Integrity degradation of UO2 pellets subjected to thermal shock

Journal of Nuclear Materials North-Holland, Amsterdam

INTEGRITY DEGRADATION Masaomi

67

127 (1985) 67-76

OF UO, PELLETS

SUBJECTED

TO THERMAL

SHOCK

OGUMA

Nippon Nuclear Fuel Development Co., Ltd., Narita - cho, Oarai - machi, Ibaraki ken, 31 I 13, Japan Received

23 February

1984; accepted

23 July 1984

Thermal shock behavior of UO, pellets has been investigated by means of out-of-pile experiments and a theoretical analysis which particularly emphasized the porosity effect on the thermal shock damage. In the experiments, specimens of porosity range 0.05-0.15 were thermal-shocked by heating and then quenching in a water bath at various quenching temperature differences (AT). Results showed that with increasing porosity, AT values cause a first damage (AT) and bring about destructive failure of the specimens increased, while the strength loss at AT, was reduced. These findings suggested that the higher the porosity, the higher the pellet integrity during rise to power. Theoretical equations expressing the thermal shock damage were introduced. Good agreements were obtained between observed and predicted values.

1. htroduction

Fracture of fuel pellets has far reaching effects on the mechanical and ther-,nal performance of fuel rods, since it can be a cause of pellet-cladding mechanical interaction (PCI) and stress-corrosion cracking of zircaloy cladding (SCC) [l]-161. In a BWR, reactor power is raised by drawing control rods up to about 60% of the full power. During the power rise, the local power increasing rate of the fuel rods is more than 1 x 10’ W/cm/h. By the time the power reaches about 20% of the full power, the thermal stress generated in the fuel pellets equals the fracture strength of sintered UO,. Therefore UO, pellet fracture occurs due to thermal shock during the first rise to power. For this reason, it is important to understand the damage behavior of UO, pellets under actual thermal conditions to evaluate the fuel performance. However, thermal shock behavior of UO, pellets has seldom been considered, except in a few related studies which dealt with thermal shock behavior of BqC pellets used for control rods [7-91 and thermal shock analysis of fuel rods in transient operations [lo]. The purpose of this study was to elucidate damage behavior of UO, pellets subjected to the thermal shock during rise to power. Out-of-pile experiments and a theoretical analysis were carried out, in particular emphasizing the porosity effect on thermal shock behavior of UO, pellets, since this factor has already been reported to have a strong effect on pellet fracture [ll]. 0022-3115/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

The method of thermal shocking was the commonly used procedure [12] of quenching heated specimens in a water bath. The degree of damage to the specimens was evaluated in terms of a change in fracture strength before and after the thermal shock experiments. Equations expressing the thermal chock damage of UO, pellets were introduced after theoretical considerations based on the Griffith theory [13] and the Hasselman unified theory (141. Comparison of calculated and observed values as related to thermal shock damage of UO, pellets was also made.

2. Theoretical

analysis

UO, pellets and other ceramics are materials with a certain amount of porosity and incipient cracks as represented by pores and grain boundaries. In this section, the thermal shock damage of the UO, pellets is discussed in terms of microstructural parameters based on the theories of Griffith [13] and Hasselman [14]. 2. I. Critical temperature

difference

Assume that a pellet with porosity P, contains N number of initial cracks per unit volume; and all initial cracks are of uniform size, with a radius of I,, and uniform distribution throughout the pellet. In the following discussion, the absence of external forces is always assumed. On fracturing of the pellet under therB.V.

mal shock condition the total energy per unit volume, W, is the sum of the elastic energy, Wr, being stored in the pellet plus the fracture energy of the cracks, Wp. And Wp is also the sum of the strain energy, W,, associated with the region if there is no crack, plus the strain energy, W,, due to the extension of the cracks. Using the theoretical solutions of Walsh’s study [l5] on the compressibility of rock, W, is given by: WC=

g&J* 16(1- ~‘1N13 9(1-2~)

’

where v is Poison’s ratio; (I, the thermal stress; I,, the radius of the initial crack; & the effective compressibility given by p = 3(1 - 2v)/E,, and E, is Young’s modulus of a crack free peilet. The term NI,, on the righthand side of eq. (l), is proportional to the crack volume per unit volume, i.e. porosity. If the cracks are assumed to be spherical, NI, becomes equal to 3P/&r. Ws and Wg are also given as follows [l&17]: W, = f@o’,

Wg = 2rrGNf,1

i

3a(l-2V)

