High Cycle Fatigue: A Mechanics of Materials Perspective part 56 pps

Moreover, since the stress gradients at sharp notches and edge-of-contact zones are steep and die out at relatively short distances from the notch surface, it is necessary to consider th

comparisons provide some useful information concerning the capabilities and accuracy

of the F s approach In all cases, the predicted threshold stress levels exceeded the experimental values However, the differences between the actual and modeled notch dimensions may have contributed to the discrepancies to some degree For example,

in notch 1, the actual notch depth was approximately 10% greater than the modeled value This would increase the effective stress concentration factor, thereby reducing the nominal threshold stress level The differences in notch dimensions were less severe in the other cases Nevertheless, it is apparent the F s method tends to underestimate the notch strength reduction in components containing very small, sharp notches with severe stress gradients However, the accuracy of the method improves in cases of more blunt notches [1]

FRACTURE MECHANICS (WORST CASE NOTCH) APPROACH

The fracture mechanics approach employed to predict the nominal HCF threshold stress,

th, assumes that microcracks can initiate relatively early in the life of a notched component due to a variety of reasons, such as the damage imparted by an initial FOD impact, the highly concentrated HCF stresses associated with either an FOD notch or edge-of-contact zone, or the intermittent occurrence of low cycle fatigue (LCF) Moreover, since the stress gradients at sharp notches and edge-of-contact zones are steep and die out

at relatively short distances from the notch surface, it is necessary to consider the unique behavior of small fatigue cracks when performing a fracture mechanics analysis The method used, termed the “Worst Case Notch” (WCN) model, enables the bound-aries between crack initiation, crack growth followed by arrest, and crack growth to failure to be delineated In this model, “true” crack initiation is assumed to actually occur

as predicted by the classical S–N approach when the applied stress range is equal to the endurance limit divided by the elastic stress concentration factor e/kt However, for sharper notches, the initiated cracks can subsequently arrest due to the unique behavior

of small cracks, which cause th to initially increase, achieve a maximum, and finally decrease as crack size increases The growth/arrest boundary is determined by equating the applied stress intensity factor (SIF) range to the threshold SIF for fatigue crack propa-gation The unique behavior of small fatigue cracks is modeled in the WCN approach by

a crack-size-dependent threshold SIF The remainder of this section summarizes several enhancements to the WCN model, and compares the model predictions with several small, sharp notch experimental results [1]

Incorporation of notch plasticity and surface cracks in the WCN model

The WCN model was originally developed for through-thickness cracks and surface cracks with one degree of freedom (DOF) emanating from notches [4] Consequently, in analyses

involving thumbnail cracks, the cracks were restricted in shape so that they remained semi-circular, and crack growth was characterized only by the SIF at the deepest point on the flaw These restrictions resulted in certain predicted crack growth behaviors that were not entirely consistent with observations To overcome these limitations, the WCN model has been extended to semi-elliptical surface cracks whose growth is characterized by two DOF, namely, the SIFs at both the surface and deepest points on the crack front In these cases, crack growth and arrest is determined by the manner in which the crack shapes evolve from the initially specified shapes, and this evolution is governed by the stress gradient ahead of a notch and the crack growth rate equation Therefore, implementation

of the WCN model for two DOF cracks requires HCF crack growth calculations to be performed To facilitate these calculations the crack growth rate behavior was described using the Walker crack growth rate equation based on an effective stress intensity factor,

Keff This general equation describes crack growth rate behavior from initiation at threshold to failure from the onset of static failure modes

To incorporate notch plasticity and associated shakedown of the mean stresses at the notch, an approximate elastic-plastic stress analysis was used to determine the local notch-tip stress state The methodology used here was based on the isotropic shakedown module for univariant stressing developed for the computer code, DARWIN™[5] The shakedown module requires as input the elastic stress field ahead of a notch, which was obtained using a modification of the method of Amstutz and Seeger [6], as described in [7] Figure E.2 compares predictions of the WCN model for a 2-D through-thickness crack both with and without plastic shakedown for a notch of 0025" (0.635 mm) depth and

Elastic stress concentration factor, kt

–1 0 1 2

3

WCN prediction (with shakedown) WCN prediction (w/o shakedown)

Δσe/kt (with shakedown)

Δσe/kt (w/o shakedown)

local stress ratio, R, at initiation of

crack growth to failure (with shakedown)

WCN predictions for 2-D initial crack Notch depth = 25 mils

Variable notch radius

Crack initiation envelopes

Crack growth and arrest Crack growth to failure

Stress ratio, R

No crack initiation

No crack initiation

Failure envelopes

Figure E.2. Comparison of WCN model predictions with and without plastic shakedown of notch-tip stresses.

