Page 190 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.1 Basic Formulations of the Particle Simulation Method 181
5
(u n ≥ u n )
n k n u n
F = (Before the normal contact bond is broken),
0 (u n < u n )
(8.5)
5
n k n u n (u n ≥ 0)
F = (After the normal contact bond is broken),
0 (u n < 0)
(8.6)
where k n is the stiffness of contact C; u n is the normal displacement at contact C; u n
is the critical normal displacement in correspondence with the normal contact bond
n
breakage at contact C; F is the normal contact force at contact C. The contact force
at contact C is assumed to be exerted from particle A on particle B.

2
2
u n = R A + R B − (x B − x A ) + (y B − y A ) , (8.7)
where R A and R B are the radii of particles A and B; x A and y A are the horizontal
and vertical coordinates expressing the position of particle A; x B and y B are the
horizontal and vertical coordinates expressing the position of particle B.
Using the definition expressed in Fig. 8.2, the position of contact C can be
described as

1
x C = x A + R A − u n n x , (8.8)
2

1
y C = y A + R A − u n n y , (8.9)
2
where x C and y C are the horizontal and vertical coordinates expressing the position
of contact C; n x and n y are the direction cosines of the normal vector with respect
to the horizontal and vertical axes respectively.
The tangential component of the contact force at contact C can be calculated in
an incremental manner as follows:
5
−k s Δu s =−k s (V S Δt)(u n ≥ u n ) (before the normal
s
ΔF =
0 (u n < u n ) contact bond is broken),
(8.10)
5
−k s Δu s =−k s (V S Δt)(u n ≥ 0) (after the normal
s
ΔF =
0 (u n < 0) contact bond is broken),
(8.11)
where k s is the tangential stiffness of contact C; Δu s is the incremental tangential
s
displacement at contact C; ΔF is the corresponding tangential component of the
contact force; V S is the tangential shear velocity at contact C; Δt is the time step in
the numerical simulation.