State Variables in Saturated-Unsaturated Soil Mechanics

D.G. Fredlund

Abstract. The description of the stress state in soils is the foundational point around which an applied science should bebuilt for engineering practice. The stress state description has proven to be pivotal for saturated soil mechanics and thesame should be true for unsaturated soil mechanics. Continuum mechanics sets forth a series of principles upon which acommon science base can be developed for a wide range of materials. The principles require that there be a clear distinctionbetween state variables and constitutive relations. Constitutive relations relate state variables and incorporate materialproperties. State variables, on the other hand, are independent of the material properties. It has been possible to maintain aclear distinction between variables of state and constitutive relations in the development of saturated soil mechanics andthe same should be true for unsaturated soil mechanics. This paper presents a description of the source and character ofstress state variables for saturated and unsaturated soils. The descriptions are consistent with the principles of multiphasecontinuum mechanics and provide an understanding of the source and importance of stress state variables.Keywords: state variables, soil suction, constitutive relations, continuum mechanics.

1. Introduction

Karl Terzaghi (1883-1963) is recognized as the “Fa-ther of Soil Mechanics”. He defined the term “effectivestress” as the variable around which the physical behaviourof a saturated soil can be described. Effective stress was de-fined as (� – uw) where � was total stress and uw was thepore-water pressure. The recognition of effective stress asthe state variable controlling the equilibrium of the soilstructure of a saturated soil elevated saturated soil mechan-ics from an art to a science. Effective stress provided ameans whereby the physical behaviour of a saturated soilcould be described. The effective stress variable was inde-pendent of the properties of the soil. The effective stressvariable has become the rallying point and the unifyingvariable for understanding volume change, distortion, shearstrength, seepage and other physical process in saturatedsoils. The use of the effective stress variable in saturatedsoil mechanics was clearly illustrated for various soil me-chanics problems in Terzaghi’s first textbook, “TheoreticalSoil Mechanics” published in 1943.

This paper revisits issues relevant to the descriptionof the stress state of particulate materials; in particular, sat-urated and unsaturated soils. The fundamental basis for useof stress state variables in soil mechanics is explained. Thefundamental basis for the use of effective stress for satu-rated soils is presented first followed by a similar descrip-tion of the stress state variables for unsaturated soils. Theemphasis is on state variables associated with stresses;however, it is recognized that there are also other state vari-ables such as those required for mapping deformations anddistortions of particulate, multiphase systems. These are re-ferred to as deformation state variables but their descriptionis outside the scope of this paper.

2. Terzaghi’s Description of Effective Stress

In 1936, Terzaghi described and justified the use ofthe effective stress variable for saturated soils as follows.Terzaghi (1936) wrote, “The stresses in any point of a sec-tion through a mass of soil can be computed from the totalprincipal stresses, �1, �2, �3, which act at this point. If thevoids of the soil are filled with water under a stress, uw, thetotal principal stresses consist of two parts. One part, uw,acts in the water and in the solid in every direction withequal intensity (emphasis added). It is called the neutral (orpore-water) pressure. The balance �1’ = �1 - uw, �2’ = �2 - uw,�3’ = �3 - uw represent an excess over the neutral stress, uw,and has its seat exclusively in the solid phase of the soil(emphasis added). All the measurable effects of a change inshearing resistance are exclusively due to changes in the ef-fective stress, �1’, �2’, �3’”

The above paragraph sets forth the meaning of the ef-fective stress variable. Effective stress is the difference be-tween the total stresses and pore-water pressure in threeorthogonal directions. The three principal directions arisefrom the Cartesian coordinate system associated withthree-dimensional space. Stresses in three directions can becombined to form a stress tensor that defines the stress stateat a point in saturated soil. The stress tensor with compo-nents, (�1 – uw), (�2 – uw), and (�3 – uw), form the variablesthat control the equilibrium of the soil structure. The words“soil structure” is used in the sense of referring to the ar-rangement of the soil solids. The effective stress variablesare associated with equilibrium conditions and can there-fore be used to describe the behaviour of the soil structureof a saturated soil. Stated another way, the effective stressvariables can be used to describe changes from the equilib-rium state in a saturated soil.

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 3

Delwyn G. Fredlund, PhD., Golder Associates Ltd., 1721 – 8th Street East, Saskatoon, S7H 0T4 SK, Canada. e-mail: del_fredlund@golder.com.Invited Article, no discussion.

There are two statements in Terzaghi’s description ofeffective stress that are noteworthy because of their consis-tency with the principles of multiphase continuum mechan-ics. The first statement notes that the water pressure “acts inthe water and in the solid in every direction with equal in-tensity”. This statement is consistent with the concept of su-perposition of equilibrium stress fields in multiphase con-tinuum mechanics (Fung, 1965). The second statementnotes that the difference between the total stress and pore-water pressure “has its seat exclusively in the solid phase ofthe soil”. In other words, the effective stress variable islinked to the soil structure (or the arrangement of soil sol-ids). It is also noteworthy that while these statements areconsistent with continuum mechanics principles, Terzaghiwrote these statements when continuum mechanics wasstill in a formative stage.

Terzaghi’s statement describing effective stress re-mains consistent with the principles of multiphase contin-uum mechanics attesting to the wisdom of his originaldescription.

3. What is Effective Stress?

• History has shown that even though the original defini-tion of effective stress was presented with clarity, theterm “effective stress” has still suffered considerablemisunderstanding. As a teacher of soil mechanics, stu-dents have sometimes inquired of the author as to thephysical meaning and source of “effective stress”. Effec-tive stress has been highly esteemed and almost treatedas a sacrament within geotechnical engineering but thequestions remains, “What is the source and physicalmeaning of effective stress?” Effective stress has beendescribed in numerous ways; some descriptions beinginaccurate while others are reasonable and acceptable.Listed below are some of the descriptive terms associ-ated with effective stress along with a brief description ofhow each term is used.

• 1) Is effective stress an equation? It can be argued that“effective stress” is an equation because it has an equalsign. However, it can also be argued the effective stresssymbol is simply the difference between two variables;that difference being called effective stress. The dictio-nary defined an equation as “a statement that the valuesof two mathematical expressions are equal” (Wikipedia,2013). The use of the word equation for the effectivestress variable may have led geotechnical engineers tothink there is additional physical meaning behind the useof the term “effective stress”. The use of the word,“equation” may have fuelled the search for refinementsto the equation for both saturated and unsaturated soils.

