On the Markov approximation of fatigue crack growth

On the Markov approximation of fatigue crack growth Wen-Fang Wu Department of Civil Engineering and Engineering Mechanics, Columbia University, New York N Y 10027, USA Some recent developments on the probabilistic modelling of fatigue crack growth are reviewed here and special attention is given to several Markov approximations. A continuously parametered discrete Markov process model is then introduced. In terms of analytical sophistication, this model represents an approach between the Markov chain model and the continuous Markov process model used by other researchers. It is easy to apply and probabilistic quantities such as the distribution function of random time reaching a certain crack size, the probability of crack exceedance at a given time, constant-probability crack growth curves and generation of the crack sample functions can also be easily obtained. Three numerical examples are illustrated using the proposed model and the results are compared with available experimental data and other researchers' results.

INTRODUCTION Even in a well-controlled laboratory environment, the results obtained from crack growth experiments under either constant-amplitude cyclic loading or a given spectrum loading usually exhibit considerable statistical variability t-3. Therefore, probabilistic analyses of the fatigue growth phenomenon are quite often employed. Traditionally, the famous Palmgren-Miner rule was extensively applied to study the damage accumulation of fatigue cracks. Statistical quantities for the damage are then determined through an assumed loading process 4' s. Recently, probabilistic and/or stochastic models basea on randomization of the Paris-Erdogan crack growth equation were proposed by many researchers. Among them, Yang considered the random factor as a random variable6'7, and Madsen assumed the random factor consists of a random variable describing variations between mean values in different specimens plus a positive random process describing variations from the mean value along the crack path within each specimen s'a. The random process is further assumed to be stationary with respect to the crack size. It may even further be assumed as a white noise or other random process which has a certain correlation function. Similar model has been used by Ortiz who also applied time series analyses in his work 9. After randomization of the Paris-Erdogan equation, various failure-related probabilities or damage indices can be calculated by integrating the ParisErdogan equation. In many cases, analytical solutions may not be obtained easily due to numerical complexity. Monte Carlo simulation can usually be applied to deal with problems of this nature as demonstrated in a paper by Itagaki and Shinozuka ~°. Instead of modelling the random factor as a spatially correlated random process, Lin and Yang added a Received September 1986. Discussion closes February 1987.

nonnegative temporal correlated stochastic process to the right-hand side of the Paris-Erdogan equation to represent the variability of the crack growth 1Lt 2. It was pointed out in another report 7 that if the stochastic process is totally independent at any two different times it will lead to a result having the smallest statistical dispersion. On the other hand, if the process is totally correlated at any two different times, it reduces to a random variable, resulting in the greatest statistical dispersion for the crack growth 7. Actual experimental data suggest that a more realistic modelling of the fatigue crack growth should lie somewhere between these two extremes. Thus, in their paper, Lin and Yang modelled the stochastic process as a stationary random pulse train which has a positive mean value and a certain correlation function. Then they applied Stratonovich's method to approximate the crack growth process as a Markov process and solve the corresponding Fokker-Planck equation ~3. Several statistical properties of the growth can then be obtained. A similar method was also used by Tsurui and Ishikawa t4. In addition to the Markov approximation, they also introduced a death point probability to represent the probability of the crack size which has already gone to infinity at the time of consideration. The initial flaw distribution, size effect and multivariate crack growth were also considered in their subsequent papers t s-t Several different ways of randomization of the ParisErdogan crack growth equation have also been discussed by Kozin and Bogdanoff1a. But their major contribution to the field is a discrete-time, discrete-state Markov chain model to represent the cumulative damage of the fatigue crack or wear growth 19-26. Their basic idea is to discretize the crack size as several pre-defined states which may include replacement and failure states, and define a duty cycle to be a representative period of operation during which damage may accumulate. The change of states and damage accumulation at the end of each cycle depends in

0266-8920/86/040224-1052.00 © 1986 Computational Mechanics Publications

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Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

