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Friction and Wear of Polyethylene in Multidirectional Motion: Quantitave Descriptions of Current Theory


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FRICTION AND WEAR OF POLYETHYLENE IN MULTIDIRECTIONAL MOTION: QUANTITATIVE DESCRIPTIONS OF CURRENT THEORY By ALISON C. DUNN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Alison C. Dunn

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This document is dedicated to the graduate students of the University of Florida. Happy is the man who finds wisdom, And the man who gains understanding; For her proceeds are better than the profits of silver, And her gain than fine gold. -Proverbs 3:13, New King James Version

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iv ACKNOWLEDGEMENTS I would like to thank my husband Nick for his unending love and unending jokes during my time in school. Great appreciation goe s to my family as well, for shaping and guiding me: Robert and Patricia Rennie, Jill, and Russell. I would also like to thank all of my lab mates for their help, advice, and ability to have fun: (in no particular order) Dan and Pam Dickrell, Luis Alvar ez, Nicole McCook, Snake Mauntler, Jeff Bardt, Vinny Lee, Jerry Bourne, Benjafriend, Jason Bares, Dave Burris, Matt Hamilton, Nick Argibay, and Jason Steffens. I would also like to thank the professors who have worked with me on other projects and taken an interest in help ing me develop professionally: Malisa Sarntinoranont, Scott Banks, and Tony Brennan. Above all, I would like to thank my committee chair and advisor Greg Sawyer fo r his friendship as well as welcoming and challenging me into the world of engineering research. Though he is busy, he is often helping others to achieve their goals. I would like to acknowledge MAKO Surgical Corporation for financial support of this work.

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v TABLE OF CONTENTS Page ACKNOWLEDGEMENTS...............................................................................................iv TABLE.......................................................................................................................... ....vii LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... ..x CHAPTER 1 INTRODUCTION........................................................................................................1 Orthopaedic Implant Development and Research........................................................1 Friction and Wear in TKR Bearings.............................................................................2 2 MOLECULAR ORIENTATION HYPOTHESIS........................................................4 Ultra-high Molecular Weight Polyethylene (UHMWPe).............................................4 Performance of UHMWPe in Sliding...........................................................................5 Current Surface Orientation Hypotheses......................................................................6 Stryker Howmedica Osteonics: Dr. A. Wang.......................................................6 Clemson University: Martine LaBerge.................................................................7 University of Leeds: John Fisher...........................................................................7 Crosslinking...........................................................................................................8 New Hypothesis: Friction Trace Insight.......................................................................8 3 EXPERIMENTAL SETUP........................................................................................12 Friction Coefficient Measurements............................................................................12 Sample Preparation.....................................................................................................14 Test Parameters...........................................................................................................14 4 MOTION PATHS.......................................................................................................18 Preliminary Motion Paths...........................................................................................18 Motion Paths...............................................................................................................19 Initial Complex Motion Paths.....................................................................................20 Archimedes Spirals....................................................................................................21

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vi 5 FRICTION RESULTS................................................................................................28 Complex Motion Path Friction Results......................................................................28 Archimedes Spiral Motion Path Friction Results......................................................29 Crossing Angle ........................................................................................................29 6 DISCUSSION.............................................................................................................35 Crossing Angle Example: Circle Motion Path...........................................................36 Friction Rise Predictions with Crossing Angle..........................................................37 7 CONCLUSION...........................................................................................................43 APPENDIX A WEAR CALCULATIONS.........................................................................................45 B RADIUS OF CURVATURE CALCULATION METHOD......................................47 LIST OF REFERENCES...................................................................................................49 BIOGRAPHICAL SKETCH.............................................................................................52

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vii TABLE Table page 3-1. A list of the test paramete rs for each motion path run.............................................17

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viii LIST OF FIGURES Figure page 1-1. Drawing of knee anatomy and anatomical changes with a total knee replacement................................................................................................................3 2-1. Molecular structure of UHMWPE.............................................................................9 2-2. Schematic of fibril orientation and rupture on surface of UHMWPE pin as it moves along a non-lin ear motion path.......................................................................9 2-3. Schematic of the delamination wear of polyethylene in dry sliding........................10 2-4. Schematic of the two-component expe rimental setup and a schematic of the wear theory with fibril elongation an d perpendicular crosslink rupture..................10 2-5. Schematic of motion of a single point on a rotating and translating motion path....11 3-1. Schematic of a block sliding on a flat with frictional and normal forces indicated, as well as motion path length and volume lost........................................15 3-2. Photograph of pin-plate contact including bovine seru m bath, heating tape, and load cell....................................................................................................................15 3-3. Schematic of multidirectional tribometer including linear stages, motors, thrusters, and frame. Courtesy of Jason Steffens....................................................16 3-4. chematic of rotation a nd linear sliding producing a nonuniform velocity profile on pin surface...........................................................................................................16 4-1. Friction results over time for dry slidi ng of 3 simpler motion paths: line circle, and lemniscate..........................................................................................................22 4-2. Calculated wear rates as the slope of the lines on a plot of volume loss vs. applied force times sliding distance for three simpler motion paths: line, lemniscate, and circle...............................................................................................23 4-3. Motion path of clinical retrieval UHMWPe bearing based on in vivo observations and advance motion mapping.............................................................23 4-4. Chirp motion path description with as pect ratio of each lemniscate from 2:1 up to 14:1.......................................................................................................................24

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ix 4-5. Motion paths and relative concentrat ions of motion direction shown as a histogram of displacement vectors...........................................................................25 4-6. Archimedes spiral wear path with eve r-increasing or ever-decreasing radius of curvature its motion intensity histogram..................................................................26 4-7. Range of radius of curvature for chirp, Fermat, and Archimedes motion paths, in the range of millimet ers up to meters.......................................................................26 4-8. Range of change of radius (spatial de rivative) of curvature for each motion path..27 5-1. A summary of friction co efficients for a variety of geometries in dry and lubricated sliding......................................................................................................30 5-2. Friction coefficient as a function of wear track position for one cycle of the curvature-modulated chirp signal in dry sliding...................................................30 5-3. Friction coefficient as a function of wear track position for one cycle of the curvature-modulated chirp si gnal in lubricated sliding........................................31 5-4. Friction coefficient in dry sliding as a function of wear track position for one cycle of the double Fermat spiral motion path.........................................................32 5-5. Friction coefficient as a function of wear track position for cycles 1000 and 14000 of the ever-increasing or ever -decreasing Archimedes spirals....................33 5-6. Friction coefficient as a function of cy cle number of the ever-increasing or everdecreasing Archimedes spirals................................................................................34 6-1. Schematic showing that is the same for two circles of different sizes ( = S = L ) because they contain the same number of position points..............................39 6-2. Schematic showing that is different for two circles of different sizes ( = S = L ) because they have constant velocity (different number of position points)....40 6-3. Plot of size ratio d / versus crossing angle produced...........................................40 6-4. Friction coefficient rise predicti ons for base friction coefficients x from 0.1 0.5 over the course of from 0 radians..............................................................41 6-5. Experimental and theoretical friction coefficient plotted ve rsus crossing angle for selected lemniscates of the dry sliding chirp motion path..................................42 A-1. A schematic describing the dimensions a nd forces used to calculate wear rate. Courtesy of Dave Burris...........................................................................................46 B-1. Schematic of curve-fitting a circle between three points on a continuous curve.....48

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x Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science FRICTION AND WEAR OF POLYETHYLENE IN MULTIDIRECTIONAL MOTION: QUANTITATIVE DESCRIPTIONS OF CURRENT THEORY By Alison C. Dunn August 2006 Chair: W. Gregory Sawyer Major Department: Mechanic al and Aerospace Engineering Multi-directional wear of UHMWPe is of particular interest due to its use as the state-of-the-art bearing com ponent polymer in orthopaedic implants. In unidirectional sliding, UHMWPe chains or fibrils are able to align on the surface forming a stable, low shear running film which leads to very low rates of wear. During directio nal changes, these running films are disrupte d, and additional energy is requi red to align the chains in the new direction of sliding, during which so me of the chains may break off as wear debris. When this process takes place brie fly in a wear path, a higher local friction coefficient results. In addition, when the process repeats conti nuously over the entire wear path, a higher average friction coefficient results. It was hypothesized that in situ friction measurement over a relevant motion path will give a better understanding of the relationship between friction and wear. A cu stom-built multidirectional tribometer was built to supply uniform velocity and pressu re under the contact, some characteristics absent from current modeling of the theory described above.

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xi Friction is observed as a function of parame ters of a curvature-modulated wear path (radius of curvature = 1 1400 mm). In general, the spatially resolved friction coefficient increased with decreasing of the sliding path, which is consistent with some orientation of the surface. Previous models are used to predict the change in friction coefficient as the wear path curvature changes. These predictions are used to verify the hypothesis that friction in multid irectional sliding is direct ly correlated to the UHMWPe surface orientation, which lead s directly to wear.

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1 CHAPTER 1 INTRODUCTION Orthopaedic implants are becoming incr easingly common for younger patients, and the finite wear lives of these joints require multiple surgeries to carry patients through long life. Consequently, the state-of-the-art bearing material, ultra-high molecular weight polyethylene (UHMWPe), has been the focus of a significant number of research efforts aimed at improving the lives of these implants. Creep, counterface roughness, third body wear, and sensitivity of the material to multidirectional motion paths have all been cited as primary contributors to premature b earing failures in UHMWPe implants. Orthopaedic Implant Development and Research In the earliest knee replacement, engi neers used an unnaturally over-constrained mechanics model; in the 1970s, Installs Condyl ar Knee was used with wide success, though high wear. Later innovations reduced wear and allowed more of the natural ligaments to stay in place. Today, the modular total knee replacement (TKR) has had great success in restoring the quality of life to those with knee impairments (Figure 1-1). The key element of a TKR is the load-beari ng polyethylene that functions both as a support and a solid lubricant. Paradoxically, this key comp onent is also the cause of failure because the forces applied over time cause wear and degradation. Many researchers have taken clinical retr ievals of these bearing components to study the wear and deformation that occu rred during the life of the bearing [1-5 ]. This wear is well-quantified on the order of m illimeters per year thickness worn away. Specifically, Fregly, Hamilton, and Laurent used the geometry of clinical retrievals along

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2 with other tools such as in vivo observations and finite-eleme nt methods to map out wear rates and the motion path seen on the UHM WPe bearings in particular cases [3-5 ]. Friction and Wear in TKR Bearings It is well accepted that higher slidi ng friction produced between UHMWPe and a metal surface will produce more wear of the pol yethylene. Friction tests are one of the primary ways to assess the effectiveness of treatments on UHMWPe such as irradiation or absorbed fluid. Various unified theories of polyethylene interac tions include friction, wear, material, and physical considerations, but all must assume constant friction coefficient or constant velocity to allow other parameters to be fit to empirical data. This thesis work entertains the suggestion that friction measure over th e course of a motion path can more adequately give an indication of the mechanisms of wear, not just a gross observation of wear over a specified interval The motivation for this work is an incomplete treatment of the coupled mechanis ms of friction and wear of polyethylene. The goal of this work is to improve unified wear theories in scope of applications and accuracy. Relevant experiments were devised and executed to explore the friction traces of various multidirectional motion paths, as well as to discuss the pertinence of the results to the current unified theories.

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3 Figure 1-1. Drawing of knee anatomy and anatomical changes with a total knee replacement.

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4 CHAPTER 2 MOLECULAR ORIENTAT ION HYPOTHESIS This chapter discusses the current theory of molecular chain orientation in special polymer samples while in sliding contact. Th e experiments to veri fy this hypothesis are very new, and include work from industry and academia. Ultra-high Molecular Weight Polyethylene (UHMWPe) Many materials including metals, ceramics and polymers (plastics) have been examined for possible use in the human body, especially for bearing components in partial or total joint replacements. The cu rrent state-of-the-art total knee replacements use metal components that attach to the fe mur and tibia bones with a polymer bearing, essentially a slab, between them made of ultra-high molecular weight polyethylene (Figure 2-1). The metal components are typically coba lt-chromium-molybdenum alloys (CoCr), and the polymer is typically ultra-high molecu lar weight polyethylen e. This special engineering polymer was chosen for many r easons: it is biologically inert, easy to manufacture, has low friction and wear, and is very durable. In conjunction with the synovial fluid of the knee, this polymer can function in a TKR for up to 15 years should the patient remain trim and only moderately active. UHMWPe is a semi-crystalline polymer. Upon processing, it cr ystallizes as it cools. It has been reported that UHMWPe can be cross-linked before or after molding. UHMWPe is less likely to wear if cross-li nked because there are more bonds holding the chains together, limiting pol ymer chain scission.

