Explicit Complex Variable Solution for the Response of Surrounding Rock in an Arbitrarily Shaped Tunnel Considering Rheological Effects

The theoretical analytical solutions for the response of surrounding rock under far-field loads typically take the form of complex variable functions involving numerous high-order variables, which leads to low computational efficiency and poor readability. In this study, a simplified approach applicable to the analytical solutions of surrounding rock responses with arbitrary-shaped tunnels is proposed. The surrounding rock containing the orifice is equivalent to an infinite plane subjected to far-field stress, and the elastic analytical solution is simplified based on the biharmonic equation in solid mechanics. This analytical solution is a polynomial fraction consisting of no more than five terms. The error between this analytical solution and the theoretical result is < 0.1%, and the error relative to the numerical result is < 3%. Additionally, the viscoelastic analytical solution has also been obtained. The analytical solution intuitively demonstrates the relationship between the response results and key material parameters. The research findings can provide valuable references for the design of underground structures and engineering practices.