If only the step tests are used in combination with the LCF tests, a tighter fatigue limit variation is obtained with more error in life than for the staircase plus LCF tests where fatig
Trang 1predictions, the RFL model, described above, was used The baseline tests with Nf< 106 cycles were selected as LCF tests that would typically be available from test programs
in addition to HCF properties The RFL model treats run-out and failure tests with 2D scatter in both life (LCF regime) and the endurance stress (HCF regime) An advantage
of the RFL model is that the 1D scatter assumptions are not required The RFL model predictions with the baseline+ step or staircase results are given in Figure 3.35
If only the step tests are used in combination with the LCF tests, a tighter fatigue limit variation is obtained with more error in life than for the staircase plus LCF tests where fatigue limit variation dominates the life error term In Figure 3.36, the average and lower bound HCF limits for all approaches are summarized If a 1D scatter in stress (s) is used, similar lower bound predictions are obtained for the step and staircase tests However, a 1D scatter in life results in a different lower bound as noted before For the cases using
(a)
60 70 80 90 100
Baseline +step at interpolated Smax
Smax
Cycles
130 120 110
Tighter fatigue limit variation and higher life error term vs staircase
(b)
60 70 80 90 100
Baseline + staircase tests
Smax
Cycles
130 120 110
90%
Fatigue limit variation dominates life error term
Figure 3.35. Random fatigue limit model predictions using baseline tests (Nf< 10 6 cycles) + step or staircase
approaches.
Trang 2HCF Limits for Ti-6Al-4V at 75 ° F and R = 0.1
40.0
60.0
80.0
100.0
Step
(s scatter)
Staircase
(s scatter)
Staircase
(Nf scatter)
LCF + step
LCF + staircase
All BAA single load data
Smax
average Smax(ksi)
lower bound Smax(ksi)
Analysis with 2D scatter Analysis with 1D scatter
Figure 3.36. Summary of predicted average and lower bound HCF limits for Ti-6Al-4V at 75F and R = 01.
the 2D scatter, a similar lower bound is observed for either step, staircase, or all data, when combined with the baseline LCF data These predictions are seen to be different than those obtained with the 1D scatter assumptions
Another important feature that distinguishes the RFL model from conventional least squares fitting (LSF) is the manner in which run-out tests are handled As pointed out in [50], the RFL model deals with the probabilities of an observation being above or below some value whereas in ordinary LSF routines, the behavior is addressed as an average In the RFL model, each specimen has its own FLS Least squares cannot deal with run-outs because there is no specific data point to evaluate, but the RFL type model can deal with the life exceeding some number at a given stress These “censored” observations, where a specimen could fail at any unknown time after the test was suspended, cannot
be included in any least squares calculations of the sum of the errors, so the LSF method breaks down if these observations are to be included For the RFL model, on the other hand, these observations can be included if exceedance is the criterion being evaluated
In the estimation of the parameters for the RFL model, a maximum likelihood approach
is used As Annis and Griffiths [50] point out, if there are no censored observations, the maximum likelihood method produces the exact same results as the least squares error method
As observed in all of the statistical approaches discussed above, and illustrated for specific cases in Figure 3.36, the average HCF limits are relatively insensitive to the assumptions However, predicted lower bound limits are highly dependent on the assumed scatter direction for the 1D scatter approaches (stress versus life scatter) Lower bound limits are also highly dependent on the assumed scatter type (1D versus 2D scatter) In the final report [24], the authors who represent the turbine engine industry conclude that
“the best approach needs to be assessed within the current design systems Additional work establishing confidence limits for the RFL model also is needed for use in design.”
