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Okada, N. & Nemat-Nasser, S. (1994). GPotechnique 44, No. 1, 1-19

Energy dissipation in inelastic flow of saturated cohesionlessgranular media

N. OKADA* and S. NEMAT-NASSER*

The results of a study of energy dissipation incohesionless granular media are presented. Therelation between the excess pore water pressure,accumulated in a water-saturated granular mass,and the corresponding external work in undrainedcyclic loading is studied experimentally, underdisplacement-controlled conditions. A micro-mechanical model of internal energy dissipationdue to slip between contacting granules is intro-duced, and the results are compared with experi-mental measurements. The specimens are subjectedto two sequences of loading with an intermediatereconsolidation to simulate reliquefaction. Externalwork per unit volume is calculated from the experi-mental results, and its correlation with the excesspore water pressure is examined. In the firstloading, a unique non-linear relation exists betweenthe excess pore water pressure and the externalwork per unit volume which is independent of theshear strain amplitude. In the second loading thisrelation is a function of strain amplitude. Based ona micromechanical model, it is shown that theinternal dissipation per unit volume in cohesionlessgranular media can be expressed in terms of thetime history of the applied effective pressure and asingle scalar parameter which depends on thedensity and strain amplitude. The model is furthervalidated by torsion tests with random variation inthe applied strain amplitude, and excellent agree-ment with the experimental results is obtained.

KEYWORDS: friction; laboratory tests; liquefaction;pore pressures;sands; torsion.

INTRODUCTIONLiquefaction is a complex phenomenon in whichfluid-saturated granular media may momentarilybehave like fluids. It is an important aspect of

Manuscript received 14 January 1992; revised manu-script accepted 11 March 1993.Discussion on this Paper closes 1 July 1994; for furtherdetails see p. ii.* University of California, San Diego.

Larticle prCsente les rbultats obtenus au toursdune Ctude sur la dissipation de Knergie dans desmilieux granulaires pulvCrulents. La relation exis-tant entre Iexc&s de pression interstitielle deau,accumul& dans une masse granulaire saturi?e, et letravail extkrieur asscn+ apparaissant au tours decycles de chargement non-drain&s, h d&placementcontrSlt, est Ctudiee expi?rimentalement. Unemod%sation microm&anique de la dissipationdbnergie interne IiCe au glissement entre grainsjointifs, est proposCe. Les rCsultats obtenus i Iaidede ce modkle sont cornpark aux mesures exp&i-mentales. Les izchantillons sont soumis g deuxcycles de chargement, s&par& par une reconsolida-tion permettant de simuler la reliqkfaction. Letravail extCrieur, par uniti: de volume, est calculb gpartir des ritsultats exp&rimentaux. La corr6lationpouvant exister avec lexc& de pression inter-stitielle est i?tudi&e.Lors du premier chargement, ilnexiste quune seule relation entre Iexctis de press-ion interstitielle et le travail extbrieur, cette rela-tion (?tant indbpendante de Iamplitude de lad&formation au cisaillement. Lors du secondchargement, cette relation devient dbpendante delamplitude de cette dtformation. I1 apparait, ipartir du micromod6le, que la dissipation interne,par unit& de volume dun milieu granulaire p&&u-lent, peut sexprimer en fonction de Ihistorique dela pression effective appliqute et dun paramktrescalaire unique dependant de la densiti! et deIamplitude de la dbformation. Le modhle estensuite valid& P Iaide dessais de torsion, pour desvariations alCatoires de Iamplitude de deforma-tion. La correspondance avec les rCsultats expicri-mentaux est excellente.

earthquake-resistant foundation design of manystructures, especially those located in coastalareas, which are often built on sand with highunderground water levels. As liquefaction takesplace under seismic loading, saturated sandbehaves more like a fluid, and therefore fails tosupport the applied loads of the building. Severedamage to the structure is often the result.Damage resulting from liquefaction has beenobserved in the aftermath of many earthquakes,including the Loma Prieta earthquake (1989), the

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2 OKADA AND NEMAT-NASSERNiigata earthquake (1964) and the Alaska earth-quake (1964).

The mechanism of liquefaction is closelyrelated to the dilatancy of granular media. Dilat-ancy was first studied by Reynolds (1885). It isdefined as the rate of volume expansion in granu-lar media per unit rate of shearing. The granulesare rearranged during shear deformation, and thisresults in a change in the total volume. If the gra-nular medium (e.g. sand) is water-saturated andundrained, the tendency towards densification(contractancy) results in an increase in the porewater pressure and hence a decrease in the corre-sponding frictional resistance of the contactinggranules. In continued cyclic shearing, the porewater pressure at the termination of each cycleincreases until it reaches a value close to theapplied hydrostatic pressure. At this stage, thecontact resistance of the granules can be regardedas essentially negligible. This process leads to aloss of the load-bearing capacity of the sand-mass, which hence ceases to behave like a solidbody. This is what is meant by liquefaction in thisPaper; see Casagrande (1975) and Seed (1979).Liquefaction has been extensively treatedexperimentally (see, for example, Silver & Seed,1971; Castro, 1975; Ishihara & Yasuda, 1975). Insuch experiments, parameters influencing theonset of liquefaction of the sand within a con-trolled volume are identified and measured. Theseparameters typically include overall density,initial packing conditions and granule size dis-tribution (Seed, 1979; Miura & Toki, 1982; Tat-suoka, Muramatsu & Sasaki, 1982). In additionto these internal characteristics, the appliedloading affects the onset and nature of liquefac-tion (Ishihara & Towhata, 1983, 1985; Symes,Gens & Hight, 1984).Previous work has in general studied liquefac-tion experimentally. Theoretical work has focusedon phenomenological considerations rather thana micromechanical approach. A unified energymodel for densification and liquefaction of cohe-sionless sand was proposed by Nemat-Nasser &Shokooh (1979), who compared its predictionswith the experimental results of Peacock & Seed(1968), Youd (1970, 1972) and DeAlba, Seed &Chan (1976), and obtained excellent agreement.The present work correlates the results of anexperimental programme with a theoreticalmodel based on micromechanics and energy prin-ciples. Models of this kind seek to relate theoverall response of granular materials to theresponse of their microconstituents. Micro-mechanical analyses of densification and liquefac-tion phenomena, which are also co-ordinatedwith experiments, are given by Nemat-Nasser(1980), Nemat-Nasser 8~ Tobita (1982) andNemat-Nasser & Takahashi (1984).

