A REGRESSION APPROACH FOR ZIRCALOY-2
IN-REACTOR CREEP CONSTITUTIVE EQUATIONS
by
YUNG LIU, Y and A. L. BEMENT
(Energy Laboratory Report No. MIT-EL 77-012)
December 1977
11
I
A REGRESSION APPROACH FOR ZIRCALOY-2 IN-REACTOR CREEP
CONSTITUTIVE EQUATIONS
by
YUNG LIU, Y.
and
A. L. BEMENT
ENERGY LABORATORY
and
Department of Nuclear Engineering
and
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Special Report for Task 2 of the Fuel
Performance Program
Sponsored
by
New England Electric System
Northeast Utilities Service Co.
under the
MIT Energy Laboratory Electric Power Program
Energy Laboratory Report No. MIT-EL 77-012
December 1977
AUTHORS'NOTES
This report contains two parts of the study on Zircaloy-2 inreactor creep constitutive equations. Part I was completed earlier
in May, 1976 in which the essentials
and the preliminary
results
of and from the analysis are given. Part II which was completed
later in November, 1976 is an addendum of Part I with additional
results and discussions. Readers are advised to follow the sequence
of presentation
in order to see the entirety
of the analysis.
A condensed version of Part I is presented at the SMiRT -4th
Conference
August,
1977 and can be found
in the conference
ceedings under Division C, paper 3/3.
*
SMiRT - Structural Mechanics in Reactor Technology
pro-
ABSTRACT
Typical data analysis procedures used in developing current
phenomenological in-reactor creep equations for Zircaloy cladding
materials are examined. It is found that the data normalization
assumptions and the curve fitting techniques generally adopted in
these procedures can make the prediction of creep strain rate from
the resulting equations highly questionable.
Multiple regression analysis performed on a set of carefully
selected Zr-2 in-reactor creep data on the basis of comparable as
fabricated metallurgical conditions indicates that models of the form
= Aaonm exp (-Q/RT) can give significant account for the data.
Both regression statistics and residual plots have provided strong
evidences for the significance of the regression equations.
ABSTRACT OF THE ADDENDUM
The addendum of this report contains the results and discussions of the analysis we made on our revised Zircaloy-2 in-reactor
creep data. Although fewer creep rate data were used, much of the
same general conclusions can be made as we did previously. The
differences between the final recommended regression Zircaloy-2
in-reactor creep equations in the report and the addendum are primarily those of the estimated coefficients. The starting model
equations continue to provide excellent correlations for the creep
strain rate data.
Our attempt to verify the additivity of thermal and irradiation
creep components directly from the creep data set show that there
is little merit in using a composite formula made up by the above
two components for ranges covered by the test parameters.
This
statement should not be looked upon as a denial to such a treatment,
however, especially in situations of combinations of high , a and
T where changes in dominating deformation mechanism are possible.
At present, we do not have reliable data in those regimes.
Part I
1;
Table of Contents
Page No.
1.
Introduction
2.
Examination of current in-reactor creep models ----------------
3
2.1
Watkins and Wood's assumptions --------------------------------
7
3.
Regression analysis ---------------------------------
10
3.1
Regression Models ---------------------------------------------
11
3.2
Regressions Programs ------------------------------------------
15
4.
Examination of Zr-2 in-reactor creep data ---------------------
18
4.1
Accuracies of data --------------------------------------------
25
-----------------------------------------------
4.1.1 Errors due to Measurement
-------------------------------------
1
25
4.1.2 Errors due to calculation -------------------------------------
26
4.2
Normalization assumptions -------------------------------------
27
4.3
Time Effects ----------------------------------------------
30
5.
Regression results ----------------------------------------
31
5.1
Residual plots ---
33
5.2
Estimated regression coefficients and 95% confidence
intervals
5.3
5.4
-----------------------------------------------------
Comparisons
of current
empirical
38
and regression
creep equations -----------------------------------------------
40
Concluding remarks --------------------------------------------
50
References ---------------------------------------
51
Appendix A:
Zircaloy-2 In-reactor Creep Data
----------------------
53
Appendix B:
The Actual Set of In-reactor Creep Data
Used in Regression Analysis ----------------------------
58
Appendix C:
Regression Statistics ----------------------------------
60
Appendix D:
Stepwise Regressions of Zr-2 In-reactor
Creep Data --------------------------------------------
70
List 'ofTables
Page No.
Table 1:
Summary of in-reactor creep models and their data
bases
Table 2:
Summary of irradiation creep data used in
assessment of Watkins and Wood's equation
Table 3:
24
Comparisons of Normalized creep rates under
two different normalization assumptions
Table 7:
20
Operating conditions of U-49 tubular creep
insert
Table 6:
19
Summary of 20% CW and stress relieved Zr-2
in-reactor creep data
Table 5:
9
Total number of data observations in each
subset
Table 4:
4
29
Multiple regressions coefficients and F
statistics
32
Table 8:
Regression coefficients and their SEE's
38
Table 9:
Overall 95% confidence intervals
39
List of Figures
Page No.
Figure 1:
Figure 2:
Comparisons
of creep
ksi, flux=10
14
Comparisons
of creep
laws at stress=7.5
2
n/cm s
5
laws at T=300 0 C, flux=
6
1O14 n/cm2s
Figure 3:
Schematic
residual
Figure 4:
A schematic
17
plots
diagram of the samll tube in22
reactor creep specimen assembly
Figure
5:
The operating conditions for the Zircaloy-2
tubular creep experiment (Ross-Ross & Hunt)
Figure
6:
Creep rate of pressure tube normalized
to 14
2
28
ksi and 1013 n/cm s vs temperature
Figure
7:
Residual plots(a), (b) for regression models
(3)
Figure
8:
34
and (5)
Residual plots(a), (b) for regression models
(6) and (10)
Figure
9:
Residual
35
(b) for regression
plots(a),
models
(11) and (12)
Figure
36
10: Residual plot for the revised Ross-Ross
Hunt's regression model
Figure
23
and
37
(14)
11: Comparison of creep laws at T250C,
flux=
5)10 1 3 n/cm 2 s
42
Figure 12: Comparison of creep laws at T=3000 C, flux=
5x1013 n/cm 2s
Figure 13: Comparison
43
of creep laws at T=350 0 C, flux=
5xl013 n/cm2s
44
Figure 14: Comparison of creep laws at
0
T=400 C, flux=
5x10 1 3 n/cm2s
Figure 15:
Comparison
45
of creep laws at stress=7.5
ksi,
flux=5x101 3 n/cm2s
Figure 16:
Comparison
of creep
flux=5x10 1 3 n/cm2s
46
laws at stress=20
ksi,
47
r
I·
1.
Introduction:
In this report results
and discussions
are given on the development
of
Zr-2 in-reactor creep constitutive equations. Although several theoretical
irradiation enhanced creep models have been proposed( '
2'
3), present fuel
rod design and fuel clad deformation analysis involving in-reactor creep
are still based largely on empirical
are often obtained
procedures.
equations.
directly from in-reactor
These
empirical
equations
creep data by certain
kinds of
Close examination of the current empirical creep models shows,
however, that various normalization assumptions are often involved, and
these assumptions are generally different among investigators. The final
forms of these equations
are arrived at by adjusting
parameters
until good
agreement is reached between the equation predicted value and the actual
creep data.
While such procedure may be adequate in obtaining an equation
which summarizes a given set of data in the test parameter ranges, the
usage of the equation for prediction purpose at other parameter values
is seriously
questionable.
large deviations
Therefore,
in the predicted
it is of no surprise
to see rather
creep strain rates from these equations
under equivalent stress, temperature and fast neutron flux conditions.
Statistical data analysis using multiple regression techniques, on the
other hand, requires
plored
no normalization
assumptions
in the original form as they are collected.
ticularly
useful in establishing
the functional
and the data can be exThe technique
relationship
is par-
between
the
dependent variable (response) and the independent variables (factors) .
In our case in-reactor creep
strain rate is the dependent
variable whereas
stress, temperature and fast neutron flux are the independent variables.
-2-
All that is required is a tentatively selected model equation which relates
the dependent variable to the independent variables with unknown coefficients.
The problem then reduces to estimating these coefficients from data.
Modern
computer statistical programs are available which not only greatly shorten
the lengthy computational effort but also provide means to check the adequacy
of the underlying
The process
model.
of model validation
is perhaps
the most
important aspect of all good data analysis and should be exercised with caution.
It is also important to point out at this time that although researchers are
aided by various means for model selection, great savings can be achieved if
one can start with a reasonably good model.
Such models may be constructed
from prior experience and/or from the understanding of the governing physical
mechanisms.
The latter may come directly from theory developments.
Starting from the next section, we shall first give a brief review on
the data analysis procedure which had been used to derive the current empirical
in-reactor creep equations.
Much of the later efforts focus on the regression
aspects of Zr-2 in-reactor creep data analysis.
The principal areas discussed
are:
(1)
The selection of tentative model equations, their physical grounds and
their use in conjunction with linear multiple regressions.
(2) The examination of Zr-2 in-reactor creep data.
ing of the data according
This includes the regroup-
to test types, material
conditions;
the removal
of improper data normalization assumptions, and the elimination of certain
improper data points.
The possible sources of errors in the data are
also discussed.
(3)
Regression results.
Regression statistics, residual plots, estimated
coefficients and confidence intervals are covered with major emphasis on
model justifications.
-3-
2.
Examination of current in-reactor creep models:
Table 1 contains five current empirical in-reactor creep models for Zr-2.
The references of the data-base upon which these models were constructed are also
indicated in Table 1.
that
One notices immediately from the list of data references
a large portion of data used for obtaining these equations really come from
the same in-reactor creep experiments. The five equations are arranged in
chronological order as they were derived so that the latter of these equations
actually use some of the more recent in-reactor creep experimental data.
Before we discuss these equations further, it would be interesting to
compare the predicted in-reactor creep rates from these equations under the
same a,
, and T values.
In (a) plots respectively
Fig. 1 and Fig. 2 show the ln(E) vs 1/T and ln(E) vs
for these
equations
radiation creel)model derived by Gittus( 4) .
and one which
is based on the ir-
In both cases the Gittus model pre-
dicts creep rates an order of magnitude lower than do the empirical models.
Among the five empirical models, reasonably good agreements are attained only at
low temperature and low stress regions.
high stresses
The deviations at high temperatures and
are 2 to 3 orders of magnitude.
An apparent
inconsistency,
therefore, exists among these models which raises serious doubts for their
applicabilities especially at high stress and high temperature regions.
In spite of the difference
in the various
constants,
the forms of the
last four equations in Table 1 look rather similar. A flux exponent of 0.85
originally obtained by Watkins and Wood for SGHWR pressure tubes in 1968 has
been used rather consistenly in recent years and is the same exponent used in
the latest model.
It is, therefore,
useful
to review the Watkins
and Wood's
The word flux is referred to the fast neutron flux (E > 1 Mev) throughout
this report.
4
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COMPARISONS OF CREEP LAWS AT STRESS=7.5 KSI,FLUX=IE+14
-iog
--- A--
.1
Af
, I
_.- (3)
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r-,
t. · t
m
01%
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-6-
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Figure-2
r
4n lt
COMPARISONS OF CREEP LAWS AT T=300 C,FLUX=IE+14
f
*_l'13.,
.
(31
..
(5:
(2
((4
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Gittus
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leo'
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LN(STRESS) KSI
T
I
-7 -
method for deriving their creep equation.
The conditions for the Ross-Ross and
Hunt model will also be examined later because they are relevant to the data
set on which the regression analysis is performed.
Documented details for the
rest of the models can be found in the references and will not be covered here.
It suffices to mention that the analytical procedures behind these models are
essentially
no different
next section.
than the one which
is going to be discussed
in the
There are certain features on the data base for these models
which should be pointed
out as follows:
(1) The constants in Pankaskie's equation are found by fitting to
Fidleris' uniaxial in-reactor creep data.
(2) Data for
ender A and Vender
B's model come from the same two
major sources, i.e., work by Watkins and Wood and those by Ibrahim,
Ross-Ross
and Hunt.
Although
it is known that the pressure tubes tested
by these investigators are different in the amount of cold work and were
provided by different manufacturers, no recognition has been given
to these aspects, and all of the data have been used to construct
a
single equation. This is unacceptable since creep depends strongly on
the materials metallurgical conditions which are functions of the
thermal-mechanical treatments.
