Geoscience Reference
In-Depth Information
G
ma
p
/
p
a
m
G
G
m
=
(13.25)
where
G
ma
are the shear modulus measured at laboratory with an effective mean stress
p
a
. The power index
m
G
typically takes the value of 0.5 for sand. The failure criterion
for sand isoftenapproximated by Mohr-Coulomb criteriongiven by
c
cot
sin
σ
1
−
σ
3
φ
f
−
σ
1
+
σ
3
2
2
=
φ
f
≡
τ
max
(13.26)
σ
1
and
σ
3
denote major and minor principal stresses,
c
and
φ
f
denote cohesion
where
and internal friction angle, and
τ
max
denotes shear strength. Other failure criteria such
as Tresca, von Mises, extended Tresca and Drucker-Prager can also be introduced. Any
of these failure criteria can be used to identify the virtual shear mechanism parameter
q
v
and
γ
v
. The resultstake the following general form:
q
v
∝
τ
max
γ
v
∝
τ
max
/
(13.27)
G
m
Thecoefficientsaregivenasanintegralofdirectioncomponentsofmultiplemechanism.
See Iai and Ozutsumi (2005) fordetails.
With the dilatancy d
ε
0
in Eq. (13.19) specified as a function of cumulative plastic shear
work (Towhata and Ishihara, 1985a; Iai et al., 1992a), the strain space multiple mech-
anism model has ten primary parameters as shown in Table 13.1; two specify elastic
properties of soil,twospecifyplastic shear behavior, and the restcontrol dilatancy.
As summarized in Eqs. (13.18) through (13.22) the strain space multiple mechanism
model characterizes a twofold structure of an assemblage of particles in granular mate-
rials. The first structure is represented by
J
sets of virtual two dimensional mechanisms,
the second by
I
sets of virtual simple shear mechanisms of one dimensional nature. Sec-
ond order fabric tensors are naturally incorporated in the direct stress-strain relationship
Table 13.1. Model parameters ofstrain space multiple mechanism model (Iai et al., 1992a)
Symbol
Type of mechanism
Parameter designation
K
ma
Elastic Volumetric
Reboundmodulus
G
ma
Elastic Shear
Shear modulus
φ
f
Plastic Shear
Internal friction angle
h
max
Plastic Shear
Upper bound for hysteretic damping factor
φ
p
Plastic Dilatancy
Phase transformation angle
p
1
Plastic Dilatancy
Parameter controllinginitial phase ofcumulative
dilatancy
p
2
Plastic Dilatancy
Parameter controlling final phase ofcumulative dilatancy
w
1
Plastic Dilatancy
Parameter controlling overall cumulative dilatancy
S
1
Plastic Dilatancy
Parameter controlling ultimate limit ofdilatancy
c
1
Plastic Dilatancy
Parameter controlling threshold limit for dilatancy