The formulation goes on to use a “universal” empirical relation between the fatigue ratio, Vr, and the normalized creep rupture strength, r, r= ut T The empirical relationship is of the
Trang 1where A is the fatigue strength at an arbitrary mean stress and depends not only on temperature but on time in order to account for the time-dependent behavior This formula
is based on the assumption that the fatigue ratio is a function of the creep rupture strength only The formulation goes on to use a “universal” empirical relation between the fatigue ratio, Vr, and the normalized creep rupture strength, r,
r= ut T
The empirical relationship is of the form
By applying this relation to data at two values of stress ratio, R= −1 (fully reversed loading at zero mean stress) and R= 0 (pulsating tension), the coefficients Ar and Rcan
be obtained From these, the fatigue strength of a smooth bar at R= −1 and R = 0 can be determined at any temperature in the creep regime At that temperature, the Haigh diagram
is plotted as shown in Figure 5.32 where the two quantities representing the fatigue strengths at two values of R are connected by a straight line Using the recommendation
of Forrest [20], the diagram is cut by a vertical line through the creep rupture strength, indicating a weak interaction between fatigue and creep damage
To apply this diagram to a notched component, consideration has to be given to the fact that stresses at the notch root = ktSm, where the geometry can be defined by an elastic stress concentration factor, kt, will relax with time toward the nominal (fictitious) mean stress level, Sm The fatigue loading is then considered to be allowable by comparing the nominal mean stress level, Sm, and the alternating stress at the notch with the FLS based on smooth bar tests as illustrated in Figure 5.33 The lower curve is that shown in Figure 5.32 when the value of ktis not known According to Forrest [20], it is reasonable
to assume that the mean stress component will not be affected by a notch and that the alternating stress component will be reduced by the fatigue notch factor, kf If ktis known,
kt can be substituted for kf for a conservative estimate This produces the upper curve in Figure 5.33 for a known kt
s–1
sy (t,T) Mean stress
s0
Figure 5.32. Haigh diagram of a smooth component in the creep regime (after [37]).
Trang 2sy (t, T )
s–1
Fictitious mean stress
ktsy (t, T )
Figure 5.33. Haigh diagram of a notched component in the creep regime (after [37]).
As pointed out by Forrest [20], the fatigue strength at elevated temperature can be represented on a Haigh diagram by a series of contours that represent failure at a given time Each point on a contour represents a combination of mean and alternating strength corresponding to that particular time For low mean stress, the alternating stress is dom-inated by the fatigue behavior (number of cycles), whereas for large mean stresses, the behavior is primarily creep The behavior thus goes from the alternating fatigue strength
R= −1 to the creep rupture strength A plot such as Figure 5.34 is for a single tem-perature so that a complete description of a material would have to include a sufficient number of Haigh diagrams, or analytical expressions, to represent the material behavior at different temperatures One method for representing the Haigh diagram for creep–fatigue behavior is to represent the mean and alternating stresses in normalized form as shown
in Figure 5.35 [20] The alternating stress is normalized with respect to the alternating fatigue strength corresponding to an appropriate (large) number of cycles as is done in
a conventional Haigh diagram The mean stress is normalized with respect to the creep rupture strength based on an acceptable (large) time to rupture for a given application Experimental data show that, for many materials, both the Goodman straight line and the Gerber parabola are overly conservative under combined creep and fatigue loading
A circular arc is suggested as being more representative of real material behavior [20]
Mean stress
t1
t2
t3
t3 > t2 > t1
Figure 5.34. Creep rupture curves at elevated temperature.
Trang 3Mean stress/Creep rupture strength
Modified Goodman law
Circular arc
Gerber parabola
Figure 5.35. Normalized creep curves (after [20]).
