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Reliability assessment by aid of probabilistic fracture mechanics
Int. J. Pres. Ves. & Piping 54 (1993) 341-352
Reliability Assessment by Aid of Probabilistic Fracture Mechanics
F r e d Nilsson Department of Solid Mechanics, Royal Institute of Technology, Stockholm, Sweden
ABSTRACT Reliability assessments of nuclear power plants are often performed by using probabilistic safety analysis. To which extent probabilistic fracture mechanics (PFM) can be of value to assess the mechanical reliability of nuclear power components (especially the reactor vessel) is discussed, together with some particular problems with PFM applications. Some simple examples are given where the principles of PFM are illustrated.
1 INTRODUCTION There are several motivations for probabilistic assessment of the failure risk of a reactor pressure vessel (RPV). Since a vessel failure is virtually the only accident mechanism that is not accounted for in the design it is of great interest to know whether this remaining risk level is acceptable or not. It seems that the probabilistic approach furnishes the only possibility for a comparison of mechanical risks with risks because of other reasons. Further, when planning in-service inspection (ISI) activities, probabilistic reasoning may provide a method of how to select the most important areas for inspection and also how to choose the inspection intervals. Probabilistic safety analysis has become a widely used tool for systematic assessment of undesired events in complicated systems. The resulting probability for an unwanted event is obtained by combining failure probabilities of the components. In most cases the component 341 Int. J. Pres. Ves. & Piping 0308-0161/93/$06.00 (~) 1993 Elsevier Science Publishers Ltd,
England. Printed in Northern Ireland
342
Fred Nilsson
failure probabilities are obtained by direct observation of the failure rate. For a component such as the RPV this approach is not possible since failures have never occurred. In order to obtain estimates of the failure probability, indirect methods have to be used. Statistical information about various mechanical quantities is combined according to the methods of mathematical statistics. Such methods are often termed probabilistic fracture mechanics (PFM). 2 GENERAL OBSERVATIONS To perform a complete PFM analysis of a RPV which takes into account all possible phenomena and aims at exact estimates is a formidable task. The methodology for performing such analyses is well developed for certain classes of problems, namely those where the quantities involved are not stochastic processes in time. For solving such problems the so-called second moment methods I can be used with advantage as well as Monte Carlo methods and there exists several computer programs that are able to perform the analyses. When the problem necessitates modelling by aid of stochastic processes, such as is the case, e.g. for fatigue crack growth under random loading, the methodological problems are more difficult and the only general route open seems to be direct simulation by Monte Carlo methods. The main problems with PFM applications to RPVs are, however, not of a methodological nature, but rather the difficulties involved in obtaining relevant data. Since nuclear equipment is by design highly reliable, failure cannot in general occur because of an outcome of the involved variables that falls within the range of normally observed data. Thus, extrapolation to ranges without any observations must usually be done. This state of affairs has, at least in the author's opinion, the consequence that it is not particularly meaningful to aim at very precise and computationally complicated estimates of the failure probabilities. Instead, it is argued that fairly simple models can be used to obtain at least partial answers to questions of whether the reliability is sufficiently high and how this is affected by changes of the assumptions. Practical experience with fracture mechanics assessments of RPVs suggests that the potential problems that contribute to the failure probability can be divided into the following classes.
2.1. Fracture due to a pre-existing defect with insignificant growth In most parts of vessel the loading conditions are such that possible
Reliability assessment by aid of probabilistic fracture mechanics
343
fatigue crack growth is very limited and can in general be neglected. If no toughness degradation processes can occur it follows that failure can only take place if the loading conditions are more severe than the previous ones the vessel has been subjected to during its operation. Thus, defects of this kind are mainly of interest when judging the failure probability under accident loading conditions.
2.2. Fracture due to a pre-existing defect with insignificant growth in a region where toughness degradation processes can occur
The typical example is irradiation induced toughness degradation. The regions where this can occur are usually not loaded in such a way that fatigue crack growth is a problem. 2.3 Fracture due to an in-service induced defect with significant growth
If the conditions are such that a fatigue or stress-corrosion crack can be initiated during service its subsequent growth will in general be so rapid that if left to itself the crack will eventually grow to the critical size. It is not claimed that the categorization made here is by any means exhaustive, but it is the author's feeling that most of the potential problems can be put into one of these categories. The advantage with this is that in each category the random nature of some of the variables can be neglected and thus allow for a simpler treatment. This is to be illustrated by some examples below. The variables involved in PFM studies and which can be of random nature can be grouped into: (a) (b) (c) (d)
Material properties: fracture toughness, fatigue crack growth data, stress corrosion cracking susceptibility, etc. Loads: transients, fatigue loading, residual stresses, etc. Defects: fabrication defects, in-service induced defects. Inspection: reliability of inspection methods, in-service inspection plans, etc.
