A REGRESSION APPROACH FOR ZIRCALOY-2 IN-REACTOR …
145
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
Transcript of A REGRESSION APPROACH FOR ZIRCALOY-2 IN-REACTOR …
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 TechnologyCambridge, Massachusetts 02139
Special Report for Task 2 of
Performance Programthe Fuel
Sponsored by
New England Electric SystemNortheast 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 in-reactor creep constitutive equations. Part I was completed earlierin May, 1976 in which the essentials and the preliminary results
of and from the analysis are given. Part II which was completedlater in November, 1976 is an addendum of Part I with additionalresults and discussions. Readers are advised to follow the sequenceof presentation in order to see the entirety of the analysis.
A condensed version of Part I is presented at the SMiRT -4thConference August, 1977 and can be found in the conference pro-
ceedings under Division C, paper 3/3.
*SMiRT - Structural Mechanics in Reactor Technology
ABSTRACT
Typical data analysis procedures used in developing currentphenomenological in-reactor creep equations for Zircaloy claddingmaterials are examined. It is found that the data normalizationassumptions and the curve fitting techniques generally adopted inthese procedures can make the prediction of creep strain rate fromthe resulting equations highly questionable.
Multiple regression analysis performed on a set of carefullyselected Zr-2 in-reactor creep data on the basis of comparable asfabricated 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 strongevidences for the significance of the regression equations.
ABSTRACT OF THE ADDENDUM
The addendum of this report contains the results and discus-sions of the analysis we made on our revised Zircaloy-2 in-reactorcreep data. Although fewer creep rate data were used, much of thesame general conclusions can be made as we did previously. Thedifferences between the final recommended regression Zircaloy-2in-reactor creep equations in the report and the addendum are prim-arily those of the estimated coefficients. The starting modelequations continue to provide excellent correlations for the creepstrain rate data.
Our attempt to verify the additivity of thermal and irradiationcreep components directly from the creep data set show that thereis little merit in using a composite formula made up by the abovetwo components for ranges covered by the test parameters. Thisstatement should not be looked upon as a denial to such a treatment,however, especially in situations of combinations of high , a andT where changes in dominating deformation mechanism are possible.At present, we do not have reliable data in those regimes.
Part I
1;
Table of ContentsPage No.
1. Introduction ----------------------------------------------- 1
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 ------------------------------------- 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% confidenceintervals ----------------------------------------------------- 38
5.3 Comparisons of current empirical and regression
creep equations ----------------------------------------------- 40
5.4 Concluding remarks -------------------------------------------- 50
References --------------------------------------- 51
Appendix A: Zircaloy-2 In-reactor Creep Data ---------------------- 53
Appendix B: The Actual Set of In-reactor Creep DataUsed in Regression Analysis ---------------------------- 58
Appendix C: Regression Statistics ---------------------------------- 60
Appendix D: Stepwise Regressions of Zr-2 In-reactorCreep Data ------------------------------------- ------- 70
List 'ofTables
Page No.
Table 1: Summary of in-reactor creep models and their data
bases 4
Table 2: Summary of irradiation creep data used in
assessment of Watkins and Wood's equation 9
Table 3: Total number of data observations in each
subset 19
Table 4: Summary of 20% CW and stress relieved Zr-2
in-reactor creep data 20
Table 5: Operating conditions of U-49 tubular creep
insert 24
Table 6: Comparisons of Normalized creep rates under
two different normalization assumptions 29
Table 7: 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: Comparisons of creep laws at stress=7.5
ksi, flux=10 1 4 n/cm2s
Figure 2: Comparisons of creep laws at T=300 0C, flux=
1O14 n/cm2s
Figure 3: Schematic residual plots
Figure 4: A schematic diagram of the samll tube in-
reactor creep specimen assembly
Figure 5:
Figure 6:
Figure 7:
Figure 8:
Figure 9:
Figure 10:
Figure 11:
Figure 12:
Figure 13:
Figure 14:
Figure 15:
Figure 16:
The operating conditions for the Zircaloy-2
tubular creep experiment (Ross-Ross & Hunt)
Creep rate of pressure tube normalized to 14
ksi and 1013 n/cm2s vs temperature
Residual plots(a), (b) for regression models
(3) and (5)
Residual plots(a), (b) for regression models
(6) and (10)
Residual plots(a), (b) for regression models
(11) and (12)
Residual plot for the revised Ross-Ross and
Hunt's regression model (14)
Comparison of creep laws at T250C, flux=
5)10 13 n/cm2 s
Comparison of creep laws at T=3000 C, flux=
5x1013 n/cm2s
Comparison of creep laws at T=350 0 C, flux=
5xl013 n/cm2s
Comparison of creep laws at T=4000 C, flux=
5x10 13 n/cm2s
Comparison of creep laws at stress=7.5 ksi,
flux=5x101 3 n/cm2s
Comparison of creep laws at stress=20 ksi,
flux=5x10 1 3 n/cm2s
5
6
17
22
23
28
34
35
36
37
42
43
44
45
46
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 equations. These empirical equations
are often obtained directly from in-reactor creep data by certain kinds of
procedures. 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. Therefore, it is of no surprise to see rather
large deviations in the predicted 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 no normalization assumptions and the data can be ex-
plored in the original form as they are collected. The technique is par-
ticularly useful in establishing the functional relationship between the
dependent variable (response) and the independent variables (factors) .
In our case in-reactor creep strain rate is the dependent variable whereasstress, 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 model. The process 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. This includes the regroup-
ing of the data according 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. One notices immediately from the list of data references
that 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. Fig. 1 and Fig. 2 show the ln(E) vs 1/T and ln(E) vs
In (a) plots respectively for these equations and one which is based on the ir-
radiation creel) model derived by Gittus( 4) . 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. The deviations at high temperatures and
high stresses 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) throughoutthis report.
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J Figure-1
COMPARISONS OF CREEP LAWS AT STRESS=7.5 KSI,FLUX=IE+14
_.- (3)
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-6-
Figure-2
COMPARISONS OF CREEP LAWS AT T=300 C,FLUX=IE+14
*_l'13-
(31
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-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 than the one which is going to be discussed in the
next section. 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 ) affects the creep rate independently.
The effect of each variable 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 3000 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. It was found that
a sinh curve can fit the normalized creep rates acceptably with
a stress coefficient of 1.15xlO - 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) sinh (1.l5xl01Oa)
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 ASSESSMENTOF WATKINS AND WOOD'S EQUATION*
Temp..Type deg C
II] .... uniax.[2] .... uniax.
[3] ..... uniax.[1 ..... tube4] ..... tube
[4] .... tube[4 ..... tube
tube[4 ..... tube[4 ..... tube[3) ..... uniax.[31 ..... uniax.
[3] ..... uniax.
[3] ..... uniax.3].....uniax.[3].....uniax.[3 ..... uniax.
Stress,psi
290 25 000300 16000325 35 000
275 25 000
285 11 500
281 11 000286 13 400
286 17 300
268 10 600
327 4 900300 30000300 20 000300 18 000
300 20 000350 30000
300 45 000220 30 000
* Reproduced from Ref.
Flux(>I MeV),n/cm sec
2.4 X 10t
2.43.1
3.42.72.73.1
3.11.15
3.10.59
0.54
0.58
0.64
0.940.840.96
Duration, Creep Rate,h. in./in./h
30008 500
5 00024005 700
11 000
3 300
3 300
29000130001 3003 600
15001 100530
800
2 100
1.17 X 10-'
3 X 10"
1.2 X 10-
5.2 X 10-'1.9 X 10-1.8 X 10-2.9 X 10-
3.6 X 10-'5.5 X 10-'9.1 X 10'8 X 10'3.5 X 10-
3 X 10 -
3 X 10-3.2 X 10-'1.2 X 10- S
1.5 X 10 T
(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 tub-
ular data. This is done by multiplying the stress by a factor of 1.15
and dividing the creep rate by the same factor. 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
Test No. andRef.
829829
829629
U2-Mk. IIIU2-Mk. IV
U2-Mk. IX
NPDU3-Mk. V
R4R6R9RX 14R X R X21R X 15
0
__
__ ___
- 10 -
of materials is retained.
