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Experimental and theoretical studies of the wear of heat exchanger tubes Magel, Eric E. 1990

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E X P E R I M E N T A L A N D T H E O R E T I C A L S T U D I E S O F T H E W E A R O F H E A T E X C H A N G E R T U B E S By Eric E . Magel B.A.Sc. (Mechanical) University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apri l 1990 © Eric E . Magel, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of //£<fA<*?scd /^^e^Ay^z The University of British Columbia Vancouver, Canada Date /fpr-;/ 2?^ /?j?&>  DE-6 (2/88) Abstract A study of heat exchanger tube wear has been completed. A simple theoretical model of elastic/plastic deformation has been developed and used in a new model of wear. Experimental results were used to corroborate the theoretical developments. A literature survey of wear mechanisms and wear models was conducted to provide the author with an opportunity to familiarize himself with current knowledge of the field of tribology. Experiments were conducted to simulate a heat exchanger tube/support wear sys-tem. For the first series of experiments, a simple impacting rig was used, while a second set was conducted using a much more accurate rig and facilities of the National Re-search Council of Canada's Tribology Laboratory. Modifications to the N R C rig were designed by the author to incorporate the specific specimen geometries. The main operating parameters of the test apparatus were varied in an effort to determine their effect on wear rates. Force and displacement data were collected and the normal and shear forces cal-culated, as was the work input. Comparison between the frictional work input and the measured wear showed that there was an approximately linear correlation between work and wear rates. Inspection of the surfaces of the worn specimens showed that a number of wear mechanisms operate in this wear system but that wear is primarily due to delamina-tion and shear fracture. Also, it was noticed that the micro-surface geometry of the worn specimens has a consistent texture, regardless of magnitude and angle of impact between the tube and ring. n A model of plastic contact deformation was developed to allow calculation of the contact parameters between two surfaces, given that the softer surface is repeatedly plastically deformed. This model says that repeated stress cycles lead to the introduc-tion of residual stresses, which combined with work hardening of the material, lead the softer material to an elastic shakedown state. Once the typical asperity contact state is known, the typical stress distribution is calculated using Hertzian line contact stress formulae. A series of computer programs were developed to calculate the stress distribution beneath a sliding contact. The depth of maximum shear stress can then be found. This depth corresponds to the expected wear particle thickness. A wear sheet was assumed to form when the frictional work input is equal to the energy required to cause failure in ductile shear. A wear equation was then developed to predict the wear rate between a heat exchanger tube and its support. . The final wear model has seen limited comparison with experimental results. The theoretical work input was found to be about 25% of the correlated bulk work. This indicates that the geometry assumptions of the model are quite reasonable. Unfortu-nately, the predicted wear rate was found to exceed the measured values by a factor of about 5000. If this empirical value is factored into the the wear model, then the predicted results are found to correspond well with the experimental values. i i i Table of Contents Abstract " List of Tables viii List of Figures i x Acknowledgement x " 1 Introduction 1 2 Literature Survey 5 2.1 Wear Mechanisms 5 2.1.1 Adhesion 5 2.1.2 Abrasion 6 2.1.3 Delamination 7 2.1.4 Shear Fracture 7 2.1.5 Corrosion or Oxidation 8 2.1.6 Fretting Wear 8 2.2 Wear Models 9 2.2.1 Adhesive Wear 9 2.2.2 Abrasive Wear 10 2.2.3 Oxidative Wear 12 2.2.4 Delamination Wear 12 2.2.5 Asperity Fracture 15 iv 2.3 Heat Exchanger Tube Wear 18 2.3.1 Wear Mechanisms 18 2.3.2 Heat Exchanger Tube Wear Models 20 3 Preliminary Investigation 22 3.1 The Test Apparatus 22 3.1.1 Data Collection 26 3.2 Analysis: Procedure 27 3.2.1 Examining Weight Loss • 27 3.2.2 Surface Inspection 29 3.2.3 Force Correlation , 41 3.3 Discussion 42 3.4 Summary 46 4 Theoretical Modelling 47 4.1 Contact Deformation Model 48 4.1.1 The Shakedown Theory of Plasticity 48 4.1.2 Mathematical Development 50 4.1.3 Extension to Multi-Asperity Model 54 4.2 A Wear Model For Heat Exchanger Tubes 58 4.2.1 Mathematical Development 59 4.2.2 Applications of the Contact Deformation - Work Model 64 5 Verification of Mathematical Models 68 5.1 Contact Deformation Model 68 5.1.1 Procedure 68 5.1.2 Results 74 v 5.1.3 Discussion 78 5.2 Wear Model 85 5.2.1 Procedure 85 5.2.2 Results 89 5.2.3 Discussion 93 5.3 Summary 98 6 Conclusions 101 Appendices 104 A Data Correlation 104 B Numerical Solution To Equation 4.11 106 C Proof of Elliptical Planform 109 D Average Pressure Pavg(x) on Tube Surface 111 E Solution Algorithm for Equation 4.20 113 F Line Contact Stress Equations 115 G Work Input From Sliding Asperity Load 116 H Tube Shear Fracture Wear Model 118 I Results of Experimental Indentation Tests 121 J Effect of Elastic Recovery in Contact Deformation Model 126 K Results of Data Correlation Program 130 vi L Tube Trajectories Bibliography List of Tables 3.1 Results of Wear Experiments: Mass Loss from Tube and Ring Specimens. 27 5.2 Values of E and K for Brass and M i l d Steel 72 5.3 Values of P0/K for Extreme Cases of Load (L) and Slip Distance (/) . . 81 5.4 Summary of Data Correlation Procedure with Model Work and Wear Predictions 91 1.5 Results of Indentation Tests - Indenter on Brass Target 122 1.6 Results of Indentation Tests - Indenter on M i l d Steel Target 123 1.7 Ratio of Predicted Radius Value to Measured Value - Indenter on Brass Target 124 1.8 Hardened Steel Indenter on M i l d Steel Target 125 J.9 New Values of r and Ratio of Measured Radius / Predicted Radius Hardened Steel When Elastic Recovery of Transverse Radius is Included. Indenter on Brass Target 128 J.10 New Values of r and Ratio of Measured Radius / Predicted Radius Hardened Steel When Elastic Recovery of Transverse Radius is Included. Indenter on M i l d Steel Target 129 vii i List of Figures 1.1 Typical C A N D U Steam Generator 3 1.2 Worn Tube in Heat Exchanger 4 2.3 Asperity Geometry for Abrasive Wear Model 11 2.4 Surface of Worn Heat Exchanger Tube [34] 20 3.5 Tube Support Specimen and Transducer Arrangement 23 3.6 Room Temperature Fretting Rig 24 3.7 Schematic of Vibration Generator 25 3.8 Typical Trajectories 25 3.9 Geometry of Test Specimens 26 3.10 Diagram of Electrical System 28 3.11 The Boundary Between the Unworn and Worn Surface 30 3.12 "Run- in" Surface 30 3.13 The Transition from Unworn to Worn Surface Texture 32 3.14 a) Initial Furrows Covered by Plastically Deformed Material b) Close-Up 33 3.15 Typical Fracture Pit 34 3.16 Typical Indentations in a Ring Specimen 34 3.17 Corrosion Pit . 35 3.18 Oxidized Flakes on Tube Surface 36 3.19 Opposing Wear Scars Showing Evidence of Material Transfer: (top) Tube Surface (bottom) Ring Surface 37 3.20 Typical Tube Wear Particles 38 ix 3.21 Typical Severe Wear Scar 39 3.22 Subsurface Damage of Tube 39 3.23 Pits on Main Wear Scar 40 3.24 Severe Wear Scar 41 3.25 Dimple Edged By Subsurface Furrow 43 3.26 Rippled and Indented Surface of Polished Tube 43 3.27 Etched Surface of Wear Scar 44 4.28 Texture of Steady-State Ring Surface . . 48 4.29 Shakedown 49 4.30 Pressure Distribution on Indentation 51 4.31 Profile of Indentation in Sliding Direction 52 4.32 Primary (p) and Secondary (rj) Radii of Curvature 54 4.33 Depth of Maximum Octahedral Shear Stress (y<72/P0) for Surface and Subsurface Peaks 60 4.34 Contour Plot of / j 2 / P „ for p, = 0.262 . 60 4.35 Location of Jj2/Po (maximum TOCT) as it Varies with p, 61 4.36 Variation of Jj2/Pa for a Line Contact p = 0.20, ( j j ) = 0.5663 62 4.37 Indentation Microsurface 65 4.38 Indentation From Polished Specimen 67 5.39 N R C Fretting Wear Rig . . 69 5.40 Dynamic Specimen Holder 69 5.41 Spherically Tipped Indenter 70 5.42 Tangential Displacement and Normal Force Wave 70 5.43 Shakedown Map For Line Contact Geometry 73 x 5.44 Typical Indentation: Photo Shown With 3D Plot and Corresponding Longitudinal Traces and Curve Fits 75 5.45 Steel: Ratio of Measured Value to Predicted Value 76 5.46 Brass: Ratio of Measured Value to Predicted Value 77 5.47 A "Double Pi t" 79 5.48 Shakedown Maps For Line Contact (top) and Circular Contact (bottom) 82 5.49 Dynamic Specimen Holder 86 5.50 Stationary Specimen Holder 87 5.51 Teflon Lined Bolt For Positioning Transducer and Tube Specimen. . . . 87 5.52 The Longitudinal (top) and Transverse (bottom) Talysurf Traces 88 5.53 The Asperity Height (top) and Radii (bottom) Distributions 90 5.54 Experimental: Wear Rate vs Work Rate 91 5.55 Experimental and Predicted Mass Loss 92 5.56 Experimental and Revised Predicted Mass Loss 93 A . 57 Vector Diagram of Forces and Displacements 104 B . 58 Plot of Exact Solution b(x) to Equation 4.11 107 C. 59 Geometry Used in Planform Derivation 110 D. 60 Sliding Load Distribution on Tube Surface I l l K.61 Test P I Waveforms 130 K.62 Test N13 Waveforms 133 xi Acknowledgement The author is grateful for the support of Atomic Energy of Canada Limited through the Natural Sciences and Engineering Research Council (NSERC) , and for the assistance and guidance of his supervisors, Drs. P .L .Ko (NRC) and H.Vaughan (UBC) . As well, the author is particularly indebted to Mr.M.Robertson (NRC) and Dr.D.Dunwoody (UBC) for their help in the experimental and analytical developments, respectively. Drs. K.L.Johnson (University of Cambridge) and J.Kalousek (NRC) were also of in-valuable aid to the author, especially in the areas of theoretical developments. And of course, I must thank my wife for her encouragement and patience. xii Chapter 1 Introduction Wear is without doubt one of the most difficult physical processes to categorically quantify. Examining even a small range of environments, materials and loading systems reveals that many different wear mechanisms can operate in any wear system at one time. Even in cases where a single wear mechanism is involved (and those cases are rare), defining exclusive limits where a wear mechanism acts has been an elusive process. In practical cases, the complex dependencies between definable variables has limited researchers' abilities to accurately model even the simplest of wear processes. Wear is typically regarded as any process which causes removal of material from a surface. Some tribologists (Rigney [1] for example) have stressed that even without the removal of wear particles, considerable damage may be inflicted upon the microsurface in the form of plastic deformation and subsurface crack initiation and propagation. For this reason, the author prefers to use the following definition for wear: Wear is any process which results in the damage or weakening of a component surface with subsequent material loss. There has been great academic and economic incentive in recent years to develop wear models. Discerning the mechanisms of wear and quantifying that wear with rea-sonable accuracy is a challenge that requires amalgamation of knowledge from the chemist, metallurgist, mechanical engineer, and applied mathematician. Any general-ized wear system must necessarily be extremely complex due to the millions of possible combinations of contact geometries, material parameters, loading conditions etc.. With 1 Chapter 1. Introduction 2 the advent of computers, theoretical wear model development has proceeded at an ac-celerated pace, but still only with highly simplified, ideal systems. Finite element modelling has allowed researchers to apply combined elasticity and plasticity theory to contact mechanics and thence develop the stress conditions in perfect solids, The effect of various geometries and material strengths on crack propagation thus becomes discernable. These efforts have been spurred on by industrial concerns. In a 1984 A C O T report [2] it was estimated that the total annual cost to industry in Canada as it relates to component wear and its associated consequences (e.g. loss of tolerances in machining processes, downtime to replace a failed component, noise pollution, replacement of contaminated lubricants etc.) amounted to 5 billion dollars. The development of wear models will not eliminate wear, but in the design phase, models can be used to optimize designs with the goal of minimizing wear within the physical and economic constraints. Surface finishes and geometries, for example, may have serious effects on the life of a component. One such industrial concern is the steam generator and heat exchanger manufac-turing industry. Heat exchangers are playing an increasingly important role in modern industry. In C A N D U (Canadian Uranium/Deuterium) nuclear reactors, for example, large amounts of heat generated by the dissociation of atoms must be transferred from heavy water contained in the reactor's closed circulation system to the feedwater which is heated to steam and passed through turbines to generate electricity (see Fig.1.1). In the interest of efficiency, feedwater flow velocities have increased. The consequent rise in turbulence leads to greater dynamic forces between the tube and its support with consequent increased wear (Fig.1.2) of the heat exchanger tubes and reduced heat ex-changer life. The downtime to replace worn tubes is phenomenal and current practice is to simply plug the tube so it no longer functions. A reasonable model of this wear Chapter 1. Introduction 3 Figure 1.1: Typical CANDU Steam Generator process would allow proper evaluation of the effects of a design change and qualified decisions could then be made in compromising short term efficiency gains in the inter-ests of extended service life. Wear models can be similarly used in many other areas of industrial design and manufacture. In modelling a wear process, the first step is to determine which parameters have the greatest influence on the wear system characteristics. To this end, a test rig was de-veloped to simulate the tube/tube support interactions in practical heat exchangers. A variety of impact/sliding force combinations were applied to the test specimens and the worn surfaces examined. Observations allowed the most important wear mechanisms to be identified and gave some insight into the relationship between the important Chapter 1. Introduction 4 If mm 111 Figure 1.2: Worn Tube in Heat Exchanger variables. The main wear mechanisms isolated were shear fracture and delamination. The cause of failure in both of these mechanisms is the development of high subsurface shear stresses, combined with extremely high near surface strains. Large plastic strains viewed in the micro-surface layer indicate that elastic contact models would be inade-quate for describing the heat exchanger wear system. Using the shakedown theory of plasticity, a contact deformation model has been developed which predicts the geome-try of a single indentation formed during a typical contact between a hard asperity and a soft, flat surface. This model was then extended to a multi-asperity situation which incorporates the tube and ring geometry of the heat exchanger tube wear system. The resulting contact parameters were then used to evaluate the stress situation beneath a sliding asperity. Evaluation of an appropriate failure criteria allowed the development of a model for heat exchanger tube wear. Chapter 2 Literature Survey 2.1 Wear Mechanisms Wear can be classified according to several different schemes. Archard and Hirst [3] simply classified wear as either mild or severe. Other authors have catagorized wear according to the particular type of mechanical interaction and relative motion. The relative motion may be unidirectional sliding, reciprocating, rolling or impacting. A more fundamental scheme was used by Burwell and Strang [4] and later modified by Burwell [5] which includes five particular types of wear: 1) adhesion or galling, 2) abrasion, 3) corrosion, 4) surface fatigue and 5) minor types (e.g. the wear of electrical contacts). 2.1.1 Adhesion One of the most fundamental wear mechanisms is known as adhesive wear. This type of wear is also sometimes known as scuffing or galling. This is the basic phenomena which occurs when two surfaces are in dry sliding contact. Regardless of how smooth surfaces may appear, they are microscopically rough. When two surfaces are pressed together, they will contact only at the tips of the surface roughness peaks, or asperities. The pressures in these small zones may be extremely high, certainly exceeding the elastic yield point of the softer material and leading possibly to cold welding of the asperity tips. The application of a tangential load further increases the local stresses, 5 Chapter 2. Literature Survey 6 aggravating the adhesive wear tendency. With relative motion, material will be torn from the softer or weaker of the two surfaces. The adhesive mechanism doesn't predict wear as an inevitable consequence. If the junction shears at the original interface, the surfaces will be unchanged. But when a weld forms between two asperities, the junction is strengthened due to work hardening and in the general case, material will be torn from weaker of the two surfaces. The presence of oxides and other contaminants, which inhibit adhesion, will thus retard the adhesive wear process. 2.1.2 Abrasion Hard asperities on one surface or hard constituents embedded in one material may score, gouge, or scratch material from the opposite surface. The hard constituents in one surface may be the product of a chemical process such as oxidation or else due to temperature effects, such as the precipitation of carbides. Erosive wear is sometimes categorized as a form of abrasive wear1. Moore [6] describes two mechanisms by which material may be removed during the abrasive wear process. Wear particles can be removed as prows which form in front of a moving asperity due to plastic deformation or as chips due to fracture with limited plastic deformation. The abrasive wear mechanism also includes third body wear. This type of wear occurs when hard, abrasive particles are trapped between the two surfaces. The hard particles may be the products of wear (i.e. work hardened wear particles) or may otherwise be introduced from outside sources (dust, sand etc). Third body wear is one of the most common types of wear and according to recent tribological surveys [2] is responsible for the bulk of wear in industrial machinery. xOther authors, such as Engel [25] have categorized erosion as a form of impact wear Chapter 2. Literature Survey 7 2.1.3 D e l a m i n a t i o n Delamination wear was first described by Suh and his co-workers [7] as the loss of metal in the form of flakes, caused by the formation and propagation of subsurface fatigue cracks running parallel to the surface. Delamination wear is often classified as a fatigue mechanism since it depends upon the incremental propagation of microcracks. Cracks may initiate by the accumulation of dislocations or the formation of voids around inclu-sions in the material. The stress reversal encountered in the subsurface during sliding contact is considered responsible for the propagation of cracks parallel to the surface. During sliding contact, an increment of damage to the material will be experienced during every asperity interaction. As the micro-cracks nucleate and propagate, they may link up with others and form a two-dimensional "penny" crack. Eventually, when the subsurface cracks encounter surface cracks, a thin wear sheet will lift from the surface. The depth of a wear particle can be calculated as the depth of the maximum reversed shear stress. Many finite element investigations have been performed which evaluate the effects of crack angle and crack face friction on the stress intensity factors at the crack tips. The stress intensity factor is a direct indicator of the relative propagation rates. Many authors have suggested that the subsurface cracks propagate due to Mode I (tensile) stresses while others have analysed Mode II (in-plane shear) [8] or combined Mode I and Mode //stresses [9]. 