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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
1/14
Manual of Validation
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: V3.04.110
SSLV110 - Elliptic fissure in an infinite medium
Summary:
It is about a test in statics for a three-dimensional problem. This test makes it possible to calculate the rate of refund
of energy total and local on the bottom of fissure by the method
.
The radii of the crowns of integration are variable along the fissure, and the rate of refund of energy
room is calculated according to 2 different methods (LEGENDRE and LAGRANGE).
The interest of the test is the validation of the method
in 3D and following points:
·
comparison between the results and an analytical solution,
·
stability of the results according to the crowns of integration,
·
comparison between 2 methods different for calculation from G local,
·
2 cases of equivalent loadings (pressure distributed and voluminal loading).
This test contains 4 different modelings.
The 3rd modeling tests the derivative of G compared to the parameters material and loading.
Code_Aster
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
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Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
2/14
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V3.04 booklet: Linear statics of the voluminal structures
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1
Problem of reference
1.1 Geometry
It is about an elliptic fissure plunged in a presumedly infinite medium. Only one eighth is modelized
of a parallelepiped:
120 mm
6 mm
25 mm
725 mm
0
125
0 m
m
P
y
Z
X
0
: melts of elliptic fissure
1.2 Properties
materials
E= 210.000.MPa
= 0.3
1.3
Boundary conditions and loadings
Symmetry compared to the 3 main plans:
U
X
= 0. in the plan X = 0.
U
Y
= 0. in the plan Y = 0.
U
Z
= 0. in the plan Z = 0. out of the fissure
The conditions of loadings are is:
P = 1 MPa in the plan Z = 1250 mm (modelings A and B)
that is to say:
FZ = 8.104 NR/mm
3
on all the elements of volume (loading are equivalent to the precedent)
(modelings C and D).
Code_Aster
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Version
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
3/14
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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
The reference solution is an analytical solution resulting from SIH [bib1] and [bib2].
It is noted that the angle
indicate here the parametric angle of the point M (angle compared to axis OX of
projected M on the circle of radius b) and not the polar co-ordinate of this point.
()
K
B
E K
B
has
K
B
has
I
=
+
=
-
1
1
2
2
2
2
1 4
2
2
1 2
sin
cos
/
/
with
()
(
)
E K
K
D
=
-
1
2
2
0
2
1 2
sin
/
/
Here:
has
= 25 mm
B
= 6 mm, therefore
K
= 0,9707728
Values of the elliptic integrals
()
K
E
are tabulées in [bib3], according to asin
()
K
who is worth
here 76,11°. One finds then:
()
K
E
= 1,0672.
From where the factor of intensity of the stresses in MPa.
mm:
4
/
1
²
cos
²
²
²
sin
0680
,
4
)
(
+
=
has
B
K
I
Then, starting from the formula of Irwin (plane deformation):
()
()
2
²)
1
(
I
K
E
G
-
=
The total rate of refund of energy
ref.
G
is calculated by integration of
()
G
:
ref.
G
= 5,76.10
- 3
J/Misters.
Derived from G (modeling C):
For the derivative of G compared to the Young modulus E, one can write:
E
G
=
(with
4
,
302
=
) thus
E
G
E
G
-
=
Code_Aster
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Version
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
4/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
In addition, while varying the Fz loading, one finds:
2
Z
F
G
=
with
9
,
2276
=
thus
Z
Z
F
F
G
2
=
2.2 Bibliography
[1]
G.C. SIH: Mathematical Theories off Brittle Fractures - FRACTURE, flight II - Academic Close -
1968
[2]
Mr. K. KASSIN and G.C. SIH: Three-dimensional stress distribution around year elliptical ace
under arbitrary loadings J. Appl. Mech., 88, 601-611, 1966.
[3]
H. TADA, P. PARIS, G. IRWIN: The Analysis Stress off Handbook Aces - Third Edition -
International ASM - 2000
Code_Aster
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
5/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
3 Modeling
With
3.1
Characteristics of modeling
With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8;
=/4)
C
1
With
B
C
C
2
C
3
Z
X
Y
Loading: Unit pressure distributed on the face of the block opposed to the plan of the lip:
P = 1.MPa in the plan Z = 1250.mm.
