COMBINED USE OF STRAIN-BASED FATIGUE …

1

COMBINED USE OF STRAIN-BASED FATIGUE METHODOLOGY AND NOTCHED MEMBER TEST DATA

Norman E. DowlingVirginia Polytechnic Institute and State UniversityBlacksburg, VA

2005 USAF Structural Integrity Program (ASIP) Conf.29 November to 1 December 2005Memphis, TN

2

Major Approaches to Fatigue Design and Evaluation

• Stress-Based

• Strain-Based

• Crack Growth (Fracture Mechanics)

3

Stress-Based Approach –

Where to Get the S-N Curve?

• Component Data

• Estimates (Shigley, Juvinall)

• Design Codes (AWS, AASHTO)

• Test Specimen Data (MILHDBK 5)

• Standard Curve Shapes

4

Matching a Component to Specimen Data

5

Equating stresses as ktS gives similar notch strain and initiation life

6

But crack growth effects cause a size effect– so also match K/ktS for total life

7

Total Life to Failure Data

8

Scaling of Standard Curve Shapes to Limited Component Data

[From Dickson et al., 1999]

9

Strain-Based Approach

10

11

cff

bf

fa NN

E)2()2( ε′+

σ′=ε

naa

a HE

′

′σ

+σ

=ε/1

Cyclic stress-strain curve

Strain-life curve

12

13

14

15

An analysis using the cyclic stress-strain curveis needed:

• Elasto-plastic finite elements

• Neubers ruleESkt

2)(=εσ

na

aat

a HE

kS

′

′σ

σ+σ=/1

21

16

Example: zero mean stress (R = −1) loading

17

0

10

20

30

40

0.000 0.006 0.012 0.018 0.024 0.030 0.036

εa, Notch Strain Amplitude

Pa,

Load

Am

plitu

de, k

N

dataNeuberFEA

t = 5.08 mm

18

0

10

20

30

40

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Ni, Nf, Cycles

Pa,

Load

Am

plitu

de, k

N

0.5 mm crack Neuberfailure FEA

t = 5.08 mm

19

Mean Stress Effects

cff

bf

f

mfa NN

E)2()2(1 ε′+

σ′σ

−σ′

=ε

( ) cbfff

bf

fa NN

E+ε′σ′+

σ′=εσ )2()2( 2

2

max

20

21

Example: zero-to-maximum (R = 0) loading

22

23

Variable Amplitude Loading

24

Component-Specific Strain-Life Curve- for surface finish and load interaction effects

0.0001

0.001

0.01

0.1

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

ε a ,

Stra

in A

mpl

itude SAE 4142 Steel (450 HB)

σ'f / E

ε'f

polished, b

modified, bs

25

Data for helicopter load spectrum Felix applied to plates with round holes (kt = 2.5) of Ti -6Al-4V

1 10 100 1000 10000 100000Flights to Failure

200

400

600

800

Max

imum

Nom

inal

Stre

ss, M

Pa

-0.10-0.105

-0.08-0.085-0.095

-0.09

b=-0.0763

Notched Titanium 6AI-4V

26

Conclusions• Consider size effects from crack-growth in using

notched specimen S-N data.

• Fatigue limits and load interaction effects are issues.

• Strain-based analysis provides an S-N curve.

• Strain-based analysis considers the effects of localyielding on mean stress.

• Fabrication detail and load interaction effects can beincluded by using a component-specific strain-life curve.

27

10

100

1000

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Nf , Cycles

S a ,

MPa

Juvinal 2000Shigley 2004Data

AISI 4130 Steelσu = 807 MPakt = 4.00, R = −1

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Goodman Sar for double-edge-notched axial specimens of 2024-T3 Al

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

Goo

dman

Sar

, M

Pa

0691382070 Fit

Mean Stress, MPa

2024-T3 Alkt = 2.15

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Smith, Watson, and Topper (SWT) Sar for double-edge-notched axial specimens of 2024-T3 Al

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

SW

T S

ar ,

MP

a

0691382070 Fit

Mean Stress, MPa

2024-T3 Alkt = 2.15

30

Bfar NAS =

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

Wal

ker S

ar ,

MP

a

069138207Fit all

Mean Stress, MPa

2024-T3 Alkt = 2.15

Walker Sar for double-edge-notched axial specimens of 2024-T3 Alfit to with γ = 0.733

31

32

33

10

100

1000

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Nf , Cycles

Sa

, Nom

. Stre

ss A

mpl

itude

, MP

a

0 Calc0 Data138 Calc138 Data

Mean Stress, MPa

7075-T6 Alkt = 4.00

COMBINED USE OF STRAIN-BASED FATIGUE METHODOLOGY AND NOTCHED MEMBER TEST DATA Norman E. Dowling Virginia Polytechnic Institute and State University Blacksburg, VA 2005 USAF Structural Integrity Program (ASIP) Conference 29 November to 1 December 2005 Memphis, TN ABSTRACT The strain-based method combines the strain-life and cyclic stress-strain curves, obtained from small, axially loaded test specimens, with estimates of notch strain to provide life estimates for notched members. If applied to a notched member of interest over a range of load levels, an S-N curve shape is generated that is specific to the member geometry and loading configuration. The analysis of local notch stress and strain permits rational handling of mean stresses by addressing the effect of prior plasticity on the local notch mean stress. High-low load sequence effects that cause early microscopic physical damage to the material can be addressed to an extent by employing a strain-life curve for specimens that have been subjected to a small number of cycles of prior plastic straining. The method does not directly address the effects of surface finish, member size, and stress gradient. But test data on actual, as-manufactured components do automatically include these effects, so that it is suggested that the strain-life curve be adjusted to obtain agreement with the component data, resulting in a component-specific strain life curve. Primarily, the strain-life curve will need to be lowered at long lives, as the fatigue life is sensitive to effects such as surface finish only at long lives. An advantage of such an approach is that the modified strain-life curve is expected to be valid for load spectra other than those for which it was developed, and also for modest changes in member geometry, although not for changes in surface condition. INTRODUCTION There are currently three major approaches to design and analysis of fatigue strength and life: stress-based approach, strain-based approach, and crack growth (fracture mechanics) approach. This paper provides discussion on the use of the first two of these. In the stress-based approach, curves relating nominal stress or other applied loading variable to fatigue life (S-N curves) are employed in engineering design and analysis. However, unless fatigue test data are available for the exact component of interest, obtaining the appropriate curve for a specific application can be a problematical process. In the treatment that follows, various methods for obtaining S-N curves in the context of a stress-based approach are first identified and some comments are made on each. The issues of mean stress adjustments and of fatigue limits are included in the discussion. Next, the strain-based method is summarized, and its use in deriving S-N curves is described. In particular, it is noted that the method provides a rational means of accounting for the effects of component geometry and mean stress effects, both of which present difficulties for stress-based methods. Also considered is the use of a component-specific strain-life curve that

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has been adjusted to obtain agreement with spectrum-loading fatigue data on a component of interest. S-N CURVES Several methods are in use for obtaining S-N curves for engineering components. These include: (1) extensively testing the component of interest, (2) estimates as in mechanical engineering design textbooks, (3) curves provided in structural design codes, (4) adapting notched laboratory specimen data, and (5) scaling of a standard curve shape to limited component data. In employing an S-N curve obtained from any of these methods, mean stress effects need to be handled, and the question of a fatigue limit is likely to arise. Comments on each of the above methods and issues follows. Obtaining S-N Curves If a number of actual components are available for testing, then it may be possible to test these over a range of stresses to obtain a complete S-N curve. This is the preferred approach, but the cost is likely to be high. Also, it cannot be employed for components still in the design stage. Mechanical engineering design books, such as Shigley (2004)* and Juvinall (2000), contain methods for estimating S-N curves. Typically, a fatigue limit, as at 106 cycles for steels, is estimated based on adjusting the elastic stress concentration factor, kt , to obtain a fatigue notch factor, kf . Additional adjustments are also made for size effect, surface finish, etc. A log-log straight line is then drawn from the fatigue limit point to a second point at 103 cycles, where a modified fatigue notch factor k'f may be employed. Finally, an additional log-log straight line can be drawn between the point at 103 cycles and an estimate of the static strength at 1 cycle. For example, Fig. 1 shows a comparison of such estimates with test data on notched specimens of an alloy steel. Such estimates rely heavily on empirical data that is well developed only for steels and to some extent for aluminum alloys. Also, they are often not very accurate. Design codes for structural applications, such as those of the American Welding Society (AWS 2004), often contain S-N curves based on test data for full size structural members containing various connection details. To apply these to a particular case, one needs to match the severity of the geometric detail to one of the family of S-N curves that is available. These design codes apply mainly to large stationary structures, with emphasis on welded structures. Hence, they are not generally applicable to other situations. Extensive test data on notched test specimens is available, as in Military Handbook No. 5 (MILHDBK 1998). It is tempting to employ such data by matching the elastic stress concentration factor, kt , between test specimen and component. However, this procedure can result in a situation where differing notch size effects between the test specimen and the component invalidate the comparison. A better method is to match the notch radius between specimen and component, and then relate stresses according to the same value of the elastically calculated stress, ktS, where S is a nominal stress consistently defined with kt . This gives similar size effects for the initiation of small cracks, as well as similar local notch stresses and strains, even if there is local yielding at the notch for short lives. ________________ *References listed at the end of this paper are identified by an author's name, or an organization's acronym, and the year of publication.

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For matching the behavior of specimen and component beyond the early initiation phase of life, it is necessary to have a similar variation of the (fracture mechanics) stress intensity factor, K, as cracks grow beyond the local region affected by the stress concentration of the notch. Figure 2 shows four notched member geometries, and Fig. 3 crack initiation data for an alloy steel for the three blunter notches. Then Figure 4 shows the approximate variation of K for cracks within the local notch field (dashed line) and beyond (solid lines). Two of the geometries have a similar variation of K and hence similar total fatigue lives, as seen in Fig. 5. See Dowling (1981) for more detail on this approach to matching specimens and components. A common procedure in the helicopter industry is to use a standard curve shape based on test specimen data for notched or unnotched specimens of a given material (Dickson 1999; Cronkhite 2000). Influences such as surface condition, environment, and stress concentration factor kt are considered, and normalization to a fatigue limit is often a prominent part of the process. The standard curve shape is scaled to fit limited component test data, as in Fig. 6, which has the obvious advantage of providing a quantitative link to component data. However, member geometry affects the shapes of S-N curves, so that beyond the range of the actual component data, there is no guarantee that the curve shape is valid. Mean Stress Effects Assuming that an S-N curve is available, its use involves making estimates of behavior for mean stresses differing from those for the curve. Note that mean stress effects are often expressed in terms of ratios R = Smin/Smax . A common approach is to use the Goodman equation applied to nominal stresses. This method can be expressed as an equation giving the amplitude of completely reversed (zero mean level) nominal stress Sar that is expected to cause the same fatigue life as a desired combination of stress amplitude Sa and nonzero mean stress Sm . The most basic form of this relationship employs the ultimate tensile strength σu and in effect includes no concentration factor for the mean stress.

u

ma

ar SS

S

σ−

=1

(1)

Entering an S-N curve for the completely reversed case with Sar then provides a life estimate for the nonzero mean stress situation. This method is generally conservative, but also often quite inaccurate. This can be seen for test data at various mean stresses on notched members of an aluminum alloy in Fig. 7. Note that Eq. 1 implies that plotting Sar versus life should consolidate all of the data onto the curve for zero mean stress. Another approach that generally gives more accurate results is the method of Smith, Watson, and Topper (Smith 1970).

2

1, maxmax

RSSSSS araar

−== (a, b) (2)

4

where Smax = Sa + Sm and R = Smin/Smax . Forms (a) and (b) are two mathematically equivalent and can be chosen as convenient. For the same set of data as above, this SWT equation is seen to provide improved correlation in Fig. 8. If component data at more than one mean stress or R-ratio are available, then the equation of Walker (1970) can be employed with a value of the adjustable parameter γ that has been fitted to the data.

