Solution of the elasto-plastic interface of circular tunnels in Hoek–Brown media subjected to non-hydrostatic stress

Abstract The Hoek-Brown (H-B) failure criterion is widely used in rock engineering. A solution of the elasto-plastic interface is presented for the plane strain problem of a pressurized circular tunnel in H-B media subjected to non-hydrostatic stress at infinity. The non-circular elastic zone in the physical plane can be transformed into the outer region of the unit circle in the imagine plane. Thus, the problem of determining the elasto-plastic interface is turned to solving the coefficients of the conformal mapping functions, which can be considered as an inverse problem in elastic mechanics. The core of analysis is that the complicated expressions of stress combination on the elasto-plastic interface are expanded into Fourier series, and the coefficients in Fourier series are determined by the linear complex equations. Then, the nonlinear complex equations containing the coefficients for conformal mapping functions are established. In order to find the elasto-plastic interface, these nonlinear equations are solved using the optimization method of mixed penalty functions. The correctness of the method is verified in both analytical and numerical ways. Based on the programmed computing, the influences of various parameters on the elasto-plastic interface are discussed.