Numerical methods in rock mechanics

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L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427
difference between the DEM and continuum-based
methods is that the contact patterns between components
of the system are continuously changing with the
deformation process for the former, but are fixed for
the latter.
The most representative explicit DEM methods are
the computer codes UDEC and 3DEC for two and
three-dimensional problems in rock mechanics [86–92].
The hybrid technique with distinct element and boundary
element methods was also developed [93] to treat the
effects of the far-field most efficiently for two-dimensional
problems. Similar but different formulations and
numerical codes with the same principles have also been
developed and applied to various problems, such as rigid
block motions [94,95], plate bending [96], and the blockspring
model (BSM) [97–99] which is essentially a subset
of the explicit DEM by treating blocks as rigid bodies.
Due mainly to its conceptual attraction in the explicit
representation of fractures, the distinct element method
has been enjoying wide application in rock engineering.
A large quantity of associated publications has been
published, especially in conference proceedings. Some of
the publications are referenced here to show the wide
range of applicability of the methods.
* Underground works [100–109].
* Rock dynamics [110,111].
* Nuclear waste repository design and performance
assessment [112–114].
* Reservoir simulations [115].
* Fluid injection [116–118].
* Rock slopes [119].
* Laboratory test simulations and constitutive model
development [120–122].
* Stress-flow coupling [123,124].
* Hard rock reinforcement [125].
* Intraplate earthquakes [126].
* Well and borehole stability [127,128].
* Rock permeability characterization [129].
* Acoustic emission in rock [130].
* Derivation of equivalent properties of fractured rocks
[131,132].
A recent book [133] includes references to a collection of
explicit DEM application papers for various aspects of
rock engineering.
The seminal DEM work for granular materials for
geomechanics and civil engineering application is in a
series of papers [134–138]. The development and
applications are mostly reported in series of proceedings
of symposia and conferences, such as in [139–140] in the
field of geomechanics. The most well-known codes in
this field are the PFC codes for both two-dimensional
and three-dimensional problems [141], and the DMC
code [142,143]. The method has been widely applied to
many different fields such as soil mechanics, the
processing industry, non-metal material sciences and
defence research. More extensive coverage of this
subject will be given in a later expanded version of
this review.
The implicit DEM is represented mainly by the
discontinuous deformation analysis (DDA) approach,
originated in [144,145], and further developed in
[146,147] for stress-deformation analysis, and in
[148,149] for coupled stress-flow problems. Numerous
other extensions and improvements have been implemented
over the years in the late 90s, with the bulk of
the publications appearing in a series of ICADD
conferences [32–35]. The method uses standard FEM
meshes over blocks and the contacts are treated using
the penalty method. Similar approaches were also
developed in [150–152] using four-noded blocks as the
standard element, and was called the discrete finite
element method. Another similar development, called
the combined finite-discrete element method [153–155],
considers not only the block deformation but also
fracturing and fragmentation of the rocks. However, in
terms of development and application, the DDA
approach occupies the front position. DDA has two
advantages over the explicit DEM: relatively larger time
steps; and closed-form integrations for the stiffness
matrices of elements. An existing FEM code can also be
readily transformed into a DDA code while retaining all
the advantageous features of the FEM.
The DDA method has emerged as an attractive model
for geomechanical problems because its advantages
cannot be replaced by continuum-based methods or by
the explicit DEM formulations. It was also extended to
handle three-dimensional block system analysis [156]
and use of higher order elements [157], plus more
comprehensive representation of the fractures [158]. The
code development has reached a certain level of maturity
with applications focusing mainly on tunnelling, caverns,
fracturing and fragmentation processes of geological
and structural materials and earthquake effects
[159–165].
Similar to the DEM, but without considering block
deformation and motion, is the Key Block approach,
initiated independently by Warburton [166,167] and
Goodman and Shi [168], with a more rigorous
topological treatment of block system geometry in the
latter (see also [169,170]). This is a special method for
analysis of stability of rock structures dominated by the
geometrical characteristics of the rock blocks and hence
the fracture systems. It does not utilize any stress and
deformation analysis, but identifies the ‘key blocks’ or
‘keystones’, which are formed by intersecting fractures
and excavated free surfaces in the rock mass which have
the potential for sliding and rotation in certain directions.
Key block theory, or simply block theory, and the
associated code development enjoy wide applications in
rock engineering, with further development considering
Monte Carlo simulations and probabilistic predictions