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Titrate:
Identification of the model of Weibull
Date:
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R. MASSON, W. LEFEVRE
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Organization (S): EDF/RNE/MTC
Handbook of Référence
R7.02 booklet: Breaking process
Document: R7.02.09
Identification of the model of Weibull
Summary
One tackles here the problem of the identification of the parameters of the model of WEIBULL on a sample of tests
representative of behavior with rupture of a fragile material (typically, ferritic steel with low
temperature). The method of regression linear and the method of the maximum of probability are both
adopted methods. One details of it the principle as well as the associated methods of resolution, resting in
two cases on an iterative process. Lastly, one shows their extension if one of the two parameters of it
model (the constraint of cleavage) depends on the temperature.
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Titrate:
Identification of the model of Weibull
Date:
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Contents
1 Introduction ............................................................................................................................................ 3
2 Recalls .................................................................................................................................................. 3
2.1 The model of WEIBULL .................................................................................................................. 3
2.2 Identification of the parameters .......................................................................................................... 3
3 Method of the linear regression .......................................................................................................... 4
3.1.1 Principle .................................................................................................................................. 4
3.1.2 Resolution .............................................................................................................................. 5
4 Method of the maximum of probability .............................................................................................. 6
4.1 Principle ............................................................................................................................................ 6
4.2 ........................................................................................................................................ Resolution 6
5 Dependence of the parameters with the temperature ................................................................................ 7
5.1 Linear regression .......................................................................................................................... 7
5.2 Maximum of probability ............................................................................................................ 8
6 Conclusion ............................................................................................................................................. 8
7 Bibliography .......................................................................................................................................... 8
Handbook of Référence
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Titrate:
Identification of the model of Weibull
Date:
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Author (S):
R. MASSON, W. LEFEVRE
Key:
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1 Introduction
When they call upon the model of WEIBULL (cf POST_ELEM [U4.81.22]), the study of modeling
brittle fracture of steels in general require a preliminary identification of the parameters of
this model. In order to avoid a hard identification “with the hand” of these parameters which would require
to start again repeatedly operation POST_ELEM with option WEIBULL, a procedure of retiming
automatic was established in Code_Aster.
In this document, one briefly recalls the equations of the model of WEIBULL then one defines it
problem of identification posed. One then describes the principle of the two adopted methods of resolution
(linear and maximum regression of probability) by including the case where one of the two parameters of
model depends on the temperature.
2 Recalls
2.1
The model of WEIBULL
One considers a structure of behavior elastoplastic subjected to a stress
thermomechanics. It is supposed that the probability of cumulated rupture of this structure follows the law of
WEIBULL [bib1] with two parameters following:
m
P
W
1
F (W) = - exp -
éq 2.1-1
U
expression in which the module of WEIBULL m > 1 described the tail of the statistical distribution
sizes of the defects at the origin of cleavage, U is the constraint of cleavage and W is the constraint of
WEIBULL which depends on the history of the principal stress field in the zone plasticized on
structure. For example, in the case of a monotonous way of loading, it is written:
V
p m
p
W = m (I)
.
éq 2.1-2
V
p
0
p
The summation relates to volumes of Vp matter plasticized, I indicating the principal constraint
maximum in each one of these volumes (V0 is a volume characteristic of material).
2.2
Identification of the parameters
In a very general way, one considers an experimental base made up of tests the different ones
natures (type 1, 2,…, N), each type of test being carried out N J time so that the total number
tests rises with:
j=n
NR = N J.
j=1
This experimental base could for example consist of tests on axisymmetric test-tubes
notched different radii of notch led to various temperatures. Taking into account nature
random of the properties with rupture of material considered, this base constitutes only one sample.
The more important the number of these samples will be, the more it will be representative of the behavior of
material considered.
Handbook of Référence
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Titrate:
Identification of the model of Weibull
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Among the various methods of identification suggested in the literature (see for example [bib2]),
we retain two of them: method of regression linear, often used, like that of
maximum of probability recommended by “European Structural Integrity Society (ESIS)” [bib3].
Note:
A comparative systematic study of the results given by these two methods [bib2] in
function of the number of sample taken by chance on a theoretical distribution showed
that method of the maximum of probability led to a better estimate of
parameters of the model of WEIBULL. Method of regression linear remaining
nevertheless very much used, we integrated it into our developments.
