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

Luo-bin Lin, Fu-quan Chen, Chang-jie Zheng, Yi-qun Huang

TL;DR

The paper presents a time-dependent complex variable method for shallow tunnelling in visco-elastic gravitational geomaterial, addressing the far-field displacement singularity by fixing the far-field ground surface. It recasts the quasi-three-dimensional problem as a plane-strain problem with a time-dependent convergence coefficient $U(t)$ and uses Verruijt conformal mapping to transform the mixed boundary value problem into a Riemann-Hilbert framework solved iteratively for numerical stability. The approach yields time-dependent stress and displacement fields that agree well with a reference finite element solution while delivering substantial computational efficiency. It also provides insights into how excavation rate and material rheology influence displacement evolution and notes limitations related to non-porous material assumptions and potential infinite-domain paradoxes.

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.

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

TL;DR

The paper presents a time-dependent complex variable method for shallow tunnelling in visco-elastic gravitational geomaterial, addressing the far-field displacement singularity by fixing the far-field ground surface. It recasts the quasi-three-dimensional problem as a plane-strain problem with a time-dependent convergence coefficient and uses Verruijt conformal mapping to transform the mixed boundary value problem into a Riemann-Hilbert framework solved iteratively for numerical stability. The approach yields time-dependent stress and displacement fields that agree well with a reference finite element solution while delivering substantial computational efficiency. It also provides insights into how excavation rate and material rheology influence displacement evolution and notes limitations related to non-porous material assumptions and potential infinite-domain paradoxes.

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.
Paper Structure (21 sections, 114 equations, 9 figures, 2 tables)

This paper contains 21 sections, 114 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formation
  3. Time-dependent mechanical model on shallow tunnelling in gravititational geomaterial
  4. Stage unification and mechanical model of static equilibrium
  5. Mixed boundary value problem and conformal mapping
  6. Riemann-Hilbert problem transformation
  7. Solution of plane-strain Riemann-Hilbert problem
  8. Solution preparation
  9. Iterative solution
  10. Time-dependent stress and displacement fields
  11. Analytical solution
  12. Numerical results
  13. Case verification
  14. Parameter selection
  15. Examination of mixed boundary conditions
  16. ...and 6 more sections

Figures (9)

  • Figure 1: Schematic diagram of time-dependent shallow tunnelling in gravititional geomaterial
  • Figure 2: Bidirectional conformal mapping of the remaining geomaterial
  • Figure 3: Stress components and deformation on the ground surface against time
  • Figure 4: Finite element model for verification
  • Figure 5: Comparisons of stress and displacement components along ground surface (a) and tunnel periphery (b) for time moment $t = t_{4}$
  • ...and 4 more figures