Page 189 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

180 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems
2
d x α
F αx = m α 2 , (8.1)
dt

2
d y α
F αy = m α , (8.2)
dt 2
2
d θ αz
M αz = I α 2 , (8.3)
dt
where F αx , F αy and M αz are the total translational force components and rotational
moment exerted on the mass center of particle α; m α and I α are the mass and princi-
pal moment of inertia with respect to the z axis that is perpendicular to the x-y plane;
x α and y α are the horizontal and vertical coordinates expressing the position of par-
ticle α; θ αz is the rotation angle of particle α with respect to the principal rotational
axis of the particle.
For a two-dimensional disk-shaped particle, the corresponding principal moment
of inertia is expressed as follows:

1 2
I α = m α R , (8.4)
α
2
where R α is the radius of particle α.
The translational forces and rotational moment expressed in the above equations
can be calculated by adding all the forces and moments exerted on the particle. It
is noted that a particle can contact several particles at the same time so that there
are several contact forces exerted on the particle. For a particular contact (i.e. con-
tact C) between two particles (i.e. particles A and B) shown in Fig. 8.2, the normal
component of the contact force can be calculated using the following formula:

→
q

R B
→
n C
R A B
A Contact
plane
y

Fig. 8.2 Definition of the x
contact between two o
particles