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

Transcription

1 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) Conf. 29 November to 1 December 2005 Memphis, TN 1

2 Major Approaches to Fatigue Design and Evaluation Stress-Based Strain-Based Crack Growth (Fracture Mechanics) 2

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 3

4 Matching a Component to Specimen Data 4

5 Equating stresses as k t S gives similar notch strain and initiation life 5

6 But crack growth effects cause a size effect so also match K/k t S for total life 6

7 Total Life to Failure Data 7

8 Scaling of Standard Curve Shapes to Limited Component Data [From Dickson et al., 1999] 8

9 Strain-Based Approach 9

10 10

11 Cyclic stress-strain curve ε a = σ a E + σ a H 1/ n Strain-life curve ε a = σ f E f b ( 2N ) + ε f (2N f ) c 11

12 12

13 13

14 14

15 An analysis using the cyclic stress-strain curve is needed: Elasto-plastic finite elements Neubers rule σ ε = 2 ( k t S) E S a = 1 k t σ 2 a + σ a E σ a H 1/ n 15

16 Example: zero mean stress (R = 1) loading 16

17 40 t = 5.08 mm P a, Load Amplitude, kn data Neuber FEA ε a, Notch Strain Amplitude 17

18 40 P a, Load Amplitude, kn mm crack Neuber failure FEA t = 5.08 mm 0 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 N i, N f, Cycles 18

19 19 Mean Stress Effects c f f b f f m f a N N E ) (2 ) (2 1 ε + σ σ σ = ε ( ) c b f f f b f f a N N E + ε + σ σ = ε σ ) (2 ) (2 2 2 max

20 20

21 Example: zero-to-maximum (R = 0) loading 21

22 22

23 Variable Amplitude Loading 23

24 Component-Specific Strain-Life Curve - for surface finish and load interaction effects 1 ε' f εa, Strain Amplitude σ' f / E SAE 4142 Steel (450 HB) polished, b modified, b s E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 24

25 Data for helicopter load spectrum Felix applied to plates with round holes (k t = 2.5) of Ti-6Al-4V 800 Notched Titanium 6AI-4V Maximum Nominal Stress, MPa b= Flights to Failure 25

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 local yielding on mean stress. Fabrication detail and load interaction effects can be included by using a component-specific strain-life curve. 26

27 1000 S a, MPa AISI 4130 Steel σ u = 807 MPa k t = 4.00, R = 1 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 N f, Cycles Juvinal 2000 Shigley 2004 Data 27

28 Goodman S ar for double-edge-notched axial specimens of 2024-T3 Al 400 Goodman S ar, MPa T3 Al k t = 2.15 Mean Stress, MPa Fit E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 28

29 Smith, Watson, and Topper (SWT) S ar for double-edge-notched axial specimens of 2024-T3 Al SWT S ar, MPa Mean Stress, MPa Fit T3 Al k t = E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 29

30 Walker S ar for double-edge-notched axial specimens of 2024-T3 Al fit to S = A with γ = ar N B f Walker S ar, MPa T3 Al k t = 2.15 Mean Stress, MPa Fit all E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 30

31 31

32 32

33 1000 S a, Nom. Stress Amplitude, MPa T6 Al k t = 4.00 Mean Stress, MPa 0 Calc 0 Data 138 Calc 138 Data 10 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 N f, Cycles 33

34 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 strainlife 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

35 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 10 6 cycles for steels, is estimated based on adjusting the elastic stress concentration factor, k t, to obtain a fatigue notch factor, k f. 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 10 3 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 10 3 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, k t, 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, k t S, where S is a nominal stress consistently defined with k t. 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. 2

36 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 k t 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 = S min /S max. 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 S ar that is expected to cause the same fatigue life as a desired combination of stress amplitude S a and nonzero mean stress S m. The most basic form of this relationship employs the ultimate tensile strength σ u and in effect includes no concentration factor for the mean stress. Sar Sa = Sm 1 σu (1) Entering an S-N curve for the completely reversed case with S ar 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 S ar 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). S ar 1 R = SmaxSa, Sar = Smax (a, b) (2) 2 3

