Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

270

In line with the observed problems with oversimplified particle shapes, the later model improvements include the use of breakable, deformable, polygonal discrete elements which, in the long run, led to the development of UDEC and 3DEC (Itasca TM Consulting Group) software packages [108]. These programs nowadays represent an indispensable tools for the calculation od intermittent (fractured and jointed) rock massed in the vicinity of underground facilities (Figure 6.21). Potyondy [141] improved this model by including time-dependent behavior by intro ducing into the parallel bond the law of damage rate that mimics stress corrosion. Mas Ivars and co-authors [142] performed an important synthesis and rounding up of the whole approach by extending the model of parallel bonds to simulate natural cracks in rock masses by including joints on scales larger than the grain scale. When 3DEC is used to study the stability of cracked rock masses, the considered block is usually divided into several discrete segments defined by the intersection of natural cracks or joints that are piecewise straight (Figure 6.21). These segments (which emulate jointed rock blocks; see Figure 6.1h) are then assigned constitutive properties. The external boundary conditions of discrete segments are set while the internal boundary conditions are calculated from contact interactions which allows explicit modeling of existing cracks in fractured rock [108]. Finally, a detailed overview of the bonded-particle method as a research tool in rock mechanics and engineering applications with current trends and future directions of development is presented by Potyondy [139]. In that reference, an exhaustive list of papers is given in which the method is applied in a wide range of phenomena in rock mechanics and beyond. Equivalence of beam lattice and DEM representations A set of nodal points (Figure 6.22a), randomly located in general, can be connected with beams and represented by the beam lattice (Figure 6.22b). Alternatively, these nodal points could be considered centers of circular/spherical particles (Figure 6.22c) connected in the manner illustrated by Figure 6.18d; this particle network is the typical DEM. In general, the stiffness between two particles engaged in contact is represented by the three types of springs (a normal, a shear and a rotational defined, respectively, by the spring constants k n , k t , k a ), which is depicted by the symbol shown in Figure 6.22d.

Figure 6.22: (a) A set of nodal points in space and the corresponding: (b) beam lattice and (c) DEM (network of circular particles). (d) The symbol for the particle contact reflecting three types of springs: a normal, a shear, and a rotational.