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


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.


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

List of Symbols


Hertzian half contact width

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

Influence coefficients relating surface traction to stresses


Influence coefficients relating plastic strain to plastic displacement


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


Effective case depth


Tangent modulus in linear hardening law


Galerkin vectors


Fatigue parameter


Film thickness

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

Initial and geometry gap between surfaces, respectively


Material hardness

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

Gear surface and core hardness, respectively


Euler transform matrix


Meyer’s hardness coefficient


Surface pressure


Hertzian maximum pressure


Fluid shear traction


Yield strength function


Equivalent radius of curvature


The deviatoric stress tensor


Composite surface roughness



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

Influence coefficients relating plastic strain to residual stress


Surface displacements


Rolling velocity


Elastic deformation


Plastic deformation


Applied normal load


Coordinates (x is parallel to rolling direction)


Pressure-viscosity constant

\(\mu ,\lambda\)

Lamé constants


Poisson’s ratio


Effective accumulative plastic strain

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

Viscosity and ambient viscosity of the lubricant, respectively


Equivalent viscosity

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

Density and ambient density of the lubricant, respectively


Plastic strain


Kronecker delta


Initial yield stress

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

Elastic stresses


Residual stresses


The ultimate strength of gear material


Von Mises equivalent stress


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



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.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanical TransmissionsChongqing UniversityChongqingChina

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