FRACTURE PROBABILITY AND LEAK
BEFORE BREAK ANALYSIS FOR THE
COLD NEUTRON SOURCE MODERATOR VESSEL
S. J. Chang
Research Reactors Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee
Presented at
ASME Pressure Vessel and Piping Conference
San Diego, California
July 26-30, 1998
“The submitted manuscript has been authored by a
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contribution, or allow others to do so, for U.S.
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Prepared by the
Research Reactors Division
OAK RIDGE NATIONAL LABORATORY
Oak Ridge, Tennessee 3783 1
managed by
LOCKHEED MARTIN ENERGY RESEARCH COW.
for the
U. S. DEPARTMENT OF ENERGY
under contract DE-AC05-960R22464
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.
FRACTURE PROBABILITY AND LEAK BEFORE BREAK ANALYSIS FOR THE
COLD NEUTRON SOURCE MODERATOR VESSEL'
Shih-Jung Chang
Senior Research Staff
Research Reactors Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee 37831
PhoneFax: (423)574-9134/8576
e-mail: [email protected]
ABSTRACT
Fracture mechanics calculations are made to ensure the safety
of the moderator vessel against failure by fracture. The 6061T6 aluminum alloy is used for the moderator vessel structure.
The fracture analysis of the moderator vessel consists of
(1) the probability of fracture calculations at the locations of
the moderator where either the primary stress or the secondary
stress assumes the highest value. (2) the vessel wall leakbefore-break analysis by applying an edge crack solution, and
(3) the crack penetration calculation as a result of radiation
embrittlement by applying the flaw assessment diagram
(FAD). The probability of fracture for the capsule is
calculated by using a direct probability integration method
(refs. 4 and 5 ) instead of the Monte Carlo simulation method
used by the PRAISE Code or the FAVOR Code developed in
Oak Ridge. The probability of fracture as a function of
radiation embrittlement is obtained. The leak before break
analysis indicates that the vessel will fail by leak before fail
by catastrophic fracture. A mass spectrometer will be installed
to monitor the leak of hydrogen circulating within the
moderator.
1. INTRODUCTION
Fracture mechanics calculations are made in order to ensure
the safety of the moderator vessel against failure by fracture.
The 6061-T6 aluminum alloy material property is based on
the experimental results of K. Farrell and others (refs. 1,2 and
3). The fracture toughness value to be used in the fracture
analysis is determined as a result of an earlier communication
with D. Alexander. The fracture analysis of the moderator
'Based on work performed at Oak Ridge National Laboratory,
managed by Lockheed Martin Energy Research COT., for the U.S.
Department of Energy under contract DE-AC05-960R22464.
Accordingly. the U.S. government retains a nonexclusive, royalty-free
license to publish or reproduce the published form of this contribution,
or allow others to do so, for U.S. government purposes.
vessel consists of (1) the probability of fracture calculations
at the locations of the moderator where either the primary
stress or the secondary stress assumes the highest value,
(2) the vessel wall leak-before-break analysis by applying an
edge crack solution, and (3) the crack penetration calculation
as a result of radiation embrittlement by applying the flaw
assessment diagram (FAD). The probability of fracture for the
capsule is calculated by using a direct probability integration
method (refs. 4 and 5 ) instead of the Monte Carlo simulation
method used by the PRAISE Code or by the FAVOR Code
developed in Oak Ridge National Laboratory (ref. 6). The
probabilities of fracture at two different locations on the
moderator capsule are obtained. It provides a quantitative
estimate on how the capsule is structurally weakened as a
result of the embrittlement of the moderator material after an
extended period of irradiation. The leak-before-break analysis
leads to the conclusion that the moderator structure will
satisfy this condition and no catastrophic fracture is possible
before leaking first appears. A mass spectrometer will be
installed to monitor the leak of hydrogen. The crack
penetration calculation is carried out by using the method of
flaw assessment diagram (FAD). The FAD method is a well
established fracture mechanics procedure (ref. 7). After a
period of neutron irradiatiorf, the result shows that the critical
crack depth of the penetrating crack does not change
significantlyas a result of radiation. The effect of irradiation
is mainly manifested by the mode of fracture that shifts
gradually from failure by rupture to failure by brittle fracture.
