Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
1/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R7.02 booklet: Breaking process
Document: R7.02.04
Lagrangian representation of variation
of field
1 Goal
·
To calculate the mechanical fields relating to a field with variable geometry, by using one
field fixes reference in way, for example, to carry out parametric studies on
several fields by using one mesh.
·
Within the framework of the breaking process, to calculate the rate of refund of energy for
various lengths of fissure (in 2D and 3D) by using only one mesh representing one
length of fissure fixes reference.
These developments are available in linear elasticity, for the elements of continuous medium 2D and
3D, in the situations where the variations of geometry do not affect the edges charged.
possible initial deformations are treated only in the medium 2D.
Any calculation using this method requires, to ensure the passage of the real field studied
area of reference, the preliminary creation of a field
, using the control
CALC_THETA
[U4.63.02].
The formulation developed in Code_Aster does not take account of the thermal terms, of
loadings on the lips of the fissure nor of the forces of volume in general, except deformations
initial which is taken into account in 2D only.
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
2/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
2
Principle of the method
The principle is double:
·
to use a bijective geometrical transformation making correspond the field of
reference (that which one nets) with the real field. For that, one is useful oneself of the field
and
of a real parameter
. In theory, any type of transformation is authorized provided that it is
regular but the shape of the field théta currently established authorizes roughly speaking only
translations of under-fields, except in 3D where by differentiating the modules from the field
théta along a border, one can simulate changes in form of this border
(application to the parametric study of fissures 3D),
·
to write the new equations of the elastic problem on the field fixes reference,
variability (translated by the parameter
) appearing in the equations and either in
field (from where a single mesh to analyze all the configurations).
With regard to the applications to the breaking process, one calculates moreover the rate of
restitution of energy relating to various lengths of fissures by using the configuration of
reference. For that, one writes the potential energy in form Lagrangienne and one derives sound
expression compared to the parameter
.
2.1
Transformation of field by the field théta.
A succession of fields is considered
I
, and an area of reference
0
. Each point
hardware pi of the various fields is identified by a point M of the area of reference in the way
following (cf [Figure 2.1-a]):
P
F
M
M
M
I
I
I
=
=
+
()
()
M
P1
P2
P3
Field of
reference
F
1
F
2
F
3
0
3
2
1
Appear 2.1-a: Representation of a succession of fields by a field
of reference.
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
3/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
2.2
Lagrangian representation of the problem of elasticity.
Mechanical fields relating to the current configuration and written on the configuration of reference
express themselves in the following way, that one will call Lagrangian expression:
~ ()
(
), ~ (
)
(
)
U M
U M
M
M
=
+
=
+
, where
U
and
are respectively the fields of
displacements and of stresses. In a general way any field C will have its representation
Lagrangian
~
C
with
~ ()
(
)
C M
C M
=
+
.
The principle of virtual work (P.T.V) is written classically, in linear elasticity, on each
field
in the following way:
With U v D
T v dS
With
v D
I
.
.
.
=
+
,
with
U
field solution of the problem,
v
field of displacement kinematically acceptable,
I
. one
field of initial deformations,
With
the tensor of elasticity and
T
forces imposed on the border of
field.
By noticing that the field of deformation is written:
()
~ () (
)
P
M
F
=
-
1
, and that
D
F D
=
det
0
, P.T.V is written on the area of reference (knowing that the support of
transformation is strictly included in
, i.e. this transformation does not affect them
edges charged):
With U
F
v F
F D
T v dS
With
v
F
F D
I
. ~ (
).
(
) det
.
.~.
(
) det
0
0
0
1
1
1
=
+
-
-
-
The discretized problem arises then in the following form (with the usual notations of
finite elements:
~ () ~
~ ()
K F
U
F Q F
= +
,
where
~
U
solution of the problem is the Lagrangian field,
~
K
the matrix of rigidity,
~
Q
the vector
corresponding to the initial deformations, both depend on the transformation (and thus on the field
and of the parameter
).
