In the original 3-D DDA formulation proposed by Shi, a first-order displacement function was utilized to describe the block movement and deformation. Therefore, stresses and strains throughout the block are assumed constant and the capability of block deformation is limited. In this paper, a third order displacement function is proposed for analysis of problems using 3-D DDA. The third-order displacement function allows nonlinear distribution of stress and strain within a discrete block, which significantly enhances the ability of 3-D DDA as an analysis technique. An illustrative example is presented to show the improvement achieved by this model. The calculated results show close agreement with the theoretical solutions.
The Discontinuous Deformation Analysis (DDA) is well-suited for investigating fractured rock mass behavior important to many geotechnical and structural problems [1, 2]. The method has the following major characteristics :
The principle of minimum total potential energy is used to calculate an approximate solution similar to FEM.
Dynamic and static problems can be solved by applying the same formulations.
Any constitutive law can be incorporated.
Any contact criterion (i.e., Mohr-Coulomb criterion), boundary conditions (i.e., constraint displacement), and loading conditions (i.e., initial stress, inertia force, volume force, etc.) can be modeled.
In the original DDA formulation, a first-order displacement function was used to model block deformations, which does not allow for variable stress/strain distribution within a block. This approximation precludes the application of the firstorder polynomial function to problems with significant stress variations within blocks. This may yield unreasonable results when the block deformation is large or when the geometry of the block is irregular. In two dimensions, to overcome this shortcoming, some approaches have been attempted. One approach adopted was to glue small blocks together in an artificial manner to form a larger block. Ma et al. , Koo et al.  and Hsuing  implemented high-order displacement functions into the DDA algorithm. Shyu , Chang  and Grayeli & Mortazavi  implemented finite element mesh into the original DDA blocks to account for stress variations within the blocks. In 3-D, there have been some published works but they use a linear displacement function as in the original 2-D DDA, so the stresses and strains within each block are constant. In this paper, the 3-D DDA with third-order displacement functions is derived. The Visual C++ .Net code for the third-order 3-D DDA has been developed and used to calculate a cantilever beam deformation under a force.
2. FUNDAMENTALS OF 3-D DDA
In the DDA method, the formulation of blocks is very similar to the definition of a finite element mesh. A problem of the finite element type is solved in which all elements are physically isolated blocks bounded by preexisting discontinuities. When the blocks are in contact, Coulomb?s law is applied to the contact interfaces, and the simultaneous equilibrium equations are formed and solved at each loading or time increment. The large displacements are the accumulation of incremental displacements and deformations at successive time steps.