2007 Tensile Strength Characteristics of Unsaturated Sands

Tensile Strength Characteristics of Unsaturated SandsNing Lu,M.ASCE1;Bailin Wu2;and Chee P.Tan3Abstract:Tensile strength characteristics of unsaturated sands are examined through a combined theoretical and experimental study.The characteristics of tensile strength in all three water retention regimes of pendular,funicular,and capillary are examined.A simple direct tensile strength apparatus is employed to determine tensile strength for sands with a broad range of particle sizes from silty sand tofine sand and medium sand over a full range of degree of saturation.Tensile strength characteristic curves͑TSCC͒are established experi-mentally for these sands and are used to validate the existing theories for tensile strength in the pendular regime.The TSCC for sand characteristically exhibits two zeros at0and near100%saturation,and two peak values occurring in the pendular and capillary regimes, respectively.A minimum tensile strength is observed in the densefine sand,indicating that either water bridges or pore pressure contributes exclusively to the tensile strength in the funicular regime of this sand.The maximum tensile strength for the silty sand is 1,448Pa,thefine sand is1,416Pa,and the medium sand parison between the soil–water characteristic curves obtained for these sands indicates that the peak tensile strength in the capillary regime is highly correlated to the air-entry pressure.Photographs of the failure surfaces clearly delineate distinct geometric characteristics for different water retention regimes.Analysis of the patterns of failure surfaces in different water retention regimes indicates that the effective stress principle is valid for tensile stress failure in unsaturated sands.DOI:10.1061/͑ASCE͒1090-0241͑2007͒133:2͑144͒CE Database subject headings:Tensile strain;Tensile strength;Sand;Measurement;Unsaturated soils.IntroductionIn granular media,tensile stress is a result of various interparticle physicochemical forces such as͑1͒van der Waals attraction,͑2͒electrical double layer repulsion or attraction,͑3͒cementation due to solute precipitation,͑4͒capillary stress due to the negative pore water pressure,and͑5͒capillary stress due to the surface tension of liquid͑e.g.,Lu and Likos2006͒.The macroscopic manifesta-tion of tensile stress is the cohesive behavior shown widely in many granular media.Thefirst four interparticle stresses can exist under either saturated or unsaturated condition,but are mostly pronounced infine granular media like clay͑e.g.,Verwey and Overbeek1948;Schubert1975;van Olphen1991͒.The last in-terparticle stress is of interfacial in nature and can only exist under unsaturated condition͑e.g.,Fisher1926;Dallavalle1943; Santamarina et al.2001;Lu and Likos2004͒.Since tensile stress is generally dependent on soil moisture content or degree of satu-ration,it is practically convenient to conceptualize the stress as a constitutive function of either degree of saturation,water content, or matric suction͑e.g.,Schubert1982;Heibrock et al.2003;Lu and Likos2004͒.Tensile stress could be considered as a part of the effective stress and thus bears an important role in stress and strain behav-ior of unsaturated granular media.For example,in processing raw kaolin material to manufacture various ceramic products,tensile stress plays a vital role in controlling the raw material’s viscosity and consistency͑e.g.,Israelachvili1992͒.In oil and gas explora-tion,tensile stress͑or capillary stress for sand͒is widely consid-ered to control mobilization of disaggregated materials in oil and gas wells͑e.g.,Urmenneta et al.1998;Wu and Tan2002͒.Sand production in wellbore is generally considered to have adverse effects on oil and gas production,and costs the petroleum indus-try tens of billions of dollars every year.In geotechnical engineer-ing,although the magnitude of tensile stress is usually less than several hundred kilopascals͑e.g.,Ingles1962;Lu and Likos 2004͒,it plays important roles in the magnitude of apparent co-hesion that greatly affects soil bearing capacity͑e.g.,Lu and Likos2004,pp.24–26͒,slope stability͑e.g.,Griffiths and Lu 2005͒,and failure of road bases and retailing walls͑e.g.,Lu and Likos2004,pp.294–301;Heath et al.2004͒.Since tensile stress is interparticle in nature,it generally de-pends on particle packing order or soil fabric.A granular assem-bly with isotropic