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WATER RESOURCES RESEARCH, VOL. 46, W05515, doi:10.1029/2009WR008646, 2010

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Full
ArticleA closed‐form equation for effective stress in unsaturated soil
Ning Lu,1 Jonathan W. Godt,2 and David T. Wu3
Received 15 September 2009; accepted 14 December 2009; published 15 May 2010.

[1] We propose that the recently conceptualized suction stress characteristic curve
represents the effective stress for the shear strength behavior of unsaturated soil.
Mechanically, suction stress is the interparticle stress called tensile stress. The working
hypothesis is that the change in the energy of soil water from its free water state is mostly
consumed in suction stress. We demonstrate that the suction stress lies well within theframework of continuum mechanics where free energy is the basis for any thermodynamic
formulation. Available experimental data on soil water characteristic curves and suction
stress characteristic curves are used to test the hypothesis, thus validating a closed‐form
equation for effective stress in unsaturated soil. The proposed closed‐form equation is
intrinsically related to the soil water characteristic curve by two pore parameters: the air
entry pressure and pore size spectrum number. Both semiquantitative and quantitative
validations show that the proposed closed‐form equation well represents effective stressfor a variety of earth materials ranging from sands to clays. Of important practical
implications are (1) the elimination of the need for any new shear strength criterion forunsaturated soil, (2) the elimination of the need for determining the Bishop’s effective
stress parameter c because the new form of effective stress is solely a function of soil
suction, and (3) the ready extension of all classical soil mechanics work on limit
equilibrium analysis to unsaturated soil conditions.
Citation: Lu, N., J. W. Godt, and D. T. Wu (2010), A closed‐form equation for effective stress in unsaturated soil, Water
Resour. Res., 46, W05515, doi:10.1029/2009WR008646.

1. Introduction[2] In recent years, the suction stress characteristic curve
has been introduced to represent the state of stress inunsaturated soil [Lu and Likos, 2004, 2006]. In a broad
sense, it is an expansion and extension of both Terzaghi’s
effective stress for saturated soil [e.g., Terzaghi, 1943] andBishop’s effective stress for unsaturated soil [Bishop, 1954,
1959]. Like previous effective stress approaches, the suction
stress approach seeks a single stress variable that is
responsible for the mechanical behavior of earth materials.
However, suction stress differs from Terzaghi’s “skeleton”
stress in that forces contributing to suction stress are self‐balanced at the interparticle level and thus do not pass on
from one particle to another. Suction stress originates fromthe available interaction energy at the soil solid surface thatcan be conceptualized to exist in the forms of van der Waals
and double‐layer forces, surface tension, and solid‐liquidinterface forces due to pore water pressure [Lu and Likos,
2006]. A macroscopic continuum representation of suction
stress is the tensile stress [e.g., Mitarai and Nori, 2006; Lu
et al., 2007] that can be simply determined from uniaxial
tensile strength tests of unsaturated earth and other granular1
Division of Engineering, Colorado School of Mines, Golden,
Colorado, USA.
2
U.S. Geological Survey, Denver, Colorado, USA.
3
Departments of Chemical Engineering and Chemistry, Colorado
School of Mines, Golden, Colorado, USA.

Copyright 2010 by the American Geophysical Union.
0043‐1397/10/2009WR008646

materials [Lu et al., 2007]. The suction stress concept differsfrom Bishop’s effective stress mainly in that it eliminates theneed to define the coefficient of effective stress c, as suction
stress is solely a function of soil suction. The uniqueness ofc and its determination have been major obstacles for thewide acceptance of Bishop’s effective stress. However,recent work has established a link between c and the air
entry pressure head [Khalili and Khabbaz, 1998; Khalili etal., 2004] and the utility of Bishop’s effective stress as aconstitutive framework for critical state soil mechanics [Nuthand Laloui, 2008]. Finally, the suction stress characteristic
curve, similar to the soil water characteristic curve, does notneed to be restricted to a single‐valued function. Hysteresis
due to different wetting states could also be treated under theframework of suction stress. For example, different parameters can be used for the air entry or water entry pressure,depending on the soil’s wetting state.
[3] Suction stress provides a framework for the examination of the state of stress in unsaturated soil that differs
radically from the more commonly known two independentstress state variable approach [e.g., Fredlund and Morgenstern,1977]. In the two independent stress state variable approach,the total stress or “net normal stress” and matric suction are
hypothesized to be the necessary and sufficient stress variables for describing the mechanical behavior of unsaturated
soil. Under this framework, shear strength needs to becompletely modified to account for its dependence on matricsuction, and this is accomplished by the introduction of anadditional friction angle, b. Despite the popularity of thetwo independent stress state variable approach, it is quite
controversial [e.g., Khalili and Khabbaz, 1998; Nuth andW05515

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Laloui, 2008]. Whether matric suction is a stress variable andthe physical basis for the additional shear strength parameter b
are disputed at the present time [e.g., Lu, 2008]. The major
theoretical and practical obstacle faced by the two independentstress state variable approach is that it cannot be reconciled
within the context of classical mechanics for saturated soil. In
classical soil mechanics, the single stress variable, effective
stress, can be used for both shear strength (e.g., limit state) anddeformation (e.g., consolidation) analyses. This philosophy has
been widely adopted as the design basis in geotechnicalpractice today.
[4] The two independent stress state variable approachhas been expanded to handle elastoplasticity, critical statesoil mechanics, and coupled yield limits with some success[e.g., Alonso et al., 1990; Wheeler and Sivakumar, 1995;
Gallipoli et al., 2003]. While these theories provide insight
into important concepts such as loading collapse and suction
increase and decrease curves, they add additional complexity by requiring additional material parameters that areoften either variable or difficult to determine experimentally.
These theories typically involve an additional variable orparameter and are most suitable for stress‐strain analysis of
postfailure or deformable soil. A thorough critical review
and analysis of these models can be found in works by Gens
et al. [2006] and Nuth and Laloui [2008].
[5] In this paper, we present our working hypothesis thatthe change in energy of soil water from its free water state ismostly consumed in suction stress and establish a thermodynamic justification for a closed‐form equation for effective stress in variably saturated soils. We then reinterpretavailable experimental results from the literature to validate
the closed‐form equation and conclude with a discussion oftheoretical and practical implications.

