A Generalized 3D-DDM for both Fracture-Type and Asymmetric Closed-Contour Problems: Modified Fully Analytical Approach and Applications

The displacement discontinuity method (DDM) has been widely used for large-scale engineering problems, e.g., hydraulic fracturing related to metal mining and hydrocarbon extraction. However, there is a lack of detailed explanation for the implementation of 3D-DDM in closed-contour boundary problem, and applications to address asymmetric closed-contour boundary problem remains limited. In this paper, we have theoretically derived and elaborated the numerical procedure, identified the caveats of DDM in solving fracture-type and closed-contour boundary problems. Subsequently, we modified the analytical integral and differential expressions for the basic solution on 3D-DDM based on triangular elements and provided a more general suite of expressions. Additionally, implementation details including element traversal criteria and precautions against rigid body movement are introduced to address the asymmetric closed-contour boundary problem. The accuracy of the generalized 3D-DDM were verified through simulations involving a penny-shaped fracture and a spherical cavity. Finally, the generalized 3D-DDM was employed to elucidate the stress redistribution and interference with respect to the tunnel excavation. This study demonstrates that: (1) the modified analytical integral and differential expressions combined with the element traversal criterion and precautions against rigid body movement provide a detailed explanation of how 3D-DDM can be applied to asymmetric closed-contour boundary problem; (2) during tunnel excavation, the arch flanks and floor would suffer from the biggest damage risk; and (3) the interaction effect between adjacent tunnels plays a role when the tunnel spacing is less than eight times the radius of the arch.