2.1.4 S h e a r F r a c t u r e Shear fracture is a relatively new wear concept and is often combined under the um-brella of fatigue or delamination wear. But the distinction between shear fracture and delamination is an important one. Chapter 2. Literature Survey 8 The mechanism of delamination is usually considered as a brittle fracture phenom-ena whereby cracks propagate beneath the surface due to a combination of cyclic Mode I and Mode II stresses. Shear fracture, on the other hand, is due to failure in duc-tile shear. Shear fracture thus includes large near surface plastic deformation while delamination usually does not. The wear products of both delamination and shear fracture will be thin and flake-like and the wear scars will appear similar. The difference is the inclusion of considerable plastic deformation in the case of shear fracture. There is little doubt that crack forma-tion and propagation will occur in the shear fracture situation due to the accumulation of dislocations and nucleation of micro-cracks around inclusions. Perhaps shear fracture and delamination can be considered different extremes of the same mechanism. 2.1.5 Corrosion or Oxidation Corrosive wear is usually a result of both chemical reactions and rubbing. Oxidation, for example, proceeds at its highest rate when a fresh surface is exposed and decreases parabolically from that point onwards. If the oxide product is not removed, the oxide film will actually protect the material from further corrosive wear. Stainless steels for example, readily form protective oxide films. Rubbing will usually remove a weakly adhering oxide product however, and the process will then repeat. In some cases a very hard surface layer is formed and may actually decrease wear. But if that hard layer is broken off, it may accelerate wear due to the abrasive third body mechanism. 2.1.6 Fretting Wear This form of wear is common when two components undergo relative oscillatory slip of very small amplitudes. It is actually a combination of many wear mechanisms, including Chapter 2. Literature Survey 9 abrasion, adhesion, corrosion, fatigue and third body. Fretting wear is usually initiated by adhesive or abrasive wear and subsequently, when the wear particles are oxidized, third body wear. According to Waterhouse [28], delamination is responsible for the final generation of wear particles. 2.2 Wear Models 2.2.1 Adhesive Wear Adhesive wear was once considered one of the most fundamental of wear mechanisms and dominated the literature. But adhesive wear has in many cases been eliminated or significantly reduced due to the increasing knowledge of lubricants and material compatibility. The reduction in the number of dry metal-metal contact situations has reduced adhesive wear from a primary to a contributing mechanism. Recent research into adhesive wear has thus been limited. Little progress has been made beyond the work of Archard, which was conducted in the 1950". Archard [11] developed a model for adhesive wear and it was later extended to most other wear mechanisms. The model relates the worn volume V , produced in a sliding distance L, to the true area of contact, A. He proposed that a real surface consists of semicircular asperities and so for each asperity which contacts the mating surface, the contact area will be 7ra 2. Further, he suggested that the depth of the material removed is proportional to the asperity contact radius a, so 8V « a 3 , and 81 « a. Since the wear rate is the volume of material removed per unit sliding distance, 8V/81 « a 2 (i.e. proportional to 8A). Integrating gives where K\ is the wear coefficient. Chapter 2. Literature Survey 10 Realizing that the real area is much less than the apparent area of contact, the contact zones were assumed to be plastically stressed. With this, the real area can be simply found from Force = pressure • Area of contact — p • A Therefore: A = Fn/H where H = the Brinell Hardness or flow pressure of the softer material. The final wear equation then is j = KiFJH This basic wear equation can be derived from abrasive, delamination and oxidation wear theory with the only difference being the factors which make up the wear coefficient K. 2.2.2 A b r a s i v e W e a r Over the past two decades several authors have developed a variety of abrasive wear models by assuming an assortment of geometrical configurations, all with limited suc-cess. Torrence [12], for example, developed a simple upper bound abrasion model for grinding by assuming that the abrading particles were three-dimensional pyramids. Challen and his co-workers [13] used two-dimensional slip line fields in developing an abrasive wear model. Suh [14] by assuming that the scouring features are conical asper-ities with hemispherical tips, found the following formula for wear volume (see Fig.2.3): V = w2 for w > 2r sin 6 -2 / „ „ \ 1 / „ „ \ - i ( /,„\2>\l~> ( ! )*-* £ H ( ! ) > ( w \ 1 2r~) Chapter 2. Literature Survey 11 Figure 2.3: Asperity Geometry for Abrasive Wear Model where r is the tip radius 8 is the cone angle w is the width of the scour The width of the scour is dependent on the depth of penetration which in turn depends on the relative hardnesses of the interacting surfaces. It then remains to determine the extent of this wear mechanism in any given wear process. The application of semi-empirical geometrical relations which utilize a constant of proportion, characteristic of the material pairs and thermal conditions, has shown to be effective in modelling abrasive wear. Chapter 2. Literature Survey 12 2.2.3 Oxidative Wear Using an equation similar to that developed by Archard, Quinn et.al [15] were able to describe oxidative wear according to L - K ^ d ~ oxid- H where 0e-[QP/RT] JC . _ til [fVPo\2v pe-[Qp/RT] — parabolic oxidation constant P IQP — constants for given material R = universal gas constant T = temperature / = fraction of oxide which is oxygen 77 = critical oxide thickness p0 = average density of oxide v — velocity of sliding 2.2.4 Delamination Wear In his first paper on the delamination theory of wear Suh [16] recognized the deficiencies in current knowledge regarding the void formation and accumulation process as well as difficulties in relating shear deformation to surface tractions. As a result, the develop-ment of a theoretical wear equation devoid of empirical constants and relations is thus far unattainable. He proceeded, then, to formulate a semi-empirical wear equation. The following assumptions were made: Chapter 2. Literature Survey 13 1. Metals wear layer by layer, each layer consisting of N wear sheets. 2. The number of wear sheets per layer is proportional to the average number of asperities in contact at any time. 3. The rate of void accumulation, crack nucleation and critical degree of shear de-formation for wear particle formation can be expressed in terms of SQ, the critical sliding distance required for the removal of a complete layer. For the case of a hard surface sliding against a soft surface, the total wear can be expressed as W = N^S/SoJArhr + N2(S/' So,)A2h2 where the subscripts 1 and 2 refer to the soft and hard metals sliding against each other. Note that SQ is likely to be lower when more compatible materials are slid against each other since a greater shear stress is likely to be established. Assuming 1) that the thickness of a wear particle is equal to the depth of the low dislocation density zone, 2) the average surface area of each delaminated sheet is proportional to the real area of contact per asperity Ar and 3) AT and the number of asperities in contact n are proportional to the applied load leads to the following result: K\G\ | K2G2 (JhS0l(l - v^S^l - v2) where K\ and K2 are constants which depend primarily on the surface topography, b is the Burgers vector and G the shear modulus. The ratio oiGjOf will in general decrease with an increase in solid solution harden-ing and with hardening by secondary phase particles, but increases with overaging. SQ is expected to increase with lubrication and when less compatible materials are tested. W = b- LS 4.7T Chapter 2. Literature Survey 14 S0 should also decrease with an increase in the density of hard particles in the matrix (due to increased likelihood of crack nucleation) [17]. From these observations, the wear rate can be expected to decrease when increasing the hardness by formation of a solid solution and possibly increase when hardening by inclusion of secondary phase particles. This equation may be written in the form presented by Archard as W = K-LS where AC is a wear factor given by K i d | K2G2 <TflS0l(l - I/ i) (ThS02(l - V2) Note that this equation, unlike Archard's wear equation, does not depend directly on hardness. For Mode II crack propagation the stress intensity factor is the greatest and accord-ing to Suh the wear volume V can be related to the normal load L, sliding distance S and the depth of crack from the surface d according to the following expression: V = S(2kn) • LSd2 where 8 is a proportionality constant ku is the Mode 7/ stress intensity factor The factor of two was inserted to account for the complete shear stress reversal when an asperity slides over the crack tip. In some materials the threshold stress intensity factor will be lower than the value of ku and so cracks will not propagate in that material. In this case delamination wear may still occur as void accumulation continues to be active. When a sufficient Chapter 2. Literature Survey 15 number of voids have joined, wear particles will form when the surrounding material fails due to the applied asperity loads. The delamination wear process may then be controlled by the rate of plastic deformation (corresponding to the applied loads), the crack nucleation rate (presence of existing cracks, second phase particles or inclusions) or the crack propagation rate (corresponding to the fracture toughness). 2.2.5 Asperity Fracture This particular model was developed by Jain and Bahadur [18] for the wear of polymeric materials. It is included here as an example of fatigue wear modelling and contains the classical elastic asperity contact model originally developed by Greenwood and Williamson [19]. Jain and Bahadur developed their model on the basis that asperity fatigue is the prominent failure mechanism in the wear of polymers. The interactions between as-perities which occur during sliding of surfaces lead to cyclic contact and reversals of the principal tensile stress. This principal stress is considered to be responsible for the nucleation and propagation of fatigue cracks. Deformation is assumed to remain elastic between random distributions of spherically tipped asperities. They report that this is in agreement with experimental studies of polymer-metal contacts. As well, they assume that the contact zones are sufficiently separated to act independently of each other and that the asperities on one surface are aligned with those on the other surface and have the same pitch in the sliding direction. For elastic contact between a sphere and flat, the following equations can be ob-tained from Hertz: Ai = ir(3u Chapter 2. Literature Survey 16 ^ 4 „ ^ i 1 where E* E\ E2 and i / is Poisson's Ratio, u> the compliance, the radius of the sphere, a\ the radius of the contact zone and E the modulus of elasticity. Greenwood and Williamson, extended the above single contact equations to the case of contact between a rough and a smooth surface to develop expressions for the number 7]0 of discrete contact zones, Ar the area of real contact and load P. By normalizing with respect to the standard deviation of the asperity height distribution, Jain and Bahadur arrived at the following: n = r,A0F0(h) (2.1) Ar = irqA0(3(rF\(h) P = \-qA0E*i3*a*F*{h) where roc Fn(h) = / (s - h)n(f>(s)ds J h h = d/cr s = z/a (2.2) and d is the separation between the reference planes of the surfaces in contact. To include surface roughness on the opposing surface, Greenwood and Tripp [20] showed that two rough surfaces in contact can be replaced by one rough surface in contact with a flat by using an equivalent radius of curvature (3 and equivalent standard Chapter 2. Literature Survey 17 deviation cr given by p = - M L . where /?i and /?2 are the average radii of curvature of the asperity tips on the two surfaces and <Ji and <T2 are the standard deviations of the asperity height distributions. Using the stress equations of Hamilton and Goodman[52], an expression for the maximum tensile stress S was developed. s = 3/cxP 2irnA0/3o-F1(h) where SlA \ l - 2 v By then representing the fatigue properties of the material by Wohler's curve (Nf = [S/So]1), the number of loading cycles Nf needed to cause fracture of an asperity is obtained as 2*S0riAop<rF1(h)Y 3/cj The asperity encounter rate, NR ( number of times any given asperity is contacted by another) for a sliding speed v can be found as NR = r)LnA0vF0(h) where ni is the line density of asperities on the moving surface. The number of wear particles formed per unit time can then be found as Nw = NR/Nf [(IvMSoBaF^h)]* Chapter 2. Literature Survey 18 The volume of wear per unit time is then simply found as the product of Nw x vp where vp is the volume of a typical wear particle. Through manipulation, this equation can be represented in a form similar to Ar-chard's equation where the wear coefficient depends on the surface topography, modulus of elasticity, fatigue properties, wear particle size and frictional coefficient. 2.3 Heat Exchanger Tube Wear Considerable work has gone into the development of test apparatus for examining heat exchanger tube wear. Utility companies such as Westinghouse, Atomic Energy of Canada, U.S. Department of Energy etc., have been supportive of efforts in this direction. Of course the incentive is an economic one with the hope that these research efforts lead to better materials and improvements in design for the purpose of prolonging service life. Ko [21], Blevins [22] and Fricker [24] have made direct investigations to determine the wear mechanisms and develop wear models for heat exchanger tubes. Numerous others have made contributions to the modelling of sliding, oblique impact and impact wear (e.g. [25] and [26]). 2.3.1 Wear Mechanisms Several authors have described the wear of heat exchanger tubes using the concept of fretting in which low amplitude oscillations lead to fretting corrosions and the devel-opment of fatigue cracks. Fretting wear is a common problem when the contact parts are held together by a tight tolerance fit or normal pressure. Chivers et al [27] describe a potential vibrating wear problem in the gas circulators of advanced gas cooled re-actors. The fretting wear properties of Inconel and other Ni-Cr alloys comonly used in heat exchangers and titanium often used in aero-engines have been investigated by Chapter 2. Literature Survey 19 Waterhouse et al [28] and many others (e.g. [29]). A major finding of a study carried out by Nishioka and Hirakawa [30] was that the points of fatigue crack nuclation were usually slightly inside the edge of component contact where the local stress was the maximum. The fatigue strength reduction was attributed to the stress concentration. Tyler, Burton and K u [31] studied contact fatigue under normal oscillatory load using a ball and flat specimen. Toroidal rings of hardened and softened material were observed both on and below the surface. These hardness variations appear to have been related to the stress pattern. They found that fatigue cracks initiate in the material immediately adjacent to the most severely hardened material. Laird [32] showed that the fatigue fracture phenomena was closely associated with cyclic plasticity since cracks nucleated in active slip bands and subsequently propagated along them. If there is sufficient clearance between the components, then damage to the surfaces may be the result of both impact-sliding wear. The various aspects of impact dynamics have been considered by Goldsmith [33] and Engel [25]. In many cases determining the maximum stress levels, the locations and the impact durations is sufficient for describing the impact process. In cases where structural vibration and external excitation exist, extensive vibration and force analysis must first be performed. In a recent study, Hogmark et.al [34] had a rare opportunity to examine heat ex-changer tubes taken from a nuclear reactor. They determined that wear occurs by "flow enhanced removal of material from the superficial layer of the oxide film". This finding is contrary to those obtained from studies with room temperature rigs in which fretting appears to be the primary wear mechanism. They were unable to find any evidence of plastic deformation and thus ruled out the possibility of mechanical action. The topography of the worn tube consists of a regular series of shallow concave areas (see F ig . 