A probabilistic method to predict fatigue crack initiation

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1 International Journal of Fracture (2006) 137:9 17 DOI /s Springer 2006 A probabilistic method to predict fatigue crack initiation SALIL. S. KULKARNI, L. SUN, B. MORAN, S. KRISHNASWAMY and J.D. ACHENBACH Department of Mechanical Engineering, Northwestern University, Evanston, IL-60208, USA Author for Correspondence. ( sail@northwestern.edu) Received 8 August 2005; accepted in revised form 6 September 2005 Abstract. A probabilistic method to predict macrocrack initiation due to fatigue damage is presented in this paper. Acoustic non-linearity is used to quantify pre-macrocrack initiation damage. This data is then used in a probabilistic analysis of fatigue damage. The probabilistic fatigue damage analysis consists of a suitably chosen damage evolution equation to model accumulated damage coupled with a procedure to calculate the probability of macrocrack initiation. The probability of macrocrack initiation is evaluated using the Monte Carlo Method with Importance Sampling. Numerical results for the probabilistic assessment of fatigue damage for a sample problem are presented and compared with experimental results. Key words: Acoustic non-linearity, crack initiation, damage evolution, fatigue, probabilistic method, structural health monitoring. 1. Introduction Conventional procedures for life prediction of components subjected to fatigue are generally based on the safe-life approach (see e.g. Suresh, 1991), coupled with Palmgren (see Palmgren, 1924) and Miner (see Miner, 1945) rules of linear cumulative damage. In the safe-life approach for metal fatigue, life prediction is based on data from fatigue testing of components. Components of a structure are replaced when the probability of failure reaches a prescribed (often small) value, even though some of them may have a significant remaining life. Hence it is a conservative approach with an economic penalty. To avoid this penalty, the damage-tolerant approach is often a suitable alternative for life predictions. This approach is especially useful when the rate of damage accumulated is well understood and can be monitored with a technique of quantitative non-destructive evaluation. However, in certain materials, such as high strength steels, like those used for drive trains of rotorcraft, critical damage in the form of a crack of detectable but very small length, often occurs late in the lifetime of a component. When a detectable crack has developed out of microscopic damage processes, it grows to an unacceptable length in a time that is short as compared to the total lifetime of the component. Periodic monitoring by a condition monitoring system can, however, significantly improve the safety of the damage tolerance approach. Particularly, if pre-crack damage can be monitored and related to crack formation by an analytical fatigue damage procedure, substantial safety and cost benefits can be gained. With this objective, a methodology to

2 10 S. S. Kulkarni et al. calculate the probability of macrocrack initiation based on periodic in situ measurements is presented in this paper. The methodology essentially consists of two parts: A structural health monitoring system to monitor the fatigue process. In the present approach, acoustic second harmonic generation (see Section 2) is used to monitor the damage accumulated during the fatigue process. A probabilistic fatigue damage procedure which takes the data from the health monitoring system as its input and calculates the probability of macrocrack initiation as its output. The probabilistic damage procedure consists of a suitably chosen damage evolution equation to model accumulated damage (see Section 3) and a procedure to calculate the probability of macrocrack initiation (see Section 4). The procedure to calculate the probability of macrocrack initiation is capable of updating its forecast based on the latest inspection data from the monitoring system. Using this methodology, the probability of macrocrack initiation is calculated for a sample problem taken from the literature, and the results are presented in Section Acoustic second harmonic generation To quantify the damage accumulation in a component undergoing fatigue, it is first necessary to relate the accumulated damage to an observable variable. The accumulated damage is caused by changes in the microstructure of the component which in turn introduce material non-linearity in the specimen. To characterize the material non-linearity, a single frequency ultrasonic wave generated by a transducer (generator) is transmitted through the specimen and the signal received by the receiving transducer is analyzed. The material non-linearity distorts the single frequency wave and leads to generation of second and higher harmonics. As a result, the signal received at the receiver not only consists of a component at the transmitting (fundamental) frequency but also second and higher harmonics whose amplitudes increase as the material non-linearity increases. Therefore, the material non-linearity and the underlying accumulated damage can be characterized by the ratio of the amplitude of the second harmonic A 2 to the amplitude of the fundamental A 1. This ratio is henceforth referred to as A 2 /A 1 and is expected to increase as damage accumulation progresses. Note that the characterization of the acoustic non-linearity by A 2 /A 1 (see Ogi et al., 2001) used here is different from the one usually found in the literature (see e.g. Moris et al., 1979). Ogi et al. (2001) have used an electromagnetic acoustic transducer to measure A 2 /A 1 in a specimen undergoing fatigue. As opposed to the procedure described earlier to measure the ratio A 2 /A 1 transmit a single (fundamental) frequency wave and measure the signal amplitude at the fundamental and the second harmonic, Ogi and his co-workers used a different but analogous procedure. They transmitted two different signals separately, one at resonance frequency f r and the other at frequency f r /2, into the specimen. For the case when the signal was transmitted at frequency f r, they measured the amplitude of the component of the received signal with frequency f r denoted A 1 (f r ). For the case when the signal was transmitted at frequency f r /2 they measured the amplitude of the component of the received signal with frequency f r denoted by A 2 (f r /2). Observe that A 2 (f r /2) is the amplitude of the second harmonic. Note that this procedure ensures that both the measured signals are measured at

