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Modified crack-analogy methodology: the equivalent theory
International Journal of Fatigue 27 (2005) 439–452 www.elsevier.com/locate/ijfatigue
Modified crack-analogy methodology: the equivalent theory S. Naboulsi* Department of Aeronautics and Astronautics, Air Force Institute of Technology (AFIT/ENY), 2950 Hobson Way, Wright-Patterson Air Force Base, OH 45433-7765, USA Received 24 April 2003; received in revised form 30 January 2004; accepted 27 July 2004
Abstract The crack initiation process of fretting fatigue is a complex phenomenon, and various approaches have been instigated to investigate fretting fatigue damage tolerance and crack initiation. The Crack-Analogy Methodology (CAM) is one approach, which is introduced by Giannakopoulos et al. It utilizes the similarity between contact mechanics and fracture mechanics to investigate fretting fatigue life. The present study is motivated by the crack-analogy methodology where a Modified Crack-Analogy Methodology (MCAM) is developed to extend CAM capabilities and to improve its prediction of crack initiation. The modified form extends CAM to include various indenter– substrate geometries as well as modifying its crack initiation parameter to include the effect of the bulk stress in the substrate. MCAM uses the change of the stress intensity factor (DK-parameter) as a fretting fatigue crack initiation parameter, since the change is consistent with the cyclic mechanism of fretting fatigue. It uses experimental data to establish DK-parameter–life curves similar to the stress–life S–N curve prototype in fatigue and to validate crack initiation for various geometric configurations under various load conditions. The results show similar trends to plain fatigue with lower damage tolerance as expected. By including the bulk stress, the crack initiation predictions of fretting fatigue show better consistency with the experimental data than without it where it exhibits fatigue characteristics consistent with S–N curves. In general, the DK-parameter–life trends show dependency on applied load and pad geometry. Reduced scatter is observed when bulk stress is included in the analyses as compared to that without it. It further demonstrates that the presence of bulk stress in general shifts fretting fatigue characteristics toward low cycle fatigue. The MCAM also predicts the limit of high cycle fatigue. The MCAM formulation shows potentials in life prediction such that it can be used as a tool in the design of components under fretting fatigue. q 2004 Published by Elsevier Ltd. Keywords: Fretting fatigue; Crack initiation parameter; Fracture mechanics; Contact mechanics
1. Introduction As the case for fatigue, fretting fatigue in general is a complex subject, where some of the problem modeling fretting fatigue may be characterized as follow [1]: † Calculations of life are generally less accurate and less dependable than strength calculations. Order of magnitude errors in life estimate are not unusual. † Fatigue characteristics of a material cannot be deduced from other mechanical properties. They must be measured directly using experiments. † Full-scale prototype testing is usually necessary to assure an acceptable model predictions. * Tel.: C1 937 255 6565. E-mail address: [email protected] 0142-1123/$ - see front matter q 2004 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2004.07.010
† Results of different but identical tests may differ widely, requiring, therefore, a statistical interpretation by designer. Furthermore, damage tolerance of pre-crack initiation in fretting fatigue is of complex nature, which is attributed to its dependency on pad geometries, surface properties, material properties, and mechanical loading conditions [2]. For example, smaller radius cylindrical pads under the same fretting load conditions cause lower damage tolerance (i.e. higher damage) where crack initiates faster than the one with larger radius, which exhibits higher damage tolerance (i.e. lower damage). In the present study, the term crack initiation is used to indicate the number of cycles of precrack initiation (i.e. number of cycles leading to the development of small cracks without including the number of cycles of post-crack growth or crack propagation of
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failure). Also, the term damage tolerance is used to quantify the resistance of the material to fretting fatigue damage due to either loading conditions or pad geometry (i.e. higher damage tolerance means more pre-crack initiation compared to lower damage tolerance). Further, the term crack initiation parameter is a norm, where its value is equivalent to an estimated damage tolerance and quantitatively predicts the number of cycles to crack initiation. The parameter is a design tool, which provides a quantitative mean of comparison of material/structure damage under various fretting fatigue scenarios. In practical applications, one needs fretting fatigue crack initiation parameter that reflects its complex process. Traditionally, the investigation of fretting fatigue crack initiation has been performed using contact mechanics [3–8]. However, the similarity of the stress state near the contact surface using Hertzian contact and asymptotic solution of fracture mechanic has been employed to interpret experimental observations of fretting fatigue crack initiation. For example, the studies [9,10] suggested the similarities between the singular fields for sharp-edged contacts [4–6] and for cracked bodies [11,12]. Nadai [13] analyzed the asymptotic stress solution within an elastic substrate in the vicinity of the corners of the punch for both normal and tangential loading. His result confirms the similarity between the contact surface stress singularity predicted by Sadowski [14], where the elastic singular fields at the sharp edges of two-dimensional contact between a rigid flat punch and planar surface were derived, and the contact surface stresses derived earlier by Hertz [15] for normal loading of a cylindrical punch. Recently, Giannakopoulos et al. [16,17] used the Crack-Analogy Methodology (CAM) to validate and interpret their experimental data. In their studies, a quantitative equivalence has been explored in detail between contact mechanics and fracture mechanics using the vast literature of both fields. This equivalence is established by identifying geometries of cracked bodies, which facilitate an analogy with an axisymmetric sharp-edge pad and rounded pad in contact with flat substrate. Their experimental data are obtained for a spherical pad in contact with substrate. They used a stress intensity factor as crack initiation prediction parameter. They assumed that fatigue crack initiates when the stress intensity factor is greater than the stress intensity threshold. Further, the crack-analogy’s results [17] for the weak adhesion case showed that the damage tolerance is insensitive to changes in the tangential load, such that prediction of fretting fatigue crack initiation using mode II stress intensity factor parameter is independent of load, which is contrary to experimental observation. For example, experimental data of cylindrical pads show dependency on radius size and tangential load [18]. One of the fundamental bases of the Modified CrackAnalogy Methodology (MCAM), which is motivated by earlier studies of CAM [16,17], is its utilization of the similarity between contact mechanics and fracture
mechanics. This allows the use of fracture mechanic metrics to characterize the contact based fretting fatigue phenomena such as stress intensity factor. MCAM includes the effect of various pad geometry and bulk stress on crack initiation, which is different from CAM. Also, contrary to the stress intensity factor used in CAM, MCAM uses the change of stress intensity factor as fretting fatigue parameter, since it is consistent with the cyclic mechanism of fatigue. It will be referred to as DK-parameter for the remaining of the paper. MCAM also establishes DK-parameter–life curve similar to S–N curve, which constitutes design information of fundamental importance to specimens subjected to repeated fatigue load.
2. Crack-analogy’s methodology The concept behind the development of MCAM and CAM [16,17] is the equivalence between the asymptotic field of two bodies in contact derived from classical contact mechanics theory and the asymptotic field at a stationary crack determined from fracture mechanics. This allows the use of fracture mechanic metrics to characterize the contact based fretting fatigue phenomena, and it involves the following. First, the identification of the cracked specimen configuration, which provides a geometric equivalence for the contact region of a pad normally pressing on planar substrate, is made, see Fig. 1. Second, its asymptotic solution to the stress and strain fields at the edge of contacts from classical contact is determined. Third, the corresponding stress and strain field solutions for the analogous cracked body from linear elastic fracture mechanics are also determined. In other words, the assumption that the cracked body is subjected to a normal compressive load, whose magnitude is P, and a tangential load, whose magnitude is Q, is the same as that of the load pressing and sliding the pad against the substrate (see Fig. 2). The scalar amplitude of the singular fields at the crack tip is the stress intensity factor. Fourth, equating the different components of the stress fields, one can solve for the stress intensity factor as a function of the load (i.e. an equivalent mode I stress intensity factor for separating or opening is used to characterize the effect of the normal load P, and an equivalent mode II stress intensity factor for slip or shear is used to characterize the effect of the tangential load, Q). Thus, the analogy between the contact mechanics and the fracture mechanics is established, the equivalency of the stress fields can be determined, and MCAM, for example, uses DK-parameter as crack initiation parameter of fretting fatigue. One of the bases of crack-analogy methodology is the singular field that exists in both the contact mechanics and fracture mechanics solutions. Theoretically, it has been demonstrated that under monotonic compressive loading of contact interface, adhesion induces a tensile square root singular stress fields at the edge of contact. Adhesion is used
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Fig. 1. Schematic of indenter–substrate configurations and their MCAM representation.
to indicate that the analogy is for a stationary crack. It is defined as that for which the two contacting surfaces are bonded and the works needed to debond the contacting surfaces are high. In the case of applied normal and tangential loads, adhesion produces a mode I tensile stress and mode II shear stress, see Fig. 2 (i.e. mode I stress intensity factor characterizes the fretting effect of the normal load P, and mode II stress intensity factor characterizes the fretting effect of the tangential load Q). In fretting contact, the work of adhesion for receding (i.e. separating or opening) contact is greater than the work of adhesion to advance (i.e. approaching or closing) contact, which is characterized as weak adhesion. Unlike the strong adhesion, a weak adhesion is insufficient to sustain the singularity, and the contact does not produce a square-root singularity where a compressive stresses along the contact are developed, see Fig. 3. For example, considering a sphere of diameter, D, elastic modulus, E, and Poisson ratio, n, contacting a planar surface of a large substrate of similar material, the contact sizes for strong and weak adhesion are as follow. In the case of monotonic normal loading from zero to, PmaxO0, under weak adhesive, the contact stress field is non-singular and the maximum contact radius, amax, is given as [17]
3Dð1 K v2 Þ Pmax amax Z 4E
1=3
:
(1)
For strong adhesion, where the forces of attraction effectively increase the contact load across the contact interface, contact produces a tensile square-root singular stress field. It is asymptotically equal to the mode I crack field at the contact perimeter [19]. The maximum contact radius, amax, is equal to 3Dð1 K v2 Þ 3pDw Pmax C amax Z 4E 2 131=3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3pDw 2 A5 C 3pDwPmax C ; ð2Þ 2 where w is the work of adhesion, and it is set equal to wz1 N/m [20]. Hence, the distinction between strong and weak adhesion is of great significance in MCAM and CAM analyses, especially that strong and weak adhesion have different formulations as will be shown subsequently. It also assumes the following. First, the materials are homogenous, isotropic and linear elastic. Second, to avoid elastic mismatch of contacting bodies, the cylinder and the flat substrate are assumed of similar materials. Third, to avoid microstructural scale effects, the contact area is assumed to cover at least several grains of the material. Fourth, twodimensional configurations, which are deduced from threedimensional configuration (see Fig. 1) are only considered in the MCAM, assuming no geometric variation through the thickness and ignoring the edge effects. Hence, quantities
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Fig. 2. Schematics demonstrating (a) the analogy between fracture mechanics and contact mechanics and (b) the configuration of including the bulk stress in MCAM.
per unit thickness will be used in MCAM formulations accordingly. Finally, the weak adhesion, the strong adhesion, and inclusion of bulk stress formulations will be presented separately as follows.
