Abstract
In this study, a three-dimensional thermo-elastic model that considers the interaction of mechanical and thermal deformation is developed using a semi-analytic method for steady-state rolling contact. Creepage types in all directions are considered in this model. For verification, the numerical analysis results of shear traction and temperature increase are compared separately with existing numerical results, and the consistency is confirmed. The analysis results include heat flux, temperature increase, contact pressure, and shear traction. Under severe rolling conditions, the thermal effect changes the behavior of the contact interface significantly. Furthermore, the effects of creepage, rolling speed, and conformity under different rolling and creep conditions are investigated.
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Abbreviations
- a 0 :
-
Hertzian contact half width in rolling direction (mm)
- A c, A c0, A st, A sl :
-
Real contact area, Hertzian contact area, stick area, and slip area (mm2)
- b 0 :
-
Hertzian contact half width in lateral direction (mm)
- C :
-
Influence coefficient
- E :
-
Elastic modulus (GPa)
- E * :
-
Equivalent elastic modulus (GPa), \(1/E = (1 - \nu _1^2)/{E_1} + (1 - \nu _2^2)/{E_2}\)
- F :
-
Friction force (N)
- G :
-
Green’s function
- h, h i :
-
Surface gap, initial gap (mm)
- k :
-
Thermal conductivity (W/(m·K))
- m, n :
-
Frequency domain coordinates corresponding to x and y
- l :
-
Characteristic length (mm)
- p :
-
Contact pressure (MPa)
- p 0 :
-
Maximum Hertzian contact pressure (MPa)
- p :
-
Dimensionless contact pressure, p = p / p0
- q, q 1 q 2 :
-
Total heat flux, and heat flux flowing to two bodies (W/m2)
- q :
-
Dimensionless total heat flux, q = qαtl / k
- Q F :
-
Total frictional heat (W), \({Q_{\rm{F}}} = \int\!\!\!\int_{{A_{\rm{c}}}} {q{\rm{d}}x{\rm{d}}y} \)
- R :
-
Radius of sphere (mm)
- \(R_y^\prime \) :
-
Equivalent radius in the lateral direction of the two bodies (mm), \(1/R_y^\prime = 1/{R_{y,1}} + 1/{R_{y,2}}\)
- Pex, Pey :
-
Peclet numbers along two axes
- ΔT :
-
Temperature increase (K)
- u x, u y, u z :
-
Surface deformations in three directions (mm)
- \(u_z^{\rm{e}},u_z^{\rm{t}}\) :
-
Elastic and thermal surface deformations in normal direction, respectively (mm)
- \(\underline {u_z^{\rm{t}}} \) :
-
Dimensionless thermal surface deformation in normal direction, \(\underline {u_z^{\rm{t}}} = u_z^{\rm{t}}/l(1 + \nu)\)
- U :
-
Lateral velocity (m/s)
- V :
-
Rolling velocity (m/s)
- ṡ :
-
Slip velocity (mm/s)
- w :
-
Distance in frequency domain, \(w = \sqrt {{m^2} + {n^2}} \)
- w′ :
-
Effective distance in frequency domain, \({w^\prime} = \sqrt {{w^2} - i(m \cdot {\rm{P}}{{\rm{e}}_x} + n \cdot {\rm{P}}{{\rm{e}}_y})} \)
- W :
-
Normal load (N)
- x, y, z :
-
Space coordinate (mm)
- x; y :
-
Dimensionless space coordinate, x = x / a0, y = y / b0
- α t :
-
Linear thermal expansion coefficient (µm/(m·K))
- γ c :
-
Contact area ratio, γc = Ac / Ac0
- γ p :
-
Maximum contact pressure ratio, γp = pmax / p0
- γ st :
-
Stick area ratio, γst = Ast / Ac
- δ :
-
Rigid body approach (mm)
- κ :
-
Thermal diffusivity (m2/s)
- μ :
-
Friction coefficient
- ν :
-
Poisson ratio
- φ :
-
Angular speed of contact surfaces for spin (rad/s)
- ψ :
-
Angular speed of contact surfaces in lateral direction (rad/s)
- ω :
-
Angular speed of contact surfaces in rolling direction (rad/s)
- ξ x :
-
Creepage ratio in rolling direction
- ξ y :
-
Creepage ratio in lateral direction
- ξ φ :
-
Spin creepage ratio (rad/mm)
- τ :
-
Shear traction (MPa)
- τ :
-
Dimensionless shear traction, τ = τ / p0
- η :
-
Heat partition coefficient
- C x :
-
Longitudinal creepage (mm/s)
- C y :
-
Lateral creepage (mm/s)
- C φ :
-
Spin creepage (rad/s)
- ≈:
-
Two-dimensional Fourier transform operator
- *:
-
Convolution operation
- e:
-
Elastic
- t:
-
Thermal elastic
- 1,2:
-
Lower and upper surfaces, respectively
- x,y,z :
-
Rolling, lateral, and normal direction, respectively
References
Xi Y H, Almqvist A, Shi Y J, Mao J H, Larsson R. A complementarity problem-based solution procedure for 2D steady-state rolling contacts with dry friction. Tribol Trans 59(6): 1031–1038 (2016)
Kanetani K, Mikami T, Ushioda K. Effect of retained austenite on sub-surface initiated spalling during rolling contact fatigue in carburized SAE4320 steel. ISIJ Int 60(8): 1774–1783 (2020)
Foo C T, Omar B, Jalil A S. A review on recent wheel/rail interface friction management. J Phys: Conf Ser 1049: 012009 (2018)
Johnson K L. Contact Mechanics. Cambridge (UK): Cambridge University Press, 1987.