EXPERIMENTAL SET-UPMot ivat ion and background

Cohesionless granular materials supportgeneral external loads through contact friction.An experimental programme must include com-pression and shearing of reproducible samples ina controlled manner with reliable data. This mayrequire complex experimental facilities, with aclosed-loop feedback system to control theexperiment and to monitor the specimen defor-mation. The specimen geometry used for thepresent investigation is a large hollow cylinder,25 cm high, with inner and outer diameters of20 cm and 25 cm respectively. This geometry issuch that in torsion, the shear stress remains(approximately) homogeneous throughout thethickness of the specimen; see Hight, Gens &Symes (1983) for a detailed examination of thisand related issues. The specimen is supported bya triaxial load frame (Fig. 1). The axial and tor-sional deformations are controlled through anMTS servohydraulic loading system. In addition,the specimen is subjected to lateral hydrostaticpressure on both its inside and outside cylindricalsurfaces. In this manner, triaxial states of stresscan be imposed on the material under controlledconditions with complete data acquisition capa-bility. This load frame, to the Authors know-ledge, is one of four that have been constructed todate, and is fully computer-controlled. Either thestress path or the strain path can be pre-programmed with automatic mode switchingcapability.

Specimen preparatio n and inst allat ionThe granular material chosen for this study isSilica 60, manufactured by U.S. Silica. This sandis chosen for its fine particle size, which is neces-sary in order to minimize membrane penetrationphenomena that would otherwise invalidate thetest results. The particle size distribution is shownin Fig. 2. The mean particle diameter is 220 urnand the specific gravity of the sand is 2.645.Depending on the packing conditions, variousvoid ratios are obtained. For Silica 60, theminimum and maximum void ratios are 0.631and 1.095 respectively, and are measured by theJSSMFE method (Committee of JSSMFE on theTest Method of Relative Density of Sand, 1979).Special fixtures are used to prepare hollowcylindrical sand specimens. These fixtures includeinner and outer moulds to which rubber mem-branes are attached. The sand is initially sup-ported on the bottom by a ring of porous metalwith six evenly spaced fins, called the pedestal,which in combination with a mating top ring,called the cap, applies the torsional load to thespecimen. The pedestal (with associated fins) is

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INELASTIC FLOW OF GRANULAR MEDIAAdjustment bar

Top plateMetal bands

Plexiglas chamber

Torque load cellTie bars

SpecimenRam of trlaxlal apparatus

BearingsPorous metalBottom plate

Electrical dial gaugePotentiometer

CapSupporltng barsCounter balanceInner membraneOuter membranePedestal

Clamp

3

Vertical clamps

Fixed counter frame

Fig. 1. Triaxial load frame

attached to the bottom support plate. The outermembrane is then slid over the inner membraneand fixed to the pedestal with o-rings. The outermould is bolted in place, and the top of the outermembrane is draped over the outer mould andheld in place by o-rings. A separate fixture isinstalled on top of the outer mould to preventsand spillage on the rest of the triaxial loadframe. This fixture also allows for an overfillamount of sand so that a desired packing condi-tion can be obtained. The excess sand is removedlater.It is well known that the initial packing condi-tion of the sand has a noticeable effect on thematerial response of the specimen (Arthur &Menzies, 1972; Oda, 1972a; Miura & Toki, 1982).The specimen preparation method must thereforeachieve a consistent initial packing condition sothat experiments are repeatable. To this end, a

