When CDF= 1, the corresponding stress defines the condition under which all specimens fail at or below Nf cycles.∗ If there is a large amount of scatter as in curve “A,” which may occur
Trang 1Mean stress
10 7 2 × 10 7 3 × 10 7
Goodman diagram Loading history
Figure 3.2. Schematic of step-loading procedure.
others do not because the test is terminated after a large number of cycles (run-out) This results in two populations of specimens, one failed and the other unfailed, which are difficult to analyze statistically Another justification for a non-constant load to determine the fatigue limit is, as Prot [14] points out, “in practice, fatigue loads are not regularly variable, but they are not uniform amplitude loads.”
One of the main concerns in establishing material allowables for HCF is the sparse amount of data available and the time necessary to establish data points for fatigue limits
at 107 cycles or beyond The conventional method for establishing a fatigue limit is to obtain S–N data over a range of stresses and to fit the data with some type of curve or straight-line approximation For a fatigue limit at 107cycles, for example, this requires a number of fatigue tests, some of which will be in excess of 107cycles This is both time consuming and costly One method for reducing the time is to use a high frequency test machine such as one of those that have appeared on the market within the last several years In addition, the use of a rapid test technique such as one developed by Maxwell and Nicholas [22] involving step loading, described above, can save considerable testing time It has been demonstrated that such a technique provides data for the fatigue limit
of a titanium alloy which are consistent with those obtained in the conventional S–N manner [22, 26]
To examine the expected outcome using the step-loading technique, consider the schematic
of Figure 3.3 One can define a fatigue limit on an S–N curve arbitrarily as Nf, even though there is no assurance that this is a true endurance limit corresponding to “infinite” life At N, there will exist an unknown cumulative distribution function (CDF) which
Trang 2CDF
Step number 0 2 4 6
B A
Figure 3.3. Schematic of S–N curve and CDF for two different degrees of scatter.
will define the failure function at that number of cycles over some range of stresses The stress corresponding to CDF= 0 defines the stress level below which there are no failures within Nf cycles When CDF= 1, the corresponding stress defines the condition under which all specimens fail at or below Nf cycles.∗ If there is a large amount of scatter as in curve “A,” which may occur if the S–N curve is very flat, then a larger number of steps
in the step-loading technique will be required to cover all of the possible values of stress where failure may occur below the cycle count being considered, Nf If, however, there is less scatter as in curve “B” or the S–N curve is steeper, which will essentially cut off the higher values of stress which cause failure at lower numbers of cycles, then the number
of steps is fewer In either case, the larger the number of steps in a test, the higher is the expected stress Thus, what might appear to be a “coaxing” effect is no more than the statistics of the distribution of material fatigue strength The actual number of steps in a step-loading experiment depends on the starting stress, the distribution function or range
in stress levels, and the size of the step
An alternate to the step-loading approach for determining the fatigue limit is to conduct tests at various values of stress up to the number of cycles corresponding to the fatigue limit Two types of data are obtained First, some specimens will fail before Nfis reached, and these will provide data for a S–N curve which can be fit and extrapolated to Nf The second type of data will be stress levels for which no failure was obtained within
Nf cycles These stress levels will be denoted as run-outs or lower bounds on the fatigue limit In conducting tests under constant stress, consider the case where the S–N curve
is relatively flat such as when the number of cycles, Nf, is very large As a hypothetical example, consider the fatigue behavior in the region between 107and 109cycles, where it
∗
It should be noted here that some mathematical representations of distribution functions can go from zero
to infinity, such as a normal distribution In those cases, we have to deal with a situation where the CDF approaches 0 or 1 within some very small probability.
