where
[퐷휀푝]=(퐾푒−퐴){
휕휀V
휕휀
}{
휕휀V
휕휀
}
푇
−퐵{휕휀V
휕휀
}{
휕휀
휕휀
}
푇−퐶{
휕휀
휕휀
}{
휕휀V
휕휀
}
푇
+ (3퐺푒−퐷){휕휀
휕휀
}{
휕휀
휕휀
}
푇
,(45b)that is,
[퐷휀푝]=
1
|퐴|
{훼퐴{
휕휀V
휕휀
}{
휕휀V
휕휀
}
푇
+퐵{휕휀V
휕휀
}{
휕휀
휕휀
}
푇+퐶{
휕휀
휕휀
}{
휕휀V
휕휀
}
푇
+퐷
훼
{
휕휀
휕휀
}{
휕휀
휕휀
}
푇
},(45c)where훼=퐾푒/3퐺푒=2(1+휇푒)/9(1−2휇푒),and휇푒is the elastic
Poisson ratio.
Hence, the total stress increment can be expressed as
{d휎}=[퐷푒]{d휀푒}=[퐷푒]({d휀}−{d휀푝})=[퐷푒]{d휀}−{d휎푝}=[퐷푒]{d휀}−[퐷휀푝]{d휀}=[퐷휀푒푝]{d휀},(46a)where
[퐷휀푒푝]=[퐷푒]−[퐷푝휀]. (46b)The duality of stress and strain is evident in (45a), (45b), (45c),
(33a), and (33b).
It should be noted that it is practically impossible to
obtain the total strain in soil, and thus the elastoplastic
matrix in stress space is more applicable. However, if we
could further extend the framework to the space of strain
increment, the practicability becomes promising.
7. Relationship with Traditional
Elastoplastic Model
7.1. General Form. The elastoplastic compliance matrix of the
traditional elastoplastic model is
[퐶휎푒푝]=[퐶푒]+
1
퐴퐻
{
휕푔
휕휎
}{
휕푓
휕휎
}
푇
, (47)where푓and푔aretheyieldfunctionandplasticpotential
function,퐴퐻 = −휕푓/휕퐻{휕퐻/휕휀푝}푇{휕푔/휕휎}is the plastic
hardening modulus, and퐻is the hardening parameter.
푓and푔are usually expressed in terms of the stress
invariants,푝, 푞, 휃.IftheeffectoftheLode’sangle휃is not
considered, the expression only concerns푝and푞,thatis,
{
휕푓
휕휎
}=
휕푓
휕푝
{
휕푝
휕휎
}+
휕푓
휕푞
{
휕푞
휕휎
}, (48a){
휕푔
휕휎
}=
휕푔
휕푝
{
휕푝
휕휎
}+
휕푔
휕푞
{
휕푞
휕휎
}. (48b)Substituting (48a)and(48b)into( 47 ), we have[퐶휎푒푝]
=[퐶푒]+
1
퐴퐻
×{
휕푔
휕푝
휕푓
휕푝
{
휕푝
휕휎
}{
휕푝
휕휎
}
푇
+휕푔
휕푝
휕푓
휕푞
{
휕푝
휕휎
}{
휕푞
휕휎
}
푇+
휕푔
휕푞
휕푓
휕푝
{
휕푞
휕휎
}{
휕푝
휕휎
}
푇
+휕푔
휕푞
휕푓
휕푞
{
휕푞
휕휎
}{
휕푞
휕휎
}
푇
}.(49)Comparing this with (33a), (33b), (34a), and (34b), it follows
that퐴=
1
퐴퐻
휕푔
휕푝
휕푓
휕푝
,퐵=
1
퐴퐻
휕푔
휕푝
휕푓
휕푞
,
퐶=
1
퐴퐻
휕푔
휕푞
휕푓
휕푝
,퐷=
1
퐴퐻
휕푔
휕푞
휕푓
휕푞
.
(50a)Equation (33a)and(33b) can be seen as a general formula for
traditional constitutive models, and ( 49 )isaspecialformof
(33a)and(33b). For associated models, when푓=푔,퐴=
1
퐴퐻
(
휕푓
휕푝
)
2
,퐵=퐶=1
퐴퐻
휕푓
휕푝
휕푓
휕푞
,
퐷=
1
퐴퐻
(
휕푓
휕푞
)
2
.(50b)Most traditional constitutive models are to determine
the relationship between푓,푔and푝,푞,whichcanbeused
to calculate the four model parameters퐴, 퐵, 퐶,and퐷
indirectly. It can be seen from (50a)and(50b)thatthe
traditional nonassociated models actually assume that퐴퐷 − 퐵퐶 = 0. (51a)The associated models still need to satisfy (51a)andalso
require퐵=퐶. (51b)Rewrite (27a)and(27b)intomatrixform{
d휀푝Vd휀푝}=[
퐴퐵
퐶퐷
]{
d푝
d푞}. (52)
Clearly (51a) requires the determinant rank of the coef-
ficient matrix in ( 52 )tobe1orrequiresd휀V푝and d휀푝to be
linearly correlated. Equation (51b) additionally requires the
coefficient matrix to be symmetric.
The traditional elastoplastic model in stress space can be
translatedtostrainspacebythedualityof (45a), (45b), (45c),