2.4) and is considered further evidence of erosive wear. Chapter 2. Literature Survey 20 Figure 2.4: Surface of Worn Heat Exchanger Tube [34] 2.3.2 Heat Exchanger Tube W e a r M o d e l s There have been a variety of approaches to modelling the wear of heat exchanger tubes. Most of these models consider only the bulk loading and geometry parameters and of-ten make use of factors involving the clearance, tube length, bending etc. Fricker [24] calculated the wear rate by first deriving an equation for the the energy input per cycle from the excitation force due to vortex sheddng. He then equated this energy to the wear energy using the adhesive theory of wear. Blevins [22] studied the fretting wear of heat exchangers in a nitrogen/air mixture at room temperature. The effects of tube/tube support clearance, eccentricity, frequency of vibration, and mid span dis-placement were studied. He was able to fit an empirical model based on his findings. In a later paper, Blevins [23] developed a new predictive model based on the impact theory of wear. He modelled the stress field using Hertzian theory for an elastic point Chapter 2. Literature Survey 21 contact. The dynamic contact force and stress were obtained by equating the potential energy associated with deformation with the kinetic energy from impacting. He pre-dicted that wear would occur when the estimated contact stress exceeded the allowable fatigue failure limit. His emphasis was on the impact mechanism and the sliding mech-anism was neglected. K o and Basista [36] on the other hand identified the importance of the shear component during impacting and showed that normal impacting alone did not cause severe wear. Rice et al [35] studied the effect of transverse sliding velocity on wear of different materials during compound impact and established that subsurface material structures are altered during these loading cycles, giving rise to wear. They indicated that the altered characteristics of the subsurface zone depend on the magnitude of these com-pound impacts together with the material properties and environment. Levi and Morri [37] related the impact angle to the impulsive wear coefficient which is a function of the volume loss per unit energy impact. They found that the wear coefficient peaks at impact angles around 50 degrees from the horizontal plane and that the effects of energy and impact angle vary according to the material pairs and the ambient temperature. Ko [38] found that the wear rate correlates well with the shear force component but not with the resultant force nor the normal force. Engel [25] considered the wear energy to be a fraction of the peak strain energy of the impact cycle. By analysing the stress distribution, he too found that the contribution of the normal component is small compared with that of the sliding component. Engel concluded that the ratio of the peak stress to the yield stress and the sliding speed are two important parameters for impact wear. Note that since the shear force and normal force are often related by the coefficient of friction, the wear rate can sometimes correlate equally well with either force component. Chapter 3 Preliminary Investigation 3.1 The Test Apparatus A simple apparatus for simulating tube-tube support interactions was developed at the Chalk River Nuclear Laboratories [39] and used by the author in his experiments. A small tube specimen is mounted near the end of a length of vertically cantilevered tubing which in turn passes through an annular ring representing practical tube supports. The ring is supported at four points by piezoelectric transducers mounted in a stainless steel housing. Clearance between the ring and the recepticle assures that all forces are transferred to the transducers. Two proximeters are mounted at ninety (90) degrees to each other at forty-five (45) degrees to the force transducers (see Fig.3.5). A n excitation device is then mounted at the very end of the tubing (Fig.3.6). This device consists of a light-weight motor housing and two stepper motors driving eccentric masses (Fig.3.7). The stepper motors are driven by a frequency generator. Various trajectories and force magnitudes can be achieved by mounting different eccentric mass combinations and varying the phase angle between masses. Examples of the trajectories obtained are shown in Figure 3.8. The base of the rig is made of brass as are the support columns. The center and top plate are constructed of aluminum. The entire apparatus sits in a waterproof trough. Tube specimens were machined from Incoloy 800 heat exchanger tubes supplied by Atomic Energy of Canade ( A E C L ) and tube support specimens were machined from 22 Chapter 3. Preliminary Investigation 23 Figure 3.5: Tube Support Specimen and Transducer Arrangement one inch stainless steel type 304 and type 410 roll stock. The geometry of the specimens is shown in Figure 3.9. The initial diametrical clearance between the ring and tube has been fixed for all tests as 0.015". Room temperature distilled water was pumped from the trough through a de-ionizing filter and then to the contact region. That water was collected and then run through a "sluice" box lined with ten micron filter paper to collect wear particles. Overflow was simply spilled to the trough. The duration of tests was either eight or sixteen hours with the motor frequency set to between 27 and 30 hertz. There are several problems with this apparatus which makes the consistency of tests difficult to maintain. The rig is relatively light and flexible and experiences beating at certain frequencies and under certain loading conditions. The simple design also makes it difficult to ensure that the ring and tube specimens are mutually perpendicular. With the sluice box filter system, only very small quantities of wear particles are actually Figure 3.6: Room Temperature Fretting Rig Chapter 3. Preliminary Investigation f v = ( m . + m , ) roi 'man ' 'x = ( m . - m ,) rcu mai 1 1 f = m . r o i 2 s i nwt - m , roi 2 s i n (cut - I 80 ) Figure 3.7: Schematic of Vibration Generator Figure 3.8: Typical Trajectories Chapter 3. Preliminary Investigation 26 Tube Specimen Incoloy 800 Ring Specimen Type 410 S.S. 0.250 Figure 3.9: Geometry of Test Specimens collected since most of the water overflows the container. The debris which is salvaged is often contaminated with either aluminum from the plates or brass from the columns. Also, since chemically different materials were in contact through water, the possibility of galvanic cell action could not be eliminated. Despite these problems, a wide range of tests were completed and the results have been helpful in determining the active wear mechanisms. 3.1.1 Data Collection The output signals from two opposing force transducers are negatives of each other (identical but of opposite sign) so one is inverted and then the two are summed to double the apparent force signal (to increase resolution) and the resulting x and y signals are then input to the digital oscilloscope. From the oscilloscope, it is possible to generate hardcopies of the plots or to export the signal to a computer for processing by means of GPIB cables, I E E E card in the computer and the appropriate software. The signal from the proximeter probe is first amplified by proximeters and then sent directly to the oscilloscope where it can then be similarly exported to the computer. Chapter 3. Preliminary Investigation 27 Test Tube Ring Time Lubricant Comments # (mg) (mg) (hrs) 2 1.17 0.25 8.0 Dry 3 0.17 0.14 8.0 Water 4 0.36 0.57 8.0 Water Chlorine and Copper Deposits 5 0.47 0.88 18.5 Water Problem with Beating 6 0.72 1.50 16.0 Water M i = 10, M 2 = 20 (Shear) 7 1.07 1.23 16.3 Water M i = 20, M 2 = 15 8 0.71 0.48 16.5 Water M i = 20, M 2 = 15 9 0.21 0.35 14.8 Water Polished - M i = M 2 = 11 (Impacting) 11 2.36 1.79 16.0 Water Polished - M i = 20, M 2 = 11 (High Shear) 12 0.13 0.45 18.5 Water Polished - Almost Pure Impacting 13 0.53 0.51 1.0 Dry Low Impact Forces Table 3.1: Results of Wear Experiments: Mass Loss from Tube and Ring Specimens. A schematic of the signal amplification and collection system is shown in Figure 3.10. 3.2 Analysis: Procedure 3.2.1 Examining Weight Loss The impact specimens were weighed before and after each test with analytical balance having resolution of ±0.01 milligrams. The results of the tests are given in table 3.1. Those tests not tabulated failed due to mechanical or other errors. These results are scattered due mainly to the wide range of test conditions. Since the primary concern of this phase of the experiment is to determine the active wear mechanisms, the actual mass losses from these wear tests are given little consideration. Some very general conclusions can still be made. Comparison between trials 11 and 12 reveals the importance of the shear force component in wear. It is also apparent that the wear volume also increases with test duration. Chapter 3. Preliminary Investigation 28 • • @ D O • O D O a • • • ® © ® @ © © °DDD <p 9 ° ^ DIGITAL SCOPES • • <o) • • D D Q D Q OO O © © © O D D D O Q • • • . (5) (o) (5) © o_ DISPLACEMENT FDRCE SIGNALS SIGNALS FX t py t _ r Y 1 ti tox SUMMING BOX DC POWER SDURCE ® O M i l (9) PROX(METERS <r o D_ Figure 3.10: Diagram of Electrical System Chapter 3. Preliminary Investigation 29 Before and after micro-hardness tests on the specimens were inconclusive in estab-lishing that hardening of the microsurface layer had taken place. 3.2.2 Surface Inspection Wear is generally considered a local surface failure process and any attempt to ac-curately model wear must be pursued on the microscale where items such as grain structure, dislocation density, and surface roughness are of paramount importance. It is critical that the surface be examined with extremely high resolution instruments. To identify the wear mechanisms involved in this process, surface inspection with the scanning electron microscope (SEM), surface profilometer, and x-ray spectrum analyser has proven fruitful. Surface Inspection: Results Micrographs of the specimens obtained with the S E M have been helpful in determining wear mechanisms. A l l failure processes have a characteristic topography which may be viewed under the electron microscope and analysed. In the case of these tube/ring interactions, a variety of wear processes have been identified. Delamination, scuffing, pitting, corrosion, scoring and erosion have all been observed in this wear system. Examination of a large series of specimens and wear particles reveals that adhesion and corrosive wear have only minor influences on the overall wear. Plastic flow and delamination appear to be the dominant wear mechanisms. The preliminary stages of wear appear to involve smoothing of the original rough surface to a consistent wavy texture (Fig.3.11). This dimpled topography is charac-teristic of the "run-in" surface and was found on virtually all the specimens. Another example is shown in Figure 3.12. Chapter 3. Preliminary Investigation 30 -Unworn Figure 3.12: "Run-in" Surface Chapter 3. Preliminary Investigation 31 A close look at the plastic waves on the smoothed surface reveals that the typical ' h i l l ' is about 40 microns long by 20 wide and 5 microns in height. These dimensions are very consistent from one end of the wear scar to the other, from specimen to specimen. The transition from unworn to worn texture is clearly revealed by the series of four photos shown in Figure 3.13. Initial peaks and furrows from the tube forming processes and handling (Fig.3.13a) have been smoothed over until only the wavy texture characteristic of all the mild wear scars remains. The fourth photo (Fig.3.13d) shows the furrows almost entirely covered by plastically deformed material. Another photo (Fig.3.14a) shows the initial furrows vanishing as they enter the wear zone while a close-up (Fig.3.14b) shows tongues of metal bridging over the gap. A prominent feature of this smooth run-in surface was the large number of shallow elliptical pits found on the ring specimen. A closeup of a typical pit (see Fig.3.15) reveals a fracture pattern at one end and suggests that a subsurface penny crack has propagated until the particle finally tears or fractures from the layer. Another surface feature found during the examinations was a series of shallow indentations on the harder stainless steel ring (Fig.3.16). These indentations are of the same shape and dimensions as the fracture pits mentioned previously but the edges are much more rounded and do not appear to be formed by the removal of material. These indentations may be due to third body particles trapped between the two contacting surfaces (due to the combined effect of sqeeze film and entrapped particles[26]). In a small number of localized areas, darker, discolored areas can be seen underneath the S E M and indicate the presence of oxide products on the surface. Corrosion pits (Fig.3.17) on the ring surface and oxidized flakes on the tube (Fig.3.18) indicate that the water environment (with the possibility of a galvanic cell operating) may be a factor. Matching wear scars on the opposing surfaces allows a large area of transferred material to be identified (Fig.3.19). X-ray spectrum analysis verified the constitution Chapter 3. Preliminary Investigation Figure 3.13: The Transition from Unworn to Worn Surface Texture Chapter 3. Preliminary Investigation 33 (b) Figure 3.14: a) Ini t ia l Furrows Covered by Plast ical ly Deformed M a t e r i a l b) Close-Up Chapter 3. Preliminary Investigation Figure 3.16: Typical Indentations in a Ring Specimen Chapter 3. Preliminary Investigation 35 Figure 3.17: Corrosion Pit of the materials to prove that transfer had taken place. Very few wear particles were identified but in all cases the tube wear particles are elliptical in shape and very thin (0.25 to 0.75 microns thick). Photographs of tube wear particles are shown in figure 3.20. Very small stainless steel (ring specimen) particles were identified but they are smaller than one micron in diameter and are very difficult to distinguish. The more severe wear regime was attained in many of the later experiments. These tests were of longer duration and the forces were higher. The micrograph of a typical wear scar (Fig.3.21) shows that larger scale delamination may be occuring. A close-up of these layers (Fig.3.22) shows that the surface is at points only losely adhering to the specimen. It appears that wear in this regime may occur either as a peeling of thin layers from the surface or else through a pitting mechanism (Fig.3.23). Another S E M photo of a small wear area (Fig.3.24) shows a severe wear scar that is just starting to develop. Notice that the scale of this photo is 2/5th" that of figure 3.21. It appears as Chapter 3. Preliminary Investigation 36 Figure 3.18: Oxidized Flakes on Tube Surface Chapter 3. Preliminary Investigation 37 Figure 3.19: Opposing Wear Scars Showing Evidence of Material Transfer: (top) Tube Surface (bottom) Ring Surface Chapter 3. Preliminary Investigation Figure 3.20: Typical Tube Wear Particles Chapter 3. Preliminary Investigation Figure 3.22: Subsurface Damage of Tube Chapter 3. Preliminary Investigation Figure 3.23: Pits on Main Wear Scar Chapter 3. Preliminary Investigation 41 Figure 3.24: Severe Wear Scar though a very thin layer is being sheared from the bulk surface. 3.2.3 Force Correlation Force and displacement signals were transferred from the digital oscilloscopes to a P C computer using the Signal Processing and Display (SPD) software. A program was then written to convert the x and y force signals into shear and normal components (see Appendix A) . With this program it is also possible to determine the rms, standard deviation etc. of the force signals, calculate the energy input and total work done. A n attempt was also made to calculate the instantaneous frictional coefficient. This calculation is extremely sensitive to the phase angle between the force and displacement signals and the inability to accurately measure that angle meant that no meaningful value of the coefficient of friction could be obtained. Chapter 3. Preliminary Investigation 42 3.3 Discussion The results of the computer program for force correlation are not being used for evalu-ation of these tests since the author was unable to collect force and displacement data for all tests and because operational difficulties were encountered with many of the experiments. Since some of the important parameters were difficult to measure, the results of the force/work correlations using this limited information would have been suspect. Examination of the tested specimens with the S E M proved most fruitful. Initial tests made with as-machined specimens revealed very little wear (in the neighborhood of 0.1 milligrams/1.75 million cycles), but significant near surface damage was incurred as a result of the impact-sliding interactions. The initial surface texture of the tube specimen in Figure 3.11 is flattened by the opposite ring specimen (and vice versa). Figure 3.19 clearly shows that material transfer from the tube to ring occurs but this test (water lubricated, high magnitude normal impact force) was only one of two where such an observation was made. Figure 3.13 shows how the deep grooves of the machined surface are covered over by the plastically deforming micro-surface. The resulting "subsurface cracks" may act as important sites for the nucleation of further cracks, leading to the rapid formation of delamination particles. It was also noticed that the width of the dimples corresponded closely with the spacing of the subsurface channels shown in Figure 3.13c. Also, it was observed that most of the dimples had at least one edge bordered by a subsurface furrow (Figs.3.15, 3.25). To determine if these furrows affected the steady state mild wear texture, some specimens were polished with 1000 grit emery cloth before being tested. The surface of the polished specimens showed the same rippling texture of the unpolished ones (Fig.3.26) after being tested. Polishing the surfaces has minimal effect on the wear Figure 3.26: Rippled and Indented Surface of Polished Tube Chapter 3. Preliminary Investigation 44 Figure 3.27: Etched Surface of Wear Scar mechanisms since initial texture is rapidly removed by plastic deformation to give a consistent rolling texture to the interacting surfaces (this could be called run-in). Polishing with emery cloth will not remove the large number of deep furrows in the tube surface which appear to influence the wear pattern in both the ring and tube specimens. There then may exist a correlation between initial surface texture and the width of the wear particles, dimples, pits and indentations found on the tube and ring specimens. The channels seen in Figure 3.13c are approximately 20 microns apart, which corresponds well with Figures 3.20 and 3.15. To determine whether microstructure is contributing to the consistent size of the dimples and pits, the wear scar was etched and then viewed under the S E M . It is evident that the grain structure shown in Figure 3.27 is quite random with respect to the indentations. Examination of the wear debris reveals very thin, "flake-like" wear particles consis-tant with the delamination theory of wear. These particles are of the same dimensions Chapter 3. Preliminary Investigation 45 as the dimples found on the tube specimens or the pits and indentations found on the ring specimens. A l l of the pits are about 40 microns long by 20 microns wide and show great consistency despite changing the impact angle and magnitude of the force vector. The effect of running in a water environment still remains inconclusive. Water often manifests its presence in the formation of corrosion products but the Incoloy 800 and 410 stainless steel have been chosen by the designers of heat exchangers for their resistance to corrosion. Oxide products were found in this system in a few localized areas. Stainless steel products form thin oxide coatings on exposure to oxygen which inhibit further oxidation. These materials will thus continue to corrode if the oxide film is removed by some sort of wear process. The presence of oxides in this system can then be accounted for by one of the two following possibilities: 1) Either surface energy or a very small local surface separation prevents proper circulation of water and with corrosion and wear, the interfacial liquid increases in acidity, further encouraging the corrosion product formation or 2) Similarly, water enters the subsurface "chambers" and with each cycle the two faces of the crack wear against each other, removing existing oxide films, increasing the acidity of the local liquid and enhancing the local corrosion process. The contribution of corrosion to wear in this system appears minimal, since in most cases the circumstances of the first possibility are avoided, and corrosion due to the second appears relatively rare. Since we suspect a large network of subsurface cracks and can see microcracks in the smooth surfaces, water is likely present in the the subsurface. This can have two effects: 1) the water lubricates the opposing faces of a crack, reducing the resistance to relative motion (increasing the tendency for crack propagation due to Mode //stresses), or 2) hydraulic pressures may be generated which encourage crack propagation due to Mode / stresses. Chapter 3. Preliminary Investigation 46 3.4 Summary A variety of tests which simulated heat exchanger tube wear were performed using a simple fretting wear test rig. A number of loading conditions and durations were exam-ined. The wear scars of all the specimens were viewed under the electron microscope and those observations suggest that heat exchanger tubes in this system wear in the following manner: 1. Plastic deformation rapidly alters the original surface texture to yield a smooth wavy texture. This procedure also produces a system of subsurface cracks by covering deep furrows with the plastically deformed material. 2. Wi th accumulated plastic strain, the subsurface cracks propagate parallel to the surface and link with other existing cracks. Penny cracks form and propagate. This propagation may be aided by the presence of water. 