Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
1/8
Organization (S): EDF-R & D/AMA
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
V7.03.103 document
HPLV103 - Calcul of KI and G thermo elastic 3D
for a circular fissure
Summary
It is about a test of breaking process into thermomechanical for a three-dimensional problem. One
consider a circular fissure plunged in an elastic thermo medium. A uniform temperature is imposed
on the lips of the fissure. This test makes it possible to calculate the total rate of refund of energy G and the factor
of intensity of the constraints room KI in various points of the bottom of fissure.
The interest of the test is the invariance of G and KI according to various crowns and the comparison with a solution
analytical.
This test contains two modelings 3D.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
2/8
1
Problem of reference
1.1 Geometry
One considers a circular fissure plunged in an elastic thermo medium. Taking into account
symmetries of the problem, only a eighth of the structure is represented:
Z
E
H
F
G
D
O
R
y
P
Q
B
C
X
Dimensions of the fissure are as follows:
COP = GOLD = 1.0
The medium is modelled by a parallelepiped of dimensions:
OB = OD = OC = 30.0
1.2
Material properties
Thermal conductivity:
= 1.
Thermal dilation coefficient:
= 10-6/°C
Young modulus:
E = 2.10+5 MPa
Poisson's ratio:
= 0.3
1.3
Boundary conditions and loadings
·
Mechanics: displacements imposed (DDL_IMPO) on the following groups of meshs:
DX = 0 on ODHE;
DY = 0 on OEFB;
DZ = 0 on PBCDRQ (i.e lower face of the parallelepiped, without the lip of the fissure).
·
Thermics: temperature imposed (TEMP_IMPO) on the following groups of meshs:
TEMP = 0 on BCGH, CDHG and EFGH (outsides of the parallelepiped);
TEMP = - 1 on OPQR.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution results from the collection of MURAKAMI [bib1]:
Z
T0 = constant = - 1
Y
O
has
X
The expression of the rate of refund of energy is as follows:
(1 2
-)
E
2
G =
K avecK =
T
F has, with = has/B and,
1
1
E
(1 -)
()
0
F () = 1 -.
0
6366 -.
0
4053 2 +.
2
0163 3 -.
0
6773 4 -.
3
8523 5 +.
4
1687 6 +.
3
2741 7.
Note:
For = 0 (infinite medium), the solution is exact. For a medium finished, uncertainty on the solution
is unknown. In this test, = 1/30.
2.2
Result of reference
The result of reference is thus: KI = 157.73 103 Pa m1/2 and G = 1.132 10-1 J/m2
2.3 References
bibliographical
[1]
Stress intensity factors Handbook (Y. MURAKAMI), box 11.39, pp. 1089-1090, the Society
off Material Science, Japan, Pergamon Press, 1987.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
4/8
3 Modeling
With
3.1
Characteristics of modeling
It is about a three-dimensional modeling. The grid was carried out using procedure GIBI of
block fissured 3D [bib1]. One represented only the eighth of the structure (and thus a quarter of the face of
the fissure), the quarter of this face being discretized in 16 sectors.
3.2
Characteristics of the grid
The grid is composed of quadratic elements
A number of meshs and types: 624 PENTA 15, 5600 HEXA 20
3.3 Functionalities
tested
Commands
THER_LINEAIRE
MECA_STATIQUE
CALC_THETA THETA_3D
CALC_G_THETA_T
CALC_G_LOCAL_T CALC_G
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
5/8
4
Results of modeling A
4.1 Values
tested
The values tested are those of the rate of refund of energy G total and the rate of refund of
energy room at points A and B starting from the various crowns of integration and the two methods
of definition of the fields:
Identification Reference
Aster %
difference
G total
Crown 1 G
8.8910-8 8.66
10-8
2.53
Crown 2 G
8.8910-8 8.68
10-8
2.31
Crown 3 G
8.8910-8 8.69
10-8
2.17
Crown 4 G
8.8910-8 8.68
10-8
2.31
G local Lagrange Legendre (degree 7)
G local 1 in A
5.66 10-8
6.13 10-8
8.31
G local 2 in A
5.66 10-8
6.19 10-8
9.37
G local 3 in A
5.66 10-8
6.42 10-8
13.46
G local Lagrange Legendre (degree 7)
G local 1 out of B
5.66 10-8 5.50
10-8
2.75
G local 2 out of B
5.66 10-8 5.51
10-8
2.62
G local 3 out of B
5.66 10-8 5.50
10-8
2.84
G local Legendre Legendre (degree 7)
G local 1 in A
5.66 10-8
5.54 10-8
2.07
G local 2 in A
5.66 10-8
5.58 10-8
1.39
G local 3 in A
5.66 10-8
5.71 10-8
1.01
G local Legendre Legendre (degree 7)
G local 1 out of B
5.66 10-8 5.51
10-8
2.62
G local 2 out of B
5.66 10-8 5.52
10-8
2.46
G local 3 out of B
5.66 10-8 5.52
10-8
2.52
Crown 1:
Rinf=0.07
Rsup=0.2
Crown 2:
Rinf=0.2
Rsup=0.4
Crown 3:
Rinf=0.4.
