High Cycle Fatigue: A Mechanics of Materials Perspective part 26 pptx

This region, on a Haigh diagram, will cover values of mean stress from zero until the value at which the maximum stress, Smax, equals the yield stress, Sy, or in terms of mean and altern

10

20

30

40

50

60

70

Smooth data Notch data

Jasper Eqn Alt stress only Bell and Benham Eqn Mean and alt stress

Mean stress (ksi)

Ti-6Al-4V

kt = 2.5

kf = 2.0

UTS

Figure 5.19. Smooth and notch fatigue data from [22] and methods for representing the notch curve.

It is often tempting to take a small amount of data that includes inherent scatter and postulate a new or modified theory that will fit the data better than existing models So, not heeding my own advice, I offer the following as a method for fitting notch data based

on the results shown in Figure 5.19 Following the concepts presented above that were put forth by Bell and Benham [18], the modification for the value of kf is further modified slightly for the following reasons If the notch behavior is purely elastic, then the value

of kf should not change for a given notch geometry This region, on a Haigh diagram, will cover values of mean stress from zero until the value at which the maximum stress,

Smax, equals the yield stress, Sy, or in terms of mean and alternating stress

Sm= S∗

where Sm∗ is defined as a transition mean stress For values of Smbelow S∗m, the fatigue notch factor is constant and equal to the value of kf at R= −1, or zero mean stress, which we denote by kf0 For mean stress values above Sm∗, we can allow the value of kfto decrease linearly up to the ultimate strength, Su This can be expressed by the equation, which I will modestly call the “Nicholas equation,” as

kf= kf0



1−Sm− S∗

m

Su− S∗ m

 +Sm− S∗

m

Su− S∗ m

(5.17)

In the above, the transition mean stress is calculated from Equation (5.16) based on the assumption that for a mild notch kf is approximately equal to kt In reality, kt relates the peak stress to the average stress so that k will govern when first yielding occurs

based on average stress values Then, first yielding would occur as per Equation (5.16) which is based on smooth bar values The concept behind proposed Equation (5.17) is that when the ultimate strength of the material with a notch is approached, the notch and the smooth bar ultimate strengths are nearly the same because both involve net-section yielding Therefore, at the ultimate strength point, kf= 1 for the notched specimen The linear variation is the simplest way to transition from the mean stress where yielding first occurs to the ultimate strength In the example shown in Figure 5.19, the value of kf

is overestimated for data at high R when using a constant value of kf If the “Nicholas equation” (5.17) is used, as shown in Figure 5.20, the resulting curve, based on a Jasper equation for representing the smooth bar data, produces an amazingly good fit to the limited experimental data points For this material, the yield strength was 135 ksi and the transition mean stress Sm∗ = 122 ksi While the fit to a limited data set in itself does not completely prove or validate any such theory, the method has an advantage over using a single value of kf With a single value, the highest mean stress that can be predicted for notch data is Su/kf0 Since smooth bar data are not available beyond a mean stress equal

to Su, the range in which predictions can be made is limited as shown in Figure 5.19 (also in Figure 5.20), where the constant kf equation terminates at a mean stress of under

75 ksi The effect of using a constant value is to shrink the smooth bar curve radially inward along lines of constant R on a Haigh diagram It makes no sense to try to extend the smooth bar equation to mean stresses beyond the ultimate strength With the new formulation, the notch curve is extended to the ultimate strength of the material without having to extrapolate smooth bar data

0

10

20

30

40

50

60

70

Smooth data

Notch data

Jasper Eqn

Kf constant

Nicholas Eqn

Mean stress (ksi)

Ti-6Al-4V

kt = 2.5

kf = 2.0

UTS

Figure 5.20. Illustration of the proposed Nicholas equation ability to fit notch data.