packing order implies that the corresponding tensile stress is isotropic,i.e.,it is directional invariant.However, for an anisotropic particle arrangement,it follows that tensile stress too could be anisotropic͑Li2003;Lu and Likos2004͒. Furthermore,the tensile stress is usually counterbalanced by in-terparticle contact forces͑e.g.,Mitchell1976;Lu and Likos 2006͒.The practical implication of this is that it is difficult to measure tensile stress directly.Alternatively,a practical way to quantify tensile stress is to measure a material’s mechanical be-havior by conducting various strength tests and thereafter deduc-1Professor,Division of Engineering,Colorado School of Mines,1600 Illinois St.,Golden,CO80401-1887.E-mail:ninglu@2Principal Research Scientist,Division of Petroleum Resources, Commonwealth Scientific and Industrial Research Organization,Perth, WA6151,Australia.3Program Leader,Division of Petroleum Resources,Commonwealth Scientific and Industrial Research Organization,Perth,WA6151, Australia.Note.Discussion open until July1,2007.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must befiled with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on June7,2005;approved on August23,2006.This paper is part of the Journal of Geotechnical and Geoenvironmental Engineer-ing,V ol.133,No.2,February1,2007.©ASCE,ISSN1090-0241/2007/ 2-144–154/$25.00.ing tensile stress from the material’s strength.For example,a direct uniaxial tensile test can be performed on compacted clay, and its tensile strength can be considered as the tensile stress existing in the clay under known moisture content͑e.g.,Heibrock et al.2003͒.In measuring unsaturated soil behavior,it is common to conduct triaxial or a direct shear strength test to obtain the apparent cohesion and frictional angle͑e.g.,Vanapalli et al.1996; Khalili et al.2004͒.From the apparent cohesion and frictional angle,it is possible to infer the magnitude of tensile stress that exists in unsaturated soils͑e.g.,Kim2001͒.An intensive philosophical debate is going on as to whether or not to consider tensile strength as a material property or an indi-cator for a stress variable.The proponents for rationalizing tensile strength as a stress state variable argue that it is highly dependent on soil’s moisture content and it is consistent with the effective stress concept for unsaturated soils.Thus,it can unify all stress states under both saturated and unsaturated conditions within one theoretical framework͑e.g.,Bishop1959;Heath et al.2004;Kha-lili et al.2004;Lu and Likos2004,2006͒.The dissenting school considers it as a material property like cohesion and advocates matric suction as an independent stress state variable indicative of the state of stress in unsaturated soils͑e.g.,Coleman1962;Fred-lund and Morgenstern1977͒.However,the exact way to connect matric suction to physical stress existing in unsaturated soils has yet to be demonstrated and established.It is the writers’opinionthat matric suction generally bears little direct relationship witheither the magnitude or direction of the physical stress existing inunsaturated soils.An illustrative practical example supporting theabove statement is that beach sand under either0or greater than100kPa of matric suction exhibits little tensile stress and thus,noapparent cohesion.However,at10kPa of matric suction͑or5–10%of gravimetric water content͒,it usually exhibits tensilestress strong enough for people to build sand castles.This paperespouses using tensile strength as a direct stress indicator or vari-able in describing the effective stress in unsaturated soils.In this paper,the magnitude and characteristics of tensile stressdue to capillary forces are investigated both experimentally andtheoretically.Sands are used to exclude tensile stress mechanismsother than capillarity.Clean sands with a broad range of particlesizes ranging from silty sand tofine sand and medium sand aretested for their tensile strength characteristics over the full rangeof degree of saturation.Tensile Strength of Unsaturated SandsSoil–Water Retention RegimesThe mechanisms contributing to the tensile strength can be iden-tified by examining water retention mechanisms in unsaturatedsoils,as depicted in Fig. 1.Soil–water characteristic curve ͑SWCC,denoted as solid line͒reflects a soil’s constitutive func-tion between matric suction͑or often called“capillary pressure”͒and degree of saturation͑or