In fluence of anisotropic stress path and stress history on stiffness of calcareous sands from Western Australia and the Philippines

Hun He ,Siyue Li ,Kosts Senetkis ,Mtthew Richr Coop ,Songyu Liu

a Institute of Geotechnical Engineering,Southeast University,Nanjing,210096,China

b Department of Architecture and Civil Engineering,City University of Hong Kong,Kowloon,Hong Kong,China

c Department of Architecture and Civil Engineering,Yeung Kin Man Academic Building,Blue Zone 6/F,City University of Hong Kong,Kowloon,Hong Kong,China

d Department of Civil,Environmental and Geomatic Engineering,University College London,London,UK

Keywords:Calcareous soils Dynamic properties Shear stiffness Stress anisotropy Stress history

ABSTRACT Investigation of dynamic properties of carbonate/calcareous soils is important in earthquake and offshore engineering as these soils are commonly encountered in large-scale projects related with energy geomechanics and land reclamation.In this study,the stiffness and stiffness anisotropy of two types of calcareous sands(CS)from the Western Australia and the Philippines were examined using bender elements con figured in different directions in stress path setups.Stiffness measurements were taken on specimens subjected to constant p′compression/extension and biaxial stress paths and additional tests were performed on three types of silica sands with different geological origins and particle shapes,which were used as benchmark materials in the study.Compared with the three brands of silica sands,the stiffness of the CS was found to be more signi ficantly in fluenced by anisotropic loading;an important observation of the experimental results was that stress anisotropy had different weighted in fluences on the stiffness in different directions,thus in fluencing stiffness anisotropy.Comparisons were made between the specimens subjected to complex loading paths,and respected model parameters as suggested from published expressions in the literature.These comparisons further highlighted that calcareous soils have different responses in terms of stiffness,stiffness anisotropy and loading history,compared with that of silica-based sands.

1.Introduction

Calcareous sands(CS)are wide-spread in shallow sea environments,especially in tropical and sub-tropical regions,such as the South China Sea,the Red Sea,the west continental platform of Australia and Bass Strait(Alba and Audibert,1999).They are often encountered in offshore projects,such as oil platforms,sub-sea infrastructures, and man-made islands (e.g. Wees and Chamberlin,1971;McClelland,1988;Brandes,2011;Wang et al.,2011).Compared with typical silica sands,CS consists of particles with lower tensile strength and higher crushability and their properties are highly variant,due to the differences in biogenic types,geological origins,gradings,depths,and/or chemistry of the environment(Fookes,1988;Coop,1990;Miao and Airey,2013).This gives uncertain characteristics to the mechanical behaviour of these soils and could,potentially,result in accidents or collapse of infrastructures founded on carbonate sediments(King and Lodge,1988;Randolph and Gourvenec,2011).Many research works have reported on the mechanical behaviour of CS with emphasis on large-deformation behaviour as well as the crushing behaviour of individual grains and assemblies of grains(Coop,1990;Airey,1993;Coop and Atkinson,1993;Coop et al.,2004;Wang et al.,2011,2017,2018,2020;Miao and Airey,2013;Shahnazari and Rezvani,2013;Wei et al.,2018;Yu,2018;Lv et al.,2019).Despite these efforts and the importance of calcareous soils in large-scale and critical projects related with energy and land reclamation,there are relatively limited number of works investigating the stiffness characteristics of these complex soils,especially under very small strains and by applying complex loading paths,which represent,more effectively,the in situ stress-state of sediments and geological materials subjected to cyclic loads from vibrations or inclined loads from foundations.

The stiffness of soils is critical to evaluate the seismic design of geo-structures and predict seismic ground response(Ishihara,1996;Kramer,1996)as it comprises a key property for soil characterization both under static and dynamic loadings(Clayton,2011).Many studies on calcareous soils have focused on their small-strain stiffness under isotropic compression stress states(He et al.,2019;Morsy et al.,2019;Liu et al.,2020)or medium-to-large deformation behaviour(e.g.Coop,1990;Airey,1993;Coop and Atkinson,1993;Coop et al.,2004;Miao and Airey,2013;He et al.,2017a).In engineering practice,soils are often subjected to anisotropic stress states,such as fresh sediments in shallow seas,subsoil beneath foundations and soil close to slopes or dams.Nonetheless,only a few studies have examined the small-strain stiffness of calcareous soils under anisotropic stress conditions.Wang and Ng(2011),Senetakis and He(2017)and Jafarian and Javdanian(2020)stressed that constant mean effective stress(p′)anisotropic loading affected notably the small-strain stiffness of crushable soils;the aforementioned studies examined completely decomposed granite and CS which,even though they are different materials in terms of origin and composition,they both have highly crushable grains.Therefore,a rather comprehensive understanding of the deformation characteristics(in terms of stiffness)of CS under anisotropic stress state is needed for enhancement of our predictive tools in engineering design as well as to provide further insights into the fundamental behaviour of these complex geo-materials.

