A modified virtual multi-dimensional internal bonds model for geologic materials

Transcription

1 47 ISSN MEHANIKA 015 Volume 1(5): 4751 A modfed vrual mul-dmesoal eral bods model for geologc maerals JFu Ke* AXag Wu** *Uversy of Scece ad Techology Bejg Bejg ha E-mal: kejfu@homalcom **Uversy of Scece ad Techology Bejg Bejg ha hp://dxdoorg/105755/j01mech Iroduco I fracure smulao feld developg ew models wh more realsc mcrosrucure s a flourshg edecy Xu ad Needlema (1994) [1] developed he cohesve surface models by roducg a seres of dscree cohesve surfaces durg he fe eleme dscrezao Those cohesve surfaces have lmed cohesve sregh ad lmed work of fracure Ad cracks are formed ad spread alog eleme boudares Gao ad Kle (1998) [] proposed he vrual eral bod (VIB) model whch s a dffere approach by roducg vrual eral bods wh srech sffess aachg erracally dspersed maeral parcles sde he sold ad sascally averagg he sochasc spaal ework of bods The VIB uses he auchy-bor rule of crysal elascy o oba macroscopc collecve behavour of he sochasc bod ework sde sold maeral ad oly cosders he srech eergy of vrual eral bod ad has oly oe ype of bod possessg oly oe kd of sffess coeffce (srech sffess) whch lead o free roao of bods Volokh ad Gao (005) [] proposed a modfed vrual eral bod (MVIB) model wh bods owg boh srech ad bed sffess I he MVIB model he Gree sra esor s decomposed o sphercal ad devaorc pars ad he srech deformao of bods are decomposed o sphercal dlaaoal deformao ad devaorc deformao o calculae bods sra eergy Zhag ad Ge (005) [4 5] proposed aoher modfed verso of VIB whch roduced a R-bod o resrc he roao freedoms of maeral parcles The ew model s called as vrual mul-dmesoal eral bods (VMIB) model ad oly ca be used o maeral wh Posso rao lower ha 05 Zhag ad Gao (01) [6] proposed a augmeed vrual eral bod (AVIB) model usg he Xu-Needlema poeal o descrbe bods eergy The AVIB decomposed bod s sra o srech sra ad shear sra ad he calculaes he bod sra eergy by he Xu-Needlema poeal Zhag e al (014) [7] used he Sllger-Weber poeal whch s a combao of wo- ad hree-body eracos o calculae bods eergy VIB model All he above-meoed models could oly be appled o maeral wh Posso rao smaller ha 05 whch s a severely resrco o he furher developme of vrual eral bod heory osuve model for o-lear elasc maeral I VIB heory he sold maerals are cossed of dscree ad radom mass parcles mcro level whle hey do have o be aoms ad vrual eral bods are added o descrbe he eracos bewee parcles (Gao ad Kle 1998) [] The he deformao of he couum dsplaceme feld s placed he deformao of he crysalle correspodely hrough he auchy-bor rule ha s o say he aomc posos are coeced o he couum felds va he local deformao grade (Tadmor e al 1996) [8] The local deformao grade F s appled o he udeformed crysal lace bass ad rebuldg he crysal hrough he alered base vecors o oba he deformed crysal srucure (see Fg 1) I hs way each couum represeave volume eleme s represeed by fe crysals uderakg uform deformao Fg 1 The auchy-bor rule I hs modfed model he vrual eral bod possesses boh srech sffess ad shear sffess Fg shows he mcrosrucure used hs modfed model Fg Mcrosrucure of modfed model Vrual eral bod refers o he radal lk bewee wo mass parcles Therefore each bod has a uque oreao If all he vrual eral bods wh oe u of volume of a maeral are rearraged by her oreaos he sphere coordae sysem (Fg ) Ad he a vrual eral bod s oreao ξ would be expressed as: ξ s cos s s cos (1) T

