Dynamic tribological response of SiC fracture surfaces

Mechanics of Materials 38 (2006) 186–202 www.elsevier.com/locate/mechmat

Dynamic tribological response of SiC fracture surfaces H. Huang, R. Feng

*

Department of Engineering Mechanics, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, United States Received 3 November 2003; received in revised form 19 January 2005

Abstract A new experimental method combining dynamic tribometry based on the torsional Kolsky bar technique and threedimensional surface profilometry that enables characterization of surface topography and wear has been applied to study the dynamic tribological response of closed silicon carbide (SiC) fracture surface pairs under normal compressive stresses up to 1900 MPa. It is found that global sliding of such a tribo-pair occurs when the interfacial shear stress reaches approximately 39% of the applied normal stress. The engaged interfacial asperity pairs mostly undergo shear-induced disengagement as the sliding initiates. However, surface wear-crushing of asperity-occurs at isolated locations and evolves, though at a decreasing rate, as the sliding progresses, resulting in fine interfacial wear debris particles that act effectively as solid lubricant. The transient behavior of the tribo-pair is affected strongly by the interfacial wear debris, whose response is sensitive to the acceleration rather than the velocity of sliding. Despite the variation of transient response from one experiment to another, the steady-state response is essentially Coulombic (linear) over the sliding velocities examined (0.05–3.4 m/s) with an effective kinetic friction coefficient of 0.36. A dynamic friction model allowing acceleration-dependent overstress and relaxation has been worked out. It is demonstrated that the model can capture the key dynamic features observed in the transient response and recover the Coulombic relation for the steady-state response. The significance of these findings for analyzing shear cracking in SiC deformed under confining stresses is also discussed.
2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic tribometry; Torsional Kolsky bar; Friction; Wear; Silicon carbide

1. Introduction

*

Corresponding author. Tel.: +1 402 472 2384; fax: +1 402 472 8292. E-mail address: [email protected] (R. Feng).

When deformed under confining stresses, brittle solids may undergo shear cracking. A unique feature of this type of shear cracking is that the fractured surfaces are closed or partially closed because of the local confining stress. Examples

0167-6636/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2005.06.009

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

include the damage of ceramic armor under ballistic impact (Viechinichi et al., 1991), failure of brittle materials under bi-axial compression(NematNasser and Horii, 1982), and intersonic shear crack propagation along bimaterial interfaces (Singh et al., 1997). Partial crack closure may occur even in the evolution of a morphologically complex crack under far-field tensile load (Swanson et al., 1987; Rodel et al., 1990). The frictional contact between a pair of closed fracture surfaces is expected to play a significant role in such a failure process. A good understanding of the tribological response of a closed fracture surface pair including the effects of interface evolution is therefore critically important to the development of model analysis that can accurately predict the damage evolution and the resulting material response. Microcracking models assuming flat fracture surfaces and Coulombic (linear) friction behavior to account for the effects of frictional contact have been proposed (Margolin, 1984; Addessio and Johnson, 1990; Theocaris et al., 1993; Rajendran, 1994; Wright, 1998). However, little has been done to verify the validity of these simplifications independently. In particular, tribological study of fracture surface pairs is lacking. Technically, time-resolved tribometry to measure the transient tribological response of such a surface pair remains a challenge. Another outstanding issue is how to correlate the dynamic tribological behavior of a surface pair with the characteristic features of surface topography and the effects of surface wear. The work by Rajagopalan and Prakash (1999) and Espinosa et al. (2000) has demonstrated that the torsional Kolsky bar (TKB) technique can be utilized for dynamic tribometry. The technique has recently been developed further to enable time-resolved measurement of the transient frictional response of surface pairs under a wellcontrolled single stroke of frictional sliding and preservation of the as-tested surfaces for post-test surface examination (Huang, 2003). It has been shown (Huang and Feng, 2004) that integrating such a dynamic tribometric experiment with optical three-dimensional (3-D) profilometric examinations of the initial and as-tested surfaces provides an effective method to characterize the dynamic

187

tribological response of fracture surface pairs under well-defined impulsive loading and in relation with the effects of surface topography and wear. In this work, this new experimental method has been applied to investigate the dynamic tribological response of polycrystalline a-6H silicon car-bide (SiC) fracture surface pairs. To correlate the tribometric measurements with the topographic features and wear evolution of the surfaces, comparative examinations of the pre- and post-test surfaces were performed using both 3-D optical surface profilometry and scanning electron microscopy (SEM). The loading conditions examined in this work cover a range of sliding velocities between 0.04 and 3.4 m/s and various normal stresses up to 1900 MPa. The typical time duration of the TKB tribometric tests is 250 ls. The compressive stresses attained in this study are moderate. However, the nominal shear strain rate in vicinity of the specimen (tribo-pair) interface (where most of the shear displacement takes place) reaches 106 s
1 or higher. The experimental results obtained are the first of this kind. Compared with the measurements from conventional pin-on-disk friction experiments (e.g., Ishigaki et al., 1986), the current results are more relevant for use in modeling brittle shear cracking and damage in ceramics deformed under confining stresses. In what follows, the experimental method will be described briefly first in Section 2. The experimental results will then be presented in Section 3 along with related discussion. Finally, the key findings of this work are summarized in Section 4.

2. Experimental method The experimental method used in this work consists of the dynamic tribometry based on the torsional Kolsky bar technique and the surface profilometry that facilitates quantitative topographical comparison between the initial and as-tested specimen surfaces. In addition, SEM is used to examine the microscopic details of tested surfaces. A brief description of the experimental method is provided below. Other details can be seen elsewhere (Huang, 2003; Huang and Feng, 2004).