Rsup=0.6.
Crown 4:
Rinf=0.07
Rsup=0.6
The supports of the local field correspond to the first three crowns of the total field.
4.2 Remarks
·
The value of reference is the value of the rate of refund of energy room: Gréf = 5.66 10-8
J/m2. The total rate of refund of energy provided by Code_Aster is:
2 A
G
= G
Aster
ref. ×
, since by reason of symmetry one models only one quarter of the plan of
8
fissure and only one lip.
·
The results of G local are not given that for the points A and B respectively located on one
symmetry plane and face of fissure. Results concerning the point B (medium of
face) reveal a variation from approximately 3% compared to the result of reference. Results
concerning point A are worse (the variation ranges between 3% and 13.5%), which is a report
usual for the estimate of G local for the points located on a symmetry plane.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
6/8
5 Modeling
B
5.1
Characteristics of modeling and the grid
It is about a three-dimensional modeling. To calculate the local KI it is necessary to pass
by command DEFI_FISS_XFEM.
A grid made up of quadratic elements, identical to that of modeling A, is used for
thermal calculation:
A number of meshs and types: 624 PENTA 15, 5600 HEXA 20
After thermal calculation the quadratic meshs are converted into linear meshs (operator
CREA_MAILLAGE, key word QUAD_LINE). The conversion of the grid is necessary because the operator
DEFI_FISS_XFEM functions for the moment only with linear elements.
A number of meshs and types: 624 PENTA 6, 5600 HEXA 8
5.2 Functionalities
tested
Commands
THER_LINEAIRE
CREA_MAILLAGE QUAD_LINE
MECA_STATIQUE
CALC_THETA THETA_3D
CALC_G_THETA_T
DEFI_FISS_XFEM
CALC_G_LOCAL_T CALC_K_G
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
7/8
6
Results of modeling B
6.1 Values
tested
The values tested are those of the total rate of refund of energy G and of the factor of intensity of
constraints room KI at points A and B starting from the various crowns of integration:
Identification Reference
Aster %
difference
G total
Crown 1 G
1.7781 10-1 1.686
10-1
- 5.2
Crown 2 G
1.7781 10-1 1.695
10-1
- 4.7
Crown 3 G
1.7781 10-1 1.696
10-1
- 4.6
Local KI Lagrange - Lagrange
Local KI 1 in A
157.73 103
158.8 103
0.7
Local KI 2 in A
157.73 103
161.0 103
2.1
Local KI 3 in A
157.73 103
161.9 103
2.7
Local KI 1 out of B
157.73 103
159.7 103
1.2
Local KI 2 out of B
157.73 103
162.1 103
2.7
Local KI 3 out of B
157.73 103
163.1 103
3.3
Crown 1:
Rinf=0.04
Rsup=0.2
Crown 2:
Rinf=0.08
Rsup=0.28
Crown 3:
Rinf=0.08
Rsup=0.36
In this modeling, the use of key word SYME_CHAR makes it possible to multiply it automatically
result by two to take into account symmetry compared to the lips of the fissure.
The results of local KI are not given that for the points A and B respectively located on a plan
of symmetry and in the middle of the face of fissure. Results concerning point A and the point B (medium of
face) are also satisfactory, with a variation from approximately 3% compared to the result of reference.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
Code_Aster ®
Version
8.2
Titrate:
HPLV103 - Calcul of G thermo elastic 3D for a circular fissure
Date:
15/02/06
Author (S):
E. Key CRYSTAL
:
V7.03.103-B Page:
8/8
7
Summary of the results
·
The passage of a quadratic grid to a linear grid for mechanical calculation decreases
precision of the result: G total have a variation of 4.8% on average with the reference for
linear grid against 2.2% for the quadratic grid.
·
The smoothing LEGENDRE-LEGENDRE led, on this case test, to the most precise results for
local values of G. Pour the calculation of K local, one advises smoothing LAGRANGE-LAGRANGE.
·
The precision on the calculation of local KI is satisfactory, the average deviation being limited to 2.3%.
Handbook of Validation
V7.03 booklet: Thermomechanical stationary linear of the voluminal systems
HT-62/06/005/A
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