The capillary-induced hardening is described by ( 10 ), while
the plastic volumetric strain hardening is given by
푝푐 0 (휀푝V)=푝푐∗ 0 exp(휐휀푝V
휆−휅
), (15)
where푝∗푐 0 is the initial preconsolidation pressure;휆is the
slope of the normal consolidation line on the휐−ln푝耠plane
of the soil under the fully saturated condition.
2.6. Effect of Deformation on the Soil-Water Characteristic
Curve.Whether or not the hydromechanical behavior of
an unsaturated soil can be effectively described at the con-
stitutive level depends largely on the characterization of
capillary hysteresis. To characterize the capillary hysteresis,
an advanced SWCC model, which can properly address the
soil-water characteristics of the soil experiencing arbitrarily
wetting/drying cycles, should be properly implemented into
a generalized mechanical constitutive framework [ 13 ].
We i a n d D e w o o l k a r [ 18 ]proposedathermodynamically
consistentmodelforcapillaryhysteresisinpartiallysatu-
ratedporousmedia.Inthismodel,thecapillaryhysteresis
phenomenon is linked to intrinsic energy dissipation pro-
cesses, which can be characterized by the series of internal
state variables,휉푓푖, and the dissipative energy is given by
휁푓푖훿휉푓푖,asin( 5 ). By virtue of the notion of the bounding
surface plasticity, a simplified model of capillary hysteresis is
developed. Provided that the main drying-wetting boundary
curves have been experimentally determined, the model
introduces only one additional parameter to describe all types
of scanning curves (primary, secondary, and higher order)
under arbitrary hydraulic paths. This model is introduced
here to describe the soil-water characteristics.
Experimental results [ 33 ] show that in a deformable soil,
the sizes and the connectivity of pores may vary with skeletal
deformation, which in turn induces change in the soil-water
characteristics. For example, the air entry value increases with
thedecreaseofvoidratio,andthesoil-watercharacteristic
curve may shift upward on the푆푟:푠푐plane. To address the
effect of deformation on the soil-water characteristic curve,
one first notes that, in general, the change in the degree of
saturation has two contributions: one is the change in the
amount of pore water due to seepage or dissipation, and the
otherisduetothechangeintheporevolume,namely,
푑푆푟=푑(
푉푤
푉V
)=
푑푉푤
푉V
−(
푉
푉V
)(
푉푤
푉V
)(
푑푉V
푉
). (16)
As the very meaning of a partial differential, the first
item of the right-hand side (r.h.s.) represents the change in
the degree of saturation under the constant-푉Vcondition,
that is, only due to the change in the amount of pore water,
while the second item describes the contribution of the
change in volumetric strain. Neglecting the effect of elastic
deformation, ( 16 )canbecastinto
푑푆푟=푑푆푟儨儨儨儨푑휀V=0+
푆푟
푛
푑휀푝V. (17)
According to Wei and Dewoolkar [ 18 ], the first term of
the r.h.s. can be described by푑푆푟儨儨儨儨푑휀V=0=
−푑푠푐
퐾푝(푠푐,푆푟,푛)̂
, (18)
wherê푛denotes the hydraulic loading direction, and its value
assumes 1 (or−1) for drying (or wetting);퐾푝is the negative
slope of the current soil-water characteristic curve (either
scanning or boundary), which is a function of푠푐,푆푟,and푛̂,
given by퐾푝(푠푐,푆푟,푛) =̂ 퐾푝(푆푟,푛) +̂
푐儨儨儨儨푠푐−푠푐(푆푟,푛)̂儨儨儨儨
푟(푆푟)−儨儨儨儨푠푐−푠푐(푆푟,푛)̂儨儨儨儨
, (19)
where퐾푝(푆푟,푛)̂ is the negative slope of the corresponding
mainboundary,whichisthemaindryingboundaryif
̂푛=1orthemainwettingboundaryif푛=−1̂ ;푐is a
positive material parameter which is used to describe the
scanning behavior;푠푐(푆푟,푛)̂ is the matric suction value on
the corresponding main boundary curve; that is,푠푐(푆푟,1) =
휅DR(푆푟)for drying and푠푐(푆푟,−1) = 휅WT(푆푟)for wetting,
where휅DR(푆푟)and휅WT(푆푟)describe the main drying and
wetting boundaries, respectively;푟(푆푟)is the current size of
the bounding zone; that is,푟(푆푟)=휅DR(푆푟)−휅WT(푆푟).
Although the effect of change in volumetric strain is
excluded in calculating푑푆푟|푑휀V=0(via( 18 )), the volumetric
deformation (or change of void ratio) may induce change in
the SWCC curve, as mentioned above. That is, function퐾푝
should depend explicitly upon the void ratio or equivalently,
thetotalplasticvolumetricstrain휀푝V(the effect of elastic strain
is neglected).
To address this issue, we adopt the following SWCC
model by Feng and Fredlund [ 47 ] to describe the main
boundaries:휅푘(푆푟)=푏푘(
1−푆푟
푆푟−푆irr푟)
1/푑푘
,푘=DR,WT, (20)where푏푘and푑푘are the positive material parameters and
assume different values for wetting and drying. The푑푘
parameter determines the curvature of the scanning curves,
while푏푘related to the air-entry value. Ignoring the influence
of the elastic volumetric strain and shear strain,푏푘and푑푘
depend on the plastic volumetric strain,휀V푝. Experimental
results [ 33 ] suggest that the skeletal deformation changes the
position of the SWCC only and leaves the shape of the curve
almost unchanged. Thus, for simplicity, we propose that푏푘=푏푘^0 +훼푘휀V푝,푑푘=푑^0 푘, (21)where푏푘^0 ,푑^0 푘,and훼푘are curving-fitting parameters,푘=DR,
WT.
Due to a lack of experimental data, it is assumed for
simplicity that parameter푐is constant, independent of the
skeletal deformation. Now, for deforming soils, ( 19 )canbe
replaced by퐾푝(휀푝V,푠푐,푆푟,̂푛)=퐾푝(휀푝V,푆푟,̂푛) +
푐儨儨儨儨푠푐−푠푐(휀푝V,푆푟,̂푛)儨儨儨儨
푟(휀푝V,푆푟)−
儨儨儨
儨儨푠푐−푠푐(휀
푝
V,푆푟,̂푛)