Page 192 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.1 Basic Formulations of the Particle Simulation Method 183
s
n
F Cx = F n x + F q x , (8.14)
t
s
n
F Cy = F n y + F q y , (8.15)
t
where F Cx and F Cy are the horizontal and vertical components of the contact force
at contact C; q x and q y are the direction cosines of the tangential vector at contact
C with respect to the horizontal and vertical axes respectively.
Consequently, the translational and rotational forces exerted on particles A and B
due to their contact at point C can be calculated using the following formulas:

F Ax =−F Cx , F Ay =−F Cy , (8.16)

F Bx = F Cx , F By = F Cy (8.17)

M Az =−[(x C − x A )F Cx +(x C − x A )F Cy +(y C − y A )F Cx +(y C − y A )F Cy ], (8.18)

M Bz = (x C − x B )F Cx + (x C − x B )F Cy + (y C − y B )F Cx + (y C − y B )F Cy , (8.19)

where F Ax , F Ay and M Az are the translational force components and rotational
moment exerted on the mass center of particle A; F Bx , F By and M Bz are the trans-
lational force components and rotational moment exerted on the mass center of
particle B.
Since a particle may have contacts with several particles, it is necessary to search
the number of contacts for the particular particle under consideration. Therefore, the
total translational forces and rotational moment exerted on a particle can be calcu-
lated by adding the contributions of all the contacts to the translational forces and
rotational moments exerted on the particle. After the total translational forces and
rotational moment are calculated in a particle by particle manner, the central finite
difference method is used to solve the equations of motion expressed by Eqs. (8.1),
(8.2), and (8.3) for each of the particles in the simulation system, so that new dis-
placements can be determined and the position of each particle can be updated. As a
result, a solution loop is formed for each time step in the particle simulation method.
This solution loop is comprised of the following four sub-steps: (1) From the posi-
tion of particles at the beginning of a calculation time step, Eqs. (8.16), (8.17) and
(8.19) are used to calculate the contributions of a contact between a particle and
each of its surrounding particles to the translational force components and rota-
tional moment exerted on the mass center of both the particle and the surrounding
particle; (2) By adding the contributions of all the contacts of a particle to the trans-
lational forces and rotational moments exerted on the particle, the total translational
forces and rotational moment exerted on the particle are calculated. (3) Sub-steps
(1) and (2) are repeated for all the particles so that the total translational forces