Geoscience Reference
In-Depth Information
In this paper, Eq. (13.8) with a term “virtual simple shear mechanism” will be used in
order to maintain the consistency with those used in the previous papers (Towhata and
Ishihara,1985b;Iaietal.,1992a).Theconceptofthemodel,however,isbetterdescribed
by Eq. (13.5), where the couples of normal components of contact forces are explicitly
written.
In order to describe the macroscopic stress-strain relationship of a granular material, the
stress contributions in Eq. (13.8) should be defined as a function of macroscopic strain
field
ε
kl
. As an assumption of the simplest in its kind, the isotropic stress contribution
˜
p
is defined as a function of
ε
=
δ
mn
(ε
mn
−
ε
0
δ
mn
)
(13.11)
q
(
i
)
is defined as a function of
and each virtual simple shear stresscontribution
˜
γ
(
i
)
=
t
(
i
)
m
n
(
i
)
n
t
(
i
)
m
n
(
i
)
n
,
˜
(ε
mn
−
ε
0
δ
mn
)
=
,
˜
ε
mn
(13.12)
δ
mn
in the right hand side of Eqs. (13.11) and (13.12) represents the
volumetric strain tensor due to dilatancy. The scalar
0
where the term
ε
γ
(
i
)
defined in Eq. (13.12) is the
projectionofmacroscopicstrainfieldintothedirectionofvirtualsimpleshearmechanism
and called “virtual simpleshear strain.”
t
(
i
)
k
n
(
i
)
l
,
˜
The incremental stress-strain relation is obtained in the similar manner as described
above and is given by
q
(
i
)
I
t
(
i
)
k
n
(
i
)
l
d
σ
kl
=
d
p
˜
δ
kl
+
d
˜
,
˜
ω
(13.13)
i
=
1
The incremental stresscontributions are given by
=
K
L
/
U
d
d
p
˜
ε
(13.14)
q
(
i
)
=
G
(
i
)
γ
(
i
)
d
˜
L
/
U
d
(13.15)
wheretheloading(L)andunloading(U)fortheisotropicandvirtualsimpleshearmech-
anisms are defined by the signs of d
γ
(
i
)
, respectively. From Eqs. (13.11) through
(13.15), the incremental constitutive equation isgiven by
ε
and d
=
D
klmn
d
d
σ
kl
(ε
mn
−
ε
0
δ
mn
)
(13.16)
I
U
t
(
i
)
m
1
G
(
i
)
t
(
i
)
k
n
(
i
)
l
D
klmn
=
K
L
/
U
δ
kl
δ
mn
+
n
(
i
)
n
,
˜
,
˜
ω.
(13.17)
L
/
i
=
By superposing these two dimensional mechanisms over
J
sets of planes, each with a
solid angle of
(
j
)
, covering a unit sphere, the macroscopic stress-strain relationship