Parametric study thus consists in dealing with several problems by using the same mesh and in
calculating each time the elementary matrices of rigidity depending on the parameter
using
the option `
RIGI_MECA_LAGR'
(available in 2D and 3D) of the operator
CALC_MATR_ELEM
[U4.41.01] and
option `
CHAR_MECA_LAGR'
(available only for the initial deformations and in 2D) of
the operator
CALC_VECT_ELEM
[U4.41.02], with in data the field
and the parameter
. The remainder of
calculation continues in a usual way: assembly and resolution (This development are not
integrated in the total control
MECA_STATIQUE
[U4.31.01], calculation thus should be broken up
with the elementary operators). The field of displacement obtained is a Lagrangian field. Of
even, the Lagrangian stresses are obtained thanks to the operator
CALC_CHAM_ELEM
[U4.61.01],
with the options `
SIEF_ELGA_LAGR'
and `
SIGM_ELNO_LAGR'
[U4.61.01].
The calculation of the stresses takes account of the possible presence of a field of initial deformations.
In particular, one can calculate with these options the field of the real stresses with
predeformation without variation of field (i.e. while taking
=0):
=
-
With
I
(
)
.
The initial deformations are obligatorily defined in the form of constants or of functions of
space, via the operator
AFFE_CHAR_MECA
[U4.25.01] with the key word
`
EPSI_INIT'
. The development does not take into account initial deformations given like
fields with the nodes or the points of Gauss.
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
4/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
2.3
Calculation of the rate of refund of energy
The field of Lagrangian displacement
~
U
being available, one can evaluate written potential energy
in Lagrangian form:
~ (~ (), ~, ~)
((),
)
W U
T
W U
T
I
I
=
, like its derivative.
The expression of the rate of refund of energy is then:
()
()
()
() (
)
()
()
()
(
)
()
G
With
U
F
F D
With
F
U
F
F D
With
U
F
F
D
With U
F
U
F
F D
With U
F
U
F
F D
With
I
I
I
I
=
+
+
-
-
-
-
-
-
-
-
-
-
-
-
~.
, det
~
.
det
~.
det
.
.
det
~.
det
,
,
,
0
0
0
0
0
1
1
1
1
1
1
1
1
()
(
)
() (
)
-
-
-
U
F
U
F
F
D
I
1
1
0
~.
det
,
3
Some examples of use
3.1 Parametric study of a structure having an inclusion with
variable positioning.
The goal is to calculate the mechanical fields of a structure containing a heterogeneity (material
of different characteristic or obviously for example,) with different
positions, by using one mesh corresponding to a position of reference
(cf [Figure 3.1-a]).
T
U
Position of
reference
Support of
field théta
r1
r2
M
P
T
U
Position
current
P
Appear 3.1-a
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
5/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
The first stage consists in choosing a support of the field théta compatible with the kinematics of
inclusion. In practice, if the current position of inclusion underwent a translation
compared to
position of reference, with
=1, one will build the field théta so that the small disc of
crown (r< r1) includes inclusion completely (
=1 for any point of inclusion) and that
<r2-r1.
From now on, in all the analysis, the points M attached to the mesh represent in fact the points
hardware P, determined by P = M +
.
With these considerations, all the positions of inclusion are not possible to analyze and
depend on the geometry of the structure.
The second stage consists in formulating the problem of Lagrangienne representation by calculating them
elementary matrices using the operator
CALC_MATR_ELEM
[U4.41.01] and the option
RIGI_MECA_LAGR
, with like data the field
and the kinematic parameter
.
The remainder of the analysis continues in a conventional way, but the fields obtained at the points of
mesh (nodes or points of Gauss) represent the mechanical state of truths material points of
variable configuration:
~ ()
(
), ~ (
)
(
)
U M
U M
M
M
=
+
=
+
.
3.2 Calculation of the rate of refund of the energy of a fissured structure
in a field of initial deformations
One considers a springy medium 2D containing a rectilinear fissure AO and being propagated (until
not O') in a field of initial deformations (cf [Figure 3.2-a]). The problem consists in calculating
the rate of refund of the energy of the structure for all the positions of the fissure.
field where is
defined a field
deformations
initial
I
T
U = 0
O
O'
With
D
Appear 3.2-a
One starts by defining the field of initial deformations in the field D:
CH = AFFE_CHAR_MECA_F (
MODEL ........,
EPSI_INIT: (
GROUP_MA: D
EPXX: f1
EPYY: f2
EPZZ: f3
EPXY: f4)
.........);
Quantities
fi
are functions of space defined as a preliminary by
DEFI_FONCTION
[U4.21.02].
Let us notice that even in plane deformations one can define a component
EPZZ
.
components
EPXZ
and
EPYZ
can be defined but will not be used for nothing in the continuation processing
since one limits oneself to the 2D. It is necessary that the fissure is not propagated apart from the field
predeformed.