2 48 Assumg ha he maeral s soropc he he bod desy ad sffess should be uform spaal So he average bod eergy for per surface area of he u sphere (Fg ) wh oe u of volume of a maeral s U k K x xdx kw U r R x xdx rw 0 0 (4) Fg Vrual eral bod s oreao ad he u sphere I hs model he deformao of each vrual eral bod s decomposed o hree pars: he sphercal srech sra he devaorc srech sra ad he shear sra The sphercal srech sra s he bod s ormal sra owg o sphercal dlaaoal deformao he devaorc srech sra s he bod s ormal sra owg o he devaorc deformao ad he shear sra s he bod s ageal sra due o he whole deformao Ad he case of small deformao he expressos of he sphercal srech sra he devaorc srech sra ad he shear sra are: 1 1 = j j ξ εξ T T T ξ ε εξ j j j j I he orgal vrual mul-dmesoal eral bods model he srech sra s he summao of he devaorc srech sra ad he sphercal srech sra Ad s shear parameer would be egave whe maeral Posso rao s greaer ha 05 Ths s maly caused by he excessve srech eergy The sphercal srech sra s he major srech sra of vrual eral bods so hs modfed model would be cosdered as a effecve srech sra for bod s srech eergy The sra eergy sored vrual eral bods cosss of srech eergy U ad shear eergy U Assumg ha he srech sffess s fuco of sphercal srech sra ad shear sffess s fuco of shear sra : k k0k r r0 R K0 R0 1 here k 0 s al srech sffess ad r 0 s al shear sffess () () here k ad r are bods average srech ad shear maeral parameers for per surface area of u sphere respecvely I quas-couum heory he sra eergy desy mus be equvalece order o make a represeave volume eleme of dscree mcrosrucure mechacally equvale o a couum volume eleme Therefore he macro sra eergy desy Φ should be equal o he summao of he ere bods srech ad shear eergy (Fg ): U Usd d (5) Based o he work cojugae prcple he auchy sress esor s (Ogde 1984) [9]: j W W j k r (6) j j j here jm s: sd d Ad he elasc modulus W W jm k j m j m (7) W r r j m j m So ha wh Eq (4) he dervaves of would be: W W W K K K 1 R W 1 R Ad he paral dervaves of Eqs (6) ad (7) would be: ad W ad W (8)

3 49 1 j j kk j + jkk a abb j j (9) 1 m j m j jm jm j m jm here 1 s cos s s cos Icremeal cosuve model uder raxal compresso Assumg ha maerals are soropc elasc uder raxal compresso cremeal deformao ad age Youg s modulus E ad age Posso rao are he maeral elasc parameers each sage of deformao These wo parameers chage value relaed wh sages of deformao I geologc maerals basg o he Duca-hag o-lear elasc model [10] E v are gve as: E E 1 R f S v v D1 1 E1 R fs (10) here E s he al age modulus S s he sress level R f s he falure rao v s he al age Posso rao 1 are he major ad mor prcpal sresses D s he maeral parameer I cremeal deformao wo age parameers k r are used o replace k r ad: K R 1 1 Subsug Eq (11) o Eq (8): W W W W Basg o FEM aalyss he esor be expressed a marx form: Ω (11) (1) jm could (1) Ad soropc lear elasc maeral he sresssra relao could be expressed as: E 1 Ω (14) symmery Subsug Eqs (9) ad (1) o Eq (7) ad he egrag Eq (7) o oba jm :

4 k r k r k r k r k r k r 0 Ω 15 6 (15) r 5 symmery r 0 5 r 5 By combg Eq (14) wh Eq (15) he relao bewee k r ad he lear elasc maeral parameers (age Youg s modulus ) could be obaed: E k 4 1 5E r 4 1 E ad age Posso rao (16) ad sra creme a he curre creme he updaes he sresses ad sae varables o he approprae values a he ed of he me creme [1] Wh he UserMa subroue some umercal smulaos of uaxal eso geologc maerals are performed ANSYS I hese smulaos a cyldrcal specme s fxed oe sde ad he appled wh a esle dsplaceme he oher sde Axsymmerc sold plae18 eleme s used o mesh he specme (Fg 4 shows he loadg codos ad he mesh of elemes) From Eq (16) boh he wo maeral parameers k r rema posve whe age Posso rao s greaer ha 05 Ad he lm Posso rao of hs model s cosse wh he maeral lm Posso rao 05 4 Applyg o he uaxal eso process brle rock maerals I cosderao of smple loadg case he olear elasc model ca be appled o smulae uaxal eso process soropc brle rock maerals Ad order o smulae hs process hose fucos K R could be chose as (Gao ad Kle 1998 Zhag ad Ge 005 Zhag ad he 009) [ ]: Fg 4 The loadg codos ad he mesh of elemes The real esle ess of Tako Sadsoe Sajome adese ad Kmach Sadsoe were doe arcle [1 14] ad he es daa were ced here o compare wh he umercal smulao resuls K exp A R exp B (17) here cosas A B are used o f maeral properes ( AB 0) The o-lear elasc cosuve Eqs (6) ad (7) are combed wh he Eqs (8) (9) ad (17) ad he embed o he ANSYS Those maeral cosuve equaos are wre o he ANSYS UserMa Subroue The UserMa subroue s used o defe a maeral s sress-sra relaoshp ad apples o ay aalyss procedure volvg mechacal behavor The subroue s called a every maeral egrao po of he elemes durg he soluo phase The program passes sresses sras ad sae varable values a he begg of he me creme Fg 5 omparg he umercal smulao resuls wh he real es daa The parameers used for umercal smulao are: Sajome adese k = 6541 GPa r = 55 GPa A = 188e-4 B = 11e-7 Tako Sadsoe k = 97 GPa r = 1691 GPa A = 1917e-4 B = 617e-7

