The effects of anisotropic surface elasticity on the contact problem in an anisotropic material

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.

Appendix

From Eq. (18) we find that

$$\begin{aligned} {{\varvec{\upsigma }}}_{2}^+ =-\mathbf{Ku}_{,11} -{{\varvec{\uptau }}}(x_{1} )\quad \hbox { on } x_{2} =0, \end{aligned}$$

(60)

where

$$\begin{aligned} \mathbf{u}_{,11} =\mathbf{{h}'}(x_{1} )+\overline{\mathbf{{h}'}(x_{1} )}\quad \hbox {on }x_{2} =0. \end{aligned}$$

(61)

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]:

$$\begin{aligned} \begin{aligned} \mathbf{u}_{,2} =\mathbf{N}_1 \mathbf{u}_{,1} +\mathbf{N}_{2} {{\varvec{\upsigma }}}_{2},\qquad -{{\varvec{\upsigma }}}_{1}=\mathbf{N}_{3} \mathbf{u}_{,1} +\mathbf{N}_{1}^\mathrm{T} {{\varvec{\upsigma }}}_{2}. \end{aligned} \end{aligned}$$

(62)

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.