3.2
Characteristics of the mesh
A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20
3.3 Functionalities
tested
Controls
DEFI_FOND_FISS LEVRE_SUP
GROUP_MA ALL
CALC_THETA FOND_3D
THETA_3D
CALC_G_THETA_T RESULT
TOUT_ORDRE
NUME_ORDRE
LIST_ORDRE
CALC_G_LOCAL_T
“THETA_LEGENDRE”
DEGREE = 7
“G_LEGENDRE”
R_INF_FO/R_SUP_FO
AFFE_CHAR_MECA FORCE_FACE
3.4 Remarks
The degree of the polynomials of LEGENDRE used to calculate G (S) is 7 (maximum value Aster).
For the 3 crowns of integration, the radii
R_INF
and
R_SUP
vary linearly along the bottom of
fissure.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
6/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
4
Results of modeling A
4.1 Values
tested
The values tested are:
·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.
The mesh includes/understands only one of the lips of the fissure, it is thus necessary to use the key word
“SYME_CHAR”
automatically to multiply by 2 in calculation Aster the rate of refund of energy calculated
by virtual extension of the single lip.
In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.
Identification Reference
Aster %
difference
G Crown C
1
1.44 10
3
1.410
10
3
- 2.1
G Crown C
2
1.44 10
3
1.451
10
3
0.8
G Crown C
3
1.44 10
3
1.424
10
3
- 1.1
G (A) crowns C
1
7.171
10
- 5
6.829 10
5
- 4.8
G (A) crowns C
2
7.171
10
- 5
7.239 10
5
0.95
G (A) crowns C
3
7.171
10
- 5
6.864 10
5
- 4.3
G (B) crowns C
1
1.721
10
- 5
1.48 10
5
- 13.8
G (B) crowns C
2
1.721
10
- 5
1.57 10
5
- 8.7
G (B) crowns C
3
1.721
10
- 5
1.90 10
5
- 6.9
G (C) crown C
1
5.215
10
- 5
4.992 10
5
- 4.3
G (C) crown C
2
5.215
10
- 5
5.124 10
5
- 1.7
G (C) crown C
3
5.215
10
- 5
5.013 10
5
- 3.9
4.2 Notice
The results are rather stable between the crowns safe at the point B where the variation of G (S) is more
large and results far away from the reference solution. One can explain this variation by the mesh
of poor quality.
Code_Aster
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Version
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
7/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
5 Modeling
B
5.1
Characteristics of modeling
With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)
C
1
With
B
C
C
2
C
3
Z
X
Y
Loading: Unit pressure distributed on the face of the block opposed to the plan of the lip:
P = 1 MPa in the plan Z = 1250 Misters.
5.2
Characteristics of the mesh
A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20
5.3 Functionalities
tested
Controls
DEFI_FOND_FISS LEVRE_SUP
GROUP_MA
CALC_THETA FOND_3D
THETA_3D
CALC_G_THETA_T
CALC_G_LOCAL_T' THETA_LAGRANGE”
“G_LEGENDRE”
DEGREE = 4
R_INF_FO/R_SUP_FO
AFFE_CHAR_MECA FORCE_FACE
5.4 Remarks
“THETA_LAGRANGE”
: the field
is discretized starting from the functions of forms of the nodes of the bottom
of fissure, but G (S) is always discretized starting from the polynomials of LEGENDRE.
The degree of the polynomials of LEGENDRE used to calculate G (S) is 4 [R7.02.01].
For the 3 crowns of integration, the radii
R_INF
and
R_SUP
vary linearly along the bottom of
fissure.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
8/14
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
6
Results of modeling B
6.1 Values
tested
The values tested are:
·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.
The mesh includes/understands only one of the lips of the fissure, it is thus necessary to use the key word
“SYME_CHAR”
automatically to multiply by 2 in calculation Aster the rate of refund of energy calculated
by virtual extension of the single lip.
In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.
