Shear stiffness of granular material at small strains: does it depend on grain size?

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1 Title Shear stiffness of granular material at small strains: oes it epen on grain size? Author(s) Yang, J; Gu, X Citation Géotechnique, 213, v. 63 n. 2, p Issue Date 213 URL Rights This work is license uner a Creative Commons Attribution- NonCommercial-NoDerivatives 4. International License.; Permission is grante by ICE Publishing to print one copy for personal use. Any other use of these PDF files is subject to reprint fees

2 Yang, J. & Gu, X. Q. (213). Géotechnique 63, No. 2, [ Shear stiffness of granular material at small strains: oes it epen on grain size? J. YANG an X. Q. GU The shear stiffness of granular material at small strain levels is a subject of both theoretical an practical interest. This paper poses two funamental questions that appear to be interrelate: whether this stiffness property is epenent on particle size; an whether the effect of testing metho exists in terms of laboratory measurements using resonant column (RC) an bener element (BE) tests. For three uniformly grae types of glass beas of ifferent mean sizes (. 195 mm,. 92 mm an mm), laboratory tests were conucte at a range of confining stresses an voi ratios, using an apparatus that incorporates both RC an BE functions an thus allows reliable an insightful comparisons. It is shown that the small-strain stiffness, etermine by either the RC or BE tests, oes not vary appreciably with particle size, an it may be practically assume to be size inepenent. The laboratory experiments also inicate that the BE measurements of small-strain stiffness are comparable to the corresponing RC measurements, with ifferences of less than 1%. Furthermore, the BE measurements for fine glass beas are foun to be consistently higher than the RC measurements, especially at large stress levels, whereas this feature becomes less evient for meium-coarse glass beas, an eventually iminishes for coarse glass beas. The stuy inicates that the characteristics of output signals in BE tests can be largely affecte by the frequency of the input signal, the mean particle size of the material an the confining stress level, an that these factors are interrelate. Improper interpretation of wave signals may lea to shear stiffness measurements that are unreasonably low, either showing a substantial increase with particle size or showing the opposite. A micromechanics-base analysis assuming the Hertz Minlin contact law is presente to offer an unerstaning of the size effect from the grain scale. KEYWORDS: ynamics; elasticity; laboratory equipment; laboratory tests; stiffness INTRODUCTION The shear stiffness of granular soils at strain levels less than. 1% usually enote as G or G max is a key parameter in major geotechnical applications involving eep excavations an tunnels, liquefaction evaluation or earthquake groun response analysis. Extensive research has been carrie out to stuy this property, mainly through well-controlle laboratory experiments, particularly using resonant column (RC) tests as they offer high reliability an accuracy at this small strain level (Harin & Richart, 1963; Iwasaki & Tatsuoka, 1977; Chung et al., 1984; Lo Presti et al., 1997). Of the factors ientifie as affecting shear stiffness, confining stress an voi ratio are recognise to be the main ones. They are now commonly accounte for using an empirical formula taking the form G ¼ AFðÞ e ó 9 n (1) p a where G is in MPa; ó9 is the effective confining stress in kpa; p a is a reference stress, usually taken as the atmospheric pressure; A an n are two best-fit parameters; an F(e) is a function of voi ratio (e) with a typical form of (e.g. Harin & Richart, 1963; Iwasaki & Tatsuoka, 1977) Manuscript receive 6 July 211; revise manuscript accepte 17 May 212. Publishe online ahea of print 16 October 212. Discussion on this paper closes on 1 July 213, for further etails see p. ii. Department of Civil Engineering, The University of Hong Kong, Hong Kong. ð Fe ðþ¼ 2 :17 eþ 2 1 þ e Recent notable work using the resonant column technique inclues that by Wichtmann & Triantafylliis (29), who conucte a structure programme of resonant column (RC) tests to examine the effect of particle-size istribution on the stiffness of san. Their ata showe that the value of G ecrease markely as the coefficient of uniformity (C u ¼ D 6 /D 1 ) of san increase, where D 6 an D 1 represent the particle sizes that 6% an 1% of the san mass are smaller than, respectively. This result is in agreement with that of Iwasaki & Tatsuoka (1977), which was also erive from a number of s on san. The experimental ata also suggeste that the small-strain stiffness was inepenent of the mean particle size of san (D 5 ). Bener element (BE) tests, which involve irect measurement of shear waves, have become a promising alternative in recent years for laboratory etermination of G values of soils (Dyvik & Mashus, 1985; Viggiani & Atkinson, 1995; Brignoli et al., 1996; Yamashita et al., 29; Clayton, 211). A significant avantage of this technique is that it can be incorporate in stanar soil mechanics apparatuses such as triaxial an oeometer evices, an the approaches for ata interpretation are relatively simple. Relying on this technique, Patel et al. (28) measure the shear wave velocity (V s ) in assemblies of glass beas of ifferent sizes. Their tests inicate that V s increase as the mean particle size of glass beas ecrease, as shown in Fig. 1. At the confining stress of kpa, the V s of glass beas with a mean iameter of.5 mm was etermine to be 32 m/s, which was about 45% higher than that of glass beas with a mean iameter of 2.5 mm. In terms of shear moulus, the G (2) 165