The local R-value is also shown to decrease with increasing notch severity as the result of plastic shakedown.

varying notch radii to give the range of kt values (based on the gross cross-sectional area) shown In this figure, the threshold stress is normalized by the limiting threshold value obtained for a notch with a high kt As indicated, the threshold values determined with and without shakedown differ significantly at the initiation of cracking, but the two sets of values differ only slightly at the threshold stress corresponding to failure The reason for this can be seen from the plot of the local stress ratio R that is also presented

in Figure E.2 At the higher cyclic stresses needed to initiate and propagate cracks to failure, shakedown has occurred at the notch tip and changed the local stress ratio at initiation so that it is no longer simply related to the remote stress ratio, specified as being

R= 05 in this example Indeed, at high ktvalues, the local stress ratio becomes negative Although these negative stress ratios will increase the cyclic threshold stress needed to initiate cracking, they clearly do not significantly influence the threshold needed to cause propagation to failure This is a consequence of the fact that the residual stress field due to shakedown is very localized at the notch tip, and cracks can readily propagate to depths where the localized stress has little influence on the applied SIF Threshold stresses that fall between the initiation and failure envelopes shown in Figure E.2 will initiate cracks either on the initiation envelope or at cyclic stress values above this envelope if shakedown occurs, but these cracks are predicted to eventually arrest

Figure E.3 compares WCN model predictions for 2-D and 3-D surface cracks Although the trends in threshold behaviors for the two cases are similar, the 3-D crack model for notches with high ktvalues predicts that failure will occur at threshold stresses less than

0

1

2

3

WCN prediction (2-D crack) WCN prediction (3-D)

Δσe/kt

WCN predictions

Variable notch radius

Crack growth to failure

No crack initiation

Crack growth and arrest Crack initiationenvelope

Failure envelopes

Figure E.3. Comparison of WCN model predictions for 2-D versus 3-D cracks All results include the effects

of plastic shakedown of notch mean stresses.

those predicted for the 2-D crack It should be noted that in the fatigue crack growth calculations the 3-D crack was allowed to grow at both the deepest and surface locations

of the crack front, so that situations could occur during growth where one location was propagating when the other crack tip location was not, and vice versa This is the reason why the crack with the additional DOF associated with the 3-D crack gives lower threshold stresses

Comparison of WCN predictions with experimental results

Threshold stress estimates from the 3-D WCN model are compared in Figure E.4 to experimental results from the small, sharp notch tests described in the previous section

In discussing these results, it is useful to separately consider blunt kt< 65 and sharp

kt> 65 notches For the case of blunt notches, the data tend to lie above the curve defined by the assumed initiation curve e/kt, regardless of notch depth However, for the sharp notches, the data tend to be layered with respect to notch depth as predicted

by the WCN model This is most clearly illustrated for the case of the notch depth of

24 mils where the data point is considerably above the initiation curve but in excellent agreement with the WCN prediction for b= 24 mils A similar trend can be seen for the case of b= 48 mils; however, in this case the distinction between the predicted initiation and failure curves is less distinct because the two curves are relatively close together Ideally, one would like to generate additional small sharp notch data at higher ktvalues for all notch depths to verify the predicted limiting values for the various notch depths

0 5 10

15

20

25

Δσe/kt

WCN, b = 24 mils, variable radius WCN, b = 48 mils, variable radius

FS, b = 8 mils

FS, b = 24 mils

FS, b = 48 mils measured, b = 8 mils measured, b = 24 mils measured, b = 48 mils

Figure E.4. Comparison of WCN and F s model predictions with small, sharp notch data for varying notch

depths.

However, generating such data is limited by the sharpest notches that can be machined.

In this set of experiments, the sharpest notch radii = 2 mils could only be produced

by EDM, since the final chem milling used for all other notches significantly reduced the notch sharpness to about = 4 mils Although the use of EDM without subsequent chem milling raised the possibility of producing notches that would initiate cracks prematurely, this does not appear to have been the case since the threshold stresses for small sharp EDM notches were significantly above the initiation curve, as well as above data on blunter notches produced by EDM plus chem milling Moreover, this trend is consistent with the WCN theory that predicts a limiting threshold stress that depends on notch depth and corresponds to the growth and arrest of microcracks in the steep stress gradient ahead

of the sharp notches

The fact that blunt notches kt< 65 also resulted in measured threshold stresses that were above the initiation curve, corresponding to e/kt, could be due to a number of factors First, this simple initiation criterion may be inaccurate for the biaxial stresses that exist at the notch surface Thus, it may be necessary to include a more accurate multiaxial crack initiation criterion such as that employed in the F s approach described

in the previous section To assess this hypothesis, the F s predictions from Table E.2 are included in Figure 4 for comparison These predictions, which use the multiaxial criterion, shift the predictions in the correct direction; i.e., they predict higher threshold stresses than those given by e/kt However, the predicted results are now greater than the measured results, as previously discussed This is an undesirable prediction for use

in HCF design since the predictions are non-conservative with respect to the measured results It may be that the multiaxial initiation criterion is accurate for blunt notches, but the surface area correction term in F s is overcompensating for notch size effects