3 SOME E X A M P L E S OF PFM ANALYSIS In this section some highly idealized examples are considered. The main objects are to illustrate the methods and problems of PFM and to
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draw some potentially useful conclusions. The following situation is assumed. In a certain region it is required to estimate the failure probability of potential cracks. A s s u m e that the shape of the cracks is known and the crack size can be characterized by a depth p a r a m e t e r a, which is distributed according to the frequency function f,. The stress state is assumed to be d e t e r m i n e d so that the stress intensity factor is K~(a). For simplicity is assumed that linear elastic fracture mechanics applies and the fracture toughness is distributed as Fr,c. With these assumptions the fracture probability because of this potential crack can be written: ps = ~
aa= o
f~(a)FK,°(Ki(a))da
(1)
3.1 Exponentially distributed crack depth We first consider an example that might be representative for the case of a pre-existing defect with insignificant growth. As specific forms for the distributions it is assumed that the crack depth is exponentially distributed 2 and the fracture toughness is Weibull distributed: 3
(-a/~t)/a FK,~= 1 -- exp ( - ( K,/ Ko)4). f~ = exp
(2) (3)
~/is the m e a n crack depth and 0-9064K0 is the m e a n fracture toughness. The relevance of these distributions is to be discussed below. We now assume that
K, = g , ( a l a ) ''2
(4) where / ~ is the stress-intensity factor corresponding to t/ and the current load situation. Insertion into eqn (1) and performing the integration lead to Pl = 1 - exp (1/4r4)(1 -
~¢=
~(1/2K2))(V'-~/2K 2)
K,,IKo
(5)
(6)
Here, • denotes the error function. Since K is usually a small quantity it is advantageous to use an asymptotic expansion of ~4 and eqn (6) can be written as Pi = - ~
( - 1 ) m x 1 x 3 x • • • x (2m - 1)22mK 8'',
K < 8 -1/4 = 0 " 5 9 7
m=l
(7) Pr is shown as function of r z in Fig. 1.
Reliability assessment by aid of probabilistic fracture mechanics 10°t lOlt
1°2t 10-3t lO-'t Pl lO-St lO-°t 10-7 ~,,/ 10 -8 ]
10 "z
345
/ / ................. 10-1 tq 2
10°
Fig. 1. Fracture probability Pr for exponentially distributed crack depth. The sensitivity of Pi on r is noted. K2 can be interpreted as being proportional to the m e a n crack d e p t h divided by the m e a n critical crack depth. Thus, values of r 2 a r o u n d 0-05 o u g h t to be typical. We n o t e that an increase of r 2 by 20% (10% increase of K) causes a doubling of the failure probability. Since r d e p e n d s linearly on stress and inversely on fracture toughness it is easily u n d e r s t o o d that it is futile to search for very accurate estimates. O f s o m e interest is also to estimate the increase in failure probabiity corresponding to the different safety factors applied in normal and accident situations according to the A S M E XI rules. T h e ratio b e t w e e n the safety factors is ~/10:2 with respect to stress which corresponds to an increase of Pl of about 6 x 102. Suppose now that the region has survived a loading situation that corresponds to a value r~ according to the definition of eqns (4) and (6). This means that the toughness distribution can be truncated at KoKl(a/~) ~/z. A load corresponding to K2 applied later, can only cause fracture if it creates a m o r e severe situation, i.e. K2 > ~:1. T a k i n g this truncation of eqn (3) into account w h e n p e r f o r m i n g the integration, eqn (1), it can easily be shown that eqns (5) and (7) remain valid also for this conditional probability provided the following substitution is made: /("-'--9' (/~4 -- /(.4)1/4
(8)
This shows that loads slightly larger than the n o r m a l ones do not cause a large failure probability while the probability of failure for very severe loads is hardly affected by the fact that the vessel has survived the previous loads. T h e s t a t e m e n t m a d e u n d e r 2.1 above is thereby supported.
Fred Nilsson
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This example can be further developed to account for degradation of the toughness by letting Ko be a decreasing function of time.
3.2 Normally distributed crack depth In the second example we assume the fracture toughness to be distributed as above while the crack depth is normally distributed with mean crack depth 5(0 and standard deviation sa(t), which are both functions of time. This may serve as an illustration for the analysis of growing cracks. The normal distribution is truncated at a = 0 to account for the non-negativity of crack depth. Thus, at a particular time instant:
fa(a;&sa)
=
(2/G(2~)l/2)exp(-(a
-
ti)Z/2sZ)(l+~(sa-~))
,a>O
(9)
Making assumptions analogous as above and setting
= (saV~)/K
(10)
Integration of eqn (1) gives p; = 1 _
(1 + K4~2)1/2(1 q'- (P(1/~))
K 4
1 + (I) ~(1 + 12~2)1/2]) exp [ 1 + ~-q-¢:] (11)
This probability is shown in Fig. 2 as function of r 2 with ¢ as parameter; r z can here be interpreted as a measure of the growth since it is proportional to the crack depth. 10
0
10-I 1o.2 10 3
0.0 1 0 .4
. . . . . . . . -2
10
i
. . . . . . .
101
1(~
2
Fig. 2. Fracture probability Pr for normally distributed crack depth.
