Asymptotic solution of a small parametered $2$-D integral equation arising from a contact problem of elasticity based on the solution of a $2$-D integral equation

Abstract:

Asymptotic solution of a 2-D integral equation of constant kernel with small parameter $\varepsilon$, \[ \int _0^\pi {\int _{ - \infty }^\infty p } dsd\psi + \varepsilon r\int _0^\pi {\int _{ - \infty }^\infty p } ds\cos \psi d\psi = G(r),\] which occurs in a more exact form of Hertz’s contact problem in elasticity, is presented in this paper based on the solution of a 2-D integral equation \[ \int _0^\pi {\int _{ - \infty }^\infty } pdsd\psi = F(r)\] with constant kernel, and the unknown function $p = p(s,\psi ) = p(t,\phi )$ is subjected to the following two constraints: \[ \begin {array}{*{20}{c}} {p(t,\phi ) = p(t)\quad \forall \phi ,} \\ {p(s,\psi ) = 0\quad {\text {for}}\;(s,\psi ) = (t,\phi ) \notin E = \{ (t,\phi )|t \leq a\} } \\ \end {array} \] where $(s,\psi )$ are local polar coordinates with origin at $M(r,0)$, with $(r,0)$ measured by global polar coordinates $(t,\phi )$ with origin at $O(0,0)$. A more exact solution of Hertz’s contact problem is found as an example.