Mathematical Physics - Volume II - Numerical Methods

6.5 Discrete Element Methods

263

the translational and rotational motion of the particle is defined as follows

N i ∑ j = 1

N i ∑ j = 1

d ω i d t

d v i d t

i j , I i

r i j

f n

i j + f t

T t

= m i g +

i j + T

m i

(6.66)

=

where I i designates the moment of inertia of particle i , while the translational ( v ) and rotational (angular) ( ω ) particle velocities are indicated on Figure 6.19a. T t i j is the torque of the tangential component of the contact force, while T r i j is the torque of the rolling friction force. In the case of multiple interactions, the forces and torques on each contact points are evaluated and added to calculate the resultant action on the discrete element. When calculating the contact forces, the contact between the particles is modeled with a pair of linear rheological models of the spring-damper-slider type [100] in both tangential and normal directions (Figure 6.19). Contact force vector f i j , which represents the action of a discrete element j on its neighbor i , can be decomposed into tangential (shear) and normal directions f i j = f n i j + f t i j ( f n i j = − h k n u n i j + η n ( v i j · n i j ) n i j i , f t i j = min { k t u t i j − η t v t i j , µ f | f n i j | t i j } (6.67) wherein both components of the force include dissipative terms, η . The detailed discussion of the this model is presented by Xiang and co-authors [121]. Obviously, Equation (6.67) with µ f = 0 and η n = η t = 0 (no energy dissipation by friction and damping ) corresponds to the simpler case of Figure 6.19c. Alonso-Marroquin and Herrmann [122] give an almost identical representation of the DEM methodology, with the discrete elements being in the form of convex polygons. For tetrahedra, one can consult, for example, Munjiza’s book [11], and for clusters [123]. Nat urally, all grain shapes more complex than circular and spherical ones require much more complex algorithms, for both contact detection and definition of appropriate interactions [122], [124], [125], which will not be subject of this elementary review. However, it should be noted that complex grain shapes are more demanding in terms of computer memory and processing time, which may reduce number of grains that can be used in modeling. The model parameters (e.g., those figuring in Equation (6.67) and Figure 6.19d) in DEM infancy were typically determined by an ad hoc procedure of calibrating the results of numerical simulations of standard laboratory tests with the corresponding experimental results (e.g., [126]). However, the parameters determined in this way depend on the size of the typical discrete element. Tavarez and Plesha [116] systematically approached the determination of meso-parameters of the model by deriving expressions for elastic stiffness coefficients k n = Et √ 3 ( 1 − ν ) , k t = 1 − 3 ν 1 + ν k n (6.68) as a function of unit cell thickness t , and the plane elastic coefficients E = E ( 2 D ) and ν = ν ( 2 D ) . It should be noted that Equations (6.68) are derived for a densely-packed, ideal triangular 2D lattice using displacement equivalence conditions. The expressions (6.68) also apply to the corresponding irregular lattice obtained by the DEM cluster consolidation process [116]. Also, it should be noted that the condition of non-negativity of