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2.1. Dynamic tribometric experiment The dynamic tribomeric experiment used in this work is based on a modified version of the torsional Kolsky (or split-Hopkinson) bar technique. The conventional torsional Kolsky bar (TKB) technique is typically used to study dynamic plasticity and adiabatic shear localization in metals (e.g., Duffy et al., 1971; Hartley et al., 1985). Chichili and Ramesh (1999) reported a modification to the standard TKB technique, in which an additional axial piston is used to enable both dynamic torsional and axial loadings. The current experimental setup combines this loading technique with a special bar design to achieve high compressive loading as well as single-stroke testing and preservation of as-tested tribo-pair surfaces. 2.1.1. Experimental setup and principle Fig. 1 presents a picture (a) and a schematic view (b) of the TKB device used in this work. The device consists essentially of two long metal bars of 1.0 in. diameter. The two bars are mounted on a leveled steel track. The bar carrying the pulley is the input bar, and the bar behind the specimen (referred to also as the tribo-pair) is the output bar. The input bar is made of 7075-T6 aluminum alloy (Al) and the output one 4041 stainless steel. The pulley is 10 in. in diameter and is clamped on the front end of the input bar. The pulley is used to convert the axial action of a hydraulic piston to a torque applied to the input bar. A two-way piston is aligned axially with the bars as the actuator for axial loading. Before each test, the friction clamp is activated first. The segment between the clamp and pulley is then twisted and compressed by activating the pulley and axial piston to prestore torsional and compressive loadings of designed magnitudes. Forcing a pre-notched bolt that locks the friction clamp to fracture releases rapidly the stored axial and torsional strain energies giving rise to a pair of equal-amplitude fast traveling axial stress waves and a pair of equalamplitude trailing torsional stress waves propagating away from the clamp in the both directions as illustrated in the distance–time (x–t) diagram shown in Fig. 1(b). The waves propagate towards the specimen are loading waves and those towards

the pulley are rarefaction (partial release) waves. The latter are reflected from the pulley (which behaves dynamically as a rigid wall) to form the complete axial and torsional unloading waves. The specimen sandwiched between the two bars consists of two cylindrical disks of the same diameter as the bars. One is glued to the input bar and the other to the output bar using high-strength epoxy. Placed in contact, the two disks form the tribo-pair. The contact surfaces are machined to annular rings of the same size. The width of the annular ring is designed to be sufficiently small so that when the tribo-pair slips under an applied torque the variation of sliding velocity across the width of the interface is minimal. The waves incident on the tribo-pair are partially transmitted and partially reflected because of the mechanical impedance mismatches resulted from the asymmetric (Al-steel) bar paring. The magnitudes of reflected and transmitted torsional waves are also affected by the relative sliding of the tribo-pair. While the average normal and shear stresses sustained at the annular interface can be determined respectively from the transmitted axial and torsional waves, the determination of average shear-sliding velocity at the interface requires the measurements of incident, reflected and transmitted torsional waves (Huang, 2003). Note that the asymmetric bar pairing makes the effective compressive load on the tribo-pair 40% higher than the magnitude of axial loading wave. 2.1.2. Design for tribo-pair separation The x–t diagram in Fig. 1(b) shows the propagation of the axial and torsional wave fronts of interest. The x coordinate measures the position of a wave front with respect to that of the clamp, where all waves are initiated, and the t coordinate measures the time elapse after the waves are initiated. A critically important feature of the current dynamic tribometric experiment is to separate the tribo-pair before the arrival of torsional unloading wave. This is achieved by letting L1/cb1 = L2/cb2, where L1 is the bar length between the clamp and pulley, L2 that of the output bar, cb1 the elastic bar wave velocity in the input bar, and cb2 that in the output bar. This allows the axial rarefaction wave resulted from the reflection of the transmitted axial compressive wave at the free end of

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189

Fig. 1. Experimental setup: (a) modified TKB apparatus, and (b) distance–time (x–t) diagram of the propagation of primary axial and torsional wave fronts in the TKB tribometric experiment.

output bar and the axial unloading wave from the pulley to arrive at the tribo-pair interface simultaneously. The interaction of the two causes each side of the specimen to move away from the other thus separating the tribo-pair and releasing both axial and shear loadings before the arrival of torsional unloading wave from the pulley. 2.1.3. Sample material and specimen preparation The material studied in this work is a hotpressed polycrystalline a-6H silicon carbide (SiC)

purchased from Cercom Inc. of Vista, California (Cercom SiC-B material). The material has been studied extensively for impact applications and there is a rich database available in the literature (e.g., Feng et al., 1998; Grady, 1999). Preparing specimen with macroscopically flat fracture surfaces is crucial for the purpose of this work. The following fabrication technique was developed and used to produce the fracture surface pairs used in this study. A 1-in. thick, 8-in. wide and 11-in. long SiC plate was cut into 1-in. square and

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11-in. long bars using a diamond saw. Each square bar was further ground into a cylindrical rod using a diamond grinding wheel. Each rod was then sliced into short cylinders of 0.9 in. in length. As shown in Fig. 2(a), a deep and sharp circumferential notch was cut in the middle of each cylinder using a 0.012-in. thick diamond wafering blade. The diameter of notch root was approximately 0.6 in. After being inscribed with surface mark(s) parallel to the cylinder axis for the purpose of alignment, each notched specimen was then loaded in a four-point bending configuration as shown in Fig. 2(b) until it fractured into two pieces. This approach can consistently produce SiC fracture surface pairs with good macroscopic planarity. The central portion of each of the fracture surfaces was further removed using a diamond end mill to form an annular-ring test surface as shown in Fig. 2(c). A scanning electron microscopy (SEM) picture of the fractography of one of the SiC fracture surfaces prepared with the above approach is shown in Fig. 2(d). Intergranular fracture is apparent. The average grain size of this material is approximately 5 lm.