First, considering the normal load, the forces of attraction effectively increase the contact load across the interface and, thus, the contact area. The contact due to monotonically loaded normal force produces a tensile square-root singular stress field, which is asymptotically equal to the mode I crack field at the contact perimeter [21]. The asymptotic stress field using local polar coordinates, (r, q, z), see Fig. 2, at the contact perimeter are KI q q 3q p ffiffiffiffiffiffiffiffi ffi snor Z 1 K sin sin ; (3) cos rr 2 2 2 2pr KI q q 3q Z pffiffiffiffiffiffiffiffiffi cos 1 C sin sin ; 2 2 2 2pr
(4)
(5)
where KI is the mode I stress intensity factor, and its maximum value is given [21,22] as Pmax KI Z pffiffiffiffiffiffiffiffiffiffiffiffi : pamax
2.1. Strong adhesion
snor zz
KI q q 3q p ffiffiffiffiffiffiffiffi ffi Z sin cos ; cos snor rz 2 2 2 2pr
(6)
Note that for a cylinder of diameter, D, in strong adhesion contact with a flat substrate under normal loading, Pmax, the contact width, amax assuming a unit thickness, is given as [23] " 1=2 # pE a2max 4amax wð1 K v2 Þ Pmax Z K2 : (7) pE 2ð1 K v2 Þ D Similarly, for Pmin, the contact width, amin assuming a unit thickness, is " 1=2 # pE a2min 4amin wð1 K v2 Þ Pmin Z K2 : (8) pE 2ð1 K v2 Þ D
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Fig. 3. Schematics illustrating weak adhesion, strong adhesion, and stationary crack representations using contact mechanic and fracture mechanic formulations.
Since fretting fatigue is of interest, one should consider the formulation for cyclic loading. Based on the above, the corresponding mode I stress intensity factor due to oscillatory normal load ranging between, PmaxRPminR0, is
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Pmax pamin K pamax Rnor ; DKI K parameter Z DKI Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p amax amin (9) where Rnor is the normal load ratio, i.e. RnorZPmin/Pmax. Second, similar to the normal case, the tangential load produces a tensile square-root singular stress field equal to the mode II crack field at the contact perimeter [24,25]. For elastic, monotonic, tangential load, QmaxO0, under constant normal load, Pmax, the asymptotic stress field using local polar coordinates (r, q, z), see Fig. 2, at the contact perimeter are KKII q q 3q tan srr Z pffiffiffiffiffiffiffiffiffi sin 2 C cos cos ; (10) 2 2 2 2pr
KII q q 3q p ffiffiffiffiffiffiffiffi ffi Z sin sin ; cos stan zz 2 2 2 2pr
(11)
KII q q 3q p ffiffiffiffiffiffiffiffi ffi Z 1 K sin sin ; cos stan rz 2 2 2 2pr
(12)
where KII is the mode II stress intensity factor, and its maximum value is given [25] as Qmax KII Z pffiffiffiffiffiffiffiffiffiffiffiffi : pamax
(13)
For cyclic tangential oscillatory load ranging between, QmaxRQminR0, DKII equals
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Qmax pamin K pamax Rtan DKII K parameter Z DKII Z ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p amax amin (14) where Rtan is the tangential load ratio, i.e. RtanZQmin/Qmax.
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Third, for rectangular punch or sharp wedge in contact with flat substrate, the width of the punch is 2aw, see Fig. 1. Assuming linear elastic isotropic material with small strain and small deformation at all times, the size of the plastic zone, rpls is small compared to the width of the contact region, i.e. rpls%aw or rpls%20aw. Also, assuming frictionless surfaces, i.e. mZ0, the singular stress fields at rectangular rigid punch or sharp edge in contact with incompressible substrates and subjected to normal and tangential loads are developed by Sadowski [14] and Nadai [13]. Using polar coordinate system (r, q), the plane strain asymptotic contact stresses at the left edge are 3 3q q w p ffiffi srr fK sin C 5 sin ; (15) 4 r 2 2
(r, q) at the left sharp edge given by Nadai [13] are 1 3q q p ffiffi f 3 cos C 5 cos ; (24) sw rr 4 r 2 2 sw qq f
sw rq
1 3q q pffiffi 3 cos K 3 sin ; 4 r 2 2
1 3q q Z pffiffi 3 sin K sin : 4 r 2 2
(16)
K3 sw ff f pffiffi
3 3q q p ffiffi fK sw cos K cos : rq 4 r 2 2
(17)
sw rf
q/ p K q Z f;
0% q% p;
(18)
and expanding the trigonometric terms of the asymptotic stresses in Eqs. (15)–(17), the angular variation of stresses reduce to 1 f w 2 f p ffiffi srr f cos 1 C sin ; (19) r 2 2 1 w 3 f sff f pffiffi cos ; r 2
(20)
1 f 2 f p ffiffi Z sw sin cos : rf r 2 2
(21)
The stress fields are identical to those derived in Ref. [26] for a mode I crack, and the mode I stress intensity factor for a sharp wedge using crack-analogy is Pmax KIw Z K pffiffiffiffiffiffiffiffi ffi: paw
(22)
DKIw K parameter Z DKIw Z
f 2 f sin cos ; r 2 2
Pmax K Pmin pffiffiffiffiffiffiffiffiffi : paw
(23)
Similarly, consider the frictionless contact between the rigid rectangular punch and the incompressible substrate subjected to normal and tangential load, the plane strain asymptotic contact stresses using polar coordinate system
(28)
1 f 2 f Z pffiffi cos 1 K 3 sin : r 2 2
(29)
The classical contact mechanic stresses [4–6] are identical to the standard mode II stress fields at the crack tip as derived from linear fracture mechanics [11,12]. From classical contact mechanics, the mode II stress intensity factor for a sharp wedge is Qmax KIIw Z pffiffiffiffiffiffiffiffi ffi: paw
(30)
For cyclic tangential oscillatory load ranging between, QmaxRQminR0, DKIIw equals DKIIw K parameter Z DKIIw Z
Qmax ð1 K Rtan Þ : pffiffiffiffiffiffiffiffiffi paw
(31)
So far the frictionless case is considered. However, for a frictional condition, where mZQ/P signifies the effective friction coefficient, the limiting case where the rigid punch adheres completely to the surface is also considered. Thus, using the shear stress for frictional complete stick condition [4], the following relations for modes I and II are deduced mZ
For cyclic oscillatory load ranging between, PmaxR PminR0, DKIw equals
(26)
Introducing the coordinate transformation in Eq. (18), the crack tip stress field reduces to 1 f 2 f p ffiffi f sin 1 K 3 sin ; (27) sw rr r 2 2
3 3q q p ffiffi sw fK Ksin C 3 sin ; qq 4 r 2 2
Using the crack-analogy transformation
(25)
KII ; KI
Qmax Pmax ð1 K 2vÞ KII Z pffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffi ; paw paw 3pð1 K vÞ
(32)
(33)
and for cyclic oscillatory load, DKII equals Qmax Pmax ð1 K 2vÞ : DKII K parameter Z DKII Z pffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffi paw paw 3pð1 K vÞ (34)
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2.2. Weak adhesion For a weak adhesion under normal loading where the maximum strain energy release rate of the contacting surfaces, Gnor max , is greater than the work needed to normally debond the two contacting surfaces locally, Gnor d , at their nor perimeter, i.e. Gnor max O Gd , the contact does not produce an asymptotically tensile square-root singular stress field as in Eqs. (3)–(5). In this case, the work of adhesion is insufficient to sustain the singularity at the maximum contact width. Consequently, the work at the contact perimeter needed to debond, Gnor d , and to produce contact width, amax, decreases. Based on [27], this work, which is needed to debond two contacting surfaces locally, Gnor d , is assumed sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Q max Gnor : (35) d Z 14:3651 0:3 C Pmax Thus, first, considering the corresponding mode I stress intensity factor due to oscillatory normal load ranging between, PmaxRPminR0, for weak adhesion, i.e. if nor 2 1=3 3 pEDðGnor d ð1 K R ÞÞ ; (36) Pmax K Pmin O 2 8ð1 K v2 Þ it is equal to DKI K parameter Z DKI 2 1=3 3 pEDðGnor d Þ Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pamax 8ð1 K v2 Þ nor 2 1=3 3 pEDðGnor d R Þ K pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pamin 8ð1 K v2 Þ
