Gear tooth flank damage prediction using high-cycle fatigue and wear models

DEPARTAMENTO DE ENGENHARIA MECˆ ANICA GEAR TOOTH FLANK DAMAGE PREDICTION USING HIGH-CYCLE FATIGUE AND WEAR MODELS JOS´ E AUGUSTO DE SOUSA FERREIRA BRAND˜ AO 2013 JOS´ E AUGUSTO DE SOUSA FERREIRA BRAND˜ AO GEAR TOOTH FLANK DAMAGE PREDICTION USING HIGH-CYCLE FATIGUE AND WEAR MODELS A THESIS SUBMITTED TO THE FACULDADE DE ENGENHARIA DA UNIVERSIDADE DO PORTO FOR THE PROGRAMA DOUTORAL EM ENGENHARIA MEC ˆ ANICA Supervisor: Prof. Jorge H. O. Seabra Co-supervisor: Prof. Manuel Jorge D. Castro DEPARTAMENTO DE ENGENHARIA MECˆ ANICA FACULDADE DE ENGENHARIA UNIVERSIDADE DO PORTO Keywords Spur gears Gear micropitting Gear wear Mixed film lubrication Fatigue crack initiation Dang Van fatigue criterion Palavras chave Engrenagens cil´ındricas Micropitting em engrenagens Desgaste em engrenagens Lubrica¸c˜ao em filme misto Inicia¸c˜ao de fendas de fadiga Crit´erio de fadiga de Dang Van Mots-cl´es Engrenages cylindriques Micropitting des engrenages Usure des engrenages Lubrication en film mixte Initiation de fentes de fatigue Crit`ere de fatigue de Dang Van v Acknowledgements I would like to express my gratitude to Prof. Jorge Seabra, my supervisor, and to Prof. Jorge Castro, my co-supervisor, for their patient guidance and support throughout the realisation of this work. I wish to thank the Laboratoire de M´ecanique des Contacts et des Structures, INSA de Lyon, and especially Dr. Fabrice Ville, for putting their lubricant traction measurement machine at my disposal. I also want to thank the Centro de Estudos de Materiais por Difrac¸c˜ao de RaiosX, Departamento de F´ısica, FCTUC, Universidade de Coimbra, and especially Prof. Ant´onio Castanhola Batista, who graciously performed X-ray diffraction measurements of residual stresses. I gratefully acknowledge the support of FEUP – Faculdade de Engenharia da Universidade do Porto and of INEGI – Instituto de Engenharia Mecˆanica e Gest˜ao Industrial. I am grateful for the funding by the European Regional Development Fund (ERDF) through the COMPETE – Competitive Factors Operational Program and by Portuguese Government Funds through FCT – Funda¸c˜ao para a Ciˆencia e Tecnologia as part of project Projecto Estrat´egico – LA 22 – 2011–2012, reference number Pest – OE / EME / LA0022 / 2011 and also the funding by Portuguese Government Funds through FCT – Funda¸c˜ao para a Ciˆencia e Tecnologia, under grant PTDC / EME – PME – 100808 – 2008 awarded to the project High Efficiency Gears and Lubricants for Windmill Planetary Gearboxes. Last but not least, my gratitude and love go to my wife and children for the forbearance they showed when this work lead me to neglect them. To my wife, Deborah and my children, Tom´as and Carolina . . . Contents Abstract xi Table of contents xvii List of figures xxi List of tables xxix List of symbols xxxi 1. General introduction 1 I. State of the art 3 2. Gear wear and gear Micropitting 5 3. Gear lubrication 13 3.1. Reynolds’sequation........................... 13 3.2. Lubricating oil rheology . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.1. Newtonian viscosity . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2. Non-Newtonian viscosity . . . . . . . . . . . . . . . . . . . . 18 3.2.3. Kinematic viscosity . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.4. Elasticity ............................ 20 3.2.5. Density ............................. 21 3.3. Hetziancontact............................. 21 3.3.1. Pointcontact .......................... 21 3.3.2. Linecontact........................... 22 3.4. Elastohydrodinamic lubrication . . . . . . . . . . . . . . . . . . . . 23 3.5. Mixed film and boundary lubrication . . . . . . . . . . . . . . . . . 28 4. Surface stresses and fatigue in rolling/sliding contact of spur gears 33 4.1. Stresses in a spur gear . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1. Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2. Elastic stresses . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2. Elastic shakedown in rolling and sliding contact . . . . . . . . . . . 36 xvii Contents 4.3. Metalfatigue .............................. 38 4.3.1. W¨ohlercurve .......................... 38 4.3.2. Fatigue crack initiation mechanism . . . . . . . . . . . . . . 40 4.3.3. Fatigue crack propagation . . . . . . . . . . . . . . . . . . . 41 4.3.4. Fatigue life duration . . . . . . . . . . . . . . . . . . . . . . 43 4.3.5. Fatigue criteria . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3.6. Dang Van high cycle fatigue criterion . . . . . . . . . . . . . 45 II. Friction properties of gear oils 51 5. Traction curves and rheological parameters for gear oils 53 5.1. Preamble ................................ 53 5.2. Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2.1. Testedoils............................ 53 5.2.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 55 5.3. Experimental results and comparison of the oils . . . . . . . . . . . 57 5.4. Simplified model for the EHD lubrication of a circular point contact. 63 5.4.1. Low shear viscosity . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.2. Non-Newtonian viscosity . . . . . . . . . . . . . . . . . . . . 64 5.4.3. Friction shear stress . . . . . . . . . . . . . . . . . . . . . . . 65 5.4.4. Temperature within an EHD film. . . . . . . . . . . . . . . . 67 5.4.5. Coupling of the thermal and friction problems. . . . . . . . . 77 5.5. Determination of the rheological parameters. . . . . . . . . . . . . . 77 5.6. Chaptersummary............................ 82 6. Stribeck curves of gear oils. 83 6.1. Preamble ................................ 83 6.2. Materials and experimental procedure . . . . . . . . . . . . . . . . . 83 6.3. Modified Stribeck Parameter . . . . . . . . . . . . . . . . . . . . . . 84 6.4. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5. Influence of the operating conditions in mixed film lubrication . . . 91 6.6. Boundary film lubrication . . . . . . . . . . . . . . . . . . . . . . . 93 6.7. Comparison of the oils . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.8. Chaptersummary............................ 97 III. Gear tooth flank damage 99 7. Models of surface damage on the tooth flanks of spur gears 101 7.1. Kinematics and normal load in spur gear teeth . . . . . . . . . . . . 101 7.2. Mixed film lubrication model . . . . . . . . . . . . . . . . . . . . . . 106 7.2.1. Preamble ............................ 106 7.2.2. Overallscheme ......................... 107 xviii Contents 7.2.3. Normal contact pressure in smooth EHD lubrication . . . . 110 7.2.4. Normal contact pressure in rough boundary lubrication . . . 110 7.2.5. Smooth EHL part of the tangential contact traction . . . . . 111 7.2.6. Rough boundary lubrication part of the tangential contact traction ............................. 112 7.3. Tooth flank micropitting model . . . . . . . . . . . . . . . . . . . . 112 7.3.1. Preamble ............................ 112 7.3.2. Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . 113 7.3.3. A simulation of gear meshing . . . . . . . . . . . . . . . . . 114 7.4. Model for wear on spur gear teeth . . . . . . . . . . . . . . . . . . . 125 7.4.1. Preamble ............................ 125 7.4.2. Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . 125 7.5. Chaptersummary............................ 127 8. Correlation of the tooth flank micropitting model with tests 129 8.1. Preamble ................................ 129 8.2. Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . 129 8.3. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.3.1. Overall procedure . . . . . . . . . . . . . . . . . . . . . . . . 130 8.3.2. Residual stresses . . . . . . . . . . . . . . . . . . . . . . . . 132 8.3.3. Roughness profiles . . . . . . . . . . . . . . . . . . . . . . . 133 8.4. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.5. Comparison of test and simulation results. . . . . . . . . . . . . . . 158 8.6. Chaptersummary............................ 162 9. Correlation of the wear model with tests 163 9.1. Preamble ................................ 163 9.2. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.3. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.4. Discussion of the wear simulation results . . . . . . . . . . . . . . . 173 9.5. Chaptersummary............................ 175 10.General conclusion and future work 177 10.1.Conclusion................................ 177 10.2.Futurework............................... 179 A. Publications 181 A.1. Articles in international peer reviewed scientific journals . . . . . . 181 A.2. Articles in Portuguese peer reviewed scientific journals . . . . . . . 181 A.3. Articles in conference proceedings . . . . . . . . . . . . . . . . . . . 182 A.4. Communications as invited speaker . . . . . . . . . . . . . . . . . . 182 A.5.Master’sThesis ............................. 183 xix List of Figures 2.1. Photograph of a micropit in a surface hardened steel gear tooth section. 7 2.2. Orientation of surface fatigue cracks according to their position on the tooth flank surface. The directions of rolling and sliding velocities, as well as gear rotation are shown. . . . . . . . . . . . . . . . . 8 2.3. Photograph of the surface of a surface hardened steel gear tooth where micropitting has occurred. The micropitted areas are surrounded by a red line. It is seen that micropitting is mainly restricted to the part of the flank below the pitch line. . . . . . . . . . . . . . 8 2.4. Shematic representation of micropitting initiation (a) and micropitting popagation (b) (taken from [1, Figure 11]): micropitting fatigue cracks propagate along the boundary between the plastic deformation region (PDR) and the dark etching region (DER). WEB stands for white etching band. . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1. Hydrodynamic lubrication of two surfaces in relative motion. . . . . 14 3.2. Typical curve of the variation of the logarithm of viscosity with pressureforliquids. ............................. 17 3.3. Hydrodynamic lubrication of two cylindrical surfaces in relative motion. ................................... 24 3.4. Comparison between the Hertzian and EHD distributions of pressure (pHertz and pEHD) and gap (hHertz and hEHD).............. 25 3.5. Example of a traction curve: the coefficient of friction µis plotted against the slide-to-roll ratio for oils P1, M1, E1, E2, E3 and T1. . . 28 3.6. Example of an “ideal” Stribeck curve: the curve progresses from fullfilm (elasto)hydrodynamic, to mixed, to boundary film lubrication as the rolling speed decreases. . . . . . . . . . . . . . . . . . . . . . 30 4.1. Elastic half-space coordinates and surface loads: pis the surface pressure, τis the tangential surface traction. . . . . . . . . . . . . . 34 4.2. Ratio τoct/p0of the octahedral stress to the maximum Hertzian stress in a tooth submitted to a Herzian pressure distribution and a coefficient of friction 0.05. ......................... 35 4.3. As in Figure 4.2 but with rough surfaces. . . . . . . . . . . . . . . . 35 4.4. Shakedown map of a Hertzian line contact taken from Williams [2]. 37 4.5. S-N diagram taken from Moore [3]: a) linear scale; b) semilogarithmicscale. ................................ 39 xxi List of Figures 4.6. Micrograph (150 ×magnification) of a specimen having endured 60000 stress cycles, taken from [4]. Slip-bands can be clearly seen. . 41 4.7. Small crack of length 2aat the geometric center of a much larger rectangular thin plate under uniform traction S............ 42 4.8. Stress cycle and hardening of a material point in pure shear stress. . 48 4.9. Position of the mesoscopic stress state of a material point during a load cycle on the pH/τmax plane. The shaded half-plane represents the area where the Dang Van criterion is violated. Thus, the parts of the cycle placed in the shaded area violate the criterion. . . . . . 49 5.1. Mini-traction machine. . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2. Experimental traction curves of the 6 oils at T0= 120 ◦C. . . . . . . 58 5.3. Experimental traction curves of the 6 oils at T0= 80 ◦C. . . . . . . 59 5.4. Experimental traction curves of the 6 oils at T0= 40 ◦C. . . . . . . 60 5.5. Friction coefficient µvs. LP parameter at p0= 1 GPa, U= 1 m/s, SRR = 0.4. Each ellipse groups data points from tests performed at the same inlet temperature. . . . . . . . . . . . . . . . . . . . . . . 62 5.6. Film thickness hand contact pressure paccording to the simplified EHD lubrication model. . . . . . . . . . . . . . . . . . . . . . . . . 66 5.7. Schema of the thermal problem. (1) disc; (2) sphere; the oil flows in the gap. ρ∗is the density of the body (∗), Cp∗its heat capacity, λ∗its heat conductivity and U∗its velocity. θis the temperature difference above the inlet temperature and Φ the viscous dissipation in the oil film. wis the contact half-width in this plane: w=√a2−y2. 68 5.8. Temperature excess in the section of symmetry (y= 0) for the case: oil P1, T0= 40 ◦C, U= 1 m/s, FN= 35 N (pavg = 0.86 GPa), SRR = 0.6. ............................... 73 5.9. Temperature excess in the section of symmetry (x= 0) for the case: oil P1, T0= 40 ◦C, U= 1 m/s, FN= 35 N (pavg = 0.86 GPa), SRR = 0.6. ............................... 74 5.10. Representative temperature excess for the case: oil P1, T0= 40 ◦C, U= 1 m/s, FN= 35 N (pavg = 0.86 GPa), SRR = 0.6......... 74 5.11. Oil temperature in the symmetry section (y= 0) of an EHD point contact. The ordinate represent the representative temperature excess above the inlet temperature. The abscissa represents the nondimensional position along the rolling direction. Several operating conditions are used, centered around T0= 40 ◦C, U= 1 m/s, FN= 35 N, SRR = 0.6: (a) shows the influence of the inlet temperature; (b) shows the influence of contact load; (c) shows the influence of the rolling speed; (d) shows the influence of the slide-to-roll ratio. 76 5.12. Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40◦C, U= 1m/s, FN= 16N (p0= 1GPa). 78 5.13. Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 80◦C, U= 1m/s, FN= 16N (p0= 1GPa). 79 xxii List of Figures 5.14. Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40◦C, U= 1m/s, FN= 35N (p0= 1.3GPa). 79 5.15. Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40◦C, U= 0.1m/s, FN= 16N (p0= 1GPa). 80 5.16. Influence of the elastic shear modulus on each oil: predicted curves for operating conditions T0= 40 ◦C, U= 1 m/s, FN= 16 N (p0= 1 GPa) are drawn including the elastic deformation (curves w/ G) and neglecting it (w/o G). ....................... 81 5.17. Slide-to roll ratio along the meshing line of an FZG type C spur gear. The solid line marks the actual relation between the meshing position and the SRR. ......................... 81 6.1. Specific film thickness under different operating conditions. . . . . . 86 6.2. Influence of the choice of parameter for the abscissa of a Stribeck curve: a) Stribeck parameter Uη/FN; b) Modified Stribeck Parameter Sp. .................................. 89 6.3. Experimental Stribeck curves of the oils . . . . . . . . . . . . . . . 90 6.4. Influence of the operating conditions . . . . . . . . . . . . . . . . . 92 6.5. Boundary film lubrication . . . . . . . . . . . . . . . . . . . . . . . 95 6.6. Comparison of the oils . . . . . . . . . . . . . . . . . . . . . . . . . 96 7.1. Position of the spur gears . . . . . . . . . . . . . . . . . . . . . . . 102 7.2. Coordinates on the surface of the pinion tooth flank. . . . . . . . . 102 7.3. Notable moments during meshing of a pair of teeth: a) the pair initiates its contacts while another is already in contact on the lefthand side; b) the pair now bears the contact alone; c) the pair is in pure rolling; d) a new pair initiates contact on the right-hand side; e) the pair ceases its contact. . . . . . . . . . . . . . . . . . . . . . 104 7.4. Notable moments of the meshing of a pair of teeth: the consecutive positions of a pair of contacting teeth are shown superimposed, as well as the share of the normal load borne by this pair of teeth as a function of the contact position along the contact line. . . . . . . . 104 7.5. Direction of sliding on a tooth surface . . . . . . . . . . . . . . . . . 106 7.6. Mixed film lubrication model: calculation of the EHD portion of the normalpressure.............................. 109 7.7. Mixed film lubrication model: calculation of the BDR portion of the normalpressure.............................. 109 7.8. Mixed film lubrication model: calculation of the normal pressure. . 109 7.9. Diagram of the numerical micropitting model . . . . . . . . . . . . 113 7.10. Some load sharing functions fΛand their associated boundary friction coefficients µBDR. ......................... 116 7.11. Simulation: values of βeq > βDV in the xz plane. . . . . . . . . . . . 117 7.12. Simulation: βeq > βDV in the part of the driving gear tooth below thepitchline............................... 117 xxiii List of Figures 7.13. Simulation: contour plot of a detail of Figure 7.12. . . . . . . . . . . 118 7.14. Simulation: contour plot of another detail of Figure 7.12. Points Q and Q0are singled out for later reference. . . . . . . . . . . . . . . . 119 7.15. Surface pressure field when the patch of Figure 7.14 undergoes its highest mixed film pressure. . . . . . . . . . . . . . . . . . . . . . . 120 7.16. History of the value of τmax +αDV ·pHin point Qplotted against the position of the contact in the meshing line. Two horizontal lines corresponding to the values of βDV and βeq are also shown. . . . . . 120 7.17. Map of the cycle undergone by point Qthat plots the mesoscopic maximum shear stress against the hydrostatic stress for the whole meshingcycle............................... 121 7.18. The cycle is shown as in Figure 7.17. The line of the Dang Van limit is rotated until it corresponds to a value of αDV =α0 DV = 0.242 such that β0 eq −β0 DV =βeq −βDV. The construction lines and points marked illustrate the geometric reasoning. . . . . . . . . . . . . . . 122 7.19. Comparison of the directions of maximum mesoscopic shear stress associated with βeq computed by the model with the directions of propagation of fatigue cracks above and below the pitch line: (a) A patch above the pitch line and the directions of τmax. (b) A patch below the pitch line and the directions of τmax. (c) The directions of propagation of contact fatigue cracks in a driving gear tooth. . . . 124 7.20. Diagram of the numerical wear model . . . . . . . . . . . . . . . . . 126 7.21. Example of calculation of the wear depth on the surface of a gear tooth................................... 127 8.1. Areas of the flank were residual stress measurements were performed. 