DEFORMATION CHARACTERISTICS OF HOSTUN AND TOYOURA SAND IN THE QUASI-ELASTIC DOMAIN

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1 Paamèts d calcul géotchiqu. aga (d.), Psss d l ENPC/LCPC, Pais DEFORATION CHARACTERISTICS OF HOSTUN AND TOYOURA SAND IN THE QUASI-ELASTIC DOAIN ODÉLISATION DU COPORTEENT DES SABLES D HOSTUN ET DE TOYOURA DANS LE DOAINE DE COPORTEENT QUASI-ÉLASTIQUE H. Goffoy, H. Di Bdtto, C. Sauzat, A. Dutti DGCB, Ecol Natioal ds Tavaux Publics, 69 Vaulx Vli, Fac ABSTRACT A tosioal hollow cylid dvic calld T4C StaDy was dvlopd to study th bhaviou of d i a wid ag of stais (fom appox. -6 to - ). Th xistc of a limit tso has b obsvd fo vy small amplitud cycls. Eight tms of th compliac tso ca b dtmid fom T4C StaDy tsts. A w hypolastic modl calld DBGS, fo this limit tso is poposd to tak ito accout som xpimtal obsvatios. Simulatios a pstd fo two kids of ds i th cas of gal stss-stai coditio. RÉSUÉ U ouvl appail d'si su épouvtt cylidiqu cus applé T4C StaDy st pésté. Il pmt d'appliqu ds cycls compssio ou/t tosio, d'amplitud d défomatio compis t qulqus -6 t qulqus -. Ls ésultats péstés fouisst difféts tms diagoaux t o diagoaux du tsu héologiqu équivalt. U ouvau modèl hypoélastiqu «DBGS» pat compt ls obsvatios xpéimtals st poposé. Ds compaaisos t simulatios t xpéimtatios sot péstés pou dux bls difféts.. Itoductio A hollow cylid xpimt pototyp has b dvlopd at th Dpatmt of Civil Egiig (DGCB) of Ecol Natioal ds Tavaux Publics d l Etat (ENTPE) thaks to a coopatio with Elcticité d Fac (EDF). This quipmt calld T4C StaDy is a appaatus of tosio, compssio, cofimt o hollow cylid mpl allowig static ad dyamic loadig. Th iducd aisotopy, th otatio of picipl stss axs ad th ifluc of th 3 dimsioal stss path loadig a studid. Th ag of masuabl stais covs th vy small to th lag stai domais. oov, som wav popagatio tsts a caid out to ivstigat th bhaviou ud dyamic loadig. This appaatus T4C StaDy is dscibd aftwads. Bhaviou of soils is stogly o-lia ad ivsibl as soo as th stai amplitud is lag (mo tha som -3 ). Howv, it sms possibl to dfi a stai domai aoud a giv stat of stss/stai, fo which th latio btw stss ad stai is quasi lastic. Expimtal sults obtaid by may authos (Lo Psti & al., 997 ; Tatsuoka & al., 997 ; Di Bdtto & al., 997 ; Cazacliu & al., 998 ; Jadi & al, 999 ; Di Bdtto & al., ) i th cas of uiaxial loadig cofim that, fo ds, th quasi lastic domai is chaactizd by stai amplitud low tha som -5 ( vy small stai). Isid this domai, th bhaviou vals to b quasi lia ad a w gal aisotopic hypolastic fomulatio ( DBGS modl) is poposd. Expimtal sults a pstd fo two difft ds, Hostu d ad Toyoua d. Compaisos btw xpimtal T4C StaDy sults ad umical simulatios with th DBGS law a poposd i this small stai domai.. Hollow cylid appaatus : T4C StaDy Th hollow cylid appaatus T4C StaDy (Figu ) allows th ivstigatio of d bhaviou fom vy small to lag stai domais. Idpdt applicatio of tosio, compssio / xtsio ad cofimt loadig is possibl fo mootoous o cyclic paths. 75

