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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
 
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