γ

γγ−

−

==2

1, max

1max

RSSSSS araar (a, b) (3)

Excellent correlation on this basis of the same set of data is seen in Fig. 9. See Dowling (2000) and Dowling (2004) for additional discussion. Note that the above discussion assumes that the significant mean stress or R-ratio is that for the nominal stress. An entirely different approach is to estimate the local mean stress at the critical stress raiser and then base the mean stress adjustment on this variable, as discussed later in this paper. Fatigue Limit Effects A fatigue limit, that is, a safe stress below which there is no fatigue damage, often appears to exist based on S-N curves from constant amplitude fatigue test data. Consequently, S-N curves from various estimates and standard curve shapes often have this feature. The concept of a fatigue limit needs to be treated with some caution. If the stress history is such that the fatigue limit is never exceeded, then the concept can be applied. However, occasional severe stress cycles can cause subsequent cycles below the apparent fatigue limit to contribute to fatigue damage. Experimental data illustrating this are shown in Fig. 10. This effect is caused by the severe cycles initiating fatigue damage in the microstructure of the material, specifically, damage that would otherwise not be able to initiate if the stress never exceeded the fatigue limit. A corrosive environment may also have a similar effect. For highly irregular loading, fatigue damage may even accumulate according to an effective damage line that is below the S-N curve even above the fatigue limit. One approach to this situation is to simply extrapolate the S-N curve below the fatigue limit by extending a straight line on a log-log plot, as suggested by the solid line in Fig. 10. Other strategies include lowering but not eliminating the fatigue limit, or providing a shallow slope to the S-N curve beyond the beginning of the apparent fatigue limit. Experimental data for several metals (Heuler 1986; DuQuesnay 1993) suggest that there is a true fatigue limit around half of the apparent value. Also, recent test data extending to 109 cycles and beyond have exhibited the surprising behavior of a drop in the stress-life curve beyond the flat region in the 106 to 107 cycles range. Such behavior has been observed in a number of steels and other engineering metals, with one set of data being shown in Fig. 11. Detailed study reveals that there are two competing mechanisms of fatigue failure, in particular, failure initiating from surface defects, versus failure initiating from internal nonmetallic inclusions. The former dominates the behavior up to around 107 cycles and exhibits the apparent fatigue limit, but the latter causes failures at lower stresses and very long lives. For more discussion and data, see that paper by Bathias (1999) and others in the same journal issue. Hence, where very large numbers of cycles are applied in service, the concept of a safe stress may not be valid even for constant amplitude loading.

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STRAIN-BASED METHOD The strain-based approach is summarized by Fig. 12. Life estimates are based on the stress and strain, σ and ε, occurring locally at the stress raiser of interest. Materials properties are needed from tests on unnotched axially loaded specimens under controlled strain, or controlled stress at long lives, as in Fig. 13. In particular, a cyclic stress-strain curve and a strain-life curve are needed. These are combined with a method of determining local notch stress and strain that considers plastic deformation behavior, such as elasto-plastic finite elements analysis, or approximate methods such as Neuber's rule. Cyclic Stress-Strain and Strain-Life Curves The cyclic stress-strain curve is usually fitted to the Ramberg-Osgood form

n

aaa HE

′

′σ

+σ

=ε/1

(4)

where stress and strain are expressed as amplitudes (half-ranges), E is the elastic modulus, and H' and n' are fitted to the data for a particular material. An example for an alloy steel is shown in Fig. 14, where curves are shown for the initial tensile behavior, for several particular numbers of cycles as this material cyclically softens, and for stable behavior from half the fatigue life. The half-life curve is usually taken as representative of the behavior for cyclic loading and on this basis employed in life estimates. For the strain-life curve, separately fitting the elastic and plastic strains and then adding these provides the equation that is usually used to represent the curve of total strain amplitude εa versus fatigue life in cycles, Nf .

cff

bf

fa NN

E)2()2( ε′+

σ′=ε (5)

where σ'f , b, ε'f , and c are fitted to the data for a particular material. Figure 15 shows strain-life curves for an aluminum alloy. One curve is for simple constant amplitude loading and the other for samples that have been subjected to several cycles of plastic deformation before testing. Figure 16 shows strain-life curves for the same alloy steel as in Fig. 10, including simple constant amplitude loading, a few cycles of initial plastic deformation, and occasional (periodic) plastic deformation. The initial plastic deformation lowers the fatigue limit, and the periodic plastic deformation appears to eliminate it. Estimating Local Notch Stress and Strain and then Fatigue Life One approach is to apply elasto-plastic finite elements analysis with the cyclic stress-strain curve to determine the local notch stress and strain. For example, the notched member of Fig. 17, made of the same material as in Fig. 14, was analyzed in this manner as described in Dowling (1979). The cyclic strains from analysis for stable (half-life) behavior over a range of completely reversed (R = −1) load levels are compared in Fig. 18 with strains measured by metal-foil strain gages. Note that this analysis used the half-life cyclic stress-strain curve from Fig. 14, which was fitted to an elastic, power hardening form of stress-strain curve with a distinct proportional limit, rather than to Eq. 4. Excellent agreement is obtained.

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As analysis by elasto-plastic finite elements is not always feasible, approximate methods such as Neuber's rule are frequently employed instead. If there is no general yielding over the entire cross section, that is, if yielding is confined to the local region of the notch, Neuber's rule takes the form

ESkt

2)(=εσ (6)

The quantities σ and ε are the local stress and strain at the notch, respectively, S is a nominal or average stress, and kt is an elastic stress concentration factor, consistently defined with S. Where the cyclic stress-strain has the form of Eq. 4, combining this with Eq. 6 gives

n

aaa

ta H

Ek

S′

′σ

σ+σ=/1

21 (7)

Hence, for a given applied cyclic load expressed as a nominal stress amplitude Sa , this equation can be solved iteratively for the local notch stress amplitude σa , and then Eq. 4 provides the local notch strain amplitude εa . The strain amplitude εa , obtained from finite elements, from Neuber's rule, or from other approximate method, can then be used with the strain-life curve, such as Eq. 5, to obtain the estimated number of cycles Nf to cause fatigue cracking. For the notched member of Fig. 17, life estimates made on this basis are compared with fatigue test data in Fig. 19. For the initiation of a small crack, excellent agreement is obtained based on the finite elements analysis. The estimates based on Nueber's rule are slightly below the data (conservative), as a result of this method overestimating the strains, as can be seen in Fig. 18. Figure 19 also shows the lives for final failure of the notched member. These lives are longer than those for the initiation for a small crack, due to the rather large amount of material that the crack must grow through to cause complete failure. This latter stage of life can of course be analyzed by fracture mechanics using a da versus ∆K curve for the material. Mean Stress Effects on the Strain-Life Curve The above procedure for making strain-based life estimates can be readily extended to cases of nonzero mean stress. This is done on the basis of the local notch mean stress, σm , as illustrated in Fig. 20. Thus, the strain-life curve needs to be generalized to include this variable. One possibility is to employ the mean stress equation of Morrow (1968).

f

ma

ar

σ′σ

−

σ=σ

1 (8)

where σar is an equivalent completely reversed stress amplitude that is expected to cause the same life as the actual (σa, σm) combination. This has the same form as the Goodman relationship, Eq. 1, but differs by the use of σ'f in place of σu and in the application to local mean stress. Incorporating the effect on life required by Eq. 8 into Eq. 5 gives

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b

f

mf

cf

bfa NNNN

E

/1

1*,*)2(*)2(−

σ′σ

−=ε′+σ′

=ε (a, b) (9)

where N* is calculated from εa using (a), as if the mean stress were zero, and then this value is employed in (b) to obtain Nf that does include the mean stress effect. The Morrow relationship is also often applied with the mean stress effect dropped from the second term of Eq. 9(a), which gives.

cff

bf

f

mfa NN

E)2()2(1 ε′+

σ′σ

−σ′

=ε (10)

The Smith, Watson, and Topper method (Smith 1970) is usually applied to the strain-life equation in the form:

( ) cb

fffb

ff

a NNE

+ε′σ′+σ′

=εσ )2()2( 22

max (11)

where σmax = σa + σm . A further option is to apply the Walker (1970) equation to the local notch stress. This can be accomplished by using Eq. 9(a) with a revised expression for calculating Nf from N*.

b

f

ba

fR

NNNN/)1(/)1(

max 21

*,*γ−−

σγ−−

−

=

σσ

= (a, b) (12)

where Rσ = σmin/σmax . Forms (a) and (b) are mathematically equivalent and can be chosen as convenient. Equation 9 generally works quite well for steels, but is often inaccurate for aluminum alloys, for which Eq. 11 should be used instead. If unnotched specimen data exist for fitting γ, Eq. 12 is the best choice. See (Dowling 2004) for more detail. Including Local Mean Stress in Notched Member Life Estimates It is useful to idealize the stress-strain behavior of the material as following the stable cyclic stress-strain curve from the beginning, thus neglecting any effect of the transient cycle-dependent hardening or softening that occurs mainly near the beginning of the fatigue life. Such an idealized material follows the stable cyclic stress-strain curve as its initial monotonic loading path (as in a tension test). Further, we neglect the additional transient effects of cycle-dependent creep and relaxation. These transient effects can be included, but the added complexity and need for additional materials constants is not generally considered to be justified by the relatively small benefit that results. However, note that the major, first order effect of local yielding on the value of the local mean stress is included by considering plastic deformation in the analysis. The procedure that is needed is illustrated in Fig. 20. First, it is necessary to have a static elasto-plastic analysis of the notched member that employs the cyclic stress-strain curve as the monotonic loading path. (Figure 18 is an example of such an analysis from finite elements.) The results of this analysis are then invoked for both the maximum load Smax during cycling (or the minimum if |Smin| > Smax) and the amplitude Sa , to obtain the corresponding local notch stresses and strains, σmax , εmax , and σa , εa .

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If the approximation of Neuber's rule is being employed, the calculations needed are equivalent to two solutions that satisfy both the σ ε = constant hyperbola of Eq. 6 and the cyclic stress-strain curve, once for Smax, and a second time for Sa . This is illustrated by Fig. 20(a). For the Eq. 4 form of stress-strain curve, this can be accomplished with Eq. 7 as follows:

nn

t HEHE

kS

′′

′σ

+σ

=ε

′σ

σ+σ=/1

maxmaxmax

/1max

max2maxmax ,

1 (a, b) (13)

First, (a) is entered with Smax and solved iteratively to obtain σmax , and then (b) gives εmax . Next, follow a similar procedure by employing the Sa value to obtain σa and εa , using

n

aaa

na

aat

a HEHE

kS

′′

′σ

+σ

=ε

′σ

σ+σ=/1/1

2 ,1

(a, b) (14)

Additional stresses and strains that are of interest are then computed.

amaa σ−σ=σε−ε=εσ−σ=σ maxmaxminmaxmin ,2,2 (15) The stress-strain response at the notch may then be plotted as shown in Fig. 20(c). To do so, first follow the cyclic stress-strain curve as the loading path to the point σmax , εmax . If Eq. 4 is represented as ε = f (σ), the subsequent cyclic loading may then be approximated as repeatedly forming a stress-strain hysteresis loop as shown, where the loop curves follow the original stress-strain curve as scaled up by a factor of two, ∆ε/2 = f (∆σ/2), with ∆ε and ∆σ being the changes relative to each loop tip. Once the above stress-strain estimates are complete, the values obtained readily provide a life estimate by solving for Nf from the various options available as Eqs. 9 to 12. For example, for zero-to maximum (R = 0) axial loading of plates with holes of 2024-T351 aluminum, estimated stress-strain responses for various load levels are shown in Fig. 21. The corresponding life estimates using Eq. 11 are shown in Fig. 22. In this case, the life data are available only as total numbers of cycles to failure, so the estimates are a bit below the data at the shorter lives, due to the crack growth phase being included in the data. (Compare to Fig. 19.) On this basis, the life estimates are quite reasonable. Additional similar life estimates are compared with test data in Fig. 23, here for fairly sharp double-edge-notches in 7075-T6 aluminum. Two cases are included, zero mean load and a relatively high constant mean load. As before, the crack growth phase is included in the life data, and the success of the estimates is good. EXTENSION TO VARIABLE AMPLITUDE LOADING

To extend the procedure just described to irregular variation of load with time, it is first useful to note some features of stress-strain behavior during irregular histories. The stress-strain behavior that might occur locally at a notch is illustrated by Fig. 24. The behavior follows stress-strain hysteresis loops, which have proportions approximately obeying the cyclic stress-strain curve from constant amplitude tests, Eq. 4. Moreover, the loop curves approximately follow this curve as scaled up by a factor of two, ∆ε/2 = f (∆σ/2), with ∆ε and ∆σ being the changes relative

9

to each loop tip. The closing of these loops is controlled by the memory effect. In particular, at points such as B' in Fig. 24(b) and (c), a hysteresis loop closes when it reaches the point of prior unloading where the loop started, such as B. At such points, the memory effect acts, and the previously established path, such as A-B-D, is continued.

