A Constitutive Model for Multi-Line Simulation of Granular Material Behavior Using Multi-Plane Pattern

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1 Joural of Compuer Scece 5 (): 8-80, 009 ISSN Scece Publcaos A Cosuve Model for Mul-Le Smulao of Graular Maeral Behavor Usg Mul-Plae Paer S.A. Sadread, A. Saed Darya ad M. Zae KN Toos Uversy of Techology, Tehra, Ira Absrac: Problem saeme: Improper udersadg of maeral behavor preves he effce ad correc usage of avalable maerals ad cosequely, creases he cosruco ad maeace coss ad eve usuable cosruco. Cosderg he ecessy of exac vesgao abou maeral behavor, several researches have bee carred ou hs feld bu he maory of hese researches dd o propose a geeral mehod for predco of graular maeral behavor. Furhermore, may of he mehods proposed by researchers are o able o prese he properes such as he oreao of falure mechasm of propagag plascy maerals. Approach: I hs sudy, a geeral mehod was proposed for mul-lamae smulao o predc he behavor of maerals. The geeral applcably of hs mehod for predco of graular maeral s oe of s sgfca advaages. The sudy was carred ou he framework of mul-plae paer whch s able o predc asoropc behavor, cosder he effec of sress ad sra axs roaos plascy, cosder he sem-mcro mechacal hsory ad fally predc he oreao of falure mechasm. The mehod was preseed a maer ha here s o lmao for dffere shapes of sress-sra curves. Resuls: I was cocluded ha usg hs mehod, fudameal properes of maeral such as maeral fracure, oreao of falure, asoropc behavor of maeral, separao of behavor several plaes ad roao of prcple axs of sress ad sra durg olear behavor ca be deermed. Cocluso/Recommedaos: Ths mehod ca be used for complcaed maeral behavor smulao uder sesmc loadg, cyclc loadg or fague effecs. For fuure works, he mehod ca be exeded by creasg he umber of plaes. Hgher-order equaos ca also be used o have a more accurae approxmao of sress-sra curve. Key words: Mechacal behavor, Mul-le model, Mul-plae paer INTRODUCTION Msudersadg of maeral behavor ad occasoal predcos wh lmed accuracy has caused several problems aalyss ad cosruco dffere egeerg braches. The use of superfluous safey facors o compesae he lack of kowledge wll cause grea coss dusry. Therefore, usg he mehods whch are able o predc maeral behavor wll eable he users o recogze all of he affecg facors ad properly cosder he effec of each facor. Sem-lear or mul-lear aalyss s oe of he smple mehods preseed for modelg he maeral behavor []. Koush ad Gree elasc models, Hypo elasc models, varable-parameer models are some saces of such model. The oable po hese modelg mehods s ha some cases, for beer udersadg of maeral behavor, s ecessary o chage he parameers a aural ad specfc maer ad he deermao of he parameers may be Correspodg Auhor: S.A. Sadread, K.N.Toos Uversy of Techology, Tehra, Ira 8 dffcul ad eve mpossble. Besdes, he lmed ably of hese models s he case of smple sress pahs. Ths fac forced researchers o cosder more complcaed mahemacs wh varous dffere compoes he ew models. Bu he ma problem, whch s he depedecy of maeral behavor o sress pah ad sress hsory [] sll exs. Because of he meoed reasos, hs sudy was red o propose a process based o a mul-le mehod he form of mul-plae paer whch do o have he meoed problems. I addo, he sgfca advaces compuer sceces have eabled researchers o erus he compuao of dffere compoes o compuers a orgazed sysem shape. I hs maer, more complcaed behavoral parameers ca be roduced o models. A sgfca characersc mul-plae paer s he ably of behavor aalyss specfc drecos. Exsece of seems soes or poso of aggregae posog sad ad alluvum sedmeao ad smlar cases s caused he maeral

