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 |  موضوع: كتاب Modeling and Analytical Methods in Tribology السبت 28 أكتوبر 2017, 12:29 am | |
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Modeling and Analytical Methods in Tribology
Ilya I. Kudish
Kettering University
Flint, Michigan
Michael J. Covitch
Lubrizol Corporation
Wickliffe, Ohio
ويتناول الموضوعات الأتية :
Contents
1 Basics of Asymptotic Expansions and Methods 1
1.1 Introduction 1
1.2 Ordering, Order Sequences, and Asymptotic Expansions . 1
1.3 Asymptotic Sequences and Expansions . 3
1.4 Asymptotic Methods . 4
1.5 Exercises and Problems . 6
2 Contact Problems for Coated and Rough Surfaces 9
2.1 Introduction 9
2.2 Some Classic Results for Smooth Elastic Solids 10
2.2.1 Formulas and Results for Elastic Half-Space 10
2.2.2 Formulas and Results for Elastic Half-Plane 11
2.3 Spatial Rough Contacts Modeled by Nonlinear Coating . 13
2.3.1 Problem Formulation . 13
2.3.2 Existence and Uniqueness of the Problem Solution 15
2.3.3 Solution Properties 18
2.3.4 Problems with Fixed Contact Region 22
2.3.5 Problems for Rough and Smooth Surfaces . 24
2.3.6 A Different Approach to Problems (2.35)-(2.37) 26
2.4 Asymptotic Analysis of Plane Rough Contacts 28
2.4.1 Plane Contact Problems with Fixed Boundaries 30
2.4.2 Plane Contact Problems with Free Boundaries 34
2.4.3 Pad´e Approximations 38
2.4.4 Analysis of Plane Rough Contacts with Fixed and Free
Boundaries 39
2.5 Numerical Methods and Results for Rough Contacts . 45
2.5.1 Numerical Methods for Problems in the Original Formulations . 45
2.5.2 Numerical Methods for Problems in the Asymptotic
Formulations . 51
2.5.3 Numerical Results for Contact Problems for Rough
Surfaces 53
2.6 Analysis of Axially Symmetric Rough Contacts 61
2.6.1 Axially Symmetric Rough Contacts with Fixed
Boundaries 66
v
vi Modeling and Analytical Methods in Tribology
2.6.2 Axially Symmetric Rough Contacts with Free Boundaries 68
2.6.3 Numerical Results for Axially Symmetric Rough Contacts 71
2.7 An Example of an Application to Roller Bearings . 73
2.8 Closure 74
2.9 Exercises and Problems . 76
3 Contact Problems with Friction 81
3.1 Introduction 81
3.2 Plane Frictional Contacts with Fixed Boundaries . 84
3.2.1 Application of Regular Perturbations. Contact with
Fixed Boundaries . 84
3.2.2 Application of Matched Asymptotic Expansions. Contact with Fixed Boundaries . 85
3.3 Plane Frictional Contacts with Free Boundaries 88
3.3.1 Application of Regular Perturbations. Contact with
Free Boundaries 88
3.3.2 Application of Matched Asymptotic Expansions. Contact with Free Boundaries 90
3.4 Plane Frictional Rough Contacts Modeled by Nonlinear Coating . 92
3.4.1 Existence and Uniqueness of the Problem Solution 94
3.4.2 Qualitative Properties of the Solution . 97
3.5 Asymptotic and Numerical Analysis for Large Roughness 99
3.5.1 Asymptotic Analysis for Large Roughness . 100
3.5.2 Numerical Solutions for Rough Contacts with Friction 101
3.6 Closure 105
3.7 Exercises and Problems . 106
4 Rheology of Lubricating Oils 109
4.1 Introduction 109
4.2 Rheology Relationships for Lubricating Oils 110
4.2.1 Definition of Viscosity 110
4.2.2 The Effect of Temperature on Viscosity 111
4.2.3 The Effect of Shear Rate on Viscosity . 113
4.2.4 The Effect of Pressure on Viscosity . 115
4.2.5 The Effect of Suspended Contaminants on Viscosity . 116
4.2.6 Density 119
4.2.7 Thermal Conductivity 120
4.3 Polymer Thickening and Shear Stability 121
4.3.1 Molecular Weight . 121
4.3.2 Dilute Solution Rheology 121
4.3.3 Shear Stability 122
4.4 Closure 124
Contents vii
4.5 Exercises and Problems . 125
5 Properties of Multi-grade Lubricating Oils 129
