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Irradiation swelling, creep and thermal fatigue behaviors of LMFBR pressure vessels
Irradiation Swelling, Creep and Thermal Fatigue Behaviors of LMFBR Pressure Vessel? by BENJAMIN
M. MA
Department of Chemical Engineering and _Nuclear Engineering Engineering Research Institute, Iowa State Universi~, Ames, Iowa
ABSTRACT: fatigue
The
induced
behaviors
of irradiation
swelling,
irradiation
by fast neutron Jluence and developed
analyzed
on the basis
swelling,
irradiation
of experimental
data.
creep and thermal-cycling
and safety surveillunce
of the pressure
From
where t is time, t, is creep rupture failure,
fatigue.
constants. fatigue
The tangential
and hydrostatic
the irradiation
swelling
of irradiation
awelling,
LMFBR
time, N is number
E ia strai,n, c, is swelling strain to failure,
pressure
tensile strain produced E aa a whole.
irradiation
the failure
creep
vessels are
of the irradiation
criterion
for the design
aa
= 1, of cycles, N, ia number
by the irradiation
creep,
of cycles to
thermal-cycling
or
thermal-cycling
coolant may be combined
The inter-relationships and
thermal-cyclkg
pressure
and a, b, c, u, v and w are parametera
pressure of the liquid sodium strain
and
the interactiona
vessels can be expressed
a(t/t,)“+b(N/N~)v+c(c/e,)W
creep
in the LMFBR
together with
between the behaviors
fatigue
developed
in
the
vessels are discmaed.
I. Introduction
In the primary system design of Liquid-Metal Fast Breeder Reactors, (LMFBR), there are two general types (or approaches) of reactor pressure vessel: (1) pipe (or piped-loop) type, (2) tank (or pot) type. In the pipe type, the pressure vessel contains only the reactor core, control elements and liquid-metal coolant. The heat exchangers, pumps and other components of the primary system are placed outside of the pressure vessel, in a design similar to the conventional nuclear power plant design. The interconnecting piping system (or closed loop) is mostly located in a large reactor building. This building must contain an inert atmosphere to prevent a sodium accident in the event of sodium leakage or fire. In the tank-type vessel the entire primary system, including the reactor vessel, heat exchangers, pumps, etc., is submerged in a large sodium tank by taking advantage of the basic sodium properties of low vapor pressure and high thermal conductivity. Both types are shown schematically in Fig. 1. The primary system designs for both the pipe- and tank-type pressure vessels have advantages and disadvantages. The main advantages of the pipe * This work was supported by the Engineering University, Ames, Iowa 50010, U.S.A.
127
Research Institute
at Iowa State
Benjamin M. Ma
Reactor COE
L ---g
Pressure vessel
” PressuW
(4 Pipetype FIG. 1. Schematic diagrams of the LMFBR
(b) Tank type
pressure vessel in the primary system.
type are (a) decoupling of the components of the primary system and (b) convenient accessibility for inspection and repair services. The main advantages of the tank type are (a) relative insensitivity to leaks in the primary system and compact plant arrangement (large volumes of shielded pipeways and equipment cells are not required) and (b) potential for low cost and built-in safety. It is obvious that the advantages of one type are the disadvantages of the other. In the pipe-type approach the proposed designs are by Westinghouse (l), Combustion Engineering (2) and Atomic International (3) in the United States, and by Russia (4) and West Germany (5) abroad. All operating sodium-cooled nuclear plants to date have utilized this approach, except the EBR-II (Experimental Breeder Reactor II) in the U.S.A. In the tank-type approach the proposed designs are by General Electric (6), and Babcock and Wilcox (7) in the U.S., and by France (8) and England (9). At present, it appears that neither approach has serious technological problems and that either approach could lead to a successful LMFBR plant design. At the same time, the pressure vessel in either type of the LMFBR plant arrangement will experience similar effects of irradiation swelling, irradiation creep and thermal-cycling fatigue. during normal operation. In general, the pressure vessel of the pipe type is close to the reactor core (or the reactor core is closely surrounded by the pressure vessel). The objectives of this paper are to (a) study the behaviors of irradiation swelling, irradiation creep and thermal-cycling fatigue and (b) analyze the interactions and inter-relationships between irradiation swelling, irradiation creep and thermal-cycling fatigue produced in the LMFBR pressure vessels during normal operation conditions. ZZ. Radiation
Damage
Radiation damage (or irradiation-induced damages) to the LMFBR pressure vessel (especially the pipe-type arrangement) can be atrributed to the bombardment, transmutation and diffusion of fast neutrons from fission
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Journal of The Franklin Institute
Fatigue
Behaviors
of LXF.BR
Pressure
Vessels
reactions in the reactor core. It is expected that for fast neutron flux 4 = Z-3(101*) neutronslcm2-see (corresponding to a fluence of 1022-1023 neutronslcm2 in 2 yr)the pressure vessel can suffer from heavy bombardment and radiation damage in the form of atomic displacements and thermal spikes. Furthermore, the fast neutrons, n, can induce the (n, (u), (n, Zn’) and (n, n’, a) transmutation reactions with elements present in vessel material such as stainless steel (for example, Ni5!(n, y) Ni5B(n, CX) Fe5”). Consequently, the neutron bombardment, nuclear trans~~utation and helium (a particle) diffusion can produce void formation, dislocation defects and gas agglomeration in the lattices of the vessel material. Neutron irradiation can, further, change the properties of the vessel material appreciably. III.
Void,
Bubble
Formation
and Irradiation
Swelling
The formation of voids and bubbles in austinitic stainless steel during fast neutron irradiation was first reported by Cawthorne and Eulton (10). In the last few years, considerable development and progress have been made toward characterizing the structural damage over a wide range of irradiation environments. From experimental data available, different nucleation and growth models of voids and bubbles have been proposed. These models may empirically predict the void density (or void number density), the void size (or void diameter), the bubble movement and the volumetric change (or density decrease) due to irradiation swelling. These variables depend mainly on the fast neutron flux #, i~adiation time t, irradiation temperature T (or effective temperature @), transmutation products and alloy composition in the austinitic stainless steel of the vessel material. At present, no nucleation and growth model of void formation has been treated analytically. Nor has the role of helium in the nucleation, void formation, dislocation and embrittlenlent been well established. In general, the precipitation of helium may form voids and bubbles (or bhsters at the inner vessel surface). Such bubbles can grow by acquiring additional helium and vaca,ncies at disloca,tions and grain boundaries. Therefore, a bubble can reach a critical size at which it is moved from the dislocation or grain boundary along the driving force due to temperature gradients (11). The void and bubble formation and the growth. of inert gases induced by neutron irradiation can result in a volume change AVIV. In other words, the irradiation swelling can cause a volume increase (by AV) in the pressure vessel material. In order to fit the experimental data obtained from stainless steels irradiated in the EBR-II, some empirical equations are developed using the least-squares technique. Some empirical equations for the void density pv (in 1O-‘5 cm-3) and void diameter d (in .k) varying with the fast neutron fluence d;t (in 1O-22 n cm-2 and neutron energy E > 0.1MeV) at different irradiation temperatures T in ( “K) or effective temperatures B ( = T - 623)are given by p,, = (#t)1-60-25’e~1*s’e*exp(lOd%- 0*015T),
d=
Vol.299,No. 2,February 1975
(#t)1315-25/8* expf
7.45-1700/T).
(1) (2)
128
Benjamin
M. Ma
The volume change AVIV (or density change Ap/p) swelling is empirically proportional to pvd3, or equal to
AV/V Introducing
AV/V
= (T-40)
due to irradiation
10-12Pvds.
(3)
the values of pv and d from Eqs. (1) and (2) into Eq. (3) yields 10-12](~t)~05-25~0+71~e*exp(32~6-5100/T+0~015T).
= [(T-40)
(4)
The results calculated from Eqs. (l)-(4) to fit the experimental data are shown in Figs. 2-4. For purposes of comparison, the calculated results obta’ined by Brager et al. (12) are shown in the dotted curves. L
lo=
AN1
ORM
370-380 9
D
0
-
460-470°c
A
8
-
590400%
A
0
5 -
WADCO
Id” 5
13
10
Neutron fluence, gt, neutrons/cd, FIG.
2.
>O.l
MeV
Void density varies with neutron fluence for annealed 304 stainless steel.
For large volume changes (AVIV > 0*005), the bulk densities or the macroscopic dimensions measured before and after neutron irradiation are generally more accurate than the transmission electron microscopic data. The empirical equation for volume change due to neutron irradiation developed by a least-squares fit of bulk density data is expressed as Av/v
= 4.35(
10-51)
(+t)l’71
101~55~104)/T-6~0(10B)/~*
(5)
(where $t = fast neutron fluence in n cm-2 and T remains in “K) which may be compared with an empirical expression obtained by Claudson et al. (13). It is noted that in the case of isotropic irradiation swelling the swelling strain is equal to $(AV/V) for practical interest. In a similar manner, the density decrease (or density change) in annealed 304 stainless steel at three different temperature ranges is shown in Fig. 5. It is seen that to a certain extent,
130
Journal of The Franklin Institute
Fatigue Behaviors of LMFBR 400 370-380 Oc 460-470 Oc 590-600 Oc cu
ANL D m
Pressure
Vessels
8
ORNL WADCO 0 0 q al . n
300 -
0 1021
I
I
2
5
1o22
I
I
2
5
Neutron fluence, @,neutrons/cm:
FIG. 3. Mean void diameter
>O.l
varies with fast-neutron stainless steel.
ld3
MeV
fluence for annealed
304
10
CD
1.0
a
1 .