2 J. Compuer Sc., 5 (): 8-80, 009 o have dffere behavor dffere drecos [-]. Mul-plae paer makes possble o deerme he maeral behavor dffere drecos. Mul-plae paer: The bass of mul-plae mehod s o deerme he umercal relaoshps bewee er-parcle behavor (mcro behavor) ad egeerg mechacal characerscs (macro behavor) he form of a esseal equao whch s obaed by soluo of umercal egrao [,]. I oher words, hs case, behavoral properes of maeral behavoral properes ad sress-sra behavor of sol ca be deermed by vesgao of erparcle behavor. Graular maerals are composed of defe umber of sold parcles whch are coac o each oher ad he reaco bewee parcles s due o he force whch exss coac surface. The aalyss of parcle behavor ad her coac surface depeds o umber, sze, shape, roughess ad he sregh of parcles hese surfaces. Cosequely, vesgao abou hese maerals uder meoed codo s more complcaed ha he case whch has couous crcumsace. Besdes, a smple vew, behavor of graular maeral ca be assumed as he combao of parcle elasc behavor ad plasc slppage coac surfaces. I hs maer, arfcal case, he hree dmesoal behavor of graular maerals ca be deermed by cosderg defe sample plaes whch slppage s possble. These plaes dvde he maeral block o a colleco of mullaeral pars whch are ex o each oher. Whe a lle shear s appled o a mullaeral par, elasc shear deformao occurs. If he shear be creased ad acheve a specfc lm, mullaeral pars move alog a boudary surface called slppage plae. Wh creasg he deformaos, ecessary shear sress for occurrece of more deformao s creased. The oal shear sress a ay me s equal o he summao of elasc shear deformao mullaeral pars ad shear deformao due o he slppage of adog pars. Whe he sress s decreased, he elasc deformao dsappears. Whe he sress s more decreased ad acheved o a specfc lm, mullaeral pars beg o slp oppose dreco. The requred shear sress for slppage depeds o ormal sress ad he slppage oly occurs whe he sress codo passes he yeld sress. Moreover, slppage us occurs slppage plaes wh drecos show Fg.. prese he chages of physcal properes o he surface of sphere. The surface of a sphere ca be esmaed by defe umber of fla plaes whch are age o dffere pos of he sphere. I hs maer, ay of meoed plaes have oe coac po wh he sphere. By lmg he umber of hese plaes, he umber of coac pos or bass pos ca be adused ad he calculao of umercal egrao gves he amou of he fuco he meoed pos. Numercal egrao of couous fuco f (x, y,z) o he surface of sphere ca be calculaed as he summao of fuco value F a dffere pos whch are mulpled o correspodg wegh coeffces. To decrease he error values, he umber of hese pos should be creased. I ca be cofrmed ha he applcao of sample pos wll decrease he error o he sxh order. (a) (b) Numercal cocep of mul-plae heory: I hs heory, he al bass s calculao of umercal egrao from a specfc mahemacal fuco o a sphere of uy radus. Ths mahemacal fuco ca 8 Fg. : Represeao of graules assembly real ad arfcal cases: (a) real ssue of graules he eral srucure of maeral (b) represeao of arfcal polyhedro pars assemblage