.Horizontal

-Wheels

160Sieve sizes: urn

Fig. 2. Particle size distribution curve

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4 OKADA AND NEMAT-NASSERtechnique has been adopted that is known as therodding method. This method consists of pouringan approximately 2 cm deep layer of sand intothe mould and then inserting a rod approx-imately l-l.5 cm into the latest layer. The rod ismoved around the circumference of the sand inan up-and-down motion for 2-3 revolutions. Thisprocedure is continued until the mould is filled.Ten layers were used to obtain loose packingconditions, and 14 layers for dense packing con-ditions. Experiments performed under loose con-ditions use the sand in a wet form, where the sandhas been air dried and then mixed with 8 wt%water before pouring into the mould. The water isneeded to prevent non-homogeneous initialpacking conditions in the loose form. The voidratio for this condition varies between 0.865 and0.874. Experiments performed with dense condi-tions use only air-dried sand with a void ratiovarying between 0.708 and 0.725.The fixture that was attached to the top of theouter mould is removed and the amount of over-filled sand is cut away. The cap is then installed;it consists of the same porous metal as the ped-estal and also has six fins. A second vacuumsystem is connected to the cap and pedestal. Thepurpose of this vacuum system is to make thespecimen rigid under atmospheric pressure.The vacuum level is maintained at 29.4 kN/m2.The first vacuum system that keeps the outermembrane fixed to the outer mould is then rel-eased. The outer mould is removed, followed bythe inner mould. A torque load cell unit is firstbolted onto the ram of the triaxial load frame(Fig. l), and then bolted onto the cap. Next, apotentiometer is attached to the load frame (Fig.1). The potentiometer measures the twist angleduring the experiment. A Plexiglas chamber withsteel bands is installed over the entire specimen,and a top plate is installed. The top plate isaffixed to the bottom plate by stainless steel tiebars, which hold the chamber firmly in place. Thechamber has three purposes: it provides confine-ment of the experiment if the sand mould losesintegrity, it holds the water that is used to applyhydrostatic pressure to the specimen, and it isused as a viewport to observe the progress of theexperiment.The specimen assembly is now complete. Theassembly is raised to the level of the MTS loadframe by a forklift. A special work frame has beenbuilt onto the MTS load frame: this allowsattachment of all connections to the specimenassembly, and provides a railway for installationand removal of the specimen into and from theMTS load frame.The MTS load frame used for this experimenthas an axial capability of 89 kN and a torsionalcapability of 565 Nm, which can be used inde-

pendently. The system uses a Digital EquipmentCorporation PDP-11 computer to control theservohydraulic actuators. The system is closed-loop, so that feedback from any selected trans-ducer can be used to control the test.

Once the triaxial load frame has been rolledinto place over the ram of the MTS load frame, itis secured in place by both vertical and horizontalclamps (Fig. 1). The hydraulics for the MTSsystem are turned on, and the MTS ram is raisedto the level of the universal joint by use of dis-placement control. An air clamp that is fixed tothe top of the MTS ram is then actuated andgrips the universal joint on the bottom of the ramof the triaxial load frame. The universal joint isrequired to accommodate any misalignmentbetween the ram of the MTS load frame and theram of the triaxial load frame.The first step in the experimental procedure isto fill the Plexiglas chamber with water until thespecimen is completely submerged. The remain-ing space above the specimen is pressurized withair to 29.4 kN/m, the same value as the vacuuminside the specimen. During this operation thevacuum in the specimen is released and waterpressurized in such a manner as to keep the effec-tive pressure in the specimen constant at 29.4kN/m2.The specimen is then water saturated asfollows. To attain full saturation, the specimen isfirst saturated with CO, gas through the porousmetal in the pedestal and cap. The flow of gas iscontinued until all air is removed from the speci-men. CO, gas is used because of its high solu-bility in water. A fixed amount (41) of de-airedwater is used to saturate the specimen. As muchas possible of the small amount of air and CO,gas remaining in the specimen must then beremoved. The pore water pressure is increased to196 kN/m as back pressure, using a burettesystem, while at the same time the externalhydrostatic pressure is increased to 225.4 kN/m2,in order to keep the effective pressure constant(29.4 kN/m2) during this procedure. The volumeof the excess gas in the specimen is hence reduceddue to the relatively high pore water pressure.For experiments of this type, the specimen isrequired to be highly saturated. The degree ofsaturation is measured by the B value. For the Bvalue to be measured, the specimen must be inthe undrained condition. This condition is met byclosing the valve to the burette, ensuring that thespecimen remains at a fixed volume. The speci-men is said to be perfectly saturated (B = 1) if anincrease in the external hydrostatic pressure hasthe effect of increasing the pore water pressure inthe specimen by an identical amount. The B valueis defined as the ratio of the incremental increaseof pore water pressure to the incremental increase

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INELASTIC FLOW OFof hydrostatic pressure. The value for all thepresent experiments is higher than 0.99.The last step of specimen preparation is toincrease the effective pressure to 196 kN/m* byreopening the valve to the burette, allowing waterto drain from the specimen. The external hydro-static pressure is thereby increased to 392 kN/m,where pore water pressure is 196 kN/m. Finally,the specimen is left undisturbed in this conditionto consolidate isotropically for a period of 3 h.

Experimental procedure and data acquisitionThe MTS load frame has a computer-operatedcontroller system. The computer operates threeindependent controllers. Each controller has threeindependent feedback channels. Controller 1 isassociated with the vertical movement of theMTS/triaxial load frame ram assembly; channel 1is used to monitor the load from the torque loadcell and channel 2 is used to monitor the verticaldisplacement of the specimen. Channel 3 is notused with any controller. Controller 2 is associ-ated with the pressure; channel 1 is used tomonitor the chamber pressure P, and channel 2 isused to monitor the pore water pressure Pi. Con-troller 3 is associated with the twist of the ramassembly; channel 1 monitors the torque from thetorque load cell and channel 2 monitors the angleof twist from the potentiometer.The experiment is conducted by use of twoclosed-loop feedback systems. The first feedbacksystem uses channel 1 of controller 1 in loadcontrol to keep the specimen in a state of hydro-static compression in accordance with the exter-nal pressure P,. The second feedback system useschannel 2 of controller 3 in displacement controlto twist the specimen cyclically to desired shearstrain amplitudes at desired shear strain rates.

