ'w[N_`Ljs\^N_w5Y

Cm[c^N_ZeouNFx)\edlN_pY:Y

CiiW:d`tx N\^ILN-x jl`uZvdgw[N>c^N_wMr\ec^dp² dZJILN_ceNrOx grlZ^Zvdgx>rluN_r4Z]\^dx

MONFw[dpatMCm

dZJ'c^dp\v\eN_`rlZ_­

Cm = 3kmJ + 2µmK,«1ll¬

'ItN>c^Nkm

rs`twµm

rlceN-c^N_Zeo<N_x)\^dp4N>Y\eItNLaL¤MOj[w[aLatZTrs`tw\^ILNZeILN_rlc'Mj[w[atpatZRjs\eILNMOrs\ec^d¦²7HJItN\^N>`tZejlcyZ

Jrl`tw

KwLN §t`LN!\^ILN'Z]\yrs`twLrlc^wtrlZeN&jlt\^ILN!jlaLce\eI[°ùjlcyw[N_cdgZvjl\ec^jloLdgx´\^N>`tZejlcF­

J =1

3i⊗i

K = I − J7'd¦\^I

i\^ILNOdwLN>`5\edp\]Yjs\eILNZeN_x j4`tw:°ùjlcyw[N>cR\eN_`tZvj4c^Zrs`uw

Iklij =

1

2(δikδjl + δilδjk)

\eILNdgw[N_`4\^d¦\]Yjs\eILNj4aLcv\^I[°ùjlcyw[N>c!\eN_`tZejlcyZ>

HJILN'N_r4Z]\^dx&\^N>`tZejlcjs<\eItN!d`tx>patZedjl`tZCi

dgZILN_ceN'w[N>§t`LN_w\^Itrs`L¤[Z\^jrZeMrs[pN_`Ls\^Iotrlc^rlMON \eN_c_eHJILdZourscyrsMON \^N>cZv\^rl`twLZjlc\^ILN!\eILdgxy¤:`LN_Z^Z#jl<rl`N_f5aLdrsN>`5\´w[jlMrsd`c^N>oLc^N_ZeN>`5\^N_w-:Yrx jlItN_Zedp4N

¶_jl`LN-MOj[w[N_!«¯'d¦\^ILjla[\R\eILdgxy¤:`LN_Z^Z^¬ <rl`tw¢ZvItjlaLgw\eILN_`rs`LdgZeI < Nx grsdM \eIur\Ci

dZRldlN_`:Y\eILNj4pj'd`Lc^N>gr\^dpj4`­

Ci = eC,'d¦\^I

C = CNEl + CT Kl,«1s|4¬

'ItN>c^N-\eILN\eN>`uZvj4c^ZEl

rs`twKl

rsc^Nc^N_ZeouNFx)\^dp4N>Y \eItNaL`Ldgr²[dgrs#Zv\ecyrsd`rsjl`tO\eILN`Ljlc^Mrsn

\ej\eILNd`5\eN_cv¯r4x N-rs`tw\eItN-pj4`Lldp\eatwLdp`trlZvItN_rscnZv\ecyrsd`­

El = n⊗ n⊗ n⊗ n«G.s¬

Kl = 2 (js ⊗ js + jt ⊗ jt)'ILN>c^N

jα = n⊗s α.«1,4¬

HJItNlNFx)\ej4c^Zsrs`uw

tw[N>§t`LNrl`jlce\eILj4`Ljlc^MrsEtr4ZvdgZ'js\^ILNoLgrs`tN\^rs`tlN>`5\n\^j\eItNx j4ILN_Zedp4N¶>jl`tNl

rl`twn⊗s α = (n⊗α+ α⊗ n)/2

´HJILNceN_rs\edjl` «1s|5¬x_rs`<NoLc^jlN_wOauZvd`L\eILNN_f5atrs\edjl`tZ« #F#¬rl`tw¡«;:¬js!\eILN /J·'ÈESdp`\^ILN N>grlZv\edgxx>r4ZvN4d1 N4 j4c

β = 1 5Ò` \^ILdgZotrlcv\^dx>aLrlc-x>rlZeNl#\eILNZ]\^ceNFZeZ

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» ¿ 4¾ ¿ s»º¯¾ º ¾ º ^¼l¼ l¾ &%

\^N>`tZejlc'r4x c^j4Z^Z!rOx>jlILNFZvdlN¶>j4`LNjs`Ljlc^Mrsn

c^N_r4wLZ_­

σ = Rcoh ⊗s n

=

((CNn⊗ n+ CT

uT ⊗ uT

‖uT‖2

)· [u]

)⊗s n

= (CNuNn+ CTuT ) ⊗s n

= CNuNn⊗s n+ CTuT ⊗s n

= CNuNEl : (n⊗s n) + CT Kl : (uT ⊗s n)

= (CNEl + CT Kl) : (uNn⊗s n+ uT ⊗s n)

= (CNEl + CT Kl) : ([u] ⊗s n).«1%4¬

? Z-aL`twLN>c^pd`LN_w:Y SdxyILN_&rl`twW:atf5aLN \ " #+%-%;&'17\^ILNZv\ecyrsd`rlx c^j4Z^ZrZvMrlpigrYlN_cjsJ\eILdgxy¤:`LN_Z^Zex_rs`uNrloLoLc^j4r4xyILN_w5YE­

ε(u) ' [u] ⊗s n

e.

