Page 205 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

196 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

of a particle with its diameter can be straightforwardly considered in the particle
simulation.
Note that in the particle simulation method such as the distinct element method,
the contact force exerted on a particle is calculated using the following formula:

F n = k n u n , (8.43)

where F n is the normal contact force at a contact between two particles; u n is the
normal displacement at the contact. It needs to be pointed out that in the linear
elastic range of the particle material, a similar relationship to Eq. (8.43) is also valid
for the shear contact force and tangential displacement at a contact between two
particles.
For a quasi-static system, two particle models of different length-scales can be
considered to establish the upscale theory. The first particle model (i.e. model one)
is of a small length-scale such as a laboratory length-scale, while the second parti-
cle model (i.e. model two) is of a large length-scale such as a regional geological
length-scale. From elasticity theory, the necessary condition, under which these two
models are similar, is that the relative displacements (i.e. strain) of the two mod-
els are identical. This results in the following similarity criterion for two particle
models of the same number of particles:

u m1 D m1 L m1
n = = , (8.44)
u m2 D m2 L m2
n
where u m1 and u m2 are the displacements of models one and two; D m1 and D m2 are
n n
the diameters of the particles of models one and two respectively; L m1 and L m2 are
the geometrical lengths of the two models.
Equation (8.44) indicates that if two particle models of different length-scales are
similar, then both the displacement ratio and the diameter ratio of the two models
are equal to their geometrical length ratio. For this reason, Eq. (8.44) is called the
first similarity criterion between two particle models of different length-scales.
Consideration of Eqs. (8.40), (8.43) and (8.44) yields the following similarity
criterion for the two particle models:

F m1 u m1 L m1
n = n = , (8.45)
F m2 u m2 L m2
n n
where F m1 and F m2 are the contact forces between two particles of models one and
n n
two respectively.
Equation (8.45) states that the similarity ratio of the contact forces of the two
models is equal to both their similarity ratio of displacements and their similarity
ratio of geometrical lengths. This equation is called the second similarity criterion
between two particle models of different length-scales.