GRANULAR MEDIA 5The imposed cyclic angular displacement thatproduces the applied shear strain has a triangulartime variation with constant strain rate, f%/minover each quarter cycle. Shear strain amplitudesare 0.2%, 0.5% and 1.0% for both loose anddense specimens. Tests at 0.4% and 2.0% shearstrain amplitudes are performed on dense speci-mens. All tests are conducted in an undrainedcondition, and continue until the excess porewater pressure reaches 95% of the initial effectivepressure, i.e. 186.2 kN/m. The tests are actuallystopped at the end of the cycle after which trans-channel 2 of controller 2 (pore pressuretransducer) reaches a value of 382.2 kN/m*. Thisentire process is defined as the first loading.The valve to the burette is then opened, and Piis reduced to its initial value of 196 kN/m2. Thespecimen is not disturbed for 3 h, for reconsolida-tion purposes. The valve to the burette is closed,and the exact procedure for the first loading isrepeated. This is called the second loading. Afterthe second loading, the experiment is disassem-bled. Care is taken to remove the sand from thespecimen and place it in an oven for drying. Thesand is dried for 24 h and then weighed.

EXPERIMENTAL RESULTSFirst loading

All the experiments described were performedunder strain-controlled conditions, in contrast tothe work of most other researchers, who con-ducted the undrained cyclic shear tests understress-controlled conditions (see, for example,Ishihara & Yasuda, 1975; Seed, 1979; Tatsuokaet al., 1982). Fig. 3 shows the relation of the shearstrain and the effective pressure for loose samples,for two strain amplitudes, 0.2% and 1.0%. Theeffective pressure decreases during each cycle; the

Effective pressure: kN/mFig. 3. Relation between shear strain and effective pressure in first loading of loosespecimens; strain amplitudes are 0.2% and l-O%

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OKADA AND NEMAT-NASSER

75-

-25-

NumberNo. Amplitude Void ratio of cycles- 40 0.2% 0.874 27---- 4, 1 O% 0.871 2

,--_, --. .

- 5 0 - I0 50 100 150Effective pressure: kN/m*

t200 250

Fig. 4. Relation between shear stress and effective pressure in first loading of loosespecimens; strain amplitudes are @2% and 19%

reduction after the first cycle is especially large.The number of cycles required for the excess porewater pressure to attain 95% of the initial effec-tive pressure depends on the employed strainamplitude: 27 cycles are needed for 0.2%, andonly two cycles for 1.0% strain amplitude. Fig. 4shows the relation of the shear stress and theeffective pressure; Fig. 5 plots the correspondingshear stress against shear strain. The peak shearstress (and the secant modulus) decreases aftereach cycle.The energy supplied through external work ismainly consumed by the frictional loss at con-tacting granules, resulting in a change of themicrostructure in the granular mass. Therefore,the external work can be used to measure thehistory of fabric change in a granular mass.

The rate of external work per unit volume(tir) can be evaluated in terms of the appliedboundary tractions z and the boundary velocityfield i s.tidSL?Dwhere a dot denotes the inner product and aD isthe boundary of the sample domain D. If it isassumed that the boundary tractions are uniform,then equation (1) can be expressed in terms of theoverall stresses and strain rates, as (Hill, 1963,1967)

(~,,> + (a,,) + (~,,>(3,e> (2)

Number

/

No. Amplitude Void ratio of cycles40 0.2% 0.874 274150 1 O% 0.871 2

I I-0.5 0.0 0.5 1.0 1.5Shear strain: %Fig. 5. Relation between shear stress and shear strain in first loading of loose speci-mens; str ain amplitudes are @2% and lQ%

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INELASTIC FLOW OF GRANULAR MEDIA ISince the specimen is isotropically pressurized bythe external pressure P, throughout the experi-ment, all three normal stresses equal P,. If p/V isthe average volumetric strain rate, equation (2)can be rewritten as

(GE) = PJ(&,) + (&,> + (&,)I + (c,&jlrCJ= P0CQ/V + (e,,>(3,e> (3)

Since, for the pressure levels used here, sand par-ticles and water can be assumed to be incom-pressible, v/V is zero during the undrainedexperiment if the sample is completely saturatedand the change in the elastic rebound of themembrane due to the reduction of the effectivelateral stress is neglected. The rate of externalwork per unit volume then becomes(&) = (a,,>(?.s> (4)

The pressure term does not contribute to the rateof external work for incompressible materials, asis evident at the outset. The external work perunit volume (wr) up to time t can then be evalu-ated by the time-integration of equation (4)

and dense specimens. The relation of the externalwork per unit volume and the excess pore waterpressure for the loose specimens is shown in Fig.6. Three strain amplitudes, 0.2%, 0.5% and l.O%,are used here. The data at the end of each cycle ineach experiment are plotted in Fig. 6, whichshows a unique non-linear relation of the externalwork and the accumulated pore water pressure;this relation is independent of the shear-strainamplitude employed, but the number of cycles to95% of the initial effective pressure does dependon the strain amplitude.

The relation of external work per unit volumeand excess pore water pressure for dense speci-mens is also shown in Fig. 6. Five strain ampli-tudes, 0.2%, 0.4%, 0.5%, 1.0% and 2.0%, areemployed. Again, there is a unique non-linearrelation of the external work and the accumu-lated pore water pressure.