3 166 YANG AND GU value of the small-size glass beas was about 11% higher than that of the large-size glass beas. The experimental work of Patel et al. (28), showing an appreciable size epenence of small-strain stiffness, oes not agree with that of Wichtmann & Triantafylliis (29) an Iwasaki & Tatsuoka (1977). It oes, however, appear to be consistent with that of Bartake & Singh (27), who performe BE tests on three ry sans with similar graation an foun that the G value increase as the mean particle size (D 5 ) of the san ecrease. The work of Sharifipour et al. (24), also using the BE technique to measure the shear wave velocity in glass beas, as further uncertainty: for glass beas of three ifferent nominal sizes (1., 2. an 3. mm), they obtaine an opposite result, showing that the value of V s increase with increasing particle size, as shown in Fig. 1. In terms of shear moulus, 3. mm glass beas at a confining stress of kpa were etermine to have a G value of about MPa, being 66% larger than that of 1. mm glass beas. Eviently, the issue of whether the small-strain shear stiffness is size epenent remains inconclusive an open to iscussion. Also, the experimental ata from BE tests in the literature seem always to inicate that particle size has an effect on small-strain shear stiffness (although opposite trens were observe in the G variation with grain size), whereas the ata seem always to suggest that there is not any size effect. In this respect, an aitional question arises: oes the testing metho have any effect on smallstrain stiffness? In s, a soil specimen is subjecte to torsional excitation typically at frequencies of several tens of Hz, an the shear stiffness (G ) is estimate from the measure resonant frequency. In BE tests, however, the shear wave velocity in a soil specimen (V s ) is measure irectly Shear wave velocity: m/s Shear wave velocity: m/s D 5 mm D 1 5 mm D 2 5 mm Confining stress: kpa D 1 mm D 2 mm D 3 mm Confining stress: kpa Fig. 1. Test ata on shear wave velocity in glass beas of ifferent sizes: Patel et al. (28); Sharifipour et al. (24) through a pair of piezoelectric transucers at frequencies from several to a few tens of khz, an is then converte to the shear moulus by G ¼ r(v s ) 2, where r is the mass ensity of the soil. The two methos inee involve ifferent principles an interpretations. With the aim of aressing these two funamentally important questions, a specifically esigne experimental programme has been carrie out using an apparatus that incorporates both RC an BE features. The apparatus allows the BE an s to be performe on an ientical specimen, thus afforing more reliable an convincing comparisons; this is both necessary an esirable, given the observe iscrepancies between the RC an BE measurements. Three types of glass beas with ifferent mean particle sizes (D 5 ) but the same uniformity (C u ) were prepare an use as analogue granular soils in the experiment. Their simple an spherical geometry, together with the same uniformity, allowe the influences of particle shape (Cho et al., 26) an particle size istribution (Wichtmann & Triantafylliis, 29) to be isolate, so that any observe ifference is attributable to the ifference in particle size. This paper presents the main results of this experimental work. Moreover, efforts are mae here to explore the possible causes of the contraictory results in the literature through a comprehensive analysis of the experimental ata. A grain-scale moel is also presente to provie an unerstaning from the micromechanical point of view. TEST EQUIPMENT AND MATERIAL Equipment set-up The overall set-up of the apparatus is schematically shown in Fig. 2, with close-up views of the key components given in Fig. 3. The apparatus has both RC an BE features an a robust signal conitioning an ata acquisition system, along with an environmental chamber. It can accommoate a soil specimen 5 mm in iameter an mm high, with an airfille cell pressure up to 1 MPa. The resonant column is of bottom-fixe an top-free configuration (the Stokoe type). Compare with the free free configuration, the fixe free configuration has the avantages of relatively high available torque an convenient access to the specimen for control of effective stress. A careful calibration of the equipment was carrie out using three aluminium bars of ifferent imensions to establish a calibration curve for the frequencyepenent mass polar moment of inertia of the rive hea. Attention shoul be pai to specimen fixity when testing very stiff materials such as highly cemente san or weak rock; more etails of this issue are beyon the scope of this paper but can be foun in, for example, Clayton et al. (29). To allow for the pulse test in the same system, a pair of bener elements has been instrumente, as shown in Fig. 3. Unlike the conventional esign, this single pair of bener elements is able to generate not only shear waves (i.e. S-waves) but also compression waves (i.e. P-waves); this has been achieve by moifying the wiring configuration of the bener elements (Lings & Greening, 21). Measuring the compression wave can facilitate the interpretation of shear wave signals in situations where the so-calle near-fiel effect is complicate, an it may also serve as a promising alternative for checking the egree of saturation of soil specimens (Yang, 22). A careful calibration of the pulse testing system was carrie out by putting the tips of the two bener elements in irect contact to etermine the system elay, incluing the response time of bener elements an the travel time in the cables. The phase relationship (i.e. initial polarisations) between the input an output signals was also checke in the calibration.

4 SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS 167 CP controller Compresse air p t USB hub BP an BV controller Cell PC an control software Data logger TT Power PWP CP: cell pressure BP: back pressure BV: back volume TT: temperature transucer PWP: pore water pressure LVDT: linear variable eformation transucer LVDT Temperature transucer in cell Fig. 2. Set-up of ynamic testing system at The University of Hong Kong Environmental chamber Accelerometer LVDT Proximeter Coil Magnet Drive hea Driving arm Specimen Metal target Bener elements Drainage tubes Bener element Fig. 3. Close-up views of apparatus (not to scale) Test material an sample preparation Three types of uniformly grae glass beas were prepare for the experiment. Fig. 4 shows their size istribution curves, along with a scanning electron microscopy (SEM) photograph showing the shape of the particles. Glass beas of type A, enote GB-A, were the coarsest, with a mean particle size (D 5 )of1. 75 mm; glass beas of type D (GB- D) were the finest, with a D 5 of.195 mm; an glass beas of type B (GB-B) with a D 5 of. 92 mm were in between (Table 1). The range of particle size was reasonably wie to cover the sizes of fine to coarse san, an all three types of glass beas share a similar graation (C u ¼ )

