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SSNV166 Cylindre fissured under multiple loadings
Date
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15/02/06
Author (S):
P. MASSIN, S. GENIAUT
Key
:
V6.04.166-A
Page:
1/20
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-62/06/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.166
SSNV166 Cylindre fissured under loadings
multiples
Summary
The purpose of this test is calculation of the stress intensity factors along the bottom of fissure for a cylinder
comprising an axisymmetric fissure.
The influence of the degree of the elements and the type of the method is studied through various modelings.
· Modeling A tests K1 and K3 with a linear mesh 3D and a method with the finite elements
conventional (FEM).
· Modeling B tests K1 and K3 with a quadratic mesh 3D (elements of Barsoum) around
melts of fissure and a FEM.
· Modeling C tests K1 and K3 with a linear mesh 3D with a conventional resolution but one
extraction of the factors of intensity based on an energy calculation.
Moreover, for each modeling, various cases of loadings are studied:
- traction (stress in mode I);
- torsion (stress in mode III);
- bending (opening of dimensioned, closing of the other) with and without taking into account of the contact.
The cases of traction and torsion do not put concerned the contact.
Although symmetries exist in certain cases (axisymetry for case 1, symmetry planes for the 2
ème
)
representation is made in 3D to make the test generalizable under multiple loading.
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SSNV166 Cylindre fissured under multiple loadings
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1
Problem of reference
1.1 Geometry
The fissure is a circular ring in an orthogonal plan with the axis of the cylinder [Figure 1.1-a].
parameters has and B determine the radius of the cylinder and the depth of the fissure. [Figure 1.1-b] is
a cut of the cylinder in the plan of fissure (plane Oyz). So that the medium is regarded as
infinite, the height of the cylinder is H = 10 B.
Appear 1.1-a: Geometry of the fissured cylinder
2a
2b
y
Z
Z
one
fissured
·
·
PFON_FIN
NODE A
Appear Plane 1.1-b: of cracking
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SSNV166 Cylindre fissured under multiple loadings
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1.2
Material properties
Young modulus: E= 205000 MPa
Poisson's ratio:
= 0.3
1.3
Boundary conditions and loadings
Three loadings will be applied in order to calculate the stress intensity factors K1 and K3 in
3D by using operator POST_K1_K2_K3.
Loading 1 tests K1, and K3.
Loading 2 tests K2 without taking into account of the contact.
Loading 3 tests K2 with taking into account of the contact.
One expects that K1 and K3 are constant along the bottom of fissure and so that K2 varies.
Note: the cases of traction and torsion can be treated indifferently with or without contact
(here, without contact) because it y forever of closing of the fissure.
Case 1: traction and
torsion
Case 2: bending without
contact
Case 3: bending with
contact
Higher face
NR
X
= 6 MN
T
X
= 3 MN
M
y
= 1.5 MN
M
y
= 1.5 MN
Table 1.3-1: Case of loadings
The preceding efforts are applied to the structure via discrete elements 3D located at
center higher face. It is noted that the point of maximum opening due to the imposed bending
(moment following OY) will be node A (see [Figure 1.1-b].
The rigid movements of body are locked by the same process with embedding of the center of
the lower face.
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SSNV166 Cylindre fissured under multiple loadings
Date
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2
Reference solution
2.1
Method of calculation used for the reference solution
For an axisymmetric fissure in a cylinder infinite length, method of the Equations
Singular integrals and of the Asymptotic Developments [bib1] makes it possible to calculate the values of
stress intensity factors.
· Case 1: Traction and Torsion
Traction induces an opening in mode 1. K1 is given by the following formula:
()
B
has
F
has
has
P
K
I
1
2
=
where P is the effort applied to the higher and lower face and F
1
a given function [Figure 2.1-a].
Torsion induces an opening in mode 3. K3 is given by the following formula:
()
B
has
F
has
has
T
K
III
3
3
2
=
where T is the moment applied to the higher and lower face and F
3
a given function
[Figure 2.1-a].
· Case 2: Bending without contact
The bending induces an opening in mode 1. The value of K1 at the point of maximum opening A is given
by the following formula:
()
B
has
F
has
has
M
K
With
I
2
3
4
=
where M is the moment applied to the higher and lower face and F
2
a given function
[Figure 2.1-a].
· Case 3: Bending with contact
There is not analytical solution with this problem. One expects on the one hand that K1 is close to
case without contact on the part of the fissure in opening, and in addition that K1 is null on the part of
the fissure in closing.
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SSNV166 Cylindre fissured under multiple loadings
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Appear 2.1-a: Functions F1, F2 and F3
These three functions come from [bib1].
