Modelo Constitutivo Elastoplástico Anisótropo en

CENTRO INTERNACIONAL DE MÉTODOS NUMÉRICOS EN INGENIERÍA
Modelo Constitutivo Elastoplástico
Anisótropo en Deformaciones Finitas
para Análisis de Materiales
Compuestos Reforzados con Fibras
E. Car
S. Oller
E. Oñate
Publicación CIMNE Nº-139, Octubre 1998
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INCREMENT: n
∆ Ui+1 = ∆ Ui +K-1 Fresid
NEXT ITERATION
Reference Configuration
E i +1 = Ei + 1 / 2(C − I )
e i +1 = φ (Ei +1 )
(σ ) = (c ) : (e
i +1 *
c
i +1
c
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Elastic Trial Stress
i
1
(σ )
i *
n
i *
3
CONSTITUTIVE
MODEL PHASE 3
COMPOUNDING 3
CONSTITUTIVE
MODEL PHASE 2
COMPOUNDING 2
CONSTITUTIVE
MODEL PHASE 1
COMPOUNDING 1
(σ )
(σ )
(σ )
i *
2
(σ )
i
(σ )
i
i
2
CONSTITUTIVE
MODEL PHASE n
(σ )
i *
1
COMPOUNDING n
(σ )
Updated Configuration
CONSTITUTIVE MODELS FOR EACH COMPOUND
3
n
RECOMPOSITION
σ i = ∑ k c (σ i )c
c
=
NO CONVERGENCE
CHECK RESIDUAL
FORCE AND
CONVERGENCE
Reference
Configuration
S = φ (σ i )
NEXT
INCREMENT: n+1
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sdud od ghqlflöq gh od hqhujðd oleuh hoävwlfd vh xwlol}d xq srwhqfldo fxdguäwlfr +Jdufðd
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whuðvwlfdv gho pdwhuldo1 Vdwlvidfhu odv frqglflrqhv gh lqyduldqfld hq hvwrv fdvrv uhvxowd
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Uhhpsod}dqgr od hf1+64, hq od hf1+5<, uhvxowd od h{suhvlöq gho sulphu sulqflslr gh
od whuprglqäplfd hq ixqflöq gh od hqhujðd oleuh gh Khopkrow}1
pf # . . @ V =H .pf u gly +Mt,
4;
pf # @ V =H .pf u gly +t, pf pf +65,
Od irupd hxohuldqd
orfdo gho sulphu
sulqflslr uhvxowd gh frqvlghudu od wudqvirupdflöq

3A
3
j = g @ a V =H/ vlhqgr g @ I
ho judglhqwh gh yhorflgdghv/ pf @ Mp od pdvd
HI
_T
| t @ Mt=
hq od frqjxudflöq uhihuhqfldo/ M @ ghw I @ _T
f
p #@ j = g . pu gly +t, p p +66,
h{suhvlöq orfdo Hxohuldqd gho sulphu sulqflslr gh od whuprglqäplfd hq ixqflöq gh od
hqhujðd oleuh gh Khopkrow}1
Ho vhjxqgr sulqflslr hvwdeohfh txh sdud xq surfhvr luuhyhuvleoh vh gheh fxpsolu txh
ho fdpelr hq od surgxfflöq lqwhuqd gh hqwursðd gheh vhu pd|ru r ljxdo txh ho fdpelr
gh hqwursðd lqwurgxflgd +Pdoyhuq/ 4<9<, +Roohu/ 4<;<,/ hvwr hv=
g
+67,
@ @?
gw
U
U 6o
vlhqgr
K ^ od hqwursðd lqwhuqd @ @ T pgY | od hqwursðd lqwurgxflgd @? @ T w gY .
7 w gV1
Wudqvirupdqgr hq od ghvljxdogdg dqwhulru odv lqwhjudohv gh vxshuflh hq lqwhjudohv
yroxpìwulfdv | frpelqdqgr frq od irupd orfdo hxohuldqd +hf15;, gho sulphu sulqflslr gh
od whuprglqäplfd uhvxowd od glvlsdflöq orfdo sru xqlgdg gh pdvd frpr +Roohu/ 4<;<,=
4
j=g
@$.
t=u
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p
Whqlhqgr hq fxhqwd od ghqlflöq gh od hqhujðd oleuh gh Khopkrow} +hf1+63,, uhvxowd
od h{suhvlöq orfdo Hxohuldqd gh od glvlsdflöq=
4
j=g
t=u
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p
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Ixqflöq gh hqhujðd oleuh gh Khopkrow}
Hq od ghqlflöq gh od hqhujðd oleuh gh Khopkrow} qr hv fruuhfwr hpsohdu frpr ydul0
deoh oleuh ho whqvru hoävwlfr gh Juhhq 0 Odjudqjh hq od frqjxudflöq uhihuhqfldo |d txh
ho plvpr vh yh dihfwdgr sru ho xmr soävwlfr d wudyìv gh od frpsrqhqwh soävwlfd gho
judglhqwh gh ghirupdflrqhv IR / sru or wdqwr hv frqyhqlhqwh ghqlu od hqhujðd oleuh gh
Khopkrow} frpr=
# @ # +he > > ,
+6:,
grqgh hv od whpshudwxud devroxwd/ he hv ho whqvru gh ghirupdflrqhv gh Dopdqvl |
xq frqmxqwr gh yduldeohv gh hvwdgr soävwlfdv/ frpr sru hmhpsor od frkhvlöq hqwuh
sduwðfxodv/ od yduldeoh gh hqgxuhflplhqwr/ ho äqjxor gh ur}dplhqwr lqwhuqr/ hwf1
4<
71616
Klsöwhvlv gh hodvwlflgdg ghvdfrsodgd
Xq fdvr sduwlfxodu gh od ixqflöq gh hqhujðd oleuh or frqvwlwx|h od klsöwhvlv gh hodvwlflgdg
ghvdfrsodgd h{suhvdgd phgldqwh od hfxdflöq=
# +he > > , @ # e +he > , . # R +> ,
+6;,
Hvwd klsöwhvlv vh edvd hq txh orv qlyhohv gh hqhujðd qhfhvdulrv sdud glvwruvlrqdu od
uhg fulvwdolqd/ uhvsrqvdeoh gh od ghirupdflöq hoävwlfd/ vrq glihuhqwhv gh orv qlyhohv gh
hqhujðd qhfhvdulrv sdud ho ghvol}dplhqwr lqwhufulvwdolqr1 Hvwd klsöwhvlv kd vlgr xwlol}dgd
sru glyhuvrv dxwruhv | frqvwlwx|h od edvh gh prghorv frpsxwdflrqdohv sursxhvwrv/ hqwuh
rwurv sru +Jdufðd Jdulqr dqg Rolyhu/ 4<<4, +Jdufðd Jdulqr dqg Rolyhu/ 4<<5, +Oxeolqhu/
4<<3, +Roohu/ 4<;<, +Oxfflrql hw do1/ 4<<9,1
Odv yduldeohv oleuhv gho sureohpd vrq od whpshudwxud | od fxrwd hoävwlfd gh od
ghirupdflöq gh Dopdqvl ghqlgd frpr=
] |
4
R
e
R
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h @h h @
h gw
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f
grqgh hR hv od ghirupdflöq soävwlfd hq od frqjxudflöq hvsdfldo/ e hv ho whqvru l}txlhugr
R
gh Fdxfk|0Juhhq/ | h uhsuhvhqwd od yduldflöq whpsrudo gh od ghirupdflöq soävwlfd hq
od frqjxudflöq hvsdfldo1
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H{suhvlöq gh od glvlsdflöq
Od yduldflöq whpsrudo gh od hqhujðd oleuh vh h{suhvd frpr=
C# e C# C# g .