Identification Reference
Aster %
difference
G Crown C
1
1.44 10
3
1.410
10
3
- 2.1
G Crown C
2
1.44 10
3
1.451
10
3
0.8
G Crown C
3
1.44 10
3
1.424
10
3
- 1.1
G (A) crowns C
1
7.171
10
- 5
7.120
10
- 5
- 0.7
G (A) crowns C
2
7.171
10
- 5
7.452
10
- 5
3.9
G (A) crowns C
3
7.171
10
- 5
7.431
10
- 5
3.6
G (B) crowns C
1
1.721
10
- 5
1.608
10
- 5
- 6.6
G (B) crowns C
2
1.721
10
- 5
1.662
10
- 5
- 3.4
G (B) crowns C
3
1.721
10
- 5
1.706
10
- 5
- 0.9
G (C) crown C
1
5.215
10
- 5
4.978
10
- 5
- 4.5
G (C) crown C
2
5.215
10
- 5
5.096
10
- 5
- 2.3
G (C) crown C
3
5.215
10
- 5
5.014
10
- 5
- 3.9
6.2 Notice
The results are better than in modeling A at the point B, but the disparity between
crowns remains strong.
Code_Aster
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Version
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Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
9/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
7 Modeling
C
7.1
Characteristics of modeling
With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)
C
1
With
B
C
C
2
C
3
Z
X
Y
Loading: Voluminal force
FZ
equivalent to a unit pressure on the face of the block opposed to
plan of the lip:
FORCE_INTERN
:
FZ
= 8.10
4
NR/mm
3
on all the elements of volume.
7.2
Characteristics of the mesh
A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20
7.3 Functionalities
tested
Controls
DEFI_FOND_FISS LEVRE_SUP
GROUP_MA
CALC_THETA FOND_3D
THETA_3D
CALC_G_THETA_T SENSITIVITY
CALC_G_LOCAL_T
“THETA_LEGENDRE”
DEGREE = 7
“G_LEGENDRE”
R_INF_FO/R_SUP_FO
AFFE_CHAR_MECA FORCE_INTERN
7.4 Remarks
The degree of the polynomials of LEGENDRE used to calculate G (S) is 7 (maximum value Aster).
For the 3 crowns of integration, the radii
R_INF
and
R_SUP
vary linearly along the bottom of
fissure.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
10/14
Manual of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
8
Results of modeling C
8.1 Values
tested
The values tested are:
·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure,
·
the derivative of G compared to E and to the loading in voluminal force Fz.
The mesh includes/understands only one of the lips of the fissure, it is thus necessary to use the key word
“SYME_CHAR”
automatically to multiply by 2 in calculation Aster the rate of refund of energy calculated
by virtual extension of the single lip.
In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.
Identification Reference
Aster %
difference
G Crown C
1
1.44 10
3
1.437
10
3
- 0.2
G Crown C
2
1.44 10
3
1.479
10
3
2.7
G Crown C
3
1.44 10
3
1.450
10
3
0.7
G (A) crowns C
1
7.171
10
- 5
6.962
10
- 5
2.9
G (A) crowns C
2
7.171
10
- 5
7.379
10
- 5
+2.9
G (A) crowns C
3
7.171
10
- 5
6.997
10
- 5
2.4
G (B) crowns C
1
1.721
10
- 5
1.509
10
- 5
12.2
G (B) crowns C
2
1.721
10
- 5
1.598
10
- 5
7.1
G (B) crowns C
3
1.721
10
- 5
1.629
10
- 5
5.2
G (C) crown C
1
5.215
10
- 5
5.085
10
- 5
2.5
G (C) crown C
2
5.215
10
- 5
5.219
10
- 5
0.1
G (C) crown C
3
5.215
10
- 5
5.107
10
- 5
2.1
DG/of crown C
1
- 6.8610 10
9
- 6.842
10
9
0.2
DG/of crown C
2
- 6.8610 10
9
- 7.041
10
9
2.7
DG/of crown C
3
- 6.8610 10
9
- 6.907
10
9
0.7
DG/dFz crown C
1
3.599 3.592
- 0.1
DG/dFz crown C
2
3.599 3.697
2.7
DG/dFz crown C
3
3.599 3.629
0.9
8.2 Notice
The results are rather stable between the crowns. One always notes worse results with
node B.
