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Code_Aster
®
Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
1/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
Document: R3.03.04
Efforts external of pressure into large
displacements
Summary:
A loading of pressure in great displacements is a following loading. By employing elements
of skin, one is brought to calculate, on the one hand, a second member to which calculation is close to that into small
displacements, and in addition, an additional term of rigidity, which is in general not symmetrical. One
chooses nevertheless to symmetrize it, discounting an appreciable saving of time even if some iterations
additional can be necessary to converge.
background image
Code_Aster
®
Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
2/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
Contents
1 Introduction ............................................................................................................................................ 3
2 virtual Work of the efforts external of pressure ................................................................................... 3
3 Variation of the virtual work of the efforts external of pressure ................................................................ 4
4 Adoption of a curvilinear parameter setting of surface .............................................................................. 5
5 Introduction into Code_Aster ............................................................................................................ 6
6 particular Cases of a structure subjected to an internal or external pressure constant ........................ 7
7 Bibliography .......................................................................................................................................... 8
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Code_Aster
®
Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
3/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
1 Introduction
The taking into account of loadings of the type pressure (key word
PRES_REP
in the control
AFFE_CHAR_MECA
[U4.25.01]) a certain number of difficulties in the absence of the assumption poses of
small displacements. Indeed, unlike the dead loads evoked with [R5.03.20], the pressure
depends on displacements since it is about an effort whose direction is normal with the field; one speaks
then of following forces, activated by the key word
TYPE_CHARGE: “SUIV”
in the control
STAT_NON_LINE
[U4.32.01]. Nevertheless, the choice of the current configuration like configuration of
reference (Lagrangian updated) led to simple expressions - with the help of some concepts of
differential geometry - work of the efforts of pressure and its variation first compared to
displacement, the latter being a nonsymmetrical bilinear form.
2
Virtual work of the efforts external of pressure
R
P (X)
Pr
N
R
p (X)
N
p
X =



(X)
= F
Appear 2-a: Configuration of reference and current configuration
In the current configuration, the virtual work of the efforts external of pressure is written simply
[Figure 2-a]:
()
()
W
.
p
p
U
v
N v
U






=
-
ds
p
éq 2-1
Moreover, one supposes henceforth that the value of the pressure does not depend explicitly on
displacement but only of the material point of application:
()
()
(
)
p
P
X
X
=
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Code_Aster
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Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
In this case, one can then express the virtual work of the efforts of pressure in the configuration of
reference:
()
()
()
(
)
W
.
P det
.
p
U
v
F F
N
v
X
R






=
-
-
T
R
R
ds
p
1
éq 2-2
On the practical level, one will use the formula [éq 2-1] to calculate the work of the efforts of pressure.
However, the formula [éq 2-2] is adapted best to a derivation compared to the displacement, of which
one will see the need in the following paragraph.
3 Variation of the virtual work of the efforts external of
pressure
In the optics of a resolution of the problem of balance of the structure by a method of Newton, one
is brought to express the variation of the virtual work of the efforts external of pressure compared to
displacement, in a way similar to what was made for the virtual work of the interior efforts with
[R5.03.20]. The field of integration being fixed in the expression [éq 2-2], derivation under the sign
nap is licit, (cf [bib2]):
()
()
[
]
W
.
.
P
det
.
.
p
U U
U
v
U
F F
U N
v
R












=
-
-
Pr
T
R
ds
1
We decide to choose like configuration of reference the current configuration, for which
F Id
=
. This choice led to a simple expression of derived from the term between hooks:
()
[
]
()
U
F F
U
U Id
U
det
.
div
-
=
-
T
T









Finally, the variation of the virtual work of the efforts external of pressure is written in the configuration
current:
()
()
[
]
()
W
.
.
p div
.
p
U U
U
v
U Id
U
N v
U















=
-
-
T
p
ds
éq 3-1
In the expression [éq 3-1] remains a difficulty. Indeed, one expects to obtain a size
primarily surface whereas the intégrande reveals terms of normal derivation with
surface. In other words, it is necessary to know the expression of virtual displacements not only on
surface field but also inside the aforementioned (in a vicinity of surface to be able
to express the derivative normals). This disadvantage is not pain-killer since in Code_Aster, for
to calculate the elementary terms due to the surface efforts, one employs elements of skin for
which a normal variation does not have a direction.
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Code_Aster
®
Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
5/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
4
Adoption of a curvilinear parameter setting of surface
To cure the problem mentioned previously, it is necessary to seek to express the relation [éq 3-1] with
the aid of surface sizes only. For that, one has recourse to elements of geometry
differential, [bib1], which one adopts the notations in particular (, one adopts the convention of summation
repeated indices where the Greek indices take values 1 and 2 while the Latin indices
take values 1 to 3).
1
N
2
p
3
S
M
Appear curvilinear 4-a: Parameter setting of the vicinity of subjected surface
with the pressure
That is to say
(
)
1
2
,
an acceptable parameter setting of surface. To describe volume made up of one
vicinity of this surface, one associates a third variable to him,
3
, which measures the progression according to
the unit normal
N
in
(
)
1
2
,
. One has thus [4-a]:
(
)
(
)
(
)
OM
OS
N
1
2
3
1
2
3
1
2
,
,
,
,
=
+
With this choice of parameter setting, the natural base covariante
(
)
G G G
1
2
3
,
,
and the metric tensor
G
are:
G
OM
G
OM
G
OM
N
G G
1
1
2
2
3
3
11
12
21
22
0
0
0
0
1
=
=
=
=
=
=




