Tribology Letters

, Volume 28, Issue 1, pp 27–38

Analysis and Convenient Formulas for Elasto-Plastic Contacts of Nominally Flat Surfaces: Average Gap, Contact Area Ratio, and Plastically Deformed Volume

  • W. Wayne Chen
  • Q. Jane Wang
  • Yuchuan Liu
  • Wei Chen
  • Jiao Cao
  • Cedric Xia
  • Raj Talwar
  • Rick Lederich
Original Paper


Interaction of nominally flat engineering surfaces that leads to a large contact area exists in many mechanical systems. Considering periodic similarity of surface geometry, a numerical three-dimensional elasto-plastic contact model can be used to simulate the contact behaviors of two nominally flat surfaces with the assistance of the continuous convolution and Fourier transform (CC-FT) algorithm. This model utilizes the analytical frequency response functions (FRF) of elastic/plastic responses of materials and provides contact performance results, including the average surface gap, the contact area ratio, and the volume of plastically deformed material, which may be defined as performance variables. Following the digital filtration technology, rough surfaces can be numerically generated with specified autocorrelation length and the first four orders of statistical moments. A group of contact simulations are conducted with various working conditions. The effects of topographic and material properties on the contact behaviors are discussed. With a multi-variables regression method, empirical formulas are developed for the performance variables as functions of surface statistical characteristics, material properties, a hardening parameter, and the applied load in terms of pressure.


Elasto-plastic contact Surface roughness Contact performance formulas 


Ac, An

Real contact area, nominal contact area


Young’s modulus, GPa


Equivalent Young’s modulus, GPa, EE/(1−ν2)


Elasto-plastic tangential modulus

g (λ)

Yield strength, MPa


Green’s functions

h, hi

Surface gap, initial gap, mm

\( L = {\ifmmode\expandafter\bar\else\expandafter\=\fi{p}} \mathord{\left/ {\vphantom {{\ifmmode\expandafter\bar\else\expandafter\=\fi{p}} Y}} \right. \kern-\nulldelimiterspace} Y \)

Dimensionless average pressure

M = E*/Y

Ratio of Young’s modulus to yield strength


Surface pressure

Rz, Rη

Autocorrelation functions of asperity heights and random sequence


Root mean square of roughness (RMS), μm

Sk, K

Skewness of roughness, Kurtosis of roughness

S = ET/E

Hardening modulus ratio

\( u_{3} ,u^{p}_{3} ,u^{r}_{3} \)

Total normal displacement, elastic and residual normal displacement


Initial yield strength without strain hardening


Asperity heights

Greek Letters

βx, βy

Correlation length in two dimensions, β≥ βy, μm

γ= βxy

Ratio of correlation length, ≥1

\( \Gamma = {\ifmmode\expandafter\bar\else\expandafter\=\fi{h}} \mathord{\left/ {\vphantom {{\ifmmode\expandafter\bar\else\expandafter\=\fi{h}} {R_{q} }}} \right. \kern-\nulldelimiterspace} {R_{q} } \)

Dimensionless average surface gap


Mesh size, μm

\( \varepsilon ^{p}_{{ij}} \)

Plastic strain component


Independent random sequence


Accumulative effective plastic strain

Λ = Ac/An

Contact area ratio


Poisson ratio

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

Cauchy stress, elastic and residual stress, MPa


von-Mises equivalent stress, MPa

χ = βy/Rq

Asperity shape ratio


Contact interference, mm

Ω = Vp/AnRq

Dimensionless plasticity volume

Special Marks


Autocorrelation function


Discrete Fourier transform


Operator of continuous convolution

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • W. Wayne Chen
    • 1
  • Q. Jane Wang
    • 1
  • Yuchuan Liu
    • 1
  • Wei Chen
    • 1
  • Jiao Cao
    • 1
  • Cedric Xia
    • 2
  • Raj Talwar
    • 3
  • Rick Lederich
    • 3
  1. 1.Department of Mechanical EngineeringNorthwestern UniversityEvanstonUSA
  2. 2.Scientific Research LaboratoriesFord Motor CompanyDearbornUSA
  3. 3.The Boeing CompanySt. LouisUSA

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