.
+73,
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C
C
grqgh ge @ O +he , 7 uhsuhvhqwd od yduldflöq whpsrudo gh od fxrwd hoävwlfd gh od ghiru0
pdflöq1 Uhhpsod}dqgr od hf1+64, hq od hf1+69, vh rewlhqh od h{suhvlöq gh od glvlsdflöq=
C# e C# C# 4
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.
.
p . = g t=u 3
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C
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Rshudqgr od h{suhvlöq gh od glvlsdflöq txhgd=
C#
C# 4
C# e
t=u 3
p
@ = g eg p p Ch
C
C
+75,
Whqlhqgr hq fxhqwd od ghvfrpsrvlflöq dglwlyd gho whqvru yhorflgdg gh ghirupdflöq
g @ ge . gR +Jdufðd Jdulqr dqg Rolyhu/ 4<<4, od hf1+75, txhgd=
C#
C#
C# 4
e
@ p e =g p
t=u 3
. = gR p
Ch
C
C
7
Ghulydgd gh Ohh1 Yhu Dsìqglfh L
53
+76,
H{suhvlöq txh uhfleh ho qrpeuh gh ghvljxdogdg gh Fodvlxv0Gxkhp1 Hvwd lqhfxdflöq
gheh vhu vdwlvihfkd sdud wrgrv orv srvleohv surfhvrv1 Sxhvwr txh wdqwr g frpr uhs0
uhvhqwdq yduldflrqhv whpsrudohv duelwuduldv gh odv yduldeohv oleuhv gho sureohpd/ sdud
jdudqwl}du ho fxpsolplhqwr gh od hf1+76, sdud xq gdgr hvwdgr whuprglqäplfr/ vxv pxo0
wlsolfdgruhv ghehq vhu lgìqwlfdphqwh qxorv1 Gh dtxð vxujh od hfxdflöq frqvwlwxwlyd gho
pdwhuldo=
C#
C#
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@
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e
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C
Odv hfv1 +77, vrq odv oh|hv frqvwlwxwlydv jhqhudohv sdud wrgr wlsr gh pdwhuldohv1
Hvwdeohfhq txh ho hvwdgr whqvlrqdo | od hqwursðd vh rewlhqhq frpr od ghulydgd gh od
ixqflöq gh hqhujðd oleuh uhvshfwr gh od fxrwd gh ghirupdflrqhv hoävwlfdv gh Dopdqvl
| od whpshudwxud uhvshfwlydphqwh/ vlhqgr qhfhvdulr sru hoor ghqlu od h{suhvlöq gh od
ixqflöq gh hqhujðd oleuh1
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H{suhvlöq gh od hqhujðd oleuh1 Sduwlfxodul}dflöq d shtxhôdv ghiru0
pdflrqhv hoävwlfdv | hvwdgrv lvrwìuplfrv
Orv pdwhuldohv frpsxhvwrv/ d h{fhsflöq gh orv pdwhuldohv hodvwöphurv/ suhvhqwdq ghiru0
pdflrqhv hoävwlfdv shtxhôdv1 Hvwr frqgxfh d txh od sduwh hoävwlfd gho whqvru judglhqwh
gh ghirupdflrqhv Ie vh dsur{lph d od xqlgdg | od sduwh hoävwlfd gho whqvru l}txlhugr
gh Fdxfk|0Juhhq +ee ,3 wlhqgh do whqvru pìwulfr hvsdfldo j1 Hq hvwh fdvr od glvwlqflöq
hqwuh frqjxudflöq lqwhuphgld | ghirupdgd fduhfh gh lpsruwdqfld1
Vlq hpedujr/ odv ghirupdflrqhv soävwlfdv frqwlqýdq vlhqgr qlwdv | hv qhfhvdulr
pdqwhqhu od suhvhqfld gho whqvru ghuhfkr gh Fdxfk|0Juhhq F hq od h{suhvlöq pdwhuldo
gho prghor frqvwlwxwlyr1 Vl hvwh whqvru qr vh wlhqh hq fxhqwd h{soðflwdphqwh vh hvwä
ghqlhqgr xq pdwhuldo glihuhqwh/ pdwhpäwlfdphqwh yäolgr shur vh slhugh ydolgh} iðvlfd1
Sdud ho fdvr gh shtxhôdv ghirupdflrqhv hoävwlfdv uhvxowd vxflhqwh fdudfwhul}du od
frpsrqhqwh hoävwlfd gh od hqhujðd oleuh phgldqwh xqd ixqflöq fxdguäwlfd gho whqvru
hoävwlfr gh ghirupdflrqhv gh Dopdqvl he / hvwr hv=
4 e j e
+78,
h =f =h
5p
Whqlhqgr hq fxhqwd od klsöwhvlv gh hodvwlflgdg ghvdfrsodgd +hf1 6;,/ od h{suhvlöq gh
od sduwh hoävwlfd gh od hqhujðd oleuh +hf1 78, | od h{suhvlöq gho whqvru frqvwlwxwlyr hq ho
hvsdflr dqlvöwursr uhdo hq ixqflöq gho whqvru lvöwursr fwlflr +hf1 58, sxhgh uhhvfuleluvh
frpr=
#e @
4
+he = f = he , . # R
5p
l
4 e k j 3
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h = +d , = f = +de , = he . # R +,
5p
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54
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flr dqlvöwursr uhdo ghqlgd hq wìuplqrv gho whqvru frqvwlwxwlyr hq ho hvsdflr lvöwursr
fwlflr/ hvwr hv=
k
l
C#
j 3
e
@
+d
,
=
f
=
+d
,
= he
Che
l
k
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j@p
+7;,
+7<,
grqgh he uhsuhvhqwd od fxrwd gh ghirupdflöq hoävwlfd gh Dopdqvl hq ho hvsdflr lvöwursr
fwlflr | j uhsuhvhqwd od whqvlöq hq ho hvsdflr lvöwursr fwlflr1
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Uhjod gh xmr1 Oh| gh hyroxflöq gh odv yduldeohv lqwhuqdv
Od oh| gh hyroxflöq hq ho wlhpsr gh od ghirupdflöq soävwlfd hvwä gdgd sru=
Cj
+83,
Cj
Whqlhqgr hq fxhqwd txh wrgd od dqlvrwursðd gho pdwhuldo hvwä frqwhqlgd hq ho whqvru
gh wudqvirupdflöq dj / od ixqflöq gh srwhqfldo soävwlfr sdud ho vöolgr dqlvöwursr uhvxowd=
R
gR @h @
j +j> , @ j +j> dj > , @ j +j> , @ N
+84,
Uhhpsod}dqgr od hf1+84, hq od hf1+83, ho lqfuhphqwr gh od fxrwd soävwlfd gh od
ghirupdflöq gh Dopdqvl uhvxowd=
R
h @
j
Cj Cj Cj Cj
@
=
@
= dj @ h = dj
Cj
Cj Cj
Cj
+85,
j
grqgh h hv ho xmr soävwlfr qrupdo do srwhqfldo lvöwursr j1
Od oh| gh hyroxflöq gh od yduldeoh lqwhuqd gh hqgxuhflplhqwr soävwlfr hvwä gdgr sru=
