Atomic Energy of Canada Limited AN ESTIMATION OF THE PROBABILITY OF FATIGUE FAILURE OF SOME GRAPHITES

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1 Atomic Energy of Canada Limited AN ESTIMATION OF THE PROBABILITY OF FATIGUE FAILURE OF SOME GRAPHITES by B.J.S. WILKINS and A.R. REICH Whifeshell Nuclear Research Establishment PinawQ, Manitoba February 1972 AECL-3958

2 AN ESTIMATION OF THE PROBABILITY OF FATIGUE FAILURE OF SOME GRAPHITES by B.J.S. Wilkins and A.R, Reich Manuscript prepared: November, 1971 Whiteshell Nuclear Research Establishment Pinawa, Manitoba February, 1972 AECL-3958

3 AN ESTIMATION OF THE PROBABILITY OF FATIGUE FAILURE OF SOME GRAPHITES by B.J.S. Wilkins and A.R. Reich ABSTRACT Engineering design of structural ceramic components has to make allowance for fatigue failures. A method has been developed to estimate the probability of such failures. This report describes the application of this method to published data for graphites. The estimates obtained are very conservative because data in the literature are not in the specialized form required by the method. Nevertheless, the estimates are useful and are used to illustrate the effect of the fatigue limit and of fracture strength and its variability for POCO and RC4 graphites. Also included is a hypothetical material, Hy-POCO graphite, which is as strong as POCO but as variable as RC4 graphite. The results indicate the great importance of the fatigue limit, and that permissible service stresses are greatest for the strongest, least variable graphite. Proof testing also is desirable because, as the proof stress is increased, so is the maximum operating stress for any given probability of failure. This is more noticeable in the more variable graphites and has practical importance for it improves their performance significantly. Manuscript prepared November, 1971 Whiteshell Nuclear Research Establishment Pinawa, Manitoba February, 1972 AECL-3958

4 Estimation de la probabilité de rupture, due à la fatigue, de quelques graphites par B.J.S. Wilkins et A.R. Reich Résumé Lorsque l'on conçoit des composants céramiques structurels il faut tenir compte des ruptures pouvant résulter de la fatigue. On a développé une méthode permettant d'estimer la probabilité de telles ruptures. On décrit dans ce rapport la façon d'appliquer cette méthode aux données publi.ées pour les graphites. Les estimations obtenues sont très conservatives car les données figurant dans la littérature ne se présentent pas sous la forme spécialisée requise par la méthode considérée. Néanmoins, ces estimations sont utiles et elles permettent d'illustrer l'effet de la limite de la fatigue et de la résistance à la fracture et sa variabilité pour les graphites POCO et RC4. Il est également question d'un matériau hypothétique s le graphite Hy-POCO, qui est aussi solide que le POCO, mais aussi variable que le graphite RC4. Les résultats montrent que la limite de la fatigue est très importante et que les contraintes de service admissibles sont maximales dans le cas du graphite le plus solide et le moins variable. Des essais sont également souhaitables parce que à mesure que la contrainte d'épreuve est augmentée, la contrainte opératoire maximale est également augmentée pour toute probabilité donnée de rupture. Cela se voit davantage dans les graphites très variables et cela est important car on peut, de cette façon, améliorer de façon significative leur performance. Manuscrit préparé en novembre 1971 L'Energie Atomique du Canada, Limitée Etablissement de Recherches Nucléaires de Whiteshell Pinawa, Manitoba AECL-3958

5 LIST OF SYMBOLS n Number of specimens in sample Probabilities P. Cumulative probability of instantaneous fracture P f Cumulative fatigue failure probability P T Total cumulative probability of fatigue failure Stresses a. Instantaneous fracture stress a a Applied stress Homologous fatigue stress - defined as GJGo' Value of a where P, = 0.5 a. n. t C Fatigue limit, i.e., value of a n where? c = 0 a P Proof stress k Ratio of applied to proof stress, i.e., a Jo : 1 > k > 0 Distributions P.(a.) Distribution of instantaneous fracture stress (a.) for virgin material. p (a ) Cumulative probability corresponding to applied stress (o ) for virgin material. P.(a ) x p P (a.) p x Cumulative probability corresponding to proof stress (a ) for virgin material. Truncated distribution of instantaneous fracture stress (a ± ) of survivors of proof stress (a ).