In the two adopted methods of retiming, one carries out the first calculation of the constraints of
WEIBULL with a play of parameter given (typically, m=20, =3000 MPa). These NR is classified
U
tests using their constraint of WEIBULL reached at the instant of the failure. One thus lays out
of an increasing list of constraints of WEIBULL (1
I
NR
W,…, W,…, W), such as for each (I), it
a number of test-tubes broken with a constraint of WEIBULL lower or equal to I
I
W is nw (in
General nor = I
I
W
). Among the various possible estimators of the probability of cumulated rupture PF
I
correspondent with I
I
W [bib2], we choose that generally recommended: PF =
.
NR + 1
Note:
In the particular case where the constraint of WEIBULL depends on the temperature, it
preceding classification must be made temperature by temperature, each temperature
correspondent with a different statistical law. The estimator of the probability of rupture
I
precedent thus becomes: Conk =
, if the test-tube (I) were broken with
NT + 1
temperature T, for which there was NT tests.
The two adopted methods of retiming are valid as long as [éq 2.1-1] remains true. If identification
is carried out on test results anisothermes whereas the constraint of cleavage is supposed
to depend on the temperature, this condition is not checked any more (cf POST_ELEM [U4.81.22]). In it
particular case, one will not be able to thus apply the developments which follow.
3
Method of the linear regression
3.1.1 Principle
The variation theory-experiment is measured by the expression:
2
1
1
LogLog (
)
- LogLog (
)
éq 3.1.1-1
1
- Pi
1 - P
I
I
F
F (W)
(“Log” indicates the Napierian logarithm). One wants to minimize this variation compared to (m, U).
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3.1.2 Resolution
The method of retiming usually used is based on successive linear regressions: with
the iteration (K), the values (mk, U (K)) of the module and the constraint of cleavage are known. It is thus
possible, with these values, to calculate the constraints of WEIBULL iW (K) at the various moments of
rupture thanks to [éq 2.1-1]. One then classifies these new constraints of WEIBULL per amplitude
increasing and one deduces the new estimates from them from the probability of rupture Pif (K) to the iteration (K).
For these values of constraints of fixed WEIBULL, the minimization of [éq 3.1.1-1] is brought back to one
1
simple linear regression on the scatter plot (Log
I
(W K)
(), LogLog (
)) since if one
1 - Conk (K)
1
defer LogLog (
)
(), one obtains a line of slope m which cuts the axis
1
according to Log
P
W
F
X-coordinates in (Log ()). The new values (m
,
) of these parameters are thus
U
K +1
U (K +1)
data by (cancellation of the derivative partial of [éq 3.1.1-1] compared to each parameter):
1 X Y
()
() - Y
X
NR
I K
J K
I (K)
I (K)
I, J
I
mk+1 = 1
éq 3.1.2-1
X X
()
() - X
2
NR
I K
J K
I (K)
I, J
I
1
1
the U.K.
exp
X
+1 =
I K - iY
(
)
()
(K
,
éq 3.1.2-2
NR
m
)
I
I
1
with X
= Log I
I (K)
(W (K)) and Y
= LogLog
I (K)
(
).
1 - Conk (K)
One repeats these iterations as long as the difference between the plays of parameter obtained with the iterations (K) and
(k+1) is significant (typically, five iterations). The measurement of this variation is given
m
1 - m
the U.K. +1 - the U.K.
by: Max
K +
K
,
(
)
()
m
K
.
U (K
)
Note:
If m is fixed, U (k+) 1 is always given by [éq 3.1.2-2]. On the other hand, if U is fixed,
X
Y
I (K) I (K)
m
I
K +1 is not given any more by [eq 3.1.2-1] but: mk +1 =
.
X
2
() - log ()
X
I K
U I (K)
I
I
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Identification of the model of Weibull
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4
Method of the maximum of probability
4.1 Principle
Let us note p F (W) the density of probability associated with the probability of cumulated rupture PF (W):
M-1
m
m
p
W
W
F (W) =
exp -
U U
U
The quantity p () D
F
W
W is equal to the probability of breaking a test-tube subjected to one
stress corresponding to a constraint of WEIBULL included/understood in the interval [W, W + dW].
The probability so that all the test-tubes of the base broke thus raises with:
p (m,) D = p
I
() D
U
W
F
W
W,
éq 4.1-1
I
p being related to probability. The method of the maximum of probability consists then with
to choose the parameters of the model of WEIBULL so that the function of definite probability
by [éq 4.1-1] (in practice rather its Napierian logarithm) that is to say maximum.