Finally, an alternative empirical relation that involves a straight line Goodman equation
on a Haigh diagram, but using the tensile strength as discussed earlier, can be used
to represent the combined creep and fatigue behavior on a Haigh diagram at elevated temperatures For each temperature, a different line represents the data as illustrated in Figure 5.36 which appears in [35] The restriction in this method of representing data is that the static stress does not exceed the creep rupture strength at any temperature Forrest claims that the straight line curves, truncated by the creep rupture stress, represent data as well as the circular arc method at high temperatures and is “a better guide to fluctuating fatigue strengths at moderate temperatures That this criterion fits the experimental data reasonably closely is an indication that in general there is little interaction between the creep and fatigue processes” (see [20], p 248)
The discussion above illustrates some of the complexities in designing for fatigue under non-zero mean stress in the creep regime where the static (creep rupture) strength depends
on time (or frequency of loading) whereas the alternating stress capability under zero
Mean stress
Alternating stress RT
T1
T2
T3
T3 > T2 > T1
Figure 5.36. Creep–fatigue Haigh diagram (after [20]).
Trang 4mean stress depends on number of cycles For a fatigue limit, both a maximum number
of cycles as well as a maximum exposure time have to be established for a particular design, whether or not the application is for a smooth component or one containing a notch or other stress raiser
REFERENCES
1 Schütz, W., “A History of Fatigue”, Engineering Fracture Mechanics, 54, 1996, pp 263–300.
2 Weixing, Y., Kaiquan, X., and Gu, Y., “On the Fatigue Notch Factor, Kf”, Int J Fatigue, 4,
1995, pp 245–251
3 Neuber, H., Theory of Notch Stresses: Principle for Exact Stress Calculations, Edwards, Ann
Arbor, Mich., 1946
4 Peterson, R E., “Notch Sensitivity”, Metal Fatigue, G Sines and J.L Waisman, eds,
McGraw-Hill, New York, 1959, pp 293–306
5 Heywood, “Stress Concentration Factors, Relating Theoretical and Practical Factors in Fatigue
Loading”, Engineering, 179, 1955, pp 146–148.
6 Dowling, N.E., in Mechanical Behavior of Materials, 2nd Edition, Prentice Hall, Upper Saddle
River, New Jersey, 1999
7 Haritos, G.K., Nicholas, T., and Lanning, D.B., “Notch Size Effects in HCF Behavior of
Ti-6Al-4V”, Int J Fatigue, 21, 1999, pp 643–652.
8 Smith, R.A., and Miller, K.J., “Fatigue Cracks at Notches”, Int J Mech Sci., 19, 1977,
pp 11–22
9 Peterson, R.E., “Stress Concentration Factors”, John Wiley & Sons, Inc., New York, 1974,
p 22
10 Smith, R.A., and Miller, K.J., “Prediction of Fatigue Regimes in Notched Components”, Int.
J Mech Sci., 20, 1978, pp 201–206.
11 Kitagawa, H and Takahashi, S., “Applicability of Fracture Mechanics to very Small Cracks
or the Cracks in the Early Stage”, Proc of Second International Conference on Mechanical Behaviour of Materials, Boston, MA, 1976, pp 627–631.
12 El Haddad, M.H., Dowling, N.F., Topper, T.H., and Smith, K.N., “J Integral Applications for
Short Fatigue Cracks at Notches”, Int J Fract., 16, 1980, pp 15–24.
13 Taylor, D., “Geometrical Effects in Fatigue: A Unifying Theoretical Model”, Int J Fatigue,
21, 1999, pp 413–420
14 Taylor, D., “A Mechanistic Approach to Critical-Distance Methods in Notch Fatigue”, Fatigue
Fract Engng Mater Struct., 24, 2001, pp 215–224.
15 Miller, K.J., “The Two Thresholds of Fatigue Behaviour”, Fatigue Fract Engng Mater Struct.,
16, 1993, pp 931–939
16 Nisitani, H and Endo, M., “Unified Treatment of Deep and Shallow Notches in Rotating
Bending Fatigue”, Basic Questions in Fatigue: Volume I, ASTM STP 924, J.T Fong and
R.J Fields, eds, American Society for Testing and Materials, Philadelphia, 1988, pp 136–153
17 Isibasi, T., Prevention of Fatigue and Fracture of Metals, Yokendo, Tokyo, 1954.
18 Bell, W.J and Benham, P.P., “The Effect of Mean Stress on Fatigue Strength of Plain and Notched Stainless Steel Sheet in the Range from 10 to 107 Cycles”, Symposium on Fatigue Tests of Aircraft Structures: Low-Cycle, Full-Scale, and Helicopters, ASTM STP 338, American
Society for Testing and Materials, Philadelphia, 1962, pp 25–46
Trang 519 Gunn, K., “Effect of Yielding on the Fatigue Properties of Test Pieces Containing Stress
Concentrations”, Aeronautical Quarterly, 6, 1955, pp 277–294.
