High Cycle Fatigue: A Mechanics of Materials Perspective part 17 pptx

Log stress, Log flaw size, a Threshold stress intensity, Kth FATIGUE REGIME σe Endurance limit stress, σe Growth below long crack threshold Growth below endurance limit El Haddad a0 c

Log stress,

Log flaw size, a

Threshold stress intensity, Kth

FATIGUE REGIME

σe

Endurance limit stress, σe

Growth below long

crack threshold

Growth below endurance limit

El Haddad a0 correction

a0

Figure 4.1. Schematic of a Kitagawa diagram.

should not grow.∗ There are two regions shown in the diagram where cracks can exist either below the endurance limit or the below the extrapolated long crack threshold In the first case, the crack behavior is dominated by fracture mechanics, not stress In the second, the small crack correction of El Haddad smoothes out the threshold curve in order to meet the endurance limit at very small crack lengths

The Kitagawa type diagram [2], shown again schematically in Figure 4.2, is a useful tool to evaluate the potential for a crack to reduce the HCF capability of a material, irrespective of crack size This plot of stress  against crack length (a) on log–log scales was first suggested for a simple edge-crack geometry where stress intensity is proportional to√

a The endurance limit when no crack is present, or the threshold stress intensity when there is a crack, represent the limits above which failure might occur and below which it will not, corresponding to some chosen number of cycles or crack growth rate, respectively The endurance limit is a horizontal line in Figure 4.2 while the threshold stress intensity is line of slope= −05 for the simple edge crack The two

∗

The terminology for the diagram with log stress plotted against log crack length [2], used subsequently for any crack geometry and stress ratio, is referred to variously in the literature as a Kitagawa–Takahashi diagram,

a K–T diagram, a Kitagawa type diagram, or most commonly, a Kitagawa diagram In this book, the common terminology “Kitagawa diagram” will be used to refer to that type of plot In a similar manner, the terminology

“El Haddad short crack correction” will be used to represent the contribution of El Haddad et al [3] The use

of this terminology is in no way meant to be disrespectful to, or to ignore the contributions of, the co-authors of Kitagawa or El Haddad in the referenced publications Rather, the respective terminologies are for convenience only.

Log stress,

Log flaw size, a

Threshold stress intensity, Kth

Endurance limit stress, se Short crack correction

a

a0

a0

Figure 4.2. Schematic of a Kitagawa diagram for any geometry Short cracks of length a behave as if they

were long cracks with length a + a 0 

lines can be connected to form a single curve for all crack lengths using the short crack correction proposed by El Haddad et al [3] In the work of Moshier et al [4], it is shown how the Kitagawa diagram can be used for any crack geometry In their application, they applied it to a surface flaw in a double-edge-notch tension specimen Of particular significance in using the Kitagawa diagram for any geometry is that the endurance limit and the value of a0 for the short crack correction are not material constants but, rather, material parameters for the particular geometry and stress ratio, R, being investigated

To describe the Kitagawa diagram mathematically for any geometry, the following analysis is presented The K solution for any crack geometry can be written in the form

where K is the stress intensity,  is the stress, a is the crack length, and Ya defines the K solution for a particular geometry If Equation (4.1) is written for the threshold condition of a long crack, and the crack length is represented by a1, then the long-crack solution for stress as a function of crack length, the threshold line in Figure 4.2, is

√

where KthLCrepresents the long-crack threshold stress intensity, a material constant for

a given stress ratio, R Denoting the intersection of the long crack threshold and the endurance limit stress in Figure 4.2 by a0, the intersection is defined in the equation

e= KthLC

where eis the endurance limit stress for the particular geometry for which the K solution

is given by Equation (4.1) The usual form for this equation solves for a0,

a0= 1





KthLC

eYa0

2

(4.4)

and is commonly found with the Ya term missing or equal to a constant For the more general case, either of these Equations (4.3) or (4.4), can be solved for a0either iteratively

or graphically Equation (4.4) for a0will be used extensively in the remainder of the book and will be repeated quite often

The short crack correction of El Haddad et al [3], shown graphically in Figure 4.2, uses the concept that a crack of length a behaves as if it had a length a+ a0, which has the long crack threshold value Noting that the crack length a1can be written as

the FLS for any true crack length a becomes, from Equations (4.2) and (4.5):

where, as the crack length goes to zero, the FLS goes to e The short crack curve in the Kitagawa diagram is easily constructed if the long crack curve exists by taking the value of a for any value of stress from Equation (4.2) and reducing it by a0 Alternately,

if the effective value of K is wanted for a short crack [5], it is easily shown that the effective K is

