Fatigue

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Figure 1. Description

Table 1. References for figure 1

Item number Description 12a Pin hole

Introduction

Horns must be designed for adequate life at the required output amplitude. This means that the designer must choose the proper horn shape, the proper material, and the proper manufacturing process. In this chapter we will describe the testing procedure for evaluating the relative lives of different materials. We will describe the results of such tests and discuss how such data can be used to roughly estimate horn life. In subsequent chapters we will describe the testing used to evaluate competing horn designs.

Fatigue overview

In this section we will describe the fatigue process by which most horn failures occur. (See Juvinall[1] for a good general discussion of fatigue.)

Ultrasonic horns can fail in a variety of modes. They can fail through excessive wear, through galling of the antinode joints, through damage to a carbide wear insert, etc. However, the preponderance of horn failures are due to fatigue cracking.

Fatigue is a process whereby a structure gradually fails because of progressive crack growth due to fluctuating stress. The stress at which failure occurs is considerably below the material's tensile strength (i.e., the stress at which the material would fail if it was stretched in a single direction, as in a tensile test).

Ultrasonic resonators are especially prone to fatigue cracking, because the resonator experiences a large number of stress cycles in a relatively short time. For instance, a 20 kHz resonator will experience one million stress cycles in only 50 seconds of operation.

The fatigue process

The fatigue process can be divided into two phases. During the first phase (the nucleation phase), many small micro-cracks are formed at locations of high stress. These locations be surface notches (holes, slots, screw threads, etc.), or internal stress concentrators (e.g., defects such as inclusions). These stress concentrators may cause localized stresses that are several times higher than the nominal stress at that cross-section. (See the chapter on "Stress".)

The micro-cracks tend to form in grains whose slip planes are oriented in the direction of maximum applied shear stress. (See Juvinall[1], p. 197.) Thus, because of the somewhat random orientation of grains in a material, certain grains are statistically more likely to be the site for crack nucleation. After No stress cycles (depending on the material and test conditions), these micro-cracks eventually unite to form a predominate macro-crack.

In the second phase (the growth phase) the macro-crack grows by a small amount during each loading cycle. When the crack has become sufficiently large (after Nf total stress cycles, depending on the material and test conditions), the structure fails. The relative time spent in the nucleation and growth phases (No/Nf) depends of the particular material. (See Grosskreutz pp. 24 - 28 for a discussion.)

It is usually assumed that the fatigue process proceeds only during that portion of the ultrasonic cycle when the net stress at the fatigue location is tensile. When the net stress is compressive the fatigue process is temporarily halted. However, Mukherjee (p. 47) shows some data that casts doubt on the assumption that the compression portion of a loading cycle can be ignored. ASTM STP 511

The performance of ultrasonic resonators is not affected during the nucleation phase. During crack growth phase, however, the adjacent walls of the crack rub against each other, which dissipates energy. This energy must be supplied by the power supply. Thus, the crack acts like a load. As the crack grows larger, the rubbing area increases and the power supply must provide more power. Eventually the crack becomes sufficiently large that the power supply can no longer supply the required power. The power supply then shuts down. At this point the horn is considered to have failed, even though it has not broken into separate pieces. (For further discussion, see appendix [].)

Fatigue surface appearance

If you break the horn apart at the fatigue crack you will usually find a darkened spot at which the crack started. Most cracks start from the horn surface, where machining irregularities have caused high local stress concentrations. (See figure f.) (Note: by "surface" I mean any portion that could be exposed to the air. For instance, this could include the threads in the stud hole.) Occasionally, however, the crack will start from an internal material defect where the stress is also high. (See As the crack grows outward from the nucleation site, it will have two distinct propogation directions. The initial growth direction will be approximately along a plane of maximum shear stress, which will be at a 45o angle to the direction of maximum tensile stress. This is called Stage I growth. The crack then changes direction so that it is substantially perpendicular to the direction of maximum tensile stress. This is called Stage II growth. (See Juvinall[1], pp. 197 - 203, and Grosskreutz, pp. 26 - 32.)

For most materials, Stage I growth represents only a very small portion of the total fatigued surface (few tenths of a millimeter), especially if the crack starts from an inclusion. (See Grosskreutz, pp. 26 - 27. Also, see below) The length of a Stage I crack increases as the stress decreases. Then the majority of crack growth will then be Stage II. This behavior is typical of titanium, steel, and most aluminums. (Figure [f] shows a typical titanium failure. Figure [f] show typical aluminum failures. In these figures the direction of maximum tensile stress is along the axis of the rod.) Thus, in most cases the crack direction provides an indication of the direction of maximum tensile stress.

One exception to this growth pattern is the nickel-based super alloys which fatigue almost entirely by Stage I cracking. (Grosskreutz, p. 28) Grosskreutz, (p. 56) indicates that planar slip materials may crack substantially in Stage I and that such materials could be substantially strengthened by properly orienting the grains parallel to the fatigue stress axis (p. 56, p. 22). However, he is not very specific about which materials are planar slip (see his footnote, p. 9).

Aerospace aluminum also appears to crack substantially in the Stage I mode. When Aerospace aluminum is ultrasonically tested parallel to the extrusion direction, the crack propogation direction always remains at a 45° angle to the maximum tensile stress -- i.e., 45° to the test direction. (See figure [f], there the maximum tensile stress is along the axis of the S-N horn.) The reason for this behavior is not known.

As the crack progresses during Stage II growth, each stress cycle leaves a small striation in the material. The progression of these semicircular striations from the nucleation site gives the appearance of a clam shell. (See Grosskreutz, pp. 30 - 31, for some good photomicrographs of aluminum crack growth.) If the material has a very coarse- grained structure, then these striations will not be evident. (See figure [f]., which shows the failure surface of a coarse-grained 2024 aluminum alloy.)

Fatigue testing

Fatigue testing has three purposes:

1) Material selection. Since horns fail by fatigue, we would like to use a material with a high resistance to fatigue cracking. Therefore, we must fatigue test a variety of materials at a specified stress to find those with the best fatigue resistance. Because material selection involves considerations other than fatigue resistance (e.g., cost, wear resistance, power loss, etc.) we may ultimately find several "best" materials from which to choose, depending on the customer's application.

2) Life estimation. Once we have found a good material, we can fatigue test this material at additional stress levels. This additional data will allow us to determine the relation between the stress and life, from which we can construct an S-N (strength-life) curve, also known as a Wöhler curve. We can then use this information to roughly estimate the lives of production horns whose stresses are known.

3) Evaluating other effects. Fatigue resistance can be dramatically changed by such effects as plating, machining, heat treatment or chemical treatment, shot-peening, etc. Fatigue testing allows us to determine how the material responds to these effects.

The s-n diagram

In this section we will discuss the relation between fatigue strength (S) and fatigue life (N). This will provide the insight required for proper design of a fatigue test method. For this and the following sections, you may want to review the chapter on "Statistics".

Statistical considerations

Although there some variations, the usual fatigue test subjects a specimen to a known alternating stress until the specimen fails, which gives the fatigue life of that particular specimen. Unfortunately, fatigue data often has significant scatter. Thus, nominally identical specimens of the same material tested under nominally identical test conditions may have significantly different fatigue lives. Lives may vary by ratios of 10:1 or even 100:1 (Juvinall[1], p. 352).

Because fatigue depends on the localized condition of the material (grain orientation, grain boundaries, localized natural and man-made defects, etc.), such data scatter cannot be avoided. This data scatter means that many specimens will have to be tested before meaningful conclusions can be reached about the median fatigue life and its associated confidence interval, and about the confidence interval of the data.

The s-n curve

If the purpose of the testing is to find materials with good fatigue lives, then much of the comparative testing can be done at a single stress level. (However, see the section on "Limitations".) Once a material has been found that seems to have reasonable fatigue life, then it may be desirable to test this material at more than one stress level so that the trend of life versus stress may be developed. This trend is usually called an S-N diagram, where "S" refers to strength (or sometimes stress) and "N" refers to the number of cycles to failure or a proportional value, such as the time to failure.

If such a diagram is well developed, it may be used to estimate the life of a horn that is being driven at a known stress.

Once you have tested a sufficiently large number of samples at many different stress levels, you can then construct the S-N diagram. The most common method of plotting an S-N diagram is to use log-log graph paper, where both the horizontal and vertical axes are logarithmic. Thus, the horizontal axis is the log of the time or number of cycles to failure, while the vertical axis is the log of the stress (or some proportional parameter, such as amplitude). In some cases the vertical axis may be plotted to a linear (i.e., nonlogarithmic) scale.

There are two reasons why the S-N diagram is plotted on log-log paper. First, this allows lives that are significantly different to be plotted on a single sheet of paper. More importantly, the resulting plot is often linear over certain ranges, which permits easier determination of the endurance limit (see below).

We would eventually like to find an equation that fits the failure data at the different stress levels so that N can be expressed as a function of S. Probably the most frequently used equation is the log-log relation:

) log(N) = b0 + b1 log(Í)

where Í = peak stress

b0, b1 = constants

This is a linear equation between log(N) and log(Í). Thus, if variables N and Í of equation 5 are plotted on log-log graph paper, the result will be a a straight line. Note that the constants b0 and b1 may only be valid over a limited range of N or Í.

Other possible fits to the S-N data are given by Spindel (p. 92). (See appendix X.) Some of these (e.g., the Weibull) are useful because they extend the range of N or S over which the S-N equation is valid. A semilog equation (log(N) versus S) is suggested by ASTM standard E739-80 as a possible alternative to the log-log relation of equation 5.

Figure [f]. shows a typical S-N diagram plotted on log-log graph paper. The center curve represents the median fatigue life -- i.e., the life at which 50% of the tested samples will have failed. This is the single curve that is most often plotted on S-N diagrams. However, such information is of limited value, however, since few products are designed where a 50% failure rate is acceptable.

Thus, for prediction of acceptable product life, additional curves should be drawn on the same diagram, showing other probabilities of failure. For example, the left and right curves of figure f represent the 95% confidence interval of the data -- i.e., the region for which 95% of the sample failures are most likely to have occur. To the left of the left curve, only 2.5% of the sample group will have failed. To the right of the right curve, only 2.5% of the sample group will have survived.

The required confidence interval will depend on the requirements of the application. For example, ball and roller bearings are often life-rated based on 10% probability of failure. (Juvinall[1], p. 342.) For our work in ultrasonic fatigue testing, we will somewhat arbitrarily choose a 2.5% lower confidence limit and a 95% confidence interval. Why not choose a very high confidence level -- e.g., 1% or 0.1% probability of failure? The reason is that many additional fatigue tests are required to assure such confidence. Also, since working horns of any particular design may experience a range of ill-defined stresses (e.g., from different boosters, converters, line voltage variations, etc.), their lives cannot be predicted with great precision even if the S-N curve is extremely well defined.

Obviously, the usefulness of a well-developed S-N curve is that it allows prediction of horn life. This assumes that the horn stresses can be reliably determined, which is not always an easy task.

Type I and Type II materials: Endurance (fatigue) limit

The exact shape of the S-N curve will depend on the material that is being tested. For many materials there is a "knee" in the S-N curve, where the slope of the S-N curve changes appreciably between high and low stress levels.

Type I materials

In some cases, the S-N curves become essentially horizontal to the right of the knee. (See figure [f].. Maennig[1] (pp. 612 - 614) calls this a type I material.) This means that if you operate at any stress below a certain cutoff level, then the material will have infinite life. This cutoff stress level is called the endurance limit or fatigue limit. Materials such as structural steels, non age hardening aluminum-magnesium alloys such as AlMg5Error! Reference source not found., and possibly titanium and its alloys have a well-defined fatigue limit. (See Maennig[1], pp. 612 - 614; Juvinall[1], pp. 208 - 210, 218; Grosskreutz, p. 42.)

BBased on figure f, the stress can be divided into three ranges. The upper range is called the range of fracture, where essentially all samples will have failed if tested long enough. The middle range is the range of transition, where some samples will fail while others will not, regardless of the length of testing. The lower range (below the endurance limit) is the range of infinite endurance, where essentially all samples will survive the testing. The exact location of these three ranges on the S-N diagram is somewhat arbitrary and will depend on the confidence required. For our work we will use the 2.5% and 97.5% probabilities of failure as the limits for these ranges.

Many materials do not have a true fatigue limit. The case for aluminum is unclear. Hessler (pp. 250, 260) reports a fatigue limit between 107 and 108 cycles for pure polycrystaline aluminum tested at 20 kHz. A number of his tests ran up to 1010 cycles without failure. In discussing the general low-frequency fatigue characteristics of aluminums, the SAE Fatigue Design Handbook (p. 56) says, "Fatigue tests that have been caried well beyond 500 million cycles [6.9 hours at 20 kHz] indicate that the [S-N] curve is essentially horizontal at this life." Grosskreutz (p. 42) says, "Other materials especially aluminum and its alloys, do not give S-N curves with strictly zero slopes at long lives. Nevertheless, the curve is nearly flat, and it is usual to speak of the endurance limit at 107 or 108 cycles". Juvinall[1] (p. 215) says that heat-treatable aluminum alloys have less tendency to approach a true endurance limit than do non-heat-treatable alloys.

Type II materials

Maennig[1] (pp. 612 - 614) designates materials which do not have a true endurance limit as type II materials. For these materials, Maennig suggests a "technological fatigue limit", which is the stress at which the S-N curve assumes a much smaller (but nonzero) slope. (See figure [f]..)

The existance of an endurance limit generally correlates with the linearity of the stress-strain curves and a sharply defined yield point. (Juvinall[1], p. 201; Grosskreutz, p. 42) If this is the case, then most aluminums should not have a fatigue limit. However, Maennig indicates (p. 614) that the high-strength age-hardening aluminum alloys may have a true fatigue limit (at conventional test frequencies) at 1011 to 1013 cycles (i.e., 1400 hours to 140,000 hours at 20 kHz) for stresses of about 20 MPa (2900 lbf/in2).

Obviously, if we are searching for the most fatigue resistant material, we would hope to find one with a true fatigue limit, which occurs at the highest stress and the greatest number of cycles.

Stress control

One characteristic of all S-N diagrams is the shallow slope of the failure curves. This means that a slight change in stress can cause a very large change in fatigue life. Therefore, the stress (or amplitude) must be carefully controlled in order to produce meaningful fatigue data.

Fatigue probability distribution

From the above S-N curves, we see that the probability of failure depends on two criteria: the stress Í and the number of cycles N at which the probability is desired. We can therefore construct two types of fatigue probability distributions: the fatigue life distribution (the distribution of fatigue lives at a specified constant stress Í) and the fatigue stress distribution (the distribution of stresses at a specified constant life N).

Fatigue life distribution

It is typical of fatigue that most failures at a given stress level occur relatively early. However, a few samples will have relatively long lives. We need a theoretical distribution to describe this behavior. We can then use this distribution to statistically analyze the failure data.

A number of possible failure distributions have been suggested. (See Maennig[1], pp. 618 - 624.) The debate among these distributions stems from two considerations. First, is the distribution theoretically compatible with the fatigue process? Second, does the distribution accurately predict the few early failures that occur, which are the main interest? We shall ignore the first consideration, as does Spindel (p. 92).

As to the second consideration, Maennig (pp. 618 - 624) concludes that the log-normal distribution is well suited to fatigue failure times. (If you consult this reference, you will see that Maennig refers to the normal distribution instead of the log-normal distribution. However, this appears to be a matter of semantics, since he uses log-normal probability paper to plot his data. See his figure 11, p. 627.) Juvinall[1] (p. 351) also suggests the log-normal distribution, although he cautions against making precise failure predictions, especially for less that 5% probability of failure. The SAE Fatigue Design Handbook (p. 40) and Spindel (p. 92) both suggest either a log-normal distribution or a Weibull distribution.

Figure [f]. shows a log-normal probability plot of 174 samples of 7075-T6 aluminum which Sinclair fatigue tested at six stress levels (low frequency tests). Although there is some scatter, most of the data at a given stress level approximated a straight line, for which Sinclair concluded (p. 869):

In general, agreement between the logarithmic-normal distribution and experimental data appeared to be reasonably close for frequencies of failure between 1 and 99 per cent.

However, the article cautions several times against extrapolation of the log-normal plot below 1% probability of failure.

Figure [f]. (Bastenaire, p. 12) shows a log-normal probability plot for fatigue tests of XC 10 steel at eight stress levels. Here we see that many (but not all) of the failures are distributed reasonably along straight lines, indicating a valid log-normal distribution. Commenting on this distribution, Bastenaire (p. 11) says:

Experience shows that the logarithm of the NCF [number of cycles to fracture] is distributed normally only at intermediate stress levels. When the stress decreases, the scatter and skewness of these distributions increases to a considerable degree ...

This is seen from the stresses labeled 27, 26, and 25. (Also see Nishijima, pp. 76 - 77.)

Figure [f]. shows a log-probability plot of _41_ S-N horn fatigue tests of Martin Marietta 7075 QQA225 aluminum at 135 MPa (152 microns). (The raw data is given in __.) This is currently the most extensive S-N horn testing done at a single stress level. The fit of this data to a straight line is quite good, indicating a log-normal failure distribution.

Where sufficient data is not available, we will assume that a particular life distribution is log-normal. Where sufficient data is available, this assumption will be checked by plotting the failure times on log-normal probability paper. (Alternately, the log of the failure times can be plotted on normal probability paper.) If the resulting log-normal plot falls reasonably close to a straight line, then we can reasonably assume that the failures are log-normally distributed.

Effect of stress level on life variability

Range of fracture

Let's first consider a material whose S-N behavior conforms to figure f. (For convenience, we are assuming a material with a sharply defined knee, although this is not necessary.) For the range of fractures in figure f, the width of the confidence interval (on a log scale) remains relatively constant, regardless of the stress level or the median life. For this type of behavior, the (log X)SD should be relatively constant regardless of the stress level. Alternately, if the life data from this type of S-N diagram is plotted on log-probability paper, then the slopes of the lines at each stress level will be nearly equal. This type of behavior is shown for the steel data of figure f for stresses between 29 and 37, all of which have reasonably constant slope. Maennig[1] (pp. 612 - 613) claims that this distribution of failures is typical of pure steels, AlMg aluminum alloys, and specimens with one sharp notch.

In many cases, however, the S-N diagram may resemble that of figure [f].. For materials with this type of S-N behavior, the confidence interval (on a log scale) increases as the stress level decreases. This means that there will be more data scatter when samples are tested at low stress than when tested at high stress. For this type of behavior, slope of the curves on log-probability paper decreases as the associated stress decreases. This behavior seems to be true of many aluminums.Error! Reference source not found. Error! Reference source not found.

Figure f (from Sinclair) shows this behavior for 7075-T6 aluminum. For instance, for a 95% confidence level (ranging from 2.5% to 97.5% on the vertical axis), the life range at 62500 psi was approximately between 16600 and 15400 cycles (i.e., a ratio of 1.08). However, for a 95% confidence level at 30000 psi, the life ranged between 20.5*106 cycles and 12.7*106 cycles (i.e., a ratio of 1.61). (See Sinclair's table 2 on page 868.)

This trend toward increased scatter has been found in Branson's tests of structural aluminums and 7-4 titanium. (Results will be given later.)

Range of transition

In the range of transition, and generally in a region of changing S-N slope, the life will no longer be log-normally distributed. (See Spindel, p. 103, Nishijima, pp. 76 - 77.) The distributions will be skewed with tails toward long lives. (See figure f.) This likely explains the data shown by Bastenaire (figure f) at stresses of 27, 26, and 25.

Fatigue strength distribution

When fatigue testing is used to determine the S-N curve, multiple specimens are tested at specified stress levels and the times to failure are recorded. The resulting data can then be analyzed as life distributions at the specified stress levels. Alternately, however, the data can also be analyzed as having strength distributions at specified lives. (See figure [f]..) This is especially appropriate for performing long life testing. (See section _Error! Reference source not found._.)

Whereas the shape of the life distribution is dependent on the stress level, the shape of the stress distribution is somewhat independent of the life (Nishijima, p. 76, Spindel, p. 103.). For seven different steels, Nishijima (pp. 81 - 85) found the strengths to be normally distributed. Juvinall[1] (pp. 351 - 354) cites literature that supports a normal strength distribution. However, Maennig[1] (pp. 634 - 635) uses the log-normal strength distribution. The SAE Fatigue Design Handbook (pp. 78 - 81) gives charts based on both normal and log-normal strength distributions. The proper choice of the strength distribution is important in analyzing the data for certain kinds of fatigue tests.

If the life distribution is known, then in some cases the strength distribution can be inferred. For this discussion, let's assume that the life at any specified stress is symmetric about its mean when plotted on a log scale.

Log-log plot. Figure f shows a portion of a log-log S-N curve where the lines denoting the confidence limits are parallel to the mean life line. Because the confidence limits are symmetric about the mean, we could conceivably draw a normal-shaped curve in either the Í or N direction. Since this S-N curve is drawn on log-log paper, we might guess that Í and N are log-normally distributed, although we have no guarantee.

Now let's look at figure f. This shows a portion of a log-log S-N curve where the confidence limits converge toward the mean life line. For any chosen stress Í, the confidence limits are symmetric about the mean in the N direction. Thus, N may possibly be log-normally distributed. However, at any chosen life N the confidence limits are not symmetric about the mean in the Í direction -- i.e., the distance from the mean to the upper confidence limit is less than the distance from the mean to the lower confidence limit. Hence, because the strength distribution is not symmetric on log-log paper, it cannot be log-normally distributed.

Log-linear plot. Figure f shows a portion of a log-linear S-N curve, where Í is plotted on a linear scale and N is plotted on a log scale. Because the confidence limits are symmetric about the mean, we could conceivably draw a normal-shaped curve in either the Í or N direction. Since N is plotted on a log scale, we might assume that it is log-normally distributed. Since Í is plotted on a linear scale, we might assume that it is normally distributed. However, we have no guarantee that these assumptions are correct.

Figure f is similar to figure f, except that Í is plotted on a linear scale. We see that the confidence limits are symmetric about the mean in the N direction. However, because the confidence limits converge toward the mean, they are not symmetric in the Í direction. Hence, while we might presume a log-normal distribution of life N, we could not assume a normal distribution of stress Í.

Thus we see that the symmetry of the confidence limits about the mean gives some indication of the distribution of a variable. In the test method where a group of samples is tested to failure at a specified stress, the life (N) distribution at the test stress Í can be statistically checked, so no assumptions about the N distribution are needed. The strength distribution is more difficult to determine. Test methods are available, but require a large number of tests. As discussed above, however, the strength distribution can sometimes be inferred from the S-N curve and the confidence limits.

Failure criterion

In the above sections we have referred to fatigue failure. However, there are several criteria by which failure might be measured:

1. Crack detection. Failure is defined to have occured when the first crack is detected. Unfortunately, this depends on the resolving power (sensitivity) of crack detection equipment. For example, the time to failure would depend on whether you used a electron microscope, light microscope, or the naked eye to detect the crack. (See Barton for comparisons of various methods.)

2. Crack length. Failure is defined to have occured when a crack grows to a predefined length. This requires equipment to measure the length of the crack within the specimen, not just at the surface.

3. Functional failure. Failure is defined to have occured when the part can no longer perform its required function.

4. Total failure. Failure is defined to have occured when the part breaks into two or more pieces. We could also call this catastrophic failure.

Each of these failure criteria would not usually give the same life value N. For example, compare the crack detection criterion to the functional failure criterion. Microcracks sometimes form initially but then fail to propagate. (Hessler, p. 246, 255, 259 - 261; Yeske, p. 372; Grosskreutz, p. 23, pp. 42 - 43.) Then the life will be finite according to the crack detection criterion but will be infinite according to the functional criterion. Also, some materials generate detectable fatigue cracks earlier in the fatigue process. (See section .) Thus, two material could have exactly the same functional life, but could have substantially different fatigue lives according to the crack detection criterion.

There are similar conflicts between the crack length criterion and the functional failure criterion. For ultrasonic fatigue, functional failure occurs when the crack becomes sufficiently large (i.e., the remaining supporting area becomes sufficiently small) that the ultrasonic stress causes the crack to open appreciably. Then the horn draws excessive power, causing the power supply to shut down. A horn with a given crack length may run acceptably at a low amplitude but may not run at a high amplitude, where the stress is sufficient to cause the crack to open. Thus, horn functional failure depends on the stress level.

For ultrasonic fatigue testing, we will use the following functional failure criterion:

AN S-N HORN HAS FAILED WHEN THE POWER SUPPLY SHUTS DOWN AND CANNOT BE RESTARTED.

Of course, this excludes restarting problems due to malfunction of converter, power supply, etc. Therefore, the horn should always be inspected for a visible crack.

Fatigue testing methods: low frequency

In this section I will discuss low frequency fatigue testing and the problems in using the resultant data for ultrasonic resonator design.

Method

Low frequency (conventional) fatigue testing can take many forms, depending on the type of load (e.g., bending, torsional, or axial loading), the mechanism by which the load is applied, and whether a static (mean) stress is super-imposed on the alternating (vibrating) stress. For testing of cylindrical specimens, the most common type of low frequency test is the R. R. Moore rotating beam test. With this type of test, a highly polished rod of standardized dimensions is subject to a pure bending load as it is rotated. Cycle rates are typically between 1500 and 10,000 cpm (cycles/minute) -- i.e., 25 to 170 Hz. (Figure [f]. shows a typical low frequency fatigue test machine.Error! Reference source not found.) During each revolution, the surface of the specimen is subjected to a sinusoidal stress, which varies from tensile to compressive and then back again.

Limitations

A great amount of fatigue data has been accumulated by many researchers using these low frequency testing methods. Unfortunately, this data is of limited use in predicting fatigue life at ultrasonic frequencies:

1. Number of Test Cycles. Because of the low frequency of R. R. Moore testing, the test is ususlly stopped if the specimen has not failed within 500 million cycles (approximately 35 days of continuous testing at 10,000 cpm). However, fatigue data at 500 million cycles (7 sonic hours at 20 kHz) is not sufficient for evaluating horns at ultrasonic frequencies, where working horns are expected to live 2000 hours (2.4*109 cycles at 20 kHz). Thus, the knowledge about 7-hour fatigue performance from conventional testing is not very useful.

2. Frequency Effect. Even when conventional and ultrasonic fatigue tests are conducted at the same stress levels, there may be a poor correlation between the two methods. This is known as the frequency effect. (There are numerous references to this effect in Ultrasonic Fatigue.) Depending on the material and test conditions, conventional testing may show longer life, shorter life, or little difference when compared to ultrasonic testing. The reasons are not well understood.

3. Test Method. For the R. R. Moore rotating beam bending test, the tensile stress is highest at the surface of the specimen and decreases to zero at some point in the interior. Axial fatigue tests (e.g., ultrasonic tests), however, produce a stress that is uniform across the stressed section. (See figure [f]..) Because of the difference in stress distribution, the fatigue life of materials tested under axial loading is generally less than for the rotating beam test, even when the test frequency is exactly the same. Unfortunately, there is no exact factor that relates the life differences between bending and axial tests. (See Juvinall[1], p. 227, for a good figure and discussion.)

Thus, if accurate data is needed on fatigue properties at ultrasonic frequencies, then the material must be tested at ultrasonic frequencies. (Note: there are some general trends which seem to hold true between low-frequency and ultrasonic fatigue tests -- e.g., increased notch sensitivity with fatigue strength, increased fatigue resistance with increased strength, etc. These will be discussed later.)

S-N horn design

In this section I will discuss the appropriate horn design for ultrasonic fatigue testing.

If we intend to conduct ultrasonic fatigue tests, then we need to design a fatigue test horn. I will henceforth refer to this horn as the S-N horn.

Design criteria

The S-N horn design criteria are:

1. Reproducibility. Because of the inherent data scatter in fatigue testing (see below), many horn tests will be needed to characterize the material fatigue characteristics. To help minimize the data scatter problem, the horn design should be simple so that it can be made repeatedly with little dimensional variation.

2. Cost. Because many horns will be made, the horn must be reasonably inexpensive.

3. Stress analysis. In order to determine the fatigue characteristics, the horn must be driven at a known ultrasonic stress. Unfortunately, ultrasonic stress is not easy to measure. However, horn amplitude is easily measured. Thus, we would like an S-N horn whose shape provides a simple mathematical relation between the stress and the amplitude. Then, after the amplitude has been measured, the stress can be easily calculated. The design must also preclude any unwanted stresses, such as flexure stresses.

4. Coarse gain adjustment. In order to establish a curve of horn life versus horn stress (the S-N curve), groups of horns will have to be tested at different stresses (amplitudes). The exact amplitudes may not be known before the test program begins. Therefore, if a common horn design is to be used for all tests, the design should permit easy machining to achieve gross changes in horn gain and, hence, in the horn output amplitude. (Note: Although the horn amplitude could be adjusted to some extent by changing the converter drive voltage, such adjustment should be minimized to prevent any additional data scatter. See below.)

5. Fine gain adjustment. Quite often, a group of S-N horns will tested simultaneously at a given amplitude. Each horn will be driven by its own converter, whose output amplitude will normally differ somewhat from the other converters being used. This difference in converter amplitude would then cause the horn amplitudes to be different. To correct this the horn should be easily machined to achieve fine changes in its gain.

For either coarse or fine gain adjustment, the horn should be designed so that the relation between stress and amplitude does not change as the horn gain is adjusted by machining.

6. High gain. To reduce possible flexure and starting problems, the S-N horn should be run without a booster. Then, to achieve sufficient amplitude to induce fatigue failure and still maintain a simple shape, the S-N horn must have high gain. With high gain (and, hence, low input amplitude at the stud end), the probability of horn-converter joint problems (heating and galling) is also greatly reduced.

7. Other properties. If possible, the horn design should also permit wave-speed and Q measurements. (See the chapters on "Elastic Material Properties" and "Inelastic Material Properties".)

S-N design selection

To accommodate the above requirements, the full wave 20 kHz horn of figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. has been designed. The maximum stress occurs in the front half-wave section at the node (1/4 wavelength from the front). (See .F.figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND..) This maximum stress that causes failure is given by equation Óe from the chapter on _Error! Reference source not found.Stress_:

) Í = (1/2) (pC) (2Ðf) Å

where Í = Peak ultrasonic stress

p = Density

C = Material wave speed

f = Frequency

Å = Horn amplitude (peak-to-peak)

Let us assume that the following units are used:

Í == Pa (i.e., Newton/m2)

p == kg/m3

C == m/sec

f == cycle/sec

Å == microns

Then, if the left and right sides are to have consistent units, the right side must be divided by a factor of 106 microns/m.

The 17.3 mm front diameter was initially chosen so that the required output amplitude could be obtained when the horn was machined from 50 mm diameter stock (i.e., 50 mm back diameter). To maintain consistency, this front diameter has been retained for all further testing, even though the back diameter has subsequently been increased for greater gain.

Determining the initial back diameter

The S-N horn back diameter must be sufficiently large to achieve the required output amplitude. To estimate the back diameter, we start with a rough estimate of the S-N horn gain, based on the horn of figure f:

Ö Back diameter Ì2

) Gain ~ 0.8 ° ------------- °

Û Face diameter ì

Since the gain is defined as the ratio of the horn output amplitude to input amplitude:

Output_amplitude

) Gain = ----------------

Input_amplitude

we can substitute equation 3 into equation 2 to estimate the approximate back diameter:

Ö Output_amplitude ̽

) Back_diameter = 1.25 Face_diameter ° ---------------- °

Û Input_amplitude ì

The input amplitude is the converter amplitude at full RF voltage. The converter amplitude is not an exact value, since it can vary by ±10% from the nominal value. Also, the converter nominal amplitude may not equal the converter average amplitude. Thus, rather than using the nominal converter amplitude in equation 5, you should probably obtain the converters that will actually be used for the fatigue tests and then measure each one's amplitude. (See the chapter on "Amplitude Measurement".) Make sure that the converter's are well-aged (at least six months) so that the amplitude will not later change significantly. From the measured converter amplitudes, increase the highest value by 10% (as a factor of safety) and use this in equation 5.

(Note: data given later will permit an estimate of the required output amplitude.)

Unless you have some prior data, you may not know exactly what output amplitude will be needed to cause fatigue failure. In this case, you should choose an over-large amplitude, which will result in a horn of excessively large back diameter. If, after some initial testing, you discover that the output amplitude is too high, you can then reduce the horn's back diameter to reduce its gain.

Frequency considerations

In the above discussion we have assumed that the horn frequency was 20 kHz. This was mainly because most early tests were conducted at 20 kHz. However, the ultrasonic fatigue tests could also have been conducted at any other reasonable ultrasonic frequency -- e.g., 40 kHz.

