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Titrate:
Law of behavior of Barenblatt and control


Date:
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Author (S):
J. LAVERNE Key
:
R7.02.11-B Page
: 1/8

Organization (S): EDF-R & D/AMA
Handbook of Référence
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Document: R7.02.11
Law of behavior of Barenblatt

Summary:

The law of behavior of Barenblatt is a law of interface making it possible to model the opening of a fissure
while taking account of a force of cohesion enters the lips of this one. An energy of surface allows
to take into account the energy cost of the opening of the fissure. The latter will be represented by
finite elements of joint type. An energy of penalization will make it possible to take into account the condition of not
interpenetration of the lips of the fissure.
We present here the form of energy and the constraint which in drift according to the jump of displacement
as well as the internal variables. The existence of instability requires a control by the elastic prediction of which one
will detail the elements specific to this law.


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Titrate:
Law of behavior of Barenblatt and control


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J. LAVERNE Key
:
R7.02.11-B Page
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Count

matters

1 Law of behavior Barenblatt ............................................................................................................ 3
1.1 Energy of surface ........................................................................................................................... 3
1.2 Constraint in the element of joint ................................................................................................... 5
1.3 Variables intern ............................................................................................................................ 6
2 Control .................................................................................................................................................. 7
3 key Words ................................................................................................................................................ 8
4 Bibliography .......................................................................................................................................... 8
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Titrate:
Law of behavior of Barenblatt and control


Date:
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Author (S):
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:
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1
Law of Barenblatt behavior

To raise the problem of infinite constraints on the face of fissure Barenblatt introduces forces of
cohesion enters the lips of the fissure. They are gravitational attractions being exerted between the particles of
leaves and other of the plan of separation of the fissure.

It is considered that the opening of the fissure costs an energy defined on the surface of discontinuity of
the fissure. It is called energy of surface. The problem is solved by minimizing the sum of this
energy and of elastic energy.
The surface of discontinuity is modelled by a finite element of joint (Doc. [R3.06.09]) only element of
code supporting this law of behavior.
The condition of noninterpenetration of the lips of the fissure is taken into account by a method of
penalization.

To preserve a local processing of the conditions of opening, one carried out a regularization of
the energy of surface for jumps close to zero. With this intention a small parameter was introduced
(key word: SAUT_C) defining the width of these “small” jumps.

1.1
Energy of surface

On the surface of discontinuity () modelled by elements of the type joint (QUAD4)
(see Doc. [R3.06.09]) one defines an energy of surface Es depend on the standard of the jump on
displacement.

E = (U) D + I + U
S
[]

([] N)
R


The first term corresponds to energy necessary for the opening of the fissure, the second is one
energy of penalization which will make it possible to take into account the condition of noninterpenetration
lips of the fissure.

+ if U < 0
The function is defined:
N
I
U
+
[(]

N)
[]
=
R

0
if [U] 0
N

In other words, to want to impose the interpenetration of the lips will cost an infinite energy. In
practical, one will approach this function by a continuous function which tends quickly towards the infinite one
when the normal jump becomes negative:

1
2
I
U
~
C U
D
.
+ ([]
with C coefficient of penalization and
negative part.
N)
[]

R

N -
2
-

Note:

C is adjusted starting from critical jump SAUT_C entered by the user, it is the same coefficient
that that taken for the regularization: cf slope of the constraint in 0+.
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Titrate:
Law of behavior of Barenblatt and control


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Author (S):
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:
R7.02.11-B Page
: 4/8

2
Moreover one poses [U] =
[U] + U so that a negative normal jump does not influence
N
[] 2t
+
the energy of surface and does not make evolve/move the threshold of the criterion.

One defines the energy of surface according to the cases of figure:

·
If [U] [SAUT_C, + [and [U] 0:
N



- C U
E = (U)
with ([U]
[]

C
G

) =G 1 - E

S
[] D

C




G

C is the rate of critical refund of energy and
C the constraint criticizes in the beginning.
K having the following properties (choice carried out starting from article [F&M]):

K

GC

[]

G
U
C


(0) = 0


concave


SAUT_C

[U]

·
If [U] [0, SAUT_C [and [U] 0:
N

E = (U)
with K continuous and derivable in
K' 0 =:
S
[] D

SAUT_C and such as
() 0


1
2
~
~
([U]) = C [U] + C
with C constant
2

G
K
C

SAUT_C
[U]
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·
If [U] < 0 and [U] = 0:
N
T