On the Markov approximation: Wen-Fan 0 a probabilistic manner only on the amount of damage present at the start of the duty cycle and the transition matrix, and is indepdendent of how damage was accumulated up to the start of the duty cycle. Thus, an initial state, a basic transition matrix or transition matrices, and a specification of damage state plus inspection and replacement, if there is any, will be able to characterize the probabilistic structure of the crack growth. A similar procedure can also be found in a paper by Ghonem and Dore where a continuously parametered discrete birth process, which will be discussed later in the present paper, was incorporated into the discontinuous Markov model 2~. It is noted that because of its simplicity the Markov process approximation has been widely used in the study of probabilistic fatigue crack growth. Therefore, in the following section, several Markov process representations of the fatigue crack growth will be discussed. Following that, a very simple Markov model will be applied to a fatigue crack growth problem. Finally, three examples will be illustrated for the proposed Markov model which will result in probabilistic quantities such as the distribution function of random time reaching a certain crack size, the probability of crack exceedance at a given time and constant-probability crack growth curves. Simulation of crack growth sample functions corresponding to the estimated statistical quantities obtained from the experimental data can also be done very easily. MARKOV APPROXIMATION

The reason for the use of Markov process approximation in a fatigue crack growth problem is its mathematical simplicity and the many nice properties it possesses. Basically, in the simplest Bogdanoff and Kozin stationary Markov chain model, an initial state distribution Po and a duty cycle independent basic transition matrix Q are sufficient to describe the entire evolutionary probabilistic structure. The probability distribution right after m cycles, P,,, is obtained as

P~=PoQ ~

(1)

where P , and Po are n-dimensional vectors representing the probability distributions with respect to predefined n states, and Q is the basic n x n probability transition matrix. The transition matrix between any different duty cycles can be constructed from the basic transition matrix. It is well known that the probability transition matrices of a stationary Markov process satisfy the following Chapman-Kolmogorov equation,

Q"'+~=Q",Q":

(2)

which can also be written in the following index form, n

~ , +.2~_ - - ~.l.~.,~.~ ~ l i j ~l j k J tlik

(2a)

in which q~.~,)denotes the probability of transition from state i to state j in exactly/~, duty cycles. It also represents the (i,j)th element of matrix Q", in equation (2). The Chapman-Kolmogorov equation reflects the simple fact that the first /~1 steps lead from state i to some intermediate state j, and that the probability of a subsequent passage from state j to state k does not depend

on the manner in which state j was reached. The stationarity of the process can also be seen from the following property of the transition probabilities.

q~2-m')=Prob{X~m2~=klX~m')=i}

(3)

That is, the transition probabilities depend only on the cyclic difference m2 - m 1. The probabilities are independent of the initial m 1th duty cycle and the present ra2th duty cycle. In many fatigue crack growth problems, a stationary Markov chain is not able to describe the complicated growth behaviour and a nonstationary Markov chain is therefore needed 23'2s'26. The nonstationarity is usually named with respect to the counting duty cycles, that is, discretized temporal nonstationarity. In this case, the transition probability on the right-hand side of equation (3) depends not only on the states but also on the counting cycles mI and m2. It is then more proper to denote the transition probability density a s qlk(ml,m2). Under this condition, the stationary Chapman-Kolmogorov equation can be generalized as & qik(ml, m2) = ~ qi~(ml, m)q ik(m, m2) J

(4)

or, in matrix form,

Q(ml,m2)=Qml+lQ~,+2...Qm~-lQm2

(4a)

Equations (4) and (4a) are valid for all m with ra1 < m < m 2. At this moment, it is not difficult to see that the Chapman-Kolmogorov equation plays an important role in an evolutionary Markov process model. Most of the Markov process-related probabilistic quantities can be derived from this equation, or at least, satisfy it. Nevertheless, it should be noted that although a Markov process should satisfy the Chapman-Kolmogorov equation, the reverse is not necessarily true. That is, there may be some non-Markovian process which also satisfy the Chapman-Kolmogorov equation 2s and it is beyond the discussion of the present paper, For the crack growth problem, unlike Bogdanoff and Kozin's Markov chain approximation which is directly applied to the fatigue crack growth phenomenon, Lin and Yang, and Tsurui and Ishikawa made the Markov process assumption after randomizing the Paris-Erdogan equation. There are two major differences between these two approaches. First, in Bogdanoff and Kozin's approach, they discretize both the time and the crack size, while in the Lin/Yang and Tsurui/Ishikawa approaches both the time and crack size are considered to be continuous variables. Secondly, since a random factor is added to the crack growth equation in the latter case, it is thus more of a dynamic problem instead of being just a static problem from the mechanics point of view. Nevertheless, owing to the Markov assumption used in both cases, one can still find similarities between them. For example, the Chapman-Kolmogorov equation for the continuously parametered (time) continuous (state) random process becomes 29

qlxltx2,zl +z2lxl)=

Ix1)q{X}(X2,T21X)dx

(5)