• 2) Is effective stress a law? This question could also bepresented in a slightly different manner as, “Should ef-fective stress strictly be referred to as a physical law?”And if effective stress is a law, then what is the physicalprocess behind the law. Maybe the word law once again

elevates the term effective stress to a perceived level thatis not justifiable. Or is the only law involved behind theterm effective stress, the summation or equilibrium offorces associated with Newtonian mechanics?

• 3) Is effective stress a constitutive equation? The wordconstitutive implies the incorporation of one or more soilproperties. Consequently, it would seem clear to come tothe conclusion that effective stress should not be referredto as a constitutive equation. However, the history of soilmechanics shows that there have been attempts to bringsoil properties into the description of effective stress. It isimportant to remember that variables associated withstate cannot be constitutive in character.

• 4) Is effective stress a concept? The concept could bestated as follows, “If one changes effective stress, itshould be anticipated that something might happen to theequilibrium of the soil structure”. For example, it is an-ticipated that there may be a change in volume of the sat-urated soil. On the converse, if effective stress is notchanged there should not be a change in volume. Thelinkage between changes in stress state and changes inequilibrium of the soil structure can be referred to as theeffective stress concept. Stated another way, the equilib-rium of the soil structure is perturbed when `effectivestresses’ in a saturated soil are changed.

• 5) Is effective stress a principle? The principle of effectivestress could also be described in a manner similar to thatused for the concept of effective stress. The principle ofeffective stress would state, “If effective stresses arechanged, something should happen to the equilibrium ofthe soil structure and if effective stresses are not changed,there should be no change in the equilibrium of the soilstructure of a saturated soil”. Changes in the saturated soilstructure may be small but equilibrium conditions havebeen perturbed when the stress state is changed.

• 6) Is effective stress a stress state variable? State vari-ables should be defined independent of material proper-ties (i.e., non-material based variables). In this sense“effective stress” qualifies as a stress state variable. Itcan also be reasoned that effective stress applies in eachof the Cartesian coordinate directions since equilibriumconditions must be satisfied in all three orthogonal direc-tions.

• 7) Is effective stress a tensor? A tensor is a 3 by 3 matrixwhich arises out of consideration of three orthogonal di-rections. The three-dimensions define space. Effectivestress can also be visualized as having three principal di-rections where there are only normal stresses and noshear stresses. The stress tensor contains effective stressvariables along the trace and as such, effective stress hasthree spatial components.

• 8) Is effective stress only for saturated soils? The termeffective stress was initially defined solely for saturatedsoils. As shown later in this paper, effective stress doesnot qualify as an adequate term to describe the stress

4 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

state of an unsaturated soil. However, the principles ofcontinuum mechanics allow for the use of more than onetensor (or combination of state variables) for the descrip-tion of stress state.

The following responses to the above-mentionedquestions should apply to the use of the words, “effectivestress”, if the strict definitions from continuum mechanicsare applied.1) Is effective stress an equation? No2) Is effective stress a law? No3) Is effective stress a constitutive equation? No4) Is effective stress a concept? Yes5) Is effective stress a principle? Yes6) Is effective stress a stress state variable? Yes7) Is effective stress a tensor? Yes8) Is effective stress only for saturated soils? Yes

There may be some room for debate regarding the us-age of some of the effective stress terms; however, as far aspossible it would appear to be best to remain consistentwith continuum mechanics’ terminology and definitions.

The question might rightfully be asked, “Where doeseffective stress come from and does it have a physical ba-sis?” There would appear to be a sound continuum mechan-ics basis for the “effective stress” variable for saturated soilbut that will be presented later in the paper. The same con-tinuum mechanics principles and definitions can also be ap-plied to unsaturated soils and the result is the realizationthat more than one independent state variable is required todescribe the stress state of an unsaturated soil.

4. Definition of Key Technical Terms fromContinuum Mechanics

There are several technical terms and definitions thatare well accepted within continuum mechanics that need tobe described in order to understand the characteristics ofstate variables (Fung, 1965; 1977). Continuum mechanicshas attempted to define a consistent set of terms that can beapplied to all types of material behaviour. The terminologyapplies to single and multiphase particulate media. It ap-plies to solids as well as fluids. It applies to elastic behav-iour, plastic behaviour and viscous behaviour. The keyvariables that need to be defined are as follows:

State Variables: Are variables independent of mate-rial properties required for the characterization of a system(e.g., pressure, temperature, volume, time, etc.). These va-riables are defined independent of the physical propertiesof the material.

Stress State Variables: Are independent of materialproperties required for the characterization of force (orstress) equilibrium conditions.

Deformation State Variables: Are independent ofmaterial properties variables required for the description ofdeformations, distortions and deviations from an initialstate.

Constitutive Relations: Single-valued equations ex-pressing the relationship between state variables. Constitu-tive relations incorporate the physical properties of a mate-rial corresponding to a particular process to be simulated.

5. The Search for an EquationThe original definition of effective stress for saturated

soil behaviour has been the single most important conceptbehind the development of saturated soil mechanics. Whilethe original description of effective stress by Terzaghi(1936) is clear, there is a subtle variation of the definitionthat has been included in virtually every soil mechanicstextbook. The variation arises out of an attempt to describethe meaning of effective stress. The variations are often putin the context of a “refinement” to the effective stress equa-tion; however, the revisions to the effective stress equationswould appear to constitute a step away from the conceptsassociated with continuum mechanics and an accurate ap-plication of soil mechanics. One of the suggested revisionsto the effective stress equation involved the use of a wavyplane passed through the points of contact in a saturatedparticulate medium.

Most soil mechanics’ textbooks have a figure thatshows a particulate media with a wavy plane passed be-tween the contact points between particles (Fig. 1). The fig-ures common to most soil mechanics textbooks portraywhat is referred to as the “wavy plane” concept. A portionof the wavy plane is shown to pass through the contactpoints between particles while the remainder of the planepasses through water in the voids. The common explana-tions suggest that static equilibrium can be applied acrossthe wavy plane over a unit area. This figure is then treatedas a free-body diagram and forces are summed in the verti-cal direction across the wavy plane.

The analysis usually goes on to justify the form of theeffective stress equation and suggests that an area correc-tion should be applied to the effective stress equation tomake is more refined and accurate. It is also often suggestedthat the area correction in the effective stress equation issmall and its usage is not necessary. It is important to fur-

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 5

State Variables in Saturated-Unsaturated Soil Mechanics

Figure 1 - Illustration of the use of the wavy plane concept.

ther investigate the wavy plane analysis to ascertainwhether the wavy plane analysis is fundamentally flawed.

