A Strain-based Non-linear Elastic Model for Geomaterials

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1 A Strai-basd No-liar Elastic Modl for Gomatrials ANDREW HEATH Dpartmt of Architctur ad Civil Egirig Uivrsity of Bath Bath, BA2 7AY UNITED KINGDOM Abstract: - Th o-liar lastic bhaviour of gomatrials has traditioally b modld usig strssdpdt lastic modls. Th approach usd by may rsarchrs is ot rigorous whr th lastic modls ca crat rgy ad ca prdict prmat dformatio udr crtai circumstacs. Othr rsarchrs hav usd rigorous hyprlastic modls whr rgy is cosrvd by th lastic portio of th modl, but ths modls hav strss-dpdt stiffss which ca slow computatio ad lad to itrativ solutios i fiitlmt or othr strai-basd computatio tchiqus. To ovrcom ths problms a rigorous lastic modl is ivstigatd which prdicts similar bhaviour to hyprlastic modls, but has th advatag of big lastic strai basd which allows mor dirct solutio i strai-basd fiit-lmt or othr computatioal cods. Ky-Words: - Elasticity, Gomatrials, No-liar, Fiit Elmt, Elastic strai Itroductio Th rsilit rspos of gomatrials has rcivd much atttio ovr th yars. Iitial modls wr liar lastic, but ths prstd a poor rprstatio of graular matrial rspos. I th lat 960s ad arly 970s bulk strss dpdt, o-liar lastic modls wr proposd []. Ths arly modls (ad may latr modls) wr basd o obsrvatios of laboratory tst rsults ad ca crat rgy ad, possibly mor importatly, prmat strais o crtai strss paths. Bcaus of th larg umbr of load applicatios i rpatd load applicatios (.g. arthquak girig), prmat strais prdictd by ad lastic modl ar cosidrd uaccptabl. To ovrcom ths limitatios, a umbr of hyprlastic modls wr dvlopd [2,3]. Ths modls do ot prdict prmat strais ad always cosrv rgy, but do ot always provid ralistic prdictios of rspos as thy ca prdict a gativ bulk modulus or icrasig shar modulus with icrasig shar strss, trds that ar ot otd durig laboratory tstig. O raso for th mismatch i thory ad obsrvd rspos is th hyprlastic modls igor hystrtic rgy dissipatio durig loadig. Most currt o-liar lastic modls ar strssbasd modls (stiffss is dfid as a fuctio of strss stat), which simplifis paramtr dtrmiatio durig strss cotrolld laboratory tsts, but ca sigificatly icras computatioal tim ad crat covrgc problms wh aalysig fild loadig coditios. Modls basd o simpl laboratory tsts ar oft uabl to rprst bhaviour udr fild loadig coditios. This papr discusss th rquirmts for a oliar lastic modl, ad ivstigats a w straibasd modl for gomatrials. 2 Problm Formulatio A xampl of a o-rigorous lastic modl is show for a modl whr th stiffss is strss dpdt []. This ad othr similar formulatios ar probably th most popular for modllig gomatrials ad th modl formulatio is charactrisd by a o-liar scat Youg s modulus (E=E R p ) ad a costat Poisso s ratio (ν). To dtrmi th ffct of complx strss paths o modl prdictios, it is asist to rwrit th modl i trms of shar modulus (G) ad bulk modulus (K), thrby ablig th shar ad volumtric bhaviour to b idpdtly cosidrd: E ER p' Δp' K = = : Δε p = 3( 2ν ) 3( 2ν ) K E ER p' Δs G = = : Δε s = 2( + ν ) 2( + ν ) 2G () (2) ISSN: Issu 6, Volum 3, Ju 2008

