Three-dimensional discontinuous deformation analysis derived from the virtual work principle with a simplex integral on the boundary

Abstract

Three-dimensional discontinuous deformation analysis (3D-DDA) is a powerful deformable discrete element method that is derived from the principle of minimum potential energy. However, the principle cannot be easily applied for all constitutive models, and certain forces do not have corresponding potential energy, which can impede DDA from being applied to more complex problems. In addition, since simplex integration in 3D can only be applied to tetrahedrons, no accurate integral is presented in 3D-DDA for polygonal boundaries of 3D blocks. Thus, face constraints have rarely been employed in 3D DDA. This study derives the DDA governing equation from the virtual work principle. The submatrices of stress, inertia, point force, face loading, body force, fixed boundary, roller boundary, bolts, implicit contact forces, and explicit contact forces are reformulated. An accurate integral on polygonal boundaries of 3D blocks is specified with the face integral of scalar fields and 2D simplex integration. Several numerical examples corroborate the correctness of the reformulation and merits of the face constraints.