3 Probabilistic method to predict fatigue crack initiation 11 Figure 1. Typical evolution of the ratio A 2 (f r /2)/A 1 (f r ) for 0.25 mass % C steel, Nf = 56,000 (from Ogi et al., 2001). the transducer resonance frequency f r, leading to more accurate measurements. For more details refer to Ogi et al. (2001). They observed that the ratio A 2 (f r /2)/A 1 (f r ) increases nearly monotonically, and shows a distinct peak at the point of macrocrack initiation (see Figure 1). This suggests that the state of damage in a specimen during fatigue can be tracked by measuring the ratio A 2 /A 1 for different cycle numbers. 3. Damage model This section presents a damage model whose evolution represents the evolution of the ratio A 2 (f r /2)/A 1 (f r ) (see Section 2) up to the point of macrocrack initiation. The state of damage in a specimen at a particular cycle during fatigue is represented by a scalar damage function D(N), where N is the number of cycles. The magnitude D =0 corresponds to no damage, and D =1 corresponds to the appearance of the first macrocrack. The following phenomenological model is assumed to represent the evolution of the damage with the number of cycles ( ) dd 1 σ/2 rc ( σ) m 1 dn = if σ/2 >r N c σ/2 (1 D) n c ( σ) (1) 0 if σ/2 <r c ( σ) Here, N c is a normalizing constant, σ is the stress range in a cycle, r c ( σ) is the endurance limit when the mean stress in a cycle is σ, and m>0 and n>0 are parameters which depend on the material and the loading conditions and are determined by correlating the evolution of the ratio A 2 /A 1 to the evolution of D. It is assumed that r c ( σ) follows the Goodman relation (see Goodman, 1899), i.e. ( r c ( σ)= r c (0) 1 σ ) σ ult where σ ult is the ultimate tensile strength of the material. Assuming that σ and σ are constant during cycling and that σ/2 is always greater than r c ( σ), Equation (1)

4 12 S. S. Kulkarni et al. can be solved to obtain [ D(N)= 1 (1 D 0 ) n+1 N N ( ) 0 σ/2 rc ( σ) m (n + 1)] 1 n+1. (2) N c σ/2 Here D 0 is the damage present in the specimen at cycle N Probability of macrocrack initiation This section describes the procedure for calculation of the probability of macrocrack initiation and is based on the model described in Section 3. To begin, some of the quantities appearing in Equation (2) are taken as random variables. The probability of macrocrack initiation P ma, which is defined as the probability that the number of cycles to macrocrack initiation, will be less than a specified number of cycles N s,is then given by P ma = 1 P r(d(n s )<1). (3) From Equation (1), it follows that D(N s )<1 implies [ (1 D 0 ) n+1 N ( ) s N 0 σ/2 rc ( σ) m (n + 1)] 1 n+1 > 0. N c σ/2 From the previous equation it is observed that D(N s )<1 is equivalent to where N s <N ini (4) N ini = N 0 + N c n + 1 (1 D 0) n+1 ( ) σ/2 m. σ/2 r c ( σ) Note that in a purely deterministic setting, N ini corresponds to the cycle at which a macrocrack appears. This can be seen by setting D = 1 in Equation (2) and solving for N. The probability of macrocrack initiation can therefore be calculated as P ma = 1 P r(d(n s )<1) = 1 Pr(N s <N ini ) = Pr(N ini <N s ). (5) To calculate P ma given by Equation (5), one first defines a limit state surface g = N ini N s. Also let X = [X 1 X 2...X k ] T denote the random quantities in Equation (2) (for example, in the present paper we consider a problem with known constant stress