Reliability assessment by aid of probabilistic fracture mechanics
For cases when
/(.4~2
347
and ¢ are small, eqn (11) reduces to Pr ~ 1 - exp
(--t'~"4)
(12)
This is the same as would have been obtained if the crack length had been assumed to be deterministic. We note from Fig. 2 that the approximative relation eqn (12) is reasonably accurate up to ¢ ~ 1 for interesting values of ~¢z. Thus, provided that the variability of the growth is not too large and the mean values sufficiently apart, the variability of the growth can be neglected. This can simplify the calculations in some cases. In many cases, however, the variability of the growth may be rather large because of the large dependence of the parameters governing growth on environmental and material conditions. A relative standard deviation exceeding unity is not uncommon judging from experimental crack growth investigations. Truncation of the fracture toughness distribution due to survival of previous loads can be accomplished by the substitution eqn (8) in the same way as for the first example. If the crack tip continues to grow, however, K will increase even though the loading conditions remain the same and the conditioning effect is less pronounced in this case. Furthermore, since the tip changes its position, the fracture toughness may vary along the prospective crack path, although in most cases this variation is probably small compared to the total uncertainty of the toughness properties. The latter conjecture does not apply to cases with irradiation damage. 3.3 Crack occurrence probability and nondestructive testing The examples considered may be further elaborated to account for the probability of crack occurrence, nondestructive testing (NDT), etc. It can thus be shown 5 that, for a region containing a number of cracks where each potential crack causes the same small fracture probability Pl, the combined fracture probability is of first order: Paot = ~/Pr-
(13)
Here, ~" is the expected number of cracks in the region. Equation (13) also holds when the occurrence of any crack is uncertain; N is then to be interpreted as the probability that one crack is present. NDT inspections can be considered as follows. Assume that the probability of not detecting a crack is p,d(a) only dependent of the crack length. The effect of n inspections on a time-invariant crack length distribution is accounted for by modifying f~ according to famod f~(a) (p.d(a))n =
(14)
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Fred Nilsson
Note here that the integral of famod with respect to crack depth is not unity. The difference represents the total probability that the potential crack is detected and does not contribute to the failure probability. The first example is again considered and we assume for simplicity that P,a = exp ( - ) , a ) , (15) and that n inspections have been performed before the considered time instant. The modified failure probability then becomes Pmoo = (1 + nZci)-'pi(r(1 + nZti)-"2), (16) with Pr defined by eqn (5) and K by eqn (6). The first factor of eqn (16) reflects the number of cracks that remain after the inspections. The much large effect becuase of the change of the argument of Pl is due to the fact that large cracks are relatively much easier to detect. The effect may be dramatic. If n~, = ti -~ the failure probability is reduced by a factor 1/16. For growing cracks the computations are more involved since the crack depth changes continuously. Assume that the crack depth is given by a(t) = C~(t), (17) where d(t) is a deterministic function and C is a random normally distributed constant with mean 1 and standard deviation ~/V~. Suppose that inspections are performed at the time instants tk with the efficiency implies by eqn (15). With the remaining assumptions as in the second example above, the probability that the crack is not detected by n inspections and causes fracture at time t is P~oo =
exp - C =0
~.~(t,)
(C; 1, ¢/V~)(1 - exp (-K(t)4C2)) dC
k=l
(18) Here, fa is given by eqn (9) and K(t) is obtained from eqns (4) and (6). The integral in eqn (18) can be obtained in closed form and an expression similar to, but more complicated than, eqn (11) results. Using the same degree of approximation as in eqn (12) the following result is obtained: Prmo~ {1 - exp ( - r 4 ) ) exp { - ~ gti(tk)}
(19)
k=l
3.4 Assessment o f detected cracks
When cracks are detected by ISis it is often of interest to assess these by probabilistic methods. Since the location and size of the crack in
Reliability assessment by aid of probabilistic fracture mechanics
349
these cases are known with some accuracy the difficulty of the probabilistic problem is greatly reduced as compared to the case when the crack is unknown. Since measurement errors are often reasonably well normally distributed, the second example above serves as a good illustration. The sizing capabilities of N D T are nowadays so good that the standard deviation should be small enough for the approximate relation eqn (12) to be applicable.
4 D A T A F O R PFM A N A L Y S E S In this section some comments will be made about the data available concerning some of the properties involved in PFM analyses. We can distinguish between two ways of specifying the input distribution functions. In the most direct one as much data as possible about the quantity are collected and histograms are formed. These then serve directly as estimates of the distributions. The direct approach has been used in several investigations. A more c o m m o n approach is to assume a functional form for the distribution and then estimate the parameters of the distribution from available data. In consequence with the arguments put forward earlier in this article, the second approach seems to be the more tractable.
4.1 Fracture toughness The statistical properties of fracture toughness have been the subject in rather a lot of investigations. These achievements have recently been reviewed by Wallin. 3 The distribution for K~c used in the examples here is, for example, discussed in Ref. 3. The distribution follows directly from a weakest link type of analysis provided true small scale yielding conditions prevail and that the microdefect distribution in the crack border vicinity is statistically homogeneous. Wallin also discussed the extension of this theory to situations where crack growth and large scale yielding occurs. Success with the verification to experimental data was also claimed. The perhaps largest weakness of this kind of model as regards their practical application is the assumption of statistical homogeneity. This assumption may be reasonable for a limited n u m b e r of well controlled specimens taken from the same region of say a plate. Gross differences due to say the manufacturing process are not likely to satisfy this assumption. This certainly widens the field of possible probability distributions.
