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8.1 Basic Formulations of the Particle Simulation Method 179
8.1 Basic Formulations of the Particle Simulation Method

The basic idea behind the particle simulation method is that either a granular mate-
rial or a solid material can be simulated using an assembly of particles. In the case
of simulating granular materials, there is no cementation between two adjacent par-
ticles so that the particles are not bonded together in the assembly. However, in the
case of simulating solid materials, particles are bonded together through cementation.
Although particles may have different shapes and sizes, they are assumed to be rigid.
Thus, the motion of a particle can be described using the motion of its mass center. A
small overlapping between two particles is allowed so that deformation of the particle
assembly can be simulated. The magnitude of the overlapping depends on both the
contact force and the stiffness of a contact between any two particles. This means
that a relationship between contact force and displacement needs to be used to cal-
culate the contact force at a contact between two particles. In the case of simulating
solid materials such as brittle rocks, the strength of the material can be simulated by
using the strength of the bond at a contact between two particles. Once the contact
force reaches or exceeds the strength of the bond at a contact between two particles,
the bond is broken so that a microscopic crack is created to represent the failure of
the material at this particular bond. This consideration is very useful for explicitly
simulating the initiation of spontaneous and random microscopic cracks in a brittle
material, because there is no need to artificially describe any microscopic flaws and
cracks in the beginning of a numerical simulation. Although the above idea was ini-
tially proposed in the discrete element method (Cundall and Strack 1979, Cundall
2001, Itasca Consulting Group, inc. 1999), it has been recently enhanced and built
into a two-dimensional Particle Flow Code. In this regard, the particle simulation
method can be viewed as a particular kind of discrete element method.
Considering a particle (i.e. particle α) shown in Fig. 8.1, the position of the par-
ticle is described using its x and y coordinates in the coordinate system shown.
According to Newton’s second law, the motion of the particle can be represented
using the motion of its mass center as follows:

F αy
R α
F αx
M
α
αz
y

x
Fig. 8.1 Definition of the motion of a particle o