- Email: [email protected]
Low-cycle fatigue data—statistical analysis and simulation
LOW-CYCLE FATIGUE DATA-STATISTICAL ANALYSIS AND SIMULATION
alit
FAiSAL H. AL-SUGAIR of Civil Engineering, King Saud University, P.0, Box SO&Riyadh 11421, Saudi Arabia and ANNE S. ~~MI~~iA~
aepartment of Civil Envying,
Stanford U~~s~,
Stanford, CA 94305, U.S.A.
Akstract-‘The analysis of low-cycle fatigue crack growth (LGFCG) data is complicated by the fact that the number of data sets availabIe is usuahy smah. In this work, techniquea are used to obtain useful nonagon from limited amaunts of data. Also, a simulation scheme is introduced which makes it possible to generate realistic replicates of data from availabie experimeuttd data sets. The analysis and simulation is demonstrated on iow-cy& fatigue data obtained by Dowling (&a& end Fracfwe ASTM STP 681,19-32 1976).
1. INTRODUCTION Tin!ram-
of fatiw crack growth has gone through several stages. Starting from the stress vs the number of cycles of loading to f&ure (S-N) approach, introduced by Wzjhter in the nineteeuth aatpryll], and eudktg with the fmcture me&a&s approach to fatigue which has ~,~p~~~ in the past 20 years. High-cyck fatigue crack growth @IC+FCG) has been modeled quite aeeuratiy by the Par& law of hn~ar-ehmtic fracture mechanica (LEFT) relating the fatigue crack growth rate, da JdT, to the range of the stress intensity factor, hK Anaiogous~y,LGFCG has bsen modeled by a law, similar in form to the Paris law, that relatea the fatigue crack growth rate to the range of F&e’s J-integ~@2]. This is done because the stress intensity factor is based on the assumption that the amount of Pasty in the crack tip region is small[3], and during LGFCG relatively large plasticity occurs in the crack tip region. In fatigue crack propagation experhnents, the data reported usually consists of the total crack iength and the uumber of cycles required to reach that crack length. In k3w-ey& fatigue experiments the range of the .&integrai, A.&is also reported. This is because, contrary to high-cycIe fatigue5 there does not exist at this time a generally acce@ed formula to compute the range of the J-integral. as in the case of the stress intensity factor range. Instead, hystemais loops of the load versus deikmtion during the testarc usedto obtain values of the J-integral range. The J-integral range has been used stuxessfully to correlate fatigue crack growth rates by Dowling and Begled4J. Since their work, several other authors have used the J-integral range for low-cycle fatigue experiments with similar success (e.g. tgl. 161and VI). In order to account for the random nature of crack growth, sevemdimwtiga~on have suggested that a random process be mukipked by the original de&minis& equation. For exampIe, Lin and Yang@]mode&f the m&mness in HC-FCG by a correh&edrandom process which is dependent on time, t. S~~ar~y, Grtii and ~~~~ pursued the same gene& approach, however, they modeled the randomness as a function of total crack length, u. In this paper, the same general approach as in[9] is used to model the randomness in the LC-FCG process. In LC-FCG modehing, a least-square tit between the crack growth rate and the J-integral range is obtained, then a statistical aualysis is performed to characterize the deviations from the least-square et. Different hypotheses have been made about the statistical properties of these deviations for high-cycle fatigue problems. In this paper, recommendations for &aracterizing the deviations for LC-FCG are made. The problem of limited amounts of replicates avaibsbie is more pronounced for LC-FCG than for WC-FCG. Therefore, a &mutation scheme is deveIoped which rises the random de~a~o~s as input to generate sample pa&s of the fatigue crack growth process. This scheme is thought to reaiisticahy simulate data sets u&g limited information,
2. DATA ANALYSIS METHOD LC-FCG cau be described by an eqnation employing I&& I-integral as foIlow$2k
where da/dT is the fatigue crack growth rate, dl is the J-integral range, and C and m are constants which can be obtainext by a least-square fit procedure to data. If the vats of C and rtaobtained from the least-square fit are ~~titu~ into eq. (I}, the values which repmsent the deviations from the original equation can be obtained. These can be model&d as a stochastic process. In general, the deviations axe modelled as a stationary stochastic process which is a function af some 691
692
Tezhnicai note