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Fred Nilsson
4.2 Crack growth data In the literature one can find suggestions for the distributions of fatigue crack growth data. It is, however, important that the statistical analysis of experimental data is performed in an appropriate way. The fine scale variation (on a length scale about the same size as the length resolution) of growth velocities that is often observed is of little interest for probabilistic assessments. Unfortunately, this variation is often erroneously utilized to determine the distributions. It is instead the variation on a large scale, i.e. between plates, welds and batches that is of interest. One of the few examples of an appropriate approach is the work by Ditlevsen and Olesen 6 where these authors analysed data from 68 fatigue growth experiments on aluminium performed by Virkler et al. 7 They found, among other things, that the exponent of Paris's law is strongly correlated to the magnitude factor, which has also been observed by other authors. No analysis of the same nature seems to be available for stress corrosion crack growth.
4.3 Defect distributions Defects that remain in a reactor vessel would in most cases have occurred during the welding process. Since such defects are mainly due to mistakes made during the welding it seems difficult to base a distribution on physical modelling. Br/ickner and Munz 2 considered a weld consisting of a number of weld beads. Elementary cracks were assumed to occur in the weld beads which may be joined into larger cracks and through Monte Carlo simulation the resulting size and number distributions were studied. The results for the crack depth distribution showed good agreement with the exponential distribution. They also proposed an equation for the expected n u m b e r of cracks per unit length of weld. The exponential distribution was also utilized in the Marshall report 8 and later by Cameron 9 for evaluation of the importance of ISis on vessel reliability. Thus, some consensus seems to exist in the literature that the exponential distribution can be used to describe the depth distribution of pre-existing cracks. As pointed out by Briickner and Munz l° several distributions, including the exponential, can be very well fitted to published data although the distributions may behave very differently outside the observed range. The size distribution of growing cracks will d e p e n d on the distribution of the crack growth data and on the loading conditions. No
Reliability assessment by aid of probabilistic fracture mechanics
351
general statement about the resulting distribution can be given, and the normal distribution in the second example above was merely chosen for illustrative purposes.
4.4 NDT detection capabilities In the Marshall report 8, as well as in the work by Cameron 9 and others, different suggestions for the crack nondetection function were discussed and compared. It seems that for small and medium crack lengths an exponential function could be fitted to most suggestions. For large crack lengths, however, it is often argued that a lower limit to the crack nondetection probability should be introduced to account for operator errors. The effect of such a cut-off was studied by Cameron 9 and he found that if the cut-off was introduced at 10 -3 the resulting failure probability was 20 times the one obtained if no lower limit was assumed.
5 CONCLUDING REMARKS In the examples considered we have chosen to work with some simple assumptions. In a more ambitious assessment, nonlinear fracture mechanics may be applied and more complicated distribution functions may be used. This will, unfortunately, preclude the analytical treatment chosen here and instead the use of numerical methods as discussed above is necessary. The behaviour of the probabilistic functions at large distances from the mean is, however, very uncertain in most cases. It is an open question whether it is worthwhile to bother with more complicated expressions that are anyway reliable only in a limited central region which may not be of large importance for the end result. It is the belief of the author that observations of the type made here regarding the relative effects of different conditions are the most valuable contributions of PFM.
REFERENCES 1. Madsen, H. O., Krenk, S. & Lind, N. C., Methods of Structural Safety, Prentice-Hall, Engelwood Cliffs, NJ, 1986. 2. Brtickner, A. & Munz, D., A statistical model of crack formation in welds. Eng. Fract. Mech., 10 (1978) 223-32.
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Fred Nilsson
3. Wallin, K., Statistical modelling of fracture in the ductile to brittle transition region. In Defect Assessment in Components--Fundamentals and Applications, eds J. G. Blauel & K.-H. Schwalbe Mechanical Engineering Publications, London, 1991, pp. 415-46. 4. Abramovitz, M. & Stegun, I., (eds), Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards, Washington, DC, 1964, p. 298. 5. Nilsson, F., Probabilistic Fracture Mechanics--Lecture Notes, Report UPTEC 86/12K (rev. 1), Department of Technology, Uppsala University, Sweden, 1989. 6. Ditlevsen, O. & Olesen, R., Statistical analysis of the Virkler data on fatigue crack growth. Eng. Fract. Mech., 25 (1986) 177-95. 7. Virkler, D. A., Hillberry, B. M. & Goel, P. K., The statistical nature of fatigue crack propagation. Trans. ASME, J. Eng. Mater. Technol., 101 (1979) 148-53. 8. An Assessment of the Integrity of PWR Pressure Vessels, Second Report of a study group chaired by Dr W. Marshall, UKAEA, Harwell, UK. 9. Cameron, R. F., Calculating PWR pressure vessel failure frequencies. In Probabilistic Methods of Solids and Structures, eds S. Eggwertz & N. C. Lind. Springer, Berlin, pp. 331-42. 10. Briickner, A. & Munz, D., Probabilistic assessment of structures with weld defects. In Probabilistic Methods of Solids and Structures, eds S. Eggwertz & N. C. Lind. Springer, Berlin, pp. 343-53.