2 // Th systms of masumt ad cotol hav a high dg of accuacy, whil still allowig o to follow th loadig up to lag stais. Ivstigatio of hological bhaviou is possibl i a wid stai domai (fom appox. -6 to - ) by cosidig ay combiatio of compssio / xtsio ad tosio, statig fom ay poit o th global stss-stai cuv. Rigiditis ad hysttic dampig at ach of ths poits ca b giv. oov, pizolctic ssos allow a dyamic ivstigatio by wav popagatio, at ay stss ad stai stat availabl with th appaatus. Cyclic ad dyamic moduli a obtaid fo quasi-static ad dyamic loadig i th vy small stai domai. Schm of "T4CStady" appaatus Systm of stais masumt Pizolctic ssos Pisto axis Suspsio of ig (x3) / // / Cofiig cll Rigs at two lvls "high" ad "low" // // // // Ital bals Top plat Alumiium pap tagt (adial stai) No-cotact ssos (x) // // // // // // // // // // Pizolctic ssos Sampl Bottom plat Suppot of sha stai masumt sso Suppot of adial stai masumt sso 3 6i 6i 6i Alumiium tagt scwd to ig Tagts fo vtical stai masumt Rig hagig by 3 stips Pai (x3) of ssos ("i" ad "out") masuig adial stai Tagts fo sha stai masumt Figu. Hollow cylid appaatus dsciptio 3. Tstig pocdu Th mpl is fist mad by pluviatio. Eith loos o ds d a cosidd. Th iitial void atio ags btw.63 ad.95. Th mpl is th cofid to a pssu P of ith 5 kpa o 8 kpa o 3 kpa. Fom this iitial isotopic stat a loadig path is applid. Th th tst cosists i patig th stags at difft ad succssiv lvls of stss (Figu ) : Stag A quasi-static loadig (volutio loadig) is applid to th mpl, which achs a pdfomd stat. Subsqutly w will call this stat th ivstigatio poit. Th volutio loadig cospods to o of th stss paths, that ca b pfomd with th hollow cylid appaatus. Fo th cosidd tst campaig, th typs of volutio loadig w applid: classical tiaxial loadig (typ C), pu tosio fom a isotopic stat (typ T) ad tosio aft axial loadig (typ K, pu tosio fom a aisotopic stat). Th, at ach ivstigatio poit, stsss a kpt costat (btw foty miuts ad som hous). Stag At ach ivstigatio poit, th bhaviou of th mpl i th vy small ad th mdium stai domais is chaactisd by a applicatio of quasi-static cyclic loadig with a stai amplitud agig fom appox. -6 to -4. Accodig to th dictio of loadig, th cyclic paths a distiguishd as i) axial, ii) tosioal, ad iii) bi-axial. Duig th axial cyclic loadig, axial stss vais at costat sha ad cofimt stsss. Th tosioal cyclic loadig cosists i shaig th mpl at costat axial ad 76

3 cofimt stsss. Lastly, duig th bi-axial cyclic path, axial ad sha stsss vay at costat cofimt stss, dscibig a sgmt i th stss spac. Stag 3 Vlocity masumts fo wavs mittd by th bd ad compssio pizolctic lmts (dyamic loadig) a also pfomd at ach ivstigatio poit. Duig th cp piod (d of stag ), viscous stai occus. Aft this piod, th viscous chaactistics of bhaviou fad ad oly o-viscous poptis a cosidd. Th, th small amplitud cycls a aly stabilizd ad povid dictly th o-viscous poptis (Di Bdtto & al., 997; Cazacliu & al., 998; Di Bdtto & al., 999). Th masud stai icmt is th th o-viscous o : δε v. Th aalysis ad modllig