Fig. 1. An example of the residuals of the fatigue crack growth law before removing the nonstationary trend. controlling variable. For example, in high-cycle fatigue analysis both the time, I, and the total crack length, a, have been used[8,9] in cases where the load amplitude is kept constant. In this paper, the total crack length was selected as the controlling parameter. Aho, since for most low-cycie fatigue tests the load amplitude varies to maintain a stable crack growth, the load amplitude range, AP, was included as a variable to characterize the deviations. Hence, the deviation process is written as (r>(u, AP); o > 0, AP 2 0), where a is the total crack length and AP is the load amplitude range. The fatigue crack growth equation is then written as da dr = C(AJ)“‘D(a,AP). (2) The deviation process D(u, AP) is assumed to be lognormally distributed with unit median, stationary in mean, and nonstationary in variance. The dependence of D(u, AP) on the load amplitude range, AP, is assumed only through AP’s effect on the J-integral range AJ. The deviation proeess is thus written as Z3(u,AJ). It shot&I be noted that A3 is a function of both a and AP. However, for convenience the deviation process will be written as I)@, A.Q As a result of performing the least-square fit procedure between ln(dajdT) and hr(df), the process which is observed is the logarithm of D(a, AJ) given as
ecu. AJ) = h@(u, U)}.
(3)
An example of a set of observations of r (dr) plotted against the total crack length, a, is shown in Fig. 1. An obvious trend is observed in the figure. This trend is assumed to be proportional to (AJ), The removal of this trend is done by assuming that the <{a,AJ) process can be separated into two parts. The first part is a stochastic process <,(a) which carries the statistical properties of the original {t > process. The second part is a deterministic function of (A.!), denoted by &(AJ), which carries the trend observed in the original {is) process. When this trend is removed from the data, the remaining process t,(u), shown in Fig. 2, can be modelled as a stationary stochastic process.
4.4; o_u
,
, am
,
, 01
,
, au
,
, lwli
,
I
0s
I
I
0.m
t
I
0)
1
I
0-U
Crack length Q (inehea). Fig. 2. An example of the residuals of the fatigue crack growth law after removing the nonstationary trend.
693
Technical note
The deviation process D(a, AJ) was assumed to be lognormally distributed with unit median; therefore, &(a, AJ) is Gaussian distributed with zero mean. Since the values of {(a, AJ) are small, a first-order approximation of the exponential function ec RZ1 + < is used for D(o, AJ). The mean, variance, autocovariance and autocorrelation functions of D(a, AJ) can be found usmg this approximation to be co-
1,
(4)
2
(5)
z~iJ-@y,
R,(a) = 1 +
&(a),
and
(6) (7)
PJGO = P&). <(a, AJ) is assumed to be separable into two parts:
56~ 4
(8)
= 6,tuMW.
The two parts are described as follows: {C,(a); a 3 0) is a Gaussian distributed stochastic process with zero mean and stationary variance ui,. On the other hand, &(AJ) is a deterministic equation that models the trend present in the deviation Process. Because the process r,(e) is Gaussian, it is fully described by the autocovariance function &(a):
&,(a) = JW,taXI (a + all
(9)
= 4 PC,(a),
(10)
where ~+~(a) is the autocorrelation function of the process. Equation (10) is true for any zero mean process, which is also the case for 4, (a). The autocorrelation function reflects the correlation between observations of a stochastic process. The area under the autocorrelation function, p<,(a). is called the scale of fluct~tion, S,,, and is given as [lo] (11) The fluctuation scale reflects the length over which <,(a) remains significantly correlated. The process {&(a), a 2 0) is assumed to be a white-noise stationary Gaussian process. Theoretically, a white-noise random process X(t) has a spectrum S,(W) with a constant value S, over the whole range of frequencies (- 03, + 03). In practice, however, a spec&mn is called white if it extends well past ah frequencim of interes@l]. One characteristic of white-noise procemcs is that values associated with nonoverlap~ng time intervals are independent. In general, when a stochastic process approaches a white noise process, the fluctuation scale, 6,, approaches zero and the variance, oz), approaches infinity in such a way that the product aft),) remains constant[fO]. The autocovariance function of a stationary white-noise process is given as[lO]: g,(a) = &!,)
W),
(12)
where d(a) is the dirac-delta function that takes the values S(a) =
1 ifa=O
I
0
otherwise.