water content͒.In general,SWCCcan be divided into three regimes according to the state of waterretention:pendular;funicular;and capillary͑e.g.,Orr et al.1975;Schubert1975͒.However,the physical processes or mechanisms contributingto tensile stress and consequently apparent cohesion or tensilestrength in each regime could be quite different.The physicalmechanisms attributing to the cohesive stress in unsaturated soilinclude:͑1͒van der Waals attraction at or near particle contacts;͑2͒electrical double layer repulsion between platy clay particles and attraction near face-to-edge contacts;͑3͒cementation due to solute precipitation;and͑4͒capillary attraction due to the exis-tence of water bridges or bodies between particles.Thefirst two physical mechanisms can operate under both saturated and unsat-urated conditions,and are most pronounced infine-grained soils such as clay,and the third mechanism could be important when soil water is of high solute concentration.At the present time,the quantitative understanding of the roles and magnitudes of each of thefirst three mechanisms in mechanical behavior of unsaturated soils is yet to be established.Advances have been made in the past several decades on the quantitative understanding of the capillary attraction mechanism in unsaturated granular media.Capillary attraction can exist in any unsaturated soil and its magnitude is generally inversely pro-portional to particle size and degree of saturation͑e.g.,Orr et al. 1975;Dobbs and Yeomans1982͒.For example,capillary stress could vary from several kilopascals in dry sand to be as high as several hundreds kilopascals in relatively dry clay͑e.g.,Ingles 1962͒.Most of the current theories on capillary attraction are based on interaction between two particles,thus restricting theo-retical predictions of tensile strength of real soils within the pen-dular regime͑e.g.,Orr et al.1975;Dobbs and Yeomans1982; Lian et al.1993;Molencamp and Nazemi2003͒.To date,only a few quantitative theories have been developed for tensile strength in the funicular and capillary regimes͑e.g.,Lechman et al.2006͒.A comprehensive picture of tensile strength within the entire unsaturated state is still largely obscure.This paper attempts to develop a comprehensive understanding of tensile strength char-acteristics within the entire saturation range by focusing on under one physical interparticle stress mechanism:capillary attraction. Sands partially saturated with distilled water are used to avoid other physical mechanisms contributing to tensile strength such as physicochemical forces and chemical cementation.Tensile Strength Characteristics of SandsWhile Schubert’s͑1975͒seminal conceptual model for tensile strength is generally applicable to all unsaturated granularmedia, Fig.1.Conceptual illustration of soil–water retention regimes and tensile strength characteristics in unsaturated sands.The boundaries between different regimes are estimated based on equal spheres with the simple cubic packing and generally depend on particle size distribution and porosity.several characteristics of sands need to be considered to establish an accurate and quantitative tensile strength characteristic curve ͑TSCC ͒for sands.These characteristics include the physical mechanisms responsible for tensile strength,the resulting magni-tude and shape of TSCC,and the physical basis for the occurrence of maximum and minimum tensile strengths.The tensile strength within the pendular regime can be as-sessed from existing theoretical works ͑e.g.,Fisher 1926;Dalla-valle 1943;Rumpf 1961;Pietsch and Rumpf 1967;Orr et al.1975;Schubert 1982;Lian et al.1993;Molencamp and Nazemi 2003͒.Recent studies ͑e.g.,Molencamp and Nazemi 2003;Lech-man and Lu,unpublished,2005͒show that most of the theories predict both water retention and capillary stress reasonably well.In this study a simple two-particle system shown in Fig.2͑a ͒is used.Tensile stress between two identical spherical particles due to a water bridge can be conveniently expressed using the theory developed by Rumpf ͑1961͒and Schubert ͑1982͒as follows:␴t =1−n n F t d 2=1−n n ␲T s d sin ␪ͫsin ͑␪+␣͒+sin ␪4ͩ1r 1*−1r 2*ͪͬ͑1a ͒r 1*=r 1d =͑1−cos ␪͒+a /d 2cos ͑␪+␣͒͑1b ͒r 2*=r 2d =sin ␪2+r 1d͓sin ͑␪+␣͒−1͔͑1c ͒where n ϭporosity;F t ϭinterparticle capillary force;d ϭparticlediameter;T s ϭsurface tension of water;␪ϭwater content fillingangle;␣ϭcontact angle;a ϭparticle separation distance;and r 1and r 2ϭradii of the water bridge shown in Fig.2͑a ͒.The water content filling angle ␪can be related to the gravimetric water content w and the specific gravity G s by ͑Pietsch and Rumpf 1967͒w =6k G s V bridge 2␲d 3=6k G s ͓ͭr 1*2+͑r 1*+r 2*͒2͔r 1*cos ͑␪+␣͒−r 1*3cos 3͑␪+␣͒3−r 1*2͑r 1*+r 2*͒ͫcos ͑␪+␣͒sin ͑␪+␣͒ͩ␲2−␪−␣ͪͬ−124͑2+cos ␪͒͑1−cos ␪͒2ͮ͑1d ͒where k ϭmean number of contact points per particle or the meancoordination number.Eqs.