To study the effect of stress anisotropy on the small-strain stiffness of calcareous soils,there are mainly two brands of anisotropic stress paths adopted by the majority of researchers,i.e.the constantp′stress path and the biaxial stress path(where“biaxial stress path”is often termed as“axisymmetric loading path”).Senetakis and He(2017)and Jafarian and Javdanian(2020)reported that under constantp′compression stress paths(wherep′is kept constant and the deviatoric stress(q)gradually increased),the small-strain shear modulus(Gmax)of CS increases.Fioravante et al.(2013)and Nanda et al.(2018)examined the small-strain stiffness of CS along biaxial stress paths,where one of the stress components is changed while keeping the others constant,to examine the sole effect of each stress component on stiffness.Previous studies stressed that the notable effect of stress anisotropy on the stiffness of CS cannot be ignored.However,the relationship and difference between the effects of different stress paths on carbonate sand stiffness,as well as the stiffness anisotropy induced by stress anisotropy,which can be critical for modelling,have been overlooked.Studies by Jovicic and Coop(1997)and He et al.(2019)showed that the over-consolidation ratio has a signi ficant in fluence on the magnitude of stiffness of CS subjected to isotropic stress paths,which was reported to be a much less in fluential factor for quartz sands.However,there is a signi ficant gap in the literature in understanding and quantifying the in fluence of anisotropic stress history on the small-strain stiffness of CS,which might be an important factor in modelling the small-strain behaviour of these complex soils.

Facing the insuf ficient understanding of the small-strain stiffness of calcareous soils subjected to anisotropic stress states,a systematic study on the small-strain stiffness of CS from two distinct origins under different stress paths was carried out in the present work.Bender element tests were performed on both loading and unloading phases of(i)constantp′and(ii)biaxial(or axisymmetric)stress paths.Three types of silica sands,which were used as benchmark materials,with different particle shapes and geological origins,were also studied.The empirical model parameters derived from the biaxial stress path tests were used to predict the shear stiffness of soils under isotropic and constantp′stress path,and the results are further discussed and analysed in the subsequent sections.

2.Materials and methods

2.1.Calcareous and silica sands studied

Two types of uncemented CS,from Western Australia(WA)and the Philippines(PH),were examined.Both types of CS originated from shallow sea environments but with different properties in terms of particle size,particle shape and composition.The grading curves and representative scanning electron microscope(SEM)images of both soils are given in Fig.1.Observations of the SEM images suggest that both soils have a biogenic,rather than chemical,origin.The Philippines CS(PH-CS),which has a slightly higher speci fic gravity than the Western Australia CS(WA-CS),predominantly consists of dead coral reefs,while the WA-CS is composed of both dead coral reefs and shells.The basic properties of both CS are listed in Table 1.The particles of the PH-CS were coarser,with a higher mean grain size(D50)value compared to the WA-CS(D50equals to 0.23 mm and 0.50 mm for the WA-CS and the PH-CS,respectively),but the coef ficients of uniformity for both soils were close(1.7 and 1.85 for the WA-CS and the PH-CS,respectively).Based on the uni fied soil classi fication system(USCS)(ASTM D2487-11,2011),both types of geo-materials are classi fied as poorly graded sands(SP).The sphericity(S)and roundness(R)values of the two CSs were quanti fied through visual comparison from SEM images with respect to the empirical chart proposed by Krumbein and Sloss(1963),which was later modi fied by Cho et al.(2006)by introducing the shape descriptor of regularity(ρr)(summarised in Table 1).Regularity equals to the arithmetic mean of roundness and sphericity so that it captures quantitatively the“averaged”in fluence of different particle shape descriptors simultaneously.With lower average values of sphericity and roundness,the PH-CS was found to have grains of more irregular shape compared to the WA-CS.It was indicated that both types of CSs are composed,predominantly,of calcium compounds(normalised weight percentage of calcium exceeding 40%)based on energy dispersive spectroscopy(EDS)analysis(Table 1).Through a series of monotonic shearing tests,the critical state friction angle(ϕcritical)of the PH-CS was found to be 39.4°,which is higher compared with that of the WA-CS(ϕcritical=36.4°).These differences are attributed predominantly to the higher angularity of the grains from the PHCS.

Table 1Basic characteristics of the WA-CS,the PH-CS and the reference silica sands(LBS,SS and CR).

Fig.1.Grading curves of the sands tested and SEM images for the WA-CS and the PHCS.The CSs are illustrated in solid lines and the silica sands are presented in dashed lines.