Carter F W. On the action of a locomotive driving wheel. Proc Roy Soc A Math Phys Eng Sci 112(760): 151–157 (1926)
Nowell D, Hills D A. Tractive rolling of dissimilar elastic cylinders. Int J Mech Sci 30(6): 427–439 (1988)
Nowell D, Hills D A. Tractive rolling of tyred cylinders. Int J Mech Sci 30(12): 945–957 (1988)
Bentall R H, Johnson K L. Slip in the rolling contact of two dissimilar elastic rollers. Int J Mech Sci 9(6): 389–404 (1967)
Kalker J J. A minimum principle for the law of dry friction, with application to elastic cylinders in rolling contact—Part 1: Fundamentals—Application to steady rolling. J Appl Mech 38(4): 875–880 (1971)
Johnson K L. The effect of a tangential contact force upon the rolling motion of an elastic sphere on a plane. J Appl Mech 25: 339–346 (1958)
Johnson K L. The effect of spin upon the rolling motion of an elastic sphere on a plane. J Appl Mech 25: 332–338 (1958)
Kalker J J. The computation of three-dimensional rolling contact with dry friction. Int J Numer Methods Eng 14(9): 1293–1307 (1979)
Kalker J J. Numerical calculation of the elastic field in a half-space. Commun Appl Numer Methods 2(4): 401–410 (1986)
Kalker J J, Johnson K L. Three-dimensional elastic bodies in rolling contact. J Appl Mech 60(1): 255 (1993)
Wang Z J, Jin X Q, Keer L M, Wang Q. A numerical approach for analyzing three-dimensional steady-state rolling contact including creep using a fast semi-analytical method. Tribol Trans 55(4): 446–457 (2012)
Polonsky I A, Keer L M. A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques. Wear 231(2): 206–219 (1999)
Liu S B, Wang Q, Liu G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear 243(1–2): 101–111 (2000)
Xi Y H, Almqvist A, Shi Y J, Mao J H, Larsson R. Linear complementarity framework for 3D steady-state rolling contact problems including creepages with isotropic and anisotropic friction for circular hertzian contact. Tribol Trans 60(5): 832–844 (2017)
Bogdanski S, Olzak M, Stupnicki J. Numerical stress analysis of rail rolling contact fatigue cracks. Wear 191(1–2): 14–24 (1996)
Liu Y M, Liu L M, Mahadevan S. Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM. Eng Fract Mech 74(17): 2659–2674 (2007)
Bijak-Żochowski M, Marek P. Residual stress in some elasto-plastic problems of rolling contact with friction. Int J Mech Sci 39(1): 15–21, 23–32 (1997)
Jiang Y Y, Xu B Q, Sehitoglu H. Three-dimensional elastic-plastic stress analysis of rolling contact. J Tribol 124(4): 699–708 (2002)
Xu B Q, Jiang Y Y. Elastic-plastic finite element analysis of partial slip rolling contact. J Tribol 124(1): 20–26 (2002)
Pletz M, Meyer K A, Künstner D, Scheriau S, Daves W. Cyclic plastic deformation of rails in rolling/sliding contact-quasistatic FE calculations using different plasticity models. Wear 436–437: 202992 (2019)
Jiang Y Y, Sehitoglu H. A model for rolling contact failure. Wear 224(1): 38–49 (1999)
Srivastava J P, Sarkar P K, Meesala V R K, Ranjan V. Rolling contact fatigue life of rail for different slip conditions. Lat Am J Solids Struct 14(12): 2243–2264 (2017)
Yang Z, Deng X Y, Li Z L. Numerical modeling of dynamic frictional rolling contact with an explicit finite element method. Tribol Int 129: 214–231 (2019)
Lai V V, Chiello O, Brunel J F, Dufrénoy P. Full finite element models and reduction strategies for the simulation of friction-induced vibrations of rolling contact systems. J Sound Vib 444: 197–215 (2019)
Yang L Q, Hu M, Zhao D M, Yang J, Zhou X. Thermomechanical analysis of train wheel-rail contact using a novel finite-element model. Ind Lubr Tribol 72(5): 687–693 (2020)