Cj 6
Cj Cj Cj
@ +k6 ,j =
+86,
@ +k ,j =
=
= dj
Cj
Cj Cj
Cj
grqgh +k6 ,j hv xq whqvru gh vhjxqgr rughq/ ixqflöq gho hvwdgr gh whqvlrqhv dfwxdol}dgr
| gh od yduldeoh gh hqgxuhflplhqwr soävwlfr wdpelìq dfwxdol}dgd txh hq ho fdvr päv
vlpsoh gh od whruðd gh od sodvwlflgdg wrpd od irupd gho whqvru gh whqvlrqhv1 Od oh| gh
hyroxflöq gh od yduldeoh lqwhuqd uhvxowd=
@ +k6 ,j =
Cj
+87,
Cj
Ho frqfhswr gh dglwlylgdg gh odv ghirupdflrqhv shuplwh h{whqghu od uhjod gh wudqv0
irupdflöq gh odv ghirupdflrqhv d od fxrwd soävwlfd gh odv ghirupdflrqhv/ hv ghflu=
@ j =
j
Cj
= dj @ de = h = dj
Cj
R
grqgh h hv od ghirupdflöq soävwlfd lvöwursd fwlfld1
R
R
h @ de =h @ de =
55
+88,
71714
Od glvlsdflöq hq ho hvsdflr lvöwursr fwlflr1 Xqlflgdg gh od glvl0
sdflöq1
Hq hvwh dsduwdgr vh ghpxhvwud txh od glvlsdflöq hv xqd lqyduldqwh gho surfhvr whuprg0
lqäplfr/ vlhqgr vx ydoru lqghshqglhqwh vl vh od fdofxod hq ho hvsdflr lvöwursr fwlflr r
hq ho hvsdflr dqlvöwursr uhdo uhvxowdqgr htxlydohqwh hvfulelu ho prghor frqvwlwxwlyr hq
ho hvsdflr uhdo dqlvöwursr r hq ho hvsdflr fwlflr lvöwursr1
Od glvlsdflöq phfäqlfd sxhgh vhu hvfulwd hq ho hvsdflr lvöwursr fwlflr whqlhqgr
hq fxhqwd odv wudqvirupdflrqhv=
j @ dj = j
+89d,
e
h @ d =h
+89e,
Od oh| gh xmr hvwä h{suhvdgd sru=
Cj
+8:,
= dj
Cj
Whqlhqgr hq fxhqwd od hf1+76, od h{suhvlöq gh od glvlsdflöq soävwlfd hq ho hvsdflr
dqlvöwursr uhdo uhvxowd=
R
h @ de =
C# 4
t=u 3
+8;,
C
Ho sulphu wìuplqr gh od hf1+8;, uhvxowd whqlhqgr hq fxhqwd odv wudqvirupdflrqhv
h{suhvdgdv hq od hf1+89d, | od hf1+89e, vh wlhqh=
k
l k
Rl
R
j =h @ +dj ,3 = j = +de ,3 =h
j6eS @ = gR p
R
j =h @ +dj ,3 = j = +de ,3 = de =
Cj
= dj
Cj
j
Cj
+8<,
@j= h
Cj
j
C# C# 6eS @ = gR p
@ j = h p
6eS 3
+93,
C
C
Gh od hf1+93, vh ghgxfh txh od glvlsdflöq hv xqd lqyduldqwh gho surfhvr whuprg0
lqäplfr/ vlhqgr vx ydoru lqghshqglhqwh vl vh fdofxod hq ho hvsdflr lvöwursr fwlflr r hq
ho hvsdflr dqlvöwursr uhdo1
Od h{suhvlöq gh od hqhujðd oleuh gh Khopkrow} hq ho hvsdflr lvöwursr fwlflr hq xq
surfhvr lvrwìuplfr uhvxowd=
R
j =h @ j =
# +he > , @
R
4
+he = f = he , . # +6 ,
5p
56
+94,
Od hf1+94, hv htxlydohqwh d od h{suhvlöq gh od hqhujðd oleuh hq ho hvsdflr dqlvöwursr
uhdo +hf1+79,,1 Od hfxdflöq frqvwlwxwlyd hq ho hvsdflr lvöwursr fwlflr uhvxowd=
C# +he > ,
@ f = he
+95,
Che
Whqlhqgr hq fxhqwd od hf1+58, | od hf1+89e, ho whqvru gh whqvlrqhv gh Fdxfk| hq ho
hvsdflr lvöwursr fwlflr hvwä gdgr sru=
k
l
j @ dj = fj = +de ,3 = +de = he ,
j@p
j @ dj = fj = L = he
j @ dj = fj = he @ dj = j
+96,
Od h{suhvlöq dqwhulru ghpxhvwud txh whqlhqgr hq fxhqwd odv wudqvirupdflrqhv hqwuh
hvsdflrv/ uhvxowd htxlydohqwh hvfulelu ho prghor frqvwlwxwlyr hq ho hvsdflr uhdo dqlvöwursr
r hq ho hvsdflr fwlflr lvöwursr1 Reyldphqwh hvfulelu orv prghorv frqvwlwxwlyrv hq ho
hvsdflr lvöwursr fwlflr shuplwh xwlol}du odv yhqwdmdv | dojrulwprv xwlol}dgrv sdud ho
fdvr gh pdwhuldohv lvöwursrv1
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Hfxdflöq frqvwlwxwlyd wdqjhqwh
Od hfxdflöq frqvwlwxwlyd wdqjhqwh vh rewlhqh frqvlghudqgr od yduldflöq whpsrudo gh od
whqvlöq gh Fdxfk|/ hvwr hv=
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Hvwd hfxdflöq hv srvleoh rewhqhuod wdpelìq frqvlghudqgr od hf1+89d, txh uhodflrqd
od whqvlöq gh Fdxfk| hq ho hvsdflr lvöwursr fwlflr | od whqvlöq gh Fdxfk| ho hvsdflr
dqlvöwursr uhdo1 Whqlhqgr hq fxhqwd hq hvwd hfxdflöq txh ho whqvru gh wudqvirupdflöq
hqwuh hvsdflrv dj vh pdqwlhqh frqvwdqwh hq ho wlhpsr | od frqglflöq gh frqvlvwhqfld
soävwlfd hv srvleoh rewhqhu od hfxdflöq frqvwlwxwlyd hq ho hvsdflr lvöwursr fwlflr/ hvwr
hv=
57
j@ +f,eR =h
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grqgh +f,eR uhsuhvhqwd ho whqvru frqvwlwxwlyr hodvwrsoävwlfr wdqjhqwh hq ho hvsdflr lvöwursr
fwlflr | hvwä gdgr sru=
Ys
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for Yj
f
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or
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Whqlhqgr hq fxhqwd od hf1+89d, txh uhodflrqd ho whqvru gh whqvlrqhv hq ho hvsdflr
dqlvöwursr uhdo | ho hvsdflr lvöwursr fwlflr hv srvleoh rewhqhu od yduldflöq gh whqvlöq
hq ho hvsdflr dqlvöwursr uhdo frpr=
3 eR @ dj&,
f&,or hor
+9:,
Frpelqdqgr od hf1+9:, frq od hf1+89e, | whqlhqgr hq fxhqwd txh ho whqvru gh wudqv0
irupdflöq gh hvsdflrv de qr ghshqgh gho wlhpsr/ vh rewlhqh od h{suhvlöq gho whqvru
frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr frpr=
3 eR e @ dj&,
f dorR^ hR^

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€
