Int. Agrophys., 2013, 27, 275-281 oi: 10.2478/v10247-012-0095-6 INTERNATIONAL Agrophysis www.interntionl-grophysis.org Visuliztion of hrteristis of the ontt network etween spheres in 3D ssemly R. Koy³k* n M. Molen Institute of Agrophysis, Polish Aemy of Sienes, Doœwizln 4, 20-290 Lulin, Poln Reeive Otoer 31, 2012; epte My 5, 2013 A s t r t. Anlysis of geometril fetures of 3D grnulr ssemlies poses iffiulties in visuliztion of the effets tking ple within the volume of the smple. This work presents the metho for visuliztion of sptil istriution of the ontt points in monoisperse; fritionl ssemly ompose of spheres n generte using three filling methos for uoil or ylinril ontiner. The visuliztion is se on the projetion of the ontt points etween prtiles on hypothetil sphere followe y their representtion on the plne. The propose metho ws foun to equtely visulize ifferenes in the sptil struture of ssemlies generte using vrious filling methos. K e y w o r s: grnulr mteril, grnulr ssemly, pking struture, istriution of ontts, ontt network INTRODUCTION Los exerte y grnulr mterils on the onstrution memers of storge n proessing equipment re the result of intertions etween prtiles n epen on their sptil istriution in the eposit. Numerous projets were onute to fin esription of mehnil ehviour of grnulr ssemly se on mehnil intertions etween prtiles. Mny experimentl investigtions were performe to interpret the phenomen tht tke ple in grnulr mterils, ut re hrly oserve in gses, liquis nor solis, suh s nisotropy of mehnil properties (Nouguier- Lehon et l., 2003; Mueller, 1993), pressure swith t the strt of silo ishrge (Moysey, 1979), or symmetry of lo istriution uring eentri ishrge of silo (Horik et l., 1993; Koy³k n Molen, 2013). One of the promising methos for nlysis of mehnil phenomen in grnulr mterils is the numeril istint element metho propose y Cunll n Strk (1979). The metho llows estimtion of the pressure istriution within n ssemly n los exerte y the eposit se on the nlysis of istriution of inter-grnulr fores n their hnges uring eformtion (Sithrm n Shimizu, 2000; Sykut et l., 2008). It is not possile to vlite suh metho in n experiment without isturne of the smple struture. At n erly stge, simultions using DEM were fouse on the pking struture (Rothenurg n Bthurst, 1989) n methos for visuliztion of the ontt network etween prtiles (Dresher n e Josselin e Jong, 1972) n were onute on 2D ssemlies of reltively low numer of irulr prtiles. In the se of 3D ssemlies, there is still no stisftory metho for visuliztion of the network of ontt points n ontt fores. One of the promising pprohes is the proposl of Fisher et l. (1987) to onnet ontt points of equl proility with isolines n to projet the otine figure on the surfe of sphere. In the se of two-imensionl prtiles, representtion of ontt fores in the form of polr istriution got wie eptne. No suh metho of wier eptne exist for visuliztion of istriution of ontt fores in the of 3D ssemlies. As we hve strte DEM numeril simultions of ehvior of grnulr systems uner vrious onitions of mehnil lo, the ttempt to elorte metho of visuliztion reporte in this work ws unertken. In this work, metho for visuliztion of istriution of vetors norml to ontt res (ontt normls) n the ontt fores in 3D ssemly of spheres is propose. Exmples of the pplition of the metho in nlysis of the ssemly struture re lso shown. MATERIALS AND METHODS Numeril simultions of filling uoil n ylinril ontiners were performe using the PAPA softwre otine from the we pge of the University of Stuttgrt *Corresponing uthor e-mil: rkoylk@ipn.lulin.pl 2013 Institute of Agrophysis, Polish Aemy of Sienes
276 R. KOBY KA n M. MOLENDA (Shwrzer, 2004) n evelope. Three filling methos were opte to form n ssemly of 3 000 spheril prtiles with imeter of 0.002 m: entri prtiles were free flling one y one long the vertil xis of the ontiner; isperse strem of prtiles ws roppe; n in-volume prtiles were generte in rnom positions within the volume of the ontiner n then llowe to fll uner grvity. The se of the ontiner ws irle or squre of equl surfe res n imeter of 20.6 prtile imeters or sie of 18.25 prtile imeters. To lulte the intertions etween prtiles the visoelsti ontt moel of Kuwr n Kono (1987) ws use s implemente y the uthors in the PAPA pkge: 3 1 F kn x 2 x 2 v, (1) where: x stns for n overlp etween two prtiles, v reltive veloity, k n spring stiffness efine s (E Young moulus): kn 4 1 E * R * 2, (2) 3 Ei E j E*, (3) Ei ( 1 v j ) E j ( 1 vi ) n the reue prtile rius (R*): Ri R j R*. (4) Ri R j Inexes i n j refer to the numer of the prtile (prtile i) n i j. Ftor is the effetive issiptive ftor relte to the ulk visosities B i, B j, i, j n the rii of urvture of the olliing oies: 1 2B * R * 2, (5) Bi B j B*. (6) Bi ( 1 j ) B j ( 1 i ) The first term of the expression (1) esrie the nonliner epenene on n overlp of the norml fore ting etween two perfet elsti spheres oring to the Hertz theory. The seon, lso nonliner issiptive prt is similr to the originl Hertz pproh se on the ulk visosity prmeters. The mehnil prmeters of the prtiles were tken s equl to the properties of rpesee (Molen n Stsik, 2002; Wi¹ek n Molen, 2011; Wojtkowski et l., 2010): Young moulus E of 8.7 MP, Poison rtio of 0.17, oeffiients of inter prtile frition pp n wllprtile frition w oth of 0.3, prmeters of mping B of 708.2 Ns m -2 n of 0.15, prtile ensity of 1 050 kg m -3. Bse on the oorintes of generte prtiles, the oe ws written to nlyze n visulize the network of the ontts etween prtiles in hosen moment of simultion s expline elow. The onition of ontt to exist is tht the istne etween entrois of two spheres is less or equl thn the sum of their rii: 2 2 2 ( x j xi ) ( y j yi ) ( z j zi ) R j Ri. i j The propose metho of the ontt visuliztion is se on the representtion of the oorintes of the ontt points trnsforme from the Crtesin to the spheril system of oorintes. The onition require to mke suh trnsformtion possile is esrie y the seon onition, whih must e heke uring the omputtion: (7) ( x j xi ) 2 ( y j yi ) 2 0. (8) i j Conition 2 (Eq. (7)) is stisfie only in the se of x i =x j n y i =y j ie the ontt point is lote on line prllel to z xis in Crtesin oorinte system, whih in the se of the spheril oorintes mens, it is lote t the pole ( 0 or unefine). Otherwise the oorintes of the ontt point my e uniquely trnsforme to the spheril oorinte system. In oth the Crtesin n spheril systems of oorintes, three oorintes re neessry to esrie the lotion of point in the spe. In the Crtesin system, these re three liner oorintes x, y, z, while in the spheril system one liner istne r n two ngles n re use (Fig. 1). Angle mesure t the XY plne is etermine s: y j yi r tn x x, (9) j i when x j - x i <0 n y j - y i 0, ngle hs to e ugmente y. In turn, when x j - x i <0 n y j - y i < 0, ngle hs to e ontrte y. In orer to otin ngle in the rnge from 0 to 2, negtive vlues nee to e ugmente y 2. Fig. 1. Trnsformtion from the Crtesin to the spheril oorinte system.
VISUALIZATION OF CHARACTERISTICS OF CONTACT NETWORK BETWEEN SPHERES IN 3D ASSEMBLY 277 Fig. 2. Projetion of the ontt point on hypothetil sphere. ontt point on the surfe of the sphere my e illustrte through its lotion in ell of the network rwn on the plne (, ). The ensity of the ontt points or the vlue of fore norml to the ontt re my e represente y the thir oorinte of the grph. The ensity of the ontt points P(, ) ws efine s set of rtios: numer of ontts in the onsiere ell ( /180, /180) ( /180, /180) of the network to the totl numer of ontts (Fig. 3). Normlize vlues of the ontt fores etween prtiles in the ontiner were shown t the plne (, ). RESULTS AND DISCUSSION Fig. 3. The plne (, ) refleting the surfe of the sphere with two y two egrees (/90 /90) mesh n mrke projetions of ontt point shown in Fig. 2. The seon ngle, mesure in the vertil plne, is etermine s follows: r os z j zi x x y y z z.(10) 2 2 2 ( j i ) ( j i ) ( j i ) The spheril system of oorintes ws opte euse only two ngulr oorintes n re require to efine the point lote on the surfe of the sphere inste of three numers neessry in the Crtesin system. In the se of the monosize ssemly of spheril prtiles onsiere, the ontt points of iniviul prtiles my e projete on one hypothetil sphere s shown in Fig. 2. To filitte visul omprehension the surfe of the hypothetil sphere my e reflete s (, ) plne. The two spheres shown in the Fig. 2 ontt eh other in one point, whih is projete on single sphere (rius of this sphere is irrelevnt). As two spheres tke prt in the ontt, the seon point representing the ontt point from the seon sphere my e otine in the symmetril trnsformtion ginst the entre of the sphere. The lotion of the Distriutions of the ontt points otine for the two shpes of the ontiner n for the three filling methos re presente in Figs 4 to 8 eh figure ontins four illustrtions: the top left is 2D istriution mp of the ontt points mesure s the rtio P of the numer of ontt points foun in the, lotion to the totl numer of ontt points; the ottom left shows the sme t