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
Outline document