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Prediction of crack-opening stress levels for 1045 as-received steel under service loading spectra
International Journal of Fatigue 25 (2003) 149–157 www.elsevier.com/locate/ijfatigue
Prediction of crack-opening stress levels for 1045 as-received steel under service loading spectra M. Khalil ∗, T.H. Topper Department of Civil Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3B8 Received 7 May 2001; received in revised form 28 March 2002; accepted 30 May 2002
Abstract A crack growth analysis based on a fracture mechanics approach was used to model the fatigue behavior of as-received 1045 steel specimens for three load spectra scaled to various maximum stress range levels. The crack growth analysis was based on an effective strain-based intensity factor, a crack growth rate curve obtained during closure-free loading cycles, and a local notch strain calculation based on Neuber’s rule. The crack-opening stresses were modeled assuming that the crack-opening stress when it is not at the constant amplitude steadystate level for a given stress cycle builds up as an exponential function of the difference between the current crack-opening stress and the steady-state crack-opening stress of the given cycle unless this cycle is below the intrinsic stress range for crack growth or the maximum stress in the cycle is below zero in which cases the crack-opening stress does not change. The crack-opening stress model was implemented in a fatigue notch model and the fatigue lives of the notched annealed 1045 steel specimens under the three different load spectra scaled to several maximum stress levels were estimated. The average measured crack-opening stresses for the various histories and levels were within between 8 and 13% of the average calculated crack-opening stresses. The fatigue life predictions based on the modeled crack-opening stresses were in good agreement with the experimentally obtained fatigue data. The averages of the measured crack-opening stresses and those calculated using the model were nearly the same for all the histories examined. When these average crack-opening stresses were used in the life prediction model they gave predictions as good as those obtained by modeling crack-opening stress on a cycle by cycle basis. The use of a crack-opening stress level corresponding to the cycle causing a reduction to a crack-opening stress reached for 1/200 of the cycles in the history gave a conservative estimate of average crack-opening stress for all the histories. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Service spectrum; Crack-opening stress; Effective strain intensity factor; Steady-state crack-opening stress
1. Introduction An empirical model for the steady-state crack-opening stress Sop under constant amplitude loading by DuQuesnay [1] gave good predictions of measured crack-opening stress levels. Many researchers have documented the effects of variable amplitude loading on crack closure [2–8]. The effect of variable amplitude loading is usually illustrated by examining the crack-opening behavior of a crack in a specimen under a load history having an
Corresponding author. Tel.: +1-519-888-4567; fax: +1-519-8886197. E-mail address: [email protected] (M. Khalil). ∗
imposed overload, between two blocks of constant amplitude loading. The application of a tensile overload can cause either an acceleration or a delay in crack growth. A post overload increase in crack closure level and crack growth retardation occurs when the applied overload is less than approximately one-half the yield stress of the material [6,9,10]. An overload of much more than one-half the yield stress of the material will decrease the crack closure level and accelerate crack growth. Pompetzki et al. [11,12] investigated periodic tensile overloads of yield stress magnitude and found an accelerated fatigue damage. They postulated an interaction damage model, based on crack closure concepts, with an effective stress as the damage parameter. Their model assumed that the crack-
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opening stress was reduced immediately following a high stress overload, and that closure then gradually built up to a steady-state level during subsequent cycling. Applying an underload causes a flattening of the asperities in the crack wake, which decreases the crack closure level, and in turn increases the effective stress intensity factor and accelerates crack propagation. The effect of the magnitude of the compressive fatigue cycle on fatigue crack growth was reported in a paper by Yu et al. [13]. They reported that an increase in the magnitude of the compressive peak stress in a compression– tension test, caused the crack propagation rate to increase and the threshold stress intensity factor to decrease. From a study of crack closure in variable amplitude loading Dabayeh and Topper [14] proposed a formula to simulate the buildup of crack-opening stress after an overload. They found that in a block history with blocks consisting of constant amplitude cycles and a tensile overload or an underload, the crack-opening stress drops immediately after the application of the underload or tensile overload. Then the crack-opening stress increases at an exponentially decreasing rate under the following constant amplitude loading. They also found that the number of cycles needed for the crack-opening stress to recover to a steady-state level increases as the stress difference between the post underload or overload crackopening stress and the steady-state crack-opening stress increases. This research is aimed at providing crack closure inputs for modeling fatigue damage or crack growth in a specimen under service load spectra. The crack-opening stress behavior of cracked specimens under service load spectra is modeled and the accuracy of the model is evaluated experimentally using measured crack closure data. This paper is a continuation of research previously presented [15] using the same load spectra in tests on a 2024-T351 aluminum alloy.
Table 2 Mechanical properties of 1045 as-received steel Mechanical properties
Units
Magnitude
Modulus of elasticity Tensile yield stress (0.2% offset) Cyclic yield stress (0.2% offset) Ultimate tensile stress True fracture stress Area reduction
MPa MPa MPa MPa MPa %
206000 761 580 781 647 40
shown in Fig. 1. An edge notch of 0.6 mm diameter was machined into one side of the specimen, at mid length as shown in Fig. 1. Using a notched specimen allows the stresses to exceed the material yield stress at the notch root without buckling or tensile yielding of the whole specimen. This notch size was small enough that, once initiated, the crack rapidly grew out of the zone of influence of the notch. The crack-opening stress was measured using a 900× power short focal length optical video microscope at given cycles before and after an overload was applied. The procedure for measuring the crack-opening stress was to stop the test at the maximum stress of the chosen cycle, and then to decrease the load manually while observing the crack tip region in the monitor attached to the optical video until the crack surfaces in the region within 0.1 mm from the crack tip started to touch each other. Two sets of readings were recorded for each given cyclic stress level, and the average was calculated. 2.1. Crack-opening stress model Dabayeh and Topper [14] measured crack-opening stress changes for various levels of tensile and compressive overloads followed by constant amplitude cycles having a variety of R-ratios. The crack-opening stress after being reduced by the overload increased to its steady-state level in an approximately exponential manner.