5 168 YANG AND GU Finer by weight: % GB-A GB-B GB-D Particle size: mm Fig. 4. Particle-size istribution curves of glass beas use in experiments Table 1. Physical properties of glass beas teste Glass bea G s D 1 :mm D 5 :mm D 6 :mm C u GB-A GB-B GB-D such that the influence of uniformity was isolate. The uniformity of the glass beas is comparable to that of Toyoura san (1. 392), teste by Yang & Gu (21) using the same apparatus. Inustrially mae glass beas may not be perfectly spherical. This efect in shape regularity was observe in all three types of glass beas. From a statistical point of view, however, it can be assume that these glass beas have a similar rounness an surface roughness, an the potential effect of shape efect or the effect of ifference in surface roughness on test results is minor. Prior to the stiffness measurements, the glass beas were oven-rie an then coole in seale containers to remove the potential influence of moisture on the particle surface. All specimens in the tests were prepare using the ry tamping metho in five layers. The metho involve pouring the material using a funnel without falling height an then performing compaction using a tamper. Note that in the preparation of specimens at the loose state, no significant tamping was use. To stan the specimens an support the weight of the rive arm of the resonant column, a suction of 25 kpa was applie to the specimens. The cell pressure was then increase an the suction was ecrease simultaneously to keep a constant isotropic effective confining stress of 25 kpa, which was taken as the initial stress level of the specimens. Test series For each type of glass beas a set of specimens were prepare in ifferent initial packing states: loose (e ¼ ), meium-ense (e ¼ ) an ense (e ¼ ) states. For each packing ensity, the BE an s were conucte at the confining stresses of 5,, 2 an 4 kpa in sequence. In bringing the specimens to a specific stress level, each specimen was first consoliate for 15 min at this stress level, an the corresponing eformation was measure by the internal high-resolution linear variable ifferential transucer (LVDT) (the reaing of the LVDT usually became stable within the time allowe); then the BE test was performe uner a range of excitation frequencies. Following the BE test, the was then performe on the same specimen for the purpose of comparison of the stiffness measurements. Table 2 summarises the test series. BENDER ELEMENT MEASUREMENTS Despite the increasing popularity of BE tests, consierable uncertainty remains in signal interpretation, an thus in the estimate shear stiffness. Clayton (211) showe a goo example of large scatter in shear wave velocities estimate from a single BE test on a specimen of natural clay, commenting that previous estimates of the accuracy of BE measurements have been optimistic. This opinion is supporte by the largely scattere results of the international parallel BE tests on uniform Toyoura san (Yamashita et al., 29). These observations, together with the contraictory results in the literature for stiffness variation with particle size, unerscore the nee for careful examination of BE tests, particularly for granular material, in which wave propagation is complex owing to its particulate nature. Here, efforts are mae to clarify several issues that have not yet been extensively stuie but are closely relate to the reliability of BE measurements: the characteristics of receive signals in both the time an frequency omains over a wie range of excitation frequencies an wavelengths; how changes of particle size alter the characteristics of receive signals; an the performance of ifferent interpretation methos uner a variety of combinations of test conitions (i.e. grain size, excitation frequency an confining stress). Effect of frequency on waveforms Figure 5 shows the waveforms generate in a specimen of coarse glass beas (GB-A, D 5 ¼ 1.75 mm) by one cycle of sinusoial signal at ifferent excitation frequencies. The specimen was at an isotropic confining stress of kpa an a voi ratio of The excitation frequencies covere a wie range, varying from 1 khz to as high as 4 khz, thus allowing a systematic examination of the frequency effect. The ashe line inicates the travel time of the shear wave in the specimen, euce from the shear wave velocity etermine by the. By comparison, the upwar triangle inicates the first arrival of the shear wave base on the waveform at the frequency of 1 khz. Eviently, as the excitation frequency increases, the receive signal tens to contain more high-frequency components. At very high excitation frequencies (2 khz an 4 khz), the waveforms become quite similar in shape. In aition, the initial signal component with negative polarity preceing the arrival of the major components marke by a ownwar triangle an representing the near-fiel effect appears to be strongest at the excitation frequency of 1 khz, an tens to fae as the excitation frequency increases. Similar observations have been obtaine on a meiumcoarse specimen (GB-B, D 5 ¼. 92 mm), which was also at an isotropic confinement of kpa an a voi ratio of. 585, an was also excite by a sinusoial input of varying frequencies (Fig. 6). Compare with the waveforms shown in Fig. 5, the receive signal at each corresponing frequency contains components of higher frequencies, implying that a change in particle size can affect the frequency content of the receive signal. This feature becomes more evient in Fig. 7, where the waveforms in a fine specimen (GB-D, D 5 ¼. 195 mm) uner various excitation frequencies are shown. Also, it is to be note that when the grain size reuces from mm to. 195 mm, the initial component with negative polarity tens to fae as well.

6 Table 2. Summary of test series SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS 169 Test series Material State (e, ó9) State 1 (e, ó9) State 2 (e, ó9) State 3 (e, ó9) State 4 (e, ó9) Note I-1 GB-A (. 56, 25) (. 56, 5) (. 56, ) (. 558, 2) (. 557, 4) Dense (. 566, 25) (. 566, 5) (. 565, ) (. 564, 2) (. 563, 4) (. 559, 25) (. 559, 5) (. 558, ) (. 557, 2) (. 556, 4) (. 558, 25) (. 558, 5) (. 557, ) (. 556, 2) (. 555, 4) (. 558, 25) (. 558, 5) (. 557, ) (. 556, 2) (. 554, 4) I-2 (.583,25) (.583, 5) (.582, ) (.581, 2) (.579, 4) Meium (.585, 25) (.585, 5) (.584, ) (.583, 2) (.581, 4) (.582, 25) (.582, 5) (.581, ) (.58, 2) (.578, 4) I-3 (.63, 25) (.63, 5) (.62, ) (.61, 2) (.6, 4) Loose (.598, 25) (.598, 5) (.597, ) (.596, 2) (.594, 4) (.595, 25) (.595, 5) (.594, ) (.593, 2) (.591, 4) II-1 GB-B (.561, 25) (.561, 5) (.56, ) (.559, 2) (.558, 4) Dense (. 562, 25) (. 562, 5) (. 561, ) (. 56, 2) (. 558, 4) (. 567, 25) (. 567, 5) (. 566, ) (. 565, 2) (. 563, 4) (. 566, 25) (. 566, 5) (. 565, ) (. 564, 2) (. 562, 4) (. 561, 25) (. 561, 5) (. 56, ) (. 559, 2) (. 557, 4) II-2 (. 584, 25) (. 583, 5) (. 583, ) (. 582, 2) (. 58, 4) Meium (. 586, 25) (. 586, 5) (. 585, ) (. 584, 2) (. 582, 4) II-3 (.63, 25) (.63, 5) (.62, ) (.61, 2) (.599, 4) Loose (.61, 25) (.6, 5) (.6, ) (.598, 2) (.597, 4) (.596, 25) (.596, 5) (.595, ) (.594, 2) (.592, 4) (.598, 25) (.598, 5) (.597, ) (.596, 2) (.594, 4) (.61, 25) (.61, 5) (.6, ) (.598, 2) (.596, 4) III-1 GB-D (.564, 25) (.564, 25) (.563, 5) (.563, 5) (.563, ) (.563, ) (.561, 2) (.562, 2) (.56, 4) (.56, 4) Dense III-2 (. 587, 25) (. 586, 5) (. 585, ) (. 584, 2) (. 582, 4) Meium (. 585, 25) (. 584, 5) (. 583, ) (. 582, 2) (. 58, 4) (. 584, 25) (. 584, 5) (. 583, ) (. 582, 2) (. 579, 4) (. 575, 25) (. 575, 5) (. 574, ) (. 573, 2) (. 571, 4) III-3 (. 63, 25) (. 62, 5) (. 61, ) (. 6, 2) (. 598, 4) Loose (. 61, 25) (. 61, 5) (. 6, ) (. 599, 2) (. 596, 4) (. 616, 25) (.623, 25) (. 615, 5) (.623, 5) (. 615, ) (.622, ) (. 614, 2) (.621, 2) (. 612, 4) (.619, 4) Note: e ¼ voi ratio; ó9 ¼ effective confining stress (in kpa); state ¼ initial state. 1 khz 1 khz 2 khz 2 khz Voltage: mv 5 khz 1 khz Voltage: mv 5 khz 1 khz 2 khz 2 khz 4 khz 4 khz 319 μs () Time: μs μs () Time: μs Fig. 5. Shear wave signals in glass beas GB-A (D mm) at various excitation frequencies (sinusoial input, ó9 kpa; e.584) Fig. 6. Shear wave signals in glass beas GB-B (D 5.92 mm) at various excitation frequencies (sinusoial input, ó9 kpa; e.585)

