- Email: [email protected]
On secondary creep of anisotropic nuclear materials
JOURNAL
OF NUCLEAR
33
MATERIALS
ON SECONDARY
(1969)
52-63.
CREEP OF ANISOTROPIC
of Waterloo,
Received
Department
26 November
anisotropic pressure tubes may be twice
half
as high
as diametral
tubes if the uniaxial about
the third
creep rates
power
test results available in-reactor
creep
is shown
at
can
least
Using
worked
or one isotropic
and diametral
that
specimens
in accordance
to
the limited
zirconium
qualitatively
alloys
it
experimental
sont
En
disponibles,
5t la
de
moins
zirconium
with
the
sur Bchantillons des tubes accord
avec le theor&me
anisotrope
qui suppose
de puissance contrainte.
Diese
dans
de fluage en rkacteur,
sol1
an
en contrainte
la
aus
anisotropen
metralen
uniaxiale,
wenn
des
theorbme
de
moduln
etwa
20%
Bcroui.
Un
pour des alliages
dans les modules
de fluage d’environ
zirconium
peuvent
les vitesses
de fluage dans le sens diametral
der sich
Btre 2 St 1,5 fois aussi Blevbes que
designed alloy
to
materials serve
pressure
in
tubes
are the
as
form core
of
analytic *
theory
zirconium
components
Research
Assistant
on leave
creep from
in
for multiBuilding
bzw.
Spannungszustand
Reaktor-
kijnnen bei
die dia-
anisotropen
Unterschiede
betragen
und
der
die
Kriech-
einachsigen
etwa mit der dritten Anhand
qualitativ
Zusammenhiinge lassen.
ge-
zum Kriechen unter
Danach
bzw.
Zirkon
halb so gross wie bei isotropen die
variieren.
zumindest
Kriech-
Bestrahlung
Weitere
mit
der
Potenz
vorhandenen
nachgewiesen, dem
Theorem
Untersuchungen
dass be-
werden
axial states of stress. Berman and Pai 7) have formulated a simplified theorem of creep for anisotropic materials assuming that in stress space the magnitude but not the direction of the creep rate vector depends on anisotropy. Here an exact analysis of anisotropic in-reactor creep is proposed for zirconium alloys considering the work of Kachanov *), Odqvist 9), Rabotnov lo), Malinin I’), Olszak and Sawczuk
and
l-4). Recently,
of anisotropic
la
vorgeschlagen.
Ross-Ross and Hunt 5) and Ibrahim 6) studied the inreactor creep behaviour of these pressure tubes experimentally and indicated the need for an nuclear power reactors
wird die
schreiben
des tubes
fabricated
Spannung
Daten
de force
Introduction Anisotropic
avec
der
kalt-bearbeitetem
Kriechgeschwindigkeiten
20%, les vitesses des tubes
anisotrope
1.
aniso-
que pour des diffhrences
dans le sens diam&ral
fluage
de ces travanx
wie sie unter
Kriechgeschwindigkeiten sein,
de fluage
anisotropes
Zr-Legierungen
doppelt
qui indique
le
du coefficient
Diskrepanzen
vorgeschlagen.
Rohren
en reacteur
pour
Es wird ein Theorem
Rohren
trope est propose
et sur eux en
ult&ieur
mit einachsigem
Druckrohren
bedingungen
mesur&
die
par
fluage
uniaxial
propose
klgren,
constitue
zirconium
qn’au
i?tre relies entre
ou des coefficients
et sur des tubes de force d’un reacteur de
montre
expkrimentaux
une variation
Un dt?veloppement
Arbeit
von
alliages
il est
& un fluage
peuvent
condition
dans un rkacteur
ces r&ultats
soumis
de force,
en
limit&
est sugg&&
theorem
les d&accords
it la fois sur des Bchantillons
Bcroui,
qualitativement
3 de
d’essais
le fluage
d’alliage
messen wurden.
des vitesses
pour
et dans le sens diam&ral
bei Proben
corr8lation
fois
la puissance
&ultats
and pressure tubes
for further work are indicated.
d’expliquer
les
uniaxiale
coefficients
tente
Canada
8. environ
utilisant
or of the anisotropy
Ce travail
Ontario,
proportionnelles
l’effort.
geschwindigkeiten
with stress. Suggestions
*
isotropes si les vitesses de fluage en condition uniaxiale
a variation
creep
MATERIALS
Waterloo,
assuming
for anisotropic
of the power-index
of
are proportional
for both uniaxial
of cold
be correlated
rates
of stress.
results of both uniaxial proposed
creep
CO., AMSTERDAM
1968; in revised form 9 April 1969
This work attempts to explain discrepancies in the correlation of in-reactor creep rates measured on both uniaxial specimens 14) and reactor pressure tubes 5, 6) made of cold worked zirconium alloys. An in-reactor creep theorem for anisotropic zirconium alloys is proposed which indicates that for differences in creep moduli of approximately 20’36, diametral creep rates of
Engineering,
of Civil
PUBLISHING
NUCLEAR
and M. J. HOLICKY
J. SCHROEDER University
0 NORTH-HOLLAND
Research 52
Institute,
GWT,
Prague,
Czechoslovakia.
ON
is) Freudenthal
SECONDARY
and
CREEP
OF
ANISOTROPIC
53
MATERIALS
B may be calculated in view of eqs. (1) and (2)
Geiringer is). The work-
ability of the proposed analysis is demonstrated
NUCLEAR
from the condition
by an attempt to correlate diametral creep rates
that
B~jm&&m =
/%” = 8,
(5)
measured in the reactor on cold worked zirconium alloy pressure tubes 596) with uniaxial
where
creep data obtained by Fidleris 14). An extensive
B and s are good measures of creep behaviour
review
of zirconium
given 2.
of
literature
on
anisotropic
creep
is
in ref. ‘9 12).
through In the following,
Cartesian tensors are used
term. &f denotes the Kronecker delta. Considering the work *-Is) the following theorem
is
proposed
iij = /9s(“-1)Asjrlskl, &t = 0, dff represents
secondary
for
(1)
steady
creep
rates, both B and m are material constants depending on temperature T and/or fast neutron flux 9j and may be found from uniaxial creep test, the Aijkl are coefhcients of anisotropy, .Q = GIN--&ah&~ designates stress tensor aij= ajr and
the deviator
8 = (A~,xIw&, is the so-called Theorem
equivalent creep
of the
(2) or effective
(1) may be developed
a dissipative
alloys,
depends
potential
stress.
such
that
,5W = J.$ daa/ represents the complementary of the dissipative strain energy density dusij = &a~~ and consequently Baj=/lo W/bsij. Choosing W = =sm+l/(m+ 1) the power law (1) is obtained. The incompressibility condition dii = 0 may be enforced by setting Ati&
= 0.
(3)
An equivalent or effective strain rate 6 is defined by postulating 16) that gi+si3= is and it follows from eqs. (1) and (2) that the theorem (1) can be rewritten in the form i = @n.
(4)
upon
how well
in terms of test data
of this work, theorems
(1)
or (4) will be restricted to cases where the axes of anisotropy are orthogonal and approximately coincident with the principal axes x1, x2, 5s of strain rate and stress. In addition, creep rate differences in tension and compression are neglected even though theorem (1) may be generalized to account for these differences by defining All11 = A$,, if a11 is tensile, Aiiaa= if ai1 is compressive, Aliz,= A&& if = A,,, all and aa2 are tensile, Anss= A&; if ai1 is tensile and a22 is compressive, etc. Under the restrictions specified above, and in view of the symmetry of both the strain rate tensor and stress tensor, problem (1) involves six coefficients of anisotropy AIIII, A22227 A33339 A1122, 4133, which are interrelated by the three constraints shown in (3). Choosing AIIII, Az222, An22 as independent, s and 6 may be expressed for these simplified conditions as A2233
s = [A1111(~11- ~33)~
by introducing W
Whether or not
eq. (4) or eq. (1).
In the remainder
as described for example l5), i.e., the orthogonal Cartesian coordinates are designated by xl, x2, x3 and the subscripts i, i, k, 1, m, n, p, q= 1, 2, 3 are summation indices if repeated in a single
where
= Amnpp
i and s can be correlated
A creep theorem for zirconium alloys
anisotropic creep zirconium alloys :
&d~~rnnAklpq
+ A2222(S22 - ~33)~ +
+ 2A1122(S33 - Sll)(S33
- S22)]*
(5)
and d= (All1lA2222 - A2~~2$t(A,,,,~2,, + 3.
A1111i222 -
+
%41122il1i22)t
(6)
Uniaxial and biaxial in-reactor creep of cold-worked zirconium-alloys
As part of a major test programme, Fidleris 14) investigated the in-reactor creep behaviour of uniaxial specimens manufactured from coldworked zirconium alloys. The specimens were made from bar stock or machined from pressure tubes such that the longitudinal axes of the specimens were parallel to the axes of the tubes.
J. SCHROEDER
54
For
bot’h types
{ lOi%}
plane
of
specimens
normals
the
were
parallel to the longitudinal
AND 111.J. HOLICKY
prismatic
predominantly
uniaxial data 14) but using normalization proposed 5).
axis of the specimen.
Zirconium
the
linear
flux
alloy pressure tubes investigated
Ross-Ross
and Hunt 5) and Ibrahim 6) studied
by
in-reactor
creep of biaxially
exhibited a t,hickness/radius ratio 0.05 -
tubes made from cold-worked
stressed pressure zirconium
alloys
ROSS-ROSS and
and have
to be classified
of the t’ype studied 14). Results
of all t’hree
r and h denote
investigations
of creep rate
and
indicate dependence
on st’ress, temperature
and fast neutron
flux
over the range tested. [In 14, 6) test data of heat treated zirconium alloys are also report’ed but because of the scatter of results encountered in the case of heat-treated uniaxial creep tests, no correlation is attempted.] In 5) in-reactor diametral creep rates of pressure tubes tested were correlated employing an empirical creep law and an attempt was made to interrelate the uniaxial results 14) and biaxial test data of the pressure tubes 5) using an isotropic analysis as an approximation since a manageable theorem of creep for anisotropic materials was not available as remarked 5). Using the isotropic approach, diametral creep rates calculated in accordance with uniaxial test data appeared to be in agreement wit’h t’est results of tubes at hoop stresses of 11 000 psi only, while at 20 000 psi t,he calculat’ed creep rates were three to six times the rat’es indicated by the tubes tested. Uniaxinl creep rat,es 14) and diametral
thickness
complicated
Hunt 5) and as thick
Ibrahim 6) shells 18);
radius of the middle of
the
tubes
surface
respectively.