Three types of silica sands from different geological environments were also tested in the current study for comparison,including two silica sands with rounded grains:Leighton Buzzard sand(LBS),Sydney sand(SS)and a sand-sized uniform fraction of crushed rock(CR)(grain sizes:1.18-2.36 mm)with angular particles.The grading curves of the silica sands are also presented in Fig.1.SS is a natural material found in New South Wales,Australia,LBS is a material derived from clastic-sedimentary rock from the UK,and the CR was derived from a basaltic-type fine-grained igneous rock.The three types of silica-based sands represent three broad types of materials of comparatively strong grains but with various particle shapes and compositions.Basic properties of the silica sands,including speci fic gravity,mean grain size,coef ficient of uniformity,coef ficient of curvature,and particle shape parameters,are listed in Table 1.

2.2.Experimental method and specimen preparation

Fig.2.Schematic illustration of the dynamic triaxial system with the local strain gauge(specimen diameter:50 mm;specimen height:100 mm).

A stress-path triaxial apparatus with maximum capacity of 1 MPa,previously described by He et al.(2017b),was used in the study.The apparatus was supplied by GDS Inc.UK.A schematic plot of the apparatus is given in Fig.2.The base pedestal and top cap were equipped with bender elements that are used to measure the shear wave velocities(Vs,vh)propagating in the vertical(along the axis of the specimen)direction.The apparatus is also equipped with a back volume controller.The radial strain gauges paired with the vertically mounted linear variable differential transformers(LVDTs)were used as an alternative technique to record the volume changes during the tests,especially for dry specimens.

A custom-modi fied Bishop and Wesley(BW)triaxial apparatus(which was introduced by Li and Senetakis(2017)and Todisco et al.(2018)).with embedded bender elements was also used in the study(a schematic plot of the apparatus is given in Fig.3).This apparatus was supplied by Imperial College London,UK.A custombuilt local radial strain gauge(after Ackerley et al.,2016)was used in the BW triaxial system to monitor the radial strains of the specimens.Both vertical and lateral bender element systems were implemented into the BW triaxial apparatus.The lateral bender element system consisted of two pairs of piezo-elements con figured in T-shape,i.e.one pair is placed vertically and the other pair is placed horizontally,which can be used to measureVs,hv(the velocity of the shear waves which travel horizontally and induce vertical particle motion)andVs,hh(the velocity of shear waves which travel horizontally and induce horizontal particle motion),respectively,and the con figuration was initially designed by Pennington et al.(2001).A schematic plot of the arrangements of the vertical and lateral bender elements is given in Fig.4.Representative illustration of the transmitted shear wave is given in Appendix A,where the first time of arrival method for signal interpretation is also shown.The validation of the signal interpretation method was presented by He and Senetakis(2016)and He et al.(2019).

The specimens were prepared in a dry state in layers using the dry compaction method.The initial void ratio of the specimen was controlled by varying the applied compaction energy.Most of the specimens were tested in a dry state,while some of the WA-CS specimens were tested in a fully saturated state.Notably,only vertical bender element tests were performed for the WA-CS specimens,while both vertical and lateral bender element tests were performed for the PH-CS specimens.

2.3.Testing program and stress paths

The details of the specimens and the applied stress paths are given in Table 2 for the WA-CS specimens and in Table 3 for the PHCS and the silica sand specimens.The anisotropic stress paths adopted in the study are illustrated in Fig.5,where the markers represent the stress states where the bender element tests were performed.Measurements of wave velocities and stiffness were performed during both loading and unloading of the anisotropic stress paths for most of the specimens,so that to explore the influence of loading history apart from the in fluence of stress path.

Apart from the constantp′stress paths,the small-strain shear moduli of the PH-CS specimens,includingGvh,GhvandGhh(shear moduli derived fromVs,vh,Vs,hvandVs,hh,respectively),were also measured along constantσ′aand constantσ′rstress paths,during which the magnitude of one stress component is kept constant while altering the other stepwise(as illustrated in Fig.5b).The constantσ′r-constantσ′atype of stress path(shown in Fig.5c)was adopted to avoid extension stress states,whereσ′r＞σ′a,along the stress path.

3.Results and discussion

3.1.Results

3.1.1.Constantp′anisotropiccompression/extensionstresspaths testresults-loadingphase

The stiffness-stress ratio relationship of a soil subjected to a constantp′anisotropic stress path test can be expressed by(Payan et al.,2016a):

whereGvhaniandGvhisoare the small-strain shear moduli under constantp′anisotropic stress state and isotropic stress state,respectively;eaniandeisoare the corresponding void ratios;and αG,vhis a model constant.The stiffness is normalised with respect to a void ratio function(Eq.(2))to eliminate possible in fluences from the changes in void ratio among different specimens.The termηonthe right side of Eq.(1)represents the stress ratio(equal toq/p′).From a set of anisotropic compression loading tests on different silica sands,Payan et al.(2016a)reported that the exponentαG,vhis a function of the coef ficient of uniformity(Cu)and particle shape(expressed with the shape descriptor of regularity,ρr),and they proposed the following equation:

The normalised shear modulus([Gvhani/f(eani)]/[Gvhiso/f(eiso)])againstη+1 plots of representative specimens from two types of CS are given in Fig.6a.Based on theCuandρrof the two types of CS(WA-CS:Cu=1.7,ρr=0.565;PH-CS:Cu=1.85,ρr=0.415),theanisotropic compression loadingαG,vhvalues for WA-CS and PH-CS are predicted to be 0.06 and 0.11,respectively,from Eq.(3).Both CS exhibited higherαG,vhexponent values in comparison with what the empirical equation predicted.Since the empirical equation proposed by Payan et al.(2016a)was developed based on quartz soils,it is understood that the shear stiffness of the CS is more greatly in fluenced by the stress anisotropy compared with quartz sands of the same grading and particle shape.It is observed that the stiffness componentGvhof the PH-CS specimens had slightly higher sensitivity to the anisotropic compression compared with the WACS specimens,which is manifested by the higher averageαG,vhexponent value of the PH-CS specimens(averageαG,vhof PH-CS equals to 0.144)compared with that of the WA-CS specimens(averageαG,vhof WA-CS equals to 0.11),as listed in Tables 4 and 5.Note that at a givenp′,the axial strains induced by the constantp′anisotropic loading for the two brands of CS were found to be close.Thus,the higherαG,vhvalues observed from the PH-CS specimens were not likely induced by strains,but were rather in fluenced by the lower grain regularity of the PH-CS in comparison with the WACS(Payan et al.,2016a).No clear effect of the saturation state(i.e.dry or fully saturated specimens)on the small-strain stiffness under anisotropic stress states was identi fied,given the close exponent values of the dry and saturated WA-CS specimens.

Table 2Specimen details of WA-CS.

For constantp′anisotropic extension tests,the stiffness of representative WA-CS specimens,which are illustrated by hollow markers in Fig.6b,dropped by around 0-5%,while the stiffness for the PH-CS specimens slightly increased(presented in solid markers in Fig.6b),asηdecreased from 0 to-0.5.In general,the stiffness componentGvhfor both CS was only slightly in fluenced by the decrease of the stress ratio(η=q/p′)under extension stress states.From Tables 4 and 5,it can be noticed that the constantp′extension αG,vhvalues for both sands were close to zero,but the average value for the WA-CS was positive,while that for the PH-CS was negative.

3.1.2.Constantp′anisotropiccompression/extensionstresspaths testresults-unloadingphase

Representative plots of normalised stiffness againstη+1 for both loading and unloading phases of the anisotropic stress states are given in Fig.7a and b for WA-CS and PH-CS.These data suggest that the stiffness of both soils under the unloading stage deviated from the loading stage,i.e.the anisotropic loading history affects notably the magnitude of the stiffness.Adopting a similar approach as the data analysis during the loading phase,the test results of the unloading phase of both sands were fitted by the power law expression given in Eq.(1),and the derived values of the unloading exponent(αUG,vh)are listed in Tables 4 and 5 The average unloading exponent values for anisotropic compression are 0.012 and-0.029 for WA-CS and PH-CS,respectively,representing a limited change in stiffness during the unloading process of the anisotropic compression.On the other hand,the unloading of the anisotropic extension affected more pronouncedly the stiffnessGvhof both sands,with the averageαUG,vhvalues for the anisotropic extension stress paths to be equal to 0.115 and 0.120 for WA-CS and PH-CS,respectively.It is therefore understood that the stress history may have a different in fluence on the stiffness in different directions,contributing in this way to further anisotropic behaviour of the calcareous sands.This may have important implications in geomechanics modelling and geophysical characterization of sediments,as this behaviour has not been reported in the literature for quartz-based sands.

Table 3Specimen details of PH-CS,LBS,SS and CR.

Fig.5.Illustration of the anisotropic stress paths followed(the data points represent the stress states where bender element tests were conducted):(a)Constant p′compression and extension stress paths on the q-p′plane,(b)Constantσ′a and constantσ′r stress paths on theσ′a-σ′r plane,and(c)Constantσ′r-constantσ′a stress path on theσ′a-σ′r plane.

Fig.6.Representative plots of normalised G vh againstη+1 for both types of CS subjected to(a)anisotropic compression stress paths and(b)anisotropic extension stress paths.