2.2
Results of reference
Numerical application:
Except contrary mention, in the continuation of this document, the parameters retained for A and B are:
= 0.4 m has
B = 0.5 m
Case 1: Traction and torsion
Case 2: Bending
K1 = 5.35 MPa.m
1/2
K3 = 11.22 MPa.m
1/2
K1
With
= 11.71 MPa.m
1/2
Table 2.2-1: Values of reference
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SSNV166 Cylindre fissured under multiple loadings
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3
Linear modeling a: Mesh, conventional formulation
3.1
Characteristics of modeling
Appear 3.1-a: Cuts mesh in the plan of the fissure
The elements are all of command 1.
The interest of this modeling is to be used as a basis for more evolved/moved formulations, and thus, of
to be able to note the contribution and the improvements of the other methods.
3.2
Characteristics of the mesh
A number of nodes: 11310
A number of meshs: 14453
Type of meshs
A number of meshs
POI1 4
SEG2 39
TRIA3 360
QUAD4 930
PENTA6 5440
HEXA8 7680
Table 3.2-1: Characteristics of the meshs
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Titrate:
SSNV166 Cylindre fissured under multiple loadings
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3.3 Functionalities
tested
Controls
DEFI_FOND_FISS
POST_K1_K2_K3
CALC_G_LOCAL_T
3.4 Notice
The calculation of the stress intensity factors is done using POST_K1_K2_K3 (method
of extrapolation of displacements on the lips of the fissure) [bib2].
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SSNV166 Cylindre fissured under multiple loadings
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4
Results of modeling A
4.1 Values
tested
Procedure POST_K1_K2_K3 makes it possible to identify the values of the stress intensity factors with
a coefficient close, by two methods. We will limit the test to the results resulting from the method 1, which
seem more stable. One recalls that this method calculates, for each couple of nodes in opposite
around the point of the bottom of fissure considered, the jump of the field of displacement squared and divides it
by R [bib2]. One limits oneself to 3 or 4 couples of nodes, and one will take the maximum value like value
result (the most constraining value).
4.1.1 Results in the case of a loading in traction (K1) and torsion (K3)
Identification Reference
Aster
% difference
K1 with node PFON_FIN
5.35 10
6
4.98
10
6
6.92
K3 with node PFON_FIN
- 11.22 10
6
- 10.12
10
6
9.80
The values of K1 and K3 must be identical [Figure 4.2-a] for all the nodes of the bottom of fissure
because there is an axisymmetric configuration. Here, we test only the values with node PFON_FIN.
4.1.2 Results in the case of a loading in bending (K1) without contact
Identification Reference
Aster
% difference
K1 with node A
11.71 10
6
10.17
10
6
13.15
One compares the value of K1 with the reference solution only to the point of maximum opening
(node A) because it is the only analytical value available in the literature.
4.1.3 Results in the case of a loading in bending (K1) with contact
Identification Reference
Aster
% difference
K1 with node A
10.17 10
6
9.29
10
6
8.66
One not compares the result obtained with that obtained by Code_Aster without taking into account of the contact (
regression). This taking into account is carried out by the method of the active stresses.
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SSNV166 Cylindre fissured under multiple loadings
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4.2
Evolutions of K1, K2, K3 along the bottom of fissure
Appear 4.2-a: K1, K2 and K3 along the bottom of fissure (in MPa.m
1/2
)
Appear 4.2-b: K1 along the bottom of fissure (in MPa.m
1/2
)
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Titrate:
SSNV166 Cylindre fissured under multiple loadings
Date
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Comments on the results:
[Figure 4.2-a] the evolution of the factors of intensity of the stresses along the bottom of fissure shows
axisymmetric fissure of depth 100 mm subjected to traction and torsion. One observes
many axisymmetric results (with the computational errors near). Moreover, one notes that the fissure is not
not solicited in mode II.
On [Figure 4.2-b], one highlights the taking into account of the contact. On half of fissure in
opening, K1 has lower values with taking into account of the contact, because the contact rigidifies
structure. On half in closing, K1 is null.
In fact, the contact does not take place on all the higher half of the fissure [Figure 4.2-c] but on one
surface a little smaller. On [Figure 4.2-c] the area in red the area of contact represents and
area in blue that of noncontact.
Appear 4.2-c: Contact
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5 quadratic Modeling b: Mesh, formulation
conventional
5.1
Characteristics of modeling
Appear 5.1-a: Mesh and core
A core is created around the fissure. The elements of the core are quadratic elements.
elements apart from the core are linear. Moreover, one uses elements of BARSOUM (nodes
mediums moved with the quarter) for the meshs having an edge pertaining to the bottom of fissure [bib3].
