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Tribology Letters

, 67:92 | Cite as

Investigation of Contact Performance of Case-Hardened Gears Under Plasto-elastohydrodynamic Lubrication

  • Ye Zhou
  • Caichao ZhuEmail author
  • Huaiju Liu
  • Hailan Song
Original Paper
  • 101 Downloads

Abstract

Case-hardening is widely used to enhance gear loading capacity. Simulation of the material gradient properties and contact characteristics are the key issues in contact fatigue analysis of case-hardened gears. In this work, a plasto-elastohydrodynamic lubrication (PEHL) model incorporating the hardness gradient and surface roughness is developed to investigate the contact performance of case-hardened gears. The generalized Reynolds equation is solved to determine film thickness and contact pressure. The plastic deformation and residual stress are obtained via the half-space eigenstrain problem solving. The Dang Van multiaxial fatigue criterion and the Euler transformation are employed to evaluate the contact fatigue parameter based on the predetermined stress field. The discrete convolution and fast Fourier transform (DC-FFT) algorithm is used for accelerating the computation. The influences of effective case depth, surface hardness and surface roughness on the contact performance are investigated. Numerical results indicate that as the surface hardness increases, the probability of fatigue crack nucleation decreases, and the depth of the crack initiation site increases. For a lower surface roughness case, the maximum von Mises stress and equivalent plastic strain appear at a deeper layer. As the surface roughness increases, the maximum values of pressure and stress increase sharply and move closer to the surface.

Keywords

Gear contact fatigue Plasto-elastohydrodynamic lubrication Hardness gradient Elastoplastic contact Surface roughness 

List of Symbols

\(b\)

Hertzian half contact width

\(C_{ij}^{n} ,C_{ij}^{t}\)

Influence coefficients relating surface traction to stresses

\(D_{3kl}\)

Influence coefficients relating plastic strain to plastic displacement

\(E_{0}\)

Effective elastic modulus, \(E_{0} = 2/\left( {\left( {1 - v_{1}^{2} } \right)/E_{1} + \left( {1 - v_{2}^{2} } \right)/E_{2} } \right)\)

ECD

Effective case depth

ET

Tangent modulus in linear hardening law

\(\varvec{F}\)

Galerkin vectors

\({\text{FP}}\)

Fatigue parameter

\(h\)

Film thickness

\(h_{0} ,h_{g}\)

Initial and geometry gap between surfaces, respectively

\(H\)

Material hardness

\(H_{\text{sur}} ,H_{\text{cor}}\)

Gear surface and core hardness, respectively

\({\mathbf{M}}_{\text{u}}\)

Euler transform matrix

\(m\)

Meyer’s hardness coefficient

\(p\)

Surface pressure

\(p_{H}\)

Hertzian maximum pressure

\(q\)

Fluid shear traction

\(Q_{y}\)

Yield strength function

\(R\)

Equivalent radius of curvature

\(\varvec{S}_{ij}\)

The deviatoric stress tensor

Su

Composite surface roughness

t

Time

\(T_{ijkl}^{\left( 0 \right)}\)

Influence coefficients relating plastic strain to residual stress

\(u_{i}^{r}\)

Surface displacements

\(u_{r}\)

Rolling velocity

Ve

Elastic deformation

Vp

Plastic deformation

\(W_{n}\)

Applied normal load

\(x,y,z\)

Coordinates (x is parallel to rolling direction)

\(z_{e}\)

Pressure-viscosity constant

\(\mu ,\lambda\)

Lamé constants

\(\nu\)

Poisson’s ratio

\(\kappa\)

Effective accumulative plastic strain

\(\eta ,\eta_{0}\)

Viscosity and ambient viscosity of the lubricant, respectively

\(\eta^{*}\)

Equivalent viscosity

\(\rho ,\rho_{0}\)

Density and ambient density of the lubricant, respectively

\(\varepsilon_{ij}^{p}\)