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5 Performance of UHMWPe in Sliding The over-arching theory is that when the polymer deforms in sliding, some geometry of the polymer like fibrils or mo lecular chains orient themselves along the direction of sliding, and then either break off or re-orient when the motion changes direction, whichever requires less energy. Th e most appropriate way to describe this change in motion is an angle that is calculated as a velo city vector change due to coupled rotation and linear sliding motions (F igure 2-2). Previous work shows that UHMWPe performs well in both dry and lubri cated sliding. Author Gonzalez-Mora has shown that since this polymer is so compliant in comparison with its counter-surface, the exact constitution of the Co Cr alloy is negligible [6 ]. In dry sliding, UHMWPe forms a thin transfer film on the counter-surface which allows for polymer-polymer sliding. These low-shear running films are formed from surface defects or high pressure causing polymer chains to be pulled into the contact and semi-permanently transferred to the counter -surface. When these chains collect and cover the contact area, the friction is lowe red because the polymer is sliding on an oriented layer of chains with the same mol ecular composition rather than sliding on the initial wear surface (usually metal or ceramic ). Even in initial sliding of a friction experiment, there can be surf ace orientation of the pin though no material has yet visibly transferred to the counter-surf ace, and therefore stable frictio n can be disrupted before a transfer film has formed. In lubricated sliding, fluid particles such as water fill small spaces within the contact and prevent this transf er film from forming. However, friction and wear in the presence of lubrication remain low if the both the polymer surface and the sliding counter-surface are very smooth (Rz ~ 5-10 nm). The primary wear mechanism of

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6 UHMWPe is commonly referred to as delamina tion, which means the wear particles are thin patches released by cracking due to subsur face shear stress (Figure 2-3). That debris helps form the transfer film, or is pushed fr ee from the contact zone. Authors have used friction and wear to screen different types of UHMWPe, wh ether it has been modified by irradiation, cross-linking, or a dditives. Some have also used other techniques such as a punch to deform the materials. Delamination or other wear debris as well as the wear surfaces are analyzed in detail with scanning electron microscopy (SEM) [7-11 ]. These techniques are very appropria te given consistent loading and velocity conditions. Current Surface Orientation Hypotheses Three main research groups have deve loped testing machines and subsequent theories about how to describe this surface or ientation for their part icular setup, and how it is manifest in friction and wear data. Stryker Howmedica Osteonics: Dr. A. Wang Dr. Wangs theory takes the perspective of a general orientation of the polymer surface and defines from that a geometry of w ear debris. This stems from the assumption that there is a preferred orie ntation direction, in which en ergy is dissipated in chain motion, but does not contribute directly to we ar. Following from this is a perpendicular direction in which chains (and cross-links ) are broken apart from each other, causing elongated fibril-shaped wear de bris (Figure 2-4). Energy in that direction does not include chain motion but bond rupture, and c ontributes directly to wear. Based on bond strengths and the work to remove a singl e fibril, a wear equation was developed (Equation 1), 0 0sin2 '1 22c cd kkMM Equation 1

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7 where k is a wear rate constant, is friction coefficient, is the C-C bond energy, Mc is the average molecular wei ght between crosslinks, M0 is the critical molecular weight between crosslinks, and is the cross-shear angle. This is multi-parameter model that is born from the wear geometry assumption; the relationships between variables have been verified, but the model as a whole has yet to fully describe a system. The apparatus used to perform the experiments was a custom-built simulator where rotation is applied by a top component (polymer pin), a nd linear reciprocation is app lied by the flat plate (metal counter-surface) [12 ]. Of particular inte rest is the added term 0 which describes the initial friction coefficient. This means that wear is related to a rise in the friction coefficient due to this fibril orientation. Clemson University: Martine LaBerge This group has built an apparatus with xand y-tables that can be simultaneously controlled, providing multi-direction path cap abilities to their UHMWPe plate sliding against a spherical steel pin. They perform 5 tests with the same mo tion path, a 5-pointed star with 5 crossing points. They comp are this to the motion seen by the tibial component in TKRs, but neglect curvature of the motion path. They conclude that there is a direct and quantitative relationship betw een the measurements of cross-shear angle and linear damage, and that the ratio of those measurements is indicative of a materials ability to resist wear in a cross-shearing configuration [13 ]. University of Leeds: John Fisher This group uses similar equations to Dr Wang, but they also map out discrete locations on the pin surface to track during thei r test (Figure 2-5). They recognize the velocity distribution and calcu late an average cross-shear ratio under the pin contact. Their conclusions are as follows: a) duri ng unidirectional sliding, an orientational

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8 hardening of molecules occurs along the sliding direction; a softening in the perpendicular direction, b) cross-linking increases bonding, retarding chain mobility and resisting breakage, and c) cross-linking provi des less orientation hardening because the molecular chains cannot move as muc h, and therefore a lower wear rate [14 ]. Crosslinking Conflicting conclusions regarding how cro sslinking affects this surface orientation theory are present in current literature. Authors Galvin [14 ] and Muratoglu have concluded that crosslinking inhi bits this important mechanis m responsible for wear, i.e. orientation and re-orientation [15 ]. Conversely, Author Ku rtz affirms that both radiation and chemical crosslinking hinder mol ecular mobility at large deformations, and hence promote strain hardening and molecula r alignment during the multiaxial loading of the small punch test [8 ]. New Hypothesis: Friction Trace Insight All manifestations of this theory so far agree that cross-linking helps to make an UHMWPe component more wear resistant, a nd generally agree that chain elongation and rupture are the mechanisms that lead to fa ilure. This work probes this theory by inspecting in situ friction traces of multidirectional motion paths. Friction coefficient is an indication of energy needed to put two surfaces in relative motion, so changes in energy dissipation caused by surface orientati on and re-orientation shoul d be discernable. It is presumed that the friction trace can also be used not just to surmise that more friction leads to increased wear, but that specific locations of increase are causing some higher percentage of wear than ot her locations. The ideal expe rimental set up would measure wear in situ for a variety of locations along a motion path. The final way a friction trace would influence understanding of UHMWPe wear is that friction as a function of either

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9 path geometry can be inserted into a unified wear theory. The aim of this work is to show how in situ friction measurements can inform surface orient ation theory, wear prediction, and unified theories of sliding wear of UHMWPe. Figure 2-1. Molecular structure of UHMWPE. Figure 2-2. Schematic of fibril orientati on and rupture on surface of UHMWPE pin as it moves along a non-li near motion path.

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10 Figure 2-3. Schematic of the delamination wear of polyethylene in dry sliding. Figure 2-4. Schematic of the two-component experimental setup and a schematic of the wear theory with fibril elongation and perpendicular crosslink rupture.

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11 Figure 2-5. Schematic of motion of a single point on a rotating and translating motion path.

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12 CHAPTER 3 EXPERIMENTAL SETUP Friction Coefficient Measurements Friction coefficient is defined as the fr iction force produced divided by the normal force applied (Figure 3-1). Because of this tribometers must be designed with force reaction paths as the foremost considerati on, as friction measurements should be made with the highest resolution available in order to illuminate subtleties. Friction is typically measured with a load cell, so there must be no other ways to react the friction force outside of the load cell. Motors and stages cannot be frictionless, so the load cell is not able to be fixed to the component that provides motion to the tribometer contact. The apparatus designe d for these experiments uses an AMTI 6channel load cell mounted to pneumatic th rusters, which apply a normal load. The sample pin is mounted directly to this lo ad cell (Figure 3-2). Two Parker ball screw stages are mounted to the table perpendicula r to each other, which provide two-axis motion to the counter-sliding surf ace (Figure 3-3). Therefore, when the stage moves, all forces produced in the contact are reacted up through the samp le and in the load cell. One key feature of this tribometer is that all points under the pin have the same velocity and pressure this is achieved by keeping a ny rotation and linear sl iding in the linear path coupled together. Other auth ors have achieved this as well [13 ,15 ,17-21 ]. This entire setup operates in a class 10,000 clean room under slight positive pressure to prevent ambient contamination.

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13 Multi-directional motion can be produced a variety of ways, from cams to linear motors. For this work ball-screw linear motors were utilized because they are capable of 50,000 pulses per revolution, up to 2 in/sec. Motion paths are crea ted by uploading a Microsoft Excel file that in terfaces with the master LabView program. To assure a continuous path, splines are creat ed from the ideal excel file. This master program both runs the experiments and measures the outcomes in a usable format. Other tribometers that have been used to create multi-directional motion have used components that necessitate a variety of assumptions which confound data analysis and theoretical application. For example, Dr Wangs apparatus that others have copied uses a rotating pin in unidirectional sliding [14 22-24 ]. In this way, his simulator has separate rotation and linear sl iding components, considered a relevant setup for hip replacement motion. However, because those motions are decoupled, the velocity under the rotating pin is not consta nt and varies radially (Fi gure 3-4). The faster-moving outside edge will dissipate more en ergy, and therefore wear more. After this happens, the pressure under th e contact is non-uniform and increases over the course of a test as the contact area decreases. In the c ourse of his analysis, pressure is assumed to be constant, causing errors in the range of values to which his other parameters can be fit. One might ar gue that the pressure does not deviate enough over the course of the test such that it cannot be considered constant, but the complications of applying theories of fibril elongation and rupture st ill exist. If the surface of the pin elongates the fibers in such a way that they are circles of constant radius emanating from the center of the pi n, it is impossible to determine and model a single sliding direction of the pin that would be perpendicular to that. This non-uniform

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14 velocity and pressure do not allow the appa ratus results to apply appropriately to UHMWPe wear in TKR bearings. One last apparatus possibility for providing multidirectional motion is a commercial knee or hip simulator. Authors Fisher and Turell have explored this option [25-27 ], with the latter attempting to get rectilinear motion by matching the rotation of both counterface and pin on a commercially available OrthoPOD pin-on-disk by Advanced Mechanical Technology, Incorporated (AMTI). Sample Preparation UHMWPe used for this work was provided by the Hospital for Special Surgery in New York City. From this sample, cylinders 0.2 inches in diameter and 0.3 inches long were cut using a programmable Minitech Machinery Corporation CNC milling machine. The running surface of the sample was prepar ed by slicing one face with a razorblade. The provided has a molecular weight of approximately 6 million. Cobalt-chromium-magnesium (CoCr) counterfaces were obtained from Encore Orthopaedics and were polished to an RMS roughness of 5-10 nm on a polishing and lapping wheel. The disk was then adhered to the plate mounted to the linear stages. Test Parameters The testing conditions varied based on th e limitations of the apparatus to hold bovine serum and maintain a full range of motion. Table 3-1 lists the various test parameters. Bovine serum is a constituted of 25 mg/mL protein from Hyclone Alpha Calf Fraction and 0.3% EDTA as a preservative. Anti-bacter ial agents such as sodium azide are omitted due to possible hazards.

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15 Figure 3-1. Schematic of a block sliding on a flat with frictional and normal forces indicated, as well as motion path length and volume lost. Figure 3-2. Photograph of pi n-plate contact including bovine serum bath, heating tape, and load cell.

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16 Figure 3-3. Schematic of multidirectional tr ibometer including linear stages, motors, thrusters, and frame. Courtesy of Jason Steffens. Figure 3-4. Schematic of rotation and linea r sliding producing a non-uniform velocity profile on pin surface.

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17 Table 3-1. A list of the test para meters for each motion path run. Geometry of wear pathsize Lubricant? Line50.8 mm No Reciprocating circle50.8 mm No Lemniscate (figure 8)50.8 mm No chirp50.8 mm No Double Fermat spiral50.8 mm No line25.4 mm Yes lemniscate25.4 mm Yes chirp25.4 mm Yes Archimedes incr. radius162.2 mm No Archimedes decr. radius162.2 mm No

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18 CHAPTER 4 MOTION PATHS This chapter describes the motion paths run on the multi-directional tribometer and discusses the motivations and advantages of each. Preliminary Motion Paths Perpendicular ball screw motor linear tables have the ability to run any desirable wear path, so motion parameters were cons idered carefully. Previous work by an undergraduate researcher was used as the basis for more complex paths1. In his work, a simple dry sliding linear reciprocating pa th was examined with UHMWPe pins as previously described on a CoCr disk to vali date the ability of th e apparatus to produce wear rates seen by other rese archers on these materials ( k = 1 x 10-7 mm3/Nm). Next a reciprocating circular path was used due to its dissimilarity to a reciprocating linear path, in that it avoids linear sliding. After this a lemniscate was examined because it has a single crossing point, along with curvature and a non-uniform as pect ratio. These tests were run up to 1 million cycles and compar ative wear of the paths with varying complexity were analyzed. Friction as a func tion of time was also analyzed, to see if the data supported a link between friction res ponse and eventual wear response of a particular material. The tests were ru n at 50.8 mm/s sliding speed under 10 MPa of pressure, as this is considered to be an av erage pressure seen on bearings in TKRs. 1 Rizwan Sajan ran these experiments in our laboratory on this testing equipment, but the data required analysis before use in this manuscript.