«G|l4¬

·&j4MLd`Ld`LON_f5atrs\edjl`tZ«¯|44¬Jrl`tw «1%5¬Y:dN>gwLZJ\^ILNoLcej4ouN_cv\]Y «Gs|5¬)HJILNj4N>cyrsN_r4Z]\^dx\^N>`tZejlc

C js\^ILNMOdx>cej5Z]\^ceaux)\eatceNOZ]\^atw[dN_wd` \^ILdgZ-otrscyrs4c^rloLIx_rs` uN

Zed¶>N_w'dp\eI\eILN-ItN>ojl TrlZeILd`rl`twW[I4\^ced¤:Mrs` " #%l|l~'p°ùdp¤4NRN_Zv\edMr\^N_Z_­

C (C

0)

=[f[C

∗(C

0)

+ Ci

]−1+ (1 − f)

[C

∗(C

0)

+ Cm

]−1]−1

− C∗(C

0) «G|6#F¬

'ItN>c^N-\eILNceNFrsrlpatNf ∈ [0, 1]

dgZ'\^ILN4jlaLMOdxcyrlx \edjl`jlidp`tx>pauZvdjl`tZTrs`tw\eILN\eN>`uZvj4c'aL`tx \edjl`C∗(C0) dgZJ5YE­

C∗(C

0)

= C0 :[S(C

0)− I].

«G|4l¬

HJItN\^N>`tZejlcS(C0) dZO\^ILN /ZeILN>:Y¥\^N>`tZejlcOc^N>gr\eNFw¥\^j¥rs` dp`ux atZvdjl`¡N_M<N_wLw[NFw dp` rs` d`[§t`Ldp\eN

MONFw[dpatM jlnN>grlZv\edgxO\eN_`tZejlcC0 HJILNljlaLMOdgxOc^r4x)\^dpj4`

fMrYuNw[N §u`LN_w:Y \eILNoLc^j:wLatx)\

eZ«¯'ILN>c^NZ

dgZnrOw[N>`tZedp\]Yotrlc^rlMN>\eN_cy¬ ©tjlcJ\^ILN-oLc^N_ZeN>`5\nx>rlZeNjsouN_`L`:YZeItrso<N_wd`tx atZedpj4`tZ_e\eN_`twLZ

\^jO¶>N>c^jrs`tw\eItN/ZvItN>5Y\eN_`tZvj4cJItrlZ!\^ILNjlpj'd`LZedpMOoLNjlc^M " SaLcyrL3#+%-,4'1­

S(C

0)

= El + Kl.«G|l~4¬

4irlcedjlauZJuj4aL`twLZnrl`twN_Zv\edMr\eNFZJjlcC

x>rl`<Nw[N_cedlNFw c^jlM «¯|#F¬Jrlx_x j4c^w[d`L\ejO\^ILNxyILj4dx>Njl\^ILNc^N N_ceN_`tx N´MONFw[dpatM

C0 < ILN_`ers`LdgZvItN_Z_5«¯dG Nl

Ci = eC → 0rs`tw

f = eZ → 0¬ F\eILN RrlZeILd`

rl`twW:I5\^cedgxy¤:MOrl`aLoLo<N>c<jlaL`uw«C0 = Cm

¬)E\^ILNSjlc^drl`twHrs`trl¤rN_Zv\edMr\eN«C0 = Cm

¬)rs`uw\^ILNOZvN_¦&x>jl`tZedgZ]\^N>`5\TNFZ]\^dpMr\^N«

C0 = C ¬Trsc^NNFf4aursrs`twldlN\^ILN\ec^dp:dgrsceNFZvaLp\_­

C = Cm

Rj&N_lN_c_4\eILN TrlZeILd`¢rl`tw¢W:I5\ec^dxy¤:Mrs`pj!N>cJ<jlaL`uwN_r4wLZ!\eju­

C −

= C (Ci) =(ZS : C

−1 + C−1m

)−1

.«G|;5¬

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~4 ¾ :)º7

QTZvd`L«¯|4~4¬!rs`uw«1s|5¬)[jl`tNT§u`twLZ

S : C−1 =

1

CNEl +

1

CTKl.

«G|4l¬

W[dp`tx>N&NOItr4NOdp`5\eN_ceNFZ]\-d` dp`tx>pauZvdjl`tZcyrs`uw[jlMOYw[dgZ]\^cedLa[\^N_wdp`Zeotrlx>Nrs`twd`j4cedN>`5\yr\edjl`#7\eILNj4N>cyrsiN>grlZv\edgx\eN>`uZvj4c

C Itr4Z\^j <NdgZejs\ec^jlotdxlrl`twc^jlM \^ILdZo<jld`5\jlR:dpN_-\eILNf5atrs`5\^d¦\]Y

S : C−1 ItrlZ\ej<N¢r4N>cyrs4N_w jlN_crlpJo<j4Z^ZvdLNj4cedN>`5\yr\edjl`uZ>HJItdZotrlcv\^dx>aLrlcr4N>cyrs4N ceNFrlwLZ" 1r\e\nN \RrlGts440'ù­

⟨S : C

−1⟩

= PJK

(S : C

−1) «G|l|4¬

'ItN>c^NPJK(A)

w[N>`tjs\eNFZT\eILNoLc^jsk]N_x \edjl`jl&rs`:Y¢j4aLcv\^Ij4c^w[N_cn\^N>`tZejlcA

jl`\^ILNtrlZeN J,K rl`tw〈A〉

dgZn\^I rlN_c^rllNjlN_cTrlp\eILNouj5ZeZedLpNjlc^dpN_`5\^r\^dpj4`tZnjlrs`:Y\eN_`tZvj4cAHJILdgZTr4Z]\f4aurs`5\edp\]Y¢dZ

NFf5atrsE\^j⟨

S : C−1⟩

=

1

3CNJ +

2

5

(1

3CN+

1

CT

)K.

«G|.s¬

<c^d¦\^dp`LC

− dp`\^ILNjlc^M

C −

= 3k −

J + 2µ −

K«G|&,4¬

'ItN>c^Nk − rs`tw

µ − rlceN'c^N_ZeouNFx)\^dp4N>Y-\eILNRjlN>cyrs[LaL¤rl`twZeILN_rlc´Mj[w[atpd15rl`twatZedp`LN_f5atr\^dpj4`

«G|;:¬)[jl`tNj4[\^rldp`uZ'r¨\eN_cnZvj4MONrlp4N>LcyrL­

k −

km=

x

1 − x

'dp\eIx =

CN

kmZ,

«G|&%4¬

µ −

µm=

y

1 − y

'd¦\^Iy =

15CNCT

4µm(CT + 3CN )Z.

