Complex variable solution on over-/under-break shallow tunnelling in gravitational geomaterial with reasonable far-field displacement

Over-/under-break excavation is a common phenomenon in shallow tunnelling, which is nonetheless not generally considered in existing complex variable solutions. In this paper, a new equilibrium mechanical model on over-/under-break shallow tunnelling in gravitational geomaterial is established by fixing far-field ground surface to form a corresponding mixed boundary problem. With integration of a newly proposed bidirectional composite conformal mapping using Charge Simulation Method, a complex variable solution of infinite complex potential series is subsequently derived using analytic continuation to tranform the mixed boundaries into a homogenerous Riemann-Hilbert problem, which is iteratively solved to obtain the stress and displacement in geomaterial. The infinite complex potential series of the complex variable solution are truncated to obtain numerical results, which is rectified by Lanczos filtering to reduce the oscillation of Gibbs phenomena. The bidirectional conformal mapping is discussed and validated via several numerical cases, and the subsequent complex variable solution is verified by examining the Lanczos filtering and solution convergence, and comparing with corresponding finite element solution and existing analytical solution. Further discussions are made to disclose possible defects of the proposed solution for objectivity.