Computations of the fluctuation scale of&(u) from the data points for low-cycle fatigue obtained by Dowling[2] have shown it to be less than the crack growth increments Aa*. The average fluctuation scale from all specimens was estimated to be 0.009 inches and the crack growth increment was found to be greater than 0.011 inches for almost ah data points with very few exceptions. Therefore, it is reasonable to assume the stochastic process <,(a) to be a white-noise Gaussian process. The method used for computing the guctuation scale is that introduced by Vamvarcke[ IO]. The method is summarized as follows: a family of moving average processes, X,(t), can be obtained from a stationary random process, X(r), with mean, nr,, and variance, af, as follows:
I+ r/z
t W)=p
s
r_T,~
-Vu)
du,
(14)
where T is the averaging time. The ~lations~p between the processes X(t) and XT(t) are shown in Fig. 3. The mean is not affected by the averaging operation; however, the variance of X,(t) is a&&ted. This can be expressed as follows:
a:, =y(T)uf.
(15)
The variance function, y(T), measures the reduction of the point variance u: under local averaging. Vanmarcke[lO] introduced the following approximation for the variance function y(T):
(16) Equation (16) was plotted against the variance function for several common, wide-band analytical forms of the autocorrelation function. They were found to converge quite rapidly toward the common asymptotic expression y(T) = 0,/T. This is illustrated in Fig. 4.
694
Technical note
Fig. 3. Sample functions of a random process X(r) with mean m and a local average process XT(t) with the same mean.
The advantage of this method is that the fluctuation scale can be found from one set alone and by a simple calculation. If other sets are available, the values found can be averaged to get a more accurate value. Other methods for tinding the fluctuation scale use spectral analysis as in [9]; however, they require that a sufhcient number of data sets be available. The values of the fluctuation scale found using Vanmarcke’s method are very close to those obtained by more sophisticated methods. For example, in the analysis presented in 191.the scale of fluctuation of the deviation process for high-cycle fatigue data was 0.012 mm using the detailed spectral analysis. The value obtained using the Vanmarcke method was 0.0117 mm. The second part of the {(a, AJ) process is the deterministic part, &(AJ). Ibis part models the trend that is present in C(u, AJ). This trend is obviously a function of the intensity of the crack tip strain field. It is therefore a function of the J-integral, since the J-integral is a measure of the intensity of the crack tip strain field during crack growth[2]. The trend is assumed to be proportional to the J-integral range raised to some power as follows: (17) where Ho is a constant. Based on the testing of LC-FCG data obtained by Dowling[2], the value of (I) chosen is I = m/2, where m is the exponent of the fatigue crack growth equation. Substituting eqs (12) and (17) into eq. (lo), the autocovariance function of ((a, AJ) is found to be R<(e) =H~(AJ.~‘2(~~+.~‘20j,Bc,b(a),
(18)
where AJ. denotes AJ(a, AP) and AJ.+. denotes AJ(u + a, AP). The stochastic process O(a, AJ) is now completely described. It is important to note here that the constant of proportionality, Ho, that appears in eq. (17) does not affect the autocovariance function R,(a) because the (r,) process is not observed directly. Instead, it is computed from the (<) process divided by the (&) as given in eq. (17). Hence, a constant of Hi will appear in the denominator of u$, and will cancel the one which appears in eq. (18).
\n t 3. SIMULATION
SCHEME
As previously mentioned, low-cycle fatigue data is usually available in small quantities. Therefore, it is proposed to use simulation in order to generate sample paths of the fatigue crack growth process. The parameters needed for simulation are obtained from the available data. The simulation scheme proposed for LC-FCG is as follows: (1) Since the {C,} process is assumed to be white-noise and stationary, the observations of the time series (<,)i can be randomly selected to produce a process with the same properties. If the number of data points in the data record are n, then the number of possible combinations is n” which is more than sullicient for the purpose of simulation.
1.0
1 ......
r(‘)=
3tT.m
as
Q2
&ST kpT
Shapes cofre5pondinp to several common wide-band
I..--.
5
B/T . . . . .
10
15
.I..
1
correlation
models
+
T/e,
Fig. 4. Variance function plotted against the normalized averaging interval 0,/T for several common wide-band correlation models (from [IO]).