A
thick shell analysis, as for example
demonstrated 9, 19), is in this case not warranted since it may improve the results by only 5o;b I*). Pressure tubes test,ed 5) passed vertically through the reactor core a,nd contained a t,cn foot high stack of six fuel assemblies cooled by pressurized water. Each fuel assembly had a different power output which decreased the neutron flux in steps from the center towards both ends of the tubes. Due to the variation of creep rates with fast neutron flux the deformation mode of axial sections of the tubes consisted of a series of bell-shaped stel)s associated with every fuel assembly as shown in fig. 3 of ref. 5). In accordance with elast’ic theory of thin shells 2o) each portion of the pressure tube surrounding a fuel assembly may be classified as a long cylindrical shell. where the deformation in the central portion is independent of the end condition of each portioll. This assumption appears to be confirmed by
creep
the test data shown in figs. 3 and 7 of ref. 5).
rates 5, 6, were normalized in 6) for temperature and fast neutron flux using the empirical linear correction 5) based on biaxial data only.
The tube sections exposed to fast neutron flux 6, also qualify as long shells and consequently the only non-zero stresses existing in accordance with thin shell bheorem in the tube sections 5, 6, where creep rates were measured are
In addition, the tubes considered purposes by a adjustment for
hoop stress of all cold-worked 596) was reduced for correlation constant fact’or of 0.5. an biaxiality based on known
texture coefficients IT), and reasonable grouping of uniaxial and biaxial data along a convex curve was obtained by plotting normalized uniaxial and diametral creep rates versus uniaxial and adjusted stress, respectively, on a log-log graph. In this work, test data of cold-worked zirconium alloys shown 14, 5, 6)> are correlated through eqs. (1) and (4) by finding in accordance with continuum mechanics B and m from
oB= p,rlh = 2~~= constant,
(7)
assuming that the end conditions of the pressure tubes represent closed-ends. go and a, designate hoop and axial stress respectively and p is the internal pressure of the tube. The effects of elastic deformation of the tube due to temperature and internal pressure will be neglected for the creep analysis since they do not change the original shape of the tubes significantly. The stresses in the central portion
ON
of each tube recalculating
SECONDARY
section
CREEP
should
OF
ANISOTROPIC
be adjusted
by
NUCLEAR
55
MATERIALS
and
h and r for creep strains larger
than 0.05% which is the upper limit for the infinitesimal theory of small displacements. But this adjustment up to the maximum measured diametral strain of about 0.2% is insignificant in comparison assumptions limitations
with the errors due to both the made above and the experimental
indicated
(9b) Similarly,
eq. (4) may be expressed
in view
of eqs. (% (I), (7) and (5) for uniaxial biaxial cases as
and
51 6, 14). (lOa) and
4.
Correlation of in-reactor cold-worked Zircaloy-2
creep data of
As stated by Fidleris 14), uniaxial
LO= ((aA,,+A,,+A,,)t/(A,,+3A&)}~~= test data
are inconclusive with respect to the creep rate dependence on fast neutron flux +, because the range of flux tested was 0.36 to 1.16 x 1013 n/cm2esec only where 11 denotes numbers of neutrons. But the results 5) indicate an almost linear dependence of diametral strain on integrated fast neutron flux if stress and temperature are constant. Because of this linearity the variation of creep rates with fast neutron flux will be accounted for by normalizing as in ref. 5) both uniaxial and biaxial creep rates over periods of approximately constant flux, stress and temperature to a reference level of 1 x 1013 n/cm2. set and normalized creep rates will be denoted by .$i = iii/no, where n0=gj/(1013 n/cm2.sec). In view of the above simplifications,
(8) theorem
(1) or (4) may be used directly to calculate normalized creep rates corresponding to a state of stress indicated in eq. (7) using cylindrical r, 8, z coordinates, i.e., setting pairs of subscripts ll=z, 22=0, 33=r. Eq. (1) may be written for both &lo and 0~1 denoting normalized creep rate and stress respectively of uniaxial specimens with axes parallel to the longitudinal z-axis of the tubes and for 8,O the normalized hoop creep rate which is identical to the diametral creep rate of the tubes. Using eqs. (S), (7) and (5) in eq. (1) Pa)
=BO[(aAZz+Aee+Aez)tag]m=r60sm.
(lob)
Since ,80and m will be evaluated in accordance with eq. (9a) for cold-worked Zircaloy-2 using uniaxial experimental data 14)shown in table Al of the appendix, it is convenient to set AZz=l.
(11)
The data of table Al were selected as fairly reliable from a temperature range of 220 to 350 “C and a stress range of 11 000 to 40 000 psi since these are approximately the ranges at which the tubes 516) were tested. [In ref. 596~14) all measured values indicated are averages but in this approximate analysis instantaneous and average values will not be differentiated.] /?Oand m may be estimated from these limited uniaxial data and used for the pressure tubes by assuming that 1. The dependence
of normalized
creep rate
on temperature indicated by ,50 does not vary significantly with stress from 11 000 to 40 000 psi; 2. The stress dependence of creep rates measured by m is not significantly affected by temperature between 220 and 350 “C and does not vary with stress between 11 000 and 40 000 psi; 3. Neither /P nor m change significantly with fast neutron flux over the range of 0.36 to 3.1 x 1013n/cm2. sec. The last assumption is necessary, since the uniaxial tests were conducted at fluxes of 0.36
56
J. SCHROEDER
AND
M.
J.
HOLICKY
to 1.16 n/cm2. set and the tubes were tested
the estimates of ,P and m exhibited
at fluxes of 1.15 to 3.1 n/crnz. sec. The above
may be less useful outside these limits.
assumptions
had to be made in order to utilize
the limited
number
of results available.
even if the following to be qualitative suggestions
correlation
As remarked
But, and
the
concept
creep 14) is used,
of
activation
energy
/l’J= B exp ( -b/T)
B
300 “C and at fast neutron fluxes greater than 1.16 x 1013 n/cm2 -sec. Fig. 6 of ref. 5) indicates strain versus integrated neutron flux at various elevations of this tube from which average normalized creep rates d, were obtained as shown in table 2. In accordance with eqs. (9b), (12) and (ll), the coefficient
and b are constants and T is temperature (“K). Employing the method of least-squares for two independent variables 22) to the logarithmic form of eq. (9a) simplified by eq. (ll), the constants, B, b and m may be estimated using the selected data of table Al in connection with the assumptions (1) to (3) such that for cold-worked Zircaloy-2 ,P=B
A = (4 + A,, + A,JQm-1) (A,, + QA&) = 4/(B04?),
exp (-b/T)=
m=2.72;
(12)
Numerical values of /?O are tabulated in table 1 for various temperatures. /?Oand m shown in eq. (12) should be considered as estimates of cold-worked
Zircaloy-2
material
constants only, since they are based on limited experimental evidence and will change if more test results become available or if different data
A values in table 2 corrected
TABLE Values 250
i
260
270
for this tempera-
ture increase using eqs. (13) and (12) indicates that ,@J may be used over the full range of temperature and fast neutron flux shown in eq. (12). [For a consistent comparison with other tubes 270 “C will be used for U-2 Mk IV at elevation 261’-4” as the average of the inlet and outlet temperatures shown in table 1 of ref. 5) instead of the weighted average temperature of 264 “C.]
are selected from ref. 14). The numbers in eq. (12) are given to three significant figures for reasons of numerical computation only. The majority of the uniaxial data on which PO and m are based correspond to temperatures of 300 and 350 “C, stresses of 20 000 and 30 000 psi and fast neutron fluxes ranging from 0.36 to 1.16 n/ems. set as shown in table Al. Consequently
T W)
(13)
should be a constant for different combinations of pressure, temperature and fast neutron flux if the anisotropy coefficients are invariants. The above measure of anisotropy A was calculated at various elevations in accordance with eqs. (13) and (12) from the average creep rates shown in table 2, first using for each elevation 264 “C? which is the temperature at elevation 461’-4” weighted with respect to time in accordance with fig. 5 of ref. 5). Estimating an increase in temperature with elevation of 3.6 “C per foot from table 1 5) the consistency of the
= 26.0 x lo-13 exp [ - 8 390/T] ; T = 222 to 350 “C, stress= 11 000 to 40 000 psi ; pi=O.36 to 3.1 x 1013 n/cm2+sec.
the
tube in fig. 6 of ref. 5) may be used to test the validity of the estimate (12) for PO below
for
where
5)
of flux by the test data. However, detailed results available for the U-2 ;Llk IV pressure
for further work may be obtained
from the analysis. If
and Hunt
effects of temperature and stress on diametral creep rates are not as well defined as the effect
is considered
only, valid information
by Ross-Ross
in eq. (12)
1 of /?”
1
280
1 290
300
1
310
320
ON
SECONDARY
CREEP
OF
ANISOTROPIC TABLE
NUCLEAR
MATERIALS
57
2
Cold-worked Zircaloy-2 pressure tube U-2 Mk IV 5, A=(*+A,+LI,,)*(~-‘)
(ABe+*Ae,),
ue= 14 000 psi
/ Elevation
1(lO_S~i~in~h_l~)
464:--g”
11.4 9.5 8.5 7.0 6.5 6.1 6.1 6.5
463’-3” 462’-0” 461’-4” 460’~0” 458’-9” 458’-3” 456’-5”
adju:d for not adjtsted for ) temperature gradient temperature gradient 1.5 1.2 1.1 0.9 0.8 0.7 0.7 too close to end
of A calculated in accordance eq. (12) for m = 2.72 for various values
Magnitudes with
of A,,and A, are shown in table 3 and indicate changes of up to 100% for differences in A, and A, of about
20%
only.