2.6061-T6 ALUMINUM ALLOY
EMBFUTTLEMENTDATA
The 6061-T6 A1 embrittlement data was reported earlier by
K. Farrell and R. T. King. New 6061-T6 A1 embrittlement
data of 20 years irradiation in HFBR was obtained in
Brookhaven National Laboratory (BNL). The BNL notchimpact tests are the first such tests for heavily irradiated
6061-T6 alloy. The 6061-T6 aluminum embrittlement data
reported by K. Farrell and R. T. King indicate that the effect
of neutron irradiation creates damage in 606 1-T6 aluminum,
which tends to increase the yield stress and to reduce the
ductility. These effects were plotted against a range of
neutron fluence in that report. It is possible that the reduction
in ductility is correlated to the fracture toughness of the
material. More often, the loss of ductility implies a decrease
in toughness. For the convenience of the present calculation,
a modified plot is made and shown in Fig. 1, that shows the
effect of irradiation to the yield strength increase and the
toughness decrease. This plot is the assumed material
property for the aluminum alloy to be used in this fracture
mechanics analysis.
In Fig. 1, for an increase in neutron fluence up to 1.8 x loz7
n/m’ the yield stress was increased from 40 ksi to 67 ksi. The
loss of ductility as reported by Farrell, et. al., reached a
saturated value for an extended period of neutron fluence.
The fracture toughness is, therefore, also assumed to reach a
saturated value in Fig. 1. For the unirradiated 6061-T6 Al, the
fracture toughness of 25 ksi was obtained by D. Alexander
(communicated on April 10, 1992) from the Metals and
Ceramics Division, ORNL. At present, Alexander
recommends an estimated value of 15 ksi at saturation. The
toughness is believed to reach the saturated value as the
fluence reaches loz6n/mz and remains constant up to 2.0 x
10” n/cm2. The reduced toughness of 15 ksi is believed to be
a rough but conservative estimate. The reduction in toughness
may also be reflected by the recent impact energy data (refs.
2 and 3) from BNL if a correlation between the toughness and
the impact energy at fracture can be established.
3. ACCEPTANCE CRITERIA ADOPTED IN
FRACTURE MECHANICS ANALYSIS
3.1 API Code recommendations
For an acceptable fracture mechanics analysis of the
moderator vessel structure, an appropriate safety margin is
needed. A consistent and up-to-date recommendation was
advanced by the American Petroleum Institute (API) Design
Code recently on the method of “Fitness for Service” in
which the margin of safety for fracture mechanics calculation
was outlined. We shall adopt the API recommendation and
use the following factors of safety:
(1) Factor of 1.6 for the design pressure.
(2) Factor of 1.4 for the choice of the critical flaw size
(3) Factor of 1.4 for the toughness values.
3.2 Factors of safety adopted in fracture mechanics
analysis
The 60% increase in the design pressure will raise the stress
distribution along the moderator vessel wall uniformly by the
. same factor because the finite element result is linear and
elastic. A 40% reduction of fracture toughness will be made
to meet the API recommendation. However, the nominal
recommended toughness value is based on the plane strain
toughness value. For the present analysis, the moderator
vessel wall has thickness values ranging between 1 mm to
5 mm. It is well known that the plane stress toughness value
will be substantially higher than the plane strain toughness
value (ref. 9). As illustrated in Fig. 2 obtained from reference
9, the energy release rate G, for 7075-T6 aluminum alloy
increases from 20 to 60 as the specimen thickness decreases
from 15 mm to 5 mm for a factor of 3. Since G, is
proportional to the square of &, the & increases a factor of
as the thickness decreases from 15 mm or greater to 5 mm. A
combination of the API recommended factor of 1.4 reduction
and the increase of plane stress factor of 1.7, the overall
increase in toughness is
1.7 KIc x
1
=
1.4
1.2 KIc
Therefore, we will adopt a 20% increase in K,, value from the
to give an
recommended nominal value of15 ksi
analysis toughness value of 18 ksi
6
6
In applying these factors of safety, it needs to be cautioned
that a factor of 60% increase in design load, combined with
a decrease of 40% in fracture toughness may provide a
condition of double use of the safety factor. Similar case may
be found in ASME Code as a result of the use of both a load
factor and a conservative allowable stress. Even it may lead
to an excessive conservatism, in the probability of fracture
calculation performed in this study, we still employ all factors
of safety as recommended by API Code.