͑1a ͒–͑1d ͒completely define the tensile stress between two identical spherical particles as a function of the water content filling angle ␪͑or gravimetric water content w ͒with parameters d ,a ,and ␣as intrinsic soil properties.The upper limit for the applicability of the two-particle system to multipar-ticle system is constrained within the pendular regime.For even-sized particles with the simple cubic ͑SC ͒packing ͑k =6͒,the porosity is 47.6%and the water content filling angle is limited to 45°if the contact angle ␣is 0,whereas with the tetrahedral ͑TH ͒packing ͑k =8͒,the porosity is 26.0%and the water content filling angle is limited to 30°if the contact angle ␣is 0.Assuming a specific gravity of sands is 2.65,Lu and Likos ͑2004,p.121͒showed that the upper limit for the gravimetric water content for SC packing is 0.063g/g,and the upper limit of the gravimetric water content for TH packing is 0.032g/g.As shown in Eqs.͑1a ͒–͑1d ͒,tensile stress is inversely propor-tional to the diameter of particle d .The dependency of tensile stress on particle size d ,contact angle ␣,and the water content w is illustrated in Fig.2͑b ͒.As shown,tensile stress in different sand size particles can vary from tens of pascals for coarse sand size to several kilopascals for fine sand size.In silts,tensile stress is typically of several tens of kilopascals.The theory expressed in Eqs.͑1a ͒–͑1d ͒also predicts that the contact angle is an important soil parameter for tensile stress.A large contact angle typically occurs in soils during wetting and has an adverse effect on the magnitude of tensile stress.As illustrated in Fig.2͑b ͒,tensile stress can decrease several times when a contact angle varies from zero ͑typical during draining process ͒to 50°͑typical in sands during wetting process ͒.A few theories have been constructed for predicting tensile strength within the capillary regime.Perhaps the most widely known theory is that of Schubert ͑1975,1984͒who theorized the tensile strength within the capillary regime for agglomerates to be␴a =SP c =Sa Ј1−n n T s d͑2a ͒where S ϭdegree of saturation;P c ϭ͑1−n ͒T s /͑nd ͒;and a Јϭconstant.For granular materials with a narrow particle size distribution,a Ј=6–8,and for granular materials with wide par-ticle size distribution,a Ј=1.9–14.5.An alternative equation is suggested by Kim ͑2001͒to account for varying nature of capil-lary pressure in both the capillary and funicularregimesFig. 2.Unsaturated sands consisting of even-sized spherical particles:͑a ͒in a two-particle system;͑b ͒theoretical tensile strength in pendular regime as a function of particle size and gravimetric water content␴a=S͑u a−u w͒͑2b͒A preliminary study by Kim͑2001͒indicates that the above equa-tion can predict the general rising-and-falling trend within the funicular and capillary regimes,but its validity remains to be tested experimentally.Tensile strength within the funicular regime can be considered as the sum of tensile stress provided by water bridges and nega-tive pore-water pressure provided in saturated pores͑Schubert 1975͒␴f=␴t+␴a͑3͒The above equation generally predicts that the tensile strength within the funicular regime follows a linear trend between the two peak values of the tensile strength in the pendular and capillary regimes.Schubert also suggested that if either the bridge model or the capillary model is exclusively involved,the superimposed lin-ear trend might not be the case and a minimum tensile strength would occur within the funicular regime.Schubert further stated that only the superimposed model͓Eq.