Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 1/6

Organization (S): EDF-R & D/AMA

Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.09

Finite elements of joint in plane 2D

Summary:

Description of the finite element of joint plane 2D allowing to model the creation of a fissure along one
predetermined path.
Presentation of the geometry, definition of the jump of displacement in the element, change of reference mark: room with
the total element/, calculation of the interior efforts as well as tangent matrix.

Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 2/6

1 Geometry

The element of joint is a quadrangle with four nodes (QUAD4) with two small sides and two large
what makes it possible to define a local reference mark in the element: N is a normal unit vector at a large side
and T a tangent vector with this one.
The local classification of the nodes must be done obligatorily as on [Figure 1-a], the side [1,2]
must correspond to a large side.
Option MODI_MAILLAGE key words ORIE_CONTACT initially developed for the elements of
contacts makes it possible to impose this classification.


2
N
T
3
1
Y
4
X

Appear 1-a: Elément of joint

The element of joint has two points of Gauss positioned as on the SEG2 of reference:
The first PG1 in - 3/3 and the second PG2 in 3/3 on the segment [- 1,1] with for weight 1
each one.

2
Change of reference mark

To be able to pass from the total reference mark (X, Y) to the local reference mark to the element (N, T) we introduce
stamp rotation R. Cette stamps applied to a vector expressed in the total reference mark gives sound
expression in the local reference mark.

cos
sin
R =

where is the angle between the two reference marks.
- sin cos

y -
-
2
y1
x2 X
one has
1
cos =
and sin = -

L
L

with L = 12 and (1
X, 1
y) and (x2, y2) co-ordinates of nodes 1 and 2.

Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 3/6

3
Jump of displacement in the element

Let us note U = (U, v)
loc
loc
loc
U
=
I
I
I
and
(U
, v
)
I
I
I
displacements with node I respectively in
total reference mark (X, Y) and in the local reference mark (N, T).

With the change of reference mark one a:
loc
Ui = RUi

One definite jumps of normal and tangent displacement in the element on each point of Gauss
starting from the components of the displacement of the four nodes in the local reference mark:

[U] g=C U U
1 C
U
U
N
G (loc -
loc
1
4) + (-
G) (loc -
loc
2
3)
[

U] g= C v
v
1 C
v
v
T
G (loc - loc
1
4) + (-
G) (loc -
loc
2
3)


with g=1,2 the list of the points of Gauss and C1 and C2 coefficients:
1
3
1
3
C
C
1 =
1+
,
2 =





1


2
3
2
3

One can rewrite the jump in matric form:

G
[


U]
U
G
[] N
loc
=










éq 3-1
[
= B D
U]
G
G
T

with loc
loc
loc
loc
loc T
D
= U
(
, v
,…, U
, v
)
1
1
4
4


C
0
1 - C
0
C - 1
0
- C
0
G
G
G
G

and B =

G




0
C
0
1 - C
0
C - 1
0
- C
G
G
G
G

Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 4/6

4 Efforts
interiors

~
That is to say R matrix 8 × 8 which makes it possible to express the components of displacement to the four nodes
in the local reference mark: loc
loc
loc
loc
loc T
D
= U
(
, v
,…, U
, v
)
1
1
4
4
starting from the components of displacement
with the four nodes in the total reference mark:
T
D = U
(, v,…, U, v)
1
1
4
4
.

R O O O


loc
~
~
O R O O
There are D
= D
R with R =

O O R O




O O O R

where R is matrix 2 × 2 of change of reference mark defined into 2) and O matrix 2 × 2 null.

The interior efforts in the element of joint are defined by a vector with eight components noted int
F
and checking the relation:

E =.
D
S
int
F

where ES =
([U]) DLL

is energy in the element of joint.
,
1
[2]
(see Doc. of the law of behavior Barenblatt [R7.02.11]).

one a:

([U])
ES =
U

,
1
[2]
[U] [] DLL

loc
=
B D DLL

according to the definition of (Doc. [R7.02.11]) and according to [éq 3-1].
,
1
[2]
=
~

loc
~
Br D DLL

since D
= D
R
,
1
[2] ~T T
=
R B D DLL


,
1
[2]

One deduces the interior efforts from them:
F =
~
RT BT DLL







éq 4-1
int
, 1 [2]

one can evaluate this integral:

L
F =
~ T T
R B with the weights of the points of Gauss = =.
int
G
G
G
1
2
G =,
1 2
2

Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 5/6

5 Matrix
tangent

The term which it is necessary to calculate in the tangent matrix is the derivative of the interior efforts by report/ratio
with displacements (matrix 8 × 8).

~
The interior efforts are given by: F
=
T
T
int
[R B DLL
,
1 2]

int
F
~ T T [U]
From where
=
R B
DLL
D
[, 12]
[U] D

[U]
loc
~
~
like [U] = data base
= B D
R then
= Br
D


and one obtains:
F
~

T
T
~
int


=
R B
Br DLL







éq 5-1

,
1
[2]
D
[U]
F
~

T
T

~
that one can evaluate:
int = R B
B R
G
G



G
D
G =,
1 2
[
U] G
L
with the weights of the points of Gauss = =
.
1
2
2
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Finite elements of joint


Date:
19/09/03
Author (S):
J. LAVERNE Key
:
R3.06.09-B Page
: 6/6

Intentionally white left page.
Handbook of Référence
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A

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