An alternative explanation for the deviation between the measured blunt notch data in Figure E.4 and the simple initiation prediction given by e/kt is that crack growth is contributing to the specimen’s life, which would effectively increase the threshold stress

in the step test It may be possible to differentiate between the above two interpretations

by carefully monitoring experiments for the presence of crack initiation, growth, or arrest This could be achieved either by interrupting multiple specimens and performing metallographic sectioning or by obtaining crack size measurements on a single specimen using either crack replication or direct observation in a cyclic loading stage of a scanning electron microscope

REFERENCES

1 Gallagher, J et al., “Advanced High Cycle Fatigue (HLF) Life Assurance Methodologies”, Report # AFRL-ML-WP-TR-2005-4102, Air Force Research Laboratory, Wright-Patterson AFB,

OH, July 2004

2 Murthy, H., Rajeev, P.T., Farris, T.N., and Slavik, D.C., “Fretting Fatigue of Ti-6A1-4V

Sub-jected to Blade/Disk Contact Loading”, Developments in Fracture Mechanics for the New

Century, 50th Anniversary of Japan Society of Materials Science, Osaka, Japan, May 2001,

pp 41–48

3 Doner, M., Bain, K.R., and Adams, J.H., “Evaluation of Methods for the Treatment of Mean

Stress Effects on Low-Cycle Fatigue”, Journal of Engineering for Power, 1981, pp 1–9.

4 Hudak, S.J., Jr., Chan, K.S., Chell, G.G., Lee, Y.-D., and McClung, R.C., “A Damage Tolerance Approach for Predicting the Threshold Stresses for High Cycle fatigue in the Presence of

Supplemental Damage”, Fatigue – David L Davidson Symposium, K.S Chan, P.K Liaw, R.S.

Bellows, T.C Zogas, and W.O Soboyejo, eds, TMS (The Minerals, Metals & Materials Society), Warrendale, PA, 2002, pp 107–120

5 Southwest Research Institute, DARWIN User’s Guide, Version 3.5, Appendix : Shakedown

Residual Stress Methodology and Validation of SHAKEDOWN Module, 2002.

6 Amstutz and Seeger, T., Accurate and Approximate Elastic Stress Distribution in the Vicinity of

Notches in Plates Under Tension Unpublished results (referenced in G Savaidis, M Dankert,

and T Seeger, “An Analytical Procedure for Predicting Opening Loads of Cracks at Notches”,

Fatigue Fract Engng Mater Struct., 18, 1995, pp 425–442).

7 Gallagher, J.P et al “Improved High Cycle Fatigue Life Prediction”, Report # AFRL-ML-WP-TR-2001-4159, University of Dayton Research Institute, Dayton, OH, January 2001 (on CD ROM)

Appendix F

Analytical Modeling of Contact Stresses

Bence B Bartha, Narayan K Sundaram and Thomas N Farris

INTRODUCTION

Closed-form analytical solutions exist for a body having an arbitrary contact geometry contacting either an infinite half space or a finite thickness plate The equations for the stress and displacement fields in the normal and tangential directions have been formulated and put into computer codes that provide numerical methods of solution of the contact problems This appendix provides the equations and the solution methods that have been incorporated into two computer codes developed at Purdue University The first code CAPRI (Contact Analysis for Profiles of Random Indenters) provides a solution method for contact of a finite body of arbitrary contact profile on a half-space when the two materials are identical The second code CARTEL (Contact Analysis of Relatively Thin Elastic Layers) provides the contact solution for an indenter on a finite thickness plate The latter problem is one that is encountered when a thin specimen is tested between two pads (indenters) in a fretting fatigue experiment Both codes are available through Purdue University, School of Aeronautics and Astronautics, West Lafayette, IN