of th o-viscous pat i th small stai domai a pstd i th followig paagaphs. Applid stss (o τ ) Cp piods ootoous loadig cuv Cycls i difft dictios (axial,tosioal ad bi-axial) Ivstigatio poits sud stai ε (o γ) Figu. Gal shap of a pfomd tst i th stai-stss axs. 4. Expimtal sis A fist xpimtal sis was caid out o mdium dy Hostu d. Th chos volutio loadig paths (paagaph 3) w classical tiaxial loadig (C typ), pu tosio (T typ) ad tiaxial loadig (to ach a aisotopic stat) followd by a pu tosio (K typ). Rctly, fou tsts w coductd o Toyoua d (Toyoua d is dscibd i Goto, 986 ; Hoqu, 996 ad Yasi, 999). Th cofiig pssu was 8 kpa. Th chos volutio loadig paths w C typ ad K typ. Th Hostu d ad th Toyoua d a fi gaid matials. Thi paticl shaps a suboud to subagula. Hostu d gais xhibit a ma diamt of.75 mm whas Toyoua d s ma diamt is aoud.9 mm. Figu 3 psts Hostu d ad Toyoua d gadig. Pct passig (%) Toyoua d Hostud,, Diamt of paticl (mm) Figu 3. Hostu ad Toyoua d s gadig 77

4 Som sults of th K typ tsts a pstd i this pap fo both ds (Tabl I). Tabl I. K typ tsts coductd Sad Typ K typ tsts Iitial dsity axial loadig (kpa) P (kpa) Toyoua K8.69_T.69 ds 8 8 K8.9_T.9 loos 8 8 K ds 5 5 K ds Hostu K ds 3 3 K5.8.8 loos 5 5 K loos 7 7 K loosd odllig of quasi-lastic bhaviou 5. Rhological aalysis fo T4C StaDy tsts I its iitial stat oly, th d is assumd to b tasvsly isotopic. Th stss ad stai filds a supposd to b homogous ad cosqutly thi tagtial compots z ad θ a ull duig th ti xpimt (Figu 4). Thus : ε ε = () τ θθ zθ τ θz zz = τ θ τ ad z ε = ε γ θθ zθ γ / θz / = ε zz εθ γ / γ / ε z oov, wh cosidig xpimtal masumts, th loadig icmt has a fiit lgth ad a scat hological tso is obtaid. This tso is dpdt o th icmt dictio ad o th histoy of loadig, but also o th stai (o stss) icmt amplitud. Fo a cyclic loadig cospodig to a sgmt i th loadig spac, half th amplituds of stai (ε ) ad stss ( ) a cosidd (Figu 4). Wh th cycl is stabilisd th quivalt matix q is itoducd : q q q q ε ε ε γ θ z = q θ q z q γ θ q θθ q zθ q γθ z q θz q zz q γz γ q θγ q zγ q γγ θ. () z τ "z" C t p =p i F a "z" z,ε z SA q θ τ zθ,γ zθ ε SA θ,ε θ,ε "" ε Figu 4. Stss ad stai stats i th hollow cylid mpl ad quivalt tso tms q dfiitio 78

5 Th lmts of th quivalt tso q cospod to scat tms, which dpd o th amplitud of th cosidd cycl. Th applicatio of paticula cyclic loadig paths (axial ad tosioal cycls, cf. paagaph 3) with T4C StaDy appaatus allows to dtmi th ight tms of th last two colums of th quivalt matix q (quatio ). 5. DBGS s w hypolastic modl Sis of small cycls w pfomd with th hollow cylid T4C StaDy at difft stai amplituds ad stss icmt dictios. Th aalysis of th sis sults shows th xistc of a limit bhaviou, that is quasi-lastic. Wh th stai amplitud is small tha appox. -5, q tds towads a limit tso, which ca b assimilatd to a lastic o. A hypolastic fomulatio, δε=(h)δ, ca b cosidd to modl th bhaviou i th quasilastic domai (h psts th histoy paamt). A w hypolastic modl, calld DBGS, was dvlopd i th ENTPE laboatoy. o dtails o this modl