523

List of Symbols

a areaa coefficient for harmonicsa cross-sectional area of a tubea crystallographic axis direction or distancea effective cluster contact areaa volumetric air contenta thermal diffusivityac effective area of interparticle contactam coefficient of compressibility with respect

to changes in water contentat coefficient of compressibility with respect

to changes in (� � ua)av coefficient of compressibility in one di-

mensional compressionA activityA areaA creep rate parameterA cross section area normal to the direction

of flowA Hamaker constantA long-range interparticle attractionsA Skempton’s pore pressure parameterA thermal diffusivityA van der Waal’s constantA� short-range attractive stressA pore pressure parameter � u /

(�1 � �3)A0 concentration of charges on pore wallA0 surface charge density per unit pore vol-

umeAc solid contact areaAƒ area of flow passagesAƒ pore pressure parameter at failureAh Hamaker constantAi state parameter in disturbed stateAi total surface area of the ith grain

0Ai state parameter at equilibrium

As specific surface area per unit weight ofsolids

A Angstrom unit � 1 � 10�10 mb coefficient of harmonicsb crystallographic axis direction or distanceb intermediate stress parameterB parameter in rate process equation �

X(kT /h)B Bishop’s pore water pressure coefficientBq grain breakage parameterBr Hardin’s relative breakage parameterc cohesionc cohesion intercept in total stressc concentrationc molar concentrationc crystallographic axis direction or distancec undrained shear strengthc velocity of lightc� cohesion intercept in effective stressc0 equilibrium solution concentration, bulk

solution concentrationc0

� cation equilibrium solution concentrationc0

� anion equilibrium solution concentrationca mid-plane anion concentrationce, c�e Hvorslev’s cohesion parametercec cation exchange capacitycic, cc mid-plane cation concentrationci 0 equilibrium solution concentrationcm mid-plane concentrationc�m mid-plane anion concentrationcu undrained shear strengthcv coefficient of consolidationcw concentration of waterC capacitanceC chemical concentrationC clay content by weightC composition

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524 LIST OF SYMBOLS

C electrical capacitanceC short-range repulsive force between con-

tacting particlesC soil compressibilityC speed of light in vacuum or in air, 3 �

108 m/secC volumetric heatC volumetric heat capacityCc compression indexC*c intrinsic compression indexCl compressibility of pore fluidCn coordination numberCR compression ratioCRR cyclic resistance ratioCs compressibility of a solidCs shape coefficientCs swelling indexCu coefficient of uniformityCu compressibility of soil skeleton by pore

pressure changeCW compressibility of waterC�, C�e coefficient of secondary compressiond diameterd distanced10 sieve size that 10% of the particles by

weight pass throughd60 sieve size that 60% of the particles by

weight pass throughdx incremental horizontal displacement at

peakdy incremental vertical displacement at peakD diameter of particleD dielectric constant, relative permittivityD diffusion coefficientD deviator stressD stress level � D /Dmax

D0 molecular diffusivity of water vapor in airD0 self-diffusion coefficientD50 sieve size that 50% of the particles by

weight pass throughDeƒƒ effective diameterDeV isothermal vapor diffusivityDmax strength at the beginning of creepDR, Dr relative densityDs characteristic grain sizeDTV thermal vapor diffusivityD* effective diffusion coefficiente electronic charge � 4.8029 � 10�10 esu

� 1.60206 � 10�10

coulombe void ratioe0 initial void ratioe*100 intrinsic void ratio under effective vertical

stress of 100 kPaec intracluster void ratio

ecs void ratio at critical stateeƒƒ void ratio at failureeg, eG void ratio of the granular phase, granular

void ratioeini initial void ratioeL void ratio at liquid limitemax maximum void ratioemin minimum void ratioep intercluster void ratioeT total void ratioE experimental activation energyE potential energyE Young’s modulusE voltage, electrical potentialE50 secant modulus at 50 percent of peak

strengthEmax small strain Young’s modulusEr rebound modulusESP exchangeable sodium percentageE(�) distribution function for interparticle con-

tact plane normalsƒ force acting on a flow unitƒ frequencyƒi fraction of particles between two sizesƒn normal forceƒt tangential forceF force of electrostatic attractionF formation factorF free energyF freezing indexF pressure-temperature parameterF tensile strengthF, F0 Faraday constant � 96,500 coulombsF partial molar free energy on adsorptionFd free energy of the double layer per unit

area at a plate spacing of 2d F free energy of activationFE electrical force per unit lengthFH hydraulic seepage force per unit length

causing flowFI fabric indexF� free energy of a single non-interacting

double layerg acceleration due to gravityG shear modulusG source-sinkG1000 shear modulus measured after 1000

minutes of constant confining pressureGg shear modulus of grainsGmax small strain shear modulusGs secant shear modulusGs specific gravity of soil solidsGSC specific gravity of clay particlesGSG specific gravity of the granular particles

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LIST OF SYMBOLS 525

h head or head lossh relative humidity of air in poresh Planck’s constant � 6.624 � 10�27 erg sechm matrix or capillary headhs osmotic or solute headH maximum distance to drainage boundaryH stress historyH thicknessH total headH water transport by ion hydrationH partial molar heat contenti gradienti unit vectoric chemical gradientie electrical gradientih hydraulic gradientit thermal gradientI electrical currentI intensityI1, I2, I3 stress invariantsIG coefficient of shear modulus increase with

timeIR dilatancy indexIv void indexJc chemical flow rateJD chemical flow rateJi flux of constituent iJi value of property i in clay-water systemJs flow rate of salt relative to fixed soil layerJv volume flow rate of solutionJw flow rate of water

0Ji value of property i in pure waterk Boltzmann’s constant � 1.38045 �

10�23 J/ �Kk hydraulic conductivity, hydraulic perme-

abilityk mean coordination number of a graink selectivity coefficientk thermal conductivityk true cohesion in a solidk0 pore shape factorkc osmotic conductivityke electro-osmotic conductivitykh hydraulic conductivityki constant characteristic of a propertykr relative permeabilityks saturated conductivityk(S) saturation dependent hydraulic conductiv-

itykt thermal conductivityk� unsaturated hydraulic conductivityK absolute permeability or intrinsic perme-

abilityK bulk modulus

K double-layer parameter �(8�n0e

2v2 /DRT)1 / 2

K pore shape factorK rate of increase in tip resistance in loga-

rithmic timeK0 coefficient of lateral earth pressure at restKa coefficient of active earth pressureKc principal stress ratioKc principal stress ratio during consolidationKd distribution coefficientKp coefficient of passive earth pressureKso stress-optical material constantK� wavelengths of monochromatic radiationl lengthl material thicknessl total number of pore classesL latent heat of fusionL lengthLij coupling coefficient or conductivity coef-

ficientLI liquidity indexLIeq equivalent liquidity indexLL liquid limitLs latent heat of fusion of waterm slope of relationship between log creep

strain rate and log timem total mass per unit total volumem total number of pore classesmc mass of clayms compressibility of mineral solids under

hydrostatic pressurem�s compressibility of mineral solids under

concentrated loadingsmv compressibilitymw compressibility of watermw mass of waterM constrained modulus or coefficient of vol-

ume changeM metal cationsM monovalent cation concentrationn concentration, ions per unit volumen harmonic numbern integern number of grains in an ideal breakage

planen porosityn total number of pore classesn unspecified atomic ration0 concentration in external solutionn1 number of bonds per unit of normal forcene effective porosityni Refractive index in i directionN Avogadro’s number � 6.0232 � 1023

mole�1

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526 LIST OF SYMBOLS

N coordination numberN monovalent cation concentrationN normal load or forceN number of moles of hydration water per

mole of ionN number of particles per cluster in a cluster

structureN number of weeks since disturbanceN total number of harmonicsN1 number of load cycles to cause liquefac-

tionNe number of load cyclesNG normalized shear modulus increase with

timeNs moles of water per unit volume of sedi-

mentNw moles of salt per unit volume of sedimentOCR overconsolidation ratiop constant that accounts for the interaction

of pores of various sizesp hydrostatic pressurep matrix or osmotic pressurep pressurep partial pressure of water vapor in pore

spacep vertical consolidation pressurep� mean effective pressurepo present overburden pressurepa atmospheric pressurepc preconsolidation pressurep�cs mean effective pressure at critical stateps osmotic or solute pressurepz gravitational pressureP areaP bond strength per contact zoneP concentration of divalent cationsP power consumptionP total gas pressure in pore spaceP total pressureP wetted perimeterPc capillary pressurePc capillary pressure at air entryPƒ injection pressure that causes clay to frac-

turePI plasticity indexPinj injection pressurePL plastic limitPN probability distribution of normal contact

forcePR peak ratioPs swelling pressurePT probability distribution of tangential con-

tact force

q degree of connectivity between water-conducting pores

q deviator stressq flow rateq hydraulic flow rateqc CPT tip resistanceqcs deviator stress at critical stateqƒ deviator stress at failureqh hydraulic flow rateqhc osmotic flow rateqhe electro-osmotic flow rateqi concentration of solidsqt heat flow rateqvap vapor flux densityqw water flow rateQ electrical chargeQ quantity of heatr pore radiusr radiusrk ratio of horizontal to vertical hydraulic

conductivitiesrp pore sizerp tube radiusR coefficient of roundnessR electrical resistanceR gas constant � 1.98726 cal/ �K-mole

8.31470 joules/�K-mole82.0597 cm3 atm/ �K-mole

R long-range repulsion pressureR ratio of cations and anionsR source or sink mass transfer termR sphere radiusR tube radiusRd retardation factorRH hydraulic radiusRp average particle radiusR(�) radius at angle �s slope of stress relaxation curvesu undrained shear strengthS entropyS fraction of molecules striking a surface

that stick to itS number of flow units per unit areaS partial molar entropyS saturationS specific surface area per unit volume of

solidsS structureS swellS partial molar entropyS0 specific surface per unit volume of soil

particlesSAR sodium adsorption ratio

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LIST OF SYMBOLS 527

St sensitivitySu undrained shear strengthSw water saturation ratioSx, Sy, Sz projected areas of interparticle contact

surfacest average thicknesst tetrahedral coordinationst timet transport numbert1 reference timetƒ time to failuretm time for adsorption of a monolayerT intercluster tortuosityT shear forceT temperatureT time factorT0 initial temperatureTc intracluster tortuosityTc temperature at consolidationTFP freezing temperatureTs surface temperatureTs temperature of shear for consolidated un-

drained direct shear testsTV time factoru excess pore pressureu ionic mobilityu midplane potential functionu pore water pressureu pore water pressure in the interparticle

zoneu pressureu thermal energyu* effective ionic mobilityu0 initial pore pressureu0 pore water pressure remote from the in-

terparticle zoneUƒ pore pressure at failureU average degree of consolidationv flow velocityv frequency of activationv ionic valancev settling velocityv specific volume � 1 � evave average flow velocityvc

0 specific volume of the pure clayvcs specific volume at critical statevh apparent water flow velocityV areaV difference in self-potentialsV electrical potentialV speedV valenceV voltage

V volumeV0 initial volumeVA attractive energyVDR volume of water drainedVGS volume of granular solidsVm total volume of soil massVp compression wave velocityVR repulsive energyVs shear wave velocityVs volume of solidsVw partial molar volume of waterVw volume of waterw water contentwL, wl liquid limitwP, wp plastic limitW water contentW widthW fluid volumeW water transportW weightx distance from the clay surfaceX distanceX friction coefficientXi driving forcey potential function � ve� /kTz direction of gravityz distance from drainage surfacez electrolytez ionic valenceZ elevation or elevation headZ number of molecules per second striking

a surfaceZ potential function � 'e�0 /kT� angle between b and c crystallographic

axes� directional parameter� disturbance factor� geometrical packing parameter� inclination of failure plane to horizontal

plane� slope of the relationship between loga-

rithm of creep rate and creep stress� thermal ratio� tortuosity factor�G normalized strain rate parameter�s thermal expansion coefficient of soil sol-

ids�ST thermal expansion coefficient of soil

structure�w thermal expansion coefficient of water� angle between a and c crystallographic

axes� birefringence ratio

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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 ratec applied shear strain or cyclic shear strain

amplituded 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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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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