For the case of a straight-line modified Goodman equation used to represent smooth bar data, this new approach would be equivalent to following the line called “brittle” in Figure 5.11 from zero mean stress up to the transition mean stress, and then continuing with a straight line to Su

5.8.1 Negative mean stresses

Notched fatigue behavior for negative values of mean stress has received essentially no attention in the open literature With the recent increased usage of compressive residual stresses for component fatigue life improvement, an increase in emphasis in this subject can be expected The use of surface treatments that produce compressive residual stresses

is discussed in Chapter 8 Here we discuss both uniform stress fields and those produced

by notches In Figure 5.21, three possible methods of extrapolating the modified Goodman equation∗ into the negative mean stress region are shown Each one is a straight line, and the three lines are denoted by A, B, and C There is no physical basis for any of the three lines, but lacking data in the negative mean stress region, these three lines cover a wide range of possible behavior in this region For each of the lines, the notch fatigue behavior is assumed to be governed by either of the two equations First, the fatigue notch factor is assumed to be linearly proportional to the mean stress so that

Mean stress

Alternating stress

0

A

B

C

A ′

B ′

C ′

kf prop to mean stress

kf alt stress only

I′

I″

A″

B″

C ″

S0

S0/kf

Figure 5.21. Representation of notch fatigue behavior for negative mean stress using three straight-line

repre-sentations of a Haigh diagram.

∗

The modified Goodman equation is a straight line from the value of the alternating stress at zero mean stress

to the ultimate stress at zero alternating stress.

the value of kf goes from its reference value at R= −1 (zero mean stress) to unity at the ultimate strength point in both tension and compression This is equivalent to using Equation (5.15) above with the added condition that Sun= Sup The alternating stress at zero mean stress is this s0/kf These curves are denoted by A, B, and C in the figure, corresponding to the un-notched curves A, B, and C, respectively The second method

of representing the notch fatigue behavior is to reduce only the alternating stress by the same factor, kf, obtained (and defined) for R= −1 The curves for this condition are denoted by A, B, and C In the positive mean stress region, the two methods produce the same curve when the straight line is used to represent the un-notched behavior, as pointed out earlier In the negative mean stress region, a range of predicted behavior can

be seen The wide variety of possible behavior, both for smooth and notched specimens, provides a compelling argument for the generation of data in the negative mean stress region before any meaningful modeling can be expected to be accomplished At the time

of this writing, no such data have been found This discussion is limited to relatively blunt notches because the possibility of very sharp notches with small root radii closing under compressive loading is a distinct possibility

An engineering approach to notches of arbitrary size and geometry which may or may not have initial cracks has been developed by Hudak et al [24] It was developed to explain the behavior of specimens subjected to FOD which can cause both craters and cracks It is based on a fracture-mechanics analysis using a concept similar to that of El Haddad, et al [12] described earlier, that attempts to predict the type of behavior shown schematically earlier in this Chapter in Figure 5.6 where there are regions of growth to failure, no initiation or growth, and an intermediate region of initiation but no growth The concepts are illustrated in Figure 5.22(a) where the regions of initiation and growth

to failure (A), initiation and arrest (B), and no initiation (C) are shown schematically The authors define a “worst case notch” as a value of kt, defined as Kw, above which further increase in ktno longer results in a decrease in threshold stress In Figure 5.22(b), the same regimes are shown as a function of crack size for a fixed value of notch depth and different values of root radius The upper and lower curves represent a large 3 and small 1 root radius, respectively, while the horizontal line represents the limiting radius, w, corresponding to Kw, above which crack arrest cannot occur (discussed earlier

in this chapter) What distinguishes this work from previous analyses is that the crack-size effect is addressed by defining a crack-size dependent threshold in the form

Ktha= KthR

 a

Stress concentration factor, Kt

Crack size, a

Kw

A

B C

A

ρ1 Crack arrest

(b) (a)

ρw Crack growth

ρ3 Crack growth

Figure 5.22. Schematic of regions of notch behavior.

where a0is the parameter defined by El Haddad in Equation (5.11) and Kthis a function

of stress ratio, R The curves shown in Figure 5.22(b) are generated for a specific geometry and far-field loading by taking into account the actual stress field ahead of the notch and calculating K as a function of crack length The analysis allows for the prediction of crack arrest or growth from a fracture-mechanics-based analysis and precludes the need

to use an empirical relationship for kf as a function of kt It further has demonstrated that the crack arrest which occurs at sharp notches depends both on the notch root radius and depth, as well as on the crack and specimen geometry The authors contend that simple correlations between kf and kt are bound to break down for very deep, sharp notches

of the type that can be produced by FOD (see Chapter 7) This same type of analysis can be used in contact fatigue problems (Chapter 6) where a complicated stress field, involving steep stress gradients, is produced by the contact loading and not by sharp notches A further advantage over empirical approaches is that residual stresses, such

as those occurring during FOD, or those produced by surface treatments such as shot peening, can be taken into account when calculating the stress intensity field ahead of the notch