Probabilistic Engineering Mechanics, 1986, Vol. I, No. 4

225

for a stationary process and

On the Markov approximation: Wen-Fang Wu [, qlxl(X2, t:-2Ix1, tl) =

jq x (x,t Ixl, t~ )q{xl(Xz, t 2Ix,

t)

dx

(6) for a nonstationary process. Equation (6) is valid for t 1 < t < t2 and equation (5) can be derived from equation (6) by letting z i = t - t ~ and ~ 2 = t 2 - t . The left-hand side of equation (6) denotes the transition probability from state x~ at time t~ to state x2 at time t2. It is interesting to note that both state x and time t are now continuous variables in both equations. The Fokker-Planck equation which reflects the dynamical behaviour of the Markov process can be derived from the Chapman-Kolmogorov equation 29. By solving this Fokker-Planck equation as Lin and Yang and Tsurui and Ishikawa did in their papers, many probabilistic and/or statistical quantities related to the crack growth can be obtained. It has been discussed that in the Bogdanoff and Kozin Markov chain approach, due to discretization of the time and crack size, many parameters have to be evaluated from experimental data in order to be able to properly represent the real growth behaviour. The finer state discretization and time step used will result in a more accurate crack growth representation at the expense that a larger Markov transition matrix and more matrix multiplications are required. Of course, the unit time step may sometimes be considered the true duty cycle. The temporal representation in the present paper will simply be called 'time' hereafter and it should be clarified that it is equivalent to the term 'duty-cycle' used in some papers. Contrary to the Markov chain approach, in the Lin/Yang and Tsurui/Ishikawa approaches, the real amount of computation is quite light. Only a few parameters need to be evaluated from experimental data. However, there are also shortcomings to this approach. First, there is no guarantee that the Fokker-Planck equation is always solvable. In many cases, approximate solutions or numerical solutions are needed. Secondly, as compared to Markov chain approch, since only a few parameters are evaluated from the experimental data, it does not capture the stastical nature of the real crack growth behaviour as well. The stochastic crack growth process originating from it may sometimes not be able to adequately represent the real growth as the Markov chain approach does. The purpose of the present paper is not to reiterate or . compare the superiority of the two Markov approaches. Instead, it is to illustrate a third Markov process approach for the fatigue crack growth problem. The proposed approach actually lies between the two discussed Markov approaches. It is within the category of a continuously parametered discrete (state) Markov process. The transition probabilities of this kind of Markov process also satisfy the following ChapmanKolmogorov equation

qik(Zl + Z2) = ~ qlj(zl)qjk('C2)

(7)

J

for a stationary process and n

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(8)

J

for a nonstationary process, in which

226

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nomenclature used in equations (7) and (8) is the same as that explained in equations (2) and (4). It is interesting to note that the total state n in the present case sometimes goes to infinity, while in the Markov chain approach, i.e., equation (2), the total state n may be infinite, but the matrix representations of the state vector and the transition matrix become more abstract. It is also noted that the change of discrete state now may occur at any time. Mathematically speaking, the stochastic process involves only coutable finite or infinite states but depends on a continuous time parameter. The famous birth and death process and Poisson process all belong to this category, but they are allowed to travel only to the neighbouring state(s) within a considerably small time interval. This small time interval is sometimes called the observation time or characteristic time. It is interesting to note that in Lin and Yang's papers, they also mention that the Markov approximation is justified only when the correlation time of the added random process is relatively short compared with the characteristic time of the growth process. The characteristic time should also be small enough for all continuously parametered Markov process approaches to ensure the correct probability measurement of the stochastic process. The Poisson process can be considered a special case of the pure birth process which allows the transition to go only from one state to the next state within the characteristic time. Its intensity may depend on the time, i.e., nonstationary (or nonhomogeneous) Poisson process, but is independent of the states. The Poisson process has been extensively studied and applied in many scientific fields. It is perhaps the simplest process which has independent increments and also satisfies the Markov properties. In the fatigue crack growth considered here, since both the crack size and the time increase in only one direction, it may then be approximated as a pure birth process or a Poisson process following analytical procedures similar to those of Bogdanoff and Kozin in their Markov chain approximation. Based on above discussions, the approach can, in certain sense, be considered parallel to Lin and Yang's and Tsurui and Ishkawa's Markov process approximations. In fact, a similar observation between the Markov chain and the semi-Markov process was made by Bogdanoff and Kreiger. But their conclusion on the use of the Poisson process was negative rather than positive 2°.

