In the original 3-D DDA formulation, first-order displacement was assumed for block deformation, which precludes the application of it to problems with significant stress variations within blocks. This may yield unreasonable results when the block deformation is large and geometry of the block is irregular. Up to now, 3-D DDA with third-order displacement functions is developed. However, there are applications that may require using polynomials greater than the third-order to achieve better accuracy. This study presents the results of an effort to develop a more general approach in which the 3-D DDA is implemented with higher-order polynomial displacement functions. In this research, formulations of stiffness and force matrices in nth-order are presented and the codes have been programmed. An illustrative example is used to validate the new formulations and codes for different orders of displacement functions. By contrast, the results calculated for the same model by use of the third order 3-D DDA are far from the theoretical solution.
Discontinuous deformation analysis (DDA) is a discrete element method growing in popularity for geomechanical simulation. The DDA method is generally formulated using the principle of minimum potential energy [1,2].
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 first order 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, several 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.
A major shortcoming of the original 3-D DDA formulation is that, similar to the 2-D DDA approach, the block deformation is modeled using a first-order polynomial displacement function, which does not allow for variable stress/strain distribution within a block. This approximation leads to large errors in problems in which stress variations within the blocks are expected to be significant. Practical examples would include cases where the sizes of rock blocks spanning the joints and discontinuities are large enough so that their deformability becomes important. Beyabanaki et al. [10, 11] has proposed the use of a third-order displacement function but this may still be inadequate for applications that require the use of polynomials greater than the thirdorder to attain better precision.
In this paper, an effort is made to develop a more general formulation of the 3-D DDA and the codes have been developed and used to calculate deformation of a simply supported beam.
Similar to 2-D, 3-D DDA analyzes a problem as an assembly of discrete blocks.