Geoscience Reference
In-Depth Information
Fig. 13.2. Stress component inthedirection of
ω
i
/
2 relative to
x
axis; (a)biaxial shear
˜
,(b) simple shear
,(c) biaxial shear with additional rotation of
n
(
i
)
k
n
(
i
)
l
t
(
i
)
k
n
(
i
)
l
˜
,
˜
,
˜
4
n
(
i
−
I
/
2
)
k
n
(
i
−
I
/
2
)
l
−
π/
˜
,
˜
q
(
i
S
in Eq. (13.5) represent the stress contributions obtained by an
average of contact forces with respect to the isotropic, biaxial shear and simple shear
mechanisms, where
q
(
i
)
F
The scalars
p
˜
,
˜
,
˜
q
(
i
F
and
q
(
i
S
are defined per unit
. In particular, the biaxial shear
due to the normal component of contact forces
F
is given by Eq. (13.6), the simple
shear due to the tangential component of contact forces
S
given by Eq. (13.7) as shown
in Figure 13.2(a) and (b). Since biaxial shear and simple shear are indistinguishable
inthetensorrepresentationexceptforthedifferenceintheorientationof
˜
˜
ω
4asshownin
Figure13.3(b)and(c),thestresscontributionofavirtualtwodimensionalmechanismin
Eq. (13.5) iswritten as
π/
I
q
(
i
)
t
(
i
)
k
n
(
i
)
l
σ
kl
=˜
p
δ
kl
+
1
˜
,
˜
ω
(13.8)
i
=
where
q
(
i
−
I
/
2
)
F
q
(
i
)
S
q
(
i
)
=˜
˜
+˜
(13.9)
Although the tensor
n
(
i
l
represents simple shear and called “virtual simple shear
mechanism,” formation of columnar structure in the assemblage of particles (e.g. Oda,
1974; Oda et al., 1985) indicates that the contributions from the couples due to normal
components of contact forces
t
(
i
)
k
,
˜
q
(
i
−
I
/
2
)
F
˜
is predominant in the shear stress contributions
q
(
i
)
.Infact,Eq.(13.8)canberewrittenintermsof“virtualbiaxialshearmechanisms”as
˜
I
q
(
i
)
n
(
i
−
I
/
2
)
k
n
(
i
−
I
/
2
)
l
σ
kl
=˜
p
δ
kl
+
1
˜
˜
,
˜
ω
(13.10)
i
=