3. When the local shear traction applied at the surface reaches a critical value (which depends on the depth of the subsurface crack and extent to which it has propa-gated) a particle will be removed. 4. As the wear process continues, larger scale delamination will occur as the geome-tries of the interacting surfaces continue to change. More and more cracks will open to the surface and material may be torn from the surface due to mechanical interlocking. Some pitting may occur when normal impacting dominates. Chapter 4 Theoretical Modelling In the wear of heat exchanger tubes, it has been shown that the worn surface of the stainless steel support rings is, in the steady state, composed of a relatively smooth base with a number of uniform 'hills' which may act as indenters against the softer Incoloy 800 tube (Fig.4.28). The tube surface meanwhile has a rippled texture in the steady state (prior to the severe wear stage) with indentations found on the surface (Fig.3.26). Even when tube wear is in a more advanced stage (Fig.3.21), the surface geometry can still be approximately characterized as a series of spherically tipped peaks whose distribution changes little with further wear. Microhardness tests show that the ring and tube have Vicker hardnesses of 315 and 220 respectively. It was shown in Section 3.2.2 that considerable plastic deformation of the heat exchanger tube surface takes place during wear of that component. To assume elastic contact between the two surfaces would obviously be inadequate. Researchers have long searched for ways to predict plastic deformation in a body due to a given loading condition. The inability to do so has placed a great stumbling block in the path of theoretical modelling of wear. Bulk loads between wearing components are commonly known but on a microscale, asperity geometry is continuously altered so that the local stress distributions become very difficult to calculate. If the local geometry can be evaluated, it is possible to determine the local stress distributions. A correlation with wear could follow. In collaboration with Professor K.L.Johnson of the University of Cambridge (U.K.) 47 Chapter 4. Theoretical Modelling 48 Figure 4.28: Texture of Steady-State Ring Surface the shakedown theory of plasticity has been used to develop the geometry of a single indentation formed when a soft target is impressed by a spherically tipped indenter. This model of a typical asperity interaction is then extended to include a multi-asperity situation. 4.1 C o n t a c t D e f o r m a t i o n M o d e l 4.1.1 T h e S h a k e d o w n T h e o r y o f P l a s t i c i t y If the two bodies are elastically similar, the stresses in rolling/sliding contact before yield will be given by the Hertz theory [41]. In the first pass beyond yield, rolling/sliding will cause plastic deformation and introduce residual stresses. In subsequent passes, the material is subject to the combined action of the contact stresses and residual stresses. After several passes, it is possible that an equilibrium will develop where the residual stresses introduced counteract the contact stresses so that only elastic Chapter 4. Theoretical Modelling 49 deformation results (see Fig.4.29). This is the process of shakedown1 whereby plastic R a t c h e t t i n g Figure 4.29: Shakedown deformation introduces residual stresses which raise the stress needed for further plastic deformation2. As well, plastic deformation leads to conforming of the two surfaces, leading to a reduction in the contact pressure by effectively increasing the contact area. In the steady state, deformation will be entirely elastic, provided the final contact stress does not exceed some threshold shakedown pressure P£. Beyond the shakedown pressure, cyclic plastic deformation, or ratchetting, takes place. The following extrapolation can be made from shakedown theory: Application of cyclic stresses beyond the elastic yield point will result in plastic deformation such that the contact stress does not exceed the 1For a similar treatment of cyclic plastic deformation using 2-D line asperities see reference [42]. 2 Strain hardening has a similar effect. Chapter 4. Theoretical Modelling 50 shakedown pressure.3 If steady state conditions prevail and, for example, the geometry of one surface is approximately constant (i.e. one surface is much harder than the other), then the geometry of the mating surface will deform in such a way that the pressure at any point at the interface does not exceed P„ (the shakedown pressure of the softer material). In this way we can determine the steady state geometry of the deformed surface. 4.1.2 M a t h e m a t i c a l D e v e l o p m e n t The model is developed to determine the geometry of a single indentation formed by applying a unidirectional tractive load to a hard (Hv = 529) spherically-tipped asperity against a flat brass or mild steel disk. When the indenter is pressed against the softer target, high pressures induced by the spherical contact may exceed the elastic yield point and plastic deformation will commence. For the first few cycles the system will be in the ratchetting regime. With continuing deformation, an indentation is formed as the material conforms to the ge-ometry of the indenter. The maximum pressure beneath the indenter decreases as the pressure distribution changes to resemble that of a line contact. At the same time residual stresses are introduced. The specimen will no longer deform if the pressure beneath the indenter does not exceed the shakedown pressure P£. If we assume that after several cycles the deformed material reaches elastic shake-down, the indentation will have a geometry such that the maximum stress at all points of contact between the indenter and the material is P<f. Also, steady state deformation will be entirely elastic (with a new, effective elastic yield point of P<f). The area of contact between the indenter and indentation will appear as a long ellipse and for an 3This is providing that the the load is not so great that the system remains continuously in the ratchetting regime in spite of the geometry changes (see section 4.1.2). Chapter 4. Theoretical Modelling 51 ellipticity ratio greater than five, the contact can be (for engineering purposes) con-sidered a line contact. The resulting pressure distribution will be as that shown in Figure 4.30. The total load applied by the indenter is represented by the area under Figure 4.30: Pressure Distribution on Indentation the pressure distribution, i.e. 7T L(x) = jPf- (2a • 2b) = P,f • nab The following line contact stress formulae for elastic contact also hold[43]: , 1 where P' is the load/unit length. a is the semi-contact width. Combining Equations 4.4 and 4.5 gives a 2Pl E* (4.3) (4.4) (4.5) (4.6) Chapter 4. Theoretical Modelling 52 where E * = E x T -1 E o and (4.7) The radius of the indenter in the longitudinal direction TL is assumed constant. The negative sign appears before the ^ since the indentation is conformal with the indenter. For a shallow indentation (Fig.4.31) which is described by a function z = f(x), the radius of curvature p is given by Figure 4.31: Profile of Indentation in Sliding Direction 1 32z or since z = 8Q — 5, p dx2 1 _ d2S p dx2 (4.8) In the transverse plane, the indentation will assume the shape of the indenter4 and b2 = (2r - 8) • 8 or, for r >> 8 8 = 2rT (4.9) 1 Assuming that the elastic rebound is insignificant compared to the plastic deformation. Chapter 4. Theoretical Modelling 53 which gives finally 1 P d2 ( b2 W \2rTJ <4-10> Substituting Equation 4.10 into Equation 4.7 and then placing the result into Equa-tion 4.6 gives the following: a = 2fl E* rjj 2rr dx2 Lastly, substituting in Equation 4.3 for a and tidying up gives the final equation J 2xr t ( i * ) ' 2rr dx2 E*L(x) (4.11) where L(x) is the load applied to the indenter and r j is the radius of the indenter in the transverse direction. This equation can be numerically solved for b(x). This is done in Appendix B where the longitudinal radius, rr,, is set equal to rr (i.e. a spherically tipped indenter). A simplification of this equation can be made by assuming that the radius of curva-ture p is constant. If so, then with the transverse radius also constant, from geometry (see Appendix C) we can show that the width of the indentation b varies elliptically with the half slip distance I, i.e. b(x) = b0\h-X* (4.12) where ba is the length of the semi-minor axis Wi th the primary and secondary radii of curvature (p and rx, respectively) of the indentation constant (see Fig.4.32) we can combine Equation 4.9 with a similar equation for the longitudinal profile to get bl I2 or 2r T 2p (4.13) Chapter 4. Theoretical Modelling 54 1 p Figure 4.32: Primary (p) and Secondary (rj) Radii of Curvature Combining Equations 4.3, 4.6, 4.12 and 4.13 then gives the following relation: If the load applied to the indenter also varies elliptically, the y ' l — x2/l2 terms will cancel and a solution for p can be easily found. 4.1.3 E x t e n s i o n to M u l t i - A s p e r i t y M o d e l The indentation model described in the previous sections can be extended to multi-asperity surfaces given the following conditions: 1. One surface is sufficiently harder than the other. This allows one to assume that the steady state asperity geometry on the harder surface is essentially unchanging while the softer surface is continually indented. L(x) = E* (4.14) Chapter 4. Theoretical Modelling 55 2. The asperity distribution on the harder surface can be represented by a series of spherically tipped indenters with a height distribution $ ( 2 ) and area density rj. 3. The softer surface can be represented as a smooth flat of material whose surface has reached shakedown. This surface will thus be plastically deformed by any contacting asperity where the contact pressure exceeds the shakedown pressure. 4. Each contacting asperity will create the shakedown indentation in one pass, al-lowing the deformation model of Section 4.1.2 to be used. For the present application to model the wear of heat exchanger tubes, the geometry of a tube inside a ring will be utilized. It is assumed in one instance that the surface roughness is negligible when compared to the bulk dimensions in order to facilitate the assumption that the bulk contact width (2a;,) can be given by the hertzian line contact Those spherical contacts where the maximum pressure does not exceed a critical pressure should not be considered to form the line contact distribution of the shakedown model. For contacts where the penetration 8 is less than some critical value 8crit, the contact area will be taken as spherical. For 8 > 8crit the contact will be taken as cylindrical. 8crit is taken to be the value of 8 where the maximum pressure in circular contact, exceeds the shakedown pressure of the softer material. For a circular contact, equations. 8 = 1 [9 F% y 2 [2(E*)2R Po so we can easily find (4.15) Chapter 4. Theoretical Modelling 56 If 8 > Scrit we will assume cylindrical contact. For a cylindrical contact, the semi-contact width a can be found from Equation 4.6. This equation can be rewritten as 2P0S  a ~ E*fl _ I) \ r p> The distribution of sliding distances (i.e indentation semi-lengths) is assumed to be directly proportional to the asperity height distribution, i.e. '••(*) = ^ i p ^ M z ) (4-16) where M * ) = [*c(*)-*c(*=o)] where a;, = x(U) and Xf = x(tf) ( see Appendix D). $ c (z) is the cumulative distribution of $(z) (the asperity height distribution). 4>c;(z) has been constructed so that $ c/(oo) — $ c,(5 = 0) = 1.0 (i.e. the distribution is normalized) and so that 4>c/(£ = 0) = 0. For an asperity with zero interference, the slip distance Z, will be zero while for the longest asperities, the slip distance will equal the contact length (\xj — xi\). Also, since Pi = lf/2Si Pi = ±;(x, - xi)'*l(z) (4.17) For individual asperity contacts, L , = rtP^aibi and so the load L on a given area A0 can be found as roo L = TxnAoP* I ab$(z)dz J z=d' or, since b2 = 2rj8{ (Eq.4.9) and rr is assumed constant for a given surface L = V2irr]A0pSri / 8*a<I>(z)dz (4.18) J z=d' where 77 is the asperity density (i.e. number of asperities in a given area) As well, the number of asperities in contact can be found from equation 2.1. Based on the findings of Section 3.2.2, $(z) is assumed constant for a given worn surface. 1 l - $ c ( £ = 0) Chapter 4. Theoretical Modelling 57 Due to the tube and ring geometry, the pressure P varies with distance x from the centre of the pressure distribution. As well, as far as the tube is concerned, the pressure distribution is moving along the surface (in the direction of 'rotation'). The average pressure Pavg(x) that a point at X sees is found to be (see Appendix D) Pavg{x) 1 rh If —1{ Ju ^ i - ( ' - j < * » a * (4.19) where U is the time where the point x is first loaded and tf the time at which load is removed from X. J is equal to yj'E*/TTRW. Equating 4.19 and 4.18 gives the following final expression J 1 rh — U Ju \ (x -x(t))2 dt a?{t) i— L r°° l V^TrnpSr? / 62a$(z)dz Jz=d' (4.20) where d' = d + £ c r , t and d is the separation distance between the planes of reference. Since we are ignoring the roughness on the softer (tube) surface, d is the distance be-tween the undeformed smooth tube and the reference plane of the asperity distribution on the hard surface. The form of Equation 4.20 allows for a numerical solution for d' and hence for <5, (since 5, = z, — d') and /?, (from Eq.4.16). A n algorithm for solving for these param-eters is developed in Appendix E . Once all of these parameters are known, the stress distribution beneath the discrete asperity contacts can be developed. From the stress distributions, we shall attempt to develop a wear model. Chapter 4. Theoretical Modelling 58 4.2 A Wear Model For Heat Exchanger Tubes The geometrical model developed in the previous section allowed the contact parameters for one plastically deforming surface to be calculated. These plastic contact parameters can then be applied to any elastic wear model, where the new elastic yield point is equal to the shakedown pressure of the softer material. These parameters could, for example, be applied to the asperity fracture model of Section 2.2.5. A model of the wear of heat exchanger tubes can be developed on a similar basis. Knowing the new contact parameters, the typical stress distribution can be calculated using Hertz's elastic equations for a sliding contact. A summary of these equations is provided in Appendix F for reference. The determination of an appropriate failure criterion is not a simple procedure. Inspection of the wear scars (Section 3.2.2) shows that both crack propagation and plastic deformation are involved in the production of wear particles, suggesting two possible failure criteria. One could consider the critical crack propagation energy. A second method involves evaluating the energy required to cause failure in ductile shear. The model which follows is based on this second criteria. In Section 4.1.2 we showed that the geometry of an indentation can be determined if we can assume that the softer material has shaken down. Since the maximum pressure at all points in the contact zone (for a saturated contact) is equal to the shakedown pressure, the stress distribution beneath the indentation will be identical at all points, moving with the loaded asperity (indenter). This means that the depth of maximum shear stress will be the same at all points beneath the indentation, suggesting that a wear particle should have the same dimensions as the indentation since the material will fail along a slip line given by the depth of the maximum shear. If some known portion of the frictional energy input by the sliding contact goes into shearing off a Chapter 4. Theoretical Modelling 59 wear particle, then we can a simple work criterion may be developed. The work input required to form a wear particle is expected to be a property of the wearing material and wil l likely be temperature and rate dependent. For a reasonable range of temperatures and strain rates, this property should be nearly a material constant. 4.2.1 M a t h e m a t i c a l D e v e l o p m e n t If one surface is much harder than the other, then the softer surface will readily conform to the asperities on the harder surface. The geometry of the softer surface can then be calculated using the multi-asperity contact deformation model developed in Section 4.1.3. S h e a r F r a c t u r e W e a r M o d e l Using the contact deformation model of Section 4.1, the contact parameters were found for one soft surface against a harder surface. In Section 4.1.3, the contact parameters for a curved surface against a flat were found by assuming that the flat has a distribution of asperities whose heights vary according to some known (or measurable) distribution $(z), while the tube is a smooth cylinder whose surface has reached the shakedown limit. These parameters could be found for any given loading cycle or displacement trajectory. For p — 0, the location of the maximum shear stress (using von Mises) is found to be (0.0,0.79a) in terms of x , z coordinates. As p increases, this location moves in the direction of the tractional force, and moves towards the surface. This trend is shown in the plot of Figure 4.35. At the same time, a stress peak forms at the surface of the contact zone (at x/a = 0.255), whose value exceeds the subsurface peak at a frictional coefficient of approximately 0.262 (see Fig.4.33). A plot of the stress contours for this value of p, is Chapter 4. Theoretical Modelling 60 shown in Figure 4.34. The depth of a wear particle is expected to be equivalent to the 0 5 0.25 0 2 S u l u f c a P a t 1 iitt 1 1 1 1 I 1 I I 1 0 005 0.1 0.15 0.2 025 O J 055 01 Frictional Coefficient p Figure 4.33: Depth of Maximum Octahedral Shear Stress ( JJ2/P0) for Surface and Subsurface Peaks. Figure 4.34: Contour Plot of \JJ2/P0 for p. = 0.262 Chapter 4. Theoretical Modelling 61 0.0 1111111111111 1111111 11111111111111111 1111111111111 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Distance x/a From Load Center Figure 4.35: Location of Jjz/Po (maximum T0CT) as it Varies with p. depth of the maximum octahedral shear stress. In the case where the maximum stress is at the surface, Bower [46] states that particles 0.1a thick can be expected. The stress cycle that results from a discrete asperity contact can be developed using Hertz's line contact equations. The material just beneath the surface of the indentation will experience fluctuating stresses due to the sliding contact. The octahedral shear stress variation for a point located 0.5663a below the surface (for p = 0.20) is shown in Figure 4.36. Since the contact width (2a,) is small in comparison with the sliding distance (2/,), each point in the indentation can be considered to experience the same stress cycle due to a sliding line load starting at —oo and going to +co. It has already been established that the maximum contact pressure at all points in the contact zone is equal to the shakedown pressure of the softer material. This stress cycle then, will vary only with frictional coefficient. When the stress exceeds the critical stress (ry), plastic yielding will commence and Chapter 4. Theoretical Modelling 62 0.35d Distance x / a of Load Center Figure 4.36: Variation of / j 2 / P e for a Line Contact p, = 0.20, (^ ) = 0.5663 the wear process will be advanced. We can assume that the material will fail along some favorable path (or slip line) given by the trajectory of the depth of the maximum shear stress. Conveniently, the contact width 5 a and the applied pressure P£ are both constant for a given contact, so this depth of maximum stress will be constant along the length of that contact. Unfortunately, it is difficult to estimate the portion of the sliding frictional work which goes into wear. We know that stresses in elastic