STATISTICAL

METHODS

THERMAL-SHOCK V.I. G.A.

OF DETERMINING

RESISTANCE

Dauknis, A.Ya. Prantskyavichyus,

THE

OF REFRACTORIES

Peras, and

UDC 666.764.2:536.49 V, L .

Yurenas

The t h e r m a l - s h o c k r e s i s t a n c e is often a decisive factor in selecting s t r u c t u r a l m a t e r i a l s or the t h e r mal cycle for furnaces and kilns. T h e r e f o r e , investigations of the t h e r m a l - s h o c k r e s i s t a n c e of r e f r a c t o r i e s and s e a r c h e s for methods of increasing it have been given special attention. It is known that the t h e r m a l - s h o c k r e s i s t a n c e of a m a t e r i a l consists of a complex of p r o p e r t i e s , and depends in turn on a n u m b e r of physical and t h e r m a l f a c t o r s such as tensile strength Crb, elastic modulus E, the P o i s s o n ratio #, the linear--thermal expansion o~, the t h e r m a l conductivity X, the t h e r m a l diffusivity, a, and others. Various t h e r m a l - s h o c k r e s i s t a n c e (spalling resistance) c r i t e r i a a r e known [1] R--

~b (I -- ~) Ec~ ' )~ab (I - - ~)

R' =

R" --

E~z

(1) '

a% (1 -- ~) E~

(2) (3)

'

designed for determining the t h e r m a l - s h o c k r e s i s t a n c e of m a t e r i a l s with r e s p e c t to their p r o p e r t i e s in different h e a t - t r a n s f e r conditions. As we see f r o m Eqs. (1)-(3) the value of the c r i t e r i a of t h e r m a l - s h o c k r e s i s t a n c e is directly p r o p o r tional to the tensile strength of the m a t e r i a l orb. Meanwhile, the strength of brittle m a t e r i a l s , including r e f r a c t o r i e s , is a statistical quantity, appreciably different f r o m specimen to specimen, even when they are p r e p a r e d in identical conditions. The theoretical principle for the s p r e a d of the strength values in brittle m a t e r i a l s has been put f o r w a r d in r e c e n t times on the b a s i s of the well-known statement r e f e r r i n g to "the hypothesis of the weak link" [2], in a c c o r d a n c e with which the cause of the spread in the s t r e n g t h lies in the v e r y nature of brittle m a t e r i a l s , and is directly connected with the p r e s e n c e in them of a large number of statistically distributed m i c r o d e f e c t s in the s t r u c t u r e . The r e g u l a r i t y of the variations in the strength, and the need to take into account this phenomenon in selecting calculation p a r a m e t e r s for m a t e r i als, has r o u s e d much i n t e r e s t in the statistics of the strength of brittle m a t e r i a l s . The distribution of strength is the object of independent studies in a number of papers [2, 3]. It has been shown that there is sufficient c o r r e s p o n d e n c e between the difference of strength tests and the wellknown exponential Weibull distribution P (o) = 1-- exp L-- \~0] J '

(4)

where P(a) is the probability of failure of the specimens f r o m the load e; (r 0 is the strength p a r a m e t e r of the m a t e r i a l according to Weibull, corresponding to the probability of failure of the specimen, equal to 0.63, and m is the heterogeneity p a r a m e t e r of the material. Since the c r i t e r i a of t h e r m a l - s h o c k r e s i s t a n c e (1)-(3) are proportional to the tensile strength, we should expect a spread in the values of the c r i t e r i o n R, close in c h a r a c t e r and magnitude to the spread in Institute of Physicotechnical P r o b l e m s of Power Engineering of the Academy of Sciences of the Lithuanian SSR. T r a n s l a t e d f r o m Ogneupory, No. 2, pp. 53-56, F e b r u a r y , 1971.