The results of a static analysis using the cyclic stress-strain curve, including the Neuber's rule approximation, can be applied to load ranges during the irregular loading where the behavior obeys the cyclic stress-strain curve, that is, ranges for which ∆ε/2 = f (∆σ/2). Hence, the static analysis can be applied to each loop, such as B-C, and also to the range locating its beginning, such as A-B. But it cannot be applied to events such as C-B-D, which spans a memory-effect event, as these do not obey ∆ε/2 = f (∆σ/2). Analyzing the permissible ranges for the entire history allows the stresses and strains to be determined for all points in the history. In addition to A-B and B-C, the following ranges need to be analyzed for this example: A-D, D-E, E-F, F-G, and E-H. Also, the starting point such as A needs to be established in the same manner as the Smax point for the constant amplitude loading in Fig 20. Hence, for the Fig. 24 example, Eq. 13 is applied to point A. Then the following equations are solved to obtain ∆σ and ∆ε for each permissible range X-Y as noted above:

n

XYXYXYn

XYXYXY

t

XYHEH

Ek

S ′′

′σ∆

+σ∆

=ε∆

′σ∆σ∆

+

σ∆

=∆ /1/12

222,

2221

2 (16)

Note that these relationships are equivalent to Eq. 14, being derived from it by substituting Sa = ∆S/2, σa = ∆σ/2, and εa = ∆ε/2. After application of Eq. 16 to the appropriate ranges, invoking the sequence of the peaks and valleys in the load history then allows the stress and strain values to be obtained for each peak and valley. In the example of Fig. 24, note that there are four closed stress-strain hysteresis loops, B-C, F-G, E-H, and A-D. Due to the memory-effect behavior, small loops occur inside larger ones, such as F-G being within E-H, which is itself within the largest loop A-D. Each closed hysteresis loop corresponds to a cycle as identified by the rainflow cycle counting method, as defined in (ASTM E1049). To proceed to a life estimate, the strain amplitude εa and the mean stress σm for each closed loop (cycle) are of interest, as is the peak (maximum) stress σmax. Taking half of each loop's strain range gives its strain amplitude, and averaging the stresses for the loop tips gives the mean stress. The life Nf corresponding to each hysteresis loop is needed and can be determined from its strain amplitude εa and mean stress σm substituted into Eq. 9 or 10, or from its peak stress σmax and strain amplitude εa substituted into Eq. 11 or 12. The life for the variable amplitude loading can then be estimated from the Palmgren-Miner rule. If a sequence of irregular loading is assumed to be repeated until failure occurs, this rule takes the form

1

.

=

∑

reponejf

jf N

NB (17)

The summation is performed for one repetition of the history. The Nj are the numbers of times each cycle (loop) occurs in one repetition of the history, with Nj = 1 applying for each of the four cycles in the Fig. 24 example. The Nf j are the corresponding numbers of cycles to failure. Once the summation is done, one can solve for Bf , which is the estimated number of repetitions of the

10

history to failure. The cycle having the largest strain range will always correspond to the range between the highest peak and lowest valley loads. More detailed descriptions of this procedure can be found in Dowling (1999) and Dowling (2003). COMPONENT-SPECIFIC STRAIN-LIFE CURVE For fatigue-critical engineering components, a few tests are sometimes run, but not generally a sufficient number to generate an entire S-N curve. Such tests have the advantage of automatically including effects such as those of size, surface finish, and fabrication detail. If a variable amplitude loading history similar to the expected service loading is applied in these tests, the results also include interaction effects between the high and low load levels. In particular, occasional severe stress levels have two effects.

First, the severe cycles affect the local mean stresses for lower level cycles. This effect is included if the strain-based approach is employed as described above for the variable amplitude loading of interest. In particular, the local mean stresses for lower level cycles, such as σm for cycle F-G in Fig. 24, are affected by the local yielding effect of the more severe loading events in the history.

Second, the more severe cycles may initiate physical damage in the material such that the lower level cycles have a greater effect than expected from constant amplitude test data. If a distinct fatigue limit exists, this is likely to be lowered or eliminated as already discussed. Even for materials with no distinct fatigue limit, the material may behave as if the strain-life curve were lower at relatively long lives, as in Fig. 15. Hence, the effect is to lower the long-life end of the elastic strain component of the strain-life curve, by replacing the slope b with a modified value bs giving a steeper slope, as illustrated in Fig. 25.

Consider a situation where some component S-N data are available from tests under variable amplitude loading, and assume that a strain-life curve is available, but not one that includes the prestrain effect. The following procedure is suggested: Adjust the strain-life curve, primarily by changing b in Eq. 5, to give a lower curve at long lives, until agreement is obtained between the S-N data and life estimates from the strain-based approach. The resulting component-specific strain-life curve would then be expected to give improved life estimates for similar situations, such as a load histories or geometries that differ only modestly from the case applied in the component S-N tests. Note that any surface finish effects will be automatically included in the new strain-life curve.

An illustration of this process is given in Fig. 26. The data points shown correspond to notched plates subjected to the standard helicopter loading spectrum Felix (Edwards 1984). Denoting the peak nominal stress in the spectrum as the 100% level, the spectrum contains occasional ground-air-ground cycles that range between −28% and 100% in each simulated flight. There are numerous additional cycles at lower levels, averaging about 16,300 in number per flight, with most of these having a relatively high tensile mean stress. A strain-life curve for the titanium 6Al-4V material (Dowling 1989) had a value of b = −0.0763 for Eq. 5. However, this gave nonconservative life estimates. Changing to b = −0.095, while leaving σ'f , ε'f , and c unchanged, gives better agreement on an average basis. Alternatively, b = −0.100 gives calculated lives at the lower limit of the test data. The component-specific strain-life curve with the latter b value is found to be similar to that for prestrained material as given in Boller (1987).

11

CLOSING REMARKS Some of the traditional features and manner of applying the stress-based approach need to be revised. Where stress-life data are available for notched specimens of a given material, considerable care is needed in applying these for a notched component. In particular, the two geometries need to be similar as to the size effect related to small cracks growing into the stress gradient beyond the notch. For estimating mean stress effects, the Goodman diagram should be replaced by more accurate methods effects, such as the Smith, Watson, and Topper equation, or the Walker equation. Emphasis on a fatigue limit is inappropriate, as stresses below this level do cause fatigue damage when combined in a load spectrum with occasional severe stress cycles. Also, if very long lives are considered, a change in fracture mechanism may cause failures below this level even in constant amplitude loading. The strain-based approach makes life estimates based on the local stresses and strains that occur at stress raisers of interest. The materials properties needed can be obtained from testing unnotched axial specimens, consisting of a strain-life curve and a cyclic stress-strain curve. An elasto-plastic analysis of the component using the cyclic stress-strain curve is needed, which can be accomplished by finite elements analysis or by approximate methods such as Neuber's rule. The strain-life curve can be obtained from specimens that have been subjected to initial or periodic severe strain cycles, so as to automatically include damage to the material due to such events in actual service. For an irregular loading history, the analysis is repeatedly applied, with due regard for the hysteresis looping and memory aspects of the material's cyclic plasticity behavior. This allows the full history of the local stress-strain response to be determined as a basis for life estimates. Use of a strain-based approach allows the effect of local yielding on the level of local mean stress to be considered. Such local yielding can occur even in relatively long life situations, due to the initial loading required to reach a high mean load level, or due to occasional severe stress cycles. The method does not automatically include the effects of fabrication detail, such as surface finish. Spectrum loading effects caused by alteration of the local mean stress is included, but those due to material damage are not included if only simple constant amplitude data are available for the strain-life curve. A useful procedure is to develop a component-specific strain-life curve that has been adjusted based on actual component data to account for the effects of fabrication detail and material damage due to spectrum loading.

12

REFERENCES AWS (2004) AWS, Structural Welding Code: Steel, 19th ed., AWS D1.1/D1.1M:2004,

American Welding Society, Miami, FL, 2004. See also Structural Welding Code: Aluminum, AWS D1.2/D1.2M:2003.

Bathias (1999) C. Bathias, “There is No Infinite Fatigue Life in Metallic Materials,” Fatigue and Fracture of Engineering Materials and Structures, Vol. 22, No. 7, 1999, pp. 559-565.

Boller (1987) C. Boller and T. Seeger, Materials Data for Cyclic Loading, 5 Vols. Elsevier Science Pubs., Amsterdam, 1987. See also Supplement 1, 1990, by A. Baumel and T. Seeger.

Cronkhite (2000)

J. D. Cronkhite, et al., "Research on Practical Damage Tolerance Methods for Rotorcraft Structures," American Helicopter Society 56th Annual Forum, Virginia Beach, VA, May 2000.

Dickson (1999)

B. Dickson, et al., "Rotorcraft Fatigue and Damage Tolerance," Report prepared for the Technical Oversight Group on Aging Aircraft (TOGAA), 1999.

Dowling (1973) N. E. Dowling, “Fatigue Life and Inelastic Strain Response under Complex Histories for an Alloy Steel,” Journal of Testing and Evaluation, Vol. 1, No. 4, July 1973, pp. 271-287.

Dowling (1979) N. E. Dowling and Wilson, W. K., "Analysis of Notch Strain for Cyclic Loading," Fifth Int. Conf. on Structural Mechanics in Reactor Technology, Vol. L, West Berlin, Germany, Paper L13/4, 1979.

Dowling (1981) N. E. Dowling and W. K. Wilson, “Geometry and Size Requirements for Fatigue Life Similitude Among Notched Members,” Advances in Fracture Research, D. Francois et al., eds. Pergamon Press, Oxford, UK 1981, pp. 581-588.

Dowling (1982) N. E. Dowling, “Fatigue Life Prediction for Complex Load Versus Time Histories,” Section 7.4 of Pressure Vessels and Piping: Design Technology—1982—A Decade of Progress, Ed. By S. Y. Zamrik and D. D. Dietrich, Book No. G00213, ASME, 1982, pp. 487-498. Also pub. in Journal of Engineering Materials and Technology, ASME, Vol. 105, July 1983, pp. 206-214, with Erratum, Oct. 1983, p. 321.

Dowling (1987) N. E. Dowling, "A Review of Fatigue Life Prediction Methods," 1987 Transactions of the Society of Automotive Engineers, Paper No. 871966, Sept. 1988. See also Durability by Design, SAE Publication No. SP-730, SAE 1987 Passenger Car Meeting and Exposition, Oct. 1987, Dearborn, MI.

Dowling (1989) N. E. Dowling and A. K. Khosrovaneh, “Simplified Analysis of Helicopter Fatigue Loading Spectra,” Development of Fatigue Loading Spectra, ASTM STP 1006, J. M. Potter and R. T. Watanabe, eds., Am. Soc. For Testing and Materials, West Conshohocken, PA, 1989, pp. 150-171.

13

Dowling (1999) N. E. Dowling, Mechanical Behavior of Materials: Engineering Methods for

Deformation, Fracture and Fatigue, 2nd edition, Prentice Hall, Upper Saddle River, NJ, 1999. 3rd edition anticipated in 2006.

Dowling (2000) N. E. Dowling and S. Thangjitham, “An Overview and Discussion of Basic Methodology for Fatigue,” Fatigue and Fracture Mechanics: 31st Volume, ASTM STP 1389, G. R. Halford and J. P. Gallagher, eds. Am. Soc. For Testing and Materials, West Conshohocken, PA, 2000, pp. 3-36.

Dowling (2003) N. E Dowling, "Local Strain Approach to Fatigue," Chapter 4.03 in Volume 4, Cyclic Loading and Fatigue, R. O. Ritchie and Y. Murakami, volume editors; part of the 10-volume set, Comprehensive Structural Integrity: Fracture of Materials from Nano to Macro, I. Milne, R. O. Ritchie, and B. Karihaloo, editors-in-chief, Elsevier Ltd., Oxford, England, 2003, Vol. 4, pp. 77-94.

Dowling (2004) N. E. Dowling, “Mean Stress Effects in Stress-Life and Strain-Life Fatigue,” SAE Paper No. 2004-01-2227, Fatigue 2004: Second SAE Brasil International Conference on Fatigue, June 2004, São Paulo, Brasil. (See also: http://www.esm.vt.edu/~ndowling)

DuQuesnay (1993) D. L. DuQuesnay, M. A. Pompetzki, and T. H. Topper, “Fatigue Life Prediction for Variable Amplitude Strain Histories,” Paper No. 930400, Fatigue Research and Applications, Pub. No. SP-1009, Society of Automotive Engineers, Warrendale, PA, 1993, pp. 175-185.

Edwards (1984) P. R. Edwards and J. Darts, “Standardized Fatigue Loading Sequences for Helicopter Rotors (Helix and Felix), Parts 1 and 2,” Report Nos. TR84084 and TR84085, Royal Aircraft Establishment, Ministry of Defense, Farnborough, Hants, England (also ICAF Document No. 1442), 1984.