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R^
Od hf1 +9;, pxhvwud od h{suhvlöq gho whqvru hodvwrsoävwlfr dqlvöwursr/ txh uhvxowd
xqd ixqflöq gho whqvru frqvwlwxwlyr hodvwrsoävwlfr wdqjhqwh hq ho hvsdflr lvöwursr fwlflr1
Ho prghor frqvwlwxwlyr suhvhqwdgr qhfhvlwd vrodphqwh d orv qhv gh prghodu ho
frpsruwdplhqwr frqvwlwxwlyr gh xq pdwhuldo frpsxhvwr dowdphqwh dqlvöwursr ghqlu odv
vljxlhqwhv fdudfwhuðvwlfdv gho pdwhuldo hq orv hvsdflrv uhdo dqlvöwursr | fwlflr lvöwursr=
Hvsdflr dqlvöwursr uhdo=
Whqvru frqvwlwxwlyr lqlfldo hq frrughqdgdv orfdohv f,JS@,
Xpeudo gh xhqfld gho pdwhuldo +e,_
Whqvru gh urwdflrqhv U
Hvsdflr lvöwursr uhdo
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Ixqflöq gh srwhqfldo soävwlfr j +j> , @ 3
Xpeudo gh xhqfld gho pdwhuldo +e,_
Hq od Iljxud 6 vh revhuydq odv rshudflrqhv txh vh ghehq oohydu d fder sdud fdgd
idvh txh frqvwlwx|h ho pdwhuldo frpsxhvwr1 Od sulphud rshudflöq frqvlvwh wudqvsruwdu ho
whqvru gh ghirupdflrqhv H gh od frqjxudflöq uhihuhqfldo d od frqjxudflöq dfwxdol}dgd
xwlol}dqgr odv rshudflrqhv sxvk iruzdug+yhu Dsìqglfh L,1 Od rshudflöq 5 frqvlvwh
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fdofxodu od whqvlöq suhglfwrud jW | wudqvsruwdu orv whqvruhv gh whqvlrqhv suhglfwrudv/
ghirupdflrqhv | frqvwlwxwlyr j W > h | f gh od frqjxudflöq dfwxdol}dgd do hvsdflr lvöwursr
fwlflr xwlol}dqgr orv whqvruhv dj | de 1 Hq ho hvsdflr lvöwursr fwlflr vh surfhgh d
yhulfdu vl odv whqvlrqhv hq ho pdwhuldo kdq vxshudgr ho oðplwh gh sursruflrqdolgdg1
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hodvwrsoävwlfr1 Od vroxflöq gho sureohpd hodvwrsoävwlfr sursruflrqd xq qxhyr hvwdgr
whqvlrqdo j1 Od rshudflöq 6 frqvlvwh hq uhwruqdu gho hvsdflr lvöwursr fwlflr do hvsdflr
dqlvöwursr uhdo
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Ho fdvr gh pdwhuldohv frq frpsruwdplhqwr lqhoävwlfr uhtxlhuh od lqwhjudflöq qxpìulfd
gh od hfxdflöq frqvwlwxwlyd hq xqd vhfxhqfld glvfuhwd gh sdvrv gh wlhpsr1+Vlpr dqg
Wd|oru/ 4<;8,
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ghqh ho whqvru gh whqvlrqhv frpr xqd ixqflöq gh od klvwruld gh ghirupdflrqhv kdvwd
ho sdvr gh wlhpsr dfwxdo1 Hvwh dojrulwpr gh lqwhjudflöq shuplwh wudwdu ho sureohpd
hodvwrsoävwlfr ixqgdphqwdophqwh frpr xq sureohpd hoävwlfr htxlydohqwh hq ho sdvr gh
wlhpsr1
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phgldqwh xqd olqhdol}dflöq gh od ixqflöq gh uhvsxhvwd frqvlvwhqwh frq ho dojrulwpr gh
lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd1 Od xwlol}dflöq gh hvwrv rshudgruhv wdqjhqwhv
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grv gh Qhzwrq1
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uhdol}d hq ho hvsdflr lvöwursr fwlflr1 Od lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd hvwä
edvdgd hq ho frqfhswr gh uhwxuq pdsslqj xwlol}dqgr ho dojrulwpr gh edfnzdug0Hxohu1
Hq od uhvroxflöq gh sureohpdv hodvwrsoävwlfrv txh lqfox|hq hfxdflrqhv frqvwlwxwlydv lq0
fuhphqwdohv od frqvlvwhqfld hqwuh ho rshudgru wdqjhqwh | ho dojrulwpr gh lqwhjudflöq
hpsohdgr hq od vroxflöq gho sureohpd lqfuhphqwdo mxhjd xq sdsho ixqgdphqwdo1 Hq ho
prghor sursxhvwr vh kdq xwlol}dgr hvwrv rshudgruhv wdqjhqwhv txh suhvhuydq od frqyhu0
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PRGHOR FRQVWLWXWLYR GHO FRPSXHVWR
Ho prghor hodvwrsoävwlfr dqlvöwursr hq uìjlphq gh judqghv ghirupdflrqhv/ suhvhqwdgr
hq ho dsduwdgr dqwhulru/ frqvwlwx|h xqr gh orv prghorv edvh txh vh xwlol}dq hq od
whruðd gh ph}fodv1 Hq sduwlfxodu sdud ho fdvr gh pdwhuldohv frpsxhvwrv uhiru}dgrv frq
eudv odujdv vh frqvlghudq xq prghor frqvwlwxwlyr sdud fdgd idvh= eud | pdwul}1
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59
REFERENCE
CONFIGURATION
Real Anisotropic Space
C; S; E
4
Pull Back
$ = ϕ(σ )
←
= ϕ(.
Push Forward
ep
e = ϕ (E )
→
)
←
ep
←
Fictitious Isotropic Space
c ;σ ; e
ep
σ = (,
) :σ
3
2
UPDATED
0 = (, ) : 0
CONFIGURATION
. = (, ) : . : ,
σ
e
ep
−1
−1
σ
−1
ep
e
σ=, :σ
0 =, :0
. = arbitrary
σ
e
Real Anisotropic Space
c;σ ; e
1
Iljxud 6= Hvtxhpd gho prghor frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr hq ghirupdflrqhv
qlwdv
5:
INCREMENT: n
∆ Ui+1 = ∆ Ui +K-1 Fresid
NEXT ITERATION
Ei +1 = Ei + 1 / 2(C − I )
Closing equation
e
i +1
(e ) = (e ) = (e )
= φ (Ei +1 )
i +1
i +1
i +1
1
(τ ) = (c ) : (e
i +1 *
c
c
i +1
− (e p )c )
2
= = (e i +1 )n
Elastic Trial Stress
COMPOUNDING 2 fiber
i *
MATRIX
(τ )
i
MATRIX
(τ )
i *
FIBER
ANISOTROPIC
ELASTOPLASTIC
CONSTITUTIVE
MODEL FOR
FIBERS
(τ )
ISOTROPIC
CONSTITUTIVE
MODEL THE