(WE) = s (e,e>(L> dr (5)f0where t, is the time at which the experiment isstarted.

The effect of specimen density on the relationdiscussed above is now considered. Both looseand dense specimens during the first loadingshow a unique relation of the external work andthe corresponding excess pore water pressure.These essentially coincide up to 130 kN/m porewater pressure, i.e. 65% of the initial effectivepressure. A significant difference appears there-after, with the loose samples developing higherpore pressure, as expected.

The external work is calculated from the To explore this interesting phenomenonexperimental results, and correlated with the further, randomly varying shear-strain amplitudesaccumulated pore water pressure for both loose up to 1.0% are applied to a dense specimen. The

No. Type Amplitude0 40 L0CSe 0.20%

0 37 Dense 0.20%l 34 Dense 0.40%Y 32 Dense 0.50%0 36 Dense 1 OO%9 38 Dense 2.00%+ 39 Dense Random

Void ratloO-674O-6650.8710.722o-7190.7060.7250.7160.719

NO of cycles27

25014

93

1 o 1:5 2:oExternal work per unit volume: kJ/m3

215 $0

Fig. 6. Relation between external work per unit volume and excess pore water pressurein first loading of both loose and dense specimens

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OKADA AND NEMAT-NASSER

Speclmen no. 39Void ratio 0.719

- ,0 i 0 160 160Effectwe pressure kN/m

260 2;o

Fig. 7. Relation between shear stress and effective pressure in first loading of densespecimen subjected to random torsional loading

relation of the shear strain and the effective pres-sure is shown in Fig. 7. The relation of the exter-nal work and the excess pore water pressure forthis random loading is shown in Fig. 6, where theaccumulated pore pressure at zero shear strain isplotted against the corresponding external work.The randomness in loading does not affect theunique non-linear relation of these two quantities.

Second loadingExperimental results for the second loading forboth loose and dense specimens are presentedbelow. After the first loading, specimens arereconsolidated under the same initial effectivepressure, 196 kN/m*, as in the first loading. Thesecond loading is then applied to the specimens.

The results where the strain amplitude in thesecond loading is the same as that in the firstloading, and then the results where the strainamplitude in the second loading is different fromthat in the first loading, are examined.The shear deformation characteristics in thesecond loading are compared with the results ofthe first loading. Fig. 8 shows the first two cyclesof the relation between the shear strain and theeffective pressure in both the first and secondloading for a dense specimen deformed at astrain amplitude of 0.5%. The excess pore waterpressure accumulated during the second loadingis much lower than in the first loading, and thenumber of cycles required to reach 95% of theinitial effective pressure in the second loading ismuch greater than in the first loading.

Strain ampktude 0.50%Void rat10 Number of cycles- First 0.706 9

----Second 0.696 36No. 32, dense speamen

0.5-

-0.5 I I I0 50 100 150 200 250Effectw pressure: kN/m

Fig. 8. Relation between shear strain and effective pressure in first and second loadingof dense specimen; strain amplitude is 05%

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INELASTIC FLOW OF GRANULAR MEDIA 9Fmt load,ng0 og..oOp Second loading00 +++++ +++++++++++++++

go: ++,++++*.0

l+. ++

Strain amplttude 0.50%NO. Void ratio Number of cycles

0 33 (FL) 0.865 60 33 (SL) 0.646 110 32 (FL) 0.706 9+ 32 (SL) 0.696 36

lb i0 i0 4bNumber of cycles

Fig. 9. Relation between number of cycles and excess pore water pressure in first andsecond loadings of both loose and dense specimens; strain amplitude is 05%

Figure 9 shows a direct comparison of the pore cases. This is due to the ordered arrangement ofwater pressure variation in loose and dense speci- the granules, attained on completion of the firstmens, deformed at a strain amplitude of 0.5%. It loading. Specifically, the contact normals after thetakes a greater number of cycles for the pore first loading tend to be oriented such that thewater pressure to reach a specified level in the specimen is better able to resist a similar shearingsecond loading than in the first loading, for both (see Oda, 1972b; Konishi, Oda & Nemat-Nasser,

200

1503Yhi;Ia, 1ocaa,kzIz6

50

(

O-5% (D) 2.0% (D)

**x,***

D = dense

NO. Type Amphtude Void Ratva Number of cycles* 37 LOWe 0.20% O-657 (26).& 34 Loose 0.50% 0.646 110 41 Loose 1 OO% 0.6520 37 Dense 0.20% O-710 (749,l 34 Dense O-40% 0.706 61w 32 Dense 0.50% 0.696 360 36 Dense 1 OO% 0.711 99 38 Dense 2.00% 0.700 4

1 12 4 6 6External work per unit volume. kJlm3

Fig. 10. Relation between external work per unit volume and excess pore water pressurein second loading of both loose and dense specimens; strain amplitude used in secondloading is the same as in first loading

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10 OKADA AND NEMAT-NASSER1982; Subhash, Nemat-Nasser, Mehrabadi & external work and the excess pore water pressureShodja, 1991). Another notable trend is that the in the second loading. In order to investigate thepore water pressure builds up faster during the true effect of the strain amplitude during thefirst loading of the dense specimen than during second loading, the specimens are subjected tothe second loading of the loose specimen. These cyclic loading at several different strain ampli-trends are seen in all results of the present study tudes, and then the second loading is performedwhen the behaviour of loose and dense specimens for each specimen at a common strain amplitude.deformed at a constant strain amplitude is com- The results are shown in Fig. 11. It is clear thatpared. However, in a strain-controlled test the the stress and strain history in the first loadingdeformation of the specimen is limited by the pre- does not affect the relation of the external workscribed strain amplitude. This prevents extensive and the excess pore water pressure in the secondparticle rearrangement, which often occurs in loading, i.e. this relation in the second loadingstress-controlled tests, once sufficiently high pore depends on the strain amplitude employed duringpressures are attained (see Nemat-Nasser & the second loading, and not on that employedTobita, 1982). during the first loading.