(6)

i

Eq. (6) indicates that the shorter the initial crack length, I,. and the higher the porosity, P, the higher is AT,. 2.2. New crack length at critical temperature d~jjeren~~ After initiation of fracture, the rate of elastic energy release exceeds the fracture energy. This excess energy is transformed into kinetic energy to propagate the cracks. The cracks continue to spread until the potential energy released equals the total fracture surface energy. Thus the final crack length, I, at A c is calculated from eq. (5) by equating the energies of the system at the initial and the final stages, W( I,,) = W( I,), and by rearranging I, =

(2)

where G is the fracture surface energy. On the other hand a sudden transient temperature difference (AT) between the surface and center of the pellet leads to the maximum tensile thermal stress (I in the pellet surface expressed by o=o*&Ar where o* is the nondimensional thermal stress which indicates the severeness of the thermal shock and is given by [18] u* = (1+4/B)-’

1+4(1-v2P

x

(7)

2.3. Residual fracture strength The fracture strength of the pellet after fracture at AT,, which is hereafter defined as residual fracture strength, S,, can be given by substituting f, obtained from eq. (7) into the Griffith relation [13], sa =

16(1 - 2v) NE2G3 3n(l

- v”)“( aAT,)‘&,

(4)

where B is the Biot Modulus B = hy,/K, and ya is the radius of the pellet; h, the heat-transfer coefficient; and XI, the thermal conductivity. Rearranging eqs. (1) to (4), the total energy per unit volume stored in the pellet ( W) can be calculated as

l/2

1’4

Eq. (8) indicates that the higher the porosity, the higher S,, so that strength loss at AT, (initial fracture strength minus S,) is gradually reduced as porosity increases. 2.4. Initial fracture strength

+ 2 vrGN1;.

(5)

The cracks are unstable under the condition of dW/df I 0. Applying this condition to eq. (5), followed by rearrangement, results in the expression for critical temperature difference, AT,, required for crack instability [13,19]:

AT,=

rrG(1 - 2~)~ ZE,ty’(l

-v’)

The initial fracture strength of the pellets, So, can be obtained by making a porosity correction for the Griffith relation [13]. An increase in porosity is simply regarded as a reduction of the effective or critical load-bearing area within the pellets. Consequently, the porosity correction is the value which expresses the relation between porosity and the effective load-bearing area of the pellets. Knudsen [20] used theoretical considerations of an idealized packing arrangement of spherical gains to show that the effective load-bearing

69

M. Oguma / Integrity degradation of UO, pelleis area varied exponentially with a linear change in porosity. Thus S,, can be represented by

Table 1 Microstructural data Avg. Avg.

Oguma [ll] investigated the relation between fracture strength and microstructure of UO, pellets whereby the porosity correction C, in eq. (9) was obtained as C, = -5.7. He also found that the initial crack which controlled UO, pellet strength was considered to be both the large pore and the grain boundaries associated with the large pore. Then I, in eq. (9) can be expressed by I,=+(PS++GS)

(10)

where PS is the large pore size and GS the mean grain size. Substitution of eq. (10) into eq. (9), results in the initial fracture strength of UO, pellets

(11) 3. Experimental 3.1. Specimen preparation UO, powder and naphthalene, as a pore forming material, were ball-mixed and then cold-pressed into green pellets at a load of about 1 x lo5 N. The amount of the naphthalene was varied in order to obtain various specimen porosities. The green pellets were fired at 1700°C for 2 h in a reduction atmosphere (984&N,2WH,). The sintered specimens were pellets (diameter: 15 mm; length: 8 mm) and essentially stoichiometric, O/U = 2.00 + 0.001. the porosities of the specimens were calculated from their density values obtained by the immersion method using the relation, P = 1 - D/D,, where D is the specimen density and D, the theoretical density of UO, (D, = 10.96 g/cm3). Specimens with porosity of 0.05, 0.10 and 0.15 were selected for the study. Information on specimen microstructures was obtained from disk specimens sliced form both ends of the pellets. These disks were mounted in an acrylic resin, polished, etched in a mixture of HNO,, H,O, and water, and then microphotographed at magnitudes of x 100 to x 400. The mean grain sizes were determined from the microphotographs of etched specimens using the conventional linear-intercept technique. The pore size distributions were analyzed from the microphotographs of as-polished specimens using an image analyzer. The pellet microstructural data are listed in table 1.

of

porosity largest pore size km)