7 17 YANG AND GU 1 khz GB-A 2 khz Voltage: mv 5 khz 1 khz Voltage: mv GB-B 2 khz GB-D 4 khz 314 μs () Time: μs Fig. 7. Shear wave signals in glass beas GB-D (D mm) at various excitation frequencies (sinusoial input, ó9 kpa; e.583) Voltage: mv Time: μs GB-A GB-B GB-D x Time: μs Fig. 8. Shear wave signals in glass beas of ifferent sizes (sinusoial input, ó9 kpa; e.584): excitation frequency 5 khz; excitation frequency 1 khz Effect of particle size on waveforms To allow a better ientification of the importance of particle size in moifying waveforms, Fig. 8 compares the waveforms generate in the three specimens uner otherwise similar conitions. The waveforms at the excitation frequency of 5 khz are compare in Fig. 8, an another three waveforms at the excitation frequency of 1 khz are compare in Fig. 8. In either case of excitation frequency, a ecrease in grain size can introuce high-frequency components to the output signal. The waveforms in the finest specimen GB-D, at either 5 khz or 1 khz, exhibit a first peak with an amplitue that is much less than the subsequent largest peak. It is the excursion of this small peak that represents the true arrival of the shear wave (marke by an upwar triangle on the waveform for specimen GB-D in Fig. 8). For purposes of comparison, the travel times of the shear waves in the three specimens were euce from the RC measurements, an are marke by three ownwar ashe arrows on the corresponing waveforms. It becomes clear that, if the excursion of the largest peak is selecte as the arrival of the shear wave (marke by the inicator x ), a stiffness value substantially lower than the RC measurement will be yiele. The existence of a small-amplitue peak preceing the largest peak in the shear wave signal was also reporte by Brignoli et al. (1996) in pulse tests on uniform Ticino san (D 5 ¼. 71 mm); it was also observe in testing Toyoura san (D 5 ¼.216 mm) using the same apparatus (Yang & Gu, 21), as shown in Fig. 9 for comparison. This feature is not evient in the waveforms generate in the coarsest specimen (GB-A; Fig. 5), but for the meium-coarse specimen (GB-B) the feature tens to appear when the excitation frequency is sufficiently high (Fig. 6). It is also foun that, for a given excitation frequency, increasing the confining stress to large levels tens to result in the occurrence of the small peak, as shown in Fig. 1 for glass beas GB-B. The waveforms in Figs 5 8 an the iscussion above show the important fining that the effect of particle size is couple with the effect of frequency in altering the characteristics of waveforms. This is unerstanable, given the particulate nature of the material. To explore their relation, the fast Fourier transform was conucte for the output signals in Figs 5 7, an the preominant frequency was ientifie for each signal. Fig. 11 shows this preominant frequency, enote as f out, as a function of the excitation frequency ( f in ) for the three specimens, together with the tren lines. There are several features in Fig. 11 that are worth noting. First, for either coarse or fine specimens, the preominant frequency tens to increase with excitation frequency in an approximately linear manner in a low-frequency range, an then tens to approach a limiting value at high excitation frequencies. In this respect, a threshol frequency marking the transition can be ientifie. Secon, this threshol frequency appears to epen on particle size, in that it takes larger