2. A Closed‐Form Equation for Effective Stressin Variably Saturated Soil2.1. Working Hypothesis[6] The effective stress principle, under the framework of
the suction stress characteristic curve, can be expressed as[Lu and Likos, 2006]
′ ¼ ð  ua Þ  s ;

ð1Þwhere ua is the pore air pressure, s is the total stress, s′ isthe effective stress, and ss is defined as the suction stresscharacteristic curve of the soil with a general functional
form ofs ¼ ðua  uw Þ

ua  uw  0;

ð2aÞ

s ¼ f ðua  uw Þ

ua  uw  0;

ð2bÞ

where uw is the pore water pressure and f is a scaling
function describing the link between suction stress andmatric suction. Lu and Likos [2004, 2006] and Lu et al.
[2007] showed that the suction stress characteristic curve,ss, could be obtained by shear strength or tensile strength
tests or by theoretical formulations. In this work we seek aclosed‐form equation for the suction stress characteristic
curve as a function of either matric suction or soil saturation
using a working hypothesis formed on the basis of experimental observations and thermodynamic justifications.

[7] The working hypothesis is that the change in energyof soil water from its free water state is mostly consumed insuction stress. Experiments show that for a variety of soils
and granular materials, there is a relationship between thesoil water characteristic curve (SWCC), plotting suction
versus saturation, and the suction stress characteristic curve(SSCC), plotting the tensile stress versus saturation. Our aim
here is to derive a first‐order approximation for this relationship within a thermodynamic framework.
2.2. Thermodynamic Justifications[8] The tensile stress can be calculated from the virtual
work of increasing the volume of a soil system. For a partiallysaturated soil held at constant temperature, T, and chemical
potential, m, the work is itself stored as the grand canonicalfree energy, F, and so the stress is given by the derivative of
the free energy with respect to volume V; that is,s ¼ @F :@V ;Tð3Þ

Assuming a constant density for water, the free energy will
have contributions from both the free capillary water and thebound residual water layers. We analyze each contribution
below.
[9] Bound residual water layers can exist because of
surface hydration attraction, extending over a layer thickness of water to the surface of the soil solid. The binding
results in a lower free energy for the water bound in thatlayer. For all N grains in a representative elementary volume(REV), the bound water layers occupy a total “residual
volume,” Vr, and have a total free energy, Fr. The remaining“free” capillary water thus has a volume Vf = Vw − Vr, where
Vw is the total water volume.
[10] Since the bound residual water is taken to have significantly lower free energy density than the free water,water added to a dry granular system up to a volume Vw ≤ Vr
will accumulate in this layer first. Since our focus will be on
Vw > Vr, we do not concern ourselves with the volume
dependence of the free energy below this residual volume,other than to identify Fr as the limiting value as Vw
approaches Vr. The remaining phases j (free capillary waterand air) having volume Vj and interfacial (or surface) areas
Ai contribute volume and surface terms to the free energy,
givingF ¼ Fr XX

uj Vj þphase j

i Ai f or Vw > Vr ;

ð4aÞinterface iwith differential
dF ¼ SdT Xjuj dVj Xj

Nj dj þX

i dAi f or Vw > Vr ;

i

ð4bÞ

where S is the entropy of the system and Nj is the number ofmolecules in phase j (omitting notation specifying speciestype). This provides the thermodynamic definition of thesurface tension for interface i,2 of 14

i ¼



@F 
@E 
¼
;
@Ai T ;V ; @Ai S;V ;Nð5Þ

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LU ET AL.: EFFECTIVE STRESS IN UNSATURATED SOIL

where the second equation expressed in terms of the totalenergy E of the system is obtained by Legendre transformation F = E − TS − mN.
[11] Specifically, for a free water phase with volume Vf =Vw − Vr and air phase with volume Va = Vv − Vw = (Vv − Vr) −Vf, where Vv is the void volume, we have
F ¼ Fr  ua Va  uw Vf þ

X

i Ai

i

¼ Fr  ua ½ðVv  Vr Þ  ðVw  Vr Þ  uw ðVw  Vr ÞXþi A ifor Vw > Vr ;stant, i.e., dSe = 0, in which case the change in water volume
is simply proportional to the change in void volume:dVw ¼ Se dV f or Vw > Vr :
X @Ai @F 
@Vw 
 ¼    ua ¼ ðua  uw Þi @V ;T@V ;T@V ;Tis

ð6Þf or Vw > Vr ;i

i Ai f or Vw > Vr :ð7Þ

We can use this expression for the free energy to derive thetensile stress or suction stress from a virtual work argument.
As the REV is subject to tension, an infinitesimal extension
with volume change dV will lead to an infinitesimal increasein the free energy, dF, corresponding to the input work. In
particular, we consider a uniaxial strain leading to a change inthe total volume, V = Vs + Vv, where Vs is the constant volume
occupied by the solid grains. Since the volumes of the rigid
grains and of the bound water layer are essentially constantfor Vw > Vr, we have dV = dVw + dVa. At constant T and m, we
thus have
X @Ai @F 
@Va 
@Vw 
¼ ua
uw
þ
i 
@V ;T@V ;T@V ;T@V ;T
i


X

@Vw 
@Ai 
¼ ua þ ðua  uw Þþfor Vw > Vr ;
i
@V 
@V 
;T

i

;T

ð8Þ


r
where we have assumed @F@V ;T ¼ 0 since the tightly boundresidual water is unperturbed by a small change in total volume.
[12] We thus need to specify how much the water volume,
Vw, and the interfacial areas, Ai, change with V under uniaxial strain. The ratio of free water volume to the total
available to it is specified by the effective saturation,
Vw Vr

Vf
S  Sr Vv Vv
Se 

¼
;
Vr
1  SrVv  Vr1
Vv

ð9Þ

or Vf = Se (Vv − Vr), with Sr being the residual saturation.
Given that dV = dVv and dVw = dVf for Vw > Vr, we havedVw ¼ Se dV þ ðVv  Vr ÞdSe f or Vw > Vr :

ð12Þ

or

or, expressed in terms of the matric suction, (ua − uw),X

ð11Þ

Accounting for the pressure, ua, provided by the surrounding atmospheric air, this leads to an expression for thetensile stress (suction stress):i