40 kHz advantages

In fact, 40 kHz actually has several advantages over 20 kHz:

1. Speed. At 40 kHz, the cycles accumulate twice as fast as at 20 kHz, so that the testing should take half as long. This assumes that the inherent fatigue life of the material is not affected by the frequency difference between 20 kHz and 40 kHz. (See disadvantages below.)

2. Material test direction. One purpose of S-N testing is to determine the life of working horns. If the material of the working horn is anisotropic (e.g., titanium bar stock), then the fatigue life may depend on the direction of vibration relative to the material direction. For instance, Bowen (7, p. 1277) has found that "considerable variations in fatigue and tensile strength can occur with changes of testing direction in large forgings of Ti-6Al-4V." (See the chapter on "Elastic Material Properties" for a discussion of directional properties of materials.)

To permit valid conclusions about the lives of working horns, the S-N horn should be vibrated in the same direction (relative to the material direction) as the working horn. For a bar horn, this means that the S-N horn should vibrate in the long-transverse material direction (i.e., LT-vibration). However, since bar stock is only about 150 mm in the long-transverse direction, a 20 kHz full-wave S-N horn (approximately [] mm) cannot be machined from bar stock in this direction.

This problem can be circumvented by using a 40 kHz horn, whose length is half that of a 20 kHz horn. Then this horn can be machined from bar stock in the long-transverse direction.

Alternately, we could use a specially designed half-wave 20 kHz S-N horn with a high-stress notch in the nodal area. The stress could then be evaluated by finite element analysis. However, such a notch may cause flexure problems and would also complicate the machining and subsequent polishing. A half-wave horn would also not permit wave speed and Q measurements.

3. Material variability. Because of the manner in which it is heat treated, the elastic properties of titanium may vary along the length of a bar. It is possible that the fatigue properties may similarly vary. Since 40 kHz horns are smaller than 20 kHz horns, the 40 kHz horns will show less variability between adjacent horns machined from the same bar. This is important proper statistical design of the fatigue experiment. (See the section .)

4. Horn cooling. The running temperature of a horn will depend on how fast it generates heat and how quickly the heat is carried away by conduction and convection to the air. For a material whose Q is independent of frequency (most acoustic materials), the heat generated in an unshaped horn depends on the horn mass and velocity. (See the chapter on "Inelastic Material Properties", equation [].)

If 20 kHz and 40 kHz S-N horns are operated at the same stress, then they will have the same velocity, for which case the heat generated will differ only by their relative masses. Thus, since a 40 kHz horn has only 1/8 the volume of a 20 kHz horn (half the length and diameter), it will generate only 1/8 as much heat. Since a 40 kHz horn has only 1/4 as much surface area as a 20 kHz horn, the heat will be carried away only 1/4 as fast. However, considering the ratio of heat generated to heat carried away, the 40 kHz horn is twice as efficient as the 20 kHz horn.

This means that the 40 kHz horn will run cooler, or that its duty cycle can be increased for the same operating temperature.

5. Cost. Since the 40 kHz horn uses only 1/8 as much material, it will cost less.

6. Space. 40 kHz power supplies are much smaller than 20 kHz power supplies. Therefore, a large number of horns can be tested simultaneously without requiring excessive laboratory space.

40 kHz disadvantages

40 kHz operation also has several disadvantages compared to 20 kHz:

1. Machining. Because the 40 kHz S-N horn is half as large as a 20 kHz horn, it will deflect twice as much under an equal machining load, and will therefore be harder to machine (e.g., for concentricity).

2. Power supply starting. Some preliminary 40 kHz tests seem to indicate that the 40 kHz power supplies may be more difficult to start than 20 kHz power supplies. Thus, for cyclic testing (e.g., 1 second on, 4 seconds off) the 40 kHz tests may have to be monitored more closely to assure that the test has not stopped before actual fatigue failure.

3. Converter heating. Comparing their relative volumes, the 40 kHz XL converters generate considerably more heat than do the 20 kHz 400 series converters. Thus, if sufficient cooling is not provided, the 40 kHz converter ceramics are more likely to become partially depolarized from overheating, which would increase the converter amplitude and invalidate the test results. This would be particularly unfortunate if it happened toward the end of a long-term (e.g., 2000 hour) test.

4. Flexure detection. The S-N horns should be designed so that they do not operate near a flexural resonance, since such flexure would introduce unaccounted stresses. At 20 kHz, flexure can often be detected by the high frequency squeal that results. At 40 kHz, however, such flexing squeal would probably not be audible, unless it generated strong subharmonics. Thus, detection of flexure problems would be less likely at 40 kHz than at 20 kHz.

General comparisons

20 kHz and 40 kHz fatigue tests may give different results. These differences do not favor either the 20 kHz and 40 kHz tests, but must be considered when applying the results to working horns at different frequencies. These differences may be due to:

1. Stock size effect. The effect of metallurgy and heat treatment may depend on the size of the raw stock, for which large stock generally gives lower fatigue life than smaller stock (at least in low frequency fatigue tests.) (See the section for further discussion.) Since 40 kHz S-N horns are typically machined from stock that is half as large as for 20 kHz horns, we might therefore expect longer lives from the 40 kHz horns, even though the 20 kHz and 40 kHz horns are otherwise identical.

2. Frequency effect. There may be some effect from the test frequency, independent of the stock size. However, since the frequency ratio is only 2:1, this effect is unlikely.

The frequency and stock size effects suggest that 40 kHz S-N horns may live longer than equivalent 20 kHz horns. Then, if we use the results of 40 kHz S-N tests to predict the lives of 20 kHz working horns, the life estimates would be too long -- i.e., the 20 kHz horns would fail earlier than predicted. Conversely, if the S-N tests were conducted at 20 kHz, then 40 kHz horns would live longer than predicted by the 20 kHz results. Thus, 20 kHz S-N tests would give conservative (safe) 40 kHz life predictions, whereas 40 kHz S-N tests would give optomistic (unsafe) 20 kHz life predictions.

Conclusions

If only rod materials are to be tested (for which either 20 or 40 kHz horns can be used), then my preference is for 20 kHz horns. However, if bar stock must be tested in the long-transverse direction, then 40 kHz testing is the only viable option (assuming that the standard S-N horn design will be used).

For the balance of this discussion, we will assume that the testing is to be done at 20 kHz, unless otherwise indicated.

Design limitations

One problem with using this full-wave horn design is that its length (approximately 10-1/2" at 20 kHz) may not allow all materials can be tested in the desired direction. For instance, titanium bar stock (from which bar horns are machined) is only about 6" long in the direction of axial vibration. Hence, the S-N horn of figure f cannot be machined from titanium bar stock in the usual direction of horn vibration.

This limitation is important in testing materials such as titanium whose properties depend on the test direction. For instance, Bowen (7, p. 1277) has found that "considerable variations in fatigue and tensile strength can occur with changes of testing direction in large forgings of Ti-6Al-4V." Also see Frost (p. 81), who gives data for steels that show that, in general, the longidudinal direction gives a higher fatigue strength than the transverse direction. The effect of directionality in aluminum is not clear. Frost cites one source with higher fatigue strength in the longitudinal direction, while another source found no fatigue directionality.

One possibility is to run the S-N tests at 40 kHz, which would reduce the length of the S-N horn to approximately 5-1/4". However, this would require additional test equipment. Also, the correlation between test results at 40 kHz and actual horns running at 20 kHz is unknown. (See the frequency effect above.) Of course, we would hope that 40 kHz horns would have exactly the same stress-life relation as 20 kHz horns.

(Note: some researchers have used half-wave "dumbbell" shaped horns for S-N testing. (See figure [f]..) While such a horn is only roughly half as long as Branson's S-N horn and is designed for high stress with low gain (hence requiring less material and machining), the stress in such a horn is not as easily calculated as an unshaped horn and such a design is more prone to flexure, which may cause early failure from additional unaccounted bending stresses.)

Machining and tuning the S-N horn

In this section we will look at the correct method for machining the S-N horn blank, for tuning it to the correct frequency, and for adjusting its gain.

Machining the S-N horn

Most fatigue failures originate from the specimen surface. Therefore, machining (surface) operations will affect the fatigue life and fatigue strength.

In machining the S-N horn, the primary requirement is to produce a surface that is reproducible (consistent) among horns. Then any variations in fatigue lives can be attributed to usual fatigue scatter, rather than to variations in horn surface preparation.

Whatever surface preparation method is used, the method must be strictly specified and carefully in order to insure reproducibility. If possible, horns should be machined on a selected NC machine to assure repeatability, especially regarding speeds, feeds, and surface finish. The handling and machining operations should be sufficiently specified that variability due to different machine operators is minimized. Whenever possible, the same machine operator should be used for machining all of the S-N horns.

Ranges of machining

Machining can be divided into three general ranges: preliminary machining, rough machining, and finishing operations. Preliminary machining gives the S-N horn a recognizible shape, but the dimensions in the critical fatigue area are still grossly oversized -- e.g., final diameter + 5 mm. The preliminary machining parameters (feeds, speeds, etc.) are left to the operator's discression.

Rough machining brings the S-N horn to its approximate final shape, with slightly oversized dimensions in the critical fatigue area. The oversized dimensions must be sufficient so that the subsequent finishing operations can remove any damaged material (microcracks, hardened or softened material, worked material, etc.) from the rough machining. Thus, the amount of oversize will depend on the abusiveness of the rough machining, which in turn depends on the material. The range of rough machining is typically between diameters of 5 mm oversized to 0.5 mm oversized. (See Swanson, p. 94; also see below.) All machining parameters are specified.

Finishing operations then bring the S-N horn to its final size, while providing the desired surface. In the following discussion, we will concentrate mainly on finishing operations.

Neutral surface

A neutral surface is one whose properties do not vary from those of the underlying material. Thus, a neutral surface will have the same hardness, degree of cold working, chemical composition, etc. as the underlying material. In addition, a neutral surface will have a sufficiently good surface finish that further improvements in finish will not increase the fatigue life. Also, the neutral surface should have no residual stress and should not cause residual stress in the underlying material.

There is no guaranteed method for producing a neutral surface. A method which is satisfactory for one material may be entirely unsuited for another.

Gentle grinding

Gentle grinding is often recommended, but even gentle grinding can leave residual stresses. For example, Zlatin (p. 499) found that gentle grinding of Ti-6Al-4V (beta rolled, 32 Rc) left a shallow (approx 0.08 mm) surface tensile stress of 40 ksi. Gentle grinding caused no microstructural or microhardness changes. Zlatin's gentle grind conditions (characterized by soft grinding wheel, low wheel speeds, very light down feeds, correct cutting fluid) were:

TABLE [t]

GENTLE GRINDING OF TI-6Al-4V

Grinding wheel C60Hv (SiC)

Wheel speed (ft/min) 2000

Down feed (in/pass) "LS"

Cross feed (in/pass) 0.050

Table speed (ft/min) 40

Grinding fluid KNO2 (1:20)

On the other hand, Heywood cites references that show that gentle grinding of steel at 0.0025-0.0127 mm (0.0001-0.0005") had little effect on fatigue. Manson (pp. 289 - 291) gives data for fatigue of Inconel 718, for which a gentle grind gave considerably higher fatigue life at 107 cycles than either electrochemical machining or electropolishing. Manson attributes this to compressive residual stresses from grinding.

Thus, it appears that gentle grinding may leave either tensile, compressive, or no residual stress, depending on the circumstances. For those cases where grinding has only caused residual surface stress, but has not affected the material properties or caused microcracking, then stress relieving may restore a neutral surface condition.

Electrolytic polishing

Electrolytic polishing (electropolishing) is sometimes recommended for producing a neutral surface. (See Heywood, p. 299, and Frost, p. 49.) Weibull (p. 90) states:

As a substitute, not to say improvement, of mechanical polishing it may be convenient, particularly when preparing a large number of notched specimens, to use electrolytic polishing, which is claimed to produce a minimum amount of resicual stress.

However, care must be exercised with electropolishing, since it may preferrentially attack certain components of a material, leaving pits. Frost (p. 50) says that aluminum alloys may be particularly affected. Pitting may account for the fatigue results of Sinclair for electropolishing two titainum alloys (cited by Heywood, p. 72):

TABLE [t]

ELECTROPOLISHING OF TITANIUM ALLOYS

Surface Condition Fatigue limit (ksi) at 107 cycles

Alloy A Alloy C

Longitudinal polish with 00 emery 87 90

Electropolish, 0.002" removed 91 79

Ground 79 --

Electropolish, 0.010" removed 66 81

Notes:

1. Alloy A: 4.08% Mn, 3.96% Al, 0.107% N2, 0.06% C,

0.02% Fe; 139 ksi tensile

2. Alloy B: 3.7% Al, 3.3% Mn, 0.03% N2, 0.01% C; 155 ksi

tensile

3. Rotating bending at 3600 rpm.

Note that the fatigue strength of alloy A decreased significantly as the depth of electropolishing increased from 0.002" to 0.010". Alloy B was relatively unaffected under the same circumstances.

Similarly, Manson (p. 291) shows Inconel 718 fatigue data for which the electropolished samples had approximately the same fatigue strength at 107 cycles as either roughing or finishing EDM, even though the surface finish for the electropolished samples was considerably better (15 µin vs 65 µin and 155 µin, AA). Note that EDM usually has a very deleterious effect on fatigue. (See the section .)

Paul Bania at TIMET (3/31/88) also mentioned that electropolishing may give off hydrogen gas, which is harmful to titanium.

Polishing

Fine polishing is often used as the final process in fatigue specimen surface preparation. The polishing process uses progressively finer abrasives, typically finishing with 600 grit. Experimental data shows that polishing may introduce slight residual surface compressive stress, which improves the fatigue life compared to a neutral surface. (See Heywook, p. 299 - 301; Frost, p. 50.)

Heywood (p. 296) says that the maximum depth of surface irregularities is more important than average or RMS roughness. There also seems to be a minimum depth of surface irregularity below which the fatigue strength shows no further improvement, although this is probably dependent on notch sensitivity of material. From Heywood's figure 14.2 (p. 299) for various steels and 2 aluminums, this limit is approximately 0.00254 mm (0.0001").

Circumferential polishing generally gives lower fatigue strength than longitudinal polishing. (See Heywood, p. 299.) This is further evidence that polishing does not produce a neutral surface. Otherwise, longitudinal and circumferential fatigue strengths would be the same.

Frost (p. 50) gives the following data comparing circumferential and longitudinal polishing for three aluminum alloys:

TABLE [t]

EFFECT OF POLISHING DIRECTION

ON FATIGUE LIFE OF ALUMINUM

Relateve Fatigue strength

Finish Roughness (µm) Alloy A Alloy B Alloy C

Rough machined 2.5 0.29 0.31

Fine machined 1.6 0.29 0.31

Circumferentially 0.23 0.31 0.33

polished

Circumferentially 200 grit 0.37

hand polished

Longitudinally 0.14 0.33 0.35

polished

Longitudinally 200 grit 0.34

hand polished

Notes:

1. Relative fatigue strength

= fatigue strength / tensile strength

2. For alloys A and B, relative fatigue strength is measured

at 107 cycles. For alloy C, at 108 cycles.

2. Alloy A: DTD 683, 550 MPa tensile

Alloy B: BS 6LI, 350-510 MPa tensile

Alloy C: 24ST (4.5% Cu), 460 MPa tensile

For alloys A and B, the fatigue strength of the longitudinally polished specimens was about 6% higher than the circumferentially polished. However, the alloy C the opposite was true, where the fatigue strength of the longitudinally polished specimens was about 8% lower than the circumferentially polished.

Thus, the exact effect of longitudinal versus circumferrential polishing is unclear. However, most investigators recommend that the final polishing operation be in the longitudinal direction. (See ASTM STP 566, pp. 92, 96.)

Production surface

In some cases, a neutral surface may not be desired. For example, you might want the S-N surface to represent that of a typical working horn. Then the S-N horn would be machined with standard production methods. As before, however, the exact type of production surface and production must be specified. Thus, cutter type, cutting fluid, speeds, feeds, etc. must all be specified. Obviously, then, a single surface specification cannot represent all production horns. For instance, a solid cylindrical production horn that is produced by turning will likely have a different surface than a bar horn that is produced by milling.

Machining specifications

The S-N horn should first be machined to a diameter that is 5.0 mm oversized in the front half-wave section. Swanson (p. 94) then recommends the following cuts, based on the British Standards Institution:

TABLE [t]

TURNING S-N HORNS

Turning Operation Diameter Depth of cut

Operation Number oversized (mm) per pass (mm)

 

Preliminary --- ·5.0 ----

Rough 1 5.0 --> 2.5 1.25

2 2.5 --> 1.0 0.75

3 1.0 --> 0.5 0.25

Finish 1 0.50 --> 0.25 0.125

2 0.25 --> 0.10 0.075

3 0.10 --> 0.00 0.050

Notes:

1. "Diameter oversized": amount by which the diameter

exceeds the finished horn dimension, during which the

associated operation is applicable.

2. During the finishing cuts, the feeds should not exceed

0.06 mm per revolution.

The above is generally applicable to steel, aluminum, and titanium alloys (see Swanson, p. 94) where grinding is not required. If grinding is required, then the British Standards Institution recommends the following (Swanson, p. 94):

TABLE [t]

GRINDING S-N HORNS

Grinding Operation Diameter Depth of cut

Operation Number oversized (mm) per pass (mm)

Preliminary --- ·5.0 ----

Rough grinding 1 5.0 --> 2.5 1.25

2 2.5 --> 1.0 0.75

3 1.0 --> 0.5 0.25

Heat treat

(if required)

Finish grinding 1 0.500 --> 0.100 0.025

2 0.100 --> 0.025 0.005

3 0.025 --> 0.000 0.0025

Notes:

1. "Diameter oversized": amount by which the diameter

exceeds the finished horn dimension, during which the

associated operation is applicable.

For grinding of unnotched cylindrical specimens, Metcut Research Associates Inc recommends the following procedure (revised 6/5/85):

TABLE [t]

GRINDING S-N HORNS

Grinding Operation Diameter Depth of cut

Operation Number oversized (mm) per pass (mm)

Roughing 1 ·0.50 0.025

Finishing 1 0.500 --> 0.050 0.0127

2 0.050 --> 0.030 0.0100

3 0.030 --> 0.000+ 0.0050

4 0.000+ --> 0.000 Spark out

Specimen material Wheel

Steels and high temp alloys 4A60J6VFM

Ti, Al, Cu 39C60J8VK

Speeds and feeds:

Work surface: 8 to 26 fpm

Table speed: 7 ipm

Wheel speed: 2900 - 3540 fpm max (12" to 16" dia wheel)

Grinding fluid: Stuart Thredkut #99, diluted 1:1 with Sohio

Factoil 39.

Tool wear should be controlled to prevent overheating of the horn's surface, which could cause residual surface stresses, and coolant should be used as required. This is especially important for titanium horns. The same coolant should be used with the S-N horn as with any normal working horns of the same material. This is especially important if the coolant contains organic oils, since these have been found to increase the fatigue strength of many materials (Swanson, p. 100).

Concentricity should be held to prevent ultrasonic flexure at high amplitudes, which would cause additional unaccounted stresses. Surface finish in the highly stressed front half-wavelength section is important.

Polishing specifications

The goal of polishing is to remove any surface scratches caused by machining, thereby reducing life variability that occurs during fatigue testing. Polishing should be viewed as a cutting operation (not a buffing or smearing operation). It should give a uniform, reproducible surface without cold working, residual stresses, overheating, etc. Swanson (p. 96) notes that complicated polishing procedures that give highly polished specimens may be no more effective at giving consistent fatigue results than a uniform standard finish.

Polishing is not always required. As Swanson (p. 96) notes:

For most nonferrous specimens a machined surface is often more desirable than a polished surface because (1) it is more easily produced, (2) it represents a practical condition, and (3) it has extremely mild stress raisers if machined carefully with a round tool.

The final surface may be polished to a prescribed finish by using progressively finer polishing operations. If a polishing operation is required, then the direction of final polishing should always be longitudinal. (Swanson, p. 96.)

Metcut Research Associates Inc. uses the following specification (revised 6/5/85):

TABLE [t]

POLISHING S-N HORNS

Process Paper Duration

Mechanical polish 40 µ 2 - 3 minutes or until all

traces of cylindrical grind

marks are removed

Mechanical polish 30 µ 2 - 3 minutes

Mechanical polish 15 µ 2 - 3 minutes

Hand polish 600 grit 1/2 - 1 minute

Notes:

1. All polishing is axial.

2. Mechanical polish is performed in Morrsion Test Specimen

Polishing Machine using Minnesota Mining Imperial Mico

finishing file, wet or dry silicon carbide 1/2" wide rolls.

3. Hand polish uses Klingspor PS12 wet or dry silicon carbide

wrapped around a rubber coated steel rod. The resulting

finish is 8 Ra.

For initial ultrasonic fatigue testing, I recommend an as-machined finish of 32 RMS or better. Such a finish is better than would be achieved on most working horns. The variability associated with other finishes (electropolishing or coarser finishes) can be evaluated separately.

Inspecting the S-N horn

The S-N horn should be handled carefully during inspection, especially in the highly stressed front half-wave section. Care should be taken not to scratch the horn at any critical sections with the measuring instruments. The diameter of the front half-wave section (which is not critical) should be checked at the end of the horn, not in the nodal region. (See figure ífÊError! Reference source not found..) If the S-N horn has been designed with a notch in the nodal region for higher stress, then the notch dimensions should be checked optically to avoid marking. After all dimensional measurements have been made, the horn should be inspected at 20x (per Swanson, p. 98) for scratches or machining marks in the front nodal region.

TUning the S-N horn

Tune the S-N blank from the small end (face) to 20,000 ± 50 Hz. The blank is most easily tuned in a milling machine by clamping the small diameter horizontally. (Tuning in a lathe is not recommended because only the horn's large diameter can be easily clamped, which allows the small end to chatter during machining.)

NOTE: USE EXTREME CARE NOT TO SCRATCH OR NICK THE HORN

ON THE SMALL DIAMETER DURING TUNING. IF THIS SHOULD HAPPEN, THEN THE HORN MAY FAILURE PREMATURELY.

For one 20 kHz aluminum S-N horn (HRD-431) of unknown back diameter, the tuning rate from the face was 3690 Hz/mm. A titanium or steel horn would tune at a somewhat faster rate.

Adjusting the S-N horn amplitude

Adjusting the horn amplitude (gain) is a two-step process. Initially we assume that the horn has excess amplitude, according to the calculations of equation 5. Then the first step is a coarse amplitude adjustement for all S-N horns that are to be run at the same stress level:

1. Tune the S-N horns as discussed above.

2. From among the fatigue test converters, choose the one with the lowest amplitude. Assemble an S-N horn to this converter.

3. The horn must be started at the lowest possible amplitude so that the fatigue data is not contaminated by any initial overstress. Turn the power supply variac down as far as possible. Try to start the horn at this variac level. If the horn won't start, adjust the power supply tuning and try again several times. If the horn still won't start, increase the variac setting slightly and repeat the above procedure. Continue until the horn starts. As soon as the horn starts, turn the variac down as far as possible.

Measure the horn amplitude at this reduced amplitude. Slowly increase the variac while measuring the horn amplitude until the horn runs at the desired amplitude. If the variac setting is less than 90% at this point, then the horn gain is too high and will have to be reduced.

4. If required, machine the back diameter of the S-N horn (per figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND.). Note that a 12.7 mm wide ring is left unmachined on the back as the gain is reduced. This ring is used to hold the horn in the lathe during machining and also retains the original spanner wrench holes.

5. Repeat steps 3 and 4 until the horn amplitude equals the required fatigue test amplitude with the variac between 90% to 100%. Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. shows a log-log plot of this process for an aluminum S-N horn whose original back diameter was 63.5 mm. (See figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND..) The log-log plot is almost exactly linear (log-log correlation = 0.9991), and the curve-fit equation is:

X) G = 0.0455 D11.29

where G = gain

D1 = new back diameter (mm)

Of course, these particular results only apply to the S-N horn of figure f. However, similar results would be expected for S-N horns with other back or front diameters.

(Note: the reason for the limited variac range is that the horn life may be somewhat sensitive to the variac setting. See section .)

figure [f].

Since the frequency change while reducing the back diameter is small, the horn will usually not have to be retuned.

6. When the required back diameter has been established, machine the remaining horns to the same dimension. At this point, no amplitude check on these horns is required.

Once the gain of the entire group is approximately correct, the gain of each individual horn can be adjusted so that it has the correct gain when running with its converter and power supply. This final gain adjustment should be done just before the particular horn is ready for fatigue testing.

1. Assemble the mated horn/converter/power-supply components. These should be exactly the components that will be used during the actual fatigue test of the chosen horn.

2. Repeat steps 3, 4, and 5 above.

The horn is now ready for fatigue testing.

Fatigue testing of an individual horn

In this section we will outline the equipment and procedures needed to test a single fatigue test horn. We assume that tests are being conducted at 20 kHz.

Equipment

The following equipment is needed. The equipment will vary somewhat depending on whether the horn will be run continuously or whether it will be cycled on and off. For the present, we will assume that the horn will be cycled. Error! Reference source not found.

1. S-N horn. See above for machining, tuning, and amplitude adjustment.

2. Converter. The converter should have been overstressed (i.e., run at excessive voltage) and should be well aged (perhaps six months) so that its amplitude does not change appreciably during the test. The 400 series converter is recommended for 20 kHz tests, since it is relatively resistant to flexure and has low loss.

3. 20 kHz power supply (e.g. 184V). The power supply should have a variac for fine amplitude adjustment and an SPM (system protection module) to shut down the power supply when horn failure occurs.

4. Power supply cycling unit. This cycles the sonics at a fixed rate (typically a second of sonics followed by 4 seconds without sonics). This is needed for S-N horn materials that would overheat if run continuously at the chosen amplitude.

5. Cycle counter. This counts the total number of times the power supply has been turned on.

6. Regulated power supply. This will control any variations in line voltage and help assure reasonably constant horn amplitude throughout the test.

7. Amplitude measuring equipment. The A-450 is recommended since it permits quick amplitude checks.

ACCURATE CALIBRATION OF THE A-450 IS CRITICAL, SINCE INCORRECT AMPLITUDE MEASUREMENTS WILL CORRUPT ALL OF

THE DATA.

See the chapter on "Amplitude Measurement" for calibration procedures. A single two-point calibration is not sufficient.

8. Cooling fan or air supply.

Amplitude adjustment

1. Determine the desired test amplitude. (This is described elsewhere.)

Equipment setup

1. Adjust the cycling unit for 1 second of ultrasonics (i.e., 1 second ON) followed by 4 seconds with no ultrasonics (i.e., 4 seconds OFF). This cycle rate is low enough to avoid excess heating in a titanium horn. Cycling rates of 1 second ON followed by 1 second OFF have been used for aluminum (which has less internal heating than titanium) so that the total testing time can be decreased. Note: more desirable to have at least 3 seconds on, which is long enough to get stabalized amplitude and frequency readings as the testing progresses.

2. Reset the cycle counter to 0 or note the initial count.

3. Position the horn and converter to the location where they will be run.

4. Direct cooling air to the horn and converter.

5. Set the variac at 90%. Tune the power supply to a dip using the power supply test switch.

6. With the power supply test switch held down, adjust the variac to achieve the required horn amplitude. The final variac setting should be between 90% and 100%. I again emphasize that careful amplitude calibration and measurement are extremely important.

7. Turn on the power supply: cycling will begin.

8. The test may occasionally stop due to SPM (system protection module) shutdown, even though the horn has not failed. Just restart.

9. If the horn fails to restart after two tries, then the horn should be checked for a crack. The crack will usually be about 50 to 70 mm from the front of the horn. For most materials, the crack will be perpendicular to the horn axis. However, for Aerospace material, the crack will be at a 45o angle to the horn axis. You may have to inspect the horn under a microscope to find the crack. If a crack is found, then the test is finished.

The horn life (in hours) is given by:

) Horn_life (hours)

= 3600 (Counter_reading) (Cycle_on_time [sec])

The horn life (in ultrasonic cycles) is given by:

) Horn_life (ultrasonic cylces)

= Frequency (Counter_reading) (Cycle_on_time [sec])

Fatigue test design

In this section we will give some consideration to proper design of the fatigue test so that the results are statistically meaningful. (Much of this information is taken from Little {12} and Miller.)

Treatments and nuisance variables

For any test program, there will be many possible parameters that may affect the results. These parameters can be divided into two categories: the treatments and the nuisance variables. The treatments are those parameters that are of specific interest in a particular test (i.e., those parameters whose effect is to be measured). For example, fatigue treatments might include such parameters as stress level, heat treatment, surface finish, etc.

Nuisance variables are those parameters which may affect the test results but which are not of primary interest in a particular test. For example, it may be possible that certain power supplies may give generally longer fatigue lives, possibly due to variations in the SPM module that controls overload shutdown. If we are not primarily concerned about how the power supply affects the test results, then this is a nuisance variable.

The decision about which parameters are treatments and which are nuisance variables will depend on the objectives of the test. For example, if we decide to study the effect of surface finish on fatigue life, then surface finish will be a treatment. However, if we are studying the effect of stress (without regard to surface finish), then surface finish becomes a nuisance variable, since its variance from a nominal value may affect the stress test results. We may also have a test where there are several simultaneous treatments (e.g., simultaneously studying the effects of stress and surface finish).

Dealing with nuisance variables

The stated objectives of the test will generally define the treatments. Then any other parameters that may possibly affect the test results must be considered nuisance variables. We would like to design the experiment so as to minimize the effects of the nuisance variables. One method is to hold all nuisance variables constant throughout the test. For example, if we are concerned about the effect of different power supplies on fatigue lives, then all tests could be performed on a single power supply. While this might not be practical, it is none-the-less possible. However, other nuisance variables have a continuous range of values and therefore cannot be completely controlled. For example, surface finish can be controlled within limits, but cannot be controlled exactly.

One problem with controlling the nuisance variables is that the generality of the results may be limited. (See Miller, p. 334.) For example, we might compare two different materials M1 and M2 (i.e., two treatments), both of which have a nominal surface finish of 32 RMS. If material M1 has X% better life than material M2 at 32 RMS, we could not necessarily conclude that it would also have X% better life at 63 RMS. In fact, we could not necessarily conclude the M1 would even be better than M2 at 63 RMS.

This problem can be partially circumvented by running some preliminary tests (called uniformity trials) to determine the relative effect of the nuisance variable. Alternately, you could conduct a literature search to see if other researchers report the effects of the nuisance variable.

If particular nuisance variables cannot be sufficiently controlled (or if such control is impractical), then we might consider the completely randomized design (CRD), where the specified nuisance variables are randomly distributed among the treatments. For example, suppose we intend to develop an S-N curve by testing a particular material at four stress levels (four treatments). Also, let's assume that the S-N horns will have surface finishes comparable to normal production horns -- say, anywhere between 32 RMS and 125 RMS. Then, for a CRD test, the individual S-N horns would be assigned at random to one of the four stress levels and testing would proceed.

Although this provides a simple method of dealing with all nuisance variables, the results are less than ideal. This is because the complete randomization of the CRD method tends to obscure the treatments that are being measured -- i.e., like introducing random noise into an electronic circuit whose parameters are being measured. Thus, the CRD method reduces the statistical power of the comparison. (Little {12}, p. 16)

The most desirable approach is a combination of the above methods, whereby all nuisance variables within a limited test block are nominally identical, but can otherwise vary among the test blocks. Each test block contains one complete set of treatments, which constitutes one complete replication. The test blocks are tested in random time order, with individual specimens within blocks tested back to back in random order.

In the above example, we might divide the S-N horns into three groups: those with finishes closest to 32 RMS, 63 RMS, and 125 RMS. (For purposes of illustration, we will assume that we end up with four horns in each group.) Thus, each of the three groups constitutes a block, because the value of the nuisance variable (surface finish) in each block is the same. In the first block, the each of the four 32 RMS horns would be randomly assigned to one of the four stress levels. The same would be done with each of the remaining blocks. Then the blocks would be chosen in random order for testing (assuming only four power supplies are available, which would permit simultaneous testing of all horns within a block). The RCB test design for this situation is shown in table [t].