695
Tech&al note
Ol#Ml
* 0.0.
__a-
coca.
8lmJ1.s.6
oata.
:
c
z
0.7 -
3 5 :* 0.0’ H Y Y
0.
4
%ooodoooaooosooosoooOuwo~--goo) 0 No. of Cyclr8. Fig. 5. An example of 15 simulated LC-FCG data sets.
(2) Each new set of the time series I{,) is then multiplied by the (&) vahtes to obtain a new set of residuals {f) of the least-squate fit between the fatigue crack growth rate, d.o/dT, and the J-integral range, dl. (3) The difIerenoe betweeu the new and old residuals (& - c), is now subtracted from the leR-hod-si& of the fatigue crack growth equation. This yields a new set of fatigue crack growth rates that would give the set of residuals {[I when the same values of the growth law coefficients are us&. (4) The values of the growth law coetllcients InC and m are sampled from their distributions which are approximately Gaussiat@2]. Using these values along with the set of residuals found above, a new set of fatigue crack growth rates is found. The new set of fatigue crack growth rates can now be integrated to yield sample paths of the process. An example of 15 simulated data sets using this procedure is shown in Fig. 5. The above procedure can also be used to generate high-cycle fatigue crack growth data by introducing the appropriate cbanga.
4. CONCLUSION In this paper we present a method for characterizing the random deviations of a low-cycle fatigue crack growth law from actually observed data. In addition, a simulation process is formulated which can be used to develop sample paths of a fatigue crack growth process. The stochastic deviation processof a fatigue crack growth taw obtained from the best fit through dataprovided by Dowlin&2] is formulated for d~ons~~on purposes. Sampk fatigue crack growth paths were simulated for this process and were compared to the sample paths of the original data. Such simulations can be used to predict the fatigue life of samples with a corresponding confidence limit. The main advantage of the simulation technique presented in this paper is that the cotidence limits of the predicted sample paths will be narrower than those obtrdned from standard Monte Carlo simulation of the process, Thus, predictions of the fatigue life of specimens subjected to low cycle replicate loads can be predicted with greater contldence. ~e~ow~dge~nts-me authors express their gratitude to Professor N. Dowling for pro~~ng the data. This work was partially supported by NSF grant ECR-8610867, and by King Saud University.
REFERENCES H. 0. Fuchs and R. I. Stevens, Metal Fatigue in Engng., John Wiley, New York (1980). N. E. Dowling, Geometry effectsand the J-integral approach to E-P fatigue crack growth. Creeks Md Fractttre ASTM STP 601,19-32 (1976). R. A. Smith, An introduction to fracture mechanics for engineers. Part I: stresses due to notches and cracks. Mu~erialr in Bagng. Applications, Vol. 1, (1978). N. E. Dowling and J. A. Regley, Fatigue, crack growth during gross plasticity and the S-integral. Mechanics ofCrack Growth, ASTM STP 590, 82-103 (1976). D. F. Mowbray, Use of a compact-type strip specimen for fatigue crack growth rate testing in high-rate regime, E-P fracture. AS?%4 STP 668, 736-752 (1979). J. A. Joyce and G. E. Sutton, An automated method of ~rnpu~r~n~~~ low-cycle fatigue crack growth testing using the elastic-plastic parameter cyclic. 3. Automated Test Metho& for Frttcture and Fatigue Crack Growth, ASTM STP 877, 227-247 (1985).
D. A. Jablonski, An experimental study of the validity of a delta-J criterion for fatigue crack growth, Proc., lrhird Int. MID. on Nonlinear Fracture Mech. Knoxville. Tennessee. (6-9 October 19861. Y. K.-Lib and J. N. Yang, On statistical moments~of fatigue crack propagation. &rgng Fracture Mech. 18, 243-256 (1983).
696
Technical note
[9] K. Ortiz and A. S. Kiremidjian, Time series analysis of fatigue crack growth rate data. Engng Fracture Me& 24, 657675 (1986). [IO] E. Vanmarcke, Random Fields: Analysis and Synthesis, The MIT Press, Boston (1983). [l l] D. E. Newland, An introduction to Random Vibrations and Spectral Analysis, Second Edition, Longman, New York (1984). [12] J. R. Benjamin and C. A. Cornell, Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill, New York (1970). (Received 5 June 1989)
alit
FAiSAL H. AL-SUGAIR of Civil Engineering, King Saud University, P.0, Box SO&Riyadh 11421, Saudi Arabia and ANNE S. ~~MI~~iA~
aepartment of Civil Envying,
Stanford U~~s~,
Stanford, CA 94305, U.S.A.