POISSON REPRESENTATION It is well-known that the mean and variance of a Poisson process are both equal to its mean value function at any time of consideration. Statistical results from the fatigue crack experimental data shown in Table 1 indicate that the growth behaviour is far away from being able to be approximated by a Poisson process. Nevertheless, one can try applying a linear transformation to a certain measurable quantity and it may result in a Poisson process. Before pursuing this any further, the crack size a will be considered as the independent variable herein, while the number of duty cycle N or time t will be the dependent variable during the growth phase. This is different from most of the other work on this subject. There are several reasons for doing this. First, the crack size a in the growth phase in reality develops continuously, while the time variable N is sometimes

Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

On the Markov approximation: Wen-Fan9 Wu Table 1. Statisticaldata from Ref 22 i

a (inches)

th~

61

1 2 3 4 5 6 7 8 9

9 11 13 17 20 26 33 42.4 49.8

0 55 700 91 100 139400 163 900 198 800 226 500 248 500 251200

0 6 580 93 500 11 140 12 400 13 830 15 210 17 430 18 310

counted as a discrete duty cycle. Secondly, the divergence of the Markov process (discussed by Tsurui and Ishikawa 14-17) can be avoided. Finally, according to Kozin and Bogdanoff's study of the randomized crack equation, it is found that the inverted form (i.e., treating a as the independent variable and N as the dependent variable) is more consistent with the real crack growth process. For statistical data such as those shown in Table 1, although N(a) cannot appropriately be approximated by a Poisson process, there may be a linear transformation

N,=TN

(9)

which makes Nt(a) closer to a Poisson process. The mean square error in making this Poisson approximation can be evaluated as e=

no

~

k,[yrh,- (76,) 2] 2

(10)

i

where the th~s and #is are the estimated means and standard deviations obtained from the experimental data shown in the third and fourth columns of Table 1; y is the scaling constant defined in equation(9); the kls are weights, i.e., emphases on the observed points; and no is the number of observed points, for example, no = 8 in Table 1 (the first 'deterministic' data point is not included). It is interesting to note that if more of these data points are considered one may obtain a more accurate approximation as in the comment made by Bogdanoff and Kozin z2. It is also interesting to note that in Lin and Yang's papers, the proper correlation time of the driving random process also has to be chosen to reproduce the experimentally comparable means and variances (see Figs 5, 6, i1 and 12 in Ref. 11). The degree of freedom in the latter case is obviously less than that of the former as discussed in a previous section. In the current approach, the minimum value ofT, hence the best approximation of Nt(a ) by a Poisson process, can easily be found as

in which a o is the initial crack size and S and D are constants which can be determined from a curve fitting technique. In fact, equation (12) is not always required in the present analysis since the real mean cycle (time) rh(a), hence the transformed raN,, as a function of the crack size can always be found from a statistical analysis of the experimental data 3°. However, the equation will be helpful in the analytical calculation of the probability of crack exceedance and simulation (reproduction) of the crack growth function as will be shown later. If the approximation of N,(a) by a Poisson process is considered appropriate, then many probabilistic descriptions related to the Poisson process can easily be determined. These descriptions can then be transformed back to represent the original fatigue crack growth process. For example, the probability distribution function of random time reaching a specified crack size a is

P[N <~n,mN,(a)]= '(~") ~ e_,.s,~, m~,(_a) j!

(13)

j=O

where l(yn) denotes the nearest integer of yn which is selected to be compatible with the Poisson approximation. For a more conservative approximation, the smallest integer which is greater than 7n can be used instead. Of course, there is no guarantee that this approximation is always appropriate. Fortunately, the numerical examples studied in the following section using this approximation all agree quite well with the experimental data and other researchers' results as will be seen later. The constant-probability crack growth curves as mentioned in Refs 6 and 27 can also be constructed by using equation (13) together with some computer algorithm. The probability distribution function of the crack size a at time (cycle) n, hence n, can be calculated from the waiting time distribution of the Poisson process Nt(a ) as a

'Ja o

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[

d~

ja~ (14)

The crack exceedance curve P* =,(a) is then calculated as P*=.(a)= 1 -PN=n(A <~a)

(15)

Finally, one can use the present nonhomogeneous Poisson process model to generate sample functions which represent the fictitious crack growth behaviour and are probabilistically/statistically consistent with the experimentally obtained data.

