Page 201 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

192 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

occurrence of any unphysical damage/crack in the particle model, whereas a frozen
period can be comprised of thousands of time steps. In the loading period, a veloc-
ity increment is applied to the loading boundary of the system, while in the frozen
period the loading boundary is fixed to allow the system to reach a quasi-static equi-
librium state after a large number of time steps. It needs to be pointed out that a pair
of load and displacement (or stress and strain) is correctly recorded at the end of a
frozen period.
Based on the above-mentioned conceptual understanding, the theoretical expres-
sion of the proposed loading algorithm can be deduced as follows. An applied force
in the quasi-static system can be divided into M loading increments.

M

P = ΔP i , (8.31)
i=1

where P is the applied force; ΔP i is the ith loading force increment and M is the
total number of loading steps.
For each loading force increment, ΔP i , it is possible to find a solution, ΔS i ,
which satisfies the following condition:

lim (ΔS ij − ΔS ij−1 ) = 0, (8.32)
j→N i

where ΔS ij is the solution at the jth time step due to the ith loading force incre-
ment; ΔS ij−1 is the solution at the (j-1)th time step due to the ith loading force
increment; N i is the total number of time steps for the particle system to reach a
quasi-static equilibrium state after the application of the ith loading force increment
to the system.
From the numerical analysis point of view, Equation (8.32) can be straightfor-
wardly replaced by the following equation:

& &
& ΔS ij − ΔS ij−1 &
max & & <δ (at j = N i ), (8.33)
& &
ΔS ij
where δ is the tolerance of the solution accuracy.
It needs to be pointed out that the value of N i can be determined using the con-
vergent condition expressed by Eq. (8.33), so that different values of N i may be
obtained in the numerical simulation. It is the convergent condition that approxi-
mately warrants the solution of the quasi-static nature, if the value of δ is not strictly
equal to zero.
Thus, the total solution S corresponding to the applied force P can be expressed as

M

S = ΔS i . (8.34)
i=1