Akstract-‘The analysis of low-cycle fatigue crack growth (LGFCG) data is complicated by the fact that the number of data sets availabIe is usuahy smah. In this work, techniquea are used to obtain useful nonagon from limited amaunts of data. Also, a simulation scheme is introduced which makes it possible to generate realistic replicates of data from availabie experimeuttd data sets. The analysis and simulation is demonstrated on iow-cy& fatigue data obtained by Dowling (&a& end Fracfwe ASTM STP 681,19-32 1976).
1. INTRODUCTION Tin!ram-
of fatiw crack growth has gone through several stages. Starting from the stress vs the number of cycles of loading to f&ure (S-N) approach, introduced by Wzjhter in the nineteeuth aatpryll], and eudktg with the fmcture me&a&s approach to fatigue which has ~,~p~~~ in the past 20 years. High-cyck fatigue crack growth @IC+FCG) has been modeled quite aeeuratiy by the Par& law of hn~ar-ehmtic fracture mechanica (LEFT) relating the fatigue crack growth rate, da JdT, to the range of the stress intensity factor, hK Anaiogous~y,LGFCG has bsen modeled by a law, similar in form to the Paris law, that relatea the fatigue crack growth rate to the range of F&e’s J-integ~@2]. This is done because the stress intensity factor is based on the assumption that the amount of Pasty in the crack tip region is small[3], and during LGFCG relatively large plasticity occurs in the crack tip region. In fatigue crack propagation experhnents, the data reported usually consists of the total crack iength and the uumber of cycles required to reach that crack length. In k3w-ey& fatigue experiments the range of the .&integrai, A.&is also reported. This is because, contrary to high-cycIe fatigue5 there does not exist at this time a generally acce@ed formula to compute the range of the J-integral. as in the case of the stress intensity factor range. Instead, hystemais loops of the load versus deikmtion during the testarc usedto obtain values of the J-integral range. The J-integral range has been used stuxessfully to correlate fatigue crack growth rates by Dowling and Begled4J. Since their work, several other authors have used the J-integral range for low-cycle fatigue experiments with similar success (e.g. tgl. 161and VI). In order to account for the random nature of crack growth, sevemdimwtiga~on have suggested that a random process be mukipked by the original de&minis& equation. For exampIe, Lin and Yang@]mode&f the m&mness in HC-FCG by a correh&edrandom process which is dependent on time, t. S~~ar~y, Grtii and ~~~~ pursued the same gene& approach, however, they modeled the randomness as a function of total crack length, u. In this paper, the same general approach as in[9] is used to model the randomness in the LC-FCG process. In LC-FCG modehing, a least-square tit between the crack growth rate and the J-integral range is obtained, then a statistical aualysis is performed to characterize the deviations from the least-square et. Different hypotheses have been made about the statistical properties of these deviations for high-cycle fatigue problems. In this paper, recommendations for &aracterizing the deviations for LC-FCG are made. The problem of limited amounts of replicates avaibsbie is more pronounced for LC-FCG than for WC-FCG. Therefore, a &mutation scheme is deveIoped which rises the random de~a~o~s as input to generate sample pa&s of the fatigue crack growth process. This scheme is thought to reaiisticahy simulate data sets u&g limited information,
2. DATA ANALYSIS METHOD LC-FCG cau be described by an eqnation employing I&& I-integral as foIlow$2k
where da/dT is the fatigue crack growth rate, dl is the J-integral range, and C and m are constants which can be obtainext by a least-square fit procedure to data. If the vats of C and rtaobtained from the least-square fit are ~~titu~ into eq. (I}, the values which repmsent the deviations from the original equation can be obtained. These can be model&d as a stochastic process. In general, the deviations axe modelled as a stationary stochastic process which is a function af some 691
692
Tezhnicai note
Fig. 1. An example of the residuals of the fatigue crack growth law before removing the nonstationary trend. controlling variable. For example, in high-cycle fatigue analysis both the time, I, and the total crack length, a, have been used[8,9] in cases where the load amplitude is kept constant. In this paper, the total crack length was selected as the controlling parameter. Aho, since for most low-cycie fatigue tests the load amplitude varies to maintain a stable crack growth, the load amplitude range, AP, was included as a variable to characterize the deviations. Hence, the deviation process is written as (r>(u, AP); o > 0, AP 2 0), where a is the total crack length and AP is the load amplitude range. The fatigue crack growth equation is then written as da dr = C(AJ)“‘D(a,AP). (2) The deviation process D(u, AP) is assumed to be lognormally distributed with unit median, stationary in mean, and nonstationary in variance. The dependence of D(u, AP) on the load amplitude range, AP, is assumed only through AP’s effect on the J-integral range AJ. The deviation proeess is thus written as Z3(u,AJ). It shot&I be noted that A3 is a function of both a and AP. However, for convenience the deviation process will be written as I)@, A.Q As a result of performing the least-square fit procedure between ln(dajdT) and hr(df), the process which is observed is the logarithm of D(a, AJ) given as
ecu. AJ) = h@(u, U)}.