Trang 3The RFL assessment for all single load Ti-6Al-4V tests is also included as a final assessment [24] Load control tests with maximum stresses above the material yield stress were not used in the final analysis The model was fit with strain and load control tests at different values of R with the equivdamage parameter defined in Equation (3.20) below
as the alternating Walker equivalent stress and is the total strain range for the cases where inelastic behavior was encountered max is the maximum stress as measured on test specimens or calculated with elastic-plastic analyses, and w is a material constant Strain control tests were used to establish the baseline half-life stress-strain properties The values of maximum stresses and strain were taken from strain control measurements near the specimen half-life The constant w= 042 was obtained with a non-linear regression
of the strain and load control tests The average and−3s RFL fits are given in Figure 3.37 The equations for the average and lower bound fit to the RFL model are:
Average fit: log Nf= −239472 log equiv− 36199 + 7629937 (3.16)
−3 fit log Nf= −239472 log equiv− 23758 + 7629937 (3.17)
In the RFL model development and analysis described in the section above, a SWT and a equiv parameter are used to consolidate data obtained at various stress ratios in order to provide a larger database The vertical axis in Figure 3.31 is the SWT parameter while Figure 3.37 uses the equiv parameter The Smith–Watson–Topper [51] parameter
is defined as
maxE
2
05
(3.18)
Ti 64 single load tests at 75F (w= 0.42)
20 30 40 50 60 70 80 90
1.E + 03 1.E + 04 1.E + 05 1.E + 06 1.E + 07 1.E + 08 1.E + 09
C ycles
σequiv
0.99 0.5 0.00135 Failures Runouts
Figure 3.37. Single load test results and random fatigue limit fits for Ti-6Al-4V.
Trang 4where max is the maximum value of stress, is the strain range, and E is Young’s modulus For purely elastic behavior, typical of the HCF regime, the formula becomes
max
2
05
= max
1− R 2
05
(3.19) where R is the stress ratio A modified version of the SWT parameter, used in industry [52], has additional flexibility to consolidate data at different values of R One form of this modified parameter, which will be denoted as SWTMODis
where w is a fitting parameter This modified version of the SWT parameter is often referred to as an equivalent stress, equiv Equation (3.20) is written for purely elastic behavior using instead of E for the more general inelastic case For the case where
w= 05, SWT and SWTMOD differ by a constant factor of √
2 for all values of R For other values of w, the difference depends on stress ratio, R Figure 3.38 shows values of SWT and SWTMODas a function of R for values of w= 075, 0.5 and 0.25 to illustrate the differences between the parameters The values of the SWT and SWTMODparameters are normalized with respect to max in the plots
Step tests at the interpolated failure stresses were not used in the final baseline fits Yet for all of the other data, one observation still puzzles those involved in analyzing the data obtained on Ti-6Al-4V Though step and single load tests produced equivalent results for R= 01, step tests potentially produced unrealistically high allowable HCF limits at
R= −1, leading some to speculate whether coaxing actually exists in this material In their final report [24], the researchers recommended that “possible issues with step tests
at negative R should be assessed in future work.”
0 0.2 0.4 0.6 0.8 1
SWT
MOD (w= 0.25)
σmax
Stress ratio, R
Figure 3.38. Normalized SWT and modified SWT parameter as function of stress ratio.
Trang 53.7 SUMMARY COMMENTS ON FLS STATISTICS
In summary, determination of the FLS and the corresponding statistics of scatter can
be accomplished in a number of ways For the Ti-6Al-4V used in the National HCF program, a number of methods and combinations thereof were used The simplest case was the use of step tests to determine the FLS by averaging the results of 5 tests Next,
26 staircase tests were conducted and the results analyzed using the Dixon and Mood method assuming a Gaussian distribution for the FLS Third, results of a large number
of LCF tests including run-outs as well as a few long life tests beyond 107cycles were used in an assessment of the RFL model which considers scatter in both stress and life Finally, the RFL model was applied to a combination of the LCF data and the staircase test data, where cycles to failure in the failed staircase tests were used as part of the database A summary of the results from the four procedures is presented in Table 3.9 All data are presented for the FLS at 107cycles and R= 01 For the normal distribution function, the−3 value is calculated For the RFL model using the SEV distribution, the value corresponding to a probability of failure of p= 001 at 107 cycles is used The results demonstrate that from four different combinations of databases and statistical procedures that a mean value is obtained that has little variability among the procedures However, a design value at the tail end of a distribution function corresponding to−3
or p= 0001∗
can have a large amount of scatter among the various techniques It is certainly not apparent what is the appropriate value of a FLS for design if the design criterion is a probability of failure of one in a hundred or one in a thousand It is not even apparent what is the appropriate distribution function with which to represent FLS based on sample sizes that do not exceed approximately 100 Another point to consider
in looking at the results presented in Table 3.9 is that the database used in the evaluations with the RFL model contain data obtained at several values of stress ratio, R, while the step tests and staircase tests were all performed at a single value of R The RFL results
Table 3.9. Summary of statistical representation of FLS at 10 7 cycles, R = 01
Data source Parameter Distribution
function
Mean stress (ksi)
−3 value
(ksi)
∗Represents p= 001 value in distribution function.
