Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
1/8

Organization (S): EDF-R & D/AMA

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
Document: V7.02.101

HPLP101 - Plaque fissured in thermoelasticity
(plane constraints)

Summary:

This test results from the validation independent of Code_Aster in breaking process (reference resulting from
Murakami: Mura11-17). It makes it possible to validate the operators of breaking process for a problem
two-dimensional (assumption of the plane constraints) in isotropic linear thermoelasticity.

This test includes/understands the first modeling in plane constraints in which are calculated:

·
the rate of refund of energy G (traditional calculation by the method théta),
·
coefficients of intensity of constraints KI and KII.

These two calculations are carried out on 6 different crowns of integration.

The interest of the test is to compare the values of G and KII compared to the reference solution and to test
the invariance of calculations compared to the various crowns of integration.

The second modeling makes it possible to calculate the derivative of G compared to the Young modulus and one
loading in voluminal forces and to compare them with an analytical solution.
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
2/8

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:

S
S
0
X
11
12
X



11
= S
S
12
22
0 +
y


y
22 ·(T - Re
T F)


0
0
S




66
xy
xy 0
S = 1 E
11
X
S = 1 E
22
y
S = - E = -
12
X
X
y E y

S = 1 G
66
xy
=
11
X
=
22
y

One limits oneself to isotropic material, as well from the thermal point of view as mechanical:

E = E
X
y = 2. 105 MPa
=
X
y = 0.3
=
X
y = 1.2 10­5 °C1
=
X
y = 54. W/m °C
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
3/8

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.)

·
complete model

UX = 0 at the item (0, L/2.)
UY = 0 at the points (- L/2.) and (L/2.)

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

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
4/8

2
Reference solution

2.1
Method of calculation used for the reference solution

Complex potential [bib1].

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.
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
5/8

2.5
Reference solution for the derivative of G (modeling B)

While varying the Young modulus and the Fy loading, one notes that:


2
G
G = F
with
3

310
.
5
-
=
that is to say
=
F

2

Y
Y
F
Y


G

G
G =
with
3

310
.
5
-
=
that is to say
= -
E
E

E

3 Modeling
With

3.1
Characteristics of modeling

For this modeling, the 3 topological parameters of the block fissure 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)

NS = 8
NC = 4
rt = 0,001 * has

The values of the higher and lower radii, to specify in command CALC_THETA are:


Crown 1
Crown 2
Crown 3
Crown 4
Crown 5
Crown
6
Rinf
3,75E5
7,500E5
1,125E4 1,500E4 1,875E4 2,250E4
Rsup 7,50E5 1,125E4 1,500E4 1,875E4 2,250E4 3,000E4

3.2
Characteristics of the grid

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

3831 nodes,
1516 elements,
884 TRI6,
632 QUA8.

3.3 Functionalities
tested

Commands



THERMAL AFFE_MODELE
PLAN
TOUT

MECHANICAL AFFE_MODELE
C_PLAN ALL

THER_LINEAIRE



MECA_STATIQUE



CALC_THETA THETA_2D



CALC_G_THETA_T OPTION
CALC_G


CALC_G_THETA_T OPTION
CALC_K_G

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
6/8

4
Results of modeling A

4.1 Values
tested

Identification Reference
Aster %
difference
KII, crown n°1
2,2347E+7 2,2814E+7
2,09
KII, crown n°2
2,2347E+7 2,2813E+7
2,08
KII, crown n°3
2,2347E+7 2,2814E+7
2,09
KII, crown n°4
2,2347E+7 2,2814E+7
2,09
KII, crown n°5
2,2347E+7 2,2817E+7
2,10
KII, crown n°6
2,2347E+7 2,2818E+7
2,11




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
G, crown n°5
2,4969E+3 2,6013E+3
4,18
G, crown n°6
2,4969E+3 2,5985E+3
4,07

4.2 Remarks

In the reference, the author supposes that KI = 0, but it does not check it a posteriori. With the sights of
deformations resulting from ASTER, coefficient KI is different from zero, but there remains very weak by report/ratio
with KII (the fissure slips more than it does not open).

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:

G

ref. = (1/E)
2
* K II

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
7/8

5 Modeling
B

5.1
Characteristics of modeling

Same characteristics as for modeling a:

·
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)

NS = 8
NC = 4
rt = 0,001 * has

The values of the higher and lower radii, to specify in command CALC_THETA are:


Crown 1
Crown 2
Crown 3
Crown 4
Crown 5
Crown 6
Rinf
3,75E5
7,500E5
1,125E4 1,500E4 1,875E4 2,250E4
Rsup 7,50E5 1,125E4 1,500E4 1,875E4 2,250E4 3,000E4

5.2
Characteristics of the grid

Even grid that for modeling a:

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

3831 nodes,
1516 elements,
884 TRI6,
632 QUA8.

5.3
Parameters materials and loading

For this modeling one took E=1Pa.
The loading is a voluminal force Fy=1N on all the structure. There is no loading
thermics.

5.4
Functionalities tested

Commands



MECHANICAL AFFE_MODELE
C_PLAN ALL

MECA_STATIQUE



CALC_THETA THETA_2D



CALC_G_THETA_T SENSITIVITY




6
Results of modeling B

6.1 Values
tested

Identification Reference
Aster %
difference
dg/of, crown n°1
- 5.3E-3
- 5.299E-3
- 7.6E-4
dg/of, crown n°2
- 5.3E-3
- 5.301E-3
0.02
dg/dFy, crown n°1
1.06E-2
1.0599E-2
- 7.6E-4
dg/dFy, crown n°2
1.06E-2
1.0602E-2
0.02
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

Code_Aster ®
Version
7.2
Titrate:
HPLP101 - Plaque fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES Key
:
V7.02.101-B Page:
8/8

7
Summary of the results

The differences between the reference solution and the results of Code_Aster do not exceed 2% on
coefficients of intensity of constraints and 4% for the rate of refund of energy. One checks
the invariance of the results compared to the various crowns of integration.

The results on the derivative of G are lower than 1% (but without thermal loading).

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A

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