Page 196 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.2 Some Numerical Simulation Issues Associated with the Particle Simulation Method 187

∂u s ∂v s 1 ∂u s ∂v s
ε x = , ε y = , γ xy = + , (8.25)
∂x ∂y 2 ∂y ∂x

where σ x and σ y are normal stresses of the solid matrix in the x and y directions; ε x
and ε y are the normal strains of the solid matrix in relation to σ x and σ y ; τ xy and γ xy
are the shear stress and shear strain of the solid matrix; u s and v s are the horizontal
and vertical displacements of the solid matrix; E and G are the elastic and shear
modulus respectively; v is the Poisson ratio of the solid matrix; f x and f y are the
body forces in the x and y directions;.
Note that Eqs. (8.20) and (8.21) represent the equilibrium equations, whereas
Eqs. (8.22), (8.23), (8.24) and (8.25) are the constitutive equations and strain-
displacement relationship equations, respectively.
By using either the variational principle or the Galerkin method (Zienkiewicz
1977, Lewis and Schrefler 1998), the above-mentioned equations can be expressed
in the finite element form:

e e e
[K] {u} ={F} , (8.26)
e
e
e
where [K] is the stiffness matrix of an element; {u} and {F} are the displacement
and force vectors of the element.
In the finite element method, the quasi-static equilibrium problem is solved in
a global (i.e. system) manner. This means that the matrices and vectors of all the
elements in the system need to be assembled together to result in the following
global equation:

[K]{u}={F}, (8.27)

where [K] is the global stiffness matrix of the system; {u} and {F} are the global
displacement and force vectors of the system.
Similarly, in the distinct element method, the quasi-static equilibrium equation
of a particle is of the following form:
p
p
p
[K] {u} ={F} , (8.28)
p
p
p
where [K] is the stiffness matrix of a particle; {u} and {F} are the relative dis-
placement and force vectors of the particle.
In order to reduce significantly the requirement for computer storage and mem-
ory, the distinct element method solves the quasi-static equilibrium equation at
the particle level, rather than at the system level. This requires that a quasi-static
problem be turned into a fictitious dynamic problem so that the explicit dynamic
relaxation method can be used to obtain the quasi-static solution from solving the
following fictitious dynamic equation:

p
p
p
p
p
[M] {¨ u} + [K] {u} ={F} , (8.29)