2.1
Watkins
and Wood's assumptions:
The basic assumption made by Watkins and Wood(6 ) in their data analysis
procedure is that each Variable (a, T, and
The effect of each variable
) affects the creep rate independently.
on creep rate is then considered
in turn by the
-8-
following assumptions:
(1) A power law relationship is assumed for fast neutron flux.
where n is a constant.
The best fit to the data is obtained with
n = 0.85, the uncertainty
in this value
being + 0.15.
(2) An activation energy form of temperature dependency is
assumed
exp( - Q/RT)
where Q is the activation
energy.
The best fit to the data is
obtained with a Q value of 14,000 cal/mole, the uncertainty
in this value being + 6000 cal/mole in the temperature range 250
to 300 0 C.
(3) Using the above expressions, the data have been normalized for
flux, temperature, and biaxiality to SGHWR conditions and are
plotted
against the remaining variable a.
a sinh curve can fit the normalized
a stress
coefficient
of 1.15xlO
-
It was found that
creep rates acceptably
with
1 /ksi and a pre-multiplier
con-
stant of 1.02xlO- ll .
(4) The entire equation then follows by joining these three separate
terms together
as
* = 1.02xlO- l l q 0.85 exp(-
14000)
1
sinh (1.l5xl0
Oa)
The set of irradiation creep data used for the construction of
the above model
is shown in Table
2.
It is interesting
to
-9-
Table 2
SUMMARY OF IRRADIATION CREEP DATA USED IN ASSESSMENT
OF WATKINS AND WOOD'S EQUATION*
Flux
Test No. and
Ref.
829
829
829
629
Temp..
Type deg C
II] .... uniax.
[2] ....
uniax.
[3]..... uniax.
tube
tube
tube
tube
tube
[4 ..... tube
[4..... tube
[3)..... uniax.
[1 .....
4].....
U2-Mk. IV [4] ....
U2-Mk. IX [4 .....
U2-Mk. III
NPD
U3-Mk. V
R4
R6
R9
RX 14
R X
R X21
R X 15
[31.....
[3].....
uniax.
uniax.
[3]..... uniax.
3].....uniax.
[3].....uniax.
[3 ..... uniax.
290
300
325
275
285
281
286
286
268
Stress,
psi
(>I MeV), Duration, Creep Rate,
25 000
16000
35 000
25 000
11 500
11 000
13 400
17 300
10 600
2.4 X 10t
2.4
3.1
3.4
2.7
2.7
3.1
3.1
1.15
3.1
0.59
0.54
0.58
0.64
0.94
327
4 900
300
300
300
300
350
300
220
30000
20 000
18 000
20 000
30000
45 000
30 000
n/cm sec
0.84
0.96
__
h.
3000
8 500
5 000
2400
5 700
11 000
3 300
3 300
29000
13000
1 300
3 600
1500
1 100
530
800
2 100
___
in./in./h
__
1.17
3
1.2
5.2
X
X
X
X
10-'
10"
1010-'
1.9 X 101.8
2.9
X 10X 10
3.6 X 10-'
5.5 X 10-'
9.1 X 10'
8
X 10'
3.5
3
3
X 10X 10 X 10-
3.2 X 10-'
1.2 X 10- S
1.5 X 10 T
* Reproduced from Ref. (6).
note that out of the seventeen creep rate data, ten of them actually came
from Fidleris'uniaxial creep tests. A biaxiality correction has been
applied to the uniaxial data so that they can be used together with the tubular data.
This is done by multiplying the stress by a factor of 1.15
and dividing the creep rate by the same factor.
0
Such a conversion method
is valid only for isotropic material, which is not the case for Zr-2.
Furthermore, uncertainty of using test results from two different sources
- 10 -
of materials is retained.
Steps (1) to
(4) above outline
the general
veloping current in-reactor creep models.
procedure
followed
in de-
The assumption that each variable
affects creep rate independently appears to be highly unsatisfactory. This
is because the creep rates used to determine the constant for one variable,
with an assumed variable dependence, contain the influences from the other
variables too.
A realistic model or data analysis procedure must, there-
fore, be able to take into account the influence from the stress, temperature, and flux simultaneously.
One convenient way for doing this is by re-
gression analysis.
3.
Regression analysis:
Regression analysis can be classified into two broad categories, i.e.,
linear and non-linear regressions.
By linear it is meant that the model
equation is linear in its coefficients whereas non-linear implies that the
coefficients may contain higher order terms.
tion, a non-linear model can be transformed
Sometimes by proper linearizainto a linear one and in that
case the model is said to be inherently linear.
Since the solution of
non-linear problems generally requires iteration and is much more involved
than linear problems, the standard practice is to start with a linear
model unless there are strong reasons to believe that the underlying model
is non-linear in nature.
The starting equation (being linear or non-
linear) is not that crucial because through the examination of results from
regression analysis such as regression statistics and residual plots,
- ll
the appropriateness
of the model
that a linear model
is inadequate,
-
will be revealed.
If the evidence
then it can be modified
shows
by either chang-
ing the variable dependence relations or considering non-linear models.
3.1 Regression Models:
The general form of a linear model with three independent variables can
be written
as
Y = Bo + Blf(X1 ) + B2 f(X2 ) + B3 f(X3) + B4 f(X1X 2 ) +
B5 f(X2 X 3 ) + B6 f(X3X 1) + B7 f(X1X 2 X 3 )
(1)
where Y is the dependent variable, f(X1 ),...,f (X1 X2 X 3) are some functional
dependences of individual and combinatorial effects of X,
on Y, and Bo,...
, B7 are the coefficients
.
£,
X1 =
model.
to be estimated.
, X 2 = a and X 3 = T, then (1) becomes our general
The task then is to specify explicitly
f(a), f(T),....,
X2 , and X3
If we let Y
-
linear creep
the functional
forms of f(4),
f (oT).
As we mentioned
earlier,
it would
be greatly
a model which has some physical meaning.
helpful
Studies
if one can choose
on materials
thermal
creep behavior have shown that in an intermediate temperature range and
moderate stress level, creep is best represented by a power law-type relationshiplo).
That is,
s = A an exp
(-
Q
RT
(2)
- 12 -
Eq. (2) is of the form which
vated rate processes.
is generally
used to describe
thermally
acti-
The exponential term or the Boltzmann frequency
factor is a measure of the probability of dislocation climbing over its
barriers.
The driving force for dislocation climb over obstacles on the
gliding plane comes from both stress and lattice thermal'fluctuations.
The activation energy, Q, is a measure of the barrier strength which may
be a complex
function
of both stress and temperature.
In irradiation creep where fast neutron flux is also present, we may
use a phenomenological equation of the following kind:
= A m an exp (- Q--)
(3)
RT
Namely, an additional flux dependent term is included.
relation has also been suggested by Fidleris.
This type of
Taking logarithms on
both sides, (3) becomes
= In A + m lnq + n lna -
ln
(4) is essentially
a linear model
Q
as in (1) with the following
tions:
Y =
f(X 1 )
n £
= in q
f(X 2) = In
f(X 3 ) = 1/T
a
; B o = In A
; B1 = m
; B2 = n
; B 3 = - Q/R
(4)
transforma-
-13
This means that if we transform
the original
dependent
and independent
variables into the forms given above, the multiple regression analysis
would enable us to find estimates for the coefficients B o to B 3 and hence
the values of A, m, n, and Q.
Two alternative models were also considered in which the activation energy is a function of stress:
= A
m
an exp(- Q °(
- /
)
(5)
)2 )
(6)
R T
and
e
= Am
an exp(- Q
(
1
R T
Again these models were obtained by superimposing a flux dependent
term to the thermal creep equations.
The mechanistic
content
of such
stress.dependent activation energies has been discussed by Ashby.and
Frost(l2 )
It does not require much effort to show that these
equations can also be transformed into linear models which lend themselves readily to examination by data analysis.
logarithms
Again taking
on both sides, we have
ln = In A +m lno + n In a -
1 +
R T
R
a
(7)
T
and
n
= In A + m lno + n In a -
qo 1 ++ Q . c2qQ
RT
RT
T
RT
2
.
(8)
T
- 14 -
effects
The last two terms in (7), (8) give the interaction
of a and
T on the creep rate respectively.
So far we have presented three models based on only three independent variables.
Other functional forms of these independent variables
could also be examined.
For example, we could use a parameter a/u instead
of a where u is the shear modulus or
diffusion
pendent
coefficient,
/D instead of
or a combination
u and D introduced
of the two.
where D is the
Both temperature
de-
in this manner will take some temperature
dependence outside the exponential term, and in some cases, as in thermal
creep analysis, have been shown to give better correlations.
The tem-
perature dependent shear modulus of Zr-2 is given by the following
formula{ 1 3)
u(ksi) = 4.77x103 - 1.906xT(°F)
(9)
It should be noted that the use of properly non-dimensionalized
parameters is a fundamentally appealing approach.
This is because of the
inconsistancies in parameter units in the previous creep equations.
the variables
are expressed
in their ordinary
4 in n/cm 2 sec, a in ksi, and T in
2mn
)'
cm sec
able.
m
(ksi)-n
-1
· hr
so that
units, e.g.,
If
in -hr 1,
K; the constant A must have unit of
units on both sides of the equation are agree-
But such a unit for A can hardly have any physical interpretation.
Since the exponential term is already dimensionless and we replace a by ao/(T),
the problem left is then with the flux term.
one may divide
*
by a reference
fast neutron
To make this term dimensionless,
flux
o and (3), (5), and (6)
15 -
become
=
A (-)m (
A (¢
=
a
)" exp(- Q)
)m ( u(T)
a )Q exp(
(T)
= A (S=AQ
t)m (
a)n
o
u(T)
(10)
RT
( 1 4)
A
exp(- Q°(1- R/T)'
R
)
(12)
(12)
T
o is taken to be 0.05x1013 n/cm2 sec.
result
(11)
1-al)
This is deduced from Ibrahim's
and gives approximately the fast flux threshold below which
there is no difference in irradiation and thermal creep behavior.
=
is, when
o
the creep equations
= A (a)n exp(-
After q and a
vibrational
frequency
rate processes.( 1 5)
reduce to
Q ), etc..
RT
have been non-dimensionalized, constant A should
unit of hr -l which
have
G
is reasonable
Regression
since we know the lattice
(or the Derbye frequency)
is involved
in the
There are, therefore, six regression models, (3),
(5), (6), (10), (11) and (12), considered
3.2
That
in this analysis.
Programs:
Details on the descriptions of linear multiple regression analysis
can be found from standard texts,(16)(17) and will not be covered
here.
Several computer statistical regression programs are available
- 16 -
with varying capabilities (18),(19),(20),(21)
One program(18)
which does least-squares only without giving residual plots is
rejected because the model cannot be accepted or denied based on
the regression statistics alone such as the multiple regression
coefficient, R2 .
Sciences)(21
)
The program, SPSS (Statistical Packages for Social
was selected to be our major analytical tool because of
its versatility in discriminatory data anlaysis.
Since the residual plots play a very important function in regression analysis, a brief discussion of their interpretation is warranted.
Now suppose a linear model is selected and the coefficients
have been estimated, the residuals are defined as the n differences
A
ei=yi-yi,
i1,2,...n
where yi is an observation
and yi is the corres-
ponding fitted value obtained by use of the fitted equation.
residuals can be plotted in a number of ways.
sideration
in examining
patterns.
Fig. 3 shows four residual plots.
the residual
plots
These
The most important con-
is to notice their overall
Except for the straight
band pattern (a), the other three patterns of the figure are all indicative of abnormalities that require corrective actions.
IF
- 17 -
0
*.Li
'·
5.
*
0
(a)·,~· .·.
-
-O..
·
R'.
. '.
b.I
(b)
I
0
0
* .01
-
5I
·
*.
(c)
Figure 3.
..
*
(d)
Schematic
Residual
Plots
- 18 -
For example, pattern (b) indicates that the variance of residuals is
dependent
on the value of the variable
plotted
in the horizontal
axis.
Pattern (c) indicates a linear relationship between residuals and the variable on the horizontal
axis.
Finally,
the arched pattern
of (d) indicates
curvilinearity that may be removed by the addition of polynomial terms to
the regression equation
The reason
Central
or, perhaps,
that a correct pattern
Limit Theorem
by a transformation
is indicated by (a) is not only due to the
but also because of the Gaussian
inherent in the least squares (16)
of the yi values.
N(O,1) error model
It is also intuitively
expected since
if the fitted equation is adequate in explaining the major portion of the
data, the residuals should be distributed randomly and evenly across
the zero residual
line.
Deviations
from such a pattern
then signals
that
the regression equation is probably inadequate.
