The results of this chapter have been published in Applied Mechanics & Materials Journal and Tribology International Journal.
The DEM, originally proposed by Cundall [80] for geotechnical applications, has been an ideal tool to simulate discontinuous, heterogeneous, anisotropic medium behavior. More recently, researchers have successfully used DEM to study damage, fracture, wear behaviors and contact mechanics between rough surfaces [94, 95].
In DEM, the solid material is simulated as spherical discrete particles, which interact with each other via contact forces. The Hertz-Mindlin with bonding contact model was used to bond particles with a finite-sized “glue” bond. The contact radius is a critical parameter as it determines the area in which the bond can be formed (Figure 3.1a). If the contact radii of two particles overlap during the bond formation time, these two particles will be bonded together. When the particles no longer overlap the bond is lost. The contact mechanics of the bond can be considered as a spring-dashpot configuration (Figure 3.1b). The basic theory of DEM has been comprehensively described in [96, 97].
82 Figure 3.1 Bond configuration between two DEM particles (a) in 2D and (b) in 3D [79]
This bond is a bilateral solid joint and can resist tangential and normal movement up to a maximum normal and tangential shear stress, at which point the bond breaks. The forces and moments are summed up, and then used to solve numerically the Newton's and Euler's equations of motion for individual simulation time steps.
Particles are bonded at the bond formation time tbond. Before this time, the particles
interact through the standard Hertz-Mindlin contact model. After bonding, the forces (Fn,t)/moments (Mn,t) on the particle are set to zero and are adjusted incrementally for
every time step according to:
𝛿𝐹𝑛 = −𝑣𝑛𝑆𝑛𝐴𝛿𝑡 ; 𝛿𝐹𝑡 = −𝑣𝑡𝑆𝑡𝐴𝛿𝑡 (3.1) 𝛿𝑀𝑛 = −𝜔𝑛𝑆𝑡𝐽𝛿𝑡 ; 𝛿𝑀𝑡= −𝜔𝑡𝑆𝑛 𝐽 2𝛿𝑡 (3.2) where: 𝐴 = 𝜋𝑅𝐵2; 𝐽 = 1 2𝜋𝑅𝐵 4 (3.3)
where RB is the radius of the “glue”, Sn,t are the normal and shear stiffness
respectively and δt is the time step. 𝑣𝑛 and 𝑣𝑡 are the normal and tangential velocities of
83 The bond is broken when the normal and tangential shear stresses exceed the predefined values: б𝑚𝑎𝑥 < −𝐹𝑛 𝐴 + 2𝑀𝑡 𝐽 𝑅𝐵 ; 𝜏𝑚𝑎𝑥 < −𝐹𝑡 𝐴 + 𝑀𝑛 𝐽 𝑅𝐵 (3.4)
The bond is lost when the particles no longer overlap. The bond cannot be reintroduced once it vanishes [98]. The bond model in this study is based on the particle contact radius which was set to 10% higher than the actual physical radius of the spheres as recommended by EDEM and in [99]. If the contact radius was too small (less than 10%), a small movement of particles can lead to a breakage of the bond, it results in the packing sample becoming loose and get broken easily as the number of contact bond will be decreased. However, if the contact radius was set too large, particles might form the bonds with the other particles which are not their neighbours; this does not reflect the realistic behavior of the whole sample and it causes the error during the packing process in DEM.
In the DEM, the interaction of the particles is treated as a dynamic process with states of equilibrium developing whenever the internal forces balance. The contact forces and displacements of a stressed assembly of particles are found by tracing the movements of the individual particles. Movements result from the propagation through disturbances of the system of particle caused by geometries movement and particle motion, externally applied forces and body forces, respectively. This is a dynamic process in which the speed of propagation depends on the physical properties of the discrete system. The calculations performed in the DEM alternate between the application of Newton’s second law to the particles and a force–displacement law at the contacts Figure 3.2.
84 Figure 3.2 Newton Law and contact mechanic of DEM bond
Newton’s second law is used to determine the translational and rotational motion of each particle arising from the contact forces, applied forces and body forces acting upon it, while the force–displacement law is used to update the contact forces arising from the relative motion at each contact. The dynamic behaviour is represented numerically by a time-stepping algorithm in which the velocities and accelerations are assumed to be constant within each time step. The DEM is based on the idea that the time step chosen should be small that, during a single time step, disturbances cannot propagate from any particle farther than its immediate neighbours. Then, at all times, the forces acting on any particle are determined exclusively by its interaction with the particles with which it is in contact. Because the speed at which a disturbance can propagate is a function of the physical properties of the discrete system, the time step can be chosen to satisfy the above constraint. In the DEM Solutions commercial software that is used in this project, the time step is calculated based on the Raleigh time step which is well demonstrated in [79]. The use of an explicit, as opposed to an implicit, numerical scheme provides the following advantage. Large populations of particles require only modest amounts of computer memory, because matrices are not stored.
Moreover, because the DEM is a fully dynamic formulation, some form of damping is necessary to dissipate some kinetic energy. This dimensionless parameter controls the
85 amount of velocity dependent damping. Without this additional damping, small vibrations of particles can persist for a long time. In real materials, various microscopic processes such as internal friction and wave scattering dissipate the kinetic energy. In the simulation model, local non-viscous damping is used by specifying a damping coefficient. From the literature review and recommendation from EDEM Solutions [92, 98, 100], a damping coefficient of 0.7 for the normal direction and 0.1 for the tangential direction will be used in this study.
The DEM can naturally describe a granular medium. However, computational resources are required to manage a large number of discrete elements. Kafui and Thornton [101] presented numerical simulations of a spherical, crystalline agglomerate impacting a wall. The effects of bond strength and impact velocity on the wall force, kinetic energy of the agglomerate and the proportion of bonds broken were analyzed. More recently, researchers have used this method to study damage in heterogeneous solids, such as concrete [102] or rock [98, 103, 104] homogeneous materials with realistic microstructure, such as ceramics [89, 90, 105], homogeneous, isotropic and perfectly brittle elastic material such as silica glass [106], perfectly elastic as fused silica [81]. Wen Li et al. [92] proposed an inter-element contact constitutive model and its microscopic parameters were determined and calibrated to reproduce continuous and discontinuous behaviours of material in DEM. The authors also proposed the method to calculate the wear of the fretting material. However, the model is still in 2D which is not realistic.