L’
. Both lines: slope=l.71 Temperature range =420-440°C
Neutron fluence,+tneutrons
/cm,2
70.1 MeV
FIG. 4. Volume change varies with neutron fluence for 304 stainless steel. ---:
AT/V
---:Av/v=
= [(T-40)10-12]
(~t)2.05-2s’~+71’eaexp (32.6-5100/T-O.OlST).
(~t)2.0~-(27/8)+(78/8~)[(~-40)~~o-~~]
expr32.6 - 0.015(T) T=“K,
$t=ncm-2x
- - -: A’V/Tr=4.85(10-51)(~t)~‘?‘lO(lGk?j T=“K,
Vol. 299, No. 2, February
1975
- (5100/T)],
10-22, e=T--623;
x 104/T)-
(6.0 x 106/T2),
+t=ncm-2.
131
Benjamin M. Ma
t 0.1
a ANL OGE 0 ORNL n WADCO
1
0.01 t 102’
_LLLllIIIII.l-uJl 1oz2
lo*
Neutron
fluence,
Ct. neutrons/cm2
>O,l
MeV
FIG. 5. Density decrease varies with neutron fluence for annealed 304 stainless steel.
the density decreases with increasing neutron fluence and irradiation temperature, according to the empirical equations for density change Ap/p obtained by the least-squares fit. For a given value of fast neutron fluence, the void density, void diameter and volume change due to irradiation swelling given in Eqs. (l)-(4) will vary with irradiation temperature T only. Figure 6 represents the typical curves, based on Eqs. (l), (2) and (4), of void density, void diameter and volume change varying with irradiation temperature in solution-treated stainless steel (types 304 and 316) irradiated to $t = 5(1022) ncm-2 (E> 0.1 MeV). The behavior of irradiation swelling manifested by the volume change in stainless steel is similar to that in most nuclear reactor materials. The maximum volume change due to irradiation swelling breakaway usually occurs in the neighborhood of 5OO’C or about 0.5 melting point temperature (“IQ In general, there are four main causes of irradiation swelling induced by fast neutron irradiation in nuclear reactor (both fission and fusion) materials (metals, metallic alloys or ceramics) in the different temperature regions : (a) Nuclear transmutation (to produce inert gases) all temperatures < 0. 2T,n, (b) Irradiation growth 0.3 - 0.5T,, (c) Void formation and growth 2 0.5T,, (d) Bubble formation and swelling is the absolute melting point temperature. The irradiation swelling where Tnap breakaway in the LMFBR pressure vessel may occur when the vessel wall is irradiated by neutrons and gamma rays in the neighborhood of 0*5T,, of the vessel material Irradiation swelling caused by inert gases (mainly helium) in the pressure vessel is apparently a complex problem. A proposed analytical model of irradiation swelling in the LMFBR oxide fuel elements has been introduced
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Institute
Fatigue Behaviors of LMFBR
I
I
400
500
Irradiation FIG.
temperature
I
Pressure Vessels
I
600
“C
6. Void density, void diameter and volume change vary with irradiation
temperature in solution-treated
type 304 and 316 stainless steel.
(10,14). The general assumptions and description of the analytical model for irradiation swelling can also be applied to irra,diation swelling in the LMFBR pressure vessel.
IV.
Heat
Generation
and Temperature
Distribution
Heat generation and temperature distribution in LMFBR pressure vessels are chiefly induced by attenuation of fast neutrons, slow neutrons, primary gamma (or X-) rays and secondary gamma rays. The radiation source is the nuclear fission reactions produced in the reactor core. For a given geometrical arrangement, the neutron flux distribution and gamma dose rate intensity are related to the nuclear and physical properties of the vessel material. The temperature distribution in the pressure vessel can be determined when the
Vol. 299, No. 2, February 1975
133
Benjamin
M. Ma
heat generation rate from the attenuation is known (15).
V. Irradiation
of the neutrons and gamma rays
Creep
The creep induced in an intensive radiation (or neutron) environment is called the “irradiation creep”. The term “irradiation creep” was introduced early in connection with the radiation and creep analysis of reactor fuel elements (16). Nuclear radia’tion to the irradiation creep in the reactor pressure vessels has two opposite behaviors: (a) irradiation hardening or helium embrittlement tends to increase hardness and mechanical strength and retard creep rate, but (b) inert gases, produced by nuclear transmutation and decay, appear to weaken mechanical strength and increase the creep rate of the pressure vessel by diffusion into the vessel wall. As the inert gases diffuse, agglomerate and accumulate in the pressure vessel, its irradiation creep rate will rapidly increase. Experimental data show that the rate of irradiation creep is highly enhanced by irradiation swelling as compared to the enhancement by elevated temperature only. At high irradiation temperatures, the behavior of irradiation creep strain E as a function of applied stress u, neutron flux + and irradiation time t can be represented by E = &ae”‘“( 1 - e-@lF) + Cm&,
(6)
where E,,is the initial creep strain, cr,,is the creep material constant, F is the reference neutron fluence (or exponential decay constant) and C is the rate constant. These constants (or parameters) can be determined from experimenta,l data of a simple irradiation creep test. Some irradiation creep curves calculated from Eq. (6) are shown in Fig. 7 (the effective strain per effective stress is defined as &Jo, = E/U in a simple tension test) (16-19). With low neutron fluence, #d is small, and Eq. (6) approaches the exponential (or hyperbolic) creep law ea/uo(1 - e-QLiF)modified by the behavior of irradiation creep (20). With high neutron fluence ($t is large), constant stress u, the last term on the right-hand side of Eq. (6) gradually dominates, and the irradiation creep strain Ebecomes linearly proportional to the neutron fluence +t (the second term in the parentheses gradually decays away, see Fig. 7). It should be noted that Eq. (B), proposed to describe the irradiation creep law at high irradiation temperature, differs from the expression suggested by Hesketh to describe the creep of helical springs of EN58B stainless steel irradiated in the Herald Reactor at relatively low temperatures (18,20). Since the LMFBR pressure vessel (pipe type), containing reactor core and liquid sodium, must operate at relatively high temperatures (about SOO“C), Eq. (6) is preferable. In most cases, it is also found that Eq. (6) fits experimenta,l data better, especially at relatively low neutron fluence. The irradiation creep rate (implicitly shown in Fig. 7) generally varies with the irradiation temperature. Figure 8 shows the irradiation creep rate
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Journal of The Franklin Institute
Fatigue Behaviors of LMFBR
Pressure
Vessels
6 304 SS (185, 200, 372 “C) Hanford K reactor o 316 SS (250 “C) Dounreayfast reactbr a EN588 ss (43 “C) Herald reactor
l
5
4
@ 316 SS (350 ‘c) EBR - II
/ /
/
0 /
/
ov 0
I
1
Neutron FIG.
I
2
I
3
fluence,~r,102’ncm~2
I
4
5 E=-0.82
MeV
7. Effective creep strain per effective stress varies with neutron fluence of austinitic stainless steel (SS).
coefficient 77,which varies with the irradiation temperature T, from experimental data of austinitic stainless steel (17, 18, 20). Basic experimental data indicated that the irradiation creep rate of nickel doubled when the irradiation temperature was lowered from 316 to 78’K. Since nickel and austinitic stainless steels (Ni-Cr alloys) have similar melting temperatures, it was inferred from Fig. 8 that the austinitic stainless steel would exhibit similar temperature-dependence behavior between 316 and 78°K. It is of importance that, in contrast to the strong increase in thermal (or thermally activated) creep rate with increasing temperatures, there is an appreciable decrease in the irradiation creep rate for austinitic stainless steel with increasing irradiation temperature. One reason for a decreasing irradiation creep rate at higher temperatures would be that the annealing effect could eliminate most voids and dislocations formed in the stainless steel. This decrease in irradiation creep rate with increasing tempera,tures in austinitic stainless steels is desirable for LMFBR pressure vessels which are to be operated at relatively high temperatures (about 600°C). This behavior can result in a higher thermal etliciency for the nuclear power plants under consideration. In general, radiation damage of nuclear materials is proportional to fast neutron energy. Experimental data show that the relative radiation damage intensity in irradiation creep studies is proportional to the average neutron energy .i? and the neutron flux 4, or the average neutron energy flux &5, of the reactor (17, 18, 20, 22, 23). The effective creep rate per unit effective
Vol. 299,No. 2, February 1975
135
Benjamin M. Ma
D Annealed EN58B, Herald reactor o
Annealed 304 SS, Hanford K reactor
0 Annealed. 316 SS, Dounreay breeder
0
-
500
1000
Irradiation temperature, T,
‘K
FIG. 8. Irradiation creep rate coefficient varies with irradiation temperature (E > 0.82 MeV).
stress normalized to 250°C by the irradiation creep rate coefficient 7 of Fig. 8 (he/va,), and varying with the average neutron energy flux & is shown in Fig. 9. (The experimental data: K -1 were conducted in a tension test of I 5-
n K-l,
4-
A K-11
175-370 C 304 SS
0 Herald, 43 C
/
o DFR, 250 C
,
v EBR-11, ROW7, 3-
I
EBR-11,
370C
Row5378c
I 1 Neutron energyflux,E$,
I’
/’
0’
0
/
,
/’ //
/G
I 2
3
(1~5MeV-n-cm~2-eec?)
FIG. 9. Effect of neutron energy flux on irradiated creep rate of annealed austinitic stainless steel.