ut visulize in 3D s olumns in given lotions with heights n olours orresponing to eh other; the top right represents the 3D view of the eposit; the ottom right shows 2D projetion of the istriution of normlize ontt fores lulte se on the Hertz theory. All istriutions shown in Figs 4 to 8 present t the () plne hrteristi onentrtion of the ontt points long the equtor of the hypothetil sphere ( / 2). This signifies tht lrge portion of ontts tke ple t the equtors of the prtiles prllel to the se of the ontiner (ie the lines joining prtiles entrois re prllel to the se of the ontiner), whih is result of the orering influene of the flt ottom of the ontiner. The other res of onentrtions of ontt points my e seen roun the ngle of pproximtely /6 to 7/36 n symmetril onentrtion (t etween 29/36 n 5/6). These intervls orrespon to the ontt ngles etween prtiles when one prtile is supporte y three or four prtiles resting elow in mutul ontt on the flt surfe the rrngement typil
278 R. KOBY KA n M. MOLENDA Fig. 4. Chrteristis of the struture of the ssemly of spheril prtiles forme in uoil ontiner y entri filling: plnr projetion of the istriution of the ontt points, 3D view of the ssemly, istriution of the ontt points 3D view, istriution of normlize vlues of the ontt fores. Fig. 5. Chrteristis of the struture of the ssemly of spheril prtiles forme in the in-volume fille uoil ontiner: plnr projetion of the istriution of the ontt points, 3D view of the ssemly, istriution of the ontt points 3D view, istriution of normlize vlues of the ontt fores.
VISUALIZATION OF CHARACTERISTICS OF CONTACT NETWORK BETWEEN SPHERES IN 3D ASSEMBLY 279 Fig. 6. Chrteristis of the struture of the ssemly of spheril prtiles forme in the ylinril ontiner y entri filling: plnr projetion of the istriution of the ontt points, 3D view of the ssemly, istriution of the ontt points 3D view, istriution of normlize vlues of the ontt fores. Fig. 7. Chrteristis of the struture of the ssemly of spheril prtiles forme ylinril ontiner y in-volume filling: plnr projetion of the istriution of the ontt points, 3D view of the ssemly, istriution of the ontt points 3D view, istriution of normlize vlues of the ontt fores.
280 R. KOBY KA n M. MOLENDA Fig. 8. Chrteristis of the struture of the ssemly of spheril prtiles forme in the ylinril ontiner y isperse filling: plnr projetion of the istriution of the ontt points, 3D view of the ssemly, istriution of the ontt points 3D view, istriution of normlize vlues of the ontt fores. of rystl lttie. The effets re lerly visile in Fig. 4, whih presents the istriution of ontt points in the se of the entrlly fille uoil ontiner. Here, the res of ense onentrtions of ontts re very regulr oth in iretions of equl height () s well s, whih is even more pronoune, long the perpeniulr iretion () where ensifitions of the ontt points pper t onstnt istnes (every /12) (Fig. 4). Another strong ensifition might e seen in lose viinity of four points: (/6,/2); (5/6, 3/2) n symmetril onentrtion (/6, 3/2); (5/6,/2). A similr pttern might e oserve in the ontt fore istriution (Fig. 4). The highest vlues of the ontt fores orrespon very well with the onentrtions of ontt normls (t the equtor, with onstnt repetitive ngle of pproximtely /12). Another hrteristi effet whih might e oserve re the four irles hving rius of /3 with entres t the equtor n horizontl ngles ( equl to /2,, 3/2 n 2This pttern my e otine when the sphere is supporte y two others remining in ontt n rottes lokwise or ntilokwise without hnging its istne to oth prtiles. The entres of suh irles ten to lign with lines prllel to vertil wlls. Suh n orere onfigurtion my e generte y entri filling where prtiles fll wn one y one n oul ttin the positions of minimum energy without eing isture y impts with other prtiles flling simultneously. The onfigurtion of the ssemly generte in the sme ontiner using the in-volume filling metho (Fig. 5) o not show suh high egree of orering. The ifferene etween these two ses is prtiulrly pronoune on the r plots for the two ssemlies (Figs 4 n 5). The onentrtions of the ontt points t the equtor re still visile, ut more isperse thn the regulr pttern presente in Fig. 4. Contrry to the ssemly generte y entri filling, no irles re visile in the se of eposit generte y in-volume filling. On the other hn, stright vertil lines t ( / 2 n 3 / 2) eme visile. Densifitions of istriutions of the ontt normls re ompnie y inrese vlues of the ontt fores. Centri filling resulte in higher numer of prtiles ontting long oplnr imeters reting the regulr rystl struture (isture y the wll size equl 18.25 prtile imeters), while in the se of in-volume filling stronger influene of the wlls ws oserve (vertil lines t pprox. / 2,, 3 / 2, 2 while in the se of entri filling only t pprox. 2 ). The eposit forme in the ylinril ontiner hs more iffuse ontt network s there is no tion of plnr wlls enforing ontts in two perpeniulr iretions. In the se of entri filling (Fig. 6), the ontts onentrte t the equtor t regulr istne of every 2/3. Similr ensifitions might e oserve t the lines t of pproximtely7/36 n 27/36 only n t the horizontl ngle rotte y /3 s ompre to the ensifition t the