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19 The results of these friction tests show that friction increases over time for all motion paths. Friction is higher for the more complex path shapes because they are constantly changing direction, a nd it is presumed that the mate rial does not have time to recover (Figure 4-1). Results also show that over time, the reciprocating circle and reciprocating lemniscate have a higher wear rate and fricti on coefficient than the linear reciprocating sliding (Figure 4-2). The specifics of the wear rate calculatio ns are described in Appendix A. These results show that higher friction coefficients in multi-directional dry sliding are indeed linked in some way to the overall wear of the ma terials under identical velocity and pressure conditions. This supports the assertion that the material wears more severely when the surface of the polymer move s in a motion deviates from reciprocation. Because the only testing parameter changed wa s the motion path, it must be that the surface responds to changes in direction, a nd therefore must have some sort of directionality itself. These data suggest that the surface cannot recover from the motions that deviate from a line, and th erefore dissipates more energy and wears at a faster rate. This is the motivation to devise a motion path that includes all features seen in the clinical retrieval: turnaro unds, high aspect ratio, and a variety of curvatures. Motion Paths The motion paths seen by TKR bearings are often complex, including high aspect ratios, stop points, turnaround poi nts, and a wide range of curvatures and changes in curvature [16 ]. Though work has been done to ascertain the typical motion path in TKRs, experimental testing is done with mo re regular paths that can be reliably reproduced and more easily analyzed. Th e majority of tri bological experiments performed on UHMWPe are recipr ocating sliding experiments, in which the polymer pin

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20 slides back and forth over the same path in a straight line. Those experiments can help to distinguish between candidate orthopaedic be aring materials, but are too simple to provide analogous wear mechanisms of polyet hylene in an implant bearing situation. The next iteration of testing moved from r eciprocating linear sliding to unidirectional linear sliding in a pin-on-dis k apparatus, in which the c ounterface is rotated under a pin mounted at some fixed radius from the center of the disk counterface. This motion is more relevant because it does not include 180-degree changes in motion every cycle; however, that is its downfall as well: it ha s no turnarounds or stops, as empirical data suggests in TKR (Figure 4-3). Author Ha milton, et al through in vivo kinematic observations and post-mortem analysis of the UHMWPe bearing determined the probably motion path of highest pre ssure, including velocity [4 ]. Later work approaches clinically relevant motion paths with the help of cams and motor-driven tables, and proposes multi-d irectional shapes such as circles [17-18 ], ovals [19 ], footballs [20-21 ] or 5-pointed stars [13 ]. All of these shapes have merit, but are not complex enough to capture subtle characteristic s of a motion path as seen in clinical retrievals. Initial Complex Motion Paths Based on the first experiments with lemn iscates and their motion flexibility, the more complex chirp signal was comprised of lemniscates of va rious aspect ratios superimposed at /6-radian angles until the wear path completed to 2 radians (Figure 44). This is called a chirp signal because its curvature modulation is analogous to the frequency-modulated signal co mmonly used in acoustics that sounds like the chirp of a bird. The radius of curvatur e of the path ranges from 1 1400 mm, and it contains some abrupt changes of direction at the center where the lemniscat es are oriented differently.

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21 The highest derivative of radius of curvat ure is 3400 mm/mm. High aspect ratio is achieved by including a long, lean lemniscate with length to width ratio of 14:1. The motion changes of the chirp path tend to be abrupt, so another motion path was chosen for its gentle changes in radius of curvature over time: a double Fermat Spiral, or parabolic spiral. The radius of curvature on this signal va ries from 1 40400 mm and the highest spatial change in radius of curvat ure is 101000 mm/mm. Each motion path has a histogram of position vectors, with higher concen trations along various axes (Figure 4-5). Archimedes Spirals To validate the hypothesis that friction co efficient is drastically affected by a change in motion path curvature, a motion path was examined called an Archimedes spiral. It is unique because motion takes pla ce in one direction on the spiral, allowing the radius of curvature to be either ever-incr easing or ever-decreasing, with a straight line return to the start in both cases (Figure 46). If the friction coefficient shows some correlation to change in motion path curvature in simpler tests, it will be magnified in tests that isolate the increasing or decreasing radius of curvature. These paths have a range of radius of curvature of 1-9700 mm. The change in curvature reaches a maximum at 785000 mm/mm. The range of radius of curvature and range of the spatial derivative are shown in Figure 4-7 and 4-8, respectively. These motion paths were run at the same basic paramete rs as the previous tests: sliding velocity 50.8 mm/s and pressure of 6 MPa. These sp ecific motion paths encompass the range of features observed in the beari ngs of TKRs, thereby giving per tinent friction results. Also the Fermat and Archimedes spirals provide isolation of the specific characteristics of gentle curvature and monotonically increasing or decreasing radius of curvature.

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22 Figure 4-1. Friction results over time for dr y sliding of 3 simpler motion paths: line circle, and lemniscate.

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23 Figure 4-2. Calculated wear ra tes as the slope of the lines on a plot of volume loss vs. applied force times sliding distance for three simpler motion paths: line, lemniscate, and circle. Figure 4-3. Motion path of clinical retrieval UHMWPe bearing based on in vivo observations and advance motion mapping.

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24 Figure 4-4. Chirp motion path description with aspect ratio of each lemniscate from 2:1 up to 14:1.

PAGE 36

25 Figure 4-5. Motion paths and relative concen trations of motion direction shown as a histogram of displacement vectors.

PAGE 37

26 Figure 4-6. Archimedes spiral wear path with ever-increasing or ever-decreasing radius of curvature its motion intensity histogram. Figure 4-7. Range of radius of curvature for chirp, Fermat, and Archimedes motion paths, in the range of millimeters up to meters.

PAGE 38

27 Figure 4-8. Range of change of radius (spati al derivative) of curvature for each motion path

PAGE 39

28 CHAPTER 5 FRICTION RESULTS Complex Motion Path Friction Results A plot of average friction coefficient ove r 100 cycles of sliding plotted versus relative motion path complexity shows that fr iction is increased for more complex path shapes (Figure 5-1). For each path, 2-D fricti on color maps were plotted to facilitate spatial geometry observations with respect to friction. This was done by taking the 3-D plot of x-position, y-position, and friction coefficient, then mapping the latter axis down onto the x-y plane with different colors to indi cate levels of friction coefficient measured (Figure 5-2). The results from a single dr y sliding curvature-modulated chirp cycle show that friction increases from the start point of the cycle (path length s), then fluctuates with each lemniscate loop before the end of the wear path. One trend is the increase in friction at the tip of each lemniscate leaf, specifically an increase in friction when entering the curve, a nd a subsequent decrease exiting the curve. This can be described by the spatial derivative of the radius of curvature along the motion path: d / ds < 0 indicates entering a curve, and d / ds > 0 indicates exiting a curve (see Appendix B for explanation of calculations ). This trend appears strongly on the lemniscates with higher aspect ra tio (#1,2,6), somewhat on the mid-range lemniscates (#3,5), and least on the wider, shorter lemnis cate (#4). Lubricated sliding produces a similar trend with overall lower friction coefficient (F igure 5-3). Because the double Fermat spiral exhibits a gradual change in curvature, it exhibits the same trends, though less severely (Figure 5-4).

PAGE 40

29 These data indicate that the extremes of friction coefficients from a single dry sliding cycle can be as low as 0.2 (broad er curvature) and as high as 0.4 (tighter curvature). It follows that regions of the motion path where the pin dissipates the most energy (i.e. highest friction coe fficient) should be the regions that would contribute more heavily to the overall wear of the system. Th e motion path is symmetr ic, but some of the areas of extreme friction are not mirrored; this could be due to uneven zeroing of the load cells before the test began. Archimedes Spiral Motion Path Friction Results At 14,000 cycles, the average cycle friction is lowest for ever-increasing radius ( = 0.199), and is highest for ever-decreasing radius ( = 0.231) (Figure 5-5). This is in agreement with the hypothesis th at increasing radius of curvature will not produce as high friction as decreasing radius of curvature (increasing curve intensity). When average friction coefficient for each cycle is plotted versus time, it can be seen that the ever-decreasing radius of curvature reaches a steady state after the 14000 cycles which is slightly higher than the initial tran sient. Conversely, the ever-increasing radius of curvature continued to decr ease to a value lower than the initial transient up to 14000 cycles (Figure 5-6). Crossing Angle In these experiments, a crossing angle can be calculated by taking the velocity vectors between consecutive points and seeing how they rotate as the motion advances. However, the plots do not yield useful inform ation for this work because of the slight scatter in the position data collected. That sc atter is magnified in velocity calculations, and does not show a clear track with fr iction coefficient at this juncture.

PAGE 41

30 Figure 5-1. A summary of fricti on coefficients for a variety of geometries in dry and lubricated sliding. Figure 5-2. Friction coefficient as a function of wear track position for one cycle of the curvature-modulated chirp signal in dry sliding.

PAGE 42

31 Figure 5-3. Friction coefficient as a function of wear track position for one cycle of the curvature-modulated chirp si gnal in lubricated sliding.

PAGE 43

32 Figure 5-4. Friction coefficient in dry sliding as a function of wear track position for one cycle of the double Fermat spiral motion path.

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33 Figure 5-5. Friction coefficient as a function of wear track position for cycles 1000 and 14000 of the ever-increasing or ever -decreasing Archimedes spirals.

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34 Figure 5-6. Friction coefficient as a function of cycle number of the ever-increasing or ever-decreasing Archimedes spirals.

PAGE 46

35 CHAPTER 6 DISCUSSION This work has shown that friction coefficient measured in situ can be a key determinant as to the mechanisms of w ear of UHMWPe in multidirectional sliding, specifically with respect to or ientation of the surface in cont act. As the derivative of radius of curvature increases, the steady-state friction is lo wer, and vice versa. This explains that there must be some orientati on of the surface, and energy expended to cause that orientation. More energy is required to re-orient more quickly, so friction coefficient is overall higher. The size scale of these moti on paths with respect to the size of the pin and the size of polyethylene molecule chai ns seems to fall in a range where these phenomena can be observed. There must exis t a sliding motion length range for which orientation and re-orientation can take place, below which molecular interactions such as Van der Waals forces are too strong, and above which surface orienta tion is negligible. An estimate of this range of relevance w ould be on the order of micrometers up to meters. Other authors have proved that UHMWPe wears more when there are more abrupt motion path changes; this confirms that the mechanisms of friction and wear are indeed related. Author Zmitrowicz has explored in detail a computational constitutive model where an initial velocity and motion path are prescribed, and then the friction trace calculated [28 ]. He concludes that for two ranges of curvature (possibly determined by parameters) the friction can either increas e or decrease with increasing radius of curvature.

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36 Crossing Angle Example: Circle Motion Path To explore restrictions of the crossing angle effect on friction, a circular motion path was chosen to analyze. According to the reigning th eory of surface elongation and perpendicular rupture, a circle with a smalle r radius of curvature would result in higher friction because the surface would not have as much time to recover in the constant change of direction; the change of direction is more severe w ith circles of smaller radius. If it is true that small ra dii of curvature produce higher friction coefficient and higher crossing angle produce higher fric tion coefficients, then the analysis of curves must be consistent in one way: the motion paths must have constant veloci ty, and constant data acquisition speed to be accurately compared. Constant velocity motion paths would allo w the spacing of points on the path to be equal. Only when the spacing of points is eq ual can the angle of crossing be adequately assessed. For example, given a circle of radius S which has a small radius of curvature, and a circle of radius L for a large radius of curvatur e, the crossing angle would be the same value for both if both circles had the sa me amount of positions (Figure 6-1). (This would require either a different acquisition rate or different slid ing velocity.) However, if the velocity remains constant, allowing an equal spacing of points for both circles, crossing angle varies with the radius of the circle (Figure 6-2). Using the latter assumption, crossing angle can be calculated for a range of radii of curvature circles, and a relati onship observed (Figure 6-3). These calculations used an initial case of radius of curvature = 20 mm, 18 position points, and 6.95 mm between position points. Crossing angle was solved for by changing the number of points on the circle until the distance d between points was equal for all circles. The relationship given here should not be specific to the circles chosen due to the non-dimensional group d/

PAGE 48

37 which is the distance between points divided by the radius of the circle motion path in question. The power fit with exponent 1.025 i ndicates that there is a nearly linear relationship between these two parameters. If there is a definitive relationship between crossing angle and friction coefficient that can be verified empiricall y, friction coefficient could be predicted for each circle as well as the friction rise due to the change in size of the circular motion path. Friction Rise Prediction s with Crossing Angle Dr. Wangs theory as described earlier relate s a change in friction coefficient to the rate of wear of UHMWPe, but he does not allow his directional energy equations to inform or quantify that change. The work in each sliding direction is defined by integrating the friction force times velocity ove r the course of time, to obtain independent equations in the x-direction (elongation), and y-direction (rupture) (Equation 2, Equation 3). The directions are different because of the dependence on the cross-shear angle, sin2 2 2xPv W Equation 2 sin2 2 2yPv W Equation 3 From the friction coefficient proportionality to the work in each direction, one can predict the friction coefficient rise in a single direction based on the rise in work in that direction (a function of the turn angle from the previous path). Taking the ratio of the work in the perpendicular directions gi ves a ratio based on the cross-shear angle (Equation 4). This is a relation of the rupture work to the elongation work.