a giv i Di Bdtto & al. () ad i Dutti (), basd o Hadi & al. (989) dvlopmt. It is assumd that th costitutiv tso dpds o th : - gomty of th assmblag, - lastic poptis of idividual paticls, - stss histoy, - cut stat of stss. Th tms of th tso dpd o th picipal stss valus,,, 3, o a pow cofficit ad o th isotopic Poisso atio ν. A fuctio F() appoximats ffcts of void atio. Th poposd hypolastic has a symmtic hological tso. Th mpl fabicatd by d dposit cats matial symmty, which should b vy clos to isotopy. Th iitial stat is th assumd to b isotopic. As a cosquc, th hypolastic law is tasvs isotopic i th picipal axs of stss (Equatio (3)). ν ν ( + ) z ν ν ( + ) (3) z =. F( ) p ν ν a ( + ) ( + ) z z z ( + ν) ( θz ) xpssd i th picipal axs of stss. Tabl II givs th stimatd valus of th DBGS modl paamts. Th vaiatios of th void atio a cosidd as gligibl ad F() is ot stimatd. P a is th atmosphic pssu. 6. Simulatios Tabl II. Paamts fo th DBGS odl Sad Typ Isotopic Poisso atio Pow cofficit Toyoua.75.4 Hostu..5 Figus 5 ad 6 pst xpimtal data ad simulatios fo th K typ tsts fo both ds (cf. Tabl I). I od to compa th sults, th atios z / zz, θz / zz, zz / zz, γz / zz, γ / γγ, Μ θγ / γγ, zγ / γγ ad γγ / γγ w plottd agaist th paamt τ/p m (P m is th ma pssu), pstativ of th stss valu fo th K typ tsts. zz ad γγ a th valus of zz ad γγ at th iitial aisotopic stat havig a ull sha stss (τ= kpa). 79

6 ,4,8, z /z,7 γ /γγ DBGS odl,6 K8.69_T,,5 K8.9_T,8,4,6,3,4 DBGS odl K8.69_T K8.9_T ν =.75 =.4,,4, θz /θz,, -,,3 θγ /γγ,,,8,6, -,,4 DBGS odl K8.69_T K8.9_T, - -,3 -,4 DBGS odl K8.69_T K8.9_T,4,3, zz /zz zγ /γγ,,,8,6, -,,4 DBGS odl K8.69_T K8.9_T, - -,3 -,4 -,5 DBGS odl K8.69_T K8.9_T,6,8 γz /zz,6 γγ /γγ,4,4,,,,8 - -,4 DBGS odl K8.69_T K8.9_T,6,4, DBGS odl K8.69_T K8.9_T -,6 τ/pm τ/pm Figu 5. Simulatio ad xpimtal data of z / zz, θz / zz, zz / zz, γz / zz, γ / γγ, Μ θγ / γγ, zγ / γγ ad γγ / γγ fo Toyoua d 8

7 3,5 z /z ν =. =.5 DBGS odl K5.65 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74,3 5,5 γ /γγ DBGS odl K5.65 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74,,5,5,5 -,5 -, -,5 -,4, θz /θz,4,3 θγ /γγ DBGS odl K5.65 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74,8,,6,4 DBGS odl K5.65 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74,6 zz /zz,4,,8,6,4 DBGS odl K5.65 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74 -, - -,3 zγ /γγ DBGS odl K5.65,5 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74,5 -,5 - -,5,5 γz /zz DBGS odl K5.65 K5.7 K5.8,6,4 γγ /γγ,5 K75.65 K7.79 K3.99 K3.74,,8,5,6 -,5 - τ/pm DBGS odl K5.65,4 K5.7 K5.8 K75.65 K7.79 K3.99 K3.74 τ/pm Figu 6. Simulatio ad xpimtal data of z / zz, θz / zz, zz / zz, γz / zz, γ / γγ, Μ θγ / γγ, zγ / γγ ad γγ / γγ fo Hostu d 8