5.1 Introduction 129
5.2 Multi-grade Lubricating Oils 130
5.2.1 Engine Startability at Low Temperature 130
5.2.2 Engine Oil Pumpability at Low Temperature . 131
5.2.3 Bearing Protection 132
5.2.4 Engine Oil Viscosity Classification SAE J300 . 132
5.2.5 Automotive Gear Oil Viscosity Classification SAE
J306 134
5.3 Viscosity Modifiers 135
5.3.1 Olefin Copolymer (OCP) Viscosity Modifiers . 135
5.3.2 Polyalkylmethacrylate (PMA) Viscosity Modifiers 137
5.3.3 Styrene-Alkylmaleate Ester (SME) Copolymers Viscosity Modifiers 138
5.3.4 Hydrogenated Styrene Diene (HSD) Copolymer Viscosity Modifiers 139
5.3.5 Star Copolymer Viscosity Modifiers . 141
5.3.6 Lubricant Applications 142
5.4 Closure 145
5.5 Exercises and Problems . 146
6 Degradation of Linear Polymers 151
6.1 Introduction 151
6.2 Kinetic Equation for Degrading Linear Polymers . 152
6.3 Probability of Scission of Linear Polymer Molecules 156
6.4 Conditional Probability of Scission for Linear Polymers 159
6.5 Lubricant Viscosity and Polymeric Molecules . 161
6.6 Some Properties of the Kinetic Equation 162
6.7 A Limiting Case of the Kinetic Equation 170
6.8 Numerical Method for the Kinetic Equation 171
6.9 Numerical Solutions of the Kinetic Equation 175
6.10 Closure 182
6.11 Exercises and Problems . 183
7 Degradation of Star Polymers 187
7.1 Introduction 187
7.2 System of Kinetic Equations for Star Polymers 187
7.3 Probabilities of Scission . 196
7.4 Forming Star Polymeric Molecules . 197
7.5 Approximation of Star Polymer Initial Distribution 200
7.6 Lubricant Viscosity and Polymer Distribution . 204
7.7 Some Properties of the System of Kinetic Equations . 205
7.8 Numerical Method for Kinetic Equations 220
viii Modeling and Analytical Methods in Tribology
7.9 Numerical Results for Lubricants with Star Polymers . 223
7.10 Closure 230
7.11 Exercises and Problems . 231
8 Review of Data on Contact Fatigue 233
8.1 Introduction 233
8.2 Contact and Residual Stresses 233
8.3 Material Defects and Lubricant Contamination 242
8.4 Bearing Fatigue Life and Contact Friction . 251
8.5 Crack Development and Material Microstructure . 254
8.5.1 Crack Initiation and Crack Propagation 254
8.5.2 Material Microstructure and Contact Fatigue Life . 260
8.5.3 Subsurface and Surface Material Cracking . 261
8.6 Some Contemporary Contact Fatigue Models . 265
8.6.1 Mathematical Models for Fatigue Life of Bearings . 266
8.6.2 Mathematical Models for Fatigue Life of Gears 275
8.7 Closure 277
8.8 Exercises and Problems . 279
9 Fracture Mechanics and Contact Fatigue 295
9.1 Introduction 295
9.2 Modeling the Vicinity of Crack Tips 296
9.2.1 Problem Formulation and Solution for a Single Crack 297
9.2.2 Multiple Cracks in an Elastic Plane . 300
9.2.3 Multiple Cracks in an Elastic Half-Plane 305
9.3 Perturbations for Multiple Cracks in a Half-Plane . 308
9.3.1 Perturbation Analysis for Multiple Cracks in an Elastic Plane 308
9.3.2 Perturbation Analysis for Multiple Cracks in an Elastic Plane 318
9.3.3 Perturbation Solution for Multiple Cracks in an Elastic Half-Plane . 323
9.3.4 Stress Intensity Factors for Multiple Cracks in an Elastic Half-Plane . 330
9.4 Contact Problem for a Cracked Elastic Half-Plane 336
9.4.1 Problem Formulation . 337
9.4.2 Problem Solution . 342
9.4.3 Contact Problem for Lubricated Solids . 348
9.4.4 Numerical Results 349
9.5 Directions of Fatigue Crack Propagation 356
9.6 Lubricant-Crack Interaction. Origin of Fatigue 360
9.6.1 General Assumptions and Problem Formulation 362
9.6.2 Problem in Dimensionless Variables 370
9.6.3 Numerical Method 371
Contents ix
9.6.4 Numerical Solutions for Subsurface and Surface
Cracks . 376