Steps (1) to (4) above outline the general procedure followed in de-
veloping current in-reactor creep models. 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, tempera-
ture, 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. Sometimes by proper lineariza-
tion, a non-linear model can be transformed into 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 will be revealed. If the evidence shows
that a linear model is inadequate, then it can be modified 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) + B2f(X2) + B3f(X3) + B4 f(X1X2) +
B5f(X2X3) + B6 f(X3X 1) + B7f(X1X2X3) (1)
where Y is the dependent variable, f(X1 ),...,f (X1X2X3) are some functional
dependences of individual and combinatorial effects of X, X2, and X3
on Y, and Bo,.... , B7 are the coefficients to be estimated. If we let Y -
£, X1 = , X 2 = a and X3= T, then (1) becomes our general linear creep
model. The task then is to specify explicitly the functional forms of f(4),
f(a), f(T),...., f (oT).
As we mentioned earlier, it would be greatly helpful if one can choose
a model which has some physical meaning. Studies 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 relation-
shiplo). That is,
s = A an exp (- Q (2)RT
- 12 -
Eq. (2) is of the form which is generally used to describe thermally acti-
vated rate processes. 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. This type of
relation has also been suggested by Fidleris. Taking logarithms on
both sides, (3) becomes
ln = In A + m lnq + n lna - Q (4)
(4) is essentially a linear model as in (1) with the following transforma-
tions:
Y = n £ ; Bo = In A
f(X1 ) = in q ; B1 = m
f(X 2) = In a ; B2 = n
f(X 3 ) = 1/T ; B 3 = - Q/R
-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 Bo to B3 and hence
the values of A, m, n, and Q.
Two alternative models were also considered in which the acti-
vation energy is a function of stress:
= A m an exp(- Q °( - / ) (5)R T
and
e = Am an exp(- Q ( 1 )2) (6)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 them-
selves readily to examination by data analysis. Again taking
logarithms on both sides, we have
ln = In A +m lno + n In a - 1 + a (7)R T R T
and
qo 1 + Q . c2qQ .n = In A + m lno + n In a - + 2 (8)RT RT T RT T
- 14 -
The last two terms in (7), (8) give the interaction effects of a and
T on the creep rate respectively.
So far we have presented three models based on only three indepen-
dent 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 /D instead of where D is the
diffusion coefficient, or a combination of the two. Both temperature de-
pendent u and D introduced 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. If
the variables are expressed in their ordinary units, e.g., in hr- 1,
4 in n/cm 2 sec, a in ksi, and T in K; the constant A must have unit of
2mn )' m (ksi)-n · hr-1 so that units on both sides of the equation are agree-
cm sec
able. 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. To make this term dimensionless,
one may divide * by a reference fast neutron flux o and (3), (5), and (6)
-15 -
become
= A (-)m ( a )" exp(- Q) (10)
= A (¢ )m ( a )Q exp( 1-al) (11)u(T) RT A
S=AQ - ) (T) R (12)= A ( t)m ( a)n exp(- Q°(1 /T)' (12)o u(T) R T
o is taken to be 0.05x1013 n/cm2 sec. This is deduced from Ibrahim's
result ( 1 4) and gives approximately the fast flux threshold below which
there is no difference in irradiation and thermal creep behavior. That
is, when = o the creep equations reduce to
= A (a)n exp(- Q ), etc..RT
After q and a have been non-dimensionalized, constant A should
have unit of hr-l which is reasonable since we know the lattice
vibrational frequency G (or the Derbye frequency) is involved in the
rate processes.( 1 5) There are, therefore, six regression models, (3),
(5), (6), (10), (11) and (12), considered in this analysis.
3.2 Regression 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. The program, SPSS (Statistical Packages for Social
Sciences)(21 ) 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 re-
gression analysis, a brief discussion of their interpretation is war-
ranted. Now suppose a linear model is selected and the coefficients
have been estimated, the residuals are defined as the n differencesA
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. These
residuals can be plotted in a number of ways. The most important con-
sideration in examining the residual plots is to notice their overall
patterns. Fig. 3 shows four residual plots. Except for the straight
band pattern (a), the other three patterns of the figure are all in-
dicative of abnormalities that require corrective actions.
IF
- 17 -
- 5. * *.Li '·
(a)·,~· .·. · -O.. R'. . '.
b.I
(b)
* .01· 5I
* . .. *
(d)
0
0
(c)
Figure 3. Schematic Residual Plots
0
0
-
I
-
- 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 var-
iable 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 or, perhaps, by a transformation of the yi values.
The reason that a correct pattern is indicated by (a) is not only due to the
Central Limit Theorem but also because of the Gaussian N(O,1) error model
inherent in the least squares (16) 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 the purpose for visual examination of abnormali-
ties, the residual plot has another important 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, it is helpful to have a way for experimenters
and/or data analysts to spot irregular points and to seek reasons for such
behavior. If it can be ascertained that the irregularity is due to error,
such data should be excluded to avoid the biasing of the result.
4. Examinations of Zr-2 in-reactor creep data:
We have recently compiled both uniaxial and tubular in-reactor creep
data of Zirconium, Zr-2, and Zr, 2.5% Nb in the literature from 1962 to
1974(22). There are in total 276 data observations and the breakdown of
- 19 -
these observations into subsets is shown in Tabl:e 3.
TABLE 3. TOTAL NUMBER OF DATA OBSERVATIONS INEACH 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 by
the same group of Canadian researchers on 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
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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 extrud-
ing (due to the smaller size), giving greater residual hot work before cold
drawing(14)
Since extensive cross references are made when using this subset
of data, it is worthwhile to designate them into a new table given by
Table 4. Our discussions of this Table will follow three general areas:
(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 wereobtained.
These experiments were conducted in NPD (Nuclear Power Demonstra-
tion) and NRU (National Research Universal) reactors with full-size pres-
sure tubes and small-size specimens. A schematic diagram of the small-
tube, in-reactor creep test assembly is shown in Fig. 4. Since 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 pres-
sure, 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.
-2 -
- 22 -
H
IMENS
SPECIMENS
MENS
UEL ROD
Figure 4. A schematic diagram of the small tubein-reactor creep specimen assembly
(reproduced from ref. [I-ll ])
- 23-
Typical 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--
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 conditionsup to 9000 hours may be obtained from ASTM Headquarters, $3.)
Reproduced from ref. (I-ll).
4wa.
1.0
EEa-
0.950
0.
0.90
5.
NE2.8
2.4
2.4
Z(0
2.0 i9500
I
- 24 -
TABLE 5 Operating conditions of U-49 tubular creepinsert. (Reproduced from ref. (14))
Time Average Average Fast neutronperiod temperature, pressure flux > 1 aMeV
(h) | (C) 0- (7N/m)) (x 0- 1 )
0 to 1232 264.4 9.24 2.551232 to 2419 256.7 8.96 2.512419 to 3516 253.4 9.09 2.343516 to 4987 260.7 9.05 2.524987 to 6053 256.7 9.08 2.526053 to 7597 257.1 9.09 2.367597 to 8910 259.2 9.14 3.128910 to 9712 260.3 9.57 2.829712 to 10747 260.4 9.17 3.39
10747 to 13626 265.6 9.54 3.4013626 to 14793 270.4 9.40) 3.0514793 to 19859 263.2 9.33 3.1819859 to 20896 279.1 9.39 3.25
Average 262 9.3 , 2.9
The purpose of this reviewing 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
- 25_
dictated by the creep tests. Therefore, each time a fuel shuffling opera-
tion takes place, it will cause changes in all local conditions. A good
example of such local changes is illustrated in Fig. 5. Data of , a
and T contained in Table 4 are then only average values obtained from the
entire operating history of each variable respectively. As will be shown
later, one or more normalization assumptions were used in some cases to re-
duce the data to the present values. While normalization is a viable pro-
cedure in data reduction, the assumptions must be carefully examined
first. Otherwise the original information may be lost or distorted.
4.1 Accuracies of data:
There are at least three major sources of error which may be intro-
duced into the values of the dependent as well as the independent variables:
(1) errors due to the use of average values (ignoring the history effects),
(2) errors of measurement and calculation, 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 from average conditions
are sufficiently small such that there are no significant history effects. There-
fore, in this section only measurement and calculational errors are dis-
cussed, and normalization errors are discussed in the next section.
4.1.1. Errorsdue to measurement: (e, T, P )
Diametral creep strains were measured both at intermediate shutdowns
and at the end of testing by linear variable differential transformer
- 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 the errors due to calculation.