Grover (1951) H. J. Grover, S. M. Bishop, and L. R. Jackson, "Fatigue Strengths of Aircraft Materials: Axial Load Fatigue Tests on Notched Sheet Specimens of 24S-T3 and 75S-T6 Aluminum Alloys and of SAE 4130 Steel with Stress-Concentration Factors of 2.0 and 4.0," NACA TN 2389, National Advisory Committee for Aeronautics, Washington, DC, 1951.

Heuler (1986) P. Heuler and T. Seeger, “A Criterion for Omission of Variable Amplitude Loading Histories,” International Journal of Fatigue, Vol. 8, No. 4, 1986, pp. 225-230.

Illg (1956) W. Illg, “Fatigue Tests on Notched and Unnotched Sheet Specimens of 2024-T3 and 7075-T6 Aluminum Alloys and of SAE 4130 Steel with Special Consideration of the Life Range from 2 to 10,000 Cycles” NACA TN 3866, National Advisory Committee for Aeronautics, Washington, DC, 1956.

Juvinall (2000) R. C. Juvinall and K. M. Marshek, Fundamentals of Machine Component Design, 3rd ed., John Wiley, New York, NY, 2000.

14

MILHDBK (1998) MILHDBK, Military Handbook: Metallic Materials and Elements for

Aerospace Vehicle Structures, MIL-HDBK-5H, 2 Vols., U. S. Dept. of Defense, MIL-HDBK-5 Coordination Activity, Wright-Patterson AFB, OH, 1998. See also: Interactive Edition, 2003, Knovel Corp., Norwich, NY, www.knovel.com.

Morrow (1968) J. Morrow, “Fatigue Properties of Metals,” Section 3.2 of Fatigue Design Handbook, Pub. No. AE-4, Society of Automotive Engineers, Warrendale, PA, 1968. Section 3.2 is a summary of a paper presented at a meeting of Division 4 of the SAE Iron and Steel Technical Committee, Nov. 4, 1964.

Naumann (1959) E. C. Naumann, H. F. Hardrath, and D. E. Guthrie, “Axial Load Fatigue Tests of 2024-T3 and 7075-T6 Aluminum Alloy Sheet Specimens Under Constant and Variable Amplitude Loads,” NASA TN D-212, National Aeronautics and Space Administration, Washington, DC, 1959.

Shigley (2004) J. E. Shigley, C. R. Mischke, and R. G. Budynas, Mechanical Engineering Design, 7th ed., McGraw-Hill, New York, NY, 2004. See also J. E. Shigley and L. D. Mitchell, 1993 reprint of the 4th edition.

Shiozawa (2001) K. Shiozawa, L. Lu, and S. Ishikara, “S-N Curve Characteristics and Subsurface Crack Initation Behavior in Ultra-Long Life Fatigue of a High Carbon Chromium Bearing Steel,” Fatigue and Fracture of Engineering Materials and Structures, Vol. 24, No. 12, 2001, pp. 781-790.

Smith (1970) K. N. Smith, P. Watson, and T. H. Topper, "A Stress-Strain Function for the Fatigue of Metals," Journal of Materials, ASTM, Vol. 5, No. 4, Dec. 1970, pp. 767-778

Topper (1970) T. H. Topper and B. I. Sandor, “Effects of Mean Stress and Prestrain on Fatigue Damage Summation,” Effects of Environment and Complex Load History on Fatigue Life, ASTM STP 462, Am. Soc. for Testing and Materials, 1970, pp. 93-104.

Walker (1970) K. Walker, "The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum," Effects of Environment and Complex Load History on Fatigue Life, ASTM STP 462, Am. Soc. for Testing and Materials, West Conshohocken, PA, 1970, pp. 1-14.

Wetzel (1968) R. M. Wetzel, "Smooth Specimen Simulation of Fatigue Behavior of Notches," Journal of Materials, ASTM, Vol. 3, No. 3, Sept. 1968, pp. 646-657.

15

Figure 1 – Estimated S-N curves for an alloy steel using the methods of two design textbooks. Figure 2 – Notched test specimens: (a) center hole, (b) compact, (c) blunt double edge notch (DEN), (d) sharp DEN. For each, w = 25.4 mm, t = thickness. [From Dowling (1981).]

10

100

1000

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

Nf , Cycles

S a ,

MPa

Juvinal 2000Shigley 2004Data

AISI 4130 Steelσu = 807 MPakt = 4.00, R = −1

16

Figure 3 – Fatigue crack initiation life data for geometries (a), (b), and (c) of Fig. 2. [From Dowling (1981).] Figure 4 – Approximate variation of the stress intensity factor K, normalized to ktS, for the geometries of Fig. 2. [From Dowling (1981).]

17

Figure 5 – Fatigue life to failure data for the geometries of Fig. 2. [From Dowling (1981).] Figure 6 – Schematic of the design S-N curve for an as-manufactured rotorcraft component, derived from a failure curve that has been scaled to fit limited test data. [From Dickson (1999).]

18

Figure 7 – Equivalent completely reversed nominal stress from the modified Goodman equation for double-edge-notched axial specimens of 2024-T3 aluminum. [From Dowling (2000); data from Grover (1951).] Figure 8 – Equivalent completely reversed nominal stress from the Smith, Watson, and Topper equation for double-edge-notched axial specimens of 2024-T3 aluminum. [From Dowling (2000); data from Grover (1951).]

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

Goo

dman

Sar

, M

Pa

0691382070 Fit

Mean Stress, MPa

2024-T3 Alkt = 2.15

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

SW

T S

ar ,

MP

a

0691382070 Fit

Mean Stress, MPa

2024-T3 Alkt = 2.15

19

Figure 9 – Equivalent completely reversed nominal stress from the Walker equation for double-edge-notched axial specimens of 2024-T3 aluminum. The curve corresponds to γ = 0.733. [From Dowling (2000); data from Grover (1951).] Figure 10 – Stress-life curve for an alloy steel showing overstrain effects. [Data from Dowling (1973).]

0

50

100

150

200

250

300

350

400

1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

Wal

ker S

ar ,

MP

a069138207Fit all

Mean Stress, MPa

2024-T3 Alkt = 2.15

20

Figure 11 – Stress-life curve extending into the gigacycle range for a bearing steel with σu = 2320 MPa, containing 1% carbon and 1.45% chromium. The specimens were tested in lab air at 52.5 Hz using a cantilever type rotating bending apparatus. (From Shiozawa (2001).] Figure 12 – The strain-based approach to fatigue. [From Dowling (1999).]

21

Figure 13 – Test specimen, extensometer, and grips for axial controlled strain testing. [Photo by G. K. McCauley, Virginia Tech.] Figure 14 – Monotonic and cyclic stress-strain curves for an alloy steel. [From Dowling (1979).]

22

Figure 15 – Effect of initial prestrain (10 cycles at εa = 0.02) on the strain-life curve of an aluminum alloy. [From Dowling (1989); data from Topper (1970).] Figure 16 – Effects of initial and periodic overstrains on the strain-life curve for an alloy steel. [Data from Dowling (1973).]

23

Figure 17 – Bluntly notched compact specimen. [From Dowling (1979).] Figure 18 – Load versus notch strain measured by metal-foil strain gages for the member of Fig. 17 made of the material of Fig. 14. Also shown are results from elasto-plastic finite elements analysis and estimates by Neuber's rule. [From data for Dowling (1979).]

0

10

20

30

40

0.000 0.006 0.012 0.018 0.024 0.030 0.036

εa, Notch Strain Amplitude

Pa,

Load

Am

plitu

de, k

N

dataNeuberFEA

t = 5.08 mm

24

Figure 19 – Load versus cycles to crack initiation and failure for the member of Fig. 17 made of the material of Fig. 14. Also shown are life estimates based on strains from elasto-plastic finite elements analysis and from Neuber's rule, as in Fig. 18. [From data for Dowling (1982).] Figure 20 – Procedure for strain-based life prediction for a notched member under constant amplitude loading.

0

10

20

30

40

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Ni, Nf, Cycles

Pa,

Load

Am

plitu

de, k

N

0.5 mm crack Neuberfailure FEA

t = 5.08 mm

25

Figure 21 – Stress-strain responses estimated using Neuber's rule for various levels of zero-to-maximum loading on plates with central round holes. [From Dowling (1987).] Figure 22 – Life estimates for the situation of Fig. 21 using the Smith, Watson, and Topper mean stress equation, and also corresponding cycles to failure data. [Data from Wetzel (1968).]

26

Figure 23 – Lives calculated by the strain-based approach compared with test data for double-edge-notched specimens of 7075-T6 aluminum. [From Dowling (2000); based on data from Grover (1951), Illg (1956), and Naumann (1959).])

Figure 24 – Analysis of a notched member subjected to an irregular load vs. time history. Notched member (a), made of 2024-T4 aluminum, is subjected to load history (b). The estimated local stress-strain response at the notch is shown in (c). [From Dowling (1999).]

10

100

1000

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Nf , Cycles

Sa

, Nom

. Stre

ss A

mpl

itude

, MP

a

0 Calc0 Data138 Calc138 Data

Mean Stress, MPa

7075-T6 Alkt = 4.00

27

Figure 25 – Elastic, plastic, and total strain versus life curves for an alloy steel, with an elastic strain slope adjustment shown as for surface finish or prestrain. Figure 26 – Strain-based life estimates using for the standard helicopter load spectrum Felix applied to plates with round holes (kt = 2.5) of mill-annealed titanium 6Al-4V. The constant b = from strain-life data on similar material gives nonconservative life estimates, but the strain-life curve adjusted with a lower value of b gives better agreement. [From Dowling (2000); based on data from Edwards (1984) and original strain-life curve from Dowling (1989).]

0.0001

0.001

0.01

0.1

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06Nf , Cycles

ε a ,

Stra

in A

mpl

itude SAE 4142 Steel (450 HB)

σ'f / E

ε'f

polished, b

modified, bs

1 10 100 1000 10000 100000Flights to Failure

200

400

600

800

Max

imum

Nom

inal

Stre

ss, M

Pa

-0.10-0.105

-0.08-0.085-0.095

-0.09

b=-0.0763

Notched Titanium 6AI-4V

Yield Zone Based Retardation ModelsUse With Caution

James A. HarterAir Force Research LaboratoryAnalytical Mechanics Branch

ASIP Conference29 Nov 2005Memphis, TN

Yield Zone Based Retardation Models

Basic Yield Zone Approach

da/dN = f(rpi /rpOL )

Plastic Zone = (Kmax/Yld)2 / (PSX * π)

Where, PSX accounts for Stress State


Life Prediction Example





Note: The closure model parameter (Cf0 = 0.653) was adjusted to match the Willenborg prediction


Beta vs. Crack Length

0

2

4

6

8

10

12

14

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Crack Length

Bet

a

CtC


Stress Intensity vs. Crack Length

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Crack Length

Stre

ss In

tens

ity (U

nit S

tress

)

CtC

(After Transition to an Oblique Crack)


Life Prediction Example - Effect of Overload Shut-off Ratio

Yield Zone Based Retardation ModelsLife Prediction Example - Hsu Model

Note: Crack arrest occurs prior to transition to an oblique crack

The closure model parameter (Cf0 = 0.4) is a typical value for aluminum


Life Prediction Example (Before Transition to an Oblique Crack)

2.02840.1253

2.02847.1252

K(a)a-length


Life Prediction Example - Wheeler Model

Note: The Wheeler model makes adjustments to crack growth rate instead of ∆K

The closure model parameter (Cf0 = 0.4) is a typical value for aluminum

Yield Zone Based Retardation ModelsSummary

• Yield zone based retardation models can be VERY sensitive when modeling cases where stress intensities values (for a given applied stress)decrease with increasing crack length.

•The frequency and magnitude of spectrum overload(s) are also important factors to consider.

• This type of sensitivity is not usually seen in closure based models since changes in opening load do not occur immediately after an overload.

• Always question your life analyses

Do your answers make sense?

Have you compared results for other retardation models?

Are your retardation model parameters reasonable?

1

2005 USAF ASIP Conference Presentation

Improved Modeling of R-RatioBehavior for the Walker Equation

Royce Forman (NASA JSC, Houston, TX)V. Shivakumar (JS ESC Group, Houston, TX)

Joachim Beek (NASA JSC, Houston, TX)

The Peabody MemphisMemphis, TN

29 November – 1 December 2005

2

Improved Modeling of R-Ratio Behavior for the Walker Equation

The accomplishments described in this presentation were motivated by the following two previous tasks performed by the presenters:

! Implementation of the Walker Equation into NASGRO

! Development of the FAA Report & Software– Fatigue Crack Growth Data Base for Damage Tolerance Analysis

Background

3


! Many software codes currently used for damage tolerance analysis are based on fatigue crack growth rate (da/dN) models developed 25 to 40 years ago

! In that time period, availability of full range and accurate da/dN data showing reliable R-ratio behavior was limited.