37 where S max = S a + S m and R = S min /S max. 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 1 γ γ 1 R Sar = Smax Sa, Sar = Smax (a, b) (3) 2 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 10 9 cycles and beyond have exhibited the surprising behavior of a drop in the stress-life curve beyond the flat region in the 10 6 to 10 7 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 10 7 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. 4

38 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 εa σa = E 1/ n σa + H (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 ε a versus fatigue life in cycles, N f. σ f b c ε a = ( 2N f ) + ε f (2N f ) (5) E 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. 5

39 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 ( S) σ ε = (6) E k t 2 The quantities σ and ε are the local stress and strain at the notch, respectively, S is a nominal or average stress, and k t 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 Sa = 1 kt 2 σa 1/ n σa + σa E H (7) Hence, for a given applied cyclic load expressed as a nominal stress amplitude S a, this equation can be solved iteratively for the local notch stress amplitude σ a, and then Eq. 4 provides the local notch strain amplitude ε a. The strain amplitude ε a, 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 N f 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, σ m, 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). σar σa = σm 1 σ f (8) where σ ar is an equivalent completely reversed stress amplitude that is expected to cause the same life as the actual (σ a, σ m ) combination. This has the same form as the Goodman relationship, Eq. 1, but differs by the use of σ' f in place of σ u and in the application to local mean stress. Incorporating the effect on life required by Eq. 8 into Eq. 5 gives 6

40 1/ b σ f b c m a (2N*) f (2N*), N f N* σ ε = + ε = 1 E (a, b) (9) σ f where N* is calculated from ε a using (a), as if the mean stress were zero, and then this value is employed in (b) to obtain N f 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. σ f σm b c ε a = 1 (2N f ) + ε f (2N f ) (10) E σ f The Smith, Watson, and Topper method (Smith 1970) is usually applied to the strain-life equation in the form: 2 ( σ f ) 2b b+ c σ max εa = (2N f ) + σ f ε f (2N f ) (11) E where σ max = σ a + σ m. 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 N f from N*. N f (1 γ ) / b (1 γ ) / b σa 1 Rσ N *, N f = N * σmax 2 = (a, b) (12) where R σ = σ min /σ max. 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 cycledependent 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 S max during cycling (or the minimum if S min > S max ) and the amplitude S a, to obtain the corresponding local notch stresses and strains, σ max, ε max, and σ a, ε a. 7

41 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 S max, and a second time for S a. 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: 1/ n 1 2 σmax Smax = σmax + σmax E, kt H εmax σmax = E 1/ n σmax + H (a, b) (13) First, (a) is entered with S max and solved iteratively to obtain σ max, and then (b) gives ε max. Next, follow a similar procedure by employing the S a value to obtain σ a and ε a, using Sa 1 = kt 1/ n 2 σa σa + σa E H, εa σa = E 1/ n σa + H (a, b) (14) Additional stresses and strains that are of interest are then computed. σ min = σmax σa, εmin = εmax 2εa, 2 σ = σ σ (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 σ max, ε max. 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 stressstrain 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 N f 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. m max a 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 stressstrain 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 8

42 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 S max 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: S XY 2 = 1 kt 2 1/ n σxy σxy E σ XY H, ε XY 2 = σxy 2E 1/ n σ XY + 2H (16) Note that these relationships are equivalent to Eq. 14, being derived from it by substituting S a = S/2, σ a = σ/2, and ε a = 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 ε a and the mean stress σ m for each closed loop (cycle) are of interest, as is the peak (maximum) stress σ max. 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 N f corresponding to each hysteresis loop is needed and can be determined from its strain amplitude ε a and mean stress σ m substituted into Eq. 9 or 10, or from its peak stress σ max and strain amplitude ε a 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 N j B f = 1 (17) N f j one rep. The summation is performed for one repetition of the history. The N j are the numbers of times each cycle (loop) occurs in one repetition of the history, with N j = 1 applying for each of the four cycles in the Fig. 24 example. The N f j are the corresponding numbers of cycles to failure. Once the summation is done, one can solve for B f, which is the estimated number of repetitions of the 9