2.1.4. Data reduction analysis The direct measurements obtained during the TKB tribometric experiment are the profiles of the stress waves incident on, reflected from and transmitted through the tribo-pair. These wave profile signals are then analyzed with the linear elastic wave theory to determine the dynamic loading conditions and the response of tribo-pair. The times for the axial and torsional waves to traverse the tribo-pair are very short (approximately 4 ls and 6 ls, respectively) compared with the rise times of the waves (typically 70–80 ls) because of the high-elastic stiffness and low density of the sample material. Hence, dynamic stress equilibrium in each of the modes is quickly attained in the specimen. In such a state, the average axial compressive stress over the tribo-pair interface, which will hereafter be referred to as the nominal
, can be related straightforwardly normal stress r to the normal strain associated with the transmitted axial wave et as (Lindhome, 1964)
¼ r

R22 E2 et ; b2
a2

ð1Þ

Fig. 2. Preparation of SiC tribo-pair: (a) picture of notched specimen, (b) four-point bending for controlled breaking of specimen, (c) a finished half of a tribo-pair showing the annular-ring test surface, and (d) SEM picture of as-fractured surface.

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

where R2 is the output bar radius, E2 the YoungÕs modulus of output bar material, and a and b respectively the inner and outer radii of the annular-ring test surface (Fig. 2(c)). Similarly, the average shear stress sustained by the tribo-pair interface, which will hereafter be referred to as the nominal shear stress
s, can be related to the shear strain associated with the transmitted torsional wave ct as (Huang, 2003)
s ¼

3R32 G2 ct ; 4ðb3
a3 Þ

ð2Þ

where G2 is the shear modulus of output bar material. The rotational velocity of the part of tribo-pair attached to the input bar is given by the difference between the rotational velocities associated the incident and reflected torsional waves and that of the part attached to the output bar by the rotational velocity associated with the transmitted torsional wave. Measured in terms of the shear strains associated with the incident, reflected and transmitted torsional waves ci, cr and ct, respectively, the relative rotational velocity of the tribo-pair is (Huang, 2003) Du ¼

cs1 ðci
cr Þ cs2 ct
; R1 R2

ð3Þ

where R1 is the input bar radius and cs1 and cs2 the shear wave speeds in the input and output bars, respectively. Hence, the average interfacial shear or sliding velocity of the tribo-pair, which will be referred hereafter as the nominal shear-sliding velocity
v, can be determined as   aþb a þ b cs1 ðci
cr Þ cs2 ct
v  Du ¼
. ð4Þ 2 2 R1 R2 During the experiment, the four strains that
,
s and
v were measured in a timedetermine r resolved manner. From that data, the histories of nominal normal stress, shear stress and shearsliding velocity were then calculated using Eqs. (1), (2) and (4). Such history profiles will be used in Section 3 to present the experimental results even though the very early portion of each profile, where the development of dynamic equilibrium is still in progress, is a rough approximation.

191

2.2. Surface examination As mentioned earlier, a unique feature of the current TKB tribometric experiment is the preservation of specimen surface after a single stroke of frictional sliding. In conjunction with non-contact optical surface profilometry, this enables a correlation between the measurement of dynamic frictional response and the surface evolution due to wear by quantitative comparison of the topographic features of the initial and as-tested surfaces. The specimen surface scans were performed using a SCANTRON Proscan 1000 system with a S38/2 chromatic sensor. The scanner is capable of measuring three-dimensional (3-D) topographical profile of specimen surface with a 1 lm/step scanning rate and a resolution of 0.1 lm. For each fracture surface pair, both the as-fractured surfaces before the test and the surfaces after the test were examined. The digital data from the 3-D profilometry were further analyzed to determine an asperity-volume-weighted average surface roughness Rv as the representative surface topography parameter. The definition of Rv is an extension of that of the usual asperity-area-weighted surface roughness Ra based on two-dimensional line scanning measurement. It should be noted that the Proscan 1000 gives a relative measurement. The height of the lowest point in the region being scanned is automatically zeroed. To examine the surface features that are too small for the profilometer to pick up, scanning electron microscopy (SEM) has also been used. The specimen surface for SEM was coated with gold to improve conductivity. The coating layers were usually less than 10 nm thick and did not alter the surface characteristics of interest.

3. Results and discussion A series of 10 TKB tribometric experiments has been conducted on the SiC fracture surface pairs. In each of the experiments, the tribo-pair was carefully assembled and aligned under an optical microscope to achieve the best conformal match of the fracture surface pair. A summary of the experimental conditions and steady-state results

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Table 1 Summary of loading conditions and steady-state results of the TKB tribometric tests on matched SiC fracture surface pairs Exp. no.

