6.2 Discontinuum Modeling
Discontinuum modeling methods treat the rock slope as a discontinuous rock mass by considering it as an assemblage of rigid or deformable blocks. The analysis includes sliding and opening of rock discontinuities controlled by the normal and shear stiffness of joints. It allows the deformation and movement of blocks relative to each other so it can model complex behaviour and mechanisms. It requires representative slope and discontinuity geometry, intact constitutive criteria, discontinuity stiffness and shear strength, groundwater characteristics and the in situ stress state. The major limitation of discontinuum modeling is that it requires representative discontinuity geometry (spacing, persistence, etc.) along with joint data and properties of each block. Discontinuities divide the problem domain into blocks that may be either rigid or deformable while continuum behavior is assumed within deformable blocks. The most widely used discrete element codes for slope stability studies are UDEC (Universal Distinct Element Code; Itasca Consulting Group, 2000) and 3DEC (3-Dimensional Distinct Element Code; Itasca Consulting Group, 2003). Several variations of the discrete element methodology are:
• Distinct element method;
• Discontinuous deformation analysis
• Particle flow codes.
Discontinuum methods treat the problem domain as an assemblage of distinct, interacting bodies or blocks subjected to external loads and expected to undergo significant motion with time. This methodology is collectively referred as the discrete element method (DEM) (Figure 5 to 7). The development of discrete-element procedures represents an important step in the modelling and understanding of the mechanical behaviour of jointed rock masses. This method of analysis permits sliding along and opening or closure between blocks or particles. The equation of dynamic equilibrium for each block in the system is formulated and repeatedly solved until the boundary conditions and laws of contact and motion are satisfied. The method thus accounts for complex non-linear interaction phenomena between blocks.
UDEC (11) uses a force-displacement law specifying interaction between the deformable joint bounded blocks and Newton’s second law of motion, providing displacements induced within the rock slope. It is particularly well suited to problems involving jointed media and has been used extensively in the investigation of both landslides and surface mine slopes. The influence of external factors such as underground mining, earthquakes and groundwater pressure on block sliding and deformation can also be simulated.
The discontinuous deformation analysis, DDA, developed by Shi (18) has also been used with considerable success in the modelling of discontinuous rock masses, both in terms of rockslides (19) and rockfalls (20).
DDA) is a type of discrete element method originally proposed by Shi in 1988. DDA is somewhat similar to the finite element method for solving stress-displacement problems, but accounts for the interaction of independent particles (blocks) along discontinuities in fractured and jointed rock masses. DDA is typically formulated as a work-energy method, and can be derived using the principle of Minimum Potential Energy (e.g., Shi) or by using Hamilton's principle. Once the equations of motion are discretized, a step-wise linear time marching scheme in the Newmark family is used for the solution of the equations of motion. The relation between adjacent blocks is governed by equations of contact interpenetration and accounts for friction. DDA adopts a stepwise approach to solve for the large displacements that accompany discontinuous movements between blocks. The blocks are said to be "simply deformable". Since the method accounts for the inertial forces of the blocks' mass, it can be used to solve the full dynamic problem of block motion.
DDA models a discontinuous material as a system of individually deformable blocks that move independently without interpenetration (Shi, 1988 and 1993). Its formulation is based on a dynamic equilibrium that considers the kinematics of individual blocks as well as friction along the block interfaces. The displacements and deformations of the blocks are the result of the accumulation of a number of small increments, corresponding to small time steps. The transient formulation of the problem, which is based on minimization of potential energy, makes it possible to investigate the progression of block movements with time.
Particle flow code allows the rock mass to be represented as a series of spherical particles that interact through frictional sliding contacts. Clusters of particles may be bonded together through specified bond strengths in order to simulate joint bounded blocks. High stresses induced in the rock slope breaks the bonds between the particles simulating the intact fracture of the rock in an approximate manner.
Figure 5: Simulation of highly intersecting joint by discontinuum code
Figure 6: Rock slope distinct-element model showing discretization of geometry blocks into finite-difference elements.
Figure 7: 3D Simulation of rock slope by 3Dec software based on finite difference method