There appears to be a fundamental flaw with theabove-mentioned analysis in the sense that a wavy planepassed through a soil mass does not represent a legitimatefree-body diagram (Fung, 1977). Even the use of a linearplane surface passed through a continuum does not repre-sent a legitimate free-body diagram. In order to qualify asan acceptable free-body diagram in static mechanics, thefree-body must have spatial variation (Fung, 1977). If awavy plane does not constitute an acceptable free-body dia-gram then any subsequent mathematical manipulations thatmay be undertaken are unacceptable.

6. Steps Taken in the Wavy Plane Analysis

The mathematical steps that are adhered to in thewavy plane analysis are presented in detail in order to ex-amine aspects of the analysis that are unacceptable. The ex-amination of the wavy plane analysis is important becauseof the significant impact that this analysis has had on at-tempting to understand and define the stress state for unsat-urated soils.

Forces are summed in the vertical direction, over aunit area with the understanding that the sum of the forcesof the parts must equal the total force (Fig. 1).

a �i + (1 – a) uw = � (1)

where a = area of contact, �i = inter-particle (or inter-granular) stress, uw = pore-water pressure, and � = totalstress,

Multiplying out the terms in Eq. 1 gives,

a �i + uw – a uw = � (2)

The term effective stress is assumed to be equivalentto the inter-granular stress multiplied by the area of contactgiving,

a �i = �’ (3)

The “a” variable is a ratio of areas but it is also a mate-rial property. The above-mentioned conversion haschanged a force to a stress. Rearranging the above equationgives the effective stress equation with an area of contactterm.

�’ = (� – uw) + a uw (4)

The effective stress equation can be written as followsprovided it is assumed that the area of contact is small.

�’ = (� – uw) (5)

At this point it is important to re-visit the above-mentioned steps and identify the shortcomings of such ananalysis.

7. Inherent Limitations Associated with theWavy Plane Analysis

There are a number of inherent limitations associatedwith the wavy plane analysis. The first question that mightbe asked is, “Why is a (wavy) plane not an acceptablefree-body diagram?” In response to this question, let usconsider the simple case of a ladder leaning against a wallas shown in Fig. 2.

The central diagram showing the forces on the lean-ing ladder is an acceptable free-body diagram. However, itis not possible to draw a free-body diagram of the planewhere the ladder touches the wall. It is possible to state that“action” is equal to “reaction” of the point where the laddertouches the wall but is not possible to apply Newton’s sec-ond law of statics. It is important to recall Newton’s threelaws of statics and the reason why an independent law of“action” is equal to “reaction” was needed.

First Law: Bodies in motion will remain in motionunless it is acted upon by an external force.

Second Law: The acceleration of an object is de-pendent upon thenet forceacting upon the object and themass of the object. Consequently, when an object is in staticequilibrium the summation of forces and moments areequal to zero.

Third Law: For every action there must be an equaland opposite reaction.

Newton’s third law can be applied to stresses on aplane passed through an object or when one object interactswith another object. Newton’s third law constitutes a state-ment of equivalence when applied to a plane. It is not possi-ble to use the third law as a basis for a static equilibriumproof. Rather, it is simply possible to let one condition beequal to another condition. The wavy plane passed througha multiphase system such as soil cannot be used as a staticequilibrium proof (i.e., Newton’s second law), becausethere is no spatial variation of forces.

The second question to address is, “Why is a wavyplane unacceptable?” A wavy plane builds bias into theanalysis. Using a wavy plane goes contrary to the basic def-inition of a continuum. Any plane passed through a multi-phase material must be unbiased (Truesdell, 1966; Fung,1969; 1977). The definition of a continuum requires that theporosity of a multiphase material must be equal with re-

6 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

Figure 2 - Free-body diagram of a ladder leaning against a wall.

spect to volume porosity (of a representative elemental vol-ume, REV), and area porosity (i.e., the sides of the REV).The definition of a continuum is described in terms of thedensity function with respect to each phase of a multiphasesystem being constant. A wavy plane creates a problem inthat it builds bias into the analysis and contradicts the defi-nition of a continuum.

The definition of a continuum requires that the sum ofthe components of a multiphase system must be the sameon a volume and area basis. A saturated soil consists of sol-ids and water with the sum of the parts equal to the whole,both in terms of volume and area. Therefore, the plane can-not be biased.

Vs + Vw = Vt (by volume) (6)

as + nw = 1 (by area) (7)

where Vs = volume of solids, Vw = volume of water, as = areaof solids, and nw = porosity.

The equilibrium of two-dimensional (or three-dimen-sional) stress fields can be used to verify the acceptabilityof the effective stress variable for saturated soils. Themulti-dimensional force equilibrium analysis illustratesthat there is no need to apply an area of contact correction tothe effective stress variable.

There have also been other attempts to refine Ter-zaghi’s original effective stress variable; however, in eachcase, a soil property has been incorporated into the stressvariable. In so doing the stress variable has become “consti-tutive” in nature and no longer rigorously qualifies as astate variable.

8. Acceptable Stress Fields for theVerification of the Stress State of MultiphaseSystems

There are two types of stress fields that can be placedon a representative elemental volume, REV, of a contin-uum; namely, i) body forces per unit volume, and ii) surfacetractions. Each phase of a multiphase system has an inde-pendent stress field. The stress fields can be viewed as be-ing superimposed when several phases are involved (Trues-dell, 1966). The continuum mechanics representation ofstress fields for a one phase solid will first be presented fol-lowed by consideration of a saturated soil (i.e., a two phasesystem). And finally, consideration will be given to thestress fields for an unsaturated soil system.

9. Representation of Stress Fields for a OnePhase Continuum

A stress field acting in a particular direction can berepresented mathematically as a linearly varying fieldacross a REV as shown in Fig. 3 (Truesdell, 1966). Whenforces are summed in the vertical direction, equilibrium ismaintained since the variation of the stress field in the y-di-rection is taken into consideration. It can also be seen that a

plane through the bottom surface of the REV cannot beused to satisfy equilibrium considerations. The same equi-librium analysis can be extended to the three coordinate di-rections as shown in Fig. 4.

Equilibrium equations for a one phase solid can bewritten by summing forces in each of the Cartesian coordi-nate directions (Fung, 1969; 1977). The equilibrium equa-tions for the x-, y-, and z-directions, respectively, are:

��

�

��

�

��

�x yx zx

x y zdxdydz� �

�

���

�� � 0 (8)

��

�

��

�

��

� xy y zy

x y zg dxdydz� � �

�

���

�� � 0 (9)

��

�

��

�

��

�xz yz z

x y zdxdydz� �

�

���

�� � 0 (10)

where �x = total normal stress in the x-direction, �y = totalnormal stress in the y-direction, �z = total normal stress inthe z-direction, �yx = shear stress on the y-plane in the x-di-rection, �xy = shear stress on the x-plane in the y-direction,�zy = shear stress on the z-plane in the y-direction, �yz = shearstress on the y-plane in the z-direction, �xz = shear stress onthe x-plane in the z-direction, and �zx = shear stress on thez-plane in the x-direction.

