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Version
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Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D

Date:
15/12/05
Author (S):
Y. WADIER, Key Mr. BONNAMY
:
V6.03.131-A Page:
1/6

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

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
Document: V6.03.131

SSNP131 ­ Identification of the energy criterion Gp
in 2D

Summary

This test of nonlinear quasi-static mechanics makes it possible to present the calculation of the Gp parameter resulting from
the energy approach of the elastoplastic rupture and the identification of the values criticize correspondent with
values of experimental tenacity given. He requires to represent the fissure by a notch and to calculate
elastic energy on the zone corresponding to the path of propagation of the notch.

Modeling is carried out with quadratic elements 2D, in plane deformation.

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
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Code_Aster ®
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Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D

Date:
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Y. WADIER, Key Mr. BONNAMY
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1
Problem of reference

1.1 Geometry

29mm

U

y imposed
has

·
H = 60mm

W=50mm
Center

Ux = 0
symmetry

L = 62.5mm

A geometry of test-tube CT25 is considered where the length of the ligament: = 27.5 mm have
(/W has = 0.55). Test-tube CT25 is modelled in 2D plane deformations. For reason of symmetry,
a half of this one is represented.

1.2
Material properties

Young modulus:
Temperature (°C)
E (Mpa)
23 220200
100 214100
150 206500
300 205000

Poisson's ratio: = 0.3

The traction diagram used is interpolated for the temperature of calculation starting from the values
presented in the following table:

Material Dated: True Stress - True Strain
T = 23°C
T = 100°C
Strain
Stress [MPa]
Strain
Stress [MPa]
0,00000E+00 0,00000E+00 0,00000E+00
0,00000E+00
4,34654E-03 8,55922E+02 3,43968E-03
7,40663E+02
6,01497E-03 9,10460E+02 4,62837E-03
8,42149E+02
7,86211E-03 9,31797E+02 6,07988E-03
8,76312E+02
1,07579E-02 9,49055E+02 7,65463E-03
8,95206E+02
1,42214E-02 9,61578E+02 1,04175E-02
9,11072E+02
1,77918E-02 9,71929E+02 1,41780E-02
9,25022E+02
2,21851E-02 9,84491E+02 1,75432E-02
9,35214E+02
2,82764E-02 1,00147E+03 2,19425E-02
9,45695E+02
3,58111E-02 1,01932E+03 2,74167E-02
9,60732E+02
4,37307E-02 1,03519E+03 3,38670E-02
9,75804E+02
5,14523E-02 1,04865E+03 4,02058E-02
9,88245E+02
5,89828E-02 1,06076E+03 4,66164E-02
1,00014E+03
6,68527E-02 1,07021E+03 5,29036E-02
1,01000E+03
5,82359E-02
1,01757E+03
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

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SSNP131 - Identification of the energy criterion Gp in 2D

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1.3
Boundary conditions and loadings

The loading is of displacement type imposed in a point located at the center of the pin which is
modelled by four indeformable angular sectors. The temperature is imposed constant on
the whole of the test-tube (T = 35°C). Half of the test-tube being modelled, a condition of
symmetry is applied to the ligament located behind the notch.

2
Reference solution

2.1
Method of calculation used for the reference solution

The notch is made of a half-circle of radius R located in bottom of fissure and of a fine zone
representing the beginning of the ligament of the defect which will be represented by a zone of grid of the type
“chips”. One determines at every moment the evolution of the quantity Gp (L
) defined by:

Gp (L
) = [
2 Welas (L
)]/L


where Welas (L
) is the elastic energy calculated on the formed zone of “chips” located behind
melts of notch and length L
. One must then calculate the maximum of this quantity compared to
L
, that one calls “Gp”.

Gp =
{
Max Gp (L
)}
L


The moment criticizes where the propagation of the defect will start is then that where Gp reached the breaking value
Gp crit”.