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Journal of The Franklin
Institute
Fatigue Behaviors of LMFBR
Pressure
Vessels
304 stainless steel in the Hanford Reactor, and K-II is the predicted rate in a Hanford reactor. The test in the Herald Reactor was performed on EN58B stainless steel in torsion. The tests in the Dounreay Faster Reactor (DFR) were conducted on 316 stainless steel, and the EBR-II were obtained from pressurized tubes of 304 stainless steel.) It is seen that the effective irradiation creep rate per unit effective stress increases monotonously with the average neutron energy flux within the range of various test results. The enhancement of irradiation creep in nuclear reactor materials was previously predicted in relation to inert-gas bubble formation and irradiation swelling occurred in the nuclear fuel (16). Very recently, experimental data for non-fissile materials (especially austinitic stainless steel) have proved that the enhanced irradiation creep in these materials is also related to the voids, dislocations, bubble formation and irradiation swelling. If there are two identical samples, and if one is tested under thermally activated creep and the other is tested under neutron-irradiated creep at the same conditions of temperature, pressure etc., the enhancement factor for irradiation creep may be determined from the samples. At the same time, the separation of temperature and neutron-energy-flux dependence may be carried out. VI.
Low-cycle
Thermal
Fatigue
Thermal cycling induced by nuclear reactor kinetics is an inherent characteristic of all nuclear reactors. Material fatigue can occur when a specimen undergoes repeated mechanical and/or thermal loading in the steady or transient state. Thermal fatigue is usually induced by thermal loading due to cyclic temperature. Low-cycle thermal fatigue (or thermal-cycling fatigue) is defined as the process whereby cracks which are incipient (microscopically or macroscopically) are propagated in materials which are subjected to cyclic temperature changes and undergo thermal strains (or stress) at elevated (or irradiation) temperatures. The behavior of low-cycle thermal fatigue in LMFBR pressure vessels is induced by the repetitive reactor period of the core. In the reactor core the neutron flux and power level always fluctuate randomly and cyclically. This fluctuating effect can be transmitted through the liquid sodium coolant to the pressure vessel (in either pipe type or tank type) ; as a result, the pressure vessel is subject to the cyclically dynamic and thermal loading. Figures 10 and 11 show the typical tangential stress distributions in the LMFBR pressure vessel for two consecutive cycles of reactor startup and shutdown (in the transient state). Figure 12 illustrates the behavior of the stress-strain loop (or U--Ecurve) in the pressure vessel during one thermalloading cycle with a hold period. Experimental data obtained from the fatigue behavior of AISA 4130 steel, and others, under strain cycling show that the relation between elastic strain per cycle ad and corresponding cyclic life (i.e. the number of cycles to failure) N, can be represented by (see Fig. 13)
-&?I
&d-E
Vol. 299, No. 2, February
1975
f'
(7) 137
Benjamin
M. Ma
-10
-2c
-30
0
1.02
l-04
1'06
Vessel thickness
1.08
1ilO
1.12
x(=/p/,,
FIG. 10. Tangential stress distribution in the pressure vessel at the first and second cycles of reactor start-up.
0> \
‘0 -
*’ /
o-
o\ 1-o
1.02
l-04
7.06
Vesselthickness,
l-as
1.10
l-12
X("dfi)
FIG. 11. Tangential stress distribution in the pressure vessel at the first and second cycles of reactor shut-down.
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Journal of The Franklin Institute
Fatigue Behaviors of LMFBR
Vessels
t
Time,
t
Time,
Strain,
Pressure
E
FIG. 12. The behavior of the stress-strain loop during one thermal loading cycle with a hold period. where E is the elastic modulus and u0 and y are the elastic material constants. Equation (7) satisfies the cyclic life range up to lo6 cycles for a large number of materials tested. Similarly, test results observed for the fatigue behavior of annealed AISA 4130 steel and other similar materials indicate that the relation between plastic strain range per cycle cPz and corresponding cyclic life can be expressed as (see Fig. 14) &PI
=
mN;
(8) The value z = - + fits
in which m and z are the plastic material constants. most materials tested.
.u z
z
I
0~001 10
1111 I III1 lo2
I IIll
lo3 Cyclic life, cycles
FIG.
I lo4
to failure,
I III
III1 lo5
lo6
P$
13. Relation between elastic strain range per cycle and cyclic life (for annealed 4130 steel and others).
Vol. 299, No. 2, February
1975
139
Benjamin M. Ma
0-2
o-1
0.04 “9 c‘ ‘iL t; .o B _m n_
0.02
0.01 0.008 0.006 o-004
0.002
0.001 00
lo2
lo3
lo4
Cyclic life,cycles to failure,
N,
FIG. 14. Relation between plastic strain per cycle and cyclic life (for 4130 steel and others).
In many practical cases, the total strain range per cycle E (or he) is of interest. Therefore, the relation between total strain range and cyclic life can be obtained from Eqs. (7) and (8) E = A& = eel+aP1 = ONu+mNj E t
(9)
where AE is the finite strain range between any two consecutive cycles at the asymptotic condition, as shown in Fig. 15. The principal need in pressure vessel design analysis and the computation of low-cycle thermal fatigue is a relation between stress range Aa and total strain range As. Such a relation can provide a means for combining the strain compatibility and stress equilibrium equations together for design analysis. Eliminating N, from Eqs. (7) and (8) and combining them with Eq. (9) yield
(10) where Aa is usually taken as the asymptotic stress range of the cyclic half-life (NJ2) of a specimen. From the logarithmic plots of elastic strain
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Journalof
The Franklin
Instit,ute
Fatigue Behaviors of LMFBR
Cyclic life,cycles to failure,
Pressure Vessels
N,
FIG. 15. Total strain range (the sum of elastic and plastic components) varies with cyclic life.
range and plastic strain range vs. cyclic life, the elastic and plastic material constants CT,,,y, m and x of the material can be determined experimentally. Hence, the strain range, stress range and cyclic life can be predicted from Eqs. (7)-(10).
VII.
Interaction Between Fatigue, etc.
Irradiation
Creep
and Low-cycle
Thermal
Operating conditions of LMFBR pressure vessels under fast neutron irradiation usually involve complex load and temperature variations (i.e. irradiation creep, cyclic thermal load, sodium hydrostatic load, etc.). The random process and time history (derived from reactor kinetics) of the resulting stresses and strains are even more complex, and differ for each part of a pressure vessel. The random process and time history of loading fluctuations and temperature variations are statistical phenomena in nature. The effects of time on the behavior of creep and fatigue, however, are referred to as the interaction between the irradiation creep and the low-cycle thermal fatigue (or, simply, creep-fatigue interaction) of the LMFBR pressure vessels. The creep-fatigue interaction relationship may be expressed (in terms of damage fractions D and interaction parameters) as D = a(DoJU+ b(D,)”
(11)
D, s t/te, D, s N/Nf,
(12)
and where D, is the creep damage, D, is the fatigue damage, D is the total damage, t is the time of creep at effective stress (Jo, t, is the creep rupture time at (TV, N is the number of cycles at the strain range AC, N, is the number of cycles
Vol. 299, No. 2, February
1975
141
Benjamin M. Ma to failure (as defined), and a, b, u and v are the interaction parameters. When a failure criterion of D = 1 is assumed for the total damage, and D, and D, are introduced from Eq. (12), the failure criterion for the creep-fatigue interaction in the pressure vessel becomes a(t/t,)” + b(N/N,)V = 1.
(13)
There are four possible cases: (1) For the creep failure only, a = u = 1, b = 0, t = t,. (2) For the fatigue failure only, b = v = 1, a = 0, N = N,. (3) For the combination of the creep and the fatigue a + 0,b f 0, u, v < 1. (4) For a linear combination of the creep and the fatigue u = v = 1. The possible forms of interaction and design curves for various values of a, b, u and v are plotted in Fig. 16. It is expected that most of the design values and experimental data for the creep-fatigue interaction will fall into region 3 of the curves.
U
1 Fatique damage,
N/N,
FIG. 16. The interaction curves of irradiation creep and thermal cycling fatigue: (1) For the creep only, a =u = 1, b = 0. (2) For the fatigue only, b = v = 1, a = 0. (3) For the creep-fatigue interaction, u, 2)f 1. (4) For the linear creep-fatigue interaction. u = v = 1.
In the same manner, the interactions between irradiation creep and irradiation swelling and low-cycle thermal fatigue and irradiation swelling can be written as a(t/tJ”
(14)
f c(E/E,)~ = 1,
b(N/N,)” + c(&,)~
= 1,
(15)
where E is the irradiation swelling strain ( 2: )AV/V), as is the irradiation swelling strain to failure (produced by a neutron fluence of about 1022-1024 n
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Fatigue Behaviors of LMFBR
Pressure
Vessels
cm-2), and c and w are other parameters in question. From Eqs. (14) and (15) the interaction and design curves for creep-swelling and fatigue-swelling, respectively, can be plotted as shown in Fig. 16 above. Since the irradiation swelling, irradiation creep and low-cycle thermal fatigue can occur simultaneously in the LMFBR pressure vessels, the failure criterion of the pressure vessels can be obtained by combining Eqs. (13)-( 15) to yield a(@,)” + b(N/N,)” + ~(E/E,)~= 1.
(16)
From safety surveillance of an LMFBR pressure vessel during reactor operation, the variables and parameters given in Eq. (16) can be obtained, and the failure criterion may be verified experimentally.
VIII.