contact will be given by Hertz's line contact equations, but since we have shown that plastic yield occurs in all contacts in a thin zone near the surface, these equations will not be valid. Bower [46] has stud-ied the relationship between ratchetting rates (incremental plastic deformation) and applied stresses and developed an equation which compared favorably with experimen-tal rates for rail steels and copper. Conceivably, such an equation could be applied to the multi-asperity model to give a relationship between work and wear. Such a 5Follows simply from equation 4.4 with p constant. Chapter 4. Theoretical Modelling 63 development is beyond the scope of this investigation. From the shakedown theory, all contacts will consist of a line contact with peak pressure equal to P / , so the portion of frictional work which goes into wear (denoted w) should be constant for a given fric-tional coefficient. This is because the stress cycle at a point below the surface will be a function only of the frictional coefficient (notice that the plots are non-dimensionalized as \JJ2IP0 where in this case PG = P / , a material property). The following is an outline of the steps used in developing a value for the expected wear volume: • Determine d, Si, pi, bi, a, for the asperity distribution (Section 4.1.3). • For each S, the indentation area Ai will be equal to rc bi U. • The work E{ required to form a wear particle of size Ai and thickness z^/a, ( with volume Vi ) is equal (Work = Force x Distance) to TyA,Z,. • As the pressure distribution slides over the surface, it performs an amount of frictional work Wi (see Appendix G) equal to pUg-i^bi maxP„ 2 • The fraction of this frictional work which goes into wear (zv) is assumed to vary only with p. • The number of wear cycles necessary to form the wear volume Vi can thus be estimated as N = = l A i i ^ „ (k) 1 wWi w piraiPS \a«/ w or conversely, the wear volume from that discrete contact is found as dV> = -E— Chapter 4. Theoretical Modelling 64 • The increments of wear volume dV{ must be summed for the asperity distribution for all locations on the wearing specimen for all time. This process is performed in Appendix H for the tube and ring geometry. The final wear equation is given below as * £ £ / a}b^{8)d8 (4.21) X=Xi N = l J ° where and RT is the tube radius, T the duration of the data (force and displacement wave-forms) sample, t the length of the test, X the point on the tube circumference, and N the number of times that each X is loaded during the sampled data time. 4.2.2 Applications of the Contact Deformation — Work Model There may be many systems that in practice exhibit the necessary characteristics for applying the contact deformation-work model. For a general system, the requirements are as follows: • The steady state geometry of one surface must remain approximately constant (i.e. one surface is much harder than the other) and the typical radius of its surface roughness is known. • The physical properties (Elastic Modulus, Vicker Hardness, Poisson's Ratio) of the two materials must be known. • The system dynamics or mechanical interactions must be sufficient for the softer surface to shake down. w v = jn^Rxw nTT Chapter 4. Theoretical Modelling 6 5 Figure 4.37: Indentation Microsurface The existence of particles that have the same size as the indentations typically found on a surface is supported by the mathematical model developed in this paper. It was shown that if shakedown has been reached, the resulting geometry changes will lead to a uniform surface pressure. Hence each subsurface point along the contact length will see a similar pressure distribution as the asperity slides along the surface. We can then conclude that since the pressure in the transverse direction is also uniform, the likelihood of subsurface shear fracture is equal at all points beneath the indentation. Thus a typical wear particle should be the same shape and dimensions as a typical indentation. For the tube and ring system, it has been speculated that the cause of wear particle formation is the process of shear fracture where large plastic strains lead to hardening of the material with subsequent fracture after sufficient strain has accumulated. If the frictional coefficient is known, then for a given contact geometry, the stress distribution can be found. The depth of the maximum octahedral shear stress can be Chapter 4. Theoretical Modelling 66 easily calculated using Hertz's equations for elastic contact. It is expected that this depth will correspond to the location of the greatest plastic strains and hence to the thickness of the expected wear particle. In Section 5.1.1, a single indenter is slid against a soft target to form an indentation. Some of these indentations were sectioned and the microstructure viewed to determine if the depth of the maximum strain corresponds to the depth of the maximum shear stress. The value calculated for test 5-3 was 0.035mm. This value corresponds to the dotted line shown in Figure 4.37. It can be seen that the maximum strain occurs much closer to the surface. This is due to surface roughness on the indenter and target. If the two surfaces were smooth (i.e. free of surface roughness) then the depth where the largest shear strains occur would be nearer to the above calculated value. A n indentation test was performed using a highly polished, flat, mild steel disk and extremely smooth titanium carbide ball with a diameter of 1.240 inches. The conditions of the experiment were similar to those of previous tests but in this case lubrication was manually applied to the interface using a syringe to ensure a low, constant frictional coefficient 6 . When the shallow indentation was sectioned (Fig.4.38) very little strain was observed, indicating perhaps that the maximum shear strain would have occurred at a greater depth than in previous tests but because of this greater depth, it requires more work for visible strain to accumulate. Although this does not prove that the thickness of a particle will correspond to the depth of the maximum octahedral shear stress, it seems a reasonable assumption and this test is at least supportive. So it is possible that the wear of heat exchanger tubes and indeed many other systems can be modelled or at least qualitatively explained using the procedure outlined 6Boundary lubrication of smooth surfaces is more difficult than for surfaces of moderate roughness since the roughness helps to 'hold' the lubrication and prevent it from being extruded from a concentrated contact. Also, if the frictional coefficient goes above .26 (for a line contact) the location of the maximum octahedral shear stress is at the surface. A lower coefficient is required if the below surface strains are to be visable Chapter 4. Theoretical Modelling 67 Figure 4.38: Indentation From Polished Specimen above, where the size of a wear particle is expected to be elliptical with major and minor axes equal to the slip distance and contact width, respectively, with a thickness equivalent to the depth of the maximum octahedral shear stress. Chapter 5 Verification of Mathematical Models Using test apparatus at the National Research Council's Tribology Laboratory, the author was able to simulate the single asperity interaction modelled in Section 4.1. A large series of experiments were run, and the results were compared to predictions made using the contact deformation model. As well, a number of wear tests were run to investigate the accuracy and reliability of the wear model of Section 4.2.1. 5.1 Contact Deformation Model 5.1.1 Procedure To test the contact deformation model, a series of experiments were run utilizing a fretting wear rig (Fig.5.39) developed at the National Research Council [47]. Briefly, the rig is controlled by input voltage signals which activate solenoids in the shakers. This rig thus controls the forces between the indenter and target in the normal and tangential directions (see Fig.5.40). A unidirectional traction load is applied to a hard, spherically-tipped indenter against brass or mild steel disks. The geometry of the indenter is shown in figure 5.41. The number of cycles was chosen as six hundred in order to ensure that a steady state indentation is reached but with minimal wear (since if wear were allowed to proceed, it could disturb the steady state geometry). Three hundred cycles was insufficient as we observed that the frictional coefficient tended to increase to a steady value not before that number of cycles. Digital force waveforms 68 Chapter 5. Verification of Mathematical Models Figure 5.40: Dynamic Specimen Holder Chapter 5. Verification of Mathematical Models 70 CENTIMETRES Figure 5.41: Spherically Tipped Indenter were generated by a computer, converted to analog signals and then amplified and sent to the shakers. A sine wave and a truncated sine wave of the same frequency (Fig.5.42) were sent to the tangential and normal shakers, respectively. For high frequencies, the Figure 5.42: Tangential Displacement and Normal Force Wave motion in the tangential direction is effectively sinusoidal and in the normal direction, the displacement is approximately zero. If the displacement leads the normal force by Chapter 5. Verification of Mathematical Models 71 ninety degrees, L(t) — Lconst + L m a x sin(a;<) for half the cycle (5.22) = Lconst for the second half of the cycle x(t) = Xm sin (^u>t — —^ = Xm cos(ut) (5.23) From Equation 5.23 we get cos(a><) = — -Xm Combining with Equation 5.22 gives sin(u;*) = yjl - (x/xm)2 = L L c o n s t l-'max and finally L = L c o n s t + Lmax^/l - (x/xm)2 (5.24) Substituting Equation 5.24 into Equation 4.14 with Lconst « 0 (i.e. the flat of the truncated sine wave has a value of approximately zero), gives i\2*i(pasy Rearranging gives the final simplified equation: p2-ClP + C2=0 (5.25) where 2*i(pasy L E* Cx=2rL+rzLrT and A n approximate value for the radius of curvature of an indentation under these loading conditions can be quickly found for any pair of / and L m a x for a given material. Chapter 5. Verification of Mathematical Models 72 Material Em. Hv SY Kvick. Krreica KvonM. (GPa) (GPa) (kg/mm2) (MPa) (MPa) (MPa) (MPa) Brass 106 90 136 271.2 222 135.6 156.5 Steel 200 213 201 474.0 329 237.0 273.5 Table 5.2: Values of E and K for Brass and M i l d Steel. By varying the magnitude of the input force waves (by means of potentiometers placed between the D / A board and the amplifiers) it is possible to change the magni-tudes of I and Lmax- Lmax is measured by observing the signal from the force transducer located beneath the target specimen while / is determined by measuring the length of the final indentation (lm). If the semi-contact width a is negligibly small compared to lm, lm is nearly equal to I. Evaluating the specimen material parameters is a much more difficult procedure. The modulus of elasticity (E) can be found in most handbooks but the Poisson's ratio (v) for brass covers a wide range (from 0.3 to 0.34) in these same books. A value of Vbrass — 0.324 [48] was used in the calculations. For friction coefficients less than about 0.25, P<f = 4K (Fig.5.43) [49, Fig.4.10] where K is the material yield strength in cyclic shear. K, in turn, is evaluated as approximately Hv/6, SV /2 (Tresca) or 0.577 Sy (von Mises). The Vickers hardness (Hv) is determined with a micro-hardness tester on a polished flat of the specimen material. The yield stress in tension (Sy) is found by performing a stress-strain test on specimens machined from the same material as the specimen disks. The elastic modulii E can also be found and compared to the handbook values. The results of the tensile tests are included in table 5.2. About fifty indentations have been generated by loading the indenter against brass specimens. A number of the mild steel specimens were also tested. The interface between indenter and specimen is at all times lubricated with either water or Tellus Chapter 5. Verification of Mathematical Models 73 C: Surface + Subsur face Floir A: Subsur face Flow . \ Repeated Plastic Flow Elast ic L imit . E l a s t i c - p e r f e c t l y Plaal ic Shakedown Limit Kinemat ic Hardening Shakedown Limit 1 -J I I I _ L 0.1 0.2 0.3 0.4 0.5 Tract ion Coefficient ^ Figure 5.43: Shakedown Map For Line Contact Geometry Chapter 5. Verification of Mathematical Models 74 34 mineral oi l . The indenter is moved at a frequency of twenty hertz and a small (approximately 10 newtons) normal load is applied to the indenter to prevent it from leaving the specimen surface and introducing an impact situation. The number of cycles run in all cases is 600. If wear particles are found the test is deemed invalid. After each test the indentation was traced using a diamond stylus profilometer. A computer program was used to view each trace, find the central (i.e deepest) trace and fit an arc to it. As well, the transverse trace is generated and similarly viewed and fit. These radius values are then tabulated and compared against the predicted values which are calculated using the experimental value of the elastic modulus E. 5.1.2 Results A photo of typical indentation is shown in Figure 5.44 with its corresponding longitu-dinal and transverse trace (with curve fits also shown). For this test Lmax — 160N and I — 1.17mm. Tables showing the results of the indentation tests are included in Appendix I. From Equation 5.25 we find that p is dependent upon the fourth power of P0" and thus the theoretical predictions will be highly dependent upon the value of K (the yield in shear). The best correlation between the experimental and predicted results was obtained for Ksteei = 305MPa and Kbraaa = 220MPa. Using these values, the bar charts shown in figures 5.45 and 5.46 were generated. Note that the top graph is plotted in order of increasing Lmax while the bottom graph is in order of increasing /. Chapter 5. Verification of Mathematical Models 75 Figure 5.44: Typical Indentation: Photo Shown With 3D Plot and Corresponding Longitudinal Traces and Curve Fits Figure 5.45: Steel: Ratio of Measured Value to Predicted Value. Chapter 5. Verification of Mathematical Models 77 Lmax (N) 0.11 03* O J I 0.90 0 J7 I Ti O J . OM OJO 0.10 ,\ao Slip Length, 1 (mm) Figure 5.46: Brass: Ratio of Measured Value to Predicted Value. Chapter 5. Verification of Mathematical Models 78 5.1.3 Discussion Experimental Difficulties The rig used for these tests has no provision for direct control of the displacement. The shakers respond to the forcing function in a way that any mechanical system does, with transient vibration and phase lag. Since it is not the tangential force that needs to lead the normal force by ninety degrees but rather the tangential displacement, to obtain the proper phase it is necessary to further advance the tangential shaker forcing function to counteract the additional phase lag between the shaker coils and the tangential displacement. This is done initially by trial and error with the normal load set to some intermediate value (since the resistance due to friction has important consequences on this phase). By changing slip distance and normal load as has been done for the bulk of these tests, the phase is likely to change. The inability to maintain an constant ninety degree phase between the tangential displacement and the normal load has reduced considerably the consistency of the results. Theoretically the longitudinal radius of curvature should be constant but in many cases, the profiles are skewed so that the point of maximum depth occurs at some point far off the midpoint of the indentation. Active feedback would be ideal but no such system has yet been incorporated into the N R C rig. A related problem is that in the first few cycles of contact, before the steady state is reached, the indenter overshoots its steady state limits. The result is a double pit (see the photo in Figure 5.47). Unfortunately if all such imperfect tests were omitted, there would not be enough data left to analyze. Still another problem occurs when a ridge of plastically deformed material builds up in front of the indenter and inhibits forward motion (i.e. sliding during the loaded half cycle). Chapter 5. Verification of Mathematical Models 79 As a result of the difficulty in precisely controlling the phase angles, the indenta-tions were sometimes not symmetrical as was expected. In these cases, curve fits were performed visually and the results cannot be considered accurate. The Model This model of plastic deformation is developed upon several premises and each one will be considered and justified. Central to this model is the statement that with initial circular contact geometry, stresses at the interface exceed the shakedown pressure leading to ratchetting (i.e. in-cremental plastic deformation). Further is the proposal that geometry changes result in a reduction in the peak pressure, causing the system to move into the elastic shake-down regime (shown in the Line contact shakedown map of Figure 5.48). Using elastic contact mechanics, the peak pressure values (P 0 ) can be determined and knowing the yield in shear K and the coefficient of friction p, a shakedown map can be used to Chapter 5. Verification of Mathematical Models 80 determine the particular regime in which the system is acting. To calculate the contact stress for circular contacts the following formula is used: '6P'(E*) 1 maximum pressure P0 = — IT _ 1 ?*\2 ~ (5.26) B? where R and E* have been defined in Section 4.1.2. For hne contacts, Equation 4.5 is again used. P', the load per unit length is in this case the load on the indenter Lmax divided by the width of the contact (2 b0). Four extreme cases are examined: • large load, small indentation length • large load, large indentation length • small load, small indentation length • small load, large indentation length For each case, a corresponding practical test is found from tables 1.5 and 1.6. Cal-culations are made for the system at x = 0, b(x) = bQ. The results of this procedure are shown in table 5.3. The indenter radius is 3.8mm. The frictional coefficient is determined experimentally as the ratio of the tangential and normal signals from the transducer. It can be seen from the circular contact shakedown map of Figure 5.48 that all points are by far in the regime of repeated plastic flow (ratchetting). The P0/K values for circular contact far exceed the shakedown limit (all of the points are off the shakedown map) so in the initial stages of indentation, the specimen will plastically deform as ratchetting proceeds. As the indentation forms, conforming (of the indentation) in the transverse direction leads to contact all across the width of the indentation while conforming in the slip direction contributes to a reduction of the peak pressure. In most cases, for line contacts the value of P0jK is below the cutoff Chapter 5. Verification of Mathematical Models Brass Steel low L low L high L high L low L low L high L high L high / low / high / low / high/ low / high / low / Reference # 1 2 3 4 5 6 7 8 Specimen 2-3 4-15 4-6 4-4 5-6 7-2 7-10 3-2 ^max(N) 100.8 55.8 305.4 314.9 49.78 165.9 303.8 345.5 /(mm) 2.15 0.5 2.3 1.18 1.93 0.69 1.91 1.05 />(mm) 190 13 45 12 140 11 35 9 2bQ 0.82 0.66 1.5 0.84 0.5 0.9 1.56 1.46 P 0.240 0.160 0.140 0.150 0.150 0.152 0.144 0.286 Spherica Contact P o/10 6(N) 1973 1304 2729 2268 1968 2257 3393 2653 Po/KHv 8.889 5.875 12.29 10.22 5.982 6.861 10.31 8.063 Po/KvonM. 12.61 8.333 17.44 14.49 7.196 8.253 12.41 9.698 Pol Krretca 14.55 9.618 20.12 16.73 8.305 9.525 14.32 11.19 Line Contact P'(kN/m) 123 84.55 203.6 374.9 99.56 184.3 194.7 236.6 P o/10 6(N) 881.3 621.1 1096 1285 944.2 1054 1264 1122 Po/KHv 3.969 2.797 4.937 5.788 2.870 3.203 3.842 3.410 Pol KvonM. 