Page 185 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

176 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

propagation driven by the maximum circumferential tensile stress (Erdogan and
Sih 1963), a theory based on propagation driven by strain energy density near
the crack tip (Sih and Macdonald 1974) and a theory based on crack propaga-
tion driven by maximum energy release rate (Rice 1968, Hellen 1975, Lorenzi
1985, Li et al. 1985), the maximum circumferential tensile propagation theory has
been widely used in the finite element analysis of crack initiation and propaga-
tion problems because it is not only physically meaningful, but it is also easily
implemented in a finite element code. It has been widely demonstrated that, for
the finite element simulation of a two-dimensional fracture mechanics problem,
the triangular quarter-point element, which is formed by collapsing one side of a
quadrilateral 8-noded isoparametric element, results in improved numerical results
for stress intensity factors near a crack tip (Barsoum 1976, 1977). Similarly, for
the finite element modelling of a three-dimensional fracture mechanics problem,
the prismatic quarter-point element, which is formed by collapsing one face of
a cubic 20-noded isoparametric element, can also lead to much better numerical
results for stress intensity factors near a crack tip (Barsoum 1976, 1977, Ingraf-
fea and Manu 1980). This kind of quarter-point element can be used to simulate
√
the 1/ r singularity in elastic fracture mechanics by simply assigning the same
displacement at the nodes located on the collapsed side or face of the element,
while it can be also used to simulate the 1/r singularity in perfect plasticity by sim-
ply allowing different displacements at the nodes located on the collapsed side or
face of the element. In order to simulate appropriately displacement discontinuities
around a crack, automatic meshing and re-meshing algorithms (Zienkiewicz and
Zhu 1991, Lee and Bathe 1994, Khoei and Lewis 1999, Kwak et al. 2002, Bouchard
et al. 2003) have been developed in recent years. However, such finite element
methods with automatic meshing and re-meshing algorithms have been applied
for solving crack generation and propagation problems in systems of only a few
cracks.
For the purpose of removing this limitation associated with the conventional
numerical method (which is usually based on continuum mechanics) in simulat-
ing a large number of spontaneously generated cracks, particle simulation meth-
ods, such as the distinct element method developed as a particle flow code (Cundall
and Strack 1979, Cundall 2001, Itasca Consulting Group, inc. 1999, Potyondy and
Cundall 2004), provides a very useful tool to deal with this particular kind of prob-
lem. Since displacement discontinuities at a contact between two particles can be
readily considered in these methods, the formulation based on discrete particle sim-
ulation is conceptually simpler than that based on continuum mechanics, because
crack generation at a contact between two particles is a natural part of the particle
simulation process.
Even though the particle simulation method was initially developed for solv-
ing soil/rock mechanics, geotechnical and other engineering problems (Cundall and
Strack 1979, Bardet and Proubet 1992, Thomton et al. 1999, Tomas et al. 1999,
Salman and Gorham 2000, Klerck et al. 2004, Owen et al. 2004, McBride et al.
2004 and Schubert et al. 2005, Zhao et al. 2006f), it has been used to deal with a
large number of geological and geophysical problems in both two and three dimen-