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Titrate:
HPLP100 - Calcul 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

Organization (S): EDF/SINETICS

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
Document: V7.02.100

HPLP100 - Calcul 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:

· traditional calculation by the method theta,
· calculation by the formula of IRWIN starting from the coefficients of intensity of constraints 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
traditional 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 modifications 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 command CREA_CHAMP
used in this modeling to build analytical fields and to relocate a grid
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
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Titrate:
HPLP100 - Calcul 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

1
Problem of reference

1.1 Geometry

It is about a fissured rectangular plate (one represents only the quarter of the structure):

Y
I
v
H
U
With
O
C
X
has

Appear rectangular 1.1-a: Plaque 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

= 5.10­6/°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.
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
6
Titrate:
HPLP100 - Calcul 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

2
Reference solution

2.1
Method of calculation used for the reference solution

The reference solution results from WILSON and YU [bib1]:

E T
K =
0 F has F
I
= 0154
.
1

has in mm

E in NR/mm2
KI = 92 0291
.

(1 - 2)
In plane deformations, the formula of IRWIN gives: G =
(K2 + K2
I
II)
E

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 =
-
38535 10 1
.

KI = 92 0291
.

KII = 0.

2.3 References
bibliographical

[1]
The Use off J-Integrals in thermal stress ace problems - International Journal 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
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
6
Titrate:
HPLP100 - Calcul 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

3 Modeling
With

3.1
Characteristics of modeling

There are 4 crowns defined by command 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 traditional 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 constraints KI
and KII (option CALC_K_G of CALC_G_THETA_T).

3.2
Characteristics of the grid

A number of nodes: 853
A number of meshs and types: 359 meshs TRIA6 and 27 meshs QUAD8

3.3 Functionalities
tested

Commands



AFFE_MODELE
THERMIQUE
PLAN
TOUT

AFFE_MODELE
MECANIQUE
D_PLAN
TOUT

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
HI-23/02/017/A

Code_Aster ®
Version
6
Titrate:
HPLP100 - Calcul 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

4
Results of modeling A

4.1 Values
tested

The values tested are those of G obtained by the traditional method and that of G_IRWIN obtained
by the formula of IRWIN starting from the coefficients of intensity of constraints:

Identification Reference
Aster %
difference
Crown 1 G
3.8535 10­1
3.6036 10­1
6.62
Crown 1 G_IRWIN
3.8535 10­1
3.5964 10­1
6.67
Crown 2 G
3.8535 10­1
3.6014 10­1
6.63
Crown 2 G_IRWIN
3.8535 10­1
3.5958 10­1
6.68
Crown 3 G
3.8535 10­1
3.6018 10­1
6.65
Crown 3 G_IRWIN
3.8535 10­1
3.5602 10­1
6.68
Crown 4 G
3.8535 10­1
3.6021 10­1
6.62
Crown 4 G_IRWIN
3.8535 10­1
3.5962 10­1
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
Obstruction memory: 8 MW
Time CPU To use: 4.22 seconds

Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
6
Titrate:
HPLP100 - Calcul 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

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
traditional 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 pointed out that this two-dimensional function theta decrease cubiquement of the value modulates (word
key MODULE) with the zero value, between the X-coordinates x1= - 50 and x2 = - 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 zone of calculation around the bottom of fissure (at the point C materializing the origin
reference mark) is modelled by the function theta fissures traditional, noted F, with Rinf= 10 and Rsup= 45.

y

Field F

Field
fissure

S

sensitivity


X



x1= - 50
x2= - 20
X R
2 inf= 10
Rsup= 45
Appear 5.1-a: Dérivée of 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 key word SENSIBILITE, to calculate the field of temperature and its
Lagrangian derivative.
Before carrying out elastic thermo calculation one affects the mechanical loadings. The activation of
operator MECA_STATIQUE having been made with key word SENSIBILITE, 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.
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
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Titrate:
HPLP100 - Calcul 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

Thereafter one reiterates this series of calculations with a grid shifted according to the law S (with
S = -
10 3) which, at a point P of the area of reference makes correspond a point M,

MR. P + S S (P)

By thus simulating a finished difference, one can distinguish in data-processing term of not-regression them
possible external modifications impacting the direct problem and/or its derivative. One tests in the last
G
G

the precision of the values of G arises (F) and of
(F) (F)

obtained, before and
S
S
s=0
after the variation of field.

5.2
Characteristics of the grid

A number of nodes: 853
A number of meshs and types: 359 meshs TRIA6 and 27 meshs QUAD8

5.3 Functionalities
tested

Commands



CREA_CHAMP
AFFE
ELNO_NEUT_F

ASSE
NOEU_DEPL_R

ASSE
ELNO_SIEF_R

DISC
ELNO_GEOM_R

EVAL
ELNO_NEUT_R

EXTR
NOEU_GEOM_R

EXTR
NOEU_DEPL_R

MODI_MAILLAGE
DEFORME
TRAN

AFFE_MODELE
THERMIQUE
PLAN
TOUT

AFFE_MODELE
MECANIQUE
D_PLAN
TOUT

THER_LINEAIRE
SENSIBILITE

MECA_STATIQUE
SENSIBILITE

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


Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

Code_Aster ®
Version
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Titrate:
HPLP100 - Calcul 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

6
Results of modeling B

6.1 Values
tested

One tests the data-processing not-regression of the values of G (on the initial grid and that shifted) and
of its derivative (on the initial grid) compared to the V6.0.19 versions of platforms SGI and SUN.
relative tolerance is thus very severe (10­ 7%).

Identification
Aster Tolerance
Initial grid
G with PESANTEUR,
­ 1.748581514 102
10­9
FORCE_INTERN, EPSI_INI
DG with PESANTEUR,
­ 4.422828409 10­1
10­9
FORCE_INTERN, EPSI_INI
G with SIGMA_INI + GREEN
3.672692719 10­1
10­9
DG with SIGMA_INI + GREEN
1.102553167 10­2
10­9
Shifted grid
G with PESANTEUR,
­ 1.748585937 102
10­9
FORCE_INTERN, EPSI_INI
G with SIGMA_INI + GREEN
3.672692982 10­1
10­9

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
Obstruction memory: 8 MW
Time CPU To use: 9.6 seconds


Machine: SUN CLI75AS
System SUNOS 5.6
Obstruction 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.
Handbook of Validation
V7.02 booklet: Thermomechanical stationary linear of the plane systems
HI-23/02/017/A

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