20 Forrest, P.G., Fatigue of Metals, Pergamon Press, Oxford, 1962 (U.S.A Edition distributed by
Addison-Wesley Publishing Co., Reading, MA)
21 Lanning, D.B., Nicholas, T., and Haritos, G.K., “On the Use of Critical Distance Theories for
the Prediction of the High Cycle Fatigue Limit in Notched Ti-6Al-4V”, Int J Fatigue, 27,
2005, pp 45–57
22 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)
23 Wagner, L., “Effect of Mechanical Surface Treatments on Fatigue Performance of Titanium
Alloys”, Fatigue Behavior of Titanium Alloys, R.R Boyer, D Eylon, and G Lutjering, eds,
The Minerals, Metals & Materials Society, 1999, pp 253–265
24 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
25 Lukas, P., Kunz, L., Weiss, B., and Stickler, R., “Non-Damaging Notches in Fatigue”, Fatigue
Fract Engng Mater Struct., 9, 1986, pp 195–204.
26 Newman, J.C., “An Improved Method of Collocation for the Stress Analysis of Cracked Plates
with Various Shaped Boundaries”, NASA Technical Note D-6376, 1971.
27 Sheppard, S.D., “Field Effects in Fatigue Crack Initiation: Long Life Fatigue Strength”, Jour.
Of Mech Design, Trans ASME, 113, 1991, pp 188–194.
28 Taylor, D and O’Donnell, M., “Notch Geometry Effects in Fatigue: A Conservative Design
Approach”, Engineering Failure Analysis, 1, 1994, pp 275–287.
29 Naik, R.A., Lanning, D.B., Nicholas, T., and Kallmeyer, A.R., “A Critical Plane Gradient
Approach for the Prediction of Notched HCF Life”, Int J Fatigue, 27, 2005, pp 481–492.
30 Atzori, B and Lazzarin, P., “Notch Sensitivity and Defect Sensitivity under Fatigue Loading:
Two Sides of the Same Medal”, Int J Fract., 107, 2000, pp L3–L8.
31 Tanaka, K., “Engineering Formulae for Fatigue Strength Reduction due to Crack-Like Notches”,
Int J Fract., 22, 1983, pp R39–R46.
32 Glinka, G., “Energy Density Approach to Calculation of Inelastic Stress-Strain Near Notches
and Cracks”, Engng Fract Mech., 22, 1985, pp 485–508.
33 Atzori, B., Lazzarin, P., and Meneghetti, G., “Fracture Mechanics and Notch Sensitivity”,
Fatigue Fract Engng Mater Struct., 26, 2003, pp 257–267.
34 Ciavarella, M., and Meneghetti, G., “On Fatigue Limit in the Presence of Notches: Classical
vs Recent Unified Formulations”, Int J Fatigue, 26, 2004, pp 289–298.
