Mathematical Physics - Volume II - Numerical Methods

6.5 Discrete Element Methods

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contact strength model (represented by the Kelvin element) and a slip model, the only difference in non-cohesive and cohesive applications is the ability transmission of the tensile force in the normal direction. As noted above, DEM models allow the particles of the cohesive material to be interconnected but also separated, if the bond-rupture criterion is met. In the case of the action of an external load such that either the tensile strength or the limit deformation or the fracture energy are exceeded, the bonds between the particles are separated and a crack is formed on the scale of the model (meso, macro). Therefore, damage modes and their interactions naturally arise from the process of gradual particle separation. The DEM approach to discretization of the computational domain is the most pronounced advantage over the continuum-based methods of computational mechanics, since common problems (such as dynamic composite response, crack singularity, crack formulation criterion itself) can be avoided due to naturally discontinuous and random representation of material meso-structure [136]. Application of DEM for rocks Rock mechanics is the discipline from which DEM originated. The basic idea is to repro duce the quasi-brittle behavior of rocks by simulating the nucleation, growth, branching and merging of local cracks. Although rocks may not look like granular materials at first glance, the main features of many types of rocks and, especially, rock massifs (Figures 6.2h,i) are the pre-existing damage and the high degree of heterogeneity and discontinuity of their structure (at various spatial scales). That is the reason why rock massifs can be considered as conglomerates of discrete blocks interconnected by different models of cohesive forces (“blocky rock systems” [99]). Therefore, the mechanical behavior of the whole jointed-rock assembly evolves from the collective contribution of these dis crete blocks during loading. Accordingly, the separation of two discrete units mimics the elementary meso-damage event, which represents the basic building-block of complex damage-evolution phenomena. Detailed reviews by Jing [137] and Jing and Stephansson [98] include techniques, advances, problems, and then predictions of future directions of development in computational rock modeling. In general, the discrete elements may represent separate rock blocks of (up to) tone and meter levels that, in 2D-DEM rock simulations, can be modeled with randomly generated circles, ellipses, or convex polygons interconnected by introducion of a specific bond into the contact area. The shapes and methods of packing of discrete elements have far-reaching effects on the distribution and intensity of interaction forces. The bond strengths may be allowed to vary from contact to contact, which may represent another source of heterogeneity in the simulated material. The very influential explicit DEM method in rock mechanics is Cundall’s method of "distinct" elements [132], [138] with quadrilateral / prismatic blocks developed in the computer programs UDEC and 3DEC (Itasca TM Consulting Group, www.itascacg.com). The focus of this brief introduction to DEM implementation for rocks will be on a simpler DEM – the bonded-particle model (also, often called the parallel-bond model) [128]. This model is based on circular / spherical discrete elements as illustrated in Figure 6.20. This approach to modeling has been generalized by Potyondy [139] and developed over the years through the commercial packages PFC 2D and PFC 3D (Itasca TM Consulting Group).