A generalized fretting wear theory

ARTICLE IN PRESS Tribology International 42 (2009) 1380–1388

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A generalized fretting wear theory Helmi Attia a,b, a b

Aerospace Manufacturing Technology Centre, Institute for Aerospace Research, National Research Council, Montreal, Quebec, Canada Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada

a r t i c l e in f o

a b s t r a c t

Available online 16 April 2009

Combined impact-sliding fretting wear is a complex phenomenon due to the random nature of the excitation force and the self-induced tribological changes. Available models, which relate wear losses to the process variables, are empirical in nature and bear no physical similarity to the actual mathematical and physical attributes of the wear process. A generalized fretting wear theory is presented to mathematically describe the fretting wear process under various modes of motion; impact, sliding and oscillatory. This theory, which is based on the findings from the fracture mechanics analysis of the crack initiation and propagation processes, takes into consideration the simultaneous action of both the surface adhesion and subsurface fatigue mechanisms. The theory also accounts for the micro-, and macro-contact configuration of the fretting tribo-system. The closed form solution requires the calibration of a single parameter, using a limited number of experiments, to account for the effect of environment and the support material. The model was validated using experimental data that were reported for Inconel 600 and Incoloy 800 materials at room and high temperature environment, and for different types of motion. The results showed that model can accurately predict wear losses within a factor of 73. This narrow range presents better than an order of magnitude improvement over the current state-of-the-art models. Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved.

Keywords: Fretting Wear theory Normal impact Oblique impact Oscillatory sliding motion

1. Introduction Accurate prediction of the fretting wear damage of mechanical components, e.g., aerospace engines and nuclear power plants is extremely critical for safe, reliable, and economical operation [1]. The fretting wear process is quite complex due to a number of factors; the competition between the wear and fatigue processes, the presence of sliding and impact motions, and the frictioninduced thermo-mechanical effects resulting from the thermal constriction phenomenon introduced by the micro- and macrocontact configuration at the contact interface. Available fretting wear models can be grouped into two categories; empirical and analytical models. In the empirical approach [2–6], the base function of the model bears no physical similarity to the actual mathematical and physical attributes of the wear process and does not reveal the internal relationship and interactions between the process variables. This deficiency makes the predictions unreliable outside the range of tested conditions. Following Archard’s equation for adhesive wear of unidirectional sliding systems [7], the concept of work rate W was adopted to relate the volumetric wear losses to the integral effect

 Corresponding author at: Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada. E-mail addresses: [email protected], [email protected] (H. Attia).

of the applied load and the relative sliding distance at the interface [8–11]: Rt _ _ N ¼ 0 F NRðtÞLðtÞdt (1) W t dt 0 _ where FN(t) is the normal contact force as a function of time t, LðtÞ is velocity of sliding during contact and t is total time over which the work rate is averaged. In relating the volumetric wear rate losses V_ to work rate _N V_ ¼ K w W

(2)

The constant Kw is the specific wear coefficient that incorporates the effects of the surface flow stress sf (hardness) of the softer material and the environment.

2. Limitations of the work rate concept 2.1. Oscillatory sliding fretting wear While Stowers et al. [12] confirmed the applicability of Archard’s work rate approach for predicting oscillatory sliding fretting wear, contradictory conclusions can, however, be drawn from the experimental data reported in [13–15]. These data, which were obtained for different material combinations, showed a

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ARTICLE IN PRESS H. Attia / Tribology International 42 (2009) 1380–1388

strong linear dependence of wear coefficient Kw on the slip amplitude. So¨derberg et al. [16] have conducted accelerated fretting wear tests on 304 austenitic stainless steel against 304 austenitic stainless steel at a frequency of 20 kHz. The tests were conducted for normal contact forces 5.4oFNo49.8 N and slip amplitudes of 13odo32 mm. These test conditions provided work rates in the range of 2.8–65 Nm/s. Analysis of the results indicated that the specific wear coefficient Kw varies by a factor of approximately 15 (Fig. 1). Instead of being constant, the wear coefficient seems to assume a linear relationship with work rate. This leads to the conclusion that even with pure oscillating sliding motion, the validity of the work rate concept in its current form is questionable. An important conclusion that can be drawn from this analysis is that for the same change in work rate, the specific wear coefficient appears to be slightly affected by variation in a