Reliability Assessment by Aid of Probabilistic Fracture Mechanics
F r e d Nilsson Department of Solid Mechanics, Royal Institute of Technology, Stockholm, Sweden
ABSTRACT Reliability assessments of nuclear power plants are often performed by using probabilistic safety analysis. To which extent probabilistic fracture mechanics (PFM) can be of value to assess the mechanical reliability of nuclear power components (especially the reactor vessel) is discussed, together with some particular problems with PFM applications. Some simple examples are given where the principles of PFM are illustrated.
1 INTRODUCTION There are several motivations for probabilistic assessment of the failure risk of a reactor pressure vessel (RPV). Since a vessel failure is virtually the only accident mechanism that is not accounted for in the design it is of great interest to know whether this remaining risk level is acceptable or not. It seems that the probabilistic approach furnishes the only possibility for a comparison of mechanical risks with risks because of other reasons. Further, when planning in-service inspection (ISI) activities, probabilistic reasoning may provide a method of how to select the most important areas for inspection and also how to choose the inspection intervals. Probabilistic safety analysis has become a widely used tool for systematic assessment of undesired events in complicated systems. The resulting probability for an unwanted event is obtained by combining failure probabilities of the components. In most cases the component 341 Int. J. Pres. Ves. & Piping 0308-0161/93/$06.00 (~) 1993 Elsevier Science Publishers Ltd,
England. Printed in Northern Ireland
342
Fred Nilsson
failure probabilities are obtained by direct observation of the failure rate. For a component such as the RPV this approach is not possible since failures have never occurred. In order to obtain estimates of the failure probability, indirect methods have to be used. Statistical information about various mechanical quantities is combined according to the methods of mathematical statistics. Such methods are often termed probabilistic fracture mechanics (PFM). 2 GENERAL OBSERVATIONS To perform a complete PFM analysis of a RPV which takes into account all possible phenomena and aims at exact estimates is a formidable task. The methodology for performing such analyses is well developed for certain classes of problems, namely those where the quantities involved are not stochastic processes in time. For solving such problems the so-called second moment methods I can be used with advantage as well as Monte Carlo methods and there exists several computer programs that are able to perform the analyses. When the problem necessitates modelling by aid of stochastic processes, such as is the case, e.g. for fatigue crack growth under random loading, the methodological problems are more difficult and the only general route open seems to be direct simulation by Monte Carlo methods. The main problems with PFM applications to RPVs are, however, not of a methodological nature, but rather the difficulties involved in obtaining relevant data. Since nuclear equipment is by design highly reliable, failure cannot in general occur because of an outcome of the involved variables that falls within the range of normally observed data. Thus, extrapolation to ranges without any observations must usually be done. This state of affairs has, at least in the author's opinion, the consequence that it is not particularly meaningful to aim at very precise and computationally complicated estimates of the failure probabilities. Instead, it is argued that fairly simple models can be used to obtain at least partial answers to questions of whether the reliability is sufficiently high and how this is affected by changes of the assumptions. Practical experience with fracture mechanics assessments of RPVs suggests that the potential problems that contribute to the failure probability can be divided into the following classes.
2.1. Fracture due to a pre-existing defect with insignificant growth In most parts of vessel the loading conditions are such that possible
Reliability assessment by aid of probabilistic fracture mechanics
343
fatigue crack growth is very limited and can in general be neglected. If no toughness degradation processes can occur it follows that failure can only take place if the loading conditions are more severe than the previous ones the vessel has been subjected to during its operation. Thus, defects of this kind are mainly of interest when judging the failure probability under accident loading conditions.
2.2. Fracture due to a pre-existing defect with insignificant growth in a region where toughness degradation processes can occur
The typical example is irradiation induced toughness degradation. The regions where this can occur are usually not loaded in such a way that fatigue crack growth is a problem. 2.3 Fracture due to an in-service induced defect with significant growth
If the conditions are such that a fatigue or stress-corrosion crack can be initiated during service its subsequent growth will in general be so rapid that if left to itself the crack will eventually grow to the critical size. It is not claimed that the categorization made here is by any means exhaustive, but it is the author's feeling that most of the potential problems can be put into one of these categories. The advantage with this is that in each category the random nature of some of the variables can be neglected and thus allow for a simpler treatment. This is to be illustrated by some examples below. The variables involved in PFM studies and which can be of random nature can be grouped into: (a) (b) (c) (d)
Material properties: fracture toughness, fatigue crack growth data, stress corrosion cracking susceptibility, etc. Loads: transients, fatigue loading, residual stresses, etc. Defects: fabrication defects, in-service induced defects. Inspection: reliability of inspection methods, in-service inspection plans, etc.
