Post on 01-Oct-2018
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
CENTRO INTERNACIONAL DE MÉTODOS NUMÉRICOS EN INGENIERÍA
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Reference ConfigurationNEXT ITERATION
NO CONVERGENCE
∆ Ui+1 = ∆ Ui +K-1 Fresid
( )ICEE −+=+ 2/11 ii
( )11 ++ = ii Ee φ
CO
MP
OU
ND
ING
1
( )*
1iσ
CO
NST
ITU
TIV
EM
OD
EL
PH
ASE
1
( )1iσ
RECOMPOSITION
( )∑=
=c
ci
ci k σσ
CHECK RESIDUALFORCE AND
CONVERGENCE
CONSTITUTIVE MODELS FOR EACH COMPOUND
NEXTINCREMENT: n+1
INCREMENT: n
( )iσφ
=S
CO
MP
OU
ND
ING
2
( )*
2iσ
CO
NST
ITU
TIV
EM
OD
EL
PH
ASE
2
( )2iσ
CO
MP
OU
ND
ING
3
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3iσ
CO
NST
ITU
TIV
EM
OD
EL
PH
ASE
3
( )3iσ
CO
MP
OU
ND
ING
n
( )*
niσ
CO
NST
ITU
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EM
OD
EL
PH
ASE
n
( )niσ
Elastic Trial Stress
Upd
ated
Con
figu
ratio
n
Ref
eren
ceC
onfi
gura
tion
( ) ( ) ( )( ) : 1*1c
picc
i eec −= ++σ
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3
REFERENCECONFIGURATION
Real Anisotropic SpaceESC ;;
UPDATEDCONFIGURATION
Real Anisotropic Spaceec ;;σ
Fictitious Isotropic Spaceec ;;σep
arbitrary
:
:
==
σ=σ σ
e
( )( )( ) eepep
e
::
:
:
1
1
1
−σ
−
−σ
==
σ=σ
4
1
23
Push Forward
( )Ee→
=ϕ
Pull Back
( )
( )epep
←
←
←
ϕ=
σϕ=
: *! ! !1
5
NEXT ITERATION
NO CONVERGENCE
∆ Ui+1 = ∆ Ui +K-1 Fresid
( )ICEE −+=+ 2/11 ii
( )11 ++ = ii Ee φ
( ) ( ) ( )( ) : 1*1c
picc
i eec −= ++τC
OM
PO
UN
DIN
G 1
mat
rix ( )*
MATRIXiτ
ISO
TR
OP
ICC
ON
STIT
UT
IVE
MO
DE
L T
HE
MA
TR
IX
( )MATRIXiτ
RECOMPOSITION
( )∑=
=c
ci
ci k ττ
CHECK RESIDUALFORCE AND
CONVERGENCE
CONSTITUTIVE MODELS FOR EACH COMPOUND
NEXTINCREMENT: n+1
INCREMENT: n
( )iτφ
=S
CO
MP
OU
ND
ING
2 f
iber ( )*
FIBERiτ
AN
ISO
TR
OP
ICE
LA
STO
-P
LA
STIC
CO
NST
ITU
TIV
EM
OD
EL
FO
RFI
BE
RS
( )FIBERiτ
Elastic Trial Stress
( ) ( ) ( ) ( )niiii 1
21
111 ++++ ==== eeee
Closing equation
: 8 ! ! & 1
! & ! ! ! &
*! ! * ! !
"
!! ! ! 1
C); D 4!! !-
, 61 1 ! !
, 61 @ ,
1 ! !
, 7 !
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J , 7 !,
!&
J
0==) =K. 4B7.8B= 6 7#.M#A O@ 8( 4 0==) =K. 4. 6 P2404C.4=: O@ 8( E! 7 ! & &
7 ! ! 1
-
J - - -
"
)%, I ! 1
&
.
C); D 4!! ! >? 4 1
& 0==) A=C. 7#6# #A 60 8#.4#0 7=8)2A=:
/O@B7=8)>? 7
>? !
1
>? 4
Y
Yj
Ys
Yj
Ys
Yj Y
Yj
?