͑3͔͒had been shown in the previous experimental data͑e.g.,Schubert1975͒.This later state-ment will be reexamined for its validity in unsaturated sands. Tensile Strength MeasurementDirect Tensile Strength ApparatusAlthough direct measurement of tensile strength offine-granular materials has been a subject of study for many years͑e.g.,Rumpf 1961;Schubert1975;Heibrock et al.2003͒,few methods are directly applicable to sands.For example,the adhesive method that is widely used in material science requires the test materials to be sufficiently strong to be bonded to the sample adapters.The Brazilian tensile splitting method widely used for rocks requires the materials to be strong and brittle.Other direct stretching meth-ods suitable for clays cannot be used for sands since these meth-ods are conducted with no confining stress.A recent development by Sture and Kim͑2002͒used a split acrylic box with an open top to directly measure tensile strength of moist sands.Although the method is capable of accurately measuring tensile strength of sands,it requires a relatively large amount of sand tofill up the test box causing challenges in obtaining multiple moist samples with a consistent porosity.A direct tensile strength apparatus recently was developed toaccurately measure the tensile strength characteristics of unsatur-ated sands͑Lu et al.2005͒.The method is simple in principle, accurate in measurement,and reliable in obtaining repeatable ten-sile strength.The basic idea is to use the weight generated by tilting a sample to split it.The apparatus,as shown in Fig.3, consists of an adjustable table for sample tilting,a digital probe for tilting angle measurement,a detachable sample tubing,and a sample tubing mounting plate.An unsaturated sand sample isfirst compacted or cored into the sample tubing mounted on the sample mounting plate to a desired porosity and water content using a vibration table.The sample tubing is then placed horizon-tally on a set of spherical ball bearings that minimize friction between the sample tubing and the table.The sample tubing comprises two detachable sections that are clamped together during sample preparation and released prior to testing.One section of the sample tubing shown as the left half in Figs.3͑b and c͒isfixed on to the table while the other section is free to slide.The test is then followed by progressive inclination of the sample to increase the pull force along the longitudinaldirection of the sample by tilting the adjustable table.The incli-nation angle is recorded during the test and thefinal reading in-dicates the angle when the sample breaks into two halves.Themeasured inclination angle␤,together with the sample and tubingweight͑the weight of the right half W͒and geometrical configu-ration͑sample cross section area A͒,provides a simple and accu-rate determination of the tensile strength of the sand,i.e.,␴=W sin␤/A.The test can be repeated for the same porosity and moist content for duplication or be conducted for other sets ofporosity and/or moisture content to obtain complete tensilestrength characteristics of the sand.Soil Testing ProgramSands commercially available from Western Australia͑WhiteSilica Mineral Sands,Cook Industrial Minerals Pty Ltd.͒are usedto cover a wide range of sand sizes.The sands arefirst sievedinto Fig.3.Photographic illustration of a simple direct tensile strength apparatus:͑a͒without sample tubing and mounting plate;͑b͒with sample tubing and mounting plate;and͑c͒connected to adjustable rate dc motor for sample tiltingthree distinct groups:silty sand;fine sand;and medium sand.Particles with sizes greater than 1mm ͑or 1,000␮m ͒are dis-garded since tensile stress in coarse sands is expected to be prac-tically insignificant,commonly less than several pascals.The standard laser particle size analysis is then conducted on these sands to give a detailed particle size distribution,as shown in Fig.4.Microphotograph analysis shows that these sands are well rounded with typical quartzitic surface features and aspect ratios less than 2.0.For the silty sand,the tensile strength is expected to be near the upper limit of sandy soils.This sand has a volumetric mean particle size of 105␮m,with most of the particle sizes falling between 50and 200␮m and trace amount of particles below 20␮m.The fine sand has a volumetric mean particle size of 167␮m and all the particles sizes fall between 80and 400␮m.The medium sand has a volumetric mean particle size of 451mm and the widest size range of all the sands,with particle sizes falling between 200and 900␮m.The choice of the sand types is intended to give a clear range of the magnitude of tensile strength and its characteristics for theoretical comparison,verification,and analysis.To enhance repeatability,the same sample preparation proce-dure was