2. Material and experimental methods The material used in this study is an SAE 1045 steel in the as-received condition, which is commonly used in the automotive industry. The chemical composition and mechanical properties of the material are given in Tables 1 and 2, respectively. All testing was carried out on round threaded specimens with a rectangular gauge length cross section. The geometry and dimensions are Table 1 Chemical composition (percentage by weight) Steel
C
Si
P
Mn
S
Fe
1045
0.46
0.17
0.027
0.81
0.023
rest
Fig. 1. Specimen geometry.
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They showed that when normalized all the closure stress versus cycles, data fell onto a single curve. However, the application of their relationship to complex load histories is complicated. The present authors have applied a simpler relationship suggested by Vormwald and Seeger [16] relating the change in crack-opening stress in a given cycle to the difference between the current opening stress Scu and the steady-state opening stress Sss. ⌬Sop ⫽ m(Sss⫺Scu)
(1)
where ⌬Sop is the increase in crack-opening stress during a load cycle and m is a material constant produced by fitting Eq. (1) to crack-opening stress build-up measurements. Crack-opening stress measurements are made for two different stress levels using the same type of block loading. In the first test, an overload cycle with peak of 500 MPa in compression and 450 MPa in tension was followed by 10,000 constant amplitude small cycles at an R ratio of 0.8. The crack length at the time of measurement was 0.92 mm and the constant amplitude small cycles had a maximum nominal stress of 450 MPa and a minimum nominal stress of 360 MPa. In the second test, the overload cycle stress peaks were 500 MPa in compression and 450 MPa in tension, followed by 10,000 constant amplitude small cycles with an R ratio of zero. The maximum and minimum stress peaks of the small cycles were 450 and zero MPa, respectively. The crack length at the time of measurement was 1.84 mm. A value of m equal to 0.002 gives a good fit to the measured crack-opening stress data as shown in Fig. 2. The crack-opening stress level drops directly after the overload to a stress level of ⫺28.7 MPa and then increases in an exponential manner until it reaches a steady-state stress level of 79.8 MPa. The steady-state stress level was reached after 5000 constant amplitude cycles following the overload. The constants for
151
DuQuesnay’s steady-state crack-opening stress model (Eq. (2)) were obtained by measuring crack-opening stresses during a load history consisting of different combinations of maximum and minimum stress. The two constants in Eq. (2) were found to be 0.55 and 0.23 for a and b respectively.
冉 冉 冊冊
SSS ⫽ aSmax 1⫺
Smax Sy
2
⫹ bSmin
(2)
where Smax and Smin are the maximum and minimum stresses and Sy is the cyclic yield stress. Using Eq. (1) the crack-opening stress levels were modeled assuming that the crack-opening stress for a given cycle instantaneously decreases to the constant amplitude steadystate level for that cycle if this steady-state crack-opening stress is lower than the current opening stress. Otherwise it follows the exponential build-up formula of Eq. (1) unless the range of stress in the cycle is below the intrinsic stress range, or the maximum stress in the cycle is below zero in which cases it does not change. After an overload the steady-state level for the cycle is calculated using Eq. (2). Then the increment in crackopening stress for the cycle is calculated using Eq. (1) where Sss is the steady-state opening stress from Eq. (2) and Scu is the current opening stress. The increment is added to the current level to give the opening stress at the end of the cycle. This procedure is repeated for each subsequent cycle until a new overload occurs or the steady state for a subsequent cycle is reached. 2.2. Crack growth analysis A crack growth analysis based on a fracture mechanics approach as presented by Dabayeh et al. [17] was used to model the fatigue behavior of the notched specimens for the given load spectra and stress ranges. The
Fig. 2. Curve fitting for the crack-opening stress build-up data.
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crack growth analysis was based on an effective strainbased intensity factor as presented by Dabayeh et al. [17], a crack growth rate curve obtained during closurefree loading cycles, and a local notch strain calculation based on Neuber’s rule [17]. To obtain closure free loading cycles for the crack growth rate curve, a load history that included periodic overloads was applied to the notched specimens as suggested by Dabayeh and Topper [14]. An overload of ⫺300 to 200 MPa was selected and kept constant for all of the tests. The maximum stress for the following small cycles was 200 MPa while the minimum was varied to obtain desired stress intensity factor ⌬K values. The number of small cycles inbetween two consecutive overloads was 300 and to ensure a fully opened crack the R ratio for the small cycles was never less than 0.75. The lack of crack closure was confirmed by a few measurements of crackopening stress throughout the loading blocks. A curve of effective stress intensity factor versus crack growth rate was obtained by applying the variable amplitude block loading history described above. The total crack growth rate due to both the overload and the small cycles was calculated. This was followed by the application of 300 overload cycles. The crack growth rate for the overload cycles was then calculated. The crack growth rate for the small cycles in the block test was obtained by subtracting the growth per block due to the overload cycle from the total growth per block and dividing by the number of small cycles in the block. A fatigue life prediction, as shown in the following flow chart, is carried out by a numerical integration using the closure-free crack growth curve between the initial and final crack lengths of ao and af, respectively. The integration is performed by making cycle by cycle calculations for a given service load spectrum. The initial crack size ao was set to an average slip band crack length of 3 µm. The failure length af was taken to be equal to one-half of the specimen thickness or the length at which the fracture stress intensity was reached, whichever occurred first. 2.3. Experimental measurements The three load spectra were scaled to various maximum stress ranges. The three spectra are the torsion channel of the Society of Automotive Engineers SAE GKN Grapple-Skidder (GSH) history which has a positive average mean stress, the reversed SAE GKN Grapple- Skidder (IGSH) history which has a negative average mean stress, and the cable channel of the SAE LogSkidder history (LSH) which has a zero average mean stress. The GSH history was supplied in the form of normalized sequential peak and valley points with a maximum value of 318 and a minimum value of ⫺238 and contains 41,124 reversals. The GSH history was multiplied by ⫺1 to produce the IGSH history which
Fig. 3.