« .4¬

·&j4MLd`Ld`Lk − rs`tw

µ − ldlNFZ\^ILNjlj'dp`LOj&N_cJuj4aL`twjlcJ\^ILN-jlN_c^rlp ijlaL`tMOj[w[aLatZ_­

E −

Em=

ξ

1 + ξ,'ILN>c^N

ξ =5

1 +4

3

CN

CT

CN

EmZ,

« .*#F¬

'ItN>c^NEm

uN_dp`t\^ILN ijlat`LMOj:wLaLpauZJjs\eILN-Mr\^cedp²7? \T\eILdgZTZv\^rllN4t\^ILNx>jlMOotrsc^dgZvj4`jlirlcedjlauZRx>r4ZvNFZ'jsx>jlILNFZvdlN-¶_jl`LNMj[w[N_ZTdgZRrs`dp`5\eN_ceNFZ]\^dp`t

dgZ^ZvaLN4©dc^Zv\eYl4jlcT¸ `tjZeILN_rlc_¸5·'È7S «d1 N4·'ÈES d`54jl:dp`L-`tj-\^rl`LlN_`4\^drltc^N_r4x)\edjl`<¬)s\^ILNTx>jl`twLd¦\^dpj4`

CT = 0ldlN_Z

ξ = 0,«.ll¬

rl`tw¢\^ILNd`Ldp\edgrs`Lj4ceMrsN>grlZv\edgx dp\]Y¢jsi\^ILNx>jlILNFZvdlN¶>jl`tNItr4N-\ejuNd`[§t`Ldp\eNdp` jlcyw[N>cR\ejjl[\yrsd`r`Lj4`¢rl`LdgZvILd`LOjlN_c^rlp<N>grlZv\edgx dp\]Yj4c'\eILNZv\ec^atx \eaLc^NliW:NFx jl`uw7[jlc-¸ `LjZepdgw[d`Lt¸<·'È7S «¯dG Nl!·'È7Sd`:lj4p:d`L`tj\yrs`L4N>`5\edgrsZedw[d`L:¬):\eItNx jl`uw[d¦\^dpj4`

CT → ∞ ldlNFZξ =

5CN

EmZ.

« .~4¬

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HJItdpcyw7:jlcJ\^ILNc^N_f5aLN>`5\Rx_rlZeNTjlNFf4aurs7`Lj4ceMrs7rl`tw\^rs`tlN>`5\^drl<N_r4Z]\^dx>d¦\]Y4:\eILNx>jl`twLd¦\^dpj4`

CN = CTldlNFZ

ξ =15

7

CN

EmZ.

« .0;5¬

5Ò` 'Itrs\jlpjnZ_&\eILNc^N>gr\^dpj4`CN = CT

dgZ r4ZeZeaLMON_w7BHJILNf5aLN_Zv\edjl`¡jlNFZ]\^dpMrs\ed`L \eILNwLN>`tZed¦\]Yotrlc^rlMN>\eN_c

Zw[N_ouN_`twLZTjl`¢\eItN-\]Y:ouNjs´MON_ZeI'ILdxyIdgZnauZvNFw7 5Ò`jlcyw[N>cn\ej dpatZv\ecyr\^N-\eILN

r4x>x>aLc^r4x Yjs\^ILNpj!N>c&uj4aL`tw«.#F¬&rs`tw \eILNceN_rs\edjl`« .0;:¬)[!NTx j4`tZedw[N_c\^ILNsª x>r4ZvNTjsrZ^f4aursc^NMONFZvILNFw¢'d¦\^Ic^N>latrlcnZef5atrlceNFZnw[d5dgw[NFw¢dp`¢j4aLcRNFf4aurs#dgZej4Z^x N>N_Z'\ec^drl`LlN_Z_ <NwLN>`Ljl\eN:Y

Lmesh\^ILNN>`ts\eIjsJ\eILNgrsc^lN>cZedgw[NjlnN_rlxyI \^cedgrs`tlN_Z_:Yp

\eItN`:aLM<N>cjs'MON_ZeILNFZ:Y Zvdgw[Njl'\eILNZ^f5atrsc^Nlt5Y

l\eItN-\ejl\^rspN_`Ls\^Ijsix j4ILN_ZedlN-¶>j4`LN «¨\^js\yrs#N>`ts\eIjl´uj4aL`twLrlcedN_Z'<N \]!N>N_`MON_ZeILN_Zy¬)

rl`tw5YN

\^ILNJ`5atM<N>c´js<MON_ZeILN_Z_©tjlc´`Lj4`rl`LdZeILd`LT\eILdgxy¤:`LN_Z^ZejsEx j4ILN_Zedp4N&¶_jl`LN4\eItN'ljlaLMOdgx

cyrlx \edjl`f

c^N_r4wLZ>­

f =le

le+ (L2mesh/4)N

,« .sl¬

rl`tw'ILN>`e→ 0

:\eILN-c^N>gr\^dpj4`f = Ze

ldlNFZ\^ILN-dpMOdp\Rx>rlZeNl­

Z =4l

L2meshN

.« .|4¬

HJItNT\^js\yrsN>`Ll\eIjs´x j4ILN_Zedp4NT¶_jl`LNldgZ'ceN_rs\eNFw \^jourscyrsMON \^N>cyZ

Lmeshrl`tw

p­

l = 2√

2Lmeshp2 + 2p(p− 1)Lmesh.

« .&.s¬©tjlc'grsc^lNMON_ZeILN_Z-«¯dG Nl´'ILN_`

p→ ∞ ¬ Ljl`LN§t`uwLZ>­

Z =2(1 +

√2)

Lmesh.