PAGE 49

38 sin(2) 2 sin(2) 2y xW W Equation 4 It is predicted that there are two components to the fric tion coefficient: a sort of established, steady-state friction coeffici ent in the initial, chain-elongation sliding direction, x, and a higher friction coefficient in the chain-rupture direction, y. It is assumed that more work is required to rupt ure a chain than to el ongate it. A general relationship between friction, work, and direc tionality can be surmised from Equation 4 (Equation 5). Simplifying will give us an expression for friction coefficient based on the single-direction friction coe fficient and the ratio of wo rk based on turning angle (Equation 6). sin2 2 sin2 2yy xxW W Equation 5 1y x xW W Equation 6 This model predicts a higher coefficient of friction for great cross-shear angles, with the highest values in the rupture di rection being equal to the initial, elongation direction friction coefficient. The maximum overall friction coefficient ismaxx2 C where C is an arbitrary constant. Figure 6-4 gives the change in friction coefficient as a function of cross-shear angle for va rious initial friction coefficients, x, when C = 1. For each of the lemniscates of the chirp motion path, both increases up to radians and decreases back to zero. Figure 6-5 shows how the friction traces compare when plotted against the calculated crossing an gle for C = 1. There is no strong evidence

PAGE 50

39 that friction coefficient increases up to twice as much as its starti ng value at a crossing angle of 90 degrees, but it does in crease slightly. The discrepa ncy can be attributed to the path continually changing motion, whic h may not allow for a fully developed elongation portion and therefore no consistent base friction coefficient from which to increase, or measure that increase. The theoretical rise in friction is not cu rrently quantifiable and does not correlate well enough to empirical data to apply it to shapes such as the circles discussed previously, but these calculati ons are indicative that with further thought, reasonable relationships can be deduced from relevant and careful testing. Figure 6-1. Schematic showing that is the same for two circles of different sizes ( = S = L ) because they contain the same number of position points.

PAGE 51

40 Figure 6-2. Schematic showing that is different for two circles of different sizes ( = S = L ) because they have constant velocity (different number of position points). Figure 6-3. Plot of size ratio d / versus crossing angle produced.

PAGE 52

41 Figure 6-4. Friction coeffici ent rise predictions for base friction coefficients x from 0.1 0.5 over the course of from 0 radians.

PAGE 53

42 Figure 6-5. Experimental and theoretical friction coeffici ent plotted versus crossing angle for selected lemniscates of the dry sliding chirp motion path.

PAGE 54

43 CHAPTER 7 CONCLUSION A multi-directional tribometer with the capabilities of uniform-velocity and uniform-pressure motion paths has been designed and constructed. It has been used to run a variety of motion paths with the aim of analyzing the resulting friction with the path parameters such as changes in curvature. It has been shown that when all other parameters remain constant, friction coefficient of UHMWPe in multi-directional sliding is affected by changes of curvature in the motion path. The chirp motion path with multiple passes of increasing and decreasing radius of curvature was run, and the highest friction is seen at times wh en the change of radius of curvature is negative ( d / ds < 0). Friction tends to increase with d / ds < 0 and tends to decrease with d / ds > 0. Due to size scales, this suggests that surface orient ation is dependent upon the length over which fibrils can align and the en ergy required to do so. Based on hypotheses of the surface orient ation of UHMWPe, it has been shown that there should be a quantifiable rise in friction each time the path motion vector changes in direction by some angle Because the experimental did not correlate well to the expected rise in friction coefficient for various lemniscates in our chirp motion path, more work is needed in modeling the rise in friction with changes in the motion path to quantify how friction can be a predictor of w ear, in conjunction with alternate methods of calculating The ramifications of using in situ friction traces in current theories of surface elongation and even unified wear th eories are sizable. Limits can be applied to the theory

PAGE 55

44 of surface orientation with resp ect to the pin size, sliding le ngth, and paths necessary to enact the phenomenon. More se vere motions on the wear pa th can be correlated to specific locations where higher we ar might take place, rather th an a general association of the two. Also, some functiona lity of friction coefficient with crossing angle can be inserted into unified wear theories rather th an simplified assumptions of constant friction through an entire complex motion path.

PAGE 56

45 APPENDIX A WEAR CALCULATIONS Wear calculations are done using gravimetri c analysis of polymer samples. This means that volume loss is calculated from a measured mass loss (scale resolution down to 10 g). Material comparison is made by the wear rate, rather than wear over a given sliding distance or time, which allows for flexibility of measurements systems (Figure A1). Wear rate is designated by k, and is th e volume lost divided by the normal force applied times the distance of sliding (Equation 7). lost nV k Fd Equation 7 The volume lost is calculated by taking th e measured mass lost and dividing by the density of sample (Equation 8), which is defi ned by the initial mass divided by the initial volume of the sample (Equation 9). Sliding di stance is constituted by the sliding distance of each cycle and the total number of cycles (Equation 10). lost s amplem V Equation 8 0 24samplem DL Equation 9 dSN Equation 10 When all these variables are combined, wear rate is give n by Equation 11. 2 04nmDL k FSNm Equation 11

PAGE 57

46 Figure A-1. A schematic desc ribing the dimensions and for ces used to calculate wear rate. Courtesy of Dave Burris.

PAGE 58

47 APPENDIX B RADIUS OF CURVATURE CALCULATION METHOD The common radius of curvature for th ree points on a continuous curve is calculated by find the circumcircle of the tr iangle made up of those three points. The distance between any point and the circumcenter of that circle is the radius of curvature, (Figure B-1). First, the lengths of the legs ( a, b, c ) of the triangle IJK are calculated using the distance formula (Equation 12). The first intermediate step is to calculate a parameter of the triangle s which is a function of the leg lengths (Equation 13). From this parameter and the lengths, the radius of curv ature is calculated (Equation 14). 22 ijijaxxyy Equation 12 2abc s Equation 13 4abc ssasbsc Equation 14

PAGE 59

48 Figure B-1. Schematic of curve-fitting a circle between three points on a continuous curve.

PAGE 60

49 LIST OF REFERENCES 1. Bennett, D., Orr, J., Beverland, D.,Baker R., The Influence of Shape and Sliding Distance of Femoral Head Movement Loci on the Wear of Acetabular Cups in Total Hip Arthroplasty. Proceedings of the Institution of Mechanical Engineers Part HJournal of Engineering Medi cine, 2002. 216(H6): p. 393-402. 2. Bragdon, C.R., Goetz, D.D., Jasty, M., Elde r, J.R., Variables Af fecting the Wear of Polyethylene Acetabular Components, from AAHKS Fifth Annual Meeting Proceedings. The Journal of Arthroplasty, 1996. 11(2): p. 229. 3. Fregly, B., Sawyer, W., Harman, M., Banks S., Computational Wear Prediction of a Total Knee Replacement from in vivo Kinematics. Journal of Biomechanics, 2005. 38(2): p. 305-314. 4. Hamilton, M., Sucec, M., Fregly, B ., Banks, S., Sawyer, W., Quantifying Multidirectional Sliding Motions in Total Knee Replacements. Journal of TribologyTransactions of the ASME, 2005. 127(2): p. 280-286. 5. Laurent, M., Johnson, T., Yao, J., Blanch ard, C., Crowninshield, R., In Vitro Lateral versus Medial Wear of a Kn ee Prosthesis. Wear, 2003. 255: p. 1101-1106. 6. Gonzalez-Mora, V., Hoffmann, M., Chiesa, R., Cigada, A., Hampshire, J., Stroosnijder, M.F., Screening Wear Tester Versus Hip Joint Simulator: Results of UHMWPe Sliding against Di fferent CoCrMo Counterf aces. Surface Engineering, 2003. 19(1): p. 45-50. 7. Hoffmann, A., Gonzalez-Mora, V., Chiesa R., Cigada, A., Stroosnijder, M., Screening of Stabilized Crosslinked Poly ethylene Using a Novel Wear Tester. Biomedical Materials and Engin eering, 2002. 12(4): p. 387-395. 8. Kurtz, S., Pruitt, L., Jewett, C., Foulds, J., Edidin, A., Radiation and Chemical Crosslinking Promote Strain Hardening Behavior and Molecular Alignment in Ultra High Molecular Weight Polyet hylene During Multi-Axial Loading Conditions. Biomaterials 1999. 20(16): p. 1449-1462. 9. Shen, F., McKellop, H., Salovey, R., Morphology of Chemically Crosslinked Ultrahigh Molecular Weight Polyethyle ne. Journal of Biomedical Materials Research, 1998. 41(1): p. 71-78.

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50 10. Turell, M., Bellare, A., A Study of the Na nostructure and Tensile Properties of Ultra-High Molecular Weight Polyethyl ene. Biomaterials, 2004. 25(17): p. 33893398. 11. Yao, J., Blanchet, T., Murphy, D., Lauren t, M., Effect of Fluid Absorption on the Wear Resistance of UHMWPe Orthopedi c Bearing Surfaces. Wear, 2003. 255: p. 1113-1120. 12. Wang, A., A Unified Theory of Wear for Ultra-High Molecular Weight Polyethylene in Multi-Directional Slid ing. Wear, 2001. 248(1-2): p. 38-47. 13. Gevaert, M., LaBerge, M., Gordon, J., DesJardins, J., The Quantification of Physiologically Relevant Cross-Shear Wear Phenomena on Orthopedic Bearing Materials Using the Max-Shear Wear Te sting System. Journal of Tribology Transactions of the ASME 2005. 127(4): p. 740-749. 14. Galvin, A., Kang, L., Tipper, J., Stone, M., Ingham, E., Jin, Z., Fisher, J., Wear of Crosslinked Polyethylene under Different Tribological Conditions. Journal of Materials ScienceMaterials in Medicine, 2006. 17(3): p. 235-243. 15. Muratoglu, O., Bragdon, C., O'Connor, D., Jasty, M., Harris, W., Gul, R., McGarry, F., Unified Wear Model for Highly Cro sslinked Ultra-High Molecular Weight Polyethylenes (UHMWPe). Biomat erials, 1999. 20(16): p. 1463-1470. 16. Wang, A., Essner, A., Polineni, V., Stark, C., Dumbleton, J., Lubrication and Wear of Ultra-High Molecular Weight Poly ethylene in Total Joint Replacements. Tribology International, 1998. 31(1-3): p. 17-33. 17. Burroughs, B., Blanchet, T., Factors Affec ting the Wear of Irradiated UHMWPe. Tribology Transactions, 2001. 44(2): p. 215-223. 18. Saikko, V., Ahlroos, T., Wear Simulation of UHMWPe for Total Hip Replacement with a Multidirectional Motion Pin-onDisk Device: Effects of Counterface Material, Contact Area, and Lubricant. J ournal of Biomedical Materials Research, 2000. 49(2): p. 147-154. 19. Saikko, V., Calonius, O., Keranen, J., Eff ect of Slide Track Sh ape on the Wear of Ultra-High Molecular Weight Polyethylene in a Pin-on-Disk W ear Simulation of Total Hip Prosthesis. Journal of Biomedic al Materials Research Part BApplied Biomaterials, 2004. 69B(2): p. 141-148. 20. Joyce, T., Monk, D., Scholes, S., Un sworth, A., A Multi -Directional Wear Screening Device and Preliminary Results of UHMWPe Articulating against Stainless Steel. Bio-medical Materials and Engineering, 2000. 10(3-4): p. 241-249.

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51 21. Sawae, Y., Murakami, T., Sawano, T., Noda, I., Shimotoso, T., Quantitative Evaluation of Wear Property of Polyethyl ene-on-Ceramic Sliding Pairs for Joint Prostheses in Multidirectional Sliding Test. Bioceramics 14, 2002. 218-2: p. 665668. 22. Wang, A., Essner, A., Stark, C., Dumble ton, J., A Biaxial Line-Contact Wear Machine for the Evaluation of Implant B earing Materials for Total Knee Joint Replacement. Wear, 1999. 229: p. 701-707. 23. Endo, M., Barbour, P., Barton, D., Fisher J., Tipper, J., Ingham, E., Stone, M., Comparative Wear and Wear Debris unde r Three Different Counterface Conditions of Crosslinked and Non-Crosslinked Ultr a High Molecular Weight Polyethylene. Bio-medical Materials and Egineering, 2001. 11(1): p. 23-35. 24. Marrs, H., Barton, D., Jones, R., Ward, I., Fisher, J., Doyle, C., Comparative Wear under Four Different Tribological Conditions of Acetylene Enhanced Cross-Linked Ultra High Molecular Weight Polyethyl ene. Journal of Materials Science Materials in Medicine, 1999. 10(6): p. 333-342. 25. Fisher, J., McEwen, H., Tipper, J., Galvin, A., Ingram, J., Kamali, A., Stone, M., Ingham, E., Wear, Debris, a nd Biologic Activity of Cro ss-Linked Polyethylene in the Knee Benefits and Potential Concer ns. Clinical Orthopaedics and Related Research, 2004(428): p. 114-119. 26. Turell, M., Wang, A., Bellare, A., Quantif ication of the Effect of Cross-Path Motion on the Wear Rate of Ultra-High Molecular Weight Polyethylene. Wear, 2003. 255: p. 1034-1039. 27. Turell, M., Friedlaender, G., Wang, A., T hornhill, T., Bellare, A., The Effect of Counterface Roughness on the Wear of UHM WPe for Rectangular Wear Paths. Wear, 2005. 259: p. 984-991. 28. Zmitrowicz, A., Models of Kinematics De pendent Anisotropic and Heterogeneous Friction. International Journal of So lids and Structures, 2006. 43: p. 4407-4451.