Nominal normal stress (MPa)

Nominal sliding velocity (m/s)

Nominal shear stress (MPa)

Inner diameter (mm)

Outer diameter (mm)

1 2 3 4 5 6 7 8 9 10

382 438 442 1595 248 1025 1614 1665 439 1897

1.62 1.19 0.70 1.35 1.37 1.12 0.04 1.09 0.05 2.15

139 174 170 582 115 359 443 636 163 650

9.53 9.53 9.53 9.53 9.53 6.38 9.53 9.53 9.53 9.53

12.70 12.70 12.64 10.53 12.7 8.26 10.53 10.53 12.70 10.53

Exp. no. 1 2 3 4 5 6 7 8 9 10

Initial roughness (lm)

Total sliding distance (mm)

Final roughness (lm)

8.3 8.1 9.0 7.8 8.7 9.3 9.2 9.8 9.8 10.7

0.60 0.47 0.31 0.50 0.47 0.42 0.037 0.42 0.073 0.76

7.2 7.8 8.2 6.5 7.8 7.6 8.1 8.1 8.6 Fracture

3.1. Typical dynamic response Fig. 3 presents the results of a typical test (Exp.
6) of this series in terms of the time histories of r (nominal normal stress),
v (nominal shear-sliding velocity) and
s (nominal shear stress). It can be seen from the figure that the axial compressive wave arrives at the tribo-pair first. In less than
rises to a plateau of approximately 100 ls, r

1.6

1000 Normal stress 800

1.2 Sliding velocity

600

0.8 400 Shear stress

0.4

200 0 0

100

200

300

400

500

600

Shear-Sliding Velocity, m/s

Table 2 Summary of initial and post-test roughness measurements

average surface roughness Rv as defined in Section 2.2 was measured before and after the tribometric test. The data are summarized in Table 2 along with the total sliding distance measurements. For Exp. 10, macroscopic fracture damage was observed on the tested surface. The roughness measurement is therefore excluded to avoid confusion.

Normal and Shear Stresses, MPa

is presented in Table 1. The loading conditions cover nominal shear-sliding velocities from 0.04 m/ s to 2.15 m/s and nominal normal stresses from 250 MPa to 1900 MPa. To study the effects of sliding velocity, three experiments were carried out under a nominal normal stress around 440 MPa for various nominal sliding velocities. The sliding velocities of four of the experiments were in a narrow range from 1.19 m/s to 1.33 m/s. The purpose of this set of experiments is to examine the effects of normal stress more carefully. Two different sizes of test surfaces were used. The specimens with a larger annular-ring test surface (9.53 mm inner diameter and 12.70 mm outer diameter) were used for the experiments conducted under a normal stress below 1500 MPa and the ones with a smaller test surface (9.53 mm inner diameter and 10.53 mm outer diameter) for those above 1500 MPa. For each specimen, the volume-weighted

0.0 700

Time, µs Fig. 3. Time-resolved measurements obtained in Exp. 6.

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

1030 MPa and then remains almost constant for rest of the experiment. The torsional wave arrives at the tribo-pair 220 ls later giving rise to a rapid interfacial shear and then sliding of the tribo-pair. The period of initial shear, during which the matched fracture surface pair rotate at almost the same speed, is about 10–20 ls. Relative sliding of the tribo-pair occurs afterwards and
v increases rapidly to its full amplitude and remains nearly constant until the arrival of unloading. The steady-state
v is approximately 1.12 m/s. In response,
s rises rapidly both in the initial elastic shear phase where
v is nearly zero and in the earlier portion of rising
v. However, while
v is still rising rapidly,
s starts to reduce its rate of increase first and then peaks. The rounding (rate reduction) in
s before the peak is related to the evolution of non-uniform sliding (slipping in some regions of the interface and stiction in the rest). The peak can be viewed as the onset of frictional sliding over the entire interface. The peak decays quickly by a small amount to a steady-state level, which lasts until the onset of release. By design, the unloading was initiated by the simultaneous arrivals of the two axial rarefaction waves at the tribo-pair (see
and
s decrease, only Section 2.1). Although both r the latter is visible. This is because the special design used for achieving tribo-pair separation causes an unavoidable axial wave interaction at the location of output-bar normal strain gauge that prevents it from an unambiguous recording
. Since the release of the unloading profile of r of frictional sliding was realized through forced tribo-pair separation while the bars were still rotating, the diminishing of frictional resistance as the tribo-pair disengaged gave rise to an acceleration of the part bonded to the input bar. This acceleration corresponds to the final rapid increase in
v (Fig. 3). Hence, both rapid release in
s and sudden increase in
v are signs of a clean tribo-pair separation. Before the test, the surface topography was measured at four evenly distributed sampling locations by the surface profilometer. Fig. 4(a) shows the digital topographical profile of one sampling area of the pre-test surface. The size of the area is 120 lm · 120 lm. The tip of the highest asperity reaches 96.8 lm from the bottom of the deepest

193

valley. This value is about 20 times the average grain size of the material (approximately 5 lm). Statistical analysis was also performed on the digital topographical data. The results are plotted in Fig. 4(b) in terms of relative asperity height with respect to the mean plane. The data in negative values are physically the relative valley depths with respect to the mean plane. The statistical distribution of relative asperity heights (and valley depths) is Gaussian-like. The important fact is that more than 70% of the surface asperities and valleys are in the range of ±9 lm from the mean plane and Rv = 9.3 lm (Table 2). Note that the specimen position with respect to a sample fixture mounted on the scanner was marked for reference so that the tested specimen can be repositioned accurately. Immediately after the test, wear debris was visible on the as-tested surfaces. The results from SEM examination are as follows. Fig. 5(a) shows the typical features of one of the tested surfaces. The bigger white spots are clusters of pulverized material. Some were in the stage of formation as asperity tips were crushed under the applied compression and shear loading and some were in the process of disintegrating, being displaced and reducing in size. The wear debris after this process was reduced to fine powders (small white dots) that spread rather uniformly over the surface. Some of the fine particles seen should come from the debris worn off from the other surface. On the same surface, there are also a few small regions where much more intense wear is visible. Fig. 5(b) shows a SEM picture of such an area. The surface appears to be covered by a dense layer of wear debris mostly in powders and some in flake-like patches. Clearly, crushing of asperity is the mechanism of surface wear. The material fragments sheared off from the surface were ground down to fine powders. However, only a fraction of engaged asperity pairs were crushed at any given moment while the rest slid over each other. It can therefore be concluded that the asperity pairs over the majority of the engaged interface underwent shear-induced disengagement at the initiation of global sliding of the tribo-pair. In other words, there was an interfacial shear dilatancy during the experiment. Therefore, the data presented in

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H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

Fig. 4. 3-D profilometric measurements of specimen surface (Exp. 6): (a) digital topographical profile, (b) statistical distribution of relative asperity heights of as-fractured surface, (c) digital topographical profile, and (d) statistical distribution of relative asperity heights of as-tested surface.

