LENGTH SCALE IN GRANULAR MEDIA

Transcription

1 LENGTH SCALE IN GRANULAR MEDIA Matthew R. Kuhn1 (Member, ASCE) and Takash Matsushma2 ABSTRACT The thckness of shear bands depends upon the sze of the grans themselves. We consder two possble sources of ths scalng: geometrc and mechancal. Because each partcle occupes space, partcle movements are constraned by ther physcal sze and shape and by ther surface curvatures at the contacts. We call these orgns geometrc. The motons of each partcle are also governed by the mechancs of rgd bodes wth complant contacts. We refer to ths orgn as mechancal. We suggest a general framework for the ncremental partcle motons of an assembly, accountng for the surface curvatures of partcles. We use two-dmensonal DEM smulatons to test the nfluence of the mechancal scalng on the shear band thckness. We alter the mechancal scalng by artfcally scalng the rad that are used n the knematc and equlbrum equatons. These alteratons dd not affect shear band thckness. We nfer that the thckness has a geometrc orgn and derves from the szes, sze dstrbuton, and shapes of the partcles. 1 INTRODUCTION Localzed deformaton features are prevalent n granular materals, and the sze of a feature often depends upon the szes of the grans themselves. Shear bands are nvestgated n ths paper, snce ther thckness s known to depend on partcle sze. Shear bands can be predcted n a contnuum settng, but the contnuum model must be enhanced to nclude some form of an nternal length scale. Enhancements to classcal contnuum models nclude gradent-dependent consttutve forms, Cosserat type mcropolar models, ntegral type non-local consttutve forms, and vsco-plastc models. Although contnuum models may produce the localzaton patterns that are observed n granular, dscrete materals, the result s somewhat artfcal: key materal parameters must be adjusted so that the desre behavor s elcted. These models provde lttle nsght nto the underlyng mechansms that produce localzaton features. The current study seeks a mcro-mechancal ratonale for the scale and thckness of shear bands. We have not yet completed the study, but the paper provdes our current thoughts about the mechancs of scalng n granular materals. We begn by hypotheszng two possble orgns of a length scale n granular materals; we present the results of smulatons that test one of the two orgns; and then we use the test results to elmnate ths orgn as a factor n the thckness of shear bands. 1 Dept. of Cvl and Env. Engrg., School of Engneerng, Unversty of Portland, 5 N. Wllamette, Portland, OR 9723, USA, kuhn@up.edu 2 Insttute of Engneerng Mechancs and Systems, Unversty of Tsukuba, 1-1-1, Tennoda, Tsukuba, Ibarak, Japan, tmatsu@kz.tsukuba.ac.jp

2 (a) (b) (c) (d) FIG. 1. Examples of a geometrc orgn of materal behavor. 2 POSSIBLE ORIGINS OF A LENGTH SCALE Two orgns are hypotheszed for a length scale n granular materals: geometrc and mechancal. Geometrc orgns are assocated wth the szes and shapes of the partcles and the consequent nterference among partcles whle an assembly s deformed. Mechancal orgns arse from the equlbrum and knematc equatons that apply to ndvdual partcles, and to the form of the contact consttutve law. Although the authors have also observed a possble nfluence of boundary condtons on the emergence and thckness of shear bands, we set asde ths nfluence n the current work. 2.1 Geometrc orgns Because partcles occupy space, ther movements are constraned by ther physcal sze, ther shapes, and ther topologcal arrangement. We gve four examples n whch assembly geometry affects materal behavor. Consder a granular materal that s smply a stack of plate-lke elastc sheets wth a range of nter-sheet frctonal characterstcs (Fg. 1a). As the assembly s sheared, the stack would, at frst, deform unformly, wth each sheet undergong an equal shearng dstorton. Once slppng begns between a sngle par of sheets, shearng wll contnue along the surface between these sheets. The shear band thckness s zero. Ths thckness s a consequence of the partcle shape, the regular arrangement of partcles, and the lack 2

3 of any rearrangement of sheets durng shearng. Consder two granular assembles that share the same topologcal arrangement of partcles, but whch have dfferent partcle shapes at ther contacts (Fg. 1b). Both arrangements may have evolved after an extended phase of ntal shearng, but ther subsequent ncremental behavors wll lkely dffer, and the shear bands that would eventually appear may also dffer n thckness. In Secton 2.2, we develop a framework for extractng the nfluence of partcle shape on the ncremental materal response. Consder the nterface between a granular materal and a wall (Fg. 1c). Incremental deformaton near the wall wll lkely depend on whether the wall s smooth or rough, and the thckness of the nterfacal shearng zone may dffer for the two condtons. Consder the two assembles n Fg. 1d: the frst assembly s entrely regular, but the second assembly contans smaller partcles nestled among the larger neghbors. If the two assembles are compressed, ther behavors wll dffer, snce the larger partcles n the second assembly wll eventually come nto contact wth the small partcles. The stuaton n Fg. 1d resembles mechansms n shear band evoluton. The partcles wthn a shear band are contnually rearranged: partcles are always comng nto contact new neghbors, whle exstng contacts are always beng dsengaged. The queston of whether a new contact wll be created between any two partcles depends upon whether they can fnd each other. Ths s prncpally a queston of assembly geometry, not of the ncremental partcle mechancs. The stuatons n Fgs. 1b and 1d dffer n the followng respect: the frst nvolves the effect of partcle shape on the ncremental response; the second nvolves the changes n assembly topology that result from the partcle szes, shapes, and arrangements. Although the geometrc effects n these examples are somewhat speculatve, experments demonstrate a strong nfluence of partcle shape and arrangement on shear band thckness. O Sullvan and Bray (23) tested carefully constructed, regular packngs of equal-sze metal balls and observed shear bands that were much thnner than those usually observed n rregular packngs. Yoshda and Tatsuoka (1997) placed metal balls n a regular arrangement between two glass plates and observed the deformatons durng baxal (2D) loadng. They found that the shear band had a thckness of three balls, whch s also much thnner than shear bands typcally observed n rregular packngs. The condtons n these tests regular partcle arrangements wth no rearrangement of the assembly topology are smlar to those of the frst example. Bag (23) and Babc et al. (199) have also demonstrated an effect of assembly regularty on the sze of localzaton phenomena, and Bag has proposed measures for quantfyng the noton of regularty n granular materals. In contrast, Vggan et al. (21) found that the thckness of shear bands n sand specmens depended on the mean gran sze but not on the sze dstrbuton. 2.2 Mechancal orgns Incremental partcle