671017.pdf

B

A

Unloading line

un

/t

Loading line

lnp

lnn lnni

Figure 12:푢푛-ln푝plots of the loading and unloading process in the
tests.

a ruled sawtooth. According to their approach, the loading
direction of the sawtooth should be

{푛}={

휕퐹

휕휎푛

휕퐹

휕휏

}

={sin훼푘+푢cos훼푘 cos훼푘−푢sin훼푘}.

(8)

Here,푢is the frictional coefficient of the interface when
the sawtooth height is equal to zero.훼푘is the topography
parameter.

4. Identification of the Model Parameters

4.1. Elastic Moduli:퐷푛and퐷푠.The normal elastic moduli
퐷푛can be determined from the loading-unloading curve of
푢푛-ln푝,asshowninFigure 12;thedisplacementoftheAB
section is deduced from the primary consolidation of the clay
under the initial normal stress, and the resilience occurring as
the initial normal stress휎푛푖is unloaded to the normal stress
휎푛. The slope coefficient of the resilience line was signified by
휅. Thus, the normal moduli can be determined as.

퐷푛=

휎푛

휅

. (9)

The hyperbolic model was validated by many test results,
which measured the shear moduli changes with shear dis-
placement. Consider

퐷푠=

푎

(푎 + 푏휀푠)

2 , (10)

where푎=1/퐷푠푖,푏=휎푛/휏ult,and휏ultis the ultimate shear
strength of the critical state in the test; thus,푏=1/휂푐.
From Figures 3 – 8 , the peak shear strength is not yet reached,
even when the shear displacement is accumulated to 30 mm,
which is the limit displacement of this test because, under
operational conditions, the accumulation of 30 mm of lateral

displacement was observed to possibly result in excessive

leakage of soil. Thus, the destructional ratio푅푓is introduced:

푅푓=

휏푓

휏ult

, (11)

퐷푠=퐷푠푖(1 − 푅푓

휏

휏푓

)

2

, (12)

where휏푓is the shear stress as the shear displacement reaches

30 mm and휏ultis determined through curve fitting.퐷푠푖are

the initial tangent shear moduli, which are described by

AnubhavP.K.andcoworkers;whereanincreaseinthenormal

stress will result in steeper shear-relative displacement curves

and a higher strength, and the values of퐷푠푖and휏ulttherefore

will increase with the increase in normal stress. This stress

dependence is taken into account by using empirical equa-

tions to represent the variation of퐷푠푖with normal stress:

퐷푠푖=퐾푃푎(

휎푛

푃푎

)

푛

, (13)

where퐾is the modulus number and푛is the modulus

exponent (both are dimensionless numbers), and푃푎is the

atmospheric pressure. However, the modulus number and

the modulus exponent must be determined through curve

fitting, which limits the application of the proposed model.

At the beginning of shear, the deformation can be assumed

to be elastic, so the initial shear modulus can be expressed

as follows. According to the relationship between the normal

elastic modulus and the shear elastic modulus and by analogy

between the behaviours of soil and the interface between soil

and structures,

퐷푠푖=

퐷푛

2 (1+])

=

휎푛

2휅(1+])

, (14)

where]is Poisson’s ratio of the soil. However, the above

equation cannot incorporate the influence of normal stress

history on the initial shear modulus. G.T. Houlsby and C.P.

Wroth performed a research on the stress history of soil; the

initial shear modulus that can account for the effect of stress

history was expressed as

퐺표푐=퐺푛푐(

휎푛푖

휎푛

)

0.7

, (15)

where퐺표푐is the initial shear modulus during overconsolida-

tion of soil and퐺푛푐is the initial shear modulus during normal

consolidation of soil. Note that the exponent has the value of

0.7 only for the situation of an overconsolidation ratio below

10.Therefore,theinitialshearmodulusoftheproposedmodel

can be given by

퐷푠푖=

휎푛

2휅(1+])

(

휎푛푖

휎푛

)

0.7

. (16)

Finally,theshearmodulusthataccountsfortheeffectof

normal stress history is

퐷푠=

휎푛

2휅(1+])

(

휎푛푖

휎푛

)

0.7

(1 − 푅푓

휏

휏푓

)

2

. (17)