132 8.2. Filtered roughness profiles of tooth 1 and toot 5 of the pinion gear: profiles taken at the end of each load sub-stage are stacked one above theother. ................................ 134 8.3. Coordinates on the surface of the pinion tooth flank. . . . . . . . . 136 8.4. Tooth surface and average velocity as a function of meshing position. 136 8.5. Radii of curvature as a function of meshing position. . . . . . . . . 137 8.6. Contact load between a pair of teeth as a function of meshing position.138 8.7. Hertzian pressure as a function of meshing position. . . . . . . . . . 138 8.8. Hertzian half-width as a function of meshing position. . . . . . . . . 139 8.9. Full film EHD central film thickness as a function of meshing position.139 8.10. Full film EHD shear rate as a function of meshing position. . . . . . 140 8.11. Local composite roughness as a function of meshing position. . . . . 141 8.12. Specific film thickness as a function of meshing position. . . . . . . 141 8.13. Coefficient of friction as a function of the meshing position in Full film EHL and Mixed film lubrication. . . . . . . . . . . . . . . . . . 142 xxiv List of Figures 8.14. Snapshot of contact pressure (pmix) and surface traction (tmix) on the pinion tooth flank surface at the instant when the nominal contact is on position x= 2290 μm for load sub-stages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . 143 8.15. Snapshot of contact pressure (pmix) and surface traction (tmix) on the pinion tooth flank surface at the instant when the nominal contact is on position x= 1100 μm for load sub-stages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . 145 8.16. Snapshot of the maximum macroscopic shear stress (τmax) under the undeformed pinion and wheel tooth flank surfaces at the instant when the nominal contact is on position x= 2290 μm for load sub-stages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . . . . . . . . . . . . . . . . 146 8.17. Snapshot of the maximum macroscopic shear stress (τmax) under the undeformed pinion and wheel tooth flank surfaces at the instant when the nominal contact is on position x= 1100 μm for load sub-stages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . . . . . . . . . . . . . . . . 148 8.18. Equivalent fully reversed shear stress (βeq) under the undeformed pinion and wheel tooth flank surfaces in an area centred on position x= 2290 μm for the end of load sub-stages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . 149 8.19. Equivalent fully reversed shear stress (βeq) under the undeformed pinion and wheel tooth flank surfaces in an area centred on position x= 1100 μm for the end of load substages: a) K3-1 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440 kcycles). . . 150 8.20. History of macroscopic stresses in the point at coordinate x= 2290 μm on the surface of the pinion tooth flank for load substages: a) K31 (10 kcycle); b) K3-2 (30 kcycles); c) K3-3 (50 kcycles); d) K6-1 (100 kcycles); e) K6-2 (400 kcycles); f) K6-3 (940 kcycles); g) K8-1 (100 kcycles); h) K8-2 (400 kcycles); i) K8-3 (940 kcycles); j) K9-1 (1440kcycles)............................... 151 xxv List of Tables p0maximum Hertzian pressure [Pa] pBDR.T normal contact pressure when supposing that the surfaces are not lubricated and the entirety of the load is borne by direct surface contact [Pa] pBDR portion of the normal contact pressure borne by direct surface contact [Pa] pEHD.T normal contact pressure when supposing that the surfaces are ideally smooth and the lubrication is in the full film EHD regime [Pa] pEHD portion of the normal contact pressure borne by the lubricant film [Pa] pHhydrostatic stress of the mesoscopic stresses [Pa] pini Hhydrostatic stress of the initial residual stresses [Pa] pini Hpini Haveraged over the fist 10 μm of depth [Pa] pMIX normal contact pressure (generally in the mixed film lubrication regime) [Pa] PT thermal coefficient of both surfaces defined as PT = √ρsCsKs Rradius [m] R∗effective radius of curvature [m] Raradius of the addendum circle of a gear [m] Rbradius of the base circle of a gear [m] Rpradius of the pitch circle of a gear [m] Rqroot mean square roughness parameter [m] sconstant for piezoviscosity calculation according to Gold Ssliding distance [m] S0Roelands temperature index Spmodified Stribeck parameter SRR slide-to-roll ratio tconstant for piezoviscosity calculation according to Gold Ttemperature [K] Tpinion driving torque [N·m] t−1fully reversed torsion fatigue limit [Pa] T0reference temperature or oil bath temperature [K] Tavg faverage lubricant temperature within the contact [K] Urolling velocity U= (U2+ U1)/2 U1tangential velocity of ball or pinion tooth surface [m/s] U2tangential velocity of disc or wheel tooth surface [m/s] V I viscosity index ZRoelands pressure index Znumber of teeth in a gear xxxii List of Tables αpressure-viscosity coefficient [Pa−1] α0operating pressure angle of a gear [◦] αDV Dang Van fatigue property αGpressure coefficient for elastic shear modulus [Pa−1] ατpressure coefficient for limiting shear stress [Pa−1] βtemperature-viscosity coefficient [K−1] βGtemperature coefficient for the elastic shear modulus [K] βτtemperature coefficient for the oil limiting shear stress [K] βDV Dang Van fatigue property [Pa] βeq equivalent reverse torsion stress [Pa] ∆Tmax fmaximum excess temperature of the lubricant [K] ∆Tmax smaximum excess temperature of the surface [K] ∆Vvolume lost by wear [m3] ˙γ∗representative shear strain rate [s−1] ˙γshear strain rate [s−1] Φ heat generation [W·m−3] φTinlet shear heating correction coefficient εcontact ratio of spur gears ηCarreau-Yasuda viscosity parameter ηdynamic low shear viscosity [Pa·s] η∗non-Newtonian viscosity [Pa·s] η0dynamic viscosity at reference temperature T0and atmospheric pressure [Pa·s] η100 dynamic low shear viscosity at 100 ◦C and atmospheric pressure [Pa·s] η40 dynamic low shear viscosity at 40 ◦C and atmospheric pressure [Pa·s] κwear coefficient [Pa−1] Λ specific film thickness λthermal conductivity [W·m−1·K−1] µcoefficient of friction µBDR coefficient of friction in boundary film lubrication µEHD coefficient of friction in EHL νPoisson’s ratio νkinematic viscosity [m2s−1] ν100 kinematic viscosity at 100 ◦C [m2s−1] ν40 kinematic viscosity at 40 ◦C [m2s−1] ρradius of curvature of a gear tooth surface [m] ρdensity [kg·m−3] ρresidual stresses [Pa] xxxiii List of Tables ρ∗Ponter’s fictitious residual stresses [Pa] ˜ρstabilized mesoscopic residual stress tensor [Pa] ρsdensity of the surface material [kg·m−3] ρ15 density at 15 ◦C [kg·m−3] σcombined root mean square roughness [m] σnormal stress [Pa] ˜σmacroscopic stress tensor [Pa] ˜ Σ mesoscopic stress tensor [Pa] σastress amplitude [Pa] [σela] tensor of elastic stress [Pa] [σini] tensor of initial stress [Pa] σmmean stress [Pa] σuultimate tensile strength [Pa] σYtensile yield stress [Pa] τshear stress [Pa] τ∗representative shear stress [Pa] τLlimiting shear stress [Pa] τL0limiting shear stress at reference temperature T0and atmospheric pressure τBDR portion of the tangential contact stress borne by direct surface contact [Pa] τEHD portion of the tangential contact stress borne by the lubricant film [Pa] τMIX tangential contact stress in mixed film lubrication [Pa] τmax maximum shear stress of the mesoscopic stresses [Pa] τoct octahedral shear stress [Pa] θtemperature excess above inlet temperature [K] θavg θaveraged over the film thickness ωangular velocity [rad·s−1] xxxiv 1. General introduction Gears have been used for power transmission since antiquity, and their importance and usefulness has only grown since; so much so that one would be hard pressed to mention a single industry were devices containing gears are not used. Because of the near omnipresence of gear transmissions in mechanical devices, improvements to gears or a better understanding of them could potentially bring benefits to many domains of human endeavour. A typical gear arrangement can consist in a reduction stage in which the driving gear (pinion) is also the smallest and the driven gear (the wheel) is the largest of the pair. The gears are generally oil lubricated: the lubricating oil is used to diminish the coefficient of friction between gear teeth and to evacuate both solid particles and heat from the contact area. The correct understanding of the phenomena arising from contact between gear teeth therefore demands that the following issues be considered: the kinematics of gear tooth meshing, the tooth surface conditions (roughness), the properties of the lubricating oil, the actual lubrication conditions during the meshing between gear teeth, the stresses transmitted from one tooth surface to the other through the lubricating oil layer and the way in which theses stresses act to cause surface damage. The present work is intended to study micropitting on the surface of the tooth flanks of spur gears. According to Hohn et al. [5], it is now the single most limiting factor of gear performance and competitiveness. Micropitting, widely acknowledged to arise through surface contact fatigue of the tooth, materialises as a multitude of microscopic pits, with sizes ranging in the tens of micrometres in width and depth, scattered mostly below the pitch line on the tooth flank surface. This document is divided into three main parts. A first part, entitled “State of the art”, introduces the subject of gear surface damage and attempts to give an overview of the state of the art regarding the many sub-fields whose results must be considered. A second part, entitled “Friction properties of gear oils”, presents results regarding the study of friction in lubricated contacts, be it in full film elastohydrodynamic, mixed or boundary lubrication. A third part, entitled “Gear tooth flank damage” addresses the simulation of surface damage on a spur gear tooth by micropitting and by wear. The first part contains Chapter 2, where the problem of gear surface damage in general and of micropitting in particular is discussed, Chapter 3, which gives an overview of the history and current state of knowledge of lubrication with particular stress on gear lubrication, and Chapter 4, where stresses and fatigue in spur gears are discussed. 1 1. General introduction The second part is composed of Chapter 5, where full film elastohydrodynamic lubrication is studied, in particular with regard to the rheological properties of oils and their coefficient of friction, and of Chapter 6, where mixed and boundary lubrication is studied and a new parameter for the comparison of oils in Stribeck curves is introduced. The third part comprises Chapter 7, which presents models for mixed film lubrication, for micropitting and for wear. It also comprises Chapter 8, where the micropitting model is correlated with a gear micropitting test, and Chapter 9, where the wear model is correlated with the same experiment. The present work was realised mainly at the Faculdade de Engenharia da Universidade do Porto but the experimental work of Chapters 5 and 6 was performed at the Laboratoire de M´ecanique des Contacts et des Structures, INSA de Lyon and the residual stresses measurements of Chapter 8 were performed by Prof. Ant´onio Castanhola Batista at the Centro de Estudos de Materiais por Difrac¸c˜ao de RaiosX, Departamento de F´ısica, FCTUC, Universidade de Coimbra. 2 Part I. State of the art 3 2. Gear wear and gear Micropitting The present work is concerned with micropitting, a type of damage caused by surface rolling contact fatigue. To put it in its proper frame, it is useful to discuss briefly the wider class of gear damages to which it belongs. Contacting surfaces which undergo relative motion are subject to alterations which often receive in the literature the generic designation of “wear”, a catch-all term which often stands for many distinct physical and chemical phenomena. In line with Faure [6], it was decided in the present work to call the generality of surface alterations under load “surface damage” rather than “wear”, keeping the latter term for a specific sub-class of surface damage. Although most types of surface damage have indeed detrimental effects on gear mechanisms, thus earning their name of “damage”, some cases exist in which these alterations are benign, as is the case with running-in. Because rubbing surfaces exist in every mechanism, the study of surface damage is of great technological importance. However, this phenomenon is complex in the extreme, which has prompted researchers to separate wear in different subclassifications to allow easier study, thus following the age-old strategy of “divide and conquer.” Faure [6] offers a catalogue of kinds of surface damage suffered in use by gear tooth flank surfaces, which he calls “tooth flank damage” : wear The progressive removal of matter from the surface of a tooth with use. This term covers a multiplicity of phenomena: running-in New gears, when used with light loads will see their roughness decrease in the first few hours. This has a beneficial effect on gear life. mild wear The unavoidable wear that comes from the mutual sliding of surfaces. It is presumed that it comes about through a combination of brittle fracture, plastic failure and fatigue as roughness features collide. scoring Salient roughness features of one tooth dig into the opposite contacting tooth leaving a score mark. This is not relevant in the case of spur gears because their roughness is markedly anisotropic: it consists in roughness ridges that run across the width of the tooth. adhesive wear Micro-welding or adhesion can occur because of the high contact pressure between the tooth surfaces. When the surfaces separate, material is torn from one surface by the other. Faure includes in this class of damage both hot and cold scuffing. 5 2. Gear wear and gear Micropitting three bodies wear Scratches and abrasion caused by solid particles in the lubricant. corrosion Water or other corrosive agents in the lubricant can react chemically with the gear tooth flank surface, thus corroding it. overheating or burning An excessive temperature in operation can cause an accidental heat treatment that lowers the surface hardness of the gear tooth. erosion by cavitation In some cases, in particular under high alternating loads, the lubricant can cavitate. Implosion of cavitation bubbles and projection of high speed droplets cause shock-waves that produce impact craters with circumferential cracks. Due to the near-instantaneous duration of these events, the material response is very brittle so that no real progressive fatigue is involved. electric erosion Removal of matter through the application of electric arcs resulting from high friction between the surfaces. plastic deformation This covers the permanent deformations of the surfaces caused by excessive contact pressure. contact fatigue Damages caused by the cyclic nature of the loads on the surface of a tooth, further subdivided into: case crushing In surface treated gears, the maximum Hertzian shear stress can sometimes be located below the surface treatment layer. In such cases, a fatigue crack may develop that, on reaching the surface, causes the surface treated layer to be removed. Surface roughness plays little or no part in this since the stresses of a rough tooth at the initiation depth are indistinguishable from those predicted by Hertzian theory. spalling A crack originates at the depth of maximum Hertzian shear and propagates to the surface. In consequence, a flake is removed from the tooth, leaving a large crater of several hundred micrometres in depth and width. As in the case of case crushing, and for the same reason, roughness plays no part in this type of damage. pitting A crack originates from the surface and propagates downwards to a depth of over 100 μm before rejoining the surface, at which point a pit is formed. Since the crack originates on the surface it depends on the complex, roughness induced, surface stress state. micropitting As in the case of pitting, a crack originates from the surface and propagates downward but only to a depth of around 20 μm. Thus, the greatest difference from a pit is its scale. As mentioned above, micropitting consists in cracks originating from the surface of a tooth and rejoining it instead of propagating in depth, thus forming a small 6 Figure 2.1.: Photograph of a micropit in a surface hardened steel gear tooth section. crater or pit. A photograph of a polished sample of a gear tooth that underwent micropitting is displayed in Figure 2.1, where a micropit can be plainly observed. While an ordinary pit or spall may often originate near the maximum shear stress depth in the sub-surface, its shape is similar to that of micropits; the latter differ from those by their extremely small size: while typical dimensions of a pit are of the same order as that of Hertzian contacts, a micropit’s dimensions are of the same order as roughness features, a few micrometres. When performing a longitudinal cut of a gear tooth, the surface fatigue cracks always intersect the cutting plane in well defined directions, as shown in Figure 2.2: they progress downward in the direction opposite to sliding. This is consistant with the widely held view that sliding ~ U2−~ U1is most harmful for the pinion in the direction of rolling ~ U2+~ U1. Micropitting fatigue cracks also follow this pattern, although micropitting spreads preferentially from the pinion tooth surface, near its dedendum where sliding is at its most negative, and only in more extreme cases does it spread above the pitch line. Portions of tooth surface affected by micropitting take a dull grey coloration, which is the reason why micropitting is sometimes called “frosting” or “grey staining” in the gearing industry [7]. Witness to this is Figure 2.3, where the photographed surface of a micropitted tooth flank surface is shown, with added red lines surrounding the micropitted areas of the surface. These dull patches consist in fact in many closely arrayed micropits. Once a micropit is established it may act as a concentrator for larger defects, possibly giving rise to full blown pits: Olver suggested that entrapment of the 7 3. Gear lubrication x y z h(x,y,t)U1 U2 Figure 3.1.: Hydrodynamic lubrication of two surfaces in relative motion. 10. The velocity of the fluid in direction of the thickness wis small compared to the transversal velocity v, in the direction yy, and to the longitudinal velocity u, in the direction xx. 11. The variation of the velocities uand vof the lubricant with zis much greater than the variation of these velocities with the remaining coordinates. 