KthSC= KthLC

 a

a+ a0

Y a

where, as in the calculation for a0, the Y terms can usually be ignored

The significance of the Kitagawa diagram with the short crack correction is that it shows, if the threshold data follow the curves, that cracks of length a0or less, have FLSs which are not significantly lower than the endurance limit for a particular geometry At

a crack length of a0, the FLS is 1/√

2 or 071 of the endurance limit If cracks are formed at LCF stresses that are higher than the HCF limit, then the cracks that eventually propagate under HCF may act as if an overload occurred which may retard the crack or increase the fatigue limit [6] Thus, it is not surprising to find that small cracks formed under fretting fatigue [7], or under LCF at notches [8], or in smooth bars, do not have a significant detrimental effect on the FLS These cases are discussed later in this chapter

4.1.1 Behavior of notched specimens

A Kitagawa type plot can be used to represent what happens in a notched bar, as has been done by Moshier et al [4, 9] A schematic of threshold stress as a function of crack length, assuming no load-history effects in the crack growth behavior, is given in Figure 4.3 for two different cases The diagram shows two curves, an upper and a lower one, which represent what is commonly termed blunt notch and sharp notch behavior, respectively Above or below each curve, there is either crack initiation and growth to failure at higher stresses, or no initiation or growth at lower stresses, respectively The two curves represent the relation between stress and crack length corresponding to the K-solution for a crack emanating from a notch The solution is corrected for small crack behavior as per the approach suggested by El Haddad et al [3] For the blunt notch case, represented by the upper curve through A, a crack that forms will continue to propagate because the stress for crack initiation or threshold crack growth is a decreasing function

of crack length However, in region I below the curve, stresses are below the endurance limit, and cracks should not form in this region unless the history of loading is such to allow formation of a crack The line through A in the diagram, which is constructed for

a specific notch geometry, is valid only for that geometry at a given stress ratio Cracks formed in region I must, therefore, form from some other loading condition The point A

in the figure, which represents the endurance limit of the notched bar, is often taken to be the smooth bar endurance limit stress, e, divided by kt Data such as those of Lanning

et al [8] indicate that this is not necessarily the case for mildly notched titanium bars While the value of stress at A can be written as e/kf, where kf is the fatigue notch factor, this is no more than a mathematical definition of the endurance limit stress for the notched body The stress at A is not a material constant, but a geometry-dependent, empirically determined quantity The subject of fatigue thresholds for cracks at notches

is discussed in more detail in Chapter 5

Log crack length, l

I II

A

B

C

D

l

No crack growth

Crack growth

to failure

Figure 4.3. Schematic of a Kitagawa type diagram for notched specimens.

A Kitagawa type diagram was used by Moshier et al [4] for mildly notched specimens which were precracked under LCF The endurance limit stress for any crack length was found to be well predicted by a load-history dependent model [6], described later in this chapter, combined with the equations that connect the long crack threshold and the notched specimen endurance stress using the small-crack correction from El Haddad

et al [3] It should be noted that the parameter a0 used in the El Haddad correction,

is not a material constant when used for notches in the manner described here Rather,

it represents the crack length where the long crack K solution for the notch, K= Kth, intersects the endurance limit given by the stress at point A in the diagram (Figure 4.3)

In region II of Figure 4.3, above the lower curve which represents the K solution for

a very sharp notch, typically one with a ktvalue in excess of approximately four or five, cracks are known to be able to initiate but not continue to grow Until the stress exceeds the value at B, and for crack lengths below that indicated by point D, a crack may initiate but arrest As in the case described above, the stress above which cracks initiate, C, is often taken to be e/kt, but it is extremely difficult to verify this and, further, this value

is of minor engineering significance What is of greater significance is the empirically determined endurance stress, B, under which a specimen will both initiate a crack and continue to propagate it to failure Again, this endurance limit is not a material constant but, rather, an empirically determined quantity that depends on the particular sharp notch geometry While cracks of lengths less than that of point D may initiate and arrest, these crack lengths are generally very small (the so-called small crack or short crack) and not detectable other than in a laboratory setting using sophisticated methods not applicable

to real engineering structures The transition from sharp notch fatigue to cracks, the latter governed by fracture mechanics, is discussed in greater detail later in this chapter