(3)
An example of a set of observations of r (dr) plotted against the total crack length, a, is shown in Fig. 1. An obvious trend is observed in the figure. This trend is assumed to be proportional to (AJ), The removal of this trend is done by assuming that the <{a,AJ) process can be separated into two parts. The first part is a stochastic process <,(a) which carries the statistical properties of the original {t > process. The second part is a deterministic function of (A.!), denoted by &(AJ), which carries the trend observed in the original {is) process. When this trend is removed from the data, the remaining process t,(u), shown in Fig. 2, can be modelled as a stationary stochastic process.
4.4; o_u
,
, am
,
, 01
,
, au
,
, lwli
,
I
0s
I
I
0.m
t
I
0)
1
I
0-U
Crack length Q (inehea). Fig. 2. An example of the residuals of the fatigue crack growth law after removing the nonstationary trend.
693
Technical note
The deviation process D(a, AJ) was assumed to be lognormally distributed with unit median; therefore, &(a, AJ) is Gaussian distributed with zero mean. Since the values of {(a, AJ) are small, a first-order approximation of the exponential function ec RZ1 + < is used for D(o, AJ). The mean, variance, autocovariance and autocorrelation functions of D(a, AJ) can be found usmg this approximation to be co-
1,
(4)
2
(5)
z~iJ-@y,
R,(a) = 1 +
&(a),
and
(6) (7)
PJGO = P&). <(a, AJ) is assumed to be separable into two parts:
56~ 4
(8)
= 6,tuMW.
The two parts are described as follows: {C,(a); a 3 0) is a Gaussian distributed stochastic process with zero mean and stationary variance ui,. On the other hand, &(AJ) is a deterministic equation that models the trend present in the deviation Process. Because the process r,(e) is Gaussian, it is fully described by the autocovariance function &(a):
&,(a) = JW,taXI (a + all
(9)
= 4 PC,(a),
(10)
where ~+~(a) is the autocorrelation function of the process. Equation (10) is true for any zero mean process, which is also the case for 4, (a). The autocorrelation function reflects the correlation between observations of a stochastic process. The area under the autocorrelation function, p<,(a). is called the scale of fluct~tion, S,,, and is given as [lo] (11) The fluctuation scale reflects the length over which <,(a) remains significantly correlated. The process {&(a), a 2 0) is assumed to be a white-noise stationary Gaussian process. Theoretically, a white-noise random process X(t) has a spectrum S,(W) with a constant value S, over the whole range of frequencies (- 03, + 03). In practice, however, a spec&mn is called white if it extends well past ah frequencim of interes@l]. One characteristic of white-noise procemcs is that values associated with nonoverlap~ng time intervals are independent. In general, when a stochastic process approaches a white noise process, the fluctuation scale, 6,, approaches zero and the variance, oz), approaches infinity in such a way that the product aft),) remains constant[fO]. The autocovariance function of a stationary white-noise process is given as[lO]: g,(a) = &!,)
W),
(12)
where d(a) is the dirac-delta function that takes the values S(a) =
1 ifa=O
I
0
otherwise.