5.631 3.968 7.004 8.211 3.452 3.853 4.622 4.102 Pol KTretca 6.498 4.580 8.084 9.477 3.984 4.446 5.333 4.734 Table 5.3: Values of P0/K for Extreme Cases of Load (L) and Slip Distance Chapter 5. Verification of Mathematical Models 82 4 C: Surface + Subsurface Flow Elastic l imi t Elastic-perfectly Plastic Shakedown Limit Kinematic Hardening Shakedown Limit 1 -0.1 0.2 0.3 0.4 0.5 Traction Coefficient / i 4 $ 2 1 1 \ \ Upper bounds Lower bounds y^Ov \ CYCLIC \ PLASTICITY V \ ELASTIC SHAKEDOWN l \ \ " 1 \ \ Subsurface - « — 1 1 1 • Surface i! i i 0.1 0.2 0.3 0.4 0.5 Traction Coefficient fx Figure 5.48: Shakedown Maps For Line Contact (top) and Circular Contact (bottom) Chapter 5. Verification of Mathematical Models 83 value of 4 (for / < 0.25) so the elastic shakedown regime has been attained in those instances. Unfortunately, calculations for brass with high loads lead to larger values of PolK. In these cases, ratchetting is likely to proceed continuously and the theoretical predictions may be unreliable. Another simplifying assumption made in Section 4.1.2 is that the radius of curva-ture (p) of the indentation is approximately constant. Solving Equation 4.11 (with L(x) given by Equation 5.22) using numerical integration techniques (see Appendix A) reveals that p = constant is very nearly an exact solution to this equation. This can be verified analytically. In this case it is seen from Equation 4.14 that if p is constant and L(x) = Lmaxsjl — x2/l2 then b(x) can only be elliptical in order to cancel the y l — x2/l2 term. Experimental results readily verify the assumption that the radius of the indentation is (nearly) the same as the radius of the indenter. The transverse trace was interpolated by computer from the longitudinal traces and in almost all cases, the transverse radius is within +15% of the indenter radius. Because of elastic recovery, the measured radius should at all times exceed the indenter radius. However, if the magnitude of the plastic deformation is much greater than the elastic deflection, there should be little difference between the two radii. To show that the amount of plastic deformation is indeed small, we will utilize the elastic contact equations of Hertz. First, the following assumptions are made: • Elastic Shakedown has been reached. • The load on the indenter is known. • The transverse radius equals the indenter radius when the full load is applied to the indenter. Chapter 5. Verification of Mathematical Models 84 The actual contact area is an elongated ellipse so the apparent increase in the inden-tation radius due to elastic recovery (known as shallowing) will lie somewhere between the limits given by circular contact geometry and line contact geometry calculations. The contact pressure is greatest for the circular geometry and so also should the elastic deformation be greatest. This means that the circular geometry is the limiting case and the actual unloaded transverse radius should be greater than the indenter radius but less than the circular contact value. A formula relating the load L to the indentation diameter (260), radius of inden-ter (r) and recovered transverse radius (r') is given by Hertz's classical equations (in reference [50]) as (5.27) where units of meters, Newtons and Pascals are used. If we take, for example, specimen 6-15 of table 1.5, where Lmax = 160N and 2ba = 0.00098m, while Ex = 89GPa, E2 = 200GPa, and r = 0.0038m, we calculate the recovered radius p to be 4.2935mm. From the curve fit shown in figure 5.44, it is seen that the experimental value (4.3mm) coincides almost exactly with this predicted value. A similar calculation for steel (specimen 5-3: 2fe0=1.4mm) results in a calculated value for r' of 3.92mm which agrees reasonably well with the measured value of 4.2mm. The inclusion of elastic deformation in the transverse plane is a simple process and has been done for the contact deformation model (see Appendix J). The improvement is minimal in most cases, while in others (especially the low load cases) the model in clearly unrealistic. For these reasons, the effect of elastic recovery in the transverse plane should not be included in the contact deformation model. It can be seen that for both steel and brass, the values for low Lmax and high I are poor. In the case of steel it appears that for values of Lmax < 150 the model 2ba = 2.22 Lrr' /_1_ r> -r \E~[ + E2 h) Chapter 5. Verification of Mathematical Models 85 predicts much lower values than those measured while in the case of brass, for large slip distances the model predicts much higher values than those measured. These discrepancies are likely due to the the fact that in the low Lmax steel tests, the contact is more circular than elliptical (hence the low P0/K values) while in the high slip distance brass situation, the system is in a state of ratchetting. If indentations formed with very low loads are neglected, then this model quite reliably predicts the radius of curvature (and consequently, the entire geometry of the indentation) for a wide range of slip distances / and loads Lmax, especially in the case of the steel. 5.2 Wear Model 5.2.1 Procedure Experiments to simulate the conditions of contact between heat exchanger tubes and their support plates were performed using the N R C Fretting Wear Pug (see Section 5.1.1) with special specimen holders designed by the author. The specimens are of the same dimensions as those used in Section 3.1, figure 3.9. The dynamic specimen holder (Fig.5.49) was constructed of Aluminum Alloy 7075-T651 (commonly used in aircraft construction for its high strength/weight ratio) and is driven by the two shakers to cause controlled impacting and sliding motion. The dynamic specimen in this case is a type 410 stainless steel ring (simulates the tube support plate). The stationary tube specimen (Fig.5.50) is cantileved horizontally through the ring specimen. Again, the frequency is about 30Hz, while the force magnitudes ranged between 200 and 800 newtons peak-to-peak. A triaxial force transducer was used to measure the x and y forces. It was held con-centric with the tube specimen by a specially constructed stainless steel bolt (Fig.5.51). Chapter 5. Verification of Mathematical Models Figure 5.49: Dynamic Specimen Holder. A thin layer of teflon was press fitted to the bolt and then turned on a lathe to the ap-propriate dimensions. As the teflon has a much lower compliance than the transducer or specimens, all of the forces on the stationary tube specimen will be transferred to the transducer. Displacements were monitored with displacement proximeters. The waveforms were stored on floppy disks using a digital oscilloscope and later transferred to computer through an I E E E interface and converted to ascii data files using a basic language computer program. The data was correlated using the method of Appendix A . The shear fracture wear model requires that the asperity height distribution and av-erage asperity tip radius be found for the harder (ring specimen) surface, as well as the asperity density. The ring specimen from test P I was sectioned and its surface traced using a profilometer. The trace was performed in both the longitudinal and transverse directions. The resulting plots are shown in Figure 5.52. A computer program was developed to perform a circular detrending on the longitudinal trace and then a linear Chapter 5. Verification of Mathematical Models Chapter 5. Verification of Mathematical Models Figure 5.52: The Longitudinal (top) and Transverse (bottom) Talysurf Traces. Chapter 5. Verification of Mathematical Models 89 detrending on both plots. The same program then found the asperity peaks and per-formed a three-point parabolic curve fit to those peaks. The resulting peak heights and corresponding radii were then exported to a spreadsheet where the frequency distribu-tion was found. The resulting height and radii distributions are shown in Figure 5.53. The distributions found for one other specimen were found to be very similar so the height and radii distributions for all specimens were thenceforth assumed constant (in accordance with the visual findings of Section 3.2.2). The frictional coefficient between the stainless steel (Type 410) asperities and the Incoloy 800 tube was found by running an indentation test as in Section 5.1.1. A stainless steel indenter was machined from the same stock as the tube support (ring) specimens. Incoloy 800 discs were machined from material provided by A E C L . Three tests were run at medium load and medium amplitude. The interface was immersed in distilled water. The normal and shear force waveforms were monitored and the ratio of the peak values was calculated. A value of p — 0.34 was obtained. A computer program was written to use the correlated force waveforms to calculate wear according to Equation 4.21. Twenty-five points around the circumference of the tube were examined. Three to six data sets of 1024 points were analysed and the results either summed (in the case of work and wear) or a weighted average taken (for force magnitude etc.). The data was then extrapolated for the duration of the test, which varied between six and forty-eight hours. 5.2.2 Results A sample of the waveforms generated by the correlation program can be found in Ap-pendix K . The numerical data is summarized in Table 5.4. A plot of the experimental wear rate vs work rate is shown in Figure 5.54. The chart shown in Figure 5.55 displays the measured and predicted wear results for the series of experiments. Chapter 5. Verification of Mathematical Models 1 1 i I i i i n T T i i r i i I I i i I I I i i i ri i i i r i i i t i I I I i -2.7l-2.46-2.22-1.98~l.73-l.49-l.2S-l.01-0 76-0.52-0.2B-O03 0.2l 0.45 0.70 0.34 1.18 1.42 1.67 1.91 2.IS -2.58-2.34-2.ID-I.B6-l.6l-l.37-I.IJ-O.Ba-0.64-0.40-O.I6DO9 O i l 0 57 0.82 1.06 I JO 1.55 1.79 2.03 2.27 Asperity Height (microns) o .a o i i i i i i i i i i i i ii ii i i i i i i i i i i i i i i i i i i i i ii ii i 23 164 306 447 589 730 871 1013 1154 129B 14371579 172D 1862 20D32I4522B6 2428 256927102852 93 235 376 518 659 601 S42 1084 1225 1367 15081650 17911932 2074 22152357 24982640 2781 2923 Asperity Radius (microns) Figure 5.53: The Asperity Height (top) and Radii (bottom) Distributions Chapter 5. Verification of Mathematical Models 91 Test Total Theo. Tube Ring Test RMS RMS RMS Time of Work/ Predicted # Work Work Wear Wear duration Mag Norm. Shear Contact Cycle Wear (joules) (joules) ('"g) (»'S) (hours) (N) (N) (N) (%) (joules) ('»«) PI 1.8E+07 3.6E+06 57.9 10.2 24.0 183.81 162.89 84.26 91.66 10.57 109215 P2 8.2E+05 2.0E+05 0.09 0.15 24.5 79.93 79.73 5.65 9.15 0.49 6433 N6 9.4E+06 2.4E+06 11.1 1.66 27.0 134.16 120.93 57.90 90.51 2.48 80534 N7 1.5E+06 6.7E+05 3.72 error 6.0 113.79 109.17 32.00 100.00 3.57 23936 N8 8.1E+06 2.5E+06 18.3 2.22 22.8 109.79 101.77 41.10 100.00 5.08 89910 N9 8.9E+06 2.1E+06 15.7 2.14 23.0 108.60 97.42 47.95 92.00 5.52 77726 NIO 9.1E+06 1.8E+06 12.6 1.99 23.8 99.57 87.34 47.60 99.60 5.45 71037 N i l 1.8E+07 4.1E+06 45.5 5.78 28.8 155.31 140.49 65.29 100.00 9.08 130687 N13 1.8E+07 2.7E+06 13.0 2.24 23.9 157.80 145.82 60.21 86.10 10.52 78509 N14 1.3E+07 2.5E+06 5.07 1.27 23.0 164.15 157.03 47.66 83.80 7.93 67399 N15 9.7E+06 3.0E+06 8.73 1.6 24.0 161.77 156.63 40.32 80.97 5.76 80393 N16 1.2E+07 3.4E+06 23.8 3.69 23.8 142.54 131.27 55.47 100.00 7.08 109485 N17 7.1E+06 9.0E+05 18.4 1.42 8.5 197.03 161.00 113.5 66.09 11.79 27338 N18 5.5E+05 1.1E+05 0.16 0.17 23.5 57.33 56.86 6.91 7.90 0.33 4163 N19 9.3E+06 3.6E+06 2.75 0.93 24.3 113.52 109.82 28.19 96.86 5.45 121500 N20 2.4E+07 6.8E+06 38.8 6.26 48.0 151.16 139.18 57.64 99.22 7.26 216135 N21 1.5E+06 6.7E+05 0.28 0.21 48.0 58.43 57.32 11.27 16.35 0.43 25839 Table 5.4: Summary of Data Correlation Procedure with Model Work and Wear Pre-dictions. o <D V) I o CK o OJ 5 0.0007 - PI • N17 m 0.0006 -0.0005 -Nl 1 o.ooo*-0.0003" N8 • N16 • N20 • 0.0002 -N6 N7 • NIO NI5 NIJ 0.0001 -Work Rate (Joules/cycle) Figure 5.54: Experimental: Wear Rate vs Work Rate Chapter 5. Verification of Mathematical Models 92 Measured Predic ted Figure 5.55: Experimental and Predicted Mass Loss Chapter 5. Verification of Mathematical Models 93 5.2.3 Discussion A quick glance at the plot of Figure 5.54 shows that there is a reasonable correlation between the work and wear rates. It should be noted that there is a very wide range of forces, impact angles and test durations are represented in this plot. Figure 5.55 shows that the theoretical model tends to predict about 5000 times as much wear as measured. A linear regression performed on the ratios of predicted/measured wear determined that an empirical constant of 5247 will best bring the predicted values into line with experimental values. If the model predictions are reduced by this factor, it is seen from the chart of Figure 5.56 that about one third of the tests are accurate to PI P2 N6 N7 N8 N9 NIO N i l HI 5 N14 NIS NI6 N17 N18 N19 N20 N2I Measured P r e d i c t e d / 5 2 4 7 Figure 5.56: Experimental and Revised Predicted Mass Loss within 20% while two thirds are accurate to within 50%. Chapter 5. Verification of Mathematical Models 94 The shear fracture wear model was particularly poor for cases P2, N18, N19 and N21. It can be seen from the trajectories of Appendix L that P2, N18, and N21 correspond to the high impact tests while from Table 5.4 we see that the rms force magnitude for the higher shear test N19 is quite low. In high impacting tests, there wil l be a squeeze film effect due to the rapid compression of the interfacial water. This squeeze film effect will tend to give exaggerated normal loads and should tend to reduce the bulk shear force. The squeeze film effect will have a significant effect for the low load, high sliding cases as well (i.e. N19). If so, in cases where the sqeeze film becomes important, the ratio of the predicted to bulk work is expected to be greater than in the high load, high sliding tests. Two of the three highest values of this ratio belong to N21 and N19. Also, as the absolute wear loss in P2, N18 and N21 is very low (fractions of a milligram), it is extremely possible that oxidation, scratches, material transfer between the test rig and specimens, and measurement errors contributed significantly to the inaccuracy and thus these tests should be disregarded. The wear model of Section 4.2.1 equates frictional work input with the energy required for ductile fracture. In high impacting cases, the slip length will be extremely small. Wi th the current setup, a data set consists of 1024 points sampled at 20kHz. Three to six data sets are collected for each test. If we consider a typical contact at a point X on the circumference of the tube, in a high sliding case the contact time is about three to five data points. In a high impacting case, the contact may consist of only one to two data points. In solving Equation 4.21, the computation procedure requires a series of integrations and linear interpolations for the given contact. Obviously, linearly interpolating and integrating over one or two points will be much less reliable than similar operations performed on larger numbers of points. Ideally, the sampling frequency should be much greater than 20kHz and the number of data points increased. This process is limited by the available equipment and the computation time available. Chapter 5. Verification of Mathematical Models 95 Also, extrapolating two to five data sets over five to 48 hour wear tests is likely to be inaccurate. Fortunately, the work rates, forces etc. were found to be quite consistent for most of the tests. Several assumptions were made to mathematically simplify this wear model. A l -though actual force data was used to determine normal loading on a point X on the circumference of the tube, it was later averaged to give an equivalent elliptical loading for use in Equation 5.25. When the dynamics of the test apparatus are considered and knowing that the forces to the dynamic specimen holder are sinusoidal, it appears there is some merit in the assumption. As well, since the contact at the ring tube interface is very close to a line contact, the point X will be exposed to an elliptically varying load distribution (see Fig.4.30) as the ring slides past the tube. Superimposed on this load pair will be fluctuations due to elastic rebounding of both the dynamic and stationary specimens. The slip length for an individual asperity was assumed to vary with height according to Equation 4.16. Although there is no way to verify this, it seems to be a reasonable assumption. The statistical model for contact assumes that the tube surface readily deforms (or indents) due to contact pressure from the harder ring specimen asperities. Any changes of the ring specimen texture were assumed to be negligible. This will only be true if the ring is very much harder than the tube. Since the ring and tube are of comparable hardness (Hring/Htuoe = 2.0 ), this assumption of negligible tube deformation is perhaps the least valid. As well, since there is a time span before the initial machined specimen surfaces develop the consistent wavy texture of Section 3.2.2, it is likely that tests where little wear occurs will be less accurate than tests where significant wear has occurred. This helps to explain the poor results for tests P2, N18 and N21. Since the tube microsurface is assumed to conform to the ring asperities according Chapter 5. Verification of Mathematical Models 96 to the shakedown theory, the discrete contacts are modelled as line contacts whose maximum pressure is P / . The frictional work input was then found as the product of the load x the sliding distance x the coefficient of friction. The reasonably close agreement between the theoretical work and the bulk work 1 (shown in Table 5.4) serves to validate the mult-asperity contact deformation model. But since the theoretical work is found using the normal load (influenced by squeeze film effect) while the bulk work is found using the shear force, there is bound to be some discrepancy between the work values. Accounting for an error of 5000 times between the predicted and the measured wear is not a simple process. Since the geometry of the surfaces appear to have been adequately accounted for, it becomes apparent that the largest error lies in evaluating the failure criteria. It was acknowledged in Section 4.2.1 that a significant portion of the frictional work will not go into wear but rather to elastic deformation, noise, heat etc.. As there currently exists no simple way to evaluate the percentage of the frictional work which goes into wear (denoted w) this factor was left open for speculation. It is very possible that this factor is of the order 10. Indeed, since all of the work input was assumed to contribute to fracture of the microsurface at the critical depth, whereas the plastic strains will actually be distributed throughout the deforming surface layer, the work that goes into fracture at the depth may be even less than 1%. The shear fracture wear model predicts that the number of passes required to yield an elliptical wear sheet of 6, wide, U long and a, thick ranges from fifty to 250 2. This seems to be reasonable but unfortunately there are no numbers with which to compare. The theoretical work predicted due to the sliding asperities is usually about one-fifth the calculated bulk work input so the significance of w is effectively reduced to an 1Found from Equation A.28 2This value was obtained by dividing the wear volume by the average volume of a wear particle x the number of asperity contacts for the test duration Chapter 5. Verification of Mathematical Models 97 order of twenty. Mathematical simplifications, experimental inaccuracies and the previous computa-tional difficulties may easily account for a tenfold inaccuracy. This still leaves a factor of at least ten for which to account. Simplifications in the statistical contact model may help to account for the remaining inaccuracy. Every asperity on the ring which