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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
Date:
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
:
V7.02.311-A
Page:
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V7.02 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Organization (S):
EDF/AMA, UTO, CS IF, IPSN












Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
V7.02.311 document



HPLP311 - Murakami 11.17 Fissures in the center of one
rectangular thin section making obstacle with one
uniform heat transfer rate in isotropic medium




Summary:

It is about an elastic thermo calculation static linear isotropic.

It is a basic test in plane 2D for a stationary thermal loading calculated by finite elements on
even mesh with an isotropic material in mode II.

Objective:
·
basic test in plane 2D, for a stationary thermal loading calculated by finite elements on
even mesh, with isotropic material, in mode II,
·
validation of the calculation of K
II
,
·
variability of G according to the topology (sectors, crowns) of the radiant mesh. Checking
invariance of the results in breaking process, at an end of fissure, compared to
mesh of the other end of the same fissure.

Calculation is tested on a complete mesh and a half-mesh. The parameters L/W and 2A/W being fixed.

One measures a relative variation on K
II
, the precision is nevertheless badly defined.
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
:
V7.02.311-A
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HT-66/02/001/A
1
Problem of reference
1.1 Geometry
Width of the plate:
W = 0,6 m
Length of the plate:
L = 0,3 m
Length of the fissure:
2a = 0,3 m

1.2
Properties of material
Notation for thermoelastic properties:
(
)
y
X
xy
y
y
X
X
y
X
ref.
xy
y
X
xy
y
X
G
S
E
E
S
E
S
E
S
T
T
S
S
S
S
S
=
=
=
-
=
-
=
=
=
-




+








=




22
11
66
12
22
11
22
11
66
22
12
12
11
1
1
1
0
0
0
0
0
One limits oneself to isotropic material, as well from the thermal point of view as mechanical:
Ex = Ey = 2.105 MPa
X =
y = 0,3
X =
y = 1,2 10-5 °C-1
X =
y = 54 W/m °C
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
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HT-66/02/001/A
1.3
Boundary conditions and loading
Two models are considered:
·
the half-model X 0
·
the complete model
Boundary conditions mechanical:
·
half-model
UX = 0 along the axis of symmetry X = 0
UY = 0 at the item (W/2,0)
·
complete model
UX = 0 at the item (0, L/2)
UY = 0 at the items (­ L/2,0) and (L/2,0)
Boundary conditions thermal:
·
half-model
T = 100°C on the edge higher Y = L/2
T = ­ 100°C on the edge lower Y = ­ L/2
null flow on the axis of symmetry, the free edge X = W/2 and on the edge of the fissure
·
complete model
T = 100°C on the edge higher Y = L/2
T = ­ 100°C on the edge lower Y = ­ L/2
null flow on the free edges X = ± W/2 and on the edge of the fissure
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
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V7.02.311-A
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HT-66/02/001/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Complex potential.
2.2
Results of reference
=
=
=
·
·
2
2
11 0
11
has
W
L
W
K
T
S
W
F
II
II
where the geometrical factor of correction FII is given according to
for each material, in
particular case
= 0,5 on the curves below.
The isotropic material being represented by curve I
2.3
Uncertainty on the solution
Nondefinite precision.
2.4 References
bibliographical
[1]
Y. MURAKAMI: Stress Intensity Factors Handbook, box 11.17, pages 1045-1047. The
Society off Materials Science, Japan, Pergamon Near, 1987.
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
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V7.02.311-A
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V7.02 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3
Modelings A, B, C, D, E and F
3.1
Characteristics of modeling
These 6 modelings correspond to 6 mesh where one varies 3 topological parameters.
table below summarizes the various studied cases:
NS = 8, NC = 4
NS = 4, NC = 3
rt = 0,001 * has
With
B
rt = 0,01 * has
C
D
rt = 0,1 * has
E
F
The topological parameters which vary are:
NS:
a number of sectors on 90°
NC:
a number of crowns
rt:
the radius of the largest crown (with half a: length of the fissure)
3.1.1 Modelings A and B
Half mesh - Modeling A
Zoom of the point of fissure - Modeling A
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Half mesh - Modeling B

Zoom of the point of fissure - Modeling B
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3.1.2 Modelings C and D
Complete mesh - Modeling C

Complete mesh - Modeling D
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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3.1.3 Modelings E and F
Complete mesh - Modeling E