9.6.5 Comparison of Analytical and Numerical Solutions for
Small Cracks . 394
9.6.6 Origin of Contact Fatigue. Fatigue Life of Drivers versus Followers . 398
9.7 Two-Dimensional Statistical Model of Contact Fatigue 417
9.7.1 Initial Statistical Defect Distribution 418
9.7.2 Crack Propagation versus Crack Initiation . 420
9.7.3 Applicability of Fracture Mechanics to Contact Fatigue 421
9.7.4 Direction of Fatigue Crack Propagation 423
9.7.5 Crack Propagation Calculations . 424
9.7.6 Crack Statistics 428
9.7.7 Local Fatigue Damage Accumulation 430
9.7.8 Survival Probability of Material as a Whole 431
9.7.9 Variable Loading and Contact Fatigue . 434
9.8 Analysis of the Pitting Model 436
9.8.1 Analytical Analysis of the Pitting Model 436
9.8.2 Numerical Analysis of the Pitting Model 440
9.9 Contact Fatigue of Rough Surfaces . 445
9.9.1 Contact Stresses in Rough Contacts 445
9.9.2 Modeling Contact Fatigue in Rough Contacts . 449
9.10 Three-Dimensional Model of Contact Fatigue . 455
9.10.1 Initial Statistical Defect Distribution 455
9.10.2 Direction of Fatigue Crack Propagation 456
9.10.3 Crack Propagation Calculations . 457
9.10.4 Crack Statistics 459
9.10.5 Local and Global Fatigue Damage Accumulation . 460
9.10.6 Examples of Torsional and Bending Fatigue 463
9.11 Contact Fatigue of Radial Thrust Bearings 465
9.11.1 Case of Axial Loading 466
9.11.2 Case of Radial Loading . 466
9.12 Closure 467
9.13 Exercises and Problems . 469
10 Analysis of Fluid Lubricated Contacts 479
10.1 Introduction 479
10.2 Simplified Navier-Stokes and Energy Equations 480
10.3 Lightly Loaded Lubrication Regimes 484
10.3.1 Problem Formulation . 485
10.3.2 Perturbation Solution of the EHL Problem 488
10.3.3 Thermal EHL Problem 494
10.3.4 Numerical Method for Lightly Loaded EHL Contacts . 496
10.4 Pre-critical Lubrication Regimes 501
x Modeling and Analytical Methods in Tribology
10.4.1 Problem Formulation . 504
10.4.2 Asymptotic Analysis of the Problem 506
10.4.3 Asymptotic Analysis of the System of Equations
(10.84)-(10.88) 513
10.5 Compressible Fluids in Heavily Loaded Contacts . 515
10.6 Over-critical Lubrication Regimes 517
10.6.1 Problem Formulation . 517
10.6.2 Structure of the Solution . 518
10.6.3 Auxiliary Gap Function hH(x) . 530
10.6.4 Solutions in the
q-Zones . 533
10.6.5 Asymptotic Analysis of the Inlet and Exit 0-Zones 536
10.6.6 Choosing Pre- or Over-critical Lubrication Regimes 542
10.6.7 Analysis of the Ertel-Grubin Method 543
10.7 Numerical Solution for EHL Contacts . 547
10.7.1 Numerical Procedure . 548
10.7.2 Some Numerical Results . 551
10.8 Numerical Solution of Asymptotic Equations . 555
10.8.1 Numerical Solution in the Inlet Zone 557
10.8.2 Numerical Solution in the Exit Zone 561
10.8.3 Some Numerical Results in the Inlet and Exit Zones . 565
10.8.4 Numerical Precision and Stability Considerations . 571
10.9 Analysis of EHL Contacts for Soft Solids 576
10.9.1 Formulation of an EHL Problem for Soft Solids 577
10.9.2 Qualitative Analysis of the EHL Problem . 578
10.9.3 Surface Velocities for Soft Solids 579
10.10 Thermal EHL Problems . 590
10.10.1 Formulation of TEHL Problem . 591
10.10.2 Analytical Approximations for Newtonian Fluids in
TEHL Contacts 595
10.10.3 Numerical Solutions of Asymptotic TEHL Problems
for Newtonian Lubricants 601
10.11 Regularized Solution of Asymptotic Problems . 604
10.12 Regularization of the Isothermal EHL Problem 609
10.12.1 Numerical Method for the Isothermal EHL Problem . 609
10.12.2 Some Numerical Solutions of the Regularized Isothermal EHL Problem 617
10.13 Numerical Validation of the Asymptotic Analysis . 620
10.14 Practical Use of the Asymptotic Solutions . 625