5 Probabilistic method to predict fatigue crack initiation 13 cycle amplitude σ while the quantities r c ( σ), D 0, m and n are considered random with known probability distribution. We then have X = [X 1 X 2 X 3 X 4 ] T = [r c ( σ)mnd 0 ] T ). Observe that < 0 : domain of macrocrack initiation g(x) = 0 : limit state surface > 0 : safe domain The probability of macrocrack initiation P ma is therefore given by P ma = f x (x)dx, (6) g(x)<0 where f X (x) denotes the joint probability distribution of X. To evaluate the integrals appearing in Equation (6), the random variables are first mapped via a Rosenblatt transformation (see Rosenblatt, 1952) into a standard Gaussian space where the random variables denoted by U = [U 1 U 2...U k ] T are independent, normally distributed and have zero mean and unit standard deviation. The modified Hasofer-Lind, Rackwitz-Fiessler (HL-RF) algorithm described in Kiureghian and Liu (1989) is used to obtain the point closest to the origin u on the surface g(u) = 0. In the modified HL-RF algorithm, one adjusts the step size during each iteration to obtain a sufficient decrease in the merit function which is based on the first-order optimality conditions. Monte Carlo integration with importance sampling, with the sampling density centered at u (see Fujita and Rackwitz, 1988) is then used to calculate the integrals in Equation (6). 5. Sample problem To demonstrate the application of the ideas presented in the paper, the probability of macrocrack initiation is calculated for a sample problem. The data, i.e. the ratio A 2 /A 1 as a function of the number of cycles, is obtained from Ogi et al. (2001). Ogi and his co-authors have performed a rotating bending fatigue test with a four point bending configuration on a 0.25% C (mass) steel and have obtained the variation of the ratio A 2 /A 1 with the number of cycles (see Figure 1). The yield strength of the material is 333 MPa specimen and it is subjected to a maximum bending stress of 280 MPa. The endurance limit of the material at zero mean stress, r c (0), is assumed to be 180 MPa. To simulate the inspection process, A 2 /A 1 is obtained from Figure 1 at different number of cycles with the interval between the cycles getting progressively shorter. Table 1 shows the ratio A 2 /A 1 measured as a function of number of cycles (see Figure 1). It is observed that the evolution of the ratio A 2 /A 1 is not strictly monotonic during the initial stages of fatigue. The raw damage is obtained by normalizing the nonlinearity measurements by the expected maximum value of the non-linearity which for the given problem is assumed to be (see Figure 1). The raw damage is tabulated in the third column in Table 2. Note that to be consistent with the assumed monotonic damage evolution (see Section 3), the raw damage is corrected and tabulated as corrected damage in the fourth column in Table 2.

6 14 S. S. Kulkarni et al. Table 1. Measured values of the ratio A 2 /A 1 during successive inspections. j Ninsp j (A 2 /A 1 ) j Table 2. Raw and corrected values of damage during successive inspections. j Ninsp j Raw damage Corrected damage Dinsp j These damage values which are based on the experimental results of Ogi et al. (2001) are used for two different purposes in the paper. In the first place, they are used to determine the statistics of parameters m and n in Equation (1) (see Section 3). Secondly, they are interpreted as damage measurements obtained by structural health monitoring system for input to the procedure to calculate the probability of macrocrack initiation (see Section 4) Prior information For calculating the probability of macrocrack initiation using the experimental data presented earlier, it is assumed that prior information regarding the parameters m and n is available. This is a reasonable assumption if appropriate experiments have been performed in the laboratory. In that case, the statistics for m and n can be calculated by fitting the damage model to the experimental datasets using non-linear regression (see e.g. Draper and Smith 1998). Notice that in the present case, only one experimental data set (Dinsp j,j = 0,...,6) is available. To calculate the statistics for m and n additional datasets need to be generated. The additional datasets are generated by assuming that at inspection cycle Ninsp j, the corrected damage has a truncated normal distribution in the range (0, 1) with mean of the parent normal