43 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 σ m 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 b s 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 = 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 = 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). 10

44 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 strainlife 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. 11

45 REFERENCES AWS (2004) Bathias (1999) Boller (1987) Cronkhite (2000) Dickson (1999) Dowling (1973) Dowling (1979) Dowling (1981) Dowling (1982) AWS, Structural Welding Code: Steel, 19th ed., AWS D1.1/D1.1M:2004, American Welding Society, Miami, FL, See also Structural Welding Code: Aluminum, AWS D1.2/D1.2M:2003. 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 C. Boller and T. Seeger, Materials Data for Cyclic Loading, 5 Vols. Elsevier Science Pubs., Amsterdam, See also Supplement 1, 1990, by A. Baumel and T. Seeger. J. D. Cronkhite, et al., "Research on Practical Damage Tolerance Methods for Rotorcraft Structures," American Helicopter Society 56th Annual Forum, Virginia Beach, VA, May B. Dickson, et al., "Rotorcraft Fatigue and Damage Tolerance," Report prepared for the Technical Oversight Group on Aging Aircraft (TOGAA), 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 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, 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 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 Also pub. in Journal of Engineering Materials and Technology, ASME, Vol. 105, July 1983, pp , with Erratum, Oct. 1983, p Dowling (1987) N. E. Dowling, "A Review of Fatigue Life Prediction Methods," 1987 Transactions of the Society of Automotive Engineers, Paper No , Sept 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

46 Dowling (1999) Dowling (2000) N. E. Dowling, Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture and Fatigue, 2nd edition, Prentice Hall, Upper Saddle River, NJ, rd edition anticipated in 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 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 Dowling (2004) DuQuesnay (1993) Edwards (1984) Grover (1951) Heuler (1986) N. E. Dowling, Mean Stress Effects in Stress-Life and Strain-Life Fatigue, SAE Paper No , Fatigue 2004: Second SAE Brasil International Conference on Fatigue, June 2004, São Paulo, Brasil. (See also: D. L. DuQuesnay, M. A. Pompetzki, and T. H. Topper, Fatigue Life Prediction for Variable Amplitude Strain Histories, Paper No , Fatigue Research and Applications, Pub. No. SP-1009, Society of Automotive Engineers, Warrendale, PA, 1993, pp 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), 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, P. Heuler and T. Seeger, A Criterion for Omission of Variable Amplitude Loading Histories, International Journal of Fatigue, Vol. 8, No. 4, 1986, pp 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, Juvinall (2000) R. C. Juvinall and K. M. Marshek, Fundamentals of Machine Component Design, 3rd ed., John Wiley, New York, NY,

47 MILHDBK (1998) Morrow (1968) Naumann (1959) Shigley (2004) Shiozawa (2001) Smith (1970) Topper (1970) Walker (1970) Wetzel (1968) 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, See also: Interactive Edition, 2003, Knovel Corp., Norwich, NY, J. Morrow, Fatigue Properties of Metals, Section 3.2 of Fatigue Design Handbook, Pub. No. AE-4, Society of Automotive Engineers, Warrendale, PA, 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, 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, J. E. Shigley, C. R. Mischke, and R. G. Budynas, Mechanical Engineering Design, 7th ed., McGraw-Hill, New York, NY, See also J. E. Shigley and L. D. Mitchell, 1993 reprint of the 4th edition. 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 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 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 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 R. M. Wetzel, "Smooth Specimen Simulation of Fatigue Behavior of Notches," Journal of Materials, ASTM, Vol. 3, No. 3, Sept. 1968, pp

48 1000 S a, MPa 100 AISI 4130 Steel σ u = 807 MPa k t = 4.00, R = 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 N f, Cycles Juvinal 2000 Shigley 2004 Data 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).] 15

49 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 k t S, for the geometries of Fig. 2. [From Dowling (1981).] 16

50 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).] 17

51 400 Goodman S ar, MPa T3 Al k t = 2.15 Mean Stress, MPa Fit E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 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).] SWT S ar, MPa Mean Stress, MPa Fit T3 Al k t = E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 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).] 18

