A new three-dimensional boundary element technique is presented for evaluating gravitational stresses in rock containing a single or multiple fractures. The boundary element analysis is based on the use of the integral equation written for the tractions on the boundaries, including internal boundaries from an excavation and existing discontinuities


The novel boundary element technique is proposed for solving three-dimensional elastostatic problems of fractured rock under gravitational load. The analysis is based on the use of the traction integral representation written for an internal point located on a plane characterized by some normal vector. This representation is obtained by using Somigliana’s displacement identity that includes an additional volume integral associated with the gravitational force. This integral is converted to a surface integral using a conventional technique that employs the so-called Galerkin vector [1]. By applying the traction operator to the resulting boundary-only-identity, the integral representation for the tractions is obtained. This representation includes the case when the part of the boundary contains crack surface. In the latter case, the corresponding representations for the regions that share the crack as a part of their boundaries are summed, so the final representation involves the displacement discontinuity on the crack surface rather than displacements, e.g. [2]. A novel element consists of the use of complex variables to create a complex combination of shear components of tractions. Thus, the resulting representation includes one real variable equation for the normal traction and one complex variable equation for the shear tractions. The development allows one to reduce the effort in analytical integration over the triangular elements as compared to real variable-based approaches [3, 4]. It is shown that all the integrals involved in the formulation can be expressed via a few basic integrals, some of which are presented in the paper.

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