2.2 References
bibliographical

[1]
WADIER Y.
: “
Brief presentation of the energy approach of the rupture
elastoplastic applied to the rupture by cleavage “, Note EDF R & D HT-64/03/001/A, January
2003.
[2]
WADIER Y., LORENTZ E.: “Breaking process in the presence of plasticity:
modeling of the fissure by a notch “. C.R.A.S.T. 332, IIb series, 2004.
[3]
LORENTZ E., WADIER Y.: “Energy approach of the elastoplastic rupture applied
with the modeling of the propagation of a notch “. REEF, Vol 13, n°5-6-7, pp. 583-592,
2004.
Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

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SSNP131 - Identification of the energy criterion Gp in 2D

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3 Modeling
With

3.1
Characteristics of modeling





The bottom of fissure is modelled by a notch of radius 100 microns. A zone of 2 mm
length is arranged behind this one in layers of 20 microns thickness elements (called
also “chips”).

3.2
Characteristics of the grid

A number of nodes: 8260

A number of meshs and types: 1864 SORTED 6, 1420 QUAD 8

3.3 Functionalities
tested

Commands



STAT_NON_LINE



CALC_THETA THETA_2D



CALC_G_THETA_T OPTION
CALC_G


POST_ELEM ENER_ELAS



CREA_TABLE




Handbook of Validation
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HT-66/05/005/A

Code_Aster ®
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Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D

Date:
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Y. WADIER, Key Mr. BONNAMY
:
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4
Results of modeling A

4.1 Values
tested

Identification Reference
Aster %
difference
GP crit. probability rupture 5%
- 0.673449 -
GP crit. probability rupture 50%
- 0.800954 -
GP crit. probability rupture 95%
- 0.916242 -

4.2 Notice

The results observed to ensure itself of the not-regression of the code are the breaking values of
energy parameter corresponding to the experimental probabilities of rupture to 5, 50 and 95%
associated the values of following tenacities: Kj (5%) = 27,2 MPam; Kj (50%) = 34 MPam;
Kj (95%) = 40 MPam

With these values correspond of the critical loadings identified by calculating the quantity G by
2
-
Théta method which is connected to tenacity via the formula of Irwin:
1
2
J =
K. Crowns
E
chosen for the Théta field are: [0.25 mm; 0.5 mm], [0.5 mm; 1.0 mm], [1.0 mm; 2.0 mm],
[2.0 mm; 5.0 mm], [5.0 mm; 10.0 mm]. With these critical loadings the values correspond
critical of the Gp parameter. One associates to them the breaking values of KGp deduced from Gp from
E
the formula of Irwin: K
=
Gp
Gp
.
2
1

A law of probability of the type of the model of Beremin is employed to define graphs of
probability according to Kj (see graphic below) it is written:

m
K
R
P (KGp)




Gp

= 1


exp -

K

Gp


0


where m = 22.673, and K
0
Gp identified such as:

R
P (K
) 05
.
0
min =
Gp
,
R
P (KGpmoy) = 0.5,
R
P (K
) 95
.
0
max =
Gp
.

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A

Code_Aster ®
Version
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Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D

Date:
15/12/05
Author (S):
Y. WADIER, Key Mr. BONNAMY
:
V6.03.131-A Page:
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1
E 0,8
ur
Pt
U 0,6
E
R
ilit
B 0,4
R
oba 0,2
P
0
0
20
40
60
80
100
Kj (MPa.mm1/2)


Various results are displayed in the file message:

· display for each moment of the transient considered and each crown of the field
theta well informed of the values G () and Kj deduced by the formula from Irwin.
· display for each moment of the transient considered of the Gp parameter calculated in function
distance to the bottom of notch.
· display for each moment of the transient considered of noted the maximum Gp parameter
and of the distance to the bottom of associated notch (L
max).
· display of the values of identifications for each tenacity and each field theta (urgent
interpolated on the transient, critical Gp, KGp deduced by Irwin),
· display for each moment of the transient given of the L
max, Gp max, KGpmax (deduced by
Irwin), Tfe (temperature in bottom of notch) and of the probability of rupture according to the evoked law
previously.

Handbook of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A