The relation of the external work per unitvolume and the excess pore water pressure in thesecond loading is now presented. Fig. 10 showsthis for loose and dense specimens. There are nosignificant differences up to half the initial effec-tive pressure, but after that the relation in thesecond loading seems to depend on the strainamplitude employed, although the same relationin the first loading does not. This is because, forthe same density, the sample packing is the samefor all samples at the start of the first loading,whereas at the start of the second loading eachsample has experienced a different stress andstrain history during its first loading, with a strainamplitude different from the other samples. It isstill not clear whether or not the strain amplitudeused in the first loading affects the relation of the

THEORETICAL MODELLINGEnergy dissipat ion in granular mediaIn the present context, the external work that issupplied at constant temperature to a materialsample is either dissipated through friction orstored in the material as strain energy. The rela-tion of the external work and the internal dissi-pation provides a basic constitutive constraint forthe flow of granular media (Rowe, 1962). For thestress levels considered in the present experi-ments, essentially the entire external work is dissi-pated by slip between contacting granules, andhence a negligible amount is stored in the gran-ules and the fluid as strain energy (Schofield &Wroth, 1968). Energy dissipation in granular

150.

3Yhi5Ig 100,EBaz

G50

0

AmplitudeNo. TYPO 1st 2nd Void ratio Number of cycles

* 40 LOOS? 0.20 0.20 0.857 (28)0 57 Loo se 1 .oo 0.20 0.853 (23)A 33 Loo se 0.50 0.50 0.846 11* 54 Loo se 0.20 0.50 0.84637 Dense 0.20 0.20 0.710 (::,t 56 Dense 1.00 0.20 0.706 (79)0 48 Dense 0.20 1 .oo 0.708 79 36 Dense 1.00 1 .oo 0.711 9+ 49 Dense 2.00 1 .oo 0.705 8

1 2 3 4 5 6 7 8External work per unit volume: kJ/m3

Fig. 11. Relation between external work and exea pore water pressure in secondloading of both loose and dense specimens; strain amplitude used in second loading isdifferent from that used in first loading

0.50% (Loose)1 1 .OO% (Dens e)I*A (Loose) +@DbQ)L 0.20% *o *

0-f.0.20% (Dense)

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INELASTIC FLOW OF GRANULAR MEDIA 11media is micromechanically modelled below, andis related to the pore water pressure built up incyclic shearing of saturated undrained samples.The low-strain-rate shear loading of a granularmass that occupies spatial region D of volume Vis considered. The region R within D is occupiedby the granules, whereas D - R is occupied bywater (the specimen is saturated). The rate ofexternal work per unit volume is expressed interms of the boundary tractions z and the bound-ary velocity field i by

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12 OKADA AND NEMAT-NASSERConsider a surface S (Fig. 12(b)-(d)) whichincludes the slip surfaces cf=r sa in D at time t,and which divides region D into M sub-regions,within each of which the velocity field is contin-uously differentiable. The Gauss theorem isapplied to each sub-region, to give

=tjl 8(V.a).ri+o:Vi]dl (10)where aDB is the boundary of subregion DB. Theequilibrium equation with no body and inertiaforces

V.a=Oand the symmetry of the stress tensor

o = 07give, from equation (10)

(11)

(12)

1=-Ja:idVVDwherei: = i[Vi + (Vri)T]

(13)

(14)is the strain-rate tensor. The left-hand side ofequation (13) is expressed as

+I;.idS+ir.idS) (15)where S+ and S- are opposite faces of the samesurface S, i.e.s+(x, n, t) = s-(x, --n, t) (16)Since the surface tractions z satisfy (Fig. 12(d))

5(X, n, t) = -t(X, --n, t) (17)the second and third terms in equation (15) cancelout for any x on (S - I:= 1 ) where the velocityfield is continuous. Equation (15) then becomes

1=- V [I t.idSaD

1=- V {S t.idSBD- .f, 1;.4x+) - 4x-11ds}1=- V (1 aDr.idS-irI;.[ti,dS) (18)

and equation (13) reduces to17 a Ds.idS

o:kdV +il j--.[i, dS) (19)

The first and second terms on the right-hand sideof equation (19) are the strain energy and fric-tional energy terms respectively. The strainenergy term in equation (19) can be decomposedto the strain energy for granules and for water,leading to

1v aD

which is the energy dissipation equation for satu-rated granular media.