The worst-case-notch concept has been extended to take into account local notch plasticity In addition, an approach that takes into account the surface area in the vicinity of

a notch, similar in concept to the critical volume or critical distance approaches to notches, has been developed and is discussed later in this chapter These two developments, that were accomplished under a recently completed HCF program within the Air Force, are described in detail in Appendix E

5.9.1 Non-damaging notches

One of the earliest attempts to quantify the size of a non-damaging notch, a notch that does not reduce the fatigue limit strength of a material, is due to Lukas et al [25] Using linear elastic fracture mechanics, they compared the stress intensity of an edge crack of length l in a plate with that of a crack of the same length, l, emanating from an elliptical notch having a notch root radius  For the former, the stress intensity is given as

whereas for the crack from a notch, they used an approximation of results from FEM analysis of Newman [26] in the form

Knotch= 112kt√

l



where ktis the elastic stress concentration factor for the notch with root radius , and  is the uniform far-field tensile stress For both equations, a crack length l= l0is introduced for which length the stress intensity is the threshold stress intensity for a long crack, and the stress is the endurance limit stress for the smooth e and notched en specimens, respectively Applying their assumption that the depth of microcracks on the surface of smooth specimens and the depth of microcracks at the notch root is the same at the fatigue limit, Equations (5.19) and (5.20) are combined to produce the following formula for the endurance limit at the notch,

en=e

kt



From a large number of experiments on specimens having various radii, the best value of

l0was determined by trial and error fitting to Equation (5.21) for the FLS to be 100 m for 15313 steel and 90 m for copper

The condition for a notch to be non-damaging with respect to the fatigue limit for the same applied stress level and the same crack size l0 is shown to be



If Equation (5.19) is used to relate l0 to the long crack threshold and the smooth bar endurance limit, then l0becomes the same as a0in a Kitagawa diagram and the criterion for a non-damaging notch takes the form



k2

This approach, which attempts to correlate the behavior of a notch with that of a smooth bar, both of which have a crack, is essentially the same as using a Kitagawa diagram for

the notch geometry (with a crack) and finding the crack length a0 where the endurance limit and long crack threshold curve intersect In this particular case, there is no El Haddad small crack correction

Continuing attempts have been made over the years to predict the fatigue limit of notched components from data on smooth bars The physical reasoning used has been that a critical volume of material must be subjected to a critical stress level, the smooth bar fatigue limit, in order for fatigue failure to occur This postulate means that stress at a point, for example at a notch root, is not the sole governing criterion for fatigue to occur The observations made that sharp notches have higher fatigue strengths than mild notches is partial evidence for such a theory Since the calculation of average stresses over a volume

is computationally intensive and impractical for many sharp notch geometries, the problem

is often simplified to one or two dimensions This still would involve determination of the average stress over some critical length The simple solution for such a problem, adopted

by many, has been to use the stress at a critical distance from the surface to represent the average stress over a larger distance Critical volume approaches have been reviewed by Sheppard [27] and Taylor and O’Donnell [28] The empirical relations between kf and

kt, in many cases, are derived from determination of stresses over a critical length and some simplifying assumptions about the distribution of stresses in order to come up with

a critical distance (see, for example, Peterson [4] who assumes a linear variation of stress from the notch root)

5.10.1 Critical distance approaches

The critical distance approach has the advantage over many other empirical approaches because it deals with both stress gradients and size effects Taylor [13] shows, in fact, that sharp notches and fatigue cracks have some commonality, as observed experimentally, that the two behave similarly when the notch has a ktor sharpness beyond which the FLS becomes coincidental with that of a crack of the same size Taylor shows that the stress determined at a critical distance a0/2, is the same for a sharp notch and a crack, where

results showing the stress distributions ahead of a circular notch of radius r, and of a crack of length a, are shown in Figure 5.23, for three different values of a/a0; 0.1, 1.0, and 10 The thick lines show the stress distribution for the notch, while the thin lines represent that of the crack The figure shows the overall distribution in (a), and a more detailed distribution near the notch or crack tip, in (b) The vertical dashed line is the location a/2 which represents the location where the local stress determines the fatigue