! With present day automated test systems, data can now be generated to more fully evaluate different da/dN models, determine their deficiencies and needed improvements.

Background (Con’t)

4


Two early period da/dN models commonly used today:

! Walker equation (1970)"Fits maximum spread in data for variations in R "Requires separate fits for positive and negative R values"Provides linear segment fits to 2 or more R values

! Newman equation (1981)"Based on plasticity-induced crack closure behavior"Realistic gradient of da/dN curves for 3 or more R values"Fits a single R value data set"No separate fit required for negative R data"Limited ability to fit wide spreads in curves for ranges of R

5


Newman Closure and Walker Equations

dadN

CfR

K

KK

KK

nth

p

c

q=−−

−

−

11

1

1∆

∆∆

max

( )f

KK

R A A R A R A R RA A R R

op= =

+ + + ≥

+ − ≤ <

max

max , 0 1 22

33

0 1

02 0

( )

n

R1K

m1CdNda

= −−

∆

where

Newman

Closure Eqn:

Walker Eqn:

6


Discrepancy between Walker & Closure Eqn’s In Modeling R-Ratio BehaviorExample: Fits (from R = .1 to .7) to NASMAT data for 7050-T7451 Al

Newman�s Closure Equation Behavior Walker Equation BehaviorCurves converge with increasing R Curves diverge with increasing R

∆ K [ksi√in]

∆ K [ksi√in]

da/dN[in/cycle]

da/dN[in/cycle]

R = .9,.6,.3, 0 R = .9,.6,.3, 0

3 5 7 10 15 20

1. ´ 10 - 7

1. ´ 10 - 6

0.00001

0.0001

C=0.6e-9

n=3.61

m=0.54

3 5 7 10 15 20

1e-5

1e-4

C=2.6e-9

n=3.61

α=0.54

Smax/σo=0.3

1e-6

1e-6

1e-4

1e-5

1e-6

1e-7

3 5 7 10 15 20

7


“Revised Walker Equation” & example fit to closure equationExample Fit: NASMAT data for 7050-T7451 AL

Rev. Walker Equation

( )n

m1bR RC1

KCdNda

−=

−∆

∆ K [ksi √in]

da/dN[in/cycle]

__ Closure Eqn

__ Rev. Walker

3 5 7 15 20

1. × 10 −7

1. × 10 −6

0.00001

0.0001

C=0.65e-9n=3.61m=0.74b=0.31CR=0.8

10

1e-7

1e-6

1e-5

1e-4

__ Closure Eqn

__ Rev. Walker

R = .9,.6,.3, 0

1e-5CR

b

8


Development of the “Extended Walker Equation”

! To model threshold region behavior

! To model crack instability & Kmax behavior

9


Problems in modeling threshold region da/dN data generated using the load shedding test method

! Thresholds are effected from multiple closure actions:1. Plastic wakes behind the crack front2. Roughness of the crack surfaces3. Oxides or fretting debris behind the crack tip

! Threshold values vary with type and size of test specimen

10


Comparison of Different Specimen Test Results for Threshold TestingR = 0.1 ∆Kth Results Using Load Shedding Method for Ti-6Al-4V MA Plt

M(T) Specimen C(T) Specimen ESE(T) Specimen∆Kth= 3.00 ∆Kth= 4.3 ∆Kth= 3.8

11


[ ]

( )

( )( ) th

thCR

CR

th ARf

Raa

aKK −+

−

−

−

+

∆=∆ 10

1

01 1/

112

1

NASGRO Threshold Equation

Cth)Cth

12


NASGRO Eqn fit : Ti-6-4 MA ; 0.25� Plt ; L-T ; C(T) Specimen Data ; Cth=1.5

13


NASGRO Eqn fit:Ti-6-4 MA ; 0.25� Plt ; L-T; R=0.1; C(T) & M(T) Data ; Cth=1.5

14


NASGRO Eqn Fit: Ti-6-4 MA ; 0.25� Plt ; R=0.1; C(T) & M(T) Data ; Cth=0

15


Reasons for making Cth = 0 the default value

Necessary for conservative analysis:

1. Thresholds obtained by load shedding are un-conservativefor some specimen types – mainly C(T) & ESE(T) types.

2. Higher thresholds are obtained in laboratory air & inhigh humidity air than in dry air.

16

Evaluation Of R = 0.1 Fracture Surface(S. Forth�s tests at NASA Langley)

17

Effect of Laboratory Environment(S. Forth�s tests at NASA Langley)

18

Observations by S. Forth

! Constant R = 0.1 threshold data generated with C(T) specimens are very sensitive to:

# Specimen Thickness# Laboratory Environment (Relative Humidity)

19


Development of Extended Walker Threshold Equation

For similar capabilities as in NASGRO threshold equation:

! NASGRO equation fits both +R & -R thresholds

! Same fitting constants used for both +R & -R

! A fanning correction factor (Cth) is needed

20


Preliminary relationship between m- and m+ constants

• Data - Mod. Walker fit (m+& m-) to crack closure thresholds

0.6 0.65 0.7 0.75 0.8

0.16

0.18

0.2

0.22

0.24

0.26

βα += +− mm

+m

−m 0.20

21


( )( )

( )( )

( )( )

( )( )( ) 0RforC1

R1K

0RforC1RC1KK

pth

pth

pth

C1m1R

m11

m1

C1R

RC1bR

1th

<−

−=

≥

−

−=

+−

−

−

+

+

+

−

+

∆

∆∆

Extended Walker Threshold Equation (Preliminary)

22

Recent Developments in Modeling R-Ratio Effects with the Walker Equation

Extended Walker Threshold Equation (Preliminary)

____ Closure Eqn____ Extended Walker

0.2 0.4 0.6 0.8 1

1

2

3

4

5

6

∆ Kth[ksi √in]

Stress Ratio, R

23

Recent Developments in Modeling R-RatioEffects with the Walker Equation

( ) q

c

pthn

mbR

KK

KK

RC

KCdNda

−

∆∆

−

−

∆= −

max1

1

1

1

Extended Walker Equation

2 5 10 20 50 1001. ´ 10 - 9

1. ´ 10 - 8

1. ´ 10 - 7

1. ´ 10 - 6

0.00001

0.0001

0.001R = .9,.6,.3,0,-.5,-1,-2

da/dN

[in/cycle]

∆ K [ksi √in]

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

2 5 10 20 50 100

24


Summary Comments

! Enhanced capabilities of Extended Walker Equation:1. Improved fits for multiple R-value data2. Improved fit for threshold & high da/dN regions3. Improved fits for some elevated temperature data

! Current Deficiencies of Extended Walker Equation1. Needs further development for negative R data2. Not capable of accurately fitting irregular trends in data

Prediction of Fatigue Crack Growth for 7075-T7351 Aluminium Alloy under Various Flight Load Spectra

Y. Yamada, J. C. Newman, Jr., T. E. LacyDepartment of Aerospace Engineering

Mississippi State UniversityMississippi State, MS, USA

ASIP 2005USAF Aircraft Structural Integrity Program

29 November – 1 December 2005Memphis, Tennessee

ASIP 2005 - # 2

OUTLINE OF PRESENTATION

• Briefly review the experimental test program

• Crack-closure analysis and crack-growth equation

• ∆K - dc/dN on 7075-T7351 aluminum alloy

• Generation of ∆Keff-dc/dN relation for 7075-T7351

• Compare measured and calculated crack-growth lives under four simulated aircraft load spectra

• Compare measured and predicted crack-growth lives under three simulated aircraft load spectra

• Concluding remarks

ASIP 2005 - # 3

FATIGUE-CRACK-GROWTH TEST PROGRAM

• Material: 7075-T7351 aluminum alloy (B = 0.25 in.)• Room Temperature and Laboratory Air• Test specimens: M(T)• Spectrum loading : (W = 4.0 in.) Major OEM

- Center wing spectrum 1 and 2 - Outer wing spectrum 1 and 2

• Spectrum loading for : (W = 3.16 or 6.0 in.)- Wing gust and maneuver spectrum*- Lateral gust loading spectrum*- Pressurized fuselage gust and maneuver spectrum*

* Reference: Brot and Matias, “An Evaluation of Several RetardationModels for Crack Growth Prediction under Spectrum Loading”, U.S.A.F. Aircraft Structural Integrity Program Conference, 2002.

ASIP 2005 - # 4

NASA FASTRAN – CRACK-GROWTH & CLOSURE MODEL

ASIP 2005 - # 5

FLAT-TO-SLANT CRACK-GROWTH CHARACTERISTICS

(∆Keff)T = 0.5 σo B1/2

ASIP 2005 - # 6

CRACK-GROWTH RATE EQUATION

])(1[)( 2max Ie

neffi KKKCdNdc i −∆=

),,( maxmax α

σOo

SRfSS =

KRSSK o

eff ∆−

−=∆

11 max

ASIP 2005 - # 7

LEFM ANALYSES OF 7075-T7351 ALUMINUM ALLOY

∆K, ksi-in1/21 10 100

dc/dN in/cycle

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

T73 R = 0.7 C(T)T73 R = 0.5, 0.3, 0.02, -0.3 M(T)T7351 R = 0.7 C(T) B = 0.5 inT7351 R = -1.0 M(T)T7351 R = 0.02 M(T)T7351 R = 0.5 M(T)

7075-T7351 Aluminum alloyB = 0.25 in

ASIP 2005 - # 8

EFFECTIVE SIF FOR 7075-T7351 ALUMINUM ALLOY

∆Keff, ksi-in1/21 10 100

dc/dN in/cycle

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

T73 R = 0.7 C(T)T73 R = 0.5, 0.3, 0.02, -0.3 M(T)T7351 R = 0.7 C(T) B = 0.5 inT7351 R = -1.0 M(T)T7351 R = 0.02 M(T)T7351 R = 0.5 M(T)∆Keff Baseline

(∆Keff)T

α1 = 2.4

α2 = 1.2

Single or doubleshear mode ?

ASIP 2005 - # 9

CENTER WING - 1 SPECTRUM

Time

S / Speak

-0.5

0.0

0.5

1.0

Smin/Speak = -0.93

ASIP 2005 - # 10

MEASURED AND CALCULATED CRACK GROWTHFOR CENTER WING - 1 SPECTRUM

Spectrum Center Wing -17075-T7351B = 0.25 inW = 4 in

N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.4

0.8

1.2

1.6

Test 1Test 2Constraint lossConstant constraint

ASIP 2005 - # 11

CENTER WING - 2 SPECTRUM

Time

S / Speak

-1.0

-0.5

0.0

0.5

1.0

Smin/Speak = -1.08

ASIP 2005 - # 12

MEASURED AND CALCULATED CRACK GROWTHFOR CENTER WING - 2 SPECTRUM

Spectrum Center Wing - 27075-T7351B = 0.25 inW = 4 in

N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.5

1.0

1.5

2.0


ASIP 2005 - # 13

OUTER WING - 1 SPECTRUM

Time

S / Speak

-1.0

-0.5

0.0

0.5

1.0

Smin / Speak = -1.25

ASIP 2005 - # 14

MEASURED AND CALCULATED CRACK GROWTH FOR OUTER WING - 1 SPECTRUM

Spectrum Outer Wing - 17075-T7351B = 0.25 inW = 4 in

N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.5

1.0

1.5

2.0


ASIP 2005 - # 15

OUTER WING - 2 SPECTRUM

Time

S / Speak

-0.5

0.0

0.5

1.0

Smin/Speak = -0.6

ASIP 2005 - # 16

MEASURED AND CALCULATED CRACK GROWTH FOR OUTER WING - 2 SPECTRUM


N, cycles

0.0 2.0e+6 4.0e+6 6.0e+6 8.0e+6 1.0e+7 1.2e+7

Cracklength,c, in.

