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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
1/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.134
SSLV134 - Fissure circular in infinite medium
Summary
This test allows, after obtaining the field of displacement by MECA_STATIQUE, the calculation of the rate of
restitution of energy room for a circular fissure plunged in a presumedly infinite medium.
For the first modeling, only a half space defined by the plan of the fissure is represented. Bottom of
fissure is then a closed curve (a circle) and is defined as such in DEFI_FOND_FISS. The rate of
restitution room and total is compared with the analytical solution of reference.
Three following modelings make it possible to calculate the stress intensity factors K1 and K3, in 3D
and axisymmetric, calculated by POST_K1_K2_K3.
·
Modeling B tests K1 for a mesh 3D,
·
Modeling C tests K1 for an axisymmetric mesh,
·
Modeling D tests the combination of K1 and K3 for a mesh 3D.
Lastly, modeling E makes it possible to validate the calculation of the bilinear form of G on the same problem, and
modeling F to validate same calculation for G local.
Code_Aster
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
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Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
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1
Problem of reference
1.1 Geometry
= 1.MPa
= 1.MPa
has
Z
X
y
The fissure is circular (penny shaped ace) of radius has, in the Oxy plan. So that the medium is
regarded as infinite, the sizes characteristic of the solid mass are about 5 times higher
with radius A.
1.2
Material properties
Young modulus: E= 2.10
5
MPa
Poisson's ratio:
= 0.3
1.3
Boundary conditions and loadings
Lower face
: uniform stress of traction
Z
= 1. MPa
Higher face
: uniform stress of traction
Z
= 1. MPa
According to modeling, one also has boundary conditions of symmetry and blocking of
movements of rigid body.
In the modeling D where only the quarter of the parallelepiped is represented, one uses conditions
with the limits of antisymetry for the loading of torsion: they amount imposing null them
tangential displacements with a face. The loading of torsion is introduced in the form of a force
surface tangential (shearing distributed) applied to the lips of the fissure.
·
Upper lip:
F
Y
has
X
= -
and
F
X
has
Y
= +
·
Lower lip:
F
Y
has
X
= +
and
F
X
has
Y
= -
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
3/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
For a circular fissure of radius in an infinite medium, subjected to a uniform traction
according to
the normal in the plan of the lips, the rate of refund of energy room
()
S
G
is independent of
the curvilinear X-coordinate S and is worth [bib1]:
(
)
G S
E
has
()
=
-
1
4
2
2
then the coefficient of intensity of K1 stress is given by the formula of Irwin:
(
)
G S
E
K
()
=
-
1
2
1
2
that is to say
K
has
1
2
=
If this fissure is subjected to a shearing distributed on the lips:
Z
R
has
=
(what is equivalent to a torsion ad infinitum), then one is in pure mode 3 and the factor of intensity of
stresses corresponding is worth:
K
has
3
4
3
=
thus by the formula of Irwin
()
(
)
G S
E K
= +
1
32
In the presence of the two combined modes, one will have:
()
(
)
(
)
G S
E
K
E
K
= -
+ +
1
1
2
12
32
The théta-method connects the rates of refund of energy total and local by the equation
variational following:
=
ds
S
S
G
G
ref.
)
(
.
)
(
)
(
.
m
where
)
(S
m
is the normal at the bottom of fissure
and
is the field speed of a virtual propagation
fissure.
If one chooses for
the normal unit field at the bottom of fissure, one obtains, since
G S
()
is
constant on all the bottom of fissure:
has
S
G
G
ref.
=
2
).
(
)
(
.
2.2
Results of reference
Numerical application (case with loading of traction only):
It is considered that the fissure is circular of radius has = 2. m
Code_Aster
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
4/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
For the loading considered, one obtains then:
G (S)
=
11.586 J/m
2
G
ref.
= 145.060
J/m
K3
=
1.5958E6 J/m
2
Numerical application (case with loading of torsion only):
G (S)
=
7.3565 J/m
2
G
ref.