Other than serving
ties, the residual
the purpose
plot has another
for visual examination
important
of abnormali-
function which
should not be
overlooked; that is, to identify stray data points. As we know human errors
or equipment malfunctions are sometimes inevitable even in well-ministered
experiments.
Accordingly,
and/or data analysts
behavior.
it is helpful to have a way for experimenters
to spot irregular points and to seek reasons for such
If it can be ascertained
that the irregularity
such data should be excluded to avoid
4.
is due to error,
the biasing of the result.
Examinations of Zr-2 in-reactor creep data:
We have recently compiled
data of Zirconium,
1974(22).
both uniaxial
and tubular
Zr-2, and Zr, 2.5% Nb in the literature
There are in total 276 data observations
in-reactor
creep
from 1962 to
and the breakdown
of
- 19 -
these observations
into subsets
TABLE 3.
is shown
in Tabl:e 3.
TOTAL NUMBER OF DATA OBSERVATIONS IN
EACH SUBLET
Zirconium
Zr-2
Zr-2.5%Nb
Uniaxial
9
89
65
Tubular
0
73
40
The subset of Zr-2 tubular in-reactor creep data was chosen for
analysis and this portion of data is included in Appendix A.
The term
"Reference" on the second column of the data Table indicates where these data
come from and can be compared with the reference list provided at the end
of this report. One notices from the fourth column that there are about
eight different cold-work and heat treatment material conditions.
Since
both cold-work and heat treatment affect the material condition and
hence the creep behavior, the data set is further subdivided according to
these material conditions.
As is shown, data taken from Reference I-11(Serial T14-T19), M-5
(Serial T39 - T56) and a portion of H-14,
I-13 (Serial T1 - T13) with
20%
C-W plus stress relief constitute the largest subdata set in this Table.
This subset is, therefore, chosen for the regression analysis.: It turns
out that all these data observations
were obtained
the same group of Canadian researchers on
by
Zr-2 pressure tubes.
The tube
fabrication method and the alloy conditions are, therefore, reasonably
similar.
The only difference is the size of the test specimens.
In
early experiments full-size pressure tubes were creep tested whereas later
for economical reasons and also for reasons of achieving higher stresses
- 20
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{
-
2
during creep testing, small-sized
specimens
(23 mm diameter)
were used.
The small-sized specimens are probably stronger and of finer grain size
than full-sized pressure tubes because of quicker cooling rates after extruding (due to the smaller size), giving greater residual hot work before cold
drawing(14)
Since extensive
are made when using this subset
cross references
them into a new table given by
of data, it is worthwhile
to designate
Table 4.
of this Table will follow three general areas:
Our discussions
(1) The accuracy of data, (2) normalization assumptions, and (3) time effects.
But before we proceed,
it is helpful to quickly
review some of the key
features of the tubular in-reactor creep experiments from which these data were
obtained.
These experiments were conducted in NPD (Nuclear Power Demonstration) and NRU (National Research Universal)
sure tubes and small-size
specimens.
tube, in-reactor creep test assembly
reactors with full-size pres-
A schematic diagram
is shown in Fig. 4.
of the smallSince the small-
size specimens were enclosed in a secondary containment, there is no
danger in damaging the reactor when tube rupture occurs, and, therefore,
higher applied stresses giving much higher creep strains are permitted.
Also, since the hot-pressurized water is maintained at rather constant pressure, the stress variable is altered by machining the outside of specimens
to give different wall thicknesses. Depending on their relative positions
in the reactor, three types of creep results were actually obtained
for the
fast-flux specimens, thermal-flux specimens, and out-of-flux specimens.
This is also evident
from Fig. 4.
- 22 -
IMENS
H
SPECIMENS
MENS
UEL ROD
Figure 4.
A schematic diagram of the small tube
in-reactor creep specimen assembly
(reproduced
from ref.
[I-ll ])
- 23Typical reactor operating histories are shown in Fig. 5 for Ross-Ross
and Hunt's full-size pressure tube experiments and in Table 5 for Ibrahim's
small-sized specimen, U-49 tubular creep insert.
5-1.0
E
E
a0.95
0
0.
4
w
a.
0.90
I
5.
N
E
2.8
2.4
2.4
Z
(0
2.0 i
00
TIME (hours)
Fig. 5. The operating
conditions for the Zircaloy-2 tubular creep experiment.
(The complete 18 by 40 in. chart covering the operating conditions
up to 9000 hours may be obtained from ASTM Headquarters, $3.)
Reproduced
from ref. (I-ll).
9500
- 24 -
TABLE 5
Operating conditions of U-49 tubular creep
insert.
(Reproduced
Time
Average
period
The purpose
|
1232
2419
3516
4987
6053
7597
8910
9712
10747
13626
14793
19859
20896
Average
Average
(14))
Fast neutron
temperature, pressure flux > 1 aMeV
(h)
0 to
1232 to
2419 to
3516 to
4987 to
6053 to
7597 to
8910 to
9712 to
10747 to
13626 to
14793 to
19859 to
from ref.
(C)
(7N/m))
264.4
256.7
253.4
260.7
256.7
257.1
259.2
260.3
260.4
265.6
270.4
263.2
279.1
9.24
8.96
9.09
9.05
9.08
9.09
9.14
9.57
9.17
9.54
9.40)
9.33
9.39
262
9.3
of this reviewing
(x -00-1
)
2.55
2.51
2.34
2.52
2.52
2.36
3.12
2.82
3.39
3.40
3.05
3.18
3.25
,
2.9
is to emphasize
that during
the entire period of creep testing (which may run up to 20,000 hrs.),
variables
,
, and T will have their
own histories
and are not constant
values throughout the test.
This is because the reactors are run accord-
ing to their fuel management
shcemes
and these schemes
are generally
not
-
dictated by the creep tests.
25_
Therefore,
tion takes place, it will cause changes
example of such local changes
A good
in all local conditions.
in Fig. 5.
is illustrated
in Table 4 are then only average
and T contained
opera-
each time a fuel shuffling
,
Data of
a
values obtained from the
entire operating history of each variable respectively. As will be shown
were used in some cases to re-
assumptions
later, one or more normalization
duce the data to the present values.
While normalization
is a viable pro-
cedure in data reduction, the assumptions must be carefully examined
first.
4.1
Otherwise
the original
information may be lost or distorted.
of data:
Accuracies
There are at least three major sources of error which may be introduced into the values of the dependent
as well as the independent
(1) errors due to the use of average values
(2) errors of measurement
and calculation,
variables:
(ignoring the history effects),
and (3) erros due to im-
proper normalization assumptions. We cannot do much about the first
type of error except to say that the deviations
are sufficiently
small such that there are no significant
fore, in this section only measurement
cussed, and normalization
4.1.1.
errors
Errorsdue to measurement:
Diametral
from average
and calculational
are discussed
conditions
history
effects.
errors are dis-
in the next section.
(e, T, P )
creep strains were measured
both at intermediate
and at the end of testing by linear variable differential
shutdowns
transformer
There-
- 26 -
(LVDT)( 5 ),
air gauges(1 4), and micrometers.
The accuracies of these strain
gauges have been discussed elsewhere(14)' (5 ) and are generally considered
satisfactory.
Pressure and temperature measurements were taken at inlet
and outlet only with standard instruments, and the measurement errors
should be rather small.
The errors of measurement are, in general, be-
lieved
to be small
when compared with
4.1.2
Errorsdue to calculation:
the errors due to calculation.
Two types of errors are introduced into the calculations of tangential
stresses
in the reference
data set H-14.
One is caused by the assumption
of
a linear pressure gradient along the tube made by Ross-Ross and Hunt( 5 )
and the other
convert
is the use of a thin shell approximation
the pressure
inside diameter
into tangential
stress where
and t is the wall thickness.
exact formula
for internally
pressurized
approximation
underestimates
stresses
wall thickness.
The pressure
tubes
formula,
t=P d/2t, to
d is chosen to be the tube
Calculations
using
Lame's
(23) show that the thin shell
by about
1 to 5% depending
drop and hence the localized
upon the
pressure
can be
calculated from the Darcy formula(24 ) but requires rather detailed knowledge
of the flow conditions, entrance effects, etc..
not available,
we merely
point
Since this information is
it out that the pressure
and hence the
stress in H-14 are based on the linear pressure gradient assumption.
Coolant temperatures at inlet and outlet to the tube were measured,
and the temperature gradient along the tube is calculated using fuel power
information.
Fast neutron flux (E>l Mev) is calculated from reactor physics
(5 )
using fuel bundle powers and geometry
factors.
these calculations
from the actual conditions
must be determined
The accuracies
of
and the
assumptions
in the reactor physics model.
In any case, errors due to measurement
and calculation
are probably
This is because
not major factors in these creep experiments.
of the averag-
ing procedures used to normalize the history effects would likely to overshadow the magnitudes
of the combined errors due to measurement
and calculation.
Normalization assumptions:
4.2
The empirical equation obtained by Ross-Ross and Huntin correlating
their in-reactor creep data is reproduced
Et = 4x10-2
4
t(T
as follows:
- 433)
(13)
where
Et:
hr-l
at: ksi (up to 20 ksi)
+ ; n/cm2sec
T ;
(0.5 to 3.5x10 1 3 n/cm 2 sec)
K (523 to 573QK)
The equation is obtained in a similar manner as that of Watkins and Wood
discussed previously. The stress and flux exponents were determined first
by using
t.sequentially
with
C
= Sot and
(both close to 1), the temperature
the creep rate is proportional
the reference
E
=
dependence
8
lqm.
After
is determined
to flux and stress, and
n, m are found
by assuming
is normalized
conditions of 14 ksi and 10 1 3 n/cm2sec by the following
to
formula:
- 28-
14 ksi
EN
=
EActx
Y
actual stress
actual
1013n/cm2sec
flux
(14)
actual
stress actual flux
This normalized creep rate is plotted against temperature in Fig. 6, and
(13) is derived
2
from the best fit.
AA
1.9
1.8
1.7
CREEP RATE OF PRESSURE TUBES NORMALIZED TO 14,000 PSI STRESS
AND Ix 1013 n/cm 2 s > I MeV vs TEMPERATURE
14,000 psi
I x 1013 n/cm 2 s
NORMALIZED
"ACTUAL ACTUAL STRESSx
ACTUAL FLUX
1.6
1.5
STRESS TE:'REACTCR
l
.503
oo
1.2
,1oAXFLUX
10
n/c
rnm
2
'
D-U3M1
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2
2'
5 ,00 : 1-
,10,50
6
3.
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TUBE
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3
I
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C.W.ZIRCALOY-2
1.0
I
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t/
-
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ct.,IJ 0.9
w
C
0.7
w
0.6
J
0.5
N
:
o
Z
C.W.Zr 2.5% Nb
/
04
0.3
77
0.2
'Zr-2 OUT-REACTOR
14,000 psi STRESS
0.1
-
_
I
*
220
Figure 6.
230
240
250
260
270
280
290
TE 1 PE RATURE °C
300
310
_
1_
_
320
330
_
Cree2 rate of pressure tube normalized to 14 ksi and
1013
n/cm sec vs temperature.
(Reproduced from ref. (5)).
340
- 29-
Again the overall procedure is questionable. The following Table illustrates
that if the correct creep model
is given by (3) with m=0.85,
n=1.2, and Q/R =4,000 cal/mole; the difference in the normalized creep
rates from two normalization
can be quite large at the average
procedures
reactor operating conditions.
Comparison of normalized creep rates under two different
normalization assumptions
TABLE 6
Temperature
Time period
(C)
m(hr)
Pressure
Flux
(MN/m 2 )
.1
2
264.4
9.24
2.55
1.169
1232-2419
256.7
8.96
2.51
0.740
2419-3516
253.4
2.34
0.612
3516-4987
260.7
9.09
9.05
2.52
0.941
4987-6053
256.7
9.08
2.52
0.742
6053-7597
257.1
9.09
2.36
0.767
7597-8910
259.2
9.14
3.12
0.836
8910-9712
260.3
9.57
2.82
0.914
9712-10747
260.4
9.17
3.39
0.887
10747-13626
265.6
9.54
3.40
1.200
13626-14793
270.4
9.40
3.05
1.598
14793-19859
263.2
9.33
3.18
1.058
19859-20896
279.1
9.39
3.25
2.509
Average or
reference value
262.0
9.30
2.90
0-1232
s1:
Normalized
strain rate using linear dependence
in
, o, and (T-160).
Normalized strain rate using -Ocm o n exp(-Q/RT) relation with m=0.85,
n=1.2, and Q/R=4,000 cal/mole
/ (
where
)-m(rn
1
/
)-n(Tr160)
exp(Q
r' ar' and Tr are the reference
- T
values.