0

1

2

3

4

5

σ0

r /a0

Notch

a /a0= 0.1

a /a0= 1.0

a /a0= 10

(b)

0

1

2

3

4

5

σ0

r /a0

Notch

a /a0= 0.1

a /a0= 1.0

a /a0= 10

Figure 5.23. Numerical results showing the stress distributions ahead of a circular notch of radius, r, and of

a crack of length, a Dashed line is at r = 05a 0 Plots (a) and (b) have different scales for r/a 0

strength based on smooth bar data at the same stress As Taylor points out, for a very sharp notch having a small radius, or a crack of the same length, the stresses at this point are identical For a large notch, the normalized stress approaches that corresponding to the stress concentration factor which is three for the circular notch Taylor also shows

that the average stress over a length equal to 2a0 provides the same numerical results

as the stresses at a point and, further, that the method produces the same prediction for small crack behavior as that of El Haddad et al [12] as given in Equation (5.11) The critical distance concept for notch fatigue may involve quantities other than uniaxial stress Lanning et al [21] review some approaches that involve use of a combination of hydrostatic and shear stress and crack propagation thresholds From a large body of notch fatigue limit data generated on Ti-6Al-4V, they explored other quantities in a critical distance approach to see if the data could be consolidated In addition to evaluating

a quantity at a critical distance, they also used both an average value of a quantity and a weighted average over some distance Quantities evaluated included stress range, mean stress, and elastic strain energy density From FEM computations to obtain these quantities for cylindrical notched specimens of varying notch depths and radii including variations of applied stress ratio, R, some of which produced local notch plasticity, they found best-fit parameters for each modeling concept The use of a single critical distance where the local stress range was evaluated produced the best overall correlation with the experimental data The authors conclude that the approach is moderately successful but point out the difficulties that occur when local plasticity produces variations in R with distance from the notch and when very small notch sizes are included in the database

A further attempt to find a quantity that can be used to consolidate notch data with smooth bar fatigue data using averaging over some critical distance was made by Naik

et al [29] Two approaches were used involving the Findley parameter, one was evaluating the parameter at a critical distance, ac, and the other used the average value of the same parameter over the same critical distance, ac They used the Findley parameter, FIN, for uniaxial stress in the form

aand maxare the maximum shear stress amplitude and maximum normal stress on a critical plane that is defined as the plane where FIN is maxi-mum The parameter was shown to fit a large body of smooth bar fatigue data on Ti-6Al-4V

at several values of stress ratio, R, as illustrated in Figure 5.24 from [29] For those data, the best fit to the data is when the parameter has the form for an S–N type curve,

To apply the parameter to notch data, the average value of the parameter over a distance

ac is called Gr, and is given by

GrN= 1

ac

ac 0

where GF is defined as a notch gradient parameter In this formulation, the critical distance is not a material constant Rather, the quantity G is found to be a constant

0 200 400 600 800 1000

R = 0.1

R = 0.5 Curve fit

Cycles to failure

k = 0.29

Figure 5.24. Findley parameter for cycles to failure in Ti-6Al-4V from smooth bar data [29].

from calculations involving a wide range of notch geometries With GF as a constant, a quantity s is computed for any value of stress ratio, R, from



1− R 2k

2

(5.27)

where k is the constant in the Findley parameter, Equation (5.24) A quantity A0 is defined as

A0=



1+2ac





(5.28)

where  is the notch root radius and acis the critical distance (not a constant) The authors derive the relation

from Equation (5.27) for any value of R Once A0 is found, the critical distance can be obtained from Equation (5.28) and used in Equation (5.26) to obtain the value of FIN from which number of cycles is determined using the equation that fits the smooth bar data – Equation (5.25) in this example The ability to consolidate notch data using this approach is illustrated in Figure 5.25 where the constant GF= 116 This constant was slightly different than the value of 1.21 obtained from a more limited set of notch fatigue data, but the correlation was quite similar For the results shown in Figure 5.25, the critical