Avg. pore density Parelm2) Avg. grain size bm)

UO, pellet specimens

I

0.05

I

0.10

I

0.15

I

1

50

1

42

1

37

1

/ 3x10s

I 4x10s

I

1

1

) 2X108 1

13

12

10

3.2. Thermal shock experiment The thermal shock experiments were made by holding the specimens for 20 min in a vertical furnace maintained at a predetermined temperature and subsequently dropping them into a water bath at room temperature. The severity of the thermal shock was controlled by adjusting the furnace temperature. A high purity inert gas from which oxygen and moisture had been removed by a gas purifier was fed into the furnace at an adequate flow rate during specimen heating. (A set of preliminary experiments in which the oxygen-touranium ratio (O/U) had been measured on UO, pellets heated in the same atmosphere revealed that the atmosphere did not cause any change in specimen composition.) The furnace temperature was raised at a slow rate of S”C/min to prevent thermal-stress fracture of the specimens during temperature increases. Both ends of the specimens had Al,O, dummy pellets attached by alumina cement, so that on quenching the reduced heat flow through the ends of the specimens would give only radial temperature distributions that closely approximated those expected in actual fuel pellets in a reactor. This was supported by temperature distribution analyses in specimens with dummy pellets during the quenching process, as determined using a two-dimensional FEM transient analysis code. Radial temperature distributions at thermal shock temperature differences of 300 and 500°C are shown in fig. 1 as analysis examples. After thermal-shocking, the specimens were taken out of the water bath and the Al,O, dummy pellets were removed. The degree of damage due to the thermal shock was evaluated by the change in fracture strength of the specimens before and after the thermal shock tests. The specimens were sliced into “disk” specimens, each approximately 2 mm thickness. The fracture strength of the specimens was determined at room temperature using a biaxial flexure technique [ll] in which the disk specimens were mounted peripherally on a supporting fixture and loaded centrally by a concentric ring tool. In the biaxial flexure test, the maximum equibiaxial tensile stress is developed on the bottom

M. Oguma / Integrity degradation of UO, pellets UO, pellets and were notched at the center by using a 0.1 mm thick diamond wheel to a predetermined depth which gives a notch depth to specimen depth ratio of 0.2. The average notch width was approximately 0.15 mm. The specimens were loaded to failure in the 3-point bending mode at room temperature. A deformation speed of 0.05 mm/min was used for all the tests. The critical stress intensity factor, K,,, is given by [22] K,, =

Y$$E

(13)

where F is the peak load to failure; L, the support span length; b, the specimen breadth; W, the specimen depth; C, the notch depth; and Y, the shape factor given by 1231 I

OO

0.5

I i

Distance from pellet center (normalized radius 1 Fig. 1. Temperature distribution of UOz pellets in thermal shock experiments. One minute after quenching in a water bath.

face in the central area of the disk specimen,

surface energy, G, is obtained

as 1241

expressed

by

“‘=~((‘-‘)~~+~l+.)I,g)

Fracture

(13)

where of is the fracture strength; P, the load al fracture; a, the upper ring tool radius; d, the lower supporting fixture radius; b, the specimen radius; and h, the specimen height. All the specimens were loaded at a constant loading rate of 5 Nfmin. When cracks resulting from thermal shock were visible on the surfaces of the disk specimens, crack patterns were examined by optical microscopy prior to the fracture strength measurements. Fracture surfaces of the specimens broken in the fracture strength measurements were scanning electron micrographed.

(ii) Indentation technique [25,26] Cubic specimens (5 X 5 X 5 mm) were cut from UO, pellets and one surface was polished flat and smooth with a series of polishing papers. Indentation was carried out at temperatures ranging from RT to 1OOO’C using a Vickers diamond pyramid indenter and microhardness tester for a predetermined load of 0.5 kg. The dimensions of the indentation and radial cracks were carefully measured by means of an optical microscope, and X,, was obtained by 123,271 0.4

exp(2.303Y) where H is the Vickers hardness; and a, the semi-diagonal of the indentation. The factor Y is given by [23] Y = - 1.59 - 0.34x - 2.02x2 + 11.23x3 - 24.79x4

3.3. Measurements of physical properties Characteristics of UO, pellets depend strongly upon the microstructure of the pellets [11,21]. Thus major physical properties which would affect thermal shock characteristics were measured for pellets selected from the same lot of UO, pellet specimens used for the thermal shock experiments. 3.3.1. Fracture surface energy (i) Single notched beam technique Plate specimens (4.5 x 8 x 15 mm) were cut from

+16.32x5,

wherex=log(c/a).