12. There is no slip between the lubricant and the surfaces. It is remarkable that many of the advances made in lubrication science since Reynolds’ days can be mapped to the lessening or the removing of some of the assumptions listed above. Studying cases where assumption 1 is not guaranteed to hold has led to the study of starvation. EHD lubrication was discovered because in highly non-conformal contacts, such as occur in rolling element bearings and gears, lubrication could only be explained by accepting that the lubricated surfaces deform elastically, thus dropping assumption 3, and by realizing that the pressure gradient in the contact is high enough to make assumption 8 untenable, since viscosity varies with pressure. In many cases, particularly that of gears, assumptions 4 and 5 cannot be guaranteed: surface roughness can be large enough to ensure that lubrication veers towards the mixed or even the boundary film lubrication regime, in which part or all of the contact load is born by direct contact between the lubricated surfaces. The hypothesis of a Newtonian behaviour of the fluid (assumption 2) is incompatible with the low friction coefficients observed in highly loaded, nonconformal and lubricated contacts. 14 3.2. Lubricating oil rheology All of the above mentioned aspects are relevant to micropitting, as was explained in Chapter 2 and will be the subject of attention of the remaining of this chapter. 3.2. Lubricating oil rheology 3.2.1. Newtonian viscosity In deriving his equation, Reynolds described the lubricant as Newtonian. The term “Newtonian fluid” was coined because of Newton’s discussion, in the ninth section of the second book of his Principia [20], of a fluid’s resistance to motion that is “proportional to the velocity, by which the parts of the fluid may be separated from each other.” It was in fact Stokes who presented in 1845 [21] a fully thought out description of the Newtonian rheology of fluids and who then derived the Navier-Stokes equations. The mathematical description of the rheology of a Newtonian fluid is: σx=−p+ 2η∂u ∂x +Çζ−2 3ηå·Ç∂u ∂x +∂v ∂y +∂w ∂z å σy=−p+ 2η∂v ∂y +Çζ−2 3ηå·Ç∂u ∂x +∂v ∂y +∂w ∂z å σz=−p+ 2η∂w ∂z +Çζ−2 3ηå·Ç∂u ∂x +∂v ∂y +∂w ∂z å τxy =ηÇ∂u ∂y +∂v ∂xå τyz =ηÇ∂v ∂z +∂w ∂y å τxz =ηÇ∂u ∂z +∂w ∂x å (3.2) This equation gives the stress tensor at any point as a function of fluid pressure p and velocity uin the xdirection, vin the ydirection and win the zdirection. Two properties of the fluid mediate the dependency: shear viscosity η, usually simply called viscosity or dynamic viscosity, and bulk viscosity ζ, generally disregarded. It follows from the equation that the units of viscosity are FL−2T, which become Pa·s in S.I. units. Viscosity is often expressed in centipoise (Cp) with 1 Cp = 10−3Pa·s. In lubrication, the term shear rate is often used for the quantities that follow: ˙γxy =∂u ∂y +∂v ∂x ˙γyz =∂v ∂z +∂w ∂y ˙γxz =∂u ∂z +∂w ∂x (3.3) 15 3. Gear lubrication So that the last three lines of Equation (3.2) can be replaced by: τxy =η˙γxy τyz =η˙γyz τxz =η˙γxz (3.4) Viscosity ηdepends on the thermodynamic parameters of state of the fluid. Any parameter of state can be defined in relation to only two, hence ηcan be defined as a function of any two parameters of state, usually pressure and temperature. The simplest, and to this day still widely used, expression of this dependency is Barus’s, proposed in 1893 [22]: ln ηT,p ηT0,0 =αp −β·(T−T0) (3.5) where ηT,p is the viscosity at pressure pand temperature T,ηT0,0is any known viscosity at ambient pressure and reference temperature T0and αand βare respectively the piezoviscosity and the termoviscosity coefficient, both independent of pressure and temperature. αis usually given in GPa−1and βin K−1. For lubricationg oils, it is found that αand βare always positive in value, which means that viscosity increases exponentially with rising pressure and decreases exponentially with rising temperature. While Barus’s equation is satisfactory for modest variations of pressure and temperature, it becomes inaccurate when used for highly loaded non-conformal lubricated contacts, such as is the case for gears, where pressure variation is in the order of gigapascals. Roelands [23] gathered other researchers’ measurements of the viscosity of many lubricating oils under pressures up to 700 MPa and temperatures up to 200 ◦C and elaborated his formula from their analysis: ln ηT,p ηR =G0 (1 + p/pR)Z (T/TR−1)S0(3.6) where ηT,p is as before the viscosity at pressure pand temperature T, which must be given in kelvin, and where G0,Zand S0are dimensionless material parameters of each oil: as was the case with Barus’s equation, only three material parameters are needed. In particular, the constants Zand S0determine the rate of variation of viscosity with pressure and temperature, respectively. Zand S0are always positive and less than 1 for lubricant oils; hence Roelands’s equation predicts a sub-exponentional increase of viscosity with pressure. Notice the presence in Equation (3.6) of the universal constants ηR= 6.31 · 10−5Pa·s, pR= 196 MPa and TR= 138.15 K. Equations (3.5) and (3.6) are the most widely used in lubrication literature, although other equations have been proposed [24], always with a greater number of material parameters. Bair [25] has been vocal in his criticism of the use of Roelands’s viscosity equation. He argues that since Bridgman’s pioneering measurements of material behaviour 16 3.2. Lubricating oil rheology ln η/η0 p inflection Figure 3.2.: Typical curve of the variation of the logarithm of viscosity with pressure for liquids. under very high pressure [26] it has been observed that for many liquids, a graph of the logarithm of their viscosity against pressure, such as is shown schematically in Figure 3.2, displays a less than exponential increase of viscosity up to a certain pressure—which corresponds to an inflection point in the graph and is usually greater than 1 GPa—a pressure above which the increase of viscosity becomes greater than exponential. The Roelands equation is well suited to describe the viscosity behaviour of these liquids under pressures less than the inflection pressure, but not so for greater pressures. Bair’s proposed solution to this problem is an equation with five parameters that only account for pressure effects on viscosity: ln ηT,p ηi =N p∞−p−M p−p−∞ (3.7) where N,M,p∞and p−∞ are material parameters with units of pressure and ηi is a parameter that is different for each temperature and has units of viscosity. While this expression is an improvement on Barus’s and Roelands’s, it has failed to gain adoption from the tribological community, essentially because it needs viscosity measurements on both sides of the inflection point. It is very challenging to obtain viscosity measurements under high pressures approaching the 1 GPa mark: laboratories were such measuerements are made [27, 28, 29] have had to develop their own high pressure viscometers, since no commercial solution is available. As will be seen in Section 3.4, the piezoviscosity coefficient αis very important in EHD lubrication, since the calculation of lubricant film thickness relies on it. In Barus’s description of viscosity the definition of the piezoviscosity coefficient is straightforward: α=∂(ln η−ln η0) ∂p =ln ηT,p −ln ηT,0 p(3.8) 17 3. Gear lubrication In this case, the piezoviscosity coefficient αis a material constant independent of both pressure and temperature. For more complex descriptions of viscosity, several competing definitions [27] have been proposed: •The tangent piezoviscosity coefficient. α(T) = ∂(ln ηT,p −ln ηT,0) ∂p p=0 (3.9) •The secant piezoviscosity coefficient. α(T) = ln ηT,p −ln ηT,0 p(3.10) where pis a chosen characteristic pressure, so that there are as many definitions of secant piezoviscosity coefficients as there are choices of characteristic pressure. •The reciprocal asymptotic isoviscous piezoviscosity coefficient. 1 α(T)=Z+∞ 0 ηT,0 ηT,p dp (3.11) As indicated by the mathematical notation, every one of these piezoviscosity coefficient definitions makes it dependent on temperature. In Barus’s description of viscosity, every one of these definitions of the piezoviscosity coefficient computes to the same value. This is not so for Roelands’s and most other proposed descriptions of the dependence of viscosity on pressure and temperature. The exact definition to be employed is still debated but most tribologists tend to ignore the debate, acting on the supposition that the differences between the several definitions of piezoviscosity are small enough to be ignored. 3.2.2. Non-Newtonian viscosity The preceding discussion of viscosity has been predicated on the idea that lubricating oils behave in a Newtonian fashion. This is how physicists describe them, as they do most other low molecular weight liquids, and this is largely true in ordinary conditions. However, ordinary conditions are not what is encountered in highly loaded, non-conformal contacts such as between gear teeth: the lubricant will easily have to sustain 1 GPa of pressure in its travel through the conjunction. If its behaviour during the journey were truly Newtonian, and given a nearly exponential increase of viscosity with pressure, very high shear stresses and consequently coefficients of friction would have to be observed in this type of contacts; indeed, lubrication would not be at all effective. This is manifestly not the case: 0.05 is a typical value of coefficient of friction for such conditions. 18 3.2. Lubricating oil rheology This is in essence what Crook [30] observed in his experiment with rollers; in fact the observed viscosity, what he called effective viscosity, fell with increasing shear rate. This behaviour is described by Hamrock [31] as that of a non-Newtonian pseudoplastic liquid. It is more common among the lubrication community to refer to this as shear-thinning behaviour. To distinguish between the Newtonian and non-Newtonian range of behaviour it is common to speak of low-shear viscosity and high shear viscosity, the former referring to the Newtonian viscosity of the oil under relatively low shear rate, the latter referring to viscosity when the effect of shear rate cannot be ignored. Several models of non-Newtonian viscosity have been proposed for oils over the years. Most share a feature in common: they model the non-Newtonian range of behaviour as generalized Newtonian. This means that Equations (3.2) remain valid but the Newtonian viscosity ηis replaced by its high shear counterpart η∗, which, in addition to varying with pressure and temperature, now also depends on the second invariant of the tensor:      ∂u ∂x −1 3∂u ∂x +∂v ∂y +∂w ∂z 1 2˙γxy 1 2˙γxz 1 2˙γxy ∂v ∂y −1 3∂u ∂x +∂v ∂y +∂w ∂z 1 2˙γyz 1 2˙γxz 1 2˙γyz ∂w ∂z −1 3∂u ∂x +∂v ∂y +∂w ∂z       (3.12) From this invariant can be constructed another invariant, a representative shear rate: ˙γ∗=Ã2 3"Ç∂u ∂x −∂v ∂yå2 +Ç∂v ∂y −∂w ∂z å2 +Ç∂w ∂z −∂u ∂xå2#+ ˙γ2 xy + ˙γ2 yz + ˙γ2 xz (3.13) The modified Carreau-Yasuda [27] equation makes use of this invariant: η∗ η= [1 + (λ˙γ∗)a]n−1 a(3.14) where the material constants aand nare dimensionless and λhas units of time. Alternatively, researchers may define the shear rate dependency of non-Newtonian viscosity in terms of viscous stresses, so that η∗is made to depend on the second invariant of the deviatoric stress tensor:    σx+p τxy τxz τxy σy+p τyz τxz τyz σz+p  (3.15) In effect, a representative shear stress is computed: τ∗=s(σx−σy)2+ (σy−σz)2+ (σz−σx)2 6+τ2 xy +τ2 yz +τ2 xz (3.16) It is then found that η∗=η∗(p, T, τ∗). In most proposed models of non-Newtonian behaviour of this type, high shear viscosity is specified in proportion to low shear 19 3. Gear lubrication or Newtonian viscosity as a function of τ∗and of a limiting shear stress which is in its turn a material property depending on pressure and temperature. The Ree-Eyring [32] model is probably the one used most often [33, 34]. η η∗=Çτ∗ τEå−1 sinh Çτ∗ τEå(3.17) Another much employed model is Bair and Winer’s [35]: η η∗=−Çτ∗ τLå−1 ln Ç1−τ∗ τLå(3.18) The latter is not strictly a viscosity model, since it incorporates a hard bound for the shear stress τLwhich corresponds to a limiting plastic behaviour of the oil. 3.2.3. Kinematic viscosity The most common viscometers, capillary viscometers, do not measure a fluid’s Newtonian dynamic viscosity ηbut rather its kinematic viscosity νby timing the fall of a prescribed quantity of lubricant through a tube. The kinematic viscosity is related to the dynamic viscosity through the equation: ν=η ρ(3.19) where ρis the fluid’s density. The kinematic viscosity, with SI units of m2/s is usually given in centistokes ( 1 cSt = 1 mm2/s). Since capillary viscometers are so widespread and simple, manufacturers often supply the kinematic viscosity instead of dynamic viscosity in their oil datasheets. 3.2.4. Elasticity In non-conformal contacts, the time that any given oil molecule spends traveling through the conjunction is measured in μs, an astonishing short time. Because of this, the contribution of an oil’s elasticity to its behaviour may become significant. Johnson and Tevaarwerk [33], as well as Hirst and Moore [34] dealt with this by treating the lubricant as a Maxwell viscoelastic liquid: ˙γxy =˙τxy G+τxy η∗ ˙γyz =˙τyz G+τyz η∗ ˙γxz =˙τxz G+τxz η∗ (3.20) where Gis the liquid’s elastic shear modulus, with units of pressure. 20 3.3. Hetzian contact 3.2.5. Density Density, the mass per unit volume, is a state variable in the thermodynamic sense; an equation of state must hence exist for each material, relating density to pressure pand temperature T. Hamrock [31] recommends the empirical expression: ρ=ρ0Ç1 + 0.6p 1+1.7på(3.21) where pmust be expressed in GPa. 3.3. Hetzian contact 3.3.1. Point contact The Hertzian theory of dry contact between elastic solids, as expounded in the foundational article [36] of 1881, is discussed here. While it may seem odd to consider a theory of dry contact in a chapter about lubricated contacts, much of Hertz’s theory of what is often called “Hertzian contact” is relevant to the study of the lubrication of non-conformal contacts. As a sign of this, one often hears the severity of the loading between contacting gear teeth evaluated in terms of “Hertzian pressure” instead of the actual contact load. This is because the contact pressure distribution in a lubricated, non-conformal contact is often very similar to that which would be encountered in a hypothetical dry contact between the same surfaces. When two convex bodies are brought into contact, they may initially touch at a single point, in which case the application of a contact load will cause the bodies to touch in a finite area, the contact area. This is termed “point contact.” They may instead initially touch along a line, in which case they will deform under load so that they will then touch along a band of finite width. This is termed “line contact.” An example of point contact is that of the balls and tracks of a rolling element bearing. An example of line contact is that of the meshing of a pair of spur gear teeth. Hetzian theory was established in order to determine the shape and size of the contact area, as well as the distribution of pressure within the contact area. It is applicable to cases in which the contact area is small when compared with the dimensions of the bodies so that they may be considered infinite for practical purposes. As a consequence, it is allowable to replace the actual body surfaces in the vicinity of the contact area by quadratic approximations; it is also reasonable to treat the “infinite” bodies as elastic half-spaces. Under these conditions, Hertz found that a point contact’s contact area is bounded by an ellipse centred on the initial contact point. If the contacting bodies are solids of revolution to begin with, then the contact area is a disc with radius agiven by: a=3 3 8 FNR∗ E∗(3.22) 21 3. Gear lubrication where FNis the normal contact force between the bodies and R∗and E∗are respectively the effective radius of curvature and the effective elastic modulus. The effective radius of curvature is obtained from the individual radii of curvature of each undeformed body, R1and R2, at the point of initial contact: 1 R∗=1 2Ç1 R1 +1 R2å(3.23) The effective elastic modulus is obtained from elastic properties of each body: their Young modulus, E1and E2, and their Poisson ratio, ν1and ν2: 1 E∗=1−ν2 1 E1 +1−ν2 2 E2 (3.24) If the origin of a coordinate system is placed at the center of the contact area, with the xy plane coinciding with the common osculating plane of the bodies, the pressure distribution on the surfaces is as follows: p(x, y) =  p0s1−x2+y2 a2when x2+y2< a2 0 otherwise (3.25) The maximum Hertzian pressure, often called simply “Hertzian pressure” is given by: p0=3 2 FN πa2(3.26) 3.3.2. Line contact The previous results pertain solely to point contact, but Hertz’s theory can readily be extended to cover the case of line contact [37]. In such cases, the contact area is approximately shaped as a rectangular band of width 2aalong the length bof the cylinders, except very near their extremities, where this approximation no longer holds. Because of this, it is reasonable to consider that the cylinders are infinite in length and the problem then becomes two-dimensional: pressure and film thickness are constant along the ydirection (paralel to the cylinders) except for the narrow areas near the borders of the bodies. While the shape of the contact area is already known, its half-width aneeds to be computed: a=2 π FN b RX E∗(3.27) where FNis as before the contact load, RXthe effective radius of curvature in the plane of the problem and E∗, the effective elastic modulus, retains the definition given in the previous section. For line contact, the effective radius of curvature is given by: 1 RX =1 2Ç1 RX1 +1 RX2å(3.28) 22 3.4. Elastohydrodinamic lubrication where RX1and RX2are the radii of curvature of the bodies within the plane of the problem. The pressure is distributed along the surface of the bodies as follows: p(x, y) =  p01−Åx aã2 when |x|< a 0 otherwise (3.29) In this case, the maximum Hertzian pressure becomes: p0=4 π FN 2ab (3.30) 3.4. Elastohydrodinamic lubrication In the years following Reynolds’s publication [19] of his hydrodynamic theory, it proved remarkably apt at predicting and explaining the behaviour of such machine elements as thrust and journal bearings. However, its application to machine elements with non-conformal contacts such as cams, gears or rolling element bearings was by no means so successful: for example, the theory predicted a load capacity much below that which was effectively sustained by gears. This remained a puzzling mystery until the 1940s when Grubin [38] conceived that the deformation of the contacting elements could not be neglected under the high loads of non-conformal contacts. Indeed, he would later show that