# PFC Overview

The PFC programs (PFC2D and PFC3D) provide a general purpose, distinct-element modeling framework that includes both a computational engine and a graphical user interface. A particular instance of the distinct-element model is referred to as a PFC *model*. The term PFC *model* refers to both the 2D and 3D models. The PFC model simulates the movement and interaction of many finite-sized particles.[1][2] The particles are rigid bodies with finite mass that move independently of one another and can both translate and rotate. Particles interact at pair-wise contacts by means of an internal force and moment. Contact mechanics is embodied in particle-interaction laws that update the internal forces and moments. The time evolution of this system is computed via the distinct-element method, which provides an explicit dynamic solution to Newton’s laws of motion. The PFC model provides a synthetic material consisting of an assembly of rigid grains that interact at contacts and includes both granular and bonded materials.

Thanks to its general design, PFC can be easily customized and applied to a very broad range of numerical investigations where the discrete nature of the systems is of interest. Since the release of the first version in 1994, PFC has been successfully used by many academic institutions and private companies around the world for problems ranging from fundamental research on soil and rock behavior at the laboratory scale to slope stability and rockfall hazard mitigation, hydraulic fracturing, rock-tool interactions, bulk flow, mixing, conveying and compaction of aggregates and powders, blast furnace modeling, etc. A large panel of references can be found in the proceedings of the past PFC and *FLAC/DEM* symposia periodically organized by Itasca.

## The PFC Model

A general particle-flow model simulates the mechanical behavior of a system made up of a collection of arbitrarily shaped particles. (Note that the term *particle*, as used here, differs from its more common definition in the field of mechanics, where it is taken as a body whose dimensions are negligible and therefore occupies only a single point in space. In the present context, the term *particle* denotes a body that occupies a finite amount of space.) The model is composed of distinct particles that displace independent of one another and interact at pair-wise contacts. If the particles are assumed to be rigid, then the mechanical behavior of such a system is described in terms of the movement of each particle and the inter-particle forces acting at each contact point. Newton’s laws of motion provide the fundamental relationship between particle motion and the forces causing that motion. The force system may be in static equilibrium (in which case, there is no motion), or it may be such as to cause the particles to flow. If the particle-interaction law models a physical contact between particles, the behavior of the contacts is characterized using a soft contact approach, in which a finite normal stiffness is taken to represent the measurable stiffness that exists at a contact, and the rigid particles are allowed to overlap in the vicinity of the contact point. More complex behavior can be modeled by allowing the particles to be bonded together at their contact points such that, when the inter-particle forces acting at any bond exceed the bond strength, that bond is broken. This allows tensile forces to develop between particles. One can then model the interaction of these bonded “blocks,” including the formation of cracks that may cause blocks to fragment into smaller blocks. The particle-interaction law can also be derived from potential energy functions and model long range interactions.

PFC provides a particle-flow model containing the following assumptions:

The particles are treated as rigid bodies.

The fundamental particle shape is a {disk with unit thickness in 2D; sphere in 3D}, denoted

*ball*.The clump logic supports the creation of rigidly attached {disks with unit thickness in 2D; spheres in 3D}, denoted

*pebbles*. Each clump consists of a set of overlapping pebbles that acts as a rigid body with a deformable boundary. Clumps may be of arbitrary shape.Particles interact at pair-wise contacts by means of an internal force and moment. Contact mechanics is embodied in particle-interaction laws that update the internal forces and moments.

Behavior at physical contacts uses a soft-contact approach where the rigid particles are allowed to overlap one another at contact points. The contacts occur over a vanishingly small area (i.e., at a point), and the magnitude of the overlap and/or the relative displacement at the contact point are related to the contact force via the force-displacement law.

Bonds can exist at contacts between particles.

Long range interactions can also be derived from energy potential functions.

The assumption of particle rigidity is a good one when most of the deformation in a physical system is accounted for by movements along interfaces. The deformation of a packed-particle assembly (or a granular assembly such as sand), as a whole, is well-described by this assumption, since the deformation results primarily from the sliding and rotation of the particles as rigid bodies and the opening and interlocking at interfaces, not from individual particle deformation. Precise modeling of particle deformation is not necessary to obtain a good approximation of the mechanical behavior for such systems.

In addition to traditional particle-flow applications, PFC can also be applied to the analysis of solids subjected to prescribed boundary and initial conditions. In such models, the continuum behavior is approximated by treating the solid as a compacted assembly of many small particles. Measures of stress and strain rate can be defined as average quantities over a representative measurement volume for such systems. This allows one to estimate interior stresses for granular materials such as soils, or solid materials such as rock or plastics formed by powder compaction.

