Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions

Abstract

An analytical solution is developed in this paper for viscoelastic axisymmetric plane problems under stress or displacement boundary condition involving time-dependent boundary regions using the Laplace transform. The explicit expressions are given for the radial and circumferential stresses under stress boundary condition and the radial displacement under displacement boundary condition. The results indicate that the two in-plane stress components and the displacement under corresponding boundary conditions have no relation with material constants. The general form of solutions for the remaining displacement or stress field is expressed by the inverse Laplace transform concerning two relaxation moduli. As an application to deep excavation of a circular tunnel or finite void growth, explicit solutions for the analysis of a deforming circular hole in both infinite and finite planes are given taking into account the rheological characteristics of the rock mass characterized by a Boltzmann or Maxwell viscoelastic model. Numerical examples are given to illustrate the displacement and stress response. The method proposed in this paper can be used for analysis of earth excavation and finite void growth.