TABLE 3 Change in A=(f+A,,+A,,)‘)“-l’(A,+Ae,) variation of ABo, A, for WL=2.72 A,, 1 1 1
/ A,, 1 1.2 0.8
1 A, -0.5 -0.3 -0.5
1 A 0.6 1.2 0.3
due to
1.0 1.0 1.0 0.9 0.9 1.0 1.0 too close to end
descriptions independent of temperature and for comparison of experiments and theorems, both sides of eqs. (9) and (10) have to be divided by 80. In order to facilitate this comparison numerical values of j30 are given in table 1. The uniaxial results shown in table Al of the appendix are plotted in fig. 1 as a,1 versus .$!J~Ousing a log-log scale and show reasonable scatter with respect to the straight line
Material type isotropic anisotropic anisotropic
(14)
Due to technical difficulties transverse normal
i.e., eq. (9a) combined with eq. (11) for values given in eq. (12). Two creep rates in fig. 1 are indicated at 45 000 psi 14) and appear to show a possible variation of m with stress. But since
strain rates of uniaxial creep specimens have not been measured and results of creep tests of uniaxial specimens with longitudinal
45 000 psi corresponds almost to first yield of the material 14) no conclusions can be drawn. Data of cold-worked Zircaloy-2 pressure
axes tangential to the circumferential direction of the tube are not available. Consequently the coefficients of anisotropy A,, and A, are unknown and direct correlation of uniaxial and biaxial test results through the power law (10) in terms of the effective strain rate and effective stress is not possible. However, in view of eq. (9) uniaxial and biaxial results from tubes should be grouped with respect to two parallel lines on a log-log graph of creep rate versus stress if the coefficients of anisotropy do not vary with tubes and m is invariant. For the following correlation, data are plotted in terms of &/~O, $//P, S/PO in order to use
tubes 5) are shown in table A2 of the appendix: the U-3 Mk V tube was not considered in this work because of low stress (about 5 000 psi) and an organic coolant. Results 6) of small cold-worked Zircaloy-2 tubes are tabulated in table A3 of the appendix and are represented with the data of table A2 in fig. 1 as $‘//?O versus c0 for comparison with uniaxial test results 14). The points associated with ref. 5) appear to be grouped with respect to a line where m is approximately one, whereas in view of eq. (9) data belonging to ref. 6) appear to be in agreement with uniaxial data if cB is greater than 20 000 psi. The value m= 1 was
58
J.
SCHROEDER
50
AND
M.
J.
HOLICKY
I I I ZIRCALOY-2
COLD-WORKED
0 *
40 rT 0 x
30
E 20b
UNIAXIAL
SPECIMEN
UNIAXIAL
SPECIMEN
A
PRESSURE
.
SMALL
.
REFERENCE
b b”
0 0
I
I
‘5.
/
Al
RX-21,
14 TUBES
TUBES
,L?= 260
TABLE
TABLE
TABLE
A2 A3
x10-‘3exp[-6390/TIoK;1
-,
/ (k
if,/ Fig.
1.
Cold-worked
represented
as
Zircaloy-2
uniaxial
or
in-reactor
hoop
stress
p”
creep versus
or data
;i
/ p”
from
uniaxial
normalized
by B”, eq.
hre’psi.e272
(psim
set cd
x IO”)
specimen
uniaxial
/
O+ 10’3.)
or
14) and tube
diametral
creep
specimens rate
5, 6)
divided
(12).
used by Ross-Ross and Hunt 5) in order to interrelate results of pressure tubes using a linear temperature correction. The disagreement with eq. (9) indicated in fig. 1 has two possible explanations : 4. The power index m varies with stress for cold-worked Zircaloy-2, or 5. The cold-worked Zircaloy-2 coefficients of anisotropy are stress dependent. Explanation (4) is feasible since only one
ends assumed in eq. (7) may not have been perfect. But since in 6) an open loop was used, i.e., perfect closed end condition existed, and the experimental data 6) show the same trend as those 5, at stresses below 20 000 psi, this explanation does not appear to be warranted. The validity of explanations (4) and (5) can be proven only by conducting more uniaxial creep tests in the react’or. The possibility that (5) is valid may be
uniaxial
investigated by considering whether or not a possible relationship between the measure of anisotropy A defined in eq. (13) is indicated
experimental result 14) exists below 18 000 psi as shown in fig. 1 and for example Kachanov 8) has indicated, that creep data are sometimes grouped along a line consisting of two segments of different slopes or along convex curves on a log-log graph of creep rate versus stress. In-reactor creep may depend on slip and/or diffusion 21) and rotation of basal planes of the hexagonal crystals under stress suggested by Hill 23) and/or dependence of magnitude and direction of diffusion on stress indicated by Garofalo 24), may account for a variation of anisotropy with stress, i.e., explanation (5). The discrepancy shown in fig. 1 could be due to a variation of stress ratio ae/az for the pressure tubes 5) with load since the condition of closed
by data 53 6) for a constant
m=2.72.
Values
for A calculated for all the tubes in accordance with eq. (12) are shown in tables A2 and A3 and are plotted versus stress in fig. 2a. There appears to be a significant dependence of A on stress and fig. 2b indicates that the overall change in A can be explained by variations in A,, or A, of reasonable magnitude which for tubes tested in the NDP reactor may be classified in the following way: A,,>l,
A,< -0.5; = -0.5;
0.8
A,,
A,>
A,= -0.5.
(15)
In view of eqs. (9b), (11) and (13) the effect
ON
SECONDARY
CREEP
OF
ANISOTROPIC
59
MATERIALS
on A of a possible variation
1 COLD-WORKED ZIRCALOY -2 . NDP PRESSURE TUBES TABLE A2 . NRU PRESSURE TUBES TABLE A2 . NRU SMALL TUBES TABLE A5 COLD-WORKED ZIRCONIW-25% ANRU PRESSURE TUBES TABLE
NUCLEAR
in Au is reduced
by a factor
of $ and may be neglected. Since (5) appears to be a possible explanation
NS A5
of the discrepancy constant
indicated
for cold-worked
in fig. 1 if m is a
Zircaloy-2
over the
stress range shown in eq. (12), it may be worthwhile to select comparison
2 011
I
IO
I
,
15
20
(0)
25
I 35
30
40
2
Fig.
2a.
eq.
(13)
Dependence on hoop
IO
12
of measure
stress for cold-worked
for anisotropy
cold-worked
14
coefficients
zirconium alloy tubes
anisotropic
strain
view of eq. (10) with respect to one straight line on a log-log graph of 9//30 versus s, if both 80 and s are adjusted for variations in A,, and A, with stress. Selecting from the curve shown in fig. 2a a value for A corresponding to the hoop stress of the tubes considered 59s), a comparison value for A,, was found from
16
of anisotropy
alloy tubes 5, 6) if m=2.72=constant. values
and (11) effective
rates do and stress s for the measured diametral creep rates and hoop stresses of the tubes 516). Because A was calculated from eq. (13) for a constant m=2.72, data from uniaxial specimens 14) and tubes 596) should be grouped in
+
08
and with
eqs. (lob)
U# (psi x IO31
06
values of A,
A, from fig. 2 and calculate in accordance
A
zirconium
b. Comparison A,,
and
5,
if m=2.72=
6)
A,
for
fig. 2b using a constant A,= - 0.5 as shown in tables A2 and A3. The effective strain rates and stresses calculated for the tubes in terms of these comparison values correlate as predicted with the uniaxial data shown in fig. 3. If different comparison
constant.
values for A,, and A,
are selected
40 30
;“I Fig. 3.
Cold-worked
represented
Zircaloy-2
as effective
in-reactor
anisotropic
p*
(psimx
IO”)
creep data from uniaxial
stress versus effective sssuming
specimen 14) and tube
anisotropic
m=2.72=constant.
strain rate divided
specimens
9
)
by PO, eq. ;12;,
60
J.
in accordance are obtained.
agreement
was
10, ref. 6) by grouping
indicated
both
a convex
curve
uniaxial
and
diametral
uniaxial
and
one
respectively.
creep
half
The hoop
Zircaloy-2
on a log-log of
graph
rates
the
hoop
of
versus stress
stresses were reduced
by one half in order to account for biaxiality in accordance with known texture coefficients assuming that creep is mainly caused by slip between grains. This reduction of the hoop stress is analogous to a correlation using an anisotropic creep theorem and the method of correlation indicated in fig. 10, ref. 6) may be simulated by setting in eq. (lob) the coefficient of $ to one and the coefficient of (T@to one half. It follows that independent of m and /30 the value of A,,= 1 and of A,,= - 1. The corresponding magnitude of A calculated for m= 2.72 from eq. (13) is 0.15 and appears to be too low even for the small tubes as shown in fig. 2b mainly because in this work a temperature dependence based on uniaxial results was used in accordance with continuum mechanics, whereas 5) a linear temperature correction based on biaxial results 5) only was employed for both uniaxial and biaxial data. COLD-WORKED ITi1
- _ _
v-l
0
ZIRCONIUM-2.5%
M.
J.
5.
in
uniaxial 14)
and biaxial 516) data of cold-worked along
AND
with fig. 2, similar correlations
Reasonable fig.
SCHROEDER
HOLICKY
Correlation of in-reactor creep data of cold-worked zirconium-2.5 wt o/o niobium Only three cold-worked
niobium
zirconium-2.5
wt, y.
uniaxial specimens were tested la), all
three at an average temperature
of 300 “C as
shown in table A4 of the appendix.