6 Distribution Constants a Weibull distribution constant for the instantaneous fracture stress (a ) distribution. It approximates to the difference between tne mean/mode and the lower bound of the distribution. m K V Weibull distribution constant that for a given mean/mode gives a measure of the variability of the distribution. Fracture stress distribution constant related to mode of loading. Fracture stress distribution constant related to volume of material under load. 6 Weibull distribution constant that for a given variability of the distribution gives a measure of the mean/mode. Fatigue Life and Distribution N Number of stress cycles to failure N Weibull distribution constant for stress cycles to. failure o (N) distribution. It approximates to the difference between the mean/mode and lower bound of the distribution.

7 -1-1. INTRODUCTION Ceramic materials such as silicon carbide and graphites are of interest as potential reactor core structural materials. They are attractive because of their low neutron cross sections and high strengths at elevated temperatures. Engineering design for these materials is not straightforward, however, for they are brittle and can exhibit a large variability in fracture strength. A statistical approach to design is required in which acceptable probabilities of fracture are defined. What probability is acceptable depends on the importance of the survival of the structure in which the material is used. Also, if the structure contains many different components, design must be based on the weakest, most essential component. The statistical approach is based on the failure mechanism involved. In brittle materials the variability in fracture strength is attributed to flaws throughout the solid at which fracture may be initiated *. Fracture strength is governed by the largest effective flaw, that is, the flaw requiring the smallest stress to propagate. Clearly, then, fracture stress is a function of the size and distribution of flaws and the distribution of applied stress. Therefore, it must also be related to the volume and manner of stressing of the material. Weibull 2, in his 'weak-link' model, has taken this into account and produced his familiar empirical relationship, P ± = 1 - e~^vv (1) where P is cumulative probability of instantaneous fracture, a. is fracture stress, a and m are distribution constants, K is a constant related to loading mode, and V is a constant related to volume of material under load. Design may not be based on the above considerations alone. It is further complicated by the phenomenon of fatigue, in which failure probabilities increase with service time to higher values than those

8 -2- based on instantaneous fracture data. Probility of failure as a function of required service life has to be known. This means finding the total cumulative probability of fatigue failure, P, which takes into account the probabilities of instantaneous failure and fatigue failure in service. An approach to calculating P for a ceramic population subjected to dynamic or static fatigue has been developed at the Whiteshell Nuclear Research Establishment (WNRE)^3' 4. This requires data in a particular form which is not available in the literature. However, Leichter and Robinson 5 have produced other data for graphite from their own work and from the literature^6" 10. The form of these data is sufficiently similar to be adapted to the WNRE method. This report describes how P may be estimated from the adapted data, and points out the statistical difficulties encountered and the limitations of- the results. 2. PUBLISHED WORK Fatigue data may be expressed as a family of curves of applied stress versus fatigue life covering the whole range of cumulative fatigue failure probabilities, P f. Leichter and Robinson 5 replaced simple stress with homologous fatigue stress, a, to eliminate mode of H loading and volume effects. They defined a u as the ratio of the applied n fatigue stress, a, to the expected instantaneous fracture stress, a.. For dynamic fatigue, first cycle strength was substituted for a.. resulting data are shown in Figure 1. The cumulative fatigue failure probability, P f, is the probability that any specimen will survive any Their given number of stress cycles, N, at any given a. Essentially, the n values of N at any given a were described by a Weibull distribution F ( «".>' (2)

9 -3- where N and m are constants. For a range of a values, N values corresponding to particular P. values were calculated to provide the curves shown in Figure 1. To find a values it was necessary to know the a. value for each specimen. Leichter and Robinson 5 found this indirectly for their own rod-shaped specimens. each specimen in two. They divided One half represented the 'mate' from which a. was determined experimentally and identified as a. for the whole specimen. The other, the fatigue specimen, was used to determine the fatigue life of the whole specimen. Then they matched an applied stress value, a, to each specimen to produce the a value desired. They did not obtain several values of N for any particular value of o_, rather they n obtained a whole series (50) of a,n pairs covering a range of a values from 0.6 to 1. They analyzed these data in two ways. dividing the ranked range of a The first involved values into convenient groups and finding the distribution of N values, i.e., the constants N Equation 2 for each group. and m in From this they drew P f contours on a plot of a versus N, i.e., Figure 1. In the second approach they assumed 1 each specimen to have a a. value equal to the mean value experimentally found for the sample of half rods. of a From the a values used, groupings values were again calculated with corresponding N values. P.. contours were estimated and drawn on the a, N plot as before. Their results show that there was no significant difference between the two methods of analyzing results. This allowed them to correlate their work with the work of others 6 ~ 1G; sample was available. where only the mean value of a. for the The main conclusion they drew from Figure 1 is that for several graphites the fatigue limit is ^47 per cent of their first cycle strength. If true this is a very significant design result. It means that if a sample of components is proof tested at a stress a, no failures will occur in service at a working stress of <0.A7a. This is because the highest ov, value experienced by any component will be <0.47. However, this is not certain for there are many difficulties with this approach which make Leichter and Robinson's estimation of the fatigue