 

TABLE [t]

RANDOMIZED COMPLETE BLOCK TEST DESIGN

----------------------Ú-----------------------------------------

Treatments (T) ° Blocks (B)

° 32 RMS 63 RMS 125 RMS

----------------------é-----------------------------------------

Stress level 1 ° Horn T1B1 Horn T1B2 Horn T1B3

°

Stress level 2 ° Horn T2B1 Horn T2B2 Horn T2B3

°

Stress level 3 ° Horn T3B1 Horn T3B2 Horn T3B3

°

Stress level 4 ° Horn T4B1 Horn T4B2 Horn T4B3

Problems

In attempting to use the RCB test design to determine the S-N curve, we may run into several problems. First, if time is considered a potential nuisance variable (see below), then it should be held constant within each block. This means that all tests within a block should begin at the same time. However, this means that the next block should not start until the current block has been completed. However, since the highest stressed horns within a block will fail quickly while the lowest stressed horns will have long lives, the equipment used for the highly stressed horns will be idle during much of the time for testing a particular block. This is poor utilization of equipment and will significantly extend the total testing time for all blocks.

Another problem with the RCB test design is that all treatments must be identified before the testing begins. Since each block contains one complete set of treatments, we cannot test a number of blocks and then later add additional treatments, as this would upset the data analysis. Unfortunately, this may be required -- for example, if we later decide that additional stress levels (treatments) will be required to adequately define S-N curve.

In the RCB test design described above, each block has the same number of test specimens. Thus, for example, with four stress levels and three blocks, each stress level will have three horns.

When there are a large number of nuisance variables, it may not be possible to set up a randomized complete block test program where all nuisance variables are held essentially constant within a test block. Then it may make sense to run some preliminary tests to determine if a particular variable is really a nuisance. If it is, then you might choose to control it throughout the testing. For example, if the setting of the power supply variac is found to affect the fatigue life (even when test amplitudes are nominally the same), then you might wish to adjust the gain of each horn so that the variac setting could be controlled within a narrow range.

Nuisance variables for ultrasonic testing

If you suspect that a parameter may contribute to the variability of the primary variables, then that parameter should be blocked as a nuisance variable. For fatigue testing, the following nuisance variables (among others) may be of concern:

Material variations

Within-bar variations. It is known that the physical properties of titanium can vary along the length of a piece of stock. Therefore, we might reasonably suspect that the fatigue properties could show similar variation.

Bar-to-bar variations. Just as properties can vary within a bar, they can also vary between different bars, even if the bars have been processed in the same batch.

Heat-to-heat variations. Variations may also occur between bars that are processed in different batches.

Manufacturer's variations. The same nominal material from different manufacturers may show variations. For example, Grosscreutz (p. 42) reports that 2024-T3 aluminum from seven different manufacturers showed differences in crack growth rate as high as 2:1.

Machining variations

Even if the specified machining procedure is carefully followed, variations in the finished horn can still occur. For example, accurately specifying the tool sharpness is difficult, so that some horns machined with a dull tool may have a burnished surface. Such a surface may affect the fatigue life. This particular problem could be reduced by specifying the maximum number of horns to be machined with each tool or insert. However, other variations may still be present.

Testing variations

Environment

The humidity is known to affect the fatigue life of aluminum alloys, steels, and most titanium alloys. (See Grosskreutz, pp. 40 - 41.) For example, the crack growth rates in an AlCuMg alloy increased by a factor of about 10 in humid air over the rate in a vacuum of 3 x 10-8 torr. (Grosskreutz)

The test temperature may also have an effect, especially if the specimen is improperly cooled.

Personnel

Wherever possible, the exact testing procedures should be explicitly defined. In some cases, however, the procedure must rely on the judgement of the personnel involved. For example, it is known that the A-450 amplitude meter gives slightly different readings depending on how firmly the probe is pressed against the end of the horn. Thus, different technicians may adjust the horn to slightly different amplitudes, even though the recorded (nominal) amplitude is the same. Similarly, if the horn specification states "no scratches at 20X magnification", then the quality inspector decide what constitutes a "scratch".

Procedural variations

Some procedures can only be specified in a qualitative manner, and are therefore open to interpretation. Quantitative procedures may be inadvertently violated (e.g., from lack of understanding) or deliberately violated (e.g., to expedite a process). For example, a machinist may inadvertently or deliberately substitute an incorrect cutting fluid.

Time

Time can cause a number of nuisance effects. For example, some materials may have different fatigue properties depending on the time elapsed after heat treatment (i.e., the effect of ageing). Thus, if a batch of S-N horns are machined from the same bar, then their fatigue lives may depend on when they are tested. Similarly, if the horns are not properly protected from the environment (e.g., oxidation) during storage, then the storage time could affect fatigue lives. Also, if testing is extended over a lengthy period, then the seasonal change in humidity may affect the fatigue lives.

Time can also cause indirect nuisance effects. For example, test instruments may drift out of calibration, equipment may fail and be subsequently replaced with slightly different equipment, personnel may change, etc.

Randomization

For RCB testing, randomization is important.

Cycled versus continuous tests

Testing can either be continuous (sonics left on all the time) or cycled (sonics switched on and off in a programmed sequence). In general, continuous testing is preferred where the horn and converter can be adequately cooled:

1. Improved timing accuracy. If the test is cycled (e.g., 1 second on, 4 seconds off), then the on-time for each test cycle must be accurately set. If the on-time is incorrect by X%, then the calculated total fatigue life will also be incorrect by X%. In contrast, continuous tests can be timed by synchronous clocks, whose accuracy is very good and whose error is random, rather than cumulative.

2. Better interpretation of results. For cycled tests, the "on" time has an inexact meaning. This problem occurs because the horn requires some finite time to reach full amplitude after the power supply is triggered. Similarly, the horn requires some finite time to return to zero amplitude after the power supply is shut off. (Figure [f]. shows the amplitude envelope for an S-N horn during cycled tests.) Thus it is unclear exactly what time should be used for the on time. (See section _Error! Reference source not found._ for further discussion.)

3. Reliability. Cycled tests require a counter to count the number of test cycles. These counters and the associated circuitry have not been especially reliable.

4. Power supply starting. Cycled tests require continued restarting of the power supply. If the horn has very high stored energy, then the power supply may not restart consistently.

5. Speed. The fatigue test con be completed much more quickly for continuous tests than for cycled tests.

Cycled tests are recommended only when the horn or converter cannot be adequately cooled during continuous running, or when the cycling itself may have some effect on the horn life. For example, it has been suggested that the initial power supply turn-on may may cause a very brief transient that overstresses the horn, thereby reducing its fatigue life. Such a transient would not be present during a continuous test, so that the continuous test would then predict an abnormally long fatigue life for working horns. (Note: such a transient has never been proved.)

Comparative fatigue resistance for different materials

In this section we will look at testing required to compare the fatigue resistance of different materials or the fatigue resistance of the same material under different conditions.

One of the primary goals of fatigue testing is to find materials with superior fatigue resistance. Thus, we must test different materials against each other.

Note: A "material" is defined by its significant attributes, which depend on the particular test being performed. Such attributes could include the vendor, the alloy, the manufacturing method (cast, wrought, extruded, etc.), the final raw form (rod, bar, plate, etc.), the heat treatment, etc. For example, depending on the test requirements, we can say that titanium is a different material than aluminum, 2024 aluminum is a different material than 7075 aluminum, 7075-T3 aluminum is a different material than 7075-T6 aluminum, 7075-T6 aluminum bar is a different material than 7075-T6 aluminum plate, etc. The specification of which attributes are significant (to determine whether two specimens are the "same" or "different") will depend on the context of the testing.

Limitations of single stress-level material comparisons

To find a superior material, different materials can be fatigue tested at the same stress level. The results may permit limited qualitative judgement about which material is superior. The results do not permit quantitative statements such as, "Material A is X% better (or worse) than material B." (See Maennig[1], pp. 630 - 632.)

Dependence on stress level

Let's assume that two materials A and B have the S-N curves given in figure [f].. (See Juvinall[1], p. 264.) However, let's also assume that you have not yet tested each material, so that the shape of the S-N curves is unknown to you. You want to determine which is the best material.

If you run comparative fatigue tests on both materials at any stress above stress level Í1 or below Í2, then material A appears to be the best. However, at stress levels between Í1 and Í2, material B appears to be the best. Thus, comparative testing at a single stress level will not necessarily establish the absolute superiority of one material to another.

This is a problem when two materials have somewhat similar S-N curves. However, if two materials give markedly dissimilar lives (considering the data scatter) when tested at a single stress level, then it is unlikely that the two S-N curves will overlap.  For instance, it two materials are tested at stress level Í1 and material A lasts 100 times longer than material B, then it is unlikely that material A would be inferior if tested at any other stress level.

Now let's assume that two materials C and D have the S-N curves given in figure [f].. Material C is clearly superior to material D at all stress levels. However, the degree of superiority again depends on the stress level at which the comparison is made. If the two materials had been compared at stress level Í1, then material C would be X% better than material B. However, if the two materials had been compared at stress level Í1, then material C would be only Y% better than material B. Thus, comparative testing at a single stress level will not even establish the relative superiority of one material to another.

Notch sensitivity

Even if we establish complete S-N curves for two materials, quantitative comparisons are difficult. This is because we have tested only unnotched specimens. If we had tested notched specimens, then failures would have occurred earlier, but not necessarily by the same amount for each material.

Theoretical stress concentration factor

We will digress slightly to discuss the theoretical stress concentration factor Kt. (See the chapter on "Stress".) For a smooth specimen (no radii, grooves, holes, etc.), the localized stress Í will be the same as the average stress Íav. However, for a specimen with a notch (stress concentrator), the maximum localized stress near the notch will be increased by an amount Kt, where:

Í

7b) Kt = ---

Íav

The value of Kt will depend on the dimensions of the notch relative to the other specimen dimensions, and also to the type of stress (axial, bending, torsional, etc.). Juvinall[1] (pp. 242 - 255) shows figures from which Kt can be determined.

Fatigue notch factor

The degree to which a material's fatigue performance is sensitive to the presence of a notch is called its fatigue notch sensitivity, and the associated factor Kf is called the fatigue notch factor (Juvinall[1], pp. 237, 250):

S

) Kf = --

Sn

where Kf = fatigue notch factor (· 1.0)

S = fatigue strength of unnotched

sample at a specified large

number of cycles

Sn = fatigue strength of notched

sample at the same number

of cycles as S

Kf may also be called the fatigue stress concentration factor or the fatigue strength reduction factor. (Juvinall[1], p. 250.)

The fatigue strength is the stress at which a specified percent of samples should have failed after N cycles. If the percent failure is not explicitly stated, we will assume a value of 50%.

Kf is similar to Kt. However, whereas Kt depends only on the part geometry and method of loading, Kf depends on the geometry and overall size; the type of material, its strength, and previous mechanical and heat treatment; the stress amplitude and cycle life N. (Wirsching, p. 88). Kf can range from 1.0 (for which the notch has no effect on the fatigue strength) to Kt (for which the fatigue strength is reduced by the full amount of the theoretical stress concentration factor).

Cast iron is an example of a material with a low Kf (approximately 1.0). This is because cast iron is permeated with graphite flakes (natural internal stress raisers), so that the addition of small man-made notches has little effect. (Juvinall[1], p. 238, p. 255.)

At the upper limit, Kf approaches Kt when the radius of the stress concentrator is large or for fine-grained, relatively homogeneous materials (Juvinall[1], pp. 250 - 251) and for high strength materials (Shigley, p. 171). Thus, since fatigue resistance is proportional (within limits) to material strength (see Juvinall, pp. 210 - 215), materials with the best fatigue resistance are generally the most notch sensitive.

This effect is shown in figure [f]. (SAE red book, p. 55) for three different aluminum alloys (low frequency tests). While the unnotched materials (dashed lines) have significantly different lives, the notched materials (solid lines) have essentially the same lives above 104 cycles. As the SAE Fatigue Design Handbook notes for aluminum (p. 56) —

The presence of a notch, including the notches associated with welds and riveted and bolted joints, tends to nullify most of the differences among the fatigue strengths of various alloys.

The same effect has been found for titanium. Note that the strengths of the notched samples are much closer together than for the unnotched samples. The Osgood data (p. 442) shows an example of heat treated Ti-6Al-4V that is superior in the unnotched condition but inferior in the notched condition, as compared to annealed Ti-6Al-4V.

Thus, while tests of unnotched S-N horns may indicate that one material is superior to another, working horns (which have stress concentrations) may exhibit much less difference. (Note: for implications related to finite element analysis of working horns, see the chapter on "Stress".)

Conclusion

Based on testing at a single stress level, we can only draw limited conclusions about comparative fatigue performance of two materials. Because the shape of S-N curves may differ significantly between materials and because of differing sensitivities to notches, we definitely cannot say that one material is X% better than another. The most we can say is that one material appears X% percent better than another under a specified set of test conditions. However, we can use single stress testing to find promising materials and weed out materials that are obviously unpromising. Additional tests at other stress levels would be needed to distinguish between materials with similar fatigue performances.

Determining the test stress level

For the following discussion, we will assume that single stress testing is being used to separate potentially good materials from those without hope. We will not try to make quantative comparisons between materials.

Ideally, we would like to perform the S-N material comparisons at "practical" stress levels -- i.e., at stresses that a working horn would actually experience while performing an application. However, since working horns should be designed for 2000 hours of life, such a stress level would require excessively long test times. Therefore, in order to achieve reasonable test times, the S-N horn stress must be higher than normally expected by most applications. The required stress level will be determined by the requirements for acceptable test times.

During a typical S-N test, there are some stress cycles which are not accurately accounted -- e.g., periodic tuning and amplitude checks. If we assume that such unaccounted sonic time might be as much as one minute over the duration of a test, then the accounted (recorded) sonic time must be substantially greater so that the unaccounted time can be neglected by comparison. Thus, we should require a minimum time to failure of the best material (within a comparison group of materials) of at least one sonic hour.

In order to limit testing to reasonable times, we could specify a desirable upper failure limit of 50 sonic hours (or possibly less). Thus, we would like to choose a stress such that the range of test lives of the best material of a comparison group would fall between one and 100 sonic hours. This gives a range of 50:1 between the longest and shortest tests. Such a range should be more than acceptable, since high-stress testing to date indicates an upper range of about 20:1 (for a 95% confidence interval) between longest and shortest lived samples of the same material.

Finding the required stress to produce failures within the required time range must be done empirically. We simply run fatigue tests of the particular material at various amplitudes until we find an amplitude where the range of failure times is acceptable. (See below.) We then run additional tests at this amplitude until we are satisfied that the results are statistically valid.

What happens, however, if we later find a significantly better material? If we run the fatigue tests on this material at the same amplitude as for the previous materials, then the required testing time might be unacceptably long. Of course, we could just increase the amplitude to reduce the test time, but then we would have no direct comparison to the previously evaluated materials. As we have seen in the previous section, however, quantitative comparisons between materials are not really possible anyway, so varying the stress (amplitude) to suit the material is not really a problem.

Once we have established the required amplitude and have accumulated sufficient test data at this amplitude for the basic material, we can then explore speical treatments to improve the basic material (e.g., surface treatments, heat treatments, etc.). The resulting failure times should still be within reason, since such treatments typically improve life by less than 10:1. (See many examples given by Manson.) Thus, the longest fatigue test time would still be less than 500 hours.

Preliminary tests

To separate "good" materials from "bad" materials, we can make the preliminary comparisons based on the materials' median fatigue lives at a single stress level. When the preliminary comparative tests have been completed, we can group the materials (somewhat arbitrarily) as follows:

1. Good materials. All materials whose median lives are greater than 1/5 as long as the "best" material. (The "best" material is that which has the longest median life at the tested stress level, based on the preliminary tests.) These materials will subsequently receive more refined testing to determine the "true" best material.

2. Bad materials. All other materials. For any material whose median life at the tested stress is less than 1/5 as good as the best material, we assume that this material is unlikely to ever be better than the best material at any other stress level.

Thus, the preliminary testing must be sufficient to distinguish between materials whose median lives differ by a factor of 5:1 or greater. If we are using the Smith-Satterthwaite test to compare the two materials, then we must test a sufficient number horns to calculate a reliable estimate of the population standard deviation for each of the tested materials. For these preliminary tests, n = 5 is sufficient. This n will be large enough to distinguish a 5:1 difference between sample median lives. (Note: in fatigue testing, we usually cannot assume the standard deviations of two materials at the test stress are approximately equal. Therefore, the Smith-Satterthwaite test is appropriate. See the chapter on "Statistics".)

The above discussion assumes that each S-N horn of the specified material is randomly selected from the general population. At a minimum, this requires that each horn should be machined from a different batch of the specified material. This will assure that a single defective batch will not completely bias the test data for the specified material. Although such defective material may be unlikely, one manufacturer claimed this problem after Branson tested his material.Error! Reference source not found.

Final tests

Once the "bad" materials have been eliminated from consideration, the remaining "good" materials should be evaluated by developing a complete S-N curve for each, so that the lives can be compared at several stress levels. The method for developing the S-N curve is discussed in section .

For the preliminary tests, we used the median lives to distinguish between "good" and "bad" materials. Comparison of median lives was sufficient because we were only interested in large (5:1) life differences. However, now that we are comparing the remaining good materials, we must use a more subtle approach. For each stress level at which two materials are compared, we must decide which material has the longest life at a specified the lower limit of the tolerance interval (CPl). For example, if CPl is 2.5%, then the best material at a particular stress is that which has the largest N at which 2.5% of its sample will have failed.

Why not just compare materials based on the median life, rather than the life at a specified lower tolerance limit? The reason is that the median life may be a poor indicator of real-world performance. After all, we are not really interested in knowing when 50% of a group of working horns have failed. We want to know then the first few failures will have occurred. Because different materials have different variability in their fatigue lives, the median life is not a good indicator of the early failures.

Consider, for example, the log-normal probability plot of figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND.. If we were to use a median life (50% failure) criterion, then material A would have poorer life than material B. However, at the 2.5% probability of failure, material A has better life because it has lower variability than material B. (See Maennig[1], pp. 630 - 631.) Thus, while material B will live longer on average, material A is actually better for working horns since its initial failures occur later than material B.

Selecting the materials

Preliminary note — for the following discussion, the fatigue-related values are median values from low-frequency testing (unless indicated otherwise).

How can we decide which materials should be tested, other by trial-and-error selection? There are several material properties that can be used to roughly gauge a material's fatigue resistance. These include tensile strength, hardness, and grain size.

Tensile strength

Tensile strength is a convenient gauge of a material's fatigue strength because tensile strength values can be taken directly from handbooks. Alternately, relatively inexpensive tensile tests can be performed on material samples. (Note: tensile strength is also called ultimate tensile strength.)

We first define the fatigue ratio (also called the endurance ratio) as:

Fatigue strength at specified N

8a) Endurance ratio = -------------------------------

Tensile strength

SN

= --

ST

For materials with a true endurance limit (type I materials), the fatigue strength SN is taken as the endurance limit -- i.e., the stress at the knee of the S-N curve for 50% failure. For materials without a true endurance limit (type II materials), the fatigue strength is usually taken at N = 5*108 cycles. (Juvinall[1], p. 213.)

If the endurance ratio and tensile strength are known, then the fatigue strength can easily be calculated.

Steel

Over a limited range of tensile strengths and for a specific material class, the endurance ratio is relatively constant. Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. shows a plot of the fatigue limit versus tensile strength for many carbon and alloy wrought steels. (The data is for small unnotched, laboratory-polished, low-frequency rotating bending tests.) For tensile strengths up to about 200 ksi, the fatigue ratio (labeled Sf/SU on the diagram) ranges approximately from 0.35 to 0.6, with a mean of about 0.5. Above 200 ksi, the fatigue limit tends to level off. Thus, as a general guideline, a steel resonator should have an ultimate tensile strength of at least 200 ksi (1400 MPa).

Since the ultimate strength and hardness of steels is determined by heat treatment, the heat treatment must be correctly specified and implemented. The heat treatment should minimize residual stresses and surface pitting, and should be uniform throughout the material.

Aluminum

As with steel, the fatigue limit of aluminum increases with the tensile strength. Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. shows this behavior for wrought aluminum alloys whose properties are developed by strain hardening, predominately those of the 5000 series (Al-Mg alloys). The fatigue ratio is approximately 0.5 up to about 40 ksi, above which the fatigue limit increases more slowly with increasing tensile strength. The range of fatigue ratios is generally between 0.40 and 0.60.

Heat-treatable wrought aluminum alloys show similar behavior. (See figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND..) For these alloys, the fatigue ratio varies between about 0.26 to 0.31.

For either the strain hardening or heat treatable aluminum alloys, the fatigue limit peaks at about 25 ksi (170 MPa).

Titanium

For unnotched Ti-6Al-4V, the fatigue strength generally increases with tensile strength. Figure [f]. from Bartlo (p. 152) shows this for various annealing temperatures, which affects the grain structure. Bartlo's experiments show that Ti-6Al-4V can have endurance ratios between 0.55 and 0.62 for fine grained alpha-beta structures or structures produced by water quenching or quenching and aging. On the other hand, for heavy sections of Ti-6Al-4V, which contain coarse plate-like or even coarse equiaxed alpha, the endurance ratios may be as low as 0.4 to 0.45. (Bartlo, pp. 152 - 153.) Thus, the endurance ratio is significantly affected by the heat treatment and section size, which affect grain size and structure. The actual fatigue limits varied between 450 MPa (65 ksi) and 100 MPa (690 ksi) at 107 cycles, depending on heat treatment.

RMI notes (p. 22) —

As with other metals, microstructure and section size may have an effect on the fatigue properties of RMI 6Al-4V. Heavy sections and coarse microstructures may develop somewhat lower fatigue properties without any reduction in the ultimate tensile strength.

Since titanium is anisotropic, the fatigue results are also affected by the test direction. Table [t] shows data from Bowen (7) on Ti-6Al-4V. The fatigue test samples were machined from 235 mm wide x 57 mm thick bar, which had been forged and then annealed for 2.5 hrs at 700 oC. Testing was run on rotating cantilevered specimens at 100 Hz. Fatigue strengths were measured at 107 cycles.

TABLE [t]

FATIGUE OF TI-6AL-4V BAR

TESTED IN THREE DIRECTIONS

 

Test Tensile Fatigue Fatigue

direction Modulus strength strength Ratio (%)

L 113.8 910 495 55

LT 129.0 985 430 43

ST 113.8 980 565 58

Notes:

1. L = longitudinal

LT = transverse

ST = short transverse

2. Modulus units: GPa

Strength units: MPa

Table [t] shows that the fatigue strength and fatigue ratio can be substantially different for different test directions in the same bar. Also, note that while the long transverse and short transverse directions have nearly the same tensile strengths, the short transverse direction has 31% higher fatigue strength. Bowen attributes these differences to the directionality of the material.

Note — Lucas tested Ti-6Al-4V and found only a very small correlation between fatigue strength and tensile strength. For his samples, a maximum 17% difference in tensile strength (due to differences in forging temperatures, amount of deformation, cooling rates, and heat treatments) gave a fatigue strength difference of greater than a 2:1. (See his figure 15, p. 2093.) Like Bowen, however, Lucas (p. 2086) did find strong directionality for the titanium, with approximately 25% greater fatigue strength in the longitudinal direction.

Caveats

The relation between increased fatigue strength with increased tensile strength must be viewed with some caution. First, there is considerable scatter in the data. For example, if we have a steel whose tensile strength is 200 ksi and if we have no other information about the steel except the data of figure f, then its fatigue limit may range anywhere from about 70 ksi to 120 ksi. Similarly, we may find that one steel with a tensile strength of 160 ksi may have a better fatigue life than another with a tensile strength of 200 ksi.

Second, the relation between tensile strength and fatigue strength is strongest for unnotched specimens. If the specimen is notched, then there may be little relation between tensile strength and fatigue strength. This is shown in the bottom data of figure f for notched Ti-6Al-4V (Kt = 3.5). Adair reports similar results for Beta III titanium (Ti-11.5Mo-6Zr-4.5Sn). Although the tensile strengths varied from 1200 MPa to 1390 MPa due to different processing techniques and the unnotched fatigue strengths correspondingly varied from 630 MPa to 860 MPa, the notched (Kt = 3.1) fatigue strength varied only between 170 MPa and 210 and showed no correlation to the tensile strengths. (p. 1809) (Note: Adair's tests were conducted for a stress ratio of 0.1.) Figure [f]. from Adair shows typical unnotched and notched S-N curves.

Figure f shows similar results for three aluminums, for which the dashed lines show fatigue strength differences for unnotched specimens, but the solid lines show no differences for notched specimens (Kt = 2.76) in the high-cycle region.

(Note: the above data is for tests with Kt between 2.76 and 3.5. These would be rather severe stress concentrations for well-designed horns. For example, slot ends in bar horns have Kt between 1.5 and 2.0, depending on the horn width. Thus, it is possible that the fatigue strength may correlate more closely with tensile strength for these lower Kt. Also note that Willertz () found no frequency effect when testing notched (Kt = 2.41) Ti-6Al-4V at 88 Hz and 20 kHz, or 17-4 PH at 100 Hz and 20 kHz. Thus, the above discussed low-frequency results may also apply to high frequency testing.)

Thus, tensile strength can only serve as a very general guideline to fatigue strength.

Hardness

Hardness is not used as a criterion for choosing candidate test materials. However, because hardness varies approximately linearly with tensile strength, it can be used as a quick check of fatigue properties. For example, hardness tests could be used to spot an inferior batch of material or to determine if a failed horn has been made from inferior material. If a hardness tester is available, this is much quicker than performing a tensile strength test.

Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. shows the approximate relation between hardness measured on the Shore, Brinell, and Rockwell C hardness scales. Also shown in this figure is the tensile strength for carbon and alloy steels. .F.figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND. from Lipp shows the strength-hardness relation for wrought aluminum alloys.

Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. shows a direct comparison for endurance limit versus hardness for four alloy steels. Note that there is an optimum hardness for each material. If the hardness exceeds this value, then the endurance limit decreases.

We are using hardness as an indicator of tensile strength. Thus, the same caveats that apply to predicting fatigue strength from tensile strength (above) also apply to using hardness as a predictor.

Grain size

In general, both a material's strength and its fatigue resistance increase as the grain size decreases for unnotched specimens. (See Manson, pp. 265 - 267, for possible explanations.) The magnitude of the grain size effect depends on the material and its basic crystalline structure, the size of the specimen, the stress concentration factor, and the orientation of the grain with respect to the tensile stress direction. (See the above section "Tensile Strength" for some discussion of titanium grain size. Bartlo discusses several heat treatments that affect the fatigue of Ti-6Al-4V, and gives photomicrographs on p. 151.)

For notched tests (Kt = 3.5) of Ti-6Al-4V, Bartlo (p. 153) found that "No significant trends were observed in comparing the notched fatigue data and microstructures." This is not surprising, since he also found no significant relation between tensile strength and notched fatigue data. Similarly, Lucas (axial load tests of Ti-6Al-4V, 45 ksi mean stress) found a very distinct relation between unnotched fatigue strength and alpha grain size, whereas there was only a small relation for notched (Kt = 1.8) specimens. He found no relation in either case for ß grain size.

As with hardness, grain size is not used as a criterion for choosing candidate test materials. However, it can be used as a very rough test to see if a material specimen is grossly defective. To examine the grain, aluminum can be etched with a solution called Kellers etch. Titanium can be etched with a solution of 1% hydrofluoric acid (HF) and 12% nitric acid (HNO3), with water as the remainder. A more active etchant can be achieved by reducing the HNO3 to as little as 3%. (42) These titanium etchants are known as Knoll's etch. Other percentages of HF and HNO3 have also been used.

Loss

Tests at Branson have consistently shown that materials with low loss during ultrasonic vibration have better fatigue properties. This is true for both aluminum and titanium, and may be related to grain size. (See Puskar for a discussion of grain size and loss in steel at 23 kHz.)

Table [t] shows measurements for 7-4 titanium rod from Timet and Mainland China. If we subtract approximately 8 watts from the stack loss to account for the converter loss, then the ratio of Mainland China loss to Timet loss is 14/44 = 0.3. We see that Mainland China life was longer than than the Timet life.

(Note: the long life of the Chineese titanium is not unusual for this material. Two other horns of this material were tested at higher amplitudes. One (81-T1) failed after 0.27 sonic hours at 305 microns. The second (81-T2) was tested for 0.74 sonic hours at 305 microns without failure. Its amplitude was then increased to 380 microns, where it failed after 1 minute. These S-N characteristics are much superior to those of Timet.)

(Note: also see the chapter on "Inelastic Material Properties" for loss measurements of titanium bar horns machined in different directions with respect to the grain. These loss directions may indicate preferrential directions for fatigue._Reference Bowden work on ti fatigue vs grain direction?_)

TABLE [t]

LOSS AND LIFE FOR 7-4 TITANIUM ROD S-N HORNS

AT 20 KHZ

Number Stack loss Life

Vendor of tests Ampl ( ) (watts) (sonic hrs)

Timet 7 203 52

Mainland 1 203 22 43.5

China

Notes:

1. Stack loss includes the 400 series converter and S-N horn.

Loss for the 10 Timet horns was estimated from measurements

of S-N horns HRD-439 (11/30/80) and HRD-299 (8/2/84). See

the chapter on "Inelastic Material Properties".

2. The Mainland China horn is designated 80-T11. Stack loss

was directly measured for this S-N horn. An additional

horn of the same design (80-T12) had a loss of 16 watts at

203 microns amd a loss of 21 watts at 229 microns. No life

data is available for this horn.

3. See __ for data on failures of the individual Timet horns.

Because aluminum has very low loss under any conditions, the power meters at Branson are not sufficiently sensitive to distinguish among the different aluminums. (See the chapter on "Inelastic Material Properties".) However, aluminums have been qualitatively judged by feeling the S-N horn temperature during testing. The temperature rise of the most fatigue resistant aluminums is greater than for those with poor fatigue resistance.

Caveats

From the above discussion, we should like to find a material with high tensile strength, high hardenss, and small grain size. (These properties are all correlated.) However, one problem with materials with these properties is that they are notch sensitive -- i.e., the presence of a notch significantly reduces their fatigue resistance compared to materials without these properties. Thus, since all practical horns have notches (stress concentrators), the materials with high tensile strength, high hardenss, and small grain size may not be as superior as they first appear from tests of unnotched S-N horns. (See the section .)

Comparative fatigue resistance for different treatments

In this section we will look at the testing needed to compare different treatments of the same material. Such treatments could include heat treatment (e.g., stress relieving), surface treatments (e.g., shot peening, nitriding, etc.), or others.

Comparisons between two treatments is much easier than comparisons between two materials. This is because the treatment effect should extend throughout the entire S-N range of the material. For example, if a treatment improves a material's life when tested at a particular stress level, then we should also expect to see improvement at other stress levels, although the degree of improvement may not be the same at all stress levels. We would not expect to see improved live at one stress level and decreased life at another stress level. Thus, testing at a single stress level is sufficient.

Determining the test stress level

We can presume that this material is one of the above "good" materials. Therefore, we should already have at least five data points for the material in the untreated condition. This gives a preliminary estimate of the median life and standard deviation at the test stress. This information is useful in establishing the initial test parameters, but will not otherwise be included in the data analysis.

We need to determine a stress level that will give acceptable test times for both the untreated and treated S-N horns. We will assume that the treatment is supposed to increase the material's fatigue life. Then, we would like to choose a stress level such that the untreated material has all failures in the one hour (minimum) to approximately 25 hour range. If we assume that the treatment will increase the fatigue life by 5:1 (at most), then the treated material will have failures in the range of 25 hours to 125 hours. Thus, the failure times for both the untreated and treated materials will range from one hour to 125 hours, which is quite acceptable.

Of course, the trick is to adjust the stress (amplitude) to achieve this failure range. The preliminary material evaluation data will indicate whether the stress needs to be raised or lowered. However, it will not indicate the degree to which the stress must be changed. (Of course, if you have developed a complete S-N curve for this material, then the desired stress level can be found directly from the curve.)

One way around this problem is to simply use the same stress level as for the preliminary material evaluation, assuming that the data meets the minimum one hour failure criterion. This may give longer than desired test times for the treated material, but would compensate by eliminating additional testing required to find a new stress level. This also provides some corroboration that the new data is valid, since you can compare the median life of the previously taken data to the median life of the new batch of untreated material. These values should agree, within reason. I recommend this approach, where feasible.

Estimated number of tests

The number of tests will depend on the variability (standard deviation XSD) of the data for a given treatment condition, the minimum acceptable life difference between treatments, and the required confidence level. To estimate the required number of tests, we will assume that XSD for both the treated and untreated horns is the same as XSD calculated from the initial material comparison tests of this material. We will also assume a confidence level of 95% -- i.e., we want to be be 95% sure that any differences between the treated and untreated horns is not due only to chance. Whether we use a nondirectional or directional confidence level will depend on the information available.