O I971 Consultants Bureau, a division o[ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

12S

the values of the strength. Results of determining the c r i t e r i o n R on improved equipment with a high a c c u r a c y [4] indicate a substantial spread in the values of the t h e r m a l - s h o c k c r i t e r i o n , that is, of the same o r d e r as that obtained f r o m strength tests (the ratio for Rma x is close to 2). With such a spread in the results the production of reliable calculation c h a r a c t e r i s t i c s for the t h e r m a l - s h o c k r e s i s t a n c e of a m a t e r i a l is possible only by statistical analysis of the test r e s u l t s , covering a sufficiently l a r g e s e r i e s of specimens, and determining the type and p a r a m e t e r s of the distribution. In a large number of published investigations only the average values are quoted for the t h e r m a l - s h o c k c r i teria, or values corresponding to 50% of the damaged s p e c i m e n s . Only in a few a r t i c l e s (for instance in [4]), do the authors indicate the maximum values (Rma x and Rmin), which is also inadequate for the c o m plete c h a r a c t e r i z a t i o n of the distribution. In recent y e a r s , in evaluating the t h e r m a l ' s h o c k r e s i s t a n c e , g r e a t e r use has been found for the statistical theory of brittle f r a c t u r e developed by Weibull [3, 5, 6]. The use of the theory would have been m o r e soundly based if it had been experimentally shown, for a l a r g e group of specimens typical of brittle r e f r a c t o r y m a t e r i a l s , that the distribution c r i t e r i o n R obeys an exponential law of the type (4), forming the b a s i s of the Weibull theory, as was done for the strength limit [2, 3]. The p r e s e n t authors, on the basis of a statistical analysis of the test r e s u l t s of a r a t h e r large batch of r e f r a c t o r y specimens made an attempt to study the type and distribution p a r a m e t e r s of the main factor of t h e r m a l - s h o c k r e s i s t a n c e for the m a t e r i a l R. A dense c e r a m i c (maximum grain size in the batch 0.002 mm) of z i r c o n i u m dioxide, stabilized with 6% calcium oxide, was selected as the object of the investigation; it had relatively low t h e r m a l - s h o c k r e sistance, ensuring the development of destructive t h e r m a l loads. The coefficient of t h e r m a l conductivity of the zirconia, and also other p r o p e r t i e s determining the magnitude of R, w e r e almost constant [1] in the t e m p e r a t u r e range for which we expected failure of the specimen (up to 500~ The round specimens of external d i a m e t e r 41 mm, internal diameter 24 mm, and height 7 ram, were made by the experimental factory of the Ukrainian Institute of R e f r a c t o r i e s . The batch consisted of 51 specimens of average density 5.43 g / c m 3, with a mean square deviation in t e r m s of density equal to 0.02 g / c m 3, and a variation factor of slightly m o r e than =~1% of the average. The t h e r m a l loading of the specimens was done with internal heating elements of diameter 5 and length 800 m m made f r o m Nichrome, on an open rig, s i m i l a r to the equipment described in [7]. The heat flow was determined by the o r d i n a r y method with an a c c u r a c y of • Simultaneously we tested 8-16 s p e c imens, the failure of which was effected by a staged increase in the thermal c u r r e n t with an i n c r e a s e in the stages within the limits of the m e a s u r e m e n t a c c u r a c y . At each stage we achieved a thermal equilibrium so as to prevent the breakdown of the specimen due to transition s t r e s s e s . The appearance of the thermal c r a c k s was r e c o r d e d visually and f r o m the clear audible effect. The t e m p e r a t u r e at the middle of the r a dius of the wall of the specimens at which breakdown o c c u r r e d was 280~ with a minimum thermal current, and 460 ~ with the maximum. The main t h e r m a l - s h o c k r e s i s t a n c e c r i t e r i o n of the m a t e r i a l R was determined f r o m the magnitude of the destructive t h e r m a l c u r r e n t Q, starting f r o m the well-known statement on the l o g a r i t h m i c distribution of t e m p e r a t u r e in the cylindrical wall at a constant ~. In this case [1]

R= ~

(5)

with a coefficient of the f o r m 4~ 1 -- YT---~ g2

--

F I

ln-

F1

where r 2 and r 1 are the external and internal radii of the specimen, respectively. The magnitude of X was established on specimens of the same s e r i e s , and in the calculation was a s sumed to be equal to 1.4 k c a l / ( m 9h" deg). On the b a s i s of the statistical p r o c e s s i n g of the results we established the average t h e r m a l - s h o c k r e s i s t a n c e c r i t e r i o n R = 43 ~ with a mean square deviation S(R) = 7.6 ~ R m a x = 62.3 ~ and Rmi n = 29.6 ~

126

The h i s t o g r a m of the results obtained, on whose ordinate was plotted the number n i of the specimens broken within the stated R value limits, is shown in Fig. 1.

