(33a), and (33b). Substituting (50a)and(50b)into( 42 ), (45a),
(45b), and (45c) yields the expression in strain space
[퐷휀푒푝]=[퐷푒]−
1
훽
×{훼
휕푔
휕푝
휕푓
휕푝
{
휕휀V
휕휀
}{
휕휀V
휕휀
}
푇
+휕푔
휕푝
휕푓
휕푞
{
휕휀V
휕휀
}{
휕휀
휕휀
}
푇+
휕푔
휕푞
휕푓
휕푝
{
휕휀V
휕휀
}{
휕휀
휕휀
}
푇
+1
훼
휕푔
휕푞
휕푓
휕푞
{
휕휀
휕휀
}{
휕휀
휕휀
}
푇
},(53a)where
훽=퐴퐻|퐴|=1
3퐺푒
휕푔
휕푝
휕푓
휕푝
+
1
퐾푒
휕푔
휕푞
휕푓
휕푞
+
퐴퐻
3퐾푒퐺푒
. (53b)
Hence, the stiffness matrix of (53a)and(53b)isobtained
using푓and푔from the traditional constitutive models,
and the transformation from stress space to strain space is
thus achieved. Obviously, the transformation only changes
the mathematical calculation method of the coefficient and
elastoplastic matrices and has no influence on a particular
model itself or the loading-unloading criterion of the model.
Therefore, the transformation is applicable to all traditional
elastoplastic models.
7.2. Modified Cambridge Model.In the modified Cambridge
model,
푓=푔=푝+
푞^2
푀^2 푝
−푝 0 푒(1+푒^0 )/(휆−휅)퐻=0, (54)
and so
휕푓
휕푝=1−
휂^2
푀^2
,
휕푓
휕푞
=
2휂
푀^2
,
퐴퐻=
1+푒 0
휆−휅
(1 −
휂^2
푀^2
)푝 0 푒(1+푒^0 )/(휆−휅)퐻,
(55)
where휂=푞/푝,퐻is the hardening parameter (=휀V푝,forthe
modified Cambridge model),푀is the stress ratio at critical
state,푝 0 is the initial mean stress,푒 0 is the initial void ratio,
휆is the slope of the normal compression line (NCL), and휅is
theslopeoftheunloadingline.Theelastoplasticcompliance
matrix in stress space is expressed as
[퐶휎푒푝]=[퐶푒]+1
퐴퐻
×{(1−
휂^2
푀^2
)
2
{휕푝
휕휎
}{
휕푝
휕휎
}
푇+
2휂
푀^2
(1 −
휂^2
푀^2
){
휕푝
휕휎
}{
휕푞
휕휎
}
푇+
2휂
푀^2
(1 −
휂^2
푀^2
){
휕푞
휕휎
}{
휕푝
휕휎
}
푇+
4휂^2
푀^4
{
휕푞
휕휎
}{
휕푞
휕휎
}
푇
}.(56)
The elastoplastic stiffness matrix in strain space is[퐷푒푝휀]=[퐷푒]−1
훽
×{
퐾푒
3퐺푒
(1 −
휂^2
푀^2
)
2
{휕휀V
휕휀
}{
휕휀V
휕휀
}
푇+
2휂
푀^2
(1 −
휂^2
푀^2
){
휕휀V
휕휀
}{
휕휀
휕휀
}
푇+
2휂
푀^2
(1 −
휂^2
푀^2
){
휕휀V
휕휀
}{
휕휀
휕휀
}
푇+
3퐺푒
퐾푒
4휂^2
푀^4
{
휕휀
휕휀
}{
휕휀
휕휀
}
푇
},(57a)where훽=1
3퐺푒
(1 −
휂^2
푀^2
)
2
+1
퐾푒
4휂^2
푀^4
+
퐴퐻
3퐺푒퐾푒
. (57b)
A new hardening parameter for the modified Cambridge
model was proposed by Yao et al. [ 12 ]as퐻=∫d퐻=∫푀^4 푓−휂^4
푀^4 −휂^4
d휀푝V=∫1
Ω
d휀푝V. (58a)in whichΩ=푀^4 −휂^4
푀푓^4 −휂^4
, (58b)where푀푓isthepotentialfailurestressratio.
Yao improved the modified Cambridge model by replac-
ing퐻=휀푝Vwith (58a)and(58b), which changes퐴퐻in the
modified Cambridge model to(1/Ω)퐴퐻.Theimprovedcon-
stitutive model is a unified hardening model and is suitable
for sandy soil, which actually replaces{d휀푝}byΩ{d휀푝}with
( 49 ), or the following expression with ( 52 ):
d휀푝V
d휀V푝儨儨儨
儨儨푐
=
d휀푝
d휀푝儨儨儨儨푐=Ω=
푀^4 −휂^4
푀푓^4 −휂^4
, (59)
where d휀푝V|푐,d휀푝|푐are the volumetric strain and shear strain,
respectively, that are calculated by the modified Cambridge
model.
This modification can be further improved. For instance,
the volumetric strain and shear strain of triaxial testing are
first calculated by the modified Cambridge model, and then
theratioofthevolumetricstrainandshearstraincanbefitted
according to the test results, that is
d휀푝V
d휀푝V儨儨
儨儨儨
푐=휉(푝,푞),
d휀푝
d휀푝儨儨儨儨푐=휁(푝,푞). (60)
Therefore, ( 52 )ismodifiedto{
d휀V푝d휀푝}=[
휉퐴 휉퐵
휁퐶 휁퐷
]{
d푝
d푞}. (61)휉and휁canbeestimatedbypolynomialfittingorother
fitting methods. Yao’s hardening model is obtained when휉=
휁=(푀^4 −휂^4 )/(푀푓^4 −휂^4 ).