3 J. Compuer Sc., 5 (): 8-80, 009 Fg. : sample pos for umercal egrao o he surface of a sphere wh radus of uy The below equao shows he relaoshp bewee umercal ad ormal egrao: f (x, y, z)dω = 4π w f (x, y, z ) () = Where: Ω = Sphere area w = Number of pos = Nodal wegh coeffce f (x,y,z ) = The value of F fucoa he po (x,y,z ) The poso of he pos o sphere s show Fg.. For ay po, a plae s defed so ha he dreco coses are he same as ormal vecors of he plae a coac pos. I hs maer, ay chage o he plae s cocercally relaed o. Sce for ay of desred pos a specfc plae s specfed, ad cosderg he po symmery, he pos are reduced o pos ad he surface of half sphere s approxmaed by plaes: f (x, y, z)dω = 8π wf (x, y, z ) () = The oreaos of he plaes are show he ceer of cube Fg.. O hs bass, f he qualy of slppage ad opeg ad closg of each plae s orgazed, he summao of hese slppages, opegs ad closgs compose he mehod of maeral moveme for a po ad by egral summao, oal moveme or deformao effecs ca be acheved for a specfc po. I hs maer, s requred o assume a rule for he value of fuco f (x, y, z ) whch deermes he movemes of each plae. Fg. : represeao of he meoed plaes Behavor deermao he form of mul-le mehod global coordaes: Idepede o he shape of sress-sra curve, each curve ca be smulaed usg some sragh les. For deermao of laeral ad ormal sra relaoshp a smlar mehod s used. Behavor deermao of he plaes: To deerme he behavor codo of each plae ad relaos bewee plae behavor ad oal behavor, he behavor esor s rasferred from global coordaes o each of he plaes. Sra vecor global coordaes ca be rasferred o a hree-compoe vecor o each plae cosss of ormal sra ad wo shear sras placed o he plae by usg rasformao marx. = l. () where, l s he rasformao marx of h plae: = l m lm m l l ll mm lm l m m m l l " " " " " " " " " ll mm lm l m m m l l The parameers,m,l,,m,l ad,m,l are respecvely he dreco coses of ormal o plae dreco ad wo perpedcular dreco o he plae. I he Eq., s sra vecor of sx compoes geeral coordaes whch s defed as followg equao: = C. σ (4) I Eq. 4, σ s sx-compoe sress vecor geeral coordaes ad c s behavor esor geeral 84

4 J. Compuer Sc., 5 (): 8-80, 009 coordaes. Subsug he Eq. 4 Eq. he followg equao s obaed: = l.c. σ (5) By hs equao, sra vecor of plaes are obaed. By kowg sra vecor of each plae ad usg basc relao of mul-plae, sra vecor geeral coordaes whch s calculaed by summao of plaes effec s deermed by followg equao: () = = 8. π w.l. where, l s rasformao marx from plae o geeral coordaes: " l ll ll " m mm mm " l = " " lm l m lm l m lm " " m m m m m " " l. l l. l l. Subsug Eq. 5 Eq., followg equao s obaed: coordaes o plaes ad vce versa, he compoes of proposed mehod are obaed. A oable po rasferrg hree-axal expermeal curves o he plaes s he mehod of creag sress ad deformao o hem. As ca be see Fg., carred ou sudes have show ha he effec of he plaes -4 s o creae sress- deformao he drecos ormal o plae ad boh shear drecos o he plae. I he plaes 5-7, he secod shear dreco wll o become acvaed. I he plae 8, he shear dreco oe s acve. I he plaes 9-, us he plae ormal dreco s acvaed ad o shear wll produced hese plaes. I he above equaos, he proporo of each plaes geeral behavor s deermed. If be ecessary, relaoshps bewee sress ad sra vecors each plae ca be deermed: = C. σ (9) J Sress vecor of plae ca be acheved by rasferrg he geeral sress vecor: σ = l. σ (0) By subsug he Eq. 9: = C.l. σ () (7) = = (8. π w.l.l.c) σ O hs bass ad usg umercal egrao of mul-plae paer, geeral sra vecor s calculaed: I oher words, he proporo of each plae geeral behavor of plae s as followg: The proporo of h plae geeral behavor: 8 π.w.l.l.c (8) I he proposed mehod, by rasferrg he behavor esor o he plaes, behavor of graular maeral s ally deermed each specfc dreco. I oher words, deformaos dffere drecos are ruled. The by havg he rule of deformao each dreco ad by umercal egrao of mul-plae paer (based o sra rasfer from plaes o geeral coordaes), collecve effec of plae behavor o geeral behavor s obaed. Kowg expermeal specme curves from udergroud behavor of maeral ad usg hs mehod, he model compoes ca be obaed. I oher words, due o raffc of sras from geeral 85 () = = 8π w.l.c.l. σ I hs maer, as was showed he equao, he proporo of h plae behavor effec o geeral behavor s as followgs: The proporo of h plae behavor effec o geeral behavor: = 8 π.w.l.c.l () J By equalzg he Eq. 8 ad, followg equaos are obaed: C.l = l.c (4) Thus he paramerc relao of geeral behavor esor ad behavor esor of each plae wll be deermed.