0.0

0.5

1.0

1.5

2.0


ASIP 2005 - # 17

RATIO OF CALCULATED TO MEASURED SPECTRA LIVES

Calculated crack length / Failure crack length (with constraint-loss)

0.0 0.2 0.4 0.6 0.8 1.0

Calculated life (with constraint-loss)

0.0

0.5

1.0

1.5

+24.0%+17.4%

-13.6%-6.77%

Center Wing -2 Outer Wing -2

Center Wing -1 Outer Wing -1 Test life

Test 1Test 2

ASIP 2005 - # 18

∆Keff, ksi-in1/21 10 100

dc/dN in/cycle

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

α1 = 2.4

α2 = 1.2

∆Keff baseline

(∆Keff)T

CALIBRATED EFFECTIVE SIF FOR 7075-T7351 ALLOY

Calibrated parametersfor 7075-T7351

Tensile constraint factors:α1 = 2.4α2 = 1.2

Constraint-loss regime:Initiated at 3.0e-5 in/cycleEnded at 1.0e-2 in/cycle

Compressive constraint factors:β = 1.0

ASIP 2005 - # 19

Time

S / Speak

-1.0

-0.5

0.0

0.5

1.0

LATERAL GUST AND MANEUVER SPECTRUM

ASIP 2005 - # 20

MEASURED AND PREDICTED CRACK GROWTH FOR LATERAL GUST SPECTRUM

N, cycles

0 1e+5 2e+5 3e+5 4e+5 5e+5 6e+5

Crack length,

c, in

0.0

0.2

0.4

0.6

0.8

1.0

TestConstraint lossConstant constraint

Brot and Matias7075-T7351M(T) B = 0.25 inLateral gust spectrum

ASIP 2005 - # 21

Time

S / Speak

0.0

0.2

0.4

0.6

0.8

1.0

PRESSURIZED FUSELAGE GUST AND MANEUVER SPECTRUM

ASIP 2005 - # 22

MEASURED AND PREDICTED CRACK GROWTH FOR FUSELAGE GUST AND MANEUVER SPECTRUM

N, cycles

0 1e+5 2e+5 3e+5 4e+5

Crack length,

c, in

0.0

0.2

0.4

0.6

0.8

1.0

1.2


Brot and Matias7075-T7351M(T) B = 0.25 inFuselage gust and maneuver spectrum

ASIP 2005 - # 23

WING GUST AND MANEUVER SPECTRUM

Time

S / Speak

-0.25

0.00

0.25

0.50

0.75

1.00

ASIP 2005 - # 24

MEASURED AND PREDICTED CRACK GROWTH FOR WING GUST LOADING SPECTRUM

N, cycles

0.0 5.0e+4 1.0e+5 1.5e+5 2.0e+5 2.5e+5 3.0e+5

Crack length, c, in

0.0

0.2

0.4

0.6

0.8

1.0

1.2


Brot and Matias7075-T7351M(T) B = 0.25 inWing gust and maneuver spectrum

ASIP 2005 - # 25

RATIO OF CALCULATED OR PREDICTED LIVESTO MEASURED LIVES

Test life, NT, cycles

1e+5 1e+6 1e+7

Calculatedlife,

NC, cycles

1e+5

1e+6

1e+7

Center Wing - 1Outer Wing - 1Center Wing - 2Outer Wing - 2Lateral GustFuselageWing Gust

7075-T7351 (LT)B = 0.25 inM(T)W = 3 or 4 in +30%

-30%

ASIP 2005 - # 26

Concluding Remarks

1. Difficulty in establishing ∆Keff -dc/dN relation from constant-amplitude data in the near threshold and fracture region due to lack of data.

2. Calculated fatigue-crack-growth lives on M(T) specimens under four simulated aircraft load spectra agreed within 30% of the test lives.

3. Predicted fatigue-crack-growth lives on M(T) specimens under the three simulated aircraft load spectra agreed within 30% of the test lives, except one prediction was -45% short of test data.

4. Compressive constraint factor, β, and interactions between tensile and compressive loads need to be studied further.

Fatigue-crack-growth calculations and predictions for FASTRAN on 7075-T7351 (B = 0.25 in.) aluminum alloy:

ASIP 2005 - # 27

THE END

Any questions?

Prediction of Fatigue Crack Growth for 7075-T7351 Aluminium Alloy Under Various Flight Load Spectra

Y. Yamada, J. Newman, Jr., T. Lacy Mississippi State University, USA

Abstract In this study, a plasticity-induced crack-closure model was used to calculate fatigue-crack-growth lives of middle-crack tension specimens made of a 7075-T7351 aluminium alloy subjected to seven simulated flight load histories. The fatigue loading histories were consistent with wing and fuselage spectra common to transport aircraft. The fatigue testing was conducted in laboratory air under load-control conditions; crack length measurements were obtained using a combination of crack gauges and enhanced visual measurements. The experimentally observed crack-growth behaviors were compared to crack-growth calculations obtained using the FASTRAN life-prediction code, employing material property data extracted from the available materials database, such as the NASGRO database and the literature. In order to account for a loss of crack-front constraint during crack growth, variable constraint factors were selected to correlate large-crack growth rate data as a function of the effective-stress-intensity factor range under constant-amplitude loading. For a given spectra, numerical life predictions that neglect the transition from plane-strain to plane-stress behaviour typically fail to capture the total number of cycles to failure. Such a disparity in the measured and predicted response may not necessarily manifest itself under constant-amplitude loading. In contrast, crack-growth calculations that accounted for the transition from flat-to-slant crack growth, through appropriate specification of constraint factors, generally resulted in ±30% difference from the test lives to failure under the various spectrum loadings. 1. Introduction In order to improve fatigue-crack-growth prediction for metallic materials under aircraft spectrum loading, a number of load-interaction models have been developed during the past 30 years. These models have been generally based on the plastic deformations that develop in the vicinity of crack tips. One of the significant phenomena that influences crack-growth behavior is crack closure, which is caused by residual plastic deformations remaining in the wake of an advancing crack [1, 2]. The discovery of the crack-closure concept led to a better understanding of the mechanisms of crack-growth retardation and acceleration under variable-amplitude loading; and also to identify several other closure effects such as roughness- [3] and oxide-induced closure [4], which appear to be more relevant in the near-threshold regime. The crack-closure concept has been used to correlate crack-growth rates under constant-amplitude loading over a wide range in rates from threshold to fracture for a large range in stress ratios and stress levels; and to predict or calculate crack growth under variable-amplitude loading. A few of these models were based on a concept like the Dugdale model [5] or strip-yield model, but modified to leave plastically deformed material in the wake of the advancing crack. The FASTRAN life-prediction model and code [6,7] was developed to account for the influence of

2

three-dimensional constraint on the crack-closure behavior. A similar model has also been incorporated into the NASGRO [8] life-prediction code. The AFGROW [9] life-prediction code, which is based on linear-elastic fracture mechanics concepts, also has an empirical crack-closure model. The purpose of this paper is to study the influence of constraint on fatigue-crack growth under seven simulated aircraft load spectra applied to 7075-T7351 aluminum alloy plate using the crack-closure model, FASTRAN. A variable constraint factor concept, which approximately accounts for the state-of-stress (i.e., plane stress and plane strain) effects on plastic-zone development and crack closure, was used in the model. The model was used to correlate fatigue-crack-growth-rate data under constant-amplitude loading over a wide range in rates and in stress ratios from the NASGRO database and the literature. Finally, comparisons were made between measured and calculated fatigue-crack-growth results for the 7075-T7351 aluminum alloy plate subjected to the seven aircraft load spectra. 2. Material and specimen configurations The constant-amplitude fatigue-crack-growth and fracture results on the 7075-T7351 aluminum alloy plate (thickness, B = 0.25 in.) were primarily obtained from the NASGRO materials database [8], but some additional data were obtained from the literature. Because of the lack of materials data, results from a similar material 7075-T73 were also used. The yield stress and the ultimate tensile strength of the 7075-T7351 alloy plate were 65 and 75 ksi, respectively; and the flow stress, σo = 70 ksi. All of the spectrum test data were obtained from the literature. Middle-crack tension M(T) specimens (W = 4 in.) tested under four of the simulated aircraft load (center-wing and outer-wing) spectra were obtained from a major OEM. M(T) specimens (W = 3 or 6 in.) tested on the other three spectra were obtained from Brot and Matias [10]. A crack-starter notch was machined in the center of each specimen and fatigue pre-cracked under constant-amplitude loading until the desired initial crack length was reached. 3. Plasticity-induced crack-closure model The plasticity-induced crack-closure model was developed for a central through crack in a finite-width plate subjected to remote uniform applied stress. The crack-closure model, during the past 25 years, has developed into the FASTRAN life-prediction code. The model employed a Dugdale-type plastic zone so that plastically deformed material would be left in the wake of the advancing crack and the crack-opening stress could be calculated from the contact stresses as a function of crack growth and applied stress history [6]. As shown in Figure 1, the FASTRAN crack-closure model is composed of an otherwise elastic body with elastic-perfectly plastic material elements in the plastic zone ahead of the crack tip and residual plastic deformation elements along the crack surface (i.e., plastic-wake behind the crack tip). Under cyclic loading, the plastic elements in the region of length, ρ, carries the stress, ασo, under the maximum applied stress (Smax). The constraint factor, α, accounts for the influence of stress state on tensile yielding at the crack front. The constraint factor α = 1 for plane-stress and α = 3 for plane-strain conditions. Elements behind the crack tip carry only compressive loads,

3

and then only if they are in contact at the minimum stress in the cyclic loading. Using the contact stresses (Fig. 1(b)), the crack-opening stress is calculated. The elements in contact yield in compression when the contact stress reaches –βσo, where β is the plastic constraint factor under compressive loading. Those elements that are not in contact do not affect the calculation of crack-surface displacements. Although it is known that the strip-yield model does not model the correct yield-zone shape for plane-strain conditions, the model with a high constraint factor is able to produce crack-surface displacements and crack-opening stresses quite similar to those calculated from three-dimensional, elastic-plastic, finite-element analyses of crack growth and closure for finite-thickness plates [11].

Figure 1. Crack-surface displacements and stress distributions in the FASTRAN model.

Linear-elastic fatigue-crack growth analyses have been found to be sufficiently adequate to assess or predict fatigue crack growth in metallic materials over a wide range in rates. In order to predict fatigue-crack growth under flight spectrum loading, ∆Keff as a function of the crack-growth rate must be obtained for the material of interest. The effective stress-intensity factor range developed by Elber was based on linear-elastic analyses and is given by

KRSSK o

eff ∆−

−=∆

11 max (1)

where Smax is the maximum applied stress, So is the crack-opening stress, and R is the load ratio. So/Smax is a function of the stress ratio (R), the applied stress level (Smax/σo), and the plastic constraint factor (α). Typically, fatigue-crack-growth-rate data, especially for aluminum alloys, shows some unique transitions due to monotonic and cyclic-plastic-zone sizes, grain sizes, and environments [12]. In order to characterize the crack-growth-rate curve more precisely, a material baseline curve can be generated by linearly connecting several points on a log-log scale. It is desirable to choose a material

4

baseline based on the effective stress-intensity factor range generated from “closure free” ∆K data, such as high stress ratios [13,14]. The selected points to characterize a crack-growth-rate curve were incorporated into FASTRAN by

])(1[)( maxq

Ien

effi KKKCdNdc i −∆= (2)

where Ci and ni are the coefficient and power for each linear segment, Kmax is the maximum stress-intensity factor, KIe is the elastic fracture toughness, which is function of crack length, specimen width, and specimen type, and q = 2. Under fatigue-crack growth, the plastic-zone size at the crack front increases as a crack grows in a metallic material under cyclic loading. Plane-strain condition should prevail at low applied stress-intensity factor levels, but as the plastic-zone size increases and becomes large comparing to the thickness of the specimen, a loss of constraint is expected. This loss of constraint has also been associated with the transition of the fracture mode on the crack surface from flat to slant crack growth associated with micro-structural material behavior [15] (i.e., slip directions of FCC material), as shown in Figure 2. Schijve [16] has shown that the transition occurs at nearly the same fatigue-crack-growth rate over a wide range in stress ratios for an aluminum alloy. Since fatigue-crack-growth rates are a function of ∆Keff, the transition observed by Schijve was expressed in terms of ∆Keff instead of the crack-growth rate. Newman [17] developed a simple expression to predict the value of ∆Keff for the transition from flat-to-slant crack growth:

BK oTeff σ5.0)( =∆ (3)

where σo is the flow stress and B is the sheet thickness. However, the range of the constraint-loss regime in terms of ∆Keff or crack-growth rate has not yet been established. Currently, the range of the constraint-loss regime is determined by trial-and-error methods, so that the test data and the fatigue-crack-growth calculations are in good agreement. In order to simulate the actual crack-growth phenomena in a simple 2-D numerical model, the 3-D flat-to-slant crack growth behavior is modeled with a constraint-loss behavior for a straight through crack. Below a given rate, the plane-strain constraint factor is selected and held constant. At a given higher rate, the plane-stress constraint factor is selected and held constant. The constraint factors (i.e., plane-strain and plane-stress conditions) remain constant for lower and higher crack-growth rates than those in the constraint-loss regime, respectively. In the constraint-loss regime, the constraint factors vary linearly on a log-log scale. The constraint factor affects the plastic-zone size and crack-surface displacements under cyclic loading. In FASTRAN, the two-parameter fracture criterion (TPFC) proposed by Newman [18] is used to predict failure of the cracked component. The TPFC is given as

)1( unIeF SSmKK −= for Sn < σys (4)

where KF and m are the two fracture parameters, KIe is the elastic stress-intensity factor at failure, Sn is the net-section stress, and Su is the plastic-hinge stress based on the ultimate tensile strength. The value of m is both a material and configuration parameter. It is a function of material, thickness, and specimen type (tension, bending, etc). For brittle

5

materials, m = 0 and the fracture toughness KF is equal to the elastic stress-intensity at failure (i.e., KIc for a brittle material). However, for very ductile materials, m = 1 and the fracture toughness KF is the elastic-plastic fracture toughness; and KF is the limiting value for a very large cracked panel at a very low failure stress. Once KF and m are known for a material, thickness, and specimen configuration, then the KIe values can be calculated as

)}(1{ nuFFIe FcSKmKK π−= (5)

for a given crack length and specimen width. Note that Fn is the usual boundary-correction factor (F) on stress-intensity factor with a net-to-gross section conversion [18], because the net-section stress is used in the equation for TPFC.