In addition to balls and clumps, the PFC particle-flow model also includes *walls*. Walls allow one to apply velocity boundary conditions to assemblies of balls or clumps for purposes of compaction and confinement. The balls, clumps, and walls interact with one another via the forces that arise at contacts. The equations of motion are satisfied for each ball and clump. However, the equations of motion are not satisfied for each wall (i.e., forces acting on a wall do not influence its motion). Instead, its motion is specified by the user and remains constant regardless of the contact forces acting on it. Also, contacts may not form between two walls; thus, contacts are either ball-ball, ball-pebble, pebble-pebble, ball-facet, or pebble-facet.

## Distinct-Element Method

PFC models the movement and interaction of stressed assemblies of rigid {circular in 2D; spherical in 3D} particles using the Distinct-Element Method (DEM). The DEM was introduced by [Cundall1971] for the analysis of rock-mechanics problems and then applied to soils by [Cundall1979b]. A thorough description of the method is given in the two-part paper by [Cundall1988] and [Hart1988], and in the UDEC manual ([Itasca2011]). PFC is classified as a discrete element code based on the definition in the review by [Cundall1992], since it allows finite displacements and rotations of discrete bodies (including complete detachment), and recognizes new contacts automatically as the calculation progresses. PFC can be viewed as a simplified implementation of the DEM because of the restriction to rigid {circular in 2D; spherical in 3D} particles. The general DEM can handle deformable polygonal-shaped particles.

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 the particle system of disturbances caused by specified wall and particle motion and/or body forces. This is a dynamic process in which the speed of propagation depends on the physical properties of the discrete system.

The dynamic behavior is represented numerically by a timestepping algorithm in which it is assumed that the velocities and accelerations are constant within each timestep. The solution scheme is identical to that used by the explicit finite-difference method for continuum analysis. The DEM is based upon the idea that the timestep chosen may be so small that, during a single timestep, disturbances cannot propagate further from any particle than its immediate neighbors. 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. Since the speed at which a disturbance can propagate is a function of the physical properties of the discrete system, the timestep can be chosen to satisfy the above constraint. The use of an explicit, as opposed to an implicit, numerical scheme makes it possible to simulate the nonlinear interaction of a large number of particles without excessive memory requirements or the need for an iterative procedure.

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. Newton’s second law is used to determine the motion of each particle arising from the contact 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 presence of walls in PFC requires only that the force-displacement law account for contacts with wall facets. Newton’s second law is not applied to walls because the wall motion is specified by the user.

References

Cundall, P. A. “A Computer Model for Simulating Progressive Large Scale Movements in Blocky Rock Systems,” in Proceedings of the Symposium of the International Society of Rock Mechanics (Nancy, France, 1971), Vol. 1, Paper No. II-8 (1971).

Cundall, P. A. “Formulation of a Three-Dimensional Distinct Element Model - Part I. A Scheme to Detect and Represent Contacts in a System Composed of Many Polyhedral Blocks,” Int. J. Rock Mech., Min. Sci. & Geomech. Abstr., 25(3), 107-116 (1988).

Cundall, P. A., and R. Hart. “Numerical Modeling of Discontinua,” J. Engr. Comp., 9, 101-113 (1992).

Cundall, P. A., and O. D. L. Strack. “A Discrete Numerical Model for Granular Assemblies,” Geotechnique, 29(1), 47-65 (1979).

Hart, R., P. A. Cundall and J. Lemos. “Formulation of a Three-Dimensional Distinct Element Model - Part II. Mechanical Calculations for Motion and Interaction of a System Composed of Many Polyhedral Blocks,” Int. J. Rock Mech., Min. Sci. & Geomech. Abstr., 25(3), 117-125 (1988).

Itasca Consulting Group Inc. UDEC (Universal Distinct Element Code), Version 5.0. Minneapolis:ICG, 2011.

Potyondy, D., and P.A. Cundall. “A bonded-particle model for rock,” Int. J. Rock Mech. & Mining Sci., 41, 1329-1364 (2004).

Potyondy, D. O. “The Bonded-Particle Model as a Tool for Rock Mechanics Research and Application: Current Trends and Future Directions,” Geosystem Engineering, 18(1), 1–28 (2015), DOI:10.1080/12269328.2014.998346.

Endnotes

This is a dummy reference to [Potyondy2004e] to kill a sphinx error. Will not be needed once references are all collected in one place. Whenever *that* happens.

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