Within the general approach to the modified Kirsch problem incorporating surface elasticity and residual surface stress (surface tension) by the original Gurtin–Murdoch model, the proper boundary conditions at an arbitrary cylindrical surface are derived in terms of complex variables for the plane strain and plane stress. In the case of the plane stress, the properties are allowed for not only of the cylindrical surface but the faces of a plate with the nanosized thickness as well. The solution of both 2-D problems on the infinite plane with the circular hole subjected to a uniform far-field load is obtained in a closed form. It is shown that under plane strain conditions, the general formulas for the stress tensor components are reduced to the modified Kirsch solution at the nanoscale in the case of the uniaxial loading. At the same time, the uniaxial remote loading alone is impossible in the case of the plane stress because of the presence of the axisymmetric surface tension at the faces of the plate. Analytical solution shows that the elastic field of the plate depends on the plate thickness. Numerical examples are given in the paper to illustrate quantitatively the effect of the plate thickness on the stress field at the cylindrical surface and the role of surface tension in the corresponding plane strain problem.

在通过原始Gurtin-Murdoch模型解决的结合表面弹性和残余表面应力（表面张力）的修正Kirsch问题的一般方法中，根据平面应变和平面的复变数，得出了任意圆柱表面的适当边界条件。压力。在平面应力的情况下，不仅对于圆柱表面，而且对于具有纳米尺寸厚度的板的表面，都允许具有该特性。以闭合形式获得在圆孔承受均匀的远场载荷的情况下在无限平面上的两个二维问题的解。结果表明，在平面应变条件下，在单轴载荷的情况下，应力张量分量的通用公式在纳米级被简化为改进的Kirsch溶液。同时，在平面应力的情况下，单轴的远程加载是不可能的，因为在板的表面上存在轴对称的表面张力。解析解表明，板的弹性场取决于板的厚度。本文中给出了数值示例，以定量说明板厚度对圆柱表面应力场的影响以及表面张力在相应的平面应变问题中的作用。