Formulation offrictional energy lossThe frictional energy term in equation (20) is for-mulated below as a linear function of the effectivepressure P. The unit contact normal to a slipplane and the unit vector in the slip direction aredenoted by n and s respectively. The traction r onthe slip surfaces If= 1 soIn D is decomposed as

T(X, n, t) = F(x, t)s(x, t) + tyx, t)n(x, t) (21)where T' and T" are the normal and shear com-ponents of the tractions respectively. The velocityjump [i](x, t) is assumed to have only a shearcomponent [ti](x, t). The frictional energy term inequation (20) then becomes21 T(X, ) . CW, t) dS(4s-1 S(X* II, I)

T(X, )[d(-% t) dS(x) (22)The frictional coefficient for If= r sil is denoted by~(x, t). Then the shear component T' is expressedin terms of the normal component T" as

T(X, t ) = p(X, t)T(X, t) (23)

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INELASTIC FLOW OF GRANULAR MEDIA 13Since the granules carry the effective pressurethrough intergranular frictional contacts, it is rea-sonable to assume that the normal tractions r ata slipping contact in I:= 1 sa are linearly depen-dent on the effective pressure P

51(x, t) = W(w, t)P(t) (24)where Y is a scalar-valued function defined atpoints where slip occurs. In general, Y dependson the size of the granules, the packing, theloading condition and other relevant factors. Sub-stitution from equations (23) and (24) into equa-tion (22) yields

r(x, t) . CW, t ) dW

x WCW, t) W-4 (25)Since the effective pressure P(t) is not a functionofx

6 4 . Cdb, t) dS(4= P(t) (26)

where

x W, t)Cirl(x, dS(x) (27)Since a negligibly small part of the rate ofexternal work is stored in the granules and thewater at the stress levels considered in the experi-ments, the strain energy terms for granules and

water in equation (20) are negligibly small com-pared to the frictional energy term, therefore(~~=d(S,.:idV+S,_~:PdV) (28)

The energy dissipation in granular media thenbecomes

17 a0s. ri dS = (E)P + (i)Cyclic torsional loadingThe energy dissipation for a cyclic torsion testperformed on a hollow cylindrical specimen isnow considered. Experiments are performedunder undrained conditions. The shear strain iscyclically applied to the isotropically consolidatedspecimen, as already described in the experimen-tal results. The rate of external work per unit

volume under these conditions is expressed interms of the average stresses and strains as

1v r(x, n, t) . Yx, t) dS = (c,e(t)> (31)Consider the energy per cycle. Integration ofequation (3 1) over the nth cycle gives

s dt*.-1=

s ,: (k(t))P(t) dt + fn (i(t)) dt (32)n L s*.-1where t, is the time at which the nth cycle is com-pleted. If equation (32) is decomposed by takingP outside the integral and accounting for theresulting error by the addition of a term E,, then

s (e,s(t)>(3,s(t)) drr.- 1fn= I. i (i(t)) dt + E, + s (e(t)) dt (33)I_ fn-L

where P,, is the average effective pressure in thenth cycle, given by1

s1.p, = -t - t,-1 &, P(t) dt

If the efective pressure changes only slightly in thenth cycle, the error E, is small. Summation ofequation (33) over all cycles up to the kth cyclegives(~,s(t)>t

Finally, equation (35) is written as

s (L+(t)> dt10

+ &,+s*(c(t)) dtn=l toP= Ck s P(t) dtLO (36)(37)

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14 OKADA AND NEMAT-NASSER0.3

I

l-00%0 Ftrst loading .t Second loadmg

O-50%

- 0 4bNumber of cycles

80 &I

Fig. 13. Variat ion of C, with number of cycles in first and second loading of loosespecimens

wherec, = 1t, f:^G) dts

Equation (37) represents the energy dissipationup to the kth cycle. C, is called the energy dissi-pation coefficient, and has dimensions of l/time.It is regarded as representing the microstructuralarrangement of the granular mass at each instant.The coefficient C, in equation (37) can be esti-mated experimentally. Its variation is plottedagainst the number of cycles in Figs 13 and 14 for

0.5-

2.00%P0.4-

1 OO%.O

loose and dense specimens respectively, andagainst the excess pore water pressure in Figs 15and 16. These results show that C, is nearly con-stant throughout the experiment, when a constantstrain amplitude is cyclically applied to the speci-men. There is only a small difference in the C,value between the first and second loadings,although there is a large difference between thecorresponding numbers of cycles. These resultssuggest that C, does not depend on the strainhistory, but does depend on the strain amplitude.The relation of C, and the strain amplitude forboth loose and dense specimens is shown in Fig.17, which indicates that C, depends on the strainamplitude and the density, but not on the strainhistory.

0 First loadmgl Second loading

o!0 20 40Number of cycles

I60 80

Fig. 14. Variation of C, with number of cycles in first a nd second loading of densespecimens

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INELASTIC FLOW OF GRANULAR MEDIA 15

0 First loadingl Second loading

1 OO%0 l P*

l

0.50% l l Of eO*el 0

0.20%l 0 l to* l P l ?**Q* nv

40 1 I80 120Excess pore pressure kN/m*

, 1160 200

Fig. 15. Variation of C, with excess pore water pressure in first and second loading ofloose specimens

From these results it seems reasonable toassume that at constant density and strain ampli-tude C, is a constant, say C, related to the inter-nal work per unit volume of a given granularmass. It can be evaluated experimentally, pro-vided the same strain amplitude is cyclicallyapplied to the specimen. The C values shown inFig. 17 are used to calculate the internal work perunit volume at the end of each cycle; these resultsare plotted in Figs 18-21, with the correspondingexternal work. Clearly, the data points matchclosely in each case.These results show that energy dissipation ingranular media for cyclic torsional loading can be

o-4

03z0Yx0 0.2

0.1

0,

o-5-

0

0 First loadingl Second loadfng

expressed in terms of the time history of the effec-tive pressure, together with a constant C valuethat depends on the strain amplitude and densityonly.