F= 5.4 N

F=9.8 N

F= 27.6 N

F= 49.8 N

Specific wear coefficient K, Pa

1.00E-14

1381

normal load as compared to the effect of displacement amplitude; a ten-fold increase in the normal load results in a change in Kw by a factor ofo2, while as d is increased by a factor of 2.5 only, the coefficient Kw increases eight-fold. Therefore, one can conclude that lumping these two variables in a single parameter, as in ‘work rate’, which assigns equal weights to sliding distance and normal load, may lead to a significant error in predicting wear rates. From the author’s experience, the frequency contents of the forcing function that excites most mechanical components undergoing fretting wear damage is mixed and contains more than one frequency component [1]. The experimental results reported by Leheup et al. [17] were produced under the simultaneous action of two different excitations: (a) high wear component characterized by high slip amplitude of 250 mm and low frequency of 15 Hz, and (b) low wear component characterized by low slip amplitude of 40 mm and high frequency of 50–250 Hz. The results of the fretting wear tests indicated that higher work rates associated with the mixedfrequency mode may result in lower wear rate coefficient Kw.

2.2. Normal and oblique (compound) impact

1.00E-15

1.00E-16 0

10

20

30

40

50

60

70

Work rate, Nm/s Fig. 1. Variation in the specific wear rate coefficient Kw with work rate, in oscillatory sliding fretting, for different normal loads F and slip amplitudes 13o do32 mm (materials: 304SS/304SS. Specimens geometry: crossed cylinders).

The author has analyzed and examined the data available in the open literature [18–24] and a large number of classified research reports that pertain to the wear rate–work rate relationship various materials used in the nuclear power plants [25]. A sample of this data is presented in Fig. 2 for Inconel I-600 and Incoloy I-800 fretted against carbon steel and stainless steel at 200–285 1C. The support geometry includes drilled holes (DH), broached holes (BH), wires, threaded rods (TR), welding rods (WR) and knife edge (KE). The relative orbital motions between the contacting bodies include oscillatory sliding, normal impact and _ Wg _ relationship is oblique impact. Fig. 2 shows clearly that the Vf not unique and may result in two to three orders of magnitude error in wear rate predictions. This is attributed to the fact that the work rate concept does not appropriately characterize the

1E-11 DH,I600/CS,265 C DH,I6OO/CS,265 C

Specific wear rate coefficient K, 1/Pa

BH,I600/CS,265 C

1E-12

BH,I600/CS,265 C DH,I600/304SS,285 C WR,I600/304SS,285 C TR,I600/304SS,285 C

1E-13

KE,I600/304SS,285 C DH,I800/304SS,285 C BH,I800/410SS,265 C

1E-14

WR,I800/304SS,285 C LB,I8OO/410SS,265 C TR,I800/304SS,285 C KE,I800/304SS,285 C

1E-15

DH,I600/CS,265 C DH,I600/SS,265 C DH, I600/CS, 200 C

1E-16 0

200

400

600

800

Work rate, mW Fig. 2. Compiled data on the relationship between the specific fretting wear coefficient and work rate for I-600 and I-800 alloys against supports of different materials, geometries and temperatures.