35 Lukas, P and Klesnil, M., “Fatigue Limit of Notched Bodies”, Mater Sci Eng., 34, 1978,
pp 61–69
36 Murakami, Y., Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions, Elsevier
Science, Ltd, Kidlington, Oxford, 2002
37 Harkegard, G., “High-Cycle Fatigue Design of Steam-Turbine Blades at Elevated
Tempera-ture”, Swedish Symposium on Classical Fatigue, N.-G Ohlson and H Norberg, eds, Uddeholm
Research Foundation, Sunne, Sweden, 1985
Trang 6Fretting Fatigue
Fretting fatigue is a type of damage occurring in regions of contact where small relative tangential motion occurs between the two bodies in contact that are under compressive load Fretting fatigue typically involves a contact region where both complete and partial slip occur as discussed below It is usually associated with loading conditions where one of the components is subjected to bulk loading, which can cause cracks formed locally near edges of contact to propagate Such conditions can lead to premature crack initiation and failure Fretting-fatigue damage in blade/disk interfaces has been indicated
as the cause of many unanticipated disk and blade failures in gas turbine engines As the magnitude of relative motion in the contact region increases, the nature of the damage changes to what is commonly referred to as “galling” or “wear.” The emphasis in this chapter will be on fretting fatigue that involves very small relative motion
Under laboratory conditions, the synergistic effects of the many parameters involved
in fretting fatigue make determination and modeling of mechanical behavior extremely difficult In particular, the stress state in the contact region involves very high peak stresses, extremely steep stress gradients, multiaxial stress states, and differing mean stresses Further, there is controversy over whether the problem is primarily one of crack initiation or one involving a crack propagation threshold, and whether or not stress states rather than surface conditions play a major role in the observed behavior All of these issues will be addressed in this chapter
Fretting fatigue is generally associated with HCF, whereas wear and galling are more concerned with LCF This is a broad generalization, but for the purposes of discussing HCF it is adequate to make such a distinction In particular, fretting fatigue under HCF conditions is normally associated with contact conditions where a part of the contact region
is in total contact (stick) while the edges of contact undergo small relative displacements
in the tangential direction (slip) Such relative displacements are typically of a magnitude
of tens of microns or less The slip region may be very small, only at the very edge of contact, or may cover a considerable percent of the contact area Additionally, due to the back and forth sliding that takes place and the general nonlinearity of contact problems, the boundary between stick and slip may move back and forth on every cycle and never reach a single constant location A schematic of fretting-fatigue contact between a pad (top) and a specimen (bottom) is shown in Figure 6.1 There, the specimen represents half
of a real specimen thickness (by symmetry) and is loaded by a bulk load T The applied
261
Trang 7Slip Slip
Fretting region
T
P Q
2a
Figure 6.1. Schematic of fretting region showing applied forces.
normal and tangential loads are denoted by P and Q, respectively Under typical stick– slip conditions, part of the contact region undergoes no relative displacements between pad and specimen (stick) while the edges undergo slip The maximum slip occurs at the deformed edge of contact, at the positions that are at a distance 2a apart in the figure Beyond those locations, there is no contact Under large shear forces, specifically when the ratio Q/P reaches the value of the average coefficient of friction (COF), the entire region may undergo slip
From both experimental observations and theoretical analysis it has been found that contact conditions in fretting change with increasing displacement amplitude [1] Three regions are normally defined under constant normal force and oscillating tangential force These regions are referred to as “stick,” “mixed stick–slip,” and “total slip.” Each cor-responds to a range of displacement amplitudes as depicted schematically in Figure 6.2 taken from [1] The damage in these three regions can be characterized as low damage fretting, fretting fatigue, and fretting wear, respectively As noted in the schematic, the lowest fatigue lives are associated with the fretting-fatigue region, the subject of this section It is noteworthy that the lowest lives are not associated with the lowest or largest slip amplitudes, but with intermediate values These values can cover a range of from about 5 to 50 m, but will depend in general not only on the materials but also on the contact stresses In addition to these three regions, a limiting region of large amplitudes
in which wear mechanisms and wear rates become identical to those in unidirectional sliding is defined as the reciprocating sliding regime [1] This occurs at the right side of the schematic of Figure 6.2 where both amplitude and fatigue life would be represented
on a logarithmic scale and amplitudes in this region could approach 1 mm It can be noted that the same experimental conditions are not necessarily used when obtaining data
on effects of slip amplitude and even the local conditions are somewhat different Very small values of slip are normally obtained under stick–slip conditions where computations
or indirect measurements are used to deduce the slip values These slip values are not constant but will vary from zero at the stick–slip boundary to a maximum at the edge
of contact Further, the boundary may move during a complete load cycle Large values
of slip, on the other hand, occur under full slip conditions and may only be obtained for
Trang 8Mixed stick and slip
Gross slip
Amplitude
Fatigue life Wear Reciprocating sliding
Figure 6.2. Schematic of fretting regimes showing relative wear and fatigue lives as a function of relative
displacement amplitude (after [1]).