= 92.44
J/m
K1
=
1.0638E6 J/m
2
2.3 Reference
bibliographical
[1]
Solution of Sneddon (1946) in G.C. SIH: Handbook off stress-intensity factors Institute off
Fracture and Solid Mechanics - Lehigh University Bethlehem, Pennsylvania
Code_Aster
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Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
5/18
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HT-62/06/005/A
3
Modeling a: Melts of fissure closed, calculation of G.
3.1
Characteristics of modeling
The interest of this modeling is to represent the entirety of the bottom of fissure which is a curve
closed, without benefitting from symmetries of the problem.
Only the loading of traction is taken into account.
3.2
Characteristics of the mesh
A number of nodes: 11114
A number of meshs and type: 2432 PENTA 15
3.3 Functionalities
tested
Controls
DEFI_FOND_FISS FOND_FERME
CALC_THETA
CALC_G_THETA_T SYME_CHAR=' SYME'
CALC_G_LOCAL_T SYME_CHAR=' SYME'
POST_K1_K2_K3 SYME_CHAR=' SYME'
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
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Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
6/18
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HT-62/06/005/A
3.4 Notice
One uses key word SYME_CHAR in operators CALC_G_THETA_T and CALC_G_LOCAL_T for
to multiply automatically by two the rate of refund of energy calculated on only one lip of
fissure.
In the same way in POST_K1_K2_K3, key word SYME_CHAR allows to calculate the factors of intensity
stresses and the rate of refund of energy
G_IRWIN
by interpolation of displacements
of a single lip of the fissure. The displacement of the nodes mediums of the edges of the elements
concerning the bottom of fissure to the quarter of these edges would allow to improve the precision of calculation.
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
G total
145.6
146.2
0.4
G local Node A - G Lagrange
11.586
11.82
2.0
G local Node B - G Lagrange
11.586
11.56
- 0.2
G local Node C - G Lagrange
11.586
11.83
2.1
G local Node D - G Lagrange
11.586
11.81
1.9
G local Node A - G Lagrange_no_no
11.586
11.71
1.0
G local Node B - G Lagrange_no_no
11.586
11.60
0.2
G local Node C - G Lagrange_no_no
11.586
11.72
1.2
G local Node D - G Lagrange_no_no
11.586
11.70
1.0
G (POST_K1_K2_K3 Method 3) - Node A
11.586
10.45
9.8
G (POST_K1_K2_K3 Method 3) - Node B
11.586
10.49
9.4
G (POST_K1_K2_K3 Method 3) - Node C
11.586
10.45
9.8
G (POST_K1_K2_K3 Method 3) - Node D
11.586
10.52
9.2
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
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Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
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5
Modeling b: Post_K1_K2_K3 in 3D
5.1
Characteristics of modeling
This modeling makes it possible to test the calculation of K1 using POST_K1_K2_K3 (method
of extrapolation of displacements on the lips of the fissure).
Parameter ABSC_CURV_MAXI of
the operator is selected so as to retain 5 nodes on the segment of extrapolation (dmax = 0,35).
Only the loading of traction is taken into account.