Computations of the fluctuation scale of&(u) from the data points for low-cycle fatigue obtained by Dowling[2] have shown it to be less than the crack growth increments Aa*. The average fluctuation scale from all specimens was estimated to be 0.009 inches and the crack growth increment was found to be greater than 0.011 inches for almost ah data points with very few exceptions. Therefore, it is reasonable to assume the stochastic process <,(a) to be a white-noise Gaussian process. The method used for computing the guctuation scale is that introduced by Vamvarcke[ IO]. The method is summarized as follows: a family of moving average processes, X,(t), can be obtained from a stationary random process, X(r), with mean, nr,, and variance, af, as follows:
I+ r/z
t W)=p
s
r_T,~
-Vu)
du,
(14)
where T is the averaging time. The ~lations~p between the processes X(t) and XT(t) are shown in Fig. 3. The mean is not affected by the averaging operation; however, the variance of X,(t) is a&&ted. This can be expressed as follows:
a:, =y(T)uf.
(15)
The variance function, y(T), measures the reduction of the point variance u: under local averaging. Vanmarcke[lO] introduced the following approximation for the variance function y(T):
(16) Equation (16) was plotted against the variance function for several common, wide-band analytical forms of the autocorrelation function. They were found to converge quite rapidly toward the common asymptotic expression y(T) = 0,/T. This is illustrated in Fig. 4.
694
Technical note
Fig. 3. Sample functions of a random process X(r) with mean m and a local average process XT(t) with the same mean.
The advantage of this method is that the fluctuation scale can be found from one set alone and by a simple calculation. If other sets are available, the values found can be averaged to get a more accurate value. Other methods for tinding the fluctuation scale use spectral analysis as in [9]; however, they require that a sufhcient number of data sets be available. The values of the fluctuation scale found using Vanmarcke’s method are very close to those obtained by more sophisticated methods. For example, in the analysis presented in 191.the scale of fluctuation of the deviation process for high-cycle fatigue data was 0.012 mm using the detailed spectral analysis. The value obtained using the Vanmarcke method was 0.0117 mm. The second part of the {(a, AJ) process is the deterministic part, &(AJ). Ibis part models the trend that is present in C(u, AJ). This trend is obviously a function of the intensity of the crack tip strain field. It is therefore a function of the J-integral, since the J-integral is a measure of the intensity of the crack tip strain field during crack growth[2]. The trend is assumed to be proportional to the J-integral range raised to some power as follows: (17) where Ho is a constant. Based on the testing of LC-FCG data obtained by Dowling[2], the value of (I) chosen is I = m/2, where m is the exponent of the fatigue crack growth equation. Substituting eqs (12) and (17) into eq. (lo), the autocovariance function of ((a, AJ) is found to be R<(e) =H~(AJ.~‘2(~~+.~‘20j,Bc,b(a),
(18)
where AJ. denotes AJ(a, AP) and AJ.+. denotes AJ(u + a, AP). The stochastic process O(a, AJ) is now completely described. It is important to note here that the constant of proportionality, Ho, that appears in eq. (17) does not affect the autocovariance function R,(a) because the (r,) process is not observed directly. Instead, it is computed from the (<) process divided by the (&) as given in eq. (17). Hence, a constant of Hi will appear in the denominator of u$, and will cancel the one which appears in eq. (18).
\n t 3. SIMULATION
SCHEME
As previously mentioned, low-cycle fatigue data is usually available in small quantities. Therefore, it is proposed to use simulation in order to generate sample paths of the fatigue crack growth process. The parameters needed for simulation are obtained from the available data. The simulation scheme proposed for LC-FCG is as follows: (1) Since the {C,} process is assumed to be white-noise and stationary, the observations of the time series (<,)i can be randomly selected to produce a process with the same properties. If the number of data points in the data record are n, then the number of possible combinations is n” which is more than sullicient for the purpose of simulation.
1.0
1 ......
r(‘)=
3tT.m
as
Q2
&ST kpT
Shapes cofre5pondinp to several common wide-band
I..--.
5
B/T . . . . .
10
15
.I..
1
correlation
models
+
T/e,
Fig. 4. Variance function plotted against the normalized averaging interval 0,/T for several common wide-band correlation models (from [IO]).
695
Tech&al note
Ol#Ml
* 0.0.