Complete mesh - Modeling F
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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3.1.4 Definition of the radii of the crowns
For these various cases, we define the values of the higher and lower radii, to specify
in the control
CALC_THETA
:
Modeling A
1 era crowns
2nd crown
3rd crown
4th crown
rinf (m)
3,75E5
7,500E5 1,125E4
1,500E4
rsup (m)
7,50E5
1,125E4 1,500E4
1,875E4

Modeling B
1 era crowns
2nd crown
3rd crown
rinf (m)
5,00E5
1,00E4
1,50E4
rsup (m)
1,00E4
1,50E4
2,00E4

Modeling C
1 era crowns
2nd crown
3rd crown
4th crown
rinf (m)
3,75E4
7,500E4 1,125E3
1,500E3
rsup (m)
7,50E4
1,125E3 1,500E3
1,875E3

Modeling D
1 era crowns
2nd crown
3rd crown
rinf (m)
5,00E4
1,00E3
1,50E3
rsup (m)
1,00E3
1,50E3
2,00E3

Modeling E
1 era crowns
2nd crown
3rd crown
4th crown
rinf (m)
3,75E3
7,500E3 1,125E2
1,500E2
rsup (m)
7,50E3
1,125E2 1,500E2
1,875E2

Modeling F
1 era crowns
2nd crown
3rd crown
rinf (m)
5,00E3
1,00E2
1,50E2
rsup (m)
1,00E2
1,50E2
2,00E2
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
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V7.02 booklet: Thermomechanical stationary linear
HT-66/02/001/A
3.2
Characteristics of the mesh
Half-mesh; mesh radiating at the right end of the fissure.
The table below gives the constitution of the studied mesh:
NS = 8, NC = 4
NS = 4, NC = 3

rt = 0,001 * has
3831 nodes,
1516 elements,
884 TRI6,
632 QUA8.
3507 nodes,
1388 elements,
820 TRI6,
568 QUA8.

rt = 0,01 * has
1179 nodes,
400 elements,
104 TRI6,
296 QUA8.
855 nodes,
272 elements,
40 TRI6,
232 QUA8.

rt = 0,1 * has
659 nodes,
240 elements,
104 TRI6,
136 QUA8.
335 nodes,
112 elements,
40 TRI6,
72 QUA8.