- 30 -
Creep strain rate data from Serial T7 to T13 in Table 4 normalized with
Ross-Ross and Hunt's assumption should, therefore, be transformed back to the
values according
to the actual stress and flux, and this is done by both (14)
and the Table insert of Fig. 6.
The back transformations
are necessary
to re-
move any bias in the data which might have occurred from improper normalization assumptions.
4.3
Time effects:
All previous creep equations of the form ~=f(o,+, T) apply only when
materials
are at a stage of constant
or steady state creep.
In thermal
creep
analysis, the creep rate is often found to be constantly diminishing with time
and the creep strain can be represented by
=
Differentiating
tm.
£
with
respect to time gives
= m
tm-l
Clearly steady state creep is possible only when m=l; for m < 1 the creep rate
is decreasing with time which is indicative of strain hardening,
In irradiation creep, although some of the early in-reactor creep experiments(6)(
hours,
results
5) did not show this kind of relationship after several hundred
obtained
by Ibrahim (1 4) over long test periods did.
The rate of
decrease in creep rate is somewhat slower for in-reactor creep than out-ofreactor creep (wiehgted mean m=0.468 for fast flux specimen and weighted mean
(14)
m=0.270 for out-of-flux specimens)
The problem
caused by this time dependence
of creep rate is as follows;
since creep strain rates must be obtained from dividing the measure diametral
creep strains
by a small time interval,
the time at which
such creep strain
31 -
measurement is taken will apparently matter.
T39 and T40 are at the same a,
For example creep rate data serial
, T values but estimated
at 5,000 and 20,000
hours correspondingly, they differ by a factor of 2.4.
When the form of time dependence cannot be incorporated readily, we must
select a convenient reference time for creep rate calculation such that the
creep rate data can be compared and analyzed
data observations
in M-5 should, thus be excluded.
tion of creep rate data were estimated
be the reference
on an equal
basis.
Since
the major
it to
time and hence the even numbers of data observations
creep data used in regression
Regression
por-
at 5,000 hours, we have chosen
20,000 hrs.) from T40 to T50 inclusive should be excluded.
5.
A number of
analysis can be found
(at
The actual set of
in Apprndix
B.
results:
Results from regression analyses using the six regression models (3),
(5), (6), (10), (11), (12) and one which
presented
in this section.
is of a similar form to (13) are
The last one is included for the purpose
strating that if such a model of regression
(T-433) P , the estimates
for the exponents
at all as obtained by Ross-Ross
should be written
analysis is given by
of demon-
= A
m an
n, m, and p will not be close to 1
and Hunt's procedures.
Instead, the equation
as
= 3.41x10- 2 6
+0.61
1.5
2
(T
-
433)2.047
(16)
Computer printouts on the details of regression statistics such as R2 , F
ratio, standard error of estimate
(SEE), etc. are given
in Appendix C.
It
- 32
to summarize
is useful
TABLE 7
Equation
and assemble
the information
into the following
R2(%)
No.
F
3
96.969
178.498
5
97.109
136.525
6
97.702
134.482
10
96.969
178.516
11
97.106
136.359
12
97.708
134.865
16
96.902
174.466
of fit or the fraction
explained by the regression equation.
hypothesis H:
Table:
Multiple regression coefficients and F statistics
R 2 stands for the goodness
of data that has been
A high F value means that the null
R=-Omust be rejected or the alternative hypothesis H1 : BiO
one or more i is true.
7.
-
Several
points are worth
noting
for
from Table
First of all, based on values of R2 and F, all regression equations can be
considered significant.
we examine
the residual
We shall further substantiate this conclusion when
plots.
Second,
R 2 increases
as we use more elaborate
forms of stress dependent activation energies, i.e., in ascending order in
groups of (3), (5), (6) and (10), (11),
(12), but the increase
is marginal.
Third, equations of non-dimensionalized parameters with a temperature-dependent
shear modulus give higher R2 than their counterparts in tteoriginal units of
a, q, and T, but the difference
In order to assess whether
is slight.
one should include product
terms
such as
o/T, and a2 /T to account for their interaction effects on a, stepwise (for-
- 33 -
ward inclusion) regressions were also performed .in which the independent variables are entered
into the equation
statistical criterion.
out the independent
one by one on the basis of a preselected
In short, this step-by-step procedure helps to screen
variables
less than the pre-selected
(be it single or product) whose partial
value.
These variables
are considered
only minor contributionsto the representation of the data.
F's are
as having
In all
stepwise cases we have studied based on an F value of 3.2, no independent
variables (single or product) have been eliminated. A more stringent criterion
is
therefore
required
if one intends to use a model with less than a maxi-
mum of five independent variables (a, , T, a/T, a2 /T).
regressions
5.1
Results from stepwise
are included in Appendix D.
Residual plots:
Fig. 7 to Fig. 10 give the standardized residual
plots for the previous regression
numbers.
models.
First we notice that the overall
look quite satisfactory.
versus standardized
They are identified
fit
by equation
patterns of these residual
plots
All residuals fall more or less into rather straight
error bands. There are no indications of "abnormalities" as discussed earlier
in Section
model
3.2.
Fig. 10 which
is based on the revised Ross-Ross
(16) shows that it is only slightly
and Hunt's
inferior to Fig. 7(a) which
based on (3) in terms of residual structures.
is
Fig. 7(a), 7(b), 8(a), 8(b),
9(a), and 9(b) show that the error band gets narrower when more
elaborate forms of stress-dependent activation energies were used.
responding
residual plots in either original parameter
less form show little difference
in their structures.
The cor-
units or in dimension-
- 34 -
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.38
Based on our results from regression statistics, residual plots, and
stepwise regressions, we conclude the followings:
(1) The ranking of regression models in terms of goodness of fit (in
descending order) is as follows:
Group
1 (equations
in original
units):
(6), (5), (3), (16)
Group 2 (equations with dimensionless parameters):(12), (11), (10).
(2) Although the corresponding regression models in Group 1 and Group 2
do not differ
very much, Group 2 models are recommended because they offer
better physical interpretations.
5.2
Estimated regression coefficients and 95% confidence invervals:
A summary of the estimated regression coefficients and their standard
error of estimates (SEE) is given for each model in Table 8.
From these
coefficients one can construct in-reactor constitutive equations by sub-
TABLE 8
Regression coefficients and their SEE's
A
Equation No.
(3)
1nA
-28.799
SEE
(5)
-29.112
SEE
(6)
-31.640
SEE
(10)
- 0.470
SEE
(11)
- 3.900
SEE
(12)
SEE
*In equation
6.323
m
n
Q/R
Qo/RT
^2
Qo/RT
0.611
1.540
5298.06
0.030
0.100
816.00
0.613
1.126
4746.49
0.030
0.345
920.96
0.609
2.759
4276.19
-108.915*
1.676
0.027
0.652
851.15
42.969
0.587
0.611
1.541
4855.29
0.030
0.100
799.55
0.613
1.130
4425.72
11.310
0.030
0.347
866.03
9.142
0.609
2.800
3484.49
-111.175*
0.027
0.659
849.12
43.323
(6) and (12), the coefficients
11.406
9.093
for a/T are in the form
---
1.701
0.591
of 2Q/RTr.
39 -
stituting
them back into the corresponding
models.
The 95% confidence
intervals for eaeh coeffiCinit are found by using the following formula:
BA(estmated
jth
B i (estimated i
coefficient) ± t (SEE)i
(17)
where t is obtained from a 95% point "Student-t" distribution with n-k degrees of freedom; n is the total number of observations
and k is the number
of coefficients to be estimated.
It is often more useful to know the overall 95% confidence
the entire regression equation, and this can be found from
{
interval for
25)
A
Yi
(fitted value)
±+ i7T
where M is the number of coefficients
(18)
(SEE)^
y
to be estimated,
f is obtained
a 95% point F-distribution with M, and n-M degrees of freedom.
ing Table contains the information which
from
The follow-
can be used to construct
the over
95% confidence intervals.
TABLE 9
Equation No.
M
Overall
95% confidence
f
7Mrf-
intervals
(SEE)A
+vf`
(SEE)-
(3)
3
2.88
2.939
0.2514
0.739
(5)
4
2.65
3.256
0.2493
0.812
(6)
5
2.49
3.528
0.2261
0.798
(10)
3
2.88
2.939
0.2514
0.739
(11)
4
2.65
3.256
0.2495
0.812
('12)
5
2.49
3.528
0.2258
0.797
_ 40
5.3
_
Comparisons of current empirical and regression creep equat lo n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
It is instructive
to plot on a single set of axes
creep rates from cur-
rent in-reactor creep models, regression equations, and the actual creep data.
Six such plots, Fig. 11 to Fig. 16, were made at a constant fast flux
n/cm2sec
to show the ln
vs 1/T and ln
vs lna relationships.
=5x1013
Since creep
rates in these figures are plotted against only one variable while keeping
the remaining two constant, it is necessary to transform the original creep
rate data to the reference values of these two remaining variables.
This can
be done by using one of the regression equations to provide the scaling facTo avoid obscuring the diagram with too many curves, we merely used
tors.
regression
model
(3) and its associated
demonstrate the difference.
overall
95% confidence
interval
to
Those lines representing the current in-reactor
creep models are identified by the corresponding equation numbers in Table 1.
Both temperature and stress ranges taken in these plots are considered to
be realistic values which can be achieved under LWR fuel clad operating conditions.
From these figures, one sees that while the Gittus' theoretical
model always underestimates creep rates in both types of plots, good agreements among other models are achieved only at low stress and low temperature
regions.
Fig. 11 and Fig. 12 which are applicable to CANDU reactor pressure
tube operating condition, except for the slightly higher fast neutron flux,
show that all creep models give acceptably good predictions below about 20 ksi,
since most of the curves fall within
the case, however,
the 95% confidence
interval.
in Fig. 13 and Fig. 14 when temperatures
350 and 4000 C respectively.
This is not
are raised
to
Greater departures are seen among these curves as
- 41 -
the temprature
is increased,
type relation,
temperature
This is again evidenced
As one might
have expected
from an Arrhenius
should have a very strong effect on creep
rate.
in Fig. 15 and Fig. 16 where the highest predicted
creep rate is about three orders of magnitude
larger than the one from the re-
gression equation.
Fig. 13, 14 and Fig. 16 warrant
arily high creep rates predicted
special attention
because
the extraordin-
by some of the models at combined
high stress
and high temperature would make it almost impossible to maintain clad geometries
even within a short period of reactor operating time.
stating that the regression model
We are not necessarily
is better than the rest of the models at
high temperature. There is no doubt about the danger in extrapolating beyond
the available range of data.
However, in situations where one has to extrapolate,
it is better if one proceeds on a sound statistical
basis.
Equations ob-
tained from parameter adjustment or curve fitting clearly fall short in that regard.
Figure 11
JO0603
COMPARISON OF CREEP LAWS AT T=250 C, FLUX=5E+13
.[10t
: 95% Confidence Interval
4
1
'
1.110
R
z
rC-)
m
m
ma
I
I
-o.
1.110
I
"l-
1.110
=--~~~~~~
~Gittus
-0
0
--
.10
-- 0.110
I.
I.OI
I
LN(STRESS)
-'
KSI
.x
C
1:i-I ,.
-..
43 -
J
JOQS1
Figure 12
COMPARISON OF CREEP LAWS AT T=300 C, FLUX=5E+13
: 95% Confidence Interval
r-
ra
m
t.IX0
I.t
'11
I
0--
Z-
1.110
Gittus
1.310
-me
*.ll~ao
,gt
,o-0§
,$g.,l
.3ir 0.30.
I
7.01
, I
.
I N(STRESS.1
I
LN(STRESS)
-1
KS
KSI
lol
.~. I
2.001
l.l
I
OC:
-
44
-
Figure 13
COMPARISON OF CREEP LAWS AT T=350 C, FLUX=5E+13
-i
JOeO
0e1
. 10
: 95% Confidence
Interval
1.X1O
z
C,)
m
m
m
1.11¢°
70
-4
rr
1.110 '
1.X007
2.,,o-,"
,.,/1o
·
l.7o
I
I
Ao~.,o
lebO., ezo
LN(STRESS)
.10
2.X
*°
I
--I--.O1
I
KSI
I
- 45
-j
-
Figure 14
JOOSt
COMPARISON OF CREEP LAWS AT T=400 C, FLUX=5E+13
.