8 SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS khz 4 GB-A (1 75 mm) GB-B ( 92 mm) 2 khz 3 GB-D ( 195 mm) Threshol frequency Voltage: mv 5 khz f out : khz 2 1 khz 1 2 khz 372 μs () Time: μs Fig. 9. Shear wave signals in ry Toyoura san at various excitation frequencies (ó9 kpa; e. 798) Voltage: mv 5 kpa kpa 2 kpa 4 kpa Time: μs Fig. 1. Shear wave signals in glass beas GB-B at various confining stresses (sinusoial input, e.585) values for fine specimens or smaller values for coarse specimens. For example, for specimen GB-A the threshol frequency is aroun 16 khz, whereas it is above 2 khz for specimen GB-D. Thir, the preominant frequency oes not appear to be sensitive to particle size in the range of low excitation frequency (say, below 5 khz); however, as the excitation frequency is further increase, it tens to become larger for fine specimens than for coarse specimens. Moreover, the limiting value of the preominant frequency at high excitation frequencies seems to increase with ecreasing particle size. The interesting features escribe above are base on the experiments conucte on uniform glass beas; further valiation using laboratory tests on ifferent granular materials woul be of benefit. The experimental ata escribe above offer evience that f in : khz Fig. 11. Output signal frequency against input signal frequency the shear wave signal generate in a granular sample, even with a simple sinusoial input, can be very complicate. Its characteristics epen on both excitation frequency an particle size, an these factors are interrelate. With regar to shear stiffness measurement, the important implication is that iverse or even contraictory results may be yiele on the shear wave velocity (V s ) or the small-strain stiffness (G ) if these factors are not properly taken into account in ata interpretation; this will be elaborate in more etail in the following section. Evaluation of various interpretation methos The largest uncertainty an ifficulty with BE tests lie in etermination of the travel time of the shear wave (Jovicic et al., 1996; Lee & Santamarina, 25; Yamashita et al., 29). The commonly use approaches, as ocumente in Yamashita et al. (29), are generally simplistic, an o not well recognise that the bounary conitions impose on the specimen can lea to consierable ivergence from the simplifie solutions. One of the common approaches is known as the start-to-start metho an another is the peakto-peak metho, both working in the time omain. The iea of the start-to-start metho is to fin the first arrival of the shear wave by visual inspection of the receive signal. A number of characteristic points have been propose in the literature as inicators for the first arrival of the shear wave. A systematic examination of ifferences in the estimate shear stiffness using these points has been conucte for both fine an coarse specimens, with particular attention pai to the size effect. To facilitate iscussion, a typical waveform is shown in Fig. 12, with all possible characteristic points marke as the first arrival of the shear wave, incluing the first inflection S1, the troughs S2 an S4, an the zero intercepts S3 an S5. The peak-to-peak metho applies typically to a sinusoial input in which the travel time is efine as the ifference between the peak of the input signal an that of the output signal. As the receive signal usually contains multiple peaks rather than a single peak, a common practice is to take the largest peak to estimate the travel time (P2 in Fig. 12). However, as will be further explore later, in many cases this largest peak oes not offer a reasonable reference for etermining the travel time of the shear wave; as an alternative, a smaller peak preceing the maximum one (P1 in Fig. 12) is also use for comparison purposes. To overcome the subjectivity an uncertainty involve in

9 172 YANG AND GU Amplitue P P2 Input Output 25 2 S-S1 S-S4 P-P2 RC S-S2 S-S5 CC-1 11%RC S-S3 P-P1 CC-2 9%RC S S1 P1 S3 S5 S2 S4 Time G : MPa 15 5 Fig. 12. Characteristic points for travel time interpretation the time-omain methos, Viggiani & Atkinson (1995) propose the cross-correlation metho, which works on the correlation of the input an receive signals. The unerlying assumption is that the two signals are of the same shape an frequency; however, this is not the case in BE tests an thus leas to significant errors, as shown later. In this metho the travel time of the shear wave is efine at the position corresponing to the largest peak of the correlation, as shown schematically in Fig. 13 (CC-2). Because the time histories of the correlation contain multiple peaks, an alternative is to select the first, small-amplitue peak (CC-1) to etermine the travel time. Using the various methos escribe above, values of G were etermine for specimens GB-A, GB-B an GB-D uner a confining stress of kpa an a voi ratio of about.584, for a wie range of excitation frequencies. The results are plotte in Fig. 14 as a function of excitation frequency. For purposes of comparison, the G value etermine by the resonant column test for each specimen is given as a benchmark, an two bounary lines marking 11% an 9% of this benchmark value are also given (i.e. 1% variations). The three plots in Fig. 14 suggest the following. The G values etermine using any one of the methos appear to be frequency epenent: that is, the estimate G value generally varies with excitation frequency. For a given excitation frequency, ifferent interpretation methos yiel ifferent G values, an the range of these values varies when particle size varies. The start-to-start metho using point S1 always gives G values that are unreasonably high in comparison with the RC measurements (the ata points are all beyon the scale of the graphs), suggesting that point S1 oes not correspon to the first arrival of the shear wave; it actually correspons to the compression wave, as confirme by the velocity measurement of this type of wave. () The cross-correlation metho, regarless of whether the largest peak (CC-2) or the first peak (CC-1) is use, usually yiels G values that are unreasonably low, particularly for coarse specimens GB-A an GB-B at high frequencies. To better ientify the effect of frequency, a ratio of travel istance (L tt ) an wavelength (º) is introuce as follows. Amplitue Largest peak (CC-2) First peak (CC-1) Fig. 13. Selection of peaks in cross-correlation metho Input Output Cross corr. Time G : MPa G : MPa Input frequency, f in : khz S-S1 S-S4 P-P2 RC S-S2 S-S5 CC-1 11%RC S-S3 P-P1 CC-2 9%RC Input frequency, f in : khz S-S1 S-S4 P-P2 RC S-S2 S-S5 CC-1 11%RC S-S3 P-P1 CC-2 9%RC Input frequency, f in : khz Fig. 14. Shear stiffness values estimate using various interpretation methos as a function of excitation frequency (ó9 kpa): GB-A, e.584; GB-B, e.585; GB-D, e.583 R in ¼ L tt º ¼ L tt V s f in (3) where the wavelength is calculate using the frequency of the sinusoial input ( f in ), an the shear wave velocity is taken as that provie by the. For each specimen the G values from BE measurements were normalise by the RC measurement, an are shown as a function of the R in ratio in Fig. 15. There are several points that are worth noting. The egree of scatter in G values ue to ifferent interpretation methos is epenent on the R in ratio, an it appears to be smallest for R in values ranging from 2 to 4. This fining is consistent with the theoretical analysis of Sanchez-Salinero et al. (1986) that at least two wavelengths shoul be maintaine between the transmitter an receiver; but it shoul be note that much higher R in values o not seem to help reuce uncertainty or increase accuracy. In this optimal range of R in values, the start-to-start