5.2
Characteristics of the mesh
A number of nodes: 6536
A number of meshs and type: 432 PENTA 15 and 987 HEXA 20
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
CALC_THETA THETA_3D
CALC_G_THETA_T
POST_K1_K2_K3
5.4 Notice
One represents only the quarter of the complete three-dimensional block and thus the quarter of the fissure. Thus, it
is necessary to divide the theoretical value of reference of the total rate of refund by 4:
G
glob
= 145.60/4 = 36.40 J/m
Code_Aster
®
Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
8/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
6
Results of modeling B
6.1 Values
tested
6.1.1 Results
of
CALC_G_THETA_T and CALC_G_LOCAL_T
Identification Reference
Aster %
difference
G local Node 49
11.59
11.74
1,30
G local Node 1710
11.59
11.76
1,49
G local Node 77
11.59
11.73
1,22
G total
36.40
36.82
1,16
6.1.2 Results
of
POST_K1_K2_K3
Identification Method
Reference
Aster %
difference
K1_MIN
Node 77
method 1
16.0E+05
15.9E+05
- 0,04
K1_MAX
Node 77
method 1
16.0E+05
16.1E+05
0,66
G_MIN
Node 77
method 1
11.6E00
11.6E00
0,08
G_MAX
Node 77
method 1
11.6E00
11.7E00
1,32
K1_MIN
Node 49
method 1
16.0E+05
15.9E+05
0,03
K1_MAX
Node 49
method 1
16.0E+05
16.1E+05
0,66
G_MIN
Node 49
method 1
11.6E00
11.6E00
0,06
G_MAX
Node 49
method 1
11.6E00
11.7E00
1,32
K1_MIN
Node 1710 method 1
16.0E+05
15.9E+05
0,07
K1_MAX
Node 1710 method 1
16.0E+05
16.1E+05
0,61
G_MIN
Node 1710 method 1
11.6E00
11.6E00
0,15
G_MAX
Node 1710 method 1
11.6E00
11.7E00
1,22
K1_MIN
Node 77
method 2
16.0E+05
15.3E+05
4,02
K1_MAX
Node 77
method 2
16.0E+05
15.9E+05
0,21
G_MIN
Node 77
method 2
11.6E00
10.7E00
7,87
G_MAX
Node 77
method 2
11.6E00
11.5E00
0,42
K1_MIN
Node 49
method 2
16.0E+05
15.3E+05
4,04
K1_MAX
Node 49
method 2
16.0E+05
15.9E+05
0,40
G_MIN
Node 49
method 2
11.6E00
10.6E00
7,92
G_MAX
Node 49
method 2
11.6E00
11.5E00
0,63
K1_MIN
Node 1710 method 2
16.0E+05
15.3E+05
4,09
K1_MAX
Node 1710 method 2
16.0E+05
15.9E+05
0,25
G_MIN
Node 1710 method 2
11.6E00
10.6E00
8,08
G_MAX
Node 1710 method 2
11.6E00
11.5E00
0,49
K1
Node 77
method 3
16.0E+05
15.5E+05
2,62
G
Node 77
method 3
11.6E00
11.0E00
5,16
K1
Node 49
method 3
16.0E+05
15.5E+05
2,63
G
Node 49
method 3
11.6E00
11.0E00
5,19
K1
Node 1710 method 3
16.0E+05
15.5E+05
2,68
G
Node 1710 method 3
11.6E00
11.0E00
5,29
Note:
Method 3 calculates a single value for each parameter (
K1_MAX
=
K1_MIN
in
file result).
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
9/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
7
Modeling C: Post_K1_K2_K3 into axisymmetric
7.1
Characteristics of modeling
This modeling makes it possible to test the calculation of K1 using POST_K1_K2_K3 (method
of extrapolation of displacements on the lips of the fissure) into axisymmetric.
Only the loading of traction is retained in this modeling.
Since one is in axisymmetric modeling, the relation between the total rates of refund of energy and
room is [R7.02.01]:
has
S
G
G
ref.
).
(
)
(
.
=
that is to say here
G
ref.
= 23.17 J/m
7.2
Characteristics of the mesh
A number of nodes: 1477
A number of meshs and type: 402 QUAD 8 and 60 SORTED 6
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.