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physical nature of the impact-sliding wear process, and does not account for the contribution of subsurface fatigue. As in sliding wear, shear stresses produced by impact are ultimately responsible for wear and detachment of wear particles by gradual onset of fatigue. In normal impact, the shear stresses reach their maxima in the depth of the material. The magnitude and position of the maximum shear stress tss in the sub-surface layer are proportional to maximum Hertzian contact pressure and the contact radius. In compound impact, if the coefficient of friction m exceeds certain value, the maximum shear stress tends to rise from the sub-surface layer to the surface. Following Palmgren’s empirical formula for fatigue failure, Engel proposed a wear theory for normal and compound impact based on the following formula [26]: V w ¼ ki ð1 þ bÞNsno ;

n9

(3)

below a certain depth from the surface [30], as supported by the experimental evidence reported in [30–35]. It should also be realized that fracture mechanics analysis of the fretting wear problem requires proper consideration of the thermo-mechanical interactions at the fretting interface. While accurate estimation of the temperature field in the subsurface layer is needed to define the micro-contact configuration, the later, in turn, influences the thermal constriction phenomenon and the temperature rise needed to overcome this resistance allowing the frictional heat to flow into the solid bodies.

3. Thermo-mechanical aspect of the fretting wear model 3.1. Micro- and macro-contact configuration

where the constant ki depends on the contact configuration, material properties and environmental conditions. The parameter b is a measure of the ratio between the surface and sub-surface damages. For normal impact, b ¼ 0. The peak contact pressure so is determined from the normal approach speed. A physical interpretation to Eq. (3) is that impact wear is the optimal stress relieving path, in which the progressively fatiguing impact area tries to adjust itself by assuming a shape (wear scar) more conforming to that of the hard indenter. Engel’s impact wear theory for zero and measurable wear has been experimentally verified [26–29]. The above discussion underlines two reasons for the failure of Archard’s work rate concept to model normal and oblique impact wear processes:

For plastically deformed contact asperities, the normal contact pressure is equal to the effective flow stress of the softer material sf, which is approximately three times the yield stress sy of the material and 4–6 times its shear yield stress ty. The contact configuration at the fretting interface is defined by the apparent area of contact Aa (macro-contact area) and the real area of contact Ar (micro-contacts). Following Tsukada’s investigation [36], the contact between two rough surfaces is mathematically replaced by a perfectly smooth surface and an equivalent rough surface. The standard deviation of the asperity heights s and the mean absolute slope jmj of the asperities of the equivalent rough surface are related to those of the original surfaces S1(s1,|m1|) and S2(s2,|m2|) by the following relations:

(a) The source of wear energy. The wear energy Ew in sliding is accountable on the basis of the work done by the Coulomb friction force. However, in normal impact Ew arises from the hysteresis work of the stress cycle. Eq. (3) shows that the wear rate is proportional to s9 (or F 3N ), being a fatigue-controlled process. If Archard’s wear law (which is linearly proportional to FN) is to be used, the effect of the impact force would be greatly underestimated. In normal impact, the relative sliding speed is zero and the sliding distance is only due to the elastic deformation of impacting solids. When this extremely small value of micro-slip is forced into Archard’s equation, the effect of the impact forces will further be reduced. Therefore, one would obtain very high specific wear rate (per unit work rate) if the work rate concept is applied to impact wear. (b) The contact/wear scar geometry: The stress-dependence of the wear process (Eq. (3)) implies that wear rate depends strongly on the contact geometry. Again, in Archard’s work rate formulation, only the total normal force is accounted for, with no consideration to the distribution of contact stresses and stress concentration effects due to misalignment, and the relative stiffness of contacting bodies /14/. Therefore, for constant contact forces and slip amplitudes, the wear rate predicted by the work rate parameter would be timeindependent. This hypothesis is not supported by experimental data. On the other hand, the peak stress in Eq. (8) is directly related to the contact geometry. Therefore, one should expect the wear rate coefficient to be sensitive to the geometry of the tube support structure and to decline with time, unless other factors are introduced and the wear mechanism is changed.

s¼

In the following section, a unified fretting wear theory is presented and formulated. This is made possible by considering the similarity between various types of fretting wear; oscillation, sliding and impact. In all cases, crack initiation is driven by the plastic deformation and void nucleation that only take place

