Page 195 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

186 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

determined from the mechanical response of the particle model having this particu-
lar set of particle-scale mechanical properties. If the resulting macroscopic mechani-
cal properties are different from what we expected, then another set of particle-scale
mechanical properties of materials are used in the particle model. This trial-and-
error process needs to be continued until a set of particle-scale mechanical prop-
erties of materials can produce the expected macroscopic mechanical properties.
In geological practice, a kilometer-length-scale specimen is often used to conduct a
biaxial compression test and to measure the related macroscopic mechanical proper-
ties, such as the elastic modulus and material strength, from the mechanical response
of the particle model with an assumed set of particle-scale mechanical properties of
rocks. However, if some mechanical properties are independent of particle size or
other size-dependent mechanical properties can be determined from an appropriate
upscale rule, then the expected particle-scale mechanical properties of materials to
be used in a particle model can be determined without a need to conduct the afore-
mentioned trial-and-error exercise.

8.2.2 Numerical Simulation Issue Arising from Using the Explicit
Dynamic Relaxation Method to Solve a Quasi-Static Problem

For the purpose of demonstrating the numerical simulation issue resulting from
using the explicit dynamic relaxation method to solve a quasi-static problem, it is
helpful to explain briefly how the finite element method is used to solve the same
kind of problem. For the sake of simplicity, a quasi-static elastic problem is used
to demonstrate the issue. Since the finite element method is based on continuum
mechanics, the governing equations of a two-dimensional quasi-static elastic prob-
lem in an isotropic and homogeneous material can be expressed as follows:

∂σ x ∂τ yx
+ = f x , (8.20)
∂x ∂y

∂τ xy ∂σ y
+ = f y , (8.21)
∂x ∂y

E(1 − ν) ν
σ x = ε x + ε y , (8.22)
(1 − 2ν)(1 + ν) 1 − ν

E(1 − ν) ν
σ y = ε x + ε y , (8.23)
(1 − 2ν)(1 + ν) 1 − ν

τ xy = τ yx = 2Gγ xy , (8.24)