Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
1/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
Organization (S):
EDF/SINETICS
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
Document: V7.02.100
HPLP100 - Calculation of the rate of refund of energy
of a plate fissured in thermo elasticity
Summary
It is about a test in thermo elasticity for a two-dimensional problem. A rectangular plate is considered
fissured and one places oneself on the assumption of the plane deformations.
In modeling A, the rate of refund of energy is calculated in postprocessing by two methods
different:
·
conventional calculation by the method theta,
·
calculation by the formula of IRWIN starting from the coefficients of intensity of stresses KI and KII.
These two calculations are carried out on 4 different crowns of integration. Their interest is to compare them
values of G and G (IRWIN) compared to the reference solution and to test the invariance of calculations by
report/ratio with the various crowns of integration.
As for modeling B, it is about a functional and data-processing test of calculation of derived from the rate from
conventional restitution of energy compared to a variation of field (controlled by a function theta
particular). One uses loadings which for the majority are analytical and which intervene only in post-
processing of the calculation of mechanics.
The architecture of the test makes it possible to simulate a finished difference, one can thus distinguish in term from
data-processing not-regression, possible external amendments impacting the direct problem and/or its
derived.
From a more anecdotic point of view, one can take as a starting point the the sequences of the control
CREA_CHAMP
used in this modeling to build analytical fields and to relocate a mesh
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
2/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
1
Problem of reference
1.1 Geometry
It is about a fissured rectangular plate (one represents only the quarter of the structure):
H
I
With
has
C
Y
O
X
U
v
Appear 1.1-a: Plates rectangular fissured
Dimensions of this plate are as follows:
Half-height of the plate:
H = 200.0 mm
Half-width of the plate:
I = 100.0 mm
Half-length of the fissure: = 50.0 mm have
1.2
Properties of material
Thermal properties:
CP = 0.
= 1.0 W/m°C
Mechanical properties:
E = 200000 MPa
= 0.3
= 5.10
6
/°C
We are on the assumption of the plane deformations
1.3
Boundary conditions and loadings
·
Temperature imposed in X = 0. : T = - 100.0°C
·
Temperature imposed in X = 100: T = + 100.0°C
·
Displacement for A < X < I, Y = 0. : U = 0.
·
Displacement for 0 < X < I, Y = H: U = 0.
·
Not fix for X = 0., Y = H: U = v = 0.
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
3/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The reference solution results from WILSON and YU [bib1]:
K
E T F has F
has
K
I
I
= -
=
=
0
1
0154
92 0291
.
.
in mm
E in NR/mm
2
In plane deformations, the formula of IRWIN gives:
(
) (
)
G
E
K
K
I
II
= -
+
1
2
2
2
that is to say numerically:
G
=
-
38535 10
1
.
2.2
Results of reference
The results of reference are those resulting from the reference solution from WILSON and YU [bib1]:
G
K
K
I
II
=
=
=
-
38535 10
92 0291
0
1
.
.
.
2.3 References
bibliographical
[1]
The Uses off J-Integrals in thermal stress ace problems - International Newspaper off Fracture
(1979) WILSON and YU.
[2]
Qualification complementary to the INCA codes/MAYA in thermo linear elasticity. Note
technique DRE/STRE/LMA 84/598
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
4/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
3 Modeling
With
3.1
Characteristics of modeling
There are 4 crowns defined by the control
CALC_THETA
:
Crown 1:
Rinf = 10.
Rsup = 40.
Crown 2:
Rinf = 15.
Rsup = 45.
Crown 3:
Rinf = 5.
Rsup = 47.
Crown 4:
Rinf = 3.
Rsup = 48.
The bottom of fissure is defined by
DEFI_FOND_FISS
, and for each crown one carries out:
·
a conventional calculation of G (option
CALC_G
of
CALC_G_THETA_T
),
·
a calculation of G by the formula of IRWIN starting from the coefficients of intensity of stresses
K
I
and
K
II
(option
CALC_K_G
of
CALC_G_THETA_T
).