Interrelationships Fatigue Behaviors
Between Zrradiation
Swelling,
Creep
and Thermal
The irradiation creep developed in the LMFBR pressure vessels during normal operation combined with inert gas (mainly helium) production, diffusion and swelling in the vessel material (material properties also change) is closely related to low-cycle thermal fatigue. The interrelations between void, dislocation and gas-bubble formation, irradiation swelling and irradiation creep induced by fast neutron fluence, irradiation hardening and helium diffusion and embrittlement at the micro- and macro-cracks in the pressure vessel can cause fatigue failure through brittle fracture. In other words, irradiation hardening and helium production, diffusion, swelling and embrittlement can spread cracks and accelerate thermal cycling fatigue to cause fracture failure of the LMFBR pressure vessel. Irradiation creep can produce stress relaxation (or stress relief.) (16). Correlations of irradiation creep for annealed austinitic stainless steel have been associated with temperature dependence, stress (or strain) state, neutron energy flux (or spectrum) and neutron Auence. That irradiation creep rate decreases with increasing temperature (as opposite to an observed increase in thermally activated creep with increasing temperature) can be attributed to the annealing effect on irradiation-produced defects, as predicted. The transient creep can be attributed to the effects of thermal cycling and dislocation climb caused by the absorption of excess point defects. The transition from transient creep to the steady creep is compatible with the creep strain recovery which follows a stress reduction. Similarly, both thermal creep and irradia,tion creep can produce stress relaxation (or stress relief,) in materials under consideration. Since interstitial atoms are preferentially absorbed at dislocations, the surviving vacancies and vacancy clusters resulting from neutron irradiation may also give rise to microscopic volume increases (in addition to the inert-gas swelling). Moreover, a large number of the interstitials can be absorbed at dislocations due to the tensile stress produced by irradiation creep and cyclic thermal loading. Thus, irradiation swelling may also be
Vol. 299, No. 2, February
1975
143
Benjamin M. Ma enhanced by the tensile stresses under consideration. Therefore, the total volume increase in the pressure vessel can be attributed to (a) the inert gas swelling, (b) the microscopic volume increases due to vacancies and vacancy clusters and (c) the tensile stress produced by irradiation creep and cyclic thermal loading. On the other hand, thermal cycling may sweep up and absorb interstitials and vacant point defects by oscillating dislocations. Large amplitude lowcycle thermal fatigue strains may absorb vacancies and voids and reduce bubbles and irradiation swelling. The possible modification of the swelling in the LMFBR pressure vessel by irradiation creep and thermal cycling fatigue (and vice versa) may be verified experimentally. The apparent effect of hydrostatic tangential tension (or hoop stress) on the dimensional change of the pressure vessel is, however, difficult to account, for. That is, the liquid sodium coolant, deposited in the pressure vessel, apparently produces hydrostatic pressure and tangential tensile stress, which cause a volume increase in the pressure vessel. Although the hydrostatic pressure of liquid sodium might reduce irradiation swelling somewhat, the hydrostatic tangential tension can unfavourably accelerate the irradiation creep and thermal cycling fatigue of the pressure vessels (this may be more severe to the tank-type LMFBR pressure vessels). The occurrence of micro- and macro-cracks in, and changes in the volume, shape and material properties of, the LMFBR pressure vessels are expected to vary with vessel materials and operation conditions. Any success of this correlation should be attributed to the fine structural, mechanical and nuclear properties of annealed austinitic stainless steels. Considerable experimental data, theoretical work and operation experience are needed in order to understand the interrelationships between the irradiation swelling, creep and thermal fatigue behaviors of the LMFBR pressure vessels.
IX.
Conclusions
From the preceding analysis of the behaviors of irradiation swelling, irradiation creep and low-cycle thermal fatigue in the LMFBR pressure vessels, the following conclusions are drawn: (1) The pressure vessel will suffer from severe radiation damage by fast neutrons, slow neutrons, primary gamma rays and secondary gamma rays. The radiation damage can change the vessel material properties appreciably. (2) The void density, mean void diameter and volume (or density) changes in the pressure vessel vary with the neutron fluence and irradiation temperature for a given neutron energy spectrum (Eqs. (l)-(4) and Figs. Z-5). The irradiation swelling breakaway in the pressure vessel may occur when the irradiation temperature is about half the melting point temperature of the vessel material. (3) Irradation swelling can highly enhance irradiation creep rate in a pressure vessel irradiated by neutron fluence (Eq. (5) and Fig. 7). The effective creep rate decreases with increasing irradiation temperature (in
144
Journal of The
Franklin
Institute
Fatigue Behaviors of LMFBR
Pressure
Vessels
contrast to the thermally activated creep) which is desirable for LMFBR pressure vessels to be operated at relatively high temperatures. The effective creep rate, however, is increased with the average neutron energy flux (Figs. 8 and 9). (4) Irradiation hardening and helium production, diffusion, swelling and embrittlement can spread cracks and accelerate thermal cycling fatigue (low-cycle thermal fatigue) to cause fracture failure of the LMFBR pressure vessels. (5) From the behaviors and interactions of irradiation swelling, irradiation creep and thermal cycling fatigue, the failure criterion for the design and safety surveillance of the pressure vessels can be expressed as (Eq. 16 and Fig. 16) a(@,)” + b(N/Nf)w + c(E/E,)~= 1. The tangential tensile strains (or hoop strain) produced by the irradiation creep, thermal cycling fatigue and hydrostatic pressure of the liquid-sodium coolant may be combined as a whole with the irradiation swelling strain E,. References (1) J. H. Wright, “Design philosophy for the Westinghouse fast breeder reactor”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. and evaluation of the combustion engineering (2) R. C. Noyes, “Development advanced lOOO-MW, LMFBR design”, Proc. International Conference on Sodium Technology and Large Fast Reactor Des., Argonne National Lab. Rep. ANL-7520, Part II, pp. 291-230, 1967. (3) R. Balent, “Liquid-metal fast breeder reactors”, Proc. Reactors and the University Conf., Rensselser Polytechnic Institute, Aug. 28-30, 1968. (4) A. Leipunsky, “Operating experience with BR-5 Reactor”, also, “BN-350 nuclear power plant construction”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (5) K. Gast, “NA-2 design 300 MW, German FCR”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (6) B. Wolfe and W. J. Clabaugh, “Preliminary work on a 300 MW, fast ceramic demonstration reactor”, Proc. Fast Reactors, National Topical Meeting, SNS-101, American Nuclear Society, San Francisco, 1967. (7) M. W. Croft, “Babcock and Wilcox 1000 MW, LMFBR follow-on study reference design”, Proc. Int. Conf. on Sodium Tech. Large Past Reactor Deisng, Argonne National Lab. Rep. ANL-7520, Part II, pp. 215-239, 1968. (8) G. Vendryes, “The Phenix reactor (French FCR)“, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (9) A. G. Frame, “Status of the UKAEA prototype fast reactor”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (10)C. Cawthorne and E. J. Fulton, “The influence of irradiation temperature on the defect structures in stainless steel”, Report AERE-R-5269, Atomic Energy Research Establishment, Harwell, 1967. Also “Voids in irradiated stainless steel”, Nature, Lond., Vol. 216, pp. 575-576, 1967. (11)B. M. Ma, “Thermal, radiation and mechanical analysis for fact reactor fuel elements”, Proc. First Int. Conf. on Structural Mech. in Reactor Tech., Vol. 2, C4/3, Berlin, West Germany, 1972.
Vol. 299, No. 2, February
1975
145
Benjamin M. Ma (12) H. R. Brager, “Irradiation produced defects in austinitic stainless steel”. USAEC Rep. WHAN-FR-16, WADCO, 1970 (13) T. T. Claudson, R. W. Barker and R. L. Fish, “The effects of fast flux irradiation on the mechanical properties and dimensional stability of stainless steel”, Nuclear Appl. and Tech., Vol. 9, No. 1, p. 10, 1970. (14) C. K. Cheng and B. M. Ma, “Thermal, radiation and mechanical analysis for unsteady-state fuel restructuring of cylindrical oxide elements in fast reactors”, Nucl. Sci. Engng, Vol. 48, pp. 139-153, 1972. (15) B. M. Ma, “Heat generation and temperature distribution in cylindrical reactor pressure vessel”, Nucl. Engng and Design, Vol. 11, pp. 1-15, 19’70. (16) B. M. Ma, “Radiation and creep analysis for strain and stress distributions in tubular fuel elements”, Nucl. Sci. Engng, Vol. 20, pp. 536-546, 1964. (17) D. Mosedale, “Irradiation creep in the Dounreay Fast Reactor (DFR)“, Nature, Lo&., Vol. 224, p. 301, 1969. (18) E. R. Gilbert and N. E. Harding, “Irradiation-induced creep in austinitic stainless steel”, Trans. Am. Nucl. Sot., Vol. 13, No. 1, pp. 143-144, 1970. Also E. R. Gilbert and L. D. Blackburn, “Irradiation-induced creep in austinitic stainless steel”, USAEC Rep. WHAN-FR-30, WADCO, 1970. (19) A. J. Lowell and R. W. Barker, “Uniaxial and biaxial creep rupture of type 316 stainless steel after fast reactor irradiation” , USAEC (Hanford Eng. Dev. Lab.) Report, WHAN-FR-30, WADCO, 1970. (20) R. V. Hesketh, “Collapse of vacancy cascades to dislocation loops”, Proc. Int. Conf. on Solid-state Physics Research with Acceleraters, Brookhaven National Lab. Rep. BNL-50083, 1967. Also “Irradiation creep in non-fissile metals”, Brittleness and Irradiation Effects, 10th Symp. on Special Metallurgy, Saclay, France, p. 185, 1967. (21) B. M. Ma, “A creep analysis of rotating solid disks”, J. Franklin Inst., Vol. 267, No. 2, pp. 149-165, Feb. 1959. (22) M. N. McElroy, R. E. Dahl, Jr. and E. R. Gilbert, “Neutron energy-dependent damage function for analysis of austinitic steel creep data”, Batelle Northwest Lab. Rep., BNWL-SA-3186, Richland, Wash., March 1970. (23) D. D. Walters, “EBR-11 in-pile creep experiments on stainless-steel tubing”, Trans. Am. Nucl. Sot., Vol. 13, p. 146, 1970.
