Page 204 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.3 An Upscale Theory of Particle Simulation for Two-Dimensional Quasi-Static Problems 195

geological medium, which may be non-homogeneous due to the presence of faults,
cracks and large geological structures, provided that the particle model of the geo-
logical medium satisfies the required geometrical similarity criterion.
For the analog of a two-circle contact with an elastic beam (Itasca Consulting
Group, inc. 1999), it is assumed that the behaviour of the contact between two parti-
cles is equivalent to that of an elastic beam with its ends at the two particle centres.
The beam is loaded at its ends by the force acting at the centre of each particle.
Under this assumption, the stress of the beam can be expressed as follows:

P Δ
σ = = Eε = E , (8.38)
D D
where σ and ε are the stress and strain of the equivalent elastic beam; P is the force
acting at the centre of each particle; E is the elastic modulus of the particle material;
Δ is the deformation of the equivalent elastic beam; D is the diameter of the particle.
Since the two particles are connected in series, the deformation of the equivalent
elastic beam can be also expressed as follows:

2P
Δ = , (8.39)
k n
where k n is the normal stiffness of a particle.
Substituting Eq. (8.39) into Eq. (8.38) yields the following equation:

k n = 2E. (8.40)

Since the contact-bond strength is expressed in the unit of force, the following
relationships exist mathematically:

b n = ασ n D, (8.41)

b s = ατ s D, (8.42)

where b n and b s are the normal and tangential bond strengths at a contact between
two particles; D is the diameter of the particle; α is a constant; σ n and τ s are the
unit normal and tangential contact bond strengths, which are defined as the normal
and tangential contact bond strengths per unit length of the particle diameter (Zhao
et al. 2007b). If α is assumed to be equal to one, then the values and units of the
unit normal and tangential contact bond strengths are exactly the same as those of
the macroscopic tensile and shear strengths of the particle material, while the values
of the normal and tangential contact bond strengths of a particle are equal to the
product of the unit normal/tangential contact bond strengths and the diameter value
of the particle. Thus, the variation of the normal/tangential contact bond strength