Page 210 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.4 Test and Application Examples of the Particle Simulation Method 201

Similarly, Eq. (8.54) indicates that if the absolute value of a shear contact force
exceeds the corresponding tangential bond strength at a contact between two parti-
cles, the tangential shear bond is broken and therefore, a shear crack is generated
at the contact. The above-mentioned crack criteria are checked for each time step
during the particle simulation of a computational model.

8.4.1 Comparison of the Proposed Loading Procedure
with the Conventional Loading Procedure

Figure 8.8 shows the effect of the loading rate and sample size on the deviatoric
stress versus axial strain curve for both the small and the large samples of 1000
particles. The deviatoric stress is defined as the axial stress minus the confining
stress in this investigation. As usual, the servo-control technique (Itasca Consulting
Group, inc. 1999) is used to apply the equivalent velocity to the loading boundary of
the particle model. The equivalent velocity is called the loading rate hereafter. It is
obvious that the simulated stress-strain curve is independent of the two loading rates
(i.e. LR = 1.0 m/s and LR = 10 m/s in this figure) and sample sizes in the elastic
response range, where there is no occurrence of any failure in the test material. It
can be found, from the stress-strain curve, that the simulated elastic modulus of the
particle material is equal to 0.5 GPa, which is identical to the desired value of the
expected macroscopic elastic modulus of the particle material. This indicates that
the upscale rule established from the analog of a two-circle contact with an elastic
beam (Itasca Consulting Group, inc. 1999) is appropriate for predicting the elastic
modulus when the proposed loading procedure is used in the simulation of a two-
dimensional particle model.
It is worth pointing out that since the equivalent velocity is simultaneously
applied to both the upper and the lower boundaries of a test sample, the strain rate of
the sample is equal to the ratio of the loading rate to the half-length of the sample. In
the case of the small sample of 1000 particles, the corresponding strain rates of the
sample are 10 (1/s) and 1.0 (1/s) for loading rates of 10 (m/s) and 1.0 (m/s) respec-
tively. However, in the case of the large sample of 1000 particles, the corresponding
strain rates of the sample are 0.01 (1/s) and 0.001 (1/s) for the same loading rates
(i.e. 10 (m/s) and 1.0 (m/s)) as those used in the small sample. It can be observed that
due to the solution uniqueness of the samples in the elastic range, all the stress-strain
curves of both the small and the large samples are identical before the first failure
takes place in these two samples. Although the post-failure stress-strain curves of
the two samples are very similar in shape, they are not identical for both the small
and the large samples, indicating that the post-failure response of a sample is also
dependent on the strain rate of the sample. This issue needs to be considered when
a simulation result is obtained from a particle model.
Next, we compare the particle simulation results obtained from using the pro-
posed loading procedure with those obtained from using the improved conventional
loading procedure. For this purpose, both the small and the large test samples of