The surface tractions found in the equilibrium equa-tion can be extracted to form a 3 by 3 matrix, (i.e., a tensor)as shown in Eq. 11. The tensor represents the stress state ata point in any one phase solid.

� � �

� � �

� � �

x yx zx

xy y zy

xz yz z

�

�

���

�

�

���

(11)

The stress tensor can also be plotted on a cube as arepresentation of the stress state in the one phase solid(Fig. 5). A one phase solid could consist of any materialranging from steel to cheese. The form of the stress statevariables is dictated largely by the number of phases in-volved.

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 7

State Variables in Saturated-Unsaturated Soil Mechanics

Figure 3 - Illustration of a multi-dimensional free-body diagramshowing a stress field.

10. Stress Fields for the Verification ofEffective Stress for a Saturated Soil

A stress field exists for each phase of a multiphase sys-tem. A saturated soil has a stress field associated with the wa-ter phase. Terzaghi (1936) clearly explained that the water

phase stress field acts in the water and through the solidphase. The solid phase consisting of an arrangement of parti-cles also has an independent stress field. In fact, it is thestress field associated with the soil solids that represents theequilibrium of soil structure. Unfortunately, the stress fieldassociated with the soil structure cannot be directly mea-sured. On the other hand, it is possible to write a stress fieldthat represents the summation of all the individual stressfields. This stress field can be referred to as the overall stressfield or the total stress field. The total stress field has thesame form as the stress field equation written for a one phasesolid. The summation of the superimposed coincident, equi-librium stress fields associated for each phase of a multi-phase system form the overall equilibrium stress field.

The stress field equation (Eq. 12), for the water phaseis illustrated by the free-body diagram shown in Fig. 6 forthe y-direction. The free-body diagram for the water phasemust also have two added body forces. One body force isfor the gravity force associated with water, nw wg, while theother body force is the seepage force, Fsy

w, which must betaken into consideration when the water phase is separatedfrom the soil solids phase.

n u

yn g Fw w

w w syw�

� � � � 0 (12)

8 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

Figure 4 - Equilibrium stress fields in three Cartesian coordinate directions for a one phase solid.

Figure 5 - Stress state at a point in a one phase solid.

Equilibrium equations for the x- and z-directions canbe written in a similar manner.

Since the sum of the individual stress fields must beequal to the overall stress field, it is possible to write theequilibrium stress field for the soil solids phase as the dif-ference between the overall stress field and the water phasestress field. These equations are shown for the x-, y-, andz-directions, respectively, are shown in Eqs. 13, 14, and 15.

� �

�

��

�

��

�

�

�

( )x w yx zx s ws s sx

wu

x y z

n u

xn g F

�� � � � � � 0 (13)

��

�

� �

�

��

�

�

� xu y w zy s w

s s syw

x

u

y z

n u

yn g F�

�� � � � �

( )0 (14)

��

�

��

�

� �

�

�

� xz yz z w s w

s s szw

x y

u

z

n u

zn g F� �

�� � � �

( )0 (15)

The surface tractions found in the equilibrium equa-tion can be extracted to form a 3 by 3 matrix, (i.e., a tensor)as shown in Eq. 16. The tensor represents the stress state ofthe soil structure at a point in the two phase material such asa saturated soil.

(

(

(

� � �

� � �

� � �

x w yx zx

xy y w zy

xz yz z w

u

u

u

�

�

�

�

�

���

�

�

���

)

)

)

(16)

A stress tensor can also be plotted on a cube to repre-sent the soil solids stress state for a saturated soil (Fig. 7).The stress tensor should represent the stress state variablescontrolling the solid phase of a two phase system.

11. Stress Fields for a Dry Soil

Equilibrium equations can also be written for a com-pletely dry soil with a pore-air pressure, ua. In this case thesoil structure equilibrium equations are the same as Eqs. 13,14 and 15 with the exception that pore-air pressure is sub-stituted for pore-water pressure. The stress tensor for thedry soil is shown as Eq. 17 and the stress state at a point isshown in Fig. 8. Even in the case of a highly compressiblepore fluid, the description of the stress state is independentof the properties of the pore fluid.

(

(

(

� � �

� � �

� � �

x a yx zx

xy y a zy

xz yz z a

u

u

u

�

�

�

�

�

���

�

�

���

)

)

)

(17)

The superposition of independent, coincident equilib-rium stress fields can also be applied to multiphase systemswith more than two phases, for example, a four phase sys-tem such as an unsaturated soil (Fredlund, 1973). A reviewof the research literature shows that historical, fundamentalstudies of unsaturated soil behaviour were primarily inter-ested in finding an equation that could be used to describethe physical behaviour of an unsaturated soil. The searchseemed to be for a single-valued equation rather than asearch for independent stress state variables.

12. The Rearch for an Effective StressEquation for Unsaturated Soils

Effective stress equations proposed for unsaturatedsoils shows the primary focus of a variety of researchers.The search started in the late 1950s and has continued to thepresent. The earliest proposed equation, and the best knownof all equations is Bishop’s effective stress equation (1959).

�’ = (� – ua) + �(ua – uw) (18)

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 9

State Variables in Saturated-Unsaturated Soil Mechanics

Figure 6 - Equilibrium stress field associate with the water phaseof a saturated soil.

Figure 7 - Stress state at a point for the solids phase of a two phasesaturated soil.

where � = effective stress, � = total stress, uw = pore-waterpressure, ua = pore-air pressure, and � = soil parameter re-lated to the degree of saturation of the soil, ranging fromzero to 1.0

Other effective stress equations have also been pro-posed (Croney et al., 1958; Jennings 1961; Aitchison,1961). All of the proposed equations are similar in form andbecome equal when the pore-air pressure is atmospheric.Also common to all equations is the incorporation of a soilproperty into the description of the stress state of an unsatu-rated soil. The soil property makes all equations take on thecharacter of a constitutive equation; thus violating the fun-damental separation of state variables and constitutive be-haviour.

With time the degree of saturation, S, of the soil hasoften been used in place of the � soil parameter, and theequations have been written in a tensor form (Jommi,2000). In so doing, the degree of saturation has become anapproximation for the � parameter but the equation still hasa constitutive nature.

Wavy planes have also been passed through an unsat-urated soil in an attempt to justify the so-called effectivestress equations for an unsaturated soil. The fallacy of suchan approach is the same as explained above for a saturatedsoil.