« .,4¬

ha[\v\^dp`LO\^ILdgZ'r4Z]\'NFf4aur\edjl`d` « .*#¬):\eItNjlpj'd`LOpj!N>cJ<jlaL`uwItjlgwLZ!jlcJ\^ILdZ'\]Y5o<N-jsMON_ZeILN_Z_­

E −

Em=

ξ

1 + ξ,'ILN_ceN

ξ =5/(1 +

√2)

2

(1 +

4

3

CN

CT

) CNLmesh

Em.

« .%4¬

HJILNrlx>x>aLcyrlx Y jsJ\eItdZpj!N>c<jlat`tw dZdpatZ]\^c^rs\eNFw jl`¥©dp4aLc^N #_¢'ILN_ceN\^ILN j4N>cyrs ijlaL`LMOj[w[aLatZ´jl<rl`N_r4Z]\^dxJZv\ec^atx)\^aLceNJd`tx atw[d`L-N_r4Z]\^dxJx j4ILN_ZedlN!¶>jl`tN!MOj[w[N_Zirs\´N_rlxyI§t`Ldp\eN'N_pN_MON>`5\^Z<jlat`twLrsc^YdgZiotpjl\v\eNFwjlc&rlcedjlauZotrscyrsMON \^N>cyZ

ξ´HJILNncyrs`L4NJjs7otrscyrsMON>\eN>c

ξdgZrluj4a[\

[0, 40]'dp\eI

CN/CT ∈ [1/3, 10/3]rl`tw

CNLmesh/Em ∈ [7 × 10−3, 51]

©tjlcJ\eItN-otrsce\edgx aLgrsc'x_rlZeNCN = CT

:\^ILNjlj'dp`Lx>cedp\eN_cedjl`x>rs`<N-'cedp\v\^N>`­

CNLmesh

Em> 21 =⇒ E

−

Em> 0.9.

«7,44¬

HJItdZx>cedp\eN_cedjl`®uN_dp`Lc^N>gr\^N_w¥\ej\eILNj!N>cOuj4aL`tw jsT\eILN¢j4N>cyrsJZv\edp¹E`LN_Z^ZOjsT\eILNZejldwdp\ dZ`tjs\eN_&j4cv\^I:Y\eItrs\_5dp`OoLcyrlx \edgx Nls\eILNn`:aLMON_cedgx>rl¦°ùtrlZeN_wx c^d¦\^N>c^dpj4`jl /&ZvoLd`Lj5Zer-rl`tw ÈErr\v\^dpN_ced " s4l~'ZeILj4aLw<N-ceN_pN_rl`5\!j4cnrs`:Y\]Y:o<NjlZedp\eatrs\edjl`tZ_[N>4N>`\eILj5ZvNlN_ceY ¯rscJc^jlM 'Itrs\J\eILN_YZv\eatw[dN_w

ËË ÏA )÷

~5 ¾ :)º7

0 10 20 30 400

0.2

0.4

0.6

0.8

1

E

−/E

m

ξ

@@I N_f<'«.%5¬

©dlaLc^N #FL­TlN_c^rlp ijlaL`L MOj:wLaLpauZjlRrs`¥N_r4Z]\^dxZ]\^ceaux)\eatceN d`tx>patwLdp`LN>grlZv\edgxx>jlILNFZvdlN¶_jl`LNMOj[w[N_Zr\N_r4xyI§t`Ldp\eNN>N>MON>`5\yZR<jlaL`uwLrsc^YjlcrlcedjlauZnotrlc^rlMN>\eN_c^Z

ξ«¯ZeN>NN_f5atr\^dpj4`«.%4¬e¬)­n`:a[°

MON_cedgx>rl7ceNFZvaLp\^Z-«¯ouj4dp`5\yZ^¬Jrl`twj!N>cJuj4aL`tw «.%4¬

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5Ò`\^ILdgZdpatZv\ecyr\^dpj4`s\^ILNRc^N>oLc^N_ZeN>`5\yr\edlN'4jlaLMONnN>N>MON_`4\Jx j4`tZvdgw[N_ceNFwdgZrZef5atrlceN''dp\eIpN_`Ls\^IL = 10µm

aL`tw[N_c-oLrl`LNOZ]\^c^rldp` I5Y:o<js\^ILN_ZedZ«GZvNFx)\^dpj4`S = 10−5m2 ¬)OHJILNMr\eN_cedgrsdgZ-rlZ^ZvatMNFw

\^j<NPRN>jl° Rj5j4¤lNFrs`¢Lc^d¦\e\eNl­ ij4aL`LMOj[w[aLatZEm = 135GPa

<hjldgZeZejl`x>j5N:( x dN>`5\νm = 0.32

x>cedp\edgx>rl:N>`LN_ce4Y-ceN_pNFrlZeN&cyr\^N

GIc = 0.5J/m2 lrs`tww[N_`tZvdp\]YNFf4aurs:\ej7800kg/m3 HJILN'atZeN_wx j4ILN °

ZedlNrDdZ!\^ILNjl`LNldlN_`j4`ZeN_x \edjl`[ ~L 'dp\eICN = CT

Lrpj±cedgx)\^dpj4`x>j5N:( x dN>`5\µ = 0.05

rl`tw`LjZvjl¨\!x>jlaLoLd`L<N \]!N>N_`x j4ILN_Zedpj4` rs`twcedgx)\^dpj4`

κ = 1 4´N>j:x>d¦\]YO<jlaL`uwLrsc^Yx>jl`twLd¦\^dpj4`tZ&rlceN

dMOouj5ZvNFw rlpj4`L¢lN_cv\^dx_rs¯rlx>N_Z«V = 2m/s

¬)rs`twrs` d`Ld¦\^drlix cyrlxy¤dgZ-d`tZvN_cv\^N_wr\\eItN x N>`5\^N>c-js\^ILNZv\ec^atx)\^aLceN-rl`two<N>c^o<N>`tw[dgx atrlcJ\ejO\eItN-pj5rlw[d`LOw[dpc^N_x \edjl`