7.3 Functionalities
tested
Controls
DEFI_FOND_FISS
CALC_G_LOCAL_T
CALC_THETA THETA_3D
CALC_G_THETA_T
POST_K1_K2_K3
Code_Aster
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Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
10/18
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
8
Results of modeling C
8.1 Values
tested
Identification
Method
Reference
Aster
% Difference
G
CALC_G_THETA
232E01
236E01
1,662
K1_MAX
POST_K1_K2_K3 method 1
160E+04
162E+04
1,514
K1_MIN
POST_K1_K2_K3 method 1
160E+04
160E+04
0,15
G_MAX
POST_K1_K2_K3 method 1
116E01
119E01
3,05
G_MIN
POST_K1_K2_K3 method 1
116E01
116E01
0,301
K1_MAX
POST_K1_K2_K3 method 2
160E+04
160E+04
0,569
K1_MIN
POST_K1_K2_K3 method 2
160E+04
150E+04
6,239
G_MAX
POST_K1_K2_K3 method 2
116E01
117E01
1,141
G_MIN
POST_K1_K2_K3 method 2
116E01
102E01
12,088
Code_Aster
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Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
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Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
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9
Modeling D: Post_K1_K2_K3 in 3D modes 1 and 3
9.1
Characteristics of modeling
The boundary conditions following are successively applied:
·
traction: as for modeling B;
·
torsion.
This modeling makes it possible to test the calculation of K1 and K3 combined using POST_K1_K2_K3
(method of extrapolation of displacements on the lips of the fissure).
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.
9.2
Characteristics of the mesh
A number of nodes: 6536
A number of meshs and type: 432 PENTA 15 and 987 HEXA 20
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.
9.3 Functionalities
tested
Controls
DEFI_FOND_FISS
CALC_G_LOCAL_T
CALC_THETA THETA_3D
CALC_G_THETA_T
POST_K1_K2_K3
AFFE_CHAR_MECA FORCE_FACE
Code_Aster
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Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
12/18
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HT-62/06/005/A
9.4 Notice
The two loading cases (traction and torsion) are taken into account. It is thus necessary to cumulate the values of G
for the two loadings. Moreover, one represents only the quarter of the complete three-dimensional block and
thus the quarter of the fissure, it is thus necessary to divide the theoretical value of reference of the rate of refund
total by 4.
Thus
G (S) = (11.586 + 7.356) = 18.943 J/m
2
G = (145.06 + 92.44)/4 = 59.37 J/m
Only traction contributes to K1, only torsion contributes to K3.
Code_Aster
®
Version
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Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
13/18
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HT-62/06/005/A
10 Results of modeling D
10.1 Values
tested
Identification
Method
Localization
Reference
Aster
% Difference
G_LOCAL
CALC_G_LOCAL Legendre
Node 49
1,8943E+01 1,9055E+01
0,59
G_LOCAL
CALC_G_LOCAL Legendre
Node 1710
1,8943E+01 1,9089E+01
0,772
G_LOCAL
CALC_G_LOCAL Legendre
Node 77
1,8943E+01 1,9074E+01
0,694
G
CALC_G_THETA