3.2
Characteristics of the mesh
A number of nodes: 853
A number of meshs and types: 359 meshs TRIA6 and 27 meshs QUAD8
3.3 Functionalities
tested
Controls
AFFE_MODELE
THERMICS
PLAN
ALL
AFFE_MODELE
MECHANICS
D_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
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
5/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
4
Results of modeling A
4.1 Values
tested
The values tested are those of
G
obtained by the conventional method and that of
G_IRWIN
obtained
by the formula of IRWIN starting from the coefficients of intensity of stresses:
Identification Reference
Aster %
difference
Crown 1
G
3.8535 10
1
3.6036 10
1
6.62
Crown 1
G_IRWIN
3.8535 10
1
3.5964 10
1
6.67
Crown 2
G
3.8535 10
1
3.6014 10
1
6.63
Crown 2
G_IRWIN
3.8535 10
1
3.5958 10
1
6.68
Crown 3
G
3.8535 10
1
3.6018 10
1
6.65
Crown 3
G_IRWIN
3.8535 10
1
3.5602 10
1
6.68
Crown 4
G
3.8535 10
1
3.6021 10
1
6.62
Crown 4
G_IRWIN
3.8535 10
1
3.5962 10
1
6.67
4.2 Remarks
The numerical values are stable compared to the various crowns of integration and almost
identical for the two methods of calculation. Nevertheless the variation with the values of reference is of
the command from 6 to 7%, which seems high.
4.3 Parameters
of execution
Version: 6.01.19
Machine: SGI CLASTER
System IRIX64 6.5
Overall dimension memory: 8 MW
Time CPU To use: 4.22 seconds
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
6/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
5 Modeling
B
5.1
Characteristics of modeling
It is about a functional and data-processing test of calculation of derived from the rate of refund of energy
conventional compared to a variation of field. This variation is controlled by a function theta
particular, noted
S
, generated via
CALC_THETA
with the key word factor
THETA_BANDE
.
It is reminded the meeting that this two-dimensional function theta decrease cubiquement of the value modulates (word
key
MODULATE
) with the zero value, between X-coordinates X
1
= - 50 and X
2
= - 30
(key words
R_INF
and
R_SUP
) of
points delimiting its vertical support. It is null everywhere else (cf [U4.82.03] §3.10).
The crown delimiting the area of calculation around the bottom of fissure (at the point C materializing the origin
reference mark) is modelized by the function theta fissures conventional, noted
F
, with R
inf
= 10 and R
sup
= 45.
Appear 5.1-a: Derived from G (
F
)
compared to a variation of field controlled by
S
After having built the models
Mo
and
moth
in modeling `
D_PLAN'
and the field theta sensitivity
S
(
thetas)
, one affects thermal loadings of temperatures type imposed on the edges
right and left of the part, for then, to carry out thermal calculation itself. This last
use
thetas
, provided via the key word
SENSITIVITY
, to calculate the field of temperature and its
Lagrangian derivative.
Before carrying out elastic thermo calculation one affects the mechanical loadings. The activation of
the operator
MECA_STATIQUE
having been made with the key word
SENSITIVITY
, one enriches the result by
the Lagrangian derivative of displacements.
One adds thereafter analytical loadings which intervene only in postprocessing of
calculation of mechanics. They make it possible to calculate two values of G, one with a force of gravity,
an internal force and a field of initial deformation (G1), the other with a stress field
initial (G2). The first calculation is carried out in small deformations, the second in deformations of
Green-Lagrange. The crown of calculation is defined by a call to
CALC_THETA
with the option
THETA_2D
.
These tests, purely data-processing and functional, have only little interest from a point of view
mechanics because the majority of the loadings do not check an equation with balance.