(37)
where Rnor is the normal load ratio. Second, for weak adhesion under tangential load, the formulation of cyclic loading of a cylindrical pad in contact with a much larger flat substrate, subjected to an oscillatory tangential load, mPmaxRQmaxR0, and under normal load, Pmax, shows that the elastic energy of the cylinder becomes unbounded and the displacements are indeterminate. The solution to this problem depends critically on the overall dimensions of the fretting specimen and the applied far-field boundary conditions [17]. Hence, the precise definition when adhesion becomes weak is ambiguous. But, in the present study, to overcome the indeterminate state, the condition for the normal load, Eq. (37), is assumed to apply for tangential weak adhesion if slip condition, QZmP, is satisfied. In other words, for slip condition, a weak adhesion under tangential load is assumed to be linearly proportional to a weak adhesion under normal load. Based on this, a weak adhesion under tangential load assumed to occur if tan 2 1=3 3 pEDðGtan d ð1 K R ÞÞ Qmax K Qmin O Am ; 2 8ð1 K v2 Þ
(38)
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where A is the proportionality constant, and the work needed to tangentially debond the two contacting surfaces locally, Gtan d , is assumed Qmax tan (39) Gnor Gd Z d : Pmax Hence, the corresponding mode II stress intensity factor due to oscillatory tangential load ranging between, QmaxR QminR0, for a weak adhesion, is equal to [17] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! tan tan Qcont Qcont Gtan max max R d ð1 K R Þ ; DKII Z min pffiffiffiffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffiffiffi ; pcmax pcmin ð1 K v2 Þ (40) where cmax and cmin are the maximum and minimum width of the stick-zone in weak adhesion condition (i.e. stickslip), which can be obtained from the following equations 2 c Qmax Z mPmax 1 K max ; (41) amax 2 c ; Qmin Z mPmin 1 K min amin
(42)
assuming adhesion is re-established along the stick-zone. Qcont max is the tangential force that is balanced in the stickzone, and it is defined [4,17] as " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! mPmax p cmax a2max K c2max cont C arc tan Qmax Z Qmax K 2 p c2max K a2max # c c2 K max 1 K max : ð43Þ amax a2max Third, the weak adhesion solution of a wedge does not exist. The present study assumes that the weak adhesion solution of modes I and II stress intensity factor for wedge indenter is equivalent to a cylindrical indenter with a large radius. This assumption is based on the fact that geometrically a cylindrical indenter approaches a flat surface as the radius approaches an infinite value. Further, for a weak adhesion, the contact does not produce an asymptotically tensile square-root singular stress field, and the solution becomes dependent critically on the overall dimensions of the fretting specimen and the applied farfield boundary conditions. Assuming that the weak solution for a sharp wedge loaded normally and tangentially is proportional to the weak solution of a cylinder with diameter, D, equal to 685.8 mm. The cyclic modes I and II stress intensity factor become nor 2 1=3 9AI pEðGnor d R Þ DKI K parameter Z DKI Z pffiffiffiffiffiffiffiffiffiffiffi ; 2 2paw ð1 K v2 Þ (44)
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DKII K parameter Z DKII
definition equals to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc ; KIIBulk Z sapp 2ðc C dÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! cont tan tan Qcont Q R Gtan max d ð1 K R Þ ; Z AII min pffiffiffiffiffiffiffiffiffiffiffiffi K pmax ffiffiffiffiffiffiffiffiffiffiffiffi ; pcmax pcmax ð1 K v2 Þ (45) where AI and AII are the proportionality constants for modes I and II, respectively. The conditions to instigate a weak adhesion for modes I and II are nor 2 1=3 9 pEðGnor d ð1 K R ÞÞ Pmax K Pmin O AI ; (46) 4 ð1 K v2 Þ 9 Qmax K Qmin O AII m 4
nor 2 1=3 pEðGnor d ð1 K R ÞÞ : ð1 K v2 Þ
(47)
where c and d are the width of the indenter and substrate, respectively. The pad thickness is assumed to be proportional to the radius of the cylindrical pad or the flat width of the wedge pad. Note that this is similar to the stress intensity factor of infinite plate, but with geometric factor. Assuming same material for the pad and the substrate, i.e. no materials mismatch, Eq. (50) reduces to sapp Ec KIIBulk Z pffiffiffiffiffiffiffiffi ; 2pr Ebd½c C d
(51)
and for cyclic bulk stress, DKIIBulk becomes Dsapp Ec : DKIIBulk K parameter Z DKIIBulk Z pffiffiffiffiffiffiffiffi 2pr Ebd½c C d
2.3. Bulk
(52)
The crack-analogy methodology, so far, has considered an indenter in contact with a plane surface under normal and tangential load. However, the loading conditions for a fretting fatigue of a dovetail include bulk stress, sapp. In general, analytically one could model the bulk stresses in the substrate by superposition of the effect of static or cyclic mechanical load applied parallel to the direction of interfacial surface [6,11]. For example, the bulk stresses in the CAM [16] are superimposed by invoking the classical T-stress concept of linear elastic fracture mechanics. The superposition of modes I and II fracture stresses and the applied bulk stresses, assuming linear elastic fracture mechanics, can be expressed as KI KII II sij Z pffiffiffiffiffiffiffiffi sIij ðqÞ C pffiffiffiffiffiffiffiffi sij ðqÞ C sapp d1i d1j : 2pr 2pr
(50)
(48)
However, the MCAM utilizes DK-parameter, and including the bulk stress requires the superposition of an energy form consistent with DK-parameter, but not the approach of superimposed T-stress. To compute the energy due to the bulk stress, the present study utilizes the closed form solution of cracked lap shear specimen with the bottom flat portion of the lap is loaded by a tensile stress or force, that is equivalent to the substrate’s applied bulk stress, see Fig. 2. This energy is then added to the modes I and II stress intensity factor assuming linear superposition applies. Thus, the total stress intensity factor, KTotal, becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bulk K Total Z KI2 C KII2 C KII2 ; (49) and the mode II stress intensity factor, KIIBulk , due to bulk stress is obtained as follows. First, note that the applied load on the lap shear specimen is equivalent to the bulk load, sapp, applied on the substrate, and it produces a mode II crack growth stress intensity factor. Second, the mode II stress intensity factor of the cracked lap shear [28] sapp by
3. Discussion and results As mentioned earlier, the characteristics of a material under fatigue cannot be deduced from other mechanical properties. They must be measured directly through experimental mean. In the present study, the DK-parameter–life curves are established using the experimental data of [18], which have two significances. First, the characteristics of these curves serve as indirect way to validate MCAM such that results have to be consistent with those observed in fatigue’s S–N curve, since the experimental matrix is designed with life ranges between low and high cycle fretting fatigue [18]. Second, it becomes a design tool to predict crack initiation of fretting fatigue. Hence, a summary of the experimental apparatus will be presented for sake of completeness. In these experiments, various configurations were used through the combination of a uniaxial servo-hydraulic fatigue testing machine and a fretting fixture with the appropriate positioning of a pair of pads (see Fig. 4). These fretting pads were clamped on to the surface of a flat specimen. The pad configurations used are 50.8, 101.6, and 304.8 mm cylindrical pad, 50.8 mm flat
Fig. 4. Experimental apparatus.
