Specimen Geometry and Material Property Uncertainty Model for Probabilistic Fatigue Life Predictions Prakash Chandra Gope*, Sandeep Bhatt**, and Mohit Pant** *Associate Professor, Mechanical Engineering Department, College of Technology, G B Pant University of Agriculture & Technology, Pantnagar-263145, India **Research Scholar E mail: pcgope@rediffmail.com, Phone +91941159916, Fax +915944233338 ABSTRACT The present paper describes an analytical model derived from energy theorem which is coupled with the uncertainty associated with material properties and geometrical parameter to estimate fatigue life and crack growth at different level of probability and confidence level. A substantial amount of published data has been used to validate the proposed model for predicting S-N curve as well as fatigue crack growth of steel, aluminum alloys, copper alloys and titanium alloys. Some of are 4340, 304 steel, 0.26 % Carbon steel, different aluminum alloys, titanium alloys etc. Different test geometry such as CT specimen, SEN specimen geometry etc has also been taken to find out the acceptability of the model. INTRODUCTION The analysis of cracks within structure is an important aspect if the damage tolerance and durability of structures and components are to be predicted. As part of the engineering design process, engineers have to assess not only how well the design satisfies the performance requirements but also how durable the product will be over its life cycle. Often cracks cannot be avoided in structures; however the fatigue life of the structure depends on the location and size of these cracks. In order to predict the fatigue life for any component, crack growth study needs to be performed. Fatigue life is related to and is affected to a great extent with the uncertainties in both the material properties and the specimen or component geometrical parameters. The intent of the work is to contribute to the fundamental understanding of fatigue life and its relation with these uncertainties. Fatigue life data exhibits wide scattered results due to inherent microstructural inhomogeneity in the material properties even if the test specimens are taken from the same lot and tested under same loading condition. Several methods [1-27] have been discussed in the literature to predict the fatigue life but the problem of uncertainty associated with material and specimen geometry has not been addressed so far. As the fatigue testing is time consuming and costly, setting up of an analytical method for prediction of fatigue life is necessary. The different monotonic material properties such as yield strength, modulus of elasticity, fracture properties such as critical stress intensity factor, threshold and crack opening stress intensity factors, loading parameters and geometry parameters such as specimen or component width, thickness, length, crack aspect ratio etc. have been considered in the model. These parameters are treated as random variable and assuming normally distributed, the associated uncertainty in these parameters are incorporated in the crack growth model. The assumptions of these variables are also verified from the simulated results. Crack growth model is derived from the energy theorem. In the present work an approximate analytical model derived from the energy theorem and probabilistic nature of material properties and specimen geometry parameters are combined and correlated to determine the associated error in the predicted fatigue life. The Error in estimating the fatigue life is derived from the expected values of fatigue life. The prediction is based on minimization of the error. The model can also be used to predict the fatigue life or crack growth at different level of probability and confidence level. Hence it can be used to draw P-S-N curve. MATERIAL AND METHOD The fatigue crack propagation rate equation based on strain energy release rate derived by Bhatt [2] is given as da dn = 0.139 αµ E. µσ y and fatigue life is obtained as, 1 ( µ ) 2 KI µ Kth 2 1 K max µ K IC (1)
a f da N f = 0.139 1 a0 ( µ K ) I µ 2 K αµ th E. µσ 2 y K 1 max µ K IC where 1.0 if φ >1.15 α = 3.0 if φ 0.27-0.75 φ 3 + 5.179φ 2 8.219φ + 4. 