followed closely.For practical reasons,only the wetting state is imposed on all samples.For each tensile strength mea-surement,approximately 1to 1.5kg of oven-dried sand was first hand mixed with a required amount of water.The mixture was then covered with plastic wrapping film in a tray for overnight curing.The mass of the moist sand required to compact into a desired porosity in the sample tubing was back-calculated and weighted.All the test samples are 56mm in diameter and 100mm in length.Three identical samples with the same mois-ture content and porosity were prepared for duplication.The two halves of the sample tubing are screwed together by the bolts attached to the sample mounting plate.Each sample was com-pacted statically into the sample tubing and then placed on the standard soil vibrator table to dynamically compact into the de-sired porosity.During the dynamic compaction,if necessary a weight is placed on the top of the sample.For tensile strength measurement,a consistent and repeatable testing procedure was adopted.The sample tubing was first placed horizontally on the adjustable table ͓Fig.3͑c ͔͒.The mounting screws securing the two halves of the sample tubing together are then released.A dc motor connected to the left end of the adjust-able table via a cable provides a sufficiently slow rate of sample tilting.The tilting was stopped when a sudden tensile failure was observed and the inclination angle shown on the digital angle probe corresponds to the failure angle.Prior to removing the right half of the sample tubing for weight measurement,the adjustable table was reclined back to the horizontal position and several runs are conducted to measure the offset inclination angle for friction correction.The inclination at tensile failure,the weight of the right half of the sample tubing and sample,the dimensions of the sample tubing,and the offset inclination angle provide necessary information for tensile strength calculation.Typically,three ten-sile tests were conducted for the same sand with the same poros-ity and moisture content in order to obtain consistent tensile strength within 10%of their mean values.A total of 159tensile tests were conducted for the three sands:silty sand ͑10saturations ϫ3tests/sample ϭ30tests ͒;fine sand ͑63tests ͒;and medium sand ͑66tests ͒with various saturations and porosities.The tensile strength characteristic curve ͑TSCC ͒for the silty sand was obtained for porosity equal to 45%,which reflects loose packing close to the simple cubic packing.For the fine sand,TSCCs for two porosity values are obtained;one rep-resenting a medium packing with a porosity of 37%,and the other representing a loose packing with a porosity of 45%.For the medium sand,two porosity values close to each other ͑37and 40%͒are used for the tensile strength measurement.The testing program for the 159tests is shown in Table 1.Experimental Results and AnalysisExperimental tensile strength results for the silty sand are shown in Fig.5͑a ͒as a function of degree of saturation.The tensile strength in the pendular regime varies from 0at oven dry to a maximum value of 1,448Pa at 20%saturation.An almost linear decreasing trend in tensile strength is observed after itspeakFig.4.Particle size distribution for three sands:͑a ͒silty sand;͑b ͒fine sand;and ͑c ͒medium sandvalue and within the funicular regime.At90%saturation,thesand is within the capillary regime and tensile strength of930Pais obtained.Tensile strength at saturation higher than90%cannotbe measured for this sand since it exhibits high liquidity,and thesample preparation procedure adopted in this study cannot beused to compact the sand into the desired porosity.Compared to the maximum value of1,448Pa,the tensilestrength of930Pa at90%saturation reflects a36%reduction inthe strength.According to the superimposed theory proposed bySchubert͓Eq.