5 51 Kmach Sadsoe k = 51 GPa r = 1810 GPa A = 165e-4 B = 788e-7 As show Fg 5 he sresssra curves gve by umercal smulao learly crease a frs ad he cocave upward o he peak sregh po afer ha he sress drops almos vercally he a dsc raso appears ad he sress sars o decrease slowly ul gradually levels off ear a very low cosa value The curve of Sajome adese s very much close o he real es daa ad fs well wh he expermeal resuls The curves of Tako Sadsoe ad Kmach Sadsoe are close o he es daa excep a pospeak par 5 oclusos The modfed vrual mul-dmesoal eral bods model proposed hs paper uses he sphercal srech sra ad he shear sra o calculae he bods srech eergy ad shear eergy respecvely so as o overcome he excessve srech eergy he VMIB model By comparg wh he Duca-hag o-lear elasc model he mcro maeral parameers k r he modfed model are obaed ad expressed by he macro maeral parameers (he age Youg s modulus E ad he age Posso rao ) These wo mcro maeral parameers rema posve whe he maeral Posso rao s greaer ha 05 The modfed model successfully expads he lm Posso rao of he VMIB model o he maeral lm Posso rao 05 A o-lear cosuve model s gve chaper 4 Uder small deformao ad smple loadg hs o-lear cosuve model s appled o smulae he uaxal eso process brle rock maerals ad fs well wh he expermeal daa excep a he pos-peak par Refereces 1 Xu X-P Needlema A 1994 Numercal smulaos of fas crack growh brle solds Joural of he Mechacs ad Physcs of Solds 4(9): hp://dxdoorg/101016/ (94) Gao H Kle P 1998 Numercal smulao of crack growh a soropc sold wh radomzed eral cohesve bod Joural of he Mechacs ad Physcs of Solds 46(): hp://dxdoorg/101016/s (97) Volokh KY Gao H 005 O he modfed vrual eral bod mehod Joural of Appled Mechacs 7(6): hp://dxdoorg/101115/ Zhag ZN Ge XR 005 Mcromechacal cosderao of esle crack behavor based o vrual eral bod coras o cohesve sress Theorecal ad Appled Fracure Mechacs 4():4-59 hp://dxdoorg/101016/jafmec Zhag ZN Ge XR 005 A ew quas-couum cosuve model for crack growh a soropc sold Europea Joural of Mechacs - A/Solds 4(): 4-5 hp://dxdoorg/101016/jeuromechsol Zhag ZN Gao H 01 Smulag fracure propagao rock ad cocree by a augmeed vrual eral bod mehod Ieraoal Joural for Numercal ad Aalycal Mehods Geomechacs 6(4): hp://dxdoorg/10100/ag Zhag ZN he YX Zheg H 014 A modfed Sllger-Weber poeal-based hyperelasc cosuve model for olear elascy Ieraoal Joural of Solds ad Srucures 51(7-8): hp://dxdoorg/101016/jjsolsr Tadmor EB Orz M Phllps R 1996 Quascouum aalyss of defecs solds Phlosophcal Magaze A 7(6): hp://dxdoorg/101080/ Ogde RW 1984 No-lear Elasc Deformaos Joh Wley ad Sos: New York 10 Duca JM hag Y 1970 Nolear aalyss of sress ad sra sols Joural of he Sol Mechacs ad Foudaos Dvso ASE 5: Zhag ZN he YQ 009 Smulao of fracure propagao subjeced o compressve ad shear sress feld usg vrual muldmesoal eral bods Ieraoal Joural of Rock Mechacs ad Mg Sceces 46(6): hp://dxdoorg/101016/jjrmms ANSYS Ic Asys 150 User Maual (User programmable feaures) 1 Fuku K Okubo S J F 1995 omplee sresssra curves of rock uaxal eso es Joural of he Mg ad Maerals Processg Isue of Japa 111(1):5-9 hp://dxdoorg/1047/shgeosoza Okubo S Fuku K 1996 omplee sress-sra curves for varous rock ypes uaxal eso Ieraoal Joural of Rock Mechacs ad Mg Sceces ad Geomechacs Absracs (6): hp://dxdoorg/101016/ (96) JFu Ke AXag Wu A MODIFIED VIRTUAL MULTI-DIMENSIONAL INTERNAL BONDS MODEL FOR GEOLOGI MATERIALS S u m m a r y Ths arcle preses a modfed verso of he vrual mul-dmesoal eral bods model I hs modfed model a dffere approach for vrual eral bod sra eergy s proposed whch uses he sphercal srech sra ad he shear sra o calculae he bods srech eergy ad shear eergy respecvely so as o overcome he excessve srech eergy he vrual muldmesoal eral bods model Ad he modfed model successfully exeds he lm Posso rao o 05 Ths ew model fs well wh he uaxal eso process of brle rock maerals Keywords: vrual mul-dmesoal eral bods augmeed vrual eral bod o-lear elascy Receved March Acceped Aprl 1 015