was of sufficient height was assumed to load the tube with a line contact whose maxi-mum contact pressure is P£. Since there will be roughness on the tube due to previous deformation, some asperities may have a greater interference with the opposing surface than they would if one surface is assumed to be smooth. On the other hand, some will have less interference and these two effects should offset each other. A more thorough statistical analysis would also account for the fact that short asperities which follow in line behind long asperities may not actually touch the opposite surface, whether it is smooth or not. This effect could be accounted for by using the autocorrelation of the line asperity height distribution to determine the likelihood of a shallower asper-ity following a taller one and thus not contacting etc. But if the number of asperity contacts is mathematically reduced by this means, then those asperities which do con-tact must carry a higher load. These two effects should also offset eachother. One last mathematical simplification was to use the average asperity radius rather than the radius distribution. If some correlation between the asperity heights and radii is used, then presumably the asperity distribution could be accounted for. Certainly a more rigorous statistical treatment could be made but it is uncertain whether this would result in better predictions of wear using the shear fracture wear model. There have been many other theoretical models of wear devised which ultimately end up utilizing an empirical wear constant. Like the shear fracture wear model, there are assumptions of geometry, material parameters and failure criteria. But the predic-tions made with the model developed in this thesis are likely to be more accurate for Chapter 5. Verification of Mathematical Models 98 cases where significant plastic deformation takes place. This is because 1) the dynamics of the system are well accounted for by using actual force and displacement data from experimental systems, and 2) for sliding asperity contacts, the geometry of indentations can be predicted by including plastic deformation according to the shakedown theory. In a Hertzian elastic model (such as the model of Section 2.2.5), the contact pressure at each asperity interface will vary with the interference between asperities. But using the shakedown theory, these pressures are all known (P<f), and so the remaining un-known factors (chiefly material parameters), are likely to be constant throughout the operating range. The expectation then is that once the statistical and computational uncertainties are treated, this model will accurately relate wear rates to work rates between two surfaces of largely different hardnesses where the primary wear mecha-nism is ductile fracture. Ideally, the resulting empirical factor will be exactly equal to w, the percentage of frictional work which goes into wear. Presumably w could be determined by solving the mixed elastic/plastic problem for a line contact sliding load over a plastically deforming half space, but such a development is beyond the scope of this investigation. 5.3 Summary The contact deformation model of Section 4.1.2 was verified by simulating a typical asperity interaction. A n elliptically varying load was applied to a hard, sliding inden-ter against a target of softer material to determine if the geometry of the resulting indentation could be reliably predicted by equations 4.13 and 4.14. Two different tar-get materials were examined and a variety of slip lengths and peak normal loads used. Comparison between the experimental and theoretical values shows that for moderate and heavy loads, the geometry can be reliably predicted. For low loads, Equation 5.25 Chapter 5. Verification of Mathematical Models 99 failed, since the assumption of a line contact condition was not valid. The single asperity model was then extended to a statistical multi-asperity model. The resulting model of contact deformation was used to predict wear by equating the energy required for wear with the frictional work done by asperities on the harder ring surface sliding against the softer, readily deformed tube surface. A computer program was developed to correlate the force and displacement data to give normal and shear force waveforms, as well as tangential velocity. The bulk work input was found from Equation A.28. The average load that a point X on the tube surface sees in a given loading cycle was found, and the distance between the smooth tube surface and the reference line of the asperity distribution (measured from a worn ring specimen) was calculated. These asperities were assumed to form indentations on the tube surface whose geometry could be found using the contact deformation model (Equation 4.14). The contact pressure between the tube and ring was assumed to be as shown in Figure 4.30. The frictional work input due to sliding of that load along the tube surface was equated with the work required to form a wear sheet whose area is equal to that of the indentation. The theoretical work input due to the sliding asperities was found consistently to be about one-fifth the calculated bulk work which is an indication that the assumptions in geometry etc. are not unreasonable. Unfortunately, the predicted wear did not nearly so well match the measured wear. The model was found to over-estimate the wear by a factor of about five-thousand. This was somewhat surprising in view of the fact that the empirical correlation between work and wear rates (Fig.5.54) was good and since the dynamics of the system are well accounted for by using actual force and displacement data taken from the experiments. The discussion of Section 5.2.3 considers a number of factors which may have contributed to the inaccuracy of the wear model. The most important consideration is that only a small portion of the frictional work Chapter 5. Verification of Mathematical Models 100 will go into causing shear fracture at a given depth below the tube surface. As well, there will be some errors introduced by necessary mathematical extrapolations while simplifications in the statistics involved may contribute to the overestimate of wear. It was found that by reducing the predicted wear values by a factor of 5240, the model could predict the expected wear in most cases within a factor of two. In cases where there was very small volumes of wear or when contact times were in the order of a single data point (many of the impacting cases) correspondence between experimental and theoretical values was poor. This is not necessarily the fault of the theoretical model but rather the computation procedure involved. But for most cases, excepting the empirical constant, the wear model is able to predict the correlation between wear and all those parameters relating to work input (frictional coefficient, sliding velocity, normal load, time of contact, impact angle) as well as the material parameters. Chapter 6 Conclusions Wear is and most likely always will be a pervasive problem in almost all mechanical systems. From heart valves and linkages in human beings to connecting rods in auto-mobiles and sucker rods in oil wells, the consequences of wear are manifest. Modelling of wear may allow designers of wear prone systems to determine the effect of geometry, dynamics etc. on wear and thus reduce to a minimum the economic consequences of wear. In an effort to improve the life of heat exchanger tubes, a study of their wear against support plates has been undertaken. A room temperature simulation of this wear system allowed observation of the wear scars and the effects of a variety of operating and material parameters to be experimentally determined. The theoretical developments that followed lay the foundation for further investigation into this fascinating problem. The following is a summary of the most important developments in this investiga-tion. 1. A series of specimens were generated through a room temperature simulation of the interactions between a heat exchanger tube and its support plate. Inspection of the wear scars indicates that material removal is due likely to mechanical interactions between the asperities on the two surfaces. Crack propagation and shear fracture appear to be the dominant mechainisms of failure while corrosion, adhesion and erosion are minor contributors. 101 Chapter 6. Conclusions 102 2. The topography of the worn ring surface appeared to be constant for a wide range of impacting conditions. 3. According to the shakedown theory of plasticity, a material which is loaded be-yond its yield point will plastically deform, conforming to the load as well as in-creasing its effective elastic yield point due to the introduction of residual stresses combined with strain hardening. If the initial load is a sliding circular Hertzian contact, then these geometry changes will, after several passes lead to a line con-tact situation with a maximum contact pressure equal to P<f, the shakedown pressure of the softer material. 4. Following the shakedown theory, the geometry of an indentation formed by a hard, sliding asperity was predicted as the initial plastic deformation gave way to a new elastic state. Hertzian line contact formulae were used to determine the radius of the conforming indentation necessary for the contact pressure not to exceed P^. The width of the indentation easily followed. 5. The single indentation model was then extended to a multi-asperity model by assuming that the asperity distribution on the harder surface did not change as the softer surface was worn. 6. A wear model based strictly on mechanical considerations was developed by equating the frictional work input due to a sliding asperity with the energy re-quired for fracture of a wear particle. The energy for fracture of that wear particle is assumed equal to the shear stress x the area of the particle x half the length of the particle in the direction of load application. 7. The experimental results showed a linear relationship between the wear rate and frictional work rate. Chapter 6. Conclusions 103 8. The wear rate predicted by the model was compared to experimental results and it was found that the predictions wear greater than the measured by a factor of approximately 5000. The primary reason for this inaccuracy is the assumption that all of the frictional work input goes into fracture of the subsurface at the depth of the maximum subsurface shear. If the predicted results are reduced by this factor, the model will predict the wear rate as a function of the work input within a factor of two. 9. Despite several decades of research into wear mechanisms and wear modelling and advances in instrumentation and control, wear still defies accurate quantifi-cation and even characterization. Wear mechanisms remain as speculations, and models are simplistic at best. It is not surprising that even the best efforts fail to describe the extremely complex system that comprise most wear situations. Unti l it can be determined with certainty which wear mechanisms act in a given operating range, a wear model for that range will serve as little better than an academic exercise. Advances in modelling procedures with improving inspection techniques and equipment, combined with the continuing integration of metal-lurgical, mechanical and chemical engineering expertise on the subject, give hope that the modelling of wear may one day yield accurate results without the need for unknown empirical factors. A p p e n d i x A D a t a C o r r e l a t i o n Four signals were collected during the wear tests - two displacement signals and two force waveforms. The wave pairs were taken from sensors located at ninety degrees to the other. These waveforms will be arbitrarily referenced with respect to an 'x' axis and a 'y' axis. We will assume that the the offset between the force and displacement signals is zero degrees. The force and displacement vectors are shown in figure A.57. Figure A.57: Vector Diagram of Forces and Displacements Using the same notation as the figure, the following relations can be determined: « = - » - ( ! ) 104 Appendix A. Data Correlation 105 C = <f>-o FN = FcosC Fs = FsinC So the four input waveforms can be easily correlated to determine the normal and shear forces F^ and Fs respectively. Notice also that the impact angle (taken from the normal to the ring surface) is equal to £ and that the coefficient of friction can be found as tan £. The frictional work input can be found as the integral of the shear force x the velocity or (A.28) where RR is the ring radius. Appendix B Numerical Solution To Equation 4.11 The solution to Equation 4.11 is developed by obtaining a pair of coupled differential equations and then numerically integrating to the solution. This process proceeds as follows: Setting gives 0 12 n l db c = —b2 = 2b— OX ox db dx c 2b Substituting Equation B.30 into 4.II 1 leads to w here 1 b K = K Therefore 2 — dx 2P(x) 47rr(P05)2 E* Kb ' F o r reference, E q u a t i o n 4.11 is P(x) 1 = 2^r (P/) a b E*P(x) (B.29) (B.30) (B.31) (B.32) 106 Appendix B. Numerical Solution To Equation 4.11 107 and the following system of coupled differential equations results: 0b_ dx dc dx c 2b Kb P(x) -2 (B.33) (B.34) At x — 0, we have a point of symmetry where b = b0 and c = 0. A numerical integration routine (Runge-Kutta for example) can be used iteratively to find bQ such that 6(x = 0) = 0. The results of this process is a series of points which define the curve b(x). This plot is shown in figure B.58. 0.5 0 . 4 5 _ 0 .4 0 . 3 5 0 .3 c o +J c d +J ti <u - 0 .2 0 . 2 5 0 0 . 1 5 TJ S 0.1 0 . 0 5 Predicted Indentation Profile 1=1.0mm, b o = 4 7 . 2 6 m m , P m a x = 1 5 0 N 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 Distance X From Center Line (mm) 1.00 1.20 Figure B.58: Plot of Exact Solution b(x) to, Equation 4.11 Appendix B. Numerical Solution To Equation 4.11 108 Wi th this accomplished, p can be easily found. Combining Equation B.29 above dc d2b2 „ 4nr(P^)2b(x) with Equation 4.8 gives dx dx2 1 P = - 2 + p(x) E*P(x) d (b2\ dx2 \2r, 2*r(P0s)2 b(x) E*P(x) (B.35) Appendix C Proof of Elliptical Planform If the longitudinal radius p is constant and the transverse radius r is also, then the resulting planform of the indentation is elliptical. This can be developed using geometry as follows: In the practical case, the build up of material around the outer edges of the indenta-tion, due to plastic deformation, may mean that the apparent planview is not perfectly elliptical. But at the plane z = 0, the planform should not be distorted. The situation can be simulated as a sphere on the end of a pendulum where the pendulum rod is pinned at (0,0,6). The sphere cuts through the half space, starting at x — — I and ending at x = I. Assuming that the radius of the sphere r is less than the radius of the indentation p then the planform can be determined using geometrical considerations. b' = y/rl - P b = b' + rT 8 = rL-b' The trajectory of the center of the sphere is (c-bf + x2=rl while along the y and z plane the path cut through the half plane is given by (z - cf +y2=r2T 109 Appendix C. Proof of Elliptical Planform and so at z = 0 V = \jTT ~ [(rl - x2)* - b]2 This equation was compared to the assumed expression bx = b o S j \ - ^ using a spreadsheet and was found to correspond to within 0.1%. (0,0,b) Figure C.59: Geometry Used in Planform Derivation Appendix D Average Pressure Pavg(x) on Tube Surface In this appendix, Equation 4.19 is derived. Figure D.60 shows schematically the load distribution of amplitude Lmax(t) where its center is at x(t). It is assumed that the contact between the tube and support plate is such that a line contact results. Lmax(0 Figure D.60: Sliding Load Distribution on Tube Surface We can easily see that X = x(ti) + a(ti) X = x(tf)-a(tf) (D.36) 111 Appendix D. Average Pressure Pavg(x) on Tube Surface 112 where for a Hertzian line contact, a(t) = 2\ (t)R wirE* For U <t <tf the point X sees a pressure P(x) = Po(t) 1 (X ~ *(t))2 a?{t) where for line loads the maximum pressure pa = yfP'E*/<KR = ^LmaxE*/TrRw. The average load Pavg(x) is then found as P3vg(x) — J ~ f iraoi(')* 1 : —1{ Ju \ a (x - x(t)y dt tf-tiJu — '\\ o?(t) where J = More simply, the average load can be found from the following equation: Pavg{x) — J tf — ti Jt{ dt The second expression is computationally simpler and gives values within 2-3% of the first expression for practical cases. Appendix E Solution Algorithm for Equation 4.20 A n algorithm for solving Equation 4.20 is developed. Equation 4.20 is stated below. J 1 fh tf — ti Jti i - ^ - f w y * i- t00 1 = V2irP^rjr^ / 8*a$dz Jz=d' where a = 2 P'R irE* This equation can be solved for a given location on the tube surface. The easiest way is to evaluate nx equally spaced points on the surface. The position x(i) is determined from the phase data (Appendix A) by simply multiplying the angle measurements by the inner radius of the ring specimen while Pmax, the normal load, is obtained using the procedure of the same appendix, tf and U are found by solving Equations D.36. For a given loading cycle of X, the left hand side will evaluate to a constant. As well, the y/2irP^ri term can be moved to the left hand side. Therefore, Equation 4.20 can be simplified to the following: roo L ((x) = / a8* $(z)dz Jz=d< where J y/2irPSnr? a?{t) 1 rh t j ^ t i i L m a ^ \ If we set 8 = z — dl then $(z) — $(8 + d'). Also, when z = d, 8 = 0. This leads to f°° i C(x) = / a8*$(8 + d')d8 Js=o 113 Appendix E. Solution Algorithm for Equation 4.20 114 To solve for d'(x) we proceed as follows: 1. Guess a trial value of d'. 2. Evaluate p(S) 3. Solve for a;(/9,) 4. Evaluate 5. If TRIAL — £(x) < e then go onto next X, else return to step 1. If TRIAL > £(x) then d! should be increased, and vice versa. It is quite possible, depending on the duration of the data collection and the fre-quency of the cycles, that the point X will see several loading cycles. It is also possible that X will see no load at all. Appendix F Line Contact Stress Equations The equations for calculating the stress distribution due to a cylinder sliding per-pendicular to its axis (for a known friction coefficient) [43] are summarized here for convenience: ° x = < m 1 + - 2z \ a ( \ m2 + nl) J Po (i z2 +n2\ cr, = m 1 a \ m2 + n21 Po (m2 - z2\ T x z = n 2 , 2 where m2 = h{(a2-x2-rz2)2+4x2z2}1>+(a2-x2 + z2)] 2 = h{(a2-x2 + z2)2 + 4x2z2}l*-(a2-x2 + z2)] and the signs of m and n are the same as the signs of z and x respectively. The onset of plastic yield is usually governed by either von Mises shear-strain energy criterion: J 2 = 1 [(ar -a2)2 + (cr2 - <r3)2 + l>3 - c^)2 or by the Tresca criteria: Y2 = * 2 = T maxfjo-! - a2\, \o2 - <r3|, |cr3 - o-il} = 2K = Y 115 Appendix G Work Input From Sliding Asperity Load For a discrete asperity sliding load, the maximum contact pressure is assumed constant at while the width and length of the distribution are 2a,- and 26,(a;), respectively. The average value of the is nbmax/4 so the load distribution thus has an average value equal to So the amount of frictional work imparted to the opposite surface, assuming no losses is equal to the value of this load x distance slid x // , or To determine the total work input during a tube/ring interaction, the increments of work must be summed over the surface for the time of the test. The number of asperities in contact is given in Section 2.2.5 as /•oo n0 = nA0 / $(z)dz J z=d Therefore, at a given time t, over a sliding distance Xj — X{ the work W being done is found as W- ^ w ^ P o \ X f _ x.