5 J. Compuer Sc., 5 (): 8-80, 009 MATERIALS AND MATHODS I he followg, for a geeral behavor of maeral, smulao s carred ou usg hs mehod o llusrae he usage of he mehod. Sress-sra carves of hs geeral case show Fg. 4. Frsly, sffess ad complace marces are calculaed each sep by equaos obaed by preseed curves. To preve he ambguy of complace marx, a very small value s assumed for he seep of les where he seeps were equal o zero. The am of wrg sffess marx was o eable he smple summao of equaos based o sress ad deermao of fal complace marx each sep. I should be oed ha durg he soluo of problem, he effec of shear sress producg some proporo of ormal sress s cosdered. I addo ormal sress equaos are cosdered boh compresso ad eso rages. (a) (b) I he followg, he correspodg equaos of each sep of carve are preseed: Par : 0 C D = 0 = 0 σ = β σ σ β τ = = β = τ = τ = Par : 0 σ τ C = D = 0 0 (c) Fg. 4: Sress-sra curve of maeral a geeral case: (a) shear sress-shear sra; (b) Normal sra-shear sra; (c) ormal sress-ormal sra 8 σ = β β( ) τ = ( ) σ β = ( β β ) τ β = τ ( β β ) σ = τ 0

6 J. Compuer Sc., 5 (): 8-80, 009 Par : C = D = 0 0 ( ) σ = β β ( ) β τ = ( ) σ β = ( β β ) τ β β = τ β σ = τ 0 Par 4: β ( β β ) 0 C = D = ( ) σ = β β ( ) β τ = ( ) 4( ) σ = β β( ) β ( ) = τ ( ) β β( ) β ( ) σ = τ 0 ( ) As ca be see, all of he above equaos are he form of followg equao, where he vecor { } 0 s he sra whou sress vecor. C 0 0 σ = 0 C 0 τ 0 0 C τ { } 0 Havg deermed sress-sra equaos for he plaes, he complace marx s rasferred o geeral coordaes usg rasfer vecors. I s obvous ha for calculao of rasfer marces, dreco coses should be calculaed ormal ad shear drecos. The values of hese coses are abulaed Table. I should be oed ha for he meoed problem, sce he resula shear sress s gve, dreco coses oe shear dreco s us used. Usg dreco coses ad meoed equaos, rasfer marces of he plaes are deermed ad dcaed Table. Usg below equaos, geeral complace marx s calculaed: ^ T C = T. C. T = T T T T T T C 0 0 T T T 0 C 0 T4 T4 T C T5 T5 T 5 T T T T T T T4 T5 T T T T T4 T5 T T T T T4 T5 T T C T C T C T C T C T C T T T T4 T5 T TC TC TC T T T T4 T5 T T4C T4C T4C T T T T4 T5 T T5C T5C T5C = TC TC TC ^ = = C... SYM...

7 J. Compuer Sc., 5 (): 8-80, 009 Table : Dreco coses of egrao pos Dreco coses of egrao pos Weghs l m * l * m * ** l 0 ** m ** (W ) Table : rasfer marces of he plaes Trasfer marces T T T T T T T T T T T T T ^ = π = C 8 w.c... SYM As ca be see, above equao are very log ad he calculao s very me cosumg. Thus for smplcy, a program s developed Excel sofware whch calculaes he compoes of geeral complace marx ha are cosa for all of he 4 maeral codos. Usg hs program, geeral complace marx for all of he 4 pars of he gve problem as dscussed above. To compare he resuls of he proposed model wh real resuls, he resuls of a expermeal es o a soe specme dffere al pressures s seleced. These resuls were publshed Lodo Geoechcal oural. Cosderg he carves gve for hs problem, he curve whch s correspodg o al pressure 00 kpa s he mos suable carve. I 88

