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8.2 Some Numerical Simulation Issues Associated with the Particle Simulation Method 189

1979, Bardet and Proubet 1992, Saltzer and Pollard 1992, Antonellini and Pollard
1995, Donze et al. 1996, Scott 1996, Strayer and Huddleston 1997, Camborde et al.
2000, Iwashita and Oda 2000, Burbidge and Braun 2002, Strayer and Suppe 2002,
Finch et al. 2003, 2004, Imber et al. 2004). In this case, it is strongly recommended
that a particle-size sensitivity analysis of at least two different models, which have
the same geometry but different smallest particle sizes, be carried out to confirm the
particle simulation result of a large-scale quasi-static geological system.

8.2.3 Numerical Simulation Issue Stemming from the Loading
Procedure Used in the Particle Simulation Method

The distinct element method is based on the idea that the time step used in the
simulation is chosen so small that force, displacement, velocity and acceleration
cannot propagate from any particle farther than its immediate neighbors during a
single time step (Potyondy and Cundall 2004). The servo-control technique (Itasca
Consulting Group, inc. 1999) is often used to apply the equivalent velocity or dis-
placement to the loading boundary of the particle model. This will pose an important
scientific question: Is the mechanical response of a particle model dependent on the
loading procedure that is used to apply “loads” at the loading boundary of the par-
ticle model? If the mechanical response of a particle model is independent of the
loading procedure, then this issue can be neglected when we extend the applica-
tion range of the particle simulation method from a small-scale laboratory test to a
large-scale geological problem.
In order to answer this scientific question associated with the distinct element
method that is used in the PFC2D (i.e. Particle Flow Code in Two Dimensions), it
is necessary to investigate how a “load” is propagated within a particle system. For
the purpose of illustrating the “load” propagation mechanism, a one-dimensional
idealized model of ten particles of the same mass is considered in Fig. 8.5. The
“load” can be either a directly-applied force or an indirectly-applied force due to
a constant velocity in this idealized conceptual model. The question that needs to
be highlighted here is that when a “load” is applied to a particle system, what is
the appropriate time to record the correct response of the whole system due to this
“load”? This issue is important due to the fact that particle-scale material properties
of a particle are employed in a particle model and that a biaxial compression test is
often used to determine the macroscopic material properties of the particle model.
If the normal stiffness coefficient between any two particles has the same value (i.e.
k i = k (i = 1, 2, 3, ... , 10)) and the time step is equal to the critical time step
(Itasca Consulting Group, inc. 1999) in the particle model, then the displacement of
particle 1 (i.e. the particle with “load” P)is P/k at the end of the first time step (i.e.
t = 1Δt, where Δt is the time step). The reason for this is that in the distinct element
method, the “load” can only propagate from a particle to its immediate neighboring
particles within a time step. Thus, during the first time step, the other nine particles
in the right part of the model are still kept at rest. This is equivalent to applying a