3. Moderator surface flaw density estimate
Both in the Marshall Report (ref. 10) and in HFlR pressure
vessel radiation embrittlement study (ref. 6, page 241), a flaw
density of 0.03 flawdcubic feet is assumed as a reasonable
estimate for pressurized water reactor (PWR) vessels. We
shall adopt this density even though in the present study the
moderator vessel is made’ of aluminum alloy rather than
pressure vessel steel. It is the project’s opinion that with the
available inspection techniques (radiography, dye penetrant,
and eddy current) to be applied to the fabrication of the
vessel, the final vessel will have this hypothetical flaw density
that will be at least as good.
The highest surface flaw density for the moderator vessel of
5 mm thick thickness at the patch area is estimated as
0.03
5 mm x inch
= 0.0005 flaws/ft2
12 inch x 25.4 mm
.
In the HFIR study (ref. 6), the surface flaw density is
determined to be 0.007, or 14 times larger than that used in
the moderator study, because the HFIR vessel wall is 3-inchthick. A total area of square feet underneath HB-3 was
assumed in HFIR study that led to a total of 0.007 flaws for
the probability calculation. Likewise, for the present study,
one square foot area is also assumed to represent the total
vessel surface in the following probability calculation.
4. METHOD OF THE PROBABILITY OF
FRACTURE ANALYSIS
The method of probability of fracture analysis developed
earlier (refs. 4 and 5) is adopted for the moderator vessel
fracture probability calculations. The following information
are required for the calculations: (1) the distribution function
of the possible cracks in the sample, (2) the fracture
toughness of the material at different levels of neutron
irradiation, and (3) the regions of maximum stress distribution
of the moderator capsule. The crack size is assumed to follow
an exponential distribution function (Marshall distribution)
and the toughness is assumed to be that shown in Fig. 1. The
fracture toughness is further assumed to vary according to a
Gaussian distribution. The maximum membrane stress is
9.1 ksi located at the lower part of the outer surface and the
maximum bending stress located at the patch area of the inner
cylindrical surface is 22.6 ksi for moderator internal pressure
of 19 bars. Along the edges of the capsule, the finite element
results show much higher stresses, but the edge welds are to
be designed separately. The finite element results do not
represent the realistic values.
The Marshall distribution function of crack size x is assumed
to have the following form:
f i x ) = 4.064 x exp (-4.064 x)
where x is expressed in inches. The Marshall distribution is
developed for the pressure vessel steel. It is not known
whether this distribution can appropriately represent the
aluminum alloy. However, this is an exponential distribution.
A variation of the numerical coefficients in the above
distribution is believed to fit the aluminum alloy. This is only
a preliminary calculation. Further improvement of the
distribution function may be made for a better representation
of the aluminum alloy. A function B(x) is often assumed as
the probability that a crack of size x cannot be detected,
B(x) = 0.05
+
0.995 x exp (-2.870 x )
The probability of fracture at a deterministic fluence value is
an integration of the joint probability of the crack size and the
fracture toughness,
To apply the above analysis procedure, it is required that
various factors of safety as described in the preceding section
are needed in order to ensure an appropriate margin of safety.
5. PROBABILITY OF FRACTURE CALCULATIONS
FOR THE MODERATOR VESSEL
The probability of fracture calculation of the moderator vessel
is based on the M I Code guidelines of the fitness for service.
The distribution function for crack density is multiplied by a
factor 1.4 to account for the crack size. The fracture
toughness is adjusted for the plane stress condition to increase
by a factor of 1.7 and decrease by a factor of 1.4 based on the
API code recommendation.
Two types of maximum stresses along the moderator surface
are considered in the probabilistic of fracture analysis.
Maximum membrane stress is selected by using the contour
stress plot for the shell midplane section, point 2 shown in the
plot. The maximum membrane stress is 9.1 ksi located at the
outer cylindrical body near pipe inlet for the moderator under
19 bar internal pressure. This location of maximum stress will
be used as a reference point for the calculation of the
probability of fracture due to membrane stress. A variation of
internal pressures from 10 to 25 bars is used to obtain the
membrane stresses at this point and an API Code load factor
of 1.6 is adopted for the membrane stresses. The results are
shown in Fig. 3.
A similar probability of fracture calculation is performed
using the maximum bending stress of 22.6 ksi at 19 bars
internal pressure. It is located on the moderator outer surface
at patch area. A variation of internal pressures from 10 bars
to 25 bars is used in the probability of fracture calculations.