Grapple Skidder history maximum stress 910 MPa.
has a negative average mean stress and normalized sequential peak and valley points with a maximum value of 238 and a minimum value of ⫺318 and contains 41,124 reversals. The LSH history was supplied in the form of normalized sequential peak and valley points with a maximum value of 7.3 and a minimum value of ⫺7.7 and contains 62,326 reversals. In scaling these histories for tests the most severe level was chosen so that the maximum and minimum stresses did not cause large scale notch plasticity as the fatigue crack grew out of the notch while the least severe level was set so that fatigue lives did not exceed 1500 blocks. In order to obtain fatigue life curves the three spectra mentioned above were scaled to various stress ranges within these limits and applied to the 1045 notched steel specimens till failure. 2.4. Results for the SAE Grapple Skidder load history For the experimental crack closure measurements the torsion channel of the Grapple Skidder spectrum which has a positive average mean stress was scaled to two different stress levels, a maximum nominal stress range of 910 MPa and a maximum nominal stress range of 751 MPa. The measured crack-opening stresses for the two different maximum stress levels are shown in Figs. 3 and 4, respectively. The figures show the applied nominal
Fig. 4.
Grapple Skidder history maximum stress 751 Mpa.
M. Khalil, T.H. Topper / International Journal of Fatigue 25 (2003) 149–157
Fig. 5. Expended view of section A in Fig. 3.
stress spectrum for each load history, the calculated crack-opening stresses using the crack-opening stress model and the experimentally measured crack-opening stresses. Fig. 5 is an expanded view of a portion of Fig. 3 that shows the modeled and experimental crack-opening stress. As expected, the crack-opening stress decreases when the specimen is subjected to a large overload then starts to build-up again during subsequent smaller cycles. The average measured crack-opening stresses were within 8 and 9% of the average calculated crack-opening stresses for the two different maximum stress levels, respectively. Fig. 6 shows a plot of the measured fatigue lives in blocks (one block is equal to 41,124 reversals) versus the maximum stress range of the Grapple Skidder load spectrum for each test and the lives predicted using the crack-opening stress model in the fatigue crack growth program. The estimated fatigue lives are in good agreement with the experimental observations. 2.5. Results for the SAE Log Skidder load history The cable channel of the Log Skidder spectrum which had an average mean stress of zero was scaled to two
Fig. 6. Fatigue life versus maximum stress (Grapple Skidder history).
Fig. 7.
153
Log Skidder history maximum stress 887 Mpa.
different stress levels for crack closure measurements. These stress levels were: a maximum nominal stress range of 887 MPa and a maximum nominal stress range of 675 MPa. The crack-opening stresses for the maximum stress ranges are shown in Figs. 7 and 8, respectively. The figures show the nominal applied spectrum for maximum stress ranges of 887 and 675 MPa, the calculated crack-opening stresses using the crackopening stress model and the experimentally measured crack-opening stresses. Fig. 9 is an expanded view of a portion of Fig. 7 that shows the modeled and experimental crack-opening stress. The average measured crack-opening stress was within 9 and 11% of the average calculated crack-opening stress for the two different maximum stress levels, respectively. Fig. 10 shows the measured fatigue lives in blocks (one block is equal to 62,326 reversals) and a curve for the calculated fatigue lives versus the maximum stress range of the Log Skidder Load spectrum for each test. The estimated fatigue lives are in good agreement with the experimental values.
Fig. 8.
Log Skidder history maximum stress 675 MPa.
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Fig. 9.
Fig. 10.
Fig. 11.
Inverse Grapple Skidder history maximum stress 1008 Mpa.
Fig. 12.
Inverse Grapple Skidder history maximum stress 675 MPa.
Expended view of section B in Fig. 7.
Fatigue life versus maximum stress (Log Skidder history).
2.6. Results for the inverse Grapple Skidder load history The inverse of the torsion channel of the Grapple Skidder spectrum which had a negative average mean stress was scaled to two different maximum stress levels. These stress levels are: a maximum nominal stress range of 1008 MPa and a maximum nominal stress range of 675 MPa. The crack-opening stresses for the two maximum stress ranges are shown in Figs. 10 and 11, respectively. Figs. 12 and 13 is an expanded view of a portion of Fig. 11 that shows the modeled and experimental crack-opening stress. The average measured crack-opening stresses were within 11 and 12% of the average calculated crack-opening stress for the two different maximum stresses levels, respectively. Fig. 14 shows the measured fatigue lives in blocks (one block is equal to 41,124 reversals) versus the maximum stress range of the inverse Grapple Skidder spectrum. The estimated fatigue lives are in good agreement with the experimental values.
Fig. 13.
Expended view of section C in Fig. 11.