4.1.2 Errorsdue 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 is the use of a thin shell approximation formula, t=P d/2t, to
convert the pressure into tangential stress where d is chosen to be the tube
inside diameter and t is the wall thickness. Calculations using Lame's
exact formula for internally pressurized tubes (23) show that the thin shell
approximation underestimates stresses by about 1 to 5% depending upon the
wall thickness. The pressure drop and hence the localized pressure can be
calculated from the Darcy formula(24 ) but requires rather detailed knowledge
of the flow conditions, entrance effects, etc.. Since this information is
not available, we merely point 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
using fuel bundle powers and geometry factors. (5 ) The accuracies of
these calculations must be determined from the actual conditions and the
assumptions in the reactor physics model.
In any case, errors due to measurement and calculation are probably
not major factors in these creep experiments. This is because of the averag-
ing procedures used to normalize the history effects would likely to over-
shadow the magnitudes of the combined errors due to measurement and calculation.
4.2 Normalization assumptions:
The empirical equation obtained by Ross-Ross and Huntin correlating
their in-reactor creep data is reproduced as follows:
Et = 4x10-2 4 t(T - 433) (13)
where
Et: hr-l
at: ksi (up to 20 ksi)
+ ; n/cm2sec (0.5 to 3.5x10 13 n/cm2 sec)
T ; 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 E = 8 lqm. After n, m are found
(both close to 1), the temperature dependence is determined by assuming
the creep rate is proportional to flux and stress, and is normalized to
the reference conditions of 14 ksi and 1013 n/cm2sec by the following formula:
- 28-
EN = EActx
14 ksi 1013n/cm2secactual stress actual flux
actual stress actual flux
This normalized creep rate is plotted against temperature in Fig. 6, and
(13) is derived from the best fit.
CREEP RATE OF PRESSURE TUBES NORMALIZED TO 14,000 PSI STRESSAND Ix 1013 n/cm 2 s > I MeV vs TEMPERATURE
14,000 psi I x 10 13 n/cm 2 sNORMALIZED " ACTUAL ACTUAL STRESS x ACTUAL FLUX
STRESS TE:'- ,1oAX FLUX REACTCR TUBE PSI C 10
_ n/c rnm- u J2 rIK l . oo 503 2
C-U2 V7 (rQ 4, ' 00 2 6IC-UŽ?.4rtXE|3,500G 2_ 3. c2-u't,'a7,'C0 2'D-U3 M1 5 ,00 : 1- 3
,PD -? F-? 7 10,50 , ,
t/
C.W. Zr 2.5% Nb/
77'Zr-2 OUT-REACTOR14,000 psi STRESS
I - _
c2C.W. ZIRCALOY-2
I -
C2
I
220 230 240 250 260 270 280 290 300 310
TE 1 PE RATU RE °C
320 330 340
Cree2 rate of pressure tube normalized to 14 ksi and 1013
n/cm sec vs temperature. (Reproduced from ref. (5)).
(14)
1.9
1.8
1.7
1.6
1.5
1.4
I 1.3o 1.2
w 1.0
c 0.9t.,IJ
wC 0.7
w 0.6NJ 0.5
: 04oZ 0.3
0.2
0.1
Figure 6.
Y
A
* _ 1_ _ _
A2
. Ic
F
- 29-
Again the overall procedure is questionable. The following Table il-
lustrates 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 procedures can be quite large at the average
reactor operating conditions.
TABLE 6 Comparison of normalized creep ratesnormalization assumptions
under two different
Time periodm(hr)
0-1232
1232-2419
2419-3516
3516-4987
4987-6053
6053-7597
7597-8910
8910-9712
9712-10747
10747-13626
13626-14793
14793-19859
19859-20896
Temperature
(C)
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
Average orreference value 262.0 9.30 2.90
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
/ ( 1 )-m(rn / )-n(Tr160) exp(Q - T
where r' ar' and Tr are the reference values.
Pressure(MN/m 2 )
9.24
8.96
9.099.059.08
9.099.14
9.57
9.17
9.54
9.40
9.33
9.39
Flux
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
. 1 2
1.169
0.740
0.612
0.941
0.742
0.767
0.836
0.914
0.887
1.200
1.598
1.058
2.509
- 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 normaliza-
tion 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 = tm. Differentiating £ 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 ex-
periments(6)( 5) did not show this kind of relationship after several hundred
hours, results 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-of-
reactor 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. For example creep rate data serial
T39 and T40 are at the same a, , 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 on an equal basis. A number of
data observations in M-5 should, thus be excluded. Since the major por-
tion of creep rate data were estimated at 5,000 hours, we have chosen it to
be the reference time and hence the even numbers of data observations (at
20,000 hrs.) from T40 to T50 inclusive should be excluded. The actual set of
creep data used in regression analysis can be found in Apprndix B.
5. Regression results:
Results from regression analyses using the six regression models (3),
(5), (6), (10), (11), (12) and one which is of a similar form to (13) are
presented in this section. The last one is included for the purpose of demon-
strating that if such a model of regression analysis is given by = A m an
(T-433)P, the estimates for the exponents n, m, and p will not be close to 1
at all as obtained by Ross-Ross and Hunt's procedures. Instead, the equation
should be written as
= 3.41x10- 26 +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 -
is useful to summarize and assemble the information into the following Table:
TABLE 7 Multiple regression coefficients and F statistics
Equation No. R2(%) 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
R 2 stands for the goodness of fit or the fraction of data that has been
explained by the regression equation. A high F value means that the null
hypothesis H: R=-O must be rejected or the alternative hypothesis H1 : BiO for
one or more i is true. Several points are worth noting from Table
7. First of all, based on values of R2 and F, all regression equations can be
considered significant. We shall further substantiate this conclusion when
we examine the residual plots. Second, R2 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 is slight.
In order to assess whether 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 var-
iables are entered into the equation one by one on the basis of a preselected
statistical criterion. In short, this step-by-step procedure helps to screen
out the independent variables (be it single or product) whose partial F's are
less than the pre-selected value. These variables are considered as having
only minor contributionsto the representation of the data. 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). Results from stepwise
regressions are included in Appendix D.
5.1 Residual plots:
Fig. 7 to Fig. 10 give the standardized residual versus standardized fit
plots for the previous regression models. They are identified by equation
numbers. First we notice that the overall patterns of these residual plots
look quite satisfactory. All residuals fall more or less into rather straight
error bands. There are no indications of "abnormalities" as discussed earlier
in Section 3.2. Fig. 10 which is based on the revised Ross-Ross and Hunt's
model (16) shows that it is only slightly inferior to Fig. 7(a) which is
based on (3) in terms of residual structures. 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. The cor-
responding residual plots in either original parameter units or in dimension-
less form show little difference in their structures.
- 34 -
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. I
i I~ i .
1 . ! _ _ _ _- _ _ _ _- _ _
i ;
I , I I I! I
* * iI* * .
I , :I I . ii . I
*.-- - - - - - 4_ ,
o
I
I _ on-I' :
.- o· N · i
i i i ' "
i ' C" 0 .a
I * , jI I . 1
. : CC
* in, -
r-
II
i
i
.t i
II
- 35 -
W 1., i !.
i 4,:a ii
0.I-
* 3 1I : . I ;, i
3- . l
] Wt
0 61% .4.
WI au Wu i s .4z': . W us S
WOOa' Z z r 4
A -0 I J .,
: .u
4 * >I U
I Z ZY1 0
- w .dO F <: .C O to Jt _r t 4t
* CS ' <
.( k
I . 1 ': ; .' '. I , I, O I , K I' I * I I I
~~~~~~~~ ~~~~~~~iii t~~~~~~~~~~~~~i·+; j~~~~~~~~~~~~~~~~~~~~~~~i Ii01 !i 6' i'' °
6)- !
:~~ ~ ~~~~ : '' I '. . ! I .~I I Io 'l I
ii' I ~~~~~~~~~,*I
:,· .r ~ (II· j ·· I ' · LI I cIL· ~ LIIII I i I ·"· ·I~~~~~~~~~~~~~~~~~~~~~or; ' I aio Sf~~~~~~~~~~~~~~~~~~~~~~~~~~~~'oI!
ii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1.!! I i .