MATRIX
COMPOUNDING 1 matrix
CONSTITUTIVE MODELS FOR EACH COMPOUND
(τ )
i
FIBER
RECOMPOSITION
τ i = ∑ k c (τ i )c
c
=
S = φ (τ i )
NO CONVERGENCE
CHECK RESIDUAL
FORCE AND
CONVERGENCE
NEXT
INCREMENT: n+1
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lvöwursr foävlfr | ho prghor frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr vh xwlol}d sdud vlpxodu
ho frpsruwdplhqwr gho uhixhu}r1
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hq ho txh xqd gh odv idvhv hvwd frqvwlwxlgd sru xq pdwhuldo dqlvöwursr | od idvh uhvwdqwh
hv xq pdwhuldo lvöwursr1
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Ghvfulsflöq jhqhudo
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hmhpsorv frqvlvwhq hq vlpxodu qxpìulfdphqwh xq hqvd|r vreuh surehwdv gh pdwhuldo
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surslhgdghv phfäqlfdv gh fdgd xqr gh orv pdwhuldohv frqvwlwx|hqwhv gho pdwhuldo frp0
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hqwdoodgxud uhvshfwr gho ghvsod}dplhqwr hq ho h{wuhpr gh od surehwd/ frpsdudgd
frq phglflrqhv h{shulphqwdohv1 Orv gdwrv h{shulphqwdohv vh kdq rewhqlgr gho
HVL Surmhfw Qe HG2;606;62UG2PV | HG2;707482UG2PV +Vwdyulqlglv/ 4<;8, |
gh orv hqvd|rv uhdol}dgrv sru od Hxurshdq Vsdfh Djhqf| +GIYOU/ 4<;6,1
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hodvwrsoävwlfr/ plhqwudv txh odv eudv vh frpsruwdq frpr xq pdwhuldo hodvwrsoävwlfr
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revhuydq odv glphqvlrqhv gh od surehwd | xq ghwdooh gh od }rqd fhqwudo hq od txh vh
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;1 Hq od plvpd vh pxhvwud od }rqd gh ruljhq gh grv sduhv gh vxudv txh qdfhq hq od
udð} gh fdgd hqwdood | vh sursdjdq hq od gluhfflöq gho hmh orqjlwxglqdo gh od surehwd1
Hq od Iljxud < vh suhvhqwdq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Vh
revhuyd txh hq od }rqd fhqwudo gh od surehwd frpsuhqglgd hqwuh dpedv hqwdoodv ho fdpsr
gh ghvsod}dplhqwr suhvhqwd xqd yduldflöq vxdyh/ h{lvwlhqgr fxdwur }rqdv fodudphqwh
glihuhqfldgdv hq odv fxdohv ho fdpsr gh ghvsod}dplhqwrv suhvhqwd xqd yduldflöq ixhuwh1
Hvwdv }rqdv frplhq}dq hq od udð} gh fdgd xqd gh odv hqwdoodv | surjuhvdq sdudohodphqwh
do hmh orqjlwxglqdo gh od surehwd frlqflghqwh frq od gluhfflöq gh odv eudv1
Hq od Iljxud 43 vh revhuydq odv lvräuhdv gh od yduldeoh lqwhuqd gh sodvwlflgdg sdud ho
frpsxhvwr txh lqglfdq odv }rqdv hq odv fxdohv vh kd surgxflgr ghirupdflöq soävwlfd1 Vh
pxhvwud txh odv }rqdv päv dihfwdgdv vh hqfxhqwudq hq od udð} gh odv hqwdoodv ghelgr d od
frqfhqwudflöq gh whqvlrqhv hq hvd }rqd | txh od sodvwlflgdg surjuhvd hq irupd sdudohod
do hmh orqjlwxglqdo gh od surehwd1
Hq od Iljxud 44 vh suhvhqwd xq frqwruqr gh odv }rqdv grqgh vh kd surgxflgr ho
ihqöphqr gh gherqglqj r ghvshjxh gh odv eudv uhvshfwr gh od pdwul}1 Xqr gh orv pr0
wlyrv gho frpsruwdplhqwr qr olqhdo gh orv pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv
vh gheh do ihqöphqr gh irupdflöq gh julhwdv hq od pdwul} | ho frqvljxlhqwh ghvol}d0
plhqwr r prylplhqwr uhodwlyr hqwuh eud | pdwul}1 Hvwh ihqöphqr gh djulhwdplhqwr gh
od pdwul} | ghvol}dplhqwr hqwuh eud | pdwul} uhgxfh od uljlgh} gho pdwhuldo h lqgxfh
67
Iljxud := Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Ghirupdgd 83=4
Iljxud ;= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Ghwdooh ghirupdgd hq od hqwdo0
odgxud
68
Iljxud <= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh ghvsod}dplhqwrv
Iljxud 43= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr yduldeoh lqwhuqd gh
sodvwlflgdg
69
ghirupdflrqhv lqhoävwlfdv r qr uhfxshudeohv1 Hvwh ihqöphqr vh wlhqh hq fxhqwd hq od
vlpxodflöq qxpìulfd olplwdqgr od fdsdflgdg gh fdujd gh od eud ghelgr d od lpsrvlelo0
lgdg gh od pdwul} gh wudqvplwluoh fdujd1 Hq wudedmrv ixwxurv vh dqdol}duä frq pd|ru
ghwdooh hvwh ihqöphqr1 Hq Iljxud 44 vh revhuyd frpr ho ihqöphqr surjuhvd ghvgh od
udð} gh odv hqwdoodv/ ghelgr d od frqfhqwudflöq gh whqvlrqhv hq od }rqd/ kdfld ho fhqwur
gh od surehwd1 Ho ihqöphqr gh ghvshjxh gh od eud uhvshfwr gh od pdwul} lpslgh txh
odv eudv dofdqfhq od whqvlöq pä{lpd gh xhqfld gh odv plvpdv |d txh od pdwul} qr hv
fdsd} gh wudqvihulu orv hvixhu}rv d od eud/ revhuyäqgrvh xq fdpelr hq od shqglhqwh gh
od fxuyd Fdujd 0 Ghvsod}dplhqwr1 +yhu Iljxud 48,
Iljxud 44= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh sodvwlflgdg hq od
eud
Hq odv Iljxud 45 | 46 vh revhuydq orv frqwruqrv gh whqvlrqhv hq od gluhfflöq [[ h
\ \ uhvshfwlydphqwh gho pdwhuldo frpsxhvwr1 Vh pxhvwud od frqfhqwudflöq gh whqvlrqhv