In fig. 4, the
power index m = 2.72 estimated for cold-worked Zircaloy-2
in eq. (12) appears to be in agreement
with
three
the
uniaxial
data
obtained
for
zirconium-2.5 wt o/o niobium and due to lack of experimental evidence this value of m and the variation of normalized creep rate with temperature shown in eq. (12) will be used such that as an est’imate for cold-worked zirconium-2.5 wt o/o niobium PO= 8.5 x IO-13 exp [- 8 390/T (OK)], m= 2.72,
(16)
where the coefficient 8.5 x lo-13 was estimated from eq. (14) in connection with the average of the three creep rates shown in table A4. Data of cold-worked zirconium-2.5 wt y. niobium pressure tubes obtained from 59 6) are shown in table A5 of the appendix and are plotted as &O//30versus u0 on a log-log scale in fig. 4a where a similar discrepancy with uniaxial data exists as indicated in fig. 1 for cold-worked Zircaloy-2. The measures of anisotropy A in table A5 are calculated in accordance
NB 25
25
x ‘g
t?
20
15
ii
A PRESS. TUBES TABLE A5
tf
IO
I
2 ;z,
/fl”
3
45
IO
or 2; /p” (psimx IO”)
s? z x ‘3 e YI
15
IO
I
2 8”/
3 6”
45
IO
(psimx IO”)
(b)
(a) Fig. 4.
Cold-worked zirconium-2.5 wt y. niobium in-reactor creep data from uniaxial specimen 14) and tube specimens 5. 6) represented in (a) as uniaxial and hoop stress versus normalized uniaxial and diametral creep rate divided by PO, eq. (16) and in (b) as effective anisotropic stress versus effective anisotropic
strain rate divided
by ,P, eq. (16) assuming
WL= 2.72=constant.
ON
SECONDARY
with eqs. (13) and (16) similar dependence
CREEP
OF
ANISOTROPIC
and appear to have a
effective
anisotropic
Ross-Ross, Atomic Energy of Canada
1) P. A. Limited
Report,
AECL-3126
(1968)
2, W. Evans, J. E. Le Surf and W. R. Thomas, At,omic
stress
Energy
AECL-2890
versus effective anisotropic strain rate divided by ,80 was established in a similar manner as
B. A.
of
Canada
Cheadle and W.
of Canada
Limited
Evans,
Report,
fig. 3 and indicates the same type of correlation.
Report,
AECL-1048
6.
P. A. Ross-Ross
and C. E.
Limited
Conclusions
of the material. If the power index for coldworked zirconium alloys is constant the variation of the coefhcients of anisotropy indicated by the variation of the so-called measure of anisotropy A shown in fig. 2a, may be used to draw conclusions regarding the basic inreactor creep mechanism 21) since A appears to approach a constant or at least a minimum value.
Mat. E.
The author wishes to acknowledge the valuable assistance and advice given by Mr. Ross-Ross, Fuels and Materials Division, Atomic Energy of Canada Limited, for both the formulation of the problem and the evaluation of the experimental data.
Report,
Atomic
Energy
AECL-2652
(1966)
Atomic
Energy
of Canada
(1960) L.
Hunt,
J. Nucl.
2
Ibrahim,
In-reactor
creep of zirconium
alloy tubes and its correlation with uniaxial data, ASTM
Symp.
delphia;
7)
I.
Berman
8 (1966)
8)
zirconium
Nov.,
and D.
L. M. Kachanov, tizdat,
and
hafnium
(Phila-
1968) H.
Pai,
Int.
J. Mech.
Sci.
341 The theory
Moscow,
Lending
1960)
Library
Boston
of creep (Fizma-
; Engl. transl., National
for Science
Spa, Yorkshire,
and
England
Technology,
(1967)
Q) K. G. Odqvist, Mathematical theory of creep and creep rupture
loI
(Oxford,
Y. N. Rabotnov, Progress
in
New York,
1966)
The Prager anniversary volume
Applied
Mechanics
(MacMillan,
1963) p. 307
11) N. I. Malinin, PMFT Zh. Prikl. Mat. Tekh. Fiz. 3 (1964)
16
12) W. Olszak and A. Sawczuk, Inelastic behaviour in shells (Noordhoff, Groningen, The Netherlands, 1967) 13
)
A. M. Freudenthal pedia of Physics
and H.
Geiringer,
(Springer,
V. Fidleris, J. Nucl.
15)
W. Prager, Introduction to mechanics of Continua
9
R. Hill,
(Ginn, New York, 17
)
B.
19
Mech.
A.
Cheadle
and
Techn.
4 (1966)
329
Groningen, )
D. H. S.
29
26 (1968)
51
1961)
J. Appl.
V. V. Novozhilov,
13)
Mat.
Encyclo-
1958) p. 229
14)
17 (1950)
C.
E.
64
Ells,
Electrochem.
Thin shell theory (Noordhoff,
The Netherlands,
1964)
Pai, Int. J. Mech. Sci. 9 (1967)
Timoshenko
Theory
Acknowledgment
Perryman,
26 (1968)
F.
Limited
(1967)
E. C. W.
The anisotropic creep theorem proposed in this work appears to be useful to analyse uniaxial and biaxial in-reactor creep behaviour of coldworked zirconium alloys. Using the limited results from uniaxial tests 14) it was shown how both uniaxial data and results of biaxially stressed tubes596) may be correlat,ed. In order to analyse the behaviour of zirconium alloys in the reactor throughly it is necessary to test assumptions (1) to (3) by conducting more uniaxial creep experiments which will automatically lead to conclusions regarding explanations (4) or (5) given in section 4. Furthermore, it appears to be essential to determine the coefficients of anisotropy Ace, A, and their variation with stress if the power index m is found to be invariant, in order to facilitate the correlation of these coefficients with texture
61
MATERIALS
References
on stress as the A values
of the cold-worked 7ircaloy-2 tubes for an 17~ of 2.72 as shown in fig. 2a. The log-log graph in fig. 4b indicating
NUCLEAR
Toronto,
of
and
plates
S.
and
335
Woinowsky-Krieger, shells
(McGraw-Hill,
1959)
21) G. R. Piercy, J. Nucl. Mat. 26 (1968) 18 22
)
E. F. Croxton and D. J. Cowden, Applied general statistics
23
24
)
1
(Prentice-Hall,
1955)
R. Hill,
The mathematical
(Oxford
Clarendon Press, 1964) p. 317
F.
Garofalo,
Fundamentals
rupture in metals p. 178
theory
of plasticity
of creep and creep
(MacMillan,
New York,
1965)
Appendix Al
TABLE
Data
based
on uniaxial
test results
Test
Temp.
no.
(“C)
R-6
of cold-worked
Stress (lO:‘,si)
Fast
flux
rate
table
/
0.54
3.5 I_t 107;
5.7
0.60
13.0 & loo/:,
5.8
300
18
0.58
3.0 *
20%
5.1
300
11
0.61
0.5 i
600,b
0.7
300
20
0.64
3.0 & 3076
4.1
320
20
0.68
6.0 & 20%
5.1
350
20
0.68
14.0 5 200,;
5.6
220
30
0.96
1.5 *
3076
11.5
260
30
0.93
5.0 & 4o”/b
14.2
350
30
0.94
32.0 + 10%
9.2
R-2
300
30
0.66
15.0 & 10%
19.9
R-4
300
20
0.58
3.5 & 30%
5.3
300
30
0.36
8.0 + lo”:,
20.0
Rx-2(
350
30
1.16
60.0 f
lOo,d
14.0
Rx-21
300
40
0.84
35.0 *
5076
36.5
1 and fig.
8 of ref. 5,
Zircaloy-2
TABLE Data
based Data
on results
based
Av. of inlet
Hoop
and outlet
stress
I temperature (“C)
selected
on results
I
tubes
A2
from
table
TABLE
A3
shown
in figs.
3 and
7 of ref. 6, ~
~ Diametral
A,,
comparison
ng
I(lo3psi) Table
264
IJ)
( x 101’)
20
Cold-worked
4 of ref.
i:1 IB” eqs. (8, 12)
‘z (1O-7 (:n;in)h-I)
20
Rx-1I
NDP
from
350
Rx-14
designation
Creep
~
1013 n/zmj.sec)/
selected
300
R-9
Tube
Zircaloy-2
A2
U-2 Mk III
281
~ 10.5 i 11.5
U-2 Mk IX U-2 Mk IV
281
13.4
3.1
270
~ 14.0
2.7
1.9
1.4
0.73
~
1.06
U-2 Mk IX
281
17.3
3.1
2.9
I
1.4
0.40
~
0.92
258
15.7
2.56
1.3
~
0.98
19.4
2.56
2.4
I I
0.55
258
2.6
258 258
23.9 29.2
i
2.56 2.56
2.7 3.6
’
2.56 2.56
1
~
~
1.15
0.54
1.1
2.7
1.8
0.9
~ 1.27 i 0.86
2.3
1.1
0.65
~
Table
258 258
I
~ 34.0
37.6
I
1
I
1.28 1.20 1.09
A3
6.1 8.8
1.4 2.9 3.9
;
6.6 9.6
I ~
0.57
0.87
0.36
0.80
0.28
0.80
0.31 0.34
0.80 0.80
ON
SECONDARY
CREEP
OF
ANISOTROPIC
NUCLEAR
63
MATERIALS
TABLE A4 Data based on uniaxial test results of cold-worked zirconium-2.5 wt ye niobium selected from table 4 of ref. 14)
R-11 R-12 R-17
~- Creep rate
Fast flux
I_
Test no.
“” (10-e (in~m)h-1)
/ / )
300 300 300
i j /
23.0 16.5 20.0
20.0 & 15% 7.0 & 10% 10.5 -& 40%
o.7
\
0.56 0.68
1 ; ;
7.7 3.4 4.2
TABLE A5 Data based on results from cold-worked zirconium-2.5 wt ye niobium pressure tubes selected from ref. 5, 6)
Tube designation
Av. of inlet and outlet temperature
EIoop stress
Diametral creep rates
Fast flux
9, (1013n/om2*sec) 1(10-7 (~~]in)h-l)l
(“C)
/ (lG:sii)
U-l Mk VIII
270 270
16.5 22.0
2.9 2.9
U-1MkV
285 285
15.5 21.0
3.1 2.0
/ /
W” eqs. (8, 12) ( X 1Olif
A eq. (13)
6.5 10.4
1.35 2.16
0.46 0.33
6.7 7.8
1.07 1.54
0.43 0.27
A,, comparison value selected from fig. 2b for A,= -0.5
1
0.96 0.80 1.01 0.80
OF NUCLEAR
33
MATERIALS
ON SECONDARY
(1969)
52-63.