10 G (BE)/ G (RC) G (BE)/ G (RC) S-S1 S-S4 S-S2 S-S5 S-S3 P-P1 P-P2 CC-1 CC R in % 1% SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS 173 S-S1 S-S4 P-P2 S-S2 S-S5 CC-1 S-S3 P-P1 CC R in 1 5 calculate using the frequency of the input sinusoial signal an the RC reference value for shear wave velocity. The three plots in Fig. 16 offer several important finings. The start-to-start metho using either point S5 or S4 tens to give G values that ecrease as the º/D 5 ratio increases: that is, for a given excitation frequency, the shear stiffness will ecrease with ecreasing particle size. The peak-to-peak metho using P P2 (i.e. the largest peak) tens to give G values that increase as the º/D 5 ratio increases; this means that, for a given excitation frequency, the shear stiffness increases with ecreasing particle size. The cross-correlation metho also prouces G values that increase with increasing º/D 5 ratio, either ue to a ecrease in particle size or to a reuction of excitation frequency. Also, the start-to-start metho using point S2 is able to consistently provie reasonable G values over a wie range of º/D 5 ratios (2 ). These G values are slightly greater than the RC benchmark values, an G (BE)/ G (RC) S-S2 S-S3 S-S4 S-S5 1% G (BE)/ G (RC) 1 5 S-S1 S-S4 S-S2 S-S5 1% S-S3 P-P1 P-P2 CC-1 CC R in Fig. 15. Shear stiffness values estimate using various interpretation methos as a function of ratio between travel istance an wavelength (ó9 kpa): GB-A, e.584; GB-B, e.585; GB-D, e.583 metho using point S2 seems to work well for the fine specimen GB-D, whereas the start-to-start metho using point S4 seems to work well for the coarse specimens GB-A an GB-B, for which point S2 appears to merge with point S4 (see Fig. 8). The performance of the cross-correlation metho using either CC-1 or CC-2 is improve in this optimal R in range, but the metho still provies G values that are more than 2% lower than the corresponing RC measurements. Bearing in min the couple effects of excitation frequency an particle size, a new inex is introuce as the ratio between wavelength (º) an mean particle size (D 5 ), an variations of the normalise G values with this inex are examine in Fig. 16. To facilitate comparison, the ata shown in Fig. 16 were all generate using the start-tostart metho, the ata in Fig. 16 were generate using the peak-to-peak metho, an Fig. 16 was prouce using the cross-correlation metho. Note that the wavelength was G (BE)/ G (RC) G (BE)/ G (RC) λ/ D Large particle size High input frequency P-P1 Small particle size Low input frequency P-P λ/ D CC-1 1% CC-2 1% λ/ D 5 Fig. 16. Shear stiffness values estimate using various interpretation methos as a function of ratio between wavelength an particle size (ó9 kpa; e.584): start-to-start metho; peak-to-peak metho; cross-correlation metho

11 174 YANG AND GU exhibit a tenency to ecrease with º/D 5 ratio, but the variation is approximately less than 1%. 3 Base on a systematic examination of the characteristics of waveforms, an taking the RC measurements as a reference, the following strategies are recommene for conucting an interpreting BE tests for stiffness measurement. A range of excitation frequencies covering the threshol frequency shoul be use, an the waveforms at various frequencies shoul be examine as a whole. The excitation frequencies shoul be selecte such that the ratio of travel istance an wavelength is between 2 an 4 (higher R in values o not seem to help reuce the uncertainty). The start-to-start metho using point S2 is recommene; for coarse materials where point S2 is not clear, point S4 is a reasonable alternative. COMPARISONS OF BE AND RC MEASUREMENTS A large number of BE an s have been performe over a range of confining stresses an voi ratios, an thereby offer an excellent opportunity to compare the stiffness values from the two methos. In oing so, the BE an RC measurements of the small-strain stiffness are shown against confining stress in Figs 17, 17 an 17 for specimens GB-A, GB-B an GB-D, respectively. It is evient that G values estimate from both the RC an BE tests show an increase with the confining stress. As inicate by the tren lines for the upper an lower bouns, the stress epenence can be well represente by a power law, with the exponent being about. 4. The ata also inicate that at a given confining stress the G value ecreases with increasing voi ratio. Using the empirical formula given in equation (1) to account for the effects of confining stress an voi ratio, the two parameters A an n have been etermine for the RC an BE measurements, respectively (see Table 3). For the purpose of comparison, values of the two parameters etermine using test ata for Toyoura san (Yang & Gu, 21) are also inclue. It is of interest to note from Fig. 17 an Table 3 that for fine granular materials (GB-D an Toyoura san) the BE measurements are apparently greater than the RC measurements, whereas for coarse granular materials (GB-A) the two methos give G values that are consistent overall. The results for the meium-coarse glass beas (GB-B) appear to be in between, with the BE measurements being slightly greater than the RC measurements. This marke feature can be foun in an alternative way, as shown by Fig. 18, where the BE measurements of G are plotte against the RC measurements. For both the fine an coarse specimens, the ifference between the BE an RC measurements is approximately within 1%, meaning that the effect of testing metho is practically negligible. Possible reasons for the fining that BE measurements of stiffness are larger than the corresponing ones from RC tests for fine an meium-coarse granular may be that: the overall strain level involve in BE tests is somehow lower than that involve in s; an the measures the overall stiffness of the specimen, whereas the BE test measures the central part of the specimen between the transmitter an receiver (Fig. 19), which tens to be stiffer than the whole specimen in the case of fine an meium-coarse granulates. Further work to explore the reasons behin this interesting observation woul be of interest. Figure 2 shows variations of stiffness values, etermine by the BE an s, with particle size at the ense, meium-ense an loose states. An alternative view of the G : MPa G : MPa G : MPa Confining stress, σ : kpa y 21 66x 432 RC ( e 559) RC ( e 585) RC ( e 598) y 23 39x 417 RC ( e 561) RC ( e 586) RC ( e 598) RC ( e 564) RC ( e 585) RC ( e 61) y 19 89x 413 BE ( e 559) BE ( e 585) BE ( e 598) y 25 4x 374 BE ( e 561) BE ( e 586) BE ( e 598) Confining stress, σ : kpa y 27 82x 399 y 25 79x 384 BE ( e 564) BE ( e 585) BE ( e 61) Confining stress, σ : kpa Fig. 17. Shear stiffness as function of confining stress: GB-A, D mm; GB-B, D mm; GB-D, D mm results is given in Fig. 21, where the BE measurements normalise by the corresponing RC measurements are shown as a function of particle size. Both BE an RC measurements show a tren that G values ecrease slightly with mean particle size D 5, particularly for specimens at the loose state. However, given that the variations are approximately within 1%, an given the uncertainty in-