F ¼ Fr  ua ðVv  Vr Þ þ ðua  uw ÞðVw  Vr Þ þ

W05515ð10Þ

Since the arrangement of soil particles within the REV isassumed to be random, there is a distribution of local grainarrangements and strengths, and we propose that the sample
will deform at a localized region. This region under deformation can draw upon both water and air in the neighboring
area, which acts as a reservoir. As a first approximation, we
can assume that the effective saturation, Se, remains con-s ¼ ðua  uw ÞSe 

X @Ai i f or Vw > Vr :@V ;T;uniaxiali

ð13Þ

If the second term (interfacial contribution) is ignored, we
have the approximation of suction stress:s ¼ ðua  uw ÞSe f or Vw > Vr :

ð14ÞAlthough the surface tension term has been neglected inprevious work [e.g., Houlsby, 1997], no convincing physicaljustification has been reported. In the capillary (occluded air
bubble state) regime, the surface tension term should besmall or zero, but it could be significant in both the pendular(discontinuous menisci state) and the funicular (continuous
menisci and pore air) regimes. Houlsby [1997] argued thatthe relative velocity of the interface and the soil solid issmall, so that the surface tension can be ignored, but theexact circumstances where such a condition applies, as wellas the quantitative role of interfacial tension in the tensilestress, have yet to be established. The work done by thetensile stress, captured by equation (13) or equation (14), issimilar to that proposed by Bishop [1954] or derived by
Houlsby [1997, equation (26)], except that the energy in theresidual water bound by surface hydration is removed fromthe tensile stress. As argued above, the residual water content is the amount of water that remains primarily in theform of thin films surrounding the soil particle surfaces atvery high suctions but has very little effect on the interparticle suction stress. The graphical representation of suction stress is the area under the normalized soil water
characteristic curve and is shown in Figure 1a. In this work,
we rely on the physical justification of the validity ofneglecting the surface tension term using experimental data,
shown in section 3.
2.3. A Closed‐Form Equation for Effective Stress[13] Bishop’s [1954, 1959] effective stress can be written as′ ¼ ð  ua Þ þ ðua  uw Þ:

ð15Þ

Bishop [1954] also suggested that for his effective stressparameter c, c = S,′ ¼ ð  ua Þ þ S ðua  uw Þ:ð16ÞLu and Likos [2006] proposed a form of suction stress thatis consistent with Terzaghi’s effective stress:
′ ¼ ð  ua Þ  s ;where ss = −(ua − uw)S.

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ð17Þ

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suction. If van Genuchten’s [1980] SWCC model is used,
the normalized degree of saturation can be expressed as

Se ¼

1
1 þ ½ðua  uw Þn11=n;ð19Þ

where n and a are empirical fitting parameters of unsaturated soil properties, with a being the inverse of air entry
pressure for water saturated soil and n being the pore size
distribution parameter. Figure 1b shows the range of valuesof the a and n parameters for various soil types.
[15] A closed‐form expression for suction stress for thefull range of saturation can be arrived at by substituting
equation (19) into equation (14) and eliminating matricsuction:s ¼ 

1nn
Se  1n
Se  10  Se  1:

ð20Þ

Similarly, a closed‐form expression for suction stress for thefull range of matric suction also can be arrived at by substituting equation (19) into equation (14) and eliminating thedegree of saturation:s ¼ ðua  uw Þs ¼ 

ua  uw  0;ðua  uw Þ
ð1 þ ½ðua  uw Þn Þ

ð21aÞ

ua  uw  0: ð21bÞðn1Þ=nIn what follows, we focus on the characteristics and validation of equation (21), as most of the data available in theliterature on shear strength and soil water characteristic
curves are exclusively expressed in terms of matric suction.
Substituting equation (21) into equation (18), the closed‐
form equation for effective stress in the entire pore waterpressure range (all saturations) is′ ¼   ua þ ðua  uw Þ

Figure 1. (a) Interrelationship between the soil water characteristic curve (SWCC) and the suction stress characteristic
curve (SSCC) and (b) illustration of suction stress regimes
for various soils.

[14] Following equation (14), we propose an effective
stress as an extension of Bishop’s and an expansion of
Terzaghi’s for all saturations by modifying the saturation
contribution to effective stress:
′ ¼ ð  ua Þ  ½Se ðua  uw ÞS  Srðua  uw Þ ¼ ð  ua Þ  s ;
¼ ð  ua Þ  
1  Sr

ð18Þ

where ss = −(ua − uw)Se. Note that equation (18) is different
than that of Bishop in the degree of saturations and canrecover Terzaghi’s effective stress s′ = s − uw when a soil issaturated. We make an additional extension of equation (18)by applying the relationship between the normalized volumetric water content or degree of saturation and matric′ ¼   ua þ

ua  uw  0;ðua  uw Þ
ð1 þ ½ðua  uw Þn Þ

ðn1Þ=nð22aÞua  uw  0: ð22bÞ

Equation (22a) is for saturated conditions, which is Terzaghi’s equation, and equation (22b) is for unsaturated
conditions. In Appendix A we show that equation (22b)continuously and smoothly approaches equation (22a) as
matric suction approaches zero or from an unsaturated state
to a saturated state.
[16] The general patterns of the SSCC defined byequation (21) and its interrelationship with the SWCC for
different types of soil are illustrated here. Figure 2a shows
SWCCs for typical sandy, silty, and clayey soils, andFigure 2b shows the corresponding SSCCs predicted using
equation (21b). Each of these three “typical” soils has uniquecharacteristics of suction stress. For sandy soil, suction stressis zero at zero matric suction (saturated condition) and atsome large value of matric suction (110 kPa for the example
here). Suction stress reaches a minimum value at a given
matric suction (suction stress of −2 kPa at matric suction of
3 kPa or 70% saturation). The down‐and‐up characteristicof suction stress can be illustrated by plotting equation