Plastic strain

\(\delta_{ij}\)

Kronecker delta

\(\sigma_{y0}\)

Initial yield stress

\({{\sigma }}_{ij}^{e}\)

Elastic stresses

\(\sigma_{ij}^{r}\)

Residual stresses

\(\sigma_{\text{uts}}\)

The ultimate strength of gear material

\(\sigma_{\text{vm}}\)

Von Mises equivalent stress

\(\sigma_{\text{H}}\)

Hydrostatic stress

\(\sigma_{ - 1} , \tau_{ - 1}\)

Fatigue limits under fully reversed bending and torsion

\(\tau_{ \hbox{max} }\)

Maximum amplitude of shear stress

\({{\varOmega }}\)

Plastic zone

Notes

Acknowledgements

This work is supported by the National Key R&D Program of China (Grant No. 2018YFB2001300), the National Natural Science Foundation of China (Grant No. 51575061), and the Fundamental Research Funds for the Central Universities (Grant No. 2018CDXYJX0019). The authors are grateful to Dr. Nicholaos Demas for his help and discussions during the research.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Moorthy, V., Shaw, B.A.: Contact fatigue performance of helical gears with surface coatings. Wear 276–277, 130–140 (2012).  https://doi.org/10.1016/j.wear.2011.12.011 CrossRefGoogle Scholar
  2. 2.
    Wang, W., Liu, H., Zhu, C., Wei, P., Tang, J.: Effects of microstructure on rolling contact fatigue of a wind turbine gear based on crystal plasticity modeling. Int. J. Fatigue 120, 73–86 (2019)CrossRefGoogle Scholar
  3. 3.
    Ren, N., Zhu, D., Chen, W.W., Wang, Q.J.: Plasto-elastohydrodynamic lubrication (PEHL) in point contacts. J. Tribol. 132(3), 031501 (2010).  https://doi.org/10.1115/1.4001813 CrossRefGoogle Scholar
  4. 4.
    Sadeghi, F., Jalalahmadi, B., Slack, T.S., Raje, N., Arakere, N.K.: A review of rolling contact fatigue. J. Tribol. 131(4), 041403 (2009).  https://doi.org/10.1115/1.3209132 CrossRefGoogle Scholar
  5. 5.
    Li, S., Kahraman, A., Klein, M.: A fatigue model for spur gear contacts operating under mixed elastohydrodynamic lubrication conditions. J. Mech. Des. 134(4), 041007 (2012).  https://doi.org/10.1115/1.4005655 CrossRefGoogle Scholar
  6. 6.
    Zhu, D., Ren, N., Wang, Q.J.: Pitting life prediction based on a 3D line contact mixed EHL analysis and subsurface von Mises stress calculation. J. Tribol. 131(4), 041501 (2009).  https://doi.org/10.1115/1.3195040 CrossRefGoogle Scholar
  7. 7.
    Li, S.: An investigation on the influence of misalignment on micro-pitting of a spur gear pair. Tribol. Lett. (2015).  https://doi.org/10.1007/s11249-015-0613-3 CrossRefGoogle Scholar
  8. 8.
    Evans, H.P., Snidle, R.W., Sharif, K.J., Shaw, B.A., Zhang, J.: Analysis of micro-elastohydrodynamic lubrication and prediction of surface fatigue damage in micropitting tests on helical gears. J. Tribol. 135(1), 011501 (2012).  https://doi.org/10.1115/1.4007693 CrossRefGoogle Scholar
  9. 9.
    Brandão, J.A., Seabra, J.H.O., Castro, J.: Surface initiated tooth flank damage. Wear 268(1–2), 1–12 (2010).  https://doi.org/10.1016/j.wear.2009.06.020 CrossRefGoogle Scholar
  10. 10.
    Liu, H., Zhu, C., Sun, Z., Song, C.: Starved lubrication of a spur gear pair. Tribol. Int. 94, 52–60 (2016)CrossRefGoogle Scholar
  11. 11.