146
Journal of The Franklin
Institute
M. MA
Department of Chemical Engineering and _Nuclear Engineering Engineering Research Institute, Iowa State Universi~, Ames, Iowa
ABSTRACT: fatigue
The
induced
behaviors
of irradiation
swelling,
irradiation
by fast neutron Jluence and developed
analyzed
on the basis
swelling,
irradiation
of experimental
data.
creep and thermal-cycling
and safety surveillunce
of the pressure
From
where t is time, t, is creep rupture failure,
fatigue.
constants. fatigue
The tangential
and hydrostatic
the irradiation
swelling
of irradiation
awelling,
LMFBR
time, N is number
E ia strai,n, c, is swelling strain to failure,
pressure
tensile strain produced E aa a whole.
irradiation
the failure
creep
vessels are
of the irradiation
criterion
for the design
aa
= 1, of cycles, N, ia number
by the irradiation
creep,
of cycles to
thermal-cycling
or
thermal-cycling
coolant may be combined
The inter-relationships and
thermal-cyclkg
pressure
and a, b, c, u, v and w are parametera
pressure of the liquid sodium strain
and
the interactiona
vessels can be expressed
a(t/t,)“+b(N/N~)v+c(c/e,)W
creep
in the LMFBR
together with
between the behaviors
fatigue
developed
in
the
vessels are discmaed.
I. Introduction
In the primary system design of Liquid-Metal Fast Breeder Reactors, (LMFBR), there are two general types (or approaches) of reactor pressure vessel: (1) pipe (or piped-loop) type, (2) tank (or pot) type. In the pipe type, the pressure vessel contains only the reactor core, control elements and liquid-metal coolant. The heat exchangers, pumps and other components of the primary system are placed outside of the pressure vessel, in a design similar to the conventional nuclear power plant design. The interconnecting piping system (or closed loop) is mostly located in a large reactor building. This building must contain an inert atmosphere to prevent a sodium accident in the event of sodium leakage or fire. In the tank-type vessel the entire primary system, including the reactor vessel, heat exchangers, pumps, etc., is submerged in a large sodium tank by taking advantage of the basic sodium properties of low vapor pressure and high thermal conductivity. Both types are shown schematically in Fig. 1. The primary system designs for both the pipe- and tank-type pressure vessels have advantages and disadvantages. The main advantages of the pipe * This work was supported by the Engineering University, Ames, Iowa 50010, U.S.A.
127
Research Institute
at Iowa State
Benjamin M. Ma
Reactor COE
L ---g
Pressure vessel
” PressuW
(4 Pipetype FIG. 1. Schematic diagrams of the LMFBR
(b) Tank type
pressure vessel in the primary system.
type are (a) decoupling of the components of the primary system and (b) convenient accessibility for inspection and repair services. The main advantages of the tank type are (a) relative insensitivity to leaks in the primary system and compact plant arrangement (large volumes of shielded pipeways and equipment cells are not required) and (b) potential for low cost and built-in safety. It is obvious that the advantages of one type are the disadvantages of the other. In the pipe-type approach the proposed designs are by Westinghouse (l), Combustion Engineering (2) and Atomic International (3) in the United States, and by Russia (4) and West Germany (5) abroad. All operating sodium-cooled nuclear plants to date have utilized this approach, except the EBR-II (Experimental Breeder Reactor II) in the U.S.A. In the tank-type approach the proposed designs are by General Electric (6), and Babcock and Wilcox (7) in the U.S., and by France (8) and England (9). At present, it appears that neither approach has serious technological problems and that either approach could lead to a successful LMFBR plant design. At the same time, the pressure vessel in either type of the LMFBR plant arrangement will experience similar effects of irradiation swelling, irradiation creep and thermal-cycling fatigue. during normal operation. In general, the pressure vessel of the pipe type is close to the reactor core (or the reactor core is closely surrounded by the pressure vessel). The objectives of this paper are to (a) study the behaviors of irradiation swelling, irradiation creep and thermal-cycling fatigue and (b) analyze the interactions and inter-relationships between irradiation swelling, irradiation creep and thermal-cycling fatigue produced in the LMFBR pressure vessels during normal operation conditions. ZZ. Radiation
Damage
Radiation damage (or irradiation-induced damages) to the LMFBR pressure vessel (especially the pipe-type arrangement) can be atrributed to the bombardment, transmutation and diffusion of fast neutrons from fission
128
Journal of The Franklin Institute
Fatigue
Behaviors
of LXF.BR
Pressure
Vessels
reactions in the reactor core. It is expected that for fast neutron flux 4 = Z-3(101*) neutronslcm2-see (corresponding to a fluence of 1022-1023 neutronslcm2 in 2 yr)the pressure vessel can suffer from heavy bombardment and radiation damage in the form of atomic displacements and thermal spikes. Furthermore, the fast neutrons, n, can induce the (n, (u), (n, Zn’) and (n, n’, a) transmutation reactions with elements present in vessel material such as stainless steel (for example, Ni5!(n, y) Ni5B(n, CX) Fe5”). Consequently, the neutron bombardment, nuclear trans~~utation and helium (a particle) diffusion can produce void formation, dislocation defects and gas agglomeration in the lattices of the vessel material. Neutron irradiation can, further, change the properties of the vessel material appreciably. III.
Void,
Bubble
Formation
and Irradiation
Swelling
The formation of voids and bubbles in austinitic stainless steel during fast neutron irradiation was first reported by Cawthorne and Eulton (10). In the last few years, considerable development and progress have been made toward characterizing the structural damage over a wide range of irradiation environments. From experimental data available, different nucleation and growth models of voids and bubbles have been proposed. These models may empirically predict the void density (or void number density), the void size (or void diameter), the bubble movement and the volumetric change (or density decrease) due to irradiation swelling. These variables depend mainly on the fast neutron flux #, i~adiation time t, irradiation temperature T (or effective temperature @), transmutation products and alloy composition in the austinitic stainless steel of the vessel material. At present, no nucleation and growth model of void formation has been treated analytically. Nor has the role of helium in the nucleation, void formation, dislocation and embrittlenlent been well established. In general, the precipitation of helium may form voids and bubbles (or bhsters at the inner vessel surface). Such bubbles can grow by acquiring additional helium and vaca,ncies at disloca,tions and grain boundaries. Therefore, a bubble can reach a critical size at which it is moved from the dislocation or grain boundary along the driving force due to temperature gradients (11). The void and bubble formation and the growth. of inert gases induced by neutron irradiation can result in a volume change AVIV. In other words, the irradiation swelling can cause a volume increase (by AV) in the pressure vessel material. In order to fit the experimental data obtained from stainless steels irradiated in the EBR-II, some empirical equations are developed using the least-squares technique. Some empirical equations for the void density pv (in 1O-‘5 cm-3) and void diameter d (in .k) varying with the fast neutron fluence d;t (in 1O-22 n cm-2 and neutron energy E > 0.1MeV) at different irradiation temperatures T in ( “K) or effective temperatures B ( = T - 623)are given by p,, = (#t)1-60-25’e~1*s’e*exp(lOd%- 0*015T),
d=
Vol.299,No. 2,February 1975
(#t)1315-25/8* expf
7.45-1700/T).
(1) (2)
128
Benjamin
M. Ma
The volume change AVIV (or density change Ap/p) swelling is empirically proportional to pvd3, or equal to
AV/V Introducing
AV/V
= (T-40)
due to irradiation
10-12Pvds.
(3)
the values of pv and d from Eqs. (1) and (2) into Eq. (3) yields 10-12](~t)~05-25~0+71~e*exp(32~6-5100/T+0~015T).
= [(T-40)
(4)
The results calculated from Eqs. (l)-(4) to fit the experimental data are shown in Figs. 2-4. For purposes of comparison, the calculated results obta’ined by Brager et al. (12) are shown in the dotted curves. L
lo=
AN1
ORM
370-380 9
D
0
-
460-470°c
A
8
-
590400%
A
0
5 -
WADCO
Id” 5
13
10
Neutron fluence, gt, neutrons/cd, FIG.
2.
>O.l
MeV
Void density varies with neutron fluence for annealed 304 stainless steel.
For large volume changes (AVIV > 0*005), the bulk densities or the macroscopic dimensions measured before and after neutron irradiation are generally more accurate than the transmission electron microscopic data. The empirical equation for volume change due to neutron irradiation developed by a least-squares fit of bulk density data is expressed as Av/v
= 4.35(
10-51)
(+t)l’71
101~55~104)/T-6~0(10B)/~*
(5)
(where $t = fast neutron fluence in n cm-2 and T remains in “K) which may be compared with an empirical expression obtained by Claudson et al. (13). It is noted that in the case of isotropic irradiation swelling the swelling strain is equal to $(AV/V) for practical interest. In a similar manner, the density decrease (or density change) in annealed 304 stainless steel at three different temperature ranges is shown in Fig. 5. It is seen that to a certain extent,
130
Journal of The Franklin Institute
Fatigue Behaviors of LMFBR 400 370-380 Oc 460-470 Oc 590-600 Oc cu
ANL D m
Pressure
Vessels
8
ORNL WADCO 0 0 q al . n
300 -
0 1021
I
I
2
5
1o22
I
I
2
5
Neutron fluence, @,neutrons/cm:
FIG. 3. Mean void diameter
>O.l
varies with fast-neutron stainless steel.
ld3
MeV
fluence for annealed
304
10
CD
1.0
a
1 .