CREEP OF ANISOTROPIC
of Waterloo,
Received
Department
26 November
anisotropic pressure tubes may be twice
half
as high
as diametral
tubes if the uniaxial about
the third
creep rates
power
test results available in-reactor
creep
is shown
at
can
least
Using
worked
or one isotropic
and diametral
that
specimens
in accordance
to
the limited
zirconium
qualitatively
alloys
it
experimental
sont
En
disponibles,
5t la
de
moins
zirconium
with
the
sur Bchantillons des tubes accord
avec le theor&me
anisotrope
qui suppose
de puissance contrainte.
Diese
dans
de fluage en rkacteur,
sol1
an
en contrainte
la
aus
anisotropen
metralen
uniaxiale,
wenn
des
theorbme
de
moduln
etwa
20%
Bcroui.
Un
pour des alliages
dans les modules
de fluage d’environ
zirconium
peuvent
les vitesses
de fluage dans le sens diametral
der sich
Btre 2 St 1,5 fois aussi Blevbes que
designed alloy
to
materials serve
pressure
in
tubes
are the
as
form core
of
analytic *
theory
zirconium
components
Research
Assistant
on leave
creep from
in
for multiBuilding
bzw.
Spannungszustand
Reaktor-
kijnnen bei
die dia-
anisotropen
Unterschiede
betragen
und
der
die
Kriech-
einachsigen
etwa mit der dritten Anhand
qualitativ
Zusammenhiinge lassen.
ge-
zum Kriechen unter
Danach
bzw.
Zirkon
halb so gross wie bei isotropen die
variieren.
zumindest
Kriech-
Bestrahlung
Weitere
mit
der
Potenz
vorhandenen
nachgewiesen, dem
Theorem
Untersuchungen
dass be-
werden
axial states of stress. Berman and Pai 7) have formulated a simplified theorem of creep for anisotropic materials assuming that in stress space the magnitude but not the direction of the creep rate vector depends on anisotropy. Here an exact analysis of anisotropic in-reactor creep is proposed for zirconium alloys considering the work of Kachanov *), Odqvist 9), Rabotnov lo), Malinin I’), Olszak and Sawczuk
and
l-4). Recently,
of anisotropic
la
vorgeschlagen.
Ross-Ross and Hunt 5) and Ibrahim 6) studied the inreactor creep behaviour of these pressure tubes experimentally and indicated the need for an nuclear power reactors
wird die
schreiben
des tubes
fabricated
Spannung
Daten
de force
Introduction Anisotropic
avec
der
kalt-bearbeitetem
Kriechgeschwindigkeiten
20%, les vitesses des tubes
anisotrope
1.
aniso-
que pour des diffhrences
dans le sens diam&ral
fluage
de ces travanx
wie sie unter
Kriechgeschwindigkeiten sein,
de fluage
anisotropes
Zr-Legierungen
doppelt
qui indique
le
du coefficient
Diskrepanzen
vorgeschlagen.
Rohren
en reacteur
pour
Es wird ein Theorem
Rohren
trope est propose
et sur eux en
ult&ieur
mit einachsigem
Druckrohren
bedingungen
mesur&
die
par
fluage
uniaxial
propose
klgren,
constitue
zirconium
qn’au
i?tre relies entre
ou des coefficients
et sur des tubes de force d’un reacteur de
montre
expkrimentaux
une variation
Un dt?veloppement
Arbeit
von
alliages
il est
& un fluage
peuvent
condition
dans un rkacteur
ces r&ultats
soumis
de force,
en
limit&
est sugg&&
theorem
les d&accords
it la fois sur des Bchantillons
Bcroui,
qualitativement
3 de
d’essais
le fluage
d’alliage
messen wurden.
des vitesses
pour
et dans le sens diam&ral
bei Proben
corr8lation
fois
la puissance
&ultats
and pressure tubes
for further work are indicated.
d’expliquer
les
uniaxiale
coefficients
tente
Canada
8. environ
utilisant
or of the anisotropy
Ce travail
Ontario,
proportionnelles
l’effort.
geschwindigkeiten
with stress. Suggestions
*
isotropes si les vitesses de fluage en condition uniaxiale
a variation
creep
MATERIALS
Waterloo,
assuming
for anisotropic
of the power-index
of
are proportional
for both uniaxial
of cold
be correlated
rates
of stress.
results of both uniaxial proposed
creep
CO., AMSTERDAM
1968; in revised form 9 April 1969
This work attempts to explain discrepancies in the correlation of in-reactor creep rates measured on both uniaxial specimens 14) and reactor pressure tubes 5, 6) made of cold worked zirconium alloys. An in-reactor creep theorem for anisotropic zirconium alloys is proposed which indicates that for differences in creep moduli of approximately 20’36, diametral creep rates of
Engineering,
of Civil
PUBLISHING
NUCLEAR
and M. J. HOLICKY
J. SCHROEDER University
0 NORTH-HOLLAND
Research 52
Institute,
GWT,
Prague,
Czechoslovakia.
ON
is) Freudenthal
SECONDARY
and
CREEP
OF
ANISOTROPIC
53
MATERIALS
B may be calculated in view of eqs. (1) and (2)
Geiringer is). The work-
ability of the proposed analysis is demonstrated
NUCLEAR
from the condition
by an attempt to correlate diametral creep rates
that
B~jm&&m =
/%” = 8,
(5)
measured in the reactor on cold worked zirconium alloy pressure tubes 596) with uniaxial
where
creep data obtained by Fidleris 14). An extensive
B and s are good measures of creep behaviour
review
of zirconium
given 2.
of
literature
on
anisotropic
creep
is
in ref. ‘9 12).
through In the following,
Cartesian tensors are used
term. &f denotes the Kronecker delta. Considering the work *-Is) the following theorem
is
proposed
iij = /9s(“-1)Asjrlskl, &t = 0, dff represents
secondary
for
(1)
steady
creep
rates, both B and m are material constants depending on temperature T and/or fast neutron flux 9j and may be found from uniaxial creep test, the Aijkl are coefhcients of anisotropy, .Q = GIN--&ah&~ designates stress tensor aij= ajr and
the deviator
8 = (A~,xIw&, is the so-called Theorem
equivalent creep
of the
(2) or effective
(1) may be developed
a dissipative
alloys,
depends
potential
stress.
such
that
,5W = J.$ daa/ represents the complementary of the dissipative strain energy density dusij = &a~~ and consequently Baj=/lo W/bsij. Choosing W = =sm+l/(m+ 1) the power law (1) is obtained. The incompressibility condition dii = 0 may be enforced by setting Ati&
= 0.
(3)
An equivalent or effective strain rate 6 is defined by postulating 16) that gi+si3= is and it follows from eqs. (1) and (2) that the theorem (1) can be rewritten in the form i = @n.
(4)
upon
how well
in terms of test data
of this work, theorems
(1)
or (4) will be restricted to cases where the axes of anisotropy are orthogonal and approximately coincident with the principal axes x1, x2, 5s of strain rate and stress. In addition, creep rate differences in tension and compression are neglected even though theorem (1) may be generalized to account for these differences by defining All11 = A$,, if a11 is tensile, Aiiaa= if ai1 is compressive, Aliz,= A&& if = A,,, all and aa2 are tensile, Anss= A&; if ai1 is tensile and a22 is compressive, etc. Under the restrictions specified above, and in view of the symmetry of both the strain rate tensor and stress tensor, problem (1) involves six coefficients of anisotropy AIIII, A22227 A33339 A1122, 4133, which are interrelated by the three constraints shown in (3). Choosing AIIII, Az222, An22 as independent, s and 6 may be expressed for these simplified conditions as A2233
s = [A1111(~11- ~33)~
by introducing W
Whether or not
eq. (4) or eq. (1).
In the remainder
as described for example l5), i.e., the orthogonal Cartesian coordinates are designated by xl, x2, x3 and the subscripts i, i, k, 1, m, n, p, q= 1, 2, 3 are summation indices if repeated in a single
where
= Amnpp
i and s can be correlated
A creep theorem for zirconium alloys
anisotropic creep zirconium alloys :
&d~~rnnAklpq
+ A2222(S22 - ~33)~ +
+ 2A1122(S33 - Sll)(S33
- S22)]*
(5)
and d= (All1lA2222 - A2~~2$t(A,,,,~2,, + 3.
A1111i222 -
+
%41122il1i22)t
(6)
Uniaxial and biaxial in-reactor creep of cold-worked zirconium-alloys
As part of a major test programme, Fidleris 14) investigated the in-reactor creep behaviour of uniaxial specimens manufactured from coldworked zirconium alloys. The specimens were made from bar stock or machined from pressure tubes such that the longitudinal axes of the specimens were parallel to the axes of the tubes.
J. SCHROEDER
54
For
bot’h types
{ lOi%}
plane
of
specimens
normals
the
were
parallel to the longitudinal
AND 111.J. HOLICKY
prismatic
predominantly
uniaxial data 14) but using normalization proposed 5).
axis of the specimen.
Zirconium
the
linear
flux
alloy pressure tubes investigated
Ross-Ross
and Hunt 5) and Ibrahim 6) studied
by
in-reactor
creep of biaxially
exhibited a t,hickness/radius ratio 0.05 -
tubes made from cold-worked
stressed pressure zirconium
alloys
ROSS-ROSS and
and have
to be classified
of the t’ype studied 14). Results
of all t’hree
r and h denote
investigations
of creep rate
and
indicate dependence
on st’ress, temperature
and fast neutron
flux
over the range tested. [In 14, 6) test data of heat treated zirconium alloys are also report’ed but because of the scatter of results encountered in the case of heat-treated uniaxial creep tests, no correlation is attempted.] In 5) in-reactor diametral creep rates of pressure tubes tested were correlated employing an empirical creep law and an attempt was made to interrelate the uniaxial results 14) and biaxial test data of the pressure tubes 5) using an isotropic analysis as an approximation since a manageable theorem of creep for anisotropic materials was not available as remarked 5). Using the isotropic approach, diametral creep rates calculated in accordance with uniaxial test data appeared to be in agreement wit’h t’est results of tubes at hoop stresses of 11 000 psi only, while at 20 000 psi t,he calculat’ed creep rates were three to six times the rat’es indicated by the tubes tested. Uniaxinl creep rat,es 14) and diametral
thickness
complicated
Hunt 5) and as thick
Ibrahim 6) shells 18);
radius of the middle of
the
tubes
surface
respectively.