1
2
Energy of penalization: E =
C

S
[U]


D

N -
2

K
GC

[U] N

1.2
Constraint in the element of joint

The constraint in the element of joint derives from the energy of surface. It is given by:






U
N
[
]
N
=
=




T

[
U]

T

·
If [U] [SAUT_C, + [and [U] 0:
N


[

U] - C [U]

N

C
G


E
C [U]

=

C

[U]


- C [U]

T
G

E
C [
C
U]




SAUT_C
[]
U

·
If [U] [0, SAUT_C [and [U] 0:
N



C

_
C
U.E.

C []
-
JUMP C
G


C
=
N

C


_
C
U.E.
C []
-
JUMP C
G



T


SAUT_C
[]
U
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Law of behavior of Barenblatt and control


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·
If [U] < 0 and [U] = 0:
N
T

N
C [U] N
=





C
0

SAUT_C
[]

U N

Note:

The regularization of energy in zero makes it possible to define a constraint in the element for one
null jump. Physically in other words, as soon as the constraint in the element of joint
will increase, a small jump will appear. The behavior of the Barenblatt type
will be carried out that when the standard of the jump in the element exceeds SAUT_C, before the joint
comprise as a spring.

1.3 Variables
interns

The law of Barenblatt behavior at summer implemented with three internal variables.
First 1
v is used for control and for the discharges, it is a threshold which corresponds to
greater jump (in standard) ever reached (see: [§2] Control).
The second v2 makes it possible to know if the element is in elastic mode ([U] < SAUT_C) or polishing substance
([U] > SAUT_C).
The third corresponds to the percentage of energy of surface dissipated during the loading:
C
v
1 exp
v
3 =
-
-
1
Gc
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Law of behavior of Barenblatt and control


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2 Control

The control recommended with the law of Barenblatt behavior is of type PRED_ELAS [R5.03.80], it
allows to follow dissipative solutions presenting of instabilities.

The implementation of this technique of control requires the resolution of the following local equation:

F U + U
- +
=
+ +
éq
2-1
el
[0] [1] (N SAUT _C)
(L
N
JUMP _ C)
C

with unknown factor which one obtains by the solution of a quadratic equation, LLC length
characteristic of the model, and N
threshold which will make it possible to keep in memory the standard of the jump with
the previous moment (stored in the variable interns No 1 of the law of Barenblatt behavior).

The threshold evolves/moves as follows:

+
If [U] N 1 + SAUT _ C then
+
= -
If not
+
= [U] N 1
+ - JUMP _ C

Control imposes ultimately that the jump of displacement continues to increase some share length
potential fissure.
The implementation of this control makes it possible to follow unstable branches of balance of the curve
total force/imposed displacements (see [Figure 2-a]).


Appear total 2-a: Courbe Force/Déplacements imposed with unstable branch
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3 Words
keys

The law of Barenblatt behavior is used in STAT_NON_LINE and DYNA_NON_LINE with the word
key BARENBLATT. This law of behavior is usable on elements of joint with
modeling PLAN_FISSURE and AXIS_FISSURE.

Two parameters are to be seized in DEFI_MATERIAU:

SIGM_C: critical stress
SAUT_C: small parameter of regularization

Commands



STAT_NON_LINE COMP_INCR
RELATION
BARENBLATT
DYNA_NON_LINE COMP_INCR
RELATION
BARENBLATT
AFFE_MODELE MODELING PLAN_FISSURE

AXIS_FISSURE

DEFI_MATERIAU RUPT_FRAG
SIGM_C

SAUT_C


This law of behavior was validated by the case test SSNP118 (see Doc. [V6.03.118]).

4 Bibliography

[1]
Mr. CHARLOTTE, J.J. MARIGO, G. FRANKFURT, L. TRUSKINOVSKY: “Revisiting brittle
fracture ace year energy minimization problem: Comparisons off Griffith and Barenblatt surfaces
energy models ", Symposium one continuous ramming and fracture.
[2]
J. LAVERNE, E. LORENTZ, J.J. MARIGO: “A model of rupture with energy of
Barenblatt: theoretical considerations and numerical establishments. 16th French congress
from Mécanique, Nice September 1-5, 2003.

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