Single or DoubleShear mode

Tensile mode

Crack front Fatigue surface

Transitionregion

c

cLw

B

Figure 2. Fatigue crack surface with transition from tensile to shear mode.

4. Constant-amplitude loading In order to obtain fatigue-crack-growth rate as a function of ∆Keff from threshold to fracture, ∆K against rate data were collected from the NASGRO database [8] and the literature [13] for 7075-T7351 alloy (B = 0.25 in.). The fatigue-crack-growth-rate data for a wide range in stress ratios is shown in Figure 3. The ∆K-rate data for M(T) specimens extracted from NASGRO are shown as the open symbols for a wide range in stress ratio (R). Since the ∆K-rate data was insufficient, especially in the near threshold regime, ∆K-rate data from compact C(T) specimens made of a thicker 7075-T7351 alloy (B = 0.5 in.) and 7075-T73 alloy in same grain (LT) orientation and same thickness were used. These data are shown by the solid symbols. While the fatigue-crack-growth-rate data from the -T73 alloy is slightly different from the -T7351 alloy, the lack of available data for the -T7351 alloy, particularly at relatively low crack-growth rates, warrants the inclusion of data for a similar material. The NASGRO data were obtained by using the ASTM standard (E647) constant R load-reduction test procedure under laboratory air conditions. The load-reduction test procedure has been shown to generate artificially high threshold values in ∆K and lower rates due to load-history effects [13,14,19]. Also,

6

C(T) specimens tend to generate higher thresholds than M(T) specimens. In addition, it is suspected that smaller wide C(T) specimens also generate lower thresholds on the aluminum alloys. It is desirable to obtain material crack-growth-rate data from the desired thickness and grain orientation without any load-history effects. The data (R = 0.7) shown by the solid diamond symbols were obtained using a compression pre-cracking constant-amplitude method, which does not have any load-history effects after the crack has grown away from the compression yield region.

∆K, ksi-in1/21 10 100

dc/dN in/cycle

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

T73 R = 0.7 C(T)T73 R = 0.5, 0.3, 0.02, -0.3 M(T)T7351 R = 0.7 C(T) B = 0.5 inT7351 R = -1.0 M(T)T7351 R = 0.02 M(T)T7351 R = 0.5 M(T)


Figure 3. Stress-intensity factor against crack-growth rate for 7075-T7351 alloy.

The data shown in Figure 3 was analyzed with the crack-closure model using equation (1). These results are shown in Figure 4 as the effective-stress-intensity factor, ∆Keff, against crack-growth rate. The ∆Keff is a function of the stress ratio (R), the applied stress level (Smax/σo), and the plastic constraint factor (α). Because the ∆K-rate data in NASGRO did not have the appropriate applied stress levels used in the tests, estimates for these stress levels had to be made. As previously discussed, the constraint-loss regime due to flat-to-slant (single or double shear) crack-growth behavior needs to be modeled to characterize the material response as a function of crack growth. Based on equation (3), the transition value of ∆Keff was calculated and shown as the vertical dashed line in Figure 4. Arguably it is estimated that the initiation of the slant-crack-growth regime is independent of plate thickness, and occurs as the plastic zone at the free surface begins to allow shear deformations. In this analysis, the constraint-loss regime for the 7075-T7351 alloy was chosen by trial-and-error to be between 3.0e-5 to 1.0e-2 in/cycle. The constraint-loss range would, of course, be a function of the material thickness. The corresponding tensile constraint factor α1 for the flat crack growth and α2 for the slant crack growth were selected to be 2.4 and 1.2, respectively, based on three-dimensional elastic-plastic finite element analyses [15].

7

∆Keff, ksi-in1/21 10 100

dc/dN in/cycle

1e-9

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

T7351 R = -1.0 M(T)T7351 R = 0.02 M(T)T7351 R = 0.5 M(T)T7351 R = 0.7 C(T) B = 0.5 inT73 R = 0.7 C(T)T73 R = 0.5, 0.3, 0.02, -0.3 M(T)∆Keff Baseline

(∆Keff)T

α1 = 2.4

α2 = 1.2


Figure 4. Effective stress-intensity factor against crack growth rate for 7075-T7351.

The ∆Keff concept correlated fatigue-crack-growth-rate data in a relatively tight band over about seven orders of magnitude in crack-growth rates. The compressive constraint factor, β, was selected as unity for this study. The ∆Keff-rate baseline relation, shown by the solid lines with large open circles (Fig. 4, Table 1), was chosen to fit these data and used as the material data for the table-lookup option in FASTRAN. In Figure 4, the crack-growth rate data for the -T7351 alloy shows slightly higher rates than the -T73 alloy in the threshold regime but the materials are reasonably the same at higher rates. Thus, the use of the -T73 alloy data for the fracture region is reasonably adequate, but its use in the threshold regime may be doubtful. Although the crack growth rate data of the double thick -T7351 alloy were used for threshold region, it is desirable to obtain the data of same thickness and temper because of the difference in plastic zone size, load conditions, and residual stress levels. In Figure 4, the ∆Keff-rate relation for the -T73 alloy, near the constraint-loss regime, were significantly lower than the chosen material baseline (solid curve). It would be of great interest to determine why these data fall at much lower rates than the –T7351 data. 5. Aircraft spectrum loading M(T) specimens were subjected to seven different simulated aircraft load spectra (Center Wing - 1, Center Wing - 2, Outer Wing - 1, Outer Wing - 2, Lateral Gust, Wing Gust and Maneuver, and Pressurized Fuselage Gust and Maneuver). Specimens were notched and fatigue pre-cracked under constant-amplitude loading until the crack length reached a specified length from the initial notch. After pre-cracking, the specimens were subjected to the spectrum loading until failure. Crack-length-against-cycles data were

8

recorded and measured from both the front and back of the specimens for each of the two crack tips. In general, two duplicate tests were conducted for each spectrum.

Table 1: Effective stress-intensity factor relation and properties for 7075-T7351 (LT).

∆Keff ksi-in1/2

dc/dN in/cycle

0.82 4.0E-11 0.91 1.0E-09 1.32 4.0E-08 3.00 3.0E-07 4.55 2.0E-06 7.73 7.1E-06 15.9 7.1E-05 37.3 0.01575

α1 = 2.4 3.0E-05 α2 = 1.2 1.0E-02 KF = 60 m = 0

To investigate the importance of accounting for constraint loss from flat-to-slant crack-growth behavior, FASTRAN was used to calculate crack length against cycles for the seven spectra using two different options. The first option was a crack-closure model analysis with a constant tensile constraint factor, α1 = 2.4, to simulate plane-strain behavior and the condition for no flat-to-slant crack-growth behavior. The second option was a crack-closure model analysis with the constraint-loss regime, as described in Figure 4. Currently, the constraint-loss regime and constraint factors are determined by trial-and-error methods, so that the fatigue-crack-growth calculations under the various spectra are in good agreement. It is considered that the ∆Keff-rate baseline relation and the constraint-loss regime is a material dependent behavior, thus the baseline relation, the constraint factors (α1 = 2.4, α2 = 1.2), and the corresponding crack-growth-rates (3.0e-5 to 1.0e-2 in/cycle) were held constant for all spectrum load applications. Finally, the material crack-growth characteristics (∆Keff-rate baseline, plastic constraint factors, and crack-growth-rates for the constraint-loss regime) determined from one set of spectrum data were used on three other spectra investigated by Brot and Matias [10] on M(T) specimens made of the 7075-T7351 alloy (B = 0.25 in.). Some typical flights of the Center Wing - 1 spectrum are shown in Figure 5. The applied stress history has been normalized by the peak stress in the spectrum. The minimum stress in some flights will reach a compressive value of about 0.93 times the peak stress. Most of the cycles are in tensile load with a small bandwidth of 20% of the peak stress and spike reverse loadings are occasionally applied. The results from the two tests under the Center Wing - 1 spectrum are shown in Figure 6 and the two tests showed very good agreement. Calculated lives from FASTRAN are shown by the solid and dashed curves in Figure 6 for the constraint-loss regime and constant constraint factor, respectively. The calculated life with the constant tensile constraint shows a rapid fracture at 1.2 million cycles, while the calculated life with the constraint-loss regime show slower crack propagations after 1.2 million cycles.

9

As a result, constraint-loss regime option could nicely capture the trend of crack propagation under spectrum Center Wing - 1.

Time

S / Speak

-0.5

0.0

0.5

1.0

Smin/Speak = -0.93

Figure 5. Part of simulated flight spectrum for Center Wing – 1.

Spectrum Center Wing -17075-T7351B = 0.25 inW = 4 in

N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.4

0.8

1.2

1.6


Figure 6. Measured and calculated fatigue-crack growth for Center Wing -1 spectrum.

Some typical flights of the Center Wing - 2, Outer Wing - 1 and Outer Wing - 2 spectra are shown in Figures 7 to 9, respectively. The applied stress history, again, has been normalized by the peak stress in the spectrum. The minimum stress in some flights of each spectrum will reach a compressive value of 1.08, 1.25 or 0.6 times the peak stress of the Center Wing - 2, Outer Wing - 1 and Outer Wing - 2 spectra, respectively. The Center Wing - 2 and Outer Wing - 1 spectra show a very similar loading pattern. Many small load cycles are in tensile loading and at high R conditions. Compressive spike loadings are often applied. However, the Outer Wing - 2 spectrum is similar to the Center Wing - 1 spectrum except that the compressive loading is less severe in the later spectrum. Two test results under the Center Wing - 2 and Outer Wing - 1 spectrum are shown in Figures 10 and 11, respectively. Again, the two tests for each spectrum show very good agreement. The calculated behavior with the constant constraint factor option

10

(dashed curve) and the calculations with the constraint-loss option (solid curve) show very similar results. The calculated lives for both options match the test results very well. Unlike the calculations for the Center Wing - 1 spectrum, the calculated life with the constraint-loss option shows a slightly shorter life than the constant constraint factor.

Time

S / Speak

-1.0

-0.5

0.0

0.5

1.0

Smin/Speak = -1.08

Figure 7. Part of simulated flight spectrum for Center Wing – 2.

Time

S / Speak

-1.0

-0.5

0.0

0.5

1.0

Smin / Speak = -1.25

Figure 8. Part of simulated flight spectrum for Outer Wing – 1.

Time

S / Speak

-0.5

0.0

0.5

1.0

Smin/Speak = -0.6

Figure 9. Part of simulated flight spectrum for Outer Wing – 2.

11

Spectrum Center Wing - 27075-T7351B = 0.25 inW = 4 in

N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.5

1.0

1.5

2.0


Figure 10. Measured and calculated fatigue-crack growth for Center Wing - 2 spectrum.


N, cycles

0.0 5.0e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6

Cracklength,

c, in

0.0

0.5

1.0

1.5

2.0


Figure 11. Measured and calculated fatigue-crack growth for Outer Wing - 1 spectrum.