864 for 0.27< φ <1.15 φ = K σ y B B is the specimen thickness, and KI = Kmax Kmin if K min > K op otherwise KI = Kmax Kop where s : Standard deviation da : Increment in crack length, mm dn : Increment in number of cycles to failure, cycles a 0 : Initial crack length, mm a f : Final crack length, mm K I : Stress intensity factor range, MPa m N f : Fatigue life, cycles µ : Population mean σ y : Yield strength, MPa µ (...) : Population mean of variable (..) s (...) : Standard deviation of variable (..) K max : Maximum stress intensity factor, MPa m K min : Minimum stress intensity factor, MPa m K op : Crack opening stress intensity factor, MPa m The opening parameters can be obtained from the following relations. S op = A 2 3 0 + A1 + A2 R + A3 R, R 0 (3) Smax S op = A0 + A1 R, 1 < R < 0 (4) Smax where 1 π α α α 2 S (0.825 0.3 0.05 ) cos max A 0 = + 2 σ 0 A ( 0.415 0.017 ) max 1 = α S σ 0 A2 = 1 A0 A1 A3 A 3 = 2A0 + A1 1 s op = Opening stress, S max = Maximum stress level The fracture is taken as the condition at which Kmax KIC.The N f so obtained from the solution of Equation (2) depends on several monotonic strength parameters like modulus of elasticity ( E ), yield strength ( σ y ), and fracture parameters like threshold stress intensity factor ( K th ), critical stress intensity factor ( K IC ).It is well established that all such strength parameters are random variable and follows normal distribution. Due to the scattered nature of these variables fatigue life obtained can also be assumed as a random variable. The error in estimating the fatigue life can be obtained from [8-10] 1 R = ± φ + 2 2 N. t. u p ψ 1 (5) n (2)
s where φ N = and X + u pψ s n 1 Γ n 1 2 ψ = 2 n Γ 2 where µ and σ are the population mean and population standard deviation, respectively, and u p is the normal deviate corresponding to the probability of survival p and n is the sample size. In this study the scattered behavior of fatigue life and crack growth is incorporated by randomizing the various fatigue parameters considering them as statistically independent random variables following normal distribution with mean values X and standard deviation s. Equations 1-5 µ used in the solution can be obtained from have been solved to predict the fatigue life. The population mean ( ) X µ X = X + K. s (6) K is a factor whose value varies between -3 to 3 for normal distribution. In the present investigation K is randomly generated. The fatigue life corresponds to the life with minimum error obtained from the equation 5. All computations are done using MATLAB 7.0. The detailed procedure is explained in reference [2]. Brief procedure is presented below. Algorithm 1. Estimate the statistical mean of model parameters from the experimental mean and standard deviation. For computing these generate random number K (Equation (6)). The limiting values of K are ± 3. 2. Select type of specimen geometry (SEN, CT, CCT etc) and estimate statistical mean values of geometry parameter similar to step 1. 3. If load history is not given generate the load history. The procedure is explained in reference [8-10]. 4. Compute crack opening parameters. 5. Solve Equation (1) or (2) and estimate fatigue life or crack length. 6. Repeat steps 1 to 5 if sample size (number of predicted life) is less than 3. 7. Estimate Error from Equation (5) for a given probability and confidence level. 8. If Error determined at step 7 is less than pre assumed tolerable error, write fatigue life as the mean value of all lives determined and corresponding stress level, otherwise go to step 1. The material properties used in the fatigue life simulation are presented in Table 1. Table 1. Mechanical & geometrical parameters of material used in fatigue life simulation Material Parameters Modulus YS K IC K K th op Aspect W ( GPa) (MPa) ratio (mm) MPa m MPa m MPa m 7075-T6 al Mean values 68.9 540 30 1.7 * 0.2 50 alloy Co-eff. of var (%) 6.8 5.8 5.9 5.9 5.3 3.2 4.9 D16 AT Al Mean values 64.27 296 31 6.39 4.2 0.2 50 Co-eff. of var (%) 5.62 5.68 5.55 6.11 3.2 3.2 4.9 2024 T3 Mean values 68.9 393 5.5 120.7 2.19 0.2 50 Co-eff. of var (%) 5.62 6.00 5.81 6.1 3.0 3.2 4.9 4340 steel Mean values 207 862 99 6.6 * 0.2 50 Co-eff. of var (%) 5.74 5.77 5.89 5.87 * 3.2 4.9 304 Steel Mean values 207 276 229 6.46 * 0.2 50 Co-eff. of var (%) 5.24 5.98 5.73 6.27 * 3.2 4.9 0.26C Mean values 201 246 189 3.45 * 0.2 50 steel Co-eff. of var (%) 4.9 4.98 4.91 4.50 * 3.2 4.9 6061 T6 Mean values 68.67 295 63 2 * 0.2 50 Co-eff. of var (%) 6.00 5.95 6.05 5.53 * 3.2 4.9 * simulated results RESULTS AND DISCUSSION Fatigue lives of different material at different stress levels, probability of failure and confidence levels are obtained using the proposed model. In the present paper only few results have been presented. The predicted results are also compared with the available experimental fatigue life, taken from the literature [1-28]. The fatigue life predictions shown in Figs.1-7 are based on minimum tolerable error of 5%. It is seen that the difference between the experimental and predicted lives are less at higher stress level compared to lower stress level. This demonstrates that higher scatter band exists at lower stress level compared to higher stress level. It is seen that in all cases studied in the present investigation the percentage difference is less than 5% for most of the stress levels and material.