,,~ ~ ~ a1~ : i,a ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~ 0 i i o·
! ' i c :i.~~~~~~~~~~~~~~ :I ! CU)!i~~~~~ :l· I iII~~~~~~~~~~~~~~~ , ; : '.I .
oc~~~ii ' 1 I ~~~~~ .I . : I~~~ iI
ii~~~~~~~~~~~~( ' ~1, !Ii' ' "i
,O o :: o ·
.0 * I
*O 6
*. I.
Ir,. !'-'
. ' i i '. I ..... ,.,. .. , M. .,.4
I.I]ALI.I *1I1-.**..I* AllI..hll.*'llEllAI,
iI
.,_l
),oNO01:I iA
IIi =* -
o.~ 0iOa .'07
I:I 60I oIo
* ,* 0
t/
o 0. 6
I: Z
I W0 3
144
Zn
: z1o3OV
; ii f
!0 0., . o ·
.. ~ .
Ii _
tO w
' *' i !, E° ni , I i, .
*6 * *F t i.-I ' ' : _I~~ ~ ~ · :t ~ I i ,o
I I : I S
* I i i 6 I
~~~~~~~~~~I 96 1 1 6 c
,I~~~ c'a t) 0
¢.___ l-.,-*.., * U
0 0 0 L* . * * . :o o o V
6 · i.* O _ , qt _t~~~
* i
'"I II I :
0'-4
cd
t
o
-Hl
U)c)
0
,o
.ct
O
0
r'l
CJ
-- - -
t
- 36 -
!.
:
, ItoI9~~~~~~ WIZ io 1,
A ' ' t i
4 Ji S . .i
d o i I
us N I·ii
z i; W0 , i,WI 8 * 0 :; IO i °, , I: ' I .
uj iU, wJ" i I0 'd ^ . C~ I I : ·
o , - I , I - -1O ~ le t i i ·
o o > i I I
Z Z I
Zr P 0 0 c --
,! w I, I ', s t,
0 O Z - ~ m N-
, 1 I o I 'N
C I '- 4 0 .i
l ! N j .4 8
I .u~~~~~~~r u · ~~~~
: I. I
I Ii I
' I
I ·
* ·I. I
·r! !·
i
I 1 · !II j *1 i', 0 I
. | ' ~ d~dI. _ _:._ ' __. __. ,0
I
!
!I'ii
I i
I
I~~~~iI ti.~~~
tI *
I
I* I
1,, ,1 . 1
I
I I8 I
1 :I I I
'I
1 e _S|l,~~~~~~ *i. .I f I I ·
I I
I
I .
' i! I-II . I !: , !
.- _ + ..2 ' 1- .I X* O C\"lr*W·~N~~~e.
I N *qI 11~~1-e I i"
,It - Id.I on'.4i 0 !
''I a-,
· § I vi PlPd-d
1d - - 1 ; - CI(
I , Sr4
,.
!oC 4Pd~~d*(~ObfI u*; 0)rr a~
Pd.4Pd ., .x) .
:o ., . .io 0
.I
* I 1·' .I 0o' : | I::L
i, n8 !,.
i: ~ ~ ~ r
iN9 fr
I 1
1 i
I I
I
L
III
i
I
i
- 37 -
U I
I-Iw 4W ,
IUo i Z <O
O4
C)
.C1[
a . "4 '. a ,
- V0 o W _ 2 IN;u·
t cc I .! i3 " 0 0 t
Z z 2 4 i I I I t . * :
aCo --> c I
:; 4 , I
'0 !
1. U. , t
i I I. .,t,.~~~~~~~~~~~~~ t =. 1 5I SI : t j; ; O lBJ~~~~~~~
* I j i j 0@,t' :n i II ,Nllbt.* i t i-;$t-o
Ni uI~~~~~~~ I aN I , ; -'
2
*' Iii' i I I I ii ;! r
I: ' I i i ; '
I * : I ,-
o !iI ' i' I;@l,I ? 1 ~~~~~~~~~~~, * i .(· ~ ~ ~ ~ ~: '-i ; ,
' iI .*0~~~~~~~~~ ~ 5' '
II
' I I ·* 4 5~~~~~~) 0 2C
*; 4 ). i I 0' I!!i i S I j·: : I j *I i l 1 : It:'~~~~~~~~~~~~~~~~~~ ! !I, ~ ~ ~ ~~~~~~~ i I 0
: ) · · 1~~~~~0 ··~~~ II ,
~~· I
id i. Id i,!i .
· it :~~~~~~~~
0rl
U)
a
-
0
CHCrI'0
CO
H Pr-,G)
ci, a
O
),p.-aa) oP £1o,~
.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-
Equation No.
(3)
SEE
(5)
SEE
(6)
SEE
(10)
SEE
(11)
SEE
(12)
SEE
TABLE 8
1nA
-28.799
-29.112
-31.640
- 0.470
- 3.900
6.323
Regression coefficients and their SEE's
m
0.611
0.030
0.613
0.030
0.609
0.027
0.611
0.030
0.613
0.030
0.609
0.027
n
1.540
0.100
1.126
0.345
2.759
0.652
1.541
0.100
1.130
0.347
2.800
0.659
Q/R
5298.06
816.00
4746.49
920.96
4276.19
851.15
4855.29
799.55
4425.72
866.03
3484.49
849.12
A ^2
Qo/RT Qo/RT
11.406
9.093 ---
-108.915* 1.676
42.969 0.587
11.310
9.142
-111.175* 1.701
43.323 0.591
*In equation (6) and (12), the coefficients for a/T are in the form 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 jthBi (estimated i coefficient) ± t (SEE)i
where t is obtained from a 95% point "Student-t" distribution with n-k de-
grees 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 interval for
the entire regression equation, and this can be found from { 25)
A
Yi (fitted value) ±+ i7T (SEE)^y
(18)
where M is the number of coefficients to be estimated, f is obtained from
a 95% point F-distribution with M, and n-M degrees of freedom. The follow-
ing Table contains the information which can be used to construct the over
95% confidence intervals.
(17)
Equation No.
(3)
(5)
(6)
(10)
(11)
('12)
TABLE 9 Overall 95% confidence
M f 7Mrf-
3 2.88 2.939
4 2.65 3.256
5 2.49 3.528
3 2.88 2.939
4 2.65 3.256
5 2.49 3.528
intervals
(SEE)A
0.2514
0.2493
0.2261
0.2514
0.2495
0.2258
+vf` (SEE)-
0.739
0.812
0.798
0.739
0.812
0.797
_ 40 _
5.3 Comparisons of current empirical and regression creep equa-t 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 =5x1013
n/cm2sec to show the ln vs 1/T and ln vs lna relationships. 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 fac-
tors. To avoid obscuring the diagram with too many curves, we merely used
regression model (3) and its associated overall 95% confidence interval to
demonstrate the difference. 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 con-
ditions.
From these figures, one sees that while the Gittus' theoretical
model always underestimates creep rates in both types of plots, good agree-
ments 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 95% confidence interval. This is not
the case, however, in Fig. 13 and Fig. 14 when temperatures are raised to
350 and 4000C respectively. Greater departures are seen among these curves as
- 41 -
the temprature is increased, As one might have expected from an Arrhenius
type relation, temperature should have a very strong effect on creep rate.
This is again evidenced 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 special attention because the extraordin-
arily high creep rates predicted 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. We are not necessarily
stating that the regression model 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 re-
gard.
Figure 11
COMPARISON OF CREEP LAWS AT T=250 C, FLUX=5E+13
: 95% Confidence Interval
41
R
II
I
1.110=--~~~~~~ ~Gittus
-0
--0
.10 -- 0.110 I. I.OI I .x
LN(STRESS) KSI
-'
JO0603 C
1.110'
r-zC-)
mm
ma
"l-
-o.1.110
.[10 t
-.. 43 -
Figure 12
COMPARISON OF CREEP LAWS AT T=300 C, FLUX=5E+13
: 95% Confidence Interval
Gittus
1.310-me
*.ll~ao ,gt ,o-0§ ,$g.,l ol l l.l. I I I I I I .~. , ..3ir 0.30 7.01 2.001 N(STRESS.1 KS
LN(STRESS) KSI
-1
J JOQS1 OC:
t.IX0I.t '11
r-
ra
m
0--
Z-
1.110
I
1:i-- ,I .