,·O-
5.5.1W
: 95% Confidence Interval
1.110~0
z
Cr
m
1.110 0"
-4
m
%h
;n
-z
;B
11
t.110
' °
_
310
Cole
=
e
.
0
7.
I0
I
I
'0llOO'O+t,l9LN.S
R*E'SS)
LN(STRESS) KSI
-I
I
2.te °'
I
to
!
-
-J
46 -
Figure 15
4000sS
COMPARISON OF CREEP LAWS AT STRESS=7.5 KSI,FLUX=5E+l3
: 95% Confidence Interval
&.llO
-D
m
m
-0
3
5
2
1.g10-"4
|.110
R
4
1
-4
m
%..
1.110
1.to14"
Gittus
1.110
.=10
1
1
l/T
(I/DEG.
K)*,1000
0O
-
-
-
11
1
47 -
.
FiZUi.e 16
-I
u
JolseS
COMPARISONS OF CREEP LAWS AT STRESS=20
KSI,FLUX=5E+13
1.Xt10
: 95% Confidence
Interval
1.110
3
R
2
1
m
--
Z
Z.
1.310
1.1i t
I
1.110
Gittus
1.X10
7.110
I
I/T (1/DEG. K)*1000
0
- 48 -
5.4
Concluding remarks:
In this report we have discussed the-analysis of Zr-2 in-reactor creep
data.
Current empirical in-reactor creep models obtained from parameter adjust-
ments have been shown to be unsatisfactory both in terms of their underlying
normalization assumptions and also from their high predicted creep strain
rates.
Regression models, on the other hand, give significant account for
the stress, fast neutron flux, and temperature on the creep rate while allowing data analysis
to be done in a well established
statistical
manner.
The relative success achieved by the regression models in representing
the data not only suggests their values but also reveals the importance of
examining the raw data.
During the process of such an examination in this study
we have both eliminated certain improper data points and removed the bias
which may be introduced due to improper normalization assumptions. Furthermore, the limitations and the inherent inaccuracies of the data
revealed in the data examination are helpful in understanding the full weight
of the data such that they can be used in the proper framework.
In-reactor creep tests at LWR operating temperature and flux conditions are needed to extend the data range for verifying the adequacy of the regression models.
At present, we recommend the following regression model to be used as
the in-reactor creep constitutve equation for 20% cold-worked and stress relieved Zircaloy 2:
A
=
4o
)m (a
)nexp(jjfl7
y
1-a/
RT
- 49 -
where
the units are
= hr l 1,
: n/cm2s,
a: ksi, T: °K
and
A = 0.020
m = 0.613
n = 1.130
+o = 0.05x1013 n/cm2 s (E> 1Mev)
Qo - 8851.5 cal/mole
T
=
391.3 ksi
u (ksi) = 4.77x10
3
- 1.906xT(OF)
The overall 95% confidence interval for
iAf (SEE)^
= 0.812 and is given
can be constructed by using +
£
by
Y
exp (n
+ 0.812)
The above equation allows the dependence of activation energy on stress.
There
is evidence that the activation energy is also a function of temperature(l).
At sufficiently high temperature when irradiation creep is no longer significant as compared
to thermal creep, there
in the creep deformation process.
is a shift in dominating
The in-reactor
mechanism
creep equation over the
I-
entire
temperaturerange can be. constructed
as a composite of multi-stage
creep mech-
- 50 -
anisms acting over each of its dominating reqimes.
Temperature threshold
(or thresholds) at which the shifting of mechanism occurs could be themselves a function of stress and fast neutron flux.
Our present set of data
does not permit
these points mainly
us to extend the analysis
to verify
cause of the shortage of in-reactor creep data at higher temperatures.
be-
-51-
References:
1.
G.R. Piercy, "Mechanisms for the In-Reactor Creep of Zirconium Alloys",
J. of Nucl. M ts.,
2.
C.C. Dollins,
26 (1968),
F.A. Nichols,
18-50.
"Mechanisms
of Irradiation
Creep in Zir-
conium-Base Alloys", ASTM-STP(551), pp. 229-248, 1974.
3.
F.A. Nichols,
Base Alloys",
4.
J.H. Gittus,
"On the Mechanisms
J. of Nucl. Mtls.,
of Irradiation Creep
37 (1970), 59-70.
"Theory of Dislocation
in Zirconium-
Creep for a Material
Subjected
to
Bombardment by Energetic Particles - Role of Thermal Diffusion", Phil.
Mag., Vol. 30, No. 4, pp. 751-764, 1974.
5.
P.A. Ross-Ross,
C.E.L. Hunt, "The In-Reactor
Creep of Cold-Worked
Zircaloy-2 and Zirconium-2.5 wt% Niobium Pressure Tubes", J. of Nucl.
Mtls.,
6.
26 (1968), 2-17.
B. Watkins, D.S. Wood, "The Significance of Irradiation-Induced Creep
on Reactor Performance of a Zircaloy-2 Pressure Tube", ASTM-STP (458),
pp. 226-240,
7.
1969.
P.J. Pankaski, "BUCKLE - An Analytical Computer Code for Calculating
Creep Buckling
of an Internally
8.
Proprietary information.
9.
Proprietary information.
10.
F.A. McClintock,
1966.
11.
V. Fidleris,
A.S. Argon,
Oval Tube",
"Mechanical
"Summary of Experimental
BNWL-B-253,
1973.
Behavior of Materials",
Results on In-Reactor
Chap.
19,
Creep and
Irradiation Growth of Zirconium Alloys", Atomic Energy Review, Vol. 13,
No. 1., pp. 51-80.
12.
M.F. Ashby, H.J. Frost, "The Kinetics of Inelastic
Deformation
above O0K " ,
Constitutive Equations in Plasticity, pp. 117-147, 1975.
13.
G.E. Lucas, Modification of LIFE-I for Prediction of Light Water
Fuel Pin Behavior,
May, 1975.
Special
Problem, 22.901,
Reactor
Dept. of Nucl. Eng., MIT,
at 260 to 300 0 C",
14.
E.F. Ibrahim, "In-Reactor Tubular Creep of Zircaloy-2
J. of Nucl. Mtls., 46 (1973), 169-182.
15.
U.F. Kocks, A.S. Argon, M.F. Ashby, Thermodynamics
Progress in Materials Science, Vol. 19, 1975.
16.
N.R. Draper, H. Smith, Applied Regression Analysis, Wiley Interscience, 1966.
17.
C. Daniel,
F.S. Wood, Fitting
Equations
and Kinetics of Slip,
to Data, Wiley
Interscience,
1971.
-52-
18.
SSP, Scientific Subroutine Package, IBM-360, Correlation and Regression.
19.
SHARE Library (Number 360D-13.6.008) and VIM Library Number G2-CALLINWOOD.
20.
TROLL PRIMER, 2nd Ed., June 1975, National Bureau of Economic Research, Inc.
21.
SPSS, Statistical
22.
A.L.
23.
Timoshenko
24.
M.M.E. Wakil, Nuclear Heat Transport, p. 223.
25.
L. Breiman, Statistics:
Bement,
Package
for the Social Sciences,
2nd Ed., McGraw-Hill,
to be published.
and Goodier, Theory
of Elasticity,
2nd Ed., p. 59.
With a View Toward Applications, Houghton Mifflin,
1973, Chap. 9.
References
in Table 1:
(E-l)
J.J. Holmes, et al., "In-Reactor
ASTM-STP-330, p. 385, 1965.
(H-1)
V. Fidleris, "Uniaxial In-Reactor Creep of Zirconium Alloys", J. of
Nucl. Mtls.,
26 (1968),
Creep of Cold-Worked
Zircaloy-2",
51-76.
(H-14)
Same as (5).
(I-11)
E.F. Ibrihim, "In-Reactor Creep of Zirconium Alloy Tubes and Its Correlation with Uniaxial Data", ASTM-STP-458, pp. 18-36, 1969.
(I-14)
Same as (6).
(K-6)
D.S. Wood, B. Watkins, "A Creep Limit Apgroach to the Design of
Zircaloy-2
1975.
Reactor
Pressure Tubes
at 275 C", J. of Nucl. Mtls., 41
(1971), 327-340.
(M-5)
Same as (14).
(M-7)
D.S. Wood, "The High Deformation Creep Behavior of 0.6 inch Diameter
Zirconium Alloy Tubes Under Irradiation", ASTM-AIME Symposium on
Zirconium in Nuclear Applicationsi Portland, Oregon, August 1973.
- 53 -
'Appendix A
Zircaloy-2 Tubular In-Reactor Creep Data
4
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- 58 -
Appendix
The Actual
B
Set of In-Reactor
Creep Data
Used in Regression Analysis
59
ID
T(°K)
q(10 3 )
n/cm s
o(ksi)
e(10
hr
1
)
39
41
43
45
47
49
51
52
53
54
55
536.000
536.000
536.-000
536.000
536.000
536.000
536.000
536.000
536.000
536.000
536.000
16,200
19.700
24.800
29,600
34.,600
37,800
16,200
19.600
24.500
29.200
34.400
2.900
2.900
2.900
2.900
?.900
2.900
0.050
0.050
0o.as050
0.050
0.050
0.140
0.200
0.220
0,300
0,500
0.700
0.010
0.017
0,033
0.039
0.071
56
14
15
16
17
18
19
4
6
536.000
531.000
531.000
531.000
531.000
531.000
531.000
554.000
554.000
39.200
15,700
19,400
23.900
29.200
34,000
37.600
13.400
17.300
0.050
2.560
?.560
?.560
?.560
?.560
?.560
3.100
3.100
0.092
0.130
0.240
0.?70
0.350
0.610
O.BO
0.P30
0.290
1
600.000
4,900
3.100
0.091
11.500
'.700
P.700
1.150
3.100
3.100
1.150
0.190
0.190
0.054
0.077
0.128
0.037
0.053
0.054
0.059
0.127
0.155
0.194
0.338
0.133
0.239
0.117
0.208
3
5
2
77
78
87
88
97
98
107
108
117
118
127
128
137
138
ID refers
554.000
543.000
537.000
573.000
618.000
525.000
546.000
525.000
546*000
543.000
565.000
543.000
565.000
525.000
561.000
513.000
530.000
to the serial
14.000
10.500
5.500
5.500
10.500
10.500
10.500
10.500
11.500
11,500
13.500
13.500
14.000
14.000
14.000
14.000
numbers
1*150
1.150
1.150
'.500
2.500
3.100
3.100
?.600
'.600
?.600
7.600
in Table
4, ID-77 to ID-138
are obtained by transforming the normalized Ross-Ross and
Hunt's data serial T7-T13 to the original values.
- 60 -
Appendix C
Regression Statistics
- 61 -
The symbol meanings
in Appendix
C and Appendix
D are:
CRL2
: 1n
SNLOG
: ln(a/u(T)), normalized stress
FLUX4
: ln({/0.05), normalized flux
TI
: 1/T
STI
: a/T
STIA
. a2 /T
S3
: lna
FLUX2
: lnq
- 62 -
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I
Part II
.1
ADDENDUM
TO
A REGRESSION APPROACH
FOR
ZIRCALOY-2 IN-REACTOR CREEP
CONSTITUTIVE EQUATIONS
by
YUNG LIU, Y.
and
A.L. BEMENT
(September, 1976)
Department of Nuclear Engineering
and
Department of Materials Science and Engineering
Massachusetts Institute of Technology
Cambridge, Mass. 02139
Table of Contents
Page No.