The effect of anisotropic stress history on the stiffness of specimens from LBS and the PH-CS(PH-01)is illustrated in Fig.8,in terms of normalisedGvhagainstη+1 plots.Gvhof the LBS specimen was less affected by the stress anisotropy compared with that of the PH-CS specimen,with a maximum shear modulus increase of the order of 4%-5%atη=1,and the anisotropic compression loading exponent αG,vhof the LBS specimen was found to be equal to 0.073.Different fromthe tests on the CS,no signi ficant effect of the anisotropic loading history was observed for the LBS specimen.The stiffness componentGvhalong the unloading path was found to be slightly lower than that of the corresponding loading phase of the anisotropic stress path(Fig.8).The lower value of the exponentαG,vhand the small effect of the anisotropic loading history on the LBS specimen are attributed,predominantly,to the less signi ficant fabric changes induced by the anisotropic stress in the LBS specimen compared with the more pronounced in fluences observed for the CS.

The stiffness component,Ghv,of the PH-CS specimens was also measured along constantp′anisotropic stress paths,and thestiffness-stress ratio relationship ofGhvwas similar to that ofGvh.By plotting the exponent values ofGvhandGhvof individual tests in Fig.9,it was revealed that theαG,hvandαG,vhexponent values for both the anisotropic compression and extension stress paths are close in magnitude.However,the compression exponentαG,hvwas always slightly lower compared with the exponentαG,vh(the averageαG,hvandαG,vhvalues equal to 0.107 and 0.144,respectively,from Table 5).These observations agree with the data presented by Gu et al.(2017),who found that theGhvcomponent was less affected by the axial compressive stress anisotropy compared to theGvhcomponent based on numerical analyses using the discrete element method(DEM).

Table 4Constant p′anisotropic stress path test details for WA-CS.

3.1.3.Stiffnessanisotropyunderconstantp′anisotropicstresspaths tests

Systematic fabric anisotropy was not observed in the reconstituted PH-CS specimens under isotropic stress conditions from the recent study by He et al.(2019).In the present study,the stiffness anisotropy induced by anisotropic stress was examined for one of the PH-CS specimens and the LBS specimen,by comparing theGhvandGhhmeasured from the two sets of lateral bender elements in the BW stress path triaxial system.The data in Fig.10a suggest that at isotropic state,whereη+1 equals 1,no signi ficant stiffness anisotropy could be observed(Ghh/Ghvis close to unity),but the stress anisotropy induced noticeable stiffness anisotropy.As η+1 reached a value of 2,the ratioGhh/Ghvdecreased to 0.83.This observation can be explained by the stiffness-stress relationship in different directions.Ghhis affected predominantly by the horizontal(radial)stress component,whileGhvis affected by both the vertical and horizontal stress components since the particle contacts in the horizontal direction tend to be weakened by the application of the anisotropic compression(e.g.Oda et al.,1985;Rothenburg and Bathurst,1992;Cheng,2018).The ratio betweenGhhandGhvduring the unloading stages of the anisotropic compression is also plotted in Fig.10a.The data suggest that after the removal of the stress anisotropy,i.e.the stress ratioηis reduced from unity to zero,the stiffness anisotropy was not completely removed for the PH-CS specimen,with the ratioGhh/Ghvreaching a value of 0.92.From DEManalyses,Gu et al.(2013)reported that the stiffness anisotropy might remain in the specimen after the anisotropic loading is removed because of the permanent fabric changes caused by the deviatoric stress.

As for the test on the LBS,it was observed that the ratioGhh/Ghvreached 0.81 atη=1,representing evident stiffness anisotropy,which is similar to what was found for the PH-CS specimen.However,the stiffness anisotropy was removed entirely for the LBS specimen after the anisotropic stress was unloaded(Fig.10b).This difference between the silica sand and the CS,in terms of the effect of anisotropic loading history on the stiffness anisotropy,has been overlooked in the literature,but the data from the present study suggest that these sands with different origins have a completely different behaviour against anisotropic loading history.

Fig.11 shows a plot of the axial strain againstη+1 for both PH-2 and LBS specimens.These data suggest that although the maximum induced strain was relatively small in magnitude(0.38%for the calcareous sand atη=1 and 0.15%for the silica sand),plastic deformations were dominant in both types of sands(PH-2 and LBS).This“residual”stiffness anisotropy of the CS may be attributed to the higher angularity and lower tensile strength of its particles,which resulted in irrecoverable fabric changes and strain induced anisotropy during the application of the anisotropic stress path.For the LBS,which has more rounded grains with higher tensile strength,the stiffness anisotropy is considered to be predominantly the result of stress induced anisotropy,since the plastic strains remained in the specimens did not lead to notable stiffness anisotropy.This hypothesis can be further supported by previous studies on the in fluence of particle shape on the stiffness-pressure relationship of granular materials subjected to anisotropic stress state(Cho et al.,2006;Payan et al.,2016a).