10.15 Approximations for Non-Newtonian Fluids . 626
10.15.1 Formulation of Isothermal EHL Problem for NonNewtonian Fluids . 627
10.15.2 Isothermal EHL Problem for Pure Rolling. Pre- and
Over-critical Lubrication Regimes. Some Numerical
Examples . 628
Contents xi
10.15.3 Isothermal EHL Problem for Relatively Large Sliding.
Pre- and Over-critical Lubrication Regimes 651
10.15.4 Choosing Pre- and Over-critical Lubrication Regimes
for Non-Newtonian Lubricants . 661
10.15.5 Non-Newtonian Lubricants and Scale Effects . 665
10.16 TEHL Problems for Non-Newtonian Lubricants 675
10.16.1 TEHL Problem Formulation for Non-Newtonian Fluids . 675
10.16.2 Asymptotic Analysis of the Problem for Heavily
Loaded Contacts . 678
10.16.3 Heat Transfer in the Contact Solids . 688
10.17 Regularization for Non-Newtonian Fluids . 695
10.18 Friction in Heavily Loaded Lubricated Contacts 698
10.19 Closure 700
10.20 Exercises and Problems . 702
11 Lubrication by Greases 709
11.1 Introduction 709
11.2 Formulation of the EHL Problems for Greases . 712
11.3 Properties of the Problem Solution for Greases 722
11.4 Greases in a Contact of Rigid Solids 724
11.4.1 Problem Formulation . 724
11.4.2 Analysis of Possible Flow Configurations 726
11.4.3 Numerical Results 731
11.5 Regimes of Grease Lubrication without Cores . 736
11.6 Closure 742
11.7 Exercises and Problems . 743
12 Lubricant Degradation in EHL Contacts 745
12.1 Introduction 745
12.2 EHL for Degrading Lubricants . 746
12.3 Lubricant Flow Topology 754
12.4 Numerical Method for EHL Problems . 756
12.4.1 Initial Approximation 757
12.4.2 Sliding Frictional Stress . 757
12.4.3 Horizontal Component of the Lubricant Velocity and
Flux 759
12.4.4 Lubricant Flow Streamlines . 759
12.4.5 Separatrices of the Lubricant Flow . 762
12.4.6 Solution of the Kinetic Equation and Lubricant Viscosity . 764
12.4.7 Solution of the Reynolds Equation . 766
12.5 Solutions for Lubricants without Degradation . 770
12.6 EHL Solutions for Lubricants with Degradation 776
12.7 Lubricant Degradation and Contact Fatigue 790
xii Modeling and Analytical Methods in Tribology
12.7.1 Model of Contact Fatigue 791
12.7.2 Elastohydrodynamic Modeling for a Degrading Lubricant 791
12.7.3 Combined Model for Contact Fatigue and Degrading
Lubricant . 793
12.7.4 Numerical Results and Discussion 794
12.8 A Qualitative Model of Lubricant Life . 800
12.9 Closure 802
12.10 Exercises and Problems . 804
13 Non-steady and Mixed Friction Problems 809
13.1 Introduction 809
13.2 Properly Formulated Non-steady EHL Problems . 809
13.2.1 A Non-steady Lubrication of a Non-conformal Contact 810
13.2.2 Properly Formulated Non-steady Lubrication Problems for Journal Bearings 817
13.3 Non-steady Lubrication of a Journal Bearing . 827
13.3.1 General Assumptions and Problem Formulation 829
13.3.2 Case of Rigid Materials . 830
13.3.3 Contact Region Transformation . 833
13.3.4 Quadrature Formula and Discretization 835
13.3.5 Iterative Numerical Scheme . 837
13.3.6 Analysis of Numerical Results 838
13.4 Starved Lubrication and Lubricant Meniscus . 851
13.4.1 Problem Formulation . 853
13.4.2 Numerical Method 857
13.4.3 Numerical Results 862
13.5 Formulation and Analysis of a Mixed Lubrication Problem 871
13.5.1 Problem Formulation . 875
13.5.2 Fluid Friction in Lightly and Heavily Loaded Lubricated Contacts 881
13.5.3 Boundary Friction 882
13.5.4 Partial Lubrication of a Narrow Contact 887
13.6 Dry Narrow Contact of Elastic Solids 892
13.6.1 Examples of Dry Narrow Contacts . 894
13.6.2 Optimal Shape of Contacting Solids 896
13.7 Closure 898
13.8 Exercises and Problems . 899
Index 905
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