7 Probabilistic method to predict fatigue crack initiation 15 distribution equal to Dinsp j and standard deviation equal to 0.1, i.e. ( ) φ D0 µ σ f(d 0 ) = [ ) ( )] 0 D 0 1 σ µ 0 σ ( 1 µ σ with µ=dinsp j and σ =0.1. Here φ and are the probability density and the cumulative distribution function of for the normal distribution. The damage model is fitted to each of the additional datasets, i.e. m and n are calculated for each of the datasets. The statistics of m and n are then compiled. For the current problem, the simulations suggest that m and n are dependent parameters with means and respectively and the coefficient of correlation equal to The standard deviations of m and n are found to be and , respectively. For calculating m and n using non-linear regression, the following values are used: σ = MPa, σ = 0 MPa, N c = 10, 000 and r c (0) = 180 MPa. The value of N c, which is a normalizing constant is assumed to be of the order of the number of cycles to failure. For this particular problem, failure is observed at 56,000 cycles. Here N c is taken to be 10, Calculation of probability of macrocrack initiation The probability of macrocrack initiation is now calculated using the data given in Table 2. The procedure essentially consists of recording the inspection data at, say, cycle Ninsp j and then calculating the corresponding value of the damage, say, Dinsp j. If Dinsp j < 1 then the probability of macrocrack initiation is calculated for N s > Ninsp j as described in Section 4. For the problem under consideration, it is assumed that σ, σ, and N c are fixed quantities while r c (0), D 0, m and n are random variables. The fixed quantities σ, σ and N c, are assumed to have the same values as those used for the non-linear regression (see Section 5.1). The random variable r c (0) is assumed to have a log-normal distribution with a mean of 180 MPa and a standard deviation of 5.4 MPa. The random variable D 0 at cycle Ninsp j is assumed to have a truncated normal distribution in the interval (0, 1) with mean equal to Dinsp j and standard deviation equal to 0.1. This distribution of D 0 is consistent with the one used to generate the statistics of m and n (see Section 5.1). The random variables m and n are assumed to have a joint lognormal distribution with the statistics given in the Section 5.1. Starting from the recording of the third inspection, the probability of macrocrack initiation calculated for different cycles is given in Table 3. It is seen from Table 3 that the probabilistic fatigue damage analysis procedure updates the probability of macrocrack initiation with successive inspections. The cumulative probability distribution for macrocrack initiation with inspections is shown in Figure 2. The numbers (1,...,6) in the figure indicate the cycle numbers at which inspection is recorded. Observe that immediately following an inspection, the probability of macrocrack initiation drops to zero. This is due to the fact that it is implicitly assumed in the probabilistic procedure that a macrocrack if present, will be definitely detected by the inspection procedure.

8 16 S. S. Kulkarni et al. Table 3. Calculation of P ma. Cycles P ma N s 3rd Insp 4th Insp 5th Insp 6th Insp (Ninsp = 26880) (Ninsp = 30800) (Ninsp = 33040) (Ninsp = 34000) 30, , , , , , , P ma --- > N --- > x 10 4 Figure 2. Cumulative probability distribution for macrocrack initiation. Note that for the fatigue problem described, the first macrocrack is observed at approximately 34,160 cycles (see Ogi et al., 2001). As seen from Table 3 and Figure 2, the formation of the macrocrack is predicted quite well. 6. Summary A methodology to predict macrocrack initiation which has the following features has been presented:

9 Probabilistic method to predict fatigue crack initiation 17 use of acoustic non-linearity to quantify damage heuristic damage growth law material/other parameters treated as random variables periodic measurements to assess state of damage and update state of damage calculation of probability of macrocrack initiation Reasonable results are obtained for a sample problem solved using the presented methodology. 7. Acknowledgements This work was supported by the Federal Aviation Administration, Rotorcraft Structural Integrity and Safety, under Award AGMT: 01-C-AW-NWU/A04 to the Center for Quality Engineering and Failure Prevention at Northwestern University. Helpful discussions with Prof. M. E. Fine are gratefully acknowledged. References Draper, N.R. and Smith, H. (1998). Applied Regression Analysis (3rd edn.), John Wiley & Sons, New York. Fujita, M. and Rackwitz, R. (1988). Updating first- and second-order reliability estimates by importance sampling. Structural Safety/Earthquake Engineering 5, Goodman, J. (1899). Mechanics Applied to Engineering. Longmans Green, London. Kiureghian, A. and Liu, P. (1989). Finite-Element reliability methods for geometrically nonlinear stochastic structures, Report No. UCB/SEMM-89/05, Dept. of Civil Engineering, University of California, Berkeley. Miner, M.A. (1945), Cumulative damage in fatigue. Journal of Applied Mechanics 12, Moris, W.L., Buck, O. and Inman, R.V. (1979). Acoustic harmonic generation due to fatigue in highstrength aluminum. Journal of Applied Physics 11, Ogi, H., Hirao, M. and Aoki, S. (2001). Noncontact monitoring of surface-wave nonlinearity for predicting the remaining life of fatigued steels. Journal of Applied Physics 1, Palmgren, A. (1924). Die Lebensdauer von Kugellagern. Zeitschrift des Vereins Deutcher Ingenieure 68, Rosenblatt, M. (1952). Remarks on a multivariate transformation. Annals of Mathematical Statistics 23, Suresh, S. (1991). Fatigue of Materials. Cambridge University Press, Cambridge.

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