background image
Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
1/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA















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



HPLP101 - Plate fissured in thermoelasticity
(plane stresses)




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 stresses) in isotropic linear thermoelasticity.

This test includes/understands the first modeling in plane stresses in which are calculated:
·
the rate of refund of energy
G
(conventional calculation by the method théta),
·
coefficients of intensity of stresses
K
I
and
K
II
.

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
K
II
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.
background image
Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
2/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/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:
(
)


X
y
xy
X
y
xy
ref.
X
y
X
X
y
y
xy
X
y
S
S
S
S
S
T T
S
E
S
E
S
E
E
S
G




=








+






·
-
=
=
= -
= -
=
=
=
11
12
12
22
66
11
22
11
22
12
66
11
22
0
0
0
0
0
1
1
1
One limits oneself to isotropic material, as well from the thermal point of view as mechanical:
E
E
X
y
=
= 2. 10
5
MPa
X
y
=
= 0.3
X
y
=
= 1.2 10
­ 5
°C
­ 1
X
y
=
= 54. W/m °C
background image
Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
3/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/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.)
·
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
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Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
4/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Complex potential [bib1].
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
F
II
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.
background image
Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
5/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
2.5
Reference solution for the derivative of G (modeling B)
While varying the Young modulus and the Fy loading, one notes that:
2
Y
F
G
=
with
3
310
.
5
-
=
that is to say
Y
Y
F
F
G
2
=
E
G
=
with
3
310
.
5
-
=
that is to say
E
G
E
G
-
=

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
Values of the higher and lower radii, to specify in the control
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 mesh
Half-mesh; mesh radiating at the right end of the fissure.
3831 nodes,
1516 elements,
884
TRI6
,
632
QUA8
.

3.3 Functionalities
tested
Controls
THERMAL AFFE_MODELE
PLAN
ALL
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
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Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
6/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
4
Results of modeling A
4.1 Values
tested

Identification Reference
Aster %
difference
K
II
, crown n°1
2,2347E+7 2,2814E+7
2,09
K
II
, crown n°2
2,2347E+7 2,2813E+7
2,08
K
II
, crown n°3
2,2347E+7 2,2814E+7
2,09
K
II
, crown n°4
2,2347E+7 2,2814E+7
2,09
K
II
, crown n°5
2,2347E+7 2,2817E+7
2,10
K
II
, 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
K
I
= 0, but it does not check it a posteriori. With the sights of
deformations resulting from ASTER, the coefficient
K
I
is different from zero, but there remains very weak by report/ratio
with
K
II
(the fissure slips more than it does not open).
With regard to the rate of refund of energy
G
, if we suppose that
K
I
= 0, we fire
value of reference starting from the formula of IRWIN in plane stresses:
(
)
2
*
/
1
II
ref.
K
E
G
=

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Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
7/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
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
Values of the higher and lower radii, to specify in the control
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 mesh
Even mesh that for modeling a:
Half-mesh; mesh 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
Controls
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
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Code_Aster
®
Version
7.2
Titrate:
HPLP101 - Plate fissured in thermoelasticity (forced plane)
Date:
11/05/04
Author (S):
X. DESROCHES
Key
:
V7.02.101-B
Page:
8/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HT-66/04/005/A
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 stresses and 4% for the rate of refund of energy. One checks
the invariance of the results compared to the various crowns of integration.
Results on the derivative of
G
are lower than 1% (but without thermal loading).