N U M E R I C A L EXAMPLES 3 ~ ' . k,rh,a2 + ,/(3 ~'o k;h,a2) 2 -8(Z~'. k,a~)(~', k,rh~)

7=

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(11) where the plus or minus sign in front of the square root will be decided by the real situation. It is found that in many cases the transformed mean curve mu,(a) can be fitted into the following expression,

mN,(a)=S{1 - e x p [ - D ( a - a o ) ] }

(12)

Three numerical examples are studied using the present nonhomogeneous Poisson process model. Each example generates results that are statistically compatible with the experimentally obtained crack growth data. In fact, all probabilistic or stochastic crack growth models can be considered to be designed as able to represent the actual crack growth phenomenon from the limited experimental data at hand. Otherwise, they are just mathematical models with little physical reality. The crack in the first example is considered in the larger size crack range while

Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

227

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THOUSAND CYCLES

Fig. 2. Probability distribution functions of random time reaching crack sizes 14.8 and 20 mm for centre-cracked specimens

1.0

0,8

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Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

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The experimental data of 7475 - T7531 aluminum fastener hole specimens subjected to a bomber load spectrum are reported in Ref. 3. Probabilistic and/or statistical studies of this data set can be found in Refs 3, 7, 11 and 12. The crack growth historical curves are similar to those of the centre-cracked specimens. Two data sets, referred to as the XWPB data and WPB data are recorded. The letters W, P and B indicate that the specimens were drilled with a Winslow Spacematric machine (W) using a proper drilling technique (P) and subjected to a given B-1 bomber load spectrum (B). The additional symbol X for XWPB data set has the meaning that the fasteners are configured to transfer 15% load whereas the WPB fasteners transfer no load. The intial crack length for both data sets is 0.004 inches. Twentytwo and sixteen specimens were tested and recorded for the XWPB and WPB data, respectively. After the optimal scaling factor y is found from equation (11), the closeness of the Poisson process assumption in the transformed time-crack domain can be judged from Fig. 6. For the current XWPB data set, nine data points which are used in the determination of the optimal scaling factor are also shown in the same figure. It i s - f o u n d that the Poisson approximation in the transformed domain fits better than the previous example, but the transformed dependent variable N,, contains fewer states than before. Again, it indicates that fewer degrees of freedom are considered as compared to a Markov chain model with more discretized states. Nevertheless, the results obtained from the present model show that the proposed approach is still applicable as will be seen in the next paragraph. Following the same procedures as in the previous example, the probability distribution of the random time reaching crack sizes a = 0.008, 0.025 and 0.07 inches are computed from equation (13). The results are shown in Fig. 7. Also shown in the same figure are the curves obtained by Lin and Yang using the continuously

Fig. 5. Sample functions (a) generated from the present model, (b) generated from the Markov chain model I a, and (c) from experimental data 2, for centre-cracked specimens

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Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

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On the Markov approximation: Wen-Fang Wu 30.

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parametered continuous Markov process model. The corresponding test results are also displayed in the figure for comparison. The crack exceedance probability at 6000 flight hours is plotted in Fig. 8 along with Lin and Yang's result and test results. The constant-probability growth curves are plotted in Fig. 9 where the first figure (a) is the direct result from the model and the second figure (b) shows the same curves after being smoothed. It is interesting to note that the smoothing procedure has also been applied by Kozin and Bogdanoff since most of their results are discrete quantities 18. In fact, even the original growth curves shown in Fig. 5(c) have been smoothed from 164 data points and straight-line segments. Finally, 100 crack growth sample functions corresponding to XWPB data set are generated shown in Fig. 10.

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Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

On the Markov approximation: Wen-Fang Wu 70.

generated sample functions (before smoothing) and those generated by Bogdanoff (e.g., Figs 2 and 3 in Ref. 21), that is, the small step jumps of the sample functions, which is caused by the discretization of the state.

.

