Particle Discrete Element Method (pDEM)#
Rigid Particle DEM#
The rigid-particle discrete element method (DEM) models a particulate assembly as individual bodies whose shapes do not deform, while their translations, rotations, contacts, and possible bonds evolve in time. It is well suited to granular packing, impact, flow, and compression problems in which bulk behavior emerges from particle-scale interactions. OpenFDEM describes its pDEM module as supporting particle packing, kinematics, collision, and linear, Hertz, cohesive-bond, and rotation-resistance contact models.
Particle motion#
For particle \(i\) with mass \(m_i\), center position \(\mathbf{x}_i\), and translational velocity \(\mathbf{v}_i\), the translational equation of motion is
where \(\mathbf{F}_{ic}\) are contact or bond forces, \(\mathbf{g}\) is gravity, and \(\mathbf{F}_i^{\mathrm{ext}}\) is other applied loading. Rotation follows the angular-momentum balance
where \(\mathbf{I}_i\) is the particle inertia tensor, \(\boldsymbol{\omega}_i\) is angular velocity, and \(\mathbf{M}\) denotes contact, bond, or applied moments. In two-dimensional calculations, the rotational equation reduces to the out-of-plane scalar form.
Contact and friction#
For a pair of touching particles, the contact force is commonly resolved into normal and tangential components, \(\mathbf{F}_c=F_n\mathbf{n}+\mathbf{F}_t\), where \(\mathbf{n}\) is the contact normal. A linear elastic-frictional model uses the overlap \(\delta_n\) to compute a trial normal force, for example
where \(k_n\) is normal stiffness, \(c_n\) is a normal damping coefficient, \(v_n\) is the relative normal velocity, and \(\mu_f\) is the friction coefficient. A tangential spring tracks relative slip, with the trial force projected back to the Coulomb limit when sliding occurs. In Hertz-type contact, the normal force instead depends nonlinearly on overlap. The exact force law, damping, and friction update are constitutive choices; they should be selected for the particle material and the intended time scale.
Time integration and interpretation#
DEM commonly advances the particle equations explicitly. The stable time step must resolve the shortest relevant contact oscillation, which is controlled by particle mass or inertia and contact stiffness. Increasing stiffness reduces overlap but also decreases the stable time step. Contact stiffness, damping, and friction therefore affect both numerical cost and the effective macroscopic response and should be calibrated rather than treated as arbitrary numerical constants.
Because particle shapes are fixed in rigid DEM, particle-scale deformation and fracture are not resolved within a body. They can be represented approximately by bonded particle assemblies or by the deformable-cluster approach described in the following section.
Irregular Deformable Cluster DEM#
An irregular deformable particle can be represented as a cluster of interacting constituent particles. The cluster geometry captures a non-circular or otherwise irregular outline, while relative motion among its constituents allows the aggregate to deform. If internal bonds soften or fail, the particle can also fragment. This differs from a rigid clump, whose constituents retain fixed relative positions and whose overall shape cannot change.
OpenFDEM’s project description identifies realistic particles as rigid or deformable, notes overlapping-particle and Fourier-Voronoi-based shape generation, and states that particle breakage is possible. The bonded-cluster equations below are a general theoretical description; the exact internal representation, bond law, and breakage criterion depend on the selected model.
Cluster kinematics and force balance#
For a cluster with constituent particles \(a=1,\ldots,N_c\), each constituent obeys its own translational and rotational balances,
where \(\mathbf{F}^{\mathrm{int}}\) and \(\mathbf{M}^{\mathrm{int}}\) are forces and moments transmitted by internal contacts or bonds, while the external terms arise from other particles, boundaries, and applied loads. If all internal bonds remain intact and infinitely stiff, relative constituent motion is suppressed and the cluster approaches rigid-clump behavior. Finite bond stiffness permits deformation; bond damage or failure permits irreversible shape change and fragmentation.
Internal bond response and breakage#
A simple bonded-contact idealization relates the local normal and tangential relative displacements \(\delta_n\) and \(\boldsymbol{\delta}_t\) to trial bond forces through
where \(k_n^b\) and \(k_t^b\) are bond stiffnesses. A chosen strength criterion may trigger damage when the normal or shear traction reaches its corresponding tensile or shear capacity. After damage, the transmitted force is reduced according to the selected softening law; after complete failure, the interface no longer carries cohesive tension or shear, although ordinary compressive contact and friction may remain. These equations illustrate the mechanism and are not a claim about one specific OpenFDEM bond implementation.
Macroscopic response and calibration#
The response of an irregular cluster assembly emerges from both external particle contacts and internal deformation. Cluster geometry affects packing, contact orientation, and interlocking; constituent size and bond properties affect the effective stiffness, strength, and breakage pattern. Resolution should be adequate to represent the target particle shape and expected deformation modes. Calibration should consider bulk observables such as packing density, stress-strain response, peak strength, dilation, and fragment size distribution, rather than fitting only individual bond parameters.
This cluster approach provides a mesoscale compromise: it can represent particle-scale deformation and breakage without resolving a continuum mesh inside every grain. Predictions remain dependent on the chosen cluster discretization, contact law, bond law, and numerical time step.