͑3͔͒,the tensile strength within the funicular regimeshould follow a linear trend.This is the case for this sand exceptthat tensile strength in the capillary regime is lower than themaximum strength in the pendular regime.This could have re-sulted from a relatively high porosity and thus a low air-entrypressure.Surface tension through water bridges and negative porepressure are concurrently present in the funicular regime in thissand.A similar linear declining trend in loose sands was alsoreported by others͑e.g.,Kim2001;Sture and Kim2002͒.The tensile strength characteristic curve for this sand is plottedas a function of gravimetric water content as shown in Fig.6͑a͒.It can be observed that within the pendular regime,the tensilestrength increases quickly with increasing water content fromzero to about1,000Pa as water content increases from0to4%.This is consistent with the fact that the pore size distribution forthis sand is relatively uniform.Experimental tensile strength results for thefine sand are plot-ted in Figs.5͑b͒and6͑b͒,respectively,as a function of saturationand gravimetric water content.As expected,the maximum tensilestrength value within the pendular regime for this sand should besmaller than that of the silty sand because the mean particle sizeof this sand͑167␮m͒is larger than that of the silty sand ͑56␮m͒.The maximum tensile strength for this sand with a po-rosity of37%is observed to be667Pa whereas the maximumtensile strength with porosity of45%is339Pa.Both maximumstrengths are measured at the saturation value about20%.Withinthe pendular regime,the tensile strength for the lower porosity ͑37%͒is generally observed to be higher than that with higher porosity͑45%͒.This observation is more clearly shown in Fig. 6͑b͒.This is also anticipated since the same sand with a lower porosity would have more contacts with other particles,or a higher coordination number,leading to higher tensile stress.It is observed that within the capillary regime for thefine sand the maximum tensile strength for porosity of37%appears to be higher than that for porosity of45%.The physical explanation for this phenomenon is that lower porosity leads to smaller pore size,higher capillary pressure,thus higher tensile strength ͓␴t=S͑u a−u w͔͒at the same degree of saturation.Perhaps the most striking shifting in tensile strength trend is within the funicular regime.For porosity of37%,the tensile strength is observed to decrease and then increase as the satura-tion or water content increases.A minimum tensile strength of 331Pa is observed at a saturation of60%or water content of 13.5%,a reduction of50%from its maximum strength within the pendular regime.The tensile strength increases for the saturation greater than60%and reaches about1,000Pa at80%saturation, the boundary between the funicular and capillary regimes.In con-trast,for porosity equal to45%,the tensile strengthfirst follows a flat trend and then begins to increase rapidly when the saturation is greater than60%.Experimental tensile strength results for the medium sand are plotted in Figs.5͑c͒and6͑c͒,respectively,as a function of satu-ration and gravimetric water content.It is obvious that the tensile strength characteristic curve for the medium sand follows the classical model developed by Schubert͑1975͒.Within the pen-dular regime,the tensile strength increases as the sand becomes wetter and reaches its maximum value of about450Pa for both porosities at30%saturation.Tensile strength increases nearly lin-early over the entire funicular regime and reaches its peak value within the capillary regime.For porosity of37%,the peak tensile strength value of810Pa is obtained at93%saturation,and for porosity equal to40%,the peak value increases slightly to890Pa at90%saturation.Overall,the difference between the two TSCCs is small since the difference in porosity is small͑3%͒.It can be speculated from the above observations that sands with a uniform particle size distribution such as thefine sand might not follow the classical trend in tensile strength character-istic curve proposed by Schubert͑1975͒.The minimum tensile strength that occurs within the funicular regime for thefine sand may be a manifestation of the small and uniform͑narrow͒particle size distribution and uniform pore size distribution.The uniformTable1.Testing Program for Tensile Strength of Unsaturated SandsSilty sand͑G s=2.65,d=50–200␮m͒Fine sand͑G s=2.65,d=80–400␮m͒Medium sand͑G s=2.65,d=200–900␮m͒Porosity Porosity Porosity0.450.450.370.400.37S a w a S w S w S w S w ͑%͒͑%͒͑%͒͑%͒͑%͒͑%͒͑%͒͑%͒͑%͒͑%͒00.0000.0000.0000.0000.00 5 1.545 1.549 1.995 1.265 1.02 10 3.0910 3.0915 3.3210 2.529 2.05 15 4.6315 4.6320 4.4315 3.7714 3.07 20 6.1720 6.1730 6.6520 5.0319 4.10 309.26309.20459.97307.5528 6.14 4513.894513.896013.304511.32429.21 6018.526018.527516.626015.095612.29 7523.167523.169521.057518.877015.36 9027.799027.7910022.169022.648418.4310030.8710025.169320.48 a Sϭsaturation;wϭgravimetric water content.。