Trang 3Number of cycles 0
1
10 9
10 7
A B C D E F
(b)
0 1
CDF
10
A
B
C
D
E
F
(a)
Figure 3.4. Schematic of CDF (a) for two different values of N f , (b) as a function of N
has been shown that the S–N curve still has a slightly negative slope for some materials [27] For illustrative purposes, the CDF for failure within a given number of cycles is shown schematically in Figure 3.4(a) for either 107or 109cycles At 107cycles, there is no failure for stresses below level “C” and all samples will fail at or above “F.” Similarly, at
109cycles, no failure occurs below “A” and all samples will fail at “E” or above Clearly,
“A” corresponds to the fatigue limit at 109cycles Consider, however, what happens in a typical experimental investigation The CDF is shown as a function of number of cycles
in Figure 3.4(b) for several stress levels depicted in Figure 3.4(a) As shown, there are
no failures at “A” while at “F” most samples will have failed below 107and none will reach 109 At “E” there is a higher probability of survival beyond 107 but all fail by
109 At some intermediate level “D,” some will fail by 107 and most will have failed by
109, but as the stress level decreases to “C” or “B,” the likelihood of failure before 109 decreases Considering the time and cost of conducting such long life tests, the likelihood
of determining the probability density functions for a number of stress levels and, in turn, defining the fatigue limit, is poor In this situation, the step-loading procedure may provide an equally good answer with fewer tests Tests conducted at constant levels of stress, separated by equal increments, are discussed later in this chapter (see Section 3.6) along with the statistics for determining fatigue limits and the corresponding scatter
Experimental data using Ti-6Al-4V forged plate material and employing the step-loading procedure [28] are shown in Figure 3.5 In that investigation, the values of the fatigue
limit for four different values of R were not known a priori Thus, the initial stress value
in the step-loading procedure was highly variable The results, plotted against the number
of steps, show no indication of a systematic increase with number of steps and, therefore,
Trang 4no evidence of coaxing On the other hand, experimental results which show an increase
in stress with number of steps are shown in Figure 3.6 where the starting stress for any of the four conditions was either the same or very similar The tests are on a notched sample with kt= 22 with one batch untested and the other subjected to LCF cycling as indicated
on the plot [21] The plots of stress versus number of steps show a linear increase Since the starting stresses are the same for each condition, the slope is related to the size of the step Thus, this increase with number of steps is not necessarily coaxing, it is probably
no more than the scatter in material behavior as described above
200 300 400 500 600 700 800 900 1000
1
R = –1
R= 0.1
R= 0.5
R = 0.8
Number of steps
Ti-6Al-4V plate
60 Hz
Figure 3.5. Fatigue limit stress vs number of steps.
250 300 350 400 450 500
Baseline R = 0.1
Baseline R= 0.5
LCF–HCF R = 0.5
Number of steps
LCF
30 cycles
430 MPa
Figure 3.6. Fatigue limit stress vs number of steps.
Trang 5bar and plate forms of Ti-6Al-4V [21] The data are shown in Figures 3.7 and 3.8 In each of the figures, the number of steps that were used for each specimen is indicated
in the legend All steps within an individual step-loading test were conducted with a constant value of R Careful study of the data shows that there does not appear to be any systematic trend which would lead one to believe that the number of steps has any
0 100 200 300 400 500 600 700 800
2 steps
3 steps
4 steps
5 steps
11 steps
Mean stress (MPa)
Ti-6Al-4V bar
70 Hz
Figure 3.7. Haigh diagram for bar material.
0 100 200 300 400 500 600 700 800
2 steps
3 steps
4 steps
6 steps
10 steps
Mean stress (MPa)
Ti-6Al-4V plate
70 Hz
Figure 3.8. Haigh diagram for plate material.
Trang 6influence on the results In fact, it is rather remarkable that the expected trend of higher strength versus number of steps from a purely statistical point of view is not observed This is probably due to the choice of starting stress for each test which was very variable because each test covered a different value of R compared to the prior test
Conventional S–N tests conducted at 420 Hz on plate material were used to determine the fatigue strength corresponding to 107 cycles by least squares fit to the S–N data obtained at lives close to 107 cycles The results are shown in Figure 3.9 for tests conducted at a number of values of stress ratio, R, from 0.5 to 0.8 It can be seen that the data lie right on top of the data from step-loading tests in the same range of R Further, there seems to be no effect of frequency in going from 70 Hz in earlier tests to 420 Hz in the present tests
Data were also obtained at R= 05 and R = 08 using the step-loading procedure
to compare with the interpolated S–N data (horizontal line) as shown in Figure 3.10 Different values of stress in the first loading block, shown on the x-axis, were used to evaluate the effect of number of blocks for the two values of R Numbers in parenthesis in the figure indicate the number of load blocks used to determine the stress corresponding
to 107 cycles In both the plate material used here and the bar material used elsewhere, the failure at R= 05 is purely fatigue, while at R = 08, it is observed that the fracture surface shows no indication of fatigue, but rather, ductile dimpling [29] This issue is discussed later In both cases, however, Figure 3.10 shows no indication of a trend with number of blocks or starting stress for the step-loading procedure
Data obtained at 1.8 kHz are presented in Figure 3.11 Three types of tests are repre-sented, conventional S–N to failure, terminated S–N producing run-outs, and step loading
at either 107 or 108 cycles While the vertical scale is blown up significantly, it can be
0 100 200 300 400 500 600 700 800
Ti-6Al-4V Plate
10 7 cycles
ML 70 Hz Step ASE 70 Hz Step
Mean stress (MPa)
Figure 3.9. Haigh diagram for plate material comparing step test and S–N data.