2 whr E R ad ar matrial costats, p is th ma ffctiv strss, s is shar strss (diffrc i vrtical ad horizotal strss udr triaxial coditios), ε p is th lastic volumtric strai, ad ε s is th lastic shar strai (diffrc btw axial ad radial strai for triaxial coditios). If th modl i Equatios () ad (2) is tak o th strss path i Fig., th modl prdicts prmat lastic shar strais as dtaild i Tabl, idicatig it is ot a rigorous lastic modl. Tabl : Elastic shar strais prdictd by Equatio (2). Loadig stag Elastic shar strais from Fig. ( sb sa )( + ν ) ' E p ( s a E 4 0 Total ( p ' a E R p R a s )( + ν ) b ' R pb ' pb ' ' a pb )( s a s b ( +ν ) As show i Tabl, th modl prdicts prmat lastic shar strais alog th strss path cosidrd, idicatig it is ot a tru lastic modl. Although ot dmostratd i this papr, th modl ) ca also crat or dissipat rgy o th strss path, thrby violatig th laws of thrmodyamics. Thr ar, howvr, umrous rigorous o-liar lastic modls which ca ovrcom this limitatio but which hav othr problms with implmtatio bcaus thy ar strss-basd. Implmtatio of o-liar lastic modls subjctd to fild loadig coditios is most likly to b prformd usig th fiit lmt mthod Fiit lmt thory is complx, ad th discussio prstd hr is ot itdd to b comprhsiv, but rathr highlights th aspcts of th thory that ar importat for modllig oliar lastic matrial modls. A mor complt discussio o th fiit lmt mthod ca b foud i umrous txts o th subjct,.g. [4]. It should b otd that throughout this papr smallstrai bhaviour is assumd which is appropriat for this fild. Th fiit lmt mthod is quivalt to th miimisatio of th total pottial rgy of th systm i trms of a prscribd displacmt fild [4]. Th mar i which this is usually achivd is through th applicatio of displacmts (strais), alog with a attmpt to miimis th total pottial rgy of th systm. For this to occur, th rgy from xtral forcs ad displacmts must b qual to th itral strai rgy, calculatd from strsss ad strais. Shar strss (s) s b Stag 2 Stag Stag 3 s a Stag 4 p a Fig. : Strss path of o-zro ara. p b Ma ffctiv strss (p ) ISSN: Issu 6, Volum 3, Ju 2008

3 Th rlatio btw th strsss ad strais i a matrial is calculatd usig a costitutiv rlatio, ad th form of this modl govrs th strai rgy of th fiit lmt problm. Th basic stps followd by a fiit lmt program ar slightly diffrt for liar ad o-liar lastic matrials, but thy ca b simplifid as show i Fig. 2. Th procdur is itratd through timstps to sur compatibility ad covrgc. As show, th procdur is strai-basd whr a strai is iputtd to th costitutiv rlatio to calculat a strss. For liar lastic matrial modls thr is a dirct approach for computatio as thr is a liar costitutiv rlatio btw strai ad strss (Fig. 3), but for o-liar lastic matrial modls th computatio is mor complx. This is particularly tru wh th o-liar lastic costitutiv modl is strss-basd, i.. wh th rlatioship btw lastic strais ad strsss is a fuctio of strss stat. This is particularly importat for origorous o-liar lastic modls (.g. Equatios () ad (2)) whr thr is a o-uiqu solutio to th problm ad thrfor solutio istability (Fig. 4). If th costitutiv rlatio is a fuctio of strss stat, a itrativ solutio is rquird whr th calculatio of strsss is dpdt o th strss lvl. If th costitutiv rlatio is strai-basd or costat (as i liar lastic matrials), th computatio will b sigificatly quickr as th itrativ solutio is ot rquird to calculat strsss from strais. Summary of