i
Abstract -------------------------------------------------------Table of contents ----------------------------------------------ii
List of tables
List of figures
iii
-----------------------------------------------------------------------------------------------------------------------
iv
1.1
Introduction
1.2
Deletion of some questionable data
1.3
Inclusion of additional creep models
1.4
Verification of whether thermal and irradiation creep components are additive -------------------------------------
4
2
Results and discussions
----------------------------------
6
2.1
Creep model comparisons
------------------------------------
6
-----
6
2.1.1 Stress dependence
------------------------------
--------------
---------------------------------
2.1.2 Temperature dependence
1
11
--------------
11
Regression analysis with revised creep data
2.3
Verification
----------------------------------------
18
-------------------------------------
2.3.2 Irradiation creep analysis
20
----------------------
32
Comparison of regression creep models
2.5
Considerations in applying Zircaloy in-reactor creep constitutive equation --------------------------------------------Summary
References
18
---------------------------------
2.4
2.6
2
-----------------------------------
2.2
2.3.1 Thermal creep analysis
1
----------------------------------------------
--
-------------------------------------------------------
38
40
42
Appendix 1 : The revised Zircaloy-2 in-reactor creep data set used
in regression analysis ---------------------------
43
Appendix 2 : Summary of current Zircaloy in-reactor creep models
and their data bases --------------------------------
45
ii
List of Tables
Table Number
Page Number
Title
A-1
Multiple Regression Coefficients, F Statistics
Using the Revised Data Set
11
A-2
Estimated Coefficients, SEE's Using the Revised
14
Data Set
A-3
Multiple Regression Coefficients, F Statistics
for Thermal Creep Analysis
19
A-4
Estimated Coefficients, SEE's from Thermal Creep
Analysis
20
A-5
Multiple Regression Coefficients, F Statistics
for Irradiation Creep Analysis
22
A-6
Estimated Coefficients, SEE's from Irradiation
Creep Analysis
23
iii
List of Figures
Figure Number
A-1
Page Number
Title
1
=5-10"
Comparison of Creep Laws at T=250C,
J
7
n/cm s (E>1 Mev)
A-2
Comparison
n/cm
A-3
2
n/cm
=5 101
3
8
s (E>1 Mev)
Comparison
2
of Creep Laws at T=3000 C,
3
of Creep Laws at T=3500 C,
=5 · 10 1
9
s (E>1 Mev)
0
3
Comparison of Creep Laws at T400
n/cm2 s (E>1 Mev)
A-5
Comparison of Creep Laws at a=7.5 ksi,
n/cm2 s (E>1 Mev)
013
=5 · 1i
12
A-6
Comparison of Creep Laws at a=20 ksi,
n/cm2 s (E>1 Mev)
=5- 11013
13
A-7
Residual Plots (a),(b) for Regression Models (3)
and (5) respectively
15
A-8
Residual Plots (a),(b) for Regression Models
and (10) respectively
(6)
16
A-9
Residual Plots (a),(b) for Regression Models
and (12) respectively
(11)
17
A-10
Residual Plots (a),(b) for Regression Models
(A-13) and (A-14) respectively
24
A-11
Residual Plots (a),(b) for Regression Models
(A-15) and (A-16) respectively
25
A-12
Residual Plots (a),(b) for Regression Models
(A-17) and (A-18) respectively
26
A-13
Residual Plots (a),(b) for Regression Models
(A-19) and (A-20) respectively
27
A-14
Residual Plots (a),(b) for Regression Models
(A-21) and (A-22) respectively
28
A-15
Residual Plots (a),(b) for Regression Models
(A-7) and (A-8) respectively
29
A-16
Residual Plots (a),(b) for Regression Models
(A-9) and (A-10) respectively
30
iv
C,
=5-10:
A-4
10
List of Figures (Continued)
A-17
Residual Plots (a),(b) for Regression Models
(A-ll) and (A-12) respectively
31
A-18
Comparison of Regression Creep Laws at T=3000 C,
+ =5-1013n/cm2 s (E>1 Mev)
33
A-19
Comparison
Creep Laws at T--3500 C,
34
Comparison of Regression Creep Laws at T=4000 C,
35
of Regression
13
4 =5-10 n/cm s (E>1 Mev)
A-20
2
0 =5'10 1 3n/cm
A-21
s (E>1 Mev)
Comparison of Regression Creep Laws at a=7.5 ksi
0 =5-10
A-22
2
13
36
n/cm 2 s (E>1 Mev)
Comparison of Regression Creep Laws at a=20
4 =5'101 3 n/cm 2 s (E>1 Mev)
v
ksi
37
*
1.1
Introduction:
Since our last report released in May, 1976(1), we have made additional
studies on the subject of Zr-2 in-reactor creep constitutuve equation.
This
is the first addendum in which further results are presented.
The overall
structure
and the methods
of analysis
essentially the same as in our previous report.
in this addendum
are
Reanalysis and modifications
were made, however, because (1) the deletion of some questionable data from
the earlier creep data set and (2)the inclusion of two additional creep models.
Finally, attempts have also been made to verify whether the total creep strain
rate can be divided into two separate and additive thermal and irradiation
creep components.
1.2
Deletion
of some questionable
data:
Although great care has been taken in examining the raw in-reactor creep
data in the previous analysis, it was recognized later that some of the data
may have been repetitively used.
The following table summarizes which data
have been deleted and why we deleted them.
contained
in Appendix
1.
ID No
4,6
The revised creep data set is
Reason
for deletion
repetitive, they can be obtained as average of
117 and 118
3
repetitive, can be obtained as average of 107,108
1,77,78
repetitive,
2,87,88
incompatible creep test duration, 13,000 hrs
repetitive, 2 can be obtained as average of 87,88
5,127,
128
incompatible creep test duration, 11,000 hrs
repetitive, 5 can be obtained as average of 127,128
97,98
incompatible creep test duration, 29,000 hrs
137,138
incompatible creep test duration, 11,000 hrs
*
can be obtained
as average
The ID numbers are referring to Appendix B(1 )
of 77,78
-2 -
Creep test durations for the majority of creep data contained in the revised
data set are of 5,000 hours.
We deleted those data with incompatible test
durations because the Zr-2 in-reactor creep rate may decrease with time(2)
1.3
Inclusion of additional creep models:
In the comparison of current Zr-2 in-reactor creep models, two additional
ones are included.
One is taken from MATPRO( 3) and the other from the Prelimi-
nary Standardized Material Property Data Set used in the EPRI program for LWR fuel
(4)
rod code evaluations(
likelihood
.
This latter model is worth mentioning because of the
it has of being incorporated
into one or more of the six leading
LWR fuel performance codes which are now under extensive evaluations.
The Zircaloy in-reactor creep constitutive equation in MATPRO is abbreviated
as CCRPR( Clad CReeP Rate Model ) and a formula by
t
t=K
(a + B e
) exp(-
10,000/R T) t- 0 .5
(A-l)
where the corresponding units are
T:
K,
t: sec,
: fast neutron
a: tangential stress, N/m ,
flus, n/m s
(E> 1 Mev)
t: tangential creep rate, m/m/sec
and
K = 5.129.10
B = 7.252102
C = 4.967.10
-8
R = 1.987 cal/mole OK
The creep equation used in the EPRI program is given by
- 3-
e=
+ 1.0110
1e
{k
-1 9
exp(- 0.004 t)}
ae exp(- 8,000/RT) +
{k2 a (25.6 - 0.026 T) exp(- 138,000/RT)}
(A-2)
where
e
: effective strain rate, hr 1
a : effective
e
t: hours
stress, ksi
0: fast neutron
( <40 ksi)
flux, n/cm 2s (E>1 Mev)
R: 1.987 cal/mole OK
T:
K ( <773 OK)
-18
kl: 0.667. (1.36-10
)
-28
k2: 0.667.(7.010 )
} Zircaloy is assumed to be isotropic
In order to change (A-2) into circumferential strain rate and tangential
stress, simplifying assumptions of isotropic material and thin shell approximation were introduced with the following transformations.
Isotropic material
ae =
1
2
2
2
(z-a )
+ (a8 ar)
+(r- az )
(
r
1/2
(A-3a)
Thin shell approximation
a
negligible;
a
= 0.5a e
(A-3b)
(A-3a) then reduces to
a
e0
= 0.866 a
The circumferential and effective strain rates relationship
(A-4)
-4-
=
Y
(ar+
az) } ~~~~~~~~e
{ ae-
(A-5)
e
with a ,
r and a
e
r
z
4
substituted from (A-3a), (A-3b), and'(A-4), Eq. (A-5)
becomes
Be= 0.866
0
e
(A-6)
e
We have written a FORTRAN program to obtain ae from (A-4) and substituting that into
(A-2) to get
e.
e is then multiplied
by a factor of
0.866 to get the circumferential creep strain rate.
It should be noted that both these two models have a time dependent
parameter, t.
By inspection one can see that the predicted creep rates are
decreasing as irradiation time increases. We have discussed this time
dependence before in our previous report and have raised the question of
whether there is a steady state creep regime for Zircaloy in reactor.
This
issue is difficult to resolve at the moment primarily because of the effect
of long term in-reactor creep tests during which the creep rate sensitivity
on time can easily be masked
parameters
due to the fluctuations
in the controlled
, a, and T.
The time dependency in these two models is
believed to be coming from
modeling of the long term effect rather than for the primary creep regime.
For purpose of comparison with other creep models, the time parameter in (A-1)
and (A-2) is taken equal to 5,000 hours.
1.4
Verification of whether thermal and irradiation creep components are
additive
:
Several current Zircaloy in-reactor creep models have had similar forms
in which the total creep strain rate is expressed as the sum of thermal and
irradiation creep components. We have made our attempt to verify this directly
from our data set based on the presumption that the two components are indeed
additive.
The verification is made possible because of the unique creep test
-5-
specimen arrangement in Ibrihim's in-reactor tests(2)
From Fig.4 in the previous report, one can see that actually there are
three portions in Ibrihim's specimens and these are designated by fast flux,
thermal flux and out-of-flux specimens respectively.
Each portion according
to its relative axial location with respect to the test reactor core receives
neutron flux of different energy spectrum ranging from ~0 to E>1 Mev.
The second column to the right of Table-4 in our previous report gives
the minimum control creep rates which are obtained from creep strain measurements for out-of-flux specimens.
Since the temperature and pressure of the
coolant in this portion are basically the same as in the other two portions;
the tube outward dimensional change is due mainly to thermal creep and the
fast neutron or irradiation creep contribution is negligible.
This makes a
very good case for the attempted verification since the total creep rate
obtained at the fast flux portion, if it were made up by thermal and irradiation components
and the two are independent
and additive,
the difference
between the total creep strain rate and that of the control(or thermal) creep
strain rate should be a strong function although not necessarily an exclusive
function of fast neutron
flux.
Regression analyses were performed progressively on (T
with using
only and then with different combinations of
E-th)starting
, a and T.
Regressions were also carried out on the control creep strain rate data
by using a family of formula of the following kind
A a
A
exp
Q
T)
exp(- Q'/R T)
Creep strain rates from both irradiation and thermal creep regression
equations
are plotted
against
a and 1/T and so are their sums.
-6-
2.
Results
and Discussions
:
2.1
Creep model comparisons :
The comparisons of current Zircaloy in-reactor creep models are contained
in Fig. A-i to Fig. A-6.
The same reference fast neutron flux value was used
for these figures as well
as for all the latter
figures
in this addendum.
2.1.1 Stress dependence:
Fig. A-1 to Fig. A-4 show the stress dependnece of the predicted creep
rates from various in-reactor creep models.
at 51013
300
The fast neutron flux is fixed
n/cm2s and temperatures are systematically increased from 250 °C to
C and 400 C.
C, 350
The broad
darkened
lines in these figures represent
our in-reactor creep model obtained from regression analysis.
Both 95% confi-
dence interval and the transformed actual creep data are plotted for comparisons.
Transformation of the actual creep data to the fast flux and temperature values
of interest is done by using the same scheme as depicted in the earlier analysis.
That is, we use our regression model as the basis for transforming the actual
data on a, T,
and
to the corresponding values of interest and the regression
model provides the necessary scaling factor.
Much of the same trend on stress dependence is observed for the two added
creep models and the regression model obtained from using the revised data set.
Good agreement in these models is found'only when temperature is relatively low
and stress is low to moderate.
As temperature increases, its effect begins to
override the stress effect and the greatest descrepancies among models are seen
in Fig. A-4 when temperature is raised to 400 °C.
Again, BUCKLE gives the
highest predicted in-reactor creep rate which is clearly unacceptable if clad
hoop stress should exceed 20 ksi.
Both model (A-I) and model (A-2) agree reasonably well with other models
at lower temperatures, 250
0
C-300
creep rates at low stress regime.
C, although they predict slightly higher
At higher temperatures, 3500 C-400°C, these
two models underpredict creep rates at high stress regime as compared to most
-7-
0.1-0
.|l1
___ : 95% confidence interval
4
1
I-
m
-IA
Wn
%Z.
ZC
&.gI1-"
.
1.010-
Gittus
.:
9. ,
0
I,
I!
",'.
I
I
I
,
I.4,
I,..el
I
,.,le`
I
,.,,,,
LN(STRESS) KSI
Figure A-1
Comparison of Creep Laws at T=2500 C, *= 5.1013 n/cm2 s (E>1 Mev)
(Numbers associated with the curves are equation numbers given
in reference 1. They can also be found in Appendix 2)
I
-84.X1e4-
1.110 4
r-
z
dm~
t)
rrt
rr
-
z:B
'--4
qemP
Z..
1.110
xs-
I.iloe
oy
Gittus
1.110o
.
