Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
1/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
Organization (S):
EDF-R & D/AMA
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
Document: R3.06.09
Finite elements of gasket in plane 2D
Summary:
Description of the finite element of gasket plane 2D allowing to modelize 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.
Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
2/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
1 Geometry
The element of gasket 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 the aforementioned.
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.
The option
MODI_MAILLAGE
key words
ORIE_CONTACT
initially developed for the elements of
contacts makes it possible to impose this classification.
4
3
2
1
N
T
X
Y
Appear 1-a: Element of gasket
The element of gasket 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
(
)
Y
X,
with the local reference mark with the element
()
T
N,
we introduce
stamp rotation
R
. This matrix applied to a vector expressed in the total reference mark gives sound
expression in the local reference mark.
-
=
cos
sin
sin
cos
R
where
is the angle between the two reference marks.
one has
L
X
X
L
y
y
1
2
1
2
sin
cos
and
-
-
=
-
=
with
12
=
L
and
(
)
1
1
, y
X
and
(
)
2
2
, y
X
co-ordinates of nodes 1 and 2.
Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
3/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
3
Jump of displacement in the element
Let us note
)
,
(
I
I
I
v
U
=
U
and
)
,
(
loc
I
loc
I
loc
I
v
U
=
U
displacements with the node
I
respectively in
total reference mark
(
)
Y
X,
and in the local reference mark
()
T
N,
.
With the change of reference mark one a:
I
loc
I
RU
U
=
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:
[]
(
) (
)
(
)
[]
(
) (
)
(
)
-
-
+
-
=
-
-
+
-
=
loc
loc
G
loc
loc
G
G
T
loc
loc
G
loc
loc
G
G
N
v
v
C
v
v
C
U
U
U
C
U
U
C
U
3
2
4
1
3
2
4
1
1
1
with g=1,2 the list of the points of Gauss and C
1
and C
2
coefficients:
-
=
+
=
3
3
1
2
1
,
3
3
1
2
1
2
1
C
C
One can rewrite the jump in matric form:
[]
[]
[]
loc
G
G
T
G
N
G
U
U
D
B
U
=
=
éq 3-1
with
T
loc
loc
loc
loc
loc
v
U
v
U
)
,
,
…
,
,
(
4
4
1
1
=
D
and
-
-
-
-
-
-
=
G
G
G
G
G
G
G
G
G
C
C
C
C
C
C
C
C
0
1
0
1
0
0
0
0
1
0
1
0
B
Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
4/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
4 Efforts
interiors
That is to say
R
~
the matrix
8
8
×
who allows to express the components of displacement to the four nodes
in the local reference mark:
T
loc
loc
loc
loc
loc
v
U
v
U
)
,
,
…
,
,
(
4
4
1
1
=
D
starting from the components of displacement
with the four nodes in the total reference mark:
T
v
U
v
U
)
,
,
…
,
,
(
4
4
1
1
=
D
.
One has
D
R
D
~
=
loc
with
=
R
O
O
O
O
R
O
O
O
O
R
O
O
O
O
R
R
~
where
R
is the matrix
2
2
×
of change of reference mark defined into 2) and
O
the matrix
2
2
×
null.
The interior efforts in the element of gasket are defined by a noted vector with eight components
int
F
and checking the relation:
int
.F
D
=
S
E
where
[]
DLL
E
S
)
(
]
2
,
1
[
U
=
is energy in the element of gasket.
(see Doc. of the law of behavior Barenblatt [R7.02.11]).
one a:
[]
()
[] []
DLL
E
S
=
]
2
,
1
[
U
U
U
DLL
loc
=
]
2
,
1
[
D
B
according to the definition of
(Doc. [R7.02.11]) and according to [éq 3-1].
DLL
=
]
2
,
1
[
~ D
R
B
since
D
R
D
~
=
loc
DLL
T
T
=
]
2
,
1
[
~
D
B
R
One deduces the interior efforts from them:
=
]
2
,
1
[
int
~
DLL
B
R
F
T
T
éq 4-1
one can evaluate this integral:
=
=
2
,
1
int
~
G
G
T
G
T
G
B
R
F
with the weights of the points of Gauss
2
2
1
L
=
=
.
Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
5/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
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:
[]
=
2
,
1
int
~
DLL
T
T
B
R
F
From where
[] []
[]
=
2
,
1
int
~
DLL
T
T
D
U
U
B
R
D
F
like
[]
D
R
B
Data base
U
~
=
=
loc
then
[]
R
B
D
U
~
=
and one obtains:
[]
=
]
2
,
1
[
int
~
~
DLL
T
T
R
B
U
B
R
D
F
éq 5-1
that one can evaluate:
[]
=
=
2
,
1
int
~
~
G
G
G
T
G
T
G
R
B
U
B
R
D
F
with the weights of the points of Gauss
2
2
1
L
=
=
.
Code_Aster
®
Version
7.2
Titrate:
Finite elements of gasket
Date:
19/09/03
Author (S):
J. LAVERNE
Key
:
R3.06.09-B
Page
:
6/6
Manual of Reference
R3.06 booklet: Machine elements and thermal for the continuous mediums
HT-66/03/005/A
Intentionally white left page.