10 -4- limit unreliable. follows: Some of the more important difficulties are as (i) Their estimation of a is not adequate for they incorrectly assume that the variability between two halves of a specimer is less than that between two whole specimens. (ii) They find distributions of N over ranges of c rather than for a given a value. This introduces extra variability into the results. This may be advantageous, however, for it should make their results more conservative. (iii) They used a distribution of o. rather than a fixed a. As a p = a /a., two of these stresses must be specified if a particular specimen and its fatigue loading are to be unique. Hence, by generalizing their a. they have introduced more variability. A may have some advantage in terms of being more simple for practical This design, but the results are unlikely to be reproducible or representative of reality. There are an infinite number of a distributions.to choose A from and it is practically impossible to specify the a distribution to A make it unique in terms of its relation to the a. distribution. Also they have assumed that fatigue life is not a function of a. This assumption is not true. o fatigue life does depend on a. Work at WNRE ^3»4»11> has sh own that at a fixed In spite of these difficulties it is useful to estimate P_ from the data of Leichter and Robinson 5. The most persuasive reason is that no other suitable data are available apart from those generated at WNRE- 1 *. Further, because a is generalized, the results are of design significance even if they are likely to be very conservative.

11 -5-3. CALCULATION OF TOTAL CUMULATIVE PROBABILITY OF FATIGUE FAILURE, 3.1 Fatigue Failure Probability, P f Engineering design requires probability of failure to be specified for a given lifetime, i.e., for a given N. the distribution of N values at a given a a Therefore, from must be found the distribution of 0 values at a given N. As with the N distribution, i.e., Equation 2, it is convenient mathematically to use the Weibull distribution. Figures 2 and 3 show typical distributions of o superimposed on Figure 1 at a specific value of IT, i.e. N 1, where a' is the value of o that n n corresponds to P, = 0.5. Figure 3 is the same as Figure 2 except that a fatigue limit Is shown at a. = C. A fatigue limit in terms of a was ( } assumed by Leichter and Robinson 5. They assumed that it was independent of a. because they did not fix a.. In ttis they are probably correct, for recently Stevens and Dutton li ' have proposed a static fatigue model which indicates that the fatigue limit is dependent only on, temperature and material constants. H P f If 0 a for any N is distributed according to Weibull, then H in terms of o T1 at any N is given by n P f «1 - e-[(v C)/e(1 -V i (3) rf where a' corresponds to P, = 0.5. To find 6 and m, the distribution H constants for Equations 3 and 4, from data in Figure 1, we have considered that the probability of a given N at a fixed a R is equal to the probability of that a at that N. For N values of interest, values of a corresponding to various P values were taken from Figure 1. The

12 -6- constants 6 and m were then found by a least squares fit of the P^,( pairs to the linearized form of Equation 3, i.e., to^^-s «nto^j + toe (5) The form of the P versus a curve for all values of o and a given N is shown in Figure Proof Testing and Design Stresses To use the information such as is shown in Figure 4 to calculate P_, the distribution of a values applied to the sample has T n to be known. This means finding first the distribution of instantaneous fracture stress values, a., for the sample. This must be done, using the same mode of testing and specimens from the same population so that the volume and mode of loading terms, V and K in Equation 1, can be ignored. To have confidence in the a. distribution, a proof stress is necessary. When a proof stress, a, is applied to a sample of ceramic specimens, the truncated distribution for the survivors, P (a ), can be related to P 1 r \ the original distribution, P.(a.), as described by Weibull z see Figure 5. Hence, and yv P = 0 ; o.<a P i P This assumes the survivors are unaffected by the proof test. dynamic applied stress, a, If now a is applied to the sample, there will be a range of a values, theoretically from 1 to 0 if o,= a. Now, the a. H A p l axis in Figure 5 may be replaced by a a axis describing how the population is being fatigued in terms of o. On the original axis at 1 i ~ A = p» we have a H = 1» as shown in Figure 6.