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Table 1 The load condition of the 2 in. cylindrical pads and their experimental fretting fatigue life [18]
Table 3 The load condition of the 12 in. cylindrical pads and their experimental fretting fatigue life [18]
Life (N), no. of cycles
Life (N), no. of cycles
26,700 31,600 53,400 70,600 86,200 91,900 118,000 121,000 124,000 220,000 371,000 672,000 2,080,000 2,560,000 3,660,000 4,140,000 50,000,000 50,000,000
Substrate axial stress smax (MPa)
smin (MPa)
635.8 699.6 551.9 566.1 686.9 424.6 537.8 416.2 685.5 528.7 686.6 581.7 412.6 685.8 419.8 540.3 506.6 410.1
K39.6 43.9 18.1 53.0 291.4 35.2 233.3 28.7 293.6 232.2 455.8 350.5 185.8 441.6 190.9 372.3 330.6 273.2
Axial load ratio R
K0.062 0.063 0.033 0.094 0.424 0.083 0.434 0.069 0.428 0.439 0.664 0.603 0.450 0.644 0.455 0.689 0.653 0.666
Tangential load Qmax (N)
Qmin (N)
698.00 498.00 427.00 649.00 458.00 547.00 280.00 467.00 405.00 191.00 472.00 547.00 516.00 560.00 512.00 601.00 618.00 467.00
K560.00 K316.00 K294.00 K574.00 K494.00 K445.00 K254.00 K512.00 K209.00 K214.00 120.00 K138.00 K67.00 K53.00 K85.00 427.00 196.00 125.00
with 50.8 mm rounded end pad, and 4.67 mm 908 wedge. A spring-loaded holding fixture attached to the load frame of the test machine was used to position the fretting pads on to the specimen surface (see Fig. 4). A load cell was used to measure the normal load applied to the specimen through the fretting pads. A constant normal fretting load is applied to all specimens, which is 1334 N. Note that this results in various peaks of Hertzian contact pressure at the pad– specimen interface. Furthermore, these various pad configurations are fatigued at various load combination and nominal stress ratio, R (i.e. ratio of minimum to maximum load). The specimen and fretting pads were both were both made of titanium alloy, Ti–6A1–4V. The longitudinal Table 2 The load condition of the 4 in. cylindrical pads and their experimental fretting fatigue life [18] Life (N), no. of cycles 23,200 53,000 57,000 59,400 62,400 124,000 135,000 162,000 286,000 340,000 361,000 710,000 1,120,000 4,890,000 50,000,000
Substrate axial stress smax (MPa)
smin (MPa)
718.9 647.7 594.8 651.7 601.6 457.2 584.2 456.1 559.9 446.0 428.5 587.9 537.4 416.9 505.8
16.0 10.9 41.4 26.9 29.4 25.8 266.7 34.1 243.1 210.1 203.9 401.3 357.8 246.4 268.2
Axial load ratio R
0.022 0.017 0.070 0.041 0.049 0.056 0.457 0.075 0.434 0.471 0.476 0.683 0.666 0.591 0.530
24,376 54,850 160,201 245,686 580,000 1,344,333 13,300,000 19,300,000
Qmin (N)
1116.00 983.00 934.00 832.00 925.00 578.00 903.00 623.00 667.00 485.00 436.00 976.00 767.00 440.00 956.00
K552.00 K649.00 K418.00 K601.00 K498.00 K512.00 98.00 K423.00 K147.00 K49.00 K21.00 516.00 311.00 K9.00 418.00
smax (MPa)
smin (MPa)
441.2 494.6 413.7 577.3 321.7 336.4 287.5 382.4
136.0 178.3 231.7 481.7 248.9 257.4 224.3 290.5
Axial load ratio R
0.308 0.361 0.560 0.834 0.774 0.765 0.780 0.760
Tangential load Qmax (N)
Qmin (N)
336.00 577.00 379.90 300.00 260.00 300.00 163.00 306.00
K224.00 K373.00 K245.00 K300.00 K185.00 K190.00 K117.00 K204.00
tensile properties of the tested material (along the loading axis) were: Young’s elastic modulus, EZ127 GPa, yield strength, syZ930 MPa, and Poisson ratio, nZ0.3. Specimen had a dogbone configuration with thickness, 2bZ1.93 mm and width, wZ6.35 mm in the gage section (see Fig. 4). The applied bulk stress, s, and tangential stress, Q, in the experiments range between maximum and minimum values, which are in-phase. The experimental tests are performed under room temperature. The experimental results are tabulated where the test life of the specimen, the nominal bulk stress ratio, R, and their bulk and tangential loading condition are shown. The results for the 50.8, 101.6, and 304.8 mm cylindrical pads are listed in Tables 1–3, respectively. Similarly, for the 50.8 mm flat with 50.8 mm rounded end and the 4.67 mm 908 wedge pads, the results are shown in Tables 4 and 5, respectively. The experimental life of the specimens range from low cycle fretting fatigue with an upper limit of approximately 104 cycles, transitional low to high cycle approximately Table 4 The load condition of the 2 in. flat with 2 in. rounded cylindrical end pads and their experimental fretting fatigue life [18] Life (N), no. of cycles
Tangential load Qmax (N)
Substrate axial stress
29,300 54,000 55,900 71,600 111,000 124,000 138,000 198,000 223,000 260,000 614,000 1,300,000 1,890,000 3,950,000 6,930,000 10,400,000 165,000,000
Substrate axial stress smax (MPa)
smin (MPa)
691.6 691.3 566.4 574.4 695.6 406.8 704.0 564.6 422.0 484.1 423.8 613.3 555.9 607.5 534.5 546.1 394.4
36.7 36.3 33.0 32.3 326.2 K10.9 337.1 231.9 15.2 173.5 178.2 381.0 356.7 390.8 356.0 242.8 180.7
Axial load ratio R
Tangential load Qmax (N)
Qmin (N)
0.053 0.053 0.058 0.056 0.469 K0.027 0.479 0.411 0.036 0.358 0.420 0.621 0.642 0.643 0.666 0.445 0.458
774.00 770.00 672.00 667.00 538.00 780.00 565.00 396.00 503.00 805.00 440.00 520.00 569.00 596.00 423.00 454.00 543.00
K796.00 K885.00 K676.00 K721.00 K400.00 K587.00 K369.00 K440.00 K534.00 K525.00 K236.00 K111.00 49.00 49.00 49.00 K356.00 71.00
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Table 5 The load condition of the 0.1875 in. flat 908 wedge pads and their experimental fretting fatigue life Life (N), no. of cycles
Substrate axial stress smax (MPa)
smin (MPa)
32,951 147,254 260,065 552,016 5,481,603
392.7 403.5 389.5 391.6 349.3
87.3 188.2 242.3 240.8 234.4
Axial load ratio R
0.222 0.466 0.622 0.615 0.671
Tangential load Qmax (N)
Qmin (N)
1720.00 980.00 670.00 1735.00 1375.00
860.00 440.00 K181.00 500.00 535.00
bounded between 104 and 106 cycles, and high cycle with a room temperature fatigue limit starting approximately at 106 cycles until failure or runout. Furthermore, the experimental loading conditions and their effect on fretting fatigue life are examined by plotting the applied range of the tangential
load and bulk stress versus specimen’s life in Fig. 5a and b, respectively. Based on these figures one deduces the following. The applied tangential load affected fretting life such that decreasing the tangential load results in a higher life (see Fig. 5a). This is within a scatter for the various pad configurations, which is due to the probabilistic characteristics of failure as a result of different metallurgical, configurationally, environmental, and operational influences. Similarly, lower applied bulk stress increases the life of specimen linearly as shown in Fig. 5b. For comparison, the applied bulk stress of plan fatigue is also plotted. It shows that plan fatigue exhibits higher damage tolerance than fretting fatigue. That is, in order for both of them to have equivalent life, the plan fatigue requires higher applied load amplitude than fretting fatigue. Further, the tensile bulk stress produces tensile residual stress field, which cause a shift in the life curve.
Fig. 5. Experimental data of (a) ranges of applied bulk stress versus life and (b) ranges of tangential load versus life.
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449
Fig. 6. DK-parameter versus life assuming strong adhesion formulation (a) without bulk stress and (b) with bulk stress.
Assuming strong adhesion, the DK-parameter is computed for the experimental data in Tables 1–5. The values are normalized. The analyses are performed with and without bulk stress, sapp. They are plotted in Figs. 6 and 7 where the former uses linear scale and the latter uses logarithmic scale. Note that this common in presenting fatigue data/results, since S–N, or DK-parameter–life in the present study, exhibits power law characteristics in linear scale and bilinear in logarithmic scale. The figures show similar trend to the S–N curve of plain fatigue with lower damage tolerance as expected. The analyses also show cyclic limit of fatigue approximately at 106 cycles. They show dependency on pad geometry and loading condition. For example, the 50.8 mm cylindrical pad exhibits lowest damage tolerance among the ones analyzed, and the wedge
pad shows the highest damage tolerance with and without bulk stress. The 304.8 mm cylindrical pad results are closer to the wedge, which is consistent with the fact that cylindrical pad with infinitely large radius approaches the characteristics of flat/wedge pad. The flat with 50.8 rounded ends shifted toward the 50.8 mm cylindrical pad without bulk stress, however, it shifted towards the wedge pad by including bulk stress. This indicates that without bulk stress, the effect of the contact stresses induced by the rounded ends on damage tolerance dominates. But, by adding the bulk stress, the effect of contact stress on damage tolerance is less significant. The weak adhesion case is also analyzed, and the results with and without bulk stress are shown in Fig. 8. They show that without bulk stress the results are geometry independent within scatter
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Fig. 7. Logarithmic representation of Fig. 6(a) and (b), respectively.
and do not show the cyclic limit. This is contrary to the ones with the bulk stress.
4. Conclusion The present study introduced MCAM, which extends CAM to include various indenter–substrate geometries and the applied bulk stress for both strong and weak adhesion formulations. The results show the following: † Assuming strong adhesion, analyses of MCAM with and without bulk stress showed similar characteristics to S–N curve of plane fatigue within reasonable scatter.
† The current MCAM results show that by including the tensile cyclic bulk stress in the analyses, the damage tolerance of fretting fatigue increased and caused the life curve to shift towards low cycle fatigue. † The DK-parameter showed dependency on the pad geometry where the 50.8 mm has the lowest damage tolerance and the wedge pad has the highest damage tolerance. The 304.8 mm cylindrical pad is in between as expected. † Assuming weak adhesion, analyses of MCAM without bulk stress did not predict the fatigue cyclic limit contrary to analysis with bulk stress. In summary, MCAM shows potential. It further demonstrates the significance of including the bulk stresses in
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Fig. 8. Logarithmic plot of DK-parameter versus life assuming weak adhesion formulation (a) without bulk stress and (b) with bulk stress.
the analysis. The advantages of the present approach are: it is easy to implement in a design process; it is reliable and captures the physical characteristics of fretting fatigue; and it is computationally efficient compared to other approaches such as finite element, which requires a very refined mesh near the contact.
References [1] Collins JA. Failure of materials in mechanical design. New York: Wiley/Interscience; 1993. [2] Suresh S. Fatigue of materials, 2nd ed. Cambridge: Cambridge University Press; 1998. [3] Mindlin RD. Force at a point in the interior of a semi-infinite solid. Physics 1936;7:195–202.
[4] Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985. [5] Gladwell GML. Contact problem in classical theory of elasticity. Alphen aan den Rijn: Sijthoff & Noordhoff; 1980. [6] Hills DA, Nowell D. Mechanics of fretting fatigue. London: Kluwer; 1993. [7] Hills DA, Nowell D, Sackfield A. Mechanics of elastic contacts. Oxford: Butterworth/Heinemann; 1993. [8] Ciavarella M. Indentation by nominally flat indenters with rounded corners. Int J Solids Struct 1999;36:5157–65. [9] Westergaard HM. Bearing pressures and cracks. J Appl Mech 1939;6: 49–52. [10] Sneddon IN. The distribution of stress in the neighborhood of a crack in an elastic solid. Proc Camb Philos Soc 1946;42:229–60. [11] Kanninen MF, Popelar CH. Advance fracture mechanics. Oxford engineering science series. Oxford: Oxford University Press; 1985. [12] Anderson TL. Fracture mechanics: fundamentals and applications, 2nd ed. Boca Raton, FL: CRC Press; 1995.