Calculation of the required number of horns requires some reverse engineering of the Student's t table (chapter on "Statisitcs", appendix A). Since we are assuming that the standard deviations are equal and that the population data is log-normally distributed, we can use the standard t-test for means. (See the chapter on "Statistics", equations óe and óeÂ.) With XSD,1 = XSD,1 = XSD amd n1 = n2 = n, the t statistic becomes:

°*Xmean,2 - *Xmean,1° ´n

8a) t = ------------------------

*XSD

and the degrees of freedom df reduces to:

8b) df = 2 (n - 1)

Note — The superscript * in the above equation means that the raw data has been transformed to produce a normal distribution in order to use the t statistic. In this case, since we have assumed that the population of fatigue lives is log-normally distributed, the transformation is a log transformation. Thus, the Xmean and XSD are calculated from the logs of the raw failure times. The result is that the Xmean in equation e8a are actually the logs of the median failure times. For example, if the median failure time for material condition 1 is 4.4 hours, then Xmean,1 for equation e8a is:

8bb) *Xmean,1 = log(Median_life)

= log(4.4)

= 0.64

Thus, we can rewrite equation e8a as:

°log(Xmedian,2) - log(Xmedian,1)° ´n

8c) t = ------------------------------------

*XSD

Ö Xmedian,2 Ì

log ° --------- ° ´n

Û Xmedian,1 ì

= ----------------------

*XSD

Now, if *XSD is estimated from the preliminary test data and the ratio of median lives is specified, then we need only find an n such that the t value from equation e8c equals the t value from the Student t table corresponding to df of equation óeÂ8b and the specified confidence level. The confidence level should be specified as directional if we have good reason to believe that *Xmean,2 will be significantly different (either larger or smaller) than *Xmean,1. For example, if we are testing the effect of shot peening, then we know from extensive literature that shot peening generally increases the fatigue life. In this case we would use the directional test. If we have no strong indication of the test results, then we should use the nondirectional test. In either case, however, the choice of directional or nondirectional test should be made before the actual testing begins.

Example

Suppose we want to detect the effect of a treatment whenever it improves the median fatigue life by 30% or more -- i.e., Xmedian,2 · 1.3 Xmedian,1. Then the factor of equation óeÂ8c that involves the median lives is:

Ö Xmedian,1 Ì

8d) log ° --------- ° = log (1.3)

Û Xmedian,2 ì

= 0.114

Let's assume that the preliminary test data showed an XSD of 0.295. Then equation e8c becomes:

0.114 ´n

8e) t = --------

0.295

= 0.386 ´n

I will use a directional test, since we may presume that prior research has shown that the treatment changes the fatigue life. If this were not the case, then we would probably not be investigating this treatment.

From equation e8e we see that the calculated t value increases as n increases. However, from the Student t table, we see that the table value decreases as n increases (where n is related to df by equation óeÂ8b). Thus, we can always find an n where the calculated t value is approximately equal to the Student t table value.

For this problem with a 95% confidence level (directional), the correct value is n = 19, for which equation e8e gives a value of 1.684 while the table gives a value of 1.689 for df = 36. (Since there is no table value for df = 36, the 1.689 value was interpolated and is nearly the same value as df = 40.) Note that this is only an estimate for the number of tests, since the *XSD's for the nontreated and treated groups may not be the same as for the preliminary tests.

The number of tests can be decreased if you are willing to make some compromises. First, you might increase the acceptable ratio between the group median lives. The following table shows the trend:

TABLE [t]

n REQUIRED FOR *XSD = 0.295

AND 95% CONFIDENCE LEVEL (DIRECTIONAL)

Median Eqn e Student t

ratio t equation n value df table value

1.3 0.386 ´n 19 1.684 36 1.689

1.4 0.495 ´n 12 1.716 22 1.717

1.5 0.600 ´n 9 1.800 16 1.746

1.6 0.691 ´n 7 1.830 12 1.782

1.7 0.781 ´n 6 1.913 10 1.812

1.8 0.865 ´n 5 1.935 8 1.860

Since we cannot guarantee that the equation value and the Student t table value will be exactly equal for a given n, we have chosen n so that (in general) the equation value is somewhat larger than the Student t table value. This makes n err slightly on the high side, based on the assumed *XSD. Notice that n decreases rapidly for a small increase in the median ratio (Xmedian,2 /Xmedian,1).

The table stops an n = 5, since this is about the minimum value required to evaluate XSD of the treated and untreated groups. Thus, any median ratio greater than 1.8 will still require five pairs of S-N horns.

The above table is only valid for XSD = 0.295.

You can also decrease n by decreasing the confidence level. For example, if you decrease the confidence interval from 95% to 90%, then for a median ratio of 1.3, n decreases from 19 to 12.

Remember that the above analysis only gives an estimate of n before the test is run. This estimate is based on the assumption that the XSD for the treated and untreated samples are both equal to the assumed value, and that the populations are log-normally distributed. After the testing has begun, n may have to be changed as indicated by the test results. (See the section .)

If several different treated-untreated comparisons are run at the same stress level, then you will begin to accumulate a relatively large data sample for the untreated horns. If this collection becomes sufficiently large, then you can assume that its statistics are the same as the population. At this point, any further comparisons do not require additional fatigue testing of untreated S-N horns.

developing the S-N curve

In this section we will discuss the test methods needed to develop a complete S-N curve.

Method

S-N curve from life distribution

The usual method of establishing the S-N curve is to test several groups of samples, each at different stress levels. For each group of samples at a specified stress, the life distribution can be determined by plotting the failure times on the appropriate probability paper. From this plot you can determine the upper and lower tolerance limits as well as other confidence values. (See the chapter on "Statistics", section .)

Figure [f]. show such curves developed by Sinclair for low frequency testing of 7075-T6 aluminum at six different stresses. Each of the curves represent the median time to failure at the specified failure probability on the ordinate. For example, let's look at the curve for the 40000 lbf/in2 tests. We see that the most likely (median) time to failure of the first 10% of the samples is 100000 cycles -- i.e., there is a 50% probability that 10% of the samples will have failed before 100000 cycles and a 50% probability that 10% of the samples will have failed after 100000 cycles.

Let's assume that we have specified lower and upper confidence levels of 2.5% and 97.5% (i.e., 2.5% and 97.5% probability of failure). If we are willing to accept a Then we can take the data Then, for example, if we look at the curve for the 40000 lbf/in2 tests, we see that the 2.5% probability of failure occurs at approximately 70000 cycles and the 97.5% probability of failure occurs at approximately 700000 cycles. The median life (50% probability of failure) occurs at approximately 210000 cycles.

At a stress of 40000 lbf/in2, the above values (70000, 210000, and 700000 cycles) can be plotted on log-log graph paper.

S-N curve from strength distribution

Maennig[1] (pp. 633 - 638) describes an alternate method (the staircase method) which is supposed to reduce the number of samples required. In this method, the tolerance limits for the strength (rather than the life) are determined. Testing is conducted at two specially chosen stress levels whose life distributions overlap. (See figure [f]..) Maennig describes how to choose these stress levels. The testing is stopped after a fixed number of cycles N. If the stresses have been correctly chosen, then the most highly stressed group will have approximately 90% failures, while the lower stressed group will have approximately 10% failures.

The cumulative probability of failure CP is calculated for each group, using equation e from the chapter on statistics. The two probabilities are then plotted on probability paper as a function of the stress level. Using a straight line to connect the two plotted points, the probability of failure at other stresses can be determined -- at a CP of 2.5%, 50%, and 97.5%. (See figure figure [f]..) Thus, we have theoretically determined a strength distribution at a fixed number of cycles.

The same type of test is repeated for other pairs of stress levels, for which each test will be stopped at a different number of test cycles. The results of all of these tests can be plotted to establish the S-N diagram. (See .f.figure [f]..)

Unfortunately, this approach has a major problem. As described above, each probability graph for a fixed number of cycles N is determined by plotting only two points, corresponding to the probability of failure (CP) of the two stress levels. In order to use a straight line through these points to estimate other CP values, we must assume a distribution for the strength -- e.g., a normal distribution, log-normal distribution, etc. If an incorrect strength distribution is assumed, then the estimated CP values may have significant error. (See the discussion of strength distribution in the section .)

Of course, we could reduce this problem by testing at intermediate stress levels to establish additional CP points for plotting. For example, we could test at stresses that would give approximately 30%, 50% and 70% failures. Then we would have five points (including the previously determined 10% and 90% failures) to plot on probability paper. This would give a rough indication of the probability distribution. However, this increases the required number of samples and negates the advantage of the staircase method. Also, if testing occurs at a life N where the slope of the S-N is very shallow and the failure scatter band is relatively narrow, then the stress levels will be very close together in order for their lives to overlap.  (See .f.figure [f]..) Therefore the stress must be very carefully controlled.

This method is useful in determining if a material has a knee in the S-N curve. See the section .

Number of tests

When horns are fatigue tested, there will always be some variability in the fatigue lives. This will be true even if all test conditions are perfectly controlled and an infinite number of tests are run. Under these perfect conditions, the tolerance interval will be XTI_. Now, if the number of tests n is reduced, the tolerance interval will grow to XTIn. We define the ratio of XTIn to XTI_ as the relative tolerance interval:

XTIn

8x) Relative tolerance interval = ----

XTI_

A graph of the relative tolerance interval is shown in figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. in the chapter on statistics. For a 95% confidence level, the break point is about 10 tests, for which the relative tolerance interval is 1.72. Thus, testing 10 samples gives a confidence interval that is 72% larger than if an infinite number of samples had been tested. If the number of tests is doubled to 20 (still at the 95% confidence level), then the relative tolerance level decreases to 1.41, an improvement of only 18%. Above 20 tests, the tolerance interval decreases very slowly.

If a certain tolerance interval is specified, then the number of tests will increase as the required degree of confidence increases. For example, whereas 10 tests at the 95% confidence level were required to achieve a 1.72 relative tolerance interval, 17 tests are required to achieve the same tolerance interval at the 99% confidence level.

Thus, a sample size between 10 and 20 seems reasonable for most testing. This agrees reasonably with the suggestions of Maennig[1] (p. 624). For our testing and analysis, we will use a confidence level of 95% and a lowest acceptable failure rate of 2.5%.

An improved statistical approach along with supporting data can be found in Appendix [].

Analyzing the data

1. Are the lives of two groups statistically different from each other?

2. Using log probability plots to determine average life and standard deviation.

Statistical analysis of fatigue data

Distinguishing between Type I and Type II S-N curves

Type I (figure f) and type II (figure f) materials both have a knee, where the slope of the S-N curve decreases substantially. From a plot of the failure points, however, it may not be evident whether the material is truly type I, for which the S-N slope goes to zero, or whether the material is type II, for which the slope beyond the knee is shallow. The distinction is very important in predicting the fatigue behavior at long lives. All other things being equal, we would certainly prefer a type I material.

Maennig[1] (pp. 638 - 642) gives a method for distinguishing between the two types of materials. Figure [f]. shows a log-log failure plot where the test lives run well beyond the knee. We assume that all testing at all stress levels will have stopped at NF cycles.

We start the analysis from NF. At this point there will be overlapping life distributions from several stress levels. Each distribution will have different numbers of failures. For each stress level we can calculate a cumulative probability (CP) value corresponding to its number of failures. (See equation óeÂ.) Then each of these CP values and its associated stress is plotted on probability paper. (Try normal probability paper first to see if the points fall reasonably on a straight line. If they do not, then try log-normal probability paper. Some scatter about the line should be expected.) Once a straight line has been established on the appropriate probability paper, then you can read off the stress values corresponding to CP values of 1%, 50%, and 90%. These stress values are then plotted on the S-N diagram at NF.

Then you move to the left on the S-N diagram until you find then next failure poing Ni among the overlapping life distributions. The above plotting procedure is repeated at Ni. This process continues, moving left from failure to failure. After a sufficient number of points have been plotted on the S-N diagram, you can see whether these points lie generally parallel to the N axis. If so, then the material has a true endurance limit. If the points slope somewhat upward as you move toward the left, then the material does not have a true endurance limit.

If the material has a endurance limit, then we would like to find the number of cycles NE at which this occurs. This is done by continuing to move to the left on the S-N diagram while plotting points as described above. Eventually you will come to a point where the 99% stress value shifts upward. (Maennig[1] suggests a 2% shift.) This upward shift is an indication that you are moving out of the range of transition into the range of fatigue life. Hence, the number of cycles at which this upward shift occurs is NE.

S-N testing

Failure distribution in S-N testing

At Branson, the most complete set of S-N data for a single stress level has been taken for S-N horns made of Martin Marietta 7075-T6 QQA225 running at 152 microns (139 MPa). The ranked data is shown in figure [f]. (Note: the cumulative probability CP has been calculated from equation [2a] in the chapter on Statistics.)

When the life of each sample (column 4 of table [t]) is plotted on normal-probability paper (figure [f]), it is obvious that the points do not fall on a straight line. Thus, the lives are not normally distributed.

However, if the logarithm of each life (column 5 of table [t]) is plotted on normal-probability paper (figure [f]), then the points lie reasonably close to a straight line. The lives may therefore be log-normally distributed.

One odd aspect of this data should be noted: it seems that there is a distinct relation between when the testing was performed and the associated lives. This can be seen by looking at the "Test #" in the table, which provides two types of information. The number preceeding the hyphen gives the test year (1981). The number after the hyphen gives the order of the test (e.g., test 81-60 was performed after 81-59, etc.). From the table you can see that the early tests (81-9 through 81-25) have relatively long lives, ranging from 4.39 hours to 14.88 hours. The later tests (81-43 through 81-67) have appreciably shorter lives, ranging from 1.21 hours to 4.04 hours. (Note: between tests 81-25 and 81-43, other unrelated S-N tests were performed.)

This life pattern does not seem to be associated with the particular bar from which the S-N horn was machined. For instance, horns made from bar #2 have a median life of 6.97 hours when tested as part of the earlier group, whereas this same bar material has a median life of only 1.40 hours when tested as part of the later group.  Thus, there seems to be some time-related difference in lives between the early tests and the later tests. For instance, there might have been some drift in the calibration of the A-450 amplitude equipment, for which the later tests were actually run at a higher amplitude than the early tests, although the indicated amplitude was the same for each. As we will see later, an amplitude difference of only []% would account for the difference in lives.

If the data in table [t] is really time related, then it may call into question our conclusion that the life data is log-normally distributed. None-the-less, all further analysis of fatigue data will be based on the assumption of a log-normal failure distribution.

TABLE [t]

S-N TESTING OF MARTIN MARIETTA 7075-T6 QQA225 ALUMINUM

AT 152 MICRONS (135 MPA)

Life

Test # Rank CP (%) Bar Heat Variac (hrs) log10(life)

81-60 1 1.8 2 1 65 1.21 0.08

81-57 2 4.6 2 1 60 1.26 0.10

81-61 3 7.3 2 1 64 1.38 0.14

81-84 (a) 4 10.1 2 1 88 1.52 0.18

81-47 5 12.8 3 1 50 1.76 0.25

81-48 6 15.6 3 1 54 1.78 0.25

81-59 7 18.3 2 1 51 1.83 0.26

81-46 8 21.1 3 1 63 1.97 0.29

81-44 9 23.9 6 3 42 2.03 0.31

81-65 10 26.6 3 1 56 2.44 0.39

81-43 11 29.4 6 3 100 2.46 0.39

81-63 12 32.1 5 2 96 2.51 0.40

81-64 13 34.9 5 2 96 2.52 0.40

81-6 14 37.6 3 1 -- 2.66 0.42

81-45 15 40.4 6 3 93 2.69 0.43

81-83 (a) 16 43.1 2 1 80 2.70 0.43

81-51 17 45.9 4 2 93 2.82 0.45

81-56 18 48.6 1 1 93 2.84 0.45

81-4 19 51.4 3 1 -- 2.93 0.47

81-5 20 54.1 3 1 -- 3.12 0.49

81-66 21 56.9 7 3 96 3.21 0.51

81-52 22 59.6 4 2 92 3.26 0.51

81-50 23 62.4 4 2 90 3.42 0.53

81-67 24 65.1 7 3 93 3.86 0.59

81-62 25 67.9 7 3 92 4.04 0.61

81-23 26 70.6 2 1 100 4.39 0.64

81-14 27 73.4 1 1 93 6.67 0.82

81-24 28 76.1 2 1 100 7.05 0.85

81-15 29 78.9 1 1 98 7.30 0.86

81-19 30 81.7 1 1 100 7.54 0.88

81-9 31 84.4 1 1 100 7.56 0.88

81-18 32 87.2 1 1 92 8.87 0.95

81-13 33 89.9 1 1 91 9.18 0.96

81-25 34 92.7 2 1 100 11.07 1.04

81-17 35 95.4 1 1 98 14.09 1.15

81-16 36 98.2 1 1 85 14.88 1.17

Notes:

1. Tests were conducted in 1981.

2. Power supply: 184V

Duty cycle: 1 second on, 4 seconds off.

Cooling: fan directed at S-N horn.

3. (a) indicates horn with 17.3 mm square front half-wave section. See figure f.

TABLE [t]

S-N TESTING OF MARTIN MARIETTA 7075-T6 QQA225 ALUMINUM

AT 152 MICRONS (135 MPA)

Life

Test # Rank CP (%) Bar Heat Variac (hrs) log10(life)

81-60 2 92 1 1.6 1.21 0.083

81-57 2 93 2 4.0 1.26 0.100

81-61 2 92 3 6.5 1.38 0.140

81-47 3 100 4 8.9 1.76 0.246

81-48 3 100 5 11.3 1.78 0.250

81-59 2 93 6 13.7 1.83 0.262

81-46 3 100 7 16.1 1.97 0.294

81-44 6 100 8 18.5 2.03 0.307

81-65 3 90 9 21.0 2.44 0.387

81-43 6 100 10 23.4 2.46 0.391

81-63 5 96 11 25.8 2.51 0.400

81-64 5 96 12 28.2 2.52 0.401

81-45 6 93 13 30.6 2.69 0.430

81-51 4 93 14 33.1 2.82 0.450

81-56 1 93 15 35.5 2.84 0.453

81-66 7 96 16 37.9 3.21 0.507

81-52 4 92 17 40.3 3.26 0.513

81-50 4 90 18 42.7 3.42 0.534

81-67 7 93 19 45.2 3.86 0.586

81-62 7 92 20 47.6 4.04 0.606

81-23 2 100 21 50.0 4.39 0.642

81-71 3 65 22 52.4 4.49 0.652

81-20 2 60 23 54.8 4.50 0.653

81-72 3 64 24 57.3 4.59 0.662

81-10 1 50 25 59.7 4.60 0.663

81-21 2 54 26 62.1 4.89 0.689

81-8 1 51 27 64.5 5.20 0.716

81-11 1 63 28 66.9 6.29 0.799

81-14 1 93 29 69.4 6.67 0.824

81-24 2 100 30 71.8 7.05 0.858

81-7 1 42 31 74.2 7.28 0.862

81-15 1 98 32 76.6 7.30 0.863

81-19 1 100 33 79.0 7.54 0.877

81-9 1 100 34 81.5 7.56 0.879

81-22 2 56 35 83.9 7.68 0.885

81-18 1 92 36 86.3 8.87 0.948

81-13 1 91 37 88.7 9.18 0.963

81-12 1 37 38 91.1 9.25 0.966

81-25 2 100 39 93.5 11.07 1.044

81-17 1 98 40 96.0 14.09 1.149

81-16 1 85 41 98.4 14.88 1.173

Notes:

1) Power supply: 184V

2) Duty cycle: 1 second on, 4 seconds off.

3) Cooling: fan directed at S-N horn.

Estimating the life of working horns

In this section we will describe how to use the S-N diagram to estimate horn life.

The purpose of the S-N diagram is to permit the horn designer to estimate the horn life. To do this the designer must know three things: 1) the horn amplitude, 2) the relation between horn stress and horn amplitude, and 3) the effect of notches on the material. In general, none of these will be known exactly, which will increase the uncertainty of a life estimate.

For all of the following analysis, we assume that the horn will be used for a dedicated application, for which a particular booster is specified.

Determining the variability in horn stress

Let's suppose you have a material whose S-N diagram is shown in figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND.. We assume that this diagram has been determined from fatigue tests of unnotched S-N horns, for which it will give correct life predictions. To use this diagram with a working horn, we must account for a number of variable factors that increase the uncertainty of life prediction.

Stress estimation

For a working horn, the stress that causes failure (ÍSN) is given by:

) ÍSN = Íh KS Kr KF

where ÍSN = stress that causes failure in

an S-N horn

Íh = highest estimated stress in

working horn

KS = size factor

Kr = stress reduction factor to

account for notch sensitivity

KF = failure theory factor to account

for multiaxial stress in working

horn

Estimated stress

If the horn is very simple (unshaped cylindrical, exponential, catenoidal, etc.), then Íh can be determined from equations. If the horn is relatively thin so that only the surface stresses are important, then Íh can be determined experimentally by using laser halography, thermography, strain gages, etc. (See the chapter on "Stress".) Of course, this requires a prototype horn. Íh can also be determined by finite element analysis (FEM) or similar methods. Regardless of the method, each has some random error.

We will assume that Íh is estimated by the FEM method is used, since this approach is reasonably common for critical application horns and does not require prototype horns. (Íh should be the Hencky von Mises equivalent stress.) There are many factors that can affect the computed FEM value of Íh. The FEM model must adequately duplicate the actual horn, including the effects of the stud and joint. It must account for static or residual stress, such as might be found at tip threads or from abusive machining operations. If constraints are imposed, they must not limit any critical vibrational modes which may cause additional horn stress.

Even if the FEM model is perfect, the stress results will have error if the incorrect material properties are used. This is especially true for an anisotropic material like titanium. Also, the material properties of titanium can vary considerably from heat to heat. Thus, for a given amplitude, different horns from different heats will have different stresses, which will not be predicted by the FEM analysis. (See the chapter on "Elastic Material Properties" for further information.)

With these considerations, the coefficient of variation of the FEM stress (Íh,CV) is unlikely to be better than 0.02 (about ±5% error). This assumes that the horn amplitude (from which the stress is calculated) is known exactly. (Note that for FEM analysis, Íh is actually the stress per unit of horn amplitude.)

(Note: in converting from coefficient of variation to percent error, I am assuming that 99% of estimates will fall within the specified error (tolerance), so that:

) Percent error = 2.6 (Coefficient of variation) 100

or alternately:

Percent error

) Coefficient of variation = -------------

2.6 * 100

where the 2.6 value is taken from standard normal tables, corresponding to 99% two-sided probability. See the chapter on "Statistics" for a definition of the coefficient of variation.)

Size effects

There may be two material size effects. (See Juvinall[1], pp. 231 - 233; Frost, pp. 54 - ; Weibull, pp. 112 - 113; SAE Fatigue Design Handbook, pp. 28 - 29). First, there is a stock size effect, whereby larger stock typically has less strength than smaller stock, even when both are processed in the same manner. As part of this effect, larger stock may have different strength between its core material and its outer material, whereas smaller stock tends to be more uniform throughout. This is especially true for quenched materials, where the outer material cools rapidly and thus has finer grain size than the core material. These size effects will depend on the material and its processing.

The second size effect relates to the volume of stressed material. Even if two specimens have identical material on a microscopic scale, the larger specimem may fail earlier because there is more stressed material and, therefore, more potential defects from which a crack may nucleate. Alternately, if two specimens are the same size but one is stressed over a greater volume of material, then its probability of failure may be higher. This may partially explain why axially loaded fatigue specimens, which are uniformly stressed across the entire cross-section, typically have lower strengths than identical bending specimens, which are highly stressed only in the surface material. The SAE Fatigue Design Handbook cites data for various notched and unnotched steels that shows that the fatigue strength decreases by about 2.5% for each doubling of volume of highly stressed material (p. 28, equation 3-17).

Let's look at some typical problems. Assume that we have fatigue data taken from 20 kHz S-N horns made from 76 mm diameter rod stock. The front diameter is machined to 17.3 mm diameter. Thus, the S-N data actually represents the fatigue strength of a 17.3 mm diameter core section from a 76 mm diameter rod.

Now let's assume that you want to determine the fatigue life of a 125 mm diameter unshaped cylindrical horn whose stresses are known. The fatigue crack will probably start somewhere along the stud axis -- i.e., at the core material, as with the S-N horn. Because the spool horn is made from 125 mm diameter stock, it will likely have lower fatigue strength than the S-N horn that was made from 75 mm stock. However, the exact decrease in fatigue strength is no known, so the exact decrease in life cannot be estimated.

What happens if we use 25.4 mm diameter rod stock to make a 17.3 mm diameter unshaped cylindrical horn? Because it is made from smaller stock than the S-N horn, it will likely have better fatigue strength, even though the failure diameters (17.3 mm) are the same for both.

Bartlo has investigated the size effect for RMI Ti-6Al-4V bar stock. The original bar diameter was 88.9 mm. A portion of this stock was forged and rolled to reduce its diameter to 15.9 mm. Both the original bar and the reduced bar were then annealed for two hours at 1350 oF and then air cooled. (Note: this is not the best heat treatment for fatigue resistance. See Bartlo.) The final S-N specimens machined from the 88.9 mm bar had the same dimensions as the specimens from the 15.9 mm bar.

Table [t] shows the results for unnotched and notched (Kt = 3.5) fatigue tests. (Note: the data from table [t] was taken from Bartlo's figure 3, p. 147 and figure 9, p. 153. Note that the hash marks on the vertical axis of his figure 3 are incorrectly labeled. They should read from 0 to 100. This can be verified by his discussion on p. 152 and his figures 8 and 9.)

Table [t] shows that the tensile strength of the smaller bar is 14% higher than the larger bar. However, the unnotched fatigue strength of the smaller bar is about 47% higher than for the larger bar. The notched fatigue strength of the smaller bar is about 22% higher than for the larger bar. Thus, we see that there is a substantial size effect. The degree of the size effect is not determined by the tensile strength and is reduced when the material is notched.

The table also shows the fatigue stress concentration factor Kf for the two stock sizes. Kf for the smaller bar is 20% higher than for the larger bar. However, at the worst (small stock), Kf is only 60% of the theoretical stress concentration factor Kt (3.5). Note that the notch radius is only 0.254 ±0.025 mm, which would not be common in highly stressed areas of horns.

TABLE [t]

SIZE EFFECT FOR TI-6AL-4V BAR

Raw stock Specimen Tensile Fatigue

dia. (mm) condition strength (MPa) strength (MPa)

15.9 unnotched 1060 Ö- 630

88.9 unnotched 930 ° 430 -Ì

Kf=2.25 À °

15.9 notched 1060 Û- 280 û Kf=1.87

88.9 notched 930 230 -ì

Notes:

1. Both sizes annealed for two hours at 1350 oF, then air

cooled.

2. Notched: Kt = 3.5

Now let's assume that you want to determine the fatigue life of a 76 mm diameter bell made from 76 mm diameter rod stock (the same as the S-N horn). The bell horn has had its front core machined away and will normally fail in its outer material. However, the available S-N data strictly applies only to the core material. Since the bell horn's outer material may be either stronger or weaker than it's core material, the life estimate for the bell horn will probably be in error.

The factor KS accounts for the size effect. As shown in the above examples, it will depend on the the size and shape of the working horn in relation to the S-N horn. It also depends on the material and its processing. Table [t] shows some data taken from RMI literature (p. 13).

Since the fatigue strength usually varies directly with the ultimate tensile strength, the above table indicates that the fatigue strength decreases as the section size increases. RMI also notes (p. 22), "Heavy sections and coarse microstructures may develop somewhat lower fatigue properties without any reduction in the ultimate tensile strength." Thus, the fatigue strength may decrease even more rapidly than indicated by the decrease in ultimate strength of table [t]. (See Juvinall[1], pp. 231 - 233 for size effect information on steels.)

Branson's titanium horns are seldom larger than 75 mm. From table [t] with diameters between 0 and 75 mm, the ultimate strength ranges from 1140 to 970 MPa, with a mean of 1055 MPa. This represents a variation of ±8%, or KS,CV = 0.03. Aluminum may show less variation for a given diameter, but horns made from aluminum are generally much larger than those made from titanium. Thus, KS,CV = 0.03 is probably valid for aluminum also.

(Note: the chapter on "Elastic Material Properties" discusses the possible effect of stock size on the wave speed and Young's modulus for 7-4 titanium and Aerospace aluminum rod stock. It appears that the wave speed and Young's modulus increase with increasing stock size for 7-4 titanium, although this is uncertain. No change was found for the Aerospace aluminum. However, any changes in these elastic properties probably do not effect the fatigue properties of the materials.)

The size effect may also influence the results of S-N testing at different frequencies. For example, 40 kHz S-N horns are often machined from stock that is 1/2 as large as for 20 kHz S-N horns (e.g., for fatigue tests of rod material). In this circumstance, we should expect that 40 kHz S-N horns would probably have longer lives than 20 kHz S-N horns.

Notch sensitivity

For working horns, the maximum stress Íh often occurs at stress concentrators (e.g., slot radii, nodal radius, stud threads, etc.). In the fatigue process, different materials have different sensitivities to these stress concentrators. For example, small notches in cast iron have negligible effect on the fatigue life, since the material already has natural internal stress risers. Other materials behave similarly, but to a lesser degree. Therefore, the full value of Íh should not be used. The factor by which Íh should be reduced is given by (see appendix []):

1 - q

11a) Kr = q + -----

Kt

where Kr = stress reduction factor

Kt = static stress concentration

factor

q = notch sensitivity

Kt depends on the horn geometry and the type of loading in the vicinity of the notch. Its lowest possible value is 1.0 (i.e., no notch). (See the chapter on "Stress" for a discussion of Kt). Kt is usually taken from diagrams from parts with simple geometry and simple loading. Where the geometry and loading are complex, Kt can be difficult to determine exactly.

q depends on the horn material and the notch radius. (See figure f.) q ranges from 0 to 1. As noted previously (section ), q is largest for fine-grained, relatively homogeneous materials. Such materials typically have the best fatigue performance. (See section .) q is generally taken from the literature, but may not be available for a particular material. If q is unknown and if the material is fine-grained, then q can be chosen as 1.0, for which Kr = 1.0. The result will be a somewhat conservative design.

Since I have no specific data on the variability of Kr, I will assume a conservative value of Kr,CV = 0.05.

Failure theory

The S-N horns are tested under uniaxial stress (i.e., the stress acts only in a single direction -- along the stud axis). This will also be the case for relatively simple working horns such as exponential and catenoidal.

However, for many working horns, the stress acts in several directions simultaneously (multiaxial stress). This is especially true at stress concentrations, such as slot ends. (See Juninall, pp. 72 - 73.) figure [f]. from Willertz (p. 126) shows the axial, hoop, and radial stress at at notch for an axially loaded fatigue test specimen. While the axial stress is highest, as expected, the other stresses cannot be neglected. Multiaxial stresses may also be significant in relatively large unslotted horns (e.g., spool horns).

There are a number of theories that attempt to correlate multiaxial failure stress with observed uniaxial failures. For fatigue of ductile materials, the Hencky von Mises theory is often used. (See the chapter on "Stress".) However, the correlation between the Hencky von Mises theory and actual failures from multiaxial stress is not exact. Thus, a particular horn design may have consistently longer (or shorter) life than is estimated from the S-N diagram by using the horn's Hencky von Mises equivalent stress.

We use the KF factor to account for the uncertainty of the failure theory. If the failure theory adequately predicts fatigue failure (e.g., for uniaxial stress), then KF should have a mean value near 1.0. Ullman (p. 110, table 2) gives recommendations for the coefficients of variation KF,CV (which he calls the statistical ratio) to account for the uncertainty of KF in particular circumstances. For good fatigue test failure data, he recommended decreasing his KF,CV values to those of table [t] (telephone conversation of 2/10/88).

TABLE [t]

KF,CV FOR VARIOUS FATIGUE FAILURE METHODS

Failure method KF,CV

Fully reversed, uniaxial, infinite-life fatigue 0.0

Fully reversed, uniaxial, finite-life fatigue 0.0

Fully reversed, multiaxial fatigue 0.1

Since most failure problems occur for horns with multiaxial stress, KF,CV = 0.1 (about ±25% error) is usually appropriate.

Amplitude estimation

In estimating the horn stress Íh, we have assumed that the horn's amplitude is known exactly. However, this is not true. The horn's amplitude can be written as:

) Åh = (ÅC Kd KV Ka) Gb Gh

where Åh = horn output amplitude

ÅC = converter mean amplitude

Kd = converter design correction

factor

Ka = converter aging correction factor

KV = voltage correction factor

Gb = booster gain

Gh = horn gain

All of the factors on the right of this equation have some uncertainty. Thus, the horn output amplitude is not a fixed value. Rather, it is a random variable.