An API Code load factor of 1.6 is also adopted for the applied
stress. The results are shown in Fig. 4. It is to be noted that
the bending stress used in the plots is the secondary stress.
Although the probability of fracture by secondary stress is
higher, it is not caused by the more detrimental primary stress.
The numerical values for the probabilities of fracture for the
maximum membrane stress and for the maximum bending
stress are calculated by using the probability integral. The
results are tabulated in the following two tables.
80
70
I
I
70
6061-T6 Aluminum
..
60
50
n
C
40
30
20
0.1
*I
.
10
1000
100
10
Fluence > 0.1 MeV
n/m2)
5
Y
0
I
I
I
I
Fig. 1.6061-T6 Aluminum alloy flow stress and fracture toughness as a function of neutron dose.
I
3I
1
200- A
I
c
100
(1-SI
-'Oog
c
I
-I/
Q l
L
-50
$
u '
¶
*
s
Grit
0
(c4
2
'
5
1
10
15
20
Specimen thickness, 8 (mm)
I
0
25
Fig. 2. (a) Variation of toughness with thickness for 7075 alloy (AL-Zn-Mg)-T6,and (b) Fracture profiles and
stress-displacementcurves typical of regions A, B, and C.
25 bars
-5
-7
-9
19 bars internal pressure
9.1 ksi maximum membrane stress
-11
10
0
.i
ftuences,
20
n/m2
30
Fig. 3. Probability of fracture due to crack located at outer body as a result of membrane stress. Internal pressure
varies from 5 bars to 20 bars.
-3
1
j
I
I
a,
m
0
v)
0
0
-
i
-4
I
I
i
E
!
3
I
I
I
.cI
0
m -5
L
*
Lc
0
-6
-7
0
/
19 bars internal pressure
22.6 ksi maximum bending stress
,
.
10
fluences,
20
n/m2
30
Fig. 4. Probability of fracture due to surface crack located at the patch area as a result of bending stress and
surface stress. Internal pressure varies from 5 bars to 20 bars.
-
1.2 -
Failure Assessment Diagram, S = 15 ksi
Unirradiated
Klc=25 ksi/in,ay=40 ksi
Irradiated, 2.0*10n n/m2
Klc=15 ksiJin,ay=70 ksi
7 -
1
/
DUGDALE
2
0.8 .
0.6
eunirradiated
+irradiated
0.4
CEGB R/H/R6
0.2
0 '
0
path to failure
a/t =0.1 to 0.8
'
I
0.2.
I
'
I
I
0.4
0.6
,
I
0.8
Sr
a
,
I
*
1
I
*
1.2
.
'
-
i
Fig. 5. Path-to-failure curves for unirradiated and irradiated moderator capsule a t location of the patch area.
These curves are generated by the ratios of crack depths to thickness of the capsule. The ratio alt varies from 0.1
to 0.8.
Table 1.Probability of fracture (in log sca1e)'at maximum membrane stress for moderator
capsule under irradiation and internal pressure varied from 15 bars through 25 bars
Table 2. Probability of fracture (in log scale) at maximum bending stress for moderator
capsule under irradiation and internal pressure varied from 15 bars through 25 bars
6. PENETRATION FRACTURE OF THE
MODERATOR WALL-LEAK BEFORE
BREAK ANALYSIS
To study the leak-before-break condition for the moderator
problem, a part through crack of variable aspect ratios will be
assumed. Cracks of elliptic shape are used for the study. This
problem has been analyzed by Neuman and Raju (ref. 8) for
variable aspect ratios and numerical results were obtained.
The condition that the part through crack will either extend its
length or will propagate toward the penetration of the wall
relies on the relative magnitudes of the stress intensities
generated at the crack tip region or at the crack front region.
The plate contains an existing part through flaw of crack
length 2c with the aspect ratio = a/c where a is the depth of
the crack. The plate has thickness t and 4 is the angle
measured from the major axis of the crack. Uniform stress is
applied along the direction perpendicular to the crack plane.
6.1 Stress intensity factors at
(leak or break)
+ = 0 and at 4 = 90
From numerical results obtained by Neuman and Raju,
normalized stress intensity factors are:
2@/x
dc
0.2
0.4
0.5
.
-
0.4
0.6
0.8
0.724
0.899
0.190
0
0.617
1
1.173
1.359
1.642
1.851
0
0.767
0.896
1.080
1.318
1
1.138
1.225
1.370
1.447
0
0.841
0.955
1.126
1.335
1
1.124
1.185
1.300
1.354
.