CONTACT STRESSES IN A HALF-SPACE, THE CAPRI CODE

The Cauchy Singular Integral Equations (henceforth referred to as SIE) governing the contacts of two similar, isotropic elastic bodies are derived from Flamant’s solution for a point load on a half-space [1] The contact geometry before deformation is illustrated in Figure F.1 If¯vAx and¯vBx are the normal displacements of the two bodies, it can be shown that within the contact

hx− ¯vAx+ ¯vBx− C0− C1x= 0 (F.1) hx is the gap function (profile) in the local (symmetric) coordinate system, C1 is the term associated with rigid-body rotation, and C0 with rigid-body translation Similar to the gap in the y direction, hx, the gap or relative initial displacements in the x direction are defined as gx

∗This document was prepared by students under the supervision of Prof Thomas Farris at Purdue University,

West Lafayette, IN as part of a major fretting fatigue research program headed by Prof Farris Dr B Bartha

is currently at the Air Force Research Laboratory at Wright-Patterson AFB, OH.

542

x,u

A

B

h(x)

Figure F.1. Schematic of contact region showing coordinate system and gap function.

The following equations [2] govern the pressure traction px and the shear traction qx

dhx

dx − C1= d

dx¯vA− ¯vB=41− v2

E

 +a

−a

ps

d

dx¯uA− ¯uB= −dgx

dx = −41− v2

E

 +a

−a

qs

x− sds+

01− v2

where ¯uA and ¯uB are the tangential displacements of the bodies and 0 is a remotely applied bulk-stress Note that the unknown tractions occur inside the integral in both

equations and “a,” the contact half-width, is unknown a priori In this two-dimensional

formulation, plane strain conditions are assumed

It is important to note that the maximum value of shear traction is limited by the coefficient of friction,  Further, the problems of most interest are the so-called partial slip problems (mixed boundary-value problems), in which the contact is divided into outer “slip” regions, whereqx = px and a central “stick” region, in which

gx

where gx is the value of the derivative when the particles first enter the stick zone [1] The partial derivative is present because of the quasi-static formulation of the problem In the most general case, this is load-history dependent and has to be obtained incrementally

Again, the size of this stick zone, denoted by c, is not known a priori.

The Equations (F.2) and (F.3) are subject to the following equilibrium boundary conditions:

 +a

 +a

 +a

where P, M, and Q are the applied normal force, moment, and shear, respectively

It is also known that for incomplete contacts, the pressure traction must vanish at the extremities, that is

In the above SIEs, the pressure equation can be solved independently of the shear equation and we proceed to do this first

Apart from the unknowns already mentioned in the problem, there are two more: the contact eccentricity “e” and the stick-zone eccentricity “ec,” which do not explicitly occur

in the above equations The contact eccentricity “e” is defined as the separation between the local (x) and global (X) coordinate systems

Thus, in the global coordinate system, the contact is from −a + e to a + e and the stick zone is from −c + ec to c+ ec

Pressure solution

For arbitrary smooth profiles hx, it is not generally possible to invert the integral Equation (F.2) to obtain a closed-form solution for px (A limited number of profiles have analytical solutions [3, 4, 5].) However, it is possible to use a trigonometric variable transformation [6] to obtain an analytical solution Murthy et al [2] develop this as follows Using the substitutions x

dh

=dh dx

dx

= −adh

the pressure SIE becomes

+ C1= −41− v2

E

  0

p sin d

Next, the pressure traction function is expressed as an infinite cosine series

p=

n = 0

pncos n

Further, from tables of integrals, the following definite integral is known

  cos nd

Substituting Equations (F.12) and (F.13) into Equation (F.11),

dh + C1

41− v2 E





n = 1

pn

(F.14)

Note that the summation starts at n= 1, not n = 0

comparison can be made between the left- and right-hand sides of the equation This is achieved by expressing the slope as a Fourier sine series

dh

=

n = 1

Finally,

pn= Ehn

p1=Ehn+ C1a

where p1is still not determined, because C1is also unknown However, using the force equilibrium condition, Equation (F.5), we find

p0= P

Similarly, from the moment equilibrium condition, Equation (F.6), we find

p1= 2M

Also, p0= 0 implies

lim →0

p0+ p1cos + p2cos 2+ · · ·

For this limit to exist, the numerator has to vanish, hence

p0= − k= 0

Similarly, using p= 0

p1= −

k = 1

West Lafayette, IN as part of a major fretting fatigue research program headed by Prof Farris Dr B Bartha

is currently at the Air Force Research Laboratory at Wright-Patterson... class="text_page_counter">Trang 7

Appendix F

Analytical Modeling of Contact Stresses

Bence B Bartha, Narayan K Sundaram and... for Profiles of Random Indenters) provides a solution method for contact of a finite body of arbitrary contact profile on a half-space when the two materials are identical The second code CARTEL