The joint polygons and block systems arc three dimensional. Most of joints are not perpendicular to a given two dimenional cross section. Therefore the two dimensional computations of jointed rocks or block systems are of limited reliability and accuracy. The three dimensional analyses of block systems are important. Three dimensional discontinuous deformation analysis (3-d DA) forms blocks directly from general polygons. The blocks can bc convex or concave. Also. the blocks can have any numbers of polygons as their faces. The discontinuous contacts between 3-d blocks arc the main part of 3-d DDA. There are too many ways of three dimensional contacts in compare with two dimensional contacts. For the friction law, the two dimensional sliding directions form only a linc; while the three dimensional sliding directions form a whole plane. The paper will also present the 3-d block matrices such as mass matrix, stiffness matrix, point load matrix, body load matrix, initial stress matrix and fixed point matrix.
Same as two dimensional discontinuous deformation analysis (2- d DDA), 3-d DDA uses time steps for both statics and dynamics. large deformation in the blocks and the large relative movements between blocks are the accumulation of many time steps. For large deformation, the statics arc the stabilized state of dynamics due to frictions or real damping. The current three dimensional DDA code treat the damping in a rather simple way: the dynamic computation inherits the full velocity at the end of the previous time step: the static computation inherits only a part of the velocity at the end of the previous time step as the initial velocity at the beginning of this time step. The DDA computation should satisfy following two Conditions at the end of each time step: (J) Each degree of freedom of each block has a equilibrium equation: The simultaneous equilibrium equations are derived by minimizing the total potential energy at the end of each time step. " All external forces acting on each block including loads and Contact forces with other blocks reach equilibrium in X, Y, Z directions andeach moment equilibrium for rotation axis X, Y,Z. The equilibrium is also achieved between the block stresses and.the external forces acting on the block. (-) The entrance theory is used to find all possible first entrance positions.The contacts occur only on the first entrance position, then the penetrations are prevented on the first entrance positions and sliding is controlled by the friction law. Within each time step, if the tensile force from the normal Contact spring exceeds the limit, this normal spring will be removed. If penetration occurs in a entrance position, a normal spring is applied. The global equations have to be solved repeatedly while selecting the closed entrance positions. The procedure of adding or removing springs and solving equilibrium equations is open-close iteration. The open-close iteration will continue until all tensile force and all penetrations arc in the limits over all the entrances.
