Page 206 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.3 An Upscale Theory of Particle Simulation for Two-Dimensional Quasi-Static Problems 197

Similarly, consideration of Eqs. (8.41) and (8.42) results in the following simi-
larity criterion for the two particle models:

b m1 b m1 D m1 L m1
n = s = = , (8.46)
b m2 b m2 D m2 L m2
n s
where b n m1 and b m2 are the normal bond strengths of the particles of models one and
n
two; b m1 and b m2 are the tangential bond strengths of the particles of models one
s
s
and two respectively.
Equation (8.46) indicates that the similarity ratios of the (normal and tangential)
bond strengths of the two particle models are equal to both their similarity ratio of
particle diameters and their similarity ratio of geometrical lengths. This equation is
called the third similarity criterion between two particle models of different length-
scales. Since the similarity ratios of the particle bond strengths are equal to the
similarity ratio of the particle contact forces, the occurrence of the first failure should
be similar for two similar particle models. This indicates that if the first failure
occurs at a particle in model one, then the first failure should occur at a similar
counterpart in model two.
For a quasi-static geological system, it is important to consider the gravity effect
in two particle models of different length-scales. At the particle level, the gravity
force exerted on a circular particle can be expressed as follows:

π 2
G p = ρ p gD T, (8.47)
4
where G p is the gravity force exerted on a particle; ρ p is the density of the particle
material; g is the gravity acceleration; T is the thickness of the circular particle.
Note that T ≡ 1 in this investigation.
From Eq. (8.47), the similarity ratio of gravity forces for the two particle models
of different length-scales can be derived and expressed as follows:

G m1 ρ m1 m1 m1 2
p p g (D )
= , (8.48)
m2 2
G m2 ρ m2 g m2 (D )
p
p
where G m1 and G m2 are the gravity forces exerted on the particles of models one
p
p
and two respectively; ρ m1 and ρ m2 are the densities of the particle materials of the
p p
two models; g m1 and g m2 are the gravity accelerations of the particles of models one
and two respectively.
In order to implement this similarity criterion in particle simulation models, it
is desirable to keep the similarity ratio of the gravity forces equal to that of the
geometrical lengths of the two similar particle models. Since the explicit dynamic
relaxation method is used to solve the equation of motion in a particle simulation
(Itasca Consulting Group, inc. 1999), the similarity ratio of the particle densities
needs to be equal to one so that it does not affect the time-step similarity of the