this deformation could exceed many times the film thickness. This is the reason for the term elastohydrodynamic (EHD) lubrication: it concerns contacts that combine hydrodynamic flow of the lubricant with elastic deformation of the contacting surfaces. As Grubin demonstrated, it is not sufficient to describe an EHD contact solely by the Reynolds equation, even in its more general form [31]: 1 12 ®∂ ∂x ñρh3 η∗ ∂p ∂xô+∂ ∂y ñρh3 η∗ ∂p ∂yô´=∂ρh ∂t +∂ ∂x ñU2+U1 2ρhô+∂ ∂y ñV2+V1 2ρhô (3.31) where no suppositions are made about the compressibility of the lubricant and where it is accepted that its viscosity is non-Newtonian and that it varies with pressure and temperature. It is also necessary to take the deformation of the surfaces into account. According to Johnson [37], contacting surfaces can be considered elastic half spaces under most practical conditions. In cases such as depicted in Figure 3.1, where the contact area initiates as a single point (this is the meaning of the term “point contact”), Johnson gives the following formula for the gap hbetween the surfaces after they have been brought into contact and hence deformed: h(x, y, t) = h0(x, y) + δ(t) + 1 πE∗Z+∞ −∞ Z+∞ −∞ p(x−ξ, y −η, t) √ξ2+η2dξdη (3.32) 23 3. Gear lubrication 10−7 10−6 10−5 10−4 10−3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 U · η· FN−1 µ boundary mixed (elasto) hydrodynamic Figure 3.6.: Example of an “ideal” Stribeck curve: the curve progresses from fullfilm (elasto)hydrodynamic, to mixed, to boundary film lubrication as the rolling speed decreases. a boundary layer [60], for their contribution to the coefficient of friction [61] and for their influence on fatigue phenomena [12]. Many researchers have proposed numerical models of mixed film lubrication through the concept of load sharing between oil film and asperities. Naturally, the type and intensity of the roughness are important considerations in these studies. In the early 1970s, Johnson presented a statistical model [62] to determine the load distribution in mixed film lubrication. The idea of load distribution has been particularly fertile, it has more recently been used for the prediction of the coefficient of friction as well as the pressure distribution [63, 64, 65]. Holmes et al. [66, 67], on the other hand, developed a fully transient numerical simulation of mixed film lubricated contacts with real measured roughness. There is also a growing body of work on manufactured surfaces and their effect on mixed and boundary film lubrication [68]. In much of the literature on mixed and boundary film lubrication, it is assumed that the contacting elements are made of steel. Although this is almost always true in the case of cams, bearings and gears, a significant number of applications make use of other materials (e.g. bronze for worm wheels). There have been studies on the use of polymers and other metals under these lubrication conditions [69, 70]. It is surprisingly difficult to find a direct comparison between different oils in mixed or boundary film lubrication conditions. Lafountain et. al. [71] studied the Stribeck curves of several unadditivated base oil blends but concentrated mostly on 30 3.5. Mixed film and boundary lubrication the transition from mixed to full film EHD lubrication. Costello’s study [72] of the effect of basestock and additive chemistry on the Stribeck curve comes close but it emphasizes the design and formulation of the oils. 31 4. Surface stresses and fatigue in rolling/sliding contact of spur gears 4.1. Stresses in a spur gear The stress state of a spur gear tooth flank may be decomposed into residual stresses and load induced stresses: [σ] = [σini]+[σela]+[ρ] (4.1) Initial residual stresses (σini) are permanent, independent of the applied loading and were induced during manufacturing. On the other hand, load induced elastic stresses σela are transient and entirely dependent on the loading. Finally, residual stresses ρare due to permanent plastic deformation induced by the loading history and account for plastic yield and stress variations at microscopic level. 4.1.1. Residual Stresses During its production process, a typical spur undergoes cutting, a surface treatment at high temperature, such as carburization, nitruration or carbo-nitruration, and grinding. All of these steps induce plastic deformations that translate into important initial residual stresses σini, which attain easily values in the order of several hundred MPa. On the other hand, loading may also induce permanent plastic deformations within a gear tooth that give rise to residual stresses ρdue to the incompatibility of the displacement field upon unloading. It is well understood that these residual stresses can be of great importance in gear contact fatigue. This was shown by Batista [73] who systematically measured residual stresses in gears (both initial residual stresses and those induced by loading) by the method of X-ray diffraction. This method consist in shooting X-rays toward the surface of a gear tooth, which has previously been cut from its gear, and measuring their angle of diffraction. This angle gives information about the distortion of the crystalline mesh of the material within a very thin layer under the surface. This allows the estimation of the deformation of the solid under study and hence, through the use of Hooke’s Law, of the residual stresses. In the case of steel irradiated with Cr-KαX-rays, the first 5 μm or so of depth are penetrated 33 4. Surface stresses and fatigue in rolling/sliding contact of spur gears τ p xx zz Figure 4.1.: Elastic half-space coordinates and surface loads: pis the surface pressure, τis the tangential surface traction. 4.1.2. Elastic stresses As explained in Chapter 3, gear teeth are generally considered to be elastic half spaces for the purpose of computing contact stresses and displacements of the tooth flank surfaces. In particular, spur gear tooth flanks, with their negligible roughness in the direction that is transversal to rolling, can be considered half-spaces in plane strain. The same holds true when computing elastic stresses in the surface and subsurface of a gear tooth flank. It may be useful to introduce the concepts of surface and subsurface at this point. Sub-surface means the volume of tooth that lies at a depth less than the Hertzian radius (or half-width) of contact. Surface means the volume in the immediate vicinity of the actual boundary, that is to say the volume within the first 30 μm or so. Pitting is mostly associated with fatigue cracks originating from the deeper parts of the sub-surface, micropitting with fatigue cracks in the surface volume. A spur gear can then be considered an elastic half space in plane strain loaded by tangential and normal tractions as in Figure 4.1 where the pressure pand tangential traction τare constant in the yy direction and where the boundary coincides with plane xy and the tooth lies in the positive direction of zz. Johnson [37] gives the formulas for computing the stresses in such cases: σx(x, z, t) = −2 πZ+∞ −∞ p(ξ, t)(x−ξ)2z Ä(x−ξ)2+z2ä2+τ(ξ, t)(x−ξ)3 Ä(x−ξ)2+z2ä2dξ (4.2) σz(x, z, t) = −2 πZ+∞ −∞ p(ξ, t)z3 Ä(x−ξ)2+z2ä2+τ(ξ, t)(x−ξ)z2 Ä(x−ξ)2+z2ä2dξ (4.3) τxz (x, z, t) = −2 πZ+∞ −∞ p(ξ, t)(x−ξ)z2 Ä(x−ξ)2+z2ä2+τ(ξ, t)(x−ξ)2z Ä(x−ξ)2+z2ä2dξ (4.4) σy(x, z, t) = ν(σx+σz) (4.5) τxy =τyz = 0 (4.6) where Eis the gear material Young’s modulus and νits Poisson’s ratio. 34 4.1. Stresses in a spur gear 0.05 0.05 0.1 0.1 0.15 0.15 0.2 0.25 x/a z/a p/p0 -3 -2 -1 0 1 2 3 5 4 3 2 1 0 0.5 1 1.5 2 τoct /p0 Figure 4.2.: Ratio τoct/p0of the octahedral stress to the maximum Hertzian stress in a tooth submitted to a Herzian pressure distribution and a coefficient of friction 0.05. x/a z/a p/p0 0.05 0.1 0.15 0.2 0.25 0.3 0.2 0.15 0.1 0.05 0.35 0.15 0.1 0.2 0.5 0.15 0.2 0.2 0.25 0.25 0.35 0.55 -3 -2 -1 0 1 2 3 5 4 3 2 1 0 0.5 1 1.5 2 τoct /p0 Figure 4.3.: As in Figure 4.2 but with rough surfaces. Figures 4.2 and 4.3 contain examples of stress distribution within the sub-surface of a tooth flank directly under the loaded part. The octahedral shear stress τoct is an invariant of the stress tensor and is proportional to the elastic deformation energy. It figures prominently in the Von Mises yield criterion and is therefore reasonably representative of the overall stress state in a material point. Figure 4.2 contains a contour map of the octahedral stress field (in proportion to the maximum Hertzian pressure) caused by a contact between perfectly smooth spur gear tooth flanks, which correspond approximately to a Hertzian pressure distribution. By contrast, Figure 4.3 shows the same information with regard to the contact between rough surfaces. It is noticeable that the subsurface stresses are similar but that considerable differences arise in the surface, where the rough contact causes complex and localized peaks (secondary maxima) in the stress field. 35 4. Surface stresses and fatigue in rolling/sliding contact of spur gears This is very typical and is one of the reasons why so much attention must be given to mixed film lubrication in connection with tooth surface damage of gears. 4.2. Elastic shakedown in rolling and sliding contact The loads on a gear tooth surface occur cyclically. Consider, for example, a tooth of the driving gear: during each rotation of the gear, the tooth meshes exactly once with a tooth of the opposing gear. As the meshing progresses, the accompanying surface pressure and traction travel along the surface of the tooth flank. This cycle is repeated each time the tooth completes a rotation and enters again into contact with a tooth of the opposing gear. How the stresses in the tooth, which undergoes rolling and sliding contact, evolve under this cyclical loading depends on the intensity of the loads. The possible outcomes have been described by Foletti and Desimone [74]: •The elastic limit is not exceeded and once the loads are removed the stresses go back to their initial level, dictated by the initial residual stresses. •The loads initially induce plastic deformation but, as cycle follows cycle, the plastic deformation per cycle decreases until elastic shakedown is attained: no new plastic deformation is produced per cycle. At that point, a new residual stress field has been induced, and the loads merely cause elastic stresses which disappear once the load is removed. •Plastic shakedown occurs, each new cycle causes plastic deformation, but the total plastic deformation is bound. •Ratcheting occurs, each new cycle causes plastic deformation, and the total plastic deformation increases without bounds. Naturally, operating conditions conducive to plastic shakedown or ratcheting lead to premature failure of the component and elastic shakedown is therefore a necessary condition for a long life. Two methods may be employed to determine the elastic shakedown limit. The computationally costly method is to numerically simulate the evolution of the plastic stresses step-by-step as cycle upon cycle is applied, until convergence to pure elastic stresses is attained (or not, as the case may be). The fast method is to use the so-called shakedown theorems, which allow the determination of the converged stress state without the need to compute intermediate stress states. There are two kinds of shakedown theorems. The first kind are the kinematic shakedown theorems, chief among them Koiter’s theorem [75]. The second kind are the static shakedown theorems, which have been much more widely used in rolling contact theory than the kinematic kind. In the case of elastic-perfectly plastic materials, Melan’s static theorem [76], applies: “If any system of self-equilibrating residual stresses ρij can be found which, 36 4.2. Elastic shakedown in rolling and sliding contact Figure 4.4.: Shakedown map of a Hertzian line contact taken from Williams [2]. in combination with the stresses due to the repeated load σij , do not exceed yield at any time, then elastic shakedown will take place.” In the case of elastic-linear kinematic hardening materials, materials whose yield surface both expands and shifts in stress-space, Ponter’s theorem [77] applies: “If any system of fictitious residual stresses ρ∗ ij can be found which, in combination with the stresses due to the repeated load, σij do not exceed yield at any time, then elastic shakedown will take place.” Note that ρ∗ ij need not be self-equilibrating. Johnson [78] studied the simple case of a line contact with a Hertzian pressure distribution and no tangential traction resulting from the rolling of a cylinder on an elastic-perfectly plastic half-space. By applying Melan’s theorem, he was able to determine the range of possible ratios p0/k, the ratio of maximum Hertzian pressure p0to yield stress in shear k, with which elastic shakedown would occur. The lower bound of this range is the onset of yielding situated at p0/k ≈3 and the upper bound is elastic shakedown limit, situated at p0/k ≈4. Johnson and Jefferis [79] later extended these results to line contacts with both Hertzian pressure distribution and coefficient of friction µ. The elastic shakedown bounds are different for each value of coefficient of friction, which is the reason why they devised shakedown maps. Such a shakedown map was obtained by Williams [2] and is shown in Figure 4.4. Its abscissa represent the coefficient of friction µand its ordinate the ratio p0/k. The plane thus defined is cut by lines that divide it into areas corresponding to the regimes of stress evolution already mentioned: 37 4. Surface stresses and fatigue in rolling/sliding contact of spur gears •below curve A, yielding occurs nowhere in the body; •between curve A and curve B, elastic shakedown occurs, whether the material is elastic-perfectly plastic or kinematic strain hardening; •between curve B and curve C, repeated plastic deformation occurs for elasticperfectly plastic materials, but elastic shakedown occurs for kinematic hardening materials; •above curve C, repeated plastic deformation occurs for both types of material. Naturally, a machine element operating under conditions corresponding to a point above line C in the shakedown map will have a short life, either because of fracture or wear [2]. Conversely, a long life is expected of machine elements whose operating conditions are situated below the elastic shakedown limit. Oil lubricated gears operate with coefficients of friction smaller or equal to 0.15 under normal operating conditions, hence only the leftmost part of the figure is of interest for lubricated spur gears. The shakedown limit was similarly determined for cases of elliptical Hertzian contact by Hills and Sackfield [80] and by Ponter et al. [81]. Research into shakedown analysis of rolling contact is still very active, in part because of its relevance to the railways industry [82]. Some authors have sought to improve the methods for getting at the converged stress cycle in the classic cases of Hertzian line and elliptic load [83, 84]. Others have extended the method to include the use of yield functions more general than the Von Mises criterion [85, 86] Shakedown analysis has also been applied to Hertzian contacts of anisotropic or inhomogeneous materials [87, 88, 89, 90]. It is important to note that the works above cited all deal with cases of Hertzian pressure distribution on the surface which is tantamount to saying that the surfaces in contact were considered ideally smooth. It was shown in Section 4.1 that the stresses near the rough surface of a tooth flank are very different from those of a smooth surface because of the difference in contact pressure distribution, although the principle of Saint-Venant ensures that stresses in the subsurface are mostly insensitive to surface roughness. Hence, the shakedown limits computed under the assumption of a Hertzian pressure distribution are inadequate for evaluating the severity of the loading on material points in the surface. 4.3. Metal fatigue 4.3.1. W¨ohler curve Fatigue, the failure of mechanical structures or parts under repeated loading, has been studied since the first half of the 19th century, which witnessed the explosive expansion of railway lines throughout the western world. This kind of large scale industrial endeavour brought to the attention of engineers a number of problems 38 4.3. Metal fatigue Figure 4.5.: S-N diagram taken from Moore [3]: a) linear scale; b) semilogarithmic scale. that had hitherto been ignored. Starting in the 1840s, W¨ohler [91, 92, 93, 94] performed extensive fatigue tests trying to make sense of the premature failure of train axles operating under conditions well within static strength limits. His specimens were tested under simple cyclic load cases such as rotational bending, fully reversed bending, rotation or axial load. He arrived at a number of conclusions about wrought iron and steel behaviour under fatigue that still stand to this day: •A specimen will fail under stresses less than the elastic limit if the loading is repeated a sufficient number of times. •There is a limit for the amplitude of stress under which its repetition will never cause rupture, which is called the fatigue limit or the endurance limit. Note that this is no true of all metals: aluminium, for example, has no such limit and will fail at any stress level if subjected to enough load cycles. •The limiting stress amplitude diminishes with the increase of the average stress. W¨ohler compiled his data in tabular form, it was left to others to devise what is now known as W¨ohler curves, otherwise called S-N curves. An example taken from 39 4. Surface stresses and fatigue in rolling/sliding contact of spur gears dislocations within crystals, the mesoscopic scale of crystal grains and the macroscopic scale of the studied component as a whole, with typical distances of: scale typical lengths physical domain macroscopic 10−3m nominal geometry mesoscopic 10−6m crystal grain microscopic 10−10 m inter-atomic distance At the macroscopic scale, the material properties vary smoothly and the material is continuous. At the mesoscopic scale, the material properties cannot be said to vary smoothly because of the difference in orientation of the grains. Nevertheless, the material may still be considered continuous because a grain contains a sufficiently large number of atoms and it thus still makes sense to talk of stresses and strains. At the microscopic scale, the material is an aggregate of atoms, and no continuity of any kind exists. The stresses discussed up to this point are all macroscopic stresses. On the other hand, it was seen in Section 4.3.2 that the initiation of a fatigue crack is widely held to be a consequence of the nucleation of dislocations within crystal grains. It is therefore