13 7 3.3 Estimation of The probability of specimens being acted on by a particular a range is found by dividing the a axis into small increments. For n n example, in a sample of size n, the number of specimens being acted on by a a value between 1 o and 2 a is n( 2 P - 1 P). Hence, ( 2 P - 1 P) is the a an probability of specimens being acted on by ( 1 a + 2 cr )/2. For that a n H H value, the corresponding cumulative fatigue failure probability (ip, + 2 P f )/2 at a given N may be found, where 1 P- and 2 P correspond to 1 a fl and 2 a^, respectively (Figure 6). These P values are found from Equation 3 or 4. To find P T, all the {( 2 P - 1 P)( 1 P f + 2 P f )/2} terms are summed over the whole population. fatigue failure, P_,, is given by Hence, the total cumulative probability of or alternatively by A J f ( V d H (8) No analytical solutions for Equations 7 and 8 were attempted. However, they have been solved numerically by computer for a range of o. values for three graphites: RC4 (extruded), POCO (moulded) and Hy-POCO. Hy-POCO is a hypothetical graphite which is as variable as RC4 but as strong as POCO. Table 1 gives the instantaneous strength distributions characterizing these different materials. For the initial calculations Equations 3 and 4 were not used. Instead, P» CT H pairs were taken from Figure 1 and fitted to the expression P e- ( V 9 > m (9)

14 -8- This has the Weibull form but a is allowed to vary from 0 to», which is H incorrect. The point (0,0) was included; hence, the fatigue limit was set effectively at zero. The resulting expression at N = 10 3 was 8It (10) Equation 10 is a reasonable approximation of the results shown in Figure 1 and also has the advantage of being simple. Figure 7 shows how P_ varies with P.(a.) at N = 10 3 when a. = a. For convenience a. has been replaced with P. (a.), i.e., A p A x A the value of P. that corresponds to a. for each graphite. Note how Hy-POCO behaves as RC4 and how P T is largest for POCO, i.e., the least variable material. 3.4 The Advantage of Proof Testing at Stresses Greater than Intended Service Stresses Figure 7 shows that if the design criterion requires P = 10~ 6 for N = 10 3, then a has to be such that P-C^O < 10~ 6. This is obviously undesirable because it limits the permissible operating stress, a. A method of increasing a needed. without affecting the P T value is To achieve this, it is necessary to increase the proof stress, a, and to put a^ < a, i.e., to put a = ka where 1 > k > 0. The effect of having o. < a is to move the (* scale in Figure 6 to the left, i.e., A p n the whole sample is fatigued at lower a., values. Values of k for a 1 range of CJ values have been computed for P = 10~ 6 and N = 10 3 for the three graphites. The results are shown in Figure 8. In this figure, p is represented as P ± (o ), i.e., the P ± value corresponding to a for each graphite. Again, it is apparent how the behaviour of Hy-POCO is similar to that of RC4. Also it is important to note that if VAa ) is i p set at 0.1, the maximum permissible working stress, a, is between 0.21 a and 0.23 a for P = 10" 6 and N = Figure 9 is a plot of a. (=ko ) versus P.(a ) calculated A p 1 p

15 -9- from the results shown in Figure 8 and the a. distributions for the graphites shown in Table 1 for P = 10~ 6 and N = This figure shows that for a given P and N value, a is increased by increasing a. x A p Therefore, for any fatigue criteria, there is an economic compromise between increased sample losses, due to increasing the proof stress, a, and the gain in the consequent increase in permissible operating stress, a A. For example, in Figure 9 for Hy-POCO, a value of P. (a ) = 0.1 seems a good compromise between minimizing proof losses an maximizing permissible operating stress. Also it should be realized that increasing P.(o ) gives the additional advantage of increasing confidence in the a, distribution. l Two other important conclusions may be drawn from Figure 9: (a) strength is important, i.e., POCO has the largest permissible a value for all values of P.(a ) < 0.5 A 1 p (b) there is a significant advantage in increasing P.(a ) for the more variable materials - the percentage increase in a between P.(a ) values 10~ 6 and 10" 1 was the largest for Hy-POCO, i.e., ^ 45%. The latter conclusion does not imply that variability is advantageous, in fact it is not. However, it does imply that the disadvantage of high variability may be nullified if P. (o ) is increased. This is illustrated at P. (a ) = 0.1 where Hy-Peeo has a similar fatigue 1 P strength to that of POCO. 3.5 The Effect of Uncertainty in Fatigue Data and in the Value of the Fatigue Limit, C The calculations shown were based on the best estimates of Leichter and Robinson 5 with the assumption of no fatigue limit. However, because of the small number of specimens they used and because of doubtful assumptions they made, these calculations are uncertain. For example, the value of the constant m in Equation 3 could be anywhere between 1 and 6. Probably, it cannot be generalized for all graphites.