L’
. Both lines: slope=l.71 Temperature range =420-440°C
Neutron fluence,+tneutrons
/cm,2
70.1 MeV
FIG. 4. Volume change varies with neutron fluence for 304 stainless steel. ---:
AT/V
---:Av/v=
= [(T-40)10-12]
(~t)2.05-2s’~+71’eaexp (32.6-5100/T-O.OlST).
(~t)2.0~-(27/8)+(78/8~)[(~-40)~~o-~~]
expr32.6 - 0.015(T) T=“K,
$t=ncm-2x
- - -: A’V/Tr=4.85(10-51)(~t)~‘?‘lO(lGk?j T=“K,
Vol. 299, No. 2, February
1975
- (5100/T)],
10-22, e=T--623;
x 104/T)-
(6.0 x 106/T2),
+t=ncm-2.
131
Benjamin M. Ma
t 0.1
a ANL OGE 0 ORNL n WADCO
1
0.01 t 102’
_LLLllIIIII.l-uJl 1oz2
lo*
Neutron
fluence,
Ct. neutrons/cm2
>O,l
MeV
FIG. 5. Density decrease varies with neutron fluence for annealed 304 stainless steel.
the density decreases with increasing neutron fluence and irradiation temperature, according to the empirical equations for density change Ap/p obtained by the least-squares fit. For a given value of fast neutron fluence, the void density, void diameter and volume change due to irradiation swelling given in Eqs. (l)-(4) will vary with irradiation temperature T only. Figure 6 represents the typical curves, based on Eqs. (l), (2) and (4), of void density, void diameter and volume change varying with irradiation temperature in solution-treated stainless steel (types 304 and 316) irradiated to $t = 5(1022) ncm-2 (E> 0.1 MeV). The behavior of irradiation swelling manifested by the volume change in stainless steel is similar to that in most nuclear reactor materials. The maximum volume change due to irradiation swelling breakaway usually occurs in the neighborhood of 5OO’C or about 0.5 melting point temperature (“IQ In general, there are four main causes of irradiation swelling induced by fast neutron irradiation in nuclear reactor (both fission and fusion) materials (metals, metallic alloys or ceramics) in the different temperature regions : (a) Nuclear transmutation (to produce inert gases) all temperatures < 0. 2T,n, (b) Irradiation growth 0.3 - 0.5T,, (c) Void formation and growth 2 0.5T,, (d) Bubble formation and swelling is the absolute melting point temperature. The irradiation swelling where Tnap breakaway in the LMFBR pressure vessel may occur when the vessel wall is irradiated by neutrons and gamma rays in the neighborhood of 0*5T,, of the vessel material Irradiation swelling caused by inert gases (mainly helium) in the pressure vessel is apparently a complex problem. A proposed analytical model of irradiation swelling in the LMFBR oxide fuel elements has been introduced
132
Journal
of The Franklin
Institute
Fatigue Behaviors of LMFBR
I
I
400
500
Irradiation FIG.
temperature
I
Pressure Vessels
I
600
“C
6. Void density, void diameter and volume change vary with irradiation
temperature in solution-treated
type 304 and 316 stainless steel.
(10,14). The general assumptions and description of the analytical model for irradiation swelling can also be applied to irra,diation swelling in the LMFBR pressure vessel.
IV.
Heat
Generation
and Temperature
Distribution
Heat generation and temperature distribution in LMFBR pressure vessels are chiefly induced by attenuation of fast neutrons, slow neutrons, primary gamma (or X-) rays and secondary gamma rays. The radiation source is the nuclear fission reactions produced in the reactor core. For a given geometrical arrangement, the neutron flux distribution and gamma dose rate intensity are related to the nuclear and physical properties of the vessel material. The temperature distribution in the pressure vessel can be determined when the
Vol. 299, No. 2, February 1975
133
Benjamin
M. Ma
heat generation rate from the attenuation is known (15).
V. Irradiation
of the neutrons and gamma rays
Creep
The creep induced in an intensive radiation (or neutron) environment is called the “irradiation creep”. The term “irradiation creep” was introduced early in connection with the radiation and creep analysis of reactor fuel elements (16). Nuclear radia’tion to the irradiation creep in the reactor pressure vessels has two opposite behaviors: (a) irradiation hardening or helium embrittlement tends to increase hardness and mechanical strength and retard creep rate, but (b) inert gases, produced by nuclear transmutation and decay, appear to weaken mechanical strength and increase the creep rate of the pressure vessel by diffusion into the vessel wall. As the inert gases diffuse, agglomerate and accumulate in the pressure vessel, its irradiation creep rate will rapidly increase. Experimental data show that the rate of irradiation creep is highly enhanced by irradiation swelling as compared to the enhancement by elevated temperature only. At high irradiation temperatures, the behavior of irradiation creep strain E as a function of applied stress u, neutron flux + and irradiation time t can be represented by E = &ae”‘“( 1 - e-@lF) + Cm&,
(6)
where E,,is the initial creep strain, cr,,is the creep material constant, F is the reference neutron fluence (or exponential decay constant) and C is the rate constant. These constants (or parameters) can be determined from experimenta,l data of a simple irradiation creep test. Some irradiation creep curves calculated from Eq. (6) are shown in Fig. 7 (the effective strain per effective stress is defined as &Jo, = E/U in a simple tension test) (16-19). With low neutron fluence, #d is small, and Eq. (6) approaches the exponential (or hyperbolic) creep law ea/uo(1 - e-QLiF)modified by the behavior of irradiation creep (20). With high neutron fluence ($t is large), constant stress u, the last term on the right-hand side of Eq. (6) gradually dominates, and the irradiation creep strain Ebecomes linearly proportional to the neutron fluence +t (the second term in the parentheses gradually decays away, see Fig. 7). It should be noted that Eq. (B), proposed to describe the irradiation creep law at high irradiation temperature, differs from the expression suggested by Hesketh to describe the creep of helical springs of EN58B stainless steel irradiated in the Herald Reactor at relatively low temperatures (18,20). Since the LMFBR pressure vessel (pipe type), containing reactor core and liquid sodium, must operate at relatively high temperatures (about SOO“C), Eq. (6) is preferable. In most cases, it is also found that Eq. (6) fits experimenta,l data better, especially at relatively low neutron fluence. The irradiation creep rate (implicitly shown in Fig. 7) generally varies with the irradiation temperature. Figure 8 shows the irradiation creep rate
134
Journal of The Franklin Institute
Fatigue Behaviors of LMFBR
Pressure
Vessels
6 304 SS (185, 200, 372 “C) Hanford K reactor o 316 SS (250 “C) Dounreayfast reactbr a EN588 ss (43 “C) Herald reactor
l
5
4
@ 316 SS (350 ‘c) EBR - II
/ /
/
0 /
/
ov 0
I
1
Neutron FIG.
I
2
I
3
fluence,~r,102’ncm~2
I
4
5 E=-0.82
MeV
7. Effective creep strain per effective stress varies with neutron fluence of austinitic stainless steel (SS).
coefficient 77,which varies with the irradiation temperature T, from experimental data of austinitic stainless steel (17, 18, 20). Basic experimental data indicated that the irradiation creep rate of nickel doubled when the irradiation temperature was lowered from 316 to 78’K. Since nickel and austinitic stainless steels (Ni-Cr alloys) have similar melting temperatures, it was inferred from Fig. 8 that the austinitic stainless steel would exhibit similar temperature-dependence behavior between 316 and 78°K. It is of importance that, in contrast to the strong increase in thermal (or thermally activated) creep rate with increasing temperatures, there is an appreciable decrease in the irradiation creep rate for austinitic stainless steel with increasing irradiation temperature. One reason for a decreasing irradiation creep rate at higher temperatures would be that the annealing effect could eliminate most voids and dislocations formed in the stainless steel. This decrease in irradiation creep rate with increasing tempera,tures in austinitic stainless steels is desirable for LMFBR pressure vessels which are to be operated at relatively high temperatures (about 600°C). This behavior can result in a higher thermal etliciency for the nuclear power plants under consideration. In general, radiation damage of nuclear materials is proportional to fast neutron energy. Experimental data show that the relative radiation damage intensity in irradiation creep studies is proportional to the average neutron energy .i? and the neutron flux 4, or the average neutron energy flux &5, of the reactor (17, 18, 20, 22, 23). The effective creep rate per unit effective
Vol. 299,No. 2, February 1975
135
Benjamin M. Ma
D Annealed EN58B, Herald reactor o
Annealed 304 SS, Hanford K reactor
0 Annealed. 316 SS, Dounreay breeder
0
-
500
1000
Irradiation temperature, T,
‘K
FIG. 8. Irradiation creep rate coefficient varies with irradiation temperature (E > 0.82 MeV).
stress normalized to 250°C by the irradiation creep rate coefficient 7 of Fig. 8 (he/va,), and varying with the average neutron energy flux & is shown in Fig. 9. (The experimental data: K -1 were conducted in a tension test of I 5-
n K-l,
4-
A K-11
175-370 C 304 SS
0 Herald, 43 C
/
o DFR, 250 C
,
v EBR-11, ROW7, 3-
I
EBR-11,
370C
Row5378c
I 1 Neutron energyflux,E$,
I’
/’
0’
0
/
,
/’ //
/G
I 2
3
(1~5MeV-n-cm~2-eec?)
FIG. 9. Effect of neutron energy flux on irradiated creep rate of annealed austinitic stainless steel.