HJILNMNFZvIx>jl`:lN_ce4N>`tx>NTdgZRZv\eatwLdpNFw\eILc^jlaL4I\^ILNMrlx>cej5Zex>jloLdgxTc^r4x)\^aLceNN_`LN>c^lY E ux j4MOouj5ZvNFw:Y\eItNw[dgZ^Zvdotr\^N_wN_`LN>c^ldN_Z&w[aLc^dp`t\eItNRcyrlx \eaLc^NnoLc^j[x N_Z^Z_5r4Z&raL`tx)\^dpj4`jl7\eILNMON_ZeIZvd¶>N

Lmesh

HJItN&§u`Ld¦\^NJN>N>MON>`5\iw[dgZex>ceN>\ed¶_r\^dpj4`dgZtrlZeN_wjl`pd`LN_rlc´w[dZeoLgrlx>N>MON>`5\\ec^drl`LlaLgrscN>N>MON_`4\yZ\eItrs\irsc^Nrlcecyrs`tlN_wd` r >x c^j4Z^ZvNFw:°G\^cedgrs`tlN Tf4aurlw[c^dpgr\^N>cyrsour\v\^N>c^`rs`uwjlaLcMON_ZeIZvd¶>NFZRx>jl`tZedwLN>c^N_wrsc^N ­0.2µm

« #Fll4MONFZvILNFZ^¬ 0.25µm

«G|;44MNFZvItN_Zy¬)0.33µm

«¯~l|4lMON_ZeILN_Zy¬rl`tw0.5µm

« #F|l4MONFZvILNFZ^¬ HJItNd`Ldp\edgrsix>jlILNFZvdlNZ]\^d¦¹E`LN_Z^Z

CNdgZ-rlwLrlo[\eNFwjlc-NFrlxyI MNFZvId` j4c^w[N_c\ejlN>c^dpY\eILN

x>cedp\eN_cedjl`«7,44¬)­CNLmesh/Em > 21

HJILN©dlatceN #&# ZeILjnZ\^ILNceato[\eaLc^NON_rs\eaLc^N_Zjs!\eILN MONFZvIZedp¶_N_Zx>jl`tZedwLN>c^N_w7 ? ¦\^ILjlaL4I \eILNx>c^r4xy¤[Z´otr\^ItZJrsc^Nn`Ljl\edgx N_rlLYOwLd¦¹EN>c^N>`5\_5\^ILNTMr4x c^j4Z^x jlotdx!cyrlx \eaLc^NRN_`LN>c^lY E dgZ&x>jl`tZv\^rl`4\&jlc!N_r4xyIMONFZvIZvd¶>N«¯ZeN>NHrlLpN#F¬ ´HJILdZ!c^N_ZeaLp\!w[N_Mj4`tZv\ecyr\eNFZi\^ILNMNFZvIZvd¶>Nd`tw[N>o<N>`uw[N>`tx>NTjl\eItNTj4N>cyrscyrlx \eaLc^NoLcej4ouN_cv\^dpNFZJjs\eILN\]!jjlpj'd`Lx j4`tw[dp\edjl`tZ'rlceNZers\edgZ]§uN_w7­

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u¬\eILN-d`Ldp\edgrs#x>jlILNFZvdlNZv\edp¹<`LNFZeZndZJ!N>#x>rlpdLcyr\^N_w auZvd`L\eILNx>cedp\eN_cedjl` «7,44¬

«¯r4¬ #F|llMON_ZeILN_Z «u¬J~4|l4MON_ZeILNFZ

«¯xF¬J|;5lMNFZvItN_Z «¯wt¬ #Fll4MONFZvILNFZ

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CNLmesh/Em > 21

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» ¿ 4¾ ¿ s»º¯¾ º ¾ º ^¼l¼ l¾ ~5

Lmesh/L 5.0× 10−2 3.3× 10−2 2.5 × 10−2 2.0 × 10−2

E/(GIcS) 21.62 21.80 22.20 22.20

HrstpN #l­ PRjlc^Mrsdp¶_N_w MOr4x c^j4Z^x j4oLdxcyrlx)\^aLc^N N_`LN>c^lYzd` aL`tx \edjl` js\eItN `Ljl` w[dMON>`tZedpj4`trsMONFZvI Zvd¶>NBj4c³rs` rlwLrlo[\eNFw"d`Ldp\edgrsx j4ILN_ZedlN Z]\^d¦¹E`LN_Z^Z atZed`L \^ILNzjlL\^rsd`LNFw x>cedp\eN_cedjl` «7,l5¬)­CNLmesh/Em > 21

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5Ò` \eItdZZeN_x)\^dpj4`7\^ILNZvMOj:js\^IN_f5atrs\edjl`tZjlMOjs\edjl` jsJrx>jlpNFx)\edjl` js

Nx j4`5\ed`5ataLM MON_w[dgrrsc^N

d`5\ec^j[w[atx NFw¢dp`r f5aLdp\eNauZvatrlZeN \e\ed`LtHJILNx>jl`5\ed`:aLaLM MON_w[daLM dgZRdgw[N>`5\^d¦§tNFwr\T\edMONt ∈ [0, T ]:Y dp\^ZnljlaLMONdp`

IRd iHJILN-d`5\eN_lN>cd = 1, 2, 3

w[N_`Ljs\^N_Z!\eItNZvotr4x Nw[dMN_`tZedpj4`Ljsdp`5\^N>c^dpj4c

Ωi(t) ⊂ IRd, i ∈ 1, . . . , Nrl`twuj4aL`twLrlceY

∂Ωi(t) 5ù

Ωi dgZrw[N>jlc^MrsLN x j4`4\^dp`:aLatMVMON_w[daLM¢\eILNNFf5atr\^dpj4`tZ-jsJMOjs\^dpj4`rlceN-d`5\ec^j[w[atx NFw\eILc^jlatlI\eILNoLc^dp`tx>dpotpNjl´5dce\eatrl#o<j&N_c^Z'd`rO§t`td¦\^NZv\ecyrsd` rs4c^rl`Lldgrs`¢ZvN>\v\^dp`Lo<N>c^MOd¦\e\ed`LrZeotr4x N °Òw[dgZex>ceN>\ed¶_rs\edjl`urlZeN_w jl`rx>jl`:lN_`5\edjl`trl§u`Ld¦\^NN_pN_MN_`5\MON \eItj:w 5ù