8 J. Compuer Sc., 5 (): 8-80, 009 he Fg. 5, he resuls of he es are compared o he resul of proposed model. RESULTS If he mehod be properly used ad he assumpos be correc, he mehod ca accuraely smulae he behavor of complcaed maerals. The followg cases are he advaages of he mehod: smulao of ay maeral behavor wh ay sress-sra curve, aalyss of fracure, deermao of falure oreao, Possbly of applcao of ay falure crera such as Mohr-Coulomb or Drucker-Prager, possbly of creasg he umber of uder sudy plaes mulplae paer ad cosderg he effecs of maeral behavor he plaes perpedcular o prcple plaes, predco of asoropc behavor of maerals ad separao of he behavor o dffere plaes, predco of sofeg ad hardeg ad so o, possbly of accurae vesgao of geomercal oleary, possbly of vesgao abou he roao of prcpal axs, predco of asoropc behavor ad plascy propagao hsory maeral. DISCUSSION The reaso of observable dffereces bewee he resuls of he model ad he resuls of expermeal es s show Fg.. As ca be see from Fg., o predc he behavor, he le s used sead of curve. So he h erao, he values of sra are dffere by he value. Also he calculao of bulk sras, sce he dffereces exs all he hree axal sras, he value of error wll be accumulaed. To preve uaccepable error value, he umber of model les should be creased. Ths s show Fg. 7. Fg. : The error value sra predco Fg. 5: Comparso bewee he resuls of he proposed model ad expermeal es 89 Fg. 7: Icreasg he umber of les o decrease he error value

9 J. Compuer Sc., 5 (): 8-80, 009 I Fg. 7, he curve s smulaed by he hree les as ca be see ad he value of error s sgfcaly decreased. CONCLUSION I hs sudy, s red o propose a mehod by whch he behavor of graular maeral ca be predced a smple ad drec form. Usg mulplae paer make possble o sudy he behavor of maeral dffere drecos. Moreover, he kd of falure ca be deermed by vesgag he sresses he plaes. Furhermore, here wll be o lmao for dffere sress-sra curves usg he preseed mul-le mehod ad hs model has o resrco, sce geeral case s suded ad so ay complcaed ad dffere behavor ca be smulaed. Ths mehod ca also be used for sudy he effec of earhquake ad cyclc loadg o specmes. REFERENCES. Desa, C.H. ad H.J. Srwardace, 984. Cosuve Laws for Egeerg Maerals. Prece Hall PTR, ISBN: 0: 07940, pp: 48.. Erge, A.C., 9. Nolear Theory of Couous Meda. McGraw-Hll Ic., USA., ISBN: 0: , pp: Desa, C.S., 97. Applcaos of he Fe Eleme. Proceedgs of he Symposum o Mehod Geoechcal Egeerg, May - 4, Vcksburg, Msssspp. hp://oa.dc.ml/oa/oa?verb=gerecord&m eadaaprefx=hml&defer=ada Duca, J.M. ad C.Y. Chag, 970. Nolear aalyss of sress ad sra sols. J. Sol Mech. Foud. Dvs. ASCE, 9: 9-5. hp://cedb.asce.org/cg/wwwdsplay.cg? Sadrezhad, S.A., 99. Mul-lamae elasoplasc model for graular meda. J. Eg. Islamc Repub. Ira, 5: -4.. Sadrezhad S.A., 00. Cosuve model for mul-lamae duced asoropc double hardeg elasc-plascy of sad. I. J. Eg., 5: 5-4. hp:// arsr=5;sadrnezhad%0s.a.;internatio NAL%0JOURNAL%0OF%0ENGINEERING; JUNE%000;5;%0(TRANSACTIONS%0A :%0BASICS);5;4 80