A
thick shell analysis, as for example
demonstrated 9, 19), is in this case not warranted since it may improve the results by only 5o;b I*). Pressure tubes test,ed 5) passed vertically through the reactor core a,nd contained a t,cn foot high stack of six fuel assemblies cooled by pressurized water. Each fuel assembly had a different power output which decreased the neutron flux in steps from the center towards both ends of the tubes. Due to the variation of creep rates with fast neutron flux the deformation mode of axial sections of the tubes consisted of a series of bell-shaped stel)s associated with every fuel assembly as shown in fig. 3 of ref. 5). In accordance with elast’ic theory of thin shells 2o) each portion of the pressure tube surrounding a fuel assembly may be classified as a long cylindrical shell. where the deformation in the central portion is independent of the end condition of each portioll. This assumption appears to be confirmed by
creep
the test data shown in figs. 3 and 7 of ref. 5).
rates 5, 6, were normalized in 6) for temperature and fast neutron flux using the empirical linear correction 5) based on biaxial data only.
The tube sections exposed to fast neutron flux 6, also qualify as long shells and consequently the only non-zero stresses existing in accordance with thin shell bheorem in the tube sections 5, 6, where creep rates were measured are
In addition, the tubes considered purposes by a adjustment for
hoop stress of all cold-worked 596) was reduced for correlation constant fact’or of 0.5. an biaxiality based on known
texture coefficients IT), and reasonable grouping of uniaxial and biaxial data along a convex curve was obtained by plotting normalized uniaxial and diametral creep rates versus uniaxial and adjusted stress, respectively, on a log-log graph. In this work, test data of cold-worked zirconium alloys shown 14, 5, 6)> are correlated through eqs. (1) and (4) by finding in accordance with continuum mechanics B and m from
oB= p,rlh = 2~~= constant,
(7)
assuming that the end conditions of the pressure tubes represent closed-ends. go and a, designate hoop and axial stress respectively and p is the internal pressure of the tube. The effects of elastic deformation of the tube due to temperature and internal pressure will be neglected for the creep analysis since they do not change the original shape of the tubes significantly. The stresses in the central portion
ON
of each tube recalculating
SECONDARY
section
CREEP
should
OF
ANISOTROPIC
be adjusted
by
NUCLEAR
55
MATERIALS
and
h and r for creep strains larger
than 0.05% which is the upper limit for the infinitesimal theory of small displacements. But this adjustment up to the maximum measured diametral strain of about 0.2% is insignificant in comparison assumptions limitations
with the errors due to both the made above and the experimental
indicated
(9b) Similarly,
eq. (4) may be expressed
in view
of eqs. (% (I), (7) and (5) for uniaxial biaxial cases as
and
51 6, 14). (lOa) and
4.
Correlation of in-reactor cold-worked Zircaloy-2
creep data of
As stated by Fidleris 14), uniaxial
LO= ((aA,,+A,,+A,,)t/(A,,+3A&)}~~= test data
are inconclusive with respect to the creep rate dependence on fast neutron flux +, because the range of flux tested was 0.36 to 1.16 x 1013 n/cm2esec only where 11 denotes numbers of neutrons. But the results 5) indicate an almost linear dependence of diametral strain on integrated fast neutron flux if stress and temperature are constant. Because of this linearity the variation of creep rates with fast neutron flux will be accounted for by normalizing as in ref. 5) both uniaxial and biaxial creep rates over periods of approximately constant flux, stress and temperature to a reference level of 1 x 1013 n/cm2. set and normalized creep rates will be denoted by .$i = iii/no, where n0=gj/(1013 n/cm2.sec). In view of the above simplifications,
(8) theorem
(1) or (4) may be used directly to calculate normalized creep rates corresponding to a state of stress indicated in eq. (7) using cylindrical r, 8, z coordinates, i.e., setting pairs of subscripts ll=z, 22=0, 33=r. Eq. (1) may be written for both &lo and 0~1 denoting normalized creep rate and stress respectively of uniaxial specimens with axes parallel to the longitudinal z-axis of the tubes and for 8,O the normalized hoop creep rate which is identical to the diametral creep rate of the tubes. Using eqs. (S), (7) and (5) in eq. (1) Pa)
=BO[(aAZz+Aee+Aez)tag]m=r60sm.
(lob)
Since ,80and m will be evaluated in accordance with eq. (9a) for cold-worked Zircaloy-2 using uniaxial experimental data 14)shown in table Al of the appendix, it is convenient to set AZz=l.
(11)
The data of table Al were selected as fairly reliable from a temperature range of 220 to 350 “C and a stress range of 11 000 to 40 000 psi since these are approximately the ranges at which the tubes 516) were tested. [In ref. 596~14) all measured values indicated are averages but in this approximate analysis instantaneous and average values will not be differentiated.] /?Oand m may be estimated from these limited uniaxial data and used for the pressure tubes by assuming that 1. The dependence
of normalized
creep rate
on temperature indicated by ,50 does not vary significantly with stress from 11 000 to 40 000 psi; 2. The stress dependence of creep rates measured by m is not significantly affected by temperature between 220 and 350 “C and does not vary with stress between 11 000 and 40 000 psi; 3. Neither /P nor m change significantly with fast neutron flux over the range of 0.36 to 3.1 x 1013n/cm2. sec. The last assumption is necessary, since the uniaxial tests were conducted at fluxes of 0.36
56
J. SCHROEDER
AND
M.
J.
HOLICKY
to 1.16 n/cm2. set and the tubes were tested
the estimates of ,P and m exhibited
at fluxes of 1.15 to 3.1 n/crnz. sec. The above
may be less useful outside these limits.
assumptions
had to be made in order to utilize
the limited
number
of results available.
even if the following to be qualitative suggestions
correlation
As remarked
But, and
the
concept
creep 14) is used,
of
activation
energy
/l’J= B exp ( -b/T)
B
300 “C and at fast neutron fluxes greater than 1.16 x 1013 n/cm2 -sec. Fig. 6 of ref. 5) indicates strain versus integrated neutron flux at various elevations of this tube from which average normalized creep rates d, were obtained as shown in table 2. In accordance with eqs. (9b), (12) and (ll), the coefficient
and b are constants and T is temperature (“K). Employing the method of least-squares for two independent variables 22) to the logarithmic form of eq. (9a) simplified by eq. (ll), the constants, B, b and m may be estimated using the selected data of table Al in connection with the assumptions (1) to (3) such that for cold-worked Zircaloy-2 ,P=B
A = (4 + A,, + A,JQm-1) (A,, + QA&) = 4/(B04?),
exp (-b/T)=
m=2.72;
(12)
Numerical values of /?O are tabulated in table 1 for various temperatures. /?Oand m shown in eq. (12) should be considered as estimates of cold-worked
Zircaloy-2
material
constants only, since they are based on limited experimental evidence and will change if more test results become available or if different data
A values in table 2 corrected
TABLE Values 250
i
260
270
for this tempera-
ture increase using eqs. (13) and (12) indicates that ,@J may be used over the full range of temperature and fast neutron flux shown in eq. (12). [For a consistent comparison with other tubes 270 “C will be used for U-2 Mk IV at elevation 261’-4” as the average of the inlet and outlet temperatures shown in table 1 of ref. 5) instead of the weighted average temperature of 264 “C.]
are selected from ref. 14). The numbers in eq. (12) are given to three significant figures for reasons of numerical computation only. The majority of the uniaxial data on which PO and m are based correspond to temperatures of 300 and 350 “C, stresses of 20 000 and 30 000 psi and fast neutron fluxes ranging from 0.36 to 1.16 n/ems. set as shown in table Al. Consequently
T W)
(13)
should be a constant for different combinations of pressure, temperature and fast neutron flux if the anisotropy coefficients are invariants. The above measure of anisotropy A was calculated at various elevations in accordance with eqs. (13) and (12) from the average creep rates shown in table 2, first using for each elevation 264 “C? which is the temperature at elevation 461’-4” weighted with respect to time in accordance with fig. 5 of ref. 5). Estimating an increase in temperature with elevation of 3.6 “C per foot from table 1 5) the consistency of the
= 26.0 x lo-13 exp [ - 8 390/T] ; T = 222 to 350 “C, stress= 11 000 to 40 000 psi ; pi=O.36 to 3.1 x 1013 n/cm2+sec.
the
tube in fig. 6 of ref. 5) may be used to test the validity of the estimate (12) for PO below
for
where
5)
of flux by the test data. However, detailed results available for the U-2 ;Llk IV pressure
for further work may be obtained
from the analysis. If
and Hunt
effects of temperature and stress on diametral creep rates are not as well defined as the effect
is considered
only, valid information
by Ross-Ross
in eq. (12)
1 of /?”
1
280
1 290
300
1
310
320
ON
SECONDARY
CREEP
OF
ANISOTROPIC TABLE
NUCLEAR
MATERIALS
57
2
Cold-worked Zircaloy-2 pressure tube U-2 Mk IV 5, A=(*+A,+LI,,)*(~-‘)
(ABe+*Ae,),
ue= 14 000 psi
/ Elevation
1(lO_S~i~in~h_l~)
464:--g”
11.4 9.5 8.5 7.0 6.5 6.1 6.1 6.5
463’-3” 462’-0” 461’-4” 460’~0” 458’-9” 458’-3” 456’-5”
adju:d for not adjtsted for ) temperature gradient temperature gradient 1.5 1.2 1.1 0.9 0.8 0.7 0.7 too close to end
of A calculated in accordance eq. (12) for m = 2.72 for various values
Magnitudes with
of A,,and A, are shown in table 3 and indicate changes of up to 100% for differences in A, and A, of about
20%
only.