__a-
coca.
8lmJ1.s.6
oata.
:
c
z
0.7 -
3 5 :* 0.0’ H Y Y
0.
4
%ooodoooaooosooosoooOuwo~--goo) 0 No. of Cyclr8. Fig. 5. An example of 15 simulated LC-FCG data sets.
(2) Each new set of the time series I{,) is then multiplied by the (&) vahtes to obtain a new set of residuals {f) of the least-squate fit between the fatigue crack growth rate, d.o/dT, and the J-integral range, dl. (3) The difIerenoe betweeu the new and old residuals (& - c), is now subtracted from the leR-hod-si& of the fatigue crack growth equation. This yields a new set of fatigue crack growth rates that would give the set of residuals {[I when the same values of the growth law coefficients are us&. (4) The values of the growth law coetllcients InC and m are sampled from their distributions which are approximately Gaussiat@2]. Using these values along with the set of residuals found above, a new set of fatigue crack growth rates is found. The new set of fatigue crack growth rates can now be integrated to yield sample paths of the process. An example of 15 simulated data sets using this procedure is shown in Fig. 5. The above procedure can also be used to generate high-cycle fatigue crack growth data by introducing the appropriate cbanga.
4. CONCLUSION In this paper we present a method for characterizing the random deviations of a low-cycle fatigue crack growth law from actually observed data. In addition, a simulation process is formulated which can be used to develop sample paths of a fatigue crack growth process. The stochastic deviation processof a fatigue crack growth taw obtained from the best fit through dataprovided by Dowlin&2] is formulated for d~ons~~on purposes. Sampk fatigue crack growth paths were simulated for this process and were compared to the sample paths of the original data. Such simulations can be used to predict the fatigue life of samples with a corresponding confidence limit. The main advantage of the simulation technique presented in this paper is that the cotidence limits of the predicted sample paths will be narrower than those obtrdned from standard Monte Carlo simulation of the process, Thus, predictions of the fatigue life of specimens subjected to low cycle replicate loads can be predicted with greater contldence. ~e~ow~dge~nts-me authors express their gratitude to Professor N. Dowling for pro~~ng the data. This work was partially supported by NSF grant ECR-8610867, and by King Saud University.
REFERENCES H. 0. Fuchs and R. I. Stevens, Metal Fatigue in Engng., John Wiley, New York (1980). N. E. Dowling, Geometry effectsand the J-integral approach to E-P fatigue crack growth. Creeks Md Fractttre ASTM STP 601,19-32 (1976). R. A. Smith, An introduction to fracture mechanics for engineers. Part I: stresses due to notches and cracks. Mu~erialr in Bagng. Applications, Vol. 1, (1978). N. E. Dowling and J. A. Regley, Fatigue, crack growth during gross plasticity and the S-integral. Mechanics ofCrack Growth, ASTM STP 590, 82-103 (1976). D. F. Mowbray, Use of a compact-type strip specimen for fatigue crack growth rate testing in high-rate regime, E-P fracture. AS?%4 STP 668, 736-752 (1979). J. A. Joyce and G. E. Sutton, An automated method of ~rnpu~r~n~~~ low-cycle fatigue crack growth testing using the elastic-plastic parameter cyclic. 3. Automated Test Metho& for Frttcture and Fatigue Crack Growth, ASTM STP 877, 227-247 (1985).
D. A. Jablonski, An experimental study of the validity of a delta-J criterion for fatigue crack growth, Proc., lrhird Int. MID. on Nonlinear Fracture Mech. Knoxville. Tennessee. (6-9 October 19861. Y. K.-Lib and J. N. Yang, On statistical moments~of fatigue crack propagation. &rgng Fracture Mech. 18, 243-256 (1983).
696
Technical note
[9] K. Ortiz and A. S. Kiremidjian, Time series analysis of fatigue crack growth rate data. Engng Fracture Me& 24, 657675 (1986). [IO] E. Vanmarcke, Random Fields: Analysis and Synthesis, The MIT Press, Boston (1983). [l l] D. E. Newland, An introduction to Random Vibrations and Spectral Analysis, Second Edition, Longman, New York (1984). [12] J. R. Benjamin and C. A. Cornell, Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill, New York (1970). (Received 5 June 1989)