th fiit lmt mthod Problm gomtry Costitutiv rlatio Problm gomtry Displacmts (u) Elastic strais (ε) Strsss (σ) Extral forcs (f) Itral work from strsss ad strais Extral work from forcs ad displacmts Fig. 2: Summary of th fiit lmt mthod Liar lastic modls Uiqu rlatio with dirct solutio Displacmts (u) Elastic strais (ε) Strsss (σ) Extral forcs (f) Itral work from strsss ad strais always qual to xtral work from forcs ad displacmts Fig. 3: Implmtatio of th fiit lmt mthod for liar lastic modls ISSN: Issu 6, Volum 3, Ju 2008

4 Strai-cratig modls (strss basd) No-uiqu rlatio with itrativ solutio Displacmts (u) Elastic strais (ε) Strsss (σ) Extral forcs (f) Itral work from strsss ad strais Extral work from forcs ad displacmts ot qual to itral work Fig. 4: Implmtatio of th fiit lmt mthod for o-rigorous o-liar lastic modls 3 Strai-basd o-liar lastic modls for gomatrials I ordr to limiat th itrativ solutio rquird to dtrmi strss from strai i a strss-basd modl, ad to improv covrgc i a fiit lmt packag, a strai-basd o-liar lastic modl ca b formulatd for gomatrials. Th modl rquirmts ar: Strai-basd for rapid computatio ad improvd covrgc i a fiit lmt aalysis. Ergy ad lastic strai cosrvig udr all rasoabl loadig coditios. Fw modl paramtrs with ach paramtr makig physical ss, prfrably rlatd to paramtrs usd for xistig strss-basd modl Abl to accuratly modl th small-strai lastic bhaviour of gomatrials. Th first two rquirmts wr addrssd by cratig a strai-basd hyprlastic modl whr a strai rgy dsity fuctio (U(ε)) is dfid ad th strsss ca b dtrmid by diffrtiatig this fuctio, as discussd i Sctio 3.. Th third ad fourth rquirmts ar addrssd i th modl formulatio which is discussd i Sctio 3.2. A compariso btw th implmtatio of a strss-basd ad strai-basd hyprlastic formulatio is prstd i Figs 5 ad 6. Strss-basd hyprlastic modls Uiqu rlatio with itrativ solutio Displacmts (u) Elastic strais (ε) Strsss (σ) Extral forcs (f) Itral work from strsss ad strais always qual to xtral work from forcs ad displacmts Fig. 5: Implmtatio of th fiit lmt mthod for strss-basd hyprlastic modls ISSN: Issu 6, Volum 3, Ju 2008

5 Strai-basd hyprlastic modls Uiqu rlatio with dirct solutio Displacmts (u) Elastic strais (ε) Strsss (σ) Extral forcs (f) Itral work from strsss ad strais always qual to xtral work from forcs ad displacmts Fig. 6: Implmtatio of th fiit lmt mthod for strai-basd hyprlastic modls 3. Ergy ad strai cosrvatio Whil it is fairly simpl to formulat a costitutiv modl that dos ot prdict prmat strais, it is also imprativ that ay modl ithr cosrvs or dissipats th applid rgy, but vr crats it. Ay rgy cratio would b a violatio of th scod law of thrmodyamics, ad will rsult i a icorrct solutio i a fiit lmt program which is usig th miimum pottial rgy to solv th problm. Durig loadig, gomatrials matrials dissipat rgy from hystrtic / plastic rspos durig loadig ad uloadig cycls. Exprimtal vidc has show this bhaviour is largly rat-idpdt i graular matrials, with th stiffss largly idpdt of loadig rat [5]. Thr ar umrous rgy-dissipatio modls for soils (.g. [6]), with most of ths origially dvlopd for arthquak girig purposs. Thy ar grally valid oly for costat ma ffctiv strss coditios ad ar thrfor of littl us for thr dimsioal modllig applicatios whr