Page 209 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

200 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

size difference between the small and large test samples, the size effect of the test
sample can be investigated through the particle simulation.
The stiffness and bond strength of particles in a test sample can be predicted
using the macroscopic mechanical properties such as the elastic modulus, tensile
and shear strength of particle materials. From the analog of a two-circle contact
with an elastic beam (Itasca Consulting Group, inc. 1999), it has been demonstrated
that there may exist an upscale rule, which states that the contact stiffness of a circu-
lar particle is only dependent on the macroscopic elastic modulus and independent
of the diameter of the circular particle. The value of the contact stiffness of a circu-
lar 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 pro-
portional to both the tensile/shear strength of the particle material and the diameter
of the circular particle. To reflect this fact, the concept of the unit normal/tangential
contact bond strength, which is defined as normal/tangential contact bond strength
per unit length of the particle diameter, is used in this study. Using this definition,
the value of the unit normal/tangential contact bond strength is equal to that of
the macroscopic tensile/shear strength of the particle material, while the value of the
normal/tangential contact bond strength of a particle is equal to the product of the
unit normal/tangential contact bond strength and the diameter of the particle. In this
way, the variation of the normal/tangential contact bond strength of a particle with
its diameter is considered in the particle simulation. Keeping the above considera-
tions in mind, the following macroscopic mechanical properties of rock masses are
used to determine the contact mechanical properties of the particle material used in
the simulation. The macroscopic elastic modulus of the particle material is 0.5 GPa,
resulting in a contact stiffness (in both the normal and the tangential directions) of
1.0 GN/m for both the small and the large test samples. The macroscopic tensile
strength of the particle material is 10 MPa, while the macroscopic shear strength of
the particle material is 100 MPa for both the small and the large test samples. The
loading period is 10 time-steps, while the frozen period is 9990 time-steps in the
numerical biaxial compression tests.
For the particle simulation associated with the distinct element method (Itasca
Consulting Group, inc. 1999), crack initiation and generation are determined using
the following crack criteria:

F Cn ≥ b n , (8.53)

|F Cs | ≥ b s , (8.54)

where F Cn and F Cs are the normal and shear contact forces at a contact between any
two particles; b n and b s are the normal and tangential bond strengths at the contact
between the two particles.
Obviously, Eq. (8.53) indicates that if a normal contact force exceeds the corre-
sponding normal tensile bond strength at a contact between two particles, the nor-
mal tensile bond is broken and therefore, a tensile crack is generated at the contact.