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(17b), as shown in Figure 2c. This behavior of suction stressis well known for sand‐sized granular media [e.g., Schubert,
1975; Kim, 2001; Lu et al., 2007]. It is important to notethat this behavior cannot be effectively described using
Bishop’s effective stress approach because the effective
stress was treated as a function of both matric suction andthe degree of saturation (equation (16)).
[17] For a typical silty soil, suction stress follows a similar
pattern to that of sandy soil (Figures 2b and 2c), except theminimum value is now on the order of 10 kPa. However,considerable suction stress remains beyond the minimum
value. The range of matric suction for which the magnitudeof suction stress remains considerable could be on the orderof several hundreds to several thousands of kilopascals. A
practical illustration of this behavior is evidenced by loesscut slopes. In these cut slopes, suction stress of several tensof kilopascals in the loess is capable of maintaining near‐vertical cuts of several meters in height in dry environmentseven in the absence of clay films or cement.
[18] For a typical clayey soil (Figures 2b and 2c), thepattern of variation of suction stress shows some distinct
characteristics. Suction stress is zero when matric suction iszero but decreases monotonically as matric suction
increases. The minimum suction stress for clayey soil could
be on the order of several hundreds of kilopascals inmagnitude. A practical example is that when clay is dry or
under high matric suction conditions, the bonding force orsuction stress is very high, up to several hundreds of kilopascals, making it difficult to break. Under very moist
conditions, clay is plastic, and under moderately moist
conditions, clay is brittle like rock.
[19] Under the proposed framework, effective stress or
suction stress is intrinsically related to the SWCC. The formof these curves is fundamentally controlled by the poregeometry and pore fluid parameters a and n. The parametera is a direct indicator of the matric suction at which pore
fluid begins to leave a drying soil water system, whereas
parameter n reflects the pore size distribution of the soil. It
can be shown (see Appendix B) that the SSCC can be in two
distinct regimes dependent on the value of n: it monotonically decreases if n ≤ 2.0 and decreases and then increases if
n > 2.0. Figure 1b illustrates these two regimes in the a − nspace. When n > 2.0, the minimum suction stress and its
corresponding value of the equivalent degree of saturationand matric suction are
1 1n jmin ¼ 
 2nsðua  uw Þjmin ¼

Se jmin ¼

Figure 2. (a) SWCC for some typical soils, (b) SSCC for
the typical soils in terms of the effective degree of saturation, and (c) SSCC for the typical soils in terms of matricsuction.

1n
n

1
n2

1
1
 n2

1n
2n

1n
n

;

1
n

;

1
n

;

ð23aÞ

ð23bÞ

ð23cÞ

respectively. It can be shown from analyzing the mathematical properties of equations (23a) and (23b) that the aparameter dominantly controls the minimum value of suction stress and the matric suction value corresponding to thatminimum, whereas the parameter n solely controls the

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Table 1. Soil Descriptions and Properties Used to Validate Closed‐Form Equation for Effective Stressa
Name
KaolinJossigny silt
(low‐plasticity clay)
Madrid clayey sand
Sandy clay 1
Compacted glacial till
Tappen Notch Hill silt

Soil Properties
wL = 63%, Ip = 33%,
percent finer than 3 mm = 70%,g dMAX = 1.4 g/cm3
wL = 37%, Ip = 18%,
clay‐size fraction = 34%,
g dMAX = 1.7 g/cm3
wL = 32%, Ip = 15%,
fine fraction = 17%,g dMAX = 1.91 g/cm3
N.D.
wL = 35.5%, Ip = 18.7%,
clay‐size fraction = 30%,g dMAX = 1.815 g/cm3
wL = 57%, Ip = 32% (for clay fraction),clay‐size fraction = 10%, silt = 85%,
N.D.
g d = 1.48 g/cm3

Sandy clay 2dDhanauri clay
compacted to
low densityd
Mature residual soil
wL = 50.7%, Ip = 18.4%,(Vista Chinesa, Brazil)
clay‐size fraction = 24.4%,
sand = 60%
Dhanauri clay
compacted tohigh densityd
Sand‐clay mixture
Speswhite kaolin

g d = 1.58 g/cm3
Clay‐size fraction = 25%,sand = 75%, g dMAX = 1.92 g/cm3
g dMAX = 1.2 g/cm3

Yellow colluvium(Vista Chinesa,
Brazil)Compacted nonplastic
silty sandd

wL = 45.7%, Ip = 22.7%,
clay‐size fraction = 42.5%,
sand = 50.3%
g dMAX = 1.89 g/cm3

Hume Dam clay
(Southeastern
Australia)

wL = 33%, Ip = 12%,
fine fraction = 74%,
sand fraction = 26%,
g d = 1.69 g/cm3
wL = 32%, Ip = 16%,clay‐size fraction = 20%,silt = 43%, sand = 37%,

Barcelona silt

Glacial tillcompacted dryof optimum
water content
Decomposed tuff
(Hong Kong)

wL = 35.5%, Ip = 18.7%,
clay‐size fraction = 30%,
silt = 42%, sand = 28%,g d = 1.73 g/cm3
N.D.

Ottawa sand
Limestone
agglomerates

N.D.
N.D.

ub = 1/a (kPa)′c′(°) (kPa)nReference

Apparatus

Group 1395b

1.20c 22

24

Khalili andKhabbaz [1998]

Modified triaxial

182b

1.54c 22b

25b

Cui andDelage [1993]

127b

1.63c 38

0

Escario and
Sáez [1986]

Triaxial apparatus
with osmotic
suction control
Modified direct shear

35b

1.59c 37d0dBlight [1967]

Modified triaxial withwater content control

Group 2153b

1.50c 25.5 10

Gan et al. [1988]

Modified direct shear

94b

1.39c 35

Krahn et al. [1989]

70b
62b

1.58c 30 0
1.43c 28.5 7.8

Modified multistage
triaxial
Modified triaxial
Modified triaxial

38b

1.63c 28.7 13.7 de Campos
Modified direct shear
and Carrillo [1995]

0Group 3127b

1.30c 29115b

1.40c 33.5 30

86b

1.42c 25

54

b

c

1.62

7.8

0

26.4 0

Maswaswe [1985]
Satija [1978]

Satija [1978]

Modified triaxial

Khalili andKhabbaz [1998]
Wheeler andSivakumar [1995]
de Campos andCarrillo [1995]

Modified triaxial
Triaxial
Modified direct shear

43

1.50c 38.7 11.5 Drumright [1989]