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s21 þ s22

and

jmj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jm1 j2 þ jm2 j2

(4)

For surface asperities heights that follow a normal distribution curve, the attributes of the micro-contact configuration are defined by the following relations [37]:

2 ¼



Ar Aa

 ¼

pc

sf

¼

1 erfcðxÞ 2

(5)

 !   N mic 1 jmj 2 exp 2x2 g¼ ¼ erfcðxÞ Aa 16 s

(6)

rffiffiffiffi  8 s¯ expðx2 ÞerfcðxÞ p jmj

(7)

r mic ¼

where e is the constriction ratio, pc is the contact pressure, g is the density of micro-contact areas, r mic is the average radius of microcontact areas and Nmic is the number of micro-contacts over the apparent area Aa. The parameter x, which appears inpthe ffiffiffiffiffiffi complimentary error function erfc(x), is defined as x ¼ Y= 2s, where Y is the separation between the median planes of contacting surfaces. The derivation of the proposed impact/sliding wear theory is based on the following assumptions: (1) The force between the interacting bodies is transmitted through identical spherical asperities that are uniformly distributed over the contact interface (Fig. 3). (2) The interaction between neighboring contact asperities is assumed to be negligible. (3) The effect of strain-rate on the wear process is assumed to be negligible. (4) The average impact force and the total number of asperity contacts remain constant for the entire wear process.

ARTICLE IN PRESS H. Attia / Tribology International 42 (2009) 1380–1388

Surface 2 Spherical contact asperity

1383

Number of columns (ncl = 3 in this case) Motion of asperities

2aw

500

2amac Surface 1

Temperature (C)

400

ac

Sa Leading asperities

300 200 100 0 0

Fig. 3. Idealization of contact configuration of impacting rough surfaces.

100 De

3.2. Thermal constriction resistance in fretting Due to the fact that the real contact area is a very small fraction of the apparent area of contact, the friction heat generated at the micro-contact area will spread out rather than taking a straight path, giving rise to the thermal constriction resistance RC defined by the following equation: RC ¼ ðT C  T m Þ=Q

(8)

50 200

pth

z (m

300 icro

0 400

ns)

500

x (mic

–50

Fig. 4. Temperature field in the middle of the HFC (material: silicon nitride, r mic ¼ 7.5 mm, f ¼ 200 Hz, Qmax ¼ 0.192 W, Sa ¼ 100 mm, e ¼ 0.15 and Fo ¼ 1200).

Material properties, type of contact

where TC is the average contact temperature over the microcontact area, Tm is the mean temperature averaged over the whole surface area, and Q is the time-dependent frictional heat flow. Assuming that each of the micro-contact areas are associated with a ‘‘heat flow channel’’ HFC of a square cross-sectional area (AHFC ¼ Sa  Sa). Therefore, the constriction ratio e is defined by the following equation:

Contact mechanics, number and load on asperities

2 ¼ Amic =AHFC ¼ pðr 1 =Sa Þ2 ¼ pc =sf

Crack propagation rate (CTSD) analysis

(9)

One of the process variables that affect the dynamic thermal constriction phenomenon in fretting is Fourier modulus Fo r 2mic Þ

Fo ¼ 4a=f ðp

(10)

where f is the frequency of the fretting process and a is the material thermal diffusivity. The dimensionless instantaneous constriction parameter c is related to the thermal constriction resistance RC by the following relation:

c ¼ 4krmic RC

(11)

where k is the material thermal conductivity. The constriction parameter is assumed to be time-independent during the major portion of a fretting cycle, and only its average value c* is to be considered. Using an analytical approach, Attia et al. [38] proposed the following relationship for the case of a microcontact asperity in the center of a square HFC: pffiffiffiffiffi pffiffiffiffiffi c ¼ 0:953 þ 0:00074 Fo þ 0:0955 Fo  1:256 (12) Detailed analysis of the calculation of the temperature field and the temperature rise at the contact interface due to friction is presented in [38]. Fig. 4 shows an example for the high temperature rise that can be established at the micro-contact area and the steep temperature gradient in the subsurface layer. Once average interface temperature is estimated, the effective flow stress of the softer material sf is determined and the parameters defining the micro-contact configuration (Eqs. (5)–(7)) are obtained.