- 44 -
-i Figure 13
COMPARISON OF CREEP LAWS AT T=350 C, FLUX=5E+13
. 10
: 95% Confidence Interval1.X1O
1.110 '
2.,,o-," I I I I,.,/1o · l.7 o Ao~.,o lebO., ezo 2.X*°.10 .O1
LN(STRESS) KSI
I
JOeO 0e1
zC,)mmm
70
-4rr
1.11¢°
1.X007
--I--
- 45 -
Figure 14
COMPARISON OF CREEP LAWS AT T=400 C, FLUX=5E+13
5.5.1W
: 95% Confidence Interval
1.110~0
1.1100"
t.110 =_
310' ° . 0 I I I 7. 0 I I Cole e '0llOO'O+t,l9LN.S R*E'SS) 2.te °' to !
LN(STRESS) KSI-I
-jJOOSt
zCr
m
-4m
%h
11;n-z; B
. ,·O-
- 46 -
Figure 15
COMPARISON OF CREEP LAWS AT STRESS=7.5 KSI,FLUX=5E+l3
: 95% Confidence Interval
-D
mm-0
-4m%..
&.llO
1.g10- 4|.110 "
352R41
1.110
1.to14"
1.110 Gittus
.=10
1
l/T (I/DEG. K)*,10001
-J
4000sS 0O
- -
- 47 -
-I FiZUi
COMPARISONS OF CREEP LAWS AT STRESS=20 KSI,FLUX=5E+13
: 95% Confidence Interval
3
R
21
1.110
Gittus
1.X107.110
I/T (1/DEG. K)*1000
.e 16
1.Xt10
1.110
JolseS 0
m
--
ZZ.1.310
1.1it
11 .
1 u
I
I
- 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 allow-
ing 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. Further-
more, 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 condi-
tions are needed to extend the data range for verifying the adequacy of the re-
gression 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 re-
lieved Zircaloy 2:
= A )m (a )nexp(- 1-a/ 4o jjfl7 y RT
- 49 -
where the units are
= hrl 1, : n/cm2s, a: ksi, T: °K
and
A = 0.020
m = 0.613
n = 1.130
+o = 0.05x1013 n/cm2s (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 byY
£ can be constructed by using +
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 signifi-
cant as compared to thermal creep, there is a shift in dominating mechanism
in the creep deformation process. The in-reactor creep equation over the
entire temperature range can be. constructed as a composite of multi-stage creep mech-I-
- 50 -
anisms acting over each of its dominating reqimes. Temperature threshold
(or thresholds) at which the shifting of mechanism occurs could be them-
selves a function of stress and fast neutron flux. Our present set of data
does not permit us to extend the analysis to verify these points mainly be-
cause of the shortage of in-reactor creep data at higher temperatures.
-51-
References:
1. G.R. Piercy, "Mechanisms for the In-Reactor Creep of Zirconium Alloys",J. of Nucl. M ts., 26 (1968), 18-50.
2. C.C. Dollins, F.A. Nichols, "Mechanisms of Irradiation Creep in Zir-
conium-Base Alloys", ASTM-STP(551), pp. 229-248, 1974.
3. F.A. Nichols, "On the Mechanisms of Irradiation Creep in Zirconium-
Base Alloys", J. of Nucl. Mtls., 37 (1970), 59-70.
4. J.H. Gittus, "Theory of Dislocation Creep for a Material Subjected toBombardment 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., 26 (1968), 2-17.
6. B. Watkins, D.S. Wood, "The Significance of Irradiation-Induced Creepon Reactor Performance of a Zircaloy-2 Pressure Tube", ASTM-STP (458),pp. 226-240, 1969.
7. P.J. Pankaski, "BUCKLE - An Analytical Computer Code for CalculatingCreep Buckling of an Internally Oval Tube", BNWL-B-253, 1973.
8. Proprietary information.
9. Proprietary information.
10. F.A. McClintock, A.S. Argon, "Mechanical Behavior of Materials", Chap. 19,1966.
11. V. Fidleris, "Summary of Experimental Results on In-Reactor 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 ReactorFuel Pin Behavior, Special Problem, 22.901, Dept. of Nucl. Eng., MIT,May, 1975.
14. E.F. Ibrahim, "In-Reactor Tubular Creep of Zircaloy-2 at 260 to 3000 C",J. of Nucl. Mtls., 46 (1973), 169-182.
15. U.F. Kocks, A.S. Argon, M.F. Ashby, Thermodynamics and Kinetics of Slip,
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 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-CAL-LINWOOD.
20. TROLL PRIMER, 2nd Ed., June 1975, National Bureau of Economic Research, Inc.
21. SPSS, Statistical Package for the Social Sciences, 2nd Ed., McGraw-Hill, 1975.
22. A.L. Bement, to be published.
23. Timoshenko and Goodier, Theory of Elasticity, 2nd Ed., p. 59.
24. M.M.E. Wakil, Nuclear Heat Transport, p. 223.
25. L. Breiman, Statistics: With a View Toward Applications, Houghton Mifflin,1973, Chap. 9.
References in Table 1:
(E-l) J.J. Holmes, et al., "In-Reactor Creep of Cold-Worked Zircaloy-2",ASTM-STP-330, p. 385- , 1965.
(H-1) V. Fidleris, "Uniaxial In-Reactor Creep of Zirconium Alloys", J. ofNucl. Mtls., 26 (1968), 51-76.
(H-14) Same as (5).
(I-11) E.F. Ibrihim, "In-Reactor Creep of Zirconium Alloy Tubes and Its Cor-relation 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 ofZircaloy-2 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 DiameterZirconium Alloy Tubes Under Irradiation", ASTM-AIME Symposium onZirconium in Nuclear Applicationsi Portland, Oregon, August 1973.
- 53 -
'Appendix A
Zircaloy-2 Tubular In-Reactor Creep Data
I Cr--n
C;
N
I*;·
c:r0
cOd
C,
00
4.O
oO
C?
ul
o
9C,
0
OU.40_0
Ca.a-U-0 r.E =
v o "
_ c
CC
:icC)
cli
i23
I.
oN
c;0C)
a.
LA
cocoN
INullcU
·C
a:L
Eu,u11a C,Scooa
CLA
5
a 4'c
h-L
5
:3
4+0o
Intr00o,
r-NN
oIn
_
N
e3
i - I1Pj
I N 'o " 00 Io 0 - N _
00
I
0%
00EC,a_
C)
0%O
E
_,02_
C)40o_O
'A
N
r,:,Un
co0)
.o .
c .
I
0
C cSta
a,E.u
ooCs.0N
0
a-)
O
3ak
(aCN
000O0%
0
o
t
cOC,N
000C3Zt0%0i-m
N.,In
12
oo
o.r0aood:Er=
10
-0oos0%
N
01sCh
_.
00CZ%f.,.en00
aI
co
000C,
0I!
N
0%Nu
o000vU00
N
N51tIn0
00oo_0%
ror='00
o.FiGm
-0m
o'o_C,
o
:E5.0
N
aC,
naDco
oCc.
Ct,,
0%C.
114C',
00=o
N
aQNxaInr-_ @O L
Z3aC'
.0
CtL_,C.0
5 3UVlkoXy
Z _ _ _ cc oa
4.
IL1-C).N
c00
cC
C!O0O
o4.
0%0N
LA00o)o
4
4.
0C?0
0
*
0C)'o6'IC)N00
0
-4.
.5.4.'A
LA
o00
C,N0
4.
a9%000
C00
g-
0%N
0
LA
_V2!
0
'..
00
a7m
In
cn__C
N
LA
0%llIC,
0
Ci0o0%
0
N
C,
o°0C)Coa0%oN0
ftS
a.,
0%
09
C:i 0
4 C
0 1
'4-d0
O I7 4)6'C p |~
U-$4
43
C
C> f:)
a a 4J Ao 3oC4 C E '
E
14
cla.
43E 4ie.
0cc
Ce)aE
04
(O
C))
ka E t0. -.. S.-
4N C
4_L U04 C
Ct'CCC43
a
'0 0 Co S
-5 14 - asC
4.O.
oUC
cIC,
4.1coon
L0
C?
.o9
OOa
OO
r
C
'ftCC,
uC,
ccIn
3S-)
D u7
c] z:!2
;:
*
C:4N
C
4.Co
I0
(CiA ;
Eaar,
N.
LU CL
54
*4
0a45
*0
0e
0
C.
C,'1
C
K
0CU
ft
-5
045*00
N
C4.
