This book describes formulations and computations of the BEM in transversely isotropic bi-material rocks. The work concentrates on the development of the BEM using the FORTRAN program, which are then applied to the two dimensional problems. This book is divided into six chapters. Chapter one defines the problem, introduce the undertaking of the study and outlines the method of approach adopted in this book. Chapter two provides an overview of linear elastic fracture mechanics studies of bi-material systems. Various solution techniques are introduced and studies concerning: (i) a crack lying along the interface, (ii) a crack terminating or crossing through an interface, and (iii) a wedge with its vertex on the interface are reviewed. In addition, a brief review of the numerical solution techniques, crack initial angle and the propagation path of the relevant literature. Chapter three provides a detailed account of the theoretical approach used to determine the stress and displacement fields using the boundary element method. This chapter includes the basic equations and fundamental solution of anisotropic elasticity, boundary element formulation, numerical discretization, stress intensity factor expression and the fracture propagation simulation. Chapter four proposes the determination of mixed mode stress intensity factors with the boundary element formulation. Numerical examples for determining the mixed mode stress intensity factors for several cracked materials are presented for isotropic and anisotropic media. Chapter five shows the experimental result of actual rocks and discussion, which includes the numerical results of the comparison with failure mechanism, the initial angle and the propagation path. Finally, Chapter six summarizes the findings and contribution of the current work.