4. Fracture mechanics-based generalized fretting wear model The sequential events leading to wear debris formation under oscillatory sliding conditions, normal impact or oblique impact involves crack initiation and propagation in the subsurface layer

rons)

Crack nucleation analysis Experiments conducted for different conditions

Crack growth direction analysis

N

In

agreement

3 data point calibration

Ye Final calibrated wear equation

Implementation of characterization Fig. 5. Hybrid approach involving FEM-based crack initiation and propagation analyses and experimental calibration of the generalized fretting wear model.

near the micro-contact asperities. The fracture mechanics analysis undertaken in [39] provides the theoretical framework for developing a generalized impact-sliding fretting wear theory. The approach to be followed is described in Fig. 5. It consists of contact mechanics, crack initiation and crack propagation analyses using FEM. The crack propagation rate per loading cycle is determined through the crack tip sliding displacement (CTSD). The latter is obtained from the residual relative displacement observed at the crack tip.

(1) Subsurface crack initiation: Under the repetitive application of normal and shear stresses that are transmitted to contacting bodies of the fretting tribo-system through the micro-contact areas, cracks are initiated from inclusions located in the subsurface layer when the sum srr of the hydrostatic pressure and the residual shear and applied stresses exceeds the coherence strength si of the inclusion–substrate interface. The crack initiation criterion and the likelihood location of crack initiation were examined in [39] for elasto-plastic materials with isotropic work hardening properties. Example for the

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likely location of crack initiation, where srr is maximum and greater than si is shown in Fig. 6(a) for rmic ¼ 5 mm, with frictionless contact m ¼ 0, and sf/ty ¼ 5. The critical depth for

amic

hcr

Crack nucleation zone

Nucleation depth from the surface, y/rmic

5 4

Zone of maximum σrr

3

d‘ ¼ ks ‘ ‘_ ¼ dN

2 1 0 4

5 6 Maximum Hertzian pressure value, τy

crack initiation hcr was found to be of the order of the contact asperity radius amic, as can be seen in Fig. 6(b). These results are confirmed by the experimental data reported in [40,41]. It was also shown that crack initiation takes place effectively after the first loading cycle, indicating that wear rate is only controlled by the crack propagation process. (2) Crack propagation: Cyclic loading of the material will cause the dynamic stresses generated in the contact region to propagate the crack parallel to the surface, following the weakest shear resistance direction. The process continues until a critical crack length ‘cr is reached [39]. At this point, the stress field seff at the tip of the crack becomes non-symmetrical and drives the crack towards the surface. This is demonstrated by Fig. 7(a–c). The criterion for estimating ‘cr is based on the supposition that a mode-I crack propagates in the direction of maximum effective stress seff [39]. Fig. 7(d) shows that the crack propagation rate is linearly related to the crack length, as supported by the experimental observations made in [42]

7

Fig. 6. Crack nucleation analysis: (a) contour plot of the interfacial stress around an inclusion srr/ty (for rmic ¼ 5 mm, m ¼ 0, and sf/ty ¼ 5), and (b) zone of maximum srr and likelyhood for crack initiation for different maximum Hertzian contact pressure.

(13)

where ks is a proportionality constant that depends on the maximum Hertzian contact pressure of the asperity sf, the material properties, the coefficient of friction at the crack surface mc and to a lesser extent on the shear traction [39]. (3) Debris formation through the process of delamination: Once the critical crack length ‘cr is reached, the wear debris will finally be formed and separate from the surface in the form of thin circular sheets, the average thickness of which is hcr and the average radius is the critical crack length ‘cr (Fig. 8).