2.6.1. Average crack-opening stress When loaded with SAE service load histories scaled to various maximum stress range levels, crack-opening stresses for all the metals examined showed a common trend. The crack-opening stress Sop decreased immediately following overloads and then increased during subsequent smaller load cycles until it was again reduced by another large load cycle. Throughout each test, Sop
M. Khalil, T.H. Topper / International Journal of Fatigue 25 (2003) 149–157
Fig. 14. Fatigue life versus maximum stress (inverse Grapple Skidder history).
did not vary greatly from its average level as increases during smaller cycles were continuously being matched by decreases due to overloads. It was observed that the average value of Sop was not very far above its minimum value for a test. This explains the success of an assumption made by DuQuesnay et al. [1] that using the lowest Sop in the history that followed the most severe, in terms of lowering Sop, overload in the history could be used as an average value in making fatigue life calculations. As expected, since this value was lower than the true average value, effective stress ranges were overestimated and fatigue life calculations were conservative. In this paper, crack-opening stress data from the application of various severities of the three SAE load histories to the 1045 as-received steel used in this investigation are examined to relate the average crack-opening stress to the stress cycle ⌬S∗ that will reduce the crack-opening stress to this average level Sop (that is to the cycle ⌬S∗ that has the same steady-state crack-opening stress as the average for a test). To enable this cycle to be characterized for different load histories and various levels of load severity, it was described by the accumulative number of occurrences Nop in the history having Sop levels equal to or lower than Sopa. If this Nop is normalized by dividing by the total number of cycles in the history it gives the fraction of the load cycles aop in the history having Sop levels equal to or lower than Sopa. Since the average opening stress Sopa is not far removed from the minimum Sop which follows the most severe overload it is expected that the cycle ⌬S∗ will be large and aop will be small. The procedure for deriving aop values is as follows: The stress cycles in the history are arranged in a table in ascending order of Sop calculated from Eq. (2) and Nop values are assigned to each cycle. The Nop value associated with the cycle having a crack-opening stress equal to the average for a test is then obtained by choos-
155
Fig. 15.
Accumulative frequency versus Sop for (Grapple Skidder history).
Fig. 16.
Accumulative frequency versus Sop for (Log Skidder history).
ing the desired cycle from the table or graphically as shown in Figs. 15–17. Each figure gives data for the highest and lowest scaling of one of the service histories applied to the1045
Fig. 17.
Accumulative frequency versus Sop for (inverse Grapple Skidder history).
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Fig. 18. Fatigue life versus stress range for (Grapple Skidder history).
as-received steel. Also shown in the plot are the average measured crack-opening stresses Sop∗ for each of the scaling levels. For the Grapple Skidder and the inverse Grapple Skidder histories with 20,562 cycles, the average closure level corresponds to the level reached approximately 100 times in the histories for the most severe case or an aop of 1/200. For the Log Skidder history applied, the average closure level is reached about 150 times in the history which gives an average aop of 1/200. Crack closure levels corresponding to the average levels and to the minimum crack-opening stress in the histories, negative 63 for the Grapple Skidder history or 1/20,000 level in Fig. 18, were implemented in the fatigue notch model and fatigue lives were estimated for the 1045 as-received steel under the three load spectra. The results are shown in Figs. 18–20. A comparison of the predicted curves with the fatigue results shows clearly that fixing the crack-opening stress level at the minimum level in the history as suggested by DuQuesnay [18] leads to conservative fatigue life estimations. A good and still conservative estimate of fatigue life occurs using the Sop for the cycle with an aop of 1/200 for all the histories.
Fig. 20.
Fatigue life versus stress range for (inverse Grapple Skidder history).
Choosing Sop for a frequency of occurrence of 1/200 is in agreement with previous research presented by the authors [15] on a 2024-T351 aluminum alloy which also found this value to give a good fatigue life estimate. This research will be extended to other metals.
3. Conclusions 1. The crack-opening stress level dropped immediately after the application of an overload and then gradually increased with subsequent small cycles. 2. The crack-opening stress can be modeled using an exponential build-up formula which is a function of the difference between the current crack-opening stress and the steady-state crack-opening stress of a given cycle. 3. The measured crack-opening stress showed that the crack-opening stress does not vary greatly from its average value for each intensity of a load history. 4. The fatigue-life of a 1045 steel component can be modeled using the crack-opening stress level corresponding to the stress cycle which reduces crackopening stress to a level reached by 1/200 of the number of cycles in the history which gave a conservative estimate of average crack-opening stress for all the histories.
References
Fig. 19.
Fatigue life versus stress range for (Log Skidder history).
[1] DuQuesnay D. Application of overload data to fatigue analysis and testing. In: Braun AA, McKeighan PC, Niclson MA, Lohr PR, editors. Application of automation technology in fatigue and fracture testing and analysis, ASTM STP 1411. West Conshohocken, PA: American Society for Testing and Materials; 2002. [2] Bernard, P.J., Lindley T.C., Richards, C.E., Mechanics of overload retardation during fatigue crack propagation, ASTM STP 595, West Conshohocken, PA: American Society for Testing and Materials. 1976, 78-97.