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CONCLUDING REMARKS A continuously parametered discrete Markov process is proposed to model a probabilistic fatigue growth problem. The method lies between the Markov chain model and the continuously parametered continuous Markov process model proposed by other researchers. It is easy to apply and many probabilistic quantities can be obtained through only a small amount of calculation. It also provides an alternative way to model a fatigue crack problem even through it may not have as many of the nice physcial interpretations as the Markov process model does, and may not carry as much information (depending

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Fig. 11. Transformed times versus crack sizes for WPB fastener holes 3. Crack growth in fastener holes - WPB data set The same procedures are also applied to the WPB data set. The Poisson representation in the transformed N, - a domain is found to be quite accurate for the current case as can be seen from Fig. 11. The probability distribution functions of the random time reaching crack sizes a = 0.01, 0.02 and 0.04 inches are plotted in Fig. 12, and the crack exceedance probability at 8000 flight hours is plotted in Fig. 13. Lin and Yang's and the corresponding test results are also shown in both figures for comparison. Finally, the constant-probability growth curves are shown in Fig. 14 and some generated sample functions are shown in Fig. 15. In Fig. 15(a) the sample functions are obtained directly from the model while in Fig. 15(b) the sample functions have been smoothed. It is interesting to note the similarity between the present

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Probabilistic Engineering Mechanics, 1986, Vol. /, No. 4

231

On the Markov approximation: Wen-Fang Wu

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ACKNOWLEDGEMENTS The a u t h o r w o u l d like to express his g r a t i t u d e t o w a r d Professor M. S h i n o z u k a for his e n c o u r a g e m e n t in writing this p a p e r . Miss I. F r a n c k ' s help in the final refinement of the text is very a p p r e c i a t e d .

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on the discretization) as the M a r k o v chain model. In a d d i t i o n , it reveals the similarities a n d differences between the M a r k o v chain a n d the M a r k o v process a p p r o x i m a t i o n s . T h e results o b t a i n e d by using the p r o p o s e d a p p r o a c h agree r e a s o n a b l y well with e x p e r i m e n t a l results a n d are also c o m p a r a b l e to the results due to o t h e r two M a r k o v a p p r o a c h e s . C h a n g e in the p r o b a b i l i s t i c structure d u e to initial flaw d i s t r i b u t i o n , repair a n d r e p l a c e m e n t , etc. can be i m p l e m e n t e d into the present m o d e l as with o t h e r M a r k o v a p p r o x i m a t i o n s . A n o n h o m o g e n e o u s P o i s s o n process was actually used in the present a p p r o a c h . In case the P o i s s o n process, even in the t r a n s f o r m e d d o m a i n , is n o t c a p a b l e of representing the crack g r o w t h b e h a v i o u r , it can further be generalized as a p u r e b i r t h process. T h e n a g r o w t h intensity d e p e n d i n g n o t only on the i n d e p e n d e n t variable but also on the d e p e n d e n t state has to be fitted for the e x p e r i m e n t a l d a t a . This n o n h o m o g e n e o u s p u r e birth process is closer to the M a r k o v chain m o d e l , b u t the fitting p r o c e d u r e a s s o c i a t e d with it b e c o m e s as c o m p l i c a t e d as t h a t needed for the M a r k o v chain approach.

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REFERENCES 15.

1

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I

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20.

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4

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~

35.

5

30.

6

25.

7

20.

8 ~-~ 15. g

9 10 5.

11 O,

I

0.

I

i

l

5,

I

I

I

l

I

I

I

I

10.

I

I

15.

l

i

i

t

20.

TH3USRND HOUR5

Fig. 15. Sample functions generated from the present model for WPB fastener holes, (a) before smoothing and (b) after smoothing