Ys
Yk6r
6j YYj
>? !
B6 0==) A=C. 7#6# #A 60 8#.4#0
7=8)2A= 7
!&
3) G , 1
.
.
5 7 & 1
4 / M G >? !
B ! B6 0==) A=C. 7#6# 4.#74=B 6 P2404C.4=
B6 0==) A=C. 7#6# 4B7.8B= 6 7#.M#
A=)
'
!
5 *%/1 $%&-
/! ! *! !& ! 0/! ! !- ! ! ! ! 0 ! % & !, ! ! 1 ""!@ '5 ! ! ,& & A # >A @ "? > 3?
! !- & ! ! / ! ! ! !, 0 * :
. 1@ ! & ! 1 @ !
. @ 0 & ( ! &!@ ( !@ ! !- ! & &! ( ! @ ! ! ( ! 0 ( ! % A4 ) / B 69","9.698A 69",9.698A >A @ "? & A # >6K0.@ "?
0 ! ! ! ! !@ ! * 1 ! ! ! >6K0.@ "? # 1 ! ! ! ! ! & ! LL ( !&@ ! & &
A * ! &! ! !! ! ( ! / ! ! ! ! ! ! ! &
5 *&8) & (%&1 /%)4(&* (&--&&* ) A4%&* ,%>)&-*
& ,( ""# * & !! & 1 /
! & &! !! ! & * & ! * ! &!
: 6! 6 &
& & , @ ! & , * ! ! & & 1 !& ! ! & @ , & ! %!- 1 1
0 1 @ @ /
0 ( ! ! A4 ) / B 69","9.698A 69",9.698A K! 5: A% . >A @"?
8 < 12
71 )
12
0 ! ! ( ! 3"
: ) (
8 < 12
71 )
12
0 ! ! !
: ) 1
5 *&8) /%)4(&* ""# ) A4%&* )%(&&* & %*/()&- @ -)$(,&-5
! ,( 1 E / &! ! , !! 0 ! ! !- !
A * ! ! ! ! ! 0 ! (
A * 1 ! ! ! 0 ! 1 & ! !- !
A % & ! ' ! &!@ &! ! &
5 " ! 1 ! & * ! 1 !1 &! ' * ! ! ! ! ! *
5 ! * 1 ( / @ * 1 / > K 1 5 3? A * 1 * !& &
3: 8 ! 1 !
E / 1 ! ! " !! ! & 1 * & /
! &! A * & ! ! ! &! @ ( & ! ! &! & !& & ! / 1
' ! * & % ! A! * & ! & & * ! /
& % ! 1 ! & 2 !, ! ! ! ! & 1 ! ! ! & &,! !! 1 ! & ! ! ! & &! 1 ! & & !
5: ''97 1 6 ! ':
": ''97 1 6 ! ,
: ''97 1 7 &!
': ''97 1 7
3
! ! ! !- ! 1 !, ! & ! / & ! ! ! ! & @ &@ % ! 1 ! & ! * 1 !(! !! * ! & & & 1 @ ! 7 , 6&! > ?
: ''97 1 7 1
0 0 ! ! ! A ! * & ! ! * & 1 & 1 ! &1 1
& / 1
5
: ''97 1 7 ;; !
: ''97 1 7 << !
"
: ''97 1 7 << 1
55 )'/%)4&1 ,'?%)>;/%'(&-
!- & ( ! ( ! % 3 ''97 1 /
0 ! ! !- 3 ! % 1 & &!
0 3 ! ! ( !@ ! ! ! , ! ! ! ! 0 !!- ! !
# ! ( ,! ! 1 ! & 0 !!- ( ! !& ! 3 &,! & A 1 ! 7=6 >7 G = 6!? !
0
5000
10000
15000
20000
25000
30000
0 50 100 150 200 250
Displacement [µm]
For
ce [N
]
Non linear model
Experimental results
Linear model
: ''97 1 7 7 , 6&!
/ % / % & ( >K ?