M. Khalil, T.H. Topper / International Journal of Fatigue 25 (2003) 149–157
[3] Wheeler OE. Spectrum loading and crack growth. J Basic Eng Trans Am Soc Mech Eng 1972;4:181–6. [4] Matsouks S, Tanaka K. Delayed retardation phenomena of fatigue crack growth in various steels and alloys. J Master Sci 1978;13:1335–53. [5] Shin CS, Fleck NA. Overload retardation in a structural steel. Fatigue Fract Eng Master Struct 1987;9:379–93. [6] Ward-Close CM, Blom AF, Ritchie RO. Mechanisms associated with transient fatigue crack growth under variable amplitude loading. Eng Fract Mech 1989;32:613–38. [7] Suresh, S., Micromechanisms of fatigue crack retardation following overloads, ASTM STP 595, 18, West Conshohocken, PA: American Society for Testing and Materials. 1983, 577-593. [8] Schijve J. The accumulation of fatigue damage in aircraft materials. AGARD, No. 157; 1972. [9] Topper TH, Yu MT. The effect of overloads on the threshold and crack closure. Int J Fatigue 1985;7(3):159–64. [10] Alzeo, W.X., Skatjr, A.C., Hillberry, B.M., Effect of single overload/underload cycles on fatigue crack propagation, ASTM STP 595, West Conshohocken, PA: American Society for Testing and Materials. 1976, 41-60. [11] Pompetzki MA, Saper RA, Topper TH. Effect of compressive underloads and tensile overloads on fatigue damage accumulation in SAE 1045 Steel. Int J Fatigue 1990;12:207–13. [12] Pompetzki MA, Topper TH, DuQuesnay DL, Yu MT. Effect of
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compressive underloads and tensile overloads on fatigue damage accumulation in 2024-T351 aluminum. J Test Eval 1990;18(1):53–61. Yu MT, Topper TH, DuQuesnay DL, Levin MS. The effect of compressive peak stress on fatigue behavior. Int J Fatigue 1986;8:9–15. Dabayeh A, Topper TH. Changes in crack-opening stress after underloads and overloads in a 2024-T351 aluminum alloy. Int J Fatigue 1995;17(4):261–9. Khalil M, DuQuesnay D, Topper TH. Prediction of crack-opening stress levels for service loading spectra. In: Braun AA, McKeighan PC, Niclson MA, Lohr PR, editors. Application of automation technology in fatigue and fracture testing and analysis, ASTM STP 1411. West Conshohocken, PA: American Society for Testing and Materials; 2002. Vormwald M, Seeger T. The consequences of short crack closure on fatigue crack growth under variable amplitude loading. Fatigue Fract Eng Mater Struct 1991;14(3):205–25. Dabayeh AA, Xu RX, Du BP, Topper TH. Fatigue of cast aluminum alloys under constant and variable-amplitude loading. Int J Fatigue 1996;18(2):95–104. DuQuesnay DL. Fatigue damage accumulation in metals subjected to high mean stress and overload cycles. Ph.D. thesis, University of Waterloo, ON, Canada, 1991.
Prediction of crack-opening stress levels for 1045 as-received steel under service loading spectra M. Khalil ∗, T.H. Topper Department of Civil Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3B8 Received 7 May 2001; received in revised form 28 March 2002; accepted 30 May 2002
Abstract A crack growth analysis based on a fracture mechanics approach was used to model the fatigue behavior of as-received 1045 steel specimens for three load spectra scaled to various maximum stress range levels. The crack growth analysis was based on an effective strain-based intensity factor, a crack growth rate curve obtained during closure-free loading cycles, and a local notch strain calculation based on Neuber’s rule. The crack-opening stresses were modeled assuming that the crack-opening stress when it is not at the constant amplitude steadystate level for a given stress cycle builds up as an exponential function of the difference between the current crack-opening stress and the steady-state crack-opening stress of the given cycle unless this cycle is below the intrinsic stress range for crack growth or the maximum stress in the cycle is below zero in which cases the crack-opening stress does not change. The crack-opening stress model was implemented in a fatigue notch model and the fatigue lives of the notched annealed 1045 steel specimens under the three different load spectra scaled to several maximum stress levels were estimated. The average measured crack-opening stresses for the various histories and levels were within between 8 and 13% of the average calculated crack-opening stresses. The fatigue life predictions based on the modeled crack-opening stresses were in good agreement with the experimentally obtained fatigue data. The averages of the measured crack-opening stresses and those calculated using the model were nearly the same for all the histories examined. When these average crack-opening stresses were used in the life prediction model they gave predictions as good as those obtained by modeling crack-opening stress on a cycle by cycle basis. The use of a crack-opening stress level corresponding to the cycle causing a reduction to a crack-opening stress reached for 1/200 of the cycles in the history gave a conservative estimate of average crack-opening stress for all the histories. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Service spectrum; Crack-opening stress; Effective strain intensity factor; Steady-state crack-opening stress
1. Introduction An empirical model for the steady-state crack-opening stress Sop under constant amplitude loading by DuQuesnay [1] gave good predictions of measured crack-opening stress levels. Many researchers have documented the effects of variable amplitude loading on crack closure [2–8]. The effect of variable amplitude loading is usually illustrated by examining the crack-opening behavior of a crack in a specimen under a load history having an
Corresponding author. Tel.: +1-519-888-4567; fax: +1-519-8886197. E-mail address: [email protected] (M. Khalil). ∗
imposed overload, between two blocks of constant amplitude loading. The application of a tensile overload can cause either an acceleration or a delay in crack growth. A post overload increase in crack closure level and crack growth retardation occurs when the applied overload is less than approximately one-half the yield stress of the material [6,9,10]. An overload of much more than one-half the yield stress of the material will decrease the crack closure level and accelerate crack growth. Pompetzki et al. [11,12] investigated periodic tensile overloads of yield stress magnitude and found an accelerated fatigue damage. They postulated an interaction damage model, based on crack closure concepts, with an effective stress as the damage parameter. Their model assumed that the crack-
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opening stress was reduced immediately following a high stress overload, and that closure then gradually built up to a steady-state level during subsequent cycling. Applying an underload causes a flattening of the asperities in the crack wake, which decreases the crack closure level, and in turn increases the effective stress intensity factor and accelerates crack propagation. The effect of the magnitude of the compressive fatigue cycle on fatigue crack growth was reported in a paper by Yu et al. [13]. They reported that an increase in the magnitude of the compressive peak stress in a compression– tension test, caused the crack propagation rate to increase and the threshold stress intensity factor to decrease. From a study of crack closure in variable amplitude loading Dabayeh and Topper [14] proposed a formula to simulate the buildup of crack-opening stress after an overload. They found that in a block history with blocks consisting of constant amplitude cycles and a tensile overload or an underload, the crack-opening stress drops immediately after the application of the underload or tensile overload. Then the crack-opening stress increases at an exponentially decreasing rate under the following constant amplitude loading. They also found that the number of cycles needed for the crack-opening stress to recover to a steady-state level increases as the stress difference between the post underload or overload crackopening stress and the steady-state crack-opening stress increases. This research is aimed at providing crack closure inputs for modeling fatigue damage or crack growth in a specimen under service load spectra. The crack-opening stress behavior of cracked specimens under service load spectra is modeled and the accuracy of the model is evaluated experimentally using measured crack closure data. This paper is a continuation of research previously presented [15] using the same load spectra in tests on a 2024-T351 aluminum alloy.