necessary to be able to evaluate the mesoscopic stresses. It must be said that the macroscopic stresses may be seen as an average of the mesoscopic stresses over a representative volume element (RVE) of the size of many grains. Hence, the mesoscopic stresses are the sum of the macroscopic stress and of a perturbation due to the difference in elastic properties caused by the difference in grain orientation, phase etc... In the general case, the mesoscopic stress tensor is related to the macroscopic stress tensor by the expression: ˜ Σ = ˜ ˜ A: ˜σ+ ˜ρ(4.16) where ˜ Σ is the mesoscopic stress tensor, ˜σthe macroscopic one, ˜ρis the stabilized mesoscopic residual stress tensor, constant in time, and ˜ ˜ Ais the fourth order localization elastic tensor. Dang Van [120], proposed a method of obtaining the mesoscopic stresses based on the elastic shakedown concept without computing the localization tensor. As he argued, it is possible to eliminate the localization tensor from Equation (4.16) by making a few reasonable assumptions. Namely, it is assumed that at least one grain, within the RVE centred on each material point considered, is so unfavourably oriented that it will slip under the stresses and that the material suffers isotropic and kinematic hardening at the mesoscopic level. Finally, it is supposed a priori that the material around each point will shakedown elastically at the local mesoscopic level, in a manner similar to global elastic shakedown. Thus Equation (4.16) reduces to: ˜ Σ = ˜σ+ ˜ρ(4.17) 46 4.3. Metal fatigue This happens because, as the cycle progresses, the yield surface grows and shifts to encompass all the stress states through which the considered material point has travelled. Once more, this is valid because of the assumption that elastic shakedown does occur locally—this is similar to conducting elastic calculations to check for yield—and that the material, at the mesoscopic scale, undergoes isotropic and kinematic hardening—a very general description of the plastic behaviour of a material and therefore a very reasonable one. In these circumstances, the stabilized mesoscopic residual stress tensor ˜ρis simply the fictitious residual stress tensor of Ponter’s theorem already discussed in Section 4.2. The yield criterion used is the Von Mises criterion, and therefore the yield surface is a hypersphere in the axes sxx/√2, syy/√2, szz/√2, sxy,syz,sxz, whose radius expands—this is isotropic hardening—and whose centre moves— this is kinematic hardening. The final position and radius of the yield hypersphere is then such that the hypersphere is the smallest one that encompasses all the stress states in the cycle for the point under consideration. The stabilized mesoscopic residual stress associated with the local shakedown (˜ρ) is then the centre of the yield sphere. From this process of obtaining ˜ρit follows necessarily that it is a purely deviatoric stress tensor. Mathematically, the yield limit is determined by solving the optimization problem: K2= min ˜ρ0Åmax tÄ−J2(˜σ(t) + ˜ρ0)äã ˜ρ:K2= max tÄ−J2(˜σ(t) + ˜ρ)ä(4.18) This is the well known geometric problem of the “smallest enclosing ball.” As an illustration of the principle, consider a body in which the external loads only induce pure shear stress, with all the stress components null except σxz and σyz, at a given point in the body. The macroscopic elastic stress history of the point in the body may be conveniently represented in two dimensions, as in Figure 4.8. Because of this the yield surface collapses into a circle. At the start of the cycle, the stress is equal to an initial stress σ0=−ρ0and the yield radius is the initial one. The cycle progresses to instant t1, at which the elastic stress tensor is σ1. Because in reaching this instant, the elastic stress tensor has moved outside the initial yield surface, it has pulled it along so that the yield surface centre has shifted to ρ1and its radius has dilated to R1in order to envelop both σ0,σ1and every intermediate stress state. The very same thing happens when the cycle progresses to σ2. Finally, when the full cycle has been gone through, the yield surface settles into its final shape and position and need no longer change with the application of further cycles. Dang Van multi-axial high-cycle fatigue criterion The mesoscopic stresses evaluated by the method presented in the previous section are those necessary to ensure the existence of elastic shakedown locally—in fact, 47 4. Surface stresses and fatigue in rolling/sliding contact of spur gears σyz σxz R0 −ρ0−ρ1 −ρ2 −ρ R1 R2 R σ(1) σ(2) σ(3) Figure 4.8.: Stress cycle and hardening of a material point in pure shear stress. to ensure near infinite life to fatigue. Whether the material has the capacity to sustain these stresses or not is another matter. Since a fatigue crack in its initial stage usually propagates along a plane of maximum shear stress, this stress is a relevant parameter of the initiation of a fatigue crack. On the other hand, a negative hydrostatic stress—a hydrostatic pressure—has been observed to benefit the resistance to fatigue of materials by closing cracks, while a positive hydrostatic stress causes the reverse effect. From these considerations, Dang Van formulated [120] the simplest possible criterion that relates these parameters: τmax +αDV ·pH≤βDV (4.19) where τmax and pHare the maximum shear stress and the hydrostatic stress—not pressure—of the mesoscopic stress tensor and αDV and βDV are fatigue material properties that can be derived from the fatigue limit in fully reversed torsion (t−1) and alternating bending tests (f−1) as follows: αDV = 3 Çt−1 f−1−1 2å(4.20) βDV =t−1(4.21) To avoid crack initiation at some material point, Equation (4.19) must be true at all times for that material point’s stress cycle. Any point where this is not the case must eventually be the origin of a fatigue crack if sufficient cycles are applied. Another interpretation of the criterion can be given from the manner of obtaining βDV . One can think of the stress cycle at each point as equivalent to a reversed torsion stress cycle such that its maximum shear stress is: βeq = max t(τmax +αDV ·pH) (4.22) 48 4.3. Metal fatigue βDV βDV αDV αDV ❢❛t✐❣✉❡ ✐♥✐t✐❛t✐♦♥③♦♥❡ τmax +αDV·pH>βDV τmax pH 0 s❛❢❡t②③♦♥❡ Figure 4.9.: Position of the mesoscopic stress state of a material point during a load cycle on the pH/τmax plane. The shaded half-plane represents the area where the Dang Van criterion is violated. Thus, the parts of the cycle placed in the shaded area violate the criterion. Thus, fatigue cracks do not initiate if this equivalent shear stress is less than βDV . βeq ≤βDV (4.23) It is expected that the crack will propagate initially in the direction of maximum mesoscopic shear stress corresponding to βeq. Because the maximum shear stress occurs in two mutually perpendicular planes, the orientation of the crack as it initiates can be either one, or even both. As an example, Figure 4.9 shows the position of the mesoscopic stress state of a material point during a hypothetical load cycle on the pH/τmax plane. It is seen that the path of the stress crosses the straight line delimiting the safety zone. 49 Part II. Friction properties of gear oils 51 5. Traction curves and rheological parameters for gear oils 5.1. Preamble It was experimentally shown by Martins et al. [123] that the choice of lubricant can have a great influence on gear micropitting. One way in which this influence can be wrought is through the EHD friction properties of the lubricant, which translates into a greater or smaller tangential traction on a gear tooth flank surface, in turn favouring or inhibiting surface contact fatigue damage. To be able to simulate the gear EHD friction two things are needed: a numerical model for computing the friction from the operating conditions, including a rheological model of the lubricants, and the rheological properties that must be used with the model. The second requirement is in practice difficult to fulfill for most oils because the only properties readily available are the kinematic viscosity and density usually displayed in their manufacturer’s datasheets. The only avenue open to the researcher is then to conduct traction experiments on the oil under study and correlate the results with the friction model in order to deduce the rheological properties of the oil. This is the route taken in the present instance, where a set of six different, fully formulated oils were studied. The experimental part of this study took place at the Laboratoire de M´ecanique des Contacts et des Structures, INSA de Lyon. The experimental procedure and results are first presented, then followed by the EHL friction model and the deduction of the rheological properties of each oil. 5.2. Experimental procedure 5.2.1. Tested oils In order to get an overview of the gear oils generally available in practical environments, a set of fully formulated gear oils, each complying with DIN 51517 part 3 (CLP) standard, was selected: •2 paraffinic mineral oil, with references M1 and T1; •1 poly-alphaolefin oil, with reference P1; •2 fully saturated ester based oils, with reference E1 and E3; 53 5. Traction curves and rheological parameters for gear oils Table 5.1.: Properties of the gear oils Gear oil T1 M1 P1 E1 E3 E2 Chemical content Zn (ppm) n/a ≈0 n/a ≈0 n/a ≈0 Ca (ppm) n/a 40 n/a ≈0 n/a ≈0 P (ppm) n/a 175 n/a 146 n/a 300 S (ppm) n/a 15040 n/a 180 n/a 5500 Biodegradability and toxicity (standards OECD 101, 202, 301 F) Ready biodegradability (%) n/a <60 n/a ≥60 n/a ≥60 Aquatic toxicity with Daphnia EL50 (mg/l) n/a >1000 n/a >100 n/a >100 Aquatic toxicity with Algae EL50 (mg/l) n/a >100 n/a >100 n/a >100 Density at 15◦C ρ15 (kg/m3) 900 897 863 925 932 955 Kinematic viscosity at 40◦Cν40 (cSt) 231.1 150 150 99.4 108.3 114.5 Kinematic viscosity at 100◦Cν100 (cSt) 18.9 14.6 19.4 14.6 15.9 17.0 Viscosity index V I 90 96 148 152 156 162 Piezoviscosity according to Gold et al. [124] α0.2 GPa (GPa−1) at 40 ◦C 21.10 19.87 14.41 12.35 12.49 12.59 α0.2 GPa (GPa−1) at 100 ◦C 14.90 14.38 10.97 9.51 9.62 9.71 •1 highly saturated ester based oil, with reference E2. M1 is a commercial, paraffin based oil with significant residual sulphur content, as shown in Table 5.1. It was formulated with an additive system designed to provide protection against conventional wear modes such as scuffing as well as micropitting fatigue. E1 and E2 were formulated with fullyand highly-saturated esters, respectively, both highly biodegradable. The lack of unsaturated bonds in these base oils leads to an excellent thermal and oxidative stability. The additives used in these oils were selected for low toxicity and environmental compatibility without, however, sacrificing gear performance. A further characteristic of oil E2 is that a very high viscosity ester oil (1000 cSt at 40◦C) was mixed into the base ester: the base ester oil totals 90% of the oil volume and the high viscosity ester, 5%. This brings to mind the way in which bright stock is used to improve mineral oils. It is important to note that oils E1 and E3 were formulated with the same basestock but distinct additive packages. Conversely, oils E3 and P1 were formulated from diverse chemical bases but share the same additive package. Table 5.1 displays some of the properties of the lubricants that can be extracted from the manufacturers’ data sheets. The piezoviscosity coefficient according to Gold et al. [124] is also included. It is interesting to note that, while no particular 54 5.2. Experimental procedure Figure 5.1.: Mini-traction machine. relation between the viscosities and the base oil types can be discerned, large differences in the piezoviscosity coefficient roughly correspond to changes in base oil types. 5.2.2. Experimental setup The traction tests were performed on a mini traction machine (MTM), whose schematic representation is shown in Figure 5.1. The basic operating principle of the MTM machine consists in pressing a 1/2 ” diameter ball against a steel disc, both made of AISI 52100 steel. The ball and the disc rotate independently, thus permitting the occurrence of rolling and sliding in the contact. A wide range of rolling speeds and slide-to-roll ratio may be selected. The machine performs each traction measurement twice in succession: once with the ball surface moving faster than the disc surface and again with the disc surface moving faster. The normal contact load and the oil bath temperature (considered to be the same as the oil inlet temperature) can also be selected. The machine can thus be made to measure both the contact friction load and the coefficient of friction that correspond to a set of operating conditions defined by the following parameters: •the inlet oil temperature T0; •the normal contact load FN(or, which is exactly equivalent, the Hertzian contact pressure p0); 55 5. Traction curves and rheological parameters for gear oils Figure 5.5.: Friction coefficient µvs. LP parameter at p0= 1 GPa, U= 1 m/s, SRR = 0.4. Each ellipse groups data points from tests performed at the same inlet temperature. The fact that the grouping by traction curves more or less follows the grouping by base oil types shows that the influence of the base oil must be acknowledged to be crucial. The fully formulated gear oils used here illustrate the point: additives make a difference to the traction curves, but the base oil type has at least an equal influence under full film EHD lubrication conditions. Another interesting point is that the traction curves grouping does not respect the values of kinematic viscosity as shown in Table 5.1. The properties that correlate well with the grouping are the viscosity index and the piezoviscosity coefficient. Thus it appears that the variation of viscosity with pressure and temperature is more important than the actual values of viscosity. Figures 5.2–5.4 also allow the observation of the influence of the operating conditions on the traction curves: it can be seen that an increase in the normal load causes an increase in the coefficient of friction. On the other hand, an increase in oil temperature causes a decrease in the coefficient of friction. 62 5.4. Simplified model for the EHD lubrication of a circular point contact. Table 5.4.: Constants for piezoviscosity calculation T1 M1 P1 E1 E3 E2 s 9.904 9.904 7.382 6.605 6.605 6.605 t 0.139 0.139 0.1335 0.136 0.136 0.136 5.4. Simplified model for the EHD lubrication of a circular point contact. 5.4.1. Low shear viscosity In the rheological model used here, it is assumed that the low shear viscosity follows the Roelands [23] equation, Equation (3.6), reproduced here in a slightly different form: ln η η0 = (ln η0+ 9.67) ×(ÇT−138 T0−138å−S0Å1 + p 0.196ãZ −1)(5.2) where η0is now the dynamic viscosity in Pa·s at the reference temperature and atmospheric pressure. The material constants η0,Zand S0can be deduced from the properties listed in Table 5.1, by using the method here described. It follows from Equation (5.2) that three dynamic viscosity measurements would be needed to determine the material parameters Zand S0. However, the only available measured properties relating to viscosity were the kinematic viscosity at 40 ◦C (ν40) and 100 ◦C (ν100) and the density at 15 ◦C (ρ15). The density at 40 ◦C (ρ40) and 100 ◦C (ρ40) can be evaluated by applying the following expression: ρ(T) = ρ(T0)·Ç1−T−T0 1250 å(5.3) Then, from the definition of kinematic viscosity, one can obtain the dynamic viscosity: η=ρν (5.4) This gives the dynamic viscosity at 40 ◦C (η40) and 100 ◦C (η100), from which the material parameter S0can be deduced by substituting in Equation (5.2) the known values of viscosity: ln η100 η40 = (ln η40 + 9.67) ·¶1.34256−S0−1©(5.5) For the determination of the Zmaterial parameter, at least one measurement of dynamic viscosity at non-atmospheric pressure would be needed or, alternatively, at least one measured value of piezoviscosity coefficient. Unfortunately, neither measurement was available during the elaboration of the present work. The piezoviscosity coefficient was evaluated by following the method described in an article by Gold et al. [124], where an empirical formula for the determination 63 5. Traction curves and rheological parameters for gear oils of the piezoviscosity coefficient α0.2was developed: α0.2=s·νt(5.6) where sand tare numerical constants that depend on the basestock of the oil and are given in Table 5.4; and where α0.2is defined as: α0.2=ln η(T, 0.2 GPa) −ln η(T, patm)] 0.2 GPa (5.7) In the context of the Roelands equation, this becomes: α0.2=(ln ηT+ 9.67)(2.02041Z−1) 0.2 GPa (5.8) Since ν40 and ν100 are available, it is possible to evaluate α0.2at 40 ◦C and 100 ◦C by applying Equation (5.6). Two different values of Zare therefore obtained, Z40 and Z100, one for each measurement of kinematic velocity; and although they are always similar, as can be seen in Table 5.5, they are not exactly the same: this is in contradiction with the definition of Zby Roelands as a material parameter approximately independent of pressure and temperature conditions. The Zchosen here is an average of the Z40 and Z100 values, defined as follows: 2.0204Z=2.0204Z40 + 2.0204Z100 2(5.9) The resulting values of Zand S0are shown in Table 5.6. 5.4.2. Non-Newtonian viscosity It has already been observed in Section 3.2.2 that knowledge of the low shear viscosity is insufficient to fully determine the rheology of an oil within an EHD contact, because the extreme conditions lead to a non-Newtonian behaviour of the lubricant. The oil was modelled as a viscoelastic Maxwell liquid (see Equation (3.20)), with the viscous part corresponding to Bair and Winer’s visco-plasticity equation [35] (see Equation (3.18)): ˙γ=d dt τ G+τ η·−ln (1 −|τ/τL|) |τ/τL|(5.10) Table 5.5.: Alternative values of Z T1 M1 P1 E1 E3 E2 Z40 0.5975 0.5953 0.4568 0.4160 0.4156 0.4145 Z100 0.6133 0.6196 0.4764 0.4364 0.4342 0.4313 64 5.4. Simplified model for the EHD lubrication of a circular point contact. Table 5.6.: Roelands low shear viscosity parameters. oil T1 M1 P1 E1 E3 E2 η0(mPa ·s) at 80 ◦C 31.2 23.0 27.5 21.5 23.6 25.8 Z0.605 0.607 0.467 0.426 0.425 0.422 S01.29 1.27 1.09 1.07 1.06 1.04 As was the case in [129], it is supposed that the dependency of the limiting shear stress τLfrom pressure and temperature follows an exponential law: ln τL τL0 =ατ·p+βτ·Ç1 T−1 T0å(5.11) where τL0is the limiting shear stress at atmospheric pressure and reference temperature T0,ατand βτparameters that account for the influence of pressure and temperature, respectively. Likewise, in the case of the elastic shear modulus, it is supposed that [130]: G= (G0+αG·p)·exp ñβG·Ç1 T−1 T0åô (5.12) where G0is the elastic shear modulus at atmospheric pressure and reference temperature T0, and αGand βGparameters that account for the influence of pressure and temperature. Parameters τL0,ατ,βτ,G0,αGand βGfully characterize the rheology of a lubricant oil subject to the above equations. With a set of these parameters and applying the EHD model described in the following sections, it is possible to compute the coefficient of friction resulting from an EHD contact and, in turn, to compare this prediction with a test conducted under the same operating conditions. 