Numerical evaluation of the effective elastic moduli of rocks

In this paper we describe a two-dimensional numerical algorithm to calculate overall elastic properties of rocks. We assume that rocks can be viewed as an assemblage of blocks divided by discontinuity surfaces such as large-scale fractures. For each block we use the concept of representative volume element. We model a representative volume element as a finite body containing multiple defects in the form of inhomogeneities, pores, and microcracks and use simplified assumptions regarding the shapes of the defects. Effective moduli for all blocks can be used as input data for a boundary element (or finite element) computer code to model behavior of rock masses at a different length scale as an assemblage of homogeneous blocks of different shapes and elastic properties. The approach is based on a numerical solution of a hypersingular integral equation for a representative volume element. The use of global approximations and a specially tailored fast multipole technique allows one to efficiently solve large-scale problems and accurately calculate the elastic properties of an equivalent homogeneous material. An additional benefit of the approach is that it allows for an accurate representation of the displacement and stress fields anywhere within the material. A series of computer experiments is conducted to demonstrate the efficiency and accuracy of the algorithm. The possible extension of the approach for three-dimensional problems is discussed.