Table 2 Mechanical properties of 1045 as-received steel Mechanical properties
Units
Magnitude
Modulus of elasticity Tensile yield stress (0.2% offset) Cyclic yield stress (0.2% offset) Ultimate tensile stress True fracture stress Area reduction
MPa MPa MPa MPa MPa %
206000 761 580 781 647 40
shown in Fig. 1. An edge notch of 0.6 mm diameter was machined into one side of the specimen, at mid length as shown in Fig. 1. Using a notched specimen allows the stresses to exceed the material yield stress at the notch root without buckling or tensile yielding of the whole specimen. This notch size was small enough that, once initiated, the crack rapidly grew out of the zone of influence of the notch. The crack-opening stress was measured using a 900× power short focal length optical video microscope at given cycles before and after an overload was applied. The procedure for measuring the crack-opening stress was to stop the test at the maximum stress of the chosen cycle, and then to decrease the load manually while observing the crack tip region in the monitor attached to the optical video until the crack surfaces in the region within 0.1 mm from the crack tip started to touch each other. Two sets of readings were recorded for each given cyclic stress level, and the average was calculated. 2.1. Crack-opening stress model Dabayeh and Topper [14] measured crack-opening stress changes for various levels of tensile and compressive overloads followed by constant amplitude cycles having a variety of R-ratios. The crack-opening stress after being reduced by the overload increased to its steady-state level in an approximately exponential manner.
2. Material and experimental methods The material used in this study is an SAE 1045 steel in the as-received condition, which is commonly used in the automotive industry. The chemical composition and mechanical properties of the material are given in Tables 1 and 2, respectively. All testing was carried out on round threaded specimens with a rectangular gauge length cross section. The geometry and dimensions are Table 1 Chemical composition (percentage by weight) Steel
C
Si
P
Mn
S
Fe
1045
0.46
0.17
0.027
0.81
0.023
rest
Fig. 1. Specimen geometry.
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They showed that when normalized all the closure stress versus cycles, data fell onto a single curve. However, the application of their relationship to complex load histories is complicated. The present authors have applied a simpler relationship suggested by Vormwald and Seeger [16] relating the change in crack-opening stress in a given cycle to the difference between the current opening stress Scu and the steady-state opening stress Sss. ⌬Sop ⫽ m(Sss⫺Scu)
(1)
where ⌬Sop is the increase in crack-opening stress during a load cycle and m is a material constant produced by fitting Eq. (1) to crack-opening stress build-up measurements. Crack-opening stress measurements are made for two different stress levels using the same type of block loading. In the first test, an overload cycle with peak of 500 MPa in compression and 450 MPa in tension was followed by 10,000 constant amplitude small cycles at an R ratio of 0.8. The crack length at the time of measurement was 0.92 mm and the constant amplitude small cycles had a maximum nominal stress of 450 MPa and a minimum nominal stress of 360 MPa. In the second test, the overload cycle stress peaks were 500 MPa in compression and 450 MPa in tension, followed by 10,000 constant amplitude small cycles with an R ratio of zero. The maximum and minimum stress peaks of the small cycles were 450 and zero MPa, respectively. The crack length at the time of measurement was 1.84 mm. A value of m equal to 0.002 gives a good fit to the measured crack-opening stress data as shown in Fig. 2. The crack-opening stress level drops directly after the overload to a stress level of ⫺28.7 MPa and then increases in an exponential manner until it reaches a steady-state stress level of 79.8 MPa. The steady-state stress level was reached after 5000 constant amplitude cycles following the overload. The constants for
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DuQuesnay’s steady-state crack-opening stress model (Eq. (2)) were obtained by measuring crack-opening stresses during a load history consisting of different combinations of maximum and minimum stress. The two constants in Eq. (2) were found to be 0.55 and 0.23 for a and b respectively.
冉 冉 冊冊
SSS ⫽ aSmax 1⫺
Smax Sy
2
⫹ bSmin
(2)
where Smax and Smin are the maximum and minimum stresses and Sy is the cyclic yield stress. Using Eq. (1) the crack-opening stress levels were modeled assuming that the crack-opening stress for a given cycle instantaneously decreases to the constant amplitude steadystate level for that cycle if this steady-state crack-opening stress is lower than the current opening stress. Otherwise it follows the exponential build-up formula of Eq. (1) unless the range of stress in the cycle is below the intrinsic stress range, or the maximum stress in the cycle is below zero in which cases it does not change. After an overload the steady-state level for the cycle is calculated using Eq. (2). Then the increment in crackopening stress for the cycle is calculated using Eq. (1) where Sss is the steady-state opening stress from Eq. (2) and Scu is the current opening stress. The increment is added to the current level to give the opening stress at the end of the cycle. This procedure is repeated for each subsequent cycle until a new overload occurs or the steady state for a subsequent cycle is reached. 2.2. Crack growth analysis A crack growth analysis based on a fracture mechanics approach as presented by Dabayeh et al. [17] was used to model the fatigue behavior of the notched specimens for the given load spectra and stress ranges. The
Fig. 2. Curve fitting for the crack-opening stress build-up data.