Converter mean amplitude ÅC

Because of variations in ceramics, the converter amplitude will vary among converters of the same design. For example, any new Branson 20 kHz converter has an allowable variation of ±2 microns.

We can, of course, measure the amplitude of a single, particular converter. We could then use this amplitude to predict the life of the particular horn that was driven by this converter. However, what we really want is a method of predicting horn life when the horn is assembled to a randomly selected converter. Hence, we cannot exactly specify the converter amplitude.

Carefully controlled amplitude measurements of 40 kHz converters (9/5/84) gave the following values:

TABLE [t]

40 KHZ CONVERTER AMPLITUDE MEASUREMENTS

Number Average Standard

Converter tested amplitude deviation ÅC,CV

XL 15 6.8 0.32 0.047

K-30 10 8.2 0.41 0.050

Thus, an estimated coefficient of variation of 0.05 seems reasonable.

Converter design correction factor Kd

An incorrect design can cause the mean converter amplitude to deviate considerably from the specification value. For example, for 40 kHz converters, the nominal converter amplitude is 9.5 microns. However, careful amplitude measurements (9/5/84) showed the following:

TABLE [t]

40 KHZ CONVERTER AMPLITUDE MEASUREMENTS

Number Average Standard % error from

Converter tested amplitude deviation specification

XL 15 6.8 0.32 -28%

K-30 10 8.2 0.41 -14%

It is not known if this degree of error is abnormal.

Such deviations from the nominal amplitude can occur in the initial design, or from later design modifications (e.g., ceramics from a different manufacturer, power supply changes that affect the converter drive voltage, etc.). From such problems, we should expect that the final converter mean amplitude may deviate as much as ±15% from the specified (nominal) amplitude (i.e., a coefficient of variation of Kd,CV = 0.06). (Note: if the converter amplitude specification is revised to "match" the amplitude of production converters, then the deviation will be much smaller.)

Aging

Aging (an increase increase in converter amplitude over time) also causes deviations from the nominal converter amplitude. For example, whereas Branson's amplitude specification for new converters is 18.5 to 22.5 microns, the specification for used (aged) converters is 18.5 to 25.0 microns. (Note: "new" converters are normally assembled with well-aged ceramics, but aging continues none-the-less.)

Although all converters age, the aging process is not unform. For instance, aging can be accelerated by operation at high temperatures and high electric fields. (See the chapter on "Converters"). Thus, aging will probably skew the amplitude distribution to look somewhat like the log-normal distribution. Thus, the aged mean converter amplitude will probably be located closer to the lower end of the aged specification. For example, the mean amplitude for 20 kHz converters might increase from 20.5 microns to 21.0 microns.

Since aging always causes the converter amplitude to increase, Ka is always greater than 1.0. The coefficient of variation Ka,CV is unknown but is probably low -- say 0.02.

Voltage correction factor KV

Additional variation may occur because different plant locations may have different line voltages, probably within ±5% of nominal -- i.e., a coefficient of variation of KV,CV = 0.02.

Booster gain

Even well-designed boosters (e.g., 500 series) may have gains that deviate by ±5% from the nominal gain. If the boosters are designed with less care (e.g., 400 series), then the gains may deviate by ±10% or more from the nominal.(See the chapter on "Boosters".

Horn gain

Horn gains may be measured or may be taken from finite element analysis (FEM). Measured horn gains seldom agree with FEM horn gains within ±5% -- i.e., a coefficient of variation of Gh,CV = 0.02.

Overall variability

Accounting for all factors, the stress ÍSN that should be used with the S-N diagram is:

Ö Ì Ö Ì

) ÍSN = ° Íh Kr KS KF ° ° (ÅC Kd KV Ka) Gb Gh °

Û ì Û ì

Remember that Íh is the horn stress per unit of horn amplitude.

Since we would like to use the nominal values of for Íh, ÅC, Gb, and Gh for all calculations, we would hope that all K values would have a mean value of 1.0. We have already noted that Ka will be greater than 1.0.

The coefficient of variation of ÍSN is the square root of the sum of the squares of all of the factors that contribute to ÍSN. (See Barry, pp. 50 - 51.) Thus:

Ö

) ÍSN,CV = ° (Íh,CV)2 + (KS,CV)2 + (Kr,CV)2 + (KF,CV)2

Û

+ (ÅC,CV)2 + (Kd,CV)2 + (Ka,CV)2 + (KV,CV)2

̽

+ (Gb,CV)2 + (Gh,CV)2 °

ì

If we substitute the previously estimated coefficient of variation values into equation e, we have:

Ö

) ÍSN,CV = ° 0.022 + 0.032 + 0.052 + 0.102

Û

+ 0.052 + 0.062 + 0.022 + 0.022

̽

+ 0.042 + 0.022 °

ì

= 0.15

The particular values in equation 15 will change somewhat, depending on the circumstances. For example, if the horn is an integral part of the converter (e.g., the loom slitter converter), then Gb,CV will be zero. None-the-less, equation 15 shows that there is considerable uncertainty in estimating the stress. The value of ÍSN,CV may be even larger than calculated above, since we have not considered such factors as residual stresses and the effect of surface finish (Juvinall[1], pp. 233 - 236), which are especially important for titanium.

Standard deviation

The standard deviation of ÍSN is:

) ÍSN,SD = ÍSN,CV ÍSN

(Note: if each of the factors on the right hand side of equation 13 is normally distributed, then ÍSN will not be normally distributed. However, for convenience in the following analysis, we will none-the-less assume that ÍSN is normally distributed. This assumption may be approximately true over a limited range of stresses.)

Factor of Safety

Let's assume that the S-N fatigue strength at any probability of failure (e.g., 2.5%) is known exactly. (Of course, this is never true.)

Probability of failure at N cycles

The S-N diagram was developed from tests of S-N horns of known (measured) amplitude. We want develop a more generalized S-N diagram that will account for the variability in working horn stress ÍSN discussed above. (See Von Alven (pp. 76 - 81) and Juvinall[1] (pp. 365 - 368) for the following analysis.)

Note: for the following analysis, we will drop the SN subscript from ÍSN. Thus, Í == ÍSN and ÍSD == ÍSN,SD.

figure [f]. shows a distribution of fatigue strength taken from an S-N diagram at N cycles. The nominal strength (corresponding to 50% probability of failure) is S. The strength standard deviation is SSD.Error! Reference source not found. Since the strength distrubution has been taken from the S-N curve, S (and possibly SSD) will depend N.

Also shown in the figure is the horn stress distribution. Í (the nominal stress) can be chosen as desired, for which ÍSD is then given by equation e. For this analysis, we initially choose Í < S so that there is a small overlap in the distributions. A good choice for Í is:

) Í = S - (1.96 SSD + 1.96 ÍSD)

= S - 1.96 (SSD + ÍSD)

so that the S and Í distributions overlap at the 95% convidence level. (Note that this choice of Í is not critical.) Since S is greater than Í and there is only a small overlap between the strength and stress, there is only a small possibility that the horn will fail.

Let Z be a distribution that represents the difference between a randomly selected strength from the strength distribution and a randomly selected stress from the stress distribution. The Z distrubution has a mean of:

17a) Zmean = S - Í

and a standard deviation of:

Ö Ì½

17b) ZSD = ° SSD2 + ÍSD2 °

Û ì

The horn will fail if Z is less than zero -- i.e., whenever the strength is less than the stress. This is true regardless of how Í and S are distrubuted (e.g, normally, log-normally, etc). We assume that the stress is normally distributed. (See above.) In order to use standard normal tables, we will also assume that strength is normally distributed (not log-normally distributed, although this will more likely be the case). The probability of failure can then be found from the area under the normal curve from -_ to the value Z, given by:

Zmean S - Í

) Z = - ----- = - ----------------

ZSD Ö Ì½

° SSD2 + ÍSD2 °

Û ì

Dividing the right side through by Í:

S/Í - 1

18a) Z = - ----------------

Ö SSD2 ÍSD2 ̽

° ---- + ---- °

Û Í2 Í2 ì

S/Í - 1

= - ---------------------------

ÖÖ SSD Ì2 Ö S Ì2 ÍSD2 ̽

°° --- ° ° --- ° + ---- °

ÛÛ S ì Û Í ì Í2 ì

S/Í - 1

= - ------------------------

Ö Ö S Ì2 ̽

° SCV2 ° --- ° + ÍCV2 °

Û Û Í ì ì

This equation can be rearranged into a quadratic equation in terms of S/Í:

Ö Ì Ö S Ì2 Ö S Ì

18b) ° 1 - Z2 SCV2 ° ° --- ° - 2 ° --- °

Û ì Û Í ì Û Í ì

Ö Ì

+ ° 1 - Z2 ÍCV2 ° = 0

Û ì

ÍCV2 has been determined above. If the S-N curve is well developed, then SCV2 is known for any number of cycles N. Z eill be determined by the required reliability. Thus, equation 18b can be solved for S/Í, which is recognized as the factor of safety FS. This can be solved by the usual method to give:

Ö Ö Ì Ö Ì̽

1 ± ° 1 - ° 1 - Z2 SCV2 ° ° 1 - Z2 ÍCV2 °°

S Û Û ì Û ìì

18c) --- = ------------------------------------------

Í 1 - Z2 SCV2

Example

Let's assume that we have a material whose SSD at a given number of cycles N is 0.08. (See Shigley, pp. 168 - 169.) What is the required factor of safety in order for the probability of failure to be less than 2.5%. From table [t] in the chapter on "Statistics", the required Z value for 2.5% failure is -1.96. Then, substituting this value and the value from equation 18 into equation 18c:

S 1 ± (1 - 0.975 * 0.914)1/2

18d) --- = --------------------------

Í 0.975

1 ± 0.330

= ---------

0.975

The plus sign is appropriate for the 2.5% probability of failure, since othewise S/Í is less than one (i.e., a factor of safety less than one). Thus, S/Í is 1.36. Therefore, in order to achieve a 2.5% probability of failure at N cycles, the median fatigue strength must be 36% higher than the mean estimated stress or, conversely, the mean estimated stress must be 73% as large as the median fatigue strength. (Note: the minus sign in equation 18d would be for a 2.5% probability of survival.)

Special case I: ÍCV = 0

What would happen if the stress was known exactly? Then, for zero stress variability ÍCV = 0 and equation 18d becomes:

S 1 ± Z SCV

18e) --- = -----------

Í 1 - Z2 SCV2

1 ± Z SCV

= -----------------------

(1 + Z SCV) (1 - Z SCV)

As before, for a 2.5% probability of failure, the plus sign is appropriate in the numerator. Therefore, canceling the common numerator and denominator terms:

S 1

18f) --- = -----------

Í (1 - Z SCV)

Substituting the appropriate values, we have S/Í = 1.19, or Í/S = 0.84. This is exactly the result we would have expected from a fatigue curve whose strength disribution has a SCV of 0.08.

Thus, we see that as the variability in the stress increases, the factor of safety must also increase to compensate. In this particular example, changing ÍCV from 0 to 0.15 increased the required factor of safety from 1.19 to 1.36.

Special case I: SCV = 0

What would happen in we had a material that had no variability in its fatigue failures -- i.e., SCV = 0. Then equation 18d would become:

Note that for this example, the variability in strength (determined by SCV) has only a slight effect on the final result compared to the effect of the variability in stress (determined by ÍCV). For instance, even if the fatigue strength is known exactly (i.e., no variability --> SCV = 1), then S/Í is reduced only from 1.36 to 1.29.

Now, assume another nominal stress Í and repeat the above process for the same number of cycles N. For this case, we choose Í greater than S. A good Í value is:

) Í = S + (1.96 SSD + 1.96 ÍSD)

= S + 1.96 (SSD + ÍSD)

which again causes the stress and strength distributions to overlap at the 95% confidence levels. (See figure [f]..) With this value of Í, there is a very large probability that the horn will fail.

Having repeated the above process using equation 19, we now have two probabilities of failure at N cycles. We can plot each of these values and its associated nominal stress on probability paper. If the strength is normally distributed, then you should use normal probability paper. If the strength is log-normally distributed, then you should use log-normal probability (or, alternately, you can plot the log of the stress on normal probability paper).

The straight line between the two plotted points now defines the strength distribution at N cycles that accounts for the variability in horn stress. From the probability graph, you can read the strength corresponding to any desired probability of failure -- e.g., the strengths for 2.5%, 50% and 97.5% probability of failure. These strength values can then be plotted on the S-N diagram at N cycles.

Now this method can be repeated at several different N values. The resulting S-N diagram will look something like figure [f]., where the corresponding probability points have been connected by lines. This S-N diagram now accounts for possible variability in horn stress. Note that the tolerance band is now wider than that of the original S-N diagram, since it now accomodates the uncertainty associated with the converter amplitude.

To use this diagram, you just specify the horn's nominal stress corresponding to a nominal converter amplitude, from which the number of cycles to failure can be found.

Note — In the above method we have assumed that the strength is normally distributed. If this is not true (e.g., the strength is log-normally distrubuted), then Z will no longer be normally distributed and we cannot use the normal probability tables to determine the probability of failure. However, the problem can be solved using Monte Carlo simulation.

Conclusion

From the above analysis, I think we must agree with Shigley (p. 168):

"When one considers all the unknowns involved in determining the actual loads the part will endure, the approximations involved in determining the stresses, and finally the material itself and the probable surface irregularities, it is clear that the life cannot be predicted with any accuracy."

Processing methods that may affect fatigue life

In this section we will look at various processes that can affect the fatigue resistance of the base material, regardless of the part design. The emphasis is on processes that give residual surface stress.

A residual stress is a stress that exists in a part even though no external forces are applied. Residual stresses can be induced by working the material (e.g., shot peening, rolling, burnishing, etc.), by chemicals, platings, heat treatment, etc. (For a good discussion, see Manson, pp. 304 - 326.)

When there are no residual stresses, and in the absence of external forces, the alternating fatigue stress is shown in figure ÍfÊ«». When a residual compressive stress is added, the alternating stress shifts lower by the amount of the compressive stress. (Figure ÍfÊ.) The amount of fatigue damage that occurs during each cycle depends (roughly) on the magnitude of the positive (tensile) stress. Thus, the addition of a compressive residual stress (which reduces the tensile stress) should improve a material's fatigue resistance. (Note: a tensile residual stress would have exactly the opposite effect.)

Of course, the compressive residual stress must be located where the part is most likely to fail (i.e., where the Hencky von Mises stress is largest). It would do little good to have the compressive residual stress in an area of low alternating stress. In a horn, for example, the compressive residual stress should be placed at the node and at stress concentrations, such as slot ends.

Because of the way residual compressive stresses are created, they are usually largest near the surface of a part and extend only a short depth below the surface. (See figure [f]., taken from Manson (p. 311)). However, this is usually sufficient to improve the fatigue life, since most fatigue cracks start from the surface. However, if a fatigue crack starts at an interior defect, then the surface compressive residual stress will have no effect.

Shot peening

In shot peening, the residual stresses are introduced by impacting the surface with steel shot or other hard materials. This causes plastic deformation of the surface layers, causing a compressive residual stress near the surface. (For a general discussion of shot peening theory and method, see the SAE Manual on Shot Peening - SAE J808a.)

Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. shows a stress distribution for a part that has been shot peened on both sides. The surface has high compressive residual stress. However, the depth of the residual compressive stress is quite small, typically less than 1/2 millimeter. (See SAE J808a (p. 23) and Juvinall[1], p. 334.) The surface compressive stresses are balanced by internal tensile stress.

Some results

Juvinall[1] (p. 336) gives some data for the improvement in endurance limits for various materials (mostly steel) with various surface conditions. The average improvement is somewhere around 50%, although improvements of a few hundred percent may be possible under optimum conditions (p. 335). (It should be remembered that a small improvement in fatigue strength can give a very significant improvement in life.) SAE J808a (p. 3) gives examples of improvement in fatigue life up to 1000%.

For Ti-6Al-6V-2Sn (STA, 42 Rc), Zlatin (p. 502, his fig. 9) gives the results of table _ for cantilevered bending at 1800 cpm. Zlatin did not specify the shot peen conditions.

TABLE _

EFFECT OF SHOT PEENING ON THE

ENDURANCE LIMIT OF Ti-6Al-6V-2Sn

Roughness (AA) Endurance Limit (ksi) at 107 cycles Percent

Without With Without With Endurance

Operation Shot Peening Shot Peening Shot Peening Shot Peening Improvement

Gentle grind 43 43 65 83 28%

Abusive grind 70 55 20 50 250%

Standard ECM 11 48 72 85 18%

Off-standard ECM 145 120 47 85 180%

Notes:

1. ECM = electrochemical machining

Effect of stress distribution

The effects of shot peening depend on the stress distribution that would normally cause failure. When these stresses are large at the surface of the part but decrease internally (i.e., a large stress gradient), then shot peening may be beneficial. Otherwise, shot peening may actually decrease the fatigue life.

Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. shows the stresses in a bending part that has been shot peened on both sides. (For all of the following discussion, the stress direction is along the axis of the part. Also, the maximum load stress is shown as about 75% of the maximum shot peen stress. In actual practice, however, this could be otherwise.)

(a) shows the vibration stress distribution due to bending the part (assuming the top surface is currently in tension). (b) shows the residual peening stress. As noted before, the residual stress is highly compressive near the surface, but is tensile in the interior of the part. (c) shows the resultant stress distribution from both the shot peening and the bending. Note that (c) has a lower maximum tensile stress than (a), so the peened part should have better fatigue performance than the unpeened part.

Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. shows a similar situation for a notched part that is vibrating axially. The stresses at the notch (a) are very high at the surface, but decrease quickly toward the interior of the part. Thus, when the residual shot peening stress (b) is superimposed on the axial stress, the maximum tensile stress is reduced (c), and better fatigue life is expected.

Now let's consider what happens for an unnotched part that vibrates axially. (Figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND..) In this case, the axial stress is uniformly distributed across the part's cross-section (a). Thus, when the residual shot peening stress (b) is superimposed on the axial stress, the surface tensile stress is reduced, but the interior tensile stress is increased (c). Thus, shot peening increases the maximum tensile stress in the part, so fatigue failure is more likely.

Heywood (p. 303) notes this effect:

"The compensating tensile stresses are thus fairly close to the surface, making it possible for failure to occur just below the surface without increasing the life. This limits the gains in fatigue strength to cases where high stress gradients are produced, as at notches, regions of fretting, and at plain specimens of small size which are subjected to bending or torsion."

Wedden gives some data for axial fatigue tests of shot peened Ti-6Al-4V. All shot peened specimens failed from "subcutaneous foci" (p. 74). This deleterious effect was reduced by reducing the shot size and intensity.

Thus, as Juvinall[1] notes (p. 335), shot peening is most effective "where surface stress gradients are high (as with small notched parts loading in bending) ..."

The above discussion is particularly important for fatigue testing of S-N horns. S-N horns are normally unnotched. However, unnotched tests will likely show a negative effect from shot peening. Therefore, if the effect of shot peening is to be determined from fatigue testing of S-N horns, then the horns must be modified to include an appropriate notch at the front node. The notch should have approximately the same stress concentration as in working horns.

Effect of Almen intensity

The amount of energy delivered to the surface is specified by the Almen intensity, which is the amount of curvature produced in standardized thin metallic strips (designated type A, N, and C) when shot peened from one side only. (See SAE J808a, pp. 14 - 16.) To convert, the N strip gives readings approximately three times greater than the A strip; the C strip gives readings approximately 1/3 as large as the A strip.  (See SAE J808a, p. 18.)

The Almen intensity is determined by the properties of the blast (the shot's size, shape, material, hardness, density, and impact velocity), the properties of the exposure (duration, angle of impact, and shot flow rate), and properties of the test strip (dimensions and mechanical properties). However, the Almen intensity does not completely describe the effect of shot peening on fatigue. (See SAE J808a, p. 19. Also see Woelfeld.) Thus, two different peening processes may produce the same Almen intensity, but produce different fatigue results when applied to the actual part. Thus, all conditions of peening must be specified, rather than just the Almen intensity.

(Note: For spherical shot, the shot size is specified by a shot number whose value corresponds roughly to the shot diameter in ten thousanths of an inch. For instance, S110 has a diameter of approximately 0.010 inch. (See SAE J808a, table 1, p. 10.))

Effect of stress level

The beneficial effects of shot peening may depend on the stress level at which the part is tested. It has been suggested that a compressive residual stress is effective in retarding crack propagation, but has little effect on crack initiation. (See Manson, pp. 310 - 312.) Thus, for parts that have high stress and short life, where the crack propagation phase accounts for the majority of life, residual compressive stress should be beneficial. For unnotched parts that have low stress and long life, where the crack initiation phase accounts for the majority of life, the residual compressive stress would have little effect.

Ebara (p. 356) tested unnotched 13Cr stainless steel at 16 kHz (axial loading) in pure water and with various NaCl concentrations. The horns were shot peened to 0.0011A Almen intensity. Test failures were recorded from 106 to 1010 cycles, depending on the stress level. Shot peening increased the horn life in the high stress (low cycle) region. However, the effect decreased as number of cycles increasesd. In the low stress, high cycle region (greater than 108 cycles, depending on the NaCl concentration) the positive effect of shot peening disappeared. In fact, Ebara's figure 3 shows that shot peening may have actually decreased life at high cycles for two NaCl concentrations. This could not be attributed to a relaxation in the residual shot-peened compressive stress, which had only decreased from 333.4 MPa at the beginning of testing to 313.8 MPa at 1010 cycles, under an ultrasonic stress of 73.5 MPa.

Lucas fatigue tested unnotched Ti-6Al-4V (low frequency, axial load). For an annealed Â-ß forging with either 0 or 45 ksi mean stress, there was a noticible increase in life from shot peening (#110 shot, 0.008A-0.012A Almen) between 105 and 107 cycles as compared to longitudinal polishing with 400 grit silicon carbide. However, above 107 cycles there was no effect from shot peening. (Testing was performed up to 108 cycles.) (Note: these results are for the median fatigue life.)

Thus, shot peening must be evaluated at the life or stress of interest. Accelerated testing may give optomistic results that are not valid at longer lives and lower stresses.

Other factors

Shot peening provides some advantages besides compressive residual stress. It can improve the surface finish or produce a more uniform surface finish. It also increases surface hardness somewhat, which can be beneficial in corrosive environments. Shot peening may also be beneficial for subsequent plating and to overcome residual tensile stress from machining. (See Manson, p. 314.)

Shot peening can be beneficial in steels where high surface hardness through heat treatment is required for good wear, but where such high surface hardness would normally cause reduced fatigue life. (See figure ífÊError! Reference source not found. from Juvinall[1], p. 212.) With shot peening, both hardness and life can be improved. (Juvinall, p. 335. Also, see the discussion of rolled threads.)

Surface treatments

Various surface treatments (nitriding, carburizing, Chrome-ite, etc.) have been used to improve fatigue performance. (See Juvinall, pp. 337 - 338; Manson, pp. 304 - 307, 324 - 326.) These processes create compressive residual stresses in the surface of the part and may also be helpful in protecting against hostile enviroments. Figure [f]. (from Manson, p. 306) shows the improvement in fatigue strength of Ti-6Al-4V for various treatments. F.Figure [f]. (from Manson, p. 307) shows the improvement in fatigue strength of two aluminums and two steels when treated with Chrome-ite.

Some surface treatments may decrease fatigue life. This is especially true of electrodeposited hard coatings such as hard chrome and nickel, for which the fatigue limit of steel parts may be reduced by 50% or more. (See Juvinall[1], p. 339.) This may occur because electroplating tends to deposit the coating in a state of residual tension. This tensile residual stress can be minimized by using low plating-current density. Plating temperature, thickness of plating, chemical composition, and subsequent heat treatment may also have an effect. (Juvinall, pp. 339 - 341.) Electroplating with softer materials (copper, cadmium, zinc, lead, and tin) generally causes little adverse effect and may be beneficial for fatigue in a corrosive environment (which includes the effects of humidity).

Electroless nickel plating of aluminum

Branson uses an electroless nickel treatment for many of its horns. The early bath was improperly controlled, which caused a pocked and very glassy skin on the horns. Table [t] shows the effect on Reynolds 2024-T351 QQA225 aluminum S-N horns tested at 116 MPa (127 microns). Figures [x]. and F.Figures [x]. show, respectively, the unplated life data and log-life data plotted on probability paper. Since neither is a straight line, the data is neither normally nor log-normally distributed. Error! Reference source not found.

The average life without plating was 1.84 hours. The average life with poor bath control was 0.10 hours, a reduction of 95%. When the bath was corrected, the life was 2.35 hours, which appears to be even better than the unplated horns. However, with only two data points for the good bath horns, the data is insufficient to draw statistically valid conclusions.

Data was also taken with Martin Marietta 7075 QQA225 aluminum. (See table [t].) This limited data indicates no difference between unplated and plated life with good bath control.

Note: if there truly is no significant difference between the unplated and electroless nickel plated horn life, then there are two possible conclusions regarding the effect of humidity:

1. Humidity has no effect on fatigue life of aluminum at 20 kHz, so that plating has no beneficial effect.

2. Humidity may have an adverse effect on fatigue life, but its effect is the same with or without the nickel coating, so that the nickel coating does nothing to protect the aluminum.

Electroless nickel plating of titanium

Padberg tested several coatings to determine the effect on the fatigue life of Ti-6Al-6V-2Sn plate. The fatigue tests were tension-tension at a frequency of 30 Hz with a maximum stress of 828 MPa (120 ksi) and a minimum stress of 83 MPa (12 ksi). Prior to coating, each specimen was steel shot-peened to 0.015 A intensity and grit-blasted with 150 mesh aluminum oxide grit, except for detonation gun sprayed specimens which were peened to 0.009 N intensity. (p. 2482)

The results are shown in figure [f]. from Padberg, p. 2484. Only the fluoride-phosphate and anodize coatings lasted nearly as long as the control specimens, which failed at approximately 3.5*106 cycles. The electroless nickel Kanigen process (General American Transportation Corp, Chicago, IL) reduced the fatigue life to approx 1.02*105 cycles (i.e., by a factor of 35:1). (Note: the number of specimens and the data scatter for each coating was not stated by Padberg.)

Note that the stress was relatively high, so that the lives were short compared to normal ultrasonic testing. These results may not be the same for tests at lower stress and longer lives. (Note: the "Kanigen" process may not be the same as used by Branson for its aluminum horns.) However, these results do indicate that care must be used in selecting coatings.

TABLE [t]

EFFECT OF ELECTROLESS NICKEL PLATING

ON REYNLODS 2024-T351 QQA225 ALUMINUM ROD

S-N HORNS AT 116 MPA (127 MICRONS)

Life # Condition Life (hrs) Log(Life)

79-26 No plating 0.78 -0.108

79-27 No plating 1.07 0.029

80-14 No plating 2.39 0.378

80-19 No plating 2.26 0.354

80-20 No plating 1.91 0.281

80-21 No plating 2.12 0.326

80-22 No plating 2.31 0.364

80-23 No plating 2.35 0.371

80-24 No plating 2.22 0.347

80-11 No plating (a) 1.77 0.248

80-12 No plating (a) 1.77 0.248

80-13 No plating (a) 1.13 0.053

1.84 0.241

(0.557) (0.161)

80-14 Nickel, poor bath 0.03 -1.522

80-15 Nickel, poor bath 0.18 -0.744

0.10 -1.134

(0.106) (0.550)

80-16 Nickel, good bath 2.38 0.377

80-17 Nickel, good bath 2.33 0.367

2.35 0.372

(0.035) (0.0065)

Notes:

1. Tests were performed in 1979 and 1980.

2. (a) indicates that four flats were machined along the length of the horn in the 17.3 mm diameter front section. See figure [f]..

3. Numbers below the solid lines are averages.

4. Numbers in parentheses are standard deviations.

TABLE [t]

EFFECT OF ELECTROLESS NICKEL PLATING

ON MARTIN MARIETTA 7075 QQA225 ALUMINUM ROD

S-N HORNS AT 138 MPA (152 MICRONS)

Life # Condition Life (hrs) Log(Life)

79-51 No plating 5.76 0.760

79-52 No plating 4.89 0.689

79-53 No plating 4.76 0.678

5.14 0.709

(0.544) (0.045)

80-32 Nickel, good bath 4.44 0.647

Notes:

1. Tests were performed in 1979 and 1980.

2. Numbers below the solid lines are averages.

3. Numbers in parentheses are standard deviations.

Rolled threads

Threads can be either cut, ground, or rolled. Internal threads are usually cut by tapping. Except in special circumstances, external threads are almost always made by rolling. Rolling can improve fatigue resistance by giving good grain flow and reducing the liklihood of stress risers due to machining imperfections. It also gives more accurate threads. Error! Reference source not found.

Threads can be rolled either before or after heat treatment. Although rolling before heat treatment is less expensive because the rolling dies last longer, rolling after heat treatment has several advantages for fatigue life. Because the threads are not subsequently exposed to high temperatures, rolling after heat treatment preserves the residual compressive stresses and the increased hardness caused by the rolling operation. The endurance limit may be increased by factors of 2 to 3 or more as compared to threads rolled before heat treatment. (See .f.figure [f]. from McCormick, p. F27. Also see Crispell, pp. 72 - 74; Walker, p. 184.)

When threads are rolled before heat treatment, the maximum screw hardness is about Rc 43, above which the fatigue strength decreases. (See Walker, p. 185.) However, there is no such hardness limit when the threads are rolled after heat treatment. (See .f.figure [f]. from McCormick, p. F27.) This same effect was noted when discussing shot peening.

The above discussion applies primarily to external threads on screws and bolts. However, internal rolled threads can also be produced by using special taps. Because these taps "plough" the threads instead of cutting, they require higher torques and are therefore more prone to break.

Screw life can also be significantly improved by changing the thread and head-fillet geometry.

Burnishing

Burnishing adds compressive residual stress, increases hardness 5 to 10 percent (or 3 points on the Rockwell C scale), and improves surface finish (10 microinches or lower). (See Westerman.) Westermann (p. 48) says that an improvement in fatigue life of about 300% can be expected from burnishing aluminum or steel parts. However, he gives no supporting data.

Electro-discharge machining

Electro-discharge machining (EDM) is a process in which the part is submerged in a dielectric fluid (usually oil) bath and a graphite electrode of the desired contour is used to electrically erode the part. EDM can also be performed with special wire to cut out complex shapes, much like a band saw. EDM produces localized heating, and affects the material both mechanically and metallurgically. (Manson, p. 289.) It also leaves a pitted surface with high stress risers. The amount of pitting and other damage can be controlled somewhat by the cutting rate (using low current and high frequency). (Zlatin, p. 500) However, Zlatin (p. 500) found that both "roughing" and "finishing" EDM operations gave high but shallow tensile stress.

EDM has been suggested as a means for machining slots. Therefore, tests have been run on S-N horns with four flats EDM'ed along the sides. For comparison, standard S-N horns and horns with end-milled flats were also tested. Table [t] shows the results for 2024-T3/T351 aluminum rod. It is obvious from looking at the data that the standard horns and the horns with the machined flats have essentially the same lives (2.51 hours average). However, the two EDM'ed horns have only half the life (1.24 hours average). Because the data scatter is small, we can conclude that such a difference is probably not due to chance and that EDM therefore has a negative effect on aluminum horn life.

Table [t] shows the results for 7-4 titanium rod. Because of the scatter, additional tests should be run. Based on the available data it does not appear that EDM affects the life of 7-4 rod. However, the nominal ultrasonic stress may have been too high to show much effect from the EDM. (See the section "S-N Testing to Determine the Effect of Residual Stress".)

Note: Manson (p. 289) says, "Different materials respond differently to this machining process [EDM], and titanium alloys are very seriously affected." Figure [f]. (from Manson, p. 290) shows the effect of EDM on the fatigue life of 5Al-2.5Sn titanium alloy, along with other machining processes. Note that EDM has the most devastating effect of all the machining processes.

For Branson's tests the EDM was performed by the manufacturer of the EDM equipment. The EDM settings (speeds, currents, etc.) are not known and may not be optimum. Therefore, the life of the EDM'ed aluminum could possibly be improved.