Crack propagation rate, μm/cycle

0.06 0.05 0.04 0.03 0.02 0.01 0 0

5

10 15 Crack length, μm

20

P > Pcr Fig. 7. Crack propagation analysis; contour plot of the effective stress around a crack for: (a) crack length ‘=r mic ¼ 0:4, (b) crack length ‘=rmic ¼ 1:0, and (c) crack length ‘=r mic ¼ 1:4. (d) CTSD growth rate for rmic ¼ 10 mm, m ¼ 0, and sf/ty ¼ 5 ¼ 5.

ARTICLE IN PRESS H. Attia / Tribology International 42 (2009) 1380–1388

1385

Asperity

R20 > R1 > R10 2amic Wear

+ R20

hcr

+

Body 1

Subsurface inclusion

R10 Body 2

cr

Fig. 8. Wear debris formation due to impact-sliding loading.

4.1. Formulation

Radial Clearance Cr

Since the number of loading cycles Ni that are required to initiate a crack is much smaller than the loading cycles Np required to propagate it to the reach the critical length ‘cr , then NpENt [39], where Nt is the total number of loading cycles that lead to debris formation. The differential relationship given by Eq. (13) can be integrated, to obtain Np, as a function of the critical and initial crack length ‘cr and ‘o , respectively, and the constant ks ‘cr ¼ ‘o expðks Np Þ

(14)

It is assumed that the initial crack length ‘o is equal to the radius of the smallest inclusion that satisfies the crack nucleation criterion based on the critical local elastic energy, ‘o ¼ 0:025 mm [43]. The wear volume Vmic associated with a single asperity contact can now be expressed as a function of the total number of excitation cycles Nx V mic ¼ phcr ð‘o expðks Np ÞÞ2



nN x Np



(15)

where n is the number of peak loads during each excitation cycle. Eq. (15) implies that the debris formation is not a continuous process, but rather one that involves an incubation period after the complete removal of a layer. From Eqs. (5), (6) and (15), the following expression for the total wear volume after Nx excitation cycles is obtained: V ¼ nmic V mic ¼ hcr ð‘o expðks Np ÞÞ2

!  nN x Np r 2mic sf P

(16)

where P is the applied peak load. The above equation represents a general fretting wear model, relating impact-sliding fretting wear losses to the process variables. The quantity ‘n’ is strongly dependent on the relative orbital motion between the bodies of the tribo-system under consideration. In the following subsections, an expression for the parameter ‘n’ is derived for different types of orbital motions. Fig. 9 shows two bodies 1 and 2 with a radial clearance Cr and initial radii R10 and R20, respectively. The elliptical orbital motion of body 1 relative to the fixed support 2 represents a general case. When the ratio of the semi-minor axis to the semi-major axis b/a, a4b, approaches the limiting values 0 and 1, the orbital motion represents normal impact and pure sliding, respectively. 4.1.1. Oblique impact For oblique impact, the number of loads ‘n’ during each excitation cycle depends on the number of times the asperities pass over the crack nucleation zone during a single excitation. This is obviously related to the average sliding distance per impact d relative to the spacing between adjacent contact asperities Sa (Fig. 3). By neglecting the sticking period during

Fig. 9. Curved body obliquely impacting a conforming support.

contact duration t, one can estimate ‘n’ from the following relation:   d (17) n¼ 1þ Sa The parameter e depends on the geometry of the interacting bodies and represents the number of times the tube contacts the support plate during a single excitation cycle. If body 1 (Fig. 9) is of the form of a cylinder that impacts an open support (a semicylindrical or a flat bar), the parameter e ¼ 1. On the other hand, if body 1 is surrounded by body 2 (e.g., cylindrical support), the parameter e ¼ 2, provided that the radial clearance Cr is small enough to allow body 1 to hit both sides of the support. Expressing the average distance between the asperities Sa as a function of the number of asperities in contact Nmic, the half Hertzian macro-contact length amac, and the axial contact width 2aw (Fig. 3) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4amac aw (18) Sa ¼ Nmic Substitution of Eqs. (17) and (18) into Eq. (16) yields the following expression for oblique impact wear volume: V oblq ¼

