2D vs. 3D models in PFC

The PFC2D “world” is two dimensional in nature (i.e., only two force components and one moment component exist in a PFC2D model, as opposed to the three force components and three moment components that exist in a three-dimensional particle assembly). The out-of-plane force component and the two in-plane moment components are not considered in any way in the equations of motion or the force-displacement laws. A PFC2D model can be considered to be simulating a collection of variable-radius cylinders with unit thickness.

The advantages of modeling in 2D and the advisability of modeling first in 2D are discussed in the topic “2D vs. 3D Models.”

Packing and Porosity

The porosity computed by PFC2D is an area-based calculation (ratio of total void area to total area), as opposed to the volume-based calculation (ratio of total void volume to total volume) used to define three-dimensional porosity. There is no clear relation between a two-dimensional porosity value and a three-dimensional porosity value for arbitrary assemblies of spherical particles. However, since porosity is a measure of particle packing, the following information about particle packing in 2D and 3D highlights and explains some of the differences. It is shown in Deresiewicz (1958) that the closest of all regular packings of uniform spheres in 3D has a porosity of 0.2595, while the closest of all regular packings of uniform circles in 2D has a porosity of 0.0931. In the absence of compressive forces acting on the assembly, these are the theoretical lowest values of porosity obtainable without particle interpenetration. In general, there is more void space remaining in a 3D assembly than in a 2D assembly. In a real material, the porosities will be higher, because the particles will have “locked-up” before reaching this optimal packing. If particle arching is known to occur in a real material, there are many more opportunities for arches to form in a 3D assembly than in a 2D assembly. Also, the PFC2D model assumes that the centroids of all particles are aligned on a single plane—an unlikely occurrence in a 3D assembly.

The coordination number \(C_n\) of a granular assembly (average number of contacts per particle) is also an important characteristic of the packing. In [Agnolin2007b], the authors show that the coordination number, rather than the porosity, dictates the small-strain mechanical response of dense, isotropic, frictional assemblies in the rigid grain limit. [Radjaï2004] suggests that the contact network fabric of such assemblies is a relevant state variable to describe hardening in the plastic régime. [Roux2000] demonstrates an upper bound on the coordination number of dense assemblies of rigid frictional particles (in the absence of hyperstaticity) of the form \(C_n \leq d(d+1)\) for the general case, and \(C_n \leq 2d\) for spherical particles, where d denotes dimensionality. For frictionless particles, the inequalities become equalities.

Another difference between 2D and 3D assemblies relates to percolation. Small particles can easily percolate through a 3D assembly consisting of larger particles, but they can never percolate through a packed PFC2D model, regardless of relative particle sizes.

For situations in which particle packing has a significant influence upon behavior, it may be necessary to perform a small number of 3D simulations using the PFC3D code in order to establish the relevant parameters needed to obtain measured physical responses, and then use these parameters while performing a larger number of parameter studies using the PFC2D code.

Stress and Strain

Stress and strain, as continuous variables, do not exist at each point in a particle assembly, because the medium is discontinuous. Therefore, an averaging procedure is employed to compute average stress and strain-rate tensors within a user-defined measurement circle. The averaging procedure for stress (described here) relates the two in-plane force components acting on each particle in the measurement circle to a force per unit length of particle boundary, which must then be divided by a thickness value in order to obtain a stress quantity. In PFC2D, the reported stress values utilize a thickness of unity.

The averaging procedure for strain rate involves no assumptions about out-of-plane thickness. However, since the porosity and fabric of 2D and 3D systems differ, volumetric strains of 2D assemblies differ from the 3D case. The importance of the out-of-plane third dimension in direct shear simulations of granular materials is stressed out in [Hazzard2003], where 3D models are reported to show larger macroscopic internal friction angles than their 2D counterparts. Similar trends are observed when comparing 2D and 3D confined compression tests ([David2007]).

Most two-dimensional continuum-based codes determine three-dimensional elastic response by enforcing a condition of either plane stress or plane strain through the constitutive relations between stress and strain. A PFC2D model, however, enforces neither of these conditions. As mentioned above, the out-of-plane force component and stresses and strains are simply not considered in the equations of motion or in the force-displacement laws. Thus, the out-of-plane constraint necessary to enforce a state of plane strain is not present.

Mass Properties

The mass of each ball in PFC2D is determined by considering the particle to be a cylinder of unit thickness. The mass is assigned based on the density and radius of the balls. The particle masses affect both the motion calculation (since the inertial properties differ for a disk and a sphere) and the gravity force applied to each particle.