52 400 Walker S ar, MPa T3 Al k t = 2.15 Mean Stress, MPa Fit all E+03 1.E+04 N f, Cycles 1.E+05 1.E+06 Figure 9 Equivalent completely reversed nominal stress from the Walker equation for doubleedge-notched axial specimens of 2024-T3 aluminum. The curve corresponds to γ = [From Dowling (2000); data from Grover (1951).] Figure 10 Stress-life curve for an alloy steel showing overstrain effects. [Data from Dowling (1973).] 19

53 Figure 11 Stress-life curve extending into the gigacycle range for a bearing steel with σ u = 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).] 20

54 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).] 21

55 Figure 15 Effect of initial prestrain (10 cycles at ε a = 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).] 22

56 Figure 17 Bluntly notched compact specimen. [From Dowling (1979).] 40 t = 5.08 mm P a, Load Amplitude, kn data Neuber FEA ε a, Notch Strain Amplitude 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).] 23

57 40 P a, Load Amplitude, kn mm crack Neuber failure FEA t = 5.08 mm 0 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 N i, N f, Cycles 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. 24

58 Figure 21 Stress-strain responses estimated using Neuber's rule for various levels of zero-tomaximum 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).] 25

59 1000 S a, Nom. Stress Amplitude, MPa T6 Al k t = 4.00 Mean Stress, MPa 0 Calc 0 Data 138 Calc 138 Data 10 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 N f, Cycles Figure 23 Lives calculated by the strain-based approach compared with test data for doubleedge-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).] 26

60 1 ε' f εa, Strain Amplitude σ' f / E SAE 4142 Steel (450 HB) polished, b modified, b s E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 N f, Cycles 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. 800 Notched Titanium 6AI-4V Maximum Nominal Stress, MPa b= Flights to Failure Figure 26 Strain-based life estimates using for the standard helicopter load spectrum Felix applied to plates with round holes (k t = 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).] 27

61 Yield Zone Based Retardation Models Use With Caution James A. Harter Air Force Research Laboratory Analytical Mechanics Branch ASIP Conference 29 Nov 2005 Memphis, TN

62 Yield Zone Based Retardation Models Basic Yield Zone Approach Plastic Zone = (Kmax/Yld) 2 / (PSX * π) Where, PSX accounts for Stress State da/dn = f(r pi /r p OL )

63 Yield Zone Based Retardation Models Life Prediction Example

64 Yield Zone Based Retardation Models Life Prediction Example

65 Yield Zone Based Retardation Models Life Prediction Example Note: The closure model parameter (Cf0 = 0.653) was adjusted to match the Willenborg prediction

66 Yield Zone Based Retardation Models Life Prediction Example (After Transition to an Oblique Crack) Beta vs. Crack Length Stress Intensity vs. Crack Length Beta Ct C Stress Intensity (Unit Stress) Ct C Crack Length Crack Length

67 Yield Zone Based Retardation Models Life Prediction Example - Effect of Overload Shut-off Ratio

68 Yield Zone Based Retardation Models Life 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

69 Yield Zone Based Retardation Models Life Prediction Example (Before Transition to an Oblique Crack) a-length K(a)

70 Yield Zone Based Retardation Models 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

71 Yield Zone Based Retardation Models Summary 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 Are your retardation model parameters reasonable? Have you compared results for other retardation models? Do your answers make sense?

72 2005 USAF ASIP Conference Presentation Improved Modeling of R-Ratio Behavior 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 Memphis Memphis, TN 29 November 1 December

73 Improved Modeling of R-Ratio Behavior for the Walker Equation Background 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 2

74 Improved Modeling of R-Ratio Behavior for the Walker Equation Background (Con t)! 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. 3

75 Improved Modeling of R-Ratio Behavior for the Walker Equation 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 4

76 Newman Closure Eqn: where Improved Modeling of R-Ratio Behavior for the Walker Equation Newman Closure and Walker Equations f da dn C 1 f = 1 R Kop = = K max K n 1 1 K K K K c th max 2 3 ( ) max R, A0 + AR 1 + A2R + A3 R R 0 A0 + AR 1 2 R < 0 q p Walker Eqn: da dn C K = 1 m ( 1 R) n 5