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[13] Nadai AI. Theory of flow and fracture of solids, vol. II. New York: McGraw-Hill; 1963. [14] Sadowski M. Z Angew Math Mech 1928;8:107. [15] Hertz H. On the contact of elastic solids. J Reine Angew Math 1882; 92:156–71. [16] Giannakopoulos AE, Lindley TC, Suresh S. Aspects of equivalence between contact mechanics and fracture mechanics: theoretical connections and life-prediction methodology for fretting fatigue. Acta Mater 1998;46:2955–68. [17] Ginnakopoulos AE, Venkatesh TA, Lindley TC, Suresh S. The role of adhesion in contact fatigue. Acta Mater 1998;47:4653–64. [18] Lykins CD. An investigation of fretting fatigue crack initiation behavior of the titanium alloy Ti–6A1–4V, PhD Thesis, Air Force Institute of Technology; 2000. [19] Barquins M, Maugis D. Adhesive contact of axisymmetric punches on an elastic half-space: the modified Hertz–Huber’s stress tensor for contacting spheres. J Mech Theor Appl 1982;1:331–57.
[20] Semenchenko VK. Surface phenomena in metals and alloys. Reading, MA: Addison-Wesley; 1962. [21] Greenwood JA, Johnson KL. The mechanics of adhesion of viscoelastic solids. Philos Mag 1981;A43(3):697–711. [22] Maugis D, Barquins M. Fracture mechanics and the adherence of viscoelastic bodies. J Phys D: Appl Phys 1978;11:1989–2023. [23] Kalker JJ. Variation and non-variational theory of frictionless adhesive contact between elastic bodies. Wear 1987;119:63–76. [24] Savkoor AR, Briggs GDA. The effect of tangential force on the contact of elastic solids in adhesion. Proc R Soc Lond 1977; A356:103. [25] Westmann RA. J Appl Mech 1965;Vp1. 87:411. [26] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech 1957;24:109–14. [27] McFarlane JS, Tabor D. Proc R Soc Lond Ser A 1950;202:244. [28] Murakami Y. Stress intensity factors handbook. Oxford: Pergamon Press; 1980.
Modified crack-analogy methodology: the equivalent theory S. Naboulsi* Department of Aeronautics and Astronautics, Air Force Institute of Technology (AFIT/ENY), 2950 Hobson Way, Wright-Patterson Air Force Base, OH 45433-7765, USA Received 24 April 2003; received in revised form 30 January 2004; accepted 27 July 2004
Abstract The crack initiation process of fretting fatigue is a complex phenomenon, and various approaches have been instigated to investigate fretting fatigue damage tolerance and crack initiation. The Crack-Analogy Methodology (CAM) is one approach, which is introduced by Giannakopoulos et al. It utilizes the similarity between contact mechanics and fracture mechanics to investigate fretting fatigue life. The present study is motivated by the crack-analogy methodology where a Modified Crack-Analogy Methodology (MCAM) is developed to extend CAM capabilities and to improve its prediction of crack initiation. The modified form extends CAM to include various indenter– substrate geometries as well as modifying its crack initiation parameter to include the effect of the bulk stress in the substrate. MCAM uses the change of the stress intensity factor (DK-parameter) as a fretting fatigue crack initiation parameter, since the change is consistent with the cyclic mechanism of fretting fatigue. It uses experimental data to establish DK-parameter–life curves similar to the stress–life S–N curve prototype in fatigue and to validate crack initiation for various geometric configurations under various load conditions. The results show similar trends to plain fatigue with lower damage tolerance as expected. By including the bulk stress, the crack initiation predictions of fretting fatigue show better consistency with the experimental data than without it where it exhibits fatigue characteristics consistent with S–N curves. In general, the DK-parameter–life trends show dependency on applied load and pad geometry. Reduced scatter is observed when bulk stress is included in the analyses as compared to that without it. It further demonstrates that the presence of bulk stress in general shifts fretting fatigue characteristics toward low cycle fatigue. The MCAM also predicts the limit of high cycle fatigue. The MCAM formulation shows potentials in life prediction such that it can be used as a tool in the design of components under fretting fatigue. q 2004 Published by Elsevier Ltd. Keywords: Fretting fatigue; Crack initiation parameter; Fracture mechanics; Contact mechanics
1. Introduction As the case for fatigue, fretting fatigue in general is a complex subject, where some of the problem modeling fretting fatigue may be characterized as follow [1]: † Calculations of life are generally less accurate and less dependable than strength calculations. Order of magnitude errors in life estimate are not unusual. † Fatigue characteristics of a material cannot be deduced from other mechanical properties. They must be measured directly using experiments. † Full-scale prototype testing is usually necessary to assure an acceptable model predictions. * Tel.: C1 937 255 6565. E-mail address: [email protected] 0142-1123/$ - see front matter q 2004 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2004.07.010
† Results of different but identical tests may differ widely, requiring, therefore, a statistical interpretation by designer. Furthermore, damage tolerance of pre-crack initiation in fretting fatigue is of complex nature, which is attributed to its dependency on pad geometries, surface properties, material properties, and mechanical loading conditions [2]. For example, smaller radius cylindrical pads under the same fretting load conditions cause lower damage tolerance (i.e. higher damage) where crack initiates faster than the one with larger radius, which exhibits higher damage tolerance (i.e. lower damage). In the present study, the term crack initiation is used to indicate the number of cycles of precrack initiation (i.e. number of cycles leading to the development of small cracks without including the number of cycles of post-crack growth or crack propagation of
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failure). Also, the term damage tolerance is used to quantify the resistance of the material to fretting fatigue damage due to either loading conditions or pad geometry (i.e. higher damage tolerance means more pre-crack initiation compared to lower damage tolerance). Further, the term crack initiation parameter is a norm, where its value is equivalent to an estimated damage tolerance and quantitatively predicts the number of cycles to crack initiation. The parameter is a design tool, which provides a quantitative mean of comparison of material/structure damage under various fretting fatigue scenarios. In practical applications, one needs fretting fatigue crack initiation parameter that reflects its complex process. Traditionally, the investigation of fretting fatigue crack initiation has been performed using contact mechanics [3–8]. However, the similarity of the stress state near the contact surface using Hertzian contact and asymptotic solution of fracture mechanic has been employed to interpret experimental observations of fretting fatigue crack initiation. For example, the studies [9,10] suggested the similarities between the singular fields for sharp-edged contacts [4–6] and for cracked bodies [11,12]. Nadai [13] analyzed the asymptotic stress solution within an elastic substrate in the vicinity of the corners of the punch for both normal and tangential loading. His result confirms the similarity between the contact surface stress singularity predicted by Sadowski [14], where the elastic singular fields at the sharp edges of two-dimensional contact between a rigid flat punch and planar surface were derived, and the contact surface stresses derived earlier by Hertz [15] for normal loading of a cylindrical punch. Recently, Giannakopoulos et al. [16,17] used the Crack-Analogy Methodology (CAM) to validate and interpret their experimental data. In their studies, a quantitative equivalence has been explored in detail between contact mechanics and fracture mechanics using the vast literature of both fields. This equivalence is established by identifying geometries of cracked bodies, which facilitate an analogy with an axisymmetric sharp-edge pad and rounded pad in contact with flat substrate. Their experimental data are obtained for a spherical pad in contact with substrate. They used a stress intensity factor as crack initiation prediction parameter. They assumed that fatigue crack initiates when the stress intensity factor is greater than the stress intensity threshold. Further, the crack-analogy’s results [17] for the weak adhesion case showed that the damage tolerance is insensitive to changes in the tangential load, such that prediction of fretting fatigue crack initiation using mode II stress intensity factor parameter is independent of load, which is contrary to experimental observation. For example, experimental data of cylindrical pads show dependency on radius size and tangential load [18]. One of the fundamental bases of the Modified CrackAnalogy Methodology (MCAM), which is motivated by earlier studies of CAM [16,17], is its utilization of the similarity between contact mechanics and fracture
mechanics. This allows the use of fracture mechanic metrics to characterize the contact based fretting fatigue phenomena such as stress intensity factor. MCAM includes the effect of various pad geometry and bulk stress on crack initiation, which is different from CAM. Also, contrary to the stress intensity factor used in CAM, MCAM uses the change of stress intensity factor as fretting fatigue parameter, since it is consistent with the cyclic mechanism of fatigue. It will be referred to as DK-parameter for the remaining of the paper. MCAM also establishes DK-parameter–life curve similar to S–N curve, which constitutes design information of fundamental importance to specimens subjected to repeated fatigue load.
2. Crack-analogy’s methodology The concept behind the development of MCAM and CAM [16,17] is the equivalence between the asymptotic field of two bodies in contact derived from classical contact mechanics theory and the asymptotic field at a stationary crack determined from fracture mechanics. This allows the use of fracture mechanic metrics to characterize the contact based fretting fatigue phenomena, and it involves the following. First, the identification of the cracked specimen configuration, which provides a geometric equivalence for the contact region of a pad normally pressing on planar substrate, is made, see Fig. 1. Second, its asymptotic solution to the stress and strain fields at the edge of contacts from classical contact is determined. Third, the corresponding stress and strain field solutions for the analogous cracked body from linear elastic fracture mechanics are also determined. In other words, the assumption that the cracked body is subjected to a normal compressive load, whose magnitude is P, and a tangential load, whose magnitude is Q, is the same as that of the load pressing and sliding the pad against the substrate (see Fig. 2). The scalar amplitude of the singular fields at the crack tip is the stress intensity factor. Fourth, equating the different components of the stress fields, one can solve for the stress intensity factor as a function of the load (i.e. an equivalent mode I stress intensity factor for separating or opening is used to characterize the effect of the normal load P, and an equivalent mode II stress intensity factor for slip or shear is used to characterize the effect of the tangential load, Q). Thus, the analogy between the contact mechanics and the fracture mechanics is established, the equivalency of the stress fields can be determined, and MCAM, for example, uses DK-parameter as crack initiation parameter of fretting fatigue. One of the bases of crack-analogy methodology is the singular field that exists in both the contact mechanics and fracture mechanics solutions. Theoretically, it has been demonstrated that under monotonic compressive loading of contact interface, adhesion induces a tensile square root singular stress fields at the edge of contact. Adhesion is used
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441
Fig. 1. Schematic of indenter–substrate configurations and their MCAM representation.
