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

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1

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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

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Elastic Trial Stress

Upd

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Con

figu

ratio

n

Ref

eren

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tion

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picc

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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

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LA

STO

-P

LA

STIC

CO

NST

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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 !

! !

, 61 ! !, !

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=:

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>? !

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 ! ! !-

A % & ! ' ! &!@ &! ! &

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 %

' ;; << A ! 1 * * ! ! &

55 )'/%)4&1 ,'?%)>;/%'(&-

!- & ( ! ( ! % 3 ''97 1 /

0 ! ! % 1 & &!

0 ! ! ( !@

: ''97 1 6

': ''97 1 7 ;;

3

: ''97 1 7 <<

! ! ! , ! ! ! ! 0 !!- ! !

A * ( ! ! # ! * , &

0 - & ! ! ! ! ) 1 ! ! ! * ! ! ! ! ! * , !

/ ! ! , ! ! ! ! ! !&@ ! @ + ! ! ! * E !& ! ! @ ! !

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&!

"

! -! !* @ ! LL* !& ! ! ! !

0 /! !1* ! & ! !, -! ! 1! ! ! !/ ! ! @ !@ + * !, ! 0 ! >7)2 !? ! ! 1 3 !! ! !

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0 ( ! ! 7 # , A# >7#A#? 6 ) 0 ( ! ! !,- ( ! M 7 ! - "6=7'''3

/?

=

Objective derivative

Z1

Z2

Z3

X1

X2

X3

x1

x2

x3

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