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52 BIOGRAPHICAL SKETCH Alison Dunn first gained an education of cultures by moving around frequently until starting at the University of Florida in August of 2000. She earned a Bachelor of Science degree in mechanical engineering in December of 2004. She has received various awards including participation in the University Scholars Program and the Caterpillar Scholarship from the Society of Women Engineers. Upon obtaining a Master of Science, she plans to join the Peace Co rps and eventually work for a biomedical company or technical inst rumentation company.


Permanent Link: http://ufdc.ufl.edu/UFE0015867/00001

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Title: Friction and Wear of Polyethylene in Multidirectional Motion: Quantitave Descriptions of Current Theory
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Copyright Date: 2008

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Permanent Link: http://ufdc.ufl.edu/UFE0015867/00001

Material Information

Title: Friction and Wear of Polyethylene in Multidirectional Motion: Quantitave Descriptions of Current Theory
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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FRICTION AND WEAR OF POLYETHYLENE IN MULTIDIRECTIONAL MOTION:
QUANTITATIVE DESCRIPTIONS OF CURRENT THEORY















By

ALISON C. DUNN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Alison C. Dunn



























This document is dedicated to the graduate students of the University of Florida.


Happy is the man who finds wisdom,
And the man who gains understanding;
For her proceeds are better than the profits of silver,
And her gain than fine gold.

-Proverbs 3:13, New King James Version















ACKNOWLEDGEMENTS

I would like to thank my husband Nick for his unending love and unending jokes

during my time in school. Great appreciation goes to my family as well, for shaping and

guiding me: Robert and Patricia Rennie, Jill, and Russell. I would also like to thank all of

my lab mates for their help, advice, and ability to have fun: (in no particular order) Dan

and Pam Dickrell, Luis Alvarez, Nicole McCook, Snake Mauntler, Jeff Bardt, Vinny Lee,

Jerry Bourne, Benjafriend, Jason Bares, Dave Burris, Matt Hamilton, Nick Argibay, and

Jason Steffens.

I would also like to thank the professors who have worked with me on other

projects and taken an interest in helping me develop professionally: Malisa

Sarntinoranont, Scott Banks, and Tony Brennan. Above all, I would like to thank my

committee chair and advisor Greg Sawyer for his friendship as well as welcoming and

challenging me into the world of engineering research. Though he is busy, he is often

helping others to achieve their goals. I would like to acknowledge MAKO Surgical

Corporation for financial support of this work.
















TABLE OF CONTENTS
Page


ACKNOWLEDGEMENTS .........v..... .............. iv
TABLE...........CKNOWLEDGEMENTS .................................................................................. vii


LIST OF FIGURES ...................................................... ................... viii

A B ST R A C T .......... ..... ...................................................................................... x

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Orthopaedic Implant Development and Research..................................................1
Friction and W ear in TKR Bearings.................................... .......................... ......... 2

2 MOLECULAR ORIENTATION HYPOTHESIS...................................................4

Ultra-high Molecular Weight Polyethylene (UHMWPe)...........................................4
Performance of UHM W Pe in Sliding................................ ......................... ....... 5
Current Surface Orientation Hypotheses ........... ................................ ...............6
Stryker Howmedica Osteonics: Dr. A. Wang ................................................. 6
Clemson University: M artine LaBerge ...................................... ............... 7
U university of Leeds: John Fisher...................................... ......................... 7
Crosslinking................. ........... .... ...... ....................... ........... 8
New Hypothesis: Friction Trace Insight .............................................. ...............8

3 E X PE R IM E N TA L SE TU P ............................................................. .....................12

Friction Coefficient M easurem ents ........................................ ........................ 12
Sam ple Preparation ................................................. ....... ............... 14
T e st P a ra m ete rs ..................................................................................................... 14

4 M O TION PA TH S .................. ........................ ......................... .. ...... .. 18

Prelim inary M otion Paths ........................................................................... 18
M option Paths ....................................................................... ........ 19
Initial C om plex M otion Paths.......................................................... ............... 20
A rchim edes' Spirals........... ...... ........................................ ................ .......... ....... 2 1



v









5 F R IC T IO N R E SU L T S ..................................................................... .....................28

Com plex M option Path Friction R results ........................................... .....................28
Archimedes' Spiral Motion Path Friction Results.....................................................29
C crossing A ngle a ................................................................29

6 D ISC U S SIO N ............................................................................... 35

Crossing Angle Example: Circle Motion Path ................................. ...............36
Friction Rise Predictions with Crossing Angle .................................. ...............37

7 C O N C L U SIO N ......... ...................................................................... ......... .. ..... .. 43

APPENDIX

A W E A R C A L C U L A TIO N S .............................................................. .....................45

B RADIUS OF CURVATURE CALCULATION METHOD ....................................47

L IST O F R E FE R E N C E S ......... .. ............. ................................................................49

B IO G R A PH IC A L SK E TCH ..................................................................... ..................52
















TABLE

Table page

3-1. A list of the test parameters for each motion path run. ..........................................17
















LIST OF FIGURES


Figure page

1-1. Drawing of knee anatomy and anatomical changes with a total knee
replace ent. .......................................... ............................. .. 3

2-1. M olecular structure of UH M W PE. ........................................... .........................9

2-2. Schematic of fibril orientation and rupture on surface of UHMWPE pin as it
moves along a non-linear motion path. ........................................... ............... 9

2-3. Schematic of the delamination wear of polyethylene in dry sliding....................... 10

2-4. Schematic of the two-component experimental setup and a schematic of the
wear theory with fibril elongation and perpendicular crosslink rupture ................10

2-5. Schematic of motion of a single point on a rotating and translating motion path.... 11

3-1. Schematic of a block sliding on a flat with frictional and normal forces
indicated, as well as motion path length and volume lost.......................................15

3-2. Photograph of pin-plate contact including bovine serum bath, heating tape, and
lo ad cell. .......................................................... ................ 15

3-3. Schematic of multidirectional tribometer including linear stages, motors,
thrusters, and frame. Courtesy of Jason Steffens. ................................................ 16

3-4. chematic of rotation and linear sliding producing a non-uniform velocity profile
on pin surface. ........................................................................ 16

4-1. Friction results over time for dry sliding of 3 simpler motion paths: line, circle,
and lem niscate. .........................................................................22

4-2. Calculated wear rates as the slope of the lines on a plot of volume loss vs.
applied force times sliding distance for three simpler motion paths: line,
lem niscate, and circle. .......................... ......................... .. ...... .... ............23

4-3. Motion path of clinical retrieval UHMWPe bearing based on in vivo
observations and advance motion mapping. ................................. .................23

4-4. Chirp motion path description with aspect ratio of each lemniscate from 2:1 up
to 14 :1 .............................................................................. 2 4









4-5. Motion paths and relative concentrations of motion direction shown as a
histogram of displacem ent vectors. ........................................ ....................... 25

4-6. Archimedes spiral wear path with ever-increasing or ever-decreasing radius of
curvature its motion intensity histogram. ....................................... ............... 26

4-7. Range of radius of curvature for chirp, Fermat, and Archimedes motion paths, in
the range of m illim eters up to m eters ............................................ ............... 26

4-8. Range of change of radius (spatial derivative) of curvature for each motion path..27

5-1. A summary of friction coefficients for a variety of geometries in dry and
lubricated sliding. .......................................................................30

5-2. Friction coefficient as a function of wear track position for one cycle of the
curvature-modulated "chirp" signal in dry sliding .............................................30

5-3. Friction coefficient as a function of wear track position for one cycle of the
curvature-modulated "chirp" signal in lubricated sliding. .....................................31

5-4. Friction coefficient in dry sliding as a function of wear track position for one
cycle of the double Fermat spiral motion path .....................................................32

5-5. Friction coefficient as a function of wear track position for cycles 1000 and
14000 of the ever-increasing or ever-decreasing Archimedes' spirals ..................33

5-6. Friction coefficient as a function of cycle number of the ever-increasing or ever-
decreasing A rchim edes' spirals ................................................................... ....... 34

6-1. Schematic showing that a is the same for two circles of different sizes (p=S,
p=L) because they contain the same number of position points. .............................39

6-2. Schematic showing that a is different for two circles of different sizes (p=S,
p=L) because they have constant velocity (different number of position points). ...40

6-3. Plot of size ratio dip versus crossing angle a produced. ........................................40

6-4. Friction coefficient rise predictions for base friction coefficients ux from 0.1 -
0.5 over the course of a from 0 7 radians..................................... .................... 41

6-5. Experimental and theoretical friction coefficient plotted versus crossing angle a
for selected lemniscates of the dry sliding chirp motion path.............................42

A-1. A schematic describing the dimensions and forces used to calculate wear rate.
C courtesy of D ave B urris................................................................ ......... ......46

B-1. Schematic of curve-fitting a circle between three points on a continuous curve.....48















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

FRICTION AND WEAR OF POLYETHYLENE IN MULTIDIRECTIONAL MOTION:
QUANTITATIVE DESCRIPTIONS OF CURRENT THEORY

By

Alison C. Dunn

August 2006

Chair: W. Gregory Sawyer
Major Department: Mechanical and Aerospace Engineering

Multi-directional wear of UHMWPe is of particular interest due to its use as the

state-of-the-art bearing component polymer in orthopaedic implants. In unidirectional

sliding, UHMWPe chains or fibrils are able to align on the surface forming a stable, low

shear running film which leads to very low rates of wear. During directional changes,

these running films are disrupted, and additional energy is required to align the chains in

the new direction of sliding, during which some of the chains may break off as wear

debris. When this process takes place briefly in a wear path, a higher local friction

coefficient results. In addition, when the process repeats continuously over the entire

wear path, a higher average friction coefficient results. It was hypothesized that in situ

friction measurement over a relevant motion path will give a better understanding of the

relationship between friction and wear. A custom-built multidirectional tribometer was

built to supply uniform velocity and pressure under the contact, some characteristics

absent from current modeling of the theory described above.









Friction is observed as a function of parameters of a curvature-modulated wear path

(radius of curvature p = 1 1400 mm). In general, the spatially resolved friction

coefficient increased with decreasing p of the sliding path, which is consistent with some

orientation of the surface. Previous models are used to predict the change in friction

coefficient as the wear path curvature changes. These predictions are used to verify the

hypothesis that friction in multidirectional sliding is directly correlated to the UHMWPe

surface orientation, which leads directly to wear.














CHAPTER 1
INTRODUCTION

Orthopaedic implants are becoming increasingly common for younger patients, and

the finite wear lives of these joints require multiple surgeries to carry patients through

long life. Consequently, the state-of-the-art bearing material, ultra-high molecular weight

polyethylene (UHMWPe), has been the focus of a significant number of research efforts

aimed at improving the lives of these implants. Creep, counterface roughness, third body

wear, and sensitivity of the material to multidirectional motion paths have all been cited

as primary contributors to premature bearing failures in UHMWPe implants.

Orthopaedic Implant Development and Research

In the earliest knee replacement, engineers used an unnaturally over-constrained

mechanics model; in the 1970s, Install's Condylar Knee was used with wide success,

though high wear. Later innovations reduced wear and allowed more of the natural

ligaments to stay in place. Today, the modular total knee replacement (TKR) has had

great success in restoring the quality of life to those with knee impairments (Figure 1-1).

The key element of a TKR is the load-bearing polyethylene that functions both as a

support and a solid lubricant. Paradoxically, this key component is also the cause of

failure because the forces applied over time cause wear and degradation.

Many researchers have taken clinical retrievals of these bearing components to

study the wear and deformation that occurred during the life of the bearing [1-5]. This

wear is well-quantified on the order of millimeters per year thickness worn away.

Specifically, Fregly, Hamilton, and Laurent used the geometry of clinical retrievals along









with other tools such as in vivo observations and finite-element methods to map out wear

rates and the motion path seen on the UHMWPe bearings in particular cases [3-5].

Friction and Wear in TKR Bearings

It is well accepted that higher sliding friction produced between UHMWPe and a

metal surface will produce more wear of the polyethylene. Friction tests are one of the

primary ways to assess the effectiveness of treatments on UHMWPe such as irradiation

or absorbed fluid. Various unified theories of polyethylene interactions include friction,

wear, material, and physical considerations, but all must assume constant friction

coefficient or constant velocity to allow other parameters to be fit to empirical data. This

thesis work entertains the suggestion that friction measure over the course of a motion

path can more adequately give an indication of the mechanisms of wear, not just a gross

observation of wear over a specified interval. The motivation for this work is an

incomplete treatment of the coupled mechanisms of friction and wear of polyethylene.