txh vh surgxfh hq od }rqd gh odv hqwdoodv dvð frpr orv fdpelrv txh vh surgxfhq hq ho
hvwdgr whqvlrqdo hq od }rqd gh od vxud1 Hq od Iljxud 46 vh revhuyd xq ghwdooh gh od
}rqd vxudgd hq od surehwd dvð frpr od uhglvwulexflöq gho hvwdgr whqvlrqdo hq od }rqd
qdo gh od vxud1
Hq od Iljxud 47 vh revhuydq xq ghwdooh hq od }rqd gh od hqwdood gh odv lvräuhdv gh
whqvlöq ho od gluhfflöq sdudohod do hmh orqjlwxglqdo gh od surehwd hq od eud1
6:
Iljxud 45= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh whqvlrqhv [[ hq
ho frpsxhvwr
Iljxud 46= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh whqvlrqhv \\ hq
ho frpsxhvwr
6;
Iljxud 47= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh whqvlrqhv \\ hq
od eud
:1614
Frpsuredflöq qxpìulfr0h{shulphqwdo
Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh
orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder
vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 3J uhvshfwr gho hmh orqjlwxglqdo gh
od surehwd1
Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv gxudqwh ho hqvd|r | orv uhvxowdgrv
rewhqlgrv hq od vlpxodflöq qxpìulfd vh revhuydq hq odv Iljxudv 48 | 491 Hq od sulphud
gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh od surehwd hq
ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1
Od Iljxudv 48 | 49 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/ orv
uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0
wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq
qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh
odv uhvsxhvwdv1
D qlyhohv gh fdujd hohydgrv vh revhuyd xq ihqöphqr qr olqhdo hq orv hqvd|rv h{shul0
phqwdohv uhodflrqdgr frq ho sureohpd gh ghvshjxh hqwuh eud | pdwul}1 Od vlpxodflöq
qxpìulfd ghwhfwd frq xq dfxhugr qrwdeoh uhvshfwr gh orv uhvxowdgrv h{shulphqwdohv ho
frplhq}r gh hvwh ihqöphqr1 Hq od Iljxud 49 vh revhuyd odv fxuydv hqwuh ho ghvsod}d0
plhqwr hq od sduwh vxshulru gh od surehwd | od dshuwxud gh od hqwdood hq od }rqd fhqwudo1
Vh ghqh frpr FRG +Fudfn Rshqlqj Glvsodfhphqw, od dshuwxud gh od hqwdood phglgd
6<
30000
25000
Force [N]
20000
15000
Non linear model
10000
Experimental results
Linear model
5000
0
0
50
100
150
200
250
Displacement [µm]
Iljxud 48= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv
hq od sduwh vxshulru1
d 4> 8pp= gho hmh orqjlwxglqdo gh od hqwdood kdfld od sduwh vxshulru | d 4> 8pp= gho hmh
orqjlwxglqdo gh od hqwdood kdfld od }rqd lqihulru gh od surehwd | d 4pp= ghvgh ho erugh
h{whulru gh od surehwd1 +Yhu Iljxud 8,
Vh revhuyd hq od Iljxud 49 txh orv uhvxowdgrv h{shulphqwdohv pxhvwudq xqd dshuwxud
lqihulru gh od vxud sdud ghvsod}dplhqwrv hq od sduwh vxshulru gh od surehwd frpsuhqgl0
grv hqwuh 3p= | 458p= | d sduwlu gh dtxð orv uhvxowdgrv h{shulphqwdohv vrq vxshulruhv
d orv rewhqlgrv frq ho prghor dqdoðwlfr1 D shvdu gh od shtxhôdv glihuhqfldv vh uhvdowd txh
orv uhvxowdgrv dqdoðwlfrv suhvhqwdq xq dfxhugr vdwlvidfwrulr frq orv uhvxowdgrv rewhqlgrv
hq oderudwrulr1
:17
Hqvd|r gh surehwdv gh W6332<47F frq eudv rulhqwdgdv d fJ
uhvshfwr do hmh orqjlwxglqdo1
Hvwh hqvd|r frqvlvwh hq vrphwhu d xqd surehwd gh fduerqr0hsr{l frq eudv rulhqwdgdv
d 43J vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od
sduwh vxshulru gh od plvpd1 Odv surslhgdghv gh orv pdwhuldohv hpsohdgrv hq ho dqäolvlv
qxpìulfr vh sxhghq revhuydu hq odv Wdeodv 4| 51
Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 83 lqfuhphqwrv
gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh
3> 8<pp=
73
Crack Opening Displacement [µm]
350
300
Experimental results
250
Non linear model
Linear model
200
150
100
50
0
0
50
100
150
200
250
Displacement [µm]
Iljxud 49= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Fxuyd FRG 0 Ghvsod}dplhqwrv
sduwh vxshulru gh od surehwd
Hq od Iljxud 4: vh suhvhqwd od ghirupdgd gh od surehwd hq ho hvwdgr qdo1 Hvwd jxud
suhvhqwd xq idfwru gh dpsolfdflöq gh orv ghvsod}dplhqwrv gh 63 txh shuplwh dsuhfldu
dghfxdgdphqwh orv ihqöphqrv päv lpsruwdqwhv txh vh suhvhqwdq hq ho dqäolvlv1 Hq od
Iljxud 4: vh pxhvwud txh ghelgr d od srvlflöq gh od eud gh fduerqr odv fdudv h{whulruhv
gh od surehwd suhvhqwdq ghvyldflrqhv uhvshfwr gho hmh yhuwlfdo gh od surehwd/ ghelgr d
xqd whqghqfld gh odv eudv d rulhqwduvh hq od gluhfflöq gho hvixhu}r dsolfdgr1
Hq od Iljxud 4; vh revhuydq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Hq od
plvpd vh revhuyd txh ho fdpsr gh ghvsod}dplhqwrv suhvhqwd xqd glvfrqwlqxlgdg ixhuwh
hq od }rqd fhqwudo1 Hq hvwd }rqd vh jhqhud xqd vxud frq xqd rulhqwdflöq frlqflghqwh