Over the same period of time when effective stressequations have been proposed for unsaturated soils, therehave also been researchers who have maintained that thestress variables, (� – ua) and (ua – uw) should be treated as in-dependent stress state variables and used as such to formconstitutive equations that are then used in theoretical for-mulations. As early as 1941, Biot used the stress state vari-ables in an independent manner in his formulation of thetheory of consolidation of an unsaturated soil. Coleman(1962), Matyas & Radakrishna (1968), and Fredlund &

Morgenstern (1977) advocated the use of independentstress state variables.

12.1. Examination of the wavy plane analysis for an un-saturated soil

A wavy plane passed through an unsaturated soil willpass through the air and water phases in addition to thesoil-to-soil points of contact as shown in Fig. 9. Once again,the wavy plane does not constitute a legitimate free-bodydiagram and is therefore unacceptable for verification pur-poses. However, let us follow the steps common to this der-ivation to observe the fallacies associated with the wavyplane analysis.

The wavy plane analysis does not even qualify as anapproximation of the physics behind the stress state of anunsaturated soil. Rather, the wavy plane analysis is a viola-tion of statics at its most fundamental level. The rationale;however, is as follows. The attempt to sum forces in thevertical direction across a plane yields the following equa-tion.

� = �ia + uw(1 – a – aa) + ua(1 – a – aw) (19)

where a = area of contact between particles along the wavyplane, aw = portion of the unit area that is in water, andaa = portion of the unit area that is in air.

Multiplying out the terms in Eq. 19, collecting termsand setting �’ equal to �i a gives the following equation.

� = �’ + uw – aauw + ua – aw ua + a uw –a ua (20)

The terms in Eq. 20 can be collected and solved forthe effective stress term after making the assumption thatthe area of contact between solid particles approaches zero.

�’ = (� – ua) + awua + (aa – 1)uw (21)

A further collection of terms gives.

�’ = (� – ua) + aw(ua – uw) (22)

Setting the area of the plane passing through waterequal to the degree of saturation of the soil gives an equa-

10 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

Figure 8 - Stress state for the soil structure of a completely dryparticulate medium.

Figure 9 - Wavy plane passed through an unsaturated soil.

tion that is similar in character to the Bishop (1959) equa-tion.

�’ = (� – ua) + S(ua – uw) (23)

The question remains, “Can a wavy plane analysis beused as a proof for an effective stress equation?” The an-swer would appear to be “no” because a plane, (and particu-larly a wavy plane), does not constitute a legitimate free-body diagram. Or it can be stated that a wavy plane does notincorporate spatial variation and as a result it cannot be con-sidered to be a legitimate free-body diagram.

The superposition of coincident equilibrium stressfields associated with the independent phases of a multi-phase medium would appear to provide a more convincingexplanation for the selection of independent stress statevariables for multiphase materials.

12.2. What is the fundamental difficulty associated withthe use of an equation to describe the stress state for anunsaturated soils?

The term, effective stress, (� – u-w), was originally de-fined for a saturated soil and became part of a stress tensorthat described the stress state for the soil structure. At-tempts to generate a so-called effective stress equation foran unsaturated soil have always resulted in the incorpora-tion of a soil property into the attempt to describe the stressstate. Soil properties are unacceptable in the description ofstress state based on the fundamental principles of contin-uum mechanics.

Incorporating soil properties into the description ofstress state also imposes serious inflexibility with respect tothe development of a range of required constitutive equa-tions and subsequent formulations. This is particularly truewhen attempting to account for behavioural effects such asnonlinearity and hysteresis. Geotechnical engineers mustdecide whether or not it is important to adhere to the funda-mental principles of continuum mechanics or strike off onanother course of action. In this sense, the decision be-comes philosophical in nature.

Morgenstern (1979) in his comments titled, Prop-erties of Compacted Soils, published in the proceedings ofthe 6th Pan-American Conference on Soil Mechanics andFoundation Engineering in Lima, Peru, explained his con-cerns about the use of an equation containing soil propertiesto describe the stress state of an unsaturated soil. He pointedout that Bishop’s effective stress equation has “proved tohave little impact on practice. The parameter, �, when de-termined for volume change behaviour was found to differwhen determined for shear strength. While originallythought to be a function of degree of saturation and hencebounded by 0 and 1, experiments were conducted in which� was found to go beyond these bounds. The effective stressis a stress variable and hence related to equilibrium consid-erations alone”. Morgenstern went on to explain that Bi-shop’s effective stress equation “contains the parameter, �,

that bears on constitutive behaviour. This parameter isfound by assuming that the behaviour of a soil can be ex-pressed uniquely in terms of a single effective stress vari-able and by matching unsaturated soil behaviour with satu-rated soil behaviour in order to calculate �. Normally, welink equilibrium considerations to deformations throughconstitutive behaviour and do not introduce constitutive be-haviour into the stress state”.

Once the effective stress equation becomes constitu-tive in character, it must be faced with the rigorous tests ofuniqueness for usage in engineering practice. It would ap-pear that the need for an effective stress equation can easilybe replaced through use of independent stress tensors con-taining stress state variables. However, there continues tobe ongoing attempts to revert to the usage of an effectivestress equation. The use of an effective stress equation forunsaturated soils violates the basic assumptions inherent inclassical continuum mechanics and places serious con-straints on subsequent formulations. These concerns can becircumvented through the use of independent stress statevariables for the proposal of shear strength, volume changeand other constitutive relations for the practice of unsatu-rated soil mechanics (Fredlund & Rahardjo, 1993;Fredlund et al., 2012).

It would appear that the issues related to the descrip-tion of the stress state in an unsaturated soil can only be re-solved through an understanding and acceptance of theindependent roles of state variables and constitutive behav-iour within the context of continuum mechanics. The basisfor stress state variables for an unsaturated soil is the sameas those explained for a saturated soil. In other words, con-sideration of force equilibrium for each phase of an unsatu-rated soil provides the basis for the selection of appropriateindependent stress state variables. The important equationsthat must be given consideration are Newton’s equilibriumequations in three coordinate directions. The search needsto focus on variable(s) that qualify as state variables. Thesevariables need to be placed into the tensor form and in sodoing will provide a strong science basis for the formula-tion of unsaturated soil mechanics theories.

