Discrete Element Methods

Discrete element methods (DEMs) comprise different techniques suitable for a simulation of dynamic behaviour of systems of multiple rigid, simply deformable (pseudo-rigid), or fully deformable separated bodies of simplified or arbitrary shapes, subject to continuous changes in the contact status and varying contact forces, which in turn influence the subsequent movement of bodies. Such problems are nonsmooth in space (separate bodies) and in time (jumps in velocities upon collisions) and the unilateral constraints (nonpenetrability) need to be considered. A system of bodies changes its position continuously under the action of external forces and interaction forces between bodies, which may eventually lead to a steady state configuration, once static equilibrium is achieved. For rigid bodies, the contact interaction law is the only constitutive law considered, while the continuum constitutive law (e.g. elasticity, plasticity, damage, fracturing) needs to included for deformable bodies. Computational modeling of multibody contacts (both the contact detection and contact resolution) represents the dominant feature in DEMs, as the number of bodies considered may be very large. If the number of potential contact surfaces is relatively small (e.g. nonlinear finite element analysis of contact problems), it is convenient to define groups of nodes, segments, or surfaces that belong to a possible contact set a priori. These geometric attributes can then be continuously checked against one another and the kinematic resolution can be treated in a very rigorous manner. Bodies that are possibly in contact may be internally discretized by finite elements and their material behaviour can essentially be of any complexity. The category of DEMs specifically refers to simulations involving a large number of bodies where the contact locations and conditions cannot be defined in advance and need to be continuously updated as the solution progresses. DEMs are most frequently applied to macroscopically discrete system of bodies (jointed rock, granular flow) but have also beeen successfully utilized in a microscopic setting, where very simple interaction laws between individual particles provide the material behaviour observed at a homogenized, macroscopic level. The DEM is most commonly defined as a computational modeling framework that allows finite displacements and rotations of discrete bodies, including complete detachment and recognises new contacts automatically, as the calculation progresses. There are many methods (e.g. DEM, RBSM (rigid block spring method), DDA (discontinuous deformation analysis), DEM/FEM (combined discrete/finite elements), NSCD (nonsmooth contact dynamics)), which belong to a broad family of DEMs. Although these methods appear under different names and each of them is developing in its own right, there are many unifying aspects and a more general framework is emerging, which allows for equivalence between these apparently different methodologies to be recognised. Possible classification may be based on the manner these methods address (i) detection of contacts, (ii) treatment of contacts (rigid, deformable), (iii) deformability (constitutive law) of bodies in contact (rigid, deformable, elastic, elasto-plastic etc), (iv) large displacements and large rotations, (v) number (small or large) and/or distribution (loose or dense packing) of interacting bodies considered, (vi) consideration of the model boundaries, (vii) possible subsequent fracturing or fragmentation, and (viii) time stepping integration schemes (explicit, implicit). DEMs are also used for problems where the discrete nature of the emerging discontinuities needs to be taken into account. Application ranges from modeling problems of a discontinuous behaviour a priori (granular and particulate materials, silo flow, sediment transport, jointed rocks, stone or brick masonry) to problems where the modeling of transition from a continuum to a discontinuum is more important. Increased complexity of different discontinuous models is achieved by incorporating the deformability of solid material and/or by more complex contact interaction laws, as well as by the introduction of some failure or fracturing criteria controlling the solid material behaviour and the emergence of new discontinuities. The chapter covers basic ideas behind the discrete element framework and ways of regularizing of nonsmooth contact conditions. It further discusses typical methods of geometrically characterizing interacting bodies, as well as algorithms for contact detection. Further topics include the imposition of contact constraints and boundary conditions, various ways of modeling block deformability. Modeling of fragmentation and transition from continuum to discontinuum is followed by an account of commonly adopted time interatation schemes. Finally, associated frameworks and unifying aspects of different frameworks belonging to the category of DEMs is discussed. Keywords: discontinuous modeling; discrete element method; discontinuous deformation analysis; granular materials; particulate materials; contact detection