Fig. 3 are the nominal measurements including the effects of the interfacial shear dilatancy. Unlike the interfacial shear dilatancy observed for rough metallic fracture surface pairs (Huang and Feng, 2004), which can cause a momentary reduction in normal stress, the interfacial shear dilatancy in
as shown the current case has little influence on r
during the of rises of
v and
s by the steadiness of r (Fig. 3). This indicates that the fine specimen surfaces (particular in the sliding regions) and the gap-filling effect of wear debris minimize the influ
. On the ence of interfacial shear dilatancy on r

other hand, such an interface evolution may significantly affect the interfacial friction. It is expected that at a sufficient accumulation, wear debris can act as a layer of interfacial solid lubricant helpful for the attainment of steady-state response. It is also expected that the accumulation of wear debris may slow down the process of further surface wearing. As a result, the generation of the wear debris is self-regulatory. The tested surface with the loose wear debris being removed was further examined using the 3-D profilometry. Its digital topographical profile

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

195

Reduction in Rv, µm

2.5 2.0 Mean value

1.5 1.0 0.5 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Sliding Distance, mm Fig. 6. Reduction in Rv as a function of total sliding distance.

Fig. 5. SEM pictures of as-tested specimen surface (Exp. 6): (a) typical surface features (high magnification) and (b) an intensely worn region (low magnification).

and relative asperity height distribution are shown in Fig. 4(c) and (d), respectively. For comparison with the surface topography before the test, the tested specimen was carefully positioned in the scannerÕs sample fixture using the reference mark so that the sampling area (120 lm · 120 lm) was in close proximity of that scanned before the test. Therefore, the results are not influenced by longwavelength surface waviness (if any). Compared with the profile in Fig. 4(a), the one in Fig. 4(c) has much fewer asperities in the highest height division and exhibits a rather flat area in the leftfront quarter of the surface. These are clear signs of significant surface wear during the tribometric test. Both the mean and summit heights of the asperities are reduced. Indeed, the statistical distribution of relative asperity heights given in Fig. 4(d) is more concentrated in the middle compared with that shown in Fig. 4(b). Fewer asperities are more than 30 lm above or below the mean plane and the full distribution bandwidth reduces from 90 lm to 70 lm. As a result, the tested surface is flatter with a lower value of Rv (Table 2).

It should be pointed out that despite the significant variations in the loading conditions (Table 1) and sliding distance (Table 2), the amount of wear as measured in terms of the reduction in Rv does not vary too much. Fig. 6 plots the reduction in Rv as a function of the total sliding distance for each experiment of this series. Though somewhat scattered, the data clearly indicate that the reduction in Rv is independent of the total sliding distance. This observation supports the following theory of a self-regulatory evolution of the wear debris. In the early stage of interfacial sliding, which encompasses non-uniform sliding during the rise time and the initial phase of global sliding, intense asperity interactions might drive a rapid accumulation of wear debris. As the wear debris accumulates, both the frequency and the intensity of asperity interaction reduce and so does the generation of additional wear debris. This theory appears to be a reasonable explanation not only for the insensitivity of the reduction in Rv to the total sliding distance but also for the observation of very steady kinetic friction response (Fig. 3), which is not intuitively expected for a closed fracture surface pair. 3.2. Effects of sliding velocity A set of tribometric tests was performed to investigate whether the sliding velocity affects the shear stress response. It includes three tests having
plateaus of about 440 MPa and covering similar r

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3.3. Effects of normal stress v = 0.05 m/s v = 0.70 m/s

0.5

v = 1.19 m/s

0.4 0.3 0.2 0.1 0.0 200

300

400

500

600

Time, µs
plateaus are close Fig. 7.
s=
r ratios for various
v values. All r to 440 MPa.

three steady-state
v values: 0.05, 0.70 and 1.19 m/s. The results are shown in Fig. 7 in terms of the time history of the ratio of nominal shear stress to nominal normal stress
s=
r of each of the tests. The
s=
r ratio from the end of rise time to the beginning of release can be considered as the nominal kinetic friction coefficient denoted hereafter by l. The data obtained in Exp. 9 with the lowest velocity
v ¼ 0:05 m=s show that l increases a little from 0.36 at the beginning to 0.38 at the end. The results from Exp. 3 with the median velocity
v ¼ 0:70 m=s are practically the same as those from Exp. 2 with the highest velocity
v ¼ 1:19 m=s. In both cases, l decays slightly from 0.40 at the beginning to 0.38 at the end. Despite the difference between the trend of l evolution under
v ¼ 0:05 m=s and that under the higher
vÕs, l in all three cases settles at the same value of 0.38 right before the onset of unloading. Note that the lowest velocity is more than an order of magnitude lower than the higher velocities. The resulting sliding distance before the onset of unloading is still in the range of non-uniform slipping, in which rising
s is expected. In contrast, slipping over the entire interface is initiated soon after the rise time in the two tests with much higher velocities. This explains why the l evolutions in these two cases are different from that observed under
v ¼ 0:05 m=s. Nevertheless, the variations are insignificant (5%) indicating that the steadystate value of l is not influenced by the nominal sliding velocity.