Trang 7500 550 600
Ti-6Al-4V
10 7 cycles
R= 0.5, 420 Hz Step tests
7 cycles (MPa)
Block 1 stress (MPa)
From S –N curve
( ) = # steps
(3)
(6)
(3)
(2)
7 cycles (MPa)
(b)
800 850 900 950
Ti-6Al-4V
R= 0.8, 420 Hz Step tests
Block 1 stress (MPa)
From S –N curve
(9) (12)
(4)
( ) = # steps
Figure 3.10. Influence of block 1 stress on FLS at 10 7 cycles in step-loading fatigue limit strtess; (a) R = 05,
(b) R = 08.
noted that there is very little scatter at R= 08 where all the tests were conducted, and no influence of a history effect due to the step-loading procedure The lower step-test data point at 108 cycles represents two independent tests which had a maximum stress within
1 MPa of each other
The data obtained at R= 08 are of particular interest in the evaluation of the validity
of the step-loading procedure In an investigation on the bar material, Morrissey et al [29] noted that at high values of R, the material accumulated strain under fatigue loading
Trang 8950 1000 1050 1100
10 10
Failure Run-out
Step test
Number of cycles
Ti-6Al-4V bar
1800 Hz
R =0.8
Figure 3.11. Fatigue limit stress results at R = 08 1800 Hz.
Tests conducted at different frequencies showed that the strain accumulation was depen-dent primarily on number of cycles, not on time, so that the phenomenon could not
be considered to be cyclic creep Rather, the strain accumulation is due to ratcheting
A similar phenomenon has been observed in the Ti-6Al-4V plate material, where cycling
at stress ratios higher than approximately 0.7 leads to strain accumulation Micrographs
of the fracture surface at various magnifications taken with a scanning electron micro-scope (SEM) are presented in Figures 3.12 and 3.13 for stress ratios, R, of 0.7 and 0.8,
00-A-95, Ti-6-4, σ =840 MPa, R= 0.7, a = 0.4 mm
Figure 3.12. Fractographs at R = 07.
Trang 900-A-91, Ti-6-4, σ = 920MPa, R= 0.8
Figure 3.13. Fractographs at R = 08.
respectively It can be observed that at R= 07 (Figure 3.12), the fracture surface looks like fatigue with well-defined faceted features and evidence of striations At R= 08 (Figure 3.13), the features are those of a tensile test with ductile dimpling in evidence and no indications of cleavage or striations The crossover point, at about R= 075, is nominally the same as in the bar material as reported by Morrissey et al [29]
Data obtained over a range of frequencies from 30 to 1000 Hz under the Air Force HCF program at various laboratories are presented in Figure 3.14 for R= 08 Including
800 850 900 950 1000
All data
Cycles
R= 0.8
Figure 3.14 S–N data obtained from 30 to 1000 Hz.
Trang 1030 40 50
Ti-6Al-4V Plate
R= 0.8
60 Hz
60 Hz run-out
200 Hz step test
Cycles to failure
Figure 3.15. Honeywell data at 60 and 200 Hz.
the Materials Laboratory (ML) data at 420 Hz, there is very little scatter over the fatigue cycle range from 105to 108 cycles, and no effect of frequency although frequencies of each data point are not shown Additional data from Honeywell are shown in Figure 3.15
at R= 08 at both 60 and 200 Hz No frequency effect is apparent, the scatter is minimal, and data using the step-test procedure at 107 cycles fall right on top of the other data From these results, as well as from the data in Figure 3.11 at 1800 Hz, it is concluded that step testing produces an accurate estimate of FLS in the 107–108 life regime for
R= 08 in the titanium plate where strain ratcheting is the dominant fatigue failure mechanism
An interesting observation was made by Moshier et al [30] when evaluating the data from the step-test method on specimens with LCF cracks compared to data on specimens with no cracks The last loading block, defined as the block of 107cycles during which failure occurred, can have a cycle count anywhere from 1 to 107 The data for number
of cycles to failure in this block are normalized with respect to 107 to show at what fraction of the block failure occurred The results, presented in Figure 3.16, show that for specimens with no prior cracks, the failure can occur anywhere in the block When cracks are present, however, failure always occurred early in the loading block These data show that there appears to be a very well defined HCF threshold for a cracked specimen for which failure occurs within a short time, typically under one million cycles,
or does not occur at all for a given applied stress (or K) Alternately, these data show that when a crack is present, we are dealing only with the propagation phase of fatigue which is small compared to the nucleation phase which dominates the HCF life in an