3 SOME E X A M P L E S OF PFM ANALYSIS In this section some highly idealized examples are considered. The main objects are to illustrate the methods and problems of PFM and to
Fred Nilsson
344
draw some potentially useful conclusions. The following situation is assumed. In a certain region it is required to estimate the failure probability of potential cracks. A s s u m e that the shape of the cracks is known and the crack size can be characterized by a depth p a r a m e t e r a, which is distributed according to the frequency function f,. The stress state is assumed to be d e t e r m i n e d so that the stress intensity factor is K~(a). For simplicity is assumed that linear elastic fracture mechanics applies and the fracture toughness is distributed as Fr,c. With these assumptions the fracture probability because of this potential crack can be written: ps = ~
aa= o
f~(a)FK,°(Ki(a))da
(1)
3.1 Exponentially distributed crack depth We first consider an example that might be representative for the case of a pre-existing defect with insignificant growth. As specific forms for the distributions it is assumed that the crack depth is exponentially distributed 2 and the fracture toughness is Weibull distributed: 3
(-a/~t)/a FK,~= 1 -- exp ( - ( K,/ Ko)4). f~ = exp
(2) (3)
~/is the m e a n crack depth and 0-9064K0 is the m e a n fracture toughness. The relevance of these distributions is to be discussed below. We now assume that
K, = g , ( a l a ) ''2
(4) where / ~ is the stress-intensity factor corresponding to t/ and the current load situation. Insertion into eqn (1) and performing the integration lead to Pl = 1 - exp (1/4r4)(1 -
~¢=
~(1/2K2))(V'-~/2K 2)
K,,IKo
(5)
(6)
Here, • denotes the error function. Since K is usually a small quantity it is advantageous to use an asymptotic expansion of ~4 and eqn (6) can be written as Pi = - ~
( - 1 ) m x 1 x 3 x • • • x (2m - 1)22mK 8'',
K < 8 -1/4 = 0 " 5 9 7
m=l
(7) Pr is shown as function of r z in Fig. 1.
Reliability assessment by aid of probabilistic fracture mechanics 10°t lOlt
1°2t 10-3t lO-'t Pl lO-St lO-°t 10-7 ~,,/ 10 -8 ]
10 "z
345
/ / ................. 10-1 tq 2
10°
Fig. 1. Fracture probability Pr for exponentially distributed crack depth. The sensitivity of Pi on r is noted. K2 can be interpreted as being proportional to the m e a n crack d e p t h divided by the m e a n critical crack depth. Thus, values of r 2 a r o u n d 0-05 o u g h t to be typical. We n o t e that an increase of r 2 by 20% (10% increase of K) causes a doubling of the failure probability. Since r d e p e n d s linearly on stress and inversely on fracture toughness it is easily u n d e r s t o o d that it is futile to search for very accurate estimates. O f s o m e interest is also to estimate the increase in failure probabiity corresponding to the different safety factors applied in normal and accident situations according to the A S M E XI rules. T h e ratio b e t w e e n the safety factors is ~/10:2 with respect to stress which corresponds to an increase of Pl of about 6 x 102. Suppose now that the region has survived a loading situation that corresponds to a value r~ according to the definition of eqns (4) and (6). This means that the toughness distribution can be truncated at KoKl(a/~) ~/z. A load corresponding to K2 applied later, can only cause fracture if it creates a m o r e severe situation, i.e. K2 > ~:1. T a k i n g this truncation of eqn (3) into account w h e n p e r f o r m i n g the integration, eqn (1), it can easily be shown that eqns (5) and (7) remain valid also for this conditional probability provided the following substitution is made: /("-'--9' (/~4 -- /(.4)1/4
(8)
This shows that loads slightly larger than the n o r m a l ones do not cause a large failure probability while the probability of failure for very severe loads is hardly affected by the fact that the vessel has survived the previous loads. T h e s t a t e m e n t m a d e u n d e r 2.1 above is thereby supported.
Fred Nilsson
346
This example can be further developed to account for degradation of the toughness by letting Ko be a decreasing function of time.
3.2 Normally distributed crack depth In the second example we assume the fracture toughness to be distributed as above while the crack depth is normally distributed with mean crack depth 5(0 and standard deviation sa(t), which are both functions of time. This may serve as an illustration for the analysis of growing cracks. The normal distribution is truncated at a = 0 to account for the non-negativity of crack depth. Thus, at a particular time instant:
fa(a;&sa)
=
(2/G(2~)l/2)exp(-(a
-
ti)Z/2sZ)(l+~(sa-~))
,a>O
(9)
Making assumptions analogous as above and setting
= (saV~)/K
(10)
Integration of eqn (1) gives p; = 1 _
(1 + K4~2)1/2(1 q'- (P(1/~))
K 4
1 + (I) ~(1 + 12~2)1/2]) exp [ 1 + ~-q-¢:] (11)
This probability is shown in Fig. 2 as function of r 2 with ¢ as parameter; r z can here be interpreted as a measure of the growth since it is proportional to the crack depth. 10
0
10-I 1o.2 10 3
0.0 1 0 .4
. . . . . . . . -2
10
i
. . . . . . .
101
1(~
2
Fig. 2. Fracture probability Pr for normally distributed crack depth.