shar strsss ad ma ffctiv strsss udrgo simultaous chags. A rigorous implmtatio of hystrtic rgy dissipatio is xtrmly complx as ay modl must sur rsilit strais ar cosrvd ad that rgy is always dissipatd (ad vr cratd). This will gratly icras solutio tim i a fiit lmt aalysis ad it is thrfor proposd that costitutiv modls for uboud gomatrials first sur that o rgy is cratd bfor icludig rgy dissipatio. Should th radr rquir mor iformatio o rigorous implmtatio of ratidpdt rgy dissipatio, this is availabl i [7]. Th asist rigorous mthod of surig rgy is cosrvd i a o-liar lastic modl is through th us of hyprlasticity. A hyprlastic costitutiv modl is drivd from a pottial fuctio ad will sur that wh implmtd i a fiit lmt cod, th itral strai rgy will qual th rgy from xtral forcs ad displacmts. Hyprlastic modls ca b ithr strss-basd or strai-basd, ad thy cosist of a arbitrary fuctio (V(σ) or U(ε)) which, wh diffrtiatd with rspct to ay strss or lastic strai dirctio (σ or ε i dirctio ), givs th corrspodig strai or strss rlatd to that variabl: V U ε = : σ ' = (3) σ ' ε Hyprlastic modls will always cosrv rgy ad strais, rgardlss of th strss path followd, ad ar thrfor cosidrd a rigorous mthod of implmtig o-liar lastic modls i a fiit lmt cod. Th us of hyprlasticity i th formulatio of a strai-basd lastic modl is discussd blow. 3. Modl formulatio May strss-basd lastic modls hav b validatd th followig rlatio btw ma ffctiv strss (p ) ad bulk lastic strai (ε p ) udr hydrostatic coditios (.g. [], [2], [3]): p' ε p = thrfor (4) K p' R ( ) p' ε (5) = K R p Ths sam modls hav a corrspodig shar modulus (udr hydrostatic coditios) with th followig form: G = G R p' (6) ISSN: Issu 6, Volum 3, Ju 2008

6 By substitutig Equatio (5) ito Equatio (6), it is possibl to obtai th gral form of a modl that will hav similar costats ad bhaviour to strssbasd hyprlastic modls, but this modl will b strai-basd which will improv solutio tim ad covrgc i a fiit lmt packag. Th complt form of th hyprlastic modl is dtrmid from th strai rgy dsity fuctio (U(ε)) that was calculatd usig th modl rquirmts listd arlir : ( ) K ε - ε ε + ( K ε ) U = G - R R p s, s, R p ε p (7) 2 Th shar strss ad ma ffctiv strss (ad thrfor scat shar ad bulk moduli) ca b dtrmid by partially diffrtiatig U with rspct to th shar strais ad volumtric strais rspctivly: ( K ε ) 2 G - R R p εs, s = (8) ( ) G ε,, ( ) s ε R s p' = K Rε p K Rε p (9) ε p Th modl dscribd abov is a thr-dimsioal isotropic hyprlastic modl, ad uss th rpatd idx summatio covtio usd i solid mchaics. Th modl appars complx, but this is maily bcaus it was formulatd so th paramtrs ar idtical to thos from othr strss-basd modls udr hydrostatic coditios [2,3]. Udr ths coditios, ach paramtr has th sam physical maig as i Equatios () ad (2) ad as usd for othr modls [,2]. Th paramtr cotrols th icras i stiffss with a icras i ma ffctiv strss, ad th paramtrs G R ad K R giv th shar ad bulk moduli of th soil at a ma ffctiv strss of o strss uit. To sur th modl paramtrs ar idpdt of uits usd, it is suggstd that th strsss b ormalisd by atmosphric prssur bfor implmtatio i a fiit lmt packag, rsultig i G R ad K R big uit idpdt. Th bhaviour prdictd by th modl, compard with th modls with thos usd by othr rsarchrs [,2] is illustratd