Geoscience Reference
In-Depth Information
7.2.1.
Inter-particle behavior
7.2.1.1.
Elastic part
The orientation of a contact plane between two particles is defined by the vector
perpendicular to this plane. On each contact plane, a local auxiliary coordinate can
be established, as shown in Figure 7.1. The contact stiffness of a contact plane
includes normal stiffness,
n
α
α
k
k
, and shear stiffness,
r
. The elastic stiffness tensor is
defined by
α
α
α
=
k
[7.1]
i j
which can be related to the contact normal and shear stiffness:
α
ααα
α αα αα
=
k nnk s s
+
+
t t
[7.2]
ij
n i j
r
i j
i j
where
n
,
s
and
t
are three orthogonal unit vectors that form the local coordinate
system. The vector
n
is outward normal to the contact plane. Vectors
s
and
t
are on
the contact plane.
The value of the stiffness for two elastic spheres can be estimated from Hertz-
Mindlin's formulation [MIN 69]. For sand grains, a revised form was adopted
[CHA 89b], as follows:
n
n
kk
=
n
kk
=
n
[7.3]
n
n
0
0
2
2
Gl
Gl
g
g
G
is the elastic modulus for the grains,
n
f
where
is the contact force in normal
direction,
l
is the branch length between the two particles, and
kk
and
n
are
material constants. For two spherical particles, the branch length is the same as the
particle size
l
=
. Let
n
= 1/3, and
,
no
ro
2/3
d
kG
12
21
=
[7.4]
n
0
g
−
ν
g
Equation [7.4] is equivalent to the Hertz-Mindlin's contact formulation
[MIN 69].
Search WWH ::
Custom Search