An analytical solution for elastic and elastic-plastic contact models

Zhiqiang Liu, Anne Neville, Robert Lewis Reuben

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)


In tribology often a closed form solution for calculation of contact stress and real contact area is required for the purposes of, for example, developing wear maps and temperature profiles at asperities. In assuming a Gaussian distribution of asperity heights it is not possible to obtain an analytical solution for the contact load and real contact area for many analytical models such as those developed by Greenwood and Williamson (elastic model), Chang, et al. (elastic-plastic model) and Horng (elliptic elastic-plastic model). In this paper, two exponential functions have been derived from a fitting procedure applied to the numeri cal results of the Gaussian height distribution thus offering an analytical expression for the above three models. It,has been demonstrated that the two exponential functions (phi(2)* and phi(4)*) can give a fair approximation to the contact load and the real contact area in the separation of 0 to 4 sigma. In addition, variations in plasticity index (psi) and effective asperity radius (gamma) do not significantly affect the approximated accuracy. The results obtained by the newly derived exponential functions have been compared with the exponential function phi(1)* suggested by Greenwood and Williamson, 1966 and it has been shown that use phi(1)* of invariably gives a larger error than using two exponential functions over two ranges of separation distances.

Original languageEnglish
Pages (from-to)627-634
Number of pages8
JournalTribology Transactions
Issue number4
Publication statusPublished - 2000
EventSTLE/ASME International Joint Tribology Conference - Seattle, United States
Duration: 1 Oct 20004 Oct 2000


  • analytical solution
  • asperity height distribution
  • elastic-plastic contact


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