Random torsional loadingEnergy dissipation for a torsional loading test inwhich the shear strain is applied randomly (ratherthan cyclically) was studied as an application ofthe above results. The experiments were per-formed using dense specimens. Shear strains wereapplied to a specimen at randomly varying strainamplitudes but at a constant shear strain rate,

2.00%l l **0

4b .CiO li0Excess pore pressure: kN/m

160 260

Fig. 16. Variation of C, with excess pore water pressure in first and second loading ofdense specimens

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16 OKADA AND NEMAT-NASSER

0

S t r a t n mp l i t u d e . %Fig. 17. C values obtained experimentally

I ,1 0 20 30Numberf c y c l e sFig. 18. Relation between external work and calculated internal work in first loading ofloose specimens

0 5 1 0 15Number of cycles

20 25

Fig. 19. Relation between external work and calculated interna l work in second loadingof loose specimens

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INELASTIC FLOW OF GRANULAR MEDIA 17z 4] 2.00% (0.421 x 1Om4) Stratn amplitude (C value)

/ 1.00% (0.335 x 10-T/ O-50% (0.267 x 1O-4)

0 External workt Predtctlon

0.20% (0.155 x 10-e)

40Number of cycles

60 60

Fig. 20. Relation between external work and calculated internal work in first loadingof dense specimens

(0.421 x 10m4)(D

*- 1.00%YE: 4-.Fzlzas= 2-

Strain amplitude (C value)0 External workl Predicton

0 r 10 20 40 60 60Number of cycles

Fig. 21. Relation between external work and calculated internal work in secondloading of dense specimens

Specimen no. 39Void ratlo 0.719

1.0 I I I I0 50 100 150 200 250Effective pressure: kN/m

Fig. 22. Relation between shear stress and effective pressure in second loading of densespecimen subjected to random torsional loading

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18 OKADA AND NEMAT-NASSER

0 i0 4b 6bNumb er of zero strain c rossqs

Fig. 23. Relation between external work and calculated internal work in first and secondloading of dense specimen subjected to random torsional loading

over each quarter cycle. The relation of the shearstrain and the effective pressure in the firstloading is shown in Fig. 7 and examined in theexperimental results for first loading. The samerelation in the second loading is shown in Fig. 22.Since the C values are essentially independentof the strain history, as discussed above, the

energy balance for a random torsional loadinggives, modification of equation (37)dt = f Ci s P dt (39)i=l ti- 1

where ti is the time of the ith zero shear strainand Ci is the ith C value. The value of Ci isdetermined as follows. First, the maximum shearstrain between L,_~ and ti is found. Then, the Cvalue corresponding to the shear strain isobtained from Fig. 17, which shows the relationof the C value and the shear strain amplitude(broken line).The internal work per unit volume is calculatedfrom the right-hand side of equation (39), usingthe experimental results, and is shown in Fig. 23together with the external work per unit volume.These two quantities are in excellent agreement;this supports the validity of equation (39).

CONCLUSIONSEnergy dissipation in the flow of cohesionlessgranular media has been considered. A theoretical

formulation is proposed, based on a simplemicromechanical model. The internal work forcyclic torsional loading is shown to depend onthe time-history of the effective pressure and anexperimentally obtainable parameter C. Theresults of a series of experiments show that C

depends on the strain amplitude and the density,but is essentially independent of the stress orstrain history. However, the effective pressureclearly depends on the strain history, as shown bythe large difference in the number of cyclesrequired to attain the same pore water pressure inthe first and second loading. Therefore, the right-hand side of equation (37) comprises a strain-history-dependent part s& P dt and astrain-history-independent part C.

ACKNOWLEDGEMENTSThe Authors wish to express their appreciationto Professor Muneo Hori of Tokyo University forhis critical suggestions and to Mr RyuichiSugimae for his assistance in carrying out theexperiments. This work has been supported bythe US Air Force Office of Scientific ResearchGrants AFOSR 87-0079 and AFOSR F49620-92-J-01 17 to the University of California, San Diego.

NOTATIONenergy dissipation coefficientregion consideredflth sub-region in Dvolume average of the rate of strain energynumber of sub-regions DB at time texterior unit normal vectornumber of slips in D at time teffective pressurepore water pressureaverage effective presure in nth cycleexternal pressureunit vector in the slip directionccth slip surfaceunion of all surfaces containing slip surfaceinstant when the nth cycle is completed

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INELASTIC FLOW OF GRANULAR MEDIA 19i velocity

[i] velocity jump[ti] shear component of velocity jumpV volume of D

(ws) volume average of the external work(tis) volume average of the rate of external workdD boundary of D

aDB boundary of Dsi: strain raten friction coefficient5 point on slip surfaceo stress tensorr tractionsz normal component of tractions acting on slip

surfacer5 shear component of tractions acting on slipsurfaceY operator connecting effective pressure P tonormal traction component T

R region occupied by granules in D

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