16 -10- Also, there is likely to be a fatigue limit, C, that lies between a - 0 and 0.4. Unfortunately, there seems to be no experimental way H of determining it, and the only indication of its value at present is due to the theoretical work of Stevens and Dutton J1, i.e., C = Therefore, further calculations were made (POCO only) to illustrate the effect of a fatigue limit and of variations in m. For these calculations it was necessary to fit the data in Figure 1 to Equations 3 and 4, i.e., a Weibull form that allowed a to vary between C and 1. Figure 10 shows for POCO how a. varies with m for P.(o ) = 0.1, A " 1. U P = 10~ 6 and N = Two sets of data are presented, i.e., for no fatigue limit (conservative) and for a fatigue limit where C =0.32 a. n. For these calculations the value of o n at P, = 0.5, i.e. a', was taken HI H to be 0.82 at N = 10 3 rather than at 0.72 as shown in"figure 1. The higher value of a' is from WNRE work^^ on RC4 graphite for dynamic fatigue at a./2 ± a./2 rather than 0 ± a. as in Figure 1. The results An A show that for the best estimate of m, i.e. 4, a = 4890 lbf/in 2 when A C = 0.32 o H and a A = 1790 lbf/in 2 when C = 0. Clearly the fatigue limit has an important effect. Also, for 1 > m > 6 when C = 0.32 a, n 3000 < a A < 5500 lbf/in 2. Though the value of m is important, it is apparently less so than the fatigue limit. It seems likely that the value of m in Equations 3 and 4, i.e., the variability in a the particular graphite concerned. at a given fatigue life should depend on As instantaneous fracture stress, o\, becomes less variable, so ought the fatigue behaviour, i.e., m should increase. However, if m = 4 is accepted as a suitable value for POCO, then at P ± (a ) = 0.1, P T = 10" 6 and N = 10 3 the maximum permissible operating stress, a^t is -\-41% of a o without a fatigue limit. with the fatigue limit and ^15% of

17 CONCLUSIONS The total cumulative probability of fatigue failure can be estimated for ceramic materials from data such as Leichter and Robinson^5' produced for graphite. The values of P obtained, however, are too conservative. The procedure for acquiring data in the required form to allow accurate estimations with known confidence levels has been developed at WNRE 3>lf and should be used for engineering design. High mean fracture strength coupled with low variability in strength is desirable for maximum fatigue resistance. High variability in strength may be compensated for by using a proof stress. Obviously, for two graphites with the same mean fracture strength the fatigue behaviour of the more variable graphite will be superior if the proof stress is greater than the mean fracture stress. Proof testings improves the fatigue behaviour of a graphite specimen population. It increases confidence in strength characterization and increases the permissible operating stress for any fatigue criterion. 5. REFERENCES 1. A.A. Griffith, The Phenomena of Rupture and Flow in Solids, Phil. Tans. Roy. Soc., A221 (1920) W. Weibull, A Statistical Theory of the Strength of Materials, Ingeniorsvetenskapadamiens Handligar (Stockholm), No.151 (1939) 1-45; The Phenomenon of Rupture in Solids, ibid, No. 153 (1939) B.J.S. Wilkins, Static Fatigue of Graphite, J. Amer. Ceram. Soc., 54 (1971) B.J.S. Wilkins, Probability of Failure of Ceramic Materials Subjected to Static Fatigue, J. Materials (to be published in June, 1972). 5. H. Leichter and E. Robinson, Fatigue Behaviour of EP-1924 Graphite and a Generalised Design Criterion, UCRL (1967), University of California, Berkeley.