5.4.3. Friction shear stress Calculation of the coefficient of friction in full film EHD lubrication follows the method already developed by Seabra et al. [131], Sottomayor et al. [132, 133] and Campos et al. [129]. For this reason, the underlying EHD model will only be briefly described. The EHD lubrication model comprises the following, Grubin like, simplifications illustrated by Figure 5.6. •The contact is an almost Hertzian circular contact: the film thickness is constant above the Hertzian contact area, a disc with a radius computed from Equation (3.22), and an elliptic pressure distribution with a maximum equal to the maximum Hertzian pressure of Equation (3.26). Equations (3.22) 65 5. Traction curves and rheological parameters for gear oils Figure 5.6.: Film thickness hand contact pressure paccording to the simplified EHD lubrication model. and (3.26) are reproduced here, slightly manipulated to take into account the fact that the contact is between a sphere and a plane of the same material: a=3 s3 2RFN 1−ν2 E(5.13) p0=3 2 FN πa2(5.14) p(x, y) =  0 ; x2+y2> a2 p0q1−(x/a)2−(y/a)2; otherwise (5.15) where Eis the Young modulus, νthe Poisson’s ratio and Rthe sphere radius. •The film thickness is computed from Hamrock and Dowson’s formula for central film thickness of a point contact [125]. It is reproduced here in a slightly different form than in Equation (3.40), again to take into account the fact that the conjunction is spherical and that both the disk and ball are made of the same material: h0= 1.93 ·(η0U)0.67 α0.53R0.464 F0.067 NÇ1−ν2 Eå0.073 (5.16) The film thickness is also corrected due to inlet shear heating according to Ghohar’s prescription [126]: 1/φT= 1 + 0.243 ·Ä1+8.33SRR0.83ä·Çβη0U2 Kfå0.64 (5.17) h0c=h0φT(5.18) where βis the thermoviscosity coefficient of the oil and Kfits thermal conductivity. 66 5.4. Simplified model for the EHD lubrication of a circular point contact. •The only components of the shear strain field considered are those in the xz plane, all other components are disregarded. Additionally, the shear strain is considered constant within the Hertzian contact area and negligible elsewhere: ˙γ= ˙γxz =U2−U1 h0c =SRR U h0c (5.19) •The only components of the shear stress field considered are those in the xz plane, all other components are disregarded. Additionally, the shear stress is considered constant along the film thickness: τ=τxz =τ(x, y) (5.20) With these simplifying assumptions, the determination of the coefficient of friction becomes the determination of the shear stress in mutually isolated “slices”, parallel to that shown in Figure 5.6, by the solution of Equation (5.10). This equation is rewritten here considering that the contact is in steady state and that, as a simplifying assumption, the oil flows at a uniform speed equal to the rolling speed U: ˙γ=U·∂ ∂x τ G+τ η·−ln (1 −|τ/τL|) |τ/τL|(5.21) The boundary condition is that the shear stress in the upstream boundary (the left side in Figure 5.6) is zero. The differential Equation (5.21) is solvable as long as the oil properties η,Gand τLwithin the contact are known. In turn, Equations (5.2), (5.11), (5.12) demand that both the pressure and temperature fields be known. 5.4.4. Temperature within an EHD film. Calculation of the temperature field is necessary to determine the lubricant properties within the contact and hence to determine the coefficient of friction. The method used here differs significantly from that in the works cited in the previous Section [131, 132, 133, 129]. For this reason, it will be described in some detail. It is assumed that the heat transfer to and from the contact area is exclusively effected by conduction in the zz direction and convection in the xx direction, in the nomenclature of Figure 5.7. This means that, as was the case with the shear stress and strain, each plane parallel to xz, or slice, as it will be called from now on, can be isolated and treated separately from all others. Figure 5.7 poses the thermal problem by displaying the relevant differential equations and boundary conditions. In short, the balance of energy, already enunciated in Equation (3.36), must be respected by the solution in both the lubricant film and the solids 1 and 2. There must also be continuity of temperature and heat conduction at the interfaces. Disregarding the shear stress in both inlet and outlet, it follows that the corresponding dissipation is also negligible and that the temperature does not change 67 5. Traction curves and rheological parameters for gear oils Figure 5.7.: Schema of the thermal problem. (1) disc; (2) sphere; the oil flows in the gap. ρ∗is the density of the body (∗), Cp∗its heat capacity, λ∗its heat conductivity and U∗its velocity. θis the temperature difference above the inlet temperature and Φ the viscous dissipation in the oil film. wis the contact half-width in this plane: w=√a2−y2. 68 5.4. Simplified model for the EHD lubrication of a circular point contact. appreciably in the inlet. With these assumptions, one can dismiss the zones outside the Hertzian contact area and keep the analysis entirely within the bounds defined thus: |y| ≤ a(5.22) |x| ≤ »a2−y2=w(5.23) An analytical solution can be found in Carslaw and Jaeger [134], but it comes not as a closed form solution but as an infinite series of convolutions, generating some numerical problems. In order to avoid this, the energy equations are solved analytically in the frequency space of their Laplace transform with regard to the xdirection. The actual solution is then computed by numerical inverse Laplace transform. Partial solution in the film. It is convenient to transform the equations so that they refer to the non-dimensional variables: X=x+w w(5.24) Z=z/h0c(5.25) It is useful to introduce the following derived quantities: κ=λw h2 0cρCpU(5.26) K=λ/h2 0c(5.27) The energy equation in the lubricant and its boundary conditions thus become non-dimensional under the following reformulation: ∂2θ(X, Z) ∂Z2−1 κ ∂θ(X, Z) ∂X =−Φ(X) K(5.28) 0< X < 2 (5.29) −1/2< Z < 1/2 (5.30) θ(0, Z) = 0 (5.31) θ(X, −1/2) = θ1(X) (5.32) θ(X, 1/2) = θ2(X) (5.33) For the method to be applicable, it is necessary to consider Φ constant across the film thickness. As stated earlier, Carslaw and Jaeger’s analytical solution is an infinite series of convolutions: this naturally suggests that the treatment by integral transforms 69 5. Traction curves and rheological parameters for gear oils should prove fruitful thanks to the convolution theorem. To this end, the Laplace transform L[∗] is used on the Xdimension so that Equations (5.28)–(5.33) become: ∂2˜ θ(p, Z) ∂Z2−p κ ∂˜ θ(p, Z) ∂X =−˜ Φ(p) K(5.34) ˜ θ(p, −1/2) = ˜ θ1(p) (5.35) ˜ θ(p, 1/2) = ˜ θ2(p) (5.36) where: ˜ θ(p, Z) = L[θ(X, Z)] (p, Z) (5.37) ˜ θ1(p) = L[θ1(X)] (p) (5.38) ˜ θ2(p) = L[θ2(X)] (p) (5.39) The equation has become a second degree ordinary differential equation in Z, with solution: ˜ θ=cosh(tZ) cosh(t/2) "˜ θ2+˜ θ1 2−1 p κ˜ Φ K#+sinh(tZ) sinh(t/2) ˜ θ2−˜ θ1 2+1 p κ˜ Φ K(5.40) where t(p) = »p/κ (5.41) The Laplace transform of the heat flux ˜qis obtained by taking the derivative of the temperature with respect to Z: −h0c λ˜q=∂˜ θ ∂Z =tsinh(tZ) cosh(t/2) "˜ θ2+˜ θ1 2−1 p κ˜ Φ K#+tcosh(tZ) sinh(t/2) ˜ θ2−˜ θ1 2(5.42) In particular, the heat fluxes across the boundaries are computed thus: ˜q1= ˜q(p, −1/2) = −λt h0c(−tanh(t/2) "˜ θ2+˜ θ1 2−1 p κ˜ Φ K#+ coth(t/2) ˜ θ2−˜ θ1 2) (5.43) ˜q2= ˜q(p, 1/2) = −λt h0c(tanh(t/2) "˜ θ2+˜ θ1 2−1 p κ˜ Φ K#+ coth(t/2) ˜ θ2−˜ θ1 2)(5.44) Partial solution in upper solid (2) Similarly to what was done for the domain of the lubricant, the equations for the upper solid (solid 2) are rendered non-dimensional: ∂2θ(X, Z) ∂Z2=1 κ2 ∂θ(X, Z) ∂X (5.45) 0< X < 2 (5.46) 70 5.4. Simplified model for the EHD lubrication of a circular point contact. Z > 1/2 (5.47) θ(0, Z) = 0 (5.48) lim Z→+∞θ(X, Z) = 0 (5.49) θ(X, 1/2) = θ2(X) (5.50) where: κ2=λ2w h2 0cρ2Cp2U2 (5.51) The Laplace transform is applied in order to obtain the following set of equations: ∂2˜ θ(p, Z) ∂Z2=p κ2 ˜ θ(p, Z) (5.52) lim Z→+∞ ˜ θ(p, Z) = 0 (5.53) −λ2 h0c ∂˜ θ ∂Z Z=1/2 = ˜q2(p) (5.54) The solution is: ˜ θ=˜q2h0c λ2t2 e−(Z−1/2)t2(5.55) t2(p) = »p/κ2(5.56) and the boundary temperature and heat flux are related through: ˜ θ2=˜q2h0c λ2t2 (5.57) Partial solution in lower solid (1). The solution in the lower solid (1) is derived easily from the preceding one by considering the symmetries and anti-symmetries of the problem. Thus: ˜ θ=˜q1h0c λ1t1 e(Z+1/2)t1(5.58) t1(p) = »p/κ1(5.59) ˜ θ1=−˜q1h0c λ1t1 (5.60) where: κ1=λ1w h2 0cρ1Cp1U1 (5.61) 71 5. Traction curves and rheological parameters for gear oils Figure 5.12.: Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40◦C, U= 1m/s, FN= 16N (p0= 1GPa). In fact, it can be seen that only a rolling speed U= 1 m/s ensures full film EHD lubrication conditions for all temperatures and loads. On the other hand, as mentioned in the introduction, the objective was to use these results in surface contact fatigue of gear teeth, which seldom endure rolling speeds as low as 0.2 m/s, and never in the actual applications studied. For these reasons, and because the method used to predict the coefficient of friction is not valid under mixed or boundary lubrication conditions, only the traction test results at rolling speeds U= 1m/s were considered for determining the rheological parameters. For example, of the tests performed with T0= 120 ◦C, only those with U= 1 m/s were used. This is similar to the approach taken by Wu and Cheng [50], who used only a rolling speed of 1.85 m/s in their disc machine traction tests to determine the rheology parameters of a SAE 80W-90 GL5 gear oil. In Figures 5.12–5.15 both the computed and the measured traction curves of each oil are plotted for a few sets of operating conditions. For example, in Figure 5.12 the coefficient of friction of each oil is plotted against the slide-to-roll ratio for the set of operating conditions defined by T0= 40 ◦C, U= 1 m/s and p0= 1.0 GPa. Figures 5.12–5.14 show a good agreement between the predicted and the computed coefficient of friction. On the other hand, Figure 5.15 shows that at U= 0.1 m/s the prediction fails, which is to be expected, since the results at U= 0.1 m/s were not used for determining the rheological parameters. An interesting aspect of the difference between the oils is illustrated by Figure 5.16. As in Figure 5.12, the traction curves of the oils are traced when 78 5.5. Determination of the rheological parameters. Figure 5.13.: Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 80◦C, U= 1m/s, FN= 16N (p0= 1GPa). Figure 5.14.: Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40 ◦C, U= 1 m/s, FN= 35 N (p0= 1.3 GPa). 79 5. Traction curves and rheological parameters for gear oils Figure 5.15.: Measured (test) and predicted (calc) traction curves of the oils with operating conditions: T0= 40 ◦C, U= 0.1 m/s, FN= 16 N (p0= 1 GPa). T0= 40 ◦C, U= 1 m/s and p0= 1.0 GPa. For each oil, a pair of predicted traction curves is drawn: one with the full elasto-viscoplastic rheology of the oil and another in which the elastic contribution has been eliminated. Let it be noted, to avoid any ambiguity, that the rheology parameters of Table 5.7 were used in both cases. It is obvious that the presence of the elastic component influences greatly the behaviour of the mineral and PAO oils, while the ester oils are much less influenced. Nevertheless, the elastic effect is restricted to very low SRR, less than 20%. Unlike axial roller bearings, where SRR are always restricted to values well below 10%, gears will most of the time be experiencing much higher values of SRR. This is illustrated by the example of Figure 5.17, where the evolution of SRR along the meshing line of an FZG type C gear is shown. As with the most gears, only a small portion of the meshing (centred on the pitch point) is accompanied by SRR values below 20%. In fact, at the beginning and at the end of the meshing of a pair of teeth, SRR attains values in excess of 100%. Thus, for gear lubrication, the elastic part of the deformation can be discarded in the majority of cases, whatever the type of oil. 80 5.5. Determination of the rheological parameters. Figure 5.16.: Influence of the elastic shear modulus on each oil: predicted curves for operating conditions T0= 40 ◦C, U= 1 m/s, FN= 16 N (p0= 1GPa) are drawn including the elastic deformation (curves w/ G) and neglecting it (w/o G). 0 0.2 0.4 0.6 0.8 1 0 20 50 100 150 200 T1P T1T2 SRR (%) meshing of teeth starts here ends here Figure 5.17.: Slide-to roll ratio along the meshing line of an FZG type C spur gear. The solid line marks the actual relation between the meshing position and the SRR. 81 5. Traction curves and rheological parameters for gear oils 5.6. Chapter summary Six fully formulated gear oils (2 minerals, 1 PAO and 3 ester based) were submitted to traction tests, the results of which were presented. Comparison of the oils with regard to their experimental full film EHD traction curves showed significant differences between the mineral oils and the ester oils, the ester oils systematically inducing a lower coefficient of friction than the minerals. In this respect, the PAO acted as transition oil, with a behaviour and rheological properties in-between those of the minerals and esters. This suggests that the base oil is at least as important as the additives as regards its influence on the traction behaviour of the oil. Some of the experimental results were used to obtain the rheological parameters of the oils, according to a simplified model of EHD traction in point contact. The rheological parameters, which define the elasto-viscoplastic behaviour of the oils, were presented. It was shown that they lead to a good prediction of the friction. It was also shown that elastic deformation only plays a significant role in EHD lubrication with the mineral and PAO oils, and only at low (for gears) slide-to-roll ratios. 82 6. Stribeck curves of gear oils. 6.1. Preamble It was discussed in Section 2 that the regime of EHD lubrication, whether full film, mixed or boundary lubrication, is important with regard to surface contact fatigue. The present chapter presents inquiries into the behaviour of fully formulated under mixed EHD and full film lubrication. In particular, Stribeck curves were drawn from experimental measurements of coefficient of friction under conditions that ensure shallow film thickness when compared to the roughness of the surfaces. A natural extension of the work presented in Chapter 5, the present chapter extends the investigation into full film lubrication to cover the behaviour of the same oils in the remaining EHL regimes with much the same methodology. 6.2. Materials and experimental procedure The oils studied in the previous chapter are again the object of focus, except for lubricating oil T1, which was abandoned. Their properties can be found in Tables 5.1, 5.4, 5.6 and 5.7 of the previous chapter. The mini traction machine was used again for this investigation with the same specimen size and material, and using much the same procedure. What did change is that the objective was no longer to obtain full film EHL traction but Stribeck curves instead. The Stribeck curves obtained from these data are useful in studying the mixed an boundary film lubrication behaviour of an oil. The traction tests, called here somewhat improperly Stribeck tests, consisted in measuring the successive coefficient of friction µand friction load obtained by keeping T0,FNand SRR constant while varying the rolling velocity U. The testing of each oil was preceded by cleaning operations already described in Section 5.2.2. Following the cleaning, the unused specimens were assembled into the machine, and the pot was filled with the oil to be tested until the disc was fully immersed. After closing the receptacle of the machine where the specimens had been placed, the testing began with a preliminary step that consisted in running the test for 10 min under steady operating conditions: T0= 40 ◦C, U= 0.1 m/s, SRR = 0, FN= 0. Finally, by applying the operating conditions listed in Table 6.1, 27 distinct curves were obtained for each one of the tested, fully formulated gear oils. The operating conditions are listed in the order in which they were applied. 83 6. Stribeck curves of gear oils. Oil E1 was submitted to this test twice, with an interval of several months, and no significant differences were found in the results. At the end of this sequence of operations, the specimens were submitted again to the cleaning procedure and their roughness was measured. Their combined RMS roughness varied between 9 nm and 13 nm. This is less than the roughness of 14 nm advertised for new specimen by the manufacturer, from which it can be deduced that the alteration of roughness during the tests was slight. Figure 6.1 shows the estimated specific film thickness for each friction measurement condition. The centre film thickness was calculated using Hamrock and Dowson’s formula [125] and it was divided by the measured composite roughness of the test specimens in order to estimate Λ. While the correction factor for film thickness alteration due to inlet shear heating was computed following Gohar’s recommendations [126], it was found to be negligible at the operating conditions of the tests. Each subfigure of Figure 6.1 represents Λ for a particular oil. As an example, subfigure M1 shows Λ for oil M1 as three straight, inclined gray bands. Each of these bands shows the variation of Λ with Uat a particular temperature. The variation of SRR and FNcauses the thickness of the bands. This happens in all subfigures and the narrowness of the bands demonstrates that FNand SRR have much less influence on Λ than Uand T0over the ranges of variation that were used in the tests. 