Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
1/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
Organization (S):
EDF/MMC
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.148
SSNV148 - Models of Weibull and Rice-Tracey
in 3D and discharge
Summary:
This test of nonlinear quasi-static mechanics makes it possible to validate the models of Weibull and Rice and Tracey
in 3D for nonmonotonous cases of mechanical loadings (cf.
POST_ELEM
[U4.61.04]).
At the temperature of 50°C, a cylindrical test-tube smoothes is first of all deformed up to 10%. After having it
slightly discharged, one maintains constant the level of deformation reaches while decreasing way
homogeneous the temperature of the test-tube until 150°C.A this new temperature, one applies one
additional deformation to reach 15% on the whole. Probability of rupture per cleavage as well as the rate of
growth of the cavities of the test-tube are calculated for the whole of the way of loading.
The modeling of the test-tube is carried out with elements 3D (HEXA20, PENTA15).
Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
2/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
1
Problem of reference
1.1 Geometry
One considers a half - cylindrical test-tube smooth.
1.2
Properties of material
One adopts an elastoplastic law of behavior of Von Mises with linear isotropic work hardening
“VMIS_ISOT_LINE”
. The deformations used in the relation of behavior are them
linearized deformations.
E
E
T
Y
The Young modulus E, the tangent module E
T
as well as the Poisson's ratio do not depend on
temperature. One takes: E=200 GPa, E
T
= 2000 MPa and
= 0,3.
The evolution of the elastic limit with the temperature is given in the following table:
Temperature [°C]
150
100
50
Y
[MPa]
750.700.650
Lastly, thermal dilation is neglected (thermal expansion factor taken equal to 0).
Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
3/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
1.3
Boundary conditions and loadings
While referring to the figure [§1.1] boundary conditions are as follows:
·
on surface SSUP BC (Y=L
0
) displacement
L
imposed according to direction OY,
·
on surface SINF OA (Y=0) displacements locked according to direction OY,
·
locked displacements of A following X and Z,
·
locked displacements of B following Z.
Evolution temporal of the temperature (presumedly homogeneous in the test-tube) and of lengthening
L
are deferred in the following table:
Time
[S]
10 20 30 40
Temperature [°C]
50
50
150
150
Displacement
0
L
L
-
[mm]
20,35 20,30 20,30 32,525
1.4 Conditions
initial
Null stresses and deformations.
2
Reference solutions
2.1
Method of calculation
In simple traction and with the assumption of the small deformations, the tensile stress
)
(U
like
the plastic multiplier
)
(U
p&
at the moment
U
are given in the case considered by:
·
if
p
T
U
1
0
:
°
-
+
=
=
-
=
E
C
L
T
L
U
p
L
L
U
L
E
U
Y
p
)
50
(
1
)
(
0
)
(
)
(
)
(
0
1
0
0
&
·
if
10
1
U
T
p
:
0
0
0
)
(
1
)
(
)
50
(
)
(
)
(
L
U
L
E
E
U
p
C
E
E
E
L
L
U
L
E
U
T
Y
T
T
&
&
-
=
°
-
-
+
-
=
,
·
if
20
10
U
:
0
)
(
)
(
)
10
(
)
10
(
)
(
0
=
-
=
-
=
=
U
p
L
U
L
U
L
E
U
U
&
,
·
if
30
20
U
:
0
)
(
)
20
(
)
(
=
=
=
U
p
U
U
&
,
·
if
40
30
U
:
0
0
)
(
1
)
(
)
20
(
)
(
)
20
(
)
(
L
U
L
E
E
U
p
L
U
L
U
L
E
U
U
T
T
&
&
-
=
=
-
+
=
=
Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
4/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
2.2 Weibull
Probability of cumulated rupture
F
P
at the moment
T
is given by (cf.
POST_ELEM
[U4.61.04]):
-
-
=
FD
m
U
I
T
U
T
F
V
FD
U
U
T
P
p
0
)
))
(
(
)
(
(
max
exp
1
)
(
.
The summation relates to volumes of matter
I
V
plasticized (as from the moment
p
T
),
)
(U
I
and
)
(U
indicating the maximum main stress and the temperature in each one of these volumes with
various moments
)
(U
. Here, volume
0
V
of reference is equal to (50
µ
m)
3
. The module of Weibull
m
is
equal to 24 while the stress of cleavage
U
depends on the temperature according to:
Temperature [°C]
50
100
150
U
[MPa]
2800 2700 2600
The probability of cumulated rupture varies according to (
)
(
),
(
T
L
T
) according to:
-
-
=
0
))
(
(
)
(
max
exp
1
)
(
V
V
U
U
T
P
m
U
T
U
T
F
p
.
2.3
Rice and Tracey
In simple traction, the Napierian logarithm of the growth rate of the cavities at the moment
T
is given by
(cf.
POST_ELEM
[U4.61.04]):
×
×
=
T
U
p
R
T
R
Log
0
0
)
(
)
5
,
0
exp (
283
,
0
)
(
&
2.4
Sizes and results of reference
F
P
and
0
R
R
for the couples (temperature, displacements = (L
0
)) following: ( 50,0°C, 20,35 mm);
( 50,0°C, 20,30 mm); ( 150,0°C, 20,30 mm) and ( 150,0°C, 32,53 mm).
2.5
Uncertainties on the solution
Analytical solution.
Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
5/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
3 Modeling
With
3.1
Characteristics of the mesh
A number of nodes: 1137
A number of meshs and types: 64 (PENTA15), 192 (HEXA20)
3.2 Functionalities
tested
Controls
DEFI_MATERIAU
WEIBULL_FO
M
VOLU_REFE
SIGM_REFE
SIGM_CONV
STAT_NON_LINE
COMP_INCR
RELATION
DEFORMATION
VMIS_ISOT_LINE
SMALL
CALC_ELEM
OPTION
EPSG_ELGA_DEPL
POST_ELEM
WEIBULL
COEF_MULT
OPTION
SIGM_ELMOY
POST_ELEM
RICE_TRACEY
OPTION
SIGM_ELMOY
Code_Aster
®
Version
6.0
Titrate:
SSNV148 - Model of Weibull in mechanical discharge
Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. BARBER
Key
:
V6.04.148-A
Page:
6/6
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A
3.3
Sizes tested and results
Reference
Code_Aster
Reference
Code_Aster
T [°C]
L
0
[mm]
F
P
F
P
% diff.
0
R
R
0
R
R
% diff.
50 20,35
0,01465
0,01481
1,1
1,0447
1,0458
0,1
50 20,30
0,01465
0,01481
1,1
1,0447
1,0458
0,1
150 20,30
0,01465
0,01481
1,1
1,0447
1,0458
0,1
150 32,525 1,0.1,0.0,0
1,068
1,0701
0,2
3.4 Parameters
of execution
Version: 6.2
Machine: SGI - ORIGIN 20 00 - R12000
Overall dimension memory: 64 Mo
Time CPU To use: 201,81s
4
Summary of the results
The results obtained by Code_Aster are very close to the analytical solutions of reference.