Ωi dZr4ZeZeaLMON_w\ej<Nr c^dp4dw®<j:wLYl!\eILNN_f5atr\^dpj4`¡jsMjl\edjl`¡'dpn<Nw[NFZex>ced<N_w®5Y r§t`Ldp\eNZeN \ jlx>j:jlcyw[dp`ur\eNFZ> 5Ò`ujl\eIx_rlZeN_Z_o<j4Z^ZedpLY r¨\eN_cr Zvotr4x N>°ùw[dgZ^x c^N \ed¶_rs\edjl`#\eILNN_f5atrs\edjl`tZjsnMOjs\^dpj4`'duNj4ceMaLgr\eNFwrl`tw\ec^N_r\^N_wdp`rOZedp`L4pN§t`td¦\^N °Òw[dpMON_`tZvdjl`ursEcyrsMON>!jlc^¤E

? MOrs\eN_cedgrsoursce\edgx NdZTw[N_Z^x c^duNFw¢5Ydp\^Zno<j4Zedp\edjl`X

dp`r ceN>N>c^N>`tx>Nc^rlMNr\t = 0

rl`tw¢:Ydp\^Z'x>aLc^ceN_`4\Jo<j4Zed¦\^dpj4`

x = ϕ(X, t)rs\&\^dpMON

ti©Lj4c'r rs4c^rl`Lldgrs` w[NFZex>cedo[\^dpj4`[&NrsgZejr4ZeZeaLMONT!N

¤:`Ljzrs\TN_rlZv\Rj4ceMrlpY\eILNaL`tx)\^dpj4`X = ψ(x, t)

-HJILNOw[dZeoLgrlx>N>MON>`5\RdgZw[N §t`tN_w:Yu(x, t) =

x−X = x−ψ(x, t)rs`tw\eILNJ4N>j:x>d¦\]Y-rl`tw-\^ILN!r4x>x>N>N>cyr\edjl`rsc^N&w[N_`Ljs\^N_w:Y

urs`tw

uSj4Zv\jlt\^ILN

rs4c^rl`Lldgrs`rsc^drlLpNFZnN>²:otceNFZeZeN_w¢d`\^N>c^MOZRjsX

rlceNw[N>`Ljl\eNFw:Yx_rsoLdp\^rlZTN \v\^N>cyZ><j4cRd`tZ]\yrs`tx>Nl

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U(X, t)jlc#\eILN!w[dZeoLgrlx>N>MON>`5\_rl`tww[N_`Ljs\^N_w-:Ypj!N>cx>rlZeN´j4c\eItN&rlZ^Zej:x>drs\eNFw/iaLN>c^drl`-rlcedgrstpNFZ>

d`O\eItdZ&x>rlZeNu(x, t)

´HJILdgZx j4`:lN>`5\^dpj4`x_rs`<NRZvatMMrlced¶>NFw:Yu(x, t) = u(ϕ(X, t), t) = U(X, t)

ÂÆ Æ:ÅÂ ® Æ E Â ¯4 W5\^rlcv\^dp`tc^jlM \eILNN_f5atrs\edjl` jlMOjl\edjl`dp` /iaLN>c^dgrs`¢x j:jlcyw[d`tr\^N_Z_

div σ(x, t) + ρ(x, t)b(x, t) = ρ(x, t)u(x, t), ∀x ∈ Ωi(t),«7,#F¬

'ItN>c^Nσ(x, t)

dgZ7\^ILNn·!rsatxyI:YTZv\ec^N_Z^Z7\^N>`tZejlcrs`twb(x, t)

dgZ\^ILN&w[N_`tZed¦\]YjsL<j[w[Yjlcyx N_Z__\eILN&oLced`tx>dpoLNjl:dcv\^atrs7o<j&N_cnZ]\yr\^N_Z!\eIur\

∫

Ωi(t)

(u(x, t) − b(x, t))v(x, t) dm(x, t) =

∫

Ωi(t)

div σ(x, t)v(x, t) dω(x, t),« ,44¬

j4crsn:dcv\^atrsn4N>j[x dp\edN_Z w[N_`Ljs\^N_w :Yv(x, t)

HJILNMON_r4ZvatceNdω(x, t)

w[N_`Ljs\^N_ZO\^ILN N_uNFZv4aLNMONFrlZeaLceNd`

IRd r\x

rs`tw\eItNMON_rlZeaLc^Ndm(x, t) = ρ(x, t) dω(x, t)

dgZJ\eItNMrlZ^Z'MON_rlZeaLc^Nl < d¦\^I\^ILN-ILN_po¢js\eILN1c^N>N_`j4ceMaLgrlZ_:\eILN-oLc^d`tx doLNjl:dcv\^atrs7o<j!N>cJdZJauZvatrlpY c^N j4ceMaLrs\eNFwr4Z

∫

Ωi(t)

(u(x, t) − b(x, t))v(x, t) dm(x, t) = −∫

Ωi(t)

σ(x, t) :∇v(x, t) dω(x, t)

+

∫

∂ΩiF

(t)

t(x, t)v(x, t) ds(x, t) +

∫

Γic(t)

r(x, t)v(x, t) ds(x, t)