13. Theoretical Basis for Independent StressState Variables

Fredlund & Morgenstern (1977) provided a theoreti-cal justification for independent stress state variables basedon a continuum mechanics approach. The independentstress state variables were then used in the formulation ofconstitutive models for all the classic application areas ofunsaturated soil mechanics. Engineering problems weresolved based on constitutive models and theoretical deriva-tions that utilized independent stress state variables. Thisinformation was subsequently synthesized in two books onunsaturated soil mechanics; namely, Soil Mechanics forUnsaturated Soils (1993) by D.G. Fredlund & H. Rahardjo,

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 11

State Variables in Saturated-Unsaturated Soil Mechanics

and Unsaturated Soil Mechanics in Engineering Practice(2012) by D.G. Fredlund et al.

It would appear that fundamentally, “state variables”are embedded within the conservative laws of physics. Theconservation of mass dictates the form and number of “de-formation state variables” required to map the movement ofindependent phases of a multiphase system. The conserva-tion of energy (or momentum) dictates the form and num-ber of “stress state variables” and “thermal state variables”required for equilibrium considerations of a multiphasesystem (e.g., an unsaturated soil).

The general procedure outlined for the determinationof the stress tensor for the soil structure of a saturated soilcan also be used for consideration of an unsaturated soil.While the saturated soil constituted a two phase system, anunsaturated soil needs to be considered as a four phase sys-tem. In addition to the soil solids (i.e., soil structure), waterphase and air phase, the contractile skin needs to be recog-nized as a fourth independent phase (Wang & Fredlund,2003). The need for the contractile skin to be recognized asa fourth phase can be visualized by observing the changesin volume that can occur (i.e., changes in the volume de-fined by the soil structure), as a soil is dried under condi-tions where total stresses remain constant. Under this con-dition, volume changes are due to the shrinkage imposed bythe air-water interface. The air-water interface is com-monly referred to as the contractile skin.

An unsaturated soil can be visualized as having twophases that behave as solids; namely, the soil solids and thecontractile skin (Wang & Fredlund, 2003). These phasesqualify as solids within the continuum mechanics sensesince they can come to equilibrium under the application ofa stress gradient. The unsaturated soil also has two phasesthat qualify as fluids; namely, water and air. These phasesqualify as fluids in the sense that they do not come to equi-librium under the application of a stress gradient.

An unsaturated soil has four phases and as a result,four independent equilibrium equations can be written. Anequilibrium equation can also be written for the overallcombination of the four phases and it will take the form ofEqs. 8, 9, and 10. There are only four independent equa-tions that can be written. Equilibrium equations can also beformed through the combination of one or more phases asillustrated by Fredlund & Morgenstern (1977). Fredlund(1973) also showed that the equations describing the equi-librium of the soil structure were yielded the same stressstate variables for the equilibrium of the contractile skin. Inother words, the stress state variables that produce equilib-rium for the soil solids (i.e., soil structure) also produceequilibrium for the contractile skin.

13.1. Equilibrium of soil structure for an unsaturatedsoil element

The equilibrium of the soil structure in the y-directioncan be written as the difference between the total equilib-

rium equation and the sum of the water, air and contractileskin equilibrium equations. The following equation is ob-tained when using the air phase as a reference phase duringthe derivation of the equilibrium equation for the soil struc-ture.

��

�

� �

�

�

�

��

�

xy y a

w ca w

zy

c

x

u

yn n f

u u

y

zn n

��

� ��

�

� �

( )( *)

( )

( sa

s s

syw

sya

c a w

u

yn g

F F n u uf

y

)

( )*

�

�

�

�

� �

� � � � 0

(24)

where na = porosity with respect to the air phase, nw = poros-ity with respect to the water phase, nc = porosity with re-spect to the contractile skin, f* = interaction function be-tween the contractile skin and the soil structure, Fsy

a =interaction body force between the solids and the air phasein the y-direction, and Fsy

w = interaction body force betweenthe solids and the water phase in the y-direction.

Similar equations can be written for the x- and z-di-rections, respectively:

� � �

�

�

�

��

�

��

�

( )( *)

( )

(

x a xy

w ca w yx

zx

u

xn n f

u u

x y

zn

�� �

�� �

� c sa

sxw

sxa

c a w

nu

xF F

n u uf

x

� � � �

� �

)

( )*

�

��

�0

(25)

where Fsx

a = interaction body force between the solids andthe air phase in the x-direction, and Fsx

w = interaction bodyforce between the solids and the water phase in the x-direc-tion.

��

�

��

�

� �

�

�

�xz yz z a

w ca w

c

x y

u

zn n f

u u

z

n n

� ��

� ��

�

�

( )( *)

( )

( sa

szw

sza

c a w

u

zF F n u u

f

y) ( )

*�

�

�

�� � � � � 0

(26)

where Fsz

a = interaction body force between the solids andthe air phase in the z-direction, and Fsz

w = interaction bodyforce between the solids and the water phase in the z-direc-tion.

The stress variables controlling the equilibrium of thesoil structure are the stress state variables that control themechanical behavior of soils. There are three independentsets of normal stresses, (i.e., surface tractions) that can beextracted from the equilibrium equations for the soil struc-ture to form the stress state variables. The three stress statevariables are: (� – ua), (ua – uw), and (ua). The stress variable,ua, is also a stress state variable but can be eliminated if thesoil particles are assumed to be incompressible. Therefore,

12 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

the stress state variables for the soil structure and the con-tractile skin in an unsaturated soil are (� – ua) and (ua – uw).

The stress state variables act in three Cartesian coor-dinate directions and the variables can be collected to formtwo independent stress tensors. The two independent stresstensors can be written as the stress state at a point in an un-saturated soil.

(

(

(

� � �

� � �

� � �

x a yx zx

xy y a zy

xz yz z a

u

u

u

�

�

�

�

�

���

�

�

���

)

)

)

(27)

( )

( )

( )

u u

u u

u u

a w

a w

a w

�

�

�

�

�

���

�

�

���

0 0

0 0

0 0

(28)

Equation 27 is referred to as the net normal stress ten-sor and Eq. 28 is referred to as the matric suction (or soilsuction) tensor. The pore-air pressure appears in both stresstensors; however, it is the difference between stress compo-nents that allows the two tensors to qualify as independentstress state variables. The stress variables in Eqs. 27 and 28can be placed on the surface of a cube to give the stress stateat a point as shown in Fig. 10.

As an unsaturated soil approaches saturation, the de-gree of saturation, S, approaches 100%. The pore-waterpressure, uw, approaches the pore-air pressure, ua, and thematric suction term, (ua – uw), goes towards zero. The sec-ond stress tensors for the unsaturated soil disappear be-cause the matric suction, (ua – uw), becomes zero. It shouldbe noted that it is not necessary for the pore-water pressureto go to zero in order for the soil to behave as a saturatedsoil. Rather, it is necessary for the pore-water pressure toincrease until it becomes equal to the pore-air pressure andthen the soil behaves as a saturated soil.