The steady-state sliding velocities in four of the experiments of this series are in a small range of 1.19–1.33 m/s. The
s histories from these tests
. are plotted in Fig. 8 to examine the effects of r An initial peak can be clearly seen in the profiles
¼ 1025 MPa and r
¼ 1665MPa. obtained under r However, there is no visible sign of such a peak
¼ 248 MPa and r
¼ in those obtained under r 438 MPa. Since
v in each of these tests is similar and so is the sliding distance, the quantity that varies during the rise time (in addition to the stress state) is the amount of wear debris generated. It appears that the response of wear debris is sensitive to the rate of sliding velocity change or the inertial effect. The accumulation of wear debris during the rise time is expected to be negligible under low combined stresses. Under high combined stresses, it can be sufficiently large for the acceleration dependence of the wear debris shear response to become visible near the end of rise time, particularly if some asperities are crushed under the compression before the arrival of shear load. Considering also the self-regulatory effect discussed earlier, wear debris appears to have strong effects on the transient tribological response of the closed SiC fracture surface pairs. Despite the significant variation in the transient response, a well-developed steady state can be observed and the steady-state
s=
r ratio is also

700

σ = 1665 MPa

600

Shear Stress, MPa

Shear Stress/Normal Stress

0.6

500

σ = 1025 MPa

400 300

σ = 438 MPa

200 100 0 100

σ = 248 MPa 200

300

400

500

600

700

Time, µs Fig. 8. Shear stress histories under various normal stresses and similar sliding velocities (1.19–1.33 m/s).

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

independent of sliding velocity with the only exception of Exp. 7, in which non-uniform sliding at an extremely low velocity (0.04 m/s) gave rise to severely heterogeneous interfacial response with a significantly lower peak
s=
r ratio than the rest. It is therefore reasonable to conjecture that the stea
only. In Fig. 9, the dy-state response depends on r steady-state
s values from the experiments other than Exp. 7 are plotted with respect to the corre
measurements. Within the experimensponding r tal accuracy (the error bar are also shown in the figure), the data appear to suggest a linear relationship. The linear best fit to the data gives a slope of 0.363. This means that the steady-state tribological response of the closed SiC fracture surface pairs can be well characterized by a Coulombic (linear) friction law with an effective kinetic friction coefficient of 0.363. Furthermore, the
s=
r ratio at the onset of global sliding in each experiment (with or without the transient peak) falls in a narrow range from 0.38 to 0.4. On average, an effective static friction coefficient of 0.39 is determined for the closed SiC fracture surface pairs. Note that nonlinear response may occur under much higher normal stresses. Extrapolating the current result to normal stresses far beyond 1900 MPa should therefore be proceeded with caution. Validation of such an extrapolation against other relevant experimental data (e.g., Feng et al., 1998) is necessary.

197

3.4. Further examination of the role of wear debris A series of reloading experiments has been conducted on the tribo-pair tested in Exp. 7. The idea is to see whether and how the dynamic tribological response of this fracture surface pair changes with the accumulation of wear debris. There was no cleaning for wear debris between the tests. However, the tribo-pair was realigned for each of the tests to the same initial relative position as in Exp. 7. The first three tests in this series were
¼ 1660– conducted under the conditions of r 1750 MPa and
v ¼ 2:8–3:4 m=s in the steady state. These rather heavy loading conditions were used to enhance the generation of wear debris. The results obtained from these tests are presented in Fig. 10 in terms of the
s=
r ratio vs. the sliding distance S. Apparently, the only significant difference is in the transient part of the frictional response. The specimen was free from wear debris before the first test. The
s=
r
S profile, shown in the figure as the thick solid line, displays an insignificant initial peak. As discussed earlier, the behavior of the interfacial wear debris is sensitive to the acceleration of sliding (which occurs only during the initial rise time) but not to the velocity of sliding. Because the specimen in the first test was initially free from wear debris and the accumulation of wear debris during the rise of
v was limited, the contribution of wear debris response to the macroscopic transient behavior of the tribo-pair

800 0.6

Steady-state data

Shear Stress/Normal Stress

Shear Stress, MPa

600

400

τ = 0 .363σ

200

0 0

500

1000

1500

2000

Normal Stress, MPa Fig. 9. Steady-state frictional response of closed SiC fracture surface pairs. The circles with error bars are the experimental data and the line is
s ¼ 0:363
r.

σ = 1660 MPa , v = 3.4 m/s σ = 1667 MPa , v = 3.2 m/s σ = 1750 MPa , v = 2.8 m/s

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Sliding Distance, mm Fig. 10.
s=
r
S relations measured in three repeated tests on a SiC fracture surface pair.