to indicate that the analogy is for a stationary crack. It is defined as that for which the two contacting surfaces are bonded and the works needed to debond the contacting surfaces are high. In the case of applied normal and tangential loads, adhesion produces a mode I tensile stress and mode II shear stress, see Fig. 2 (i.e. mode I stress intensity factor characterizes the fretting effect of the normal load P, and mode II stress intensity factor characterizes the fretting effect of the tangential load Q). In fretting contact, the work of adhesion for receding (i.e. separating or opening) contact is greater than the work of adhesion to advance (i.e. approaching or closing) contact, which is characterized as weak adhesion. Unlike the strong adhesion, a weak adhesion is insufficient to sustain the singularity, and the contact does not produce a square-root singularity where a compressive stresses along the contact are developed, see Fig. 3. For example, considering a sphere of diameter, D, elastic modulus, E, and Poisson ratio, n, contacting a planar surface of a large substrate of similar material, the contact sizes for strong and weak adhesion are as follow. In the case of monotonic normal loading from zero to, PmaxO0, under weak adhesive, the contact stress field is non-singular and the maximum contact radius, amax, is given as [17]
3Dð1 K v2 Þ Pmax amax Z 4E
1=3
:
(1)
For strong adhesion, where the forces of attraction effectively increase the contact load across the contact interface, contact produces a tensile square-root singular stress field. It is asymptotically equal to the mode I crack field at the contact perimeter [19]. The maximum contact radius, amax, is equal to 3Dð1 K v2 Þ 3pDw Pmax C amax Z 4E 2 131=3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3pDw 2 A5 C 3pDwPmax C ; ð2Þ 2 where w is the work of adhesion, and it is set equal to wz1 N/m [20]. Hence, the distinction between strong and weak adhesion is of great significance in MCAM and CAM analyses, especially that strong and weak adhesion have different formulations as will be shown subsequently. It also assumes the following. First, the materials are homogenous, isotropic and linear elastic. Second, to avoid elastic mismatch of contacting bodies, the cylinder and the flat substrate are assumed of similar materials. Third, to avoid microstructural scale effects, the contact area is assumed to cover at least several grains of the material. Fourth, twodimensional configurations, which are deduced from threedimensional configuration (see Fig. 1) are only considered in the MCAM, assuming no geometric variation through the thickness and ignoring the edge effects. Hence, quantities
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Fig. 2. Schematics demonstrating (a) the analogy between fracture mechanics and contact mechanics and (b) the configuration of including the bulk stress in MCAM.
per unit thickness will be used in MCAM formulations accordingly. Finally, the weak adhesion, the strong adhesion, and inclusion of bulk stress formulations will be presented separately as follows.
First, considering the normal load, the forces of attraction effectively increase the contact load across the interface and, thus, the contact area. The contact due to monotonically loaded normal force produces a tensile square-root singular stress field, which is asymptotically equal to the mode I crack field at the contact perimeter [21]. The asymptotic stress field using local polar coordinates, (r, q, z), see Fig. 2, at the contact perimeter are KI q q 3q p ffiffiffiffiffiffiffiffi ffi snor Z 1 K sin sin ; (3) cos rr 2 2 2 2pr KI q q 3q Z pffiffiffiffiffiffiffiffiffi cos 1 C sin sin ; 2 2 2 2pr
(4)
(5)
where KI is the mode I stress intensity factor, and its maximum value is given [21,22] as Pmax KI Z pffiffiffiffiffiffiffiffiffiffiffiffi : pamax
2.1. Strong adhesion
snor zz
KI q q 3q p ffiffiffiffiffiffiffiffi ffi Z sin cos ; cos snor rz 2 2 2 2pr
(6)
Note that for a cylinder of diameter, D, in strong adhesion contact with a flat substrate under normal loading, Pmax, the contact width, amax assuming a unit thickness, is given as [23] " 1=2 # pE a2max 4amax wð1 K v2 Þ Pmax Z K2 : (7) pE 2ð1 K v2 Þ D Similarly, for Pmin, the contact width, amin assuming a unit thickness, is " 1=2 # pE a2min 4amin wð1 K v2 Þ Pmin Z K2 : (8) pE 2ð1 K v2 Þ D
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Fig. 3. Schematics illustrating weak adhesion, strong adhesion, and stationary crack representations using contact mechanic and fracture mechanic formulations.
Since fretting fatigue is of interest, one should consider the formulation for cyclic loading. Based on the above, the corresponding mode I stress intensity factor due to oscillatory normal load ranging between, PmaxRPminR0, is
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Pmax pamin K pamax Rnor ; DKI K parameter Z DKI Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p amax amin (9) where Rnor is the normal load ratio, i.e. RnorZPmin/Pmax. Second, similar to the normal case, the tangential load produces a tensile square-root singular stress field equal to the mode II crack field at the contact perimeter [24,25]. For elastic, monotonic, tangential load, QmaxO0, under constant normal load, Pmax, the asymptotic stress field using local polar coordinates (r, q, z), see Fig. 2, at the contact perimeter are KKII q q 3q tan srr Z pffiffiffiffiffiffiffiffiffi sin 2 C cos cos ; (10) 2 2 2 2pr
KII q q 3q p ffiffiffiffiffiffiffiffi ffi Z sin sin ; cos stan zz 2 2 2 2pr
(11)
KII q q 3q p ffiffiffiffiffiffiffiffi ffi Z 1 K sin sin ; cos stan rz 2 2 2 2pr
(12)
where KII is the mode II stress intensity factor, and its maximum value is given [25] as Qmax KII Z pffiffiffiffiffiffiffiffiffiffiffiffi : pamax
(13)
For cyclic tangential oscillatory load ranging between, QmaxRQminR0, DKII equals
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Qmax pamin K pamax Rtan DKII K parameter Z DKII Z ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p amax amin (14) where Rtan is the tangential load ratio, i.e. RtanZQmin/Qmax.
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Third, for rectangular punch or sharp wedge in contact with flat substrate, the width of the punch is 2aw, see Fig. 1. Assuming linear elastic isotropic material with small strain and small deformation at all times, the size of the plastic zone, rpls is small compared to the width of the contact region, i.e. rpls%aw or rpls%20aw. Also, assuming frictionless surfaces, i.e. mZ0, the singular stress fields at rectangular rigid punch or sharp edge in contact with incompressible substrates and subjected to normal and tangential loads are developed by Sadowski [14] and Nadai [13]. Using polar coordinate system (r, q), the plane strain asymptotic contact stresses at the left edge are 3 3q q w p ffiffi srr fK sin C 5 sin ; (15) 4 r 2 2
(r, q) at the left sharp edge given by Nadai [13] are 1 3q q p ffiffi f 3 cos C 5 cos ; (24) sw rr 4 r 2 2 sw qq f
sw rq
1 3q q pffiffi 3 cos K 3 sin ; 4 r 2 2
1 3q q Z pffiffi 3 sin K sin : 4 r 2 2
(16)
K3 sw ff f pffiffi
3 3q q p ffiffi fK sw cos K cos : rq 4 r 2 2
(17)
sw rf
q/ p K q Z f;
0% q% p;
(18)
and expanding the trigonometric terms of the asymptotic stresses in Eqs. (15)–(17), the angular variation of stresses reduce to 1 f w 2 f p ffiffi srr f cos 1 C sin ; (19) r 2 2 1 w 3 f sff f pffiffi cos ; r 2
(20)
1 f 2 f p ffiffi Z sw sin cos : rf r 2 2
(21)
The stress fields are identical to those derived in Ref. [26] for a mode I crack, and the mode I stress intensity factor for a sharp wedge using crack-analogy is Pmax KIw Z K pffiffiffiffiffiffiffiffi ffi: paw
(22)
DKIw K parameter Z DKIw Z
f 2 f sin cos ; r 2 2
Pmax K Pmin pffiffiffiffiffiffiffiffiffi : paw
(23)
Similarly, consider the frictionless contact between the rigid rectangular punch and the incompressible substrate subjected to normal and tangential load, the plane strain asymptotic contact stresses using polar coordinate system
(28)
1 f 2 f Z pffiffi cos 1 K 3 sin : r 2 2
(29)
The classical contact mechanic stresses [4–6] are identical to the standard mode II stress fields at the crack tip as derived from linear fracture mechanics [11,12]. From classical contact mechanics, the mode II stress intensity factor for a sharp wedge is Qmax KIIw Z pffiffiffiffiffiffiffiffi ffi: paw
(30)
For cyclic tangential oscillatory load ranging between, QmaxRQminR0, DKIIw equals DKIIw K parameter Z DKIIw Z
Qmax ð1 K Rtan Þ : pffiffiffiffiffiffiffiffiffi paw
(31)
So far the frictionless case is considered. However, for a frictional condition, where mZQ/P signifies the effective friction coefficient, the limiting case where the rigid punch adheres completely to the surface is also considered. Thus, using the shear stress for frictional complete stick condition [4], the following relations for modes I and II are deduced mZ
For cyclic oscillatory load ranging between, PmaxR PminR0, DKIw equals
(26)
Introducing the coordinate transformation in Eq. (18), the crack tip stress field reduces to 1 f 2 f p ffiffi f sin 1 K 3 sin ; (27) sw rr r 2 2
3 3q q p ffiffi sw fK Ksin C 3 sin ; qq 4 r 2 2
Using the crack-analogy transformation
(25)
KII ; KI
