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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
1/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Organization (S):
EDF-R & D/AMA, CS IF
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear of the structures
axisymmetric
V7.01.311 document
HPLA311 - Murakami 11.39. Fissure circular with
center of a sphere subjected to a temperature
uniform on the lips
Summary:
This test results from the validation independent of version 3 in breaking process.
It is about a basic static test into axisymmetric under stationary thermal loading calculated by
finite elements on the same mesh of a limited field.
The behavior is thermoelastic linear isotropic.
It includes/understands two axisymmetric modelings for which one varies the report/ratio has/B, has being the radius of
the fissure interns circular in the horizontal plane xoy and B radius of the sphere.
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
2/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
a: radius of the fissure interns circular in the horizontal plane xoy
b: radius of the sphere
the radii must check the condition has/B < 0,5, with B = 2,5E-3
1.2
Properties of material
Young modulus
E = 2 1011 AP
Poisson's ratio
v = 0,3
linear expansion factor
= 1,2 10-5 °C-1
1.3
Boundary conditions and loadings
UX = ur = 0 on the axis of revolution X = R = 0
UY = uz = 0 in the horizontal plane Y = Z = 0, apart from the lips has R B
The lips are supposed to be free stresses (not closing partial of the fissure).
Null temperature on the surface of the sphere.
Temperature of null reference (temperature to which the thermal deformations are considered
null).
Uniform and negative temperature T =
-
Tf on the lips of the fissure, melts of fissure included/understood.
stationary thermal problem (of Dirichlet type) must be solved beforehand by finite elements
on the same mesh as that intended for mechanical calculation. Tf = 100 °C
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
3/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Analytical calculation by transform of Hankel.
2.2
Results of reference
(
)
= <
= - · ·
= -
-
+
-
-
+
+
has
B
K
E T
F has
F
I
F
I
I
0 5
1
1 0 6366
0 4053
2 0163
0 6773
3 8523
4 1687
3 2741
2
3
4
5
6
7
,
,
,
,
,
,
,
,
2.3
Uncertainty on the solution
Badly definite, exact if
= 0
2.4 References
bibliographical
[1]
Y. MURAKAMI: Stress Intensity Factors Handbook, box 11.39, pages 1089-1090. The
Society off Materials Science, Japan, Pergamon Near, 1987.
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
4/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
Complete mesh
Zoom of the point of fissure
Code_Aster
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Version
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Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
5/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3.1.1 Definition of the radii of the crowns
For these various cases, we define (table below) the values of the higher radii and
inferiors, to specify in the control
CALC_THETA
:
1ère crown
2nd crown
3rd crown
4th crown
case has
rinf
1.E-6
2.5E-5
5.E-5
7.5E-5
rsup 2.5E-5
5.E-5
7.5E-5
1.E-4
case B
rinf
2.5E-5
2.75E-5 3.E-5 3.25E-5
rsup 2.75E-5
3.E-5
3.25E-5
3.5E-5
3.2
Characteristics of the mesh
1756 nodes and 569 elements including 529 QUA8 and 40 TRI6
3.3 Functionalities
tested
Controls
AFFE_MODELE
THERMICS
AXIS
ALL
AFFE_CHAR_THER
TEMP_IMPO
AFFE_MODELE
MECHANICS
AXIS
ALL
AFFE_CHAR_MECA
TEMP_CALCULEE
CALC_THETA
THETA_2D
CALC_G_THETA
OPTION
CALC_G
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
6/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
G, crown n°1 (J.m
2
)
1,0231E+2 9,7012E+1 *
-
5,17
G, crown n°2 (J.m
2
)
1,0231E+2 1,0051E+2 *
-
1,75
G, crown n°3 (J.m
2
)
1,0231E+2 1,0055E+2 *
-
1,71
G, crown n°4 (J.m
2
)
1,0231E+2 1,0055E+2 *
-
1,71
4.2 Remarks
·
To calculate G
ref.
, one uses the formulas of IRWIN in plane deformations:
(
)
Gref
E
K
K
I
II
= -
+
1
2
2
2
, KII = 0
·
For the low values of the report/ratio has/B, the solution must asymptotically approach
reference solution calculated for
= 0, are FI = 1, which is then exact (see MURAKAMI 11.23,
page 1069).
Code_Aster
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Version
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Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
7/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
5 Modeling
B
5.1
Characteristics of modeling
Complete mesh
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
8/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Zoom
Zoom of the point of fissure
5.2
Characteristics of the mesh
2095 nodes and 680 elements including 640 QUA8 and 40 TRI6
5.3 Functionalities
tested
Controls
AFFE_MODELE
THERMICS
AXIS
ALL
AFFE_CHAR_THER
TEMP_IMPO
AFFE_MODELE
MECHANICS
AXIS
ALL
AFFE_CHAR_MECA
TEMP_CALCULEE
CALC_THETA
THETA_2D
CALC_G_THETA
OPTION
CALC_G
Code_Aster
®
Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
9/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
6
Results of modeling B
6.1 Values
tested
Identification Reference
Aster %
difference
G, crown n°1 (J.m
2
)
1,0505E-4 * 1,0387E-4
-
1,129
G, crown n°2 (J.m
2
)
1,0505E-4 * 1,0388E-4
-
1,112
G, crown n°3 (J.m
2
)
1,0505E-4 * 1,0388E-4
-
1,111
G, crown n°4 (J.m
2
)
1,0505E-4 * 1,0387
-
1,116
*
In the case of axisymmetric calculations, to obtain the total rate of refund, report/ratio of
variation of energy to the variation of surface of the fissure, it is necessary to divide the rate of refund
obtained by radian with ASTER by R
fissure
(Cf Reference material [R7.02.01] - page18).
6.2 Remarks
·
To calculate G
ref.
, one uses the formulas of IRWIN in plane deformations:
(
)
Gref
E
K
K
I
II
= -
+
1
2
2
2
, KII = 0
·
For the low values of the report/ratio has/B, the solution must asymptotically approach
reference solution calculated for
= 0, are FI = 1, which is then exact (see MURAKAMI 11.23,
page 1069).
·
As it is awaited, the relative error on G is the double of that on K.
Code_Aster
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Version
5.0
Titrate:
HPLA311 - Murakami 11.39. Fissure circular in the center of a sphere
Date:
05/11/02
Author (S):
S. GRANET, I. CORMEAU, E. LECLERE
Key
:
V7.01.311-A
Page:
10/10
Manual of Validation
V7.01 booklet: Thermomechanical stationary linear
HT-66/02/001/A
7
Summary of the results
·
The calculation of K and G into axisymmetric in the presence of a stationary thermal loading, gives
good results since the maximum change for G is 1,75% (out first crown) for
= 0,4.
·
We check that the results of K and G for
=0,01 is better than for
= 0,4.