Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
1/6

Organization (S): EDF/MMC

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.148

SSNV148 - Modèles de 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).

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
2/6

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.

And
Y
E


The Young modulus E, the tangent module And as well as the Poisson's ratio do not depend on
temperature. One takes: E=200 GPa, And = 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 dilation coefficient taken equal to 0).
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
3/6

1.3
Boundary conditions and loadings

While referring to the figure [§1.1] boundary conditions are as follows:

· on surface SSUP BC (Y=L0) displacement L imposed following direction OY,
· on surface SINF OA (Y=0) displacements blocked according to direction OY,
· blocked displacements of A following X and Z,
· blocked 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 L - 0
L [mm]
20,35 20,30 20,30 32,525

1.4 Conditions
initial

Null constraints 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 p& (U) at the moment U is given in the case considered by:

L U
() - L0
p

Y (- °
50 C)
· if
p
0 U T
U
() = E
p & U
() = 0l T
() = L
1
0 1 +
1:

L0

E

L (U) - L E - E

E
T
H T (
& U
· if T p U 10
0
)
(U) = E
+
(- 50 C
°) p




& (U) = 1

1
:
T
Y
,

0
L
E


E 0
L
L (U =10) - L (U)
· if 10 U 20: (U) = (U =10) - E
p& (U) = 0




,

L0

· if 20 U 30: (U) = (U =
)
20 p& (U) = 0,
L (U) - L (U = 20)

And L (&u)
· if 30 U 40: (U) = (U = 20) + E
p



& (U) = 1
T
-


L

0


E L0
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
4/6

2.2 Weibull

The probability of cumulated rupture PF at the moment T is given by (cf POST_ELEM [U4.61.04]):


m



U
()

I
FD
PF T () = 1

exp - max (
)
.
p





(U
())
V
FD T C
U

0

The summation relates to volumes of matter I
V plasticized (as from the moment T p), (U)
I
and (U)
indicating the maximum principal constraint and the temperature in each one of these volumes with
various moments (U). Here, volume 0
V of reference is equal to (50 µm) 3. The module of Weibull m is
equal to 24 while the constraint 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:


m


(U) V
P (T)
F
= 1 - exp - max

.

p




T C ((U))
U

0
V




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]):

R T ()
T
Log
=,
0.283× exp (
)
5
,
0
×




p & U () of
R0
0

2.4
Sizes and results of reference

R
PF and
for the couples (temperature, displacements = (l-l0)) following: (­ 50,0°C, 20,35 mm);
0
R
(­ 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.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
5/6

3 Modeling
With

3.1
Characteristics of the grid


A number of nodes: 1137
A number of meshs and types: 64 (PENTA15), 192 (HEXA20)

3.2 Functionalities
tested

Commands




DEFI_MATERIAU
WEIBULL_FO
M
VOLU_REFE
SIGM_REFE
SIGM_CONV
STAT_NON_LINE
COMP_INCR
RELATION
VMIS_ISOT_LINE
DEFORMATION
PETIT
CALC_ELEM
OPTION
EPSG_ELGA_DEPL
POST_ELEM
WEIBULL
COEF_MULT
OPTION
SIGM_ELMOY
POST_ELEM
RICE_TRACEY
OPTION
SIGM_ELMOY
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

Code_Aster ®
Version
6.0
Titrate:
SSNV148 - Modèle de Weibull in mechanical discharge

Date:
19/08/02
Author (S):
R. MASSON, W. LEFEVRE, G. Key BARBER
:
V6.04.148-A Page:
6/6

3.3
Sizes tested and results


Reference
Code_Aster
Reference
Code_Aster
T [°C]
lL0 [mm]
P
% diff.
F
PF
% diff.
R
R


0
R
0
R
­ 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

Obstruction 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.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-26/02/009/A

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