,,
1 4
I
I
III
I
I
.
LN(STRESS) KSI
Figure A-2
Comparison of Creep Laws at TI300°C, b= 5.1013n/cm2s (E>1 Mev)
I
-9
-
a
rr
95- CRfidence Interal
Io.le-t
2
4
A
ru-
X_
I
,
814
lzf
s Xs*-
5··1
#.XI*-"
C~~~~~~~~~itt
yle
Cas
I
I
"""·~~4.80"
I
s**
I
1
WNSTRESS)
KS,1
Figure A-3
Compa
ris 0of
~ Creep Laws
at
T-3500C ip 0
510 13
2m
n/m
(E~l Mev)
-4
- 10 -
4oI
--
0-
rn
rr
m
-4
1.210
1.310
1.
.
-7."10
.
-a
--
f"-+
I
I
S
LN(STRESS)
Figure A-4
I
I
.,41
I *1
KSI
Comparison of Creep Laws at T=4000C,
= 510 13n/cm2 s
(E> 1 Mev)
I
r
-
of the other models.
2.1.2 Temperature dependence:
Fig. A-5 and Fig. A-6 give the 1/T dependence
of the in-reactor
rates at fixed fast neutron flux and two levels of stresses.
creep
Again one can
observe similar trends on T dependence for the various curves.
Model descre-
pancies become more evident when temperature is raised to a higher level.
(A-I) and (A-2) also underpredict creep rates at high temperatures as compared
to other models.
2.2
Regression analysis with revised creep data set:
As a result from the deletion oftquestionable creep data, sixteen out of
Using the remaining twenty-two
thirty-eight previous data were eliminated.
data, we have rerun the analysis with the same six starting model equations
(1)
Table A-1 and Table A-2 show the regression statistics, estimated coefficients
and the standard error of estimates for these models.
We shall first look at
the regression statistics given in Table A-1.
Table A-1
Multiple Regression Coefficients,F Statistics Using the Revised
Data Set
R2 (%)
F
3
97.712
126.638
5
98.381
128.086
6
98.381
10
97.713
126.716
11
98.380
128.014
12
98.380
96.394
Equation No*
96.443
* : All equation numbers without prefix refer to equations in
reference
(1)
- 12 -
.@t
: 95% confidence
interval
____o
0.i10
'
-@01
3
5
A-7
2
4
A-1
3
m
m
A-2
1
1.310-
-I
'O
m
1.10
3:
.47
1.310
Gittus
1.X10
I.I10 .
.
1/T (I/DEG.
Figure A-5
Comparison of Creep Laws at
K)*1000
a=7.5 ksi,
= 510 13n/cm2s (E>1 Mev)
- 13
.·-..
.-
n-
-
interval
,-ofadence
3
5
'1
I .Xi 4
4
A-7
A-1
-A-2
r,
1I
m
z2"
-
l.li@
1 .51
1.1
9
.3.0
-
.
,7
1.4..4
|
Lz
1
t I
I
I
1.~
I
I
I
I
I
I.
0
I/T /EG.
r
K)*100 0
1
Figure A-6
at a=20 ksi,
Comparison of Creep Laws
5
10
n /c m
s (E>1 Mev)
- 14 -
The meanings
of R 2 and F are explained before.
the following general
conclusions
: (1) Based on R
can be considered significant; (2) R
2
We are thus able to make
and F, all regressions
increases as more complicated model
equations are used.
The fact that the respective
R2 values
for Eq.(5),
(6) and Eq.(11),
(12)
are the same tells that there is probably little or no gain at all to use a
second power in Eq.(6) and (12) for the stress dependent activation energies.
Although Eq.(10) has a slightly higher R2 than Eq.(3) by normalizing stress
with respect to temperature dependent shear modulus, the reverse is true for
pairs of equations
(5), (11) and (6), (12).
As we shall discuss
later that
the more complicated models may not be adequate under certain situation. We
shall next examine another important piece of evidence, the residual plots.
Fig. A-7 to Fig. A-9 contain
two residual plots
six creep regression model equations.
in each figure for the
Although the error bands are narrower
for the more complicated equations, all their residual patterns look quite
satisfactory
Table A-2
and there are no signs of model
abnormalties.
Estimated Coefficients, SEE's Using the Revised Data Set
Eq. No.
lnA
m
n
Q/R
3
SEE
-21.484
---
0.597
0.033
1.827
0.181
9519.05
2310.16
-----
5
SEE
-20.382
---
0.5847
0.029
-0.795
1.007
6937.05
2229.72
58.290
22.118
6
SEE
-20.502
---
0.5848
0.031
-0.744
3.934
6917.09
2717.10
55.794*
185.01
10
SEE
8.735
---
0.597
0.033
1.827
0.181
9019.03
2281.32
-----
11
SEE
-10.731
---
0.5847
0.029
-0.789
1.007
7160.30
2101.44
58.152
22.109
12
SEE
-10.028
---
0.585
0.031
-0.660
3.933
7074.85
3316.06
51.903*
185.047
* In Eq.(6) and (12), the coefficients
_Q/R
Qo/R
2
--0.026
1.927
---
--0.066
1.928
for o/T are in form of 2Qo/R
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- 18 -
It is apparent from Table A-2 that except for creep models (3) and
(10), the estimated coefficients for n which is the stress exponent are
all negative,
We tentatively rejected those models having negative stress
exponent because they contradict to whatis believed to be the
stress exponent in power law creep.
that the resultant
It should be pointed out, however,
creep rate is still
increasing
in these models
increases despite of the negative stress exponent.
as stress
This is because of the
overriding effect of the stress-temperature interaction factor included in the
exponential term of these models.
Eq.(10) is finally selected as our regression creep model for the
slightly better regression statistics it has over (3) and more importantly
its more logical and consistent physical interpretations.
The explicit
formula for (10) will be given at the end of this addendum.
2.3
Verification:
In section 1.4, we have already discussed the assumptions and data
characteristics for the attempted verification of whether thermal and
irradiation creep components are additive.
We shall proceed first in the
following with the thermal creep analysis.
2.3.1 Thermal creep analysis:
The family of regression model equations for thermal creep analysis
on the minimum control creep rate data are the followings:
th
=
A' a
tth = A' a
th
A'
exp(-
Q'/RT)
exp{- Qo(1-
exp{
th = A' (ao/(T))'n
Q(1-
(A-7)
a/f)/RT}
/f)
/RT}
exp(- Q'/RT)
(A-8)
(A-9)
(A-10)
- 19 -
Cth
Cth
=
A'
(/p(T))
n '
exp{- Q(1-
a/t)/RT}
A'
(a/p(T))
n
expf- Q
a/)2/RT}
o(1-
(A-ll)
(A-12)
Table A-3 shows the R ,F values for the six thermal creep models above
and Table A-4 gives the estimated coefficients and standard error of estimates.
Multiple regression coeffcients, F statistics for thermal
creep analysis
Table A-3
Equation No
R2 ()
F
A-7
94.654
81.798
'A-8
96.169
73.833
A-9
96.790
63.038
A-10
94.658
81.858
A-11
96. 168
73.812
A-12
96.778
62.777
We shall postpone the examinations of residual plots from these models
until later but using the same arguements as we did in the previous section
that based on information contained in Table A-3 and Table A-4, (A-10) was
selected as our regression thermal creep equation.
by the following
eth -
This equation is given
formula:
A' (a/ 1 (T))
exp(- Q'/RT)
(A- 10)
where the units are
eth:
and
h1r , : ksi, T:
K, R=1.987 cal/mole "K
- 20 -
3 - 1.906T(F)
u(ksi)=4.7710
A'= 13200.37
n'- 2.441
Q'i 15476.28 cal/mole
Table A-4
Estimated Coefficients, SEE's From Thermal Creep Analysis
Eq. No.
lnA'
n'
Q'/R
A-7
- 9.390
2,441
8456.90
SEE
---
0.205
2649.68
A-8
- 8.618
-0.528
5513.53
66.291
SEE
---
1.143
2569.92
25.198
-7.445
8291.71
403.547*
3.998
2875.32
189.194
2.441
7788.77
0.--205
2616.40
-0.521
5663,45
66.158
1,142
2424.62
25.189
-7.379
10300,87
400.600*
A-9
SEE
A-10
7.282
___
9.488
SEE
A-11
-12.651
SEE
A-12
-49.917
SEE
---
4,008
3473.53
Q'/Rt
-0--
189.77
2
Q'/Rf
-0-
---
-3.557
1.979
-3.528
1.986
* In (A-9) and (A-12), the coefficients for a/T are in the form of 2Q'/R?
2,3,3 Irradiation creep analysis:
In this part of analysis as we mentioned earlier, the differences between
minimum in-reactor creep rate and minimum control creep rate were taken from
our revised creep data set and regressions were performed on these differences
in the following
progressive
steps.
="lA"
T -th
T=
/
A" (
th
T - eth
T
(A-13)
Allm
anc
(A-15)
th
T
/ O( )m
th =A"
T -P
T
th
th
T
= A"
=th
*m
= A"
(
-t
T
(A-16)
( c/p(T))
n" exp(- Q"/RT)
(A-17)
n"
)
A" mA an
(/m
(/(T))n
exp(- Q"/RT)
exp{- Q"(1-a/t)/RT}
(A-18)
(A-19)
th
- =( /)m
A"
T
(A-14)
mo
-
T
th
th
= A"
ma
th
A"(
( a/p(T))
exp{-Q"(1-a/t)/RT}
exp{- Q"(1-a/t) /RT}
o
0
(A-20)
(A-21)
) (cr/x(T))pn
exp{-o"(l-/t)2/RT}
(A-22)
0
Table A-5 and Table A-6 give the regression statistics,estimated coefficients, SEE's for (A-13) to (A-22), The residual plots are contained in Fig.A-10
to Fig,A-14 for irradiation creep analysis and Fig,A-15 to Fig.A-17 for thermal
creep analysis.
The information can be summarized as follows.
all possible combinations of
and in dimensionless forms.
In short, we have considered
,a and T both in their original parameter units
A number of observations can be made from Table A-5
and Table A-6 for the irradiation creep analysis.
First of all, in using
alone,
we are able to reproduce the fast neutron flux exponent m"=0.859 which has been
so idiscriminatively
proposed it in 1969(5 )
used in other creep models since Watkins and Woods first
However, as we turn to look at its R2 value as compared
to others, it is clearly inferior,
But more importantly when we look at the
- 22 -
Table A-5
Multiple Regression Coefficients, F Statistics For
Irradiation Creep Analysis
R2
Equation No
)
F
A-13
92.076
94.699
A-14
92.076
94.699
A-15
97.276
140.879
A-16
97,358
145.453
A-17
98.269
140,654
A-18
98.270
140.763
A-19
98.768
139,412
A-20
98.766
139,237
A-21
98,831
109.248
A-22
98.834
109.525
residual plots in Fig,A-10, there are distinct patterns of residual seggregations from which we know such models (A-13), (A-14) are not acceptable,
Secondly, although the residual structures improve considerably when more
parameters are introduced, e,g,, with more complicated models having stress
dependent activation energies; we either obtain negative stress exponents as
in (A-19), (A-20) or we have the associated standard error of estimates which
are sometimes even larger than the estimated coefficients. What this means is
that the uncertainties in the estimated coefficients are larger than the magnitudes of the estimated coefficients themselves and these models should also be
regarded as unacceptable,
choose from.
our model
We are, therefore, left with (A-15) to (A-18) to
Using the same arguements as before, (A-18) has been selected as
for the irradiation
creep component
and its explicit
formula
is given
by
err
QT
T
ir
eh=
(ap(T))
th =_ A"
A" (
( mo) i"(RT)
(a/p(T))
exp(- Q"/RT)
exp(-
(A-18)
- 23 -
where the units are defined as before and
A" = 6,234
m" - 0.905
n" = 1.482
Q" = 13870.47 cal/mole
Estimated Coefficients, SEE's From Irradiation Creep Analysis
Table A-6
Eq
No.,
m"
lnA"
nIT
Q"/R
A-13
-41.758
0.859
SEE
---
0.088
A-14
-18.618
0.859
SEE
---
0,088
A-15
-46,569
0.906
1,101
SEE
---
0,055
0,203
A-16
-12,982
0.9065
1.126
SEE
---
0.054
0.203
A-17
-34,012
0.905
1,482
7385.63
SEE
---
0.045
0.213
2536.23
0,905
1.482
6980.61
0.213
2499,91
A-18
1.830
< R
Q"/R
--
SEE
---
0,045
A-19
-31,955
0.877
-1,280
4928.84
SEE
---
0,041
1.179
2447.49
A-20
-18.142
0.8776
-1,270
62.061
SEE
---
0.041
1.179
5287.48
2302.13
A-21
-40,693
0,878
2.384
-1i21.330*
1.988
SEE
---
0,042
4.551
3247,37
3191.34
221.670
2,383
0.878
2,522
-128.000*
2,059
0,042
4.546
2504,38
3970,95
221.520
2.383
A-22
SEE
*
2.157
---
In (A-21),
(A-22) the coefficients
62.284
26.265
26,261
---
for alT are in the form of 2Q"/RT
241 -
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- 32 -
2.4 Comparison of regression cree
models:
Fig.A-18 to Fig.A-22 give the comparisons of creep models obtained from
regressions.