The interest of the use of a mesh of the type BARSOUM is obtaining more precise results.
5.2
Characteristics of the mesh
A number of nodes: 20030
A number of meshs: 16449
Type of meshs
A number of meshs
POI1 2000
SEG3 39
TRIA3 360
QUAD4 610
QUAD8 320
PENTA6 4800
PENTA15 640
HEXA8 5760
HEXA20 1920
Table 5.2-1: Characteristics of the meshs
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SSNV166 Cylindre fissured under multiple loadings
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The nodes mediums of the edges of the elements touching the bottom of fissure are moved with the quarter of these
edges, to obtain a better precision.
5.3 Functionalities
tested
Controls
DEFI_FOND_FISS
CALC_G_LOCAL_T
POST_K1_K2_K3
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6
Results of modeling B
6.1 Values
tested
6.1.1 Results in the case of a loading in traction (K1) and torsion (K3)
Identification Reference
Aster
% difference
K1 with node PFON_FIN
5.35 10
6
5.19
10
6
2.93
K3 with node PFON_FIN
- 11.22 10
6
- 11.08
10
6
1.25
The values of K1 and K3 must be identical [Figure 6.2-a] for all the nodes of the bottom of fissure
because there is an axisymmetric configuration. Here, we test only the values with the last node of
fissure (PFON_FIN).
6.1.2 Results in the case of a loading in bending (K1) without contact
Identification Reference
Aster
% difference
K1 with node A
11.71 10
6
10.59
10
6
9.54
One compares the value of K1 with the reference solution only to the point of maximum opening
(Node A) because it is the only analytical value available in the literature.
6.1.3 Results in the case of a loading in bending (K1) with contact
Identification Reference
Aster
% difference
K1 with node A
10.59 10
6
9.82
10
6
7.30
One compares the result obtained with that obtained by Aster calculation without taking into account of the contact
(not-regression). The method of resolution of the contact is that of the active stresses.
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SSNV166 Cylindre fissured under multiple loadings
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6.2
Evolutions of K1, K2, K3 along the bottom of fissure
Appear 6.2-a: K1, K2 and K3 along the bottom of fissure (in MPa.m
1/2
)
Appear 6.2-b: K1 along the bottom of fissure (in MPa.m
1/2
)
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SSNV166 Cylindre fissured under multiple loadings
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Note:
When one calculates the jumps of displacements along the bottom of fissure (POST_K1_K2_K3), it
first point met on the basis of the node on the bottom is the node moved with the quarter of the element
of adjacent Barsoum. However for this node with the quarter, the management of the conventional contact is not valid.
Indeed, the method of the active stresses with linearization with the nodes mediums implies that it
displacement of the nodes mediums is worth the average of displacements of the adjacent nodes. However here,
the node medium being moved with the quarter, this approximation is not licit.
As the contact is not taken into account correctly for these nodes with the quarter, it appears
judicious not to estimate K1 at it, especially when there is contact. On the following nodes, the jump of
displacement is quite correct and K1 is worth zero (see [Figure 6.2-b]).
This is not quite serious in practice bus when there is contact, one knows that K1 must be null.
Moreover, a solution consists in tracing K1_MAX when the fissure opens, and K1_MIN when it
firm (it is what is made on [Figure 6.2-b]. In the case of closing, K1_MIN is quite null since it
is based on the values of K1 given by the jumps of displacements of the following conventional nodes
the nodes with the quarter, which are correctly treated with regard to the contact.
·
·
·
·
·
·
·
·
·
·
·
·
·
Melts of fissure
Nodes with the quarter
Node of the bottom of
fissure considered
False K1 if contact
Correct K1
Appear 6.2-c: Configuration of calculation close to the bottom of fissure
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7
Modeling C: Linear mesh, conventional formulation and
energy method
7.1
Characteristics of modeling
The modeling of the problem is the same one as that used in A. All the elements are of command 1.
7.2
Characteristics of the mesh
The mesh is similar to that used in A.
A number of nodes: 13630
A number of meshs: 17013
Type of meshs
A number of meshs
POI1 4
SEG2 39
TRIA3 360
QUAD4 1090
PENTA6 5760
HEXA8 9760
Table 7.2-1: Characteristics of the meshs
7.3 Functionalities
tested
Controls
DEFI_FISS_XFEM
CALC_G_LOCAL_T
OPTION CALC_K_G
Option CALC_K_G was introduced into control CALC_G_LOCAL_T. This option allows
to calculate the stress intensity factors by an energy method. This method is more
general that the method of extrapolation of displacements (POST_K1_K2_K3) because it can be used
in the case of an unspecified fissure (not-plane fissure, at bottom not-right). This method is freed
thus controls DEFI_FOND_FISS [U4.82.01] and POST_K1_K2_K3 [U4.82.05] but requires
order DEFI_FISS_XFEM [U4.82.08] which defines two functions of levels (level sets) for
to characterize the fissure.