Modified triaxial

Group 477.5e

1.37e 29

19

Khalili et al. [2004]

Modified triaxial

14.9e

1.13e 280

Vaunat et al. [2002]

Modified direct shear

Group 540.9e

1.46e 23

0

Vanapalli et al. [1996] Modified multistage
direct shear

46.9e

1.89e 390Fredlund et al. [1995]

Group 6
2.20e
1.12e

2.50e N.D. N.D Kim [2001]
12.68e N.D. N.D. Schubert [1984]

Modified multistage
direct shear
Tensile strength tests
Tensile strength tests

N.D., no data; wL, liquid limit; Ip, plasticity index; g d, dry density; g dMAX, maximum dry density.
Estimates reported by Khalili and Khabbaz [1998].
Obtained by fitting suction stress characteristic curve to shear strength test data.
dData taken from Khalili and Khabbaz [1998].
eObtained using RETC v.6.0 code [van Genuchten et al., 1991].
abcequivalent degree of saturation corresponding to the minimum suction stress.

3. Experimental Validation[20] Thanks to the increasing experimental effort in thepast 3 decades, experimental validation of the closed‐form

equation for effective stress (equation (22)) can be achieved
using data from shear strength or tensile strength tests.
Experimental validation of the proposed effective stressequation (equation (21b)) under the tensile total stressregime has been demonstrated by Lu et al. [2009] for various sandy materials. Because most results reported in theliterature are from shear strength tests performed in the

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pare test data with the theoretical equations (equation (19)for the SWCC and equation (21b) for SSCC). If equation(21b) is valid, a unique pair of parameters a and n is
identified. For those soils for which only suction stress data
are available, we perform a “semiquantitative” analysis. We
identify the parameters a and n from those data and use
them to predict the corresponding SWCCs. We compare
these curves with the typical range of curves for different
soils to see whether they follow the expected trend, as
described below.
3.1. Semiquantitative Validation[22] We have compiled suction stress data from 14 different soils for the semiquantitative validation. These soils
cover a wide spectrum of soil types, from sandy to clayey,
as listed in Table 1. Shear strength data were reduced toyield suction stress data as follows. For direct shear testing,the Mohr‐Coulomb failure criterion can be written asf ¼ c′ þ ½ð  ua Þ  s  tan ′;

ð24Þ

where t f is shear stress at failure at a given matric suction,c′ is the drained cohesion, and ′ is drained friction angle
at saturated state. Suction stress at a given matric suction isthen reduced from equation (24):s ¼ 

f  c0  ð  ua Þ tan 0:tan 0

ð25Þ

For triaxial testing, suction stress at a given matric suction
can be obtained as follows:s ¼ 

ð1  ua Þ  ð3  ua Þ tan2

þ

0
2

 2c0 tan4
0
þtan 0
2 tan
4 2

4

þ

0
2

;

ð26Þ

Figure 3. Semiquantitative validation of the closed‐form
equation for effective stress for Group 1 soils: (a) measured
and fitted SSCCs for kaolin, Jossigny silt, Madrid clayey
sand, and sandy clay 1 and (b) predicted SWCCs for thesesoils.

compressive regime, we conduct our validation using shear
failure data for which the total normal stresses are compressive. These tests were typically conducted using either adirect shear or triaxial shear apparatus modified so thatmatric suction can be controlled. A total of 20 different soils
covering the range of sand, silt, and clay was used forvalidation purposes (see Table 1).
[21] A true validation for a given soil requires both a
SWCC and a SSCC. Suction stress characteristic curves canbe experimentally obtained from a series of shear strength
tests under different matric suctions. However, because of
the paucity of both SWCC and SSCC for the same soil, thefollowing strategy for validation is adopted. For those soils
for which SWCC and SSCC are available, we directly com-

where s1 and s3 are the total principle stresses in the verticaland horizontal directions, respectively.
[23] For clarity of presentation, we have divided the soils
for which the semiquantitative validation was performed
into three groups (Table 1). Figure 3a shows the measured
suction stress characteristic data from the four soils in Group 1.
This group consists of kaolin, Jossigny silt, Madrid clayey
sand, and sandy clay 1. In general, the SSCCs for thesesoils follow a trend of monotonic decrease in suction stresswith increasing matric suction. At matric suctions less than
the air entry value of the soil, suction stress strictly follows
the line ss = −(ua − uw), where Terzaghi’s effective stressprinciple is valid. Suction stress begins to deviate from thisline in the air entry regime and continues to do so as
matric suction increases. The air entry pressures, ub, forthese soils were estimated by Khalili and Khabbaz [1998]
and are reported in Figure 3 and in Table 1. The air entry
pressure, ub, can also be inferred from the SSCC test dataas the point at which the SSCC deviates from the saturated
line, although real soils do not have a clear deviating point
or air entry value but rather a range of pressures in which
the largest pores begin to drain. For example, the air entry
pressure from Figure 3a for the kaolin is between 300 and
400 kPa (versus the reported value of 395 kPa), and thatfor the sandy clay 1 is around 30 kPa (versus the reported

