Abstract
This paper reports new developments on the complex variables boundary element approach for solving three-dimensional problems of cracks in elastic media. These developments include implementation of higher order polynomial approximations and more efficient analytical techniques for evaluation of integrals. The approach employs planar triangular boundary elements and is based on the integral representations written in a local coordinate system of an element. In-plane components of the fields involved in the representations are separated and arranged in certain complex combinations. The Cauchy- Pompeiu formula is used to reduce the integrals over the element to those over its contour and evaluate the latter integrals analytically. The system of linear algebraic equations to find the unknown boundary displacement discontinuities is set up via collocation. Several illustrative numerical examples involving a single (penny-shaped) crack and multiple (semi-cylindrical) cracks are presented.
1. INTRODUCTION
Accurate three-dimensional modeling of fracture is of key importance for rock mechanics applications such as simulation of mining in faulty rock or computer simulation of hydraulic fracturing. In both applications, the Boundary Element Method (BEM) can serve as an efficient tool for realistic modeling of mechanical deformation of fractured rock. In previous papers [1, 2] we introduced a new approach for solving three-dimensional crack problems. The approach featured the following new elements:
• The use of complex variables to create various combinations of the fields, e.g. in-plane components of tractions, displacement discontinuities, as well as geometric parameters.
• The use of triangular elements and analytical integration over those elements.
• The use of the "limit after discretization" procedure, i.e. enforcing the boundary conditions after the discretization and analytical handling of the internal fields.
The approach was illustrated in [1] for a simple case of constant approximations of the unknowns. In [2], the method was further extended to include higher order approximations of unknowns. In addition, a new technique is employed to analytically evaluate the involved integrals. It is based on the reduction of the area boundary integrals over an element to those over its contour using complex integral representations. The technique allows for enrichment of the library of boundary elements by including those bounded by combination of straight lines and circular arcs.