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SSNV166 Cylindre fissured under multiple loadings
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8
Results of modeling C
8.1 Values
tested
L `option CALC_K_G of control CALC_G_LOCAL_T makes it possible to extract the values from the factors
of intensity of stresses by the G-théta method by using the bilinear form of G (generalization with
3D of the case already existing 2D in Code_Aster). As for the conventional G-théta method, it is necessary
to give a value for Rinf and Rsup (radii lower and higher of the core being used as support than
field théta).
8.1.1 Results in the case of a loading in traction (K1) and torsion (K3)
Identification Reference
Aster
% difference
Max (K1)
5.35 10
6
5.11
10
6
4.47
Max (K3)
11.22 10
6
10.52
10
6
6.24
The values of K1 and K3 must be identical [Figure 8.2-a] for all the nodes of the bottom of fissure
because there is an axisymmetric configuration. Here, we test the maximum of K1 and K3 for all them
points of the bottom of fissure.
8.1.2 Results in the case of a loading in bending (K1) without contact
Identification Reference
Aster
% difference
K1 with node A
11.71 10
6
10.32
10
6
11.88
One compares the value of K1 with the reference solution only to the point of maximum opening bus
it is the only analytical value available in the literature. This point is not any more one “node” but one
“not” of the bottom of fissure, it should then be identified by its number in the list of the points of the bottom of
fissure. It is the point identified by NUM_PT=11.
8.1.3 Results in the case of a loading in bending (K1) with contact
Identification Reference
Aster
% difference
K1 with node A
10.32 10
6
9.43
10
6
8.57
One compares the result obtained with that obtained by Code_Aster without taking into account of the contact
(not-regression). This taking into account is carried out by the method of the active stresses.
[Figure 8.2-b] compares the values of K1 along the bottom of fissure in the case of bending with and
without contact.
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SSNV166 Cylindre fissured under multiple loadings
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8.2
Evolutions of K1 and K3 along the bottom of fissure
Appear 8.2-a: K1 and K3 along the bottom of fissure (in MPa.m
1/2
)
Bending
- 1,50E+07
- 1,00E+07
- 5,00E+06
0,00E+00
5,00E+06
1,00E+07
1,50E+07
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
Curvilinear X-coordinate norm ée
without contact
with contact
Appear 8.2-b: K1 along the fissure (in MPa.m
1/2
)
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Note:
It is noted that if the contact is taken into account (see [Figure 8.2-b]), K1 is not
really no one on the segments of the bottom of fissure where there is closing. That comes owing to the fact that
energy method of calculation of K projects the field of solution displacement on the fields
singular auxiliaries of displacement of an infinitely long fissure in opening. However these fields
auxiliaries are not compatible with the mode of closing present.
9
Summaries of the results
The objectives of this test are achieved:
·
It is a question of validating the taking into account of the contact on the lips of the fissure with elements
quadratic (and of the elements of Barsoum). The results better, are compared with those
obtained with a linear mesh.
·
This test shows the interest of the method “G-theta” for calculation of the factors of intensity of
stress. This energy method has the advantage of being more general than that
using the jump of displacements (POST_K1_K2_K3) because it can apply to fissures
of unspecified geometry, whereas POST_K1_K2_K3 is restricts with the plane fissures. Of
more, the method “G-theta” gives better results (compared with the analytical solution)
that POST_K1_K2_K3 for the same linear mesh.
·
In addition, this test constitutes a reference and will allow the validation of the introduction of
method X-FEM in Code_Aster applied to calculations of the factors of intensity of
stresses.
Code_Aster
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Titrate:
SSNV166 Cylindre fissured under multiple loadings
Date
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Author (S):
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V6.04.166-A
Page:
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Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
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10 Bibliography
[1]
TADA, PARIS, IRWIN: The Analysis Stress Off Handbook Aces, Del Research Corporation,
Hellertoxn, Pennsylvania (1973).
[2]
PROIX: Calculation of the factors of intensity of the stresses by extrapolation of the field of
displacements, Manual of reference of Code_Aster, R7.02.08
[3]
CORNELIU: Quarter-point elements for curved ace faces, Computers & Structures vol. 17,
No 2, pp. 227-231, 1983