VISUALIZATION OF CHARACTERISTICS OF CONTACT NETWORK BETWEEN SPHERES IN 3D ASSEMBLY 281 equtor. Now only two irles re visile (si n resulting from the symmetry). The entres of these irles representing orienttion of the struture of the ssemly re not uniquely tie with the horizontl oorintes. This effet reflets the ft tht uring filling, when the first lyer on the plne surfe of the floor is forme, prtiles orient themselves long n ritrry horizontl iretion etermining further evelopment of the struture. Unlike in the se of the uoil ontiner, here the onentrtions of ontt fores o not perfetly follow the onentrtions of the ontt normls. Aprt from the firly regulr pttern t the equtor n the irles, the remining istriution seems more isperse. The eing in the in-volume fille ylinril ontiner (Fig. 7) shows the onentrtion of the ontt normls t the equtor s the result of the influene of the flt ottom. Weker n more lurre lines of the onentrtions of the ontt points t of pproximtely /6 n 5/6 n n itionl line t of pproximtely 7/36 with more ggregte ensifition of the ontt points re oserve. No hrteristi irles re forme. The istriution of ontt fores present hoti pttern, prt from the inrese in the vlues t the equtor n their onentrtions (with no inrese in the vlues) roun the lines orresponing to rystl lttie (Fig. 7). Disperse filling of the ylinril ontiner (Fig. 8) resulte in the most irregulr sptil struture. In this se, the ontt normls onentrte t the equtor s result of the influene of the flt floor lthough it is presente y the wekest n the most lurre line (Fig. 8). The other res of wek onentrtion my e oserve roun lines of pproximtely /6 n 5/6. No irles hrteristi for entri filling re present. Aprt from some inrese in vlues t the equtor, the istriution of ontt fores presents rther hoti pttern (Fig. 8). CONCLUSIONS 1. Bse on results of numeril tests the propose metho my e regre promising proposition in serh for effiient wy of visuliztion of the network of ontt fores in 3D ssemly of spheril prtiles. In the ses of influenes of the filling metho n the ontiner shpe on the sptil struture of the ssemly the metho prove to e useful tool. 2. The flt ottom of the ontiner introue orer (rystl-like struture) to the ssemly, whih resulte in the onentrtion of the ontt points t the equtor ( / 2) n rete itionl lines of onentrtion t in rnge from /6 to 7/36 n symmetril onentrtion ( in the rnge from 29/36 to 5/6). The wlls of the ontiners lso introue orering n generte regulr pttern of ontts tht were epenent on the wll shpe. 3. The filling metho h strong influene on the resulting ontt network. Centri filling of oth ontiners generte mteril of high egree of orering, while in-volume filling resulte in more hoti struture. The struture lo- sest to the rnom pking ws otine y isperse filling of the ylinril ontiner. The vlues of ontt fores in the uoil ontiner were foun higher in the regions of the inrese ensity of the ontt points, while no suh effet ws oserve in the ontiner with ylinril wlls. 4. The isrete element metho ws foun to simulte properly the effets ourring uring filling of the ontiner with spheril prtiles using three methos known from physil testing. The effet of one of repose forming t the free surfe of the eing uring filling ws firly well reproue in simultions s well. REFERENCES Cunll P.A. n Strk O.D., 1979. A isrete numeril moel for grnulr ssemlies. Geotehnique, 29(1), 47-65. Dresher A. n e Josselin e Jong G., 1972. Photoelsti verifition of mehnil moel for the flow of grnulr mteril. J. Meh. Phys. Solis, 20(4), 337-351. Fisher N.I., Lewis T., n Emleton B.J.J., 1987. Sttistil Anlysis of Spheril Dt. Cmrige Univ. Press, UK. Horik J., Molen M., Thompson S.A., n Ross I.J., 1993. Asymmetry of in los introue y eentri ishrge. Trns. ASAE, 36(2), 577-582. Koy³k R. n Molen M., 2013. 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