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Figure 4. Semiquantitative validation of the closed‐form
equation for effective stress for Group 2 soils: (a) measured
and fitted SSCCs for compacted glacial till, Tappen Notch
Hill silt, Dhanauri clay compacted to low density, sandy
clay 2, and mature residual soil and (b) predicted SWCCs
for these soils.
value of 35 kPa). Using these air entry pressures from thework of Khalili and Khabbaz [1998], we attempt to fit theexperimental data for each soil in order to obtain a value
for the pore size parameter n, which is reported in Figure 3.
The SWCCs are then calculated (using equation (19)) using
the air entry parameter, a (a being the inverse of air entry
pressure ub), and pore size parameter, n, and plotted in
Figure 3b.
[24] In general, the finer‐grained soils have larger air
entry pressures and support higher matric suctions at a given
degree of saturation compared with the coarser‐grained soils
that have smaller air entry pressures. Thus, the soil water
characteristic curves should shift to the right in Figure 3b
with increasing fine‐grained fraction and larger air entry
pressures. Figure 3b shows that equation (21) captures thisW05515expected “right shifting” trend for the Group 1 soils. The
magnitude of suction stress generally increases as the soils
become finer (Figure 3a and Table 1). For example, suction
stress varies from slightly more than about 250 kPa at amatric suction value of 800 kPa for the Madrid clayey sand(fine fraction is 17%) to about 350 kPa for the Jossigny silt
(clay‐size fraction is 34%). For clayey soils (e.g., kaolin),suction stress could be as much as several hundreds ofkilopascals. The predicted SWCCs for the soils in Group 1
are qualitatively within the ranges that would be expected
for soils with similar air entry pressures and grain size
distributions.
[25] Figure 4 shows the measured suction stress data fromfive additional soils, defined here as Group 2. This groupconsists of compacted glacial till, Tappen Notch Hill silt,
sandy clay 2, Dhanauri clay compacted to low density, and amature residual soil. The best fit SSCCs and the air entry
pressures (ub) from the work of Khalili and Khabbaz [1998]
and fit n parameters are shown in Figure 4a and Table 1.
Suction stress at a matric suction of 500 kPa varies fromabout 100 to 250 kPa moving from the relatively coarse
residual soil to the finer‐grained glacial till. The ub and nparameters are used in equation (19) to predict thecorresponding SWCCs, as shown in Figure 4b. For example, the leftmost SWCC is that for the mature residual soil,
and it reflects its relatively large sand‐size fraction (60%)(Table 1). The rightmost SWCC is that for the relatively finegrained (clay‐size fraction is 30%) (Table 1) and dense compacted glacial till. Again, these estimated SWCCs are qualitatively within the ranges that would be expected for soils withsimilar air entry pressures and grain size distributions.
[26] The final group for this semiquantitative validationconsists of five additional soils: Dhanauri clay compacted tohigh density, a sand‐clay mixture, speswhite kaolin, yellow
colluvium, and compacted nonplastic silty sand. The measured and fitted suction stress data are shown in Figure 5aand listed in Table 1, and the corresponding SWCCs fromthe best fit parameters are plotted in Figure 5b. Suction
stress at a matric suction of 200 kPa varies from about 85 to
150 kPa moving from the relatively coarse colluvial soil tothe dense Dhanauri clay. The predicted SWCCs for the soils
in Group 3 are qualitatively within the range expected forsoils with similar grain size distributions and bulk densities.
The rightward shifting trend of the SWCCs for the finer anddenser soils with greater air entry pressures is evident andqualitatively supports equation (21) for the suction stresscharacteristic curve.
3.2. Quantitative Validation[27] We identified six soils for which data on both the
SWCC and SSCC are available from the literature. For
clarity we have divided the soils into three groups. The first,
Group 4, includes Hume Dam clay [Khalili et al., 2004] andBarcelona silt [Vaunat et al., 2002]. Test results and the best
fit SWCCs are shown in Figure 6a. The curves were fit
using a least squares regression and the RETC code [vanGenuchten et al., 1991]. The parameters a and n werethen used to predict the SSCCs (Figure 6b). Comparisons of
the predicted curves with the suction stress data show thatthe closed‐form equations (19) and (21) predict the measured stresses within a few tens of kilopascals for both the
Hume Dam clay and the Barcelona silt. This comparison

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agglomerates [Schubert, 1984]. The test data and the best fit
for the SWCCs are shown in Figure 8a. As with the othertwo groups, the best fit a and n parameters were then used
to predict the corresponding SSCCs (Figure 8b). Close
matches between the measured and predicted SSCCs forboth soils support the validity of the proposed closed‐form
equation (22) for effective stress in unsaturated soils. Worthnoting is the “peak” behavior when parameter n > 2.0 (seeAppendix B), an important characteristic of suction stress incoarse‐grained materials; suction stress is at a minimum of−1.6 kPa at about 3 kPa of matric suction for the Ottawa

Figure 5. Semiquantitative validation of the closed‐form
equation for effective stress for Group 3 soils: (a) measured
and fitted SSCCs for Dhanauri clay compacted to high density, sand‐clay mixture, speswhite kaolin, yellow colluvium,
and compacted nonplastic silty sand and (b) predicted
SWCCs for these soils.
indicates that the closed‐form equation (22) for effective
stress is valid for the two soils.
[28] Group 5 consists of glacial till compacted dry of
optimum water content [Vanapalli et al., 1996] anddecomposed tuff [Fredlund et al., 1995]. The soil water
characteristic and suction stress characteristic curve data areshown in Figures 7a and 7b, respectively. As for Group 4,
the RETC model was used to identify a and n from the soil
water characteristic data (Figure 7a), and the best fit valuesof a and n are used in equation (21) to predict the SSCCs(Figure 7b). Again, equation (21) provides an accurate
prediction (within a few percent) of the measured suction
stress data for both the glacial till and the decomposed tuff.
[29] Group 6 contains two relatively coarse‐grainedmaterials: fine (Ottawa) sand [Kim, 2001] and limestone

Figure 6. Quantitative validation of the closed‐form equation for effective stress for Group 4 soils: (a) measured andfitted SWCCs for Barcelona silt and Hume Dam clay and (b)measured and predicted SSCCs for these soils.

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useful for science and engineering problems it must provide
fundamental insight and also be simple and provide resultsof sufficient accuracy. Darcy’s law for fluid flow andTerzaghi’s equation for effective stress in saturated soils
are examples of physical observations and mathematicalabstraction that have had a profound impact on science andengineering practice. This general philosophy provides
guidance in the current search for an effective stress equationfor variably saturated soil. In what follows, we point out a
few theoretical and practical implications of this work.
[31] We have shown that suction stress equation (21) isgenerally valid for a wide range of soils: from clay to

Figure 7. Quantitative validation of the closed‐form equation for effective stress for Group 5 soils: (a) measured andfitted SWCCs for decomposed tuff and glacial till and(b) measured and predicted SSCCs for these soils.
sand and a minimum of about −0.85 kPa at 1 kPa of matricsuction for the limestone agglomerates. This behavior cannot be predicted using either the classical effective stressapproach of Bishop (equation (16)) or the extended Mohr‐
Coulomb criterion under the framework of two independentstress state variables, as they are not defined as a solefunction of matric suction or saturation, but is well describedby the proposed closed‐form effective stress equation (22)when parameter n > 2.0.