specific experimental configurations and loading conditions including various aspects of how load or displacements are controlled
The author’s first real exposure to fretting fatigue was in the early part of a major HCF program where baseline long life fatigue tests were being conducted in the laboratory Under load control, using tapered cylindrical dog bone type specimens, a number of failures occurred in the grip region as depicted in a two-dimensional schematic of the test
in Figure 6.3 The failures, as shown, were soon identified as fretting fatigue, and occurred
at average stress levels in the grip area that were considerably lower than the nominal fatigue strength of the titanium alloy being tested Only by reducing the nominal stress by thinning the gage section of the specimen were these failures eventually eliminated This experience was certainly not unique since fretting in the grips is a common occurrence
in testing laboratories This experience, however, and the related observations eventually led to the design of a fretting-fatigue apparatus used by Hutson and co-workers (see [2] for example) described later in Section 13 (Figure 6.47) This apparatus made use of the propensity for fretting-fatigue failures in a contact region under cyclic loading when gross sliding is not present
Trang 9P
N
A
Figure 6.3. Schematic of uniaxial fatigue test showing fretting fatigue at grip.
One of the more common applications where fretting fatigue is a design issue and where numerous failures have occurred is the dovetail joint in an engine where the blade
is inserted into a slot as shown schematically in Figure 6.4 Here, the contact interface is normally composed of two flat surfaces in contact with blending radii in both the blade and the disk The general problem is three-dimensional in nature with loading taking place in the directions shown in the two-dimensional schematic of Figure 6.4 as well as out of the plane of the figure The loading can be a combination of LCF due to start up and shut down of the engine, producing primarily an axial load in the blade as shown, and HCF due to vibratory motion of the blade, producing lateral cyclic loading as well
as axial or bending (not depicted here) contributions The contact region, where the flat surfaces mate, sees both normal and tangential loads as well as bulk loads in both the blade and the disk At a contact interface, the normal and shear stresses at any point
Blade
Disk
region
blade disk
Crack in A Crack in
B Contact region Contact
Figure 6.4. Schematic of blade/disk contact region.
Trang 10are equal and opposite, but the bulk stresses can be different in the two bodies at the interface For this region, fretting-fatigue failures can occur in either the blade or the disk
as depicted schematically in the blowup of the contact region in Figure 6.4 For the blade, the highest bulk load parallel to the interface occurs at the upper edge of contact, shown
as point A in the figure In the disk, however, the highest bulk load occurs at the lower edge of contact, point B in the figure The two potential failures occur at two different locations so that generally either one or the other occurs first in reality But in design
or analysis, both locations have to be examined as potential sites for fretting-fatigue failure
In addition to the location of crack initiation and the stresses needed to cause initiation, the subsequent propagation of the crack also has to be considered In the dovetail region
in an engine, cracks initiated in the blade nearly always propagate due to the severity
of the bulk load from centrifugal forces However, cracks initiated in the disk frequently arrest when they are still very short [3]
Another application where fretting-fatigue failures are known to occur is at fastener joints where a rivet or bolt is used to join two members that are subjected to oscillatory loading Riveted joints in airframe construction are one of the most common places where fretting fatigue can occur A schematic of a riveted lap joint is shown in Figure 6.5 where
a symmetrically loaded joint is depicted While the joint is designed to function with essentially no relative motion anywhere, small relative displacements can occur in the region between the rivet and the plate as indicated by the solid lines in the figure This very small relative motion, produced by fatigue loading of the joint, is due primarily to the elastic strains in the adjacent materials at the contact interface The high stresses that develop in this contact region can lead to fretting-fatigue crack nucleation and subsequent propagation due to the bulk loads in the plate as depicted in the figure
Another common example of a structural configuration where fretting fatigue can occur
is where a hub is press-fitted onto a shaft A characteristic of this and other applications where fretting fatigue occurs is the small amplitude of the relative motion between
P
P
2P
Figure 6.5. Schematic of rivet joint showing fretting fatigue region.