Geoscience Reference
In-Depth Information
inthe three dimensional space isobtained as follows:
J
I
q
(
ij
)
t
(
ij
)
k
1
4
n
(
ij
)
l
ω
(
j
)
σ
kl
=
p
δ
kl
+
,
(13.18)
π
j
=
1
i
=
1
d
σ
kl
=
D
klmn
d
(ε
mn
−
ε
0
δ
mn
)
(13.19)
D
klmn
=
K
L
/
U
δ
kl
δ
mn
U
t
(
ij
)
t
(
ij
)
J
I
1
4
G
(
ij
)
L
n
(
ij
)
l
n
(
ij
)
ω
(
j
)
+
,
,
(13.20)
m
n
/
k
π
j
=
1
i
=
1
d
p
=
K
L
/
U
d
ε
(13.21)
G
(
ij
)
L
d
q
(
ij
)
=
γ
(
ij
)
U
d
(13.22)
/
where
n
(
ij
)
k
and
t
(
ij
)
k
n
(
i
k
and tangential direction
t
(
i
k
defined
in the
j
-th plane and the loading (L) and unloading (U) for the isotropic and virtual
simple shear mechanisms are defined by the signs of d
denote the contact normal
˜
t
(
ij
)
n
(
ij
)
γ
(
ij
)
ε
and d
=
,
ε
mn
,
m
n
respectively.
When the inherent soil fabric is assumed to be isotropic, the virtual simple shear mecha-
nism isdefined bya hyperbolic relation under aconstant confining stressas follows:
γ
(
ij
)
/γ
v
q
(
ij
)
=
γ
(
ij
)
/γ
v
q
v
(13.23)
1
+
where
q
v
and
γ
v
are the parameters for defining the hyperbolic relationship and called
the virtual shear strength and virtual reference strain. Substitution of Eq. (13.23) into
Eq. (13.22) yields
1
q
v
γ
v
G
(
ij
)
L
=
(13.24)
γ
(
ij
)
/γ
v
2
1
+
Hysteresischaracteristicsareassignedbyappropriatelyspecifyingthetangentialstiffness
forunloadingandreloadingbyusinganextendedMasingrule(Iaietal.,1990,1992a,b)
for representing realistic behavior of sands such as those given by Hardin and Drnevich
(1972). If no memory is given to the set of
q
(
ij
)
and other Masing variables, the material
becomes isotropic again once the applied stress is removed. The anisotropy in inherent
soilfabriccanbeintroducedbyspecifyingthevirtualshearstrengthandvirtualreference
strain as
q
(
ij
)
,γ
(
ij
v
that are specific to
i
-th mechanism in
j
-thplane.
The parameters
q
v
and
v
γ
v
can be determined by the shear modulus at small strain level
and failure criterion of soil (Iai et al., 1992a; Iai and Ozutsumi, 2005). In particular, the
shear modulus at smallstrain level isgiven by
