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Tribology Letters

, Volume 28, Issue 1, pp 27–38 | Cite as

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

Abstract

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.

Keywords

Elasto-plastic contact Surface roughness Contact performance formulas 

Nomenclature

Ac, An

Real contact area, nominal contact area

E

Young’s modulus, GPa

E*

Equivalent Young’s modulus, GPa, E E/(1−ν 2)

ET

Elasto-plastic tangential modulus

g (λ)

Yield strength, MPa

G

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

p

Surface pressure

Rz, Rη

Autocorrelation functions of asperity heights and random sequence

Rq

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

Y

Initial yield strength without strain hardening

z

Asperity heights

Greek Letters

βx, βy

Correlation length in two dimensions, β ≥ β y , 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

Δ

Mesh size, μm

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

Plastic strain component

η

Independent random sequence

λ

Accumulative effective plastic strain

Λ = Ac/An

Contact area ratio

ν

Poisson ratio

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

Cauchy stress, elastic and residual stress, MPa

σVM

von-Mises equivalent stress, MPa

χ = βy/Rq

Asperity shape ratio

ω

Contact interference, mm

Ω = Vp/AnRq

Dimensionless plasticity volume

Special Marks

ACF

Autocorrelation function

DFT

Discrete Fourier transform

*

Operator of continuous convolution

Notes

Acknowledgements

The authors would like to acknowledge financial supports from National Science Foundation, Office of Naval Research, Department of Energy, Ford Motor Company and The Boeing Company. The author would also like to thank Dr. Fan Wang, Profs. Leon Keer and Daniel Nelias, and Mr. Vincent Boucly for helpful discussions.

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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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