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H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

was small and so is the resulting initial peak. In the second test, however, a significant amount of the wear debris accumulated at the end of the first test was contained in the tribo-pair. In other words, the acceleration–sensitive response of wear debris should have a strong influence on the macroscopic transient behavior. Indeed, as shown in the figure (the dashed line), a significant
s=
r peak appears at the end of initial rise time. The peak value is as high as 0.5. Note that the first and second tests
and very similar
v values had essentially the same r (see the legends in Fig. 10). The steady-state results are also seen to be very close. The initial wear debris is the only significant difference that can be attributed to the difference between the two transient profiles. The
s=
r
S profile of the third test (the thin solid line) indicates the same. Despite the significant difference in the transient response, the steady-state response changes little from one test to another. The steady-state
s–
r results of all of the reloading tests are presented in Fig. 11 as the triangles. Clearly, they do not significantly differ from those of the tests on as-fractured surface pairs (the squares). In fact, the best linear fit to the two sets of data together has a slope of 0.368, which is practically the same as the result for the as-fractured surface pairs (0.363, Fig. 9) considering the error bars of the

experiments. Clearly, even after repeated heavy loadings, the interfacial condition in the steady state remains essentially the same. However, sufficient interfacial wear debris can result in a significantly higher frictional resistance to a sudden acceleration of sliding than that against sliding under a steady velocity even if it is a high velocity. It should be pointed out that the sliding distance in each of the experiments is within 1.0 mm. This is very different from the loading condition of conventional pin-on-disk wear testing, where the frictional sliding is much greater and measured in number of rotation cycles. This could be the reason why the type of material removal observed in conventional wear testing (e.g., power law like abrasion rate, Harris and Weiner, 1998) was not seen in the current work (see Fig. 6). From the results presented so far, the following recommendations are reached for modeling shear cracking in the SiC deformed under confining stresses. An effective static friction coefficient of 0.39 should be used for analyzing the crack initiation and the early-stage crack extension. An effective kinetic friction coefficient of 0.36 should be used for analyzing a running shear crack that has mesoscopically zigzag morphology. The effect of shear-induced interfacial dilatancy is included in this effective parameter. When such a shear crack is subjected to a sudden reloading, the transient frictional resistance can reach as high as 50% of the normal compressive stress.

800

Shear Stress, MPa

4. Phenomenological modeling

Data on as-fractured surface pairs Data from reloading tests

600

400

τ = 0.368σ

200

0 0

400

800

1200

1600

2000

Attempt has also been made to develop a phenomenological dynamic friction model to describe the observed dynamic tribological response of the SiC fracture surface pairs. In what follows, a description of the model will be presented first. The performance of the model will then be examined by comparing the model simulations with the corresponding experimental data.

Normal Stress, MPa Fig. 11. Comparison of steady-state frictional response of asfractured surface pairs and that of a fracture surface pair under various reloadings. The symbols are experimental data and the line is
s ¼ 0:368
r.

4.1. Dynamic friction model When a ceramic fracture surface pair is subjected to dynamic compression and shear, various

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

199

interfacial processes take place ranging from stiction and simple sliding over to asperity fracture, fragment crushing, and debris particle motion relative to both of the surfaces (or relative third-body motion). Many are difficult to characterize quantitatively. To keep the modeling as simple as possible, however, we divide local contacts into two types. The contacts that can slide according to the Coulomb friction law without involving timedependent interfacial processes such as generation, fracture or relative third-body motion of interfacial particle are of the first type. Included in this type are also the contacts through the interfacial particles that are stationary with respect to one of the surfaces. The rest of the contacts are of the second type and the stiction-to-sliding transition of such a contact is treated as a time-dependent kinetic process. Specifically, we define Ms as the effective stiction modulus (a parameter to be determined). Then, the time history of nominal interfacial shear stress prior to sliding is

second types of contacts, respectively. Since the limiting shear stress depends on p, the accumulation of wear debris at time t depends primarily on x(t) and p. Hence, we may consider the partition functions in the following simple form:

sðtÞ ¼ M s xðtÞ;

Clearly, the value of the second (dynamic) term in Eq. (9b) increases with the value of partition function for the second type of contacts but vanishes with diminishing v_ .

ð5Þ

where x(t) is the shear displacement at time t. When Msx(t) > lkp where lk is the effective kinetic friction coefficient and p the applied nominal normal stress, the first type of contacts are ready to slide and the corresponding local shear stress remains as lkp. However, sliding will not take place until the local shear stress associated with the second type of contacts reaches lt(t)p with lt(t) being the effective transient friction coefficient, which is defined as lt ðtÞ  lk þ T R v_ ðtÞ=vðtÞ;

ð6Þ

where v(t) is the shear-sliding velocity at time t, v_ ðtÞ the time derivative of v(t), and TR the reference time scale to be determined. Eq. (6) is an extension of the Coulomb friction law to allow the consideration of acceleration-dependent overstress and relaxation. Hence, when sliding occurs, the nominal shear stress is the weighted sum of local limiting shear stresses of the two types of contacts, i.e., sðtÞ ¼ ½lk F 1 ðtÞ þ lt ðtÞF 2 ðtÞp;

ð7Þ

where F1(t) and F2(t) = 1
F1(t) are the timedependent partition functions of the first and

F 1 ðtÞ ¼ g exp½
pxðtÞ=e

ð8aÞ

and F 2 ðtÞ ¼ 1
F 1 ðtÞ ¼ 1
g exp½
pxðtÞ=e;

ð8bÞ

where e is the reference energy dissipation per unit area (a parameter to be determined) and g the initial partition of the first type of contacts. If the tribo-pair is initially debris-free, g = 1. Combining Eqs. (5)–(7) gives  M s xðtÞ if M s xðtÞ < slim ðtÞ; sðtÞ ¼ ð9aÞ slim ðtÞ otherwise with slim ðtÞ  lk p þ pT R v_ ðtÞf1
g exp½
pxðtÞ=eg=vðtÞ. ð9bÞ

