Page 203 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

194 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

ing boundary of the particle model can be calculated so that the value of the desired
load increment is determined. At the same time, other quantities of interest such
as a displacement increment, strain increment and so forth can be also determined.
Thus, a point relating stress to strain in a stress-strain curve is obtained at the end
of this step. For this reason, this step is called the result acquisition step. (4) Steps
1–3 are repeated for every desired load increment until the final stage of the particle
simulation is reached. As a result, many points relating stress to strain in a stress-
strain curve have been obtained at the end of the particle simulation. (5) Finally,
all the obtained points relating stress to strain are connected to generate a stress-
strain curve, which should represent the true quasi-static behavior for the particle
simulation model.

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

For the purpose of establishing an upscale theory associated with the particle sim-
ulation of two-dimensional quasi-static systems, it is necessary to understand the
particle-scale mechanical properties and their relations to macroscopic mechanical
properties, which are available from either a laboratory test or an in-situ measure-
ment. If circular particles of unit thickness are used in the simulation of a particle
assembly, the particle-scale mechanical properties such as the stiffness and bond
strength of the particle are needed for a contact-bond model. Since it is very diffi-
cult, if not impossible, to directly measure particle-scale mechanical properties from
laboratory tests, it is common practice to determine these particle-scale mechanical
properties from the macroscopic mechanical properties such as the elastic modulus,
tensile and shear strength of particle materials. From the analog of a two-circle con-
tact with an elastic beam (Itasca Consulting Group, inc. 1999), it has been demon-
strated that there may exist an upscale rule, which states that the contact stiffness of
a circular particle is only dependent on the macroscopic elastic modulus and inde-
pendent of the diameter of the circular particle. The value of the contact stiffness
of a circular particle is equal to twice that of the macroscopic elastic modulus of
the material. On the other hand, the contact bond strength of a circular particle is
directly proportional to both the tensile/shear strength of the particle material and
the diameter of the circular particle.
In order to facilitate the derivation of the corresponding similarity criteria, it is
assumed that the problem domain is comprised of a homogeneous medium. Since
a heterogeneous medium can be divided into many sub-domains of homogeneous
materials, the derived similarity criteria in this investigation is, as demonstrated later
by the test and application example in this chapter, valid and applicable for any two
similar particle models of heterogeneous media, as long as each homogeneous sub-
domain of the two similar particle models satisfies the required geometrical similar-
ity criterion. For this reason, the proposed upscale theory associated with the particle
simulation is also applicable for the particle simulation of a geometrically-similar