A straight-forward critical crack length calculation is shown
in the following. A crack of length 2c is assumed to be
located along the wall of the moderator capsule. The crack is
assumed to be small and the capsule wall is approximately
assumed to be a flat plate. Under these conditions, the stress
intensity factor K is
K=0@
where u is the applied stress along the direction perpendicular
to the crack plane.
For the moderator vessel design pressure of 19 bars, the
maximum membrane stress, section point 2 shown in Fig. 2 of
previous section, is approximately 9.1 ksi of primary stress.
A 60% increase will be 9.1 x 1.6 = 14.6 ksi. Recall that a
plane stress toughness of has been regarded as a reasonable
toughness value in compliance with the API Code
recommendation. It follows, therefore, the critical crack
length for a catastrophic extension of the crack can be solved
from
18.0 = 14.6
for the critical crack length
a/t
0:2
6.2. Critical crack length along the moderator wall
The normalized stress intensity factor is defined by the
following equation:
K n = - K.f
6
The above numerical values for K,, at crack front (4 = 90) and
at crack tip (0 = 0) indicate that for a/c = 0.2 through 0.5 and
a/t = 0.2 through 0.8.
This implies that part-through cracks of the above geometrical
dimensions will propagate toward breaking the moderator
wall rather than the extension of the crack size along its major
axis.
cb=(14.6)
6
x -- 0.48 inch .
By using a very conservative assumption that the maximum
primary stress is located at the patch area of the thickest
vessel wall, the leak-before-break analysis shown in the
previous section indicates that an a/c value of 0.5 still satisfies
the leak condition rather than the break condition from dt =
0.2 through as much as 0.8 of the wall thickness. For a/c value
of 0.5 with 5 mm vessel wall thickness, the corresponding
half-crack length is
5 mm x,inch
c, = 25.4 mm x 0.5 = 0.39 inch
.
The above crack length of 0.39 inch remains to be a
subcritical crack length for breaking, whereas, the vessel
already leaks, This is also consistent with the result that cb=
0.48 inch and c, > c,. As a result, leak of the vessel will begin
before break.
7. FAILURE ASSESSMENT DIAGRAM ANALYSIS
OF CRACK PENETRATION FRACTURE
A fracture mechanics analysis procedure, the failure
assessment diagram (FAD), was developed by the Central
Electricity Generation Board (CEGB) in U.K. and by the U.S.
Nuclear Regulatory Commission (NRC) for predicting
structure fracture. The method requires a set of material
constants including fracture toughness, yield stressmd elastic
constants.
The method is based on the theoretical elastic-plastic crack
model of Dugdale (ref. 7) and its extended version. This
model can be expressed by a path to failure curve in a two
dimensional plane with x-axis representing the applied force
and y-axis the material toughness. Any point that is enclosed
within the curve represents the condition of no fracture. Both
CEGB W6 procedure and the NRC procedure are based on
this theory but adapted to realistic conditions. This procedure
is used in this analysis to determine the critical crack depth of
a part through crack that will penetrate the moderator vessel
wall under the present moderator internal working pressure of
15 bars. Main emphasis is on the effect of radiation
embrittlernent. The result shows that after a sustained period
of irradiation the critical crack length remains to be 0.7 of the
wall thickness but the mode of fracture is gradually shifted
from failure by rupture to failure by brittle fracture
Neuman and Raju (ref. 6) made an extensive calculation to
determine the stress intensity factor Kffor a given surface
crack on the surface of a rectangular plate. Their numerical
solution will be used here to analyze the elliptic crack front
stress intensity. If the crack front stress intensity reaches the
KI,value, then the part through crack will extend and
propagate toward the direction to penetrate the moderator
wall.
F = 0.144
(:)*
1.015
+
(7)
+
0.956
where t i s the thickness of the plate.
The above expression is compared to the numerical results
obtained in their paper and shown in the following table:
F
(Neuman and
Raju)
alt
0.1
1.059
0.2
1.173
0.3
1.165
1.273
0.4
1.359
0.5
1.385
1SO0
0.6
1.642
0.7
1.617
1.737
0.8
1.851
1.860
The parametric representation for the path to failure curve in
FAD is
In Newman and Raju formulation, the Kfvalue is represented
in the following form,
I
K,=S
JFF.
