Abstract

We consider the Kirsch problem, taking into account the surface stresses at the boundary of the circular hole and on the front surfaces of the plate, in the framework of the original Gurtin–Murdoch model. The boundary conditions on the cylindrical surface of a circular hole in a nanoplate are derived in terms of a complex variable in the case of the plane stress state. The solution of the two-dimensional problem for an infinite plane with a circular hole under remote loading is explicitly obtained. Based on the analytical solution, we investigated the dependence of the elastic stress field on the nanosised plate thickness and dimension of the hole. Numerical examples are given in the paper to illustrate quantitatively the effect of the plate thickness at the nanoscale on the stress field at and near the cylindrical surface. The results are presented graphically as the dependence of the components of the stress tensor on the polar angle.

Abstract

To address the issue of balancing complexity and accuracy in existing analytical solutions for seepage fields, this study considers the influence of lining thickness and employs the method of images to obtain analytical solutions for the seepage field in deeply buried underwater tunnels. The advantage of this solution lies in its simplicity and relatively high accuracy. Building upon this, the seepage field is applied to the equilibrium equation of the lining in the form of volume forces, and the stress solution for the lining is obtained using the stress function method. Furthermore, the validity of the analytical solution is established by examining tunnel engineering cases and comparing them with existing solutions. The study's findings highlight that enhancing the impermeability of the tunnel lining effectively reduces water infiltration but leads to increased external water pressure on the lining. Moreover, the relationship between water inflow and external water pressure shows approximate linearity with the burial depth of the tunnel. Furthermore, as the lining's permeability decreases and the burial depth increases, the radial normal stress within the lining intensifies. Particularly noteworthy is the observation that the circumferential normal stress experiences a more rapid increase over distance.