Only the first stress tensor is left to represent thestress state for a saturated soil once the pore-water pressureis equal to pore-air pressure. The pore-air pressure term inthe first stress tensor becomes equal to the pore-water pres-

sure, uw, as the soil becomes saturated. The stress state at apoint in a saturated soil was illustrated in Fig. 7. The stresstensor for a saturated soil is consistent with that presentedby Terzaghi (1936) as the effective stress variable, (� – uw).

13.2. Other combinations of stress state variables

The soil structure equilibrium equations can havethree different forms depending upon the reference phaseused during the derivation of the equilibrium equations.Equations 24, 25, and 26 are the resulting equations whenusing the air phase as the reference phase during the deriva-tion. Other forms of the derivation can use the water phaseor else the total stress field as a reference. Each form of theequilibrium equations contains a combination of two stressstate variables. In other words, any two of three possiblestress variables, (i.e., (� – ua), (� – uw), and (ua – uw)), can beused to describe the stress state for the soil structure andcontractile skin in an unsaturated soil.

14. The Principle or Concept Associated withStress State Variables

The principle or concept associated with stress statevariables can be stated as follows: “If one or more of thestress state variables are changed, the equilibrium of thesystem is disturbed and changes in the system can be antici-pated, and vice versa”. The same principle applies to boththe soil structure and the contractile skin.

15. Visualization of the World of SoilMechanics Based on Stress State Variables

Figure 11 illustrates the transition in the descriptionof the stress state when moving from a saturated soil to anunsaturated soil (and vice versa), (Fredlund, 1994). Thetransition is shown to occur at the water table or the point atwhich the pore-water pressures change from a positivevalue to a negative value. There is a zone immediatelyabove the water table referred to as the capillary zone. Thiszone may be essentially saturated; however, it is suggested

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 13

State Variables in Saturated-Unsaturated Soil Mechanics

Figure 10 - Stress state at a point for the soil structure and the con-tractile skin in an unsaturated soil.

Figure 11 - Visualization of the world of soil mechanics in termsof the description of the stress state for saturated and unsaturatedsoils (Fredlund, 1996).

that the transition between saturated soil mechanics and un-saturated soil mechanics is best based on whether the pore-water pressures are positive or negative.

Numerous problems in soil mechanics are analyzedusing a one-dimensional analysis with total stresses beingchanged in the vertical direction. Consequently, an increasein stresses in the vertical direction produces a tendency fordeformation in the horizontal direction as well as the(downward) vertical direction. On the other hand, a de-crease in the pore-water pressure will produce a tendencyfor shrinkage in all three directions. Only under isotropictotal stress conditions will the deformation directions besimilar for total stress and pore-water pressure changes.Consequently, it would appear to be reasonable to acknowl-edge the independent behaviour of changes in the totalstress field from that of the pore-water pressure field whenmoving above the water table. The above conceptual pic-ture is a reminder that changes in total stresses and changesin pore-water pressures can produce different deformationfields in an unsaturated soil. Therefore, the net normalstress variable needs to be considered as being independentof the matric suction variable for analytical purposes. Also,the magnitude of deformations associated with changes innet total stress may be different from the magnitude of de-formations associated with changes in matric suction. Fig-

ure 12 provides a visualization tool that illustrates the dif-ference in the stress state at a point when the soil issaturated and unsaturated.

16. Recommendations for the OngoingDevelopment of Unsaturated Soil Mechanics

Following are a series of suggested recommendationsfor the development of a continuum mechanics based ap-proach to the development of unsaturated soil mechanics.

1) That the term “effective stress” should be limited to theoriginal definition described by Terzaghi (1936) forsaturated soils.

2) The concepts of stress state variables as defined in con-tinuum mechanics should be adopted as the sciencebasis for unsaturated soil mechanics.

3) Stress state variables for any soil should be kept inde-pendent of the material properties.

4) There can be multiple stress state variables (and defor-mation state variables) for a multiphase material.

5) Temperature and time should also be considered as statevariables, in addition to deformation state variables.

14 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

Figure 12 - Visualization of the stress state at a point for saturated and unsaturated soils (Fredlund, 1996).

17. Incorporation of Soil Properties into theDefinition of Stress State

Skempton (1961) felt that it was of “philosophical in-terest to examine the fundamental principles of effectivestress, since it would seem improbable that an expression ofthe form” proposed by Terzaghi, “is strictly true.” He wenton to say, “We may therefore anticipate that, even for fullysaturated porous materials, the general expression for ef-fective stress is more complex, and that Terzaghi’s equa-tion has the status of an excellent approximation in thespecial case of saturated soils.” He goes on to say that “theeffective stress is actually the intergranular stress acting be-tween the particles” and the effective stress should be cor-rected for the area of contact between the particles.

Skempton’s (1961) statements did not recognize theneed for multiple stress state variables in some situations(e.g., the case of a saturated soil with compressible soil par-ticles). His search opened the way for introducing materialproperties into the description of stress state. The so-calledrefinement associated with the “area of contact” betweensoil particles did not have a significant impact in saturatedsoil mechanics but it set the stage for other considerationsrelated to effective stress for volume change and shearstrength of saturated and unsaturated soils.

Nur & Byerlee (1971) went through a similar exercisein attempting to develop an “exact effective stress law forelastic deformation of rock with fluids”. In coming to theconclusion that Terzaghi’s equation “is an excellent ap-proximation”, they failed to realize the need for multiplestress state variables for the situation involving compress-ible materials.

17.1. Soil properties in effective stress equations for sat-urated soils

Consideration of the area of contact between soil par-ticles set the stage for suggestions that there was a “morecorrect expression” for effective stress equation that couldbe written for analyzing shear strength problems. Equation29 was suggested as the effective stress equation for “fullysaturated materials” (Skempton, 1961).

� ��

�'

tan

tan '� � �

�

���

��1

auw (29)

where a = area of contact between particles, � = angle ofintrinsic friction of the solid particles, and �’ = angle of in-ternal friction between the soil particles.

An independent effective stress equation was sug-gested for the analysis of volume change problems involv-ing saturated soils (Skempton, 1961).

� �' � � ��

��

�1

C

Cus

w (30)

where Cs = compressibility of the solids, and C = compress-ibility of the porous material or the soil structure.

Laughton (1955) performed experiments on materialswith compressible particles. The results can be interpretedas showing the need for a second independent state variable(i.e., uw), when dealing with compressible materials.