12 SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS 175 Table 3. Best-fit parameters for shear stiffness measurements Material Test metho Parameters F(e) A n GB-A G (BE) G (RC) F(e) ¼ (2 :17 e) GB-B G 1 þ e (BE) G (RC) GB-D G (BE) G (RC) Toyoura san G (BE) G (RC) Note: A an n are the two parameters in equation (1). volve in the experiments an ata interpretations, this moest size effect may, to a first approximation, be neglecte: that is, the shear stiffness is assume to be size inepenent for the range of particle sizes examine. MICROMECHANICAL CONSIDERATIONS From the micromechanical point of view, the small-strain shear stiffness of an assembly of particles shoul be closely relate to the properties of iscrete particles that interact with each other. To make this point, a grain-scale analysis is presente here. Note that this analysis is not aime to explain the iverse features of the waveforms observe in the BE tests. Consier a simple cubic array of ientical spherical particles with imension l, an subjecte to an isotropic stress ó9, as shown in Fig. 22. The number of spheres in each imension is m, an the sphere is characterise by its raius R, Young s moulus E g, an Poisson s ratio í g or shear moulus G g (¼ E g /2(1 + í g )). Assuming that the Hertz Minlin contact law (Minlin & Deresiewicz, 1953; Duffy & Minlin, 1957) applies, the normal stiffness between two spheres that are in contact can be given as k t n ¼ N v ¼ 3 4 R 1=2 æ! 2=3 N 1=3 æ ¼ 3(1 í2 g ) (5) 4E g where N is the normal contact force between the two spheres, an v is the normal isplacement. Similarly, the shear contact stiffness of the two spheres can be given by k t s ¼ T ä ¼ (4) 4G g ðærnþ 1=3 1 T 1=3 (6) 2 í g ìn where T is the shear contact force, ä is the tangential isplacement between the spheres uner shear, an ì is the friction coefficient between the two spheres. Note that for the cubic array the normal contact force N can be calculate by N ¼ 4ó 9R 2 (7) The normal contact stiffness an the tangential contact stiffness can be further expresse as k t n ¼ A 1ðó 9Þ 1=3 R (8) k t s ¼ A 2ðó 9Þ 1=3 R 1 T 1=3 (9) ìn where " A 1 ¼ 3 3(1 í2 g ) # 2=3 (1) E g A 2 ¼ 2 h i 1=3 12G 2 g 2 í (1 í g) (11) g To etermine the shear stiffness of the cubic array, a small shear stress increment ô is applie to the array, an the corresponing shear strain is then calculate from ª ¼ 4m l ¼ 4m l ôr 2 k t s A 2 Rðó 9 ôr 2 Þ 1=3 ½1 ðt=ìnþš 1=3 (12) The shear stiffness of the simple cubic array can then be given by G ¼ ô ª ¼ 1 2 A 2ðó 9Þ 1=3 1 T 1=3 (13) ìn It is evient from equation (13) that the shear stiffness of the array is inepenent of the raius of the sphere. In other wors, the particle size has no effect on the stiffness of the array. SUMMARY AND CONCLUSIONS This paper has ientifie an aresse two funamental questions on the small-strain shear stiffness of granular material: particle size epenence an the effect of testing metho. For three uniform types of glass beas with mean grain sizes varying from.195 mm to 1.75 mm, series of BE an s have been performe for a range of confining stresses an voi ratios. Micromechanical moelling has also been conucte to offer an unerstaning from the grain scale. The key finings from the laboratory experiments are summarise as follows. The BE measurements of small-strain stiffness are comparable to the RC measurements, with ifferences being less than about 1%. The BE measurements for the fine glass beas ten to be consistently larger than the resonant column measurements, especially at high confining stress levels, whereas this feature becomes less evient for the meium-coarse glass beas, an eventually iminishes for the coarse glass beas. The BE an s both show a tren that the smallstrain stiffness (G ) ecreases slightly with mean particle

13 176 YANG AND GU 35 3 G (BE): MPa G (BE) 1 1 G (RC) G (BE) 9 G (RC) Bener element G (RC): MPa 35 Chains of contact forces Bener element 3 G (BE): MPa G (BE): MPa G (BE) 1 1 G (RC) G (BE) 1 1 G (RC) G (RC): MPa G (BE) 9 G (RC) G (RC): MPa G (BE) 9 G (RC) Fig. 18. Comparisons of shear stiffness measurements from BE an s: GB-A, D mm; GB-B, D 5.92 mm; GB-D, D mm size (D 5 ), particularly for glass beas at the loose state. Given that the variations in the measure stiffness values are generally small, an given the uncertainty in the laboratory experiments, it may be practically assume that small-strain stiffness is size inepenent. This result is confirme by the micromechanics-base analysis using the Hertz Minlin contact law. The waveforms generate in a granular specimen in BE Fig. 19. Schematic illustration of wave paths an force chains in bener element tests tests are complex, epening largely on the excitation frequency, the size of particles, an the confining stress level. The preominant frequency of the output signal tens to increase with excitation frequency, an then approach a limiting value at high frequencies. A threshol frequency marking this transition seems to exist, an both the threshol frequency an the limiting frequency show a tenency to increase with ecreasing particle size. () Uner otherwise similar conitions, a ecrease in particle size can result in high-frequency components in the output signal. The waveforms generate in fine granular specimens isplay a small-amplitue peak preceing the largest peak, an this feature tens to be enhance at high frequencies an at large confining stresses. It is the excursion of the small peak rather than the largest peak that represents the true arrival of the shear wave. (e) The conventional start-to-start metho (using either point S5 or S4) tens to yiel low stiffness values that ecrease as the º/D 5 ratio increases, meaning that the stiffness ecreases with ecreasing particle size for a given excitation frequency. The conventional peak-to-peak metho (using point P2) an the cross-correlation metho also give low stiffness values that, however, ten to increase as the º/D 5 ratio increases, meaning that for a given excitation frequency the stiffness increases with ecreasing particle size. ( f ) The egree of scatter in stiffness measurements ue to ifferent interpretation methos for BE tests appears to epen on the travel istance-to-wavelength ratio (R in ). For R in values ranging from 2 to 4, the scatter seems to be the smallest; higher R in values o not seem to help reuce uncertainty or improve accuracy. In this optimal range of values the start-to-start metho using point S2 works well for fine granular material, whereas the start-to-start metho using point S4 works well for coarse specimens in which point S2 appears to merge with point S4. R in ACKNOWLEDGEMENTS The work presente in this paper was supporte by the University of Hong Kong uner the See Funing for Basic

