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Organization (S): EDF/AMA, UTO, CS IF, IPSN
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V7.02.311 document

HPLP311 - Murakami 11.17 Fissure in the center of one
rectangular thin section making obstacle with one
uniform heat flow 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 grid with an isotropic material in mode II.

Objective:

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

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

One measures a relative variation on KII, the precision is nevertheless badly defined.
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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:

11
12
0

X
S
S
X

11





y = S
S
12
22
0

y +



22 (
T - ref.
T
)






0
0

xy
S

66
xy 0
S11 = 1 Ex
S22 = 1 Ey
S12 = - X Ex = - y Ey

S66 = 1 Gxy
11 = X
22 = y

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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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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2
Reference solution

2.1
Method of calculation used for the reference solution

Complex potential.

2.2
Results of reference

2a
= W

L
=

W
T
W
K = 11 0 ·
· F
II
S
2
II
11

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 Press, 1987.
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3
Modelings A, B, C, D, E and F

3.1
Characteristics of modeling

These 6 modelings correspond to 6 grids 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 grid - Modélisation A



Zoom of the point of fissure - Modélisation A
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Half grid - Modélisation B



Zoom of the point of fissure - Modélisation B
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3.1.2 Modelings C and D



Complete grid - Modélisation C



Complete grid - Modélisation D

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3.1.3 Modelings E and F


Complete grid - Modélisation E



Complete grid - Modélisation F
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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 command 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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3.2
Characteristics of the grid

Half-grid; grid radiating at the right end of the fissure.

The table below gives the constitution of the studied grids:


NS = 8, NC = 4
NS = 4, NC = 3

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

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

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

3.3 Functionalities
tested


Variation of the result according to the topological and geometrical parameters of the radiant grid
(a many crowns, number sectors, diameter of the largest crown)

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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 grid
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 grid
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 grid
NS= 8
NC= 4
Modeling C
KII, crown n°1
2,2347E+7
2,166 107 3,058
KII, crown n°2
2,2347E+7
2,214 107 0,919
KII, crown n°3
2,2347E+7
2,214 107 0,919
KII, crown n°4
2,2347E+7
2,214 107 0,919
Radiant grid
NS= 4
NC= 3
Modeling D
KII, crown n°1
2,2347E+7
2,214 107 0,919
KII, crown n°2
2,2347E+7
2,214 107 0,919
KII, crown n°3
2,2347E+7
2,214 107 0,919
Diameter crowns external = 0,1 * has



Radiant grid
NS= 8
NC= 4
Modeling E
KII, crown n°1
2,2347E+7
2,2632 107 1,276
KII, crown n°2
2,2347E+7
2,2572 107 1,009
KII, crown n°3
2,2347E+7
2,2572 107 1,008
KII, crown n°4
2,2347E+7
2,2564 107 0,972
Radiant grid
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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Identification Reference
Aster %
difference
Diameter crowns external = 0,001 * has


Radiant grid
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 grid
NS= 4
NC= 3
Modeling B
G, crown n°1
2,4969E+3
2,600 103 4,134
G, crown n°2
2,4969E+3
2,5996 103 4,114
G, crown n°3
2,4969E+3
2,5996 103 4,111
Diameter crowns external = 0,01 * has



Radiant grid
NS= 8
NC= 4
Modeling C
G, crown n°1
2,4969E+3
2,451 103 1,842
G, crown n°2
2,4969E+3
2,475 103 0,858
G, crown n°3
2,4969E+3
2,475 103 0,858
G, crown n°4
2,4969E+3
2,475 103 0,858
Radiant grid
NS= 4
NC= 3
Modeling D
G, crown n°1
2,4969E+3
2,475 103 0,858
G, crown n°2
2,4969E+3
2,475 103 0,858
G, crown n°3
2,4969E+3
2,475 103 0,858
Diameter crowns external = 0,1 * has



Radiant grid
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 grid
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 draw
value of reference starting from the formula of IRWIN in plane constraints:

Gref = (1/E) * KII2
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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 grid

5.2
Characteristics of the grid

Complete model, with grid radiating only at the right end of the fissure and grid
regular, not refined, at the left end.

The grid consists of 1718 nodes and 568 elements, including 464 elements QUA8 and 104 elements
TRI6.

5.3 Functionalities
tested


Independence of KII at the end of straight line compared to the grid of the end of left.

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6
Results of modeling G

6.1 Values
tested

Identification Reference
Aster %
difference
Diameter crowns external = 0,01 * has



Radiant grid
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 grid
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 constraints and 4% for the rate of refund of energy. Invariance is checked
results compared to the various crowns of integration.

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