An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann–Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin–Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin–Murdoch model with a non-zero surface tension and for the general Steigmann–Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss–Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

研究了在 Steigmann-Ogden 形式存在表面能的情况下刚性印章与半空间无摩擦接触的轴对称问题。Boussinesq 势的方法用于获得应力和位移的积分表示。使用 Hankel 变换，问题被简化为具有附加条件的接触区间上的第一类单一积分方程。研究积分方程的可解性。结果表明，对于没有表面效应的经典问题和没有表面张力的Gurtin-Murdoch表面能问题，得到的方程代表了柯西奇异积分方程。同时，对于具有非零表面张力的 Gurtin-Murdoch 模型和一般的 Steigmann-Ogden 模型，该问题相应地导致具有弱奇异核或连续核的第一类积分方程。因此，接触问题在这些情况下是不适定的。使用 Gauss-Chebyshev 求积求积分近似求解具有附加条件的第一类积分方程。报告了各种参数值的数值结果。