TABLE [t]

EFFECT OF ELECTRO-DISCHARGE MACHINING (EDM)

ON CONALCO 2024-T3/T351 QQA225 ALUMINUM ROD

S-N HORNS AT 116 MPA (127 MICRONS)

Life # Condition Bar Life (hrs) Log(Life)

79-7 Standard S-N horn B 2.59 0.413

79-8 Standard S-N horn B 2.48 0.394

79-9 Standard S-N horn B 2.43 0.386

79-10 Standard S-N horn A 3.06 0.486

79-11 Standard S-N horn A 2.21 0.344

79-12 Standard S-N horn A 2.41 0.382

80-39 4 machined flats B 2.49 0.396

80-40 4 machined flats B 2.57 0.410

80-41 4 machined flats B 2.33 0.367

2.51 0.398

(0.238) 0.039

80-42 4 EDM'ed flats B 1.20 0.079

80-43 4 EDM'ed flats B 1.27 0.104

1.24 0.092

(0.049) (0.018)

Notes:

1. Tests were performed in 1979 and 1980.

2. Horn finish: 32 - 63 micro-inch (approx)

Duty cycle: 1 sec on, 4 sec off

Horn cooling: Fan

Power Supply: Unknown

3. It is not known if the bars A and B were from the same heat.

4. Numbers below the solid lines are averages. Numbers in parentheses are standard deviations.

TABLE [t]

EFFECT OF ELECTRO-DISCHARGE MACHINING (EDM)

ON TIMET 7-4 TITANIUM ALUMINUM ROD

S-N HORNS AT 285 MPA (203 MICRONS)

Life # Condition Life (hrs) Log(Life)

79-T5 Standard S-N horn 2.47 0.393

79-T6 Standard S-N horn 3.01 0.479

79-T7 Standard S-N horn 3.21 0.507

80-T10 Standard S-N horn 3.40 0.531

80-T13 4 machined flats 2.75 0.439

80-T15 4 machined flats 2.46 0.391

80-T17 4 machined flats 15.56 1.192

4.69 0.562

(4.80) 0.283

80-42 4 EDM'ed flats 2.04 0.310

80-43 4 EDM'ed flats 11.32 1.054

6.68 0.682

(6.56) (0.526)

Notes:

1. Tests were performed in 1979 and 1980.

2. Numbers below the solid lines are averages. Numbers in parentheses are standard deviations.

S-N testing to determine the effect of residual stress

When a material is subjected to large alternating stresses, any residual stresses tend to relax (decrease). (See the SAE Fatigue Design Handbook, pp. 30, 64; Juvinall[1], p. 330.) The degree of relaxation decreases as the alternating stress decreases. For fatigue tests near the fatigue limit (i.e., long life), there is practally no relaxation of the residual stress. Thus, in order to truly evaluate the effect of residual stress, the S-N test should simulate true service conditions. Accelerated testing should not be used.

Since most working horns will last at least 50 sonic hours, I recommend an S-N stress level that gives a minimum life of at least 50 sonic hours (3.6*109 ultrasonic cycles at 20 kHz). Since lower stresses will cause even less relaxation, this test will give a conservative estimate the degree by which the life will be increased. (Of course, testing at several stress levels is desirable, if possible.)

Application to working horns

We have discussed several methods for deliberately causing compressive residual stresses. We can call these deliberate residual stresses. In S-N horns which have been carefully machined, these will be the only residual stresses.

Working horns may have other residual stresses, especially machining-related residual stresses. For example, titanium bar horns may develop residual tensile stresses if the material is overheated during machining -- e.g., from rapid milling of slot ends with inadequate cooling, or from cutting with a dull tool. Machining-related residual stresses are almost always tensile.

If machining-related residual tensile stresses are present in a critical section of the working horn, then they will detract from the deliberate residual stresses. The working horn will then have less life improvement than would have been predicted from the S-N test results. The best way around this problem is to stress-relieve the working horns prior to application of deliberate residual stresses.

SAE Fatigue Design Handbook:

"Aluminum alloys generally show a decided response to mechanical prestressing techniques such as shot peeening and surface rolling." p. 52. Cites reference G. A. Butz and J. O. Lyst, "Improvement in Fatigue Resistance of Aluminum Alloys by Surface Cold-Working", Materials Research and Standards, vol. 1 (Dec. 1961), pp. 951 - 956. "In certain situations, the fatigue strength of a specimen type may be more than doubled by changing surface stresses from tension to compression." p. 52

Coaxing (understressing)

Manson defines (p. 329) coaxing as "the progressive improvement in the fatigue strength of a material induced by applying a a progressively increaing stress, starting below the fatigue limit, and increasing the load to a value well above the fatigue limit." figure ÍfÊERROR! REFERENCE SOURCE NOT FOUND.ERROR! REFERENCE SOURCE NOT FOUND. shows results for coaxing of 1045 steel. (Juvinall[1], p. 218) This material has a fatigue limit of approximately 43,000 lbf/in2 at 2*106 cycles (dashed line). Coaxing was begun by stressing the material at 40,000 lbf/in2, which was below the fatigue limit. For the right-most coaxed curve, the stress was then increased in increments of 1000 lbf/in2 and for each stress increment (step) the specimen was stressed an additional 107 cycles. (Note: at first glance, it appears that the testing times for each step are progressively shorter. However, this is not true, since the horizontal axis is a log scale.)

Using this coaxing process, the total time to failure was approximately 2.5*108 cycles, which is 125 times greater than the usual fatigue-limit life. Also, the failure stress was 60,000 lbf/in2, which is 40% higher than the usual fatigue limit. Less improvement was found when the length of coaxing for each step was reduced to 2*106 (the left coaxed curve). Manson (p. 329) indicates that the optimum loading sequence and the degree of coaxing depend on the material. Improper coaxing may actually reduce fatigue resistance. Also see Grosskreutz, p. 37.

Some very preliminary (possibly unreliable) data for 7-4 titanium rod indicates that prestressing at 250 MPa (178 microns) may have significantly improved the fatigue life when the S-N horns were subsequently life tested at 285 MPa (203 microns).

Miscellaneous fatigue tests

Effect of square edges

For slotted cylindrical and rectangular horns, the slots are commonly radiused along their entire length, as well as at the end. It was supposed that this might reduce any stress concentration due to the sharp edges. However, Juvinall[1] (p. 241) points out that cracks parallel to the stress direction have practally no stress concentration. Thus, if slot edges are regarded as cracks running parallel to the direction of ultrasonic stress, then the shape of the slot edges should have no effect on slot failures. Of course, this conclusion would not apply to the slot ends.

Table [t] shows data for a large number of Martin Marietta 7075-T6 QQA225 aluminum S-N horns that were tested at 135 MPa (152 microns). Included in that data are two horns (81-83 and 81-84) that were machined with a 17.3 mm square front half-wave section. See figure f. The lives of these two horns (2.70 and 1.52 hours) fall somewhat toward the lower end of the spectrum. However, they are not significantly different than the other horns at the XX% confidence level. (Also see the data of tables [t] and [t] which shows fatigue tests for partially flatted horns. See figure f.)

Thus, radiusing the edges of slots or edges of horns is not inherently beneficial for fatigue. However, radiusing may have some advantage since a radiused edge is less likely to be damaged during handling, where such damage might lead to fatigue.

Improving fatigue life

If material is anisotropic, design the horn if possible so that the largest tensile stresses are aligned with the longitudinal grain direction. Cite results by Lucas and Bowen. Also, see Frost, p. 81. Note: Frost cites ref. which says no fatigue directionality for age hardening aluminum alloys.

Appendix []— Derivation of Kr

In this appendix, we will determine the fatigue stress reduction factor Kr, which is defined as:

ÍSN

1a) Kr = ---

ÍH

where ÍH = Hencky von Mises stress

ÍSN = stress to be used with S-N

diagram

The relation between the fatigue stress-concentration factor Kf, the static stress concentration factor Kt, and the notch sensitivity q is given by Juvinall[1] (p. 254) as:

1) Kf = 1 + (Kt - 1) q

Kt is defined as:

Hencky von Mises stress at notch

2) Kt = --------------------------------

Nominal stress at notch

ÍH

= ----

Ínom

where the nominal stress at the notch is given by:

Average force

3) Ínom = ----------------------

Specimen area at notch

Kf can be defined in terms of strength as (Juvinall[1], p. 250):

Sn for unnotched specimen

4) Kf = -------------------------

Sn for notched specimen

where Sn is either the endurance limit, or the fatigue strength at an arbitrarily large number of cycles. Thus, Kf is a factor by which the fatigue strength is reduced due to the presence of a notch.

Juvinall[1], p. 287: "Factors Kf and Kf' can be treated either as stress-concentration factors or as fatigue-strength-reduction factors." "Whether Kf and Kf' are treated as stress increasers or as strength reducers has no bearing on the results of fatigue calculations."

Alternately, Kf can be defined in terms of stress (Shigley, p. 170):

ÍSN

5) Kf = ----

Ínom

Dividing Kf (equation e) by Kt (equation e) gives Kr:

Kf ÍSN

6) ---- = --- = Kr

Kt ÍH

Solving equation e for Kf/Kt (i.e., Kr):

1 - q

7) Kr = q + -----

Kt

Appendix []

TABLE [t]

SIZE EFFECT FOR RMI STA 6AL-4V TITANIUM:

GUARANTEED MINIMUM PROPERTIES

Diameter or Ultimate

thickness Diameter or tensile

as rolled or thickness as Max. cross- strength

forged (mm) heat treated (mm) section (mm2) (MPa)

0 --> 25.4 0 --> 12.7 2580 1140

12.7+ --> 25.4 2580 1100

25.4+ --> 50.8 0 --> 25.4 10300 1070

24.5+ --> 50.8 2580 1030

24.5+ --> 50.8 10300 970

50.8+ --> 76.2 0 --> 25.4 10300 1030

24.5+ --> 50.8 3870 1000

50.8+ --> 76.2 7740 970

76.2+ --> 101.6 0 --> 25.4 10300 1030

24.5+ --> 50.8 10300 970

50.8+ --> 101.6 10300 900

Heat treatment (STA = Solution Treated and Aged):

1750 oF for 1 hr; Water quench (time from furnace to water should be less than 6 seconds); 900 - 1100 oF for 4-8 hrs; Air cool.

Data taken from RMI literature (p. 13). "Facts about Titanium: RMI 6Al-4V" Ti-6Al-4V

Also see TIMET fig 8, p. 7.

Appendix — data scatter

Data scatter is a condition in which "identical" test conditions produce different results from sample to sample. Data scatter depends heavily on both the care with which the testing is performed (reliability) and the phenomena that is being tested.

Consider the example of IQ (intelligence quotient) testing, which will illustrate the problems involved. If an IQ test is poorly controlled (unreliable), then an individual who takes the same test on several different days may produce significantly different scores, even though his intelligence has actually varied as indicated by the scores. However, no matter how well the test is controlled, different people who are otherwise closely matched will still have different scores. This is simply the nature of IQ testing.

In these respects, fatigue testing is similar to IQ testing. For best results, the testing must be very carefully controlled. However, even when "identical" test conditions are imposed on "identical" specimens, the lives will not be the same.

Data scatter due to uncontrolled (unreliable) test conditions

Amplitude variations

Amplitude problems fall into two categories: amplitude control and amplitude uncertainty.

1. Amplitude control. As can be seen from a typical S-N curve, a small change in amplitude will cause a large change in horn life. For example, a 5% increase in amplitude significantly decreases the the life of aluminum and titanium tested by the BSP Acoustic Research Group:

Material Decrease in life (%)

7-4 Titanium rod 36

Aerospace Aluminum 76

Thus, test horns that were thought to be run at identical amplitudes (but whose actual amplitudes were slightly different) will have significantly different lives.

Unfortunately, the equipment currently used by BSP for fatigue testing does not permit precise amplitude control. Converter amplitude (which controls horn amplitude) may vary due to fluctuations in line voltage to the power supply {see F. Dibble for data} or because of converter heating during running, which causes the converter amplitude to increase{see E. Hallabeck, G. Coles, R. Berkemann}.

2. Amplitude uncertainty. Even if the horn amplitude could be exactly controlled (i.e., once set, the amplitude would not vary), the actual value of the amplitude would still not be precisely known. This is because the actual amplitude may be different than the the measured amplitude due to calibration errors and nonlinearity errors in the amplitude meter. In addition, the measured amplitude depends on the care with which the operator zeros and reads the meter. Also, the A-450 amplitude meter is sensitive to how hard the amplitude probe is pressed against the horn face.

These considerations mean that if the amplitude meter indicates 125 microns, the actual amplitude could be either lower or higher. Since the actual amplitude is not precisely known, the horn stress will also not be precisely known. Thus, when the data is analyzed, some uncertainty will exist as to exactly what stress caused the horn to fail.

Given these factors, it is unlikely that the horn stress will be known more closely than +/- 5% {verify}.

Variac setting

Some limited testing indicates that variac setting of the power supply has an effect on fatigue life. (The data is given in table  .) It can be seen that higher variac settings give somewhat longer life. Life may be improved at higher variac settings because the power supply has more power available to extend the fatigue crack to a greater length, while at lower variac settings the power supply shuts down earlier (i.e., at shorter crack lengths).

Using 184V power supplies (20 kHz, watts), tests run after approximately 2/81 (much of the 7075 aluminum testing) have used variac settings limited to between 90 and 100%. All of the earlier testing (which includes the titanium testing) did not have this restriction.

Other effects

Humidity (oxidation at the crack tip: probably in D. Lovetts book on fatigue), surface finish, bar to bar and heat to heat variations (perhaps should be given below)

Data scatter due to the inherent fatigue mechanism

When test conditions are very carefully controlled, significant data scatter will still occur due to the inherent nature of the fatigue mechanism. This is because fatigue is to some extent a random process that depends not only on the gross (overall) properties of a material, but also on properties at very localized levels:

"At the atomic and crystalline levels of association where fatigue damage is initiated, all metals are heterogeneous {define}. The fatigue resistance of one small elemental volume will differ from the resistance of another owing to the presence of such factors as inclusions, differences in grain size, orientation, spacing of hardening particles, microresidual stresses, and so on. Thus any real metal may be considered to have a large number of weak points of varying severity distributed throughout its volume. ... Because of this chance distribution of nucleation points [where microcracks start], the fatigue limit {define} and the fatigue life {define} at a given stress will vary from specimen to specimen." {Ref. 1}

Juvinall[1] {p. 352} states that "even under controlled laboratory conditions the lives of presumably identical samples can easily vary by ratios of 10 or even 100 to 1."

Figure [17.7, p. 353, Juvinall[1]] shows the data scatter of 75 S-T aluminum tested by conventional means at low frequency. It can be seen from this graph that the amount of scatter depends (for this particular material) on the stress level at which the horn is tested. For instance, at a stress of 62 ksi (thousand pounds per square inch), the lives between 5% probability of failure and 95% probability of failure vary only between 1.4x104 and 1.9x104 cycles, a difference of 50 thousand cycles. However, when the stress is decreased to 30 ksi, the lives vary between 2.5x106 and 8x107 cycles, a difference of about 75 million cycles. {See Manning.}

Based on the above figure, Juvinall states that "the standard deviation of fatigue strength in the finite-life range tends to diminish as life is reduced. As a limit, short-life fatigue tests approach static tests, for which statistical variations are relatively small." [p. 351] For the aluminum and titanium tested at BSP, the data scatter appears to follow this trend (less data scatter at higher stress and lower life). For aluminum, this may occur because of the quasi-knee in the S-N curve. {See Juvinall, fig. 11.13, p. 216}

Machining dependence

TABLE [t] : S-N VARIAC TESTS

Purpose

These tests were run to determine the effect of power supply variac setting on horn life.

Conclusion (95% confidence level)

Higher variac settings produce longer horn life.

Test Conditions

Material: Martin Marietta 7075 QQA/225 (T6) Rod

Finish: 30 micro-inch (approximate)

Amplitude: 152 microns (6.0 mils)

Duty cycle: 1 sec on, 4 sec off

Horn cooling: Fan

Power Supply: 184V

Testing period: 1/81 - 2/81 (approx.)

Variac Life Log

Description Test # Bar # (%) (hrs) Life

High Variac 81-13 1 91 9.18 0.962

81-14 1 93 6.67 0.824

81-15 1 98 7.30 0.863

81-16 1 85 14.88 1.173

81-17 1 98 14.09 1.149

81-18 1 92 8.87 0.948

81-9 1 100 7.56 0.879

81-19 1 100 7.54 0.877

81-23 2 100 4.39 0.642

81-24 2 100 7.05 0.848

81-25 2 100 11.07 1.044

96 8.96 0.928

(0.152)

Low Variac 81-7 1 42 7.28 0.862

81-8 1 51 5.20 0.716

81-12 1 37 9.25 0.966

81-10 1 50 4.60 0.662

81-11 1 63 6.29 0.799

81-20 2 60 4.50 0.653

81-21 2 54 4.89 0.689

81-22 2 56 7.68 0.885

52 6.21 0.779

(0.116)

Notes:

1. Numbers in each column directly below the solid line are averages.

2. Numbers in parentheses are standard deviations.

TABLE [t] : S-N BAR AND HEAT TESTS

Purpose

To determine if different heats (NOT different heat treatments) have an effect on horn life.

Conclusion

Within the scope of this testing, different heats do not seem to have an effect on horn life.

Test Conditions

Material: Martin Marietta 7075 QQA/225 (T6) Rod

Finish: 30 micro-inch (approximate)

Amplitude: 152 microns (6.0 mils)

Duty cycle: 1 sec on, 4 sec off

Horn cooling: Fan

Power Supply: 184V

Variac: > 90%

Testing period: 1/81 - 6/81 (approx.)

Variac Life Log

Test # Bar # Heat # (%) (hrs) Life

81-56 1 1 93 2.84 0.453

81-57 2 1 93 1.26 0.100

81-59 2 1 93 1.83 0.262

81-60 2 1 92 1.21 0.083

0.148

(0.099)

81-46 3 1 100 1.97 0.294

81-47 3 1 100 1.76 0.246

81-48 3 1 100 1.78 0.250

81-65 3 1 90 2.44 0.387

0.294

(0.066)

81-50 4 2 90 3.41 0.533

81-51 4 2 93 2.82 0.450

81-52 4 2 92 3.26 0.513

81-61 5 2 92 1.38 0.140

81-63 5 2 96 2.51 0.400

81-64 5 2 96 2.52 0.401

0.314

(0.150)

81-43 6 3 100 2.46 0.391

81-44 6 3 100 2.03 0.307

81-45 6 3 93 2.69 0.430

0.376

(0.062)

81-62 7 3 92 4.04 0.606

81-66 7 3 96 3.21 0.507

81-67 7 3 93 3.86 0.587

0.567

(0.053)

Notes:

1. Numbers in each column directly below the solid line are averages.

2. Numbers in parentheses are standard deviations.

Notes

If a material has significantly different fatigue strength in 2 directions, but principal stress doesn't act in one of these directions, then may be difficult to predict fatigue life. e.g., titanium with diagonal crack at slot end.

Fatigue: A process where a material fails because of progressive crack growth due to repeated cyclic stress. The stress at which failure occurs is considerably below the material's normal breaking strength, and quite often below the yield strength.

Fatigue strength (fatigue limit): The stress level below which a material {Material refers to an S-N test, not a fatigue test of the final part.} can be stressed an infinite number of cycles without failure. For steel and titanium, this is a specific value. Most nonferrous alloys do not have a fatigue strength. A "pseudo-fatigue strength" {See Manning, Ultrasonic Fatigue} is defined as the stress at which a material will live a given number of cycles, usually 500 million. {Verify.}

Fatigue limit:

Notes:

1a. 3 extruded aluminum S-N horns broke in the radius. 1 Alcoa Cressona (81-76) and 2 MM. Any other radius breaks out of all testing? Does this indicate that extruded material is especially notch sensitive?

1b. Grosskreutz, p. 24, "One of the more important environmental constituents is water vapor which has a strong effect on the fatigue of aluminum and its alloys."

1c. For aluminum powdered metal alloys 7090 and 7091, "The two PM alloys also have 35 to 40% higher notched axial fatigue strengths than 7050, 7075, and 2024 alloys at 106 or more cycles." Robert H. Graham, "Putting More Muscle in Aluminum Alloys", Machine Design, January 12, 1984, p. 126.

1d. "In general, cracks develop sooner in the more ductile materials. This trend is best seen by comparing the notched 2014 and 2024 aluminum with either the 7075 aluminum or the 4340 steel in Table I." Grosskreutz, p. 24. "Lower da/dN is obtained by raising the modulus and ultimate tensile strength. Also, materials which have a high rate of strain hardening have a greater resistance to fatigue crack growth." p. 28

1e. Elementary linear regression assumes that the data scatter remains constant. However, this is not true for many fatigued materials. Therefore, must use care when using linear regression. According to ASTM standard E739-80, "An assumption of constant variance is usually reasonable for notched and joint specimens up to about 106 cycles to failure. The variance of unnotched specimens generally increases with decreasing stress (strain) level." (Little, p. 132) Young (p. 58) says that unnotched specimens show less variance than notched.

Spindel, p. 95: If standard deviation is not constant, then "the regression and conficence limits calculated wil be distorted."

Spindel, p. 105: "The least squares method of fitting S-N curves with a cutoff point is, therefore, unsatisfactory for data for which the sloped part of the S-N curve is not significantly determined."

Least squares does not give physically the most acceptable line, especially in changing from relatively steep slope to a flat slope. Therefore, use additional criterion: XSD on log scale remains const along the curve. Spindel, p. 112:

1f. Distortion energy correlation with fatigue fracture in ductile materials for torsional load, where biaxial stresses are involved: Juvinall[1], p. 230. "It seems reasonable that the distortion energy theory should corelate well with the fatigue fracture of ductile materials because (1) this theory fairly accurately predicts the beginning of static yielding of ductile materials, and (2) fatigue fracture of ductile materials begins with highly localized yielding."

1g. Coaxing. Juvinall[1], p. 218. Grosskreutz, p. 40.

1h. Juvinall[1], p. 216. Testing of Al wrought alloys in the stress range of 16 - 25 ksi is in the range of technological transition. (Fig. 11.13) Therefore, could have much scatter.

1i. log(S)-log(N) slope in range of finite life:

3.75 for welded steel joints (Haibach, p. 29, 35, Little book.)

Spindel (Little book):

7.75 (fig 2, p. 93)

3.75 - 4.08, with confidence limits of 3.12 --> 5.05 (fig 4, p. 94)

Additional data, pp. 104 - 113. Slopes generally in the range of 3 - 7.

Maennig[1]:

Suggests slopes of 6 --> 9 "preferably with higher values" (p. 625)

1l. Swanson, p. 91 "Fatigue in a homogeneous material is essentially a surface dependent phenomenon and the great majority of fatigue failures originate at the surface. Too much emphasis, therefore, cannot be placed on the importance of specimen preparation in this respect." "... no ASTM standards for fatigue specimen preparation are available at the present time ..." p. 92

Storage: "In contrast to the deleterious effects of corrosive environments, the fatigue strength of many materials is increased by protective chemical coatings or testing in so-called inert atmospheres. Organic oils and greases usually are found to have this effect, particularly those that tend to form tightly adherent surface films. These effects emphasize the importance of maintaining surface conditions constant during fatigue testing and the desirability in some cases of duplicating service conditions in the laboratory to obtain accurate estimates of service performance." p. 100

Cooling: be careful not to introduce contaminates.

Detection of fatigue cracks:

U/S: sensitivity of +/- 0.010 inch. p. 130

Eddy current probes.

Corrosive environments:

"With some materials such as the aluminum alloys, the fatigue behavior is influenced to a marked extent by the amount of water vapor in the air. For this reason, it is well to control the humidity of the fatigue test environment within reasonably; close limits in order to minimize scatter of the results due to this factor." p. 136 "The frequency of the testing machine, which is of minor importance in conventional tests, has a marked effect on the fatigue life under corrosion fatigue conditions."

Fretting:

"Fretting is the phenomenon that occurs when the surfaces of two solids are pressed togehter by a normal force and are caused to undergo small cyclic relative sliding motion. Usually, the normal force is large enough and the cyclic sliding motion small enough to significantly restrict the flow of debris away form its place of origin at the interface. Fretting usually gives rise to one or more forms of damage including (1) fretting corrosion, (2) fretting wear, or (3) fretting fatigue or both." p. 147 Damage associated with fretting corrosion and fretting wear include discoloration, pitting, oxide layer build-up, loss of fit, and seizing.

1m. May not make sense to have extermely carefully controlled fatigue tests if the field failure data (e.g., exact ampl. of horn) is limited. However, accurate fatigue data at least assures that unexpected field failure variability is not due to variability in the material -- i.e., the failures are caused by some other problem.

1p. Much check diameters to make sure that they are not close to flexural resonance. Direct dimensional scaling between frequencies may not be acceptable, since flexure half-wavelength does not vary inversely with the frequency.

1q. Fatigue of Ti bar horns at slots: tensile stress at slots is neither in longitudinal nor long-transverse material directions -- i.e., crack is at angle to stud axis. Therefore, neither the longitudinal nor LT S-N tests may give exactly correct fatigue predictions for working Ti bar horns.

3. Effect of environment: corrosion (caustics), water and humidity. See Grosskreutz, pp. 40 - 42.

4. Based on the desired confidence level, how to determine the required number of samples to test.

5. Fig. 11.13 (Juvinall[1]) shows a mild knee for aluminums. Therefore, straight line extrapolation of log-log data may be risky unless you know on which portion of the curve you are operating. I believe the data from 2024 does not plot well as a straight line on log-log paper. However, using a straight line extrapolation will always cause error on the conservative side -- i.e., the life will always be longer than predicted by the straight line extrapolation. Also see Maennig[1], fig. 2, p. 613.

7. Miner's Rule for fatigue at multiple stress levels. -- e.g., for testing 1 horn at multiple amplitudes.

8. Difficulty in estimating horn life is the final booster selection is unknown -- do not know the working amplitude.

10. However, stresses are often not easily determined (strain gages, photoelasticity, and other experimental approaches only measure surface stresses) while finite element analysis is not available to everyone and is not extremely accurate (perhaps +/- 5%). Also, reliable data on the material fatigue properties may not be available (S-N curve may not be fully developed, or the material may have changed since the curve was developed).

12. Hencky von-Mises stress (octahedral shear stress):

most indicative of fatigue failure. Give equation for this stress in terms of principle stress or stress in any direction (i.e., when shear is present.)

a. Red SAE book:

Crack initiation -- p. 31

Yield -- p. 33

13. Effect of welding (static) load as compared to U/S stress: negligible. Also, note that the weld load places the horn in compression (at least for simple horns), which should actually increase fatigue resistance.

14. Computer estimation of life based on computer calculated stress: Life testing may reveal other problems (joint problems, flexure, etc.). Note that horns fail because of stress, not amplitude. Thus, 2 horns running at the same amplitude (even though the horns may be of similar design -- e.g. frito 13") may have vastly different lives.

16. Premature failures: Juvinall[1], p. 363, "It occasionally occurs that a single straight line cannot be made to fit with even reasonable accuracy all the test points. Most commonly, the first few failures will occur prematurely with respect to the life pattern established by the later failures. This usually indicates a contaminated population, i.e., a population containing a certain percentage of defects which are distinct from the statistical variations in strength existing within the population as a whole."

18. Distribution of fatigue strengths for a given number of cycles:

Shigley, p. 169: distribution is normal.

Others?

20. Maennig[1] (p. 624):

1. Test 10 samples at the desired stress. The shortest life for the test group will be nmin and the longest life will be nmax.

2. If the ratio of nmax to nmin is greater than 10:1, then increase the number of samples to 15.

3. If the ratio of nmax to nmin is greater than 30:1, then increase the number of samples to 20.

4. If higher reliability is needed, then the number of samples should be doubled.

22. Saunders, ASTM STP 511, p. 186 - 187

Comments on establishing fatigue tolerance intervals for variable loads with Miner's rule.

Variables affecting life prediction for production horns:

4. Manufacturing and use variations (e.g., nicks, torquing of tips)

24. Juvinall[1], p. 201. "... a perfectly brittle material (having no ability to slip, even on a submicroscopic level) should never fail in fatigue. Experimentally, this tendency has been observed in glass and in certain ceramics."

25. Anton Puskar, p. 225. For steels, the crack propogation rate (mm/cycle) were about 10x higher at 70 Hz than at 22 kHz.

26. Kuz'menko, p. 27, fig. 19: Measures loss by temperature increase of cooling fluid. Shows very sudden increase in loss of Al at failure. Similar results reported by Mason?

27. Wirsching, pp. 89 - 90: Comments on accuracy of Palmgren-Miner (PM) hypothesis for fatigue life under variable load.

28. Bittner, p. 265: "Alloy content, decarburization, and hardness were almost completely overshadowed by surface condition, and endurance limits were only a fraction of the intrinsic values."

p. 266: steels hardened to around 400 Brinell.

29. Manson, pp. 259 - 262: effect of ductility, modulus, strength and hardness on fatigue life.

29. Coffin, Ultrasonic Fatigue, discusses effect of environment vs test frequency. Especially see summary, p. 438 - 439.

MATERIALS

Ceramics -- E. Harris ceramic rod.

Xmean,1 - Xmean,2

_) t = -------------------------------------------------

ÖÖ (N1 - 1) XSD,1 + (N2 - 1) XSD,2 ÌÖ 1 1 Ì̽

°° ------------------------------- °° -- + -- °°

ÛÛ N1 + N2 - 2 ìÛ N1 N2 ìì

where Xmean = mean fatigue life of specified

material

XSD = standard deviation of fatigue

life of specified material

N = number of samples of specified

material

1, 2 = specified material

Notes — Detramental residual stresses

Study of Hard Coating for Aluminum Alloys, F. G. Gillig, Cornell Aeronautical Laboratory, Inc., WADC (Wright Air Development Center) Technical Report 53-151, USAF Contract 18(600)-98, May 1953:

Tests of Martin Hard Coat (MHC) of Glenn L. Martin Co., Baltimore, MD. Also mentions 2 other processes: Alumilite Hard Coating process, ALCOA, and Hardas process, Hard Aluminum Surfaces Ltd., Glasgow, Scotland. All are electrochemical, producing a thick layer of aluminum oxide. These differ from normal anodizing in that they are performed at higher current densities and low temperatures with considerable agitation. (p. 3) Coating thicknesses: MHC: up to 6 mils; Hardas: up to 10 mils; usual: 2 to 4 mils. (Normal anodizing: typically 0.1 mil to 0.8 mil.) (p. 5)

Wear resistance of MHC, as measured by Taber Abraser: better than hardened steel or hard chrome plate. (p. 6) Coating hardness: Approximately file hard; approx. that of nitrided steel. (p. 7)

7 aluminum alloys were tested, including 24S-T4, 61S-T6, and 75S-T6.

Fatigue: sheet, reversed bending until failure or until 10 million cycles. Coating very brittle, with a maze of cracks. (p. 19) Reduction in fatigue strength does not depend on thickness -- 1 mil has about same effect as 5 mils. (p. 20)

Endurance strength

at 106 cycles (psi)

Alloy Uncoated Coated % decrease

24S 19000 15000 21

61S 15000 6000 60

75S 22000 9000 59 (p. 21)

Fatigue curves are given.

Notes — Shot peening and beneficial residual stresses

SAE Fatigue Design Handbook:

"Near the fatigue limit (that is, long fatigue life) the residual stress remains practically unchanged by the fatigue loading. At stresses above the fatigue limit, residual stresses may relax as an accompaniment of the fatigue process, thes effect being greater in 'soft' materials and at stresses will above the fatigue limit." "A 'hard' material is one that has been hardened by working or by heat treatment. Conversely, a 'soft' material is one that has been annealed or softened." p. 30 (Also see Juvinall[1], p. 330)

"The significance [of residual stress] depends upon the stress concentration, the initial magnitude, and the amount of relaxation or change in residual stress during fatigue." p. 64

"Aluminum alloys generally show a decided response to mechanical prestressing techniques such as shot peeening and surface rolling." p. 52. Cites reference G. A. Butz and J. O. Lyst, "Improvement in Fatigue Resistance of Aluminum Alloys by Surface Cold-Working", Materials Research and Standards, vol. 1 (Dec. 1961), pp. 951 - 956. "In certain situations, the fatigue strength of a specimen type may be more than doubled by changing surface stresses from tension to compression." p. 52

notes — Possible reasons for poor BSP Ti strength

1. Inferior Ti. Improperly fabricated or heat treated. Could explain why Chinese 7-4 was better than Timet 7-4, which cannot be explained by any of the reasons below. Note: Chinese Ti ran noticibly cooler.

2. Frequency effect (lower fatigue strength at higher freq.) However, Willertz did not find a freq. effect for Ti-6Al-4V.

3. Residual machining stress. Could explain why EDM had same life as non-EDM.

4. Surface finish. (32 rms vs polished) Ti is very notch sensitive. Juvinall[1], p. 235 "It should be noted that polished, ground, and machined surfaces give significantly higher endurance strengths when the surface markings produced are parallel to the loading."