77 Improved Modeling of R-Ratio Behavior for the Walker Equation Discrepancy between Walker & Closure Eqn s In Modeling R-Ratio Behavior Example: Fits (from R =.1 to.7) to NASMAT data for 7050-T7451 Al R =.9,.6,.3, 0 R =.9,.6,.3, 0 1e-4 1e e-5 1e da/dn [in/cycle] 1e-6 1e-6 C=2.6e-9 n=3.61 α=0.54 S max /σ o =0.3 da/dn [in/cycle] 1e e C=0.6e-9 n=3.61 m= K [ksi K [ksi in] Newman s Closure Equation Behavior Walker in] Equation Behavior Curves converge with increasing R Curves diverge with increasing R 6

78 Improved Modeling of R-Ratio Behavior for the Walker Equation Revised Walker Equation & example fit to closure equation Example Fit: NASMAT data for 7050-T7451 AL da/dn [in/cycle] 1e e-5 Closure Eqn Rev. Walker R =.9,.6,.3, 0 da dn = C K 1 ( C ) R R b C R 1 m n 1e e-7 C=0.65e-9 n=3.61 m=0.74 b=0.31 C R =0.8 Rev. Walker Equation K [ksi in] 7

79 Improved Modeling of R-Ratio Behavior for the Walker Equation Development of the Extended Walker Equation! To model threshold region behavior! To model crack instability & K max behavior 8

80 Improved Modeling of R-Ratio Behavior for the Walker Equation 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 front 2. Roughness of the crack surfaces 3. Oxides or fretting debris behind the crack tip! Threshold values vary with type and size of test specimen 9

81 Improved Modeling of R-Ratio Behavior for the Walker Equation Comparison of Different Specimen Test Results for Threshold Testing R = 0.1 K th Results Using Load Shedding Method for Ti-6Al-4V MA Plt M(T) Specimen C(T) Specimen ESE(T) Specimen K th = 3.00 K th = 4.3 K th =

82 Improved Modeling of R-Ratio Behavior for the Walker Equation NASGRO Threshold Equation K th = K a a + a R 1 f ( 1+ R C th ) / 1 0 [ R] 1 th ( ) ( 1 R ) Cth A 0 C th 11

83 Improved Modeling of R-Ratio Behavior for the Walker Equation NASGRO Eqn fit : Ti-6-4 MA ; 0.25 Plt ; L-T ; C(T) Specimen Data ; Cth=1.5 12

84 Improved Modeling of R-Ratio Behavior for the Walker Equation NASGRO Eqn fit:ti-6-4 MA ; 0.25 Plt ; L-T; R=0.1; C(T) & M(T) Data ; Cth=1.5 13

85 Improved Modeling of R-Ratio Behavior for the Walker Equation NASGRO Eqn Fit: Ti-6-4 MA ; 0.25 Plt ; R=0.1; C(T) & M(T) Data ; Cth=0 14

86 Improved Modeling of R-Ratio Behavior for the Walker Equation Reasons for making C th = 0 the default value Necessary for conservative analysis: 1. Thresholds obtained by load shedding are un-conservative for some specimen types mainly C(T) & ESE(T) types. 2. Higher thresholds are obtained in laboratory air & in high humidity air than in dry air. 15

87 Evaluation Of R = 0.1 Fracture Surface (S. Forth s tests at NASA Langley) 16

88 Effect of Laboratory Environment (S. Forth s tests at NASA Langley) 17

89 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) 18

90 Improved Modeling of R-Ratio Behavior for the Walker Equation 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 (C th ) is needed 19

91 Improved Modeling of R-Ratio Behavior for the Walker Equation Preliminary relationship between m - and m + constants Data - Mod. Walker fit (m + & m - ) to crack closure thresholds m m =α m + +β m + 20