« ,l~4¬

'ItN>c^NA :B = AijB

ij dgZn\^ILNw[jlaLtpNOx j4`4\^c^r4x)\^N_w\eN_`tZejlcoLcej[w[aux)\_rl`twt(x, t) = σ(x, t).n(x, t)dgZ\eItN rsoLoLdN_wjlcyx N_Zj4`\^ILN<jlaL`twtrsc^YjsJjla[\]Jrscyw`Lj4ceMrl

nrs`tw

r(x, t)\^ILNceNFrlx \edjl`j4c^x>N_Z

wLaLN\ej¢\^ILNx>jlILNFZvdlNd`5\eN>ce¯rlx>NlHJILN MON_r4ZvaLc^Nds(x, t)

dgZ-\^ILN #N><N_ZelaLNMON_r4ZvatceN rs\x ∈ ∂Ωi HJItNZeY:MMON>\ec^Yjsi\^ILN·!rsatxyI:YZ]\^ceNFZeZR\eN_`tZejlcdp` rstZeN>`ux Njs&w[N_`tZed¦\]YjlMOjlMON>`5\eatM rspjnZTjl`LN

\^jOdp`5\ec^j[w[atx>NT\^ILNZeY5MOMON \^cedgxw[N j4ceMrs\edjl`c^rs\eN\eN_`tZvj4c_

D(x, t) =1

2(∇T v(x, t) + ∇v(x, t)) «7,;:¬

N_r4w[d`L\ej\eILNZv\^rl`twLrlc^wN ²[oLc^N_Z^Zvdjl`js\eILN-:dcv\^atrs7o<j&N_cJjs\eILN-d`5\eN>c^`trlEjlcyx N_Z!jlx>jlILNFZvdjl`

Pint = −∫

Ωi(t)

σ(x, t) :∇v(x, t) dω(x, t) = −∫

Ωi(t)

σ(x, t) : D(x, t) dω(x, t)«7,5l¬

5Ò`j4c^w[N_c´\ejjlc^MaLgr\^NJ\eILNnotced`tx doLNnjs7:dcv\^atrsLo<j!N>cid` r\ejl\^rs #rllcyrs`L4drl`cyrsMON>!jlc^¤E\eILNZeN_x>jl`twhidjlgrÀ2dpcyxyILILjl¹\^N>`tZejlcF

S(X, t) = F−1 det(F )σTF−T

dgZTdp`5\ec^j[w[atx>N_w77'ILN_ceNF =

∂x

∂X=∂ϕ(X, t)

∂X

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Pint = −∫

Ωi(t)

σ(x, t) :∇v(x, t) dω(x, t) = −∫

Ωi(0)

S(X, t) : L(X, t) dΩ(X, 0)«7,l|5¬

ÉÊËɨÌ

» ¿ 4¾ ¿ s»º¯¾ º ¾ º ^¼l¼ l¾ ~*.

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X[dZ

∫

Ωi(0)

(U(X, t) −B(X, t))V (X, t) dM(X, 0) = −∫

Ωi(0)

S(X, t) : L(X, t) dΩ(X, 0)

+

∫

∂Ωi(0)

T (X, t)V (X, t) dS(X, t) +

∫

Γic(0)

R(X, t)V (X, t) dS(X, t)

« ,.s¬

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R(X, t) = S(X, t).F T (X, t).N(X, t)=5ù\ dZO`Ljl\eN_&j4cv\^I5Y \^Itr\\^ILNd`4\^N>cyrlx \edjl`tZ<N \]!N>N>`

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U(X, t) =∑

h

Nh(X, t)Uh(t), u(X, t) =∑

h

Nh(X, t)Uh(t), U(X, t) =∑

h

Nh(X, t)Uh(t),

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'ItN>c^NM(Uh)

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V = v(x, t) = vO(t) + ω(t) × (x − xO), ∀x ∈ Ωi(t), «7%5l¬'ItN>c^N

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Y:dN>gwLZ∫

Ωi(t)

(u(x, t) − b(x, t))v(x, t) dm(x, t) =

∫

∂Ωi(t)

t(x, t)v(x, t) ds(x, t), ∀v(x, t) ∈ V «7%4~4¬

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d

dt

∫

Ωi(t)

u(x, t) dm(x, t) =

∫

Ωi(t)

b(x, t) dm(x, t) +

∫

∂Ωi(t)

t(x, t) ds(x, t),

d

dt

∫

Ωi(t)

(x− xO) × u(x, t) dm(x, t) =

∫

Ωi(t)

(x − xO) × g(x, t) dm(x, t)

+

∫

∂Ωi(t)

(x− xO) × t(x, t) ds(x, t).

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) + ωi(t) × (ωi(t) × (x− xGi)).

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dt

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]=

[fext(t)mext(t)

],

« %l|4¬

'ItN>c^N

M =

∫

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dm(x, t) =

∫

Ωi(0)

dM(X, 0),

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(x− xGi)T (x− xGi

) dm(x, t),=

∫

Ωi(0)

(X −XGi)T (X −XGi

) dM(X, 0),

fext(t) =

∫

Ωi(t)

b(x, t) dm(x, t) +

∫

∂Ωi(t)

t(x, t) ds(x, t),

mext(t) =

∫

Ωi(t)

(x− xGi) × b(x, t) dm(x, t) +

∫

∂Ωi(t)

(x− xGi) × t(x, t) ds(x, t).