The goal of this work is to improve unified wear theories in scope of applications and

accuracy. Relevant experiments were devised and executed to explore the friction traces

of various multidirectional motion paths, as well as to discuss the pertinence of the results

to the current unified theories.











Patella


Cartilage
Meniscus

Anterior
Cruciate
Ligament






UHMWPE
Bearing
Tibial
Platform

Figure 1-1. Drawing of knee anatomy and anatomical changes with a total knee
replacement.














CHAPTER 2
MOLECULAR ORIENTATION HYPOTHESIS

This chapter discusses the current theory of molecular chain orientation in special

polymer samples while in sliding contact. The experiments to verify this hypothesis are

very new, and include work from industry and academia.

Ultra-high Molecular Weight Polyethylene (UHMWPe)

Many materials including metals, ceramics, and polymers (plastics) have been

examined for possible use in the human body, especially for bearing components in

partial or total joint replacements. The current state-of-the-art total knee replacements

use metal components that attach to the femur and tibia bones with a polymer bearing,

essentially a slab, between them made of ultra-high molecular weight polyethylene

(Figure 2-1).

The metal components are typically cobalt-chromium-molybdenum alloys (CoCr),

and the polymer is typically ultra-high molecular weight polyethylene. This "special

engineering polymer" was chosen for many reasons: it is biologically inert, easy to

manufacture, has low friction and wear, and is very durable. In conjunction with the

synovial fluid of the knee, this polymer can function in a TKR for up to 15 years should

the patient remain trim and only moderately active.

UHMWPe is a semi-crystalline polymer. Upon processing, it crystallizes as it

cools. It has been reported that UHMWPe can be cross-linked before or after molding.

UHMWPe is less likely to wear if cross-linked because there are more bonds holding the

chains together, limiting polymer chain scission.









Performance of UHMWPe in Sliding

The over-arching theory is that when the polymer deforms in sliding, some

geometry of the polymer like fibrils or molecular chains orient themselves along the

direction of sliding, and then either break off or re-orient when the motion changes

direction, whichever requires less energy. The most appropriate way to describe this

change in motion is an angle a that is calculated as a velocity vector change due to

coupled rotation and linear sliding motions (Figure 2-2). Previous work shows that

UHMWPe performs well in both dry and lubricated sliding. Author Gonzalez-Mora has

shown that since this polymer is so compliant in comparison with its counter-surface, the

exact constitution of the CoCr alloy is negligible [6].

In dry sliding, UHMWPe forms a thin transfer film on the counter-surface which

allows for polymer-polymer sliding. These low-shear running films are formed from

surface defects or high pressure causing polymer chains to be pulled into the contact and

semi-permanently transferred to the counter-surface. When these chains collect and

cover the contact area, the friction is lowered because the polymer is sliding on an

oriented layer of chains with the same molecular composition rather than sliding on the

initial wear surface (usually metal or ceramic). Even in initial sliding of a friction

experiment, there can be surface orientation of the pin though no material has yet visibly

transferred to the counter-surface, and therefore stable friction can be disrupted before a

transfer film has formed.

In lubricated sliding, fluid particles such as water fill small spaces within the

contact and prevent this transfer film from forming. However, friction and wear in the

presence of lubrication remain low if the both the polymer surface and the sliding

counter-surface are very smooth (Rz 5-10 nm). The primary wear mechanism of









UHMWPe is commonly referred to as delamination, which means the wear particles are

thin patches released by cracking due to subsurface shear stress (Figure 2-3). That debris

helps form the transfer film, or is pushed free from the contact zone. Authors have used

friction and wear to screen different types of UHMWPe, whether it has been modified by

irradiation, cross-linking, or additives. Some have also used other techniques such as a

punch to deform the materials. Delamination or other wear debris as well as the wear

surfaces are analyzed in detail with scanning electron microscopy (SEM) [7-11]. These

techniques are very appropriate given consistent loading and velocity conditions.

Current Surface Orientation Hypotheses

Three main research groups have developed testing machines and subsequent

theories about how to describe this surface orientation for their particular setup, and how

it is manifest in friction and wear data.

Stryker Howmedica Osteonics: Dr. A. Wang

Dr. Wang's theory takes the perspective of a general orientation of the polymer

surface and defines from that a geometry of wear debris. This stems from the assumption

that there is a preferred orientation direction, in which energy is dissipated in chain

motion, but does not contribute directly to wear. Following from this is a perpendicular

direction in which chains (and cross-links) are broken apart from each other, causing

elongated fibril-shaped wear debris (Figure 2-4). Energy in that direction does not

include chain motion but bond rupture, and contributes directly to wear. Based on bond

strengths and the work to remove a single fibril, a wear equation was developed

(Equation 1),

d(p o>0) n ( sin 2a
k =k d( (M -Mo) 1 2a Equation 1
2/, ) 2a )









where k'is a wear rate constant, / is friction coefficient, y is the C-C bond energy, Mc is

the average molecular weight between crosslinks, Mo is the critical molecular weight

between crosslinks, and a is the cross-shear angle. This is multi-parameter model that is

born from the wear geometry assumption; the relationships between variables have been

verified, but the model as a whole has yet to fully describe a system. The apparatus used

to perform the experiments was a custom-built simulator where rotation is applied by a

top component (polymer pin), and linear reciprocation is applied by the flat plate (metal

counter-surface) [12]. Of particular interest is the added term uo which describes the

initial friction coefficient. This means that wear is related to a rise in the friction

coefficient due to this fibril orientation.

Clemson University: Martine LaBerge

This group has built an apparatus with x- and y-tables that can be simultaneously

controlled, providing multi-direction path capabilities to their UHMWPe plate sliding

against a spherical steel pin. They perform 5 tests with the same motion path, a 5-pointed

star with 5 crossing points. They compare this to the motion seen by the tibial

component in TKRs, but neglect curvature of the motion path. They conclude that there

is a direct and quantitative relationship between the measurements of cross-shear angle

and linear damage, and that the ratio of those measurements is indicative of a material's

ability to resist wear in a cross-shearing configuration [13].

University of Leeds: John Fisher

This group uses similar equations to Dr. Wang, but they also map out discrete

locations on the pin surface to track during their test (Figure 2-5). They recognize the

velocity distribution and calculate an average cross-shear ratio under the pin contact.

Their conclusions are as follows: a) during unidirectional sliding, an orientational









'hardening' of molecules occurs along the sliding direction; a 'softening' in the

perpendicular direction, b) cross-linking increases bonding, retarding chain mobility and

resisting breakage, and c) cross-linking provides less orientation hardening because the

molecular chains cannot move as much, and therefore a lower wear rate [14].

Crosslinking

Conflicting conclusions regarding how crosslinking affects this surface orientation

theory are present in current literature. Authors Galvin [14] and Muratoglu have

concluded that "crosslinking inhibits this important mechanism responsible for wear, i.e.

orientation and re-orientation ..." [15]. Conversely, Author Kurtz affirms that "both

radiation and chemical crosslinking hinder molecular mobility at large deformations, and

hence promote strain hardening and molecular alignment during the multiaxial loading of

the small punch test" [8].

New Hypothesis: Friction Trace Insight

All manifestations of this theory so far agree that cross-linking helps to make an

UHMWPe component more wear resistant, and generally agree that chain elongation and

rupture are the mechanisms that lead to failure. This work probes this theory by

inspecting in situ friction traces of multidirectional motion paths. Friction coefficient is

an indication of energy needed to put two surfaces in relative motion, so changes in

energy dissipation caused by surface orientation and re-orientation should be discernable.

It is presumed that the friction trace can also be used not just to surmise that more friction

leads to increased wear, but that specific locations of increase are causing some higher

percentage of wear than other locations. The ideal experimental set up would measure

wear in situ for a variety of locations along a motion path. The final way a friction trace

would influence understanding of UHMWPe wear is that friction as a function of either








path geometry can be inserted into a unified wear theory. The aim of this work is to

show how in situ friction measurements can inform surface orientation theory, wear

prediction, and unified theories of sliding wear of UHMWPe.



C -C-C --C -C -(
carbon I I I


. hydrogen


Figure 2-1. Molecular structure of UHMWPE.


perpendicular


initial pin
surface


fibril
elongation


motion path

Figure 2-2. Schematic of fibril orientation and rupture on surface of UHMWPE pin as it
moves along a non-linear motion path.






10


Normal
Load
vpo y mr z-A0.48s
polymer pin


subsurface
cracks

Stel ounerac


wear
debris
7\


* cracks propagating


delamination


Figure 2-3. Schematic of the delamination wear of polyethylene in dry sliding.

carbon-carbon cross-link
Sor fibrils on running surface

J





or

S*- elongation axis -*


Figure 2-4. Schematic of the two-component experimental setup and a schematic of the
wear theory with fibril elongation and perpendicular crosslink rupture.






11







plate
mid-position

Figure 2-5. Schematic of motion of a single point on a rotating and translating motion
path.














CHAPTER 3
EXPERIMENTAL SETUP

Friction Coefficient Measurements

Friction coefficient is defined as the friction force produced divided by the normal

force applied (Figure 3-1). Because of this, tribometers must be designed with force

reaction paths as the foremost consideration, as friction measurements should be made

with the highest resolution available in order to illuminate subtleties.

Friction is typically measured with a load cell, so there must be no other ways to

react the friction force outside of the load cell. Motors and stages cannot be frictionless,

so the load cell is not able to be fixed to the component that provides motion to the

tribometer contact. The apparatus designed for these experiments uses an AMTI 6-

channel load cell mounted to pneumatic thrusters, which apply a normal load. The

sample pin is mounted directly to this load cell (Figure 3-2). Two Parker ball screw

stages are mounted to the table perpendicular to each other, which provide two-axis

motion to the counter-sliding surface (Figure 3-3). Therefore, when the stage moves, all

forces produced in the contact are reacted up through the sample and in the load cell.

One key feature of this tribometer is that all points under the pin have the same velocity

and pressure this is achieved by keeping any rotation and linear sliding in the linear

path coupled together. Other authors have achieved this as well [13,15,17-21]. This

entire setup operates in a class 10,000 clean room under slight positive pressure to

prevent ambient contamination.









Multi-directional motion can be produced a variety of ways, from cams to linear

motors. For this work ball-screw linear motors were utilized because they are capable of

50,000 pulses per revolution, up to 2 in/sec. Motion paths are created by uploading a

Microsoft Excel file that interfaces with the master LabView program. To assure a

continuous path, splines are created from the ideal excel file. This master program both

runs the experiments and measures the outcomes in a usable format.

Other tribometers that have been used to create 'multi-directional' motion have

used components that necessitate a variety of assumptions which confound data analysis

and theoretical application. For example, Dr. Wang's apparatus that others have copied

uses a rotating pin in unidirectional sliding [14, 22-24]. In this way, his simulator has

separate rotation and linear sliding components, considered a relevant setup for hip

replacement motion. However, because those motions are decoupled, the velocity under

the rotating pin is not constant and varies radially (Figure 3-4). The faster-moving

outside edge will dissipate more energy, and therefore wear more.

After this happens, the pressure under the contact is non-uniform and increases

over the course of a test as the contact area decreases. In the course of his analysis,

pressure is assumed to be constant, causing errors in the range of values to which his

other parameters can be fit. One might argue that the pressure does not deviate enough

over the course of the test such that it cannot be considered 'constant,' but the

complications of applying theories of fibril elongation and rupture still exist. If the

surface of the pin elongates the fibers in such a way that they are circles of constant

radius emanating from the center of the pin, it is impossible to determine and model a

single sliding direction of the pin that would be perpendicular to that. This non-uniform









velocity and pressure do not allow the apparatus results to apply appropriately to

UHMWPe wear in TKR bearings. One last apparatus possibility for providing

multidirectional motion is a commercial knee or hip simulator. Authors Fisher and Turell

have explored this option [25-27], with the latter attempting to get rectilinear motion by

matching the rotation of both counterface and pin on a commercially available OrthoPOD

pin-on-disk by Advanced Mechanical Technology, Incorporated (AMTI).

Sample Preparation

UHMWPe used for this work was provided by the Hospital for Special Surgery in

New York City. From this sample, cylinders 0.2 inches in diameter and 0.3 inches long

were cut using a programmable Minitech Machinery Corporation CNC milling machine.

The running surface of the sample was prepared by slicing one face with a razorblade.

The provided has a molecular weight of approximately 6 million.

Cobalt-chromium-magnesium (CoCr) counterfaces were obtained from Encore

Orthopaedics and were polished to an RMS roughness of 5-10 nm on a polishing and

lapping wheel. The disk was then adhered to the plate mounted to the linear stages.