 Nx ðF oblq Þ Np

(19)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   p2 mc ln 8a3w E =RP =2aw E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8pamac aw pr 2mic =3P

(20)

Phcr ð‘o expðks Np ÞÞ2 2 f r mic

s



where

F oblq ¼ 1 þ

v cosðyÞ

and E ¼



1  n21 1  n22 þ pE1 pE2

1

and

R¼

R20 R1 R20  R1

(21)

4.1.2. Normal impact In the special case of normal impact, where d ¼ 0 as y approaches p/2, Eq. (16) is reduced to V impact ¼ 

Phcr ð‘o expðks Np ÞÞ2 2 f r mic

s



Nx Np



(22)

It should, however, be noted that the position of the crack nucleation from the surface hcr and the crack propagation constant ks are different from those encountered in oblique impact.

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4.1.3. Fretting sliding wear In the case of a cylindrical body sliding inside a cylindrical support, the sliding distance is 2pCr. With the assumption that the asperities are evenly spaced over the surface, the number of columns of asperities ncl is determined from the ratio of the Hertzian macro-contact length and the average spacing between asperities (Fig. 3): ncl ¼

2amac Sa

The quantity ‘n’ in Eq. (16) that defines the number of times a cracked zone is passed by an asperity can therefore be expressed as n¼

pC r ncl amac

amac

sliding

oblique impact

Upper limit:

10.0

1.0

0.1 Lower limit: 0.0

(23)

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2PR ¼ paw E

normal impact

Predicted wear volume, mm3

1386

(24)

0.0

0.1 1.0 10.0 Measured wear volume, mm3

Fig. 10. Generalized model predictions of the fretting wear of I-600 tubes against carbon steel drilled hole support in water at room temperature under normal impact, oblique impact and sliding conditions [44].

The wear equation for fretting sliding is, therefore, reduced to    Phcr ð‘o expðks Np ÞÞ2 N x  V sliding ¼ F sliding (25) 2 N p sf rmic where Fsliding is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u u 2PR P pðR2  R1 Þ 2pE aw t F sliding ¼ PR 2 paw r 2mic sf pE aw

(26)

In the generalized model expressed by Eq. (16) and its derivatives for oblique impact, normal impact and sliding (Eqs. (19), (22) and (25), respectively) the average peak impact or sliding force P, the number of impacts per excitation cycle n, and the average sliding distances d are determined from the dynamic analysis of the tubesupport system or from force and displacement measurements. Given the geometric configuration of the system (R and Cr), the quantities Nmic, Sa, r mic , amac, and ncl are estimated from the contact mechanics analysis. The quantities ‘cr , hcr, and Np can be estimated from the fracture mechanics model. The parameter ks has to be calibrated from a limited number of tests that simulate the environmental condition (medium, temperature and chemistry), and the tube support material. The starting point in calibrating the proportionality constant ks is to estimate its value from the fracture mechanics analysis conducted in [39], assuming that ‘cr ¼ hcr ¼ amic ¼ 10 mm. Using the results published in [39], the values of the crack tip sliding distance CTSD, which correspond to the crack propagation rate at the critical crack length ‘cr are estimated to be 4.0  109 m/cycle for normal impact and 8.0  109 m/cycle for oblique and sliding motion. The corresponding proportionality constants ks are 4.0  104 and 8.0  104 cycle1, respectively. By randomly selecting any three experimental data points, preferably in the low, medium and high wear rate ranges, the CTSD, value is adjusted by matching the theoretical and experimental results in a least square sense. Using Eq. (14), the calibrated value of the proportionality constant ks is determined.