\ [°° aiiibimaxq>(z)dz Z J z—d This work increment must be integrated over the circumference of the tube for the duration of the test. If we evaluate nx points along the tube surface, then each Vi(x) must be multiplied by 2ivRT/(nx\xi — 116 Appendix G. Work Input From Sliding Asperity Load 117 Assuming that sets of data are taken such that each point x is loaded Nx times during the time period T, then the frictional work input to the wearing surfaces WT can be found as t *»» r)wpir3pSRT. . [°° _, w WT = m E E o 1*' ~ X ' l / ailibimaX$(z)dz 1 X=XlN=l Z T l x J z = d If WT equals the work input found from the bulk calculations of Appendix A , then the assumption of asperities loaded at the shakedown pressure would seem to be vindicated. Appendix H Tube Shear Fracture Wear Model A wear model will be developed to predict the volume of wear produced during inter-actions between a heat exchanger tube and its support plate. The following parameters have already been developed in Section 4.2.1 and Ap-pendix G : • Pavg(x): The average pressure that the point X on the tube surface sees. • Xj — X{\ The distance over which the highest asperities slide when the point X is loaded. • tf — t^. The time interval over which the point X is loaded. • S{, pi, fy: Geometrical parameters of the indentation. • Ai'. the area of each indentation. • Vi'. the volume of the wear particle expected for each indentation. • Wi'. the frictional work expended by the sliding asperity contact. • Eii the energy required to produce a wear particle of volume Vi. • dVi'. the increment of wear produced during this small sliding cycle. Since Ai — 7r6,7, 118 Appendix H. Tube Shear Fracture Wear Model 119 Vi Ei Wi = (^j diirbili = TyAiU = Ty-nbilf maxPo we can solve for dVi. dV = ViZoWj Ei [(%) a,7r6,/,] [w(/*f t-o t-7F 26,-m a jP 0 s/2)] TyAili afwfi^biP^ 2r„ (H.37) The number of asperities in contact can be found using Equation 2.1 where the contact area is equal to 2af,w and 2a*, is taken equal to |«,- — x / | . Therefore A0 = w\xi — Xf\ which gives where 7 T 2 f°° y. - 7—wrj\xi - x / | / a2bi$(8)d8 2. Jo (H.38) 7 • © ZOfl fn(p) Equation H.38 gives the wear volume produced by the ring sliding a distance \xf — x,| over the point X on the tube with an average pressure equal to Pavg- This wear increment must be integrated over the circumference of the tube for the duration of the test. If we evaluate Equation H.38 at nx points along the tube surface, then each Vi(x) must be multiplied by 2TrRT/(nx\xi — x / | ) . Appendix H. Tube Shear Fracture Wear Model 120 Assuming that sets of data are taken such that each point x is loaded A7* times during the time period T, then the predicted wear Wv volume for a test of length t is given as ff.^EE^/'W (H.39) The two indentation geometry parameters, a,- and 6,- depend on the separation distance d between the reference plane of the asperity distribution on the support plate and the surface of the smooth tube. The value of d for each contact at the point x is found using the procedure of Appendix E . Appendix I Results of Experimental Indentation Tests A series of experiments were conducted to test the accuracy of the contact deformation model of Section 4.1. The results of these tests are shown in tables 1.5 and 1.6 along with a series of predicted values determined by using three different values of K. Note that the K values are in M P a . Examinination of the data reveals that the best correlation between theoretical and experimental values is found for values of K for brass and steel equal to 222 M P a and 306 M P a , respectively. Tables 1.7 and 1.8 show again the values of all the tests, with the predicted values now calculated using the single K value. As well, a ratio of the measured value over the predicted value has been tabulated. These ratios are plotted in figures 5.46 and 5.45 first in order of increasing Pmax and secondly in order of increasing slip distance I. There is evidently a large, but not intolerable scatter in the results, considering the experimental difficulties. The following values were used in the calculations: Ebrasg = 89.9 G P a Esteel = 213 G P a 'indenter = 200 G P a ^brass = 0.324 Vsteel = 0.30 121 Appendix I. Results of Experimental Indentation Tests 122 Series #2 - Brass in Water Test Pmax I Pmeat Ppredicted (mm) (N) (mm) (mm) K = 150 K = 190 K = 222 3 100.8 2.15 190 107.2 21.3 32.7 4 143.5 2.12 120 55.2 13.4 19.1 5 147.4 0.75 20 12.1 6.0 7.0 6 78.5 0.90 30 36.0 10.4 13.9 7 79.1 0.84 35 31.9 9.7 12.8 8 79.0 0.84 32 32.0 9.7 12.8 10 97.6 0.86 25 24.0 8.3 10.6 11 97.3 0.85 25 23.7 8.3 10.5 12 95.2 0.49 10 12.2 6.1 7.0 14 96.0 0.51 9.5 12.7 6.1 7.2 15 333.1 1.77 15 12.7 6.1 7.2 Series #4 - Brass in Oil 2 330.1 1.68 13 12.1 6.0 7.0 4 314.9 1.18 12 9.1 5.3 6.0 5 307.9 2.28 45 18.9 7.4 9.1 6 305.4 2.30 45 19.3 7.5 9.2 7 211.7 2.28 80 32.6 9.8 13.0 8 210.6 2.26 85 32.4 9.8 13.0 9 203.6 1.20 19.7 14.2 6.5 7.7 10 195.9 1.21 20 15.0 6.6 7.9 11 156.5 1.25 25.5 20.9 7.8 9.7 12 158.8 1.28 29 21.2 7.8 9.8 13 74.5 1.34 75 78.4 17.0 25.2 14 74.3 1.37 76 82.1 17.6 26.2 15 55.8 0.50 13 24.6 8.4 10.8 16 55.9 0.49 13 23.9 8.3 10.6 Series #6 - Brass in Oil 7 110.6 9.20 21.4 22.1 8.0 10.0 8 109.4 9.00 22.3 21.8 7.9 9.9 9 161.0 7.60 9 11.2 5.8 6.7 10 160.0 7.40 9.3 11.0 5.8 6.6 11 159.6 1.00 14 15.3 6.7 8.0 12 160.1 0.90 14.4 13.5 6.3 7.4 13 159.0 1.10 22.5 17.3 7.1 8.6 14 159.9 1.10 25 17.1 7.0 8.6 15 160.0 1.77 58 34.0 10.0 13.4 16 160.0 1.77 62 34.0 10.0 13.4 Table 1.5: Results of Indentation Tests - Indenter on Brass Target. Appendix I. Results of Experimental Indentation Tests 123 Series #3 - Steel in Water Test Pmax / Pmeas Ppredicted (mm) (N) (mm' (mm) K = 250 K = 270 K = 329 1 337.9 1.00 9 10.7 6.6 7.9 2 345.5 1.05 9 11.0 6.7 8.0 3 61.3 1.74 60 415.0 117.2 202.1 4 61.7 1.79 80 433.9 122.3 211.1 5 113.2 1.22 23 66.2 22.8 35.3 6 111.8 1.17 25 62.8 21.9 33.6 Series #5 - Steel in Oil 1 334.2 1.16 14.37 12.5 7.2 8.9 2 322.9 1.16 14.5 13.0 7.4 9.1 3 202.8 1.50 21 34.9 14.0 20.1 4 201.4 1.25 25 26.5 11.6 16.0 5 49.9 1.41 170 412.0 116.4 200.7 6 49.8 1.93 140 767.7 212.2 370.6 Series #7 - Steel in Oil 1 169.2 0.70 14 15.3 8.2 10.3 2 165.9 0.69 11 15.4 8.2 10.4 3 161.4 1.00 20 26.5 11.6 16.0 4 161.7 0.97 18.8 25.2 11.2 15.4 5 161.1 1.11 27 31.1 12.9 18.3 6 159.7 1.11 30 31.6 13.1 18.5 10 303.8 1.91 35 27.1 11.8 16.3 Table 1.6: Results of Indentation Tests - Indenter on M i l d Steel Target. Appendix I. Results of Experimental Indentation Tests 124 Series #2 - Brass in Water Test Pmax I Pmeas Ppredicted Ratio (N) (mm) (mm) K = 222 3 100.8 2.15 190.0 110.9 1.713 4 143.5 2.12 120.0 57.0 2.106 5 147.4 0.75 20.0 12.3 1.624 6 78.5 0.90 30.0 37.1 0.809 7 79.1 0.84 35.0 32.8 1.067 8 79.0 0.84 32.0 32.9 0.972 10 97.6 0.86 25.0 24.7 1.013 11 97.3 0.85 25.0 24.4 1.026 12 95.2 0.49 10.0 12.5 0.802 14 96.0 0.51 9.5 12.9 0.736 15 333.1 1.77 15.0 12.9 1.163 Series #4 - Brass in Oil 2 330.1 1.68 13.0 12.3 1.059 4 314.9 1.18 12.0 9.2 1.300 5 307.9 2.28 45.0 19.3 2.329 6 305.4 2.30 45.0 19.8 2.276 7 211.7 2.28 80.0 33.5 2.385 8 210.6 2.26 85.0 33.4 2.548 9 203.6 1.20 19.7 14.5 1.355 10 195.9 1.21 20.2 15.3 1.316 11 156.5 1.25 25.5 21.4 1.192 12 158.8 1.28 29.0 21.7 1.336 13 74.5 1.34 75.0 81.0 0.926 14 74.3 1.37 76.3 84.9 0.899 15 55.8 0.50 13.0 25.3 0.514 16 55.9 0.49 13.0 24.5 0.530 Series #6 - B rass in Oil 7 110.6 9.20 21.4 22.7 0.943 8 109.4 9.00 22.3 22.3 0.998 9 161.0 7.60 9.0 11.4 0.789 10 160.0 7.40 9.3 11.2 0.832 11 159.6 1.00 14.0 15.6 0.897 12 160.1 0.90 14.4 13.7 1.048 13 159.0 1.10 22.5 17.7 1.274 14 159.9 1.10 25.0 17.5 1.425 15 160.0 1.77 58.0 35.0 1.656 16 160.0 1.77 62.0 35.0 1.770 Table 1.7: Ratio of Predicted Radius Value to Measured Value - Indenter on Brass Target. Appendix I. Results of Experimental Indentation Tests 125 Series #3 - Steel in Water Test Pmax / Pmeat Ppredicted Ratio (N) (mm' (mm) K = 305 MPa 1 337.9 1.00 9.0 9.4 0.960 2 345.5 1.05 9.0 9.6 0.939 3 61.3 1.74 60.0 312.5 0.192 4 61.7 1.79 80.0 326.6 0.245 5 113.2 1.22 23.0 51.3 0.448 6 111.8 1.17 25.0 48.8 0.512 Series #5 - Steel in Oil 1 334.2 1.16 14.4 10.8 1.327 2 322.9 1.16 14.5 11.2 1.295 3 202.8 1.50 21.0 27.8 , 0.756 4 201.4 1.25 25.0 21.5 1.162 5 49.9 1.41 170.0 310.2 0.548 6 49.8 1.93 140.0 576.4 0.243 Series #7 - Steel in Oil 1 169.2 0.70 14.0 13.0 1.080 2 165.9 0.69 11.0 13.0 0.844 3 161.4 1.00 20.0 21.5 0.932 4 161.7 0.97 18.8 20.5 0.916 5 161.1 1.11 27.0 25.0 1.081 6 159.7 1.11 30.0 25.3 1.185 10 303.8 1.91 35.0 21.9 1.598 Table 1.8: Hardened Steel Indenter on M i l d Steel Target. Appendix J Effect of Elastic Recovery in Contact Deformation Model In this appendix, elastic recovery in the transverse plane is included in the contact deformation model of Section 4.1. If we replace TT in Equation 4.13 with r' (of Equation 5.27) and proceed as in Section 5.1.1 we get where C1-2rL + rrL — — •* max1-1 while C2 remains equal to r\. If p is first found from Equation 5.25, bQ can then be found from Equation 4.13 and r' then follows from Equation 5.27. A new value of p can then be found from Equation J.41. Using this procedure, the values in tables 1.5 and 1.6 have been re-calculated and the results are tabulated in tables J.9 and J.10. The results give a direct comparison between the measured transverse radius and the predicted radius. In most cases using Equation 5.27 gives reasonable predictions for the unloaded transverse radius. But in some cases, the predicted value is much too large with the predicted longitudinal radius p being much too low. The general result is that the predicted radius is decreased by including transverse shallowing, but the improvement is small and in some cases, this (J.40) Rearranging gives again: p2-C[p + C2 = 0 (J.41) 126 Appendix J. Effect of Elastic Recovery in Contact Deformation Model 127 procedure leads to erroneous values. This comparison shows that the effects of elastic recovery are best neglected. Appendix J. Effect of Elastic Recovery in Contact Deformation Model 128 Series #2 - Brass in Water Test rmeat r' Pmeasl Ppred.l Ppred.l Ratioi Ratio2 (mm) (mm) (mm) (mm) (mm) (mm) 3 5.20 4.29 190.0 110.9 120.2 1.71 1.58 4 4.30 4.05 120.0 57.0 58.7 2.11 2.05 5 4.50 4.46 20.0 12.3 14.3 1.62 1.40 6 4.00 4.97 30.0 37.1 45.3 0.81 0.66 7 4.32 5.02 35.0 32.8 40.3 1.07 0.87 8 4.40 5.03 32.0 32.9 40.4 0.97 0.79 10 4.40 4.65 25.0 24.7 28.4 1.01 0.88 11 4.10 4.67 25.0 24.4 28.1 1.03 0.89 12 4.25 5.85 10.0 12.5 16.6 0.80 0.60 14 4.30 5.67 9.5 12.9 16.8 0.74 0.56 15 4.71 3.91 15.0 12.9 14.0 1.16 1.07 Series #4 - Brass in Oil 2 4.60 3.92 13.0 12.3 13.5 1.06 0.97 4 4.80 4.01 12.0 9.2 10.9 1.30 1.11 5 4.40 3.88 45.0 19.3 19.9 2.33 2.26 6 4.56 3.88 45.0 19.8 20.3 2.28 2.22 7 4.50 3.93 80.0 33.5 33.9 2.38 2.36 8 4.50 3.93 85.0 33.4 33.7 2.55 2.52 9 4.49 4.06 19.7 14.5 15.7 1.36 1.25 10 4.43 4.06 20.2 15.3 16.6 1.32 1.22 11 4.30 4.12 25.5 21.4 22.8 1.19 1.12 12 4.44 4.11 29.0 21.7 23.0 1.34 1.26 13 4.26 4.86 75.0 81.0 98.3 0.93 0.76 14 4.40 4.86 76.3 84.9 103.0 0.90 0.74 15 4.45 8.49 13.0 25.3 46.9 0.51 0.28 16 4.20 8.65 13.0 24.5 46.1 0.53 0.28 Series #6 - Brass in Oil 7 4.30 4.47 21.4 22.7 25.4 0.94 0.84 8 4.12 4.49 22.3 22.3 25.2 1.00 0.89 9 4.50 4.41 9.0 11.4 13.3 0.79 0.68 10 4.40 4.45 9.3 11.2 13.1 0.83 0.71 11 4.20 4.21 14.0 15.6 17.1 0.90 0.82 12 4.40 4.27 14.4 13.7 15.4 1.05 0.94 13 4.30 4.16 22.5 17.7 19.1 1.27 1.18 14 4.20 4.16 25.0 17.5 19.0 1.43 1.32 15 4.40 4.03 58.0 35.0 36.1 1.66 1.61 16 4.20 4.03 62.0 35.0 36.1 1.77 1.72 Table J.9: New Values of r and Ratio of Measured Radius / Predicted Radius Hardened Steel When Elastic Recovery of Transverse Radius is Included. Indenter on Brass Target. Appendix J. Effect of Elastic Recovery in Contact Deformation Model 129 Series #3 - Steel in Water Test rmeo» r> Ofneasl Ppred.l Ppred.l Ratioi Ratio2 (mm) (mm) (mm) (mm) (mm) (mm) 1 4.20 4.07 9.0 9.4 9.7 0.96 0.93 2 4.30 4.04 9.0 9.6 9.8 0.94 0.92 3 3.90 6.75 60.0 312.5 321.6 0.19 0.19 4 3.90 6.68 80.0 326.6 332.9 0.24 0.24 5 3.92 4.50 23.0 51.3 37.8 0.45 0.61 6 3.90 4.53 25.0 48.8 36.3 0.51 0.69 Series #5 - Steel in Oil 1 4.38 4.01 14.4 10.8 10.4 1.33 1.38 2 4.30 4.01 14.5 11.2 10.6 1.30 1.37 3 4.40 4.04 21.0 27.8 20.4 0.76 1.03 4 4.40 4.09 25.0 21.5 16.7 1.16 1.50 5 4.70 11.21 170.0 310.2 524.9 0.55 0.32 6 4.90 10.91 140.0 576.4 954.5 0.24 0.15 Series #7 - Steel in Oil 1 4.51 14.0 13.0 12.1 1.08 1.16 2 4.10 4.53 11.0 13.0 12.1 0.84 0.91 3 3.90 4.27 20.0 21.5 17.1 0.93 1.17 4 4.10 4.28 18.8 20.5 16.5 0.92 1.14 5 4.10 4.22 27.0 25.0 19.2 1.08 1.41 6 3.90 4.23 30.0 25.3 19.4 1.19 1.55 10 4.10 3.92 35.0 21.9 16.5 1.60 2.12 Table J.10: New Values of r and R a t i o of Measured Radius / Predicted Radius H a r d -ened Steel W h e n Elast ic Recovery of Transverse Radius is Included. Indenter on M i l d Steel Target. Appendix K Results of Data Correlation Program A C-language program was generated to correlated the input force and displacement waveforms according to the method of Section A . As well, the work input was calculated as the integral of the shear force x the displacement. For a variety of components, such as force, frictional coefficient etc, the mean, standard deviation, R M S , etc. were determined. The numerical data is summarized in table 5.4. The results of a typical high sliding test (Test PI) and a high impact test (Test 13) are shown in the following figures. X-Displacement Y-Displacement Tima (sec) Time (soc) Figure K.61: Test P i Waveforms 130 Appendix K. Results of Data Correlation Program 131 X-Force Y-Force Time (sec) Displacement Magnitude Force Magnitude Time (sec) Time (sec) Displacement Phase Force Phase \ V \ \ \ \ \ \ \ \ \ \ \ Time (JOC) Test P I Waveforms Appendix K. Results of Data Correlation Program 132 Normal Force Shear Force Time (joe) Tima (stc) Test P I Waveforms Appendix K. Results of Data Correlation Program 133 X-Displacement Time (sec) X-Force 1 Tima (sec) Displacement Magnitude Time (sac) Y-Displacement Tim* (soc) Y-Force Tims (sec) Force Magnitude Tim* (soc) Figure K.62: Test N13 Waveforms Appendix K. Results of Data Correlation Program 134 Displacement Phase Hma (soc) Force Phase .,_ t — t ft— k m Si ' 37 •JM Time (sec) Normal Force Time (sec) Work Input ill Time (sac) Shear Force Time (sac) Total Work Test N13 Waveforms Appendix L Tube Trajectories The trajectories of the tube as it impacts against the ring are shown on the following pages for tests N6 through N21 and P I and P2. These trajectories are all normalized to have a maximum radius of 1.0. 135 Appendix L. Tube Trajectories 136 Bibliography [1] Rigney,D.A., "Fundamentals of Friction and Wear of Materials", 1980 ASM Ma-terials Science Seminar, Pittsburgh, A S M , 1980, Page 2. [2] "Economic Losses Due to Friction and Wear, Research and Development Strate-gies", (A Workshop Report). Associate Committee on Tribology, National Research Council, Canada, Nov. 1984. [3] Archard,J.F. and Hirst,W., "The Wear of Metals Under Unlubricated Conditions", Proc. Royal Society (London), Vol.236A, 1956, p.397. [4] Burwell,J.T. and Strang,C.D., "Metallic Wear", Proc. Royal Society (London), Vol.212A, May 1953, pp.470-477. [5] Burwell,J.T., "Survey of Possible Wear Mechanisms", Wear, Vol.1, 1957/1958, pp.119-141. [6] Moore,M.A., "Abrasive Wear", Proc. ASM Materials Science Seminar on "Fun-damentals of Friction and Wear of Materials", Pittsburgh, 1980. [7] Suh,N.P., " A n Overview of the Delamination Theory of Wear", Wear, Vol.44 #1, Aug.1977, pp.1-16. [8] McClintock,F.A. , "Plastic Flow Around a Crack Under Friction and Combined Stress", Fracture, Vol.4, 1977. [9] Bower,A.F., "The Influence of Crack Face Friction and Trapped Fluid on Surface Initiated Rolling Contact Fatigue Cracks", Transactions of the ASME, Vol.110, 139 Bi bliography 140 October, 1988. [10] Waterhouse,R.B. and Taylor,D.E., "Fretting Debris and the Delamination Theory of Wear", Wear, Vol.29,1974, pp.337-344. [11] Archard,J.F., "Contact and Rubbing of Flat Surfaces", Journal of Applied Physics, Vol 24, 1953, p.24. [12] Torrance,A.A., " A n Approximate Model of Abrasive Cutting", Wear,118, (1987) 217-232. [13] Challen,J.M. and Oxley,P.L.B., " A Slip Line Field Analysis of the Transition From Local Asperity Contact to Ful l Contact in Metallic Sliding Friction", Wear, 100 (1984) 171-193. [14] Suh,N.P., Sin,H.C. and Saka,N. "Effect of Abrasive Cut Size on Abrasive Wear", M I T Rep. to D A R P A , 1978, Proc. Tribology Conf, M I T Press, Cambridge, Mass., 1979. [15] Quinn,T.F.J . and Sullivan,J.L, " A Review of Oxidational Wear", Proc. of Inter-national Conference on "Wear of Materials", A S M E , 1977, pp.110-115. [16] Suh,N.P., "The Delamination Theory of Wear", Wear,25 (1973), 111 - 124 [17] Suh,N.P., "New Theories of Wear and Their Implications for Tool Materials", Wear, 62, (1980) 1-20. [18] Ja in ,V.K. and Bahadur,S.,"Development of a Wear Equation for Polymer-Metal Sliding in Terms of the Fatigue and Topography of Sliding Surfaces", Wear, Vol 60 (1980), pgs 237-248. Bibliography 141 [19] Greenwood,J.A. and Williamson,J.B.P., "Contact of Nominally Flat Surfaces", Proc.R.Soc.London,Ser.k, 295 (1442)(1966) 300-319. [20] Greenwood,J.A. and Tripp,J.H., "The Contact of Two Nominally Flat Surfaces", Proc.Inst.Mech.Eng., London, 185 (48/71)(1970-71) 625-633. [21] Ko ,P .L . , Tromp,J.H. and Weckwerth,M.K., "Heat Exchanger Tube Fretting Wear: Correlation of Tube Motion and Wear", American Society for Testing and Mate-rials, 1982. 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[27] Chivers ,T .C, Gordelier,S.C, and Roy,J., "Vibration Data and its Employment for Component Life Projections in A G R Circulators", Proc.International Conference on "Vibration in Nuclear Plants", Keswick, U K , May 1978. Bibliography 142 [28] Waterhouse,R.B. " Fretting Corrosion", Pergamon Press, 1972. [29] Kayaba,T., Iwabuchi,A., and Kato,K. , "Fretting Wear of Ni-Cr alloy at High Tem-perature", J.of JSLE International Ed . , Vol.4, 1984, pp.47-52. [30] Nishioka,K., Hirakawa,K., " Fundamental Investigations of Fretting Fatigue", Bul-letin of JSME, Vol.12, #51, 1969. [31] Ty l e r , J .C , Burton,R.A. and K u , P . M . , "Contact Fatigue Under Oscillatory Normal Load", Trans.ASLE, Vol.6, 1963, pp.255-269. [32] L a i r d , C , "Plastic Deformation of Metals and Alloys", Treatise on Materials Sci-ence and Technology: Plastic Deformation of Materials, ed. by Arsenault, R . J . , 1975, pp.101-162. [33] Goldsmith,W., "Impact", Edward Arnold, London, 1960. [34] Hogmark,S., 6berg,A., Stridh,B., "On the Wear of Heat Exchanger Tubes", Proceedings of the JSLE International Tribology Conference, July 8-10,1985, Tokyo, Japan. [35] Rice,S.L., Wayne,S.F. and Nowotny,H., "Material Transport Phenomena in the Impact Wear of Titanium Alloys", Wear, Vol.65, #2, 1980, pp. 215-226. [36] K o , P . L . and Basista,H., "Correlation of Support Impact Force and Fretting Wear for a Heat Exchanger Tube", Trans. ASME, Journal of Pressure Vessel Technology, Vol.106, #1, Feb.1984, pp.69-77. [37] Levi ,G. and Morr i , J . , "Impact Wear in CCVbased Environments", Wear, Vol.106, #1-3, Nov.1985, pp. 97-138. Bibliography 143 [38] K o , P . L . "The Significance of Shear and Normal Force Components on Tube Wear Due to Fretting and Periodic Impacting", Wear, Vol.106, pp.261-281, 1985. [39] Ko ,P .L . , "Experimental Studies of Tube Frettings in Steam Generators and Heat Exchangers", Journal of Pressure Vessel Technology, Vol.101 (May, 1979) 125-133 [40] Ko ,P .L . , "Impact and Fretting Wear Studies Part2 - Room Temperature Test Results (Single Impact per Cycle)", National Research Council (Canada) Division of Mechanical Engineeering (1988) [41] Hertz,H. tjber die Beruhrung fester elastischer Korper (On the Contact of Elastic Solids) J.reine und angewandte Mathematik, pgs 156-171 (For English translation see Miscellaneous Papers by H.Hertz, Eds.Jones and Schott,London:Macmillan,1896) [42] Johnson,K.L., Shercliff,H.R., Kopalinsky,E. "Shakedown of 2-Dimensional Asperi-ties in Sliding Contact", Cambridge University, Technical Report, November,1989. [43] Johnson,K.L. "Contact Mechanics", Cambridge University Press (1983) [44] Griffith,A.A. "The Phenomena of Rupture and Flow in Solids" Phil.Trans.Roy.Soc.of London, A221 (1921) pp.163-197. 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