5. Stock size effect. See RMI Ti-6Al-4V lit, p. 10. Size effect may account for why 40 kHz cyl horns give better fatigue than 20 kHz of equivalent design. Note, however, that the min ultimate tensile strength of RMI Ti-6Al-4V is 130 ksi, 50% of which still gives 65 ksi fatigue strength.

6. Undetected flexure, which adds additional unaccounted stress.

7. Load const CL. Standard tests are rotating bending. For steel, endurance limits for axial loading are 90% of those for rotating bending (i.e., CL = 0.9). However, this still only reduces the 65 ksi fatigue limit given above to 58.8 ksi.

8. Other: cutting fluid reacted with Ti

9. Different length of tests. Most low frequency tests are stopped if failure does not occur by 107 cycles = 0.14 hrs at 20 kHz. However, see Willertz work.

10. Incorrect amplitude measurement --> incorrect stress.

11. Cycled tests (startup and shut-down) may cause some type of transient instability, which causes higher stress -- similar to starting a spring-mass system, which then may pass through resonances before attaining final operating frequency. This possibility could be determined by analyzing data from cycled vs continuous tests.

12. Chinese vs TIMET 7-4: Possibly tested at different times of year. Therefore, different humidities?

Notes — Etching

For titanium:

Bowen (7), p. 1273: 10% HF, 25% HNO3, 65% H20

p. 1278: 1% HF, 12% HNO3, 87% H2O

Marshall Kessler from RMI (2/11/88):

3-5% HF 30-35% HNO3 Remainder H2O

Timet (Properties and Processing of Ti-6Al-4V, 1986):

For pickling, 7:1 ratio of HNO3:HF (e.g., 5% HF ;35% HNO3 ; 60% H2O), p. 26

For pickling, 7:1 ratio of HNO3:HF, with 2-5% HF and 15-35% HNO3: inhibits hydrogen absorption and brightens metal, p. 12

For metallographic etching, 1% HF 12% HNO3, with water as the remainder. A more active etchant can be achieved by reducing the HNO3 to as little as 3%. These titanium etchants are known as Knoll's etch. p. 8

notes — Low frequency testing cycle rate

Bittner, p. 272: resonant testing of steel supported at node (Rayflex machine) at 6000 to 18000 cycles/minute. "A bar supported at the nodes is vibrated at its natural frequency (from 6000 to 18,000 cycles per minute) and the stress in claculated from the amplitude of vibration." Probably flex vibration?

Sinclair, p. 868: 8000 rpm.

Bartlo, p. 147: 8000 rpm rotating beam.

Bowen (7), p. 1272: 6000 rpm rotating cantilevered beam.

L. E. Willertz (U/S Fatigue, p. 128-130): 88 Hz (5280 rpm)

Reactive Metals Inc. literature: 1850 cpm (cycle/min)

notes — Data on Ti fatigue

Aerospace Structural Metals Handbook, vol. 4

Ti-13V-11Cr-3Al:

Code 3713, p. 20, Fig. 3.051: 0.678" bar, ST [probably Solution Treated], rotating beam:

Unnotched at approx 2*106 cycles: ~65 ksi, S-N curve nearly horizontal but no runouts

Kt = 3.9 at approx 4*107 cycles: ~30 ksi, 1 runout

Ti-6Al-2Sn-4Zr-2Mo:

Code 3718, p. 83, Fig. 3.055: 1-1/8" dia bar, Duplex anneal: 1775F, 1 hr, AC + 1100F, 8 hr, AC; Tensile = 146 ksi, unnotched, rotating beam

at room temp, 107 cycles: ~74 ksi, 2 runouts

at 900F, 107 cycles: ~63 ksi, 2 runouts

Ref. 15: D. N. Torrell, "Data on Ti-679 and Ti-6Al-2Sn-4Zr-2Mo Titanium Alloys", Pratt and Whitney Aircraft, West Palm Beach, Florida, 12 August 66, Referenced in DMIC Dat Sheet January 1967.

Code 3718, p. 84, Fig. 3.056: 1-1/8" dia bar, Duplex anneal: 1650F, 1 hr, AC + 1100F, 8 hr, AC; Tensile = 146 ksi, unnotched, rotating beam

at 900F, 107 cycles: ~63 ksi, 2 runouts

Ref. 15

Code 3718, p. 84, Fig. 3.057: 3/4" dia bar, duplex anneal: 1750F, 1 hr, AC + 1100F, 8 hr, AC; Tensile = 146 ksi, unnotched, unnotched, 13.0 kHz

at room temp, 5*108 cycles: ~38 ksi, 0 runouts, curve approx flat from 105 cycles

at 900F, 107 cycles: ~29 ksi, 0 runouts

at 900F, 109 cycles: ~27 ksi, 0 runouts

Ref. 24, Andrew Conn, Hrdronautics. (See below.)

Note possible frequency effect.

Ti-11.5Mo-6Zr-4.5Sn:

Code 3722, p. 38: 1/2" dia bar, hot rolled + 1420F, WQ + 900F, 8 hr (tensile = 200 ksi)

Axial load, tension (sinusoidal), R = 1.0 [should be -1.0?], 1800 cmp: approx 150 ksi at 107 cycles, 3 runouts

Rotating beam, R = -1.0, 10000 cpm: approx 78 ksi 5*107 cycles, 3 runouts

Note possible frequency effect.

-------------------

Andrew F. Conn and A. Thiruvengadam, (Hydronautics, Inc., Rindell School Road, Laurel, MD, (301) 792-8654) "Experimental Research on High Frequency Fatigue and Dynamic Tensile Tests at Elevated Temperatures", NASA contract NAS3-11168, July 1969.

Ti-6Al-2Sn-4Zr-2Mo, STA (p. 1). Timet, heated to 1750F for one hr and air cooled to room temp, followed by 1100F for eight hrs and air dooled to room temp.

Static E = 16.0*106 psi (data from TIMET), dynamic E = 16.7*106 psi (from high freq fatigue measurements). Table 2, no page #.

Time to failure: crack causes increased power, so that power supply could no longer drive specimen. Crack length approx 1/2 of cross-section at shutdown. Test stopped if specimen survived 109 cycles. pp. 11-12.

Predicted life calculated using four-point correlation method and method of universal slopes. Calculations made using both static and high-strain rate tensile strengths. pp. 14-15

Specimen: 0.375" major dia, 0.125" gage dia, 0.625" R in gage section, 32 finish. Resonant freq 13.0 kHz. Fig 3. Strains calculated with Meppiras eqn and verified with strain gage measurements. p. 23. End (antinode) ampl measured with voice coil, calibrated against microscope with accuracy of ±0.02*10-3 inch. Claimed error of less than 1%. pp. 24-25.

Specimens cooled with flow of tap water on node. "By comparing the fatigue behavior using distilled water as a coolant, it has been shown that tap water has no influence on the fatigue life of the metals tested." p. 25. Calculations showed thermal gradients of less than 90F across the section. p. 26.

Stress amplitudes: 38.1 (4 samples), 39.3 (4), 40.5 (3), 42.5 (4), 44.3 (4), 45.8 (3) ksi. Table 3c. Extreme scatter in data, typically 102 to 103 at each stress level. Table 3c and fig. 12. Life essentially independent of stress. [flex?]

Torrell's work (see above) on the same alloy is noted, but Conn does not mention the fact that Torrell's fatigue strengths were much higher.

-------------------

Anton Puskar, U/S Fatigue, p. 223:

Low carbon unalloyed steel (0.07%C): axial fatigue tests performed at 22 kHz and below: fatigue limit does not depend on the type of water used (technical water, distilled water or water treated with a corrosion inhibitor). However, fatigue limit strongly decreased if urn in 3% NaCl in water or in 0.5% HCl in water.

Fatigue limit decreases with increased diameter (3.5 to 6.5 mm). [Not known if this is the raw stock dia.]

-------------------

L. J. Bartlo (Mgr. Product Development Dept., RMI), Fatigue at High Temperature, ASTM STP 459, 1969.

Material: Ti-6Al-4V 3.5" dia, forged to 1.25" dia at 1750F, conditioned, and hot rolled to 0.625" dia at 1650F. (p. 145)

Specimens polished with 600 grit silicon carbide paper (p. 145). A reference is given.

Note: vertical axis labeling of 3rd graph of fig 3 is incorrect.

Reactive Metals 6Al-4V bar, various conditions: Fatigue limit varied from 65-100 ksi at 107 cycles, depending on heat treatment.

For specimen made from original 3.5" diameter bar and annealed at 1350F and air cooled: coarse grained plate-like alpha (p. 152). Unnotched: fatigue strength of 63 ksi (fig 8) [61 ksi, fig 9] compared to 91 ksi (fig 3) [90 ksi, fig 9] for 0.625" dia. Tensile strength of 135 ksi compared to 154 ksi for 0.625" dia.

For specimens made from 0.625" dia bar: Best were annealed at just below beta transus temperature (1840F) and water quenched (fatigue strength ~ 100 ksi). Aging for 4 hrs at 1000F did not affect fatigue strength but did increase tensile strength from 160 ksi to 172 ksi. Worst was annealing just above beta transus and then furnace cooling (fatigue strength ~ 65 ksi). Next worse was annealing just above the beta transus and then air cooling (fatigue strength ~ 73 ksi).

In all cases there was a sharp decrease in ductility (% elongation) when the annealing temperature passed the beta transus temperature.

"Each of the S-N curves exhibited a fatigue limit." p. 147

"Endurance ratios varied from 0.42 to 0.62 for the unnotched condition." p. 152

Referring to the original 3.5" diameter material and heat treatments which produced similar structure (air or furnace cooling from the beta annealed condition): "It is therefore likely that heavy sections of Ti-6Al-4V, which contain coarse plate-like or even coarse equiaxed alpha, can be expected to have endurance ratios of 0.4 to 0.45."

"Fine grained alpha-beta structures or structures produced by water quenching or quenching and aging can be expected to have endurance ratios between 0.55 and 0.62." p. 152-153

"The data included in the upper band also show that the fatigue strength generally increases with increasing tensile strength. Weingerg and Hanna [3] observed a similar trend in other titanium alloys. This trend does not appear to hold true for the notched condition as evidenced by the scatter present in the lower band. The notched endurance ratios varied between 0.17 and 0.3." p. 153

"No significant trends were observed in comparing the notched fatigue data and microstructures."

Notch sensitivity: Fig 3, 1350F for 2 hrs (see fig 9), then air cooled: For original 3.5" diameter sample, a Kt of 3.5 lowered the fatigue strength from 62 ksi to 34 ksi ==> Kf = 1.8. For 0.625" diameter sample, a Kt of 3.5 lowered the fatigue strength from 91 ksi to 40 ksi ==> Kf = 2.3.

For Kt = 3.5, Kf = 2.3 .. 2.6 (See my calcs in Bartlo article.)

--------------------

Juvinall[1], p. 218:

"Titanium and titanium alloys differ from other nonferrous alloys in that they tend to exhibit a true endurance limit in the range of 106 to 107 cycles, with S-N curves similar in shape to those for steel. Furthermore, the endurance ratio for titanium tends to be even higher than for steel, usually ranging from 0.45 to 0.65 [2e, 6, 9]. This means that several of the stronger titanium alloys have endurance limits in the 90-ksi range."

"Reliable fatigue data have been relatively difficult to obtain with titanium alloys. Machining problems plus a high sensitivity to surface irregularities have made it difficult to produce consistent titanium specimens for testing. Also, the fatigue strength of these materials appears to be significantly affected by heating due to hysteresis and by cyclic stress redistribution due to localized yielding."

--------------------

Kuz'menko:

Best Ti alloy tested (designated VT22, ·85% Ti, 2.5% Al, 7.5% Mo, 1.0% Fe; heated for 2 hours at 800oC, cooled in air): endurance limit of approx 60-62 kg/mm2 at frequencies between 16 Hz and 10 kHz. (His fig. 5 and table 3.) Note: kg/mm2 is not units of stress. Possibly should multiply by 9.8 m/sec2, which then gives 590-610 MPa.

--------------------

A. W. Bowen (7): (Ti Science and Technology)

Forged 9-1/4" wide x 2-1/4" thick Ti-6Al-4V bar, annealed for 2.5 hrs at 700 oC, tested at 100 Hz, rotating cantilevered: approx fatigue strength at 107 cycles:

Test Tensile Fatigue

direction Hardness Modulus strength strength %

Longitudinal ? 16.5 132 ksi 72 ksi 55

Transverse ? 18.7 143 ksi 62 ksi 43

Short transv ? 16.5 142 ksi 82 ksi 58

Variations in hardness from point to point, which Bowen attributed primarily to crystallographic differences (p. 1274). Also, variations in texture along and through the thickness of the bar, "due to inhomogeneous deformation caused by temperature gradients or inhomogeneous working introduced into the bar during forging". (p. 1274)

Extent of stage I crack growth seemed to be dictated by the applied stress amplitude for all 3 test directions. (p. 1274) 2 supporting references given.

Differences in fatigue lives for different directions were attributed to differences in crack initiation and stage I growth, rather than differences in stage II growth. (p. 1276) Life differences were caused by differing texture and grain shape in each direction.

"... test pieces oriented with a large proportion of prism planes approx parallel to the stress axis had shorter fatigue lives than those with a large spread of orientations or with a large proportion of basal planes approx parallel to the stress axis." (p. 1276)

Scatter: "The considerable scatter in Fig 2 can be attributed to the variable texture and coarse microstructure common to all three directions. Variable surface finish is not the cause of scatter ..." (p. 1279) "Small local variations in oxygen content are also a possible source of scatter." (p. 1279)

--------------------

L. E. Willertz (U/S Fatigue, p. 128-130): Ti-6Al-4V (RMI heat #896887, 593 oC (1100 oF) for 10 hrs in vacuum after machining), tested in air at room temp at 88 Hz, runouts at 107 cycles:

Test Tensile Fatigue

direction Hardness Modulus strength strength %

? 36.2 Rc 17.2E6 143 ksi ~ 50 ksi 35

--------------------

L. E. Willertz (U/S Fatigue, p. 333-348): Ti-6Al-4V bar, tested in pure water at 80 oC at both 100 Hz and 20 kHz (little life difference), max life at approx 2*108 --> 109 cycles. See fig. 7, p. 341, table III, p. 343, table IV, p. 345

Test Tensile Fatigue

direction Hardness Modulus strength strength %

(a) -- -- 919 MPa 444 MPa 48

(133 ksi) (64 ksi)

(b) -- -- 980 MPa 510 MPa 52

(142 ksi) (74 ksi)

(c) -- -- 914 MPa 480 MPa 53

(133 ksi) (70 ksi)

(d) -- -- -- 470 MPa

(68 ksi)

(a) = Coarse grained equiaxed Â-ß, produced by air cooling from the Â-ß region of the phase diagram. (p. 342)

(b) = Fine grained lamellar Â-ß, produced by quenching from the ß region of the phase diagram. (p. 342)

(c) = Bimodal

(d) = Mill annealed

All Ti are stress relieved.

Above data from tables I and III and p. 342.

"In both cases it is apparent that the low frequency and high frequency data points form continuous fatigue curves for each [material] structure." (p. 342)

Above properties at 80 oC

--------------------

R. Ebara (U/S Fatigue, p. 349-364): Ti-6Al-4V, tested in distilled water 15 kHz, max life at approx 109 cycles (Fig 10, p. 361):

Test Tensile Fatigue

direction Hardness Modulus strength strength %

(?) -- 113.27GPa 954.5 MPa ~550 MPa 58

(138 ksi) (80 ksi)

C = 6070 m/sec

Shot peening over length with 0.0011A Almen intensity.

--------------------

Facts about Titanium -- RMI 6Al-4V, RMI Company, Niles, OH 44446

p. 22 "The smooth bar endurance limit of RMI 6Al-4V is 50% or more of the ultimate tensile strength ..."

p. 22 "As with other metals, microstructure and section size may have an effect on the fatigue properties of RMI 6Al-4V. Heavy sections and coarse microstructures may develop somewhat lower fatigue properties withoug any reduction in the ultimate tensile strength."

p. 10 "In heavy sections, it is not possible to remove heat rapidly enough on quenching to achieve significant aging response throughout the cross section." Fig 18 (p. 10) shows a decrease in ultimate strength of 6Al-4V STA bar from ~165 ksi to ~142 ksi (about 14%) as the section size increases from 1" to 3".

Fig 55 (p. 22) shows an endurance limit of 100 ksi for 6Al-4V STA bar at 107 cycles; 74 ksi for annealed Ti-6Al-4V bar at 107 cycles.

-------------------

Titanium Metals Handbook (probably), data sheets supplied by C. Marshall Esler of RMI

p. 5-4:72-23 Annealed Ti-6Al-4V rolled bar, 1-1/4 inches in dia., axial fatigue at 1750 cpm in room temp air, at 107 cycles; values from fig 5-4:27:

Test Tensile Fatigue

direction Hardness Modulus strength strength %

unnotched -- -- 941.4 MPa ~503 MPa 53

(136.5 ksi) (73 ksi)

notched -- -- 941.4 MPa ~283 MPa 30

(Kt = 3.3) (136.5 ksi) (41 ksi)

Unnotched: polished longitudinally with 240, 400 and 600 emery belts.

Kf = 1.78, q = 0.34. Notch radius = 0.010" with 60o flank angle, notch root polished with 600 grit slurry and rotating copper wire.

notes — Ti data

Marshall Kessler

Customer Technical Service

Reactive Metals Inc.

(216) 652-9951

2/11/88

1. Best material for fatigue, available in rod, bar, and plate?

Ti-6Al-4V vs 7-4?

Kessler recommended Ti-6Al-4V. Said that Ti-6Al-4V fatigue strength should always be at least 50% of ultimate strength spec (i.e., 65 ksi min). He could not not understand any lower values, but said might be due to microstructure, raw stock size, oxygen content, etc.

Was not sure about problems with 7-4. Possibly less ductility, more creep.  Also, may be difficult to get uniform moly throughout.

Could not give any specification details for processing Ti-6Al-4V to give the best fatigue strength. Pratt & Whitney uses well-worked Â-ß per their internal spec. E-72

Various grades of Ti-6Al-4V determined by oxygen content:

13% max oxygen

15-20% oxygen: used for turbine blades

More oxygen ==> less ductile.

2. Fracture toughness: better for annealed or Solution Treated and Aged (STA)?

Annealed much better:

for 13% max oxygen: K1c = 60 ksi

for 15-20% oxygen: K1c = 50-55 ksi

where K1c = fracture toughness parameter (RMI p. 12)

B1 bomber required K1c of 70 ksi, which RMI could meet only 75% of time.

3. Aging (p. 11, fig 20): does not affect properties, so why use?

Kessler could not explain.

Industry standard is 1000oF for 8 hours.

4. Recommendations for preparing S-N samples.

Machining depth of cut: Kessler did not know

Surface finish: probably < 16 RMS, then polish with jewelers rouge

Stress relieving: RMI does not stress relieve for fatigue. If stress relieving is used, then 15 min per inch of thickness at 1100oF, air cool. However, possible problem with surface oxidation: may need to polish.

5. Variability of mech properties: No info.

Within same rod or bar?

Withis same plate?

From bar-to-bar in same heat?

From heat-to-heat?

6. Additional info on stock size effect? Did not ask.

7. Additional info on ti microstructure? (alpha/beta phases) Did not ask.

8. Etch solution: 3-5% HF 30-35% HNO3 Remainder H2O

9. Fatigue limit occurs at (X?) cycles. Did not ask.

10. Variability in fatigue data (strength or life)? Did not ask.

3Al-8V-6Cr-4Mo-4Zr:

For samples machined from 6" dia forging transverse and tested at R = 0.1: 87 ksi at 107 cycles. Kessler did not think that this material would be as good as Ti-6Al-4V.

Kessler will send S-N info on Ti-6Al-4V from Ti Alloys Handbook.

-------------------

W. Crichlow

"For Ti-6Al-4V forging, the oxygen content appears to have no significant effect on the static yield and ultimate tensile strengths. It appears from Figure 1, however, that the fatigue strength for Ti-6Al-4V forging, for R = 0.1, increased when the oxygen content decreases, for Kt = 1 and 2.76." p. 1258. Cites his ref 1. His fig 1 shows fatigue strength at 5e6 cycles for oxygen content between 0.10 and 0.19. Decreasing the oxygen content from 0.18 to 0.10 increases the fatigue strength at 5*106 cycles by about 40% for unnotched.

His fig. 11, p. 1268 shows fatigue lives for spectrum loading of large aircraft ring forging. Ti-6Al-4V STA (Â-ß) and Ti-6Al-4V ANN (ß) have about equal lives. Ti-6Al-4V ANN (Â-ß) has about 45% of the life of the other 2 titaniums.

"The scatter between the two titanium alloys [Ti-8Al-1Mo-1v, Ti-6Al-4V] is similar, but for both alloys there is a trend of decreasing scatter in going from single annealed to duplex annealed processing. For Ti-6Al-4V there is a further trend to decreasing scatter when going from duplex annealed to solution treated and aged heat treatments. However, relatigely few test results were available for duplex annealed and solution treated and aged specimens. When more test results become available, the scatter may increase." p. 1266, referring to table III, p. 1267 for notched ti coupons.

References are given, including:

1. Kaplan, M. P., "Evaluation of Effects of Oxygen Content on Mechanical Properties of Ti-6Al-4V Forged Bar," LR 20511, June 2, 1967, Lockheed Aircraft Corp, Burbank, CA.

2. Van Orden, J. M. and Soffa, L. L., "Evaluation of Beta Forged Ti-6Al-4V Forgings," LR 21363, Feb 20, 1968, Lockheed Aircraft Corp, Burbank, CA.

3. Simenz, R. F., Macoritto, W. L., "Evaluation of Large Ti-6Al-4V and IMI 679 Forgings," AFML-TR-66-57, April 1966, US Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio.

5. Beck, E., "Effect of Processing and Fabrication on Axial Loading Fatigue Behavior of Titanium," MCR-68-257 (issue 3), January 1969, Martin Marietta Corp.

11. Simenz, R. F., Macoritto, W. L., "Evaluation of Large Titanium Alloy Forgings," AFML-TR-65-206, July 1965, US Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio.

-------------------

M. Tiktinsky, "Fatigue Characteristics of Titanium Alloy Forgings for Rotary Wing Vehicles", The Science, Technology and Application of Titanium, R. I. Jaffee and N. E. Promisel (eds), Pergamon Press, NY, pp. 1013 - 1021.

Ti-6Al-4V

Only small variations in mechanical properties with grain orientation or section size, in annealed die forging and forged billets up to 6" thick. pp. 1013 - 1014. Fig. 2 shows graph of properties from center to edge of material, in both longitudinal and transverse directions. No general trends.

Fatigue tests: spectrum side loading with mean vertical loading. His fig 3 seems to show same load configuration as Crichlow's fig. 10. Note that Crichlow's co-author (T. Lunde) also worked at Lockheed in Burbank, so these test programs may have been the same. Also, same materials evaluated (Ti-6Al-4V, 4340, IMI 679).

EB welding (p. 1016): For smooth annealed specimens with R = 0.1 under constant amplitude load, annealed: 20% drop in fatigue strength compared to unwelded samples between 105 - 107 cycles, with all failures occurring in the heat affected zone. Fig 7 shows drop from about 56 ksi to about 34 ksi at 107 cycles, which is 39% drop.

Duplex annealing (pp. 1016 - 1017): Specimens taken from 1" to 5" thick material, R = 0.1, Kt = 2.67, transverse grain, unspecified load conditions (spectrum or const ampl?): duplex annealed had about 3 ksi better fatigue strength (compared to mill annealed) at 107 cycles. (about 18.7 ksi vs 15.9 ksi from fig 9). Note Crichlow's comments on duplex annealed scatter.

Effect of forge reductions on Ti-6Al-6V-2Sn (pp. 1017 - 1019): Highest forge reductions gave highest tensile properties. Probably no significant different in endurance limits for various forge reductions (36% - 76%), although 76% Â-ß hammer forging may have slightly better fatigue strength. (See fig 12, p. 1019) Test conditions: STA (1.5" thick coupons, 1625F 1 hr, water quench, age 1100F for 4 hr), center notched with Kt = 2.76, R = 0.1, unknown loading type.

Surface finish (p. 1019): limited data with wide scatter; spectrum testing of Ti-6Al-4V at 50 and 100 RMS, unnotched: 50 RMS gave better fatigue life and less scatter.

Shot peening (p. 1020): limited data with wide scatter; "These curves indicate that only small size shot and low intensities should be used for titanium alloy forgings. Heavy shot or high intensities deleteriously affect the fatigue characteristics of the titanium." 170 shot, 0.007A gave best results.

Ti-6Al-4V beta forged material compared to Â-ß forged (pp. 1020 - 1021): longitudinal direction, annealed material, fatigue strength at 107 cycles, Kt = 1.0, R = 0.1, const ampl tests: equivalent fatigue (69 ksi for Â-ß, 66 ksi for ß); much better EB weld for beta forged material (45 ksi for Â-ß, 66 ksi for ß), which indicates that (for ß forging) "the strength of the weld joint is equal to the base metal". (p. 1021) The strength drop for Â-ß is about 35%.

-------------------

Carl Osgood

Fig 4.49, p. 442: S-N diagram for various Ti, up to 107 cycles, rotating beam, Kt = 1.0:

Material Fatigue strength

6Al-4V 76 ksi

6Al-4V (low oxygen, value not specified) 92 ksi (1)

2Fe-2Cr-2Mo 81 ksi

5Al-1.2Fe-2.8Cr 88 ksi (1)

8 Mn 90 ksi

5Al-1.5Fe-1.4Cr-1.2Mo 100 ksi (1)

4Al-4Mn 105 ksi (1)

4Al-3Mo-1V 124 ksi (1)

(1) ==> only 1 data pt shown at 107 cycles.

4 S-N curves shown in fig 4.49 show 0 slope between 106 and 107 cycles.

A reference number (p. 439) is cited for extensive fatigue data, but the actual reference is not given.

-------------------

John J. Lucas

Ti-6Al-4V:

Axial load S-N specimens had 0.812" radius and 0.280" dia at center, which Lucas says has a Kt = 1.0. (Fig 2, p. 2083)

For an annealed Â-ß forging with either 0 or 45 ksi mean stress, there was no effect from shot peening (#110, 0.008-0.012 A [probably Almen]) on fatigue life above 107 cycles (per his fig 3) as compared to longitudinal polishing with 400 grit silicon carbide. Fig 3 goes up to 108 cycles. However, a noticible effect between 105 and 107 cycles. For 0 mean stress, fatigue strength of 48 ksi at 108 cycles and 49 ksi at 107 cycles.

Fatigue strength of annealed Ti-6Al-4V Â-ß at 107 cycles, no mean stress, per fig. 1:

Material Fatigue strength

Small dia barstock 67 ksi

Small test forgings 55 ksi

Component forgings (well worked) 58 ksi

Component forgings (slightly worked) 39 ksi

Tube extrusions (10.5" OD x 6" ID x 35' long) of annealed Ti-6Al-4V Â-ß. Extruded at 1700F with a 12:1 area ratio, which produced both crystallographic and primary alpha grain directionality. Length/width for primary alpha was as high as 30:1. The extrusion with the smallest primary alpha width had the best fatigue strength. From fig 5 at 0 mean stress:

Material Fatigue strength

Large extrusion, longitudinal direction 61 ksi

Small dia barstock 61 ksi

Large extrusion, radial direction 48 ksi

Small well-worked forgings 55 ksi

Large slightly-worked forgings 41 ksi

Referring to above: "The fatigue strength in the radial and tangential directions was found to be 25% below the longitudinal direction but still higher than that obtained from slightly worked large forging material." p. 2086

ß forging from 1900F (with 12% reduction), then annealed gave 60% better fatigue strength at 107 cycles with 45 ksi load compared to Â-ß forging. p. 2086 and fig 7, p. 2088. However, with 0 mean stress, the Â-ß forging was somewhat better (62 ksi (average of both Sikorski and Martin Marietta data) vs 57 ksi (Martin Marietta forged at 1950F)). p. 2086 and fig 8, p. 2088. Crack initiation fatigue strength for quenched ß forging was substantially greater than for air cooled ß forging. p. 2086 and fig 11, p. 2091.

Effect of heat treatment (annealed vs STA (Solution Treated and Aged) vs STOA (Solution Treated and OverAged)):

1-5/8" sections of a slightly worked Â-ß large component forging with STA at 45 ksi mean stress: 100% strength improvement compared to the same sized annealed material at 107 cycles. However, when starting from 4" thick sections, STOA gave slight improvement and STA gave only slightly more improvement compared to annealed at 107 cycles. This was because the section size was "too large for good through response." However, a 50% improvement was obtained with the ß STOA treatment (from approx 17 ksi to 26 ksi at 108 cycles). p. 2089, fig. 12 p. 2091 Curves flatten out at 108 cycles.

Treatment

Annealed 1300F for 3 hrs, air cool

STA 1750F for 1-3/4 hrs, water quench;

1000F for 3 hrs, air cool

STOA 1750F for 1-3/4 hrs, water quench;

1300F for 3 hrs, air cool

ß annealed 1850F for 1-1/2 hrs, air cool;

1350F for 3 hrs, air cool

ß STOA 1850F for 1-1/2 hrs, water quench;

1350F for 3 hrs, air cool

1850F is apparently 50F above the beta transus temperature. p. 2081.

For second 4" thick section test, slightly worked component forging, per above but 20 ksi mean stress: p. 2089, fig 13 p. 2091

Tensile Fracture Fatigue

Treatment strength % elongation toughness strength

Annealed 140 25% 60 21

STA 175 15% 50 23

ß STOA 155 5% 70 52

All strengths in ksi. Fracture toughness in ksi ´in.

Fatigue strength at 108 cycles.

Annealed and STA were shotpeened. ß STOA was polished.

"The ß STOA heat treatment provides increased response thourgh thicker sections sizes and results in improved crack initaiation and fracture toughness properties and reduced crack propagation rates." p. 2092

Effect of grain size:

Very distinct relation between fatigue strength and alpha grain size for unnotched Ti-6Al-4V with 45 ksi mean stress. Only small relation for notched (Kt = 1.8) Ti-6Al-4V. fig 14, p. 2093. No relation for ß grain size.

Effect of tensile strength for unnotched: little correlation. "... although there was only a 17% difference in tensile strength, greater than a 2 to 1 difference in fatigue strength was obtained between the lowest strength Ti-6Al-4V forging material and the highest value which was for the barstock." "Because of the small differences in tensile strength and large differences in fatigue strength observed in this material, it does not appear reasonable to reliably predict fatigue strength on the basis of tensile test values." p. 2098, fig 15 p. 2093. However, also cites Bartlo ref. which shows a direct relation between tensile and fatigue strengths for unnotched Ti-6Al-4V.

References:

3. Jenning, H. J., Beta Forging of Titanium Alloys, DMIC Report S-24, August 1, 1968.

4. Beck, E., Effects of Beta Processing and Fabrication on Axial Loading Fatigue Behavior of Titanium, AFML-TR-69-108, June 1969.

5. Coyne, Heitman, McClain and Sparks, The Effect of Beta Forgings on Several Titanium Alloys, Amer. Soc. Met. Tech. Report C7-14.4, Cleveland, Ohio, October 1967.

6. Bartlo, L. J., Effect of Microsturcture on the Fatigue Properties of Ti-6Al-4V Bar, Amer Soc. Test. Mater., STP-459, pp. 144 - 154.

8. Lucas, J. J., and Konieczny, p. P., Relationship Between Alpha Grain Size and Fatigue Strength of Ti-6Al-4V, Met. Trans., ASM, pp. 911 - 912, March, 1971.

-------------------

A. W. Bowen and C. A. Stubbington, "The Effect of  + ß working on the Fatigue and Tensile Properties of Ti-6Al-4V Bars".

4-1/2" dia rod:

1. As received condition (annealed). p. 2098

2. ß annealed at 1120C (2050F) to remove previous working, and then  + ß worked in the temperature range 850C-910C to achieve total reductions of 3, 8, 32 and 90:1. (p. 2097)

9-1/4" wide x 2-1/4" thick bar:

1. As received condition (annealed). p. 2098

2. 2-1/4" square section annealed at 1120C (2050F), then  + ß worked to achieve reductions of 10 and 50:1. (p. 2097)

All above (except as received?) had square final cross sections. All worked bars were annealed at 700C at 1 hr per inch of cross section.