i Figs 3 ad 4. As show i Fig. 7, bcaus th modl formulatios ar quivalt udr hydrostatic coditios, thr is o diffrc btw th strai basd modl ad th strss basd modls usd by othr rsarchrs. Thr is, howvr, a diffrc wh shar is applid (Fig. 8) whr th strai-basd modl prdicts a dcras i stiffss with icrasd shar. This trd i scat bhaviour has b obsrvd durig tstig [5]. Th dcras i stiffss with sharig is oft attributd to rgy dissipatio, but i th cas of th modl prstd hr, o rgy is dissipatd. Th icras i stiffss with icrasd shar for th rigorous strss-basd modl [2] is ot obsrvd i practic. ISSN: Issu 6, Volum 3, Ju 2008

7 3500 Rsilit modulus, E (atm) Hydrostatic coditios G R =400 K R =00 =0.7 Proposd modl Rfrc [] Rfrc [2] Ma ffctiv strss, p' (atm) Fig. 7: Modl bhaviour udr hydrostatic coditios (o shar) Rsilit modulus, E (atm) p' = atm (0 kpa) G R =400 K R =00 =0.7 Proposd modl Rfrc [] Rfrc [2] Shar strss, s (atm) Fig. 8: Modl bhaviour udr costat ma ffctiv strss ISSN: Issu 6, Volum 3, Ju 2008

8 3.2 Combiig with othr strai-basd modls Th most commo modls for rgy dissipatio of gomatrials udr rpatd loadig ar dscribd usig modulus dgradatio whr th rductio i stiffss is a rsult of shar strais, such as th modl proposd by Hardi ad Drvich [6]. Whil ths commo modls ar vry simplistic ad should ot b cosidrd a rigorous implmtatio of hystrtic rspos, th us of a strai-basd rgy cosrvig lastic modl allows asy combiatio with a strai-basd rgy dissipatig modl ad allows rapid computatio of strsss from total strais. Th currt mthod of usig a mixd modl with a strss-basd rgy cosrvig modl ad a strai-basd rgy dissipatig modl rsults i a complicatd ad slow ovrall solutio tchiqu. Th rgy dissipatio modl proposd by Hardi ad Drvich [6] is a mpirical modl prstd i th form of modulus dgradatio whr th ratio btw th scat shar modulus (G) ad hydrostatic shar modulus (G max ) is giv by th followig quatio: G = (0) G + γ γ max r Whr γ is th girig shar strai (twic th pur shar strai, i.. γ=2ε s ), ad γ r is a rfrc strai rlatd to th matrial strgth ad hydrostatic stiffss. Th origial form of th modl providd a mpirical rlatioship whr G max could b calculatd from th voids ratio ad ma ffctiv strss. Howvr, this rquird th ma ffctiv strss to rmai uchagd durig aalysis ad ay gratd por prssurs or vrtical acclratios wr thrfor igord i th aalysis. It is possibl to us th rgy dissipatio modl with othr modls for th pak stiffss, ad for this it was assumd that a soil had a pak frictio agl of 30 dgrs. Udr simpl shar coditios with a ma ffctiv strss of 0 kpa, th paramtr γ r proposd i rfrc [6] is for stiffss paramtrs i Fig. 8. Covrtig Equatio 0 to giv shar strss as a fuctio of shar strai as rquird i a fiit lmt solutio (s Fig. 2), th followig is obtaid: Gmax s = 2Gε s, = 2εs, () + 2ε s, If a strss-basd modl is usd to dtrmi G max (.g. Equatio 2), th followig is obtaid: ER p' s = 2εs, (2) 2( + ν )( + 2εs, ) This is a mixd formulatio whr th strsss prdictd ar basd o both th strss stat ad strais, ad it is thrfor impossibl to obtai a dirct solutio ad a itrativ solutio is rquird. If th strai-basd modl i Equatio 8 is usd, a solutio basd solly o strais is obtaid: ( K ε ) G R R p s = 2εs, (3) + 2ε s, - Although this formulatio appars complicatd, its implmtatio i a strai-basd fiit lmt cod is simpl. If a saturatd sad with th iitial coditios i Fig. 8 is modld with Equatio 3, th rspos i Fig. 9 is obtaid. I this figur th pur shar strais ar covrtd to girig shar strais to allow compariso wit th origial rfrc, which producs a idtical rsult if th sam valu of G max is usd. ISSN: Issu 6, Volum 3, Ju 2008

