Far-field displacement singularity elimination for time-dependent complex variable method on quasi-three dimensional gravitational shallow tunnelling

TL;DR

This paper presents a method to eliminate displacement singularities in time-dependent complex variable analysis of quasi-three dimensional shallow tunnelling, improving accuracy and stability in geomechanical modeling.

Contribution

It introduces a novel approach to remove displacement singularities by fixing the far-field ground surface and solving a mixed boundary value problem with iterative methods.

Findings

Good agreement with finite element solutions.

High numerical stability and efficiency.

Insights into excavation rate and viscosity effects.

Abstract

This paper identifies the nonzero resultant and consequent unique displacement singularity of time-dependent complex variable method on quasi-three dimensional shallow tunnelling in visco-elastic and gravitational geomaterial. The quasi-three dimensional problem is equivalently simplified into a plane-strain one using a time-dependent coefficient of convergence confinement method to simulate the progressive release of initial stress field. The unique displacement singularity is thereby eliminated by fixing the far-field ground surface to produce corresponding counter-acting force to equilibriate the nonzero resultant to formalize a strict equilibrium mechanical model. The mixed boundaries of fixed far-field ground surface and nearby free segment form a homogenerous Riemann-Hilbert problem with extra constraints of the virtual traction along tunnel periphery, which is simultaneously solved using an iterative linear system with good numerical stability. The mixed boundary conditions along the ground surface in the whole excavation time span are well satisfied, and detailed comparisons with corresponding finite element solution are conducted. The comparison results are in good agreements, and the proposed solution illustrates high efficiency. More discussions are made on excavation rate, viscosity, and solution convergence. A latent paradox is additionally disclosed for objectivity.