4.2 Resolution
An iterative process again is used. There still, with the iteration (K), (m
I
K, U (K)) as well as W (K)
are known. For these values of constraints of fixed WEIBULL, the maximization of Log (p) led
with a new couple (mk+1, U (k+1)) given by:
i= NR
I
mk
(
) +1 Log
I
(
)
NR
W (K)
W (K
i= NR
)
F (m
)
1
1 =
+
Log
I
(
)
- NR i=
K +
= 0 éq 4.2-1
m
W (K)
i= NR
K +1
i=1
I
mk
(
+
1
W (K)
)
i=1
i= NR
1
m
=
(I
) mk
K
+
+
1
1
.
éq 4.2-2
1
(K +)
W (K)
NR i=1
With each step, the resolution of [éq 4.2-1] can be carried out using the method of Newton, it
gradient of F (m) being given by:
i= NR
i= NR
i= NR
2
I
m
() Log2 I
I
m
I
m
() () - () Log I
W
W
W
W
(W)
df
1 =1
=1
=1
(m) = - NR
I
I
I
+
.
DM
m2
i= NR
2
I
m
(
W)
i=1
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Note:
If m is fixed,
is given by [4.2-2]. On the other hand, if is fixed, m
is not any more
U (K +)
1
U
K +1
solution of [4.2-1] but of:
NR
I
I
i= NR
F m
=
+
Log
W (K)
W (K)
mk
(
)
(
) (1
1
0
1
- (
) +) =
K +
m
.
K
I
+
=
1
1
U
U
This equation can be again solved using the method of Newton, the gradient
being now given by:
df
NR
I
I
i= NR
(m
W
m
) = -
2
2 -
(
) Log
W
(
)
DM
m
.
i=1
U
U
5
Dependence of the parameters with the temperature
If one wishes to fix independently the two parameters temperature by temperature, it is enough to
to break up the base of tests into as much of under - bases by temperature and to apply to each one of
these subbases preceding methods. If, on the other hand, one only wishes to vary
forced cleavage with the temperature, one proceeds in the following way.
U
5.1 Regression
linear
The estimate of the probabilities of rupture being now carried out temperature by temperature
(cf notices [§2.2]), it is enough to fix the constraint of cleavage on each associated scatter plot
at the various temperatures (T). The equation [éq 3.1.2-2] thus becomes:
1
1
the U.K. 1 = exp
X
+
I K - iY
(
)
()
(K
NR
)
T
m
I T
I T
(NT indicating the number of tests for the subbase corresponding to the temperature (T)), the module
WEIBULL being given by:
1 X Y
()
()
Y X
NR
I K
J K
I (K)
I (K
-
)
T
T I T
, J T
I
mk+1 =
.
1 X X
()
()
X 2
NR
I K
J K
I (K
-
)
T
T I T
, J T
I
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5.2
Maximum of probability
The constraint of cleavage is given for each temperature (T) considered by:
1
I
m
m
+1
the U.K.
(T)
1
= +1
(W (K (T K
K
(
)
))
+
,
NR
)
T I T
m being solution of:
K +1
I
mk
(
) +1 Log
I
(
)
NR
W (K)
W (K
i= NR
)
F (m
)
1 =
+
Log
I
(
)
-
NR I T
K +
= 0.
m
W (K)
T
I
m
K
K
+
+
1
1
i=1
T
(
W (K))
I T
6 Conclusion
Command RECA_WEIBULL of Code_Aster makes it possible to carry out the chock of the parameters of the model
WEIBULL [U4.82.06].
The user gives in input of this command the concepts results associated with various calculations
nonlinear carried out. The possible dependence of the constraint of cleavage with the temperature is
implicitly specified when different temperatures are associated each one of these concepts
results (if all these temperatures are identical or if they are not specified, there is not
dependence with the temperature of this parameter).
The user can carry out this retiming by the method of the maximum of probability (METHODE:
“MAXI_VRAI”) or that of the linear regression (METHODE: “REGR_LIN”).
The sizes determined by command RECA_WEIBULL are deferred in a table in
which one finds the value of the identified parameters, the probabilities of rupture estimated from
experimental results as well as the probabilities of theoretical rupture calculated with the parameters
identified.
7 Bibliography
[1]
F. BEREMIN, “A local criterion for cleavage fracture off has nuclear presses vessel steel”,
Metall. Trans. 14A, p 2277-2287, 1981.
[2]
A. KHALILI, K. KROMP, “Statistical properties off weibull estimators”, Journal off Material
Science, 26, p 6741-6752, 1991.
[3]
ESIS, TC 1.1 one “Local Approach”, Procedure to local measure and calculate approach
criteria using notched tensile specimens “, P6, 1998.
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