∗
For a normal distribution, a probability of failure of 0.001 corresponds to −3090 Conversely, −3 corresponds to p = 000135 Similarly, p = 001 equates to −2326 while −2 equates to p = 00227.
Trang 6are interpolated for R= 01 using the SWT or equivparameters The scatter in the results reflects both the inherent material variability as well as any inability of the models to consolidate data at various values of R into a single parameter
Another example of the possible problems that may be encountered in step testing,
in addition to those mentioned above for Ti-6Al-4V at R= −1, are associated with data obtained on Ti-17 under the National High Cycle Fatigue program [24] A limited number of tests were conducted on this alloy at two stress ratios, R= 01 and R = −1 Both conventional S–N tests, using single specimens at fixed stress levels, and step tests where the specimen is reused until it fails, were conducted The test matrix did not constitute a statistically designed experiment, but the data obtained revealed some interesting features At R= 01, Figure 3.39, S–N tests showed failures at stresses above approximately 105 ksi and run-outs at stresses below that Yet step tests produced slightly higher values of the FLS at 107cycles, with one test starting at 110 ksi and not failing on the first block at that stress level where other samples had failed at cycle counts below
105cycles This anomalous behavior was attributed to a combination of material scatter, possible variances in batches of material preparation/machining, and the possibility of a coaxing phenomenon in this material Even more perplexing are the results obtained at
R= −1 shown in Figure 3.40 Here, a similar phenomenon is observed as in the tests
at R= 01, but the number of specimens and the extent of the phenomenon seems to
be more prominent While speculation abounded about the specimens coming from two distinct batches that were prepared differently, no such conclusion could be drawn based
on incomplete records of the history of the individual specimens The data, however, seem to imply that there are two populations of specimens producing two distinct sets of behavior While the duality in S–N curves has been observed in gigacycle fatigue and
60 70 80 90 100 110 120 130
S –N run-out
Step run-out Step tests
σmax
N
Ti-17
R= 0.1
350 Hz
Figure 3.39. Fatigue data for Ti-17 at R = 01.
Trang 750 60 70 80 90 100
S –N run-out
Step run-out Step tests
σmax
N
Ti-17
R= –1
350 Hz
Figure 3.40. Fatigue data for Ti-17 at R = −10.
has been associated with internal initiations at long lives, and surface initiations at shorter lives in individual specimens,∗ there were no internal initiations observed in any of the specimens inspected after failure in the longer life (or higher strength) population of the test results
Another observation that is somewhat puzzling has been made in reviewing HCF limit stress data on titanium As seen in Figure 2.36 in Chapter 2 as an example, limited data show an unusual amount of scatter for tests conducted at R= −1, fully reversed loading, corresponding to zero mean stress Bellows et al [26] noted that the scatter in both step tests and conventional tests was highest at R= −1 In addition to this being their only test that went into compression, comparison with their other tests showed that these tests had the highest stress (or strain) range and had the lowest value of maximum stress No readily observable differences were seen between the fracture surfaces of the step tests and conventional tests at the same stress ratio
The question remains as to whether fatigue lives at stresses above the fatigue limit (infinite life) provide any information about the FLS or the FLS corresponding to a large but finite number of cycles Inherent in this discussion has to be the question about the exact shape of the S–N curve and the extrapolation of this curve to large numbers of cycles or to infinite life Added difficulties arise from a statistics point of view because the distribution function of lives at stresses producing finite lives and of fatigue limit strengths at long lives or even at the fatigue limit for infinite life have to be known An evaluation of some aspects of this problem was made by Loren [53], who compared two
∗
See discussion in Chapter 2.