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z

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∂zi= Qi(z, t), i ∈ 1 . . . n, « #_#F¬

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αT ]α∈1...ν

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],

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= Hα(q)q,« #F&,4¬

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ÂOÂ L Æ Æ Æ Æ Â Æ Â L # Â Â Â E ¯ [ Æ HJILN!lN_pj[x>d¦\]Yv = q

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[u]+(t) = H(q)v+(t) [u]α,+(t) = Hα(q)v+(t)dp = HT (q)dP =

∑αH

α,T (q)dPα

« #-#_~5¬

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dν : [0, T ] → IRν 'dp\eI¥c^N_ZeouNFx)\-\^jdν

HJILN d`tx atZedpj4`dr ∈ K

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r′(t) ∈ K, dνrsMOj4Zv\'N>lN_ceY:'ILN_ceN

.« #&#F|4¬

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NTIR+(uN )(u

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−dPT ∈ ∂(µ|dPN | ‖ u+T ‖). « #-#+,4¬

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−dPN + dPcohN ∈ NT

IR+ (uN )(u+N + eu−N)

« #-#+%5¬

−dPT + dPcohT ∈ ∂(µκ(β)|dPN + dPcoh

N | ‖ u+T ‖) « #s4¬

dPcoh = Rcohdt, R

coh = K(β) · [u] g(β, β, uN , uT ) = 0,« ##F¬

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« #l4¬'ItN>c^N

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−RN + RcohN ∈ NT

IR+ (uN )(u+N + eu−N)

−RT + RcohT ∈ ∂(µκ(β)|RN + Rcoh

N | ‖ u+T ‖)

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dt

°&rlpMOj4Zv\'N>4N>c^Y5'ItN>c^N « #s~4¬

rl`tw−PN ∈ NT

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q(tk+1) = q(tk) +

∫ tk+1

tk

v+ dt .

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dv7&NItrlN ∫

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M dv = M

∫

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dv = M (v+(tk+1) − v+(tk))Pnjs\^N\^Itr\

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]tk ,tk+1]

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M (v(tk+1) − v(tk)) =

∫ tk+1

tk

F (t, q, v+) dt =

∫

]tk ,tk+1]

dp,

q(tk+1) = q(tk) +

∫ tk+1

tk

v+ dt

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qk ≈ q(tk)rl`tw

qk+1 ≈ q(tk+1)vk ≈ v+(tk)

rl`twvk+1 ≈

v+(tk+1)trs`uw

pk+1 ≈∫

]tk,tk+1]

dp.

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tk

F (t, q, v) dt ≈ hF (tk, qk, vk, tk+1, qk+1, vk+1, tk+θ, qk+θ, vk+θ)« #-.l¬

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, θ ∈ [0, 1]« #F&,4¬

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F (tk , qk, vk, tk+1, qk+1, vk+1, tk+θ, qk+θ, vk+θ) =1

2K : [E(qk) +E(qk+1)] :F (qk+1/2)

« #%5¬

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θ

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qk+1 = qk + h vk+θ .HJILNj4pj'd`L\^dpMONw[dgZ^x c^N \ed¶>NFwN_f5atr\^dpj4`jlMOjs\^dpj4`dgZ'j4[\^rldp`LNFw7­M(vk+1 − vk) + hF (tk, qk, vk, tk+1, qk+1, vk+1, tk+θ , qk+θ, vk+θ) = pk+1

qk+1 = qk + hvk+θ

« #F~l5¬

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(M + hθCt + θ2h2Kt)(vk+1 − vk) + φ(tk, qk, vk) = pk+1

qk+1 = qk + hvk+θ

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dp

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[u]k+1 = H(qk+1)qk+1,« #F~44¬

pk+1 = HT (qk+1)Pk+1 =∑

α

Hα,T (qk+1)Pαk+1.

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qk+1 = qk + h(γ1vk + γ2vk+1)« #F~;:¬

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[0, 1]

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−PN,k+1 + PcohN,k+1 ∈ NT

IR+ (uN k+1)([u]N,k+1 + e[u]N,k)

−PT,k+1 + PcohT,k+1 ∈ ∂(µκ(βk+1)|PN,k+1 + Pcoh

N,k+1| ‖ [u]T,k+1 ‖)

Pcoh

k+1 = hRcohk+1, R

cohk+1 = K(βk+1) · [u]k+1 g(βk+1, βk+1, βk, [u]N,k+1, [u]T,k+1) = 0« #F~44¬

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\eILN_`PN,k+1 − Pcoh

N,k+1 = 0

dpuN k+1 6 0,

\eILN_`0 6 PN,k+1 − Pcoh

N,k+1 ⊥ [u]N,k+1 + e[u]N,k > 0

−PT,k+1 + Pcoh

T,k+1 ∈ ∂(µκ(βk+1)|PN,k+1 + Pcoh

N,k+1| ‖ [u]T,k+1 ‖)

Pcohk+1 = hRcoh

k+1, Rcohk+1 = K(βk+1) · [u]k+1 g(βk+1, βk+1, βk, [u]N,k+1, [u]T,k+1) = 0« #F~.l¬

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[u]k+1 = [u]free +WPk+1« #F~&,5¬

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W = H(qk+1)(M + hθCt + θ2h2Kt)−1)HT (qk+1),

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vfree = H(qk+1)(vk − (M + hθCt + θ2h2Kt)−1φ(tk , qk, vk))

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[u]k+1 = [u]free +WPk+1

−PN,k+1 + PcohN,k+1 ∈ NT

IR+ (uN k+1)([u]N,k+1 + e[u]N,k)

−PT,k+1 + PcohT,k+1 ∈ ∂(µκ(βk+1)|PN,k+1 + Pcoh

N,k+1| ‖ [u]T,k+1 ‖)

Pcohk+1 = hRcoh

k+1, Rcohk+1 = K(βk+1) · [u]k+1 g(βk+1, βk+1, βk, [u]N,k+1, [u]T,k+1) = 0« #F~&%5¬

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−PN,k+1 = −PN,k+1 + Pcoh

N,k+1,−PT,k+1 = −PT,k+1 + Pcoh

T,k+1« #+;45¬

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cohk+1

βk+1rl`tw

[u]k+1

[u]k+1 = [u]free +WPk+1

−PN,k+1 ∈ NTIR+ (uN k+1)([u]N,k+1 + e[u]N,k)

−PT,k+1 ∈ ∂(µκ(βk+1)|PN,k+1| ‖ [u]T,k+1 ‖)

« #+;#¬

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[u]k+1

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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)

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ISSN 0249-6399