Fracture surface energy, G, can be obtained stituting K,, into eq. (14).

by sub-

3.3.2. Thermal expansion Thermal expansion was measured on a plate specimen (5 x 5 x 8 mm) using a high-temperature microscope. These specimens were also cut from the UO, pellets. A transducer was attached to each specimen mounted on the sample stage and then the temperature was raised by steps up to 1000°C in a vacuum. During

IU. Oguma / fntegrity degradation of V@ peilets the rise to the desired temperatures,

the cross-hairs in the microscope were adjusted to a predetermined position on the specimen by shifting the sample stage. The thermal expansion of the specimen at every temperature step was obtained from the transducer output with an accuracy of about l/1000 mm. 3.3.3. Thermal conductivity The thermal diffusivity, D,, was measured on a disk specimen by the laser flash technique at temperatures ranging from RT to 1OOO“C under a vacuum atmosphere. The thermal conductivity was then obtained by substituting D, into the following relation, K = C. D, . Cr, *D, where Cp is the specific heat; D the specimen density; and C a constant. The Cr, value used was that proposed in MATPRO-V09 [ZS]. 4. Results and discussion 4.1. Morphology of thermaf shock damage Typical thermal shock damages of the specimens with a porosity of 0.05 are shown in fig. 2 for different thermal shock temperature differences, AT. At low AT, a small number of straight cracks runs from the periphery toward the center of the specimen, while at high AT a large number of irregular cracks, (i.e. crooked) are generated over a wide area inside the specimen. For the former AT, the region where thermal stress generated in the specimen (a) exceeds the fracture strength of

71

UOz(or) is limited to within a narrow area in the periphery of the specimen, and the pre-existing large pores in the region are considered to be the initial cracks. The failure therefore starts at these large pores which play the role of nuclei for propagation. On the other hand since the region of a < at extends over a wider range to the specimen center as AT increases, not only the large pores but also grain boundaries for which surface fracture energy is relatively low, can be regarded as the initial cracks. Therefore, many cracks propagate coincidently from a large number of crack nuclei. Fig. 3 shows two scanning electron micrographs of fracture surfaces for specimens with a porosity of 0.1 fractured at (a) AT = 130°C and (b) AT= 55O’C. The thermal shock cracking is predo~n~tly tr~s~an~~ fracture at low AT (fig. 3(a)) and intergranular fracture at high AT (Fig. (b)). Fig. 4 compares typical crack patterns observed in a specimen subjected to thermal shock at AT = XWC and an actual fuel pellet which has experienced a similar severity of thermal shock during rise to power. The specimen’s crack pattern which consists of a few straight cracks toward the center and a number of thin cracks in the periphery is closely similar to that observed in the actual fuel pellet. 4.2. Thermal shock temperuture strength

difference and fracture

Fig. 5 shows the relationship between AT and fracture strength of the specimens with a minimum porosity

Fig. 2. Thermal shock temperature difference (AT) influence on crack patterns of UOz pellets, (a) AT = 200°C and (b) AT = 8OO’C.

72

M. Oguma / Integrity degradation of c/O, pellets

Fig. 3. Scanning electron micrographs of fractured surfaces of UO, pellets quenched at different thermal shock (AT). (a) AT = 130°C and (b) AT = 550°C.

temperature

differences

(P = 0.05). Over the range 0 I AT < lOO”C, the specimens exhibit the initial fracture strength suggesting that thermal-shocks over this AT range have no effect. As AT increases to approximately lOO”C, the fracture strength suddenly falls off. This sudden loss in strength is due to formation of new cracks resulting when the elastic energy at AT exceeds a critical value. Thus this AT corresponds to the critical temperature difference AT,. The fracture strength retained, S,, is about one-half of the initial fracture strength (S,,). This suggests that

once fracturing initiated a significant amount of damage is sustained by low porosity pellets. The residual fracture strength at AT, is kept constant through a small range 100 s AT < 140°C. Since the elastic energy stored on fracturing is proportional to .!$ [17,29], for the specimens with low porosity and hence high S,, the rate of elastic energy release at fracture initiation is in considerable excess of the surface fracture energy. This excess energy is transformed into the kinetic energy of crack propagation. Thus the crack length instanta-

Fig. 4. Comparison of crack patterns in UO, pellets (a) quenched into water (AT = 550°C) and (b) irradiated in a reactor.