6.3. Modified Stribeck Parameter It is useful as this point to discuss in some detail the choice of the abscissa variable in the Stribeck curves. The Stribeck curve was developed for journal bearings [57, 58, 59], so it was originally used to show the transition from hydrodynamic to mixed to boundary film lubrication conditions. This is reflected in the fact that the abscissa parameter in the Stribeck curve is usually Uη/FN, in which ηis the dynamic viscosity of the oil at the oil temperature and atmospheric pressure. Gears very often operate under elastohydrodynamic lubrication in full, mixed or boundary film regimes, where the piezoviscosity of the lubricant plays an important part in terms both of film thickness and of coefficient of friction.To account for this, a new abscissa parameter, the Modified Stribeck Parameter, was developed to include in its formula the piezoviscosity coefficient (α). Also, as shown in Table 5.1 and remarked by Gold et al. [124], oils of different basestock (mineral, PAO, ester...), as in the present work, may have significantly different αeven if they have the same kinematic viscosity. The inclusion of αin this Modified Stribeck Parameter is thus also a way of accounting for the different nature of the base oils of each fully formulated lubricant considered. It is also desirable for the parameter to be dimensionless. Any dimensionless 84 6.3. Modified Stribeck Parameter Table 6.1.: Stribeck tests operating conditions inlet normal load contact slide-to-roll rolling temperature (or Hertzian pressure) ratio speed T0(◦C) FN(N) (or p0(GPa) ) SRR U (mm/s) steady running at 40 ◦C and U= 0.1 mm/s for 600s 40 6 (or 0.7) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 16 (or 1.0) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 35 (or 1.3) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 80 6 (or 0.7) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 16 (or 1.0) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 35 (or 1.3) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 120 6 (or 0.7) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 16 (or 1.0) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 35 (or 1.3) 0.1 150 to 0 0.3 150 to 0 0.5 150 to 0 85 6. Stribeck curves of gear oils. 0.1 1 2 3 5 10 T0=40◦C T0=80◦C T0=120◦C M1 0.1 1 2 3 5 10 T0=40◦C T0=80◦C T0=120◦C P1 0.1 1 2 3 5 10 T0=40◦C T0=80◦C T0=120◦C E1 Λ Λ 0.1 1 2 3 5 10 T0=40◦C T0=80◦C T0=120◦C E3 100101102 0.1 1 2 3 5 10 T0=40◦C T0=80◦C T0=120◦C E2 u(mm/s) 100101102 u(mm/s) Figure 6.1.: Specific film thickness under different operating conditions. 86 6.3. Modified Stribeck Parameter group related to the lubricated contact of the ball and disc has the form: TnTUnuSRRnSRR FnFN NηnηαnαRnR(E∗)nE∗σnσ·Y i Xni i(6.1) where Tis the oil temperature, Uis the rolling speed, SRR is the slide-to-roll ratio, FNis the normal contact load, ηand αare respectively the viscosity and the piezoviscosity coefficient at atmospheric pressure and test temperature, Ris the ball radius, E∗is the effective elastic modulus, σis the composite RMS rougnhess of the ball and disc and Xistands for other physical quantities that may influence the process, such as properties of the lubricant that pertain to non-Newtonian rheology. All tests where performed on specimens of the same size and material and which exhibited substantially the same roughness before and after having been submitted to the tests. What did change, from test to test, where the operating conditions (SRR,U,p,T) and the oil. Disregarding the effect of non-Newtonian rheology, the physical quantities Xiremained the same from test to test, since these hypothetical parameters are not among those that were altered. One must therefore conclude that the parameters that drive the difference of behaviour between tests must be looked for among those that alter from test to test; and that the relevant dimensionless groups must consequently be constructed from these parameters: U,SRR,FN,η,α. It is considered that the temperature is accounted for by its influence on the values of αand ηand thus need not be used explicitly. The formula for a dimensionless group consequently takes the form: UaηbαcFd NSRRe(6.2) SRR is already dimensionless and can therefore be isolated from the other quantities. The dimensional analysis of the remaining parameters gives: ÇL Tåa ·ÇFT L2åb ·ÇF L2å−c ·Fd= 1 (6.3) where L,Tand Frespectively stand for unit of length, time and force. This equation can be rewritten: La−2b+2c·Tb−a·Fb−c−d= 1 (6.4) Since every exponent must be 0, the following system of equations must be solved to find the one dimensionless parameter that can be extracted from the previous equation: a·     1 −1 0     +   −220 1 0 0 1−1 1  ·     b c d     (6.5) which yields the solution : b/a = 1, c/a = 1/2 and d/a =−1/2. 87 6. Stribeck curves of gear oils. to distinguish between individual Stribeck curves. In the case of oils M1 and P1, it can be observed that the envelope of the “data cloud” becomes progressively narrower as the left hand extremity of each corresponding subfigure is approached, and this regardless of the operating conditions. It appears that there exists for gear oils M1 and P1 a narrow range of possible values of the boundary film lubrication coefficient of friction (µBDR), almost independent of the operating conditions: •gear oil M1, 0.085 < µBDR <0.095. •gear oil P1, 0.095 < µBDR <0.105. In the case of gear oil E1, the value of µBDR is in the range of 0.105 < µBDR < 0.115 but at 120 ◦C and low load (FN= 6 N) that value may be smaller (≈0.09). Oil E1 is biodegradable and was formulated with low toxicity additivation (%P= 146 ppm and %S= 180 ppm, see Table 5.1). In the case of oil E3, the situation is not clear. At 120 ◦C the results seem to indicate a µBDR in a narrow range (0.090 < µBDR <0.095). At 80 ◦C some results seem to indicate significantly higher values. At 40◦C, they diverge sharply and attain a value of 0.14. In fact, the Stribeck curves at this temperature do not reach into the values of the Stribeck parameter corresponding to boundary film lubrication. These curves lack the plateau on their left extremity that would mark the end of mixed film lubrication. As was discussed in the section on mixed film lubrication, this may indicate that the additives are unable to fulfill their duty at that temperature, which would lead to a behaviour similar to dry sliding contact. This marked difference of behaviour between oils E1 and E3 in the boundary film lubrication regime is particularly significant because, as stated in Section 5.2.1, they were formulated from the same basestock but with different additive packages. The difference in their behaviour must therefore be attributed to the additives in their formulation. It must be remembered that oil E1 is a biodegradable and low-toxicity fluid, with low content of sulphur and phosphorus. In the case of oil E2, the Stribeck curves divide into two groups as boundary film lubrication is approached. The upper group, composed of the curves at 40 and 80 ◦C, converges towards a narrow range of values (0.105 < µBDR <0.115). The curves at 120◦C compose the second group and show a general trend of increasing coefficient of friction from 0.06 to 0.09 as the load increase from 6 N to 35 N. This unusual behaviour of oil E2 is matched by its unusual formulation: it consists in a mixture of 90% of low viscosity ester basestock with 5% of high viscosity ester (1000 cSt at 40◦C) (the remnant is additives), thereby ensuring biodegradability and low toxicity. 6.7. Comparison of the oils Figure 6.6 gives every Stribeck curve obtained at SRR = 0.5, an interesting value in what concerns gear lubrication; the other values of SRR are omitted because 94 6.7. Comparison of the oils 10−9 10−8 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 M1 U·η·α1/2 ·F−1/2 µ 10−9 10−8 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 P1 U·η·α1/2 ·F−1/2 µ 10−9 10−8 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 E1 U·η·α1/2 ·F−1/2 µ 10−9 10−8 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 E3 U·η·α1/2 ·F−1/2 µ 10−9 10−8 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 E2 U·η·α1/2 ·F−1/2 µ T0= 40◦C, FN= 6 N, SRR = 0.3 T0= 40◦C, FN= 6 N, SRR = 0.5 T0= 40◦C, FN= 16 N, SRR = 0.3 T0= 40◦C, FN= 16 N, SRR = 0.5 T0= 40◦C, FN= 35 N, SRR = 0.3 T0= 40◦C, FN= 35 N, SRR = 0.5 T0= 80◦C, FN= 6 N, SRR = 0.3 T0= 80◦C, FN= 6 N, SRR = 0.5 T0= 80◦C, FN= 16 N, SRR = 0.3 T0= 80◦C, FN= 16 N, SRR = 0.5 T0= 80◦C, FN= 35 N, SRR = 0.3 T0= 80◦C, FN= 35 N, SRR = 0.5 T0= 120◦C, FN= 6 N, SRR = 0.3 T0= 120◦C, FN= 6 N, SRR = 0.5 T0= 120◦C, FN= 16 N, SRR = 0.3 T0= 120◦C, FN= 16 N, SRR = 0.5 T0= 120◦C, FN= 35 N, SRR = 0.3 T0= 120◦C, FN= 35 N, SRR = 0.5 Figure 6.5.: Boundary film lubrication 95 6. Stribeck curves of gear oils. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 T0= 40◦C, FN= 6 N, SRR = 0.5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 T0= 80◦C, FN= 6 N, SRR = 0.5 µ 10−9 10−8 10−7 10−6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 T0= 120◦C, FN= 6 N, SRR = 0.5 T0= 40◦C, FN= 16 N, SRR = 0.5 T0= 80◦C, FN= 16 N, SRR = 0.5 10−9 10−8 10−7 10−6 T0= 120◦C, FN= 16 N, SRR = 0.5 U·η·α1/ 2·F −1/ 2 T0= 40◦C, FN= 35 N, SRR = 0.5 M1 P1 E1 E3 E2 T0= 80◦C, FN= 35 N, SRR = 0.5 10−9 10−8 10−7 10−6 T0= 120◦C, FN= 35 N, SRR = 0.5 Figure 6.6.: Comparison of the oils SRR has little effect in mixed and boundary film lubrication conditions, as was mentioned earlier. Each subfigure correspond to a particular combination of operating conditions and is labeled accordingly. As an example, the subfigure labeled 96 6.8. Chapter summary “T0= 40◦C, FN= 6 N, SRR = 0.5”, in the bottom left corner, shows five Stribeck curves obtained under these operating conditions, each curve corresponding to one of the gear oils. This makes the comparison of the oils easier. It was remarked in Section 5.3 that, in full film lubrication, the oils could always be grouped in the same way with regard to their coefficient of friction: oil M1 with a higher µ; oils P1, E1, and E3, mutually very similar, with an intermediate µand finally oil E2 with a lower µ. This is confirmed by Figure 6.6: this exact grouping can be observed when the lubrication regime is undeniably full film EHD (Sp>10−7). Figure 6.6 also shows that neither the grouping nor the order mentioned above are maintained as the lubrication regime progresses towards boundary film lubrication. The Stribeck curves in the central subfigure, labeled “T0= 80◦C, FN= 16 N, SRR = 0.5” are a good example of this: the Stribeck curve of E2, which starts from Sp≈10−7on the right with the lowest coefficient of friction ends at Sp≈ 6×10−10 with the second highest; the Stribeck curve of M1, which started with the highest coefficient of friction, ends with the lowest. In the subfigure labeled “T0= 120◦C, FN= 16 N, SRR = 0.5”, the curves follow a similar pattern. Although they start further into the mixed film lubrication regime (Sp≈3×10−8), the seriation inherited from the full film lubrication is still visible but the order is quickly subverted as the curves progress further into the mixed film lubrication regime. One may ask why the coefficients of friction of the ester oils, which are consistently lower in full film lubrication, tend to become higher in mixed and boundary film lubrication. It is difficult to give a categorical answer without a thorough knowledge of the chemistry of the oils, but a hypothesis may be advanced: while ester oils E1 and E2 were designed for biodegradability and low toxicity, the mineral oil M1 and PAO P1 were not, which removes a constraint from the choice of additives. Thus oils M1 and P1 could benefit from the effect of additives which could not be included in the formulation of the ester oils. This would therefore give them an advantage in the mixed and boundary lubrication regimes, where the additives become more preponderant. 6.8. Chapter summary Five oils, one mineral oil (M1), one PAO oil (P1) and three ester oils (E1, E3, E2), where submitted to traction tests performed with constant temperature, contact load and slide-to-roll ratio. The standard Stribeck parameter was modified in order to include the piezoviscosity coefficient of each base oil considered. The modified parameter, Sp= U·η·α1/2·F−1/2 N, is dimensionless and places all curves with regard to the abscissa so that there are definite positions in xx for each regimes of lubrication, even when considering Stribeck curves for different oils. Stribeck curves were traced from the experimental results and were presented here 97 6. Stribeck curves of gear oils. using the modified Stribeck parameter mentioned above. The operating conditions, namely the oil temperature (T0), the contact load (FN) and the slide-to-roll ratio (SRR), were discussed regarding their influence on the coefficient of friction in mixed and boundary film lubrication. It was found that: •SRR is of little importance in these regimes. •when T0is increased on its own while keeping all other operating conditions fixed, the value of Spis lowered and the regime of lubrication is shifted towards boundary lubrication. •when comparing tests performed with similar values of Spbut distinct temperatures, T0is irrelevant for M1, P1 and E1 but is of considerable importance in the case of E3 and E2, probably because of effects on the ability of the additives to promote a boundary layer. •FNhas an effect on the coefficient of friction that is the reverse, in mixed film lubrication, of what is observed in full film lubrication: in mixed film lubrication, an increase in FNleads to a decrease in the coefficient of friction. It was also found that the relative performance of the oils regarding their coefficient of friction is markedly dissimilar in full film, on one side, and mixed and boundary film lubrication, on the other. This suggests that the base oil is most important for friction in full film lubrication, while the additives can be said to become determinant in boundary film lubrication. 98 Part III. Gear tooth flank damage 99 7. Models of surface damage on the tooth flanks of spur gears 7.1. Kinematics and normal load in spur gear teeth In the vast majority of cases, the active part of a gear tooth flank profile is manufactured as an involute curve; for this reason, the present analysis of the geometry, kinematics and loading on spur gears will be restricted to involute spur gears. Figure 7.1 is a schematic representation of such a gear, composed of a pinion (driving wheel) with Z1teeth and a wheel (driven wheel) with Z2teeth. The figure shows only those teeth that actually participate in contact, at an instant when two pairs of teeth are meshing at once. The following notable points and characteristic dimensions are also represented: •O1and O2are respectively the centres of the pinion and of the tooth. •a0is the operating centre distance, •α0is the operating pressure angle, •Rb1, Rp1, Ra1are respectively the radii of the base circle, the operating pitch circle and the addendum circle of the pinion. •Rb2, Rp2, Ra2are respectively the radii of the base circle, the operating pitch circle and the addendum circle of the wheel. •ω1, ω2are respectively the angular velocity of the pinion and of the wheel. •T1and T2are the extremities of the contact line of length T1T2. •Pand P0are the contact points of two consecutive pair of teeth and lie on the contact line T1T2. •Both pitch circles, the segment O1O2and the contact line T1T2intersect at pitch point C. It can be shown that the nominal contact point between any two pairs of teeth at any instant in time always falls on the contact line T1T2: not only Pand P0, but any contact point must fall on the contact line. Also, for any contact point P, the tooth flank profiles are perpendicular to the contact line at Pand the radius 101 7. Models of surface damage on the tooth flanks of spur gears α’ driving gear (1) driven gear (2) a’ Ra1 Rp1 Rb1 ω1 Rb2 Rp2 Ra2 ω2 T2 O1 O2 α’ α’ operating pitch circle adendum circle base circle C T1 T1 P P’ P’ Figure 7.1.: Position of the spur gears pitch cylinder x z 0 Figure 7.2.: Coordinates on the surface of the pinion tooth flank. 102 7.1. Kinematics and normal load in spur gear teeth of curvature of the pinion tooth profile at Pis T1Pand that of the wheel tooth profile is T2P. Figure 7.2 proposes a local curvilinear reference coordinate fixed on a pinion tooth flank: xis the arc length on the surface measured from the pitch circle towards the root of the tooth, zis the depth under the surface and yis perpendicular to the page and point toward the reader. If the radius of curvature of the tooth surface is denoted ρ, it can be proved that the tooth flank surface is described by the equation: x=T1C2−ρ2 2Rb1 z= 0 (7.1) In order to clarify further discussions of gear meshing, it is useful to consider the particular case of a gear for which contact occurs at times between two pairs of teeth and at other times between one pair: a gear with a contact ratio εsuch that 1<ε<2. This is the case for the gear transmission that will be the object of the next few sections. Figure 7.3 attempts to gives a movie-like account of the meshing of a pair of teeth by showing in chronological order snapshots of five different instants during which notable events occur. Because more than one pair can be in contact, two other pairs of teeth are also represented, one on each side of the pair under analysis; hence the pair under focus is always the central one. Figure 7.3a shows the instant when the teeth of the central pair first come into contact at point A, where the addendum circle of the wheel intersects the contact line. At this instant, another pair is already in contact on the left-hand side at point Dso that the contact load is shared between the two pairs of teeth. Figure 7.3b shows the instant when the teeth of the left-hand side pair end their contact at point E, while the central pair is touching at point B. From this moment onward, the central pair bears the full contact between the wheels. Figure 7.3c shows the instant when the teeth of the central pair of teeth touch at the pitch point C, which lies at the intersection of the centreline with the contact line. Figure 7.3d shows the instant when a new pair of teeth comes into contact at point A, on the right-hand side, while the central pair is touching at point D. From then on, the contact is borne by two pairs of teeth. Figure 7.3e shows the instant when the teeth of the central pair end their contact at point E, while the right-hand side pair is touching at point B. Every pair of meshing teeth go through the history depicted in Figures 7.3a–7.3e. Hence, the successive contact points travel along segment AE on the contact line. This is illustrated in Figure 7.4, where the successive states of the central pair are superimposed. The pinion receives a driving torque Tand transmits it to the wheel as a normal contact force equal to T/Rb1. This contact force is shared between all pairs of teeth simultaneously in contact. Between points Aand B, it is shared between two 103 7. Models of surface damage on the tooth flanks of spur gears 5. Determine the fraction of the mixed film lubrication pressure contributed by pBDR.T : pBDR (x, t) = Ä1−fΛ(Λ)ä·pBDR.T (x, t) (7.18) 6. Add the EHL and boundary fractions to obtain pMIX : pMIX (x, t) = pEHD (x, t) + pBDR (x, t) (7.19) From the procedure outlined and the approximate equality of g(Λ) and fΛ(Λ) it can be deduced that pEHD can be taken as an approximation to the actual portion of the mixed film contact pressure borne by the lubricating film and pBDR as approximation to the portion borne by the direct contact between the surfaces. Tangential contact traction Having determined the mixed film lubrication normal contact pressure pMIX, there remains to be determined the tangential contact traction. This is done following the method outlined here: 1. Determine the smooth EHL tangential contact traction τEHD (x, t) as described in Section 7.2.5. 