5,937E+01 5,9849E+01
0,567
K1_MAX
POST_K1_K1_K3 Method 1
Node 49
1,5958E+06 1,5988E+06
0,189
K1_MIN
POST_K1_K1_K3 Method 1
Node 49
1,5958E+06 1,5952E+06
0,038
K1_MAX
POST_K1_K1_K3 Method 2
Node 49
1,5958E+06 1,5924E+06
0,21
K1_MIN
POST_K1_K1_K3 Method 2
Node 49
1,5958E+06 1,5620E+06
2,116
K1_MAX
POST_K1_K1_K3 Method 1
Node 1710
1,5958E+06 1,5990E+06
0,202
K1_MIN
POST_K1_K1_K3 Method 1
Node 1710
1,5958E+06 1,5953E+06
0,029
K1_MAX
POST_K1_K1_K3 Method 2
Node 1710
1,5958E+06 1,5925E+06
0,202
K1_MIN
POST_K1_K1_K3 Method 2
Node 1710
1,5958E+06 1,5618E+06
2,129
K1_MAX
POST_K1_K1_K3 Method 1
Node 77
1,5958E+06 1,5982E+06
0,155
K1_MIN
POST_K1_K1_K3 Method 1
Node 77
1,5958E+06 1,5945E+06
0,077
K1_MAX
POST_K1_K1_K3 Method 2
Node 77
1,5958E+06 1,5918E+06
0,249
K1_MIN
POST_K1_K1_K3 Method 2
Node 77
1,5958E+06 1,5610E+06
2,176
K3_MIN
POST_K1_K1_K3 Method 1
Node 49
1,0638E+06 1,0564E+06
0,704
K3_MAX
POST_K1_K1_K3 Method 1
Node 49
1,0638E+06 1,0589E+06
0,464
K3_MIN
POST_K1_K1_K3 Method 2
Node 49
1,0638E+06 9,4420E+06
11,246
K3_MAX
POST_K1_K1_K3 Method 2
Node 49
1,0638E+06 1,0387E+06
2,361
K3_MIN
POST_K1_K1_K3 Method 1
Node 1710
1,0638E+06 1,0564E+06
0,703
K3_MAX
POST_K1_K1_K3 Method 1
Node 1710
1,0638E+06 1,0589E+06
0,464
K3_MIN
POST_K1_K1_K3 Method 2
Node 1710
1,0638E+06 9,4421E+05
11,245
K3_MAX
POST_K1_K1_K3 Method 2
Node 1710
1,0638E+06 1,0387E+06
2,361
K3_MIN
POST_K1_K1_K3 Method 1
Node 77
1,0638E+06 1,0563E+06
0,708
K3_MAX
POST_K1_K1_K3 Method 1
Node 77
1,0638E+06 1,0589E+06
0,468
K3_MIN
POST_K1_K1_K3 Method 2
Node 77
1,0638E+06 9,4413E+05
11,253
K3_MAX
POST_K1_K1_K3 Method 2
Node 77
1,0638E+06 1,0387E+06
2,366
G_MIN
POST_K1_K1_K3 Method 1
Node 49
1,8943E+01 1,8866E+01
0,406
G_MAX
POST_K1_K1_K3 Method 1
Node 49
1,8943E+01 1,8884E+01
0,313
G_MIN
POST_K1_K1_K3 Method 2
Node 49
1,8943E+01 1,6896E+01
10,804
G_MAX
POST_K1_K1_K3 Method 2
Node 49
1,8943E+01 1,8551E+01
2,069
G_MIN
POST_K1_K1_K3 Method 1
Node 1710
1,8943E+01 1,8868E+01
0,395
G_MAX
POST_K1_K1_K3 Method 1
Node 1710
1,8943E+01 1,8887E+01
0,296
G_MIN
POST_K1_K1_K3 Method 2
Node 1710
1,8943E+01 1,6893E+01
10,819
G_MAX
POST_K1_K1_K3 Method 2
Node 1710
1,8943E+01 1,8553E+01
2,058
G_MIN
POST_K1_K1_K3 Method 1
Node 77
1,8943E+01 1,8856E+01
0,457
G_MAX
POST_K1_K1_K3 Method 1
Node 77
1,8943E+01 1,8875E+01
0,358
G_MIN
POST_K1_K1_K3 Method 2
Node 77
1,8943E+01 1,6882E+01
10,881
G_MAX
POST_K1_K1_K3 Method 2
Node 77
1,8943E+01 1,8541E+01
2,12
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
14/18
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
11 Modeling E: Calculation of the bilinear form of G
11.1 Characteristics of modeling
The mesh is identical to that of preceding calculations, but only the eighth of the block is retained
(Oxyz quadrant)
1) Loading 1: Idem modeling B.
2) Loading 2: Face X = 10. : uniform stress of traction
Z
= 1,
Face Z = 10. : uniform stress of traction
X
= 1 (shearing).
3) Loading 3: Loading 1 + Loading 2.
4) Loading 4: Loading 2 Chargement 1.
Four calculations are static are carried out respectively producing displacements U, v, u+v, and
v-u.