Five regression creep models are included.
These are (10),
(A-10), (A-18), the sum of (A-10), (A-18) and one based on the revised Ross-
Ross and Hunt's model.givenin the previous report
This last
model has
different estimated coeffcients because the revised creep data set was used
in the analysis.
Notice that all except the thermal creep curves (A-10) are bounded by the
95% confidence interval derived from (10), The curves of total creep rate
obtained as the sum of (A-10) and (A-18) to that of the irradiation creep rate (A-18)
are often indistinguishable on these figures because the thermal creep contribution
is
,gtligible,i
.e., at least
two orders
of magnitudes
irradiation creep component based on the above studies.
lower than the
There are differences
between the total creep rate and the one obtained from (10) in which no efforts
have been made to divide the thermal and irradiation creep components and treat
them separately, but the differences are slight.
It should be pointed out
though that the numbers of data which we used in
obtaining regression creep models for
data were used for
th and
irr are different.
th whereas nineteen were used for
irr'
Twenty-two
This inconsistency
arises because in taking the difference between the minimum in-reactor creep
rate and the minimum control creep rate, three of the differences come out to
be negative which means that the control or thermal creep rates at those three
points are larger than the corresponding in-reactor creep rates.
The origins
for such inconsistencies could either due to data recording mistakes, instrument
errors, etc. but we have not been able to verify the exact causes.
These three
negative differences are removed because they cause problem in logarithmic
operations.
Such removals can be partially justified from the fact that in all
figures, Fig.A-18 to Fig.A-22, there are only minor differences between the
total creep rate and the one obtained directly from (10).
- 33 -
: 95% confidence
_
_
interval
_ _
l..B
.
I OmW
(10
(16
(A-lO)+(A-18
(A-18
IZ
(A-10
2C
W
m
z
ziZ
Iron
ZI
I.Ue
0
1.210@
1.1
1
m
6.
,.,,x3
s.
I
* u
I
v u
I
I
Y u 1
I.I
I
a
?
LN(STRESS)
I
I
I.!
4.441
KSI
l
Figure A-18
Comparison of Regression Creep Laws at T=300 C, = 510 13n/cm2s
(E>1 Mev), (Numbers associated with the curves are equation
numbers
and they
can be found
in Appendix
2)
- 34 -
-
: 95% confidence interval
10
1.X11
(16
(A-10)+(A-18
(A-18
(A-10
r-
rz
-o
1.11
'
-0
m
1.X1i
4
emlJ
:3
1.110
I.Ilt tl
I )1
.IO4
1.. l*0 .
I
I
I
I
I
LN(STRESS)
Figure A-19
4.110
CN.O a
KSI
0
Comparison of Regression Creep Laws at T=350 C,
(E>1 Mev)
1
3310
= 51013n/cm 2s
- 35 -
: 95% confidence
interval
(
o.a
(1
"
(A-1O)+(A-1
(A-1
C,
I.310-
m
m
01%
"O
-
i
I.U
W
M
la
M30
Mt
Ag
1.106t
.,
I-IS0I
sexl r
I
I
I
I
10
I..1
i. IOi
't
I.IllII..
LN(STRESS) KSI
1
Figure A-20
Comparison of Regression Creep Laws at T400'C,
(E>1 Mev)
*=51013n/cm2s
I4
Clolle
- 36 -
1 8
I .O
: 95% confidence
interval
1.110
,
.
v
-ale
1.310
1.310
aNO-
I-xI*-
I
I/T (l/DEG. K1000
r
Figure A-21
Comparison of Regression Creep Laws at o=7.5 ksi,
(E>1 Mev)
4= 510 13n/cm2s
- 37 -
1.I10
: 95% confidence
A
interval
,.
.
1
1.S14
m
m
1.310
m
-
z
z
i.l
1
1.X10
,.x"
L I
1.4
I
I
I
I
I
1.1
I
I
I
I
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1.0
I
I
I
I
I
1.7
I
I
I
I
I
1.0
I
I
I
I 1.I
I
I/T [1/DEG. K)*1000
r
Figure A-22
Comparison of Regression Creep Laws at
(E>1 Mev)
=20 ksi, 4= 510 13n/cm s
- 38 -
Our tentative conclusion regarding to whether thermal and irradiation creep
rates are additive
is that at the particular
test parameter
ranges of
and T
covered by the creep data set, they probably can be treated as additive provided
the appropriate formula
have been derived for each of the two separate components.
The above conclusion is based primarily on the regression statistics and the
observations we have from Fig.A-18 to Fig.A-22.
We should give our preference
as to which final regression creep model we are going to recommend.
derived from the revised creep data set is our choice.
Eq,(10)
The basis for such choice
is made clearer when we examine the respective residual plots.
It is apparent from Fig,A-8(b), Fig.A-12(b) and Fig.A-16(b) that both the
error band and residual structure are better for (10) than either (A-10) or (A-18).
In the latter
two cases,
Fig.A-12(b)
which
is from irradiation
creep analysis
shows a clear residual clustering and Fig,A-16(b) which is from thermal creep
analysis gives the largest residual scattering.
a less random nature
2,5
than the residuals
Both of these features are of
in Fig.A-8(b)
which
is obtained
from
(10).
Considerations in applying Zircaloy in-reactor creep constitutive equation:
At least two important considerations should be kept in mind in applying
the Zircaloy in-reactor creep constitutive equations for LWR fuel clad:
(1)
Caution must be taken in selecting the in-reactor creep models for LWR clad
deformation evaluation,
underprediction
Comparatively speaking among the different creep models,
of creep rate means
that it is less conservative
overprediction means perhaps more conservative.
whereas
For design purposes, it is
probably safer to use a in-reactor creep equation that gives higher predicted
creep rate.
However, such equation should not be derived in the improper
manner as we discussed in our last report and the cost of the likely overconservatism
(2)
is always at stake,
It has been experimentally determined that Zircaloy exhibits a so called
- 39 -
"Strength Differential (SD)" effect in the yielding phenomenon(6 '7 '8 ).
For Zircaloy, the yield strength in compression is about
greater than the yield strength in tension.
effect may exist for creep as well,
at MIT to study this possibility.
10%
It is suspected that the SD
Currently, investigations are underway
We merely point it out because the SD
effect certainly warrants attention in that during the early to mid-stage
of a LWR fuel element life, the clad stresses are essentially compressive
under the high external coolant pressure.
The creep equations that we have obtained are based on in-reactor
creep tests in which Zircaloy-2 tubes are internally pressurized and
therefore the hoop stress is tensile throughout the tests.
This stress
state does not correspond to the stress state of Zircaloy clad in service.
However, if the same statement can be made that creep in compression is
more difficult than creep in tension as is the SD effect for yielding, the
use of creep equation derived from tensile creep data would be more
conservative.
Lastly but by no means leastly, we would like to emphasize again
that the users of these semi-empirical constitutive equations should
always realize the inherent danger in extrapolations.
While one can
try his best to utilize the results from statistical data anlysis,
there is no substitute for sound experiments.
- 40 -
2.6
(1)
Summary:
Comparisons of current Zircaloy in-reactor creep models were made
at several a and T levels corresponding to the operating conditions
of a in-service
LWR Zircaloy
clad.
The results show that there are
great departures among the model predicted creep rates especially
at high stresses and high temperatures.
The two additional creep
models included in this addendum generally predict lower creep
rates than most of the other models.
(2) Regression analyses were performed on the revised creep data set
using the same six starting model equations as those in the previous
report.
Based on regression statistics, residual patterns
and physical interpretation, Eq.(10) was selected as our recommended
Zr-2 in-reactor creep constitutive equation.
(3)
Both thermal and irradiation creep analyses were carried out on the
minimum control and minimum in-reactor creep rates respectively from
our revised
creep data set.
Although
the two separate
studies
do
satisfy the acceptance criteria for such treatments, we prefer Eq.(1O)
than the sum of
(A-10) and
(A-18) for reason of its better
residual
characteristics.
(4)
The explicit
formula
et = A (
for Eq.(10)
m)
/
is given as follows:
( a/p(T))
exp(- Q/RT)
(10)
where the units are
t
: tangential
creep strain rate, hr -
: fast neutron
C
: ksi
T
: OK
flux, n/cm2s
1
(E>1 Mev)
- 41 -
and
$o = 0.05'103
n/cm2s
(E>1 Mev)
A - 6218.47
m - 0.597
n
= 1.827
Q
= 17920.81 cal/mole
p(ksi) = 4770 - 1.906T(F)
The overall 95% confidence interval can be constructed by using
+ /M-f (SEE)=
±/33.05
+
y
exp(
n
t ± 0.812)
t
0.2685 = + 0.812
and is given by
-42-
References in the Addendum:
1.
Yung Liu, Y., A.L. Bement, "A Regression Approach for Zircaloy-2 InReactor Creep Constitutive Equations", Part I of this Report.
2.
Same as ref.
3.
P.E. MacDonald (Ed.), MATPRO Materials Properties - Library Version 005,
Aeroject Nuclear Company Report, RAM-93-75, June, 1975.
4.
M.G. Andrews, et al., Light Water Reactor Fuel Rod Modeling Code
Evaluation, Phase-II Topical Report, Appendix A: Standardized Properties
and Models for Phase-II, April, 1976.
5.
Same as ref.
6.
G.E. Lucas, A.L. Bement, "Temperature Dependence of the Zircaloy-4
Strength
7.
(14) in Part I.
(6) in Part I.
Differential",
J. of Nucl.
G.E. Lucas, The Strength Differential
Mtls.,
in Zirconium
Clad Creep Down Analysis, MS Thesis, Nucl.
8.
58 (1975), No. 2, p. 163.
and Its Effect
on
Eng. Dept., MIT, 1974.
G.E. Lucas, A.L. Bement, "The Effect of a Zirconium Strength Differential
on Cladding Collapse Predictions", J. of Nucl. Mtls., 55 (1975), No. 3,
p. 246.
- 43 -
Appendix
1
The Revised Zircaloy-2 In-Reactor Creep Data Set
Used In Regression Analysis
- 44 -
ID*
39
41
43
45
47
49
51
52
53
54
55
56
14
15
16
17
18
19
107
10H
117
118
T(°K)
o(ksi)
536.000
536.000
536.000
536.000
536.(000
536.000
536.000
536.000
536.000
536.000
536.000
536.000
531.000
531.000
531.000
531.000
531.000
531.000
543.000
565.000
543.000
565.000
16.200
1.,700
24,800
29.600
34,600
37.800
16.200
19.600
24.500
29.200
34.400
39.200
15,700
19.400)
23.900
9,200
34.000
37.600
11.500
1l1.0)
13.500
13,500
p(10 3 )
n/cm 2 s (E>1 Mev)
?. 00
2.900
2.900
?.900
?.900
2.900
0.050
0.050
O.USO
O.05
O.uU
0.U
,b U(
?.b60
?2.60
.560
2.b0
2.oubO
2.00
?.bo0
3.100
3.100
1
h(10
(10
)hr1
-hr
min in-reactor min control
creep rate
creep rate
0.140
0.200
0.220
0.300
0.S00
O.700
0,010
0.017
0.033
0.039
0.071
0.092
0.130
0,.40
0.270
0.360
0.610
O0
0 .127
n0.185
0.194
0.338
0.010
0.010
0.027
0,041
0.059
0.09S
0.010
0.010
0.027
0.041
0.05R
0.09S
0.01n
0.010
0.048
0.030
0.056
0 116
0.n08c
0.013
0.0085
0.013
* ID refers to the serial numbers in Table 4(1)
ID-107 to ID-118 are
obtained by transforming the normalized Ross-Ross and Hunt's data
serial T-10 and T-11 respectively to the original values.
- 45 -
Appendix
2
Summary of Current Zircaloy In-Reactor Creep Models
And Their Data Bases
- 46 --
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