3.1.4.Testresultsofconstantσ′aandconstantσ′rstresspath

Under biaxial and true triaxial loading,it has been reported that the predominant stress components which in fluence shear modulus are the principal stress along the wave propagation direction,denoted asσ′i,and the principal stress along the particle motion direction,denoted asσ′j,while the out of plane stress has a negligible in fluence(Roesler,1979;Knox et al.,1982;Stokoe et al.,1985;Bellotti et al.,1996;Santamarina and Cascante;1996;Fioravante et al.,2013).Based on the studies by Roesler(1979),Yu and Richart(1984),Ni(1987)and Hardin and Blandford(1989),the following general empirical formula has been proposed:

whereAij,ni,andnjare the constants;whileniandnjdescribe the sensitivity ofGmaxto the stress along the wave propagation direction and the particle motion direction,respectively.

Five specimens from the PH-CS,one specimen from the silica SS and one specimen from the silica CR were tested following constant σ′aor constantσ′rstress paths so that to examine the effect ofσ′iand σ′j,individually,on the different stiffness components.Representative results on the PH-CS are illustrated in Fig.12a and b for constant σ′rand constantσ′atests,respectively.For the PH-CS,the averageni(=0.353)was found to be higher than the averagenjvalue(=0.278),which suggests that the stress in the wave propagation direction has a higher impact on stiffness in comparison with the in fluence of the stress in the particle motion direction.The data in the unloading phases in Fig.12 show that the stiffness was increased by the anisotropic stress loading history and that the exponent values are smaller in magnitude during the unloading process(uniandunj)compared with the loading process(niandnj).A summary of the data for the loading and unloading exponent values for all the constantσ′aand constantσ′rstress path tests is given in Table 6.

The(ni,nj)values for the SSand the CRspecimens were found to be equal to(0.207,0.188)and(0.348,0.265),respectively,as illustrated in Fig.13.The data in this figure suggest that the exponent valuesniof thesilica sands were also greater in magnitude compared with their correspondingnjvalues(Table 6).Note that the horizontal axis of Fig.13 representsσ′awhen depicting the results along the constantσ′rstress path(lower curve in the figure),while it representsσ′rwhen depicting the results along the constantσ′astress path(upper curve in the figure).Although for the two types of silica sands and the PH-CS calcareous sand,thenivalues are always higher than thenjvalues,it was observed that compared with the angular sands,the difference between theniandnjof the SS was much smaller(speci fically,the averageni-njvalues for the PH-CS,the CRand the SS were 0.075,0.083 and 0.019,respectively).The positiveni-njvalues imply that the stress in the wave propagation direction has a stronger in fluence on smallstrain shear stiffness than the stress in the direction of particle motion.In the studies by Roesler(1979),Knox et al.(1982)and Viggiani and Atkinson(1995),the magnitude ofniwas reported to be greater than that ofnj.However,Yu and Richart(1984),Santamarina and Cascante(1996),Fioravante(2000)and Fioravante et al.(2013)observed the opposite trends.Yu and Richart(1984)indicated that the end restraint of their resonant column may also have an in fluence,while the difference betweenniandnjvalues is found to be dependent on the mineralogy/composition and particle shape from the current study.

Table 5Constant p’anisotropic stress path test details for PH-CS.

Table 6Exponent values from PH-CS,SS and CR tests.

Fig.7.Representative plots of normalised G vh againstη+1 for both types of CS during loading and unloading of constant p′anisotropic compression and extension:(a)WA-02(anisotropic compression)and WA-12(anisotropic extension),and(b)PH-01(anisotropic compression)and PH-06(anisotropic extension).

The loading and unloading exponent valuesni,uni,njandunjare compared in Fig.14.As illustrated in Fig.13a and b,during the unloading process,the sensitivity of the stiffness to the stress component is reduced compared with that in the loading process.Bothuniandunjdecreased more than 65%compared with the corresponding values in the loading process for the PH-CS,with the exponentuni(averageuni=0.111)to be greater than the exponentunj(averageunj=0.075).Examining a washed quartz mortar sand,Knox et al.(1982)did not observe any signi ficant drop of the exponent values from their constantσ′rand constantσ′aloading-unloading tests.

3.2.Discussion

3.2.1.Comparisonbetweenthesummationofniandnjwiththe isotropicexponentnG

At isotropic stress state,the stress components are equal in magnitude(σ′i=σ′j=p′),and the following expression can be derived from Eq.(4):

Fig.8.Comparison of normalised G vh of constant p′compression tests during loading and unloading stress paths of LBS and PH-CS.

Fig.9.αG,vh andαG,hv against p′for four tests of PH-CS.

The right side of the equation is widely adopted to describe the stiffness of soils under isotropic stress paths,and the term(ni+nj)can be denoted asnG.Thus,theoretically,the exponentnGfor isotropic compression should be equal to the summation ofniandnj.