    Zhou, Y., Zhu, C., Gould, B., Demas, N.G., Liu, H., Greco, A.C.: The effect of contact severity on micropitting: simulation and experiments. Tribol. Int. 138, 463–472 (2019).  https://doi.org/10.1016/j.triboint.2019.06.020 CrossRefGoogle Scholar
  12. 12.
    Chiu, Y.: On the stress field due to initial strains in a cuboid surrounded by an infinite elastic space. J. Appl. Mech. 44(4), 587–590 (1977)CrossRefGoogle Scholar
  13. 13.
    Jacq, C., Nélias, D., Lormand, G., Girodin, D.: Development of a three-dimensional semi-analytical elastic-plastic contact code. J. Tribol. 124(4), 653 (2002).  https://doi.org/10.1115/1.1467920 CrossRefGoogle Scholar
  14. 14.
    NĂŠlias, D., Antaluca, E., Boucly, V., Cretu, S.: A three-dimensional semianalytical model for elastic-plastic sliding contacts. J. Tribol. 129(4), 761–771 (2007)CrossRefGoogle Scholar
  15. 15.
    Zhou, K., Chen, W.W., Keer, L.M., Wang, Q.J.: A fast method for solving three-dimensional arbitrarily shaped inclusions in a half space. Comput. Methods Appl. Mech. Eng. 198(9–12), 885–892 (2009).  https://doi.org/10.1016/j.cma.2008.10.021 CrossRefGoogle Scholar
  16. 16.
    Wang, F., Keer, L.M.: Numerical simulation for three dimensional elastic-plastic contact with hardening behavior. J. Tribol. 127(3), 494 (2005).  https://doi.org/10.1115/1.1924573 CrossRefGoogle Scholar
  17. 17.
    Chen, W.W., Wang, Q.J., Wang, F., Keer, L.M., Cao, J.: Three-dimensional repeated elasto-plastic point contacts, rolling, and sliding. J. Appl. Mech. 75(2), 021021 (2008).  https://doi.org/10.1115/1.2755171 CrossRefGoogle Scholar
  18. 18.
    Chaise, T., NĂŠlias, D.: Contact pressure and residual strain in 3D elasto-plastic rolling contact for a circular or elliptical point contact. J. Tribol. 133(4), 041402 (2011)CrossRefGoogle Scholar
  19. 19.
    Wang, Z., Wang, W., Hu, Y., Wang, H.: A numerical elastic-plastic contact model for rough surfaces. Tribol. Trans. 53(2), 224–238 (2010).  https://doi.org/10.1080/10402000903177908 CrossRefGoogle Scholar
  20. 20.
    Wang, Z., Jin, X., Zhou, Q., Ai, X., Keer, L.M., Wang, Q.: An efficient numerical method with a parallel computational strategy for solving arbitrarily shaped inclusions in elastoplastic contact problems. J. Tribol. 135(3), 031401 (2013)CrossRefGoogle Scholar
  21. 21.
    Liu, S., Jin, X., Wang, Z., Keer, L.M., Wang, Q.: Analytical solution for elastic fields caused by eigenstrains in a half-space and numerical implementation based on FFT. Int. J. Plast. 35, 135–154 (2012).  https://doi.org/10.1016/j.ijplas.2012.03.002 CrossRefGoogle Scholar
  22. 22.
    He, T., Wang, J., Wang, Z., Zhu, D.: Simulation of plasto-elastohydrodynamic lubrication in line contacts of infinite and finite length. J. Tribol. 137(4), 041505 (2015)CrossRefGoogle Scholar
  23. 23.
    He, H., Liu, H., Zhu, C., Wei, P., Sun, Z.: Study of rolling contact fatigue behavior of a wind turbine gear based on damage-coupled elastic-plastic model. Int. J. Mech. Sci. 141, 512–519 (2018).  https://doi.org/10.1016/j.ijmecsci.2018.03.044 CrossRefGoogle Scholar
  24. 24.