G
G
G
G
G
ij
I
J
.
In this curvilinear parameter setting, the intégrande [éq 3-1] has as an expression:
[
]
-
-
p G N
U
v
U
v
ij
I
K K
J
J K K
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Code_Aster
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Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
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R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
This term is simplified considerably. Indeed, one can already note that when
J K
=
, the term enters
hook is null. Moreover, in the adopted curvilinear system, the components contravariantes of
N
are:
N
N
N
1
2
3
0
0
1
=
=
=
,
,
. Lastly, by taking account of the particular form of
G
, variation of
work is written simply:
()
()
W
.
.
p
p
U U
U
v
U






=
-
-




U
v
U
v
ds
p
3
3
éq 4-1
On this expression, one notes that only intervene of the surface differential operators
(derivation covariante compared to
1
and
2
only), which is well the required goal. In
introducing the base contravariante
(
)
G G G
N
1
2
3
,
,
=
, also called bases dual and which is expressed to leave
base covariante by
[]
G
G
G
I
ij
J
=
-
1
, one can free oneself from the curvilinear components:
()
(
)
(
)
()
W
.
.
p
.
.
.
.
p
U U
U
v
U G
v N
U N v G
U




















=
-


-






p
ds
éq 4-2
It is henceforth the expression [éq 4-2] which will be used to calculate the variation of the virtual work of
efforts of pressure.
5
Introduction into Code_Aster
In Code_Aster, finite elements of skin (surface elements plunged in a space
three-dimensional) are employed to discretize real and virtual displacements intervening in
surface expressions such as [éq 2-1] and [éq 4-2]. These last make it possible to express
respectively the vector second member and the matrix of rigidity due to the pressure, of which employment by
the algorithm of
STAT_NON_LINE
is specified in [R5.03.01] and which calls some note:
·
The calculation of the virtual work of the efforts of pressure [éq 2-1] is in fact identical to that carried out
in small displacements, with the help of a preliminary reactualization of the geometry. Let us recall
that it is carried out with each iteration.
·
The calculus of the variation of the virtual work of the efforts of pressure [éq 4-2], carried out with each
construction of the matrix of rigidity, proves a little more delicate insofar as it
require the knowledge of metric of the element of skin of each one of its points of
Gauss. If one calls
NR
N
functions of form and
X
N
the position of the nodes of the element,
then the metric one is calculated as follows:
[]
[]
G
X
N
G
G
G
G
G
G G
G
G
G
µ µ
=
=

=
=
-
NR
N
N
N
1
2
1
2
1
.
Moreover, this variation behaves like a term complementary to the matrix of
tangent rigidity; in general, it is not symmetrical (except particular case of a structure
subjected to an internal or external pressure constant, cf [§6]). It is then desirable
to spread out the strategy of resolution. Initially, only the part is considered
symmetrical of this complementary term: the problem remains symmetrical, even if it requires
(perhaps) some additional iterations. It is the choice carried out in Code_Aster.
In the event of problems of convergence, one could consider this complementary term in
its integrality while being ready to pay the price of a nonsymmetrical resolution.
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Code_Aster
®
Version
3
Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
7/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
6 particular Cases of a structure subjected to one
internal or external pressure constant
N
p
p
p
p
N



v = 0
Appear 6-a: Structure under internal or external pressure constant
In the particular case of a constant pressure in a cavity [Figure 6-a], one shows that the efforts
from pressure derive from a potential which is not other than the product of the pressure by the volume of
cavity. This result extends to the case from a structure plunged in a fluid with constant pressure.
()
P
=
=
p
D
p
D
p
p
R
R
det F
Again, one chooses like configuration of reference the current configuration. Variation of
P
conduit then well with the virtual work of the efforts external of pressure:
()
P
=
U v
v
v N
v












p
D
p
dS
p
p
div
W
p
=
-
=
In this particular case, the variation of virtual work is also the second variation of the potential
P
,
i.e. a symmetrical bilinear form:
()
()
W
.
.
.
.
p
U U
U
v
U U
U
v












=
2
P
2
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Code_Aster
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Version
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Titrate:
Efforts external of pressure in great displacements
Date:
05/02/96
Author (S):
E. LORENTZ
Key:
R3.03.04-A
Page:
8/8
Manual of Reference
R3.03 booklet: Boundary conditions and loadings
HI-75/96/001/A
7 Bibliography
[1]
Fung Y.C.: Foundations off solid mechanics. Prentice Hall. 1965, p 31-57.
[2]
Mialon P.: Calculation of derived from a size compared to a bottom of fissure by
method
.
EDF - Bulletin of the Management of the Studies and Search - Series C - n° 3. 1988,
p 1-28.