Page 191 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

182 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

Fig. 8.3 Definition of the →
shear velocity at a contact q F
By
between two particles R
S V B
F
Bx
M
F Bz B
Ay →
R n
A
F
M Ax
Az
A
y
x
o

As shown in Fig. 8.3, the magnitude of the tangential shear velocity, V S , at con-
tact C can be determined using the relative motion of particles A and B.

%
dx B dx A dy B dy A 2 2
V S =− − n y + − n x − ω Bz (x B − x C ) + (y B − y C )
dt dt dt dt
,
%
2
− ω Az (x A − x C ) + (y A − y C ) 2
(8.12)
where ω Az and ω Bz are the rotational angular speeds of particles A and B with respect
to their rotational axes, which are parallel to the z axis and passing through the
corresponding mass centers of each of the two particles respectively.
It is noted that the first two terms in Eq. (8.12) represent the contributions of the
relative translational motion to the relative shear velocity between the two particles,
whereas the last two terms denote the contributions of the relative rotational motion
to the relative shear velocity between the two particles.
For any given time instant, t, the tangential component of the contact force at
contact C can be calculated by adding the contact force increment expressed in
Eq. (8.11) into the tangential component at the previous time, t − Δt.

s
n
s
F = F t−Δt − k s (V S Δt) ≤ μF , (8.13)
t
s
s
where F and F t−Δt are the tangential components of the contact force at t and
t
t − Δt respectively; μ is the friction coefficient at contact C. It needs to be pointed
out that Eq. (8.13) holds true only when the normal component of the contact force
is greater than zero.
The normal and tangential components of the contact force at contact C can
be straightforwardly decomposed into the horizontal and vertical components as
follows: