##### Document Text Contents

Page 2

Fundamentals of

Soil Behavior

Third Edition

James K. Mitchell

Kenichi Soga

JOHN WILEY & SONS, INC.

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 252

SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS 275

Table 9.5 Typical Range of Flow Parameters for Fine-Grained Soilsa

Parameter Symbol Units Minimum Maximum

Porosity n — 0.1 0.7

Hydraulic

conductivity

kh m s

�1 1 � 10�11 1 � 10�6

Thermal

conductivity

kt W m

�1 K�1 0.25 2.5

Electrical

conductivity

�e siemens m

�1 0.01 1.0

Electro osmotic

conductivity

ke m

2 s�1 V�1 1 � 10�9 1 � 10�8

Diffusion

coefficient

D m2 s�1 2 � 10�10 2 � 10�9

Osmotic

efficiencyb

" — 0 1.0

Ionic mobility u m2 s�1 V�1 3 � 10�9 1 � 10�8

aThe above values of flow coefficients are for saturated soil. They may be

much less in partly saturated soil.

b0 to 1.0 is the theoretical range for the osmotic efficiency coefficient. Values

greater than about 0.7 are unlikely in most fine-grained materials of geotechnical

interest.

gradients of different types. A gradient of one type Xj

can cause a flow of another type Ji, according to

J � L X (9.57)i ij j

The Lij are termed coupling coefficients. They are prop-

erties that may or may not be of significant magnitude

in any given soil, as discussed later. Types of coupled

flow that can occur are listed in Table 9.6, along with

terms commonly used to describe them.6

Of the 12 coupled flows shown in Table 9.6, several

are known to be significant in soil–water systems, at

least under some conditions. Thermoosmosis, which is

water movement under a temperature gradient, is im-

portant in partly saturated soils, but of lesser impor-

tance in fully saturated soils. Significant effects from

thermally driven moisture flow are found in semiarid

and arid areas, in frost susceptible soils, and in expan-

sive soils. An analysis of thermally driven moisture

6 Mechanical coupling also occurs in addition to the hydraulic, ther-

mal, electrical, and chemical processes listed in Table 9.6. A common

manifestation of this in geotechnical applications is the development

of excess pore pressure and the accompanying fluid flow that result

from a change in applied stress. This type of coupling is usually most

easily handled by usual soil mechanics methods. A few other types

of mechanical coupling may also exist in soils and rocks (U.S. Na-

tional Committee for Rock Mechanics, 1987).

flow is developed later. Electroosmosis has been used

for many years as a means for control of water flow

and for consolidation of soils. Chemicalosmosis, the

flow of water caused by a chemical gradient acting

across a clay layer, has been studied in some detail

recently, owing to its importance in waste containment

systems.

Isothermal heat transfer, caused by heat flow along

with water flow, has caused great difficulties in the

creation of frozen soil barriers in the presence of flow-

ing groundwater. Electrically driven heat flow, the Pel-

tier effect, and chemically driven heat flow, the Dufour

effect, are not known to be of significance in soils;

however, they appear not to have been studied in any

detail in relation to geotechnical problems.

Streaming current, the term applied to both hydrau-

lically driven electrical current and ion flows, has im-

portance to both chemical flow through the ground

(advection) and the development of electrical poten-

tials, which may, in turn, influence both fluid and ion

flows as a result of additional coupling effects. The

complete roles of thermoelectricity and diffusion and

membrane potentials are not yet known; however, elec-

trical potentials generated by temperature and chemical

gradients are important in corrosion and in some

groundwater flow and stability problems.

Whether thermal diffusion of electrolytes, the Soret

effect, is important in soils has not been evaluated;

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 253

276 9 CONDUCTION PHENOMENA

Table 9.6 Direct and Coupled Flow Phenomena

Gradient X

Flow J Hydraulic Head Temperature Electrical

Chemical

Concentration

Fluid Hydraulic

conduction

Darcy’s law

Thermoosmosis Electroosmosis Chemical

osmosis

Heat Isothermal heat

transfer or

thermal filtration

Thermal

conduction

Fourier’s law

Peltier effect Dufour effect

Current Streaming current Thermoelectricity

Seebeck or

Thompson effect

Electric

conduction

Ohm’s law

Diffusion and

membrane

potentials or

sedimentation

current

Ion Streaming current

ultrafiltration

(also known as

hyperfiltration)

Thermal diffusion

of electrolyte or

Soret effect

Electrophoresis Diffusion Fick’s

law

however, since chemical activity is highly temperature

dependent, it may be a significant process in some

systems. Finally, electrophoresis, the movement of

charged particles in an electrical field, has been used

for concentration of mine waste and high water content

clays.

The relative importance of chemically and electri-

cally driven components of total hydraulic flow is il-

lustrated in Fig. 9.20, based on data from tests on

kaolinite given by Olsen (1969, 1972). The theory for

description of coupled flows is given later. A practical

form of Eq. (9.57) for fluid flow under combined hy-

draulic, chemical, and electrical gradients is

H log(C /C )

EB Aq � �k A � k A � k Ah h c eL L L

(9.58)

in which kh, kc, and ke are the hydraulic, osmotic, and

electroosmotic conductivities,

H is the hydraulic head

difference,

E is the voltage difference, and CA and CB

are the salt concentrations on opposite sides of a clay

layer of thickness L.

In the absence of an electrical gradient, the ratio of

osmotic to hydraulic flows is

q k log(C /C )hc c B A� � (

E � 0) (9.59)� �q k

Hh h

and, in the absence of a chemical gradient, the ratio of

electroosmotic flows to hydraulic flows is

q k

Ehe e� (

C � 0) (9.59a)� �q k

Hh h

The ratio (kc /kh) in Fig. 9.20 indicates the hydraulic

head difference in centimeters of water required to give

a flow rate equal to the osmotic flow caused by a 10-

fold difference in salt concentration on opposite sides

of the layer. The ratio ke /kh gives the hydraulic head

difference required to balance that caused by a 1 V

difference in electrical potentials on opposite sides of

the layer. During consolidation, the hydraulic conduc-

tivity decreases dramatically. However, the ratios kc /kh

and ke /kh increase significantly, indicating that the rel-

ative importance of osmotic and electroosmotic flows

to the total flow increases. Although the data shown in

Fig. 9.20 are shown as a function of the consolidation

pressure, the changes in the values of kc /kh and ke /kh

are really a result of the decrease in void ratio that

accompanies the increase in pressure, as may be seen

in Fig. 9.20c.

These results for kaolinite provide a conservative es-

timate of the importance of osmotic and electroosmotic

flows because coupling effects in kaolinite are usually

smaller than in more active clays, such as montmoril-

lonite-based bentonites. In systems containing confined

clay layers acted on by chemical and/or electrical gra-

dients, Darcy’s law by itself may be an insufficient

basis for prediction of hydraulic flow rates, particularly

if the clay is highly plastic and at a very low void ratio.

Such conditions can be found in deeply buried clay

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 504

528 LIST OF SYMBOLS

� disturbance factor

� geometrical packing parameter

� rotation angle of yield envelope

�0, �i constant characteristic of the property and

the clay

� Bishop’s unsaturated effective stress pa-

rameter

� clay plate thickness measured between

centers of surface layer atoms

� deformation parameter in Hertz theory

� displacement, distance

� solid fraction of a contact area

� relative retardation

�p particle eccentricity distance

� dielectric constant, permittivity

� porosity

� strain

�̇ strain rate

�0 permittivity of vacuum, 8.85 �

10�12 C2/(Nm2)

�1 axial strain

�̇a vertical strain rate in one dimensional

consolidation

�ƒ strain at failure

�̇min minimum strain rate

�rd volumetric strain that would occur if

drainage were permitted

�s deviator strain

�̇s deviator strain rate

�v volumetric strain

�̇v volumetric strain rate

E energy dissipated per cycle per unit vol-

ume

� friction angle

� local electrical potential

�� friction angle in effective stress

�b angle defining the rate of increase in shear

strength with respect to soil suction

�c characteristic friction angle

��crit friction angle at critical state

�e, ��e Hvorslev friction parameter

��ƒ friction angle corrected for the work of

dilation

��m peak mobilized friction angle

��r residual friction angle

�repose angle of repose

�v apparent specific volume of the water in

a clay/water system of volume V

��, ��� intergrain sliding friction angle

# dissipation function

activity coefficient

angle between a and b crystallographic

axes

unit weight

̇ shear strain rate

c applied shear strain or cyclic shear strain

amplitude

d dry unit weight

% double layer charge

% specific volume intercept at unit pressure

� dynamic viscosity

� fraction of pore pressure that gives effec-

tive stress

�0 initial anisotropy

! swelling index

!� real relative permittivity

!� polarization loss, imaginary relative per-

mittivity

� compression index

� correction coefficient for frost depth pre-

diction equation

� damping ratio

� decay constant

� pore size distribution index

� separation distance between successive

positions in a structure

� wave length of X ray

� wave length of light

�cs critical state compression index

� chemical potential

� coefficient of friction

� dipole moment

� fusion parameter

� Poisson’s ratio

� viscosity

( critical state stress ratio

� Poisson’s ratio

�b Poisson’s ratio of soil skeleton

� osmotic or swelling pressure

� angle of bedding plane relative to the

maximum principal stress direction

� contact angle

� geometrical packing parameter

� liquid-to-solid contact angle

� orientation angle

� volumetric water content

�m volumetric water content at full saturation

�r residual water content

�s volumetric water content at full saturation

� bulk dry density

� charge density

� mass density

�d bulk dry density

�T resistivity of saturated soil

�w density of water

�W resistivity of soil water

� area occupied per absorbed molecule on

a surface

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 505

LIST OF SYMBOLS 529

� double-layer charge

� electrical conductivity

� entropy production

� normal stress

� surface tension of water

� surface charge density

� total stress

�� effective stress

��0 initial effective confining pressure

�1 major principal total stress

�1 tensile strength of the interface bond

��1 major principal effective stress

�1c major principal stress during consolida-

tion

�1ƒ major principal stress at failure

��1ƒƒ major principal effective stress at failure

��2 intermediate principal effective stress

�3 minor principal total stress

��3 minor principal effective stress

�3c minor principal stress during consolida-

tion

��3ƒƒ minor principal effective stress at failure

��a axial effective stress

��ac axial consolidation stress

�as interfacial tension between air and solid

�aw interfacial tension between air and water

�c crushing strength of particles

�c tensile strength of cement

�e electrical conductivity

��e equivalent consolidation pressure

�eƒƒ effective AC conductivity

�ƒ partial stress increment for fluid phase

��ƒ effective normal stress on shear plane

�ƒƒ normal total stress on failure plane

��ƒƒ normal effective stress on failure plane

�h electrical conductivity due to hydraulic

flow

��h0 initial horizontal effective stress

��i effective stress in the i-direction

��i intergranular stress

��i isotropic consolidation

�iso isotropic total stress

�max maximum principal stress

�min minimum principal stress

��n effective normal stress

��p preconsolidation pressure

�r radial total stress

��r radial effective stress

��rc radial consolidation stress

�s conductivity of soil surface

�s partial stress increment for solid phase

�s tensile strength of the sphere

�T electrical conductivity of saturated soil

�T , ��t tensile strength of cemented soil

�v vertical stress

��v vertical effective stress

�v0 overburden vertical effective stress

��v0 overburden effective stress

��vm maximum past overburden effective stress

��vp vertical preconsolidation stress

�W electrical conductivity of pore water

�ws interfacial tension between water and

solid

�y yield strength

�� circumferential stress

� shear strength

� shear stress

� surface tension

� swelling pressure or matric suction

� undrained shear strength

�a apparent tortuosity factor

�c applied shear stress

�c contaminant film strength

�cy undrained cyclic shear stress

�d drained shear strength

�ƒƒ shear stress at failure on failure plane

�i shear strength

�i shear strength of contact

�m shear strength of solid material in yielded

zone

�peak applied shear stress at peak

�� initial static shear stress

' mass flow factor

' cation valence

� distance function � Kx, double-layer the-

ory

� ratio of average temperature gradient in

air filled pores to overall temperature gra-

dient

� dilation angle

� electrical potential

� intrinsic friction angle

� matric suction

�0 surface potential of double layer

�d displacement pressure

� electrical potential

� state parameter

� total potential of soil water

�0 electrical potential at the surface

�s gravitational potential

�m matrix or capillary potential

�p gas pressure potential

�s osmotic or solute potential

" angular velocity

" frequency

" osmotic efficiency

true electroosmotic flow

� zeta potential

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Fundamentals of

Soil Behavior

Third Edition

James K. Mitchell

Kenichi Soga

JOHN WILEY & SONS, INC.

Co

py

rig

ht

ed

M

at

er

ia

l

Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 252

SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS 275

Table 9.5 Typical Range of Flow Parameters for Fine-Grained Soilsa

Parameter Symbol Units Minimum Maximum

Porosity n — 0.1 0.7

Hydraulic

conductivity

kh m s

�1 1 � 10�11 1 � 10�6

Thermal

conductivity

kt W m

�1 K�1 0.25 2.5

Electrical

conductivity

�e siemens m

�1 0.01 1.0

Electro osmotic

conductivity

ke m

2 s�1 V�1 1 � 10�9 1 � 10�8

Diffusion

coefficient

D m2 s�1 2 � 10�10 2 � 10�9

Osmotic

efficiencyb

" — 0 1.0

Ionic mobility u m2 s�1 V�1 3 � 10�9 1 � 10�8

aThe above values of flow coefficients are for saturated soil. They may be

much less in partly saturated soil.

b0 to 1.0 is the theoretical range for the osmotic efficiency coefficient. Values

greater than about 0.7 are unlikely in most fine-grained materials of geotechnical

interest.

gradients of different types. A gradient of one type Xj

can cause a flow of another type Ji, according to

J � L X (9.57)i ij j

The Lij are termed coupling coefficients. They are prop-

erties that may or may not be of significant magnitude

in any given soil, as discussed later. Types of coupled

flow that can occur are listed in Table 9.6, along with

terms commonly used to describe them.6

Of the 12 coupled flows shown in Table 9.6, several

are known to be significant in soil–water systems, at

least under some conditions. Thermoosmosis, which is

water movement under a temperature gradient, is im-

portant in partly saturated soils, but of lesser impor-

tance in fully saturated soils. Significant effects from

thermally driven moisture flow are found in semiarid

and arid areas, in frost susceptible soils, and in expan-

sive soils. An analysis of thermally driven moisture

6 Mechanical coupling also occurs in addition to the hydraulic, ther-

mal, electrical, and chemical processes listed in Table 9.6. A common

manifestation of this in geotechnical applications is the development

of excess pore pressure and the accompanying fluid flow that result

from a change in applied stress. This type of coupling is usually most

easily handled by usual soil mechanics methods. A few other types

of mechanical coupling may also exist in soils and rocks (U.S. Na-

tional Committee for Rock Mechanics, 1987).

flow is developed later. Electroosmosis has been used

for many years as a means for control of water flow

and for consolidation of soils. Chemicalosmosis, the

flow of water caused by a chemical gradient acting

across a clay layer, has been studied in some detail

recently, owing to its importance in waste containment

systems.

Isothermal heat transfer, caused by heat flow along

with water flow, has caused great difficulties in the

creation of frozen soil barriers in the presence of flow-

ing groundwater. Electrically driven heat flow, the Pel-

tier effect, and chemically driven heat flow, the Dufour

effect, are not known to be of significance in soils;

however, they appear not to have been studied in any

detail in relation to geotechnical problems.

Streaming current, the term applied to both hydrau-

lically driven electrical current and ion flows, has im-

portance to both chemical flow through the ground

(advection) and the development of electrical poten-

tials, which may, in turn, influence both fluid and ion

flows as a result of additional coupling effects. The

complete roles of thermoelectricity and diffusion and

membrane potentials are not yet known; however, elec-

trical potentials generated by temperature and chemical

gradients are important in corrosion and in some

groundwater flow and stability problems.

Whether thermal diffusion of electrolytes, the Soret

effect, is important in soils has not been evaluated;

Co

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 253

276 9 CONDUCTION PHENOMENA

Table 9.6 Direct and Coupled Flow Phenomena

Gradient X

Flow J Hydraulic Head Temperature Electrical

Chemical

Concentration

Fluid Hydraulic

conduction

Darcy’s law

Thermoosmosis Electroosmosis Chemical

osmosis

Heat Isothermal heat

transfer or

thermal filtration

Thermal

conduction

Fourier’s law

Peltier effect Dufour effect

Current Streaming current Thermoelectricity

Seebeck or

Thompson effect

Electric

conduction

Ohm’s law

Diffusion and

membrane

potentials or

sedimentation

current

Ion Streaming current

ultrafiltration

(also known as

hyperfiltration)

Thermal diffusion

of electrolyte or

Soret effect

Electrophoresis Diffusion Fick’s

law

however, since chemical activity is highly temperature

dependent, it may be a significant process in some

systems. Finally, electrophoresis, the movement of

charged particles in an electrical field, has been used

for concentration of mine waste and high water content

clays.

The relative importance of chemically and electri-

cally driven components of total hydraulic flow is il-

lustrated in Fig. 9.20, based on data from tests on

kaolinite given by Olsen (1969, 1972). The theory for

description of coupled flows is given later. A practical

form of Eq. (9.57) for fluid flow under combined hy-

draulic, chemical, and electrical gradients is

H log(C /C )

EB Aq � �k A � k A � k Ah h c eL L L

(9.58)

in which kh, kc, and ke are the hydraulic, osmotic, and

electroosmotic conductivities,

H is the hydraulic head

difference,

E is the voltage difference, and CA and CB

are the salt concentrations on opposite sides of a clay

layer of thickness L.

In the absence of an electrical gradient, the ratio of

osmotic to hydraulic flows is

q k log(C /C )hc c B A� � (

E � 0) (9.59)� �q k

Hh h

and, in the absence of a chemical gradient, the ratio of

electroosmotic flows to hydraulic flows is

q k

Ehe e� (

C � 0) (9.59a)� �q k

Hh h

The ratio (kc /kh) in Fig. 9.20 indicates the hydraulic

head difference in centimeters of water required to give

a flow rate equal to the osmotic flow caused by a 10-

fold difference in salt concentration on opposite sides

of the layer. The ratio ke /kh gives the hydraulic head

difference required to balance that caused by a 1 V

difference in electrical potentials on opposite sides of

the layer. During consolidation, the hydraulic conduc-

tivity decreases dramatically. However, the ratios kc /kh

and ke /kh increase significantly, indicating that the rel-

ative importance of osmotic and electroosmotic flows

to the total flow increases. Although the data shown in

Fig. 9.20 are shown as a function of the consolidation

pressure, the changes in the values of kc /kh and ke /kh

are really a result of the decrease in void ratio that

accompanies the increase in pressure, as may be seen

in Fig. 9.20c.

These results for kaolinite provide a conservative es-

timate of the importance of osmotic and electroosmotic

flows because coupling effects in kaolinite are usually

smaller than in more active clays, such as montmoril-

lonite-based bentonites. In systems containing confined

clay layers acted on by chemical and/or electrical gra-

dients, Darcy’s law by itself may be an insufficient

basis for prediction of hydraulic flow rates, particularly

if the clay is highly plastic and at a very low void ratio.

Such conditions can be found in deeply buried clay

Co

py

rig

ht

ed

M

at

er

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l

Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 504

528 LIST OF SYMBOLS

� disturbance factor

� geometrical packing parameter

� rotation angle of yield envelope

�0, �i constant characteristic of the property and

the clay

� Bishop’s unsaturated effective stress pa-

rameter

� clay plate thickness measured between

centers of surface layer atoms

� deformation parameter in Hertz theory

� displacement, distance

� solid fraction of a contact area

� relative retardation

�p particle eccentricity distance

� dielectric constant, permittivity

� porosity

� strain

�̇ strain rate

�0 permittivity of vacuum, 8.85 �

10�12 C2/(Nm2)

�1 axial strain

�̇a vertical strain rate in one dimensional

consolidation

�ƒ strain at failure

�̇min minimum strain rate

�rd volumetric strain that would occur if

drainage were permitted

�s deviator strain

�̇s deviator strain rate

�v volumetric strain

�̇v volumetric strain rate

E energy dissipated per cycle per unit vol-

ume

� friction angle

� local electrical potential

�� friction angle in effective stress

�b angle defining the rate of increase in shear

strength with respect to soil suction

�c characteristic friction angle

��crit friction angle at critical state

�e, ��e Hvorslev friction parameter

��ƒ friction angle corrected for the work of

dilation

��m peak mobilized friction angle

��r residual friction angle

�repose angle of repose

�v apparent specific volume of the water in

a clay/water system of volume V

��, ��� intergrain sliding friction angle

# dissipation function

activity coefficient

angle between a and b crystallographic

axes

unit weight

̇ shear strain rate

c applied shear strain or cyclic shear strain

amplitude

d dry unit weight

% double layer charge

% specific volume intercept at unit pressure

� dynamic viscosity

� fraction of pore pressure that gives effec-

tive stress

�0 initial anisotropy

! swelling index

!� real relative permittivity

!� polarization loss, imaginary relative per-

mittivity

� compression index

� correction coefficient for frost depth pre-

diction equation

� damping ratio

� decay constant

� pore size distribution index

� separation distance between successive

positions in a structure

� wave length of X ray

� wave length of light

�cs critical state compression index

� chemical potential

� coefficient of friction

� dipole moment

� fusion parameter

� Poisson’s ratio

� viscosity

( critical state stress ratio

� Poisson’s ratio

�b Poisson’s ratio of soil skeleton

� osmotic or swelling pressure

� angle of bedding plane relative to the

maximum principal stress direction

� contact angle

� geometrical packing parameter

� liquid-to-solid contact angle

� orientation angle

� volumetric water content

�m volumetric water content at full saturation

�r residual water content

�s volumetric water content at full saturation

� bulk dry density

� charge density

� mass density

�d bulk dry density

�T resistivity of saturated soil

�w density of water

�W resistivity of soil water

� area occupied per absorbed molecule on

a surface

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

Page 505

LIST OF SYMBOLS 529

� double-layer charge

� electrical conductivity

� entropy production

� normal stress

� surface tension of water

� surface charge density

� total stress

�� effective stress

��0 initial effective confining pressure

�1 major principal total stress

�1 tensile strength of the interface bond

��1 major principal effective stress

�1c major principal stress during consolida-

tion

�1ƒ major principal stress at failure

��1ƒƒ major principal effective stress at failure

��2 intermediate principal effective stress

�3 minor principal total stress

��3 minor principal effective stress

�3c minor principal stress during consolida-

tion

��3ƒƒ minor principal effective stress at failure

��a axial effective stress

��ac axial consolidation stress

�as interfacial tension between air and solid

�aw interfacial tension between air and water

�c crushing strength of particles

�c tensile strength of cement

�e electrical conductivity

��e equivalent consolidation pressure

�eƒƒ effective AC conductivity

�ƒ partial stress increment for fluid phase

��ƒ effective normal stress on shear plane

�ƒƒ normal total stress on failure plane

��ƒƒ normal effective stress on failure plane

�h electrical conductivity due to hydraulic

flow

��h0 initial horizontal effective stress

��i effective stress in the i-direction

��i intergranular stress

��i isotropic consolidation

�iso isotropic total stress

�max maximum principal stress

�min minimum principal stress

��n effective normal stress

��p preconsolidation pressure

�r radial total stress

��r radial effective stress

��rc radial consolidation stress

�s conductivity of soil surface

�s partial stress increment for solid phase

�s tensile strength of the sphere

�T electrical conductivity of saturated soil

�T , ��t tensile strength of cemented soil

�v vertical stress

��v vertical effective stress

�v0 overburden vertical effective stress

��v0 overburden effective stress

��vm maximum past overburden effective stress

��vp vertical preconsolidation stress

�W electrical conductivity of pore water

�ws interfacial tension between water and

solid

�y yield strength

�� circumferential stress

� shear strength

� shear stress

� surface tension

� swelling pressure or matric suction

� undrained shear strength

�a apparent tortuosity factor

�c applied shear stress

�c contaminant film strength

�cy undrained cyclic shear stress

�d drained shear strength

�ƒƒ shear stress at failure on failure plane

�i shear strength

�i shear strength of contact

�m shear strength of solid material in yielded

zone

�peak applied shear stress at peak

�� initial static shear stress

' mass flow factor

' cation valence

� distance function � Kx, double-layer the-

ory

� ratio of average temperature gradient in

air filled pores to overall temperature gra-

dient

� dilation angle

� electrical potential

� intrinsic friction angle

� matric suction

�0 surface potential of double layer

�d displacement pressure

� electrical potential

� state parameter

� total potential of soil water

�0 electrical potential at the surface

�s gravitational potential

�m matrix or capillary potential

�p gas pressure potential

�s osmotic or solute potential

" angular velocity

" frequency

" osmotic efficiency

true electroosmotic flow

� zeta potential

Co

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com