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For cases when
/(.4~2
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and ¢ are small, eqn (11) reduces to Pr ~ 1 - exp
(--t'~"4)
(12)
This is the same as would have been obtained if the crack length had been assumed to be deterministic. We note from Fig. 2 that the approximative relation eqn (12) is reasonably accurate up to ¢ ~ 1 for interesting values of ~¢z. Thus, provided that the variability of the growth is not too large and the mean values sufficiently apart, the variability of the growth can be neglected. This can simplify the calculations in some cases. In many cases, however, the variability of the growth may be rather large because of the large dependence of the parameters governing growth on environmental and material conditions. A relative standard deviation exceeding unity is not uncommon judging from experimental crack growth investigations. Truncation of the fracture toughness distribution due to survival of previous loads can be accomplished by the substitution eqn (8) in the same way as for the first example. If the crack tip continues to grow, however, K will increase even though the loading conditions remain the same and the conditioning effect is less pronounced in this case. Furthermore, since the tip changes its position, the fracture toughness may vary along the prospective crack path, although in most cases this variation is probably small compared to the total uncertainty of the toughness properties. The latter conjecture does not apply to cases with irradiation damage. 3.3 Crack occurrence probability and nondestructive testing The examples considered may be further elaborated to account for the probability of crack occurrence, nondestructive testing (NDT), etc. It can thus be shown 5 that, for a region containing a number of cracks where each potential crack causes the same small fracture probability Pl, the combined fracture probability is of first order: Paot = ~/Pr-
(13)
Here, ~" is the expected number of cracks in the region. Equation (13) also holds when the occurrence of any crack is uncertain; N is then to be interpreted as the probability that one crack is present. NDT inspections can be considered as follows. Assume that the probability of not detecting a crack is p,d(a) only dependent of the crack length. The effect of n inspections on a time-invariant crack length distribution is accounted for by modifying f~ according to famod f~(a) (p.d(a))n =
(14)
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Note here that the integral of famod with respect to crack depth is not unity. The difference represents the total probability that the potential crack is detected and does not contribute to the failure probability. The first example is again considered and we assume for simplicity that P,a = exp ( - ) , a ) , (15) and that n inspections have been performed before the considered time instant. The modified failure probability then becomes Pmoo = (1 + nZci)-'pi(r(1 + nZti)-"2), (16) with Pr defined by eqn (5) and K by eqn (6). The first factor of eqn (16) reflects the number of cracks that remain after the inspections. The much large effect becuase of the change of the argument of Pl is due to the fact that large cracks are relatively much easier to detect. The effect may be dramatic. If n~, = ti -~ the failure probability is reduced by a factor 1/16. For growing cracks the computations are more involved since the crack depth changes continuously. Assume that the crack depth is given by a(t) = C~(t), (17) where d(t) is a deterministic function and C is a random normally distributed constant with mean 1 and standard deviation ~/V~. Suppose that inspections are performed at the time instants tk with the efficiency implies by eqn (15). With the remaining assumptions as in the second example above, the probability that the crack is not detected by n inspections and causes fracture at time t is P~oo =
exp - C =0
~.~(t,)
(C; 1, ¢/V~)(1 - exp (-K(t)4C2)) dC
k=l
(18) Here, fa is given by eqn (9) and K(t) is obtained from eqns (4) and (6). The integral in eqn (18) can be obtained in closed form and an expression similar to, but more complicated than, eqn (11) results. Using the same degree of approximation as in eqn (12) the following result is obtained: Prmo~ {1 - exp ( - r 4 ) ) exp { - ~ gti(tk)}
(19)
k=l
3.4 Assessment o f detected cracks
When cracks are detected by ISis it is often of interest to assess these by probabilistic methods. Since the location and size of the crack in
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these cases are known with some accuracy the difficulty of the probabilistic problem is greatly reduced as compared to the case when the crack is unknown. Since measurement errors are often reasonably well normally distributed, the second example above serves as a good illustration. The sizing capabilities of N D T are nowadays so good that the standard deviation should be small enough for the approximate relation eqn (12) to be applicable.
4 D A T A F O R PFM A N A L Y S E S In this section some comments will be made about the data available concerning some of the properties involved in PFM analyses. We can distinguish between two ways of specifying the input distribution functions. In the most direct one as much data as possible about the quantity are collected and histograms are formed. These then serve directly as estimates of the distributions. The direct approach has been used in several investigations. A more c o m m o n approach is to assume a functional form for the distribution and then estimate the parameters of the distribution from available data. In consequence with the arguments put forward earlier in this article, the second approach seems to be the more tractable.
4.1 Fracture toughness The statistical properties of fracture toughness have been the subject in rather a lot of investigations. These achievements have recently been reviewed by Wallin. 3 The distribution for K~c used in the examples here is, for example, discussed in Ref. 3. The distribution follows directly from a weakest link type of analysis provided true small scale yielding conditions prevail and that the microdefect distribution in the crack border vicinity is statistically homogeneous. Wallin also discussed the extension of this theory to situations where crack growth and large scale yielding occurs. Success with the verification to experimental data was also claimed. The perhaps largest weakness of this kind of model as regards their practical application is the assumption of statistical homogeneity. This assumption may be reasonable for a limited n u m b e r of well controlled specimens taken from the same region of say a plate. Gross differences due to say the manufacturing process are not likely to satisfy this assumption. This certainly widens the field of possible probability distributions.