    Zhang, B., Liu, H., Bai, H., Zhu, C., Wu, W.: Ratchetting–multiaxial fatigue damage analysis in gear rolling contact considering tooth surface roughness. Wear 428–429, 137–146 (2019).  https://doi.org/10.1016/j.wear.2019.03.003 CrossRefGoogle Scholar
  25. 25.
    Wang, W., Liu, H., Zhu, C., Wei, P., Wu, W.: Micromechanical analysis of gear fatigue-ratcheting damage considering the phase state and inclusion. Tribol. Int. 136, 182–195 (2019)CrossRefGoogle Scholar
  26. 26.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  27. 27.
    Mura, T.: Micromechanics of defects in solids. Springer Science & Business Media, Berlin (2013)Google Scholar
  28. 28.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241(1226), 376–396 (1957)CrossRefGoogle Scholar
  29. 29.
    Mura, T., Shodja, H., Hirose, Y.: Inclusion problems. Appl. Mech. Rev. 49(10S), S118–S127 (1996)CrossRefGoogle Scholar
  30. 30.
    Yu, H., Sanday, S.: Elastic field in joined semi-infinite solids with an inclusion. Proc. R. Soc. Lond. A 434(1892), 521–530 (1991)CrossRefGoogle Scholar
  31. 31.
    Liu, S., Wang, Q.: Elastic fields due to eigenstrains in a half-space. J. Appl. Mech. 72(6), 871 (2005).  https://doi.org/10.1115/1.2047598 CrossRefGoogle Scholar
  32. 32.
    Mindlin, R.D.: Force at a point in the interior of a semi-infinite solid. Physics 7(5), 195–202 (1936)CrossRefGoogle Scholar
  33. 33.
    Wang, Z., Jin, X., Keer, L.M., Wang, Q.: Novel model for partial-slip contact involving a material with inhomogeneity. J. Tribol. 135(4), 041401 (2013).  https://doi.org/10.1115/1.4024548 CrossRefGoogle Scholar
  34. 34.
    Wang, Z., Jin, X., Liu, S., Keer, L.M., Cao, J., Wang, Q.: A new fast method for solving contact plasticity and its application in analyzing elasto-plastic partial slip. Mech. Mater. 60, 18–35 (2013).  https://doi.org/10.1016/j.mechmat.2013.01.001 CrossRefGoogle Scholar
  35. 35.
    Liu, S., Wang, Q.: Studying contact stress fields caused by surface tractions with a discrete convolution and fast fourier transform algorithm. J. Tribol. 124(1), 36 (2002).  https://doi.org/10.1115/1.1401017 CrossRefGoogle Scholar
  36. 36.
    Yang, P., Wen, S.: A generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic lubrication. J. Tribol. 112(4), 631–636 (1990)CrossRefGoogle Scholar
  37. 37.
    Zhou, Y., Zhu, C., Liu, H., Song, C., Li, Z.: A numerical study on the contact fatigue life of a coated gear pair under EHL. Ind. Lubr. Tribol 70(1), 23–32 (2018)CrossRefGoogle Scholar
  38. 38.
    Gu, Z., Zhu, C., Liu, H., Du, X.: A comparative study of tribological performance of helical gear pair with various types of tooth surface finishing. Ind. Lubr. Tribol. 71(3), 474–485 (2019).  https://doi.org/10.1108/ILT-01-2017-0013 CrossRefGoogle Scholar
  39. 39.
    Liu, H., Liu, H., Bocher, P., Zhu, C., Sun, Z.: Effects of case hardening properties on the contact fatigue of a wind turbine gear pair. Int. J. Mech. Sci. 141, 520–527 (2018).  https://doi.org/10.1016/j.ijmecsci.2018.04.010 CrossRefGoogle Scholar
  40. 40.
    Wei, P., Zhou, H., Liu, H., Zhu, C., Wang, W., Deng, G.: Modeling of contact fatigue damage behavior of a wind turbine carburized gear considering its mechanical properties and microstructure gradients. Int. J. Mech. Sci. 156, 283–296 (2019)CrossRefGoogle Scholar