11.2 Characteristics of the mesh
A number of nodes:
2774
A number of meshs and type: 392 HEXA20 and 216 PENTA15
11.3 Functionalities
tested
Controls
DEFI_GROUP CREA_GROUP_NO
DEFI_MATERIAU ELAS
DEFI_LIST_REEL
“MECHANICAL” AFFE_MODELE
“3D”
AFFE_MATERIAU ALL
AFFE_CHAR_MECA FORCE_FACE
GROUP_MA
PRES_REP
NEAR
STAT_NON_LINE
DEFI_FOND_FISS
CALC_THETA
CALC_G_THETA_T COMP_ELAS
“CALC_G”
CALC_G_THETA_T COMP_ELAS
“CALC_G_BILI”
11.4 Notice
One represents only the eighth of the complete three-dimensional block and thus the eighth of the fissure.
Thus, it is necessary to divide the theoretical value of reference of the total rate of refund by 8.
The bilinear form
)
,
(v
U
G
check the following properties:
)
1
.
(
)
(
)
,
(
form
U
G
U
U
G
=
)
2
.
.
(
4
)
(
)
(
)
,
(
form
v
U
G
v
U
G
v
U
G
-
-
+
=
from where
)
3
.
(
)
(
)
(
4
)
2
(
)
2
(
)
,
(
form
v
G
U
G
v
G
U
G
v
U
v
U
G
-
=
-
-
=
+
-
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
15/18
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
12 Results of modeling E
12.1 Values
tested
Identification Reference
Aster %
difference
G total: G (U)
1.82 10
+1
1.8273 10
+1
0.4
G bilinear: G (U, U)
1.82 10
+1
1.8273
10
+1
0.4
G total: G (v)
6.8612
G bilinear: G (v, v)
form.1
6.8612
0.
G total: G (u+v)
4.7526 10
+1
G bilinear: G (u+v, u+v)
form.1
4.7526 10
+1
0.
G total: G (UV)
2.7428
G bilinear: G (UV, UV)
form.1
2.7428
0.
G bilinear: G (v, U)
form.2
1.1195 10
+1
0.
G bilinear: G (UV, u+v)
form.3
1.1412 10
+1
0.
One indicates by U displacement corresponding to loading 1, and v corresponding displacement
with loading 2. Loadings 3 and 4 agent with displacements (u+v) and (considering).
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
16/18
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
13 Modeling F: Calculation of the bilinear form of G local
13.1 Characteristics of modeling
The mesh is identical to that of modeling E.
1) Loading 1: Idem modeling B.
2) Loading 2: Face X = 10. : uniform stress of traction
Z
= 1,
Face Z = 10. : uniform stress of traction
X
= 1 (shearing).
3) Loading 3: Loading 1 + Loading 2.
4) Loading 4: Loading 2 Chargement 1.
Four calculations are static are carried out respectively producing displacements U, v, u+v, and
v-u.
13.2 Characteristics of the mesh
A number of nodes:
2774
A number of meshs and type: 392 HEXA20 and 216 PENTA15
13.3 Functionalities
tested
Controls
DEFI_GROUP CREA_GROUP_NO
DEFI_MATERIAU ELAS
DEFI_LIST_REEL
“MECHANICAL” AFFE_MODELE
“3D”
AFFE_MATERIAU ALL
AFFE_CHAR_MECA FORCE_FACE
GROUP_MA
PRES_REP
NEAR
STAT_NON_LINE
DEFI_FOND_FISS
CALC_G_LOCAL_T COMP_ELAS
“CALC_G”
CALC_G_LOCAL_T COMP_ELAS
“G_BILINEAIRE”
13.4 Notice
One represents only the eighth of the complete three-dimensional block and thus the eighth of the fissure.
Thus, it is necessary to divide the theoretical value of reference of the total rate of refund by 8.
The bilinear form
)
,
(
v
U
G
check the following properties:
)
1
.
(
)
(
)
,
(
form
U
G
U
U
G
=
)
2
.
.
(
4
)
(
)
(
)
,
(
form
v
U
G
v
U
G
v
U
G
-
-
+
=
from where
)
3
.