In the extreme case where a notch starts to behave as a crack, the crack length at point

D approaches zero as the notch radius approaches zero, and the geometry becomes that of

a crack extending from an existing crack To illustrate this, a numerical example is used

to produce the results shown in Figure 4.4 where an endurance limit of 300 MPa and a threshold K of 5 MPa√

m are assumed For initial crack (sharp notch) lengths of c= 5 or

500 m, the figure shows the relation between crack extension from the sharp notch and stress required to exceed Kth with and without the short crack correction of El Haddad For the initially very short crack (c= 5 m), the calculated endurance limit (using the short crack correction) corresponds to a value of kf= 103 whereas for the 500 m initial crack, kf= 258 As the initial crack length gets longer, kf increases so that for an initial crack length of 1 mm, kf= 351 In this example of a sharp notch represented as a crack, assuming no load-history effects such as those due to overloads, and using Linear elastic fracture mechanics (LEFM), there is no tendency to form a crack that will arrest Also,

as seen in Figure 4.4, as the crack length increases the computations show that all data follow the same long crack threshold

10 100 1000

c= 5 μ m, short crack

c= 5 μ m, long crack

c= 500 μ m, short crack

c= 500 μ m, long crack

Crack length, a( μ m)

Kth= 5 MPa √ m

Figure 4.4. Kitagawa type plot for a sharp notch that is treated as a crack.

The Kitagawa type diagram can also be used to examine the effects of defects on the FLS Materials such as cast aluminum have initial porosity that can be treated as initial defects or cracks of the size of the pores Assuming that the initial crack size is known, the stress level below which crack propagation should not occur can be predicted from a knowledge of the long crack threshold, the endurance or FLS for the defect free material, and the Kitagawa diagram corrected for short crack effects as shown earlier If the K solution for the specific specimen geometry is

then the critical or transition crack length is given [see Equation (4.4) in previous section] as

a0= 1





KthLC

eYa0

2

(4.9)

If the dimensionless quantities

¯a = a

Kth/√a

are introduced, then the Kitagawa diagram is as shown in Figure 4.5 in log coordinates

If however, the plot is made in linear coordinates, the resultant figure takes the form shown in Figure 4.6 Plots of critical stress versus initial crack size should have this general shape unless the assumptions in the analysis of the material with initial defects are

10 0

10 –1

0.707

2 1

a /a0

Figure 4.5. Kitagawa plot in dimensionless coordinates.

0 0.2 0.4 0.6 0.8 1

a /a0

Figure 4.6. Kitagawa diagram in dimensionless coordinates on linear scale.

violated Caton [10], in his study of 319 aluminum, has attributed the observed behavior being different than that depicted in Figure 4.6 to factors such as small crack effects, crack closure in threshold measurements, and interactions of crack tip plasticity with microstructural features His data on three different microstructural conditions denoted

as low, medium, or high SDAS are shown in Figure 4.7 The initial pore size, measured

on the fracture surface of each specimen, is taken as being equivalent to an initial crack length Thus, the plot should have the features shown in Figure 4.6 In order for all three microstructures to be compared properly as in Figure 4.7, the endurance limit and

0 200 400 600 800 1000 0

20 40 60 80

100

Low SDAS Medium SDAS

High SDAS

Initiating pore size (microns)

Figure 4.7. Threshold stress levels for step-tested W319-T7 aluminum as a function of initiating pore size.

threshold value of K would have to be the same for all three conditions This was found not to be true for the values of the threshold stress intensity The endurance limit for the material without defects was not obtained Nonetheless, the similarity of the general shape of Figure 4.7 to the dimensionless schematic, Figure 4.6, is surprising For the two conditions marked medium and high SDAS, however, the data points indicate a threshold stress that is nominally independent of initial crack (pore) size This led the author to examine other models to relate initial pore size to threshold stress