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
6/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
More precisely, it is necessary to be able to define the quantity:
~ ()
(
)
I
I
M
M
=
+
, and thus to make sure that the material point M +
is in the field
D. If it is not the case, the user will prolong the field D with zero values of
I
.
The second stage consists in defining the support of the field théta, by taking care that the aforementioned is such
that:
+
=
M
M
D
OO
,
,
'
max
max
with
.
This is carried out with the operator
CALC_THETA
[U4.63.02]. The command of the first two stages is
indifferent.
One calculates then the elementary matrices of rigidity:
mel = CALC_MATR_ELEM (OPTION: “RIGI_MECA_LAGR”
THETA: théta
PROPAGATION: alpha ......
and second elementary members:
vel = CALC_VECT_ELEM (OPTION: “CHAR_MECA_LAGR”
THETA: théta
PROPAGATION: alpha ......
The user will endeavor to indicate the same field théta and the same parameter alpha in both
operators under penalty of serious vexations.
The field théta must be compatible with the maximum extension of the fissure, i.e.
max
'
=
<
-
OO R
R
2
1
, with
R
R
2
1
and
the higher and lower radius of the attach ring of
field théta (this precaution must be taken independently of the presence of deformations
initial).
The assembly and the resolution are carried out in a conventional way.
One then can, if required to calculate the residual stresses for each strain of fissure,
for example in the absence of external forces T, this thanks to the operator
CALC_CHAM_ELEM
[U4.61.01], with the options `
SIEF_ELGA_LAGR'
(stresses at the points of Gauss) or
`
SIGM_ELNO_LAGR'
(stresses with the nodes).
The calculation of the rate of refund of energy is carried out with the operator
CALC_G_THETA
[U4.63.03] and
the option `
CALC_G_LAGR'
. The same precaution that previously must be taken, namely
coherence of the field théta and the parameter alpha (by defect this parameter is worth zero and one calculates G
conventional with initial deformations). It is also necessary to take care not to forget the key word
“LOAD”
so that the field of initial deformations is taken into account.
The elastic behavior being linear, the history of the variation of the field does not intervene. One can
thus very well on the basis of the configuration of reference (fissure length AO), to analyze all them
fissures of which the length varies between AO-OO' and AO+OO'. For the fissures of which the length is
lower than AO, one will consider a negative field théta in bottom of fissure,
positive remainder (one
“shortens” the fissure to some extent).
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
7/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
3.3 Parametric study of a structure 3D fissured, the parameter being
the fissure shape.
One considers a linear elastic body comprising a three-dimensional fissure of which geometry of
bottom is a priori unspecified but regular. The goal is to calculate the rate of refund of energy
total and room for the various shapes of fissure. Let us consider the face of fissure of reference
0 and it
current face
(cf [Figure 3.3-a])
.
The transformation binding the two faces is built using one
definite field théta local on all the face.
For that, one defines initially the face with the operator
DEFI_FOND_FISS
[U4.63.01], then the field
théta with the operator
CALC_THETA
[U4.63.02] and the key word
THETA_3D
[U4.63.02], the module being
defined in each node in order to build
from
0.
0
Appear 3.3-a
The processing takes place then as in 2D, i.e. by the construction of the elementary matrices
with the option `
RIGI_MECA_LAGR'
of the operator
CALC_MATR_ELEM
[U4.41.01]. The option
`
CHAR_MECA_LAGR'
of
CALC_VECT_ELEM
[U4.41.02] (only to treat the deformations
initial) is not planned for the 3D. The form of the various faces is defined by the field théta, it
parameter alpha creating a homothety for a given form.
The total rate of refund of energy for the configuration
is calculated with the option
`
CALC_G_LAGR'
of
CALC_G_THETA
[U4.63.03].
The rate of refund of energy room is calculated with the option `
CALC_G_LGLO'
of
CALC_G_LOCAL
[U4.63.04]. An important restriction of this development relates to this option: normals
between the various faces must be preserved (for example the parametric study of fissures
circulars does not pose problems on this subject). This restriction does not relate to the calculation of G
total.
Code_Aster
®
Version
3.6
Titrate:
Lagrangian representation of variation of field
Date:
29/11/96
Author (S):
G. DEBRUYNE
Key:
R7.02.04-A
Page:
8/8
Manual of Reference
R7.02 booklet: Breaking process
HI-75/96/048/A
Intentionally white left page.