671017.pdf

The above-mentioned modification method can be
extended to other elastoplastic models. The correction coeffi-
cients휉and휁canbefittedbasedonthetriaxialtestingresults
or other testing results. The obtained matrix can improve the
calculation accuracy of existing elastoplastic models or be
used to establish new modified models. Note that there is
no physical mechanism involved in the framework, and thus
the loading-unloading criterion and the evolution of internal
state variables of the original model should still be employed
in the modified one.

8. Application: A Simple Model

Theoretically, the constitutive model of soil would be estab-
lished if the parameters퐴,퐵,퐶,and퐷in (27a), (27b), (33a),
and (33b) are obtained by experiments such as conventional
triaxial test, isotropic compression test, and푝=Const. test.
For example, the equations for the tangent modulus퐸푡
and tangential Poisson ratio휇푡are obtained by fitting the
curve from triaxial testing, that is

퐸푡=

휕(휎 1 −휎 3 )

휕휀 1

, (62a)

휇푡=−

휕휀 3

휕휀 1

=

1

2

(1 −

휕휀V

휕휀 1

). (62b)

퐸푡and휇푡can be curve fitted by a polynomial or by
the formulae in the Duncan-Chang E-휇model [ 1 ]. However,
two supplementary equations are needed as there are four
unknown parameters in (27a)and(27b). Therefore, an
isotropic compression test or푝=Const. test should be
conducted, or the assumptions퐴퐷 − 퐵퐶 = 0and퐵=퐶
are made.
In conventional triaxial test,휎 3 =Const, d푝 = (1/3)d휎 1 ,
d푞=d휎 1 ,d휀V=d휀 1 +d휀 2 +d휀 3 =(1−2휇푡)d휀 1 ,andd휀=
(2/3)(1 + 휇푡)d휀 1. According to (62a), (62b), (27a), and (27b)
and the assumption in (51a)and(51b), we have

퐴=

퐾^2 푒푝

휔

,퐵=퐶=

퐾푒푝퐺푒푝

휔

,퐷=

퐺^2 푒푝

휔

, (63a)

where

퐾푒푝=

1−2휇푡

퐸푡

−

1−2휇푒

퐸푒

,

퐺푒푝=

2(1+휇푡)

3퐸푡

−

2(1+휇푒)

3퐸푒

,

휔=퐺푒푝+

1

3

퐾푒푝=

1

퐸푡

−

1

퐸푒

.

(63b)

퐸푒istheelasticmodulus,andtheelasticPoissonratio휇푒is
generally taken as 0.3 for soil. It is obvious that휔>0is always
fulfilled in (63a)and(63b).
By substituting (63a)and(63b)into(33a), (33b), or ( 35 ),
the elastoplastic compliance matrix in stress space or the
stiffness matrix in strain space is obtained. For convenience,
wecallthismodelthemultiplepotentialsurfacemodel(MPS
model).
It should be noted that, contrary to the Duncan-Chang
model (DC model),퐸푡and휇푡in the MPS model are not

0

5

10

15

0 2 4681012

Axial strain (%)

Calculation result

Experimental result

3 =344.^5

3 = 206.^7 kPa

3 =68.^9 kPa

kPa

Deviation stress (

100

kPa)

(a)

0 2 4681012

Axial strain (%)

Vo l

umetric strain (%)

MPS model

Experimental result

DC model

3 =344.^5 kPa

3 = 206.^7 kPa

2

0

− 2

− 4

− 6

− 8

(b)

Figure 1: Calculation and test results for triaxial test of Ottawa silica

sand.

limitedbythegeneralizedHooke’slaw;thatis,theMPSmodel

is still available when휇푡> 0.5and the stiffness matrix of the

model is not singular. Actually,퐸푡and휇푡in the new model

are not the traditional modulus and Poisson ratio, but just the

slope of the curves.

Figure 1shows the calculation and test results for triaxial

testing of Ottawa silica sand conducted by Wu [ 37 ]. The

unitweightofthesandis16.8kN/m^3 (=107 pcf ). The test

is a conventional consolidated-drained triaxial compression

test (CD test). The confining pressures were 68.9, 206.7, and

344.5kPa(=10,30,and50psi,resp.).DuringtheCDtest,

confining pressure was firstly applied and then the specimen

was consolidated. Deviation stress(휎 1 −휎 3 )was applied in

the axial direction after consolidation. Variations of deviation

stress and volumetric strainversusaxial strain can be acquired

in the test.

The calculations were made using the MPS model as well

astheDuncan-Changmodel,duringwhich퐸푡, 퐸푒,and휇푡

were calculated by the method proposed by Duncan and

Chang [ 1 ], that is

퐸푡=[1−

푅푓(1 −sin휙)(휎 1 −휎 3 )

2푐cos휙+2휎 3 sin휙

]

2

퐾푃푎(

휎 3

푃푎

)

푛

,

퐸푒=퐾푢푟푃푎(

휎 3

푃푎

)

푛

,

(64)