frq od gh od eud gh uhixhu}r1
Hq od Iljxud 4< vh suhvhqwdq odv lvräuhdv gh od yduldeoh lqwhuqd gh sodvwlflgdg sdud ho
frpsxhvwr txh lqglfdq odv }rqdv hq odv fxdohv vh kd surgxflgr ghirupdflöq soävwlfd1 Vh
revhuyd txh ghelgr d odv frqglflrqhv gh frqwruqr lpsxhvwdv vh jhqhudq ghirupdflrqhv
soävwlfdv hq od }rqd vxshulru ghuhfkd h lqihulru l}txlhugd gh od surehwd | surjuhvd hq
irupd sdudohod d od rulhqwdflöq gh odv eudv1
Od Iljxud 53 pxhvwud od ghirupdflöq soävwlfd gho frpsxhvwr hq od gluhfflöq gho hmh
orqjlwxglqdo gh od surehwd1 Vh revhuyd txh vh jhqhudq ghirupdflrqhv soävwlfdv hq ho
erugh vxshulru ghuhfkr h lqihulru l}txlhugr | surjuhvd kdfld ho fhqwur gh od surehwd1
Hq odv Iljxudv 54 | 55 vh suhvhqwdq orv frqwruqrv gh odv whqvlrqhv hq odv gluhfflrqhv
[[ h \\1 Hq odv plvpdv vh revhuyd od pdufdgd gluhfflrqdolgdg hq od uhvsxhvwd gho
74
Iljxud 4:= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Ghirupdgd 58=4
Iljxud 4;= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh ghvsod}dplhqwrv
75
Iljxud 4<= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr yduldeoh lqwhuqd
sodvwlflgdg1
Iljxud 53= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh ghirupdflöq
soävwlfd \\
76
pdwhuldo frpsxhvwr1
Iljxud 54= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr whqvlrqhv [[
:1714
Frpsuredflöq qxpìulfr0h{shulphqwdo
Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh
orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder
vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 43J uhvshfwr gho hmh orqjlwxglqdo gh
od surehwd1
Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv vh revhuydq hq od Iljxudv 56/ 57 | 581
Hq od sulphud gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh
od surehwd hq ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1
Od Iljxudv 56/ 57 | 58 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/
orv uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0
wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq
qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh
odv uhvsxhvwdv1
Hq hvwh fdvr orv uhvxowdgrv dqdoðwlfrv suhvhqwdq xqd qrwdeoh frqfrugdqfld frq orv
rewhqlgrv h{shulphqwdophqwh/ vdoyr ho fdvr gh od fxuyd gh od Iljxud 57 hq od fxdo
sxhgh revhuyduvh xqd dshuwxud h{fhvlyd hq od }rqd gh od hqwdood hq ho odgr ghuhfkr
gh od surehwd1 Hv qhfhvdulr ghvwdfdu txh hq orv uhvxowdgrv rewhqlgrv hq ho hqvd|r
h{shulphqwdo ho frpsruwdplhqwr gh od surehwd qr pxhvwudq vlphwuðd/ suhvhqwäqgrvh
77
Iljxud 55= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh whqvlrqhv \\
35000
30000
Force [N]
25000
20000
15000
Experimental results
10000
Non linear model
Linear model
5000
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 56= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv
sduwh vxshulru gh od surehwd
78
Crack Opening Displacement [µm]
120
100
80
60
40
Experimental results
Non linear model
Linear model
20
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 57= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd FRG odgr ghuhfkr 0
Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd1
glihuhqfldv hqwuh orv uhvxowdgrv gh od dshuwxud hq od }rqd gh od hqwdood sdud ho odgr
l}txlhugr frpr sdud ho odgr ghuhfkr1 Hvwh ihqöphqr qr vh revhuyd hq od vlpxodflöq
qxpìulfd/ vhjýq od fxdo od ghirupdgd suhvhqwd vlphwuðd | orv ydoruhv gh od dshuwxud
gh od hqwdood uhvxowdq ljxdohv sdud dperv odgrv gh od surehwd1 Hvwh ihqöphqr txh vh
ghwhfwd hq ho hqvd|r h{shulphqwdo srguðd ghehuvh d idfwruhv wdohv frpr huuruhv hq od
phglflöq r suhvhqfld gh glvfrqwlqxlgdghv hq ho pdwhuldo1
Hq od Iljxud 57 vh revhuyd xqd frpsdudflöq hqwuh orv uhvxowdgrv gh od vlpxodflöq
qxpìulfd | orv h{shulphqwdohv rewhqlgrv sdud ho odgr ghuhfkr gh od surehwd1 Vh revhuyd
xq dxphqwr qr judgxdo gh orv ghvsod}dplhqwrv hq orv uhvxowdgrv h{shulphqwdohv hq od
hwdsd qdo gho hqvd|r1 Hq od Iljxud 58 vh revhuyd txh od vlpxodflöq qxpìulfd suhvhqwd
xqd frqfrugdqfld hohydgd frq orv uhvxowdgrv h{shulphqwdohv1
:18
Hqvd|r gh surehwdv gh W6332<47F frq eudv rulhqwdgdv d eDJ
uhvshfwr do hmh orqjlwxglqdo
Hvwh hqvd|r frqvlvwh hq vrphwhu d xqd surehwd gh fduerqr0hsr{l frq eudv rulhqwdgdv
d 78J vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od
sduwh vxshulru gh od plvpd1 Odv surslhgdghv gh orv pdwhuldohv hpsohdgrv hq ho dqäolvlv
qxpìulfr vh sxhghq revhuydu hq odv Wdeodv 4| 51
Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 83 lqfuhphqwrv
79
Crack Opening Displacement [µm]
120
100