TABLE 3 Change in A=(f+A,,+A,,)‘)“-l’(A,+Ae,) variation of ABo, A, for WL=2.72 A,, 1 1 1
/ A,, 1 1.2 0.8
1 A, -0.5 -0.3 -0.5
1 A 0.6 1.2 0.3
due to
1.0 1.0 1.0 0.9 0.9 1.0 1.0 too close to end
descriptions independent of temperature and for comparison of experiments and theorems, both sides of eqs. (9) and (10) have to be divided by 80. In order to facilitate this comparison numerical values of j30 are given in table 1. The uniaxial results shown in table Al of the appendix are plotted in fig. 1 as a,1 versus .$!J~Ousing a log-log scale and show reasonable scatter with respect to the straight line
Material type isotropic anisotropic anisotropic
(14)
Due to technical difficulties transverse normal
i.e., eq. (9a) combined with eq. (11) for values given in eq. (12). Two creep rates in fig. 1 are indicated at 45 000 psi 14) and appear to show a possible variation of m with stress. But since
strain rates of uniaxial creep specimens have not been measured and results of creep tests of uniaxial specimens with longitudinal
45 000 psi corresponds almost to first yield of the material 14) no conclusions can be drawn. Data of cold-worked Zircaloy-2 pressure
axes tangential to the circumferential direction of the tube are not available. Consequently the coefficients of anisotropy A,, and A, are unknown and direct correlation of uniaxial and biaxial test results through the power law (10) in terms of the effective strain rate and effective stress is not possible. However, in view of eq. (9) uniaxial and biaxial results from tubes should be grouped with respect to two parallel lines on a log-log graph of creep rate versus stress if the coefficients of anisotropy do not vary with tubes and m is invariant. For the following correlation, data are plotted in terms of &/~O, $//P, S/PO in order to use
tubes 5) are shown in table A2 of the appendix: the U-3 Mk V tube was not considered in this work because of low stress (about 5 000 psi) and an organic coolant. Results 6) of small cold-worked Zircaloy-2 tubes are tabulated in table A3 of the appendix and are represented with the data of table A2 in fig. 1 as $‘//?O versus c0 for comparison with uniaxial test results 14). The points associated with ref. 5) appear to be grouped with respect to a line where m is approximately one, whereas in view of eq. (9) data belonging to ref. 6) appear to be in agreement with uniaxial data if cB is greater than 20 000 psi. The value m= 1 was
58
J.
SCHROEDER
50
AND
M.
J.
HOLICKY
I I I ZIRCALOY-2
COLD-WORKED
0 *
40 rT 0 x
30
E 20b
UNIAXIAL
SPECIMEN
UNIAXIAL
SPECIMEN
A
PRESSURE
.
SMALL
.
REFERENCE
b b”
0 0
I
I
‘5.
/
Al
RX-21,
14 TUBES
TUBES
,L?= 260
TABLE
TABLE
TABLE
A2 A3
x10-‘3exp[-6390/TIoK;1
-,
/ (k
if,/ Fig.
1.
Cold-worked
represented
as
Zircaloy-2
uniaxial
or
in-reactor
hoop
stress
p”
creep versus
or data
;i
/ p”
from
uniaxial
normalized
by B”, eq.
hre’psi.e272
(psim
set cd
x IO”)
specimen
uniaxial
/
O+ 10’3.)
or
14) and tube
diametral
creep
specimens rate
5, 6)
divided
(12).
used by Ross-Ross and Hunt 5) in order to interrelate results of pressure tubes using a linear temperature correction. The disagreement with eq. (9) indicated in fig. 1 has two possible explanations : 4. The power index m varies with stress for cold-worked Zircaloy-2, or 5. The cold-worked Zircaloy-2 coefficients of anisotropy are stress dependent. Explanation (4) is feasible since only one
ends assumed in eq. (7) may not have been perfect. But since in 6) an open loop was used, i.e., perfect closed end condition existed, and the experimental data 6) show the same trend as those 5, at stresses below 20 000 psi, this explanation does not appear to be warranted. The validity of explanations (4) and (5) can be proven only by conducting more uniaxial creep tests in the react’or. The possibility that (5) is valid may be
uniaxial
investigated by considering whether or not a possible relationship between the measure of anisotropy A defined in eq. (13) is indicated
experimental result 14) exists below 18 000 psi as shown in fig. 1 and for example Kachanov 8) has indicated, that creep data are sometimes grouped along a line consisting of two segments of different slopes or along convex curves on a log-log graph of creep rate versus stress. In-reactor creep may depend on slip and/or diffusion 21) and rotation of basal planes of the hexagonal crystals under stress suggested by Hill 23) and/or dependence of magnitude and direction of diffusion on stress indicated by Garofalo 24), may account for a variation of anisotropy with stress, i.e., explanation (5). The discrepancy shown in fig. 1 could be due to a variation of stress ratio ae/az for the pressure tubes 5) with load since the condition of closed
by data 53 6) for a constant
m=2.72.
Values
for A calculated for all the tubes in accordance with eq. (12) are shown in tables A2 and A3 and are plotted versus stress in fig. 2a. There appears to be a significant dependence of A on stress and fig. 2b indicates that the overall change in A can be explained by variations in A,, or A, of reasonable magnitude which for tubes tested in the NDP reactor may be classified in the following way: A,,>l,
A,< -0.5; = -0.5;
0.8
A,,
A,>
A,= -0.5.
(15)
In view of eqs. (9b), (11) and (13) the effect
ON
SECONDARY
CREEP
OF
ANISOTROPIC
59
MATERIALS
on A of a possible variation
1 COLD-WORKED ZIRCALOY -2 . NDP PRESSURE TUBES TABLE A2 . NRU PRESSURE TUBES TABLE A2 . NRU SMALL TUBES TABLE A5 COLD-WORKED ZIRCONIW-25% ANRU PRESSURE TUBES TABLE
NUCLEAR
in Au is reduced
by a factor
of $ and may be neglected. Since (5) appears to be a possible explanation
NS A5
of the discrepancy constant
indicated
for cold-worked
in fig. 1 if m is a
Zircaloy-2
over the
stress range shown in eq. (12), it may be worthwhile to select comparison
2 011
I
IO
I
,
15
20
(0)
25
I 35
30
40
2
Fig.
2a.
eq.
(13)
Dependence on hoop
IO
12
of measure
stress for cold-worked
for anisotropy
cold-worked
14
coefficients
zirconium alloy tubes
anisotropic
strain
view of eq. (10) with respect to one straight line on a log-log graph of 9//30 versus s, if both 80 and s are adjusted for variations in A,, and A, with stress. Selecting from the curve shown in fig. 2a a value for A corresponding to the hoop stress of the tubes considered 59s), a comparison value for A,, was found from
16
of anisotropy
alloy tubes 5, 6) if m=2.72=constant. values
and (11) effective
rates do and stress s for the measured diametral creep rates and hoop stresses of the tubes 516). Because A was calculated from eq. (13) for a constant m=2.72, data from uniaxial specimens 14) and tubes 596) should be grouped in
+
08
and with
eqs. (lob)
U# (psi x IO31
06
values of A,
A, from fig. 2 and calculate in accordance
A
zirconium
b. Comparison A,,
and
5,
if m=2.72=
6)
A,
for
fig. 2b using a constant A,= - 0.5 as shown in tables A2 and A3. The effective strain rates and stresses calculated for the tubes in terms of these comparison values correlate as predicted with the uniaxial data shown in fig. 3. If different comparison
constant.
values for A,, and A,
are selected
40 30
;“I Fig. 3.
Cold-worked
represented
Zircaloy-2
as effective
in-reactor
anisotropic
p*
(psimx
IO”)
creep data from uniaxial
stress versus effective sssuming
specimen 14) and tube
anisotropic
m=2.72=constant.
strain rate divided
specimens
9
)
by PO, eq. ;12;,
60
J.
in accordance are obtained.
agreement
was
10, ref. 6) by grouping
indicated
both
a convex
curve
uniaxial
and
diametral
uniaxial
and
one
respectively.
creep
half
The hoop
Zircaloy-2
on a log-log of
graph
rates
the
hoop
of
versus stress
stresses were reduced
by one half in order to account for biaxiality in accordance with known texture coefficients assuming that creep is mainly caused by slip between grains. This reduction of the hoop stress is analogous to a correlation using an anisotropic creep theorem and the method of correlation indicated in fig. 10, ref. 6) may be simulated by setting in eq. (lob) the coefficient of $ to one and the coefficient of (T@to one half. It follows that independent of m and /30 the value of A,,= 1 and of A,,= - 1. The corresponding magnitude of A calculated for m= 2.72 from eq. (13) is 0.15 and appears to be too low even for the small tubes as shown in fig. 2b mainly because in this work a temperature dependence based on uniaxial results was used in accordance with continuum mechanics, whereas 5) a linear temperature correction based on biaxial results 5) only was employed for both uniaxial and biaxial data. COLD-WORKED ITi1
- _ _
v-l
0
ZIRCONIUM-2.5%
M.
J.
5.
in
uniaxial 14)
and biaxial 516) data of cold-worked along
AND
with fig. 2, similar correlations
Reasonable fig.
SCHROEDER
HOLICKY
Correlation of in-reactor creep data of cold-worked zirconium-2.5 wt o/o niobium Only three cold-worked
niobium
zirconium-2.5
wt, y.
uniaxial specimens were tested la), all
three at an average temperature
of 300 “C as
shown in table A4 of the appendix.
In fig. 4, the
power index m = 2.72 estimated for cold-worked Zircaloy-2
in eq. (12) appears to be in agreement
with
three
the
uniaxial
data
obtained
for
zirconium-2.5 wt o/o niobium and due to lack of experimental evidence this value of m and the variation of normalized creep rate with temperature shown in eq. (12) will be used such that as an est’imate for cold-worked zirconium-2.5 wt o/o niobium PO= 8.5 x IO-13 exp [- 8 390/T (OK)], m= 2.72,
(16)
where the coefficient 8.5 x lo-13 was estimated from eq. (14) in connection with the average of the three creep rates shown in table A4. Data of cold-worked zirconium-2.5 wt y. niobium pressure tubes obtained from 59 6) are shown in table A5 of the appendix and are plotted as &O//30versus u0 on a log-log scale in fig. 4a where a similar discrepancy with uniaxial data exists as indicated in fig. 1 for cold-worked Zircaloy-2. The measures of anisotropy A in table A5 are calculated in accordance
NB 25
25
x ‘g
t?
