A complex variable solution for a deforming circular tunnel in an elastic half-plane.

*(English)*Zbl 0894.73127Summary: An analytical solution is presented of problems for an elastic half-plane with a circular tunnel, which undergoes a certain given deformation. The solution uses complex variables, with a conformal mapping onto a circular ring. The coefficients in the Laurent series expansion of the stress functions are determined by a combination of analytical and numerical computations. As an example, the case of a uniform radial displacement of the tunnel boundary is considered in some detail. It appears that a uniform radial displacement is accompanied by a downward displacement of the tunnel as a whole. This phenomenon also means that the distribution of the apparent spring constant is strongly non-uniform.

##### MSC:

74L10 | Soil and rock mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

##### Keywords:

uniform radial displacement of tunnel boundary; conformal mapping; circular ring; Laurent series expansion; stress functions
PDF
BibTeX
XML
Cite

\textit{A. Verruijt}, Int. J. Numer. Anal. Methods Geomech. 21, No. 2, 77--89 (1997; Zbl 0894.73127)

Full Text:
DOI

##### References:

[1] | Some Basic Problems of the Mathematical Theory of Elasticity (translated from the Russian by J. R. M. Radok), Noordhoff, Groningen, 1953. |

[2] | Mathematical Theory of Elasticity, 2nd edn, McGraw-Hill, New York, 1956. |

[3] | Jeffery, Trans. Roy, Soc. Ser. A 221 pp 265– (1920) |

[4] | and , Photo-elasticity, Cambridge University Press, Cambridge, 1931. · JFM 57.1080.07 |

[5] | ’Stress distribution around a tunnel’, Trans. ASCE, 1117-1153 (1940). |

[6] | Sagaseta, Géotechnique 37 pp 301– (1987) |

[7] | and , Theory of Elasticity, 2nd edn, McGraw-Hill, New York, 1951. |

[8] | Complex Variable Solutions of Elastic Tunneling Problems, Geotechnical Laboratory, Delft University of Technology (available upon request), 1996. |

[9] | Duddeck, Die Bautechnik 10 pp 349– (1980) |

[10] | and , ’Surface settlements due to deformation of a tunnel in an elastic half plane’, Géotechnique, to be published. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.