/2 /9

The s p r e a d in the c r i t e r i o n R is large, and close in degree to the spread observed in the mechanical testing of r e f r a c t o r y m a t e r i a l s . Thus, the variation coefficient is 30,

'~O

50

6'0 deg

Fig. 1. Histogram for the distribution of the t h e r mal-shock resistance c r i t e r i a of specimens of dense z i r c o n i a c e r a m i c s .

Kv = s~ (R) lOO= 17.5%,

close to the value of Kv obtained usually in mechanical testing.

In examining the results attention was drawn to the vague c o r r e l a tion between the density of the specimens and the values of R. The coefficient of c o r r e l a t i o n calculated by the usual technique [8] proved to be equal to 0.54, which confirms the connection between the change in the density observed within the limits of a single batch of specimens (in this case • and the c r i t e r i o n R. Apparently, the slight changes in the density which are noted in the specimens of one batch r e f l e c t the change in the defectiveness of their s t r u c t u r e . An i n c r e a s e in the number of defects and m i c r o d e f e c t s in the s t r u c t u r e inthe specimens affects the p r o p e r ties that determine the magnitude of the c r i t e r i o n R, and in the f i r s t case the strength [9]. The c o r r e s p o n d e n c e of the r e s u l t s obtained to the exponential distribution of Weibull R m P(R)=l--exp[--(~) ]

(5)*

was checked by calculating the P e a r s o n agreement c r i t e r i o n X2, according to the well-known relationship [8] ~ (n~-- NpO2 i= 1

Np~

'

where k is the number of d i s c h a r g e s for which all the r e s u l t s obtained were developed, n~ is the number i of values of R in each d i s c h a r g e , N is the total number of specimens tested, and Pi is the theoretical value of the probability of failure, corresponding to the ~volume" of the d i s c h a r g e s taken. The values of the experimentally determined distribution p a r a m e t e r s (5) required for calculating Pi were determined f r o m the following equation [2]: dN m = 2.3S (lg R)' - -

yN

IgR0 = lg R + .-2-.-.-.-.-.-.-.-~m' and d N andy N are tabulated values [2]. As a r e s u l t of the calculations we obtained: m = 9.4, and R 0 = 44.7. The c r i t e r i o n • equals 2.72, which in two stages of free distribution can be considered [8] as a sign of completely s a t i s f a c t o r y c o r r e s p o n d e n c e between the experimentally determined statistical distributions and the theoretical values. The f o r m of the h i s t o g r a m (see Fig. 1) and also the c l o s e n e s s of the d i s t r i b u tion (5) to the n o r m a l suggests that the distribution obtained experimentally is sufficiently close to the n o r mal. CONCLUSIONS A statistical analysis of the test results for the thermal-shock resistance of a large batch of specimens of zirconium dioxide established that there is satisfactory correspondence between the experimental data and the exponential distribution of Weibull. The appreciable spread in the thermal-shock resistance criteria R for brittle refractory materials should be considered as regular, and therefore to obtain reliable calculation characteristics for thermal-shock resistance it is necessary to treat the test results statistically. It is proposed that one of the causes of variation in the thermal-shock resistance criterion R is the slight (unavoidable even within the limits of a single batch of specimens) change in the density of the material. * Equation numbered

as in Russian

original -- Consultants

Bureau.

127

LITERATURE

lo

2. 3o

4. 5. 6. 7. 8. 9.

128

CITED

W.D. Kingery, J. Amer. Ceram. Soc., 38, No. 1, 3 (1955). G.S. Pisarenko et al., Strength of Cermet--s and Alloys at Normal and High Temperatures [in Russian], Akad. Nauk Ukrainian SSR, Kiev (1962), p. 192. V. Weibull, Prac. Royal Swed. Inst. Eng. Research, No. 151 (1939). G.A. Gogotsi et al., Ogneupory, No. 11, 54 (1968). S.S. Manson and R.W. Smith, J. Amer. Ceram. Soc., 38, No. 1, 18 (1955). G.A. Gogotsi et al., Teplofiz. Vysokikh. Temp., No. 3, 515 (1969). G.A. Prantskyavichyuset al., Trudy Akad. Nauk, Lithuanian SSR, Series B, 3, 50, 121 (1967). E.S. Vent-tsel', Theory of Probability [in Russian], GIFML (1962), p. 144. G.A. Prantskyaviehyus, Ogneupory, No. 5, 45 (1968).