Previous works(e.g.Roesler,1979;Knox et al.,1982;Viggiani and Atkinson,1995;Fioravante et al.,2013)found rather good agreements between the summation ofniandnjwith the exponentnG.In the current study,the summation ofniandnj(ni+nj=0.4)for SS was found to be close to thenGvalue reported by Payan et al.(2016b)(taking into account the given particle shape of the SS,would predict a valuenG=0.43).However,the summation of the averageniandnjof the PH-CS and the CR in this study are 0.631 and 0.613,respectively,which are both much higher than the average exponent values(PH-CS:nG=0.48,CR:nG=0.53)that would be predicted from the proposed expressions by He et al.(2019)and Payan et al.(2016b)for the condition of isotropic stress path.Under an anisotropic stress state,the fabric of soils with angular particles is hypothesized to be altered more pronouncedly compared to soils with rounded particles,which may be,predominantly,because of a higher level of interlocking,lower coordination number and stress redistribution(Gu et al.,2013,2017).This could explain the differences between the summation ofniandnjand the isotropicnGfor the PH-CS and the silica CR.

Fig.11.Axial strain againstη+1 during constant p′anisotropic compression tests for specimens LBS and PH-2.

3.2.2.Predictionofthestiffnessofthesandssubjectedtoconstantp′stresspathswiththemeasuredniandnjvalues

With the knowledge of theniandnjvalues of a geo-material,the shear modulus at any given stress state should,theoretically,be predicted from Eq.(4).At an anisotropic stress state,the smallstrain shear modulus(Gvhani)could be expressed by the following expression based on Eq.(4):

whereσ′i,aniandσ′j,aniare the effective stresses in the direction of wave propagation and in the direction of particle motion at anisotropic stress state.Substituting the anisotropic and isotropic stiffness(Gvhani,Gvhiso)of the left side of Eq.(1)with the calculation given in Eq.(6),the following expression can be derived:

Fig.10.G hh/G hv againstη+1 during constant p′anisotropic compression tests for(a)a representative specimen of the PH-CS,and(b)an LBS specimen.

Fig.12.Representative results of constantσ′a and constantσ′r tests:(a)Normalised shear modulus G vh/f(e)against effective axial stressσ′a for specimen PH-10,and(b)Normalised shear modulus G vh/f(e)against effective radial stressσ′r for specimen PH-11.

An attempt was made to predict the stiffness-stress ratio relationship of the PH-CS under constantp′stress paths based on the measuredniandnjvalues from Eq.(7),and the predicted values are compared with the constantp′stress path test results(based on the laboratory measurements). Taking constantp′anisotropic compression as an example,when the stress ratio(η=q/p′)equals 1.2,σ′i,ani=1.8σ′i,isoandσ′j,ani=0.6σ′j,iso,under which cases,Eq.(7)can be re-written as

Taking the averageni(=0.353)andnj(=0.278)exponent values into the equation,the normalised stiffness([Gvhani/f(eani)]/[Gvhiso/f(eiso)])is predicted to be 1.068 atη=1.2,i.e.the normalised stiffness with respect to void ratio would increase by 6.8%as the stress ratio increases from 0 to 1.2.From the set of constantp′stress path tests,the measured averageαG,vhis 0.144,which makes the right side of the equation equal 1.12,suggesting that the measured stiffness would increase by 12%as the stress ratio increases from 0 to 1.2.For SS,the stiffness componentGvhshould increase by 2.6%asηincreases from 0 to 1.2 based on the measuredni(=0.201)andnj(=0.188)values,and this predicted magnitude of stiffness increase matches well with the predicted values based on the proposed expression by Payan et al.(2016a).

Fig.13.Representative results of constantσ′a and constantσ′r tests:(a)Normalised shear modulus G vh/f(e)againstσ′a andσ′r for Sydney sand specimen,and(b)Normalised shear modulus G vh/f(e)againstσ′a andσ′r for crushed rock specimen.

Fig.14.Average loading and unloading ni,nj,uni and unj values of the PH-CS with the corresponding standard deviation illustrated.

The results from the constantσ′aand constantσ′rstress path tests suggest that the unloading exponent values(uniandunj)were notably different from the loading exponent values(niandnj).Considering thatσ′rdecreases during a constantp′compression stress path andσ′adecreases during a constantp′extension stress path,the unloading exponent values(uniorunj),theoretically,should be adopted for the stiffness prediction where the stress component decreases along the stress paths.A flow chart is plotted in Fig.15a to illustrate the predictions of change of the stiffness components(GvhandGhv)under both constantp′anisotropic compression and extension with different combinations ofni,nj,uniandunjvalues.The predicted values of the percentage of stiffness change are compared with the“measured”values in Fig.15b,where the“measured”change of shear stiffness under anisotropic stress states was calculated with the averageαGvalues derived based on constantp′stress path tests.It is noticed that the predictions ofGvhunder constantp′extension stress state,as well asGhvunder both constantp′compression and extension stress states,based only on loading exponents(ni,nj)notably deviate from the measured values.However,adopting the“loading and unloading exponents”to predict the stiffness under constantp′anisotropic stress states yielded rather satisfactorily accurate results.

