Page 199 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

190 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

k
P 1
(t = 1 Δ t)
k k
P 1 2
(t = 2 Δ t)
k k k
P 1 2 3
(t = 3 Δ t)
k k k k
P 1 2 3 4

(t = 4 Δ t)
k k k k k
P 1 2 3 4 5
(t = 5 Δ t)
k k k k k k
P 1 2 3 4 5 6

(t = 6 Δ t)
k k k k k k k
P 1 2 3 4 5 6 7

(t = 7 Δ t)
k k k k k k k k
P 1 2 3 4 5 6 7 8
(t = 8 Δ t)
k k k k k k k k k
P 1 2 3 4 5 6 7 8 9

(t = 9 Δ t)
k k k k k k k k k k
P 1 2 3 4 5 6 7 8 9 10

(t = 10 Δ t)
Fig. 8.5 Force propagation in a ten-particle system

fixed boundary condition at the right end of spring 1, as shown in Fig. 8.5. Similar
considerations can be made for the consecutive time steps (see Fig. 8.5). Ideally,
the “load” propagates through the whole system at the end of t = 10Δt, resulting
in a displacement of 10P/k for particle 1. In general, if this one-dimensional ide-
alized model is comprised of n particles of equal normal stiffness and mass, then
the displacement of particle 1 (i.e. the particle with “load” P)is nP/k at the end of
t = nΔt. Clearly, if one uses the record of the “load” and displacement at the end of
the immediate loading step to determine the elastic modulus of this one-dimensional
idealized particle system, then the determined elastic modulus will be exaggerated