A COMPLEX VARIABLE SOLUTION FOR A DEFORMING CIRCULAR TUNNEL IN AN ELASTIC HALF-PLANE

Abstract

SUMMARY An analytical solution is presented of problems for an elastic half-plane with a circular tunnel, whichundergoesacertaingivendeformation.Thesolutionusescomplexvariables,withaconformalmappingontoa circular ring. The coeƒcients in the Laurent series expansion of the stress functions are determined bya combination of analytical and numerical computations. As an example the case of a uniform radialdisplacement of the tunnel boundary is considered in some detail. It appears that a uniform radial displace-mentisaccompaniedbyadownwarddisplacementofthetunnelasawhole.Thisphenomenonalsomeansthatthedistribution oftheapparentspring constantis strongly non-uniform. ( 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods geomech., vol. 21, 77 — 89 (1997)(No. of Figures: 7 No. of Tables: 1 No. of Refs: 10) Key words: elasticity; tunnel; complex variables INTRODUCTIONIn this paper the stresses and displacements in an elastic half-plane due to the deformation of acircular tunnel are considered. The method used is the complex variable method. 1 The boundaryconditions arethat the upper boundaryof the half-planeisfreeof stress, andthat at theboundaryofthetunnelthedisplacementisprescribed.Thisisusuallycalledthesecondtypeofboundarycondition.In order to solve the problem, a conformal transformation onto a circular ring is used, and in thetransformed plane the complex stress functions are represented by their Laurent series expansions.IntheclassicaltreatisesofMuskhelishvili1andSokoliniko⁄2ontheapplicationofthecomplexvariable method in elasticity, the class of problems studied here, involving a multiply connectedregion and conformal mapping onto a circular ring, is briesy mentioned, but it is stated thatOdiƒcultiesO arise in the solution of these problems, and it is suggested to use another method ofsolution, such as the method using bipolar co-ordinates.3