Page 213 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

204 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

conventional loading procedure is used to produce the simulation results. However,
the mechanical responses of both the small and the large test samples of 1000 par-
ticles are clearly dependent on the loading rate in the elastic range of the particle
material. In the case of a loading rate of 10 m/s, there is an oscillatory behaviour
in the stress-strain curve. Such an oscillatory behaviour does not occur in Fig. 8.8,
where the mechanical responses of both the small and the large test samples of
1000 particles are obtained from using the proposed loading procedure. Neverthe-
less, the oscillatory behaviour of the mechanical response obtained from using the
improved conventional loading procedure is greatly reduced when the smaller load-
ing rate (i.e. LR = 1.0 m/s) is used in the particle simulation, indicating that the
use of the improved conventional loading procedure in a particle simulation may
produce some useful results as long as the loading rate is kept very small in the
simulation. In the case of a loading rate of 10 m/s, the maximum yielding strength
obtained from using the improved conventional loading procedure is almost twice
that obtained from using the proposed loading procedure, implying that the maxi-
mum yielding strength can be overestimated when using the improved conventional
loading procedure. It is interesting to note that the mechanical response obtained
from using the improved conventional loading procedure exhibits stronger ductile
behavior (Fig. 8.9), while the mechanical response obtained from using the pro-
posed loading procedure exhibits stronger brittle behavior (Fig. 8.8) for exactly the
same test sample. This demonstrates that in addition to conceptual soundness, the
proposed loading procedure is more appropriate than the improved conventional
loading procedure in dealing with the numerical simulation of the brittle behavior
of crustal rocks.

8.4.2 The Similarity Test of Two Particle Samples
of Different Length-Scales

The same two samples of different length-scales are considered here. The first test
sample is of small size (1 by 2 m) and is simulated using 1000 randomly-distributed
particles. The maximum and minimum radii of the particles used in the particle sam-
ple are approximately 0.01724 m and 0.01149 m, resulting in an average radius of
0.01437 m. On the other hand, the second test sample is of large size (1 by 2 km)
and is also simulated using 1000 randomly-distributed particles. The second test
sample is artificially designed to validate the proposed upscale theory in this study.
The maximum and minimum radii of the particles used in the particle sample are
approximately 17.24 m and 11.49 m, resulting in an average radius of 14.37 m.
Since the similarity ratio (i.e. 1/1000) of particle diameters is equal to that of geo-
metrical lengths for the two samples, the first similarity criterion is satisfied between
these two test samples. The initial porosity of both the small and the large test sam-
ples is set to be 0.17 in the particle simulation. The density of the particle mate-
3
rial is 2500 kg/m and the friction coefficient of the particle material is 0.5, while
the confining stress is taken as 10 MPa in the following numerical experiments.
The macroscopic elastic modulus of the particle material is 0.5 GPa, resulting in a