236
Journal of The Franklin
Institute
Fatigue Behaviors of LMFBR
Pressure
Vessels
304 stainless steel in the Hanford Reactor, and K-II is the predicted rate in a Hanford reactor. The test in the Herald Reactor was performed on EN58B stainless steel in torsion. The tests in the Dounreay Faster Reactor (DFR) were conducted on 316 stainless steel, and the EBR-II were obtained from pressurized tubes of 304 stainless steel.) It is seen that the effective irradiation creep rate per unit effective stress increases monotonously with the average neutron energy flux within the range of various test results. The enhancement of irradiation creep in nuclear reactor materials was previously predicted in relation to inert-gas bubble formation and irradiation swelling occurred in the nuclear fuel (16). Very recently, experimental data for non-fissile materials (especially austinitic stainless steel) have proved that the enhanced irradiation creep in these materials is also related to the voids, dislocations, bubble formation and irradiation swelling. If there are two identical samples, and if one is tested under thermally activated creep and the other is tested under neutron-irradiated creep at the same conditions of temperature, pressure etc., the enhancement factor for irradiation creep may be determined from the samples. At the same time, the separation of temperature and neutron-energy-flux dependence may be carried out. VI.
Low-cycle
Thermal
Fatigue
Thermal cycling induced by nuclear reactor kinetics is an inherent characteristic of all nuclear reactors. Material fatigue can occur when a specimen undergoes repeated mechanical and/or thermal loading in the steady or transient state. Thermal fatigue is usually induced by thermal loading due to cyclic temperature. Low-cycle thermal fatigue (or thermal-cycling fatigue) is defined as the process whereby cracks which are incipient (microscopically or macroscopically) are propagated in materials which are subjected to cyclic temperature changes and undergo thermal strains (or stress) at elevated (or irradiation) temperatures. The behavior of low-cycle thermal fatigue in LMFBR pressure vessels is induced by the repetitive reactor period of the core. In the reactor core the neutron flux and power level always fluctuate randomly and cyclically. This fluctuating effect can be transmitted through the liquid sodium coolant to the pressure vessel (in either pipe type or tank type) ; as a result, the pressure vessel is subject to the cyclically dynamic and thermal loading. Figures 10 and 11 show the typical tangential stress distributions in the LMFBR pressure vessel for two consecutive cycles of reactor startup and shutdown (in the transient state). Figure 12 illustrates the behavior of the stress-strain loop (or U--Ecurve) in the pressure vessel during one thermalloading cycle with a hold period. Experimental data obtained from the fatigue behavior of AISA 4130 steel, and others, under strain cycling show that the relation between elastic strain per cycle ad and corresponding cyclic life (i.e. the number of cycles to failure) N, can be represented by (see Fig. 13)
-&?I
&d-E
Vol. 299, No. 2, February
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(7) 137
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M. Ma
-10
-2c
-30
0
1.02
l-04
1'06
Vessel thickness
1.08
1ilO
1.12
x(=/p/,,
FIG. 10. Tangential stress distribution in the pressure vessel at the first and second cycles of reactor start-up.
0> \
‘0 -
*’ /
o-
o\ 1-o
1.02
l-04
7.06
Vesselthickness,
l-as
1.10
l-12
X("dfi)
FIG. 11. Tangential stress distribution in the pressure vessel at the first and second cycles of reactor shut-down.
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Fatigue Behaviors of LMFBR
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t
Time,
t
Time,
Strain,
Pressure
E
FIG. 12. The behavior of the stress-strain loop during one thermal loading cycle with a hold period. where E is the elastic modulus and u0 and y are the elastic material constants. Equation (7) satisfies the cyclic life range up to lo6 cycles for a large number of materials tested. Similarly, test results observed for the fatigue behavior of annealed AISA 4130 steel and other similar materials indicate that the relation between plastic strain range per cycle cPz and corresponding cyclic life can be expressed as (see Fig. 14) &PI
=
mN;
(8) The value z = - + fits
in which m and z are the plastic material constants. most materials tested.
.u z
z
I
0~001 10
1111 I III1 lo2
I IIll
lo3 Cyclic life, cycles
FIG.
I lo4
to failure,
I III
III1 lo5
lo6
P$
13. Relation between elastic strain range per cycle and cyclic life (for annealed 4130 steel and others).
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Benjamin M. Ma
0-2
o-1
0.04 “9 c‘ ‘iL t; .o B _m n_
0.02
0.01 0.008 0.006 o-004
0.002
0.001 00
lo2
lo3
lo4
Cyclic life,cycles to failure,
N,
FIG. 14. Relation between plastic strain per cycle and cyclic life (for 4130 steel and others).
In many practical cases, the total strain range per cycle E (or he) is of interest. Therefore, the relation between total strain range and cyclic life can be obtained from Eqs. (7) and (8) E = A& = eel+aP1 = ONu+mNj E t
(9)
where AE is the finite strain range between any two consecutive cycles at the asymptotic condition, as shown in Fig. 15. The principal need in pressure vessel design analysis and the computation of low-cycle thermal fatigue is a relation between stress range Aa and total strain range As. Such a relation can provide a means for combining the strain compatibility and stress equilibrium equations together for design analysis. Eliminating N, from Eqs. (7) and (8) and combining them with Eq. (9) yield
(10) where Aa is usually taken as the asymptotic stress range of the cyclic half-life (NJ2) of a specimen. From the logarithmic plots of elastic strain
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Fatigue Behaviors of LMFBR
Cyclic life,cycles to failure,
Pressure Vessels
N,
FIG. 15. Total strain range (the sum of elastic and plastic components) varies with cyclic life.
range and plastic strain range vs. cyclic life, the elastic and plastic material constants CT,,,y, m and x of the material can be determined experimentally. Hence, the strain range, stress range and cyclic life can be predicted from Eqs. (7)-(10).
VII.
Interaction Between Fatigue, etc.
Irradiation
Creep
and Low-cycle
Thermal
Operating conditions of LMFBR pressure vessels under fast neutron irradiation usually involve complex load and temperature variations (i.e. irradiation creep, cyclic thermal load, sodium hydrostatic load, etc.). The random process and time history (derived from reactor kinetics) of the resulting stresses and strains are even more complex, and differ for each part of a pressure vessel. The random process and time history of loading fluctuations and temperature variations are statistical phenomena in nature. The effects of time on the behavior of creep and fatigue, however, are referred to as the interaction between the irradiation creep and the low-cycle thermal fatigue (or, simply, creep-fatigue interaction) of the LMFBR pressure vessels. The creep-fatigue interaction relationship may be expressed (in terms of damage fractions D and interaction parameters) as D = a(DoJU+ b(D,)”
(11)
D, s t/te, D, s N/Nf,
(12)
and where D, is the creep damage, D, is the fatigue damage, D is the total damage, t is the time of creep at effective stress (Jo, t, is the creep rupture time at (TV, N is the number of cycles at the strain range AC, N, is the number of cycles
Vol. 299, No. 2, February
1975
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Benjamin M. Ma to failure (as defined), and a, b, u and v are the interaction parameters. When a failure criterion of D = 1 is assumed for the total damage, and D, and D, are introduced from Eq. (12), the failure criterion for the creep-fatigue interaction in the pressure vessel becomes a(t/t,)” + b(N/N,)V = 1.
(13)
There are four possible cases: (1) For the creep failure only, a = u = 1, b = 0, t = t,. (2) For the fatigue failure only, b = v = 1, a = 0, N = N,. (3) For the combination of the creep and the fatigue a + 0,b f 0, u, v < 1. (4) For a linear combination of the creep and the fatigue u = v = 1. The possible forms of interaction and design curves for various values of a, b, u and v are plotted in Fig. 16. It is expected that most of the design values and experimental data for the creep-fatigue interaction will fall into region 3 of the curves.
U
1 Fatique damage,
N/N,
FIG. 16. The interaction curves of irradiation creep and thermal cycling fatigue: (1) For the creep only, a =u = 1, b = 0. (2) For the fatigue only, b = v = 1, a = 0. (3) For the creep-fatigue interaction, u, 2)f 1. (4) For the linear creep-fatigue interaction. u = v = 1.
In the same manner, the interactions between irradiation creep and irradiation swelling and low-cycle thermal fatigue and irradiation swelling can be written as a(t/tJ”
(14)
f c(E/E,)~ = 1,
b(N/N,)” + c(&,)~
= 1,
(15)
where E is the irradiation swelling strain ( 2: )AV/V), as is the irradiation swelling strain to failure (produced by a neutron fluence of about 1022-1024 n
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Vessels
cm-2), and c and w are other parameters in question. From Eqs. (14) and (15) the interaction and design curves for creep-swelling and fatigue-swelling, respectively, can be plotted as shown in Fig. 16 above. Since the irradiation swelling, irradiation creep and low-cycle thermal fatigue can occur simultaneously in the LMFBR pressure vessels, the failure criterion of the pressure vessels can be obtained by combining Eqs. (13)-( 15) to yield a(@,)” + b(N/N,)” + ~(E/E,)~= 1.
(16)
From safety surveillance of an LMFBR pressure vessel during reactor operation, the variables and parameters given in Eq. (16) can be obtained, and the failure criterion may be verified experimentally.
VIII.