92 21 Improved Modeling of R-Ratio Behavior for the Walker Equation ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 0 R for C 1 R 1 K 0 R for C 1 R C 1 K K p th p th p th C 1 m 1 R m 1 1 m 1 C 1 R R C 1 b R 1 th < = = Extended Walker Threshold Equation (Preliminary)

93 Recent Developments in Modeling R-Ratio Effects with the Walker Equation Extended Walker Threshold Equation (Preliminary) 6 5 Closure Eqn Extended Walker 4 K th [ksi in] Stress Ratio, R 22

94 Recent Developments in Modeling R-Ratio Effects with the Walker Equation Extended Walker Equation e-3 R =.9,.6,.3,0,-.5,-1,-2 da/dn [in/cycle ] 1e e e da dn = C K th ( b) 1 m q 1 C R Kmax R n K 1 K 1 K c p 1e e e K [ksi in] 23

95 Improved Modeling of R-Ratio Behavior for the Walker Equation Summary Comments! Enhanced capabilities of Extended Walker Equation: 1. Improved fits for multiple R-value data 2. Improved fit for threshold & high da/dn regions 3. Improved fits for some elevated temperature data! Current Deficiencies of Extended Walker Equation 1. Needs further development for negative R data 2. Not capable of accurately fitting irregular trends in data 24

96 Prediction of Fatigue Crack Growth for 7075-T7351 Aluminium Alloy under Various Flight Load Spectra Y. Yamada, J. C. Newman, Jr., T. E. Lacy Department of Aerospace Engineering Mississippi State University Mississippi State, MS, USA ASIP 2005 USAF Aircraft Structural Integrity Program 29 November 1 December 2005 Memphis, Tennessee

97 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 K eff -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 # 2

98 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 Retardation Models for Crack Growth Prediction under Spectrum Loading, U.S.A.F. Aircraft Structural Integrity Program Conference, ASIP # 3

99 NASA FASTRAN CRACK-GROWTH & CLOSURE MODEL ASIP # 4

100 FLAT-TO-SLANT CRACK-GROWTH CHARACTERISTICS ( K eff ) T = 0.5 σ o B 1/2 ASIP # 5

101 CRACK-GROWTH RATE EQUATION ni dc dn = C ( K ) [1 ( K i eff max K Ie ) 2 ] K eff = So 1 1 max S S = o S R max K Smax f ( R, σ O, α) ASIP # 6

102 LEFM ANALYSES OF 7075-T7351 ALUMINUM ALLOY 1e-2 1e T7351 Aluminum alloy B = 0.25 in 1e-4 dc/dn in/cycle 1e-5 1e-6 1e-7 1e-8 1e K, ksi-in 1/2 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 in T7351 R = -1.0 M(T) T7351 R = 0.02 M(T) T7351 R = 0.5 M(T) ASIP # 7

103 EFFECTIVE SIF FOR 7075-T7351 ALUMINUM ALLOY 1e-2 α2 = 1.2 ( K eff ) T dc/dn in/cycle 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 1e-9 α1 = Keff, ksi-in 1/2 Single or double shear mode? 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 in T7351 R = -1.0 M(T) T7351 R = 0.02 M(T) T7351 R = 0.5 M(T) Keff Baseline ASIP # 8

104 CENTER WING - 1 SPECTRUM S / S peak S min /S peak = Time ASIP # 9

105 MEASURED AND CALCULATED CRACK GROWTH FOR CENTER WING - 1 SPECTRUM Crack length, c, in Spectrum Center Wing T7351 B = 0.25 in W = 4 in Test Test 2 Constraint loss Constant constraint e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6 N, cycles ASIP # 10

106 CENTER WING - 2 SPECTRUM S / S peak S min /S peak = Time ASIP # 11

107 MEASURED AND CALCULATED CRACK GROWTH FOR CENTER WING - 2 SPECTRUM Crack length, c, in Spectrum Center Wing T7351 B = 0.25 in W = 4 in Test 1 Test 2 Constraint loss Constant constraint e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6 N, cycles ASIP # 12