Test Parameters

The testing conditions varied based on the limitations of the apparatus to hold

bovine serum and maintain a full range of motion. Table 3-1 lists the various test

parameters. Bovine serum is a constituted of 25 mg/mL protein from Hyclone Alpha

Calf Fraction and 0.3% EDTA as a preservative. Anti-bacterial agents such as sodium

azide are omitted due to possible hazards.











VLost


Figure 3-1. Schematic of a block sliding on a flat with frictional and normal forces
indicated, as well as motion path length and volume lost.


6-channel load celk.

PT^.,J~


sample holder


polymer sample heating taple
holder


sample


Figure 3-2. Photograph of pin-plate contact including bovine serum bath, heating tape,
and load cell.











pneumatic thrusters


LVDT


Figure 3-3. Schematic of multidirectional tribometer including linear stages, motors,
thrusters, and frame. Courtesy of Jason Steffens.


linear sliding


resultant velocity profile


Figure 3-4. Schematic of rotation and linear sliding producing a non-uniform velocity
profile on pin surface.


rotation


+


I~C~r






17


Table 3-1. A list of the test parameters for each motion path run.
Geometry of wear path size Lubricant?
Line 50.8 mm No
Reciprocating circle 50.8 mm No
Lemniscate (figure 8) 50.8 mm No
"chirp" 50.8 mm No
Double Fermat spiral 50.8 mm No
line 25.4 mm Yes
lemniscate 25.4 mm Yes
"chirp" 25.4 mm Yes
Archimedes' incr. radius 162.2 mm No
Archimedes' decr. radius 162.2 mm No














CHAPTER 4
MOTION PATHS

This chapter describes the motion paths run on the multi-directional tribometer and

discusses the motivations and advantages of each.

Preliminary Motion Paths

Perpendicular ball screw motor linear tables have the ability to run any desirable

wear path, so motion parameters were considered carefully. Previous work by an

undergraduate researcher was used as the basis for more complex paths1. In his work, a

simple dry sliding linear reciprocating path was examined with UHMWPe pins as

previously described on a CoCr disk to validate the ability of the apparatus to produce

wear rates seen by other researchers on these materials (k = 1 x 10-7 mm3/Nm). Next a

reciprocating circular path was used due to its dissimilarity to a reciprocating linear path,

in that it avoids linear sliding. After this, a lemniscate was examined because it has a

single crossing point, along with curvature and a non-uniform aspect ratio. These tests

were run up to 1 million cycles and comparative wear of the paths with varying

complexity were analyzed. Friction as a function of time was also analyzed, to see if the

data supported a link between friction response and eventual wear response of a

particular material. The tests were run at 50.8 mm/s sliding speed under 10 MPa of

pressure, as this is considered to be an average pressure seen on bearings in TKRs.




1 Rizwan Sajan ran these experiments in our laboratory on this testing equipment, but the data required
analysis before use in this manuscript.









The results of these friction tests show that friction increases over time for all

motion paths. Friction is higher for the more complex path shapes because they are

constantly changing direction, and it is presumed that the material does not have time to

recover (Figure 4-1).

Results also show that over time, the reciprocating circle and reciprocating

lemniscate have a higher wear rate and friction coefficient than the linear reciprocating

sliding (Figure 4-2). The specifics of the wear rate calculations are described in

Appendix A. These results show that higher friction coefficients in multi-directional dry

sliding are indeed linked in some way to the overall wear of the materials under identical

velocity and pressure conditions. This supports the assertion that the material wears more

severely when the surface of the polymer moves in a motion deviates from reciprocation.

Because the only testing parameter changed was the motion path, it must be that the

surface responds to changes in direction, and therefore must have some sort of

directionality itself. These data suggest that the surface cannot recover from the motions

that deviate from a line, and therefore dissipates more energy and wears at a faster rate.

This is the motivation to devise a motion path that includes all features seen in the

clinical retrieval: turnarounds, high aspect ratio, and a variety of curvatures.

Motion Paths

The motion paths seen by TKR bearings are often complex, including high aspect

ratios, stop points, turnaround points, and a wide range of curvatures and changes in

curvature [16]. Though work has been done to ascertain the typical motion path in

TKRs, experimental testing is done with more regular paths that can be reliably

reproduced and more easily analyzed. The majority of tribological experiments

performed on UHMWPe are reciprocating sliding experiments, in which the polymer pin









slides back and forth over the same path in a straight line. Those experiments can help to

distinguish between candidate orthopaedic bearing materials, but are too simple to

provide analogous wear mechanisms of polyethylene in an implant bearing situation.

The next iteration of testing moved from reciprocating linear sliding to unidirectional

linear sliding in a pin-on-disk apparatus, in which the counterface is rotated under a pin

mounted at some fixed radius from the center of the disk counterface. This motion is

more relevant because it does not include 180-degree changes in motion every cycle;

however, that is its downfall as well: it has no turnarounds or stops, as empirical data

suggests in TKR (Figure 4-3). Author Hamilton, et al through in vivo kinematic

observations and post-mortem analysis of the UHMWPe bearing determined the probably

motion path of highest pressure, including velocity [4].

Later work approaches clinically relevant motion paths with the help of cams and

motor-driven tables, and proposes multi-directional shapes such as circles [17-18], ovals

[19], footballs [20-21] or 5-pointed stars [13]. All of these shapes have merit, but are not

complex enough to capture subtle characteristics of a motion path as seen in clinical

retrievals.

Initial Complex Motion Paths

Based on the first experiments with lemniscates and their motion flexibility, the

more complex "chirp" signal was comprised of lemniscates of various aspect ratios

superimposed at 21/6-radian angles until the wear path completed to 271 radians (Figure 4-

4). This is called a "chirp" signal because its curvature modulation is analogous to the

frequency-modulated signal commonly used in acoustics that sounds like the chirp of a

bird. The radius of curvature of the path ranges from 1 1400 mm, and it contains some

abrupt changes of direction at the center where the lemniscates are oriented differently.









The highest derivative of radius of curvature is 3400 mm/mm. High aspect ratio is

achieved by including a long, lean lemniscate with length to width ratio of 14:1.

The motion changes of the chirp path tend to be abrupt, so another motion path was

chosen for its gentle changes in radius of curvature over time: a double Fermat Spiral, or

parabolic spiral. The radius of curvature on this signal varies from 1 40400 mm and the

highest spatial change in radius of curvature is 101000 mm/mm. Each motion path has a

histogram of position vectors, with higher concentrations along various axes (Figure 4-5).

Archimedes' Spirals

To validate the hypothesis that friction coefficient is drastically affected by a

change in motion path curvature, a motion path was examined called an Archimedes

spiral. It is unique because motion takes place in one direction on the spiral, allowing the

radius of curvature to be either ever-increasing or ever-decreasing, with a straight line

return to the start in both cases (Figure 4-6). If the friction coefficient shows some

correlation to change in motion path curvature in simpler tests, it will be magnified in

tests that isolate the increasing or decreasing radius of curvature.

These paths have a range of radius of curvature of 1-9700 mm. The change in

curvature reaches a maximum at 785000 mm/mm. The range of radius of curvature and

range of the spatial derivative are shown in Figure 4-7 and 4-8, respectively. These

motion paths were run at the same basic parameters as the previous tests: sliding velocity

50.8 mm/s and pressure of 6 MPa. These specific motion paths encompass the range of

features observed in the bearings of TKRs, thereby giving pertinent friction results. Also

the Fermat and Archimedes' spirals provide isolation of the specific characteristics of

gentle curvature and monotonically increasing or decreasing radius of curvature.








0.3





S0.2 -


0
2 line path
S0.1 circle path
o lemniscate path

1 million cycles
200 N normal load
100 mm cycle sliding distance
0.0 1 1 1 1i
0 5 10 15 20 25
Normal Load sliding distance, F*d*106 [Nm]

Figure 4-1. Friction results over time for dry sliding of 3 simpler motion paths: line,
circle, and lemniscate.











lemniscate path
k = 1.8 x 10-7 + 6.2 x 10-9 mm3/Nm


5-



4-


E
E 3-



E 2-

0
1 -



0-


0 5x106 10x106 15x106
Force distance [Nm]


20x 106 25x 106


Figure 4-2. Calculated wear rates as the slope of the lines on a plot of volume loss vs.
applied force times sliding distance for three simpler motion paths: line,
lemniscate, and circle.


0

-5

" -10
0

0
0-20
a.
o -20

.T3 7C


-30


motion path (mm)


-1


-0.5 0
medial-lateral axis


mn


Figure 4-3. Motion path of clinical retrieval UHMWPe bearing based on in vivo
observations and advance motion mapping.


circular path
k = 1.5 x 10-7 + 3.8 x 10-9 mm3/Nm





line path
k = 1.5 x 10-8 + 5.4 x 10-9 mm3/Nm
_ El L I


4


1.5 mm- I-.*--












lemn #2


lemn #3



lemn #4



lemn #5


lemniscate aspect ratio

1 14:1

2 10:1

3 5:1

4 2:1

5 5:1

6 10:1


Figure 4-4. Chirp motion path description with aspect ratio of each lemniscate from 2:1
up to 14:1.


lemn #1










counts


reciprocating



o0 t/2 n

circular












Fermat spiral~ 1






chirpI



0 t/2

Figure 4-5. Motion paths and relative concentrations of motion direction shown as a
histogram of displacement vectors.














O

0






k < 90 0 7/2 n
x-track position [mm] radial position [rad]

Figure 4-6. Archimedes spiral wear path with ever-increasing or ever-decreasing radius
of curvature its motion intensity histogram.

chirp 11400 mm

Fermat spiral 40400 mm

Archimedes'spiral i 9700 mm

0 10 20 30 40 50
range of radius of curvature, p (x103) [mm]

Figure 4-7. Range of radius of curvature for chirp, Fermat, and Archimedes motion paths,
in the range of millimeters up to meters.






27


-3400 3400

-101000 101000

-785000 785000

-106 -104 -102 0 102 104 106
range of changes in radius of curvature, dp [mm/mm]
dS

Figure 4-8. Range of change of radius (spatial derivative) of curvature for each motion
path














CHAPTER 5
FRICTION RESULTS

Complex Motion Path Friction Results

A plot of average friction coefficient over 100 cycles of sliding plotted versus

relative motion path complexity shows that friction is increased for more complex path

shapes (Figure 5-1). For each path, 2-D friction color maps were plotted to facilitate

spatial geometry observations with respect to friction. This was done by taking the 3-D

plot of x-position, y-position, and friction coefficient, then mapping the latter axis down

onto the x-y plane with different colors to indicate levels of friction coefficient measured

(Figure 5-2). The results from a single dry sliding curvature-modulated "chirp" cycle

show that friction increases from the start point of the cycle (path length s), then

fluctuates with each lemniscate loop before the end of the wear path.

One trend is the increase in friction at the tip of each lemniscate leaf, specifically an

increase in friction when entering the curve, and a subsequent decrease exiting the curve.

This can be described by the spatial derivative of the radius of curvature along the motion

path: dp/ds < 0 indicates entering a curve, and dp/ds > 0 indicates exiting a curve (see

Appendix B for explanation of calculations). This trend appears strongly on the

lemniscates with higher aspect ratio (#1,2,6), somewhat on the mid-range lemniscates

(#3,5), and least on the wider, shorter lemniscate (#4). Lubricated sliding produces a

similar trend with overall lower friction coefficient (Figure 5-3). Because the double

Fermat spiral exhibits a gradual change in curvature, it exhibits the same trends, though

less severely (Figure 5-4).









These data indicate that the extremes of friction coefficients from a single dry

sliding cycle can be as low as 0.2 (broader curvature) and as high as 0.4 (tighter

curvature). It follows that regions of the motion path where the pin dissipates the most

energy (i.e. highest friction coefficient) should be the regions that would contribute more

heavily to the overall wear of the system. The motion path is symmetric, but some of the

areas of extreme friction are not mirrored; this could be due to uneven zeroing of the load

cells before the test began.

Archimedes' Spiral Motion Path Friction Results

At 14,000 cycles, the average cycle friction is lowest for ever-increasing radius (t

= 0.199), and is highest for ever-decreasing radius (t = 0.231) (Figure 5-5).

This is in agreement with the hypothesis that increasing radius of curvature will not

produce as high friction as decreasing radius of curvature (increasing curve intensity).

When average friction coefficient for each cycle is plotted versus time, it can be seen that

the ever-decreasing radius of curvature reaches a steady state after the 14000 cycles

which is slightly higher than the initial transient. Conversely, the ever-increasing radius

of curvature continued to decrease to a value lower than the initial transient up to 14000

cycles (Figure 5-6).

Crossing Angle a

In these experiments, a crossing angle a can be calculated by taking the velocity

vectors between consecutive points and seeing how they rotate as the motion advances.

However, the plots do not yield useful information for this work because of the slight

scatter in the position data collected. That scatter is magnified in velocity calculations,

and does not show a clear track with friction coefficient at this juncture.









increasing complexity --
0.301


0.20-



0.10-


A water-lubricated sliding
N dry sliding


0.001
lines lemniscates "chirps"
circles
Fermat spirals


Figure 5-1. A summary of friction coefficients for a variety of geometries in dry and
lubricated sliding.


cycle 50

10.40
0.25 t




10.350

0.20


-15mm 0


15mm


Figure 5-2. Friction coefficient as a function of wear track position for one cycle of the
curvature-modulated "chirp" signal in dry sliding.