" + ß reduction caused the as received mocrostructures to become increasingly elongated. Working not only reduced the  grain size but, more significantly, reduced and eventually eliminated the prior ß grain size; after heavy working each  grain became oriented differently from its neighbour [sic]." p. 2102

S-N test pieces cut in longitudinal direction. Tested at 6000 cpm on rotating cantilevered test machine. p. 2098

See chapter on "Elastic Material Properties" for property data on 4-1/2" dia rod (as-received) across its diameter.

Properties of reduced 4-1/2" dia rod:

Yield Tensile Modulus

Reduction Strength (ksi) Strength (ksi) (x106 psi)

3:1 129.3 137.0 17.1

8:1 127.8 135.9 16.5

32:1 131.7 142.3 16.2

90:1 134.6 146.3 16.2

Fatigue of 4-1/2" dia rod: (figs 1 and 2, p. 2099)

Fatigue strength

Condition at 108 cycles (ksi)

As received, surface of bar ~79

As received, center of bar ~74

3:1 Â-ß reduction, annealed at 700C ~66

8:1, 32:1, 90:1 Â-ß reductions, ~82

annealed at 700C

The as-received bar showed evidence of overheating at the bar center. p. 2102

For reductions 8:1, 32:1, 90:1, note no (approx) increase in fatigue strength even though tensile strength increased 7.6%.

Fatigue of 2-1/4" thick bar: (figs 3, p. 2100)

Fatigue strength

Condition at 108 cycles (ksi)

As received, annealed at 700C ~72

10:1, 50:1 Â-ß reduction, ~82

annealed at 700C

Bowen reports (p. 2098) that increasing the reduction from 10:1 to 50:1 only increased the low-cycle fatigue, but have a "very noticible" effect on reducing scatter.

Fatigue crack initiation and propagation: (p. 2104)

For a given stress amplitude, there was little difference in the extent of stage I crack growth between the worked and unworked bars. For stage II, "identical" growth rates, regardless of differences in prior ß grain sizes and textures. Therefore, "it can be concluded that the differences in smooth fatigue life ... are due to microsturcutual and textural influences on the crack initiation and stage I growth only."

"The low fatigue strength of the as received bars was due to the early initiation and growth of stage I cracks caused by localized plastic strain in large prior ß grains. The texture of the bars caused this strain to be restircted to the few favourably; oriented prior ß grains at the surface. The microstructural and property differences dependent on test piece position in the as received 4-1/2 in dia bar are considered to be caused by temperature gradients in the bar during working. These temperature gradients can be built up by inhomogeneous deformation, which in turn can lead to different degrees of recrystallization. As the section size decreases, temperature gradients are reduced, and microstructural and textural variations are therefore minimized." p. 2105

"The reduced scatter in fatigue life due to  + ß working, evident in the results shown in Fig 1, is due to these microstructural and textural modifications." p. 2107

" + ß working beyond reductions of 32:1 for the 4-1/2 in dia bar, and 10:1 for the 2-1/4 in thick bar did not increase the high cycle fatigue strength of these bars. The absence of any further improvements in strength with increasing reduction is attributed to the inability of further working to increase the misorientation between  grains and thus to inhibit the growth of stage I cracks to the critical length that will give the necressary stress intensity for stage II growth." p. 2107. I do not understand this. Bowen gives additional discussion on pp. 2105 - 2107.

References:

1. C. F. Hickey Jr., "Mechinaical Properties of Titanium Alloys as a Function of Heat Treatment and Section Size", Proc. ASTM, Vol 61 (1961), pp. 866 - 878.

2. R. M. Duncan and C. D. T. Minton, "The Role of Depth Hardenability in the Selection of High Strength Alloys for Aircraft Applications", The Science, Technology and Application of Titanium, R. I. Jaffee and R. E. Promisel (eds.), Pergamon, Oxford, 1970, pp. 945 - 957.

7. A. J. Hatch, "Texture Strengthening of Titanium Alloys", Trans. Met. Soc. AIME, vol 233, Jan 1965, pp. 44 - 50.

-------------------

Attwell M. Adair, Walter H. Reimann, and Richard F. Klinger, "The Influence of Thermomechanical Processing on the Fatigue Behavior of Extruded Beta III Titanium", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffed and H. M. Burte (ed), vol. 2(?), Plenum Press, 1973, pp. 1801 - 1812.

Ti Beta III alloy = Ti-11.5Mo-6Zr-4.5Sn; all tests on extrusions.

Notched: Kt = 3.1, with 0.007" root radius.

All tests at R = +0.1. Fatigue data from 104-->2*107 cycles.

Unnotched:

For unnotched in all cases, significant difference between water quench and air cool for preheat billet temps (before extrusion) of 1400F, 1500F, and 1800F. Aged only specimens (as opposed to STA) have much shallower S-N slope, which "indicated either a delayed crack initialtion or a slower sub-critical crack growth rate." p. 1802

From fig 4, p. 1804 for unnotched:

Unnotched Notched

Tensile Endurance Endurance

Condition strength (ksi) strength (ksi) strength (ksi)

1400F WQ+A 201 125 (60) 30 (15)

1800F WQ+A 185 125 (68) 28 (15)

1500F WQ+STA 181 120 (66) 25 (14)

1800F WQ+STA 182 115 (63) 30 (16)

1800F AC+STA 172 100 (58) 30 (17)

1500F AC+STA 174 95 (55) 28 (16)

1400F AC+STA 176 92 (52) 28 (16)

WQ = water quench

AC = air cooled

A = aged

STA = solution treated and aged

Numbers in parentheses are fatigue ratios

"Again, note that the endurance limit stress of the 1800F water quenched and aged extrusion specimens equals that for the much higher strength 1400F water quenched and aged one ..." p. 1805

1400F WQ+A had non-recrystallized structure and very fine uniform distribution of alpha, as compared with 1400F AC+STA which had recrystallized structure and coarser, less uniform alpha distribution. p. 1805.

Notched:

Only slight difference between water quenched and air cooled. "The direct benefits from thermo-mechanical processing of these Beta III extrusions are thus most effective in retarding crack initiation [referring to results of unnotched tests] and relatively ineffective in inhibiting the propagation of the crack under fatigue loading." p. 1805 Notes similar results reported by Reimann and Brisbane (6) for 7075 aluminum. "The fatigue behavior of notched specimens from the Beta III Ti extrusions was essentially the same for all variations in processing history." p. 1809

"Failures sometimes originated near the center of the specimens ..." p. 1809

References:

6. W. H. Reimann and A. W. Brisbane, "Improved Fracture Resistance of 7075 Aluminum through Thermomechanical Processing", Air Force Materials Laboratory, Wright-Patterson Air Frce Base, Ohio, to be presented at the Fracture and Fatigue Symposium, George Washington University, Washington, DC, May 3-5, 1972.

-------------------

Robert N. Shoemaker, "New Surface Treatments for Titanium", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffed and H. M. Burte (ed), vol. 2(?), Plenum Press, 1973, pp. 2501 - 2514+.

Tiduran process (cyanide-based molten salt bath, by Kolene Corp., Detroit MI) on Ti-6Al-4V for 2 hrs at 1480F gave diffusion depth of 0.002" and surface hardness of Rc 55. pp. 2510, 2513. Substantial improvement in wear resistance. pp. 2513, fig 11, p. 2514.

Some loss of fatigue strength from stress relieving effect of the 1480F treatment. However, glass beading following the heat treatment produced acceptable high cycle fatigue values. Substantial fatigue improvements by overaging for 10 hrs at 1060F and glass beading following the Tiduran treatment. p. 2513

-------------------

Donulus J. Padberg, "Fretting Resistant Coatings for Titanium Alloys", (Air Force Contract F33615-70-c-1538) Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffed and H. M. Burte (ed), vol. 2(?), Plenum Press, 1973, pp. 2475 - 2486.

"Previous testing had shown the nickel and chromium electroplating processes reduced the fatigue life at a given stress level by 50% or more." p. 2481. Therefore, tested electroless nickel ("Kanigen" process, General American Transportation Corp, Chicago, IL).

Chemical coatings:

Anodic ("Tiodize II", Tiodize Do., Inc., Burbank, CA);

Fluoride-phosphate conversion coating.

Spray deposited coatings:

Plasma sprayed copper-nickel-indium (Metco Powder #58, Metco Inc., Westbury, Long Island, NY.)

Plasma sprayed aluminum bronze (Metco Powder #51)

Detonation gun sprayed tungsten carbide (LW-1N40, Union Carbide Corp., Indianapolis, IN)

Ion vapor deposition of aluminum.

Diffusion coatings: not used because of possible harmful effect of high temperatures needed for their application.

Tension-tension tests of Ti-6Al-6V-2Sn: max stress = 120 ksi, min stress = 12 ksi, f = 30 Hz. Prior to coating, each specimen was steel shot-peened to 0.015 A intensity and grit-blasted with 150 mesh aluminum oxide grit, except for detonation gun sprayed specimens which were peened to 0.009 N intensity. p. 2482

Results shown in fig. 6 (p. 2484): only the fluoride-phosphate and anodize coatings lasted nearly as long as the control specimens, which failed at approx. 3.5*106 cycles. The electroless nickel reduced the fatigue life to approx 1.02*105 cycles. Number of test specimens is not stated. Fatigue samples were 0.250 thick plate, with surface finish of 10 RHR. p. 2476, fig 1 p. 2477.

Fretting fatigue with Molykote 106 lubricant (based on its acceptance at McDonell Aircraft Co.). When using fluoride-phosphate and anodize coatings in conjunction with shot-peening/grit-blasting and Molykote 106, fretting fatigue life increased at least 10 fold. p. 2486.

notes — Kf and Kt for Ti

Material Notch Notch Notch Strength Strenght

Ref condition radius (mm) angle depth unnotched notched Kt Kf q (calc)

1 1 0.533 60 1.37 ~340 ~180 2.70 1.94 0.55

2 2a 0.254 60 2.90 ~630 ~280 3.50 2.25 0.50

2 2b 0.254 60 2.90 ~430 ~230 3.50 1.87 0.35

2 2c 0.254 60 2.90 ~620 ~310 3.50 2.00 0.40

2 2d 0.254 60 2.90 ~690 ~320 3.50 2.16 0.46

2 2e 0.254 60 2.90 ~680 ~340 3.50 2.00 0.40

3 3a ? ? ? 550 310 3.20 1.77 0.35

3 3b ? ? ? 655 290 3.20 2.26 0.57

4 4a ? ? ? ~690 ~400 3.50 1.72 0.29

4 4b ? ? ? ~510 ~230 3.50 2.22 0.49

References

1. Willertz, Ultrasonic Fatigue, pp. 119 - 133. Test type: 88 Hz on SF-4 test machine at room temp in lab air. Notched test also performed at 20 kHz. No difference between 88 Hz and 20 kHz notched tests. Fatigue limit at 107 cycles for 88 Hz tests and 109 cycles for 20 kHz tests.

2. Bartlo. Test type: Krouse rotating beam at 133 Hz. Fatigue limit taken at stress for which 3 samples ran 107 cycles w/o failure. Data taken from figures.

3. Osgood, p. 442, table 4.14. Test type: axial loading (low frequency). Fatigue strength at 106-->107 cycles.

4. "Facts about Titanium: RMI 6Al-4V", RMI Co, fig 55, p. 22. Test type: low frequency, probably at 31 Hz (see figs 57 - 59). Test to 107 cycles.

Material condition

1. Mill annealed Ti-6Al-4V (pp. 128 - 129), RMI heat #896887, heated to 1100F for 10 hrs in vacuum after machining.

Original stock size = ?

Tensile = 143 ksi = 986 MPa

Yield = 136 ksi = 938 MPa

Hardness = 36.2 Rc

Modulus = 17.2E6 = 119 GPa

2a. 15.9 mm dia RMI Ti-6Al-4V bar stock (pp. 145, 153) annealed for two hours at 1350 oF, then air cooled

See fig 3:

tensile = 154 ksi = 1060 MPa

yield = 138 ksi = 950 MPa

Hardness = ?

Modulus = ?

2b. 88.9 mm dia RMI Ti-6Al-4V bar stock (pp. 145, 153) annealed for two hours at 1350 oF, then air cooled

See fig 3:

tensile = 135 ksi = 930 MPa

yield = 125 ksi = 860 MPa

Hardness = ?

Modulus = ?

2c. 15.9 mm dia RMI Ti-6Al-4V bar stock (see fig 4) annealed at 1550 oF, then furnace cooled at 175F/hr

Tensile = 145 ksi = 1000 MPa

Yield = 134 ksi = 925 MPa

Hardness = ?

Modulus = ?

2d. 15.9 mm dia RMI Ti-6Al-4V bar stock (see fig 4) annealed at 1750 oF, then water quenched

Tensile = 160 ksi = 1100 MPa

Yield = 138 ksi = 950 MPa

Hardness = ?

Modulus = ?

2e. 15.9 mm dia RMI Ti-6Al-4V bar stock (see fig 5) annealed at 1750 oF, then water quenched, then aged 4 hrs at 1000F

Tensile = 172 ksi = 1180 MPa

Yield = 154 ksi = 1060 MPa

Hardness = ?

Modulus = ?

3a. Ti-6Al-4V, annealed

Tensile = 136 ksi = 938 MPa

Yield = 128 ksi = 883 MPa

Hardness = ?

Modulus = ?

3b. Ti-6Al-4V, heat treated

Tensile = 170 ksi = 1170 MPa

Yield = 160 ksi = 1100 MPa

Hardness = ?

Modulus = ?

4a. Ti-6Al-4V bar, STA

Tensile = ?

Yield = ?

Hardness = ?

Modulus = ?

4a. Ti-6Al-4V bar, annealed

Tensile = ?

Yield = ?

Hardness = ?

Modulus = ?

Notes:

1. Kt = theoretical stress conc (1-dimensional)

2. For ref 2, various annealing temperatures were available for measuring Kt. The value in the table is for the annealing temp that gave the best fatigue life below the beta transus. (Above the beta transus, the ductilitly suddenly decreases.)

notes — Stress concentration Kf

Depending on how Kt and q are determined will determine the value of Kr.

I believe that all stress concentration factors are for uniaxial loading, although this might not produce uniaxial stress. If the loading is not uniaxial, then the actual stresses in each direction (accounting for stress concentrations) can be combined in Mohr's circle.

Juvinall[1], pp. 266 - 267:

When calculating Kf, should theoretically use stress concentration charts for Kt that account for 2-D stress, which would lower Kt. However, with 2-D stress, q (notch sensitivity) increases so that there is little net effect on Kt (as opposed to calculating from principal stresses and just using normal charts for Kt and q).

Shigley (p. 141): "A theoretical, or geometric, stress-concentration factor Kt or Kts is used to relate the actual maximum stress at the discontinuity to the nominal stress."

Is it possible to define Kt as:

Ínom (no stress conc) to produce failure

Kt = ------------------------------------------

Ínom (with stress conc) to produce failure

Strength (no stress conc)

= ---------------------------

Strength (with stress conc)

For the numerator, is Ínom:

1. The stress required for failure in a uniaxial tensile test.

2. The theoretical stress (possibly biaxial or triaxial, depending on the part shape) required for failure in the actual part, but excluding any effects of stress concentration.

Wirsching "Fatigue Failure in the Real World", Machine Design, p. 88:

Fatigue notch factor:

fatigue linit, unnotched

Kf = ------------------------

fatigue limit, notched

"In general, Kf varies with type of material; strength; previous treatment; the geometric shape, overall size, and dimensions of the notch; stress amplitude; and cycle life. The stress concentration factor Kt provides a quantitative measure of the stress increase at the stress riser and is the ratio of unnotched strength to notched strength of the metal in static tension."

Fatigue notch factor:

Kf - 1

q = ------

Kt + 1

"The numerator represents the effect of the notch in fatigue and the denominator represents the effect under static, pure elastic tension."

Notes — Frequency effect

Kuz'menko:

p. 23 "It should be noted that the fatigue strength of alloys does not depend on the loading frequency (in the sonic range) when the energy dissipation in the material is small and the specimens are not heated during longigudinal vibration."

p. 26 "The tests confirm that the effect of the loading frequency on the endurance diminishes as the strength of the material becomes greater."

Kuz'menko graphs show little frequency effect for some Al alloys, but significant effect for others. Same for Ti alloys.

--------------------

L. E. Willertz (U/S Fatigue, p. 333-348): Ti-6Al-4V bar, tested in pure water at 80 oC at both 100 Hz and 20 kHz (little life difference), max life at approx 2*108 --> 109 cycles:

Test Tensile Fatigue

direction Hardness Modulus strength strength %

(a) -- -- 919 MPa 444 MPa 48

(133 ksi) (64 ksi)

(b) -- -- 980 MPa 510 MPa 52

(142 ksi) (74 ksi)

(c) -- -- 914 MPa 480 MPa 53

(133 ksi) (70 ksi)

(a) = Coarse grained equiaxed Â-ß, produced by air cooling from the Â-ß region of the phase diagram. (p. 342)

(b) = Fine grained lamellar Â-ß, produced by quenching from the ß region of the phase diagram. (p. 342)

(c) = Bimodal

All Ti are stress relieved.

Above data from tables I and III and p. 342.

"In both cases it is apparent that the low frequency and high frequency data points form continuous fatigue curves for each [material] structure." (p. 342)

Above properties at 80 oC

Notes — David Ullman, &Quot;Less Fudging On Fudge Factors&Quot;

David Ullman, "Less Fudging on Fudge Factors", Machine Design, October 9, 1986, pp. 107 - 111:

Phone conversation of 2/10/88 -- (503) 754-2336:

Statistical ratio from Table 2 (p. 110):

Values in table depend on reliability of available data. For example, if good uniaxial fatigue data is available, then the statistical ratio for fully reversed, uniaxial, finite-life fatigue should be ~0. However, if the finite life is estimated based on the material's ultimate strength, then a statistical ratio of 0.05 is appropriate.

Ullman did not know the amount of fundamental error in the distortion energy theory -- i.e., assuming stresses and material strength is known exactly, the distortion energy theory may still give some error. He suggested that, under these conditions and with fully-reversed multiaxial stress, start with statistical ratio of 0.1.

Ullman could not give any supporting data or refs for the values in the table.

Ullman could not define exactly when the uniaxial stress became multiaxial stress, in terms of the ratio of principal stresses -- e.g., multiaxial stress occurs when P2 · 0.1 P1. Suggested using software in his book with various values of P2 and P1 to see results.

Book available with IBM software:

Mechanical Design Failure Analysis ISBN #0-8247-7534-1 $95

Marcel Dekker (publisher)

(212) 696-9000

Box 5005

Monticello, NY 12701

Notes — Current BSP testing proposal

1. For each specimen, record its bar and heat numbers, and its location in the original bar stock. Indicate grain direction. Date on which specimen was machined, to track aging effects or effects of improper machine tooling, etc. Also, mark each with order of machining (1st, 2nd, 3rd, etc.) Specify machining oil to be used.

2. Make wave-speed measurements on broken specimens. Requires recording initial and final tuning data.

3. Check hardness of failed samples to see if fatigue life is correlated to hardness. Could be used as a quick BSP in-house test for a batch of material, or BSP could specify a certain minimum hardness.

References

1. Wolfgang-Werner Maennig, "Planning and Evaluation of Fatigue Tests in Regard to Ultrasonic Frequency Tests", Ultrasonic Fatigue, Joseph M. Wells, Otto Buck, Lewis D. Roth, John K. Tien (eds.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 611 - 645.

2. G. M. Sinclair and T. J. Dolan, "Effect of Stress Amplitude on Statistical Variability in Fatigue Life of 75S-T6 Aluminum Alloy", Transactions of the ASME, July, 1953, pp. 867 - 872.

3. Robert C. Juvinall, Stress, Strain, and Strength, McGraw-Hill Book Company, New York, NY, 1967.

4. Fatigue Design Handbook, James A Graham (ed.), Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, PA., 1968.

5. J. C. Grosskreutz, "Fatigue Mechanisms in the Sub-Creep Range", Metal Fatigue Damage -- Mechanism, Detection, Avoidance, and Repair, S. S. Manson (ed.), ASTM Special Technical Publication 495, American Society for Testing and Materials, Philadelphia, PA, 1971, pp. 5 - 60.

6. Ultrasonic Fatigue, Joseph M. Wells et al (ed.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982.

7. A. W. Bowen, "The Effect of Testing Direction on the Fatigue and Tensile properties of a Ti-6Al-4V Bar", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2, Plenum Press, 1973, pp. 1271 - 1280.

8. J. E. Spindel and E. Haibach, "Some Considerations in the Statistical Determination of the Shape of S-N Curves", Statistical Analysis of Fatigue Data, R. E. Little and J. C. Ekvall (eds), ASTM Special Technical Publication 744, 1979, pp. 89 - 113.

9. Joseph Edward Shigley, Mechanical Engineering Design, McGraw-Hill Book Company, New York, 1963.

10. S. S. Manson, "Avoidance, Control, and Repair of Fatigue Damage", Metal Fatigue Damage -- Mechanism, Detection, Avoidance, and Repair, S. S. Manson (ed.), ASTM Special Technical Publication 495, 1971.

11. Handbook of Fatigue Testing, S. Roy Swanson (ed.), ASTM Special Technical Publication 566, 1974.

12. R. E. Little, Manual on Statistical Planning and Analysis of Fatigue Experiments, ASTM Special Technical Publication 588, 1975.

13. W. Hessler, H. Müllner, B. Weiss, H. Schmidt, "Fatigue Limits of Cu and Al up to 1010 Loading Cycles", Ultrasonic Fatigue, Joseph M. Wells et al (ed.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 245 - 263.

14. J. R. Barton and F. N. Kusenberger, "Fatigue Damage Detection", Metal Fatigue Damage -- Mechanism, Detection, Avoidance, and Repair, S. S. Manson (ed.), ASTM Special Technical Publication 495, American Society for Testing and Materials, Philadelphia, PA, 1971, pp. 123 - 227.

15. R. A. Yeske and L. D. Roth, "Environmental Effects on Fatigue of Stainless Steel at Very High Frequencies", Ultrasonic Fatigue, Joseph M. Wells et al (ed.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 365 - 385.

16. S. Nishijima, "Statistical Fatigue Properties of Some Heat-Treated Steels for Machine Structural Use", Statistical Analysis of Fatigue Data, R. E. Little and J. C. Ekvall (eds), ASTM Special Technical Publication 744, 1979, pp. 75 - 88.

17. Robert Lipp, "Relating Strength and Hardness of Aluminum Alloys", Machine Design, March 24, 1983, p. 107.

18. Anton Puskar, "Cyclic Stress-Strain Curves and Internal Friction of Steel at Ultrasonic Frequencies", Ultrasonics, May, 1982, pp. 118 - 122.

19. B. Mukherjee and E. J. Burns, "Regression Models for the Effect of Stress Ratio on Fatigue-Crack Growth Rate", Probabilistic Aspects of Fatigue, Robert A. Heller (symposium chairman), ASTM Special Technical Publication 511, 1972, pp. 43 - 60.

20. John A. Strommen, "A New Look at Metal Fatigue", Machine Design, July 11, 1974, pp. 131.

21. William J. Westerman, "Industry Rediscovers Roller Burnishing", Machine Design, August 25, 1983, pp. 44 - 48.

22. SAE Manual on Shot Peening - SAE J808a, Handbook Supplement HS84, 1967.

23. Paul G. Field and Daniel E. Johnson, "Advanced Concepts of the Process", Shot Peening for Advanced Aerospace Design, SAE SP-528, 1982, pp. 19 - 22.

24. B. Austin Barry, Engineering Measurements, John Wiley & Sons, New York, 1964.

25. David Ullman, "Less Fudging on Fudge Factors", Machine Design, October 9, 1986, pp. 107 - 111.

26. Corey Crispell, "New Data on Fastener Fatigue", Machine Design, April 22, 1982, pp. 71 - 74.

27. Richard A. Walker and Gerhard Meyer, "Design Recommendations for Minimizing Fatigue in Bolts", Machine Design, September 15, 1966, pp. 182 - 186.

28. Doug McCormick, "A Guide to Fastening and Joining", Design Engineering Fastening Guide, pp. F5 - F27.

29. RMI Company, "Facts about Titanium: RMI 6Al-4V", Niles, Ohio, 44446.

30. Carl C. Osgood, Fatigue Design, Wiley Interscience, New York, 1970.

31. Donulus J. Padberg, "Fretting Resistant Coatings for Titanium Alloys", (Air Force Contract F33615-70-C-1538) Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2, Plenum Press, 1973, pp. 2475 - 2486.

32. Ryuichiro Ebara, Yoshikazu Yamada, Akira Goto, "Corrosion Fatigue Behaviour of 13Cr Stainless Steel and Ti-6Al-4V at Ultrasonic Frequency", Ultrasonic Fatigue, Joseph M. Wells, Otto Buck, Lewis D. Roth, John K. Tien (eds.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 349 - 364.

33. John J. Lucas, "Improvements in the Fatigue Strength of Ti-6Al-4V Forgings", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2, Plenum Press, 1973, pp. 2081 - 2095.

34. L. E. Willertz and L. Patterson, "Stress Distributions in Notched Specimens Loaded Statically and Dynamically", Ultrasonic Fatigue, Joseph M. Wells et al (ed.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 119 - 133.

35. Paul H. Wirsching and John E. Kempert, "Fatigue Failure in the Real World", Machine Design, August 26, 1976, pp. 86 - 90.

36. Michael M. Woelfel, "Shot Peening -- Control and Measurement", Shot Peening for Advanced Aerospace Design, SP-528, SAE, October, 1982, pp. 15 - 18.

37. R. B. Heywood, Designing Against Fatigue of Metals, Reinhold Publishing Corp., New York, 1962.

38. p. R. Wedden and F. Laird, "Design and Development Support for Critical Helicopter Applications in Ti-6Al-4V Alloy", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 1, Plenum Press, 1973, pp. 69 - ?.

39. W. Weibull, Fatigue Testing and Analysis of Results, Pergamon Press, New York, 1961.

40. N. E. Frost, K. J. Marsh, L. p. Pook, Metal Fatigue, Clarendon Press, Oxford, 1974.

41. Norman Zlatin and Michael Field, "Procedures and Precautions in Machining Titanium Alloys", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 1 (?), Plenum Press, 1973, pp. 489 - 504.

Currently unused references:

--. V. A. Kuz'menko, "Fatigue Strength of Structural Materials at Sonic and Ultrasonic Loading Frequencies", Ultrasonics, January, 1975, pp. 21 - 30.

--. E. T. Bittner, "Alloy Spring Steels", Transactions of the ASM, vol. 40, 1948, pp. 263 - 280.

--. Walter J. Crichlow, "High Cycle Fatigue Properties of Titanium in Aircraft Application", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2?, Plenum Press, 1973, pp. 1257 - 1270.

--. A. W. Bowen and C. A. Stubbington, "The Effect of  + ß working on the Fatigue and Tensile Properties of Ti-6Al-4V Bars", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2(?), Plenum Press, 1973, pp. 2097 - 2108.

--. Attwell M. Adair, Walter H. Reimann, and Richard F. Klinger, "The Influence of Thermomechanical Processing on the Fatigue Behavior of Extruded Beta III Titanium", Titanium Science and Technology, The Metallurgical Society of AIME, R. I. Jaffee and H. M. Burte (ed), vol. 2(?), Plenum Press, 1973, pp. 1801 - 1812.

--. L. J. Bartlo , "Effect of Microstructure on the Fatigue Properties of Ti-6Al-4V Bar", Fatigue at High Temperature, ASTM STP 459, 1969, pp. 144 - 154.

--. L. E. Willertz, T. M. Rust, V. p. Swaminathan, "High and Low Frequency Corrosion Fatigue of Some Steam Turbine Blade Alloys", Ultrasonic Fatigue, Joseph M. Wells et al (ed.), The Metallurgical Society of the AIME, Warrendale, Pa, 1982, pp. 333 - 348.

--. Properties and Processing of Ti-6Al-4V, TIMET Corp., 1986.

Books and articles

Grosskreutz lists 3 articles about planar and wavy slip materials. (His reference numbers 2, 5, 6.)

R. E. Peterson, Stress Concentration Factors, John Wiley & Sons, NY, 1973. According to Ultrasonic Fatigue (p. 125), Peterson says that crack initiation in ductile materials is governed by the von Mises stress concentration factor.

Abramowitz and Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964. (per HP statistics calculator manual)

Joseph Marin, Mechanical Behavior of Engineering Materials, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. Referenced by Juvinall, p. 318, who says Marin has developed a modified distortion energy theory for anisotropic materials.

From Manson in Metal Fatigue Damage, referenced from figure 34, p. 290 for effect of machining processes on fatigue life of titanium:

Harmsworth, C. L., "Design Criteria and Test Techniques" Air Force Materials Symposium, Report AFML-TR-65-29, AD-463572, Air Force Systems Command, May 1965, pp. 831 - 853.

Rooney, R. J., "The Effect of Various Machining Processes on the Reversed-Bending Fatigue Strength of A-110AT Titanium Alloy Sheet," Report WADC-TR-57-310, AD-142118, Wright Air Development Center, Nov. 1957.

[Approx. relation between fatigue strength and ultimate tensile strength. Only approx. Does not hold strongly for Al -- e.g., extruded versus wrought 7075.]

[Where to include a discussion of material resistance to fatigue -- e.g., grain size, direction of loading with respect to grain direction, etc? Should be in this chapter or the chapter on life?]

[Chapter on life: reduce maximum horn stress and make any competing stresses at least X% less than the maximum stress. Otherwise, if several locations have the same max stress, then the probability of failure is increased. This is because the material is not microscopically homogeneous, so that some locations are weaker than others in fatigue. Therefore, the more locations that share the max stress, the more likely that one of these will coincide with a local weakness. Note: The X% can be found by analyzing the stress distribution in an S-N horn, for which only a few failed at the nodal radius, even though the nominal stress is higher in the front half-wave section.]

Fatigue is localized process -- depends on stress and grain orientation.

Our testing indicates that 7075 alloys are generally superior to 2024 alloys. (However, note tests of extruded 2024.) However, in comparing these alloy groups to the new aluminum-lithium alloys, Kubel (p.44) seems to indicate that 2024-T351 is better than 7075-T651, both for fatigue crack growth and fatigue crack initiation. (Does the 1 on the end of the designation mean that the material is extruded?)

Edward J. Kubel, Jr., "Al-Li Alloys: New Hope for Weight Watchers", Materials Engineering, April 1985, pp. 41-44. (My collection of lit on materials in big black notebook.)

Appendix [] — Cause of power supply shut-down in fatigue failures

Branson's power supplies can shut down for two reasons: if the load is greater than the available power or if the frequency changes beyond the power supply limit. It is often supposed that the power supply shuts down because the crack causes a drastic decrease in the horn frequency (due to increased compliance) or because the increased temperature of the horn a drastic frequency drop.  Neither of these frequency-related causes seems valid, although they may contribute to final failure. Considering frequency drop due to compliance change: many horns crack at the slot ends. The compliance at this location is not sufficiently large that a change would affect horn frequency. This is verified when the slot is extended completley to the horn face, which would be equivalent to a very severe crack. However, the frequency change is not significant.

As for temperature change, P&G ti horns have continuously run at []o C without any shutting down the power supply. Also, at least in S-N horns, the final failure seems to occur rather suddenly. For instance, an S-N horn may run for many hours without appreciable power increase. Then, within a few minutes the power (as indicated by the front panel meter) will suddenly increase until the meter is off-scale. This is not sufficient time for the horn to heat to a sufficient temperature to cause the power supply to shut down.

[Could make calculation of how long a power supply would have to put out 1000 watts in order for the average temp. of a Ti S-N horn to increase to the required shut-down temp.]

How to decide on candidate materials for fatigue testing:

1) Material cost.

2) Relatively low loss.

3) Wear resistance.

4) Machinability.

5) Generally high strength -- fatigue resistance increases generally with strength.

appendix []— Proposed equations for the s-n curves

1870 Wöhler log(N) = a - bS for S · E

1910 Basquin log(N) = a - b log(S) for S · E

1914 Strohmeyer log(N) = a - b log(S - E)

1924 Palmgren log(N + d) = a - b log(S - E)

1949 Weibull log(N + d) = a - b log[(S - E)/(R - E)]

1955 Strüssi log(N) = a - b log[(S - E)/(R - S)]

1963 Bastenaire log(N) = a - 1*log(S - E) + [(S - E)/b]C

where E = endurance limit

R = ultimate strength

a, b, c, d = parameters

Taken from Spindel, p. 92.

The last equation is doubtful because of the 1*log(S - E) term. See Spindel, p. 94, Haibach, p. 44. Also see Bastenaire, ASTM STP 511, pp. 14 - 16.

 

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