108 OUTER WING - 1 SPECTRUM S / S peak S min / S peak = Time ASIP # 13

109 MEASURED AND CALCULATED CRACK GROWTH FOR OUTER WING - 1 SPECTRUM Crack length, c, in Spectrum Outer Wing T7351 B = 0.25 in W = 4 in Test 1 Test 2 Constraint loss Constant constraint e+5 1.0e+6 1.5e+6 2.0e+6 2.5e+6 N, cycles ASIP # 14

110 OUTER WING - 2 SPECTRUM S / S peak S min /S peak = -0.6 Time ASIP # 15

111 MEASURED AND CALCULATED CRACK GROWTH FOR OUTER WING - 2 SPECTRUM Crack length, c, in Spectrum Outer Wing T7351 B = 0.25 in W = 4 in Test Test 2 Constraint loss Constant constraint e+6 4.0e+6 6.0e+6 8.0e+6 1.0e+7 1.2e+7 N, cycles ASIP # 16

112 RATIO OF CALCULATED TO MEASURED SPECTRA LIVES 1.5 Center Wing -2 Outer Wing Calculated life (with constraint-loss) Test life 0.5 Center Wing -1 Outer Wing % +17.4% -6.77% -13.6% Test 1 Test Calculated crack length / Failure crack length (with constraint-loss) ASIP # 17

113 CALIBRATED EFFECTIVE SIF FOR 7075-T7351 ALLOY 1e-2 1e-3 1e-4 dc/dn 1e-5 in/cycle 1e-6 1e-7 1e-8 α2 = 1.2 α1 = 2.4 ( K eff ) T K eff baseline Calibrated parameters for 7075-T7351 Tensile constraint factors: α 1 = 2.4 α 2 = 1.2 Constraint-loss regime: Initiated at 3.0e-5 in/cycle Ended at 1.0e-2 in/cycle Compressive constraint factors: β = 1.0 1e Keff, ksi-in 1/2 ASIP # 18

114 LATERAL GUST AND MANEUVER SPECTRUM 1.0 S / S peak Time ASIP # 19

115 MEASURED AND PREDICTED CRACK GROWTH FOR LATERAL GUST SPECTRUM Crack length, c, in Brot and Matias 7075-T7351 M(T) B = 0.25 in Lateral gust spectrum Test Constraint loss Constant constraint e+5 2e+5 3e+5 4e+5 5e+5 6e+5 N, cycles ASIP # 20

116 PRESSURIZED FUSELAGE GUST AND MANEUVER SPECTRUM S / S peak Time ASIP # 21

117 MEASURED AND PREDICTED CRACK GROWTH FOR FUSELAGE GUST AND MANEUVER SPECTRUM Crack length, c, in Brot and Matias 7075-T7351 M(T) B = 0.25 in Fuselage gust and maneuver spectrum Test 1 Test 2 Constraint loss Constant constraint e+5 2e+5 3e+5 4e+5 N, cycles ASIP # 22

118 WING GUST AND MANEUVER SPECTRUM S / S peak Time ASIP # 23

119 MEASURED AND PREDICTED CRACK GROWTH FOR WING GUST LOADING SPECTRUM Crack length, c, in Brot and Matias 7075-T7351 M(T) B = 0.25 in Wing gust and maneuver spectrum 0.4 Test 1 Test Constraint loss Constant constraint e+4 1.0e+5 1.5e+5 2.0e+5 2.5e+5 3.0e+5 N, cycles ASIP # 24

120 RATIO OF CALCULATED OR PREDICTED LIVES TO MEASURED LIVES 1e T7351 (LT) B = 0.25 in M(T) W = 3 or 4 in +30% -30% Calculated life, N C, cycles 1e+6 1e+5 1e+5 1e+6 1e+7 Test life, N T, cycles Center Wing - 1 Outer Wing - 1 Center Wing - 2 Outer Wing - 2 Lateral Gust Fuselage Wing Gust ASIP # 25

121 Concluding Remarks Fatigue-crack-growth calculations and predictions for FASTRAN on 7075-T7351 (B = 0.25 in.) aluminum alloy: 1. Difficulty in establishing K eff -dc/dn relation from constantamplitude 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. ASIP # 26

122 THE END Any questions? ASIP # 27