In the USAF HCF program, it was recognized early on that the understanding of LCF– HCF interactions was not only important but, further, that LCF–HCF interactions had not been addressed to any substantial level prior to the program In response to the lack of research in this area, several series of tests were conducted over the years at the AFRL Materials Directorate (AFRL/ML) where specimens were subjected to LCF loading prior

to HCF testing to obtain an FLS [11] In these types of tests, the LCF stress corresponding

to a life in the typical range 104–105cycles is determined first from a series of LCF tests and then interpolating on a S–N curve Some fraction of the LCF life is then applied

to the specimen after which it is tested in HCF using the step-loading technique [12] where load-history effects and coaxing have been shown not to be important in Ti-6Al-4V [13] Two things should be noted in this type of experiment involving preloading under LCF First, the LCF life is a statistical variable, so testing up to a predetermined fraction

of predicted life has an inherent error associated with what fraction of life it really is

Second, if LCF testing and HCF testing are performed at the same value of stress ratio, R, then the LCF test will be at a higher stress than the HCF test for an uncracked specimen

In the event that a crack is formed during LCF, application of a lower load during HCF testing can amount to having an overload effect, so the fatigue limit may tend to be higher than that obtained without an overload effect Such a phenomenon was demonstrated by Moshier et al [4] and is discussed later in this chapter

Data obtained from different tests on smooth bars subjected to LCF, then HCF, are summarized in Figure 4.8, where the FLS is normalized against the value obtained for the same specimens tested without any prior loading history Data indicated as “Maxwell” are from the authors’ investigations with his colleague David Maxwell while data designated

“Park” and “Morrissey” appear in [14] and [15], respectively The horizontal axis is the number of LCF cycles divided by the expected life at the stress level and R used in the LCF tests Noting that from a design perspective that the allowable LCF life would be much less than the average, LCF cycle ratios in practice should not be expected to exceed 0.5 of the average, and even less when factors of safety are included Thus, any reasonable number of LCF cycles that might be encountered are expected to be confined only to the left portion of Figure 4.8 The data show, that within reasonable scatter, the HCF limit does not appear to be degraded by any significant amount in tests covering a range of LCF stress levels and corresponding lives, and stress ratios in both LCF and HCF Additional data to evaluate the effect of LCF on the HCF threshold in Ti-6Al-4V were obtained in smooth bar tests in two separate but parallel investigations [14, 15] In the work by Mall et al [14], the HCF limit stress corresponding to 107cycles to failure was obtained at 420 Hz at R= 01 05, and 0.8 These specimens were subjected to initial cycling at the same frequency as in the HCF portion of the tests, but at stress levels

0 0.2 0.4

σe

0.6 0.8 1 1.2

Park

Morrissey Maxwell

Maxwell (bar)

N /Nf

Ti-6Al-4V Cylindrical specimens

Figure 4.8. Normalized endurance limit stress as function of LCF history.

and values of R that produced shorter lives to failure These initial cycles, referred to

as LCF, were applied at three different conditions: a maximum stress of 700 MPa with

R= 01 (Case A), 790 MPa with R = 05 (Case B), and 900 MPa with R = 05 (Case C) These conditions were chosen to cover a range of maximum stresses and cycles to failure The cycles to failure for cases A, B, and C were approximately 1×106 25×106, and 025× 106, respectively The relative amplitudes of these LCF conditions are shown schematically in Figure 4.9 The LCF pre-damage was selected as nominally 20% of life but went from about 15 to 50% of LCF life for Case A at R= 01 The HCF fatigue limit stresses corresponding to a life of 107cycles are also shown schematically in Figure 4.9

It can be seen that the LCF preloading can result in either an overload or underload (negative overload) condition depending on the combination of LCF loading condition and value of R for the subsequent HCF test

After applying these pre-damage conditions, the step-loading technique was used to determine the fatigue strength for 107 cycles to compare with those without pre-damage These results are summarized in Figure 4.10 Since a 10% variation is not uncommon in fatigue data, the results clearly suggest that there was practically no effect of pre-damage from LCF on the HCF fatigue strength Examination of Figure 4.9 shows that for HCF testing at R= 05, the LCF pre-damage occurred at stress levels which were above the HCF stress levels If cracks formed during LCF, these cracks could be considered to be overloaded when subsequently tested under HCF, so the stress levels might be expected

Stress, MPa

200

400

600

800

1000

R= 0.5

R= 0.8 HCF

LCF

R= 0.1

R= 0.5

R= 0.5

700

790

900

534 639

910

(A)

(B)

(C)

Figure 4.9. Schematic showing amplitudes of LCF and HCF fatigue limit stresses.