Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 1/8

Organization (S): EDF-R & D/AMA
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
V3.04.311 document

SSLV311 - Murakami 9.39. Fissure in quarter of ellipse
with the corner of a thick disc in rotation

Summary:

This test results from the validation independent of the version in breaking process.

Applicability:
Linear breaking process
Type of analysis:
Statics
Type of behavior:
Isotropic linear rubber band
Type of model:
Three-dimensional
A number of modelings:
1
Objective:
Basic test into three-dimensional for isotropic elastic materials, in
field limited in three directions, in the presence of a voluminal loading.
Explored parameters:
-
Fixed parameters:
Reports/ratios has/T, B/has, R2/R1, T/R1
Precision of the results:
Average standard deviation of 3% with the analytical reference solution


Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 2/8

1
Problem of reference

1.1 Geometry

B
With

Internal radius:
R1 = 0,1 m
External radius:
R2 = 0,6 m
Thickness:
T = 0,2 m
Half large axis:
= 0,05 m have
Small half centers:
B = 0,0125 m

1.2
Properties of material


Young modulus
E = 2.105 MPa
Poisson's ratio = 0.3

Density
= 7800 kg/m3

1.3
Boundary conditions and loading

The model will be limited to the part of the thick disc located in the half space Y 0, the plan of
fissure vertical being a symmetry plane.

In the absence of nodes on the axis of revolution, a rigid mode will be blocked by a linear relation
between degrees of freedom.
That is to say points:
With (R1,0, T) B
(- R1,0, T)

Blocking of the translation in X: UX (A) + UX (B) = 0
Blocking of the translation in Y: UY = 0 in plan XOZ, except for the lips of the fissure.
Blocking of the translation in Z: UZ (A) = 0

Blocking of rotation around OX: ensured by the C.L of symmetry in plan XOZ
Blocking of rotation around OY: UZ (B) = 0
Blocking of rotation around OZ: ensured by the C.L of symmetry in plan XOZ

Loading: stationary angular velocity = 500 radians/second
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 3/8

2
Reference solution

2.1
Method of calculation used for the reference solution

Method of integral equation of border.

2.2
Results of reference

3 +

2
1 -
2
2
K =
· R +
R
B F where the geometrical factor of correction is given,
I
2
1
I
4


3 +

·
·
according to the parametric angle of the ellipse, with the figure below.


The report/ratio has/T selected corresponds to the higher curve (squares).

2.3
Uncertainty on the solution

The maximum change enters the points marked and the curve being of 2%, the misreading on the curve
is lower than the announced maximum error (5%).


2.4 References
bibliographical

[1]
Y. MURAKAMI: Stress Intensity Factors Handbook, box 9.39, pages 786-791. The Society
off Materials Science, Japan, Pergamon Press, 1987.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 4/8

3 Modeling
With

3.1
Characteristics of modeling




Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 5/8




3.2
Characteristics of the grid

The initial grid consists of 8890 nodes and 2203 elements, including 1264 elements CU20 and
939 elements PR15.
After the conversion of the grid of quadratic to linear, the number of nodes is reduced to 2230.
This conversion is made essential by the use of the operator DEFI_FISS_XFEM, who
function for the moment that with linear elements.

3.3
Functionalities tested

Calculation of the factors of intensity of the constraints buildings, in all the nodes of the bottom of fissure, by
method THETA.

The factors of intensity of the constraints buildings are calculated on a crown of lower radius
Rinf=0,00075 m and of radius higher Rsup = 0,0025 Mr.

Commands



CREA_MAILLAGE QUAD_LINE



AFFE_CHAR_MECA FORCE_INTERN


FORMULE


DEFI_FISS_XFEM



CALC_G_LOCAL_T' CALC_K_G”



SYME_CHAR



EXCIT



Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 6/8

4
Results of modeling A

4.1 Values
tested

Identification
Reference (Pa.m)
Aster (Pa.m)
% difference
KI, = 0 degrees
5,657E+07 5,789E+07 - 2,33
KI, = 1,4 degrees
5,945E+07 5,360E+07 9,84
KI, = 2,8 degrees
6,292E+07 6,596E+07 - 4,84
KI, = 4,3 degrees
6,638E+07 6,606E+07 0,48
KI, = 5,9 degrees
6,984E+07 6,902E+07 1,18
KI, = 7,6 degrees
7,273E+07 7,289E+07 - 0,22
KI, = 9,5 degrees
7,562E+07 7,597E+07 - 0,47
KI, = 11,6 degrees
7,908E+07 8,053E+07 - 1,83
KI, = 14,4 degrees
8,197E+07 8,261E+07 - 0,78
KI, = 16,9 degrees
8,543E+07 8,695E+07 - 1,78
KI, = 20,5 degrees
8,889E+07 8,785E+07 1,17
KI, = 25,1 degrees
9,178E+07 9,190E+07 - 0,13
KI, = 31,1 degrees
9,466E+07 9,173E+07 3,09
KI, = 39,5 degrees
9,640E+07 9,562E+07 0,81
KI, = 51,5 degrees
9,755E+07 9,510E+07 2,51
KI, = 68,5 degrees
9,755E+07 9,824E+07 - 0,71
KI, = 90 degrees
9,640E+07 9,720E+07 - 0,83

The parametric angles of the values tested correspond to the position of the 17 points of the bottom of
fissure. The figure below makes it possible to compare the result of calculation with the reference solution.
The average standard deviation is very satisfactory:
(K ref. - K Aster 2
I
I
) ds
Average standard deviation =

=
= 3,11%
(Kref 2
I
) ds

100
95
90
85
0.5)
80
75
70
65
K1 (MPa/m^
60
K1 - Référence
55
K1 - Aster
50
0
10
20
30
40
50
60
70
80
90
Angle (°)


Note:

The voluminal loading is introduced here using key word FORCE_INTERN (command
AFFE_CHAR_MECA) and of FORMULE. The results are equivalent if the key word is used
ROTATION.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 7/8

5
Summary of the results

Within sight of the precision announced on the results of reference (5%) and the average standard deviation
obtained (3,11%), the results provided by Code_Aster are satisfactory.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

Code_Aster ®
Version
8.2
Titrate:
SSLV311 - Murakami 9.39. Fissure in quarter of ellipse to the corner of a disc
Date:
15/02/06
Author (S):
E. CRYSTAL
Key: V3.04.311-A Page: 8/8

Intentionally white left page.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-62/06/005/A

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