4.2. Results of model simulations Numerical simulations with the phenomenological dynamic friction model described above have been carried out. In these calculations, lk = 0.368 was used. First, parametric study against the data obtained from Exp. 6 (a test on asfractured surface pair, Fig. 3) was conducted to calibrate model parameters e, Ms, and TR. The nominal normal stress used was as measured (1025 MPa) and the loading portion of the measured v(t) (which saturates at approximately 1.12 m/s, Fig. 3) was used as the dynamic shear loading input. In Fig. 12, the simulated nominal shear stress history with e = 0.2 J/mm2, Ms = 18 000 MPa/ mm, TR = 16 ls, and g = 1 (the dashed line) is compared with the experimental data (the solid line). For this case, the model does a quite good job in capturing the most important dynamic features of the experimental data considering the simplicity of the model. Though not pursued,

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H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

5. Conclusions

1000 Experimental data Model simulations

Shear Stress, MPa

800 600

Reloading with wear debris (1667 MPa, 3.2 m/s)

400 As-fractured specimen (1025 MPa, 1.12 m/s)

200 0 200

300

400 Time, µs

500

600

Fig. 12. Comparison of model simulations with experimental data. The initial interfacial condition, nominal normal stress and plateau sliding velocity of each experiment are specified below the data.

using the measured steady-state shear stress to determine lk may further improve the match. Next, the same set of parameters were used to simulate the reloading experiment with a nominal normal stress of 1667 MPa and a sliding velocity plateau of approximately 3.20 m/s (the dashed line in Fig. 10). Because the test was done with preexisting wear debris accumulated during the previous loading, g = 0.7 was used in Eq. (8a). The simulated nominal shear stress history is also shown in Fig. 12 and compared with the measurement. The model prediction is seen to capture the magnitude of initial transient peak, which is the key dynamic feature of the data, even though the predicted relaxation is somewhat stronger than the measurement. It is particularly encouraging to see that the simple model is sensitive to the effects of pre-existing wear debris. The results shown in Fig. 12 further verify our interpretation for the experimental observation, i.e., the transient tribological response of the fracture surface pairs is sensitive to the acceleration of sliding and the interfacial condition during the acceleration but not to the magnitude of sliding velocity. The Coulombic relation is the right description for the steady-state response. In fact, the phenomenological dynamic friction model recovers the Coulomb friction law when the sliding velocity becomes steady.

The dynamic tribological response of the closed SiC fracture surface pairs has been studied experimentally using a newly developed tribometric technique. The experimental method integrates dynamic tribometry based on the torsional Kolsky bar technique with non-contact 3-D surface profilometry. It is shown that the method enables dynamic tribometric testing of surface pairs under well-controlled single-stroke frictional sliding and preservation of as-tested surfaces. Quantitative comparison of the topographic measurements of initial and as-tested surfaces can correlate the interface evolution (including wear and accumulation of wear debris) with the resulting tribological behavior. The experimental results on matched pairs of as-fractured SiC surfaces with a volume-weight average surface roughness between 7.8 and 10.7 lm indicate that for normal compressive stresses up to 1900 MPa, such a closed fracture surface pair will undergo global interfacial sliding when the shear stress reaches approximately 39% of the applied normal stress. Although crushing of asperity occurs at many isolated contact points as the global sliding initiates, shear-induced asperity pair disengagement (interfacial shear dilatancy) dominates the majority of the interface. The measured 0.39 shear-compression ratio is therefore the effective static friction coefficient. As the global sliding progresses, wear debris in the form of fine powders accumulates, begins to lubricate the interface, and eventually slows down further wearing. Due to this self-regulatory nature, the amount of total wear is insensitive to the total sliding distance for the loading duration and the sliding velocities examined in these experiments. The data from the experiments with higher normal stresses (above 1000 MPa) and from the reloading experiments with combinations of high normal stress (1660–1750 MPa) and high sliding velocity (2.8–3.4 m/s) reveal a peculiar role of the wear debris in the dynamic response of the fracture surface pairs. For the fracture surface pairs containing significant wear debris, the transient response is sensitive to the acceleration rather than the velocity of sliding. A pronounced

H. Huang, R. Feng / Mechanics of Materials 38 (2006) 186–202

transient peak with a shear stress as high as 50% of the applied normal stress may develop near the end of rise time of the rapidly imposed shearing loading. Despite significant variations in the initial transient response, a well-developed steady state has been consistently observed. The steady-state response is essentially Coulombic (linear) over the entire range of sliding velocities covered in this study (0.05–3.4 m/s) and can be well characterized by an effective kinetic friction coefficient of 0.36. Caution needs to be taken, however, if the results are extrapolated to compressive stresses much higher than 2000 MPa. Validation against other relevant experimental data is necessary. A phenomenological dynamic friction model has also been developed to describe the observed dynamic tribological response of the fracture surface pairs. It is demonstrated, by comparing the model simulations with the corresponding experimental data, that the simple model with acceleration-dependent overstress and relaxation can capture the basic dynamic features observed in the transient part of response including the effects of pre-existing wear debris. For steady sliding, the model recovers the Coulombic relation thus predicting the steady-state response accurately.

Acknowledgments The materials reported in this paper are from the work supported by the US Army Research Office (ARO) through Grant No. DAAD19-99-10117. The assistances of the ARO Solid Mechanics Program Managers, Drs. B. LaMattina and M. A. Zikry in conducting this research are gratefully acknowledged. Additional machine shop support was provided by the College of Engineering and Technology at the University of Nebraska-Lincoln. The authors would like to thank Mr. Mark Clark for his excellent work in machining the ceramic specimens and Dr. M. Negahban for enlightening discussion on acceleration-dependent material behavior. The assistance provided by Mr. Y. Hu in the laboratory is also greatly appreciated.

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