Under constantp′anisotropic stress states,the stiffness-stress ratio(η)relationship of sand depends,predominantly,on the relationship betweenni(oruni)andnj(orunj)of the soil.For a given material,if theniis much larger than theunj,the shear stiffness componentGvhwill increase along the constantp′compression stress path(whereq/p′increases).Otherwise(whenniis smaller thanunj),the stiffness componentGvhwill be not sensitive to constantp′anisotropic compression.A series of theoretical plots of normalised stiffness([Gvhani/f(eani)]/[Gvhiso/f(eiso)])against the stress ratio(η+1)is presented in Fig.16,where the effect of the change ofunjon the stiffness-stress ratio relationship under constantp′compression stress path is illustrated.Asunjincreased from 0.1 to 0.3 while keepingniconstant at 0.2,the normalised stiffness decreased under constantp′anisotropic compression stress states,which suggests lowerαGvalues.Based on Eq.(3),Payan et al.(2016a)reported thatαGvalues were found to be higher for sands with more angular particles and/or with a higher coef ficient of uniformity,and higherαGvalues further imply that theniexponent should be greater than theunjexponent from the analysis in Fig.13.This analysis is supported by the experimental work conducted by Payan and Chenari(2019).

Fig.15.(a)Flow chart of prediction of the percentage of stiffness components(G vh and G hv)change under constant p′anisotropic compression and extension stress states with different combinations of exponent values(ni,nj,uni and unj),and(b)Bar chart of the comparison between the predicted and measured percentage of stiffness components change.

Fig.16.Theoretical normalised G vh-(η+1)plot with different ni and nj combinations.

4.Conclusions

The stiffness at small strains of two CS,i.e.the WA-CS and the PH-CS,subjected to different types of stress paths was examined.The results stemming from bender element tests on the two CSs were compared with additional tests on three silica sands from different geological origins and different particle shapes,and these silica sands were used as benchmark materials in the study.The conclusions can be drawn as follows:

(1)Under constantp′compression stress paths,the stiffness componentGvhof the PH-CS appeared to increase more pronouncedly compared with the WA-CS because of the higher particle angularity of the PH-CS.Gvhof both sands was slightly affected by constantp′anisotropic extension stress states.The anisotropic stress history in fluenced notably the stiffness of both CSs.

(2)Stiffness anisotropy was observed under constantp′stress anisotropy for the PH-CS through the measurements of the stiffness ratioGhh/Ghv,and after the removal of the anisotropic stress,the stiffness anisotropy was partly remained due to strain-induced fabric anisotropy.For the LBS,which is a kind of silica sand with rounded grains,the stiffness anisotropy was also evident under anisotropic stress states,but the stiffness anisotropy was negligible after the stress anisotropy was removed.

(3)Through constant axial stress(σ′a)or radial stress(σ′r)stress path tests(also referred to as biaxial stress path tests),the exponentni,which expresses the sensitivity of stiffness to the stress component in the direction of wave propagation,andnj,which expresses the sensitivity of stiffness to the stress component in the direction of particle motion,were measured.

(4)It was observed that although the values of the exponentniwere higher than those of the exponentnjfor the PH-CS,the SS and CR specimens,the difference in magnitude between theniandnjvalues of the SS was the smallest among the three soils,due to the rounder grain shape of this sand compared with the PH-CS and CR.

(5)Along the unloading of the biaxial stress paths,the CS exhibited much lower unloading exponent values(uniandunj)compared with the values observed during the loading process(niandnj).

(6)Comparing the anisotropic stress path and the isotropic stress path test results,the summations ofniandnjof the PH-CS and CR,which both have angular particles,were found to be greater than the correspondingnGvalues derived fromisotropic stress path tests.Nevertheless,theni+njvalues of the SS,which consists of rounded particles,matched with thenGvalues as predicted from expressions proposed in the literature.

(7)It was proved necessary to adopt the unloading exponent values(uniandunj)derived from the biaxial tests,where the stress component decreases,to predict the stiffness satisfactorily along the constantp′compression or extension stress paths.Based on the analysis of the present study together with the existing data published in the literature,it is suggested that for silica sands with angular particles or high coef ficients of uniformity,the exponentniof the soil should be higher in magnitude compared with itsunjvalues,which,in turn,indicates that the stiffness is sensitive to constantp′anisotropic loading.

Declarationofcompetinginterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgments

The work described in this paper was fully supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region,China(Grant No.CityU 11210419),the Natural Science Foundation of Jiangsu Province (Grant No.BK20200405),and the National Natural Science Foundation of China(Grant No.52008098).

AppendixA.Supplementarydata

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2021.03.015.