The errors on the derivative of G are comparable with those on G.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
11/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
9 Modeling
D
9.1
Characteristics of modeling
With = N01099 (S = 0.)
B = N01259 (S = 26.68)
C = N01179 (S = 17.8)
C
1
With
B
C
C
2
C
3
Z
X
Y
Loading: Voluminal force
Fz
equivalent to a unit pressure distributed on the face of the block
opposed to the plan of the lip:
FORCE_INTERN: FZ
= 8.10
4
NR/mm
3
on all the elements of volume.
9.2
Characteristics of the mesh
A number of nodes: 1716
A number of meshs and types: 304 PENTA15 and 123 HEXA20
9.3 Functionalities
tested
Controls
DEFI_FOND_FISS LEVRE_SUP
GROUP_MA
CALC_THETA FOND_3D
THETA_3D
CALC_G_THETA_T
CALC_G_LOCAL_T' THETA_LAGRANGE”
“G_LEGENDRE”
DEGREE = 7
R_INF_FO/R_SUP_FO
AFFE_CHAR_MECA FORCE_INTERN
9.4 Remarks
“THETA_LAGRANGE”
: the field
is discretized starting from the functions of forms of the nodes of the bottom
of fissure, but G (S) is always discretized starting from the polynomials of LEGENDRE.
The degree of the polynomials of LEGENDRE used to calculate G (S) is 7.
For the 3 crowns of integration, the radii
R_INF
and
R_SUP
are supposed to vary linearly on
bottom of fissure.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
12/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
10 Results of modeling D
10.1 Values
tested
The values tested are:
·
the total rate of refund of energy G,
·
the rate of refund of energy room G in all the nodes of the bottom of fissure.
The mesh includes/understands only one of the lips of the fissure, it is thus necessary to use the key word
“SYME_CHAR”
automatically to multiply by 2 in calculation Aster the rate of refund of energy calculated
by virtual extension of the single lip.
In the same way, G total calculated here corresponds to the quarter of G of reference defined previously, only one
eighth of parallelepiped being represented.
Identification Reference
Aster %
difference
G Crown C
1
1.44 10
3
1.437
10
3
- 0.2
G Crown C
2
1.44 10
3
1.479
10
3
2.7
G Crown C
3
1.44 10
3
1.450
10
3
0.7
G (A) crowns C
1
7.171
10
- 5
7.259
10
- 5
1.2
G (A) crowns C
2
7.171
10
- 5
7.597
10
- 5
5.9
G (A) crowns C
3
7.171
10
- 5
7.575
10
- 5
5.7
G (B) crowns C
1
1.721
10
- 5
1.636
10
- 5
4.9
G (B) crowns C
2
1.721
10
- 5
1.992
10
- 5
1.7
G (B) crowns C
3
1.721
10
- 5
1.734
10
- 5
0.7
G (C) crown C
1
5.215
10
- 5
5.071
10
- 5
2.7
G (C) crown C
2
5.215
10
- 5
5.192
10
- 5
0.4
G (C) crown C
3
5.215
10
- 5
5.108
10
- 5
2.1
10.2 Notice
The results are better than in modeling C at the point B.
Code_Aster
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Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
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HT-62/06/005/A
11 Summary of the results
Calculation of G local:
·
2 methods (LEGENDRE and LAGRANGE) give the same results appreciably
(less than 5% of error compared to the analytical solution) except at the point B (not end
ellipse on the large axis) where the Lagrange method is most precise,
·
loading case: the values obtained with the voluminal loading are slightly
higher than those obtained with imposed stresses (including for the values of G).
The differences are tiny and due to numerical integrations different on the term from
volume and the term of edge.
Calculation of derived from G:
·
the errors on the derivative of G are weak and comparable with those on G.
Code_Aster
®
Version
8.2
Titrate:
SSLV110 - Elliptic fissure in an infinite medium
Date:
15/02/06
Author (S):
E. CRYSTAL, X. DESROCHES
Key
:
V3.04.110-C
Page:
14/14
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V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A
Intentionally white left page.