4. Theoretical and Practical Implications[30] A goal of theoretical inquiry is to describe physicalbehavior using mathematics. For such inquiry to be widely

Figure 8. Quantitative validation of the closed‐form equation for effective stress for Group 6 soils: (a) measured andfitted SWCCs for limestone agglomerates and Ottawa sandand (b) measured and predicted SSCCs for these soils.

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limestone agglomerates (Table 1). If equation (21) for suction stress in unsaturated soil is valid in a manner similar tothat of Terzaghi’s effective stress for saturated soil, all limit
equilibrium theories, such as those for lateral earth pressures, bearing capacity, and slope stability, in use today canbe readily expanded for design and analysis under unsaturated soil conditions by simply replacing pore water pressure with the suction stress characteristic curve.
[32] Theoretically, the closed‐form equation for suction
stress (equation (21)) has several potentially far‐reaching
implications. First, it avoids the common theoretical andpractical impediment embedded in Bishop’s effective stressequation since there is no need to determine the coefficientof effective stress c. Effective stress can be determined
simply by reducing suction stress from shear strength test
results (equation (26)) or by measuring soil water characteristic curves to identify parameters a and n. Equation (21)also captures the highly nonlinear and peak behavior ofeffective stress in sandy and silty soils. This behavior has
been widely observed in the results of unsaturated shear
strength experiments [e.g., Vanapalli et al., 1996; Kim,
2001] that show a nonlinear increase‐then‐decrease dependence of shear strength with increasing matric suction (asshown in Figure 8b). This behavior is also well known fromfield examples such as the observation of a sudden collapseof unsaturated loess slopes upon wetting [e.g., Higgins andModeer, 1996]. This work, for the first time, reconciles such
behavior under the framework of effective stress. It also
bridges the gap between the two independent stress variable
framework and the effective stress framework by upscaling
matric suction to suction stress for which matric suction isconceptualized as the controlling stress state variable. Matricsuction, as demonstrated by Lu [2008], is not a stress at a
typical REV level in soils and thus should not be defined asa stress variable. Distinctions between stress variables andstress state variables as well as their implications in theoretical soil mechanics are discussed by Lu [2008]. A thermodynamic distinction between the two independent stressvariable frameworks and the effective stress frameworks ishighlighted below.
[33] Under the two independent stress variable frameworks[Fredlund and Morgenstern, 1977], shear strength is conceptualized as a nonlinear function of both soil suction andsaturation. For example, Vanapalli et al. [1996, equation (18)]
proposed the following form for a modified Mohr‐Coulomb
criterion:f ¼ c′ þ ð  ua Þ tan ′ þ ðua  uw Þ tan ′S  Sr:1  Srð27aÞ

In light of the effective stress equation (equation (18)),
equation (27a) can be rearranged asS  Srtan 0 :f ¼ c0 þ ð  ua Þ þ ðua  uw Þ
1  Srð27bÞ

[34] Equation (27a) bears two fundamental differenceswith the proposed closed‐form effective stress equation(equation (22)). First is a simple mathematical difference;equation (27a) or equation (27b) is a function of suction andsaturation, whereas equation (22) is solely a function of

W05515suction. The second difference rests on a thermodynamic
consideration. In equation (22), we state that effective stressvaries with suction and that changes in suction stress lead tochanges in energy in soils. In equation (27a), shear strength
is conceptualized as the product of suction and saturation.
From a continuum mechanics perspective, stress, such as
suction stress, is part of energy (i.e., specific energy equals
half stress times strain). Material strength, such as shear
strength, is unambiguously not a stress and thus does notcontribute to energy stored, released, or dissipated in themedium. This distinction places our effective stress equation(equation (22)) well within the framework of continuum
mechanics where free energy is the basis for any thermodynamic formulation.
[35] We also seek a common basis for both fluid flow andeffective stress in unsaturated soil. The SWCC (e.g.,
equation (19)) is commonly used in solutions to the governing equation (Richards equation) for variably saturated
fluid flow in porous media. Fields of matric suction calculated in such a manner can be used directly in equations (21)and (22) for stress field and stability analysis under the sameclassical soil mechanics framework as saturated soils.
Analyses of many practical problems, such as the state ofstress in steep soil‐mantled hillslopes during infiltration,require such a coupled hydromechanical model. As shownin equations (19) and (21), soil moisture and suction stressare fundamentally governed by a single variable: matricsuction through the same pore fluid and solid and size
distribution parameters, namely, a and n [van Genuchten,
1980].
[36] Beyond the well‐established utility of Terzaghi’s
effective stress principle for strength behavior, it is also
considered to be “effective” for some deformation problems
such as the consolidation settlement. Strength and deformation behaviors of both saturated and unsaturated soil havebeen theoretically examined within the framework of criticalstate soil mechanics. Because the proposed effective stressequation (equation (18)) is governed by matric suction andis consistent with Terzaghi’s effective stress, it can also be
incorporated into elastic‐plastic constitutive relations andthe critical state soil mechanics frameworks.

5. Summary and Conclusions[37] A closed‐form equation (equation (22)) for effective
stress in unsaturated soil is proposed and validated. The
closed‐form equation requires only two controlling parameters: the inverse of the air entry pressure a and the poresize spectrum number n. With them, effective stress inunsaturated soils ranging from sand to silt to clay can be
accurately described. These two parameters are identical tothose commonly used in the soil water characteristic curve
equations proposed by van Genuchten [1980]. Therefore,
the proposed closed‐form equation for effective stress canbe considered to be a unified description for phenomena offlow and stress in porous granular materials.
[38] The proposed closed‐form equation for the suction
stress characteristic curve is an expansion of Terzaghi’s
effective stress principle into unsaturated conditions and aunification of Bishop’s unsaturated effective stress with
Terzaghi’s effective stress. The proposed closed‐form
equation (equation (22)) is an extension of Bishop’s effec-

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tive stress, as the effective stress is solely a function of either
matric suction or the equivalent degree of saturation,
whereas Bishop’s effective stress demands knowledge of thecoefficient of effective stress c and concurrent knowledgeof soil suction and the degree of saturation. Under the
proposed equation, the transition from saturated to unsaturated states is continuous and smooth, ensuring mathematical consistency between Terzaghi’s effective stress and theeffective stress equation (equation (22)).
[39] Suction stress is the tensile stress. We show that theconception of it as an effective stress is thermodynamicallyjustifiable when the surface tension contribution to the stresscan be neglected, which apparently is the case in thefunicular and capillary regimes. This justification leads to thestatement that suction stress is an effective stress under theconditions of no external stress and is the energy consumed
by capillary pore water. The proposed closed‐form equationfor effective stress and the justification for neglecting thesurface tension term are validated against published testing data for a wide range of soils. We conclude that the
proposed effective stress or effective stress equation isvalid within the inherent errors of current experimental
techniques.
[40] An important practical implication of using the proposed effective stress equation is that there is no need forany new shear strength criterion for unsaturated soil and allclassical soil mechanics work on limit analysis can bereadily extended to unsaturated soil conditions. Of equal
practical importance is the development of a thermodynamically consistent framework for analyzing coupled hydromechanical problems such as the state of stress in hillslopes
during infiltration.