Rodríguez-Tembleque L, Abascal R, Aliabadi M H. A boundary element formulation for wear modeling on 3D contact and rolling-contact problems. Int J Solids Struct 47(18–19): 2600–2612 (2010)
Antaluca E, Nélias D. Contact fatigue analysis of a dented surface in a dry elastic-plastic circular point contact. Tribol Lett 29(2): 139–153 (2008)
Haidari A, Hosseini-Tehrani P. Fatigue analysis of railway wheels under combined thermal and mechanical loads. J Therm Stress 37(1): 34–50 (2014)
Ekberg A, Kabo E. Fatigue of railway wheels and rails under rolling contact and thermal loading—An overview. Wear 258(7–8): 1288–1300 (2005)
Ertz M, Knothe K. Thermal stresses and shakedown in wheel/rail contact. Arch Appl Mech 72(10): 715–729 (2003)
Böhmer A, Ertz M, Knothe K. Shakedown limit of rail surfaces including material hardening and thermal stresses. Fatigue Fract Eng Mater Struct 26(10): 985–998 (2003)
Liao N T, Lin J F. Rolling-sliding analysis in ball bearing considering thermal effect. Tribol Trans 49(1): 1–16 (2006)
Hao X, Yun X H, Han Q K. Thermal-fluid-solid coupling in thermal characteristics analysis of rolling bearing system under oil lubrication. J Tribol 142(3): 031201 (2020)
Gao S, Chatterton S, Naldi L, Pennacchi P. Ball bearing skidding and over-skidding in large-scale angular contact ball bearings: Nonlinear dynamic model with thermal effects and experimental results. Mech Syst Signal Process 147: 107120 (2021)
Liu G, Wang Q, Liu S B. A three-dimensional thermal-mechanical asperity contact model for two nominally flat surfaces in contact. J Tribol 123(3): 595–602 (2001)
Liu S B, Wang Q, Harris S J. Surface normal thermoelastic displacement in moving rough contacts. J Tribol 125(4): 862–868 (2003)
Yu H, Liu S, Wang Q J, Chung Y W. Influence of temperature-dependent yield strength on thermomechanical asperity contacts. Tribol Lett 17(2): 155–164 (2004)
Liu S B, Wang Q. Transient thermoelastic stress fields in a half-space. J Tribol 125(1): 33–43 (2003)
Chen W W, Wang Q J. Thermomechanical analysis of elastoplastic bodies in a sliding spherical contact and the effects of sliding speed, heat partition, and thermal softening. J Tribol 130(4): 041402 (2008)
Zhang X, Wang Q J. Thermoelastic contact of layered materials with interfacial imperfection. Int J Mech Sci 186: 105904 (2020)
Tian X, Kennedy F E. Prediction and measurement of surface temperature rise at the contact interface for oscillatory sliding. Proc Inst Mech Eng Part J: J Eng Tribol 209(1): 41–51 (1995)
Tian X F, Kennedy F E Jr. Maximum and average flash temperatures in sliding contacts. J Tribol 116(1): 167–174 (1994)
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology of the Korean government (Grant No. NRF-2019R1A6A3A01097117).
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Yonghun YU. He received his Ph.D. degree in 2019 from Pusan National University, Republic of Korea. He is now working as a post-doctoral researcher at Pusan National University. His research focuses on thermo-mechanical contact problems, rolling contact fatigue, and tribology.
Junho SUH. He received his Ph.D. degree in 2014 from Texas A&M University, USA. He is now working as an assistant professor in the School of Mechanical Engineering, Pusan National University. His research interests include rotordynamics, vibration, and tribology.
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Yu, Y., Suh, J. Numerical analysis of three-dimensional thermo-elastic rolling contact under steady-state conditions. Friction 10, 630–644 (2022). https://doi.org/10.1007/s40544-021-0525-1
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DOI: https://doi.org/10.1007/s40544-021-0525-1