Al. Oguma / integrity

73

degradation of lJC3, pellets

i i?

72 t

/ -temper-at&e difference; Critical

Thermal shock temperature difference (“C 1 Fig. 5. Change

in fracture

strength

of U02 pellets (porosity

= 0.05) subjected

neously changes to a new value, as given by eq. (7). The new crack length is now subcritical to AT,, so that AT must be increased to a new value before the cracks again continue to propagate. As a result, in the range 100 5 AT < 140°C, no change in strength is observed. At AT 2 140°C, the fracture strength decreases gradually with increasing AT, suggesting that crack growth occurs quasi-statically over the AZ’ range. At AT =

Thermal Fig. 6. Change

in fracture

shock

strength

temperature

difference

of UO, pellets (porosity

to thermal

shock at various temperature differences.

400°C, the fracture strength is S = 0, which means the specimens are completely fractured. Fig. 6 shows the relationship between AT and S of the specimens with an intermediate porosity value (P = 0.10). As porosity increases, the thermal shock behavior differs as compared to that observed in the lower porosity specimens. With higher P, the strength loss at AT, is reduced and the AT at which S equals zero is higher. These features become

(“C )

= 0.10) subjected

to thermal

shock at various

temperature

differences.

M. Oguma/ Inlegrirydegradationof UO, peliels

Thermat shock temperature difference (“C 1 Fig. 7.

Change in fracture strength of UQ, peueC.s(porosity ;= 0.15) subjected to thermal shock at vatiaus temperature differences.

even more prominent in the specimens having the highest porosity (P = 0.15) as shown in fig. 7. Though S, de-

4.3. Comparison oj calculated and observed values

craves with increased porosity, i.e. changing by about a factor of two over the porosity range 0.05-0.15, the strength loss at LIT, a&wdecreases, so that the difference

Values of S,, AT, and S, which characterize thermal shock behavior of UO, pellets are calculated using the relations introduced in section 2 for comparison with values observed in the thread shock experiments. In the calculations, the measured values are used for the material properties of G. K and a as iltustrated in table 2, while the literature values [ZS] are used for other material properties such as E, v, etc. The initial crack length of UO, pellets, I,, is calculated by substituting the microstructural data illustrated in table 1 into eq, (IO). The results of the comparison are summarized in

between S,, and S, essenti&y vanishes for the specimens with P = O.IS. The AT, for the specimens of P = 0.15 is approximately 130°C. white the AT, is about KWC for the low porosity specimens (P = WE), and the former specimens still have a low strength value after thermal shocking at AT- 600°C. as shown in fig. 7, suggesting that the UO, pellet retains its integrity under severe thermal shock condition.

Table 2 Properties of UO, pellets used in thermal shwk analysis (J~m2~ :G=d

Fracture surface energy

~Wlm”C):K~(0.117+2,65XlO-Y) f2.14X10-“(T+273\3

Thermal conduct~vj~

(tl”C)

Thermal expansion

(Pa)

Young’s modulus

(*I)

:E=2.26Xi0~i(1-1.131X10-4T~ (I-2.62P3

($2)

(W/m2‘Cl : h = 4. ?87 X V_?

( * 1) Measured value ( * 2) Reference No. 26 i * 31 Reference No. 18

i*tj

:~=8.37X10-*+4.06X10-~

: v=o.3tFJ

Poisson’s ratio Surface heat transfer cc&f.

i*il

T : Tempwiture % 1 P : Porosity

( *21 ( *3)

M. Oguma / Integrity degradation of UO, pellets Table 3 Comparison

of observed and calculated values

* Initial crack radius =- i [Largest pore size+$Avg.

grain size)

table 3. It can be seen from the table that the observed data support broadly the theoretical predictions that the higher the porosity, the higher the AT,, the lower the S,, and the smaller the difference between S,, and S, at AT,. Comparing the calculated and observed values of S,,, AT, and S, under the same porosity level, a slight discrepancy comes up between them. This can probably be attributed to the fact that the calculated values were obtained by using the averaged initial crack length derived from only one cross section of each specimen, however, in practice, the values of S,,, AT, and S, are strongly affected by a few large pores and/or rnicrocracks existing in the specimens. The agreement between calculated and observed values can thus be considered good when these points are borne in mind. The effect of porosity on thermal shock damage of UO, pellets is shown in fig. 8, in which the degree of thermal

51

0 measured

0 .E ‘2 ‘a:,0

1 00

0.05

I

I

0.10

0.15

Porosity Fig. 8. Relation between thermal shock damage and porosity of UO, pellets.