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crack growth analysis was based on an effective strainbased intensity factor as presented by Dabayeh et al. [17], a crack growth rate curve obtained during closurefree loading cycles, and a local notch strain calculation based on Neuber’s rule [17]. To obtain closure free loading cycles for the crack growth rate curve, a load history that included periodic overloads was applied to the notched specimens as suggested by Dabayeh and Topper [14]. An overload of ⫺300 to 200 MPa was selected and kept constant for all of the tests. The maximum stress for the following small cycles was 200 MPa while the minimum was varied to obtain desired stress intensity factor ⌬K values. The number of small cycles inbetween two consecutive overloads was 300 and to ensure a fully opened crack the R ratio for the small cycles was never less than 0.75. The lack of crack closure was confirmed by a few measurements of crackopening stress throughout the loading blocks. A curve of effective stress intensity factor versus crack growth rate was obtained by applying the variable amplitude block loading history described above. The total crack growth rate due to both the overload and the small cycles was calculated. This was followed by the application of 300 overload cycles. The crack growth rate for the overload cycles was then calculated. The crack growth rate for the small cycles in the block test was obtained by subtracting the growth per block due to the overload cycle from the total growth per block and dividing by the number of small cycles in the block. A fatigue life prediction, as shown in the following flow chart, is carried out by a numerical integration using the closure-free crack growth curve between the initial and final crack lengths of ao and af, respectively. The integration is performed by making cycle by cycle calculations for a given service load spectrum. The initial crack size ao was set to an average slip band crack length of 3 µm. The failure length af was taken to be equal to one-half of the specimen thickness or the length at which the fracture stress intensity was reached, whichever occurred first. 2.3. Experimental measurements The three load spectra were scaled to various maximum stress ranges. The three spectra are the torsion channel of the Society of Automotive Engineers SAE GKN Grapple-Skidder (GSH) history which has a positive average mean stress, the reversed SAE GKN Grapple- Skidder (IGSH) history which has a negative average mean stress, and the cable channel of the SAE LogSkidder history (LSH) which has a zero average mean stress. The GSH history was supplied in the form of normalized sequential peak and valley points with a maximum value of 318 and a minimum value of ⫺238 and contains 41,124 reversals. The GSH history was multiplied by ⫺1 to produce the IGSH history which
Fig. 3.
Grapple Skidder history maximum stress 910 MPa.
has a negative average mean stress and normalized sequential peak and valley points with a maximum value of 238 and a minimum value of ⫺318 and contains 41,124 reversals. The LSH history was supplied in the form of normalized sequential peak and valley points with a maximum value of 7.3 and a minimum value of ⫺7.7 and contains 62,326 reversals. In scaling these histories for tests the most severe level was chosen so that the maximum and minimum stresses did not cause large scale notch plasticity as the fatigue crack grew out of the notch while the least severe level was set so that fatigue lives did not exceed 1500 blocks. In order to obtain fatigue life curves the three spectra mentioned above were scaled to various stress ranges within these limits and applied to the 1045 notched steel specimens till failure. 2.4. Results for the SAE Grapple Skidder load history For the experimental crack closure measurements the torsion channel of the Grapple Skidder spectrum which has a positive average mean stress was scaled to two different stress levels, a maximum nominal stress range of 910 MPa and a maximum nominal stress range of 751 MPa. The measured crack-opening stresses for the two different maximum stress levels are shown in Figs. 3 and 4, respectively. The figures show the applied nominal
Fig. 4.
Grapple Skidder history maximum stress 751 Mpa.
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Fig. 5. Expended view of section A in Fig. 3.
stress spectrum for each load history, the calculated crack-opening stresses using the crack-opening stress model and the experimentally measured crack-opening stresses. Fig. 5 is an expanded view of a portion of Fig. 3 that shows the modeled and experimental crack-opening stress. As expected, the crack-opening stress decreases when the specimen is subjected to a large overload then starts to build-up again during subsequent smaller cycles. The average measured crack-opening stresses were within 8 and 9% of the average calculated crack-opening stresses for the two different maximum stress levels, respectively. Fig. 6 shows a plot of the measured fatigue lives in blocks (one block is equal to 41,124 reversals) versus the maximum stress range of the Grapple Skidder load spectrum for each test and the lives predicted using the crack-opening stress model in the fatigue crack growth program. The estimated fatigue lives are in good agreement with the experimental observations. 2.5. Results for the SAE Log Skidder load history The cable channel of the Log Skidder spectrum which had an average mean stress of zero was scaled to two
Fig. 6. Fatigue life versus maximum stress (Grapple Skidder history).
Fig. 7.
153
Log Skidder history maximum stress 887 Mpa.
different stress levels for crack closure measurements. These stress levels were: a maximum nominal stress range of 887 MPa and a maximum nominal stress range of 675 MPa. The crack-opening stresses for the maximum stress ranges are shown in Figs. 7 and 8, respectively. The figures show the nominal applied spectrum for maximum stress ranges of 887 and 675 MPa, the calculated crack-opening stresses using the crackopening stress model and the experimentally measured crack-opening stresses. Fig. 9 is an expanded view of a portion of Fig. 7 that shows the modeled and experimental crack-opening stress. The average measured crack-opening stress was within 9 and 11% of the average calculated crack-opening stress for the two different maximum stress levels, respectively. Fig. 10 shows the measured fatigue lives in blocks (one block is equal to 62,326 reversals) and a curve for the calculated fatigue lives versus the maximum stress range of the Log Skidder Load spectrum for each test. The estimated fatigue lives are in good agreement with the experimental values.
Fig. 8.
Log Skidder history maximum stress 675 MPa.
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Fig. 9.
Fig. 10.
Fig. 11.