15 mm



'- {



0
0 0







-15 mm
-9 mm


: cycle 50



..
m l
L .


0


9mm


x-position

Figure 5-3. Friction coefficient as a function of wear track position for one cycle of the
curvature-modulated "chirp" signal in lubricated sliding.


0.100

t



0
0
4-0.025

0.025


I_

















5 r -I


0
x-position


Figure 5-4. Friction coefficient in dry sliding as a function of wear track position for one
cycle of the double Fermat spiral motion path.


30 mm-


S0.23
I


0.19
I
I

I 0.15


-30 mrr


'4


30 mm


"ftft%-b


.











ever-increasing radius

20 cycle 1000
S= 0.224


E
E
o
o
4-


_-n !


................-
20 0 2(
x-position [mm]


ever-decreasing radius
cycle 1000
20. j 0.225






0.





-20 . .. . . .


0
x-position [mm]


-20 0
x-position [mm]


E
E
C0
4,
0


cycle 14000
= 0.231




C,


20 -20


x-position [mm]


Figure 5-5. Friction coefficient as a function of wear track position for cycles 1000 and
14000 of the ever-increasing or ever-decreasing Archimedes' spirals.


20


0.275


0.250


0.225


0.200


0.175









0.24




0.22
0

0

t 0.20

S- decreasing radius of curvature
Increasing radius of curvature
0.18 ,-,-,i i -
0 2x10 6x10 1x104 1.4x104
cycle number

Figure 5-6. Friction coefficient as a function of cycle number of the ever-increasing or
ever-decreasing Archimedes' spirals.















CHAPTER 6
DISCUSSION

This work has shown that friction coefficient measured in situ can be a key

determinant as to the mechanisms of wear of UHMWPe in multidirectional sliding,

specifically with respect to orientation of the surface in contact. As the derivative of

radius of curvature increases, the steady-state friction is lower, and vice versa. This

explains that there must be some orientation of the surface, and energy expended to cause

that orientation. More energy is required to re-orient more quickly, so friction coefficient

is overall higher. The size scale of these motion paths with respect to the size of the pin

and the size of polyethylene molecule chains seems to fall in a range where these

phenomena can be observed. There must exist a sliding motion length range for which

orientation and re-orientation can take place, below which molecular interactions such as

Van der Waals forces are too strong, and above which surface orientation is negligible.

An estimate of this range of relevance would be on the order of micrometers up to

meters.

Other authors have proved that UHMWPe wears more when there are more abrupt

motion path changes; this confirms that the mechanisms of friction and wear are indeed

related. Author Zmitrowicz has explored in detail a computational constitutive model

where an initial velocity and motion path are prescribed, and then the friction trace

calculated [28]. He concludes that for two ranges of curvature (possibly determined by

parameters) the friction can either increase or decrease with increasing radius of

curvature.









Crossing Angle Example: Circle Motion Path

To explore restrictions of the crossing angle effect on friction, a circular motion

path was chosen to analyze. According to the reigning theory of surface elongation and

perpendicular rupture, a circle with a smaller radius of curvature would result in higher

friction because the surface would not have as much time to recover in the constant

change of direction; the change of direction is more severe with circles of smaller radius.

If it is true that small radii of curvature produce higher friction coefficient and higher

crossing angle produce higher friction coefficients, then the analysis of curves must be

consistent in one way: the motion paths must have constant velocity, and constant data

acquisition speed to be accurately compared.

Constant velocity motion paths would allow the spacing of points on the path to be

equal. Only when the spacing of points is equal can the angle of crossing be adequately

assessed. For example, given a circle of radius S which has a 'small' radius of curvature,

and a circle of radius L for a 'large' radius of curvature, the crossing angle would be the

same value for both if both circles had the same amount of positions (Figure 6-1). (This

would require either a different acquisition rate or different sliding velocity.) However, if

the velocity remains constant, allowing an equal spacing of points for both circles,

crossing angle varies with the radius of the circle (Figure 6-2).

Using the latter assumption, crossing angle can be calculated for a range of radii of

curvature circles, and a relationship observed (Figure 6-3). These calculations used an

initial case of radius of curvature p = 20 mm, 18 position points, and 6.95 mm between

position points. Crossing angle a was solved for by changing the number of points on the

circle until the distance between points was equal for all circles. The relationship given

here should not be specific to the circles chosen due to the non-dimensional group d/p,









which is the distance between points divided by the radius of the circle motion path in

question. The power fit with exponent 1.025 indicates that there is a nearly linear

relationship between these two parameters. If there is a definitive relationship between

crossing angle and friction coefficient that can be verified empirically, friction coefficient

could be predicted for each circle, as well as the friction rise due to the change in size of

the circular motion path.

Friction Rise Predictions with Crossing Angle

Dr. Wang's theory as described earlier relates a change in friction coefficient to the

rate of wear of UHMWPe, but he does not allow his directional energy equations to

inform or quantify that change. The work in each sliding direction is defined by

integrating the friction force times velocity over the course of time, to obtain independent

equations in the x-direction (elongation), and y-direction (rupture) (Equation 2, Equation

3). The directions are different because of the dependence on the cross-shear angle, a.

2/pPvf sin(2a)"
AWx = 2 a+(a)- Equation 2
0) 2

2pPv sin(2a}
AW- -2Pv- a-sin Equation 3
0) 2

From the friction coefficient proportionality to the work in each direction, one can

predict the friction coefficient rise in a single direction based on the rise in work in that

direction (a function of the turn angle from the previous path). Taking the ratio of the

work in the perpendicular directions gives a ratio based on the cross-shear angle

(Equation 4). This is a relation of the rupture work to the elongation work.










AW a( sin(2a)
--- -( Equation 4
AW, a sin(2a)
2

It is predicted that there are two components to the friction coefficient: a sort of

established, "steady-state" friction coefficient in the initial, chain-elongation sliding

direction, px, and a higher friction coefficient in the chain-rupture direction, py. It is

assumed that more work is required to rupture a chain than to elongate it. A general

relationship between friction, work, and directionality can be surmised from Equation 4

(Equation 5). Simplifying will give us an expression for friction coefficient based on the

single-direction friction coefficient and the ratio of work based on turning angle

(Equation 6).


Ssin (2a)
u AW 2
oc AW = Equation 5
x AW a + sin (2a)
a+ -'
2


AW,
/k=/ -- +W 1 Equation 6

This model predicts a higher coefficient of friction for great cross-shear angles,

with the highest values in the rupture direction being equal to the initial, elongation

direction friction coefficient. The maximum overall friction coefficient is kmax = 2C/,,

where C is an arbitrary constant. Figure 6-4 gives the change in friction coefficient as a

function of cross-shear angle for various initial friction coefficients, ux, when C = 1.

For each of the lemniscates of the chirp motion path, a both increases up to 7n

radians and decreases back to zero. Figure 6-5 shows how the friction traces compare

when plotted against the calculated crossing angle for C = 1. There is no strong evidence









that friction coefficient increases up to twice as much as its starting value at a crossing

angle of 90 degrees, but it does increase slightly. The discrepancy can be attributed to the

path continually changing motion, which may not allow for a fully developed

"elongation" portion and therefore no consistent base friction coefficient from which to

increase, or measure that increase.

The theoretical rise in friction is not currently quantifiable and does not correlate

well enough to empirical data to apply it to shapes such as the circles discussed

previously, but these calculations are indicative that with further thought, reasonable

relationships can be deduced from relevant and careful testing.














Figure 6-1. Schematic showing that a is the same for two circles of different sizes (p=S,
p=L) because they contain the same number of position points.











sX


cLg> cIL


Figure 6-2. Schematic showing that a is different for two circles of different sizes (p=S,
p=L) because they have constant velocity (different number of position
points).


y = 60.5x 1.025
R2 = 0.9997


0.1
size ratio d/p


Figure 6-3. Plot of size ratio dip versus crossing angle a produced.






41


1.00. 1 x= 0.5
1/

U lx= 0.4
-C 0.75-

o Px= 0.3
c 0.50
0 0.2
4-X
0.25 = 0.1


o
0.00


cross-shear angle, a [radians]

Figure 6-4. Friction coefficient rise predictions for base friction coefficients 'x from 0.1
0.5 over the course of a from 0 t radians.


























cross-shear angle [degrees]


42



Experimental values
analytical


0.6



0
I 0.4




90 0.20





0.6



0.4
^ 0.4
' I loj


'0 -
;i -
- _..


cross-shear angle [degrees]


cross-shear angle [degrees]


cross-shear angle [degrees]


Figure 6-5. Experimental and theoretical friction coefficient plotted versus crossing
angle a for selected lemniscates of the dry sliding chirp motion path.














CHAPTER 7
CONCLUSION

A multi-directional tribometer with the capabilities of uniform-velocity and

uniform-pressure motion paths has been designed and constructed. It has been used to

run a variety of motion paths with the aim of analyzing the resulting friction with the path

parameters such as changes in curvature. It has been shown that when all other

parameters remain constant, friction coefficient of UHMWPe in multi-directional sliding

is affected by changes of curvature in the motion path. The chirp motion path with

multiple passes of increasing and decreasing radius of curvature was run, and the highest

friction is seen at times when the change of radius of curvature is negative (dp/ds < 0).

Friction tends to increase with dp/ds < 0 and tends to decrease with dp/ds > 0. Due to

size scales, this suggests that surface orientation is dependent upon the length over which

fibrils can align and the energy required to do so.

Based on hypotheses of the surface orientation of UHMWPe, it has been shown

that there should be a quantifiable rise in friction each time the path motion vector

changes in direction by some angle a. Because the experimental did not correlate well to

the expected rise in friction coefficient for various lemniscates in our chirp motion path,

more work is needed in modeling the rise in friction with changes in the motion path to

quantify how friction can be a predictor of wear, in conjunction with alternate methods of

calculating a.

The ramifications of using in situ friction traces in current theories of surface

elongation and even unified wear theories are sizable. Limits can be applied to the theory






44


of surface orientation with respect to the pin size, sliding length, and paths necessary to

enact the phenomenon. More severe motions on the wear path can be correlated to

specific locations where higher wear might take place, rather than a general association of

the two. Also, some functionality of friction coefficient with crossing angle can be

inserted into unified wear theories rather than simplified assumptions of constant friction

through an entire complex motion path.















APPENDIX A
WEAR CALCULATIONS

Wear calculations are done using gravimetric analysis of polymer samples. This

means that volume loss is calculated from a measured mass loss (scale resolution down to

10 lig). Material comparison is made by the wear rate, rather than wear over a given

sliding distance or time, which allows for flexibility of measurements systems (Figure A-

1).

Wear rate is designated by k, and is the volume lost divided by the normal force

applied times the distance of sliding (Equation 7).


k = Vlot Equation 7
Fnd

The volume lost is calculated by taking the measured mass lost and dividing by the

density of sample (Equation 8), which is defined by the initial mass divided by the initial

volume of the sample (Equation 9). Sliding distance is constituted by the sliding distance

of each cycle and the total number of cycles (Equation 10).


Vost = Equation 8
Psample


Psampe- Equation 9
D2sample DL

d=S*N Equation 10

When all these variables are combined, wear rate is given by Equation 11.

AmrD2L
k = A Equation 11
4F1SNmo






46


F












Figure A-1. A schematic describing the dimensions and forces used to calculate wear
rate. Courtesy of Dave Burris.














APPENDIX B
RADIUS OF CURVATURE CALCULATION METHOD

The common radius of curvature for three points on a continuous curve is

calculated by find the circumcircle of the triangle made up of those three points. The

distance between any point and the circumcenter of that circle is the radius of curvature, p

(Figure B-1).

First, the lengths of the legs (a, b, c) of the triangle IJK are calculated using the

distance formula (Equation 12). The first intermediate step is to calculate a parameter of

the triangle s, which is a function of the leg lengths (Equation 13). From this parameter

and the lengths, the radius of curvature is calculated (Equation 14).

a= (x -x)2 (y y )2 Equation 12

a+b+c
s = Equation 13
2

abc
P= a b Equation 14
4s (s a) (s b) (s c)






48


point 2 point 3
point 1





b a






circumcenter








Figure B-l. Schematic of curve-fitting a circle between three points on a continuous
curve.
















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BIOGRAPHICAL SKETCH

Alison Dunn first gained an education of cultures by moving around frequently

until starting at the University of Florida in August of 2000. She earned a Bachelor of

Science degree in mechanical engineering in December of 2004. She has received

various awards including participation in the University Scholars' Program and the

Caterpillar Scholarship from the Society of Women Engineers. Upon obtaining a Master

of Science, she plans to join the Peace Corps and eventually work for a biomedical

company or technical instrumentation company.