Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
1/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
Organization (S):
EDF-R & D/AMA, AUSY France
Manual 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 parameter G
p
resulting from
the energy approach of the elastoplastic rupture and the identification of the values criticize agent with
values of experimental tenacity given. He requires to represent the fissure by a notch and to calculate
elastic energy on the area corresponding to the path of propagation of the notch.
Modeling is carried out with quadratic elements 2D, in plane deformation.
Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
2/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
1
Problem of reference
1.1 Geometry
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 modelized in 2D plane deformations. For reason of symmetry,
a half of the aforementioned 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
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
T = 23°C
T = 100°C
·
has
L = 62.5mm
H = 60mm
29mm
W=50mm
Center
symmetry
Ux = 0
U
y
imposed
Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
3/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
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
modelized 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 modelized, 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 area
representing the beginning of the ligament of the defect which will be represented by an area of mesh of the type
“chips”. The evolution of the quantity at every moment is determined
()
L
Gp
defined by:
()
()
[
]
L
L
W
L
Gp
elas
=
/
2
where
()
L
W
elas
is the elastic energy calculated on the formed area 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
”.
()
{
}
L
Gp
Max
Gp
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, Flight 13, n°5-6-7, pp. 583-592,
2004.
Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
4/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
3 Modeling
With
3.1
Characteristics of modeling
The bottom of fissure is modelized by a notch of radius 100 microns. An area of 2 mm
length is arranged behind the aforementioned in layers of 20 microns thickness elements (called
also “chips”).
3.2
Characteristics of the mesh
A number of nodes: 8260
A number of meshs and types: 1864 SORTED 6, 1420 QUAD 8
3.3 Functionalities
tested
Controls
STAT_NON_LINE
CALC_THETA THETA_2D
CALC_G_THETA_T OPTION
CALC_G
POST_ELEM ENER_ELAS
CREA_TABLE
Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
5/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
4
Results of modeling A
4.1 Values
tested
Identification Reference
Aster %
difference
G
P
crit. probability rupture 5%
- 0.673449 -
G
P
crit. probability rupture 50%
- 0.800954 -
G
P
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 MPa
m; Kj (50%) = 34 MPam;
Kj (95%) = 40 MPa
m
With these values correspond of the critical loadings identified by calculating the quantity G by
Théta method which is connected to tenacity via the formula of Irwin:
2
2
1
K
E
J
-
=
. Crowns
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 parameter
Gp
. One associates the values to them criticize
Gp
K
deduced from
Gp
from
the formula of Irwin:
2
1
-
=
Gp
E
K
Gp
.
A law of probability of the type of the model of Beremin is employed to define graphs of
probability according to K
J
(see graphic below) she is written:
()
-
-
=
m
Gp
Gp
Gp
R
K
K
K
P
0
exp
1
where m = 22.673, and
0
Gp
K
identified such as:
(
)
05
.
0
min
=
Gp
R
K
P
,
(
)
5
.
0
moy
=
Gp
R
K
P
,
(
)
95
.
0
max
=
Gp
R
K
P
.
Code_Aster
®
Version
8.2
Titrate:
SSNP131 - Identification of the energy criterion Gp in 2D
Date:
15/12/05
Author (S):
Y. WADIER, Mr. BONNAMY
Key
:
V6.03.131-A
Page:
6/6
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/05/005/A
0
0,2
0,4
0,6
0,8
1
0
20
40
60
80
100
Kj (MPa.mm1/2)
P
R
oba
B
ilit
E
R
U
Pt
ur
E
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 parameter
Gp
calculated in function
distance to the bottom of notch.
· display for each moment of the transient considered of the parameter
Gp
noted maximum
and of the distance to the bottom of associated notch
(
)
max
L
.
· display of the values of identifications for each tenacity and each field theta (urgent
interpolated on the transient,
Gp
critical,
Gp
K
deduced by Irwin),
· display for each moment of the transient given of
max
L
,
max
Gp
,
max
Gp
K
(deduced by
Irwin), Tfe (temperature in bottom of notch) and of the probability of rupture according to the evoked law
previously.