Inverse Grapple Skidder history maximum stress 1008 Mpa.
Fig. 12.
Inverse Grapple Skidder history maximum stress 675 MPa.
Expended view of section B in Fig. 7.
Fatigue life versus maximum stress (Log Skidder history).
2.6. Results for the inverse Grapple Skidder load history The inverse of the torsion channel of the Grapple Skidder spectrum which had a negative average mean stress was scaled to two different maximum stress levels. These stress levels are: a maximum nominal stress range of 1008 MPa and a maximum nominal stress range of 675 MPa. The crack-opening stresses for the two maximum stress ranges are shown in Figs. 10 and 11, respectively. Figs. 12 and 13 is an expanded view of a portion of Fig. 11 that shows the modeled and experimental crack-opening stress. The average measured crack-opening stresses were within 11 and 12% of the average calculated crack-opening stress for the two different maximum stresses levels, respectively. Fig. 14 shows the measured fatigue lives in blocks (one block is equal to 41,124 reversals) versus the maximum stress range of the inverse Grapple Skidder spectrum. The estimated fatigue lives are in good agreement with the experimental values.
Fig. 13.
Expended view of section C in Fig. 11.
2.6.1. Average crack-opening stress When loaded with SAE service load histories scaled to various maximum stress range levels, crack-opening stresses for all the metals examined showed a common trend. The crack-opening stress Sop decreased immediately following overloads and then increased during subsequent smaller load cycles until it was again reduced by another large load cycle. Throughout each test, Sop
M. Khalil, T.H. Topper / International Journal of Fatigue 25 (2003) 149–157
Fig. 14. Fatigue life versus maximum stress (inverse Grapple Skidder history).
did not vary greatly from its average level as increases during smaller cycles were continuously being matched by decreases due to overloads. It was observed that the average value of Sop was not very far above its minimum value for a test. This explains the success of an assumption made by DuQuesnay et al. [1] that using the lowest Sop in the history that followed the most severe, in terms of lowering Sop, overload in the history could be used as an average value in making fatigue life calculations. As expected, since this value was lower than the true average value, effective stress ranges were overestimated and fatigue life calculations were conservative. In this paper, crack-opening stress data from the application of various severities of the three SAE load histories to the 1045 as-received steel used in this investigation are examined to relate the average crack-opening stress to the stress cycle ⌬S∗ that will reduce the crack-opening stress to this average level Sop (that is to the cycle ⌬S∗ that has the same steady-state crack-opening stress as the average for a test). To enable this cycle to be characterized for different load histories and various levels of load severity, it was described by the accumulative number of occurrences Nop in the history having Sop levels equal to or lower than Sopa. If this Nop is normalized by dividing by the total number of cycles in the history it gives the fraction of the load cycles aop in the history having Sop levels equal to or lower than Sopa. Since the average opening stress Sopa is not far removed from the minimum Sop which follows the most severe overload it is expected that the cycle ⌬S∗ will be large and aop will be small. The procedure for deriving aop values is as follows: The stress cycles in the history are arranged in a table in ascending order of Sop calculated from Eq. (2) and Nop values are assigned to each cycle. The Nop value associated with the cycle having a crack-opening stress equal to the average for a test is then obtained by choos-
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Fig. 15.
Accumulative frequency versus Sop for (Grapple Skidder history).
Fig. 16.
Accumulative frequency versus Sop for (Log Skidder history).
ing the desired cycle from the table or graphically as shown in Figs. 15–17. Each figure gives data for the highest and lowest scaling of one of the service histories applied to the1045
Fig. 17.
Accumulative frequency versus Sop for (inverse Grapple Skidder history).
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Fig. 18. Fatigue life versus stress range for (Grapple Skidder history).
as-received steel. Also shown in the plot are the average measured crack-opening stresses Sop∗ for each of the scaling levels. For the Grapple Skidder and the inverse Grapple Skidder histories with 20,562 cycles, the average closure level corresponds to the level reached approximately 100 times in the histories for the most severe case or an aop of 1/200. For the Log Skidder history applied, the average closure level is reached about 150 times in the history which gives an average aop of 1/200. Crack closure levels corresponding to the average levels and to the minimum crack-opening stress in the histories, negative 63 for the Grapple Skidder history or 1/20,000 level in Fig. 18, were implemented in the fatigue notch model and fatigue lives were estimated for the 1045 as-received steel under the three load spectra. The results are shown in Figs. 18–20. A comparison of the predicted curves with the fatigue results shows clearly that fixing the crack-opening stress level at the minimum level in the history as suggested by DuQuesnay [18] leads to conservative fatigue life estimations. A good and still conservative estimate of fatigue life occurs using the Sop for the cycle with an aop of 1/200 for all the histories.
Fig. 20.
Fatigue life versus stress range for (inverse Grapple Skidder history).
Choosing Sop for a frequency of occurrence of 1/200 is in agreement with previous research presented by the authors [15] on a 2024-T351 aluminum alloy which also found this value to give a good fatigue life estimate. This research will be extended to other metals.
3. Conclusions 1. The crack-opening stress level dropped immediately after the application of an overload and then gradually increased with subsequent small cycles. 2. The crack-opening stress can be modeled using an exponential build-up formula which is a function of the difference between the current crack-opening stress and the steady-state crack-opening stress of a given cycle. 3. The measured crack-opening stress showed that the crack-opening stress does not vary greatly from its average value for each intensity of a load history. 4. The fatigue-life of a 1045 steel component can be modeled using the crack-opening stress level corresponding to the stress cycle which reduces crackopening stress to a level reached by 1/200 of the number of cycles in the history which gave a conservative estimate of average crack-opening stress for all the histories.
References
Fig. 19.
Fatigue life versus stress range for (Log Skidder history).
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