Qmax Pmax ð1 K 2vÞ KII Z pffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffi ; paw paw 3pð1 K vÞ
(32)
(33)
and for cyclic oscillatory load, DKII equals Qmax Pmax ð1 K 2vÞ : DKII K parameter Z DKII Z pffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffi paw paw 3pð1 K vÞ (34)
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2.2. Weak adhesion For a weak adhesion under normal loading where the maximum strain energy release rate of the contacting surfaces, Gnor max , is greater than the work needed to normally debond the two contacting surfaces locally, Gnor d , at their nor perimeter, i.e. Gnor max O Gd , the contact does not produce an asymptotically tensile square-root singular stress field as in Eqs. (3)–(5). In this case, the work of adhesion is insufficient to sustain the singularity at the maximum contact width. Consequently, the work at the contact perimeter needed to debond, Gnor d , and to produce contact width, amax, decreases. Based on [27], this work, which is needed to debond two contacting surfaces locally, Gnor d , is assumed sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Q max Gnor : (35) d Z 14:3651 0:3 C Pmax Thus, first, considering the corresponding mode I stress intensity factor due to oscillatory normal load ranging between, PmaxRPminR0, for weak adhesion, i.e. if nor 2 1=3 3 pEDðGnor d ð1 K R ÞÞ ; (36) Pmax K Pmin O 2 8ð1 K v2 Þ it is equal to DKI K parameter Z DKI 2 1=3 3 pEDðGnor d Þ Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pamax 8ð1 K v2 Þ nor 2 1=3 3 pEDðGnor d R Þ K pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pamin 8ð1 K v2 Þ
(37)
where Rnor is the normal load ratio. Second, for weak adhesion under tangential load, the formulation of cyclic loading of a cylindrical pad in contact with a much larger flat substrate, subjected to an oscillatory tangential load, mPmaxRQmaxR0, and under normal load, Pmax, shows that the elastic energy of the cylinder becomes unbounded and the displacements are indeterminate. The solution to this problem depends critically on the overall dimensions of the fretting specimen and the applied far-field boundary conditions [17]. Hence, the precise definition when adhesion becomes weak is ambiguous. But, in the present study, to overcome the indeterminate state, the condition for the normal load, Eq. (37), is assumed to apply for tangential weak adhesion if slip condition, QZmP, is satisfied. In other words, for slip condition, a weak adhesion under tangential load is assumed to be linearly proportional to a weak adhesion under normal load. Based on this, a weak adhesion under tangential load assumed to occur if tan 2 1=3 3 pEDðGtan d ð1 K R ÞÞ Qmax K Qmin O Am ; 2 8ð1 K v2 Þ
(38)
445
where A is the proportionality constant, and the work needed to tangentially debond the two contacting surfaces locally, Gtan d , is assumed Qmax tan (39) Gnor Gd Z d : Pmax Hence, the corresponding mode II stress intensity factor due to oscillatory tangential load ranging between, QmaxR QminR0, for a weak adhesion, is equal to [17] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! tan tan Qcont Qcont Gtan max max R d ð1 K R Þ ; DKII Z min pffiffiffiffiffiffiffiffiffiffiffiffi K pffiffiffiffiffiffiffiffiffiffiffi ; pcmax pcmin ð1 K v2 Þ (40) where cmax and cmin are the maximum and minimum width of the stick-zone in weak adhesion condition (i.e. stickslip), which can be obtained from the following equations 2 c Qmax Z mPmax 1 K max ; (41) amax 2 c ; Qmin Z mPmin 1 K min amin
(42)
assuming adhesion is re-established along the stick-zone. Qcont max is the tangential force that is balanced in the stickzone, and it is defined [4,17] as " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! mPmax p cmax a2max K c2max cont C arc tan Qmax Z Qmax K 2 p c2max K a2max # c c2 K max 1 K max : ð43Þ amax a2max Third, the weak adhesion solution of a wedge does not exist. The present study assumes that the weak adhesion solution of modes I and II stress intensity factor for wedge indenter is equivalent to a cylindrical indenter with a large radius. This assumption is based on the fact that geometrically a cylindrical indenter approaches a flat surface as the radius approaches an infinite value. Further, for a weak adhesion, the contact does not produce an asymptotically tensile square-root singular stress field, and the solution becomes dependent critically on the overall dimensions of the fretting specimen and the applied farfield boundary conditions. Assuming that the weak solution for a sharp wedge loaded normally and tangentially is proportional to the weak solution of a cylinder with diameter, D, equal to 685.8 mm. The cyclic modes I and II stress intensity factor become nor 2 1=3 9AI pEðGnor d R Þ DKI K parameter Z DKI Z pffiffiffiffiffiffiffiffiffiffiffi ; 2 2paw ð1 K v2 Þ (44)
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DKII K parameter Z DKII
definition equals to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dc ; KIIBulk Z sapp 2ðc C dÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! cont tan tan Qcont Q R Gtan max d ð1 K R Þ ; Z AII min pffiffiffiffiffiffiffiffiffiffiffiffi K pmax ffiffiffiffiffiffiffiffiffiffiffiffi ; pcmax pcmax ð1 K v2 Þ (45) where AI and AII are the proportionality constants for modes I and II, respectively. The conditions to instigate a weak adhesion for modes I and II are nor 2 1=3 9 pEðGnor d ð1 K R ÞÞ Pmax K Pmin O AI ; (46) 4 ð1 K v2 Þ 9 Qmax K Qmin O AII m 4
nor 2 1=3 pEðGnor d ð1 K R ÞÞ : ð1 K v2 Þ
(47)
where c and d are the width of the indenter and substrate, respectively. The pad thickness is assumed to be proportional to the radius of the cylindrical pad or the flat width of the wedge pad. Note that this is similar to the stress intensity factor of infinite plate, but with geometric factor. Assuming same material for the pad and the substrate, i.e. no materials mismatch, Eq. (50) reduces to sapp Ec KIIBulk Z pffiffiffiffiffiffiffiffi ; 2pr Ebd½c C d
(51)
and for cyclic bulk stress, DKIIBulk becomes Dsapp Ec : DKIIBulk K parameter Z DKIIBulk Z pffiffiffiffiffiffiffiffi 2pr Ebd½c C d
2.3. Bulk
(52)
The crack-analogy methodology, so far, has considered an indenter in contact with a plane surface under normal and tangential load. However, the loading conditions for a fretting fatigue of a dovetail include bulk stress, sapp. In general, analytically one could model the bulk stresses in the substrate by superposition of the effect of static or cyclic mechanical load applied parallel to the direction of interfacial surface [6,11]. For example, the bulk stresses in the CAM [16] are superimposed by invoking the classical T-stress concept of linear elastic fracture mechanics. The superposition of modes I and II fracture stresses and the applied bulk stresses, assuming linear elastic fracture mechanics, can be expressed as KI KII II sij Z pffiffiffiffiffiffiffiffi sIij ðqÞ C pffiffiffiffiffiffiffiffi sij ðqÞ C sapp d1i d1j : 2pr 2pr
(50)
(48)
However, the MCAM utilizes DK-parameter, and including the bulk stress requires the superposition of an energy form consistent with DK-parameter, but not the approach of superimposed T-stress. To compute the energy due to the bulk stress, the present study utilizes the closed form solution of cracked lap shear specimen with the bottom flat portion of the lap is loaded by a tensile stress or force, that is equivalent to the substrate’s applied bulk stress, see Fig. 2. This energy is then added to the modes I and II stress intensity factor assuming linear superposition applies. Thus, the total stress intensity factor, KTotal, becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bulk K Total Z KI2 C KII2 C KII2 ; (49) and the mode II stress intensity factor, KIIBulk , due to bulk stress is obtained as follows. First, note that the applied load on the lap shear specimen is equivalent to the bulk load, sapp, applied on the substrate, and it produces a mode II crack growth stress intensity factor. Second, the mode II stress intensity factor of the cracked lap shear [28] sapp by
3. Discussion and results As mentioned earlier, the characteristics of a material under fatigue cannot be deduced from other mechanical properties. They must be measured directly through experimental mean. In the present study, the DK-parameter–life curves are established using the experimental data of [18], which have two significances. First, the characteristics of these curves serve as indirect way to validate MCAM such that results have to be consistent with those observed in fatigue’s S–N curve, since the experimental matrix is designed with life ranges between low and high cycle fretting fatigue [18]. Second, it becomes a design tool to predict crack initiation of fretting fatigue. Hence, a summary of the experimental apparatus will be presented for sake of completeness. In these experiments, various configurations were used through the combination of a uniaxial servo-hydraulic fatigue testing machine and a fretting fixture with the appropriate positioning of a pair of pads (see Fig. 4). These fretting pads were clamped on to the surface of a flat specimen. The pad configurations used are 50.8, 101.6, and 304.8 mm cylindrical pad, 50.8 mm flat
Fig. 4. Experimental apparatus.
