Equivalent inclusion method for arbitrary cavities or cracks in an elastic infinite/semi-infinite space

Abstract

Shallow tunnels free of or subjected to the surface effects excavated in an elastic rock mass are usually simplified as an elastic infinite or semi-infinite plane problem with cavities in mechanics, respectively. In this paper, the equivalent inclusion method (EIM), a classic solution due to Eshelby usually used for predicting the elastic field caused by inhomogeneities embedded in a heterogeneous material, is extended to predict the elastic field induced by an arbitrarily shaped cavity. Similar to that of the inhomogeneity problems, the implementation of the EIM for the cavity problem is conducted by treating a cavity whose elastic modulus is zero as an inclusion having identical material properties to the matrix but containing eigenstrains. Based on an elementary solution for the eigenstress arising from the eigenstrains representing the cavity of an arbitrary shape can be calculated with the help of numerical discretization and superposition of contributions from each discretized element. A circular or rectangular cavity can be reduced to a crack when the distance between their upper and lower surfaces approaches an infinitesimal but not a zero value, hence the elastic field caused by cracks can be also resolved with the proposed solution method. The results obtained with the proposed method and the finite element method (FEM) for both the cavity and the crack are in good agreements. Parametric analyses on the effects of depth, length of the crack and the interactions among multiple cavities and cracks on the elastic field demonstrates the EIM as a good potential application in some significant applications in fracture behaviors for the excavated material.