A 3 * ( ! ! 1 &! ! , 4 4 * ( ! ! # *+ *
5 *&8) /%)4(&* ""# ) A4%&* )%(&&* &
%*/() &- @ -)$(,&-5
! ,( 1 E / &! ! !! 0 ! ! !-
A % & ! ' ! &!@ &! ! &
'
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
3: ''97 1 7 7=6 , 6&!
5 ! 1 1 !1 &! ' * ! ! ! ! ! * 5 ! * 1 ( / @ 1 &
" ! &! !! * ! &! & & 1 1 &
! * & % ! A * ! ! & % &* ! 1
0 ' ! ! ! / A * ! % &* %
;; << !! !
5: ''97 1 6 ! :
": ''97 1 7 &!
: ''97 1 7
': ''97 1 7 ! <<
! !
: ''97 1 7 ;;
55 )'/%)4&1 ,'?%)>;/%'(&-
!- & ( ! ( ! % 3 ''97 1 /
0 ! @ ! % 1 & &!
0 @ ! ! ( !@ ! ! ! , ! ! ! ! 0 !!- ! !
( !!@ ( & % * ( ! ! ! ! ! @
: ''97 1 7 <<
0
5000
10000
15000
20000
25000
30000
35000
0 50 100 150 200 250 300 350
Displacement [µm]
For
ce [N
]
Experimental results
Non linear model
Linear model
: ''97 1 7 7 , 6&!
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
: ''97 1 7 7=6 % ,6&!
& &* ! % ! !!- @ E ! ! ! ! * ( ! ! ! !
! !!- ( ! % A ! &! ( ! 1 * ! !- ( !
5 *&8) /%)4(&* ""# ) A4%&* )%(&&* &
%*/() &- @ -)$(,&-
! ,( 1 E / &! ! !! 0 ! ! !-
A % & ! ' !
3
0
20
40
60
80
100
120
0 50 100 150 200 250 300 350
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
: ''97 1 7 7=6 &* ,6&!
&!@ &! ! &
3 5 ! ! & 1 !1, &! * ! ! ! 1
6 1 & / & ! 1 &
" ! &! !! & ! & * & /
! ( !@ * !! 1 1 &
!! * ! ! & ! ! * / @ 1 ! %% * 1
5
3: ''97 1 6 ! :
5: ''97 1 6 !
"
": ''97 1 7 &!
& ! & * ! @ - ! 1 ! &
55 )'/%)4&1 ,'?%)>;/%'(&-
!- & ( ! ( ! % 3 ''97 1 /
0 ! @ ! % 1 & &! 0 @ ! ! ( !@ ! ! ! , ! ! ! ! 0 !!- ! !
( !! * ( ! ! ! !
: ''97 1 7
': ''97 1 7 1
'
: ''97 1 7 ;;
: ''97 1 7 <<
0
500
1000
1500
2000
2500
3000
0 50 100 150 200 250 300 350
Displacement [µm]
For
ce [N
]
Experimental results
Non linear model
Linear model
: ''97 1 7 7 , 6&!
! !- ! !- @ E ! ! ! & &!
5 *&8) /%)4(&* ""# ) A4%&* )%(&&* &
%*/() &- @ -)$(,&-
! ,( 1 E / &! ! !! 0 ! ! !-
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3 5 ! , ! & ! 1 !1 &! '' * ! ! ! 1
3 * & !
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300 350
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
: ''97 1 7 7=6 > %? ,6&!
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300 350
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
: ''97 1 7 7=6 > &* ?, 6&!
3: ''97 1 6 ! '':
5: ''97 1 6 !
& / !! 1 &
": ''97 1 7 &!
! A * ! !& & & 1 %
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!- & ( ! ( ! % 3 ''97 1 /
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3
: ''97 1 7 <<
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5
0
500
1000
1500
2000
2500
3000
3500
0 50 100 150 200 250
Displacement [µm]
For
ce [N
]Experimental results
Non linear model
Linear model
: 7 7 , 6&!
0
10
20
30
40
50
60
70
80
90
0 50 100 150 200 250
Displacement [µm]
Cra
ck O
peni
ng D
ispl
acem
ent [
µm]
Experimental results
Non linear model
Linear model
: 7 7=6 , 6&!
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