-'U
30
0:2- .0
a UccCJ
0r0O0S4593
N
N~a
-S.C
a ~4(LA 4
'0Cs
45C.0lra.Q
_
a
Ct
N0%
04
a'.5%'u
? L
-C
a
SC>a,
Xcl
i:3
c I.-
-
rc0
LA
a.
3 ZCLIL
Si.
).c.)LI
CO
Z,?:3.3C)
.4
'Ccc
IV
C
(a
c,
:6:cl
a
4-C,
'A
C
a
L4.(.
LAIn
:xj..10
c,
=1
ccLA
:3
Nl
cx'-S
:I-Ze-
iFa.
3
I
--
I
I
.
I
r
IS
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- 58 -
Appendix B
The Actual Set of In-Reactor Creep Data
Used in Regression Analysis
59
ID T(°K) o(ksi) q(10 3 ) e(10 1 )n/cm s hr
39 536.000 16,200 2.900 0.14041 536.000 19.700 2.900 0.20043 536.-000 24.800 2.900 0.22045 536.000 29,600 2.900 0,30047 536.000 34.,600 ?.900 0,50049 536.000 37,800 2.900 0.70051 536.000 16,200 0.050 0.01052 536.000 19.600 0.050 0.01753 536.000 24.500 0o.as050 0,03354 536.000 29.200 0.050 0.03955 536.000 34.400 0.050 0.071
56 536.000 39.200 0.050 0.09214 531.000 15,700 2.560 0.13015 531.000 19,400 ?.560 0.24016 531.000 23.900 ?.560 0.?7017 531.000 29.200 ?.560 0.35018 531.000 34,000 ?.560 0.61019 531.000 37.600 ?.560 O.BO4 554.000 13.400 3.100 0.P306 554.000 17.300 3.100 0.2901 600.000 4,900 3.100 0.0913 554.000 11.500 '.700 0.1905 543.000 14.000 P.700 0.1902 537.000 10.500 1.150 0.05477 573.000 5.500 3.100 0.07778 618.000 5.500 3.100 0.12887 525.000 10.500 1.150 0.03788 546.000 10.500 1*150 0.05397 525.000 10.500 1.150 0.05498 546*000 10.500 1.150 0.059107 543.000 11.500 '.500 0.127108 565.000 11,500 2.500 0.155117 543.000 13.500 3.100 0.194118 565.000 13.500 3.100 0.338127 525.000 14.000 ?.600 0.133128 561.000 14.000 '.600 0.239137 513.000 14.000 ?.600 0.117138 530.000 14.000 7.600 0.208
ID refers to the serial numbers in Table 4, ID-77 to ID-138are obtained by transforming the normalized Ross-Ross andHunt'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
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FLUX4 : ln({/0.05), normalized flux
TI : 1/T
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Appendix D
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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 ContentsPage No.
Abstract --------------------------------------------------------i
Table of contents -----------------------------------------------ii
List of tables -------------------------------------------iii
List of figures ---------------------------------------- iv
1.1 Introduction -------------------------------------- 1
1.2 Deletion of some questionable data ------------------------- 1
1.3 Inclusion of additional creep models ------ -------------- 2
1.4 Verification of whether thermal and irradiation creep compo-nents are additive ------------------------------------- 4
2 Results and discussions ---------------------------------- 6
2.1 Creep model comparisons ------------------------------------ 6
2.1.1 Stress dependence --------------------------------- ----- 6
2.1.2 Temperature dependence ----------------------------------- 11
2.2 Regression analysis with revised creep data -------------- 11
2.3 Verification ---------------------------------------- 18
2.3.1 Thermal creep analysis ------------------------------------- 18
2.3.2 Irradiation creep analysis --------------------------------- 20
2.4 Comparison of regression creep models ---------------------- 32
2.5 Considerations in applying Zircaloy in-reactor creep constitu-tive equation --------------------------------------------- 38
2.6 Summary ---------------------------------------------- -- 40
References ------------------------------------------------------- 42
Appendix 1 : The revised Zircaloy-2 in-reactor creep data set usedin regression analysis --------------------------- 43
Appendix 2 : Summary of current Zircaloy in-reactor creep modelsand their data bases -------------------------------- 45
ii
List of Tables
Table Number
A-1
A-2
A-3
A-4
A-5
A-6
Title Page Number
Multiple Regression Coefficients, F Statistics 11Using the Revised Data Set
Estimated Coefficients, SEE's Using the Revised 14Data Set
Multiple Regression Coefficients, F Statistics 19for Thermal Creep Analysis
Estimated Coefficients, SEE's from Thermal Creep 20Analysis
Multiple Regression Coefficients, F Statistics 22for Irradiation Creep Analysis
Estimated Coefficients, SEE's from Irradiation 23Creep Analysis
iii
List of Figures
Figure Number
A-1
A-2
A-3
A-4
A-5
A-6
A-7
A-8
A-9
A-10
A-11
A-12
A-13
A-14
A-15
A-16
Title
1Comparison of Creep Laws at T=250C, =5-10"n/cm s (E>1 Mev)
Comparison of Creep Laws at T=3000 C, =5 101
n/cm 2 s (E>1 Mev)
Comparison of Creep Laws at T=3500 C, =5 · 10 1n/cm 2 s (E>1 Mev)
Comparison of Creep Laws at T400 0C, =5-10:n/cm2s (E>1 Mev)
Comparison of Creep Laws at a=7.5 ksi, =5· 1in/cm2s (E>1 Mev)
Comparison of Creep Laws at a=20 ksi, =5- 11n/cm2 s (E>1 Mev)
Residual Plots (a),(b) for Regression Modelsand (5) respectively
Residual Plots (a),(b) for Regression Modelsand (10) respectively
Residual Plots (a),(b) for Regression Modelsand (12) respectively
Residual Plots (a) ,(b) for Regression Models(A-13) and (A-14) respectively
Residual Plots (a),(b) for Regression Models(A-15) and (A-16) respectively
Residual Plots (a),(b) for Regression Models(A-17) and (A-18) respectively
Residual Plots (a),(b) for Regression Models(A-19) and (A-20) respectively
Residual Plots (a),(b) for Regression Models(A-21) and (A-22) respectively
Residual Plots (a),(b) for Regression Models(A-7) and (A-8) respectively
Residual Plots (a),(b) for Regression Models(A-9) and (A-10) respectively
3
3
3
013
013
(3)
(6)
(11)
Page Number
7
8
9
10
12
13
15
16
17
24
25
26
27
28
29
30
iv
J
List of Figures (Continued)
A-17 Residual Plots (a),(b) for Regression Models 31(A-ll) and (A-12) respectively
A-18 Comparison of Regression Creep Laws at T=3000C, 33+ =5-1013n/cm2 s (E>1 Mev)
A-19 Comparison of Regression Creep Laws at T--3500C, 344 =5-101 3 n/cm2s (E>1 Mev)
A-20 Comparison of Regression Creep Laws at T=4000C, 350 =5'101 3n/cm 2 s (E>1 Mev)
A-21 Comparison of Regression Creep Laws at a=7.5 ksi 360 =5-10 13n/cm 2s (E>1 Mev)
A-22 Comparison of Regression Creep Laws at a=20 ksi 374 =5'101 3 n/cm 2 s (E>1 Mev)
v
*
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 in this addendum are
essentially the same as in our previous report. 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. The revised creep data set is
contained in Appendix 1.
ID No Reason for deletion
4,6 repetitive, they can be obtained as average of117 and 118
3 repetitive, can be obtained as average of 107,108
1,77,78 repetitive, can be obtained as average of 77,78
2,87,88 incompatible creep test duration, 13,000 hrsrepetitive, 2 can be obtained as average of 87,88
5,127, incompatible creep test duration, 11,000 hrs128 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
are referring to Appendix B(1 )* The ID numbers
-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( . This latter model is worth mentioning because of the
likelihood 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= K (a + B e ) exp(- 10,000/R T) t-0 .5 (A-l)t
where the corresponding units are
T: K, t: sec, : fast neutron flus, n/m s (E> 1 Mev)
a: tangential stress, N/m , 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-
= {k + 1.0110 -1 9 exp(- 0.004 t)} ae exp(- 8,000/RT) +e 1e
{k2 a (25.6 - 0.026 T) exp(- 138,000/RT)} (A-2)
where
: effective strain rate, hr 1e
a : effective stress, ksi ( <40 ksi)e
t: hours
0: fast neutron flux, n/cm 2s (E>1 Mev)
R: 1.987 cal/mole OK
T: K ( <773 OK)-18
kl: 0.667. (1.36-10 ) } Zircaloy is assumed to be isotropic-28
k2: 0.667.(7.010 )
In order to change (A-2) into circumferential strain rate and tangential
stress, simplifying assumptions of isotropic material and thin shell approxi-
mation were introduced with the following transformations.