5. Experimental validation of the generalized fretting wear theory The data published in [22,44,45] on the fretting wear of Inconel 600 and Incoloy 800 tubes were used for validation of the theory. A comparison between the experimental data reported by Ko et al. [44] for normal impact, oblique impact and sliding, and the calibrated model is shown in Fig. 10. These data were generated

Fig. 11. Generalized model predictions of the fretting wear of I-600 tubes against 405 stainless steel drilled hole support in water at 200 1C, under normal impact, oblique impact and sliding conditions [22].

for Inconel 600 tubes fretting against drilled holes supports made of carbon steel in water at room temperature. The calibration values of the constant ks were estimated to be 2.11 104, 5.33  104, and 7.77  104 cycle1, for normal impact, oblique impact and sliding motion, respectively. The volume measurement error was not reported, but it is estimated to be 70.012 mm3. This error band is applied to the reported wear results. Fig. 10 shows that the maximum discrepancy between the model predictions and measured wear volume is within a factor of 73. The experimental data reported by Hofmann et al. [22] for I-600 tubes fretted against drilled holes supports made of 405 stainless steel in water at 200 1C were also compared to model predictions and are presented in Fig. 11. The wear volume error is estimated to be 70.005 mm3. Following the same calibration procedure described above, the figure shows that all experimental data generated for various types of tube orbital motion were well predicted within a factor of 73. At this point, it is instructive to indicate that the error margin in predicting impact/sliding fretting wear rate using the conventional work rate parameter is greater than one order of magnitude (factor of 33). This demonstration underlines the improvement in the proposed model over the direct correlation of data using work rate concept, which is widely used by the nuclear industry.

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Acknowledgement

0.100 Predicted wear volume, mm3

1387

The financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada is gratefully acknowledged.

Upper limit: +1.75

References

0.010

Lower limit: +1.75

0.001 0.001

0.010

0.100

Measured wear volume, mm3 Fig. 12. Generalized model predictions of the fretting wear of Incoloy-800 tubes against 405 stainless steel drilled hole support in pressurized water at 200 1C, under normal impact [45].

The third set of data used in this validation was reported by Hofmann et al. [45] for normal impact of Incoloy 800 against 405 stainless steel in pressurized water at 200 1C. With a calibrated value of ks ¼ 4.37  105 cycle1, a comparison between the model predictions and the measured wear losses is given in Fig. 12. The figure shows that the ratio between the model predictions and the actual experimental data is better than 71.75. This excellent agreement shows the capability of the generalized model to describe the extremities of sliding and pure impact conditions. The validation test cases presented in these sections clearly demonstrate the validity of the proposed theory to describe the physical phenomena involved in the wear process. This model provides a viable option to tribologists at the design stage, where only limited data base is available for some material combinations and environmental conditions, particularly if the data fall outside the dynamic characteristics of the tube-support system under consideration.

6. Conclusions A generalized fretting wear theory was developed to predict fretting wear losses for various modes of motion; impact, sliding and oscillatory. This model, which is based on the findings from the fracture mechanics analysis of the process, takes into consideration the subsurface fatigue mechanisms and the micro, and macro-contact configuration of the tube-support system. The closed form solution requires the calibration of a single parameter, using a limited number of experiments, to account for the effect of environment and the support material. The model was validated using experimental data that are generated for Inconel 600 and Incoloy 800 tube materials at room and high temperatures, and for different types of motion. The results showed that model can accurately predict wear losses within a factor of o73. This narrow range presents an order of magnitude improvement over the current state-of-the-art, where work rate is used to correlate fretting wear losses to the energy imparted to the system. The good agreement between the model predictions and the experimental results shows that the model correctly describes the physical phenomena involved in the fretting wear process of heat exchanger tubes undergoing complex orbital motion. The model provides a useful tool for predicting the life and the tribological characteristics of heat exchanger tubes at the design stage.

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