Interrelationships Fatigue Behaviors
Between Zrradiation
Swelling,
Creep
and Thermal
The irradiation creep developed in the LMFBR pressure vessels during normal operation combined with inert gas (mainly helium) production, diffusion and swelling in the vessel material (material properties also change) is closely related to low-cycle thermal fatigue. The interrelations between void, dislocation and gas-bubble formation, irradiation swelling and irradiation creep induced by fast neutron fluence, irradiation hardening and helium diffusion and embrittlement at the micro- and macro-cracks in the pressure vessel can cause fatigue failure through brittle fracture. In other words, irradiation hardening and helium production, diffusion, swelling and embrittlement can spread cracks and accelerate thermal cycling fatigue to cause fracture failure of the LMFBR pressure vessel. Irradiation creep can produce stress relaxation (or stress relief.) (16). Correlations of irradiation creep for annealed austinitic stainless steel have been associated with temperature dependence, stress (or strain) state, neutron energy flux (or spectrum) and neutron Auence. That irradiation creep rate decreases with increasing temperature (as opposite to an observed increase in thermally activated creep with increasing temperature) can be attributed to the annealing effect on irradiation-produced defects, as predicted. The transient creep can be attributed to the effects of thermal cycling and dislocation climb caused by the absorption of excess point defects. The transition from transient creep to the steady creep is compatible with the creep strain recovery which follows a stress reduction. Similarly, both thermal creep and irradia,tion creep can produce stress relaxation (or stress relief,) in materials under consideration. Since interstitial atoms are preferentially absorbed at dislocations, the surviving vacancies and vacancy clusters resulting from neutron irradiation may also give rise to microscopic volume increases (in addition to the inert-gas swelling). Moreover, a large number of the interstitials can be absorbed at dislocations due to the tensile stress produced by irradiation creep and cyclic thermal loading. Thus, irradiation swelling may also be
Vol. 299, No. 2, February
1975
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Benjamin M. Ma enhanced by the tensile stresses under consideration. Therefore, the total volume increase in the pressure vessel can be attributed to (a) the inert gas swelling, (b) the microscopic volume increases due to vacancies and vacancy clusters and (c) the tensile stress produced by irradiation creep and cyclic thermal loading. On the other hand, thermal cycling may sweep up and absorb interstitials and vacant point defects by oscillating dislocations. Large amplitude lowcycle thermal fatigue strains may absorb vacancies and voids and reduce bubbles and irradiation swelling. The possible modification of the swelling in the LMFBR pressure vessel by irradiation creep and thermal cycling fatigue (and vice versa) may be verified experimentally. The apparent effect of hydrostatic tangential tension (or hoop stress) on the dimensional change of the pressure vessel is, however, difficult to account, for. That is, the liquid sodium coolant, deposited in the pressure vessel, apparently produces hydrostatic pressure and tangential tensile stress, which cause a volume increase in the pressure vessel. Although the hydrostatic pressure of liquid sodium might reduce irradiation swelling somewhat, the hydrostatic tangential tension can unfavourably accelerate the irradiation creep and thermal cycling fatigue of the pressure vessels (this may be more severe to the tank-type LMFBR pressure vessels). The occurrence of micro- and macro-cracks in, and changes in the volume, shape and material properties of, the LMFBR pressure vessels are expected to vary with vessel materials and operation conditions. Any success of this correlation should be attributed to the fine structural, mechanical and nuclear properties of annealed austinitic stainless steels. Considerable experimental data, theoretical work and operation experience are needed in order to understand the interrelationships between the irradiation swelling, creep and thermal fatigue behaviors of the LMFBR pressure vessels.
IX.
Conclusions
From the preceding analysis of the behaviors of irradiation swelling, irradiation creep and low-cycle thermal fatigue in the LMFBR pressure vessels, the following conclusions are drawn: (1) The pressure vessel will suffer from severe radiation damage by fast neutrons, slow neutrons, primary gamma rays and secondary gamma rays. The radiation damage can change the vessel material properties appreciably. (2) The void density, mean void diameter and volume (or density) changes in the pressure vessel vary with the neutron fluence and irradiation temperature for a given neutron energy spectrum (Eqs. (l)-(4) and Figs. Z-5). The irradiation swelling breakaway in the pressure vessel may occur when the irradiation temperature is about half the melting point temperature of the vessel material. (3) Irradation swelling can highly enhance irradiation creep rate in a pressure vessel irradiated by neutron fluence (Eq. (5) and Fig. 7). The effective creep rate decreases with increasing irradiation temperature (in
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Fatigue Behaviors of LMFBR
Pressure
Vessels
contrast to the thermally activated creep) which is desirable for LMFBR pressure vessels to be operated at relatively high temperatures. The effective creep rate, however, is increased with the average neutron energy flux (Figs. 8 and 9). (4) Irradiation hardening and helium production, diffusion, swelling and embrittlement can spread cracks and accelerate thermal cycling fatigue (low-cycle thermal fatigue) to cause fracture failure of the LMFBR pressure vessels. (5) From the behaviors and interactions of irradiation swelling, irradiation creep and thermal cycling fatigue, the failure criterion for the design and safety surveillance of the pressure vessels can be expressed as (Eq. 16 and Fig. 16) a(@,)” + b(N/Nf)w + c(E/E,)~= 1. The tangential tensile strains (or hoop strain) produced by the irradiation creep, thermal cycling fatigue and hydrostatic pressure of the liquid-sodium coolant may be combined as a whole with the irradiation swelling strain E,. References (1) J. H. Wright, “Design philosophy for the Westinghouse fast breeder reactor”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. and evaluation of the combustion engineering (2) R. C. Noyes, “Development advanced lOOO-MW, LMFBR design”, Proc. International Conference on Sodium Technology and Large Fast Reactor Des., Argonne National Lab. Rep. ANL-7520, Part II, pp. 291-230, 1967. (3) R. Balent, “Liquid-metal fast breeder reactors”, Proc. Reactors and the University Conf., Rensselser Polytechnic Institute, Aug. 28-30, 1968. (4) A. Leipunsky, “Operating experience with BR-5 Reactor”, also, “BN-350 nuclear power plant construction”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (5) K. Gast, “NA-2 design 300 MW, German FCR”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (6) B. Wolfe and W. J. Clabaugh, “Preliminary work on a 300 MW, fast ceramic demonstration reactor”, Proc. Fast Reactors, National Topical Meeting, SNS-101, American Nuclear Society, San Francisco, 1967. (7) M. W. Croft, “Babcock and Wilcox 1000 MW, LMFBR follow-on study reference design”, Proc. Int. Conf. on Sodium Tech. Large Past Reactor Deisng, Argonne National Lab. Rep. ANL-7520, Part II, pp. 215-239, 1968. (8) G. Vendryes, “The Phenix reactor (French FCR)“, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (9) A. G. Frame, “Status of the UKAEA prototype fast reactor”, Proc. Fast Reactors, National Topical Meeting, ANS-101, American Nuclear Society, San Francisco, 1967. (10)C. Cawthorne and E. J. Fulton, “The influence of irradiation temperature on the defect structures in stainless steel”, Report AERE-R-5269, Atomic Energy Research Establishment, Harwell, 1967. Also “Voids in irradiated stainless steel”, Nature, Lond., Vol. 216, pp. 575-576, 1967. (11)B. M. Ma, “Thermal, radiation and mechanical analysis for fact reactor fuel elements”, Proc. First Int. Conf. on Structural Mech. in Reactor Tech., Vol. 2, C4/3, Berlin, West Germany, 1972.
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Benjamin M. Ma (12) H. R. Brager, “Irradiation produced defects in austinitic stainless steel”. USAEC Rep. WHAN-FR-16, WADCO, 1970 (13) T. T. Claudson, R. W. Barker and R. L. Fish, “The effects of fast flux irradiation on the mechanical properties and dimensional stability of stainless steel”, Nuclear Appl. and Tech., Vol. 9, No. 1, p. 10, 1970. (14) C. K. Cheng and B. M. Ma, “Thermal, radiation and mechanical analysis for unsteady-state fuel restructuring of cylindrical oxide elements in fast reactors”, Nucl. Sci. Engng, Vol. 48, pp. 139-153, 1972. (15) B. M. Ma, “Heat generation and temperature distribution in cylindrical reactor pressure vessel”, Nucl. Engng and Design, Vol. 11, pp. 1-15, 19’70. (16) B. M. Ma, “Radiation and creep analysis for strain and stress distributions in tubular fuel elements”, Nucl. Sci. Engng, Vol. 20, pp. 536-546, 1964. (17) D. Mosedale, “Irradiation creep in the Dounreay Fast Reactor (DFR)“, Nature, Lo&., Vol. 224, p. 301, 1969. (18) E. R. Gilbert and N. E. Harding, “Irradiation-induced creep in austinitic stainless steel”, Trans. Am. Nucl. Sot., Vol. 13, No. 1, pp. 143-144, 1970. Also E. R. Gilbert and L. D. Blackburn, “Irradiation-induced creep in austinitic stainless steel”, USAEC Rep. WHAN-FR-30, WADCO, 1970. (19) A. J. Lowell and R. W. Barker, “Uniaxial and biaxial creep rupture of type 316 stainless steel after fast reactor irradiation” , USAEC (Hanford Eng. Dev. Lab.) Report, WHAN-FR-30, WADCO, 1970. (20) R. V. Hesketh, “Collapse of vacancy cascades to dislocation loops”, Proc. Int. Conf. on Solid-state Physics Research with Acceleraters, Brookhaven National Lab. Rep. BNL-50083, 1967. Also “Irradiation creep in non-fissile metals”, Brittleness and Irradiation Effects, 10th Symp. on Special Metallurgy, Saclay, France, p. 185, 1967. (21) B. M. Ma, “A creep analysis of rotating solid disks”, J. Franklin Inst., Vol. 267, No. 2, pp. 149-165, Feb. 1959. (22) M. N. McElroy, R. E. Dahl, Jr. and E. R. Gilbert, “Neutron energy-dependent damage function for analysis of austinitic steel creep data”, Batelle Northwest Lab. Rep., BNWL-SA-3186, Richland, Wash., March 1970. (23) D. D. Walters, “EBR-11 in-pile creep experiments on stainless-steel tubing”, Trans. Am. Nucl. Sot., Vol. 13, p. 146, 1970.
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