9 Proposd modl Rfrc [6] G/G max Egirig shar strai, γ (%) Fig. 9: Prdictd modl rspos wh combid with Hardi ad Drvich [6] modulus dgradatio curv. 4 Coclusio This papr has show th rquirmts for implmtig o-liar lastic modls ito a fiit lmt cod to modl fild coditios. May prvious o-liar lastic modls ca prdict prmat strais o crtai strss paths which is a major cocr for rpatd load applicatios. Ths modls ca also crat rgy which violats th laws of thrmodyamics ad prvts accurat implmtatio ito a fiit lmt cod. Strss-basd hyprlastic modls will vr crat rgy ad will vr prdict prmat strais ad ar rigorous costitutiv modls that ca b corrctly implmtd ito a fiit lmt cod. As th modl is strss-basd, its implmtatio ito a strai-basd fiit lmt cod rquirs a itrativ solutio, rsultig i slow computatio ad possibl lack of covrgc. I additio, may strss-basd hyprlastic modls prdict a icras i rsilit modulus with a icras i shar, a trd opposit to obsrvd bhaviour. As a rsult, a strai-basd hyprlastic modl is proposd. Th modl has th sam matrial paramtrs ad prdicts th sam bhaviour udr hydrostatic coditios as popular strss basd modls [,2], but th bhaviour udr sharig is diffrt with th w modl prdictig a dcras i rsilit modulus with icrasd shar, as otd durig laboratory tstig. Th us of a strai-basd modl also improvs covrgc ad sigificatly rducs computatio tim i a fiit lmt aalysis. A strai basd lastic modl ca asily combid with a strai-basd modulus dgradatio modl such as that origially proposd by Hardi ad Drvich [6]. This allows th dirct computatio of strsss from strais without th d for a itrativ solutio. Thr ar som problms whr this approach will ot work, for xampl wh a strss-basd hystrtic or yildig modl is rquird, but th us of strai-basd hyprlastic modls i modllig gomatrials shows promis for crtai applicatios. Rfrcs: [] Hicks, R.G. ad C.L. Moismith, Factors iflucig th rsilit rspos of graular matrials. Highway Rsarch Rcord 345, pp. 5-3, 97 ISSN: Issu 6, Volum 3, Ju 2008

10 [2] Boyc, H.R. A o-liar modl for th lastic bhaviour of graular matrials udr rpatd loadig, I: Itratioal Symposium o Soils udr Cyclic ad Trasit Loadig, Swasa, Wals, pp , 980 [3] Lort, B. ad M.P. Luog, A doubl dformatio mchaism for sad. I: 4th Itratioal Cofrc o Numrical Mthods i Gomchaics, pp , 982. [4] Zikiwicz, O.C. ad R.L. Taylor, Th fiit lmt mthod, 5th ditio, ISBN , Buttrworth-Hima, Oxford, [5] Sd, H.B., F.G. Mitry, C.L. Moismith, ad C.K. Cha, Prdictios of pavmt dflctio from laboratory rpatd load tsts. Tchical Rport TE-65-6, Soil Mchaics ad Bitumious Matrials Rsarch Laboratory, Uivrsity of Califoria, Brkly, CA, 965. [6] Hardi, B.O. ad V.P. Drvich, Shar modulus ad dampig i soils: Dsig quatios ad curvs. Joural of Soil Mchaics ad Foudatios Divisio, ASCE. 98(7): , 972. [7] Houlsby, G.T. ad A.M. Puzri, A thrmomchaical framwork for costitutiv modls for rat-idpdt dissipativ matrials, Itratioal Joural of Plasticity, 6, pp , ISSN: Issu 6, Volum 3, Ju 2008