Taking the limit of (ua − uw) → 0 for both the derivatives
shown in equations (A1) and (A2), we have


ds

¼ 1
dðua  uw Þðua uw Þ!0


ds

¼
Lim
dðu  u Þ
Lim

a

Lim 
þ
¼

w

ðua uw Þ!þ0

1
n1

f1 þ ½ðua  uw Þn g n!

ðn  1Þ½ðua  uw Þn
f1 þ ½ðua  uw Þn g n

2n1



1
n1

f1 þ ½0n g nþðua uua Þ!0ðn  1Þ½0n

!

¼ 1:

f1 þ ½0n g n2n1

Taking the limit of (ua − uw) → 0 for both the suction stressshown in equations (21a) and (21b), we haveLims jðua uw Þ!0 ¼ ðua  uw Þ ¼ 0
ðua  uw ÞLims jðua uw Þ!þ0 ¼ 

ðn1Þ=nð1 þ ½ðua  uw Þn Þ
ðua  uw Þ¼
¼ 0:
ðn1Þ=nð1 þ ½0n Þ

Since at zero matric suction, both suction stress and its
derivatives have unique values, the closed‐form equation(21) is smooth at the point where matric suction is zero.

Appendix B: Solution Regimes for the Suction
Stress Characteristic Curve
Appendix A: Proof of a Smooth Transition
for Suction Stress at Zero Matric Suction[41] This section shows the mathematical consistency
between Terzaghi’s effective stress for saturated state andthe proposed effective stress equation (equation (22)) for
unsaturated state. We show here that suction stress and its
first derivative in the closed‐form expression are continuous
as matric suction varies between the partially saturated caseto the fully saturated case. Thus, taking the derivative of
equation (17a) with respect to matric suction, we have
s

d
¼ 1
dðua  uw Þ

ua  uw  0:[42] This section shows the theoretical possibility of the“peak” behavior of effective stress in unsaturated soils. To
find out the maxima and minima of the closed‐form
equation for suction stress, we can take the derivative of
equation (20) with respect to the equivalent degree ofsaturation and set the resulting equation to zero:1n S 1  n
1n1 nnnds
1  1n1eSe  1 
Se1n  1
Se1n ¼ 0
¼

1n
dSe
n
 n
1n
n
1n S 1 Se1n  1
n1  1nn Se1n
e
 n

¼0

Se  1 
 n Se1n  1 1  n Se


ðA1Þ

n

Se1n

Taking the derivative of equation (17b) with respect tomatric suction, we have1
¼0
1   n1n1
n
Se  1 nnð1  nÞ Se1n  1  Se1n ¼ 0

ds
1
¼
n1
dðua  uw Þ
f1 þ ½ðua  uw Þn g nþðua  uw Þn 2n1n

f1 þ ½ðua  uw Þ g

n1
n

n

ð2  nÞSe1n þ ð1  nÞ ¼ 0

n1

1

 f1 þ ½ðua  uw Þn g n n½ðua  uw Þn1 
s
d
1
¼
n1
dðua  uw Þ
f1 þ ½ðua  uw Þn g nðn  1Þ½ðua  uw Þnþua  uw  0: ðA2Þ
2n1
f1 þ ½ðua  uw Þn g n

Se jmin ¼

1n
2n

1n
n

:

ð23c′Þ

It can be deduced that for equation (23c) to have a real
solution, n must be >2.0.

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[43] Substituting equation (23c) into (21), we have theminimum suction stress value:s jmin ¼ 

1 1n
 2n

1n
n

1
n2

1
n

:

ð23a′Þ

The matric suction where the minimum suction stress is
attained can also be found by substituting equation (23c)into the soil water characteristic curve (equation (19)):


n1
n
1
Se jmin ¼
1 þ ½ðua  uw Þnn
1
Se jmin n1 ¼
1 þ ½ðua  uw Þnn

1 þ ½ðua  uw Þn ¼ Se jmin 1nðua  uw Þjmin ¼

i1n
n
1h
Se jmin 1n 1


2(
14
1n
ðua  uw Þjmin ¼2nðua  uw Þjmin ¼

1
ðua  uw Þjmin ¼

1n
n

n
)1n

31n
15

1n
1n
1
2n
1
1
 n2

1
n

ð23b′Þ

Thus, we show that when n ≤ 2.0, the closed‐form equation(21) for suction stress is a monotonically decreasing function. When n > 2.0, the closed‐form equation (21) for suction stress has a minimum suction stress value described byequation (23a), and it occurs when the equivalent degree ofsaturation is at the value given by equation (23c) and matricsuction is at the value given by equation (23b).
[44] Acknowledgments. We would like to thank William Schulz andHoma Lee of the U.S. Geological Survey for their critical, thorough, andstimulating reviews. The funding for this research is provided by a grantfrom National Science Foundation (NSF‐CMMI‐0855783).

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N. Lu, Division of Engineering, Colorado School of Mines, 1600 Illinois
St., Golden, CO 80401, USA. (ninglu@mines.edu)
D. T. Wu, Department of Chemical Engineering, Colorado School ofMines, 423 Alderson Hall, Golden, CO 80401, USA. (dwu@mines.edu)

J. W. Godt, U.S. Geological Survey, PO Box 25046, MS 966, Denver,
CO 80225, USA. (jgodt@usgs.gov)

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