X
2
X
1
= - 50
Field
S
sensitivity
Field
F
fissure
X
2
= - 20
X
y
R
inf
= 10
R
sup
= 45
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
7/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
Thereafter one reiterates this series of calculations with a mesh shifted according to the law
S
(with
S
=
-
10
3
) which, at a point
P
area of reference makes correspond a point
M,
()
M
P
P
+
S
S
By thus simulating a finished difference, one can distinguish in data-processing term of not-regression them
possible external amendments impacting the direct problem and/or its derivative. One tests in the last
the precision of the values arises of
()
G
F
and of
()
()
G
G
S
F
S
F
S
=
0
obtained, before and
after the variation of field.
5.2
Characteristics of the mesh
A number of nodes: 853
A number of meshs and types: 359 meshs TRIA6 and 27 meshs QUAD8
5.3 Functionalities
tested
Controls
CREA_CHAMP
AFFE
ELNO_NEUT_F
ADZE
NOEU_DEPL_R
ADZE
ELNO_SIEF_R
DISC
ELNO_GEOM_R
EVAL
ELNO_NEUT_R
EXTR
NOEU_GEOM_R
EXTR
NOEU_DEPL_R
MODI_MAILLAGE
DEFORM
TRAN
AFFE_MODELE
THERMICS
PLAN
ALL
AFFE_MODELE
MECHANICS
D_PLAN
ALL
THER_LINEAIRE
SENSITIVITY
MECA_STATIQUE
SENSITIVITY
CALC_THETA
THETA_BANDE
CALC_THETA
THETA_2D
CALC_G_THETA_T
OPTION
CALC_G
CALC_G_THETA_T
OPTION
CALC_DG
CHARGE
CALC_G_THETA_T
OPTION
CALC_DG
ETAT_INIT
CALC_G_THETA_T
OPTION
CALC_DG
GREEN
Code_Aster
®
Version
6
Titrate:
HPLP100 - Calculation of the rate of refund of the energy of a fissured plate
Date:
20/08/02
Author (S):
O. BOITEAU
Key
:
V7.02.100-B
Page:
8/8
Manual of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A
6
Results of modeling B
6.1 Values
tested
One tests the data-processing not-regression of the values of G (on the initial mesh and that shifted) and
of its derivative (on the initial mesh) compared to the V6.0.19 versions of platforms SGI and SUN.
relative tolerance is thus very severe (10
7
%).
Identification
Aster Tolerance
Initial mesh
G with
GRAVITY,
FORCE_INTERN, EPSI_INI
1.748581514 10
2
10
9
DG with
GRAVITY,
FORCE_INTERN, EPSI_INI
4.422828409 10
1
10
9
G with
SIGMA_INI + GREEN
3.672692719 10
1
10
9
DG with
SIGMA_INI + GREEN
1.102553167 10
2
10
9
Shifted mesh
G with
GRAVITY,
FORCE_INTERN, EPSI_INI
1.748585937 10
2
10
9
G with
SIGMA_INI + GREEN
3.672692982 10
1
10
9
6.2 Remarks
These numerical values vary if the parameters of the functions theta are modified because one uses
mechanical loadings in postprocessing of elastic thermo calculation. They thus do not respect
of equation to balance. By using only loadings intervening during all the process
these instabilities are reduced considerably, while remaining about the percent.
6.3 Parameters
of execution
Version: 6.0.19
Machine: SGI CLASTER
System IRIX64 6.5
Overall dimension memory: 8 MW
Time CPU To use: 9.6 seconds
Machine: SUN CLI75AS
System SUNOS 5.6
Overall dimension memory:8 MW
Time CPU To use: 25.6 seconds
7
Summaries of the results
At the time of the first modeling, the variation with the values of reference is 6 to 7%. Validation
independent of the breaking process batch should bring brief replies on the validity
G in thermo elasticity.
The second modeling carrying out of the tests of functional and data-processing not-regression of
calculation of derived from G compared to a variation of field, its results must be
scrupulously respected, from where very severe criteria of tolerance.