Page 207 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

198 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

two models, as discussed below. For this reason, it is necessary to use the following
alternative relationships:

g m1 D m2 ρ m1
p
= , = 1, (8.49)
g m2 D m1 ρ m2
p
m1 2
G m1 ρ m1 m1 (D ) D m1
p g
p
= = . (8.50)
G m2 ρ m2 g m2 (D ) D m2
m2 2
p p
Clearly, Eq. (8.49) states that in order to maintain the similarity of two particle
models of different length-scales, the similarity ratio of the gravity accelerations
should be equal to the inverse of the similarity ratio of the geometrical lengths for the
two similar particle models. In this regard, Eq. (8.49) is called the fourth similarity
criterion between two particle models of different length-scales.
For the particle simulation based on the distinct element method, the numeri-
cal solution to the equation of motion is conditionally stable because the explicit
dynamic relaxation solver is used. The critical time-step, which is required to result
in a stable solution, can be expressed as follows (Itasca Consulting Group, inc.
1999):

6
m
Δt critical = . (8.51)
k
where Δt critical is the critical time-step; m is the mass of a particle and k is the
stiffness between two particles.
It is immediately noted from Eq. (8.51) that the critical time-steps, which are
used in two similar particle models of different length-scales, satisfy the following
similarity criterion:

Δt m1 ρ m1 m1 D m1
p D
critical = = , (8.52)
Δt m2 ρ m2 D m2 D m2
critical p
where Δt m1 and Δt m2 are the corresponding critical time-steps used in models
critical critical
one and two respectively.
Equation (8.52) provides an auxiliary similarity criterion, which is a direct result
from the above-mentioned fourth similarity criterion, for the two similar particle
models of different length-scales.
It needs to be pointed out that the proposed upscale theory associated with par-
ticle simulation methods is strictly valid when the mechanical response of a two-
dimensional particle assembly is within the elastic range. If the loading increment
is small enough to prevent any unphysical damage/crack from occurring within a
two-dimensional particle assembly and the number of time-steps is large enough to
enable the particle assembly to reach a quasi-static state during this loading period,
which can be achieved using the newly-proposed loading procedure associated with