Two test results under the Outer Wing - 2 spectrum are shown in Figure 12 and both tests show good agreement with each other. Calculated lives from FASTRAN are shown as the solid and dashed curves for the constraint-loss option and constant constraint, respectively. The calculated results with the constant constraint showed a rapid increase in rates at about 10 million cycles, while the calculated life with the constraint-loss option showed a slower rate of crack growth after 10 million cycles. The shape of the calculated crack-length-against-cycles curve with the constant-loss option is similar to the test results. Compared with the test data, the calculated behavior showed slower crack growth in the early stages. As a result, FASTRAN over predicted the fatigue-crack- growth life under the Outer Wing - 2 spectra. While the source of the

12

disparity in the calculated and measured crack-growth behavior remains unclear, one possibility is crack-growth rate data in the near threshold regime may need further study. Recall that only C(T) data was available and one set had the –T73 temper and the other set of data was from a thicker material (B = 0.5 in.).


N, cycles

0.0 2.0e+6 4.0e+6 6.0e+6 8.0e+6 1.0e+7 1.2e+7

Cracklength,c, in.

0.0

0.5

1.0

1.5

2.0


Figure 12. Measured and calculated fatigue-crack growth for Outer Wing - 2 spectrum.

Figure 13 shows a comparison of the measured and calculated fatigue-crack-growth lives, accounting for constraint loss, for the four simulated aircraft loading spectra, previously discussed. All of the calculated-life-to-test-life ratios are nearly parallel to the normalized crack-length ratios. This implies that the shape of the crack-length- against-cycles behavior is similar to the test results with nearly a constant percentage difference. The calculated lives under spectrum Center Wing - 1, Center Wing - 2, and Outer Wing - 1 fell within about +/- 10% of the test lives, whereas the calculated lives under the Outer Wing - 2 spectrum showed approximately 30% over prediction. Brot and Matias [10] considered two crack configurations and five different simulated aircraft load spectra in their investigation on several different materials. For the 7075-T7351 alloy, they conducted tests using only three spectra: (1) Wing gust and maneuver spectrum, (2) Lateral gust loading spectrum, and (3) Pressurized fuselage gust and maneuver spectrum. Specimens were notched and fatigue pre-cracked under constant-amplitude loading to grow a crack from an initial notch to a specified crack length. After pre-cracking, the specimens were then subjected to the spectrum loading until failure. Crack-length-against-cycles data were recorded in the tests. In most cases, two duplicate tests were conducted. In the following, FASTRAN has been used to “predict” the crack length against cycles results for the three spectra applied to M(T) specimens. The ∆Keff-rate relation and the constraint factors, previously developed, were used to make the crack-growth predictions. Again, the predictions were made using both a constant constraint and the constraint-loss option. The first predictions were made with a constant constraint factor α = 2.4. This option is a simple application of the crack-closure model because the

13

constraint-loss regime is not needed. The other predictions were with the constraint-loss option, as shown in Figure 4. Note that the location of constraint-loss regime, shown in Figure 4, was adjusted based on the previous spectrum load test results. However, it is suspected that the baseline ∆Keff-rate results could be applied to other spectra without modification.

Calculated crack length / Failure crack length

0.0 0.2 0.4 0.6 0.8 1.0

Cal

cula

ted

life

/ Tes

t life

0.0

0.5

1.0

1.5

Outer Wing - 1

Outer Wing - 2

Center Wing - 1

Center Wing - 2

Test 1Test 2

Figure 13. Comparison of measured and calculated fatigue-crack-growth lives.

The details on the wing gust and maneuver spectrum are given in Reference 10. This spectrum simulates the gust and maneuver loading on a wing of a transport aircraft. The spectrum is composed of seven levels of loading with a mean value of R of 0.6. The minimum stress in some flights of the spectrum will reach a compressive value of about 0.23 times the peak stress. For the M(T) specimens subjected to the wing gust and maneuver spectrum, the two test specimens showed slightly different behavior, as shown in Figure 14. In the FASTRAN life-prediction code, the standard K-solution for the M(T) specimen was used. The constant constraint option produced faster crack growth than the constraint-loss option. However, both predictions fell short of the test results, especially in the early stages of crack growth. For larger crack lengths, the shape of the predicted curve is very similar to the test results. The predicted results with the constraint-loss option (solid curve) show regions of slow and rapid crack growth as the variable-amplitude loads are applied. This crack-growth behavior is similar to what is seen in the test with the longest life. The details on the lateral gust-loading spectrum are also given in Reference 10. This spectrum simulates the lateral gust loading, which affects the vertical tail of an aircraft. Again, the spectrum is composed of seven levels of loading, but in totally reversed loading (R = -1).

14

For the lateral-gust spectra, only one M(T) specimen was tested and these results are shown in Figure 15. The two crack-closure options agreed quite well with each other and with the test data. For the two crack-closure options, the high loads produced residual plastic deformations, which shield the crack front (with high crack-closure loads) from the damaging effects of the compressive loading, and produce longer fatigue-crack-growth lives than linear-cumulative damage calculations [20]. Again, the details on the pressurized fuselage gust and maneuver spectrum are given in Reference 10. This spectrum simulates the gust and maneuver loading on a fuselage of a transport aircraft. The fuselage is pressurized once per flight and then fifteen levels of gust and maneuver loading are superimposed, producing a mean R ratio of 0.83. Of course, the minimum stress in the flight spectrum is zero.

N, cycles

0.0 5.0e+4 1.0e+5 1.5e+5 2.0e+5 2.5e+5 3.0e+5

Crack length, c, in

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Test 1Test 2FASTRAN:Constraint lossConstant constraint

Brot and Matias7075-T7351M(T) B = 0.25 inWing gust and maneuver spectrum

Figure 14. Measured and predicted crack growth under wing gust and maneuver spectrum.

The test results and calculations made on the M(T) specimens subjected to the pressurized fuselage gust and maneuver spectrum are shown in Figures 16. For the high R spectra, both constant constraint and constraint-loss life-calculations produced nearly the same crack-length-against-cycle results. Here, the crack-opening stresses for the majority of the cycles are below the minimum applied stresses, and the cracks are fully opened. In summary, a comparison of the measured and calculated (or predicted) fatigue- crack-growth lives, accounting for constraint loss, for the seven simulated aircraft load spectra applied to M(T) specimens is shown in Figure 17. Most of the calculated or predicted fatigue-crack- growth lives fell within +/- 30 percent of the test lives, but the predicted results on the wing gust and maneuver spectrum fell nearly 50 percent short of the test results on one of the tests. The analyses made on these particular spectra demonstrate that for thicker materials and higher constraint factors (such as α = 2.4), the constraint-loss regime is not as important as in thin-sheet (lower constraint) materials [17].

15

N, cycles

0 1e+5 2e+5 3e+5 4e+5 5e+5 6e+5

Crack length,

c, in

0.0

0.2

0.4

0.6

0.8

1.0

TestFASTRAN:Constraint lossConstant constraint

Brot and Matias7075-T7351M(T) B = 0.25 inLateral gust spectrum

Figure 15. Measured and predicted crack growth for lateral gust loading spectrum.

N, cycles

0 1e+5 2e+5 3e+5 4e+5

Crack length,

c, in

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Test 1Test 2FASTRAN:Constraint lossConstant constraint

Brot and Matias7075-T7351M(T) B = 0.25 inFuselage gust and maneuver spectrum

Figure 16. Measured and predicted crack growth under pressurized fuselage gust and

maneuver spectrum.

6. Concluding remarks A “plasticity-induced” crack-closure model, FASTRAN, was used to correlate fatigue- crack-growth-rate data on 7075-T7351 aluminum alloy plate (B = 0.25 in.) under constant-amplitude loading over a wide range of stress ratios (R = -1 to 0.8) and

16

crack-growth rates from near threshold to fracture. Test data on the T7351 temper and thickness was lacking in the threshold and near threshold regime on tensile loaded middle-crack tension specimens, but test data was found on compact tension C(T) specimens. The C(T) specimens, however, were either thicker than the spectrum test specimens or made of a different temper (T73). Constraint factors, which account for the three-dimensional state-of-stress effects in the crack-front region, were used in determining the effective-stress-intensity-factor- range-against-crack-growth-rate relation. Based on some of the spectrum crack-growth results, a constraint-loss regime was selected, which also roughly corresponds to the flat-to-slant crack growth region. Comparisons made between measured and calculated fatigue-crack-growth lives under seven simulated aircraft load spectra agreed quite well (within +/- 30 %) of the test results.

Test life, NT, cycles

1e+5 1e+6 1e+7

Calculatedlife,

NC, cycles

1e+5

1e+6

1e+7

Wing GustLateral GustFuselageCenter Wing - 1Center Wing - 2Outer Wing - 1Outer Wing - 2

7075-T7351 (LT)B = 0.25 inM(T)W = 3 to 6 in

+30%

-30%

Figure 17. Measured and calculated fatigue-crack-growth lives under seven spectra.

References [1] Elber, W., “Fatigue crack closure under cyclic tension,” Engineering Fracture

Mechanics, 1970, Vol. 2, pp. 37-45. [2] Elber, W., “The significance of fatigue crack closure,” In: Damage tolerance aircraft

structures, ASTM STP-486, 1971, pp. 230-242. [3] Walker, N. and Beevers, C.J., “A Fatigue crack closure mechanism in titanium,” In:

Fatigue and Fracture of Engineering Materials and Structures, 1979. Vol. 1, No. 1, pp. 135-148.

[4] Paris, P.C., Bucci, R.J., Wessel, E.T., Clark, W.G., and Mager, T.R., “Extensive study of low fatigue crack growth rates in A533 and A508 steels,” ASTM STP-513,

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1972, pp.141-176. [5] Dugdale, D.S., “Yielding steel sheets containing slits,” Journal of Mechanics and

Physics of Solids, 1960, Vol. 8, pp. 100-104. [6] Newman, Jr., J.C., “A crack-closure model for predicting fatigue crack growth

under aircraft spectrum loading,” In: Methods and models for predicting fatigue crack growth under random loading, ASTM STP-748, 1981, pp. 53-84.

[7] Newman, Jr., J.C., “FASTRAN II-A fatigue crack growth analysis program,” NASA TM 104159, 1992.

[8] NASGRO Reference Manual (version 4.02), NASA Johnson Space Center and Southwest Research Institute, 2002.

[9] AFGROW Reference Manual (version 4.02), Wright-Patterson Air Force Base, AFRL/VASM, 2003.

[10] Brot, A. and Matias, C., “An evaluation of several retardation models for crack growth prediction under spectrum loading,” In: USAF Aircraft Structural Integrity Program Conference, Savannah, GA, December 2002.

[11] Blom, A.R., Wang, G.S. and Chermahini, R.G., “Comparison of crack closure results obtained by 3D elastic-plastic FEM and modified Dugdale model,” In: Aliabadi M.H. et al., editors, Localized damage: computer-aided assessment and control. Berlin, Springer-Verlag, 1990, pp. 57-68.

[12] Yoder, G.R., Cooley, L.A. and Crooker, T.W., “On microstructural control on near-threshold fatigue crack growth in 7000-series aluminum alloys,” Scripta Metallurgy, 1982, Vol. 16, pp. 1021-1025.

[13] Forth, S.C., Newman, Jr., J.C. and Forman, R.G., “On generating fatigue crack growth thresholds,” International Journal of Fatigue, Vol. 25, 2003, pp. 9-15.

[14] Newman, Jr., J.C., Schneider, J.A., Daniel, A. and McKnight, D., “Compression pre-cracking to generate near threshold fatigue-crack-growth rates in two aluminum alloys,” International Journal of Fatigue, Vol. 27, 2005, pp. 1432-1440.

[15] Newman, Jr., J.C., Bigelow, C.A. and Shivakumar, K.N., “Three-dimensional elastic-plastic finite-element analyses of constraint variations in cracked bodies,” Engineering Fracture Mechanics, Vol. 46, 1993, pp. 1-13.

[16] Schijve, J., “Significance of fatigue cracks in micro-range and macro-range,” In: Fatigue crack propagation, ASTM STP-415, 1967, pp. 415-459.

[17] Newman, Jr., J.C., “Effects of constraint on crack growth under aircraft spectrum loading,” In: Beukers, et al., editors, Fatigue of aircraft materials, Delft University Press, 1992, pp. 83-109.

[18] Newman, Jr., J.C., “Fracture analysis of various cracked configurations in sheet and plate materials,” In: Properties related to fracture toughness, ASTM STP-605, 1976, pp. 104-123.

[19] Newman, Jr., J.C., “Analyses of fatigue crack growth and closure near threshold conditions for large-crack behavior,” In: Fatigue crack growth thresholds, ASTM STP-1372, 1998, pp. 227-251.

[20] Newman, Jr., J.C., Brot, A. and Matias, C., “Crack-growth calculations in 7075-T7351 aluminum alloy under various load spectra using an improved crack-closure model,” Engineering Fracture Mechanics, Vol. 71, 2004, pp. 2347-2363.

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