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Table 1 The load condition of the 2 in. cylindrical pads and their experimental fretting fatigue life [18]
Table 3 The load condition of the 12 in. cylindrical pads and their experimental fretting fatigue life [18]
Life (N), no. of cycles
Life (N), no. of cycles
26,700 31,600 53,400 70,600 86,200 91,900 118,000 121,000 124,000 220,000 371,000 672,000 2,080,000 2,560,000 3,660,000 4,140,000 50,000,000 50,000,000
Substrate axial stress smax (MPa)
smin (MPa)
635.8 699.6 551.9 566.1 686.9 424.6 537.8 416.2 685.5 528.7 686.6 581.7 412.6 685.8 419.8 540.3 506.6 410.1
K39.6 43.9 18.1 53.0 291.4 35.2 233.3 28.7 293.6 232.2 455.8 350.5 185.8 441.6 190.9 372.3 330.6 273.2
Axial load ratio R
K0.062 0.063 0.033 0.094 0.424 0.083 0.434 0.069 0.428 0.439 0.664 0.603 0.450 0.644 0.455 0.689 0.653 0.666
Tangential load Qmax (N)
Qmin (N)
698.00 498.00 427.00 649.00 458.00 547.00 280.00 467.00 405.00 191.00 472.00 547.00 516.00 560.00 512.00 601.00 618.00 467.00
K560.00 K316.00 K294.00 K574.00 K494.00 K445.00 K254.00 K512.00 K209.00 K214.00 120.00 K138.00 K67.00 K53.00 K85.00 427.00 196.00 125.00
with 50.8 mm rounded end pad, and 4.67 mm 908 wedge. A spring-loaded holding fixture attached to the load frame of the test machine was used to position the fretting pads on to the specimen surface (see Fig. 4). A load cell was used to measure the normal load applied to the specimen through the fretting pads. A constant normal fretting load is applied to all specimens, which is 1334 N. Note that this results in various peaks of Hertzian contact pressure at the pad– specimen interface. Furthermore, these various pad configurations are fatigued at various load combination and nominal stress ratio, R (i.e. ratio of minimum to maximum load). The specimen and fretting pads were both were both made of titanium alloy, Ti–6A1–4V. The longitudinal Table 2 The load condition of the 4 in. cylindrical pads and their experimental fretting fatigue life [18] Life (N), no. of cycles 23,200 53,000 57,000 59,400 62,400 124,000 135,000 162,000 286,000 340,000 361,000 710,000 1,120,000 4,890,000 50,000,000
Substrate axial stress smax (MPa)
smin (MPa)
718.9 647.7 594.8 651.7 601.6 457.2 584.2 456.1 559.9 446.0 428.5 587.9 537.4 416.9 505.8
16.0 10.9 41.4 26.9 29.4 25.8 266.7 34.1 243.1 210.1 203.9 401.3 357.8 246.4 268.2
Axial load ratio R
0.022 0.017 0.070 0.041 0.049 0.056 0.457 0.075 0.434 0.471 0.476 0.683 0.666 0.591 0.530
24,376 54,850 160,201 245,686 580,000 1,344,333 13,300,000 19,300,000
Qmin (N)
1116.00 983.00 934.00 832.00 925.00 578.00 903.00 623.00 667.00 485.00 436.00 976.00 767.00 440.00 956.00
K552.00 K649.00 K418.00 K601.00 K498.00 K512.00 98.00 K423.00 K147.00 K49.00 K21.00 516.00 311.00 K9.00 418.00
smax (MPa)
smin (MPa)
441.2 494.6 413.7 577.3 321.7 336.4 287.5 382.4
136.0 178.3 231.7 481.7 248.9 257.4 224.3 290.5
Axial load ratio R
0.308 0.361 0.560 0.834 0.774 0.765 0.780 0.760
Tangential load Qmax (N)
Qmin (N)
336.00 577.00 379.90 300.00 260.00 300.00 163.00 306.00
K224.00 K373.00 K245.00 K300.00 K185.00 K190.00 K117.00 K204.00
tensile properties of the tested material (along the loading axis) were: Young’s elastic modulus, EZ127 GPa, yield strength, syZ930 MPa, and Poisson ratio, nZ0.3. Specimen had a dogbone configuration with thickness, 2bZ1.93 mm and width, wZ6.35 mm in the gage section (see Fig. 4). The applied bulk stress, s, and tangential stress, Q, in the experiments range between maximum and minimum values, which are in-phase. The experimental tests are performed under room temperature. The experimental results are tabulated where the test life of the specimen, the nominal bulk stress ratio, R, and their bulk and tangential loading condition are shown. The results for the 50.8, 101.6, and 304.8 mm cylindrical pads are listed in Tables 1–3, respectively. Similarly, for the 50.8 mm flat with 50.8 mm rounded end and the 4.67 mm 908 wedge pads, the results are shown in Tables 4 and 5, respectively. The experimental life of the specimens range from low cycle fretting fatigue with an upper limit of approximately 104 cycles, transitional low to high cycle approximately Table 4 The load condition of the 2 in. flat with 2 in. rounded cylindrical end pads and their experimental fretting fatigue life [18] Life (N), no. of cycles
Tangential load Qmax (N)
Substrate axial stress
29,300 54,000 55,900 71,600 111,000 124,000 138,000 198,000 223,000 260,000 614,000 1,300,000 1,890,000 3,950,000 6,930,000 10,400,000 165,000,000
Substrate axial stress smax (MPa)
smin (MPa)
691.6 691.3 566.4 574.4 695.6 406.8 704.0 564.6 422.0 484.1 423.8 613.3 555.9 607.5 534.5 546.1 394.4
36.7 36.3 33.0 32.3 326.2 K10.9 337.1 231.9 15.2 173.5 178.2 381.0 356.7 390.8 356.0 242.8 180.7
Axial load ratio R
Tangential load Qmax (N)
Qmin (N)
0.053 0.053 0.058 0.056 0.469 K0.027 0.479 0.411 0.036 0.358 0.420 0.621 0.642 0.643 0.666 0.445 0.458
774.00 770.00 672.00 667.00 538.00 780.00 565.00 396.00 503.00 805.00 440.00 520.00 569.00 596.00 423.00 454.00 543.00
K796.00 K885.00 K676.00 K721.00 K400.00 K587.00 K369.00 K440.00 K534.00 K525.00 K236.00 K111.00 49.00 49.00 49.00 K356.00 71.00
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Table 5 The load condition of the 0.1875 in. flat 908 wedge pads and their experimental fretting fatigue life Life (N), no. of cycles
Substrate axial stress smax (MPa)
smin (MPa)
32,951 147,254 260,065 552,016 5,481,603
392.7 403.5 389.5 391.6 349.3
87.3 188.2 242.3 240.8 234.4
Axial load ratio R
0.222 0.466 0.622 0.615 0.671
Tangential load Qmax (N)
Qmin (N)
1720.00 980.00 670.00 1735.00 1375.00
860.00 440.00 K181.00 500.00 535.00
bounded between 104 and 106 cycles, and high cycle with a room temperature fatigue limit starting approximately at 106 cycles until failure or runout. Furthermore, the experimental loading conditions and their effect on fretting fatigue life are examined by plotting the applied range of the tangential
load and bulk stress versus specimen’s life in Fig. 5a and b, respectively. Based on these figures one deduces the following. The applied tangential load affected fretting life such that decreasing the tangential load results in a higher life (see Fig. 5a). This is within a scatter for the various pad configurations, which is due to the probabilistic characteristics of failure as a result of different metallurgical, configurationally, environmental, and operational influences. Similarly, lower applied bulk stress increases the life of specimen linearly as shown in Fig. 5b. For comparison, the applied bulk stress of plan fatigue is also plotted. It shows that plan fatigue exhibits higher damage tolerance than fretting fatigue. That is, in order for both of them to have equivalent life, the plan fatigue requires higher applied load amplitude than fretting fatigue. Further, the tensile bulk stress produces tensile residual stress field, which cause a shift in the life curve.
Fig. 5. Experimental data of (a) ranges of applied bulk stress versus life and (b) ranges of tangential load versus life.
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Fig. 6. DK-parameter versus life assuming strong adhesion formulation (a) without bulk stress and (b) with bulk stress.
Assuming strong adhesion, the DK-parameter is computed for the experimental data in Tables 1–5. The values are normalized. The analyses are performed with and without bulk stress, sapp. They are plotted in Figs. 6 and 7 where the former uses linear scale and the latter uses logarithmic scale. Note that this common in presenting fatigue data/results, since S–N, or DK-parameter–life in the present study, exhibits power law characteristics in linear scale and bilinear in logarithmic scale. The figures show similar trend to the S–N curve of plain fatigue with lower damage tolerance as expected. The analyses also show cyclic limit of fatigue approximately at 106 cycles. They show dependency on pad geometry and loading condition. For example, the 50.8 mm cylindrical pad exhibits lowest damage tolerance among the ones analyzed, and the wedge
pad shows the highest damage tolerance with and without bulk stress. The 304.8 mm cylindrical pad results are closer to the wedge, which is consistent with the fact that cylindrical pad with infinitely large radius approaches the characteristics of flat/wedge pad. The flat with 50.8 rounded ends shifted toward the 50.8 mm cylindrical pad without bulk stress, however, it shifted towards the wedge pad by including bulk stress. This indicates that without bulk stress, the effect of the contact stresses induced by the rounded ends on damage tolerance dominates. But, by adding the bulk stress, the effect of contact stress on damage tolerance is less significant. The weak adhesion case is also analyzed, and the results with and without bulk stress are shown in Fig. 8. They show that without bulk stress the results are geometry independent within scatter
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Fig. 7. Logarithmic representation of Fig. 6(a) and (b), respectively.
and do not show the cyclic limit. This is contrary to the ones with the bulk stress.
4. Conclusion The present study introduced MCAM, which extends CAM to include various indenter–substrate geometries and the applied bulk stress for both strong and weak adhesion formulations. The results show the following: † Assuming strong adhesion, analyses of MCAM with and without bulk stress showed similar characteristics to S–N curve of plane fatigue within reasonable scatter.
† The current MCAM results show that by including the tensile cyclic bulk stress in the analyses, the damage tolerance of fretting fatigue increased and caused the life curve to shift towards low cycle fatigue. † The DK-parameter showed dependency on the pad geometry where the 50.8 mm has the lowest damage tolerance and the wedge pad has the highest damage tolerance. The 304.8 mm cylindrical pad is in between as expected. † Assuming weak adhesion, analyses of MCAM without bulk stress did not predict the fatigue cyclic limit contrary to analysis with bulk stress. In summary, MCAM shows potential. It further demonstrates the significance of including the bulk stresses in
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Fig. 8. Logarithmic plot of DK-parameter versus life assuming weak adhesion formulation (a) without bulk stress and (b) with bulk stress.
the analysis. The advantages of the present approach are: it is easy to implement in a design process; it is reliable and captures the physical characteristics of fretting fatigue; and it is computationally efficient compared to other approaches such as finite element, which requires a very refined mesh near the contact.
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