18 L.L. France and V. Kachue, American Institute of Aeronautics, 5th Aerospace Sciences Meeting, New York, AIAA Paper No V.N. Barabanov, Yu.P. Anufriev, G.G. Zaitsev and M.Ya. Pimkin, Zavodskaya Lab. (Indust. Lab., English translation) 31^ (1964) J.W. Dally, Evaluation of Thermal Protective Systems for Advanced Aerospace Vehicles, Air Force Tech. Doc., ML-TDR , Vol. 3 (August, 1965) 9. L. Green, Behaviour of Graphite under Alternating Stress, J. Appl. Mech., JL8 (1951) Cited in General Electric Pyrolitic Graphite Engineering Handbook. 11. R.N. Stevens and R. Dutton, The Propagation of Griffith Cracks at High Temperatures by Mass Transport Processes, Mater. Sci. Eng., JJ (1971) R. Stevens and T.D. Clausen, Strength Distribution and Fracture Behaviour of Structural Ceramics of Low Neutron Absorption Cross-Section, AECL-3422 (1969), Atomic Energy of Canada Limited. TABLE I WEIBULL DISTRIBUTIONS OF INSTANTANEOUS FRACTURE STRESS CHARACTERIZING GRAPHITES POCO P ± = 1 - Hy-POCO P i - I - e- [<J i /(a o>?oco ]8 RC4 P i e-cv (ff o>rc4 I 7 ' 3 < < <*< POCO = 13,400 lbf/in 2 )RC4 = 2,631 lbf/in 2

19 i 111 mi] r-rr i i 111 T-I i i i mill i i 1111in i i i iimi i i 11 mi 0.4 III I ' I ' I I I "ill 1 I I mill I I I Ilinl I I I 11 Mil KM ilinl I I I I Life -cycles N Reference Material Ave. strength (lhf/in 2 ) Leichter and Robinson 5 France and Kachur^6 + Dally K * f. Green (9) Cocciotti..do) vlu '' AXF Carbitex CFW * f(with grain), i (7) molded graphite, Barabanov et al. ' 1 o / v against grain) p = 1.8 g/cm 3 AUF pyro graphite Open symbols indicate run-out (no failure) data. 13,500 55,000 2,100 2,000-3,000 4,000 11,000 Figure 1: Plot of applied homologous fatigue stress, a, vs fatigue life, expressed as cycles to failure, N. This plot if from Leichter and Robinson ^ and includes their data and that due to others^6" 10.

20 -14- Figure 2: Distribution of applied homologous fatigue stress, <? at N 1 cycles to failure, i.e. dp_/do" H (frequency of! ) versus a at N 1 The plot is superimposed on a plot of a versus cycles to failure N ri a is the value corresponding to where the fatigue limit is zero. n H P,,» 0.5. \ T dp f /doh \ "P7= I1 Fatigue Limit N' Figure 3: As Figure 2 except that a fatigue limit at a u = C is shown H Now, the frequency of a is zero at both o\, = 1 and C. n H N

21 Figure 4: Cumulative probability of fatigue failure, P_, versus applied homologous fatigue stress, a, with and without a fatigue limit. Virgin Material Distribution = cr. Figure 5: Distribution of instantaneous fracture stress, a., for virgin material and material proofed at cr.

22 -16- CTH Figure 6: Truncated distribution of instantaneous fracture stress, o., now replaced as a distribution of homologous fatigue stress, a u, for a given applied stress, a..

23 ^~ ==^^^^mm P0C0 io Hy-POCO / / io- 4 - io- 5 N = I0 3 cycles i I , 0-3 IO- 2 IO- 1 Pi«r A ) Figure 7: Total cumulative probability of fatigue failure, P, versus the probability corresponding to applied stress, a, where a, equals proof stress, a.

24 N = 10 P T = 10-6 cycles ~ " 10-s IO' S Pj{CTp) IO" Z 10 -I Figure 8: k, the ratio of operating stress, a., to proof stress, a, vs probability corresponding to a. At P ± (a ) = 10" 1, k (POCO) a- 0.23, k (Hy-POCO) «u 0.21, k (RC4) ~ 0.23.

25 " - N P T 1 = I0 3 cycles = IO' e * POCO o or (Ibf/ln 2 ) 2000 ( t " " Hy-POCO - '. - " I n» ^ 1 RC4 - T I ' Figure 9: Permissible operating stress, a, vs probability corresponding to proof stress, a

26 r i r Pf (CT p ) = P T = IO" S N = to or k(j p (lbf/in z ) 5000 = 0i32 _ OI790 <T H =0-82 C = 0 IQQQl i I I I I I II l O m Figure 10: Permissible operating stress, o, versus variability of fatigue data, i.e. m in Equations 3 and 4, for Poco graphite.

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