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| Complex variable solutions for elliptical hole involving time-dependent boundary in viscoelastic infinite plane |
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Abstract The structures involving time-dependent boundary regions are commonly encountered in engineering, i.e. structures under construction in civil engineering. For tunnel excavations in rheological rock mass, this paper presents the analytical solutions of problem with elliptical hole in viscoelastic infinite plane by using the complex variable method and corresponding principle of time-dependent boundary problem in combination. First, basic formulations of complex variable method are established for general viscoelastic problems involving time-dependent boundary regions. Then, based on the derived potentials with respect to ? in the reference, the potentials in z-plane are obtained by introducing the inverse mapping function, and therefore the variable t used in Laplace transformation is decoupled with the t which comes from mapping function. At last, the expressions of displacement and stress are derived for the general cases of viscoelasticity. In addition, the Boltzmann viscoelastic model is chosen as an example to obtain the exact solutions of stress and displacement in integral form by substituting the material parameters into the general expressions. The comparison between the specific analytical and FEM solutions is made to validate the correctness of the derivation and further analyses are performed to illustrate the influence of boundary varying process on the relationship between stress (displacement) and time. The results show that the variations of displacement and stress are correlated with boundary varying speeds. The solutions can be used in mechanical analysis and preliminary design of underground elliptical tunnel excavation. Furthermore, the method in this paper is also suitable for the analysis of the underground excavation problems in arbitrary sharp.
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Received: 26 November 2012
Published: 28 February 2014
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2014, Vol. 35 
