Abstract
We study the contribution of surface elasticity to the two-dimensional contact problem in a generally anisotropic material using the Stroh sextic formalism. Surface elasticity is incorporated into the model of deformation using an anisotropic version of the continuum-based surface/interface model of Gurtin and Murdoch. Full-field analytic solutions are obtained in terms of exponential integrals for an anisotropic half-space when the contact surface is subjected to two particular types of loading: first, we consider the case of a uniform load (shearing and pressure) applied to an infinitely long strip of the contact surface and second, by reducing the strip to zero width, we deduce the corresponding result for a concentrated line force acting on the contact surface. The analysis indicates that the surface deformation gradient is finite in the first case of uniform loading of the strip and exhibits a weak logarithmic singularity at the location of the applied concentrated line force in the second case.
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Acknowledgments
The authors are grateful to two reviewers whose comments and suggestions have greatly improved the presentation of the work. This work is supported by the National Natural Science Foundation of China (Grant No: 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (No. RGPIN 155112).
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Appendix
Appendix
From Eq. (18) we find that
where
Once \(\mathbf{u}_{,1} \) and \({{\varvec{\upsigma }}}_{2} \) on \(x_{2} =0\) are known, \(\mathbf{u}_{,2} \) and \({{\varvec{\upsigma }}}_{1} \) on \(x_{2} =0\) can be further obtained from the following relations [12]:
It is observed from Eq. (24) that \(\mathbf{{h}'}(z)\) also does not contain the Stroh eigenvalues and eigenvectors. Thus, \({{\varvec{\upsigma }}}_{2}^+ \) in Eq. (60) and consequently \({{\varvec{\upsigma }}}_{1} \) in Eq. (62) are also valid for mathematically degenerate materials.
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Wang, X., Schiavone, P. The effects of anisotropic surface elasticity on the contact problem in an anisotropic material. J Eng Math 101, 141–151 (2016). https://doi.org/10.1007/s10665-016-9851-0
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DOI: https://doi.org/10.1007/s10665-016-9851-0