The need for separate effective stress equations whenanalyzing shear strength and volume change problemswould appear to be surprising if the equations are truly a de-scription of “stress state”. The volume change analysis ofmaterials with compressible solids can be analyzed withoutthe need for a new effective stress equation if it is realizedthat two independent stress state variables are required;namely, the Terzaghi effective stress variable (� – uw) andpore-water pressure, (uw). There is no need for two effectivestress equations (i.e., one for shear strength and another forvolume change). All material properties must be incorpo-rated at the constitutive behaviour level of formulationsthat describe soil behaviour.

Nur & Byerlee (1971) incorporated the compressibil-ity properties into their attempt to develop a more “exactexpression for an effective stress law” for the study of elas-tic deformations of rock materials. Their statement that“there is a great deal of disagreement on the theoretical ac-curacy and validity of Terzaghi’s relation” would appear tobe an overstatement. Bringing compressibility values intothe effective stress equation is not necessary if it is recog-nized that a compressible particulate soil or rock requiresan independent stress state variable in this situation;namely, an isotropic pore-water pressure tensor.

A summary of so-called proposed refinements to Ter-zaghi’s effective stress equation has been given by Gens(2005). The proposed refinements are not repeated hereinbecause there appears to be no fundamental reason to ac-cept soil properties into the description of stress state.

17.2. Soil properties in effective stress equations for un-saturated soils

With a history of incorporating soil properties into ef-fective stress equations for saturated soils, it is not that sur-prising that soil properties should find their way into aneffective stress equation for unsaturated soils. Bishop(1959) introduced the � parameter into his proposed effec-tive stress equation for unsaturated soils. The � parameterwas found to generally be a nonlinear function of degree ofsaturation. Later, the degree of saturation variable, S, wassubstituted for the � parameter by some researchersBishop’s (1959) equation became referred to as the averageskeleton stress.

�’ = (� – ua) + S (ua – uw) (31)

The history of various forms for the effective stressfor an unsaturated soil has been summarized by Gens(2005) and is not repeated herein.

Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016. 15

State Variables in Saturated-Unsaturated Soil Mechanics

Khalili & Khabbaz (1998) suggested that the � pa-rameter could be related to the air-entry value for an unsatu-rated soil. The soil-water characteristic curve, SWCC, fordesorption of a soil was set as a demarcation between satu-rated and unsaturated soil behaviour. Unsaturated soil be-haviour was related to degree of saturation through use of asecond soil parameter, m, which was set to 0.55.

It is apparent that the search for an effective stressequation for unsaturated soil behavior had its origin in theincorporation of soil properties into Terzaghi’s effectivestress equation. This constitutes a fundamental deviationfrom the context of continuum mechanics.

There have been several researchers who have pro-posed and advocated the use of independent stress statevariables. Fredlund & Morgenstern (1977) used equilib-rium considerations of a multiphase system to illustrate thesource for acceptable, independent stress state variables.Null type laboratory tests were also used to confirm thesuitability of two independent stress state variables for un-saturated soils. There are other researchers who previouslysuggested using independent stress state variables. Biot’s(1941), in his proposed theory of consolidation advocatedthe use of two independent stress variables for unsaturatedsoils. Others who supported this approach are Coleman(1962) and Matyas & Radhakrishna (1968). Bishop &Blight (1963) also presented laboratory measured constitu-tive data using two independent stress state variables.

In his notes on effective stresses Gens (2005) con-cludes, “The description of the behaviour of unsaturatedsoils requires the use of two independent stress variables”.The author agrees with the direction advocated and wouldadd that there ought to not be an ongoing search for a uni-versal effective stress expression for unsaturated soils.Such a search appears to always lead to the inclusion of soilproperties, making the stress state expressions “constitu-tive” in character.

Theoretical formulations for unsaturated soils prob-lems have been proposed that maintained the separation ofstate variables and constitutive equations. These formula-tions have also been solved for a variety of saturated-unsaturated soils problems common to geotechnical engi-neering (Fredlund & Rahardjo (1993) and Fredlund et al.(2012)). The solutions have been used extensively in solv-ing geotechnical engineering problems. It is the soil-watercharacteristic curves, SWCCs, (i.e., desorption and adsorp-tion branches) that have proved to provide the primary un-saturated soils information required for each of the formu-lations. Sheng et al. (2008) also demonstrated that it waspossible to formulate a critical state model for unsaturatedsoils while maintaining the separation of state variables andconstitutive behaviour.

18. Grounds for a Connection Between theTwo Approaches to FormulatingUnsaturated Soil Mechanics

It is of interest to examine the possible grounds for acompromise between the two procedures that have beenproposed for formulating unsaturated soil mechanics prob-lems. The relationship between the two approaches can beviewed as follows from an elementary standpoint.

Let us start with the use of two independent stressstate variables for an unsaturated soil and then try to formu-late an acceptable constitutive relationship. Let us considerthe volume change of an unsaturated soil where the differ-ence in behaviour associated with changing net total stressand matric suction is uniquely related to the degree of satu-ration of the soil. It would then be possible to write the fol-lowing general volume change equation with constant,linear soil properties.

dv = m1

s(� – ua) + m1

s S (ua – uw) (32)

where dv = soil structure volume change, (� – ua) = net totalstress state variable, (ua – uw) = matric suction variable,m1

s = compressibility of the soil structure with respect to the(� – ua) stress state variable, and S = degree of saturation ofthe soil.

Equation 32 states that the relationship betweenchanging the net total stress and the matric suction vari-ables can be approximated through use of the degree of sat-uration variable. Equation 32 is an acceptable form for aconstitutive relationship. However, it may not be suffi-ciently general to embrace all unsaturated soil behaviour.Consequently, the use of such an equation may prove to betoo restrictive in engineering practice.

Equation 32 also suggests that the degree of satura-tion is known at all times when simulating a particular pro-cess. However, in reality, an independent analysis is gener-ally required to predict water movement in or out of thesoil. A transient seepage analysis also requires two nonlin-ear soil properties for its solution; namely, the hydraulicconductivity and the water storage function. The hydraulicproperties in turn may or may not be affected by both stressstate variables. It should also be noted that the overall vol-ume and the volume of water in the soil must both be knownin order to compute the degree of saturation.

A similar equation can be written for the characteriza-tion of shear strength but once again the form may prove tobe not sufficiently general for engineering practice. Thereare also other physical relationships such as the water con-tent constitutive relationship (i.e., the soil-water character-istic curve, SWCC), that does not have a unique response tothe degree of saturation.

In general, it would appear to be better to separatestate variables from constitutive behaviour in order to pro-vide the greatest flexibility and accuracy in defining unsat-urated soil behaviour.

16 Soils and Rocks, São Paulo, 39(1): 3-17, January-April, 2016.

Fredlund

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