Trang 8models for obtaining information on the fatigue limit (infinite life) and the endurance limit defined as the fatigue strength at a given (long) life The models were evaluated based on numerical simulations of a staircase test procedure For the fatigue limit model,
he assumed the fatigue limit is a random variable having a normal distribution of the logarithm of the stress If stresses are below the fatigue limit, they have infinite lives In the second model, the endurance limit (fatigue strength corresponding to a given number
of cycles) also is a random variable with a normal distribution of logarithm of stress If
a specimen does not fail at a stress below the endurance limit, it has a life greater than the cycle count at which the staircase testing is terminated The lives are greater than this cycle count, but not necessarily infinite A large number of numerical simulations of staircase testing were conducted corresponding to the statistical distribution assumptions for each model The results were analyzed using maximum likelihood procedures that consider both censored (run-outs) and uncensored (failures) data Both methods show that
it is possible to estimate the distribution of the fatigue limit, but the fatigue limit itself is impossible to observe The finite lives were found to give additional information on the fatigue limit or the endurance limit in some cases, while in others they simply confirm that the lives are finite The results depend on the distribution functions as well as on the life where the staircase tests are terminated The statistical mathematics for conducting these simulations are presented in the paper [53]
3.8 CONSTANT STRESS TESTS
If step or staircase tests are not used to provide information about the fatigue limit, tests
at constant stress levels can be used to determine the shape of the S–N curve This is particularly useful for LCF where the cycle counts are lower than in HCF However, as the S–N curve becomes more horizontal near the fatigue limit, the scatter in lives increases The choice of test method depends on, among other factors, the accuracy desired on both the mean and the variance, the number of available samples, and the test time available
If constant stress level testing is chosen, the number of tests at each level, the specific values of the stress levels, and the stress increment between test levels have to be chosen While guidelines exist for test planning purposes [54], they are very restrictive and pertain only to S–N relationships that are linear on the proper coordinates, that variance in fatigue life is the same at the various stress levels, and that there are no run-outs For optimum results on real data, the number of tests at each level does not have to be a constant Beretta et al [55] conducted a very large number of tests on a 0.43% carbon steel to establish the fatigue properties and scatter for the material They then conducted
a statistical analysis to determine confidence levels of test sequences where the number
of tests at each stress level was not constant The statistical analysis took into account run-outs As they point out, most existing recommendations are based on assumptions of
Trang 9the lognormal distribution of lives, the absence of run-outs and, more restrictively, the same variance at each stress level In their work, they assumed a lognormal distribution
of lives at each stress level based on test results, but the distribution was different at each stress level The largest scatter, as expected, was at the lowest stress level Three statistical models were considered which have different relationships between the standard deviation and the applied log-stress: exponentially variable according to an exponential model by Nelson [56], linearly variable, and constant The equations describing these models are shown below For the mean, , the three models use
= X1+ X2
while the three models use the following functions for :
= expX3+ X4
where S is the applied stress, is the mean value and is the standard deviation of the (log) fatigue life, while log S is the arithmetic average of all the values of log S in the test series The constants X1 X2 X3, and X4 are fitting parameters for each statistical model based on maximum likelihood estimates While the same function, Equation (3.21), is used for the mean, , for each of the models, the values of the parameters X1 and X2 are slightly different for each model because of the different distribution functions which they represent
Using maximum likelihood analysis on the results of fatigue tests they obtained on the 0.43% carbon steel including run-outs, the parameters in the equations were determined for the three models The models are shown, without the experimental data, in Figure 3.41 (a) through (c) where the mean (50% probability of failure) and 2.5 and 97.5% probability
of failure curves are shown for each of the models The differences in the distributions can be clearly seen, particularly at the lower stress levels
To evaluate the results of using a small sample size, confidence levels were determined for various sample sizes randomly extracted from the total population of tests run for each model The first point noted was that the model using a constant standard deviation, , Equation (3.24), independent of stress level (Figure 3.41c), provided the worst fit to the data and was not used in subsequent computations It has been noted that the assumption
of constant is the most commonly used, and virtually all statistical computer programs for curve fitting make this assumption because the mathematical theory and numerical computations are then simpler [56] They observed that the Nelson model that uses the exponential variation of (Figure 3.41a) provides the best estimates of model parameters
Trang 10400 500
Cycles to failure
600
97.5%
(a)
400
500
Cycles to failure
600
(b)
400 500
Cycles to failure
600
(c)
Figure 3.41. Models for fatigue life with standard deviation a function of stress: (a) exponential (Nelson),
(b) linear, (c) constant.
...105cycles This anomalous behavior was attributed to a combination of material scatter, possible variances in batches of material preparation/machining, and the possibility of a coaxing... 107 cycles is used The results demonstrate that from four different combinations of databases and statistical procedures that a mean value is obtained that has little variability among... equivparameters The scatter in the results reflects both the inherent material variability as well as any inability of the models to consolidate data at various values of R into a