232

12 13

Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 4

Virkler. D. A., Hillberry. B. M. and God, P. K. The statistical nature of fatigue crack propagation, Journal of Enoineering Materials and Technology,ASME, 1979, 101(2),148-153 Virkter, D. A., Hillberry, B. M. and God, P. K. The statistical nature of fatigue crack propagation, Air Force Flight Dynamics Laboratory, Technical Report AFFDL-TR-78-43, WPAFB, 1978 Norohna, P. J. et al. Fastener hole quality, Vol I, Air Force Flight Dynamics Laboratory, Technical Report AFFDL-TR-78206, WPAFB, 1978 Shinozuka, M. Application of stochastic processes to fatigue, creep and catastrophic failures, Seminar in the Application of Statistics in Structural Mechanics, University of Pennsylvania, November 1966 Madsen, H. O., Krenk, S. and Lind, N. C. Methodsof Structural Safety, Prentice-Hall, Englewood Cliffs, New Jersey, 1986, Chapter 9 Yang, J. N., Salivar, G. C. and Annis, C. G. Statistical modeling of fatigue-crack growth in a nickel-base superalloy, Enoineering Fracture Mechanics, 1983, 18(2), 257-270 Yang, J. N., Salivar, G. C. and Annis, C. G. The statistics of crack growth in engine materials, Vol. I]: Spectrum loading and advanced model, Air Force Wright Aeronautical Laboratory, Technical Report AFWAL-TR-82-4040, WPAFB, January 1983 Madsen, H. O. Stochastic modeling of fatigue crack growth, presented at the ASME Petroleum Workshop, Tulsa, Oklahoma, September 1983 Ortiz, K. On the stochastic modeling of fatigue crack growth, PhD Dissertation, Stanford University, 1985 Itagaki, H. and Shinozuka, M. Application of the Monte Carlo technique to fatigue-failure analysis under random loading, Special Technical Publication 511, ASTM, 1972, 168-184 Lin, Y. K. and Yang, J. N. On statistical moments of fatigue crack propagation, EngineeringFractureMechanics, 1983, 18(2), 243-256 Lin, Y. K. and Yang, J. N. A stochastic theory of fatigue crack propagation, AIAA Journal, 1985, 23(1), 117-124 Stratonovich, R. L Topics in the Theory of Random Noise, translated by R. A. Silverman, Gordon and Breach, New York, 1967, I, 89-99

On the M a r k o v approximation: Wen-Fang Wu 14 15

16 17

18 19 20 21

22

Tsurui, A. and Ishikawa, H. On the reliability degradation of fatigue crack propagation, Transactions, JSME, 1985, 51(461), 17-22 (in Japanese) Tsurui, A. and Ishikawa, H. Theoretical study on the distribution of fatigue crack propagation life under stationary random loading, Transactions, JSME, 1985, 51(461), 31-37 (in Japanese) Ishikawa, H. and Tsurui, A. A stochastic model of fatigue crack growth in consideration of random propagation resistance, Transactions, JSME, 1984, 50(454), 1309-1315 (in Japanese) Tsurni, A., Ishikawa, H., Utsumi, A. and Sako, A. The size effect on statistical properties of fatigue crack propagation process, Journal of the Japanese Society of Material Science, 1986, 35(393), 578-582 (in Japanese) Kozin, F. and Bogadanoff, J. L. A critical analysis of some probabilistic models of fatigue crack growth, Engineering Fracture Mechanics, 1981, 14(1), 59-89 Bogadanoff, J. L. A new cumulative damage model, Part 1, Journal of Applied Mechanics, ASME, 1978, 45(2), 246-250 Bogadanoff, J. L. and Krieger, W. A new cumulative damage model, Part 2, Journal of Applied Mechanics, ASME, 1978, 45(2), 251-257 Bogadanoff, J. L. A new cumulative damage model, Part 3, Journal of Applied Mechanics, ASME, 1978, 45(4), 733-739 Bogadanoff, J. L. and Kozin, F. A new cumulative damage

23 24 25 26 27 28 29 30

model, Part 4, Journal of Applied Mechanics, ASME, 1980, 47(1), 40-44 Bogadanoff, J. L. and Kozin, F. On nonstationary cumulative damage models, Journal of Applied Mechanics, ASME, 1982, 49(1), 37-42 Kozin, F. and Bogadanoff, J. L. Probabilistic fatigue crack growth and design, Proceedings, ASM E Design and Production Meeting, Dearborn, Michgan, September 1983, 39-45 Kozin, F. and Bogadanoff, J. L. An approach to accelerated testing, Canadian Aeronautics and Space Journal, 1983,29(1), 6076 Kozin, F. and Bogadanoff, J. L. On life behavior under spectrum loading, Engineering Fracture Mechanics, 1983, 18(2), 271-283 Ghonem, H. and Dore, S. Probabilistic description of fatigue crack growth in polycrystaUine solids, Engineering Fracture Mechanics, 1985, 21(6), 1151-1168 Feller, A. An Introduction to Probability Theory and Its Applications, John Wiley and Sons, New York, 1968, I, 3rd edn, 372-482 Lin, Y. K. Probabilistic Theory of Structural Dynamics, McGraw-Hill, 1967, reprinted by R. E. Krieger, Melbourne, Florida, 1976 McCartney, L. N. and Cooper, P. M. A numerical method of processing fatigue crack propagation data, Engineering Fracture Mechanics, 1977, 9, 265-272

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