305 300 295 95% Confidence 99% Confidence 290 285 280 275 270 265 260 255 150000 200000 250000 300000 350000 400000 450000 Cycles to failure Fig. 1 S-N curve for 0.26% carbon steel at 50% Probability and different confidence level, R=-1 290 Stress Amplitude(MPA) 270 250 230 210 190 95% Confidence 99% Confidence 170 150 0 200000 400000 600000 800000 1000000 1200000 1400000 Cycles to Failure Fig. 2. S-N curve for 304 stainless steel at 50% Probability and different confidence level, R=-1
250 230 210 190 170 95% Confidence 99% Confidence 150 0 1000000 2000000 3000000 4000000 Cycles to failure Fig. 3. S-N curve for 2024 T3 Aluminum alloy at 50% Probability and different confidence level, R=-1 250 230 50% Probability 90% Probability 210 190 170 150 0 1000000 2000000 3000000 4000000 Cycles to failure Fig. 4. S-N curve for 2024 T3 Aluminum alloy at 90% confidence level and different probability, R=-1
1400 1200 1000 800 600 400 50% Probability 90% Probability 200 0 100000 200000 300000 400000 500000 600000 700000 800000 Cycles to Failure Fig. 5 S-N curve for 4340 steel at 90% confidence level and different probability, R=-1 350 300 250 200 150 50% Probability 90% Probability 100 0 50000 100000 150000 200000 250000 300000 350000 400000 Cycles to failure Fig. 6 S-N curve for 7075-T6 Aluminum alloy at 90% confidence level and different probability, R=-1
650 600 95% Confidence 99% Confidence 550 500 450 0 300000 600000 900000 1200000 1500000 1800000 Cycles to failure Fig. 7 S-N curve for Ti-6Al-4V Titanium alloy at 50% Probability and different confidence level, R=-1 The model was used to study the crack growth. The variation of crack growth with number of cycles is shown in figure 8-11. The model was also verified with the ASTM procedure described in its standard E 739-91. It is found that present model can be used successfully to find out the S-N curve at different probability and confidence level, only from the monotonic and fracture properties of the material. 0.044 Crack Length a(m) 0.034 0.024 0.014 Predicted 0.004 0 10000 20000 30000 40000 50000 60000 70000 80000 Number of Cycles Fig. 8 Crack length vs. Number of Cycles Plot at constant maximum load (Pmax=8829N), 90%probability, 90% confidence level, R=0 data (Kumar & Pandey, 1990)
0.044 Crack Length a(m) 0.034 0.024 0.014 0.004 0 20000 40000 60000 80000 100000 120000 140000 160000 Number of Cycles Predicted Fig. 9 Crack length vs. Number of Cycles Plot at constant maximum load (Pmax=8829N), 90%probability, 90% confidence level, R=0.3 data (Kumar & Pandey, 1990) 0.044 Crack Length a(m) 0.034 0.024 0.014 0.004 Predicted 0 10000 20000 30000 40000 50000 60000 70000 Number of Cycles Fig. 10. Crack length vs. Number of Cycles Plot at constant load range ( P=8829N), 90% probability, 90% confidence level, R=0 data (Kumar & Pandey, 1990)
0.044 Crack Length a(m) 0.034 0.024 0.014 0.004 0 10000 20000 30000 40000 50000 60000 Number of Cycles Predicted Fig. 11. Crack length vs. Number of Cycles Plot at constant load range ( P=8829N), 90% probability, 90% confidence level, R=0.1 data (Kumar & Pandey, 1990) Conclusions The following conclusions can be drawn from the present study. 1. Fatigue life is greatly influenced by the uncertainty associated with material properties and specimen geometry parameter. 2. Material properties and geometry parameters follows normal distribution. 3. Proposed model can be used to predict fatigue life as well as crack growth rate under given confidence and probability of failure with tolerable error. References 1. Bachmann V., and Munz D., (1975), Crack closure in fatigue of a Titanium alloys, International Journal of Fracture, 11, 713. 2. Bhatt S., Specimen geometry and material property uncertainty model for probabilistic fatigue life prediction, M. Tech Thesis, G B Pant University of Ag & Technology, Pantnagar, India, 2006. 3. Boyer Howard E., Atlas of fatigue curves, American Society for Metals, Ohio 1968. 4. Brown R. D., and Weertman, J., (1978), Mean stress effect on crack propagation rates and crack closure in 7075- T76 aluminum alloy, Engng Fracture Mechanics, 10, 757. 5. Chand S. and Garg S. B. L., (1983), Crack closure studies under constant amplitude loading, Engng Fracture Mechanics, 18, 333. 6. Fatigue Design Handbook AE-10, Society of Automotive Engineers, 1988. 7. Forman, R.G., Kearney, V.E., and Engle, R.M., (1967), Numerical Analysis of Crack Propagation in a Cyclic- Loaded Structure, Trans. ASME, J. Basic Eng., Vol. D89, No. 3, pp. 459-464. 8. Gope P. C, (1999), Determination of sample size for estimation of fatigue life by using Weibull and log normal distribution, International journal of fatigue, 18 (8), 1999, 745. 9. Gope P. C, (2002), Determination of minimum number of specimen in S-N testing, Journal of Engg. Materials and Technology, ASME, ASME Trans, 2002, April 2002. 10. Gope P.C., (2002) Determination of minimum number of specimens in S-N testing, J. Engineering Material &Technology, pp 1-7. 11. Handbook of fatigue crack,(1994), Propagation in metallic structures, Elsevier, Netherlands. 12. Jelaska, Glodez, S., Podrug, S., (2003). Closed form expression for fatigue life prediction at combined HCF/LCF Loading, Vol. 3, pp.635-646. 13. Koutsourelakis P.S.,Kuntiyawichai and Schueller G.I,(2006) Effect of material uncertainties on fatigue life calculations of aircraft fuselages: A cohesive element model, Engineering Fracture Mechanics.
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