key futures of this theory are that neither yield surface nor
plastic potential surface needs to be defined explicitly, and
consistency law is not required to determine plastic modulus.
In the theory, the total strain increment is divided into elastic
and plastic components.
Consider푑휀 = 푑휀푒+푑휀푝, (6)
where푑휀푒and푑휀푝= elastic and plastic strain increments,
respectively.
The relationship between strain and stress increments is
expressed as
푑휎 =D푒푝:푑휀, (7)
whereD푒푝is the elastoplastic stiffness tensor given as
D푒푝=D푒−
D푒:n푔퐿/푈:n푇:D푒
퐻퐿/푈+n푇:D푒:n푔퐿/푈, (8)
whereD푒,n푔퐿/푈,n,and퐻퐿/푈are elastic stiffness tensor,
plastic flow direction vector, loading direction vector, and
plastic modulus under loading or unloading conditions,
respectively.
The loading direction vectornis used to judge the loading
and unloading conditions:
푑휎푇푒⋅n>0 loading,푑휎푒푇⋅n=0 neutral loading,푑휎푒푇⋅n<0 unloading.(9)
Then, the elastoplastic stiffness tensor D푒푝 can be
obtained corresponding to the loading and unloading con-
ditions.
In the framework of generalized plasticity theory, all the
components of the elastoplastic constitutive matrix are deter-
mined by the current state of stress and loading/unloading
condition.
2.2.2. Pastor-Zienkiewicz-Chan Model.This model was pre-
sented by Pastor et al. [ 19 ]. The relationships between elastic
volumetric and shear strain increments and stress increments
are defined as
푑푝耠=퐾푒V푑휀V푒,푑푞=3퐺푒푠푑휀푒푠, (10)
where퐾푒V,퐺푒푠are tangential bulk and shear moduli, respec-
tively, and they are assumed to be
퐾푒V=퐾푒푠표
푝耠
푝표
,퐺푒푠=퐺푒푠표
푝耠
푝표
, (11)
where퐾푒푠표,퐺푒푠표,and푝표are model parameters.
In order to determine the plastic stiffness tensor, variables
n푔퐿/푈,n,and퐻퐿/푈need to be defined.n푔퐿/푈andnare
expressed as follows:n푔퐿=(푑푔
√1+푑^2 푔
,
1
√1+푑^2 푔
)
푇,n=(푑푓
√1+푑^2 푓
,
1
√1+푑^2 푓
)
푇.(12)
The dilatancy푑푔and stress ratio휂=푞/푝are related as
follows:푑푔=
푑휀푝V
푑휀푝푠
=(1+훼푔)(푀푔−휂). (13)
And푑푓has a similar expression as푑푓=(1+훼푓)(푀푓−휂), (14)
where훼푓,훼푔are model parameters and푀푔/푀푓is equal
to relative density. If푑푓=푑푔, associated flow rule is used,
otherwisenonassociatedflowruleisused.
In the case of unloading, the unloading plastic flow
direction vectorn푔푈is defined asn푔푈=(−儨儨儨
儨儨
儨儨儨
儨儨儨
儨儨
푑푔
√1+푑^2 푔
儨儨儨
儨儨
儨儨儨
儨儨儨
儨儨
,
1
√1+푑^2 푔
)
푇. (15)
The loading plastic modulus퐻퐿is proposed as퐻퐿=퐻 0 푝耠퐻푓(퐻V+퐻푠)퐻퐷푀, (16)
where퐻푓=(1−휂/휂푓)^4 limits the possible state and휂푓=(1+
1/훼푓)푀푓,퐻V=1−휂/푀푔accounts for phase transformation;
퐻푠=훽 0 훽 1 exp(−훽 0 휉)considers soil degradation and휉is the
accumulated plastic shear strain;퐻퐷푀=(휍MAX/휍)훾accounts
for past history and휍=푝[1−훼푓휂/(1 + 훼푓)/푀푓](−1/훼)푓 which
is the mobilized stress function; and퐻 0 ,훽 0 ,훽 1 ,훾are model
parameters.
Under unloading condition, the plastic modulus is
defined as퐻푈=퐻푢0(
푀푔
휂푢
)
훾푢
,푀푔
휂
>1,
퐻푈=퐻푢0,
푀푔
휂
≤1,
(17)
respectively, where퐻푢0,훾푢are model parameters and휂푢is the
stress ratio from which unloading takes place.