  41. 41.
    Evans, H.P., Snidle, R.W., Qiao, H.: Comparison of fatigue model results for rough surface elastohydrodynamic lubrication. Proc. Inst. Mech. Eng. Part. 222(3), 381–393 (2008).  https://doi.org/10.1243/13506501jet347 CrossRefGoogle Scholar
  42. 42.
    Osman, T., Velex, P.: A model for the simulation of the interactions between dynamic tooth loads and contact fatigue in spur gears. Tribol. Int. 46(1), 84–96 (2012).  https://doi.org/10.1016/j.triboint.2011.03.024 CrossRefGoogle Scholar
  43. 43.
    Donzella, G., Mazzù, A., Petrogalli, C.: Failure assessment of subsurface rolling contact fatigue in surface hardened components. Eng. Fract. Mech. 103, 26–38 (2013)CrossRefGoogle Scholar
  44. 44.
    Atzori, B., Meneghetti, G., Susmel, L.: Material fatigue properties for assessing mechanical components weakened by notches and defects. Fatigue Fract. Eng. Mater. Struct. 28(1–2), 83–97 (2005)Google Scholar
  45. 45.
    Li, S.: Lubrication and Contact Fatigue Models for Roller and Gear Contacts. The Ohio State University, Columbus (2009)Google Scholar
  46. 46.
    Lang, O.: The dimensioning of complex steel members in the range of endurance strength and fatigue life. Z. fuer Werkst. 10, 24–29 (1979)CrossRefGoogle Scholar
  47. 47.
    MackAldener, M., Olsson, M.: Tooth interior fatigue fracture—computational and material aspects. Int. J. Fatigue 23(4), 329–340 (2001)CrossRefGoogle Scholar
  48. 48.
    Thomas, J.: Flankentragfähigkeit und Laufverhalten von hartfeinbearbeiteten Kegelrädern. Technische Universität München, München (1998)Google Scholar
  49. 49.
    Al, B.C., Patel, R., Langlois, P.: Comparison of tooth interior fatigue fracture load capacity to standardized gear failure modes. In: FTM (Fall Technical Meeting) 2016Google Scholar
  50. 50.
    Witzig, J.: Tooth flank fracture—limited gear load carrying capacity below the flank surface (in German: Flankenbruch - Eine Grenze der Zahnradtragfähigkeit in der Werkstofftiefe). Technical University of Munich, Munich (2012)Google Scholar
  51. 51.
    Pavlina, E.J., Van Tyne, C.J.: Correlation of yield strength and tensile strength with hardness for steels. J. Mater. Eng. Perform. 17(6), 888–893 (2008).  https://doi.org/10.1007/s11665-008-9225-5 CrossRefGoogle Scholar
  52. 52.
    Cahoon, J., Broughton, W., Kutzak, A.: The determination of yield strength from hardness measurements. Metall. Trans. 2(7), 1979–1983 (1971)Google Scholar
  53. 53.
    Tabor, D.: The hardness and strength of metals. J. Inst. Metals 79, 1 (1951)Google Scholar
  54. 54.
    Cahoon, J.: An improved equation relating hardness to ultimate strength. Metall. Mater. Trans. B 3(11), 3040–3040 (1972)CrossRefGoogle Scholar
  55. 55.
    Wang, W., Liu, H., Zhu, C., Du, X., Tang, J.: Effect of the residual stress on contact fatigue of a wind turbine carburized gear with multiaxial fatigue criteria. Int. J. Mech. Sci. 151, 263–273 (2019)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.State Key Laboratory of Mechanical TransmissionsChongqing UniversityChongqingChina

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