8 It ca b oticd fom th sults pstd abov, that : th xpimtal tms zγ ad γz fit wll o th thotical cuv. Thy pst a simila volutio ad clos valus. This tds to cofim th symmty assumptio, xpimtal data ad simulatio show good agmt fo loadig ad uloadig bachs fo zz ad γγ, which a spctivly th ivs of th Youg modulus E ad of th sha modulus G; th modl taslats quit coctly th volutio of th oth xpimtal tms ( z, θz, θ ) with th stss stat. Oly γ simulatio xhibits a divgc fom xpimtal poits. It ca b assumd that th masumts a ot vy pcis bcaus th xpimtal tms a clos to zo. It ca b cocludd that th simulatios taduc coctly th xpimtal sults obtaid fom about foty ivstigatio poits. It has to b udlid that th volutio loadig paths cospod to lag cyclic paths fo most of th tsts. Th foty ivstigatio poits a th locatd ith o th loadig o o th uloadig bach of ths lag cycls. 7. Coclusio Th hollow cylid pototyp T4C StaDy has vald to b a powful xpimtal tool fo th ivstigatio of d bhaviou fom small to lag stai domais. Spcific loadig paths w applid with th hollow cylid. Small cyclic loadig w cosidd as thy allow us to obtai xpimtal ifomatio o th quasi-lastic bhaviou fo gal stss coditios. With th hlp of th T4C StaDy data, a w hypolastic law was fomulatd fo th oviscous pat. Th complt aisotopic hypolastic tso which is dvlopd, ca b obtaid fo ay loadig with o without otatio of th axs. Som compaisos btw xpimtal sults fom th hollow cylid dvic ad umical simulatios a poposd fo dy Hostu RF d ad fo Toyoua d. Thy show that th DBGS modl is abl to tascib coctly th xpimtal obsvatios. 8. Rfcs Cazacliu, B. & Di Bdtto, H. (998). Bhaviou of d i th small stai domai obsvd with a hollow cylid appaatus, Euopa Cofc o Eathquak Egiig, Pais. Di Bdtto, H., Cazacliu, B., Bouti, C., Doah, T. & Tout, J.P. (997). Compotmt ds bls avc otatio d axs : ouvl appail couvat quat décads d défomatio. Poc. XIVth It. Cof. O Soil ch. ad Foud. Eg., Hambug. Balkma. Di Bdtto, H., Cazacliu, B., Goffoy, H. & Sauzéat, C. (999). Sad bhavio at vy small to mdium stai-ifluc of stss. I Jamiolkowski & al (d.), Poc. of IS o P-failu Df. Cha. of Gomat., Toio. Balkma. Di Bdtto, H., Goffoy, H., Sauzéat, C. & Dutti A. (). Etud du compotmt cycliqu ds bls ptits t moys défomatios, Rappot ND 7 GC. ENTPE. Dutti, A. (). Etud du compotmt cycliqu ds bls ptit t moy défomatios. DEA, ENTPE. Goto S. (99). Stgth ad dfomatio Chaactistics of gaula matials i tiaxial tsts, Ph.D Thsis, IIS - Uivsity of Tokyo. Hadi, B.O. & Bladfod, G. (989). Elasticity of paticulat matials. Joual of Gotchical Egiig, ASCE, 5(6): Hoqu, E. (996). Elastic dfomatio of ds i tiaxial tsts. PhD, Uivsity of Tokyo. Jadi R.J. Kuwao R., Zdavkovic L., Thoto C. (999). Som fudamtal aspcts of th p-failu bhaviou of gaula soils, Kyot lctu i IS o P-failu Dfomatio Chaactistics of Gomatials,Toio. Lo Psti D., Jamiolkowski., Pallaa O., Cavallao A., Poi S. (997). Sha modulus ad dampig of soils, Gotchiqu, Vol. XLVII, No 3. Tatsuoka, F., Jadi, R.J., Lo Psti, D., Di Bdtto, H. & Kohata, Y. (997). Tstig ad chaactisig p-failu dfomatio of gomatials. Kyot lctu i XIVth It. Cofc o Soil chaics ad Foudatio Egiig, Hambug. Balkma. Yasi S.J.., Umtsu K., Tatsuoka F (999). Plai stai stgth ad dfomatio of ds affctd by batch vaiatios ad difft appaatus typs, Gotchical Tstig Joual, Vol, pp