I

shock damage is expressed in terms of the fractional change in strength (SJS,,) at AT,. It seems that the calculated values agree well with the observed values, though there exists a small tendency for the calculated values to deviate from the observed values in the low porosity region. Fig. 8 confirms that SJS,, increases with increasing porosity suggesting that the higher the porosity, the lower the thermal shock damage. From the above results, it can be concluded that the thermal shock damage of UO, pellets during rise to power can be reduced by increasing porosity, thus lowering the density of the pellets.

5. Conclusion

Out-of-pile experiments and a theoretical analysis were performed to elucidate the thermal chock damage of UO, pellets during rise to power. In the study, a particular emphasis was laid on the effect of porosity, since this is critical to the understanding of pellet fracture behavior. The method of thermal shocking used in the experiments was that of quenching heated specimens (porosity range 0.05-0.15) into a water bath. The critical temperature difference, AT,, and AT at which pellets sustained destructive damages increased and catastrophic failure at AT, was reduced with increasing porosity. These results suggested that low density pellets possessed higher thermal shock resistance, and thus higher pellet integrity during rise to power, than high density pellets. Equations representing thermal chock damage of UO, pellets were introduced based on theoretical considerations. The experimental values supported well the values calculated under assumptions that the large pores and the grain boundaries

M. Oguma / Integrity degradation of c/O, pellets

16

associated with the large pores initial cracks in UO, pellets.

were regarded

as the

Acknowledgment The author wishes to thank Messrs. H. Masuda, S. Ando, I. Tanabe for their assistance in the preparation of the specimens and also the performance of the thermal shock experiments, and Dr. T. Murata for his encouragement of the work.

References PI Y. Mishima

et al., Trans. Am. Nucl. Sot. 20 (1975) 222.

121 K. Ito, et al., Res. Mechanics, 2 (1981) 109. 131 M. Oguma, Nucl. Eng. Des. 76 (1983) 35. [41 B. Brzoska eta 1.. Trans. 5th SMIRT, D2/1 (1979). I51 D.D. Lanning, Nucl. Technol. 56 (1982) 565. (1980). PI A.D. Appelhans et al., NUREG/CR-1425 [IIS. Sato et al., Carbon, 13 (1975) 309. PI G.W. Hollenberg, Am. Ceram. Sot. Bull. 59 (1980) 538. 191G.W. Hollenberg and J.A. Basmajian, J. Am. Ceram. Sot. 65 (1982) 4.

[lo] A. Carpinteri and E. Lorenzimi, Nucl. Eng. Des. 61 (1980) 1. [11] M. Oguma, J. Nucl. Sci. Technol. 19 (1982) 1005. [12] T.K. Gupta, ibid. [55,5] (1972) 249. [13] A.A. Griffith, Phil. Trans. Roy. Sot. (1920) 221. [14] D.P.H. Hasselman, J. Am. Ceram. Sot. 52 (1969) 600. [15] J.B. Walsh, J. Geophys. Res. 70 (1965) 381. [16] J.P. Berry, J. Mech. Phys. Solids 8 (1960) 194. [ll] D.P.H. Hasselman, J. Am. Ceram. Sot. 53 (1970) 490. [18] W.d. Kingrey, J. Am. Ceram. Sot. 38 (1955) 3. [19] J.A. Coppola et al., J. Am. Ceram. Sot. 55 (1972) 481. [20] F.P. Knudsen, J. Am. Ceram. Sot. 42 (1959) 376. [21] K.C. Radford, J. Nucl. Mater. 84 (1979) 222. 1221 K.C. Radford, J. Nucl. Mater. 84 (1979) 222. [22] L.A. Simpson, J. Am. Ceram. Sot. 56 (1973) 7. [23] M. Srinivasan and S.G. Seshadri, Fracture Mechanics for Ceramics, Rocks, and Concrete, ASTM-STP 745 (1980) 46. [24] P.S. Maiya, J. Nucl. Mater. 40 (1971) 57. 1251 I. Inoue and H. Matzke, J. Nucl. Sci. Technol. 17 (1980). [26] H. Matzke and T. Inoue, ibid 91 (1980) 205. [27] S.S. Smith et al., ibid (1980) 33. [28] MATPRO-V09, TREE-NUREG-1005 (1976). [29] R.D. Smith et al., Ceram. Bull. 55 (1976) 979.