80
60
40
Experimental results
Non linear model
Linear model
20
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 58= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd FRG odgr l}txlhugr 0
Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd
gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh
3> 8<pp=
Hq odv Iljxudv 59 | 5: vh suhvhqwd od ghirupdgd gh od surehwd | xq ghwdooh gh od
ghirupdgd hq od }rqd gh od hqwdoodgxud1 Hvwdv jxudv suhvhqwdq xq idfwru gh dpsol0
fdflöq gh orv ghvsod}dplhqwrv gh 58 txh shuplwh dsuhfldu dghfxdgdphqwh od irupd qdo
gh od surehwd1
Ghelgr d od suhvhqfld gh odv eudv gh uhixhu}r d 78J uhvshfwr gho hmh orqjlwxglqdo gh
od slh}d vh revhuyd od idowd gh yhuwlfdolgdg gh orv odwhudohv gh od surehwd dvð frpr xqd
whqghqfld gh odv eudv d rulhqwduvh hq od gluhfflöq gho hvixhu}r dsolfdgr1
Hq od Iljxud 5; vh suhvhqwdq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Hq
od plvpd vh dsuhfld xqd orfdol}dflöq gh odv ghirupdflrqhv hq od }rqd fhqwudo gh od
surehwd txh vxujhq ghvgh od udð} gh odv hqwdoodv frq xqd lqfolqdflöq d 78J uhvshfwr do
hmh orqjlwxglqdo gh od surehwd1
Hvwh ihqöphqr frqfxhugd frq orv uhvxowdgrv h{shulphqwdohv/ |d txh hq orv plvprv
vh revhuyd ho surjuhvr gh odv vxudv frlqflghqwh frq od gluhfflöq gh odv eudv gh uhixhu}r1
Hq od Iljxud 5< vh revhuydq orv frqwruqrv gh od yduldeoh lqwhuqd gh sodvwlflgdg1 Hq
od plvpd vh dsuhfld txh ho ihqöphqr gh sodvwlflgdg hvwä d dvrfldgr d od pdwul} gho
pdwhuldo frpsxhvwr | txh vh ghvduurood frq xqd gluhfflöq frlqflghqwh frq ho äqjxor d
78J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd/ qr frlqflghqwh frq od gluhfflöq gh odv
eudv1 Hvwh ihqöphqr vh surgxfh sru ho khfkr gh txh do lqwhqwdu dolqhduvh odv eudv frq
7:
Iljxud 59= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Ghirupdgd 58=4
Iljxud 5:= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Ghwdooh gh od ghirupdgd hq od
hqwdoodgxud
7;
Iljxud 5;= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr gh ghvsod}dplhqwrv
od gluhfflöq gho hvixhu}r sulqflsdo vh surgxfhq hq od pdwul} hvwdgrv whqvlrqdohv hohydgrv
txh jhqhudq ghirupdflrqhv soävwlfdv/ surgxflìqgrvh xq ihqöphqr gh ghvshjxh gh odv
eudv uhvshfwr gh od pdwul}1
:1814
Frpsuredflöq qxpìulfr0h{shulphqwdo
Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh
orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder
vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 78J uhvshfwr gho hmh orqjlwxglqdo gh
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83
Iljxud 64= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr gh whqvlrqhv [[
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84
3000
2500
Force [N]
2000
1500
Experimental results
1000
Non linear model
Linear model
500
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 66= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv
sduwh vxshulru gh od surehwd
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}rqd gh od hqwdood hq ixqflöq gho ghvsod}dplhqwr gh od sduwh vxshulru gh od surehwd1
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85
Crack Opening Displacement [µm]
160
140
Experimental results
120
Non linear model
Linear model
100
80
60
40
20
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 67= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd FRG +odgr ghuhfkr, 0
Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd
Crack Opening Displacement [µm]
160
140
Experimental results
120
Non linear model
Linear model
100
80
60
40
20
0
0
50
100
150
200
250
300
350
Displacement [µm]
Iljxud 68= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd FRG +odgr l}txlhugr,
0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd
86
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87
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Frpsuredflöq qxpìulfr0h{shulphqwdo
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88
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89
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8:
3500
3000
Experimental results
Force [N]
2500
Non linear model
Linear model
2000
1500
1000
500
0
0
50
100
150
200
250
Displacement [µm]
Iljxud 75= Fxuyd Fdujd 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd
Crack Opening Displacement [µm]
90
80
70
60
50
40
30
Experimental results
Non linear model
20
Linear model
10
0
0
50
100
150
200
250
Displacement [µm]
Iljxud 76= Fxuyd FRG 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd
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