14 SHEAR STIFFNESS OF GRANULAR MATERIAL AT SMALL STRAINS kpa 24 2 kpa G : MPa 16 kpa 5 kpa 8 BE (5 kpa) RC (5 kpa) BE ( kpa) RC ( kpa) BE (2 kpa) RC (2 kpa) BE (4 kpa) RC (4 kpa) Mean particle size, D 5 : mm 24 4 kpa G : MPa 16 2 kpa kpa 8 5 kpa BE (5 kpa) RC (5 kpa) BE ( kpa) RC ( kpa) BE (2 kpa) RC (2 kpa) BE (4 kpa) RC (4 kpa) Mean particle size, D 5 : mm 24 4 kpa G : MPa 16 2 kpa kpa 8 5 kpa BE (5 kpa) RC (5 kpa) BE ( kpa) RC ( kpa) BE (2 kpa) RC (2 kpa) BE (4 kpa) RC (4 kpa) Mean particle size, D 5 : mm Fig. 2. Variation of shear stiffness with particle size: ense state; meium-ense state; loose state

15 178 YANG AND GU G (BE)/ G (RC) G (BE)/ G (RC) G (BE)/ G (RC) kpa kpa 2 kpa 4 kpa Mean particle size, D 5 : mm kpa kpa 1 2 kpa 4 kpa 1 Mean particle size, D 5 : mm kpa kpa 2 kpa 4 kpa Mean particle size, D 5 : mm Fig. 21. Variation of shear stiffness ratio with particle size: ense state; meium-ense state; loose state l σ ( m m m) spheres Research scheme ( ) an the Outstaning Young Researcher Awar scheme (26 27). This support is gratefully acknowlege. NOTATION A parameter in equation (1) A 1, A 2 constants in equations (8) an (9) σ T N T N R Fig. 22. A cubic array of spheres: Hertz Minlin contact moel C u coefficient of uniformity D 1 particle size that 1% weight of soil are smaller than D 5 mean particle size D 6 particle size that 6% weight of soil are smaller than E g Young s moulus of grains e voi ratio F(e) voi ratio function f in frequency of input signal in BE test f out preominant frequency of output signal in BE test G g shear moulus of grains G (or G max ) soil small-strain shear moulus k t n normal contact stiffness k t s shear contact stiffness l length of simple cubic array L tt travel istance of shear wave m number of spheres in each imension in simple cubic array N normal contact force n stress exponent p a reference stress R raius of sphere R in ratio of wave travel istance to wavelength base on input frequency T shear contact force V s shear wave velocity ª shear strain inuce by shear stress increment ä tangential isplacement between spheres uner shear º wavelength ì friction coefficient between spheres í g Poisson s ratio of grains r mass ensity ó9 effective confining stress ô shear stress increment v normal isplacement between spheres uner compression REFERENCES Bartake, P. P. & Singh, D. N. (27). Stuies on the etermination of shear wave velocity in sans. Geomech. Geoengng 2, No. 1, Brignoli, E. G. M., Gotti, M. & Stokoe, K. H. II. (1996). Measurement of shear waves in laboratory specimens by means of piezoelectric transucers. Geotech. Test. J. 19, No. 4, Cho, G., Dos, J. & Santamarina, J. C. (26). Particle shape effects on packing ensity, stiffness an strength: natural an crushe sans. J. Geotech. Geoenviron. Engng ASCE 132, No. 5, Chung, R. M., Yokel, F. Y. & Drnevich, V. P. (1984). Evaluation of ynamic properties of sans by resonant column testing. Geotech. Test. J. 7, No. 2, Clayton, C. R. I. (211). Stiffness at small strain: research an practice. Géotechnique 61, No. 1, 5 38, 168/geot Clayton, C. R. I., Priest, J. A., Bui, M., Zervos, A. & Kim, S. G. (29). The Stokoe resonant column apparatus: effects of stiffness, mass an specimen fixity. Géotechnique 59, No. 5, , Duffy, J. & Minlin, R. D. (1957). Stress-strain relations an vibrations of a granular meium. J. Appl. Mech. 24, Dyvik, R. & Mashus, C. (1985). Lab measurements of G max using bener element. Proceeings of the ASCE convention on avances in the art of testing soils uner cyclic conitions, Detroit, MI, pp Harin, B. O. & Richart, F. E. (1963). Elastic wave velocities in granular soils. J. Soil Mech. Foun. Engng Div. ASCE 89, No. SM1, Iwasaki, T. & Tatsuoka, F. (1977). Effect of grain size an graing on ynamic shear mouli of san. Soils Foun. 38, No. 1, Jovicic, V., Coop, M. R. & Simic, M. (1996). Objective criteria for etermination of G max from bener element tests. Géotechnique 46, No. 2, , Lee, J. S. & Santamarina, J. C. (25). Bener elements: perform-