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4.2 Crack growth data In the literature one can find suggestions for the distributions of fatigue crack growth data. It is, however, important that the statistical analysis of experimental data is performed in an appropriate way. The fine scale variation (on a length scale about the same size as the length resolution) of growth velocities that is often observed is of little interest for probabilistic assessments. Unfortunately, this variation is often erroneously utilized to determine the distributions. It is instead the variation on a large scale, i.e. between plates, welds and batches that is of interest. One of the few examples of an appropriate approach is the work by Ditlevsen and Olesen 6 where these authors analysed data from 68 fatigue growth experiments on aluminium performed by Virkler et al. 7 They found, among other things, that the exponent of Paris's law is strongly correlated to the magnitude factor, which has also been observed by other authors. No analysis of the same nature seems to be available for stress corrosion crack growth.
4.3 Defect distributions Defects that remain in a reactor vessel would in most cases have occurred during the welding process. Since such defects are mainly due to mistakes made during the welding it seems difficult to base a distribution on physical modelling. Br/ickner and Munz 2 considered a weld consisting of a number of weld beads. Elementary cracks were assumed to occur in the weld beads which may be joined into larger cracks and through Monte Carlo simulation the resulting size and number distributions were studied. The results for the crack depth distribution showed good agreement with the exponential distribution. They also proposed an equation for the expected n u m b e r of cracks per unit length of weld. The exponential distribution was also utilized in the Marshall report 8 and later by Cameron 9 for evaluation of the importance of ISis on vessel reliability. Thus, some consensus seems to exist in the literature that the exponential distribution can be used to describe the depth distribution of pre-existing cracks. As pointed out by Briickner and Munz l° several distributions, including the exponential, can be very well fitted to published data although the distributions may behave very differently outside the observed range. The size distribution of growing cracks will d e p e n d on the distribution of the crack growth data and on the loading conditions. No
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general statement about the resulting distribution can be given, and the normal distribution in the second example above was merely chosen for illustrative purposes.
4.4 NDT detection capabilities In the Marshall report 8, as well as in the work by Cameron 9 and others, different suggestions for the crack nondetection function were discussed and compared. It seems that for small and medium crack lengths an exponential function could be fitted to most suggestions. For large crack lengths, however, it is often argued that a lower limit to the crack nondetection probability should be introduced to account for operator errors. The effect of such a cut-off was studied by Cameron 9 and he found that if the cut-off was introduced at 10 -3 the resulting failure probability was 20 times the one obtained if no lower limit was assumed.
5 CONCLUDING REMARKS In the examples considered we have chosen to work with some simple assumptions. In a more ambitious assessment, nonlinear fracture mechanics may be applied and more complicated distribution functions may be used. This will, unfortunately, preclude the analytical treatment chosen here and instead the use of numerical methods as discussed above is necessary. The behaviour of the probabilistic functions at large distances from the mean is, however, very uncertain in most cases. It is an open question whether it is worthwhile to bother with more complicated expressions that are anyway reliable only in a limited central region which may not be of large importance for the end result. It is the belief of the author that observations of the type made here regarding the relative effects of different conditions are the most valuable contributions of PFM.
REFERENCES 1. Madsen, H. O., Krenk, S. & Lind, N. C., Methods of Structural Safety, Prentice-Hall, Engelwood Cliffs, NJ, 1986. 2. Brtickner, A. & Munz, D., A statistical model of crack formation in welds. Eng. Fract. Mech., 10 (1978) 223-32.
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3. Wallin, K., Statistical modelling of fracture in the ductile to brittle transition region. In Defect Assessment in Components--Fundamentals and Applications, eds J. G. Blauel & K.-H. Schwalbe Mechanical Engineering Publications, London, 1991, pp. 415-46. 4. Abramovitz, M. & Stegun, I., (eds), Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards, Washington, DC, 1964, p. 298. 5. Nilsson, F., Probabilistic Fracture Mechanics--Lecture Notes, Report UPTEC 86/12K (rev. 1), Department of Technology, Uppsala University, Sweden, 1989. 6. Ditlevsen, O. & Olesen, R., Statistical analysis of the Virkler data on fatigue crack growth. Eng. Fract. Mech., 25 (1986) 177-95. 7. Virkler, D. A., Hillberry, B. M. & Goel, P. K., The statistical nature of fatigue crack propagation. Trans. ASME, J. Eng. Mater. Technol., 101 (1979) 148-53. 8. An Assessment of the Integrity of PWR Pressure Vessels, Second Report of a study group chaired by Dr W. Marshall, UKAEA, Harwell, UK. 9. Cameron, R. F., Calculating PWR pressure vessel failure frequencies. In Probabilistic Methods of Solids and Structures, eds S. Eggwertz & N. C. Lind. Springer, Berlin, pp. 331-42. 10. Briickner, A. & Munz, D., Probabilistic assessment of structures with weld defects. In Probabilistic Methods of Solids and Structures, eds S. Eggwertz & N. C. Lind. Springer, Berlin, pp. 343-53.