20
15
ii
A PRESS. TUBES TABLE A5
tf
IO
I
2 ;z,
/fl”
3
45
IO
or 2; /p” (psimx IO”)
s? z x ‘3 e YI
15
IO
I
2 8”/
3 6”
45
IO
(psimx IO”)
(b)
(a) Fig. 4.
Cold-worked zirconium-2.5 wt y. niobium in-reactor creep data from uniaxial specimen 14) and tube specimens 5. 6) represented in (a) as uniaxial and hoop stress versus normalized uniaxial and diametral creep rate divided by PO, eq. (16) and in (b) as effective anisotropic stress versus effective anisotropic
strain rate divided
by ,P, eq. (16) assuming
WL= 2.72=constant.
ON
SECONDARY
with eqs. (13) and (16) similar dependence
CREEP
OF
ANISOTROPIC
and appear to have a
effective
anisotropic
Ross-Ross, Atomic Energy of Canada
1) P. A. Limited
Report,
AECL-3126
(1968)
2, W. Evans, J. E. Le Surf and W. R. Thomas, At,omic
stress
Energy
AECL-2890
versus effective anisotropic strain rate divided by ,80 was established in a similar manner as
B. A.
of
Canada
Cheadle and W.
of Canada
Limited
Evans,
Report,
fig. 3 and indicates the same type of correlation.
Report,
AECL-1048
6.
P. A. Ross-Ross
and C. E.
Limited
Conclusions
of the material. If the power index for coldworked zirconium alloys is constant the variation of the coefhcients of anisotropy indicated by the variation of the so-called measure of anisotropy A shown in fig. 2a, may be used to draw conclusions regarding the basic inreactor creep mechanism 21) since A appears to approach a constant or at least a minimum value.
Mat. E.
The author wishes to acknowledge the valuable assistance and advice given by Mr. Ross-Ross, Fuels and Materials Division, Atomic Energy of Canada Limited, for both the formulation of the problem and the evaluation of the experimental data.
Report,
Atomic
Energy
AECL-2652
(1966)
Atomic
Energy
of Canada
(1960) L.
Hunt,
J. Nucl.
2
Ibrahim,
In-reactor
creep of zirconium
alloy tubes and its correlation with uniaxial data, ASTM
Symp.
delphia;
7)
I.
Berman
8 (1966)
8)
zirconium
Nov.,
and D.
L. M. Kachanov, tizdat,
and
hafnium
(Phila-
1968) H.
Pai,
Int.
J. Mech.
Sci.
341 The theory
Moscow,
Lending
1960)
Library
Boston
of creep (Fizma-
; Engl. transl., National
for Science
Spa, Yorkshire,
and
England
Technology,
(1967)
Q) K. G. Odqvist, Mathematical theory of creep and creep rupture
loI
(Oxford,
Y. N. Rabotnov, Progress
in
New York,
1966)
The Prager anniversary volume
Applied
Mechanics
(MacMillan,
1963) p. 307
11) N. I. Malinin, PMFT Zh. Prikl. Mat. Tekh. Fiz. 3 (1964)
16
12) W. Olszak and A. Sawczuk, Inelastic behaviour in shells (Noordhoff, Groningen, The Netherlands, 1967) 13
)
A. M. Freudenthal pedia of Physics
and H.
Geiringer,
(Springer,
V. Fidleris, J. Nucl.
15)
W. Prager, Introduction to mechanics of Continua
9
R. Hill,
(Ginn, New York, 17
)
B.
19
Mech.
A.
Cheadle
and
Techn.
4 (1966)
329
Groningen, )
D. H. S.
29
26 (1968)
51
1961)
J. Appl.
V. V. Novozhilov,
13)
Mat.
Encyclo-
1958) p. 229
14)
17 (1950)
C.
E.
64
Ells,
Electrochem.
Thin shell theory (Noordhoff,
The Netherlands,
1964)
Pai, Int. J. Mech. Sci. 9 (1967)
Timoshenko
Theory
Acknowledgment
Perryman,
26 (1968)
F.
Limited
(1967)
E. C. W.
The anisotropic creep theorem proposed in this work appears to be useful to analyse uniaxial and biaxial in-reactor creep behaviour of coldworked zirconium alloys. Using the limited results from uniaxial tests 14) it was shown how both uniaxial data and results of biaxially stressed tubes596) may be correlat,ed. In order to analyse the behaviour of zirconium alloys in the reactor throughly it is necessary to test assumptions (1) to (3) by conducting more uniaxial creep experiments which will automatically lead to conclusions regarding explanations (4) or (5) given in section 4. Furthermore, it appears to be essential to determine the coefficients of anisotropy Ace, A, and their variation with stress if the power index m is found to be invariant, in order to facilitate the correlation of these coefficients with texture
61
MATERIALS
References
on stress as the A values
of the cold-worked 7ircaloy-2 tubes for an 17~ of 2.72 as shown in fig. 2a. The log-log graph in fig. 4b indicating
NUCLEAR
Toronto,
of
and
plates
S.
and
335
Woinowsky-Krieger, shells
(McGraw-Hill,
1959)
21) G. R. Piercy, J. Nucl. Mat. 26 (1968) 18 22
)
E. F. Croxton and D. J. Cowden, Applied general statistics
23
24
)
1
(Prentice-Hall,
1955)
R. Hill,
The mathematical
(Oxford
Clarendon Press, 1964) p. 317
F.
Garofalo,
Fundamentals
rupture in metals p. 178
theory
of plasticity
of creep and creep
(MacMillan,
New York,
1965)
Appendix Al
TABLE
Data
based
on uniaxial
test results
Test
Temp.
no.
(“C)
R-6
of cold-worked
Stress (lO:‘,si)
Fast
flux
rate
table
/
0.54
3.5 I_t 107;
5.7
0.60
13.0 & loo/:,
5.8
300
18
0.58
3.0 *
20%
5.1
300
11
0.61
0.5 i
600,b
0.7
300
20
0.64
3.0 & 3076
4.1
320
20
0.68
6.0 & 20%
5.1
350
20
0.68
14.0 5 200,;
5.6
220
30
0.96
1.5 *
3076
11.5
260
30
0.93
5.0 & 4o”/b
14.2
350
30
0.94
32.0 + 10%
9.2
R-2
300
30
0.66
15.0 & 10%
19.9
R-4
300
20
0.58
3.5 & 30%
5.3
300
30
0.36
8.0 + lo”:,
20.0
Rx-2(
350
30
1.16
60.0 f
lOo,d
14.0
Rx-21
300
40
0.84
35.0 *
5076
36.5
1 and fig.
8 of ref. 5,
Zircaloy-2
TABLE Data
based Data
on results
based
Av. of inlet
Hoop
and outlet
stress
I temperature (“C)
selected
on results
I
tubes
A2
from
table
TABLE
A3
shown
in figs.
3 and
7 of ref. 6, ~
~ Diametral
A,,
comparison
ng
I(lo3psi) Table
264
IJ)
( x 101’)
20
Cold-worked
4 of ref.
i:1 IB” eqs. (8, 12)
‘z (1O-7 (:n;in)h-I)
20
Rx-1I
NDP
from
350
Rx-14
designation
Creep
~
1013 n/zmj.sec)/
selected
300
R-9
Tube
Zircaloy-2
A2
U-2 Mk III
281
~ 10.5 i 11.5
U-2 Mk IX U-2 Mk IV
281
13.4
3.1
270
~ 14.0
2.7
1.9
1.4
0.73
~
1.06
U-2 Mk IX
281
17.3
3.1
2.9
I
1.4
0.40
~
0.92
258
15.7
2.56
1.3
~
0.98
19.4
2.56
2.4
I I
0.55
258
2.6
258 258
23.9 29.2
i
2.56 2.56
2.7 3.6
’
2.56 2.56
1
~
~
1.15
0.54
1.1
2.7
1.8
0.9
~ 1.27 i 0.86
2.3
1.1
0.65
~
Table
258 258
I
~ 34.0
37.6
I
1
I
1.28 1.20 1.09
A3
6.1 8.8
1.4 2.9 3.9
;
6.6 9.6
I ~
0.57
0.87
0.36
0.80
0.28
0.80
0.31 0.34
0.80 0.80
ON
SECONDARY
CREEP
OF
ANISOTROPIC
NUCLEAR
63
MATERIALS
TABLE A4 Data based on uniaxial test results of cold-worked zirconium-2.5 wt ye niobium selected from table 4 of ref. 14)
R-11 R-12 R-17
~- Creep rate
Fast flux
I_
Test no.
“” (10-e (in~m)h-1)
/ / )
300 300 300
i j /
23.0 16.5 20.0
20.0 & 15% 7.0 & 10% 10.5 -& 40%
o.7
\
0.56 0.68
1 ; ;
7.7 3.4 4.2
TABLE A5 Data based on results from cold-worked zirconium-2.5 wt ye niobium pressure tubes selected from ref. 5, 6)
Tube designation
Av. of inlet and outlet temperature
EIoop stress
Diametral creep rates
Fast flux
9, (1013n/om2*sec) 1(10-7 (~~]in)h-l)l
(“C)
/ (lG:sii)
U-l Mk VIII
270 270
16.5 22.0
2.9 2.9
U-1MkV
285 285
15.5 21.0
3.1 2.0
/ /
W” eqs. (8, 12) ( X 1Olif
A eq. (13)
6.5 10.4
1.35 2.16
0.46 0.33
6.7 7.8
1.07 1.54
0.43 0.27
A,, comparison value selected from fig. 2b for A,= -0.5
1
0.96 0.80 1.01 0.80