Practical solutions for complex variable function for radius of tunnel plastic zone under non-hydrostatic pressures and their application extension

In order to address the non-circular plastic boundary of rock circular tunnels under non-hydrostatic pressures, a simple complex variable function for the stress in an elastic zone is firstly introduced by using the analogy method. Based on the elastic perfectly-plastic model and the Mohr-Coulomb criterion, a practical solution for complex variable function for the radius of tunnel plastic zone is then derived under the stress continuity at the elastic-plastic boundary of surrounding rocks. Additionally, an extended solutions for complex variable function for the radius of tunnel plastic zone is proposed by adopting the elastic-brittle-plastic model and the unified strength theory to take into consideration both the post-peak brittle decrease and the effect of the intermediate principal stress of rock strength. Finally, the application conditions for the proposed solutions are provided, and they are compared with the available theoretical solutions and the measured data in the literatures. The results show that the practical solution for complex variable function has the advantages of explicitly analytical formulation as well as convenient calculation and analysis, whose validity and accuracy are sufficiently demonstrated by the perturbed solution and high-precision solution for complex variable function in the literatures. The extended solution retains all the advantages of the original practical solution and has good consistency with the measured loosening depth of a deep tunnel. Moreover, the extended solution has a broad application prospect due to the fact that it reasonably accounts for the post-peak brittle decrease and the effect of the intermediate principal stress of rock strength.