Isotropic material
1 2 2 2 1/2a = ( -a ) + (a ar) +( -a ) (A-3a)e z ( 8 r r z
Thin shell approximation
a negligible; a = 0.5a e (A-3b)
(A-3a) then reduces to
a = 0.866 a (A-4)e0
The circumferential and effective strain rates relationship
-4-
Y ~~~~~~~~e (A-5)= { ae- - (ar+ az)}
e 4
with a , r and a substituted from (A-3a), (A-3b), and'(A-4), Eq. (A-5)e r z
becomes
Be= 0.866 e (A-6)0 e
We have written a FORTRAN program to obtain ae from (A-4) and substi-
tuting 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 due to the fluctuations in the controlled
parameters , 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 areadditive :
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 measure-
ments 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 irradia-
tion 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 E-th) starting
with using only and then with different combinations of , 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 exp Q T)A 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. The fast neutron flux is fixed
at 51013 n/cm2s and temperatures are systematically increased from 250 °C to
300 C, 350 C and 400 C. 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 0C-300 C, although they predict slightly higher
creep rates at low stress regime. At higher temperatures, 3500C-400°C, these
two models underpredict creep rates at high stress regime as compared to most
-7-
0.1-0
___ : 95% confidence interval
4
A
.
1.010-
Gittus
.: 0 I I , ! I I I I.4, I I I9. , ",'. , I,..el ,.,le` ,.,,,,
LN(STRESS) KSI
Figure A-1 Comparison of Creep Laws at T=2500C, *= 5.1013 n/cm2s (E>1 Mev)(Numbers associated with the curves are equation numbers givenin reference 1. They can also be found in Appendix 2)
.|l1
I-
-I-mWn
%Z.
ZC
1
&.gI1-"
-8-
4.X1e4-
1.1104
r-zdm~t)rrtrr
-
z:B'--4
1.110qemP xs-Z..I.iloe o y
Gittus
1.110o
. ,, 41 I III I I I I .
LN(STRESS) KSI
Figure A-2 Comparison of Creep Laws at TI300°C, b= 5.1013n/cm2s (E>1 Mev)
rr 95- CRfidence Interal
a
Io.le-t
ru-
X_ I
, 814
s Xs*-
Compa ris0~ of Creep Laws at T-3500C ip 0 510 13 2mn/m (E~l Mev)
Figure A-3
24A
-9 -
lzf
5··1
#.XI*-" -C~~~~~~~~~itt yle
Cas I I I"""·~~4.80" s**
I1
WNSTRESS) KS,1
- 10 -
4oI --
0-
rnrrm-4
1.210
1.310
1. . .
-7."10 -- f"-+ S I I I I I I-a .,41 *1
LN(STRESS) KSI
Figure A-4 Comparison of Creep Laws at T=4000C, = 510 13n/cm2 s (E> 1 Mev)
---4
-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 creep
rates at fixed fast neutron flux and two levels of stresses. 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
thirty-eight previous data were eliminated. Using the remaining twenty-two(1)
data, we have rerun the analysis with the same six starting model equations
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 RegressionData Set
Equation No*
3
5
6
10
11
12
* : All equation numbersreference (1)
Coefficients,F Statistics Using the Revised
R2 (%)
97.712
98.381
98.381
97.713
98.380
98.380
F
126.638
128.086
96.443
126.716
128.014
96.394
without prefix refer to equations in
- 12 -
.@t
0.i10'
-@01
: 95% confidence interval____o
35A-724A-1A-2
11.310-
1.10
.471.310
Gittus
1.X10
I.I10 -
1/T (I/DEG. K)*1000
Figure A-5 Comparison of Creep Laws at a=7.5 ksi, = 510 13n/cm2s (E>1 Mev)
3mm
-I'Om
3:
. .
- 13 -n- , --ofadence interval
354A-7
A-1-A-2
1 I
9
.3.0 -
. Lz | t I I I 1.~ I I I I I I.
,7 1 0 ..41.4
I/T /EG. K)*1000 r
Figure A-6 Comparison of Creep Laws at a=20 ksi,
5 1 0 n /c m s (E>1 Mev)
. -..· .-
I .Xi '1 4
l.li@1 .51
r,
m
z-2"
1.1
1
- 14 -
The meanings of R2 and F are explained before. We are thus able to make
the following general conclusions : (1) Based on R and F, all regressions
can be considered significant; (2) R2 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 in each figure for the
six creep regression model equations. Although the error bands are narrower
for the more complicated equations, all their residual patterns look quite
satisfactory and there are no signs of model abnormalties.
Table A-2 Estimated Coefficients, SEE's Using the Revised Data Set
Eq. No. lnA m n Q/R _Q/R Qo/R 2
3 -21.484 0.597 1.827 9519.05 ---SEE --- 0.033 0.181 2310.16 ---
5 -20.382 0.5847 -0.795 6937.05 58.290SEE --- 0.029 1.007 2229.72 22.118 ---
6 -20.502 0.5848 -0.744 6917.09 55.794* 0.026SEE --- 0.031 3.934 2717.10 185.01 1.927
10 8.735 0.597 1.827 9019.03 --- ---SEE --- 0.033 0.181 2281.32 ---
11 -10.731 0.5847 -0.789 7160.30 58.152SEE --- 0.029 1.007 2101.44 22.109 ---
12 -10.028 0.585 -0.660 7074.85 51.903* 0.066SEE --- 0.031 3.933 3316.06 185.047 1.928
* In Eq.(6) and (12), the coefficients for o/T are in form of 2Qo/R
I I~~~~~ I · 15
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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. It should be pointed out, however,
that the resultant creep rate is still increasing in these models as stress
increases despite of the negative stress exponent. 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 exp(- Q'/RT) (A-7)
tth = A' a exp{- Qo(1- a/f)/RT} (A-8)
th A' exp{ Q(1- /f) /RT} (A-9)
th = A' (ao/(T))n' exp(- Q'/RT) (A-10)
- 19 -
Cth A' (/p(T)) n ' exp{- Q(1- a/t)/RT}
Cth = A' (a/p(T)) n expf- Q o(1- a/) 2/RT}
(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.
Table A-3 Multiple regression coeffcients, F statistics for thermalcreep analysis
Equation No
A-7
'A-8
A-9
A-10
A-11
A-12
R2 ()
94.654
96.169
96.790
94.658
96. 168
96.778
F
81.798
73.833
63.038
81.858
73.812
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. This equation is given
by the following formula:
eth - A' (a/1(T))
where the units are
exp(- Q'/RT) (A- 10)
eth: h1r , : ksi, T: K, R=1.987 cal/mole "K
and
- 20 -
A'= 13200.37
n'- 2.441
Q'i 15476.28 cal/mole
u(ksi)=4.77103 - 1.906T(F)
Table A-4 Estimated Coefficients, SEE's From Thermal Creep Analysis
Eq. No. lnA' n' Q'/R Q' /Rt Q'/Rf2-0-- -0-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
A-9
SEE
A-10
SEE
A-11
SEE
A-12
SEE
7.282
___
-7.445
3.998
9.488 2.441
0.--- 205
-12.651 -0.521
1,142
-49.917 -7.379
--- 4,008
8291.71
2875.32
403.547*
189.194
-3.557
1.979
7788.77
2616.40
5663,45
2424.62
10300,87
3473.53
66.158
25.189
400.600*
189.77
-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.