(
)
(
)
(
4
)
2
(
)
2
(
)
,
(
form
v
G
U
G
v
G
U
G
v
U
v
U
G
-
=
-
-
=
+
-
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
17/18
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
14 Results of modeling F
14.1 Values
tested
Node Smoothing R_inf R_sup
Identification
Reference Code_Aster % variation
N2667 Lag-Lag
0.1
1.0
G local: G (U)
5.79
5.746
0.748
N2667 Lag-Lag
0.1
1.0
G bilinear: G (U, U)
5.79
5.746
0.748
N2667 Lag-Lag
0.1
1.0
G max: G (U, U)
-
1.4946E+01
-
N2667 Leg-Leg
0.5
1.5
G local: G (v)
-
2.1737
-
N2667 Leg-Leg
0.5
1.5
G bilinear: G (v, v)
-
2.1737
-
N2773 Leg-Leg
0.5
1.5
G max: G (v, v)
-
6.0334
-
N2667 Lag-Lag
0.2
2.0
G total: G (u+v)
-
1.5150E+01
-
N2667 Lag-Lag 0.2
2.0 G bilinear
: G (u+v,
u+v)
- 1.5150E+01 -
N2667 Lag-Lag
0.2
2.0
G max: G (u+v, u+v)
-
3.8910E+01
-
[Figure 14.1-a] below shows the values of G local in bottom of fissure obtained from
bilinear form of G. It is the first loading which is applied (simple traction), it requests it
first mode of opening of the fissure. Fields
as G are discretized according to the method
of Lagrange. The radii inferior and superior delimiting the crown in which fields
decrease linearly are respectively equal to 0.1 mm and 1mm. the value of reference is equal
with 5.79 J/m
2
, cf modeling A.
0,00E+00
1,00E+00
2,00E+00
3,00E+00
4,00E+00
5,00E+00
6,00E+00
0,
00E
+
00
1,
96E
-
01
3,
93E
-
01
5,
89E
-
01
7,
85E
-
01
9,
81E
-
01
1,
18th
+
00
1,
37E
+
00
1,
57E
+
00
1,
77E
+
00
1,
96E
+
00
2,
16th
+
00
2,
36E
+
00
2,
55E
+
00
2,
75E
+
00
2,
94E
+
00
Curvilinear X-coordinate
G_
biline
has
I
Re
G_BILI_LOCAL
G_REF
Appear 14.1-a: G bilinear local according to the curvilinear X-coordinate
One indicates by U displacement corresponding to loading 1, and v corresponding displacement
with loading 2. Loading 3 corresponds to displacement (u+v).
Code_Aster
®
Version
8.1
Titrate:
SSLV134 - Calculation of G, fissures circular in infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, J.M. PROIX
Key
:
V3.04.134-B
Page:
18/18
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
15 Summaries of the results
Objective triple of this test is reached:
·
It is a question of validating the definition of the closed funds of fissure and installations consequent
calculation of G local. One checks in particular the independence of G local with respect to the angle
for an axisymmetric fissure and a loading. One notes a variation of less than 2% on
the whole of the bottom of fissure by two methods “LAGRANCE” and “LAGRANGE_NO_NO”.
·
Moreover, this test makes it possible to validate the control POST_K1_K2_K3 which makes it possible to calculate them
stress intensity factors by exploiting the jump of displacements on the lips of
fissure. This method, less precise than G_THETA, makes it possible to obtain here (with a mesh
adapted: nodes mediums of the edges touching the bottom of fissure moved with the quarter of these
edges) of the values of K1 and K3 to less than 1% of the reference (for the method
of extrapolation number 1). With the method of extrapolation number 2, the variation can go
up to 8%. Precision of the method of extrapolation number 3, tested in modeling
B, lies between those of the two preceding methods. Method 3 is however
interesting because it provides a single value of the stress intensity factors and not
not a maximum value and a minimal value.
·
Lastly, one validates the calculation of the bilinear form of G.