Mathematical Physics - Volume II - Numerical Methods

6.5 Discrete Element Methods

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the following criteria for removal of beam elements

b ) i | , | M

( b ) j | ) max

( | M (

F ( b ) A ( b ) ±

σ ( b )

ζ

= σ cr ,

e f f =

W ( b )

(6.63)

F ( b ) / A ( b ) σ cr

Q ( b ) | / A ( b ) σ cr

+ |

> 1

where W ( b ) and ζ designate the section modulus of the beam, and the fitting parameter; while, σ cr and τ cr are the tensile strength and the shear strength, respectively. With beam lattices, it is possible to reproduce very complex patterns of damage. They are able to simulate the nucleation and propagation of microcracks, crack branching, crack curvature, bridging and coalescence of cracks etc., which results in a complete picture of macroscopic damage and fracture. The model also makes it possible to “capture” the effect of sample size. The advantages of this approach are simplicity and direct insight into the fracture process at the level of the microstructure. The recent development of lattice models and their “peculiarities” are reviewed recently by Nikolic´ and co-authors [97]. DEM is a Lagrangian technique of computational simulations in which the computational model is made of discrete (rigid or deformable) elements of the same (or even higher) Euclidean dimensionality as the analyzed problem. These discrete elements of different shapes interact through contact algorithms (smooth or rough contacts) [98]. The material is, therefore, modeled by a set of Voronoi cells (mimicking grains, granules, particles, aggregates) representative from the point of view of heterogeneity of material texture, whose meso-scale dynamic interactions determine its macroscopic behavior. Obviously, the contact algorithms are at the physical core of a group of computational techniques custom-made to solve problems characterized by extremely large discontinuities in the internal structure or geometry of materials or both [99]-[101]. DEM was introduced by Cundall [99] to analyze the intermittent progressive fracture of rocky slopes, to be later applied to the analysis of granular assemblies by Cundall and Strack [100]. Although DEM is now widely used for modeling different classes of materials (such as geomaterials, biomaterials, composites), the most natural applications are for simulations of deformation or flow of material systems that have the same topology as the representative group of discrete elements (e.g, Figure 6.1a,c-g). Cundall and Hart [102] summarized succinctly the methodological approach by defining DEM as a method that allows finite translations and rotations of discrete bodies 3 (rigid, solid, breakable), including complete separation of their mutual contacts, as well as automatic recognition of newly established contacts during simulation. The mechanical behavior of the whole 3 Throughout this chapter, the term “particle” (grain, as well) is used, when convenient, for the DEM building block – the discrete element; it should not be confused with particles used occasionally in relation with other CMD models in this introduction, which are material points.

6.5 Discrete Element Methods 6.5.1 Basic idea of DEM