$
1
s, = --
uy 1 - a/t
It is readily recognized that the quantities Q and F are the
scaling factors with respect to a simple crack depth 2a under
the plane stress condition subject to a tensile stress of S,
K,=Sfi.
(z)
1
- so 1 - a/t
where uyis the yield stress of the material.
The two constants KO and,So is determined by the specific
problem to be analyzed,
The factor Q is recommended in the paper as
Q
= 1 + 1.464
1'65
S
so==Y
period of radiation makes it easier to fracture at the crack
front and penetrate the moderator vessel wall by fracture
rather by rupture failure:
the moderator capsule wall only after an extended period of
radiation.
For the moderator capsule subjected to 15 bars internal
pressure, the maximum surface stress is located at the inner
capsule surface with patches. The maximum stress is
8. CONCLUSION
S = 15 hi.
Other material constants and the moderator capsule
dimensions required to determine the two constants KOand So
are
t =
K,, =
KI, =
q =
or =
0.1575 in.(4 mm)
25 ksi G(unirradiated at room temperature)
15 ksi G(irradiated at room temperature)
40 ksi(unirradiated at 373 OK)
70 ksi(irradiated to 2.0 x lo2’ n/m2 at room
temperature)
Q = 1 + 1.464 (0.2)’.6’= 1.10286
The probability of fracture is obtained as a measure of the
embrittlement condition for the moderator capsule. An
alternative approach but the same method is used in the
nuclear industry. The leak-before-break is satisfied by the
moderator design structure. This implies that no catastrophic
failure may be initiated before leaking begins. A mass
spectrometer will be installed to detect the leaked hydrogen.
The path-to-failure calculation indicates that, after a period of
irradiation, the failure mode of the moderator capsule wall
will change from failure by rupture to failure by brittle
fracture.
9. REFERENCES
1 Farrell, K. and King, R. T.,“Tensile Properties of
Neutron-Irradiated 606 1-T6 Aluminum Alloy,” Effects
of Radiation on Structural Materials, ASTM STP 683,
Ed. J. A. Sprague and D. Krammer, American Society
for Testing and Materials, 1979, pp. 440-449.
From the above constants, the two constants KO and So are
listed in the following table.
Unirradiated
Irradiated
K O
0.33944
0.6574
S O
0.3750
0.2143
Also, for ah from 0.1 to 0.8, the parameter values for the
path-to-failure curve are listed in the following table:
aft
&Z F(a/t)
l l ( a - dt)
0.1
0.316
1.111
0.2
0.521
1.250
0.3
0.697
1.429
0.4
0.876
1.667
0.5
1.061
2.000
0.6
1.253
2.500
0.7
1.453
3.333
0.8
1.664
5.000
The path-to-failure curves for the unirradiated and fully
irradiated moderator capsule at the location of the inner patch
are plotted in Fig. 4. The curve of the irradiated moderator is
at the left side from that of the unirradiated moderator. Also,
for the unirradiated case at dt = 0.7, the moderator fails by
rupture, not by fracture. The crack can fracture and penetrate
2 Weeks, J. R.,Czajkowski, C. J., and Tichler, P. R.,
“Effects of High Thermal and High Fast Fluence on the
Mechanical Properties of Type 6061 Aluminum on the
HFBR,” Effect of Radiation on Materials, ASTM STP
1046, Ed. N.H. Packan, R.E. Stoller, and A.S. Kumar,
American Society for Testing and Materials,
. Philadelphia, 1990, pp.441-452.
3
Czajkowski, C. J., Schuster. M. H., Roberts, T. C.. and
Milian, L. W., “Tensile and Impact Testing of an HFBR
Control Rod Follower,” BNL Report, Nuclear Waste and
Material Technology Division, Brookhaven National
Laboratory, New York, August 1989.
4
Chang, S. J., “Probability of Fracture for HFIR Pressure
Vessel Caused by Random Crack Size or by Random
Toughness,” ASME J. Pressure Vessel Technology, vol.
116, pp. 24-29, 1994
5
Chang, S. J., “Probability of Fracture and Life Extension
Estimate of the High Flux Isotope Reactor Vessel,”
ASME J. Pressure Vessel Technology, vol. 120, to
appear, 1998
6
Cheverton, R. D., Merkle, J. G.and Nanstad, R. K.,
“Evaluation of HFIR Pressure Vessel Integrity
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