3.3 Functionalities
tested
Variation of the result according to the topological and geometrical parameters of the radiant mesh
(a many crowns, number sectors, diameter of the largest crown)
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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:
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V7.02 booklet: Thermomechanical stationary linear
HT-66/02/001/A
4
Results of modelings A, B, C, D, E and F
4.1 Values
tested
Identification Reference
Aster %
difference
Diameter crowns external = 0,001 * has
Radiant mesh
NS= 8
NC= 4
Modeling A
KII, crown n°1
2,2347E+7
2,2814E7
2,09
KII, crown n°2
2,2347E+7
2,2813E7
2,08
KII, crown n°3
2,2347E+7
2,2814E7
2,09
KII, crown n°4
2,2347E+7
2,2814E7
2,09
Radiant mesh
NS= 4
NC= 3
Modeling B
KII, crown n°1
2,2347E+7
2,282E7
2,10
KII, crown n°2
2,2347E+7
2,282E7
2,10
KII, crown n°3
2,2347E+7
2,281E7
2,09
Diameter crowns external = 0,01 * has
Radiant mesh
NS= 8
NC= 4
Modeling C
KII, crown n°1
2,2347E+7
2,166 10
7
3,058
KII, crown n°2
2,2347E+7
2,214 10
7
0,919
KII, crown n°3
2,2347E+7
2,214 10
7
0,919
KII, crown n°4
2,2347E+7
2,214 10
7
0,919
Radiant mesh
NS= 4
NC= 3
Modeling D
KII, crown n°1
2,2347E+7
2,214 10
7
0,919
KII, crown n°2
2,2347E+7
2,214 10
7
0,919
KII, crown n°3
2,2347E+7
2,214 10
7
0,919
Diameter crowns external = 0,1 * has
Radiant mesh
NS= 8
NC= 4
Modeling E
KII, crown n°1
2,2347E+7
2,2632 10
7
1,276
KII, crown n°2
2,2347E+7
2,2572 10
7
1,009
KII, crown n°3
2,2347E+7
2,2572 10
7
1,008
KII, crown n°4
2,2347E+7
2,2564 10
7
0,972
Radiant mesh
NS= 4
NC= 3
Modeling F
KII, crown n°1
2,2347E+7
2,255E7
0,932
KII, crown n°2
2,2347E+7
2,2568E7
0,988
KII, crown n°3
2,2347E+7
2,2568E7
0,987
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Titrate:
HPLP311 - Murakami 11.17 Fissures in the center of a thin section
Date:
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Author (S):
S. GRANET, E. SCREWS, I. CORMEAU, E. LECLERE
Key
:
V7.02.311-A
Page:
12/14
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V7.02 booklet: Thermomechanical stationary linear
HT-66/02/001/A
Identification Reference
Aster %
difference
Diameter crowns external = 0,001 * has
Radiant mesh
NS= 8
NC= 4
Modeling A
G, crown n°1
2,4969E+3
2,5984E+3
4,07
G, crown n°2
2,4969E+3
2,5990E+3
4,09
G, crown n°3
2,4969E+3
2,5992E+3
4,10
G, crown n°4
2,4969E+3
2,5993E+3
4,10
Radiant mesh
NS= 4
NC= 3
Modeling B
G, crown n°1
2,4969E+3
2,600 10
3
4,134
G, crown n°2
2,4969E+3
2,5996 10
3
4,114
G, crown n°3
2,4969E+3
2,5996 10
3
4,111
Diameter crowns external = 0,01 * has
Radiant mesh
NS= 8
NC= 4
Modeling C
G, crown n°1
2,4969E+3
2,451 10
3
1,842
G, crown n°2
2,4969E+3
2,475 10
3
0,858
G, crown n°3
2,4969E+3
2,475 10
3
0,858
G, crown n°4
2,4969E+3
2,475 10
3
0,858
Radiant mesh
NS= 4
NC= 3
Modeling D
G, crown n°1
2,4969E+3
2,475 10
3
0,858
G, crown n°2
2,4969E+3
2,475 10
3
0,858
G, crown n°3
2,4969E+3
2,475 10
3
0,858
Diameter crowns external = 0,1 * has
Radiant mesh
NS= 8
NC= 4
Modeling E
G, crown n°1
2,4969E+3
2,5624E3
2,627
G, crown n°2
2,4969E+3
2,5503E3
2,139
G, crown n°3
2,4969E+3
2,5499E3
2,124
G, crown n°4
2,4969E+3
2,5489 E3
2,084
Radiant mesh
NS= 4
NC= 3
Modeling F
G, crown n°1
2,4969E+3
2,5470 E3
2,006
G, crown n°2
2,4969E+3
2,5497 E3
2,117
G, crown n°3
2,4969E+3
2,5491 E3
2,094
4.2 Remarks
In the reference, the author supposes that KI = 0, but it does not check it a posteriori.
With regard to the rate of refund of energy G, if we suppose that KI = 0, we fire
value of reference starting from the formula of IRWIN in plane stresses:
G
ref.
= (1/E) * KII
2
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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HT-66/02/001/A
5 Modeling
G
5.1
Characteristics of modeling
For this modeling, we use the complete model with the best parameters NS, NC and rt
calculated in preceding modelings. We thus used the following values:
·
NS = 8,
·
NC = 4,
·
rt = 0,01 * A.
Complete mesh

5.2
Characteristics of the mesh
Complete model, with mesh radiating only at the right end of the fissure and mesh
regular, not refined, at the left end.
The mesh consists of 1718 nodes and 568 elements, including 464 elements QUA8 and 104 elements
TRI6.

5.3 Functionalities
tested
Independence of K
II
at the end of straight line compared to the mesh of the end of left.
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HPLP311 - Murakami 11.17 Fissures in the center of a thin section
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HT-66/02/001/A
6
Results of modeling G
6.1 Values
tested
Identification Reference
Aster %
difference
Diameter crowns external = 0,01 * has
Radiant mesh
NS= 8
NC= 4
KII, crown n°1
2,2347E+7
2,2640E7
1,31
KII, crown n°2
2,2347E+7
2,2640E7
1,31
KII, crown n°3
2,2347E+7
2,2640E7
1,31
KII, crown n°4
2,2347E+7
2,2641E7
1,31
Identification Reference
Aster %
difference
Diameter crowns external = 0,01 * has
Radiant mesh
NS= 8
NC= 4
G, crown n°1
2,4969E+3
2,5620E3
2,610
G, crown n°2
2,4969E+3
2,5626E3
2,631
G, crown n°3
2,4969E+3
2,5627E3
2,635
G, crown n°4
2,4969E+3
2,5628E3
2,640





7
Summary of the results
The differences between the reference solution and the results of Code_Aster do not exceed 3% on
coefficients of intensity of stresses and 4% for the rate of refund of energy. Invariance is checked
results compared to the various crowns of integration.