Large deformations and rotations of rock blocks may occur under high stresses around deep underground rock engineering works such as deep mines and deep tunnels. The original discontinuous deformation analysis (DDA) was developed by Shi and Goodman to analyze large deformations, rotations and displacements of rock blocks by accumulating small components of these quantities in a time-marching scheme. The small rotation angle approximation adopted in the original DDA may induce block expansion with rotation (free expansion). Some methods have been used to study and reduce the rotation errors, including the Taylor series method and trigonometric method. Based on mechanics, the free expansion is caused by the error due to the approximation of the real behavior using a displacement function, and the theory used to describe the geometrical relationship between movement (displacement and rotation) and deformation. The original DDA uses the geometrical relationship that is based on small strain theory. Small strain theory gives a linear relationship between displacement and strain that does not consider the high order components and the decomposition of rotation and displacement. The finite deformation theory, however, decomposes the displacement and rotation and gives a nonlinear relationship between displacement and strain. Therefore the finite deformation theory can handle the block rotation problem more correctly. In our previous work, we compared the displacement field around a circle tunnel obtained from the finite deformation theory and small strain theory. The results show that the difference increases as the deformation increases. The rotation error due to the small strain assumption and the validity of the finite deformation theory were also studied by Chen using an analytical method. In this paper, we use the finite deformation theory to adjust the displacement and strain components in DDA to study the free expansion of blocks in a model in which a rock block falls down a slope. The area of the block is monitored during the fall. The expansion of the block computed by DDA modified by the finite deformation theory, and the original DDA will be compared. The result shows that the DDA modified with finite deformation theory can eliminate the free expansion.
The discontinuous deformation analysis (DDA) developed by Shi and Goodman was a 2-D numerical method for the statics and dynamics of discontinuous block systems. Shi adopted the small angle approximations (sin r0˜0, cos r0˜1) in the original DDA formulation for the displacements of a point within the block. This simplification is convenient for formula derivation and computationally efficient and fast, particularly for small rotation of the blocks. However, the linearization of the rotational displacements causes the blocks to expand with every increment of rotation and results in distortion of stress and velocity fields. Ke proposed to use a postadjustment with a maximum rotation limit, 0.1 radian, set up to reduce the errors in computing contact forces.This approach is easy to implement and can prevent the blocks from expanding. Koo and Chern also proposed to use linear displacement function and post-correction.