T -th A" ="l (A-13)
T = th A" ( / mo (A-14)
T - eth Al lm anc (A-15)T th
T th =A" O( / )m ( c/p(T)) (A-16)
T -P =th = A" *m n" exp(- Q"/RT) (A-17)T th
n"th = A" ( ) (/m (/(T))n exp(- Q"/RT) (A-18)
T -t A" mA an exp{- Q"(1-a/t)/RT} (A-19)T th
- = A" ( /)m ( a/p(T)) exp{-Q"(1-a/t)/RT} (A-20)
= - A" ma exp{- Q"(1-a/t) /RT} (A-21)T th o
th A" ( ) ( cr/x(T))pn -exp{- o"(l-/t)2/RT} (A-22)T th 0 0
Table A-5 and Table A-6 give the regression statistics,estimated coeffi-
cients, 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. In short, we have considered
all possible combinations of ,a and T both in their original parameter units
and in dimensionless forms. 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 used in other creep models since Watkins and Woods first
proposed it in 1969(5 ) 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 ForIrradiation Creep Analysis
Equation No
A-13
A-14
A-15
A-16
A-17
A-18
A-19
A-20
A-21
A-22
R2 )
92.076
92.076
97.276
97,358
98.269
98.270
98.768
98.766
98,831
98.834
F
94.699
94.699
140.879
145.453
140,654
140.763
139,412
139,237
109.248
109.525
residual plots in Fig,A-10, there are distinct patterns of residual seggrega-
tions 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 magni-
tudes of the estimated coefficients themselves and these models should also be
regarded as unacceptable, We are, therefore, left with (A-15) to (A-18) to
choose from. Using the same arguements as before, (A-18) has been selected as
our model for the irradiation creep component and its explicit formula is given
by
err QT eh= A" ( mo) (ap(T)) exp(- i"(RT)ir T t h =_ A" ( (a/p(T)) exp(- Q"/RT) (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
Table A-6 Estimated Coefficients, SEE's From Irradiation Creep Analysis
lnA" m"
-41.758 0.859
--- 0.088
-18.618 0.859
--- 0,088
-46,569 0.906
--- 0,055
-12,982 0.9065
--- 0.054
-34,012 0.905
--- 0.045
1.830 0,905
--- 0,045
-31,955 0.877
--- 0,041
-18.142 0.8776
--- 0.041
-40,693 0,878
--- 0,042
2.157 0.878
--- 0,042
nIT
1,101
0,203
1.126
0.203
1,482
0.213
1.482
0.213
-1,280
1.179
-1,270
1.179
2.384
4.551
2,522
4.546
Q"/R
--
7385.63
2536.23
6980.61
2499,91
4928.84
2447.49
5287.48
2302.13
3247,37
3191.34
2504,38
3970,95
< R Q"/R
62.284
26.265
62.061
26,261 ---
-1i21.330* 1.988
221.670 2,383
-128.000* 2,059
221.520 2.383
* In (A-21), (A-22) the coefficients for alT are in the form of 2Q"/RT
Eq No.,
A-13
SEE
A-14
SEE
A-15
SEE
A-16
SEE
A-17
SEE
A-18
SEE
A-19
SEE
A-20
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A-21
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A-22
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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.given in 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 contri-
bution is ,gtligible,i .e., at least two orders of magnitudes lower than the
irradiation creep component based on the above studies. 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 th and irr are different. Twenty-two
data were used for th whereas nineteen were used for irr' 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_ _ _ _
(10(16
(A-lO)+(A-18(A-18(A-10
0
,.,,x3 6. I I I I Is. * u v u Y u 1 a ? I. I I I
LN(STRESS) KSI
Figure A-18 Comparison of Regression Creep Laws at T=300 C, = 510 13n/cm2s
(E>1 Mev), (Numbers associated with the curves are equationnumbers and they can be found in Appendix 2)
I OmW
I-Z2C
mzziZIronZI
W
I.Ue1.210@
1.1 1
I.!4.441
. l..B
m
l
- 34 -
- : 95% confidence interval
10(16
(A-10)+(A-18(A-18(A-10
I.IO4 )1 13310 4.110
CN.O a
LN(STRESS) KSI
Figure A-19 Comparison of Regression Creep Laws at T=3500C, = 51013n/cm 2s
(E>1 Mev)
1.X11
r-rz
-o-0
m
:3
1.11 '
1.X1iemlJ4
1.110
I.Ilt tl
1.. l*0 . I I I I I
- 35 -
: 95% confidence interval
((1
(A-1O)+(A-1(A-1
II..I.Ill't I4Clolle
LN(STRESS) KSI1
Figure A-20 Comparison of Regression Creep Laws at T400'C, *= 51013n/cm2s(E>1 Mev)
o.a "
I.310-
I.U
1.106t
C,
mm-01%"O
WiMtM laM-30Ag
.,
I..i. IOi1I-IS0I I I I I
sexl r 10
- 36 -
I .O 1 8
: 95% confidence interval
1.110
-ale
1.310
1.310aNO-
I/T (l/DEG. K1000
Figure A-21 Comparison of Regression Creep Laws at o=7.5 ksi, 4= 510 13n/cm2s(E>1 Mev)
r
, . v
I-xI*- I
- 37 -
1.I10
: 95% confidence interval
1.S14
1.310
i.l 1
1.X10
,.x" L I1.4
I I I I I I I I I I I I I I I I I I I I I I I I I1.1 1.0 1.7 1.0 1.I
I/T [1/DEG. K)*1000r
Figure A-22 Comparison of Regression Creep Laws at =20 ksi, 4= 510 13n/cm s(E>1 Mev)
m
m-m
zz
,. 1A .
- 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. Eq,(10)
derived from the revised creep data set is our choice. 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. Both of these features are of
a less random nature than the residuals in Fig.A-8(b) which is obtained from (10).
2,5 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, Comparatively speaking among the different creep models,
underprediction of creep rate means that it is less conservative whereas
overprediction means perhaps more conservative. 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 over-
conservatism is always at stake,
(2) 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 10%
greater than the yield strength in tension. It is suspected that the SD
effect may exist for creep as well, Currently, investigations are underway
at MIT to study this possibility. 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 Summary:
(1) 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 for Eq.(10) is given as follows:
et = A ( / )m ( a/p(T)) exp(- Q/RT) (10)
where the units are
: tangential creep strain rate, hr- 1
t
: fast neutron flux, n/cm2s (E>1 Mev)
C : ksi
T : OK
- 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 0.2685 = + 0.812 and is given byy
exp( n t ± 0.812)t
-42-
References in the Addendum:
1. Yung Liu, Y., A.L. Bement, "A Regression Approach for Zircaloy-2 In-Reactor Creep Constitutive Equations", Part I of this Report.
2. Same as ref. (14) in Part I.
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 CodeEvaluation, Phase-II Topical Report, Appendix A: Standardized Propertiesand Models for Phase-II, April, 1976.
5. Same as ref. (6) in Part I.
6. G.E. Lucas, A.L. Bement, "Temperature Dependence of the Zircaloy-4Strength Differential", J. of Nucl. Mtls., 58 (1975), No. 2, p. 163.
7. G.E. Lucas, The Strength Differential in Zirconium and Its Effect on
Clad Creep Down Analysis, MS Thesis, Nucl. Eng. Dept., MIT, 1974.
8. G.E. Lucas, A.L. Bement, "The Effect of a Zirconium Strength Differentialon 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* T(°K) o(ksi) p(10 13) h(10 )hr1 (10 -hr-
n/cm2s (E>1 Mev) min in-reactor min controlcreep rate creep rate
39 536.000 16.200 ?. 00 0.140 0.01041 536.000 1.,700 2.900 0.200 0.01043 536.000 24,800 2.900 0.220 0.02745 536.000 29.600 ?.900 0.300 0,04147 536.(000 34,600 ?.900 0.S00 0.05949 536.000 37.800 2.900 O.700 0.09S51 536.000 16.200 0.050 0,010 0.01052 536.000 19.600 0.050 0.017 0.01053 536.000 24.500 O.USO 0.033 0.02754 536.000 29.200 O.05 0.039 0.04155 536.000 34.400 O.uU 0.071 0.05R56 536.000 39.200 0.U 0. 092 0.09S14 531.000 15,700 ,b U( 0.130 0.01n15 531.000 19.400) ?.b60 0,.40 0.01016 531.000 23.900 ?2.60 0.270 0.04817 531.000 9,200 .560 0.360 0.03018 531.000 34.000 2.b0 0.610 0.05619 531.000 37.600 2.oubO O0 0 116
107 543.000 11.500 2.00 0 .127 0.n08c10H 565.000 1l1.0) ?.bo0 n0.185 0.013117 543.000 13.500 3.100 0.194 0.0085118 565.000 13,500 3.100 0.338 0.013
* ID refers to the serial numbers in Table 4(1) ID-107 to ID-118 areobtained by transforming the normalized Ross-Ross and Hunt's dataserial 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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