2. Determine the rough boundary tangential contact traction τBDR (x, t). 3. Add the two parts to obtain the mixed film lubrication friction contact stress: τMIX (x, t) = τEHD (x, t) + τBDR (x, t) (7.20) 7.2.3. Normal contact pressure in smooth EHD lubrication In order to obtain pEHD.T , an approach similar to Grubin’s [38] was taken. It is assumed, mostly correctly as was seen in Section 3.4, that the contact normal pressure distribution in a smooth EHL contact is close to the Hertzian pressure distribution. Hence, pEHD.T is: pEHD.T =p01−Åx aã2 (7.21) where p0is the Hertzian maximum pressure and ais the Hertzian half-width computed according to Equations (3.27) and (3.30). 7.2.4. Normal contact pressure in rough boundary lubrication In boundary film lubrication, the full brunt of the load is born by direct contact of the surfaces, although still retains some lubricating capacity: it is reasonable to treat the contact problem as dry disregarding the tangential stress for the purpose of computing the contact pressure. 110 7.2. Mixed film lubrication model The equations that govern this problem are given by Johnson [37]: h(x, t) = h0(x, t)−2 πE∗Z+∞ −∞ ln  x−x0 L(t) pBDR.T (x, t)dx0(7.22) Z+∞ −∞ pBDR.T (x, t)dx =FN(t) b(7.23) ∀x, t :h(x, t)≥0∧pBDR.T (x, t)≥0 (7.24) where the unknowns are: •h(x, t), the separation between the surfaces; •pBDR.T (x, t), the contact pressure; •L(t), an integration constant. and where the parameters have the meaning: •h0(x, t), the undeformed separation between the surfaces, not to be confused with the central film thickness; •E∗, the effective elastic modulus. A number of numerical algorithms exist for solving this problem [144]. The one used here was proposed by Polonsky and Keer [145]: it is unconditionally convergent and quite fast. 7.2.5. Smooth EHL part of the tangential contact traction A method for the determination of tangential contact traction in full film EHL of a point contact has already been presented in Section 5.4. The method employed here to the case of line contact is the same. The method of Section 5.4 relies on the simplifying assumption that side leakage of both lubricant and heat is small enough to allow the consideration of longitudinal “slice” for analysis. This is even more true in the case of line contact, where only one “slice” need be considered at all. At this stage, the reduced full film EHL pressure field pEHD has been computed, so that no uncertainties remain as to the influence of pressure on the lubricant material properties. As described in Section 5.4, this becomes a matter of solving the coupled problems of determination of temperature and shear rate in the lubricant film. Once both temperature and shear rate (or, equivalently shear rate) have converged, the sought result, the smooth EHL tangential contact traction, is the shear stress in the lubricant film. 111 7. Models of surface damage on the tooth flanks of spur gears 7.2.6. Rough boundary lubrication part of the tangential contact traction The rough boundary lubrication regime has until now been treated as if the lubricant had no effect. While it is true in the case of the pressure distribution, it is very much otherwise when it comes to determining the contact tangential stress. It was observed in Chapter 6 that the studied gear oils attained a well defined coefficient of friction essentially µBDR independent of operating conditions. Hence the boundary coefficient of friction is essentially a property of the lubricant oil and of the surfaces and must be determined experimentally. Supposing that the constancy of the boundary coefficient of friction, that concerns the relation between the overall contact and friction force also holds for the pressure and tangential traction distribution, we then have: τBDR (x, t) = µBDR (t)·pBDR (x, t) (7.25) 7.3. Tooth flank micropitting model 7.3.1. Preamble A numerical model for the prediction of surface initiated damage on gear tooth flanks caused by rolling contact surface fatigue is presented. This is related to micropitting because, as was said in Chapter 2, micropitting is widely acknowledged to be caused by fatigue cracks initiating in the surface. It is useful at this point to enunciate the assumptions that underly the model: •Because gear teeth roughness is essentially uni-directional, constituted of ridges perpendicular to the rolling direction, it is considered that the problem is two-dimensional and that the teeth are in plane strain. •It is assumed that any tooth on a gear (driving and driven) is representative of all other teeth on the same gear. •It is also assumed that one meshing between a pair of teeth is representative of all subsequent meshings between the same pair, so that the stress cycle of one meshing can be used for all subsequent meshings, and the full stress history of any point is approximated by a ceaseless repetition of the stress history obtained from the computed meshing. •It was seen in Chapter 2 that micropitting occurs sooner in the driving gear teeth than in the driven gear teeth, so that it is generally sufficient to check for micropitting initiation in the driving gear teeth, which are the weakest link. 112 7.3. Tooth flank micropitting model geometry, kinematics, loads mixed lubrication model pMIX, τMIX elastic stresses residual stresses mesoscopic residual stresses mesoscopic stress history Dang Van criterion elastic stresses contact fatigue criterion iterate over time Figure 7.9.: Diagram of the numerical micropitting model 7.3.2. Numerical Model As shown in Figure 7.9, where a diagram of the numerical model is shown, it may be divided into two main parts: 1. The determination of the elastic stresses in each material point under the surface of a driving gear tooth at each instant of the cycle. 2. The application of the contact fatigue criterion to each of these points. In the first part of the model the geometry of the contact—which must include the measured roughness of the gear tooth flanks in the rolling direction—and the contact force are determined, following which the contact loads between the driving and driven gear teeth are obtained through the application of the mixed film lubrication model, already described in Section 7.2. These surface tractions cause stresses within the bulk of the material, which are computed from Equations (4.2)– (4.6) to a depth of 30 μm under the surface of the driving gear tooth flank under analysis, since micropitting occurs in the first few tens of μm of depth. This procedure must be repeated for every instant of the loading cycle, in this case one meshing of gear teeth. When each instant has been examined, the entire history of the stresses induced by contact is known at each point. In the second part of the model the residual stresses present before the cycle and the elastic stresses induced by the contact cycle are added. At this point, 113 7. Models of surface damage on the tooth flanks of spur gears Table 7.1.: Residual stresses. depth (μm) 5 15 35 55 1000 residual σx(MPa) -287 -366 -463 -164 0 Table 7.2.: Mineral gear oil properties. Parameter Method Desig. Lubricating Oils Density at 15 ℃DIN 51757 ρ15 0.894 g cm−3 Kinematic Viscosity at 40 ℃DIN 51562 ν40 153.6 cSt Kinematic Viscosity at 100 ℃DIN 51562 ν100 14.4 cSt Viscosity Index DIN ISO 2909 VI 96 Pour point DIN ISO 3106 −27 ℃ the macroscopic stress history of each material point under the tooth flank of the driving gear is known. Next, the Dang Van criterion, described in Section 4.3.6, is applied at each material point to verify if fatigue life is infinite at that point. It was seen in Section 4.3.6 that a smallest enclosing ball problem must be solved for each material point under analysis in order to apply the criterion. Dang Van et al. proposed in [120] an approximate method for the solution of the smallest enclosing ball problem, a method that is also detailed by Ciavarella et al. [146]. A more recent algorithm, both faster and more accurate, had been made available by G¨artner [147] in the context of computational geometry and is used in the numerical implementation of this model. 7.3.3. A simulation of gear meshing The use of the model is made clearer in the present section where an example is presented to illustrate the application of the model to a realistic case. The simulation of an actual experimental case, as well as its comparison with the experiment, will be presented in Chapter 8. Simulation parameters In an approach not unlike that of Tao et al. [141], this section presents the results of a simulation of the meshing of gears with a FZG type C geometry made of DIN 20MnCr5 carburizing steel. The angular velocity of the driving gear is n1= 2250 RPM and that of the driven gear is n2= 1500 RPM, and the total contact load to be shared among meshing pairs of teeth is T/Rb1= 5073 N. Values of residual stresses for surface treated gears were taken from Gon¸calves’ PhD thesis [148] and are shown in Table 7.1. Because of surface treatment, the residual stresses vary with depth below the surface. The lubricant simulated is an ISO VG 150 paraffinic mineral oil. Some of its properties are listed in Table 7.2. Those are nevertheless insufficient to model the 114 7.3. Tooth flank micropitting model Table 7.3.: Rheological parameters of the gear oil. Roelands viscosity Bair and Winer viscoplasticity T0=363 K τL0=25 MPa η0=1.56 ×10−2Pa·s 1/ατL=588 MPa S0=1.28 βτL=0 K Z=0.608 rheology of the oil with the needed rigour, since it does not provide sufficient data to model the variation with pressure and temperature of the relation between the shear stress and the shear rate in the oil. To use the mixed lubrication model, the parameters for the Roelands viscosity equation, Equation (5.2), must also be known. The equation is reproduced here: ln η η0 = (ln η0+ 9.67) (ÇT−138 T0−138å−S0Å1 + p 196ãZ −1)(7.26) where Zand S0are non-dimensional parameters of the oil, T0is a reference temperature in K, η0is the dynamic viscosity in Pa·s at the reference temperature and atmospheric pressure, Tis the temperature of the lubricant in K, pis the pressure in GPa and ηis the dynamic viscosity in Pa·s. In addition, the Bair and Winer viscoplasticity model of Equation (5.10) is used, whose limiting shear stress also is pressure and temperature dependant, a dependency described by Equation (5.11), reproduced here: ln τL τL0 =ατL·p+βτLÄT−1−T0 −1ä(7.27) where τL0is the limiting shear stress at a reference temperature T0and atmospheric pressure and ατLand βτLare constant parameters of the oil. Furthermore, the boundary friction coefficient µBDR must be known, as must also the load sharing function fΛ. It is assumed that the load sharing function can be approximated by a formula of the type: f(Λ) = tanh ÄaΛ·ΛbΛä(7.28) The members of this family of functions have all the properties required from a load sharing function as described in Section 7.2, while also allowing a considerable degree of freedom in the shape of the function, all of this with only two parameters (aΛand bΛ). Neither the boundary friction coefficient, nor the rheological parameters of the gear oil, nor information that can help determine the load sharing function are available in the open literature, so that they must be obtained independently. This was attempted through a least-square data fitting of the friction coefficients, obtained through the mixed lubrication model already presented, to the estimates of friction 115 7. Models of surface damage on the tooth flanks of spur gears 00.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Λ fΛ µBDR=0.08 µBDR=0.1 µBDR=0.12 µBDR=0.14 Figure 7.10.: Some load sharing functions fΛand their associated boundary friction coefficients µBDR. coefficient obtained in gear power loss tests using the same lubricant [140]. It was possible to estimate the parameters to introduce into Equations (7.26) to (7.27). These are displayed in Table 7.3. However, this approach was unsuccessful in determining both µBDR and fΛ. Nevertheless, it was possible to associate a load sharing function to each boundary friction coefficient, as shown in Figure 7.10, so that the numerical and experimental friction power loss results fit. The pairings of a few boundary friction coefficient to the corresponding load sharing function formulas are as follows: µBDR = 0.08 ⇒f(Λ) = tanh Ä1.318 ·Λ0.270ä µBDR = 0.10 ⇒f(Λ) = tanh Ä1.614 ·Λ0.243ä µBDR = 0.12 ⇒f(Λ) = tanh Ä1.800 ·Λ0.213ä µBDR = 0.14 ⇒f(Λ) = tanh Ä1.925 ·Λ0.187ä (7.29) Finally, the gear material parameters to be introduced in the Dang Van criterion (αDV and βDV ) must be known. A value of αDV = 0.987 found in Gon¸calves’ PhD thesis [148] was chosen. Thus, the values µBDR and βDV are still missing. For the present goal, that of providing a qualitative understanding of the events associated with surface initiated tooth flank damage it is sufficient to arbitrate likely values, in this case: µBDR = 0.14 βDV = 440 MPa (7.30) Simulation results In Figure 7.11, the values of βeq that violate the Dang Van fatigue criterion (βeq > βDV ) are shown for the whole pinion tooth flank. The figure was obtained by 116 7.3. Tooth flank micropitting model −5 −4 −3 −2 D −1 C 1 B 2 −4 −2 0 2 4 6 8 βeq[MPa] x[mm] z[µm] 600 800 1000 1200 1400 Figure 7.11.: Simulation: values of βeq > βDV in the xz plane. C 1 B 2 −4 −2 0 2 4 6 8 βeq[MPa] x[mm] z[µm] 600 800 1000 1200 1400 Figure 7.12.: Simulation: βeq > βDV in the part of the driving gear tooth below the pitch line. 117 7. Models of surface damage on the tooth flanks of spur gears 0.60.62 0.64 0.66 0.68 0.7 -2 0 2 4 6 8 10 x[mm] z[µm] βeq[MPa] 440 560 680 800 920 920 1040 920 Figure 7.13.: Simulation: contour plot of a detail of Figure 7.12. “straightening” the nominal profile of the tooth until it became a straight line while conserving the roughness variations. Thus, the xx coordinates measure the position on the surface of the tooth flank and the zz coordinates the depth within the tooth. Notable points in the meshing are marked on the abscissa as B, C and D: the point where the tooth becomes the only one of the driving gear teeth in the meshing is marked as B , the pitch point is marked as C and the point where another pair of teeth initiates contact as D. It is interesting to note that the points where the fatigue criterion is violated come in patches. It is immediately apparent in Figure 7.11 that the part of the tooth below the pitch line (from A to C) suffers the most from contact fatigue, according to the Dang Van criterion: it has a greater concentration of violation patches and these tend to have higher values of βeq. Figure 7.12 shows the same information but only for the part of the tooth below the pitch line. It is puzzling at first to remark that not all salient roughness peaks give rise to severe contact fatigue initiation, as well as that some valleys do give rise to contact fatigue initiation. To explain this, one has to remember that the opposing driven gear tooth also has a roughness. This means that the film thickness above a driving gear tooth roughness peak is only shallow when a sufficiently salient driven gear tooth roughness feature opposes the driving gear tooth roughness peak. This is essentially a matter of chance, although the probability of fatigue initiation under a roughness peak is much higher than anywhere else, as can be seen in Figure 7.12. Figure 7.13 shows with greater detail the patch around the point with higher βeq. This patch is rather atypical in that it is very wide (around 0,1 mm). It is composed of a number of patches that coalesced into one, as can be deduced from the fact that a number of local maxima are present on the surface. In Figure 7.14 is shown another patch where the Dang Van criterion is violated. This is a simpler one, were a single maximum is present. This type of patch is far 118 7.3. Tooth flank micropitting model 1.442 1.444 1.446 1.448 1.45 -2 -1 0 1 2 3 4 5 440 480 520 600 680 760 840 880 x[mm] z[µm] βeq[MPa] Q Q′ Figure 7.14.: Simulation: contour plot of another detail of Figure 7.12. Points Q and Q0are singled out for later reference. more frequent than the one shown in Figure 7.13. In fact, from the perusal of Figure 7.12, one can conclude that the more complex patches are an agglomeration of several simple patches such as this one caused by the proximity of several roughness peaks. Therefore, the patch shown in Figure 7.14, can legitimately be considered typical. It is interesting to note its dimensions: roughly 10 µm wide by 5 µm deep. This is very similar to the size of a micropit. The points Qand Q0are singled out for later use. It is most instructive to see what happens in the area of the tooth around this typical patch of fatigue initiation, and the remainder of this section will be devoted to this. The first step is to observe the pressure field when the patch of Figure 7.14 undergoes its most intense contact pressure. This is shown in Figure 7.15, where the outlines of the patches within which βeq > βDV are represented along with the pressure distribution on the surface. Note that the tooth surfaces represented in Figure 7.15 include both the nominal geometry of the teeth and the elastic deformations, unlike those in Figures 7.11–7.14. It is striking that the pressure peak over the patch is very intense (around 8 GPa) and very localized: no more than 10 µm in width. The surface shear is not represented in the figure. It is proportional to the surface pressure and has the same direction as the sliding velocity U2−U1 depicted in the figure. It is interesting to turn ones attention to a single point and see what happens to it during the meshing cycle. The point in question is marked as Qin Figure 7.14. The point on the surface of the tooth on the vertical of Qis marked Q0on the same figure. The history of point Qis shown in Figures 7.16–7.18 In Figure 7.16, the history of the value of τmax +αDV ·pHin point Qis shown, as well as two horizontal lines corresponding to the values of βDV and βeq. Time is 119