Page 214 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.4 Test and Application Examples of the Particle Simulation Method 205

contact stiffness (in both the normal and the tangential directions) of 1.0 GN/m for
each particle in both the meter-scale and the kilometer-scale test samples. Since the
same value of the contact stiffness is used in both the meter-scale and kilometer-
scale samples, the second similarity criterion is satisfied between these two test
samples.
To satisfy the third similarity criterion, it is assumed that 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 meter-scale and the kilometer-scale
test samples. In the case of α = 1, the values of the unit normal and tangential
contact bond strengths are equal to the macroscopic tensile and shear strengths of
the particle material (Zhao et al. 2007b). Since the normal and tangential contact
bond strengths are directly proportional to the particle diameter, the third similarity
criterion is satisfied between these two test samples. Thus, all the necessary simi-
larity criteria are satisfied for these two samples of different length-scales. For the
purpose of testing the upscale theory in a wide parameter space, effects of three
important parameters, such as the confining stress, the normal bond strength and
the shear bond strength, are investigated using the two similar samples of different
length-scales.
Figure 8.10 shows the effect of the confining stress on the deviatoric stress versus
axial strain curve for both the meter-scale and the kilometer-scale samples of 1000
particles. Keeping other parameters unchanged, three different values of the confin-
ing stress, namely CS = 0.1, 1 and 10 MPa shown in Fig. 8.10, are considered in the
particle simulation of the two similar test samples. Note that the deviatoric stress is
defined as axial stress minus confining stress in this investigation. Because it is diffi-
cult to directly apply a stress boundary condition to the boundary of a particle model,
the servo-control technique (Itasca Consulting Group, inc. 1999) is used, as an alter-
native, to apply the equivalent velocity of the applied stress to the loading boundary
of the particle model. Using the newly-proposed loading procedure associated with
the distinct element method (Zhao et al. 2007b), the equivalent velocity of 1 m/s is
applied to both the upper and the lower boundaries of the two similar test samples. It
is obvious that the simulated stress-strain curve is dependent on the confining stress.
In particular, the maximum values of the failure stresses of the particle samples are
significantly different for three different confining stresses. The general trend of the
confining stress effect is that the higher the confining stress, the greater the max-
imum value of the failure stress of the particle sample. Since both the meter-scale
sample and the kilometer-scale sample are similar, the simulated stress-strain curves
are similar for these two samples of different length-scales, especially in the elas-
tic response range of the particle assemblies. This demonstrates that the proposed
upscale theory is appropriate and useful for establishing an intrinsic relationship
between two similar particle systems of different length-scales.
To examine whether or not the similar particle samples can reproduce the rock
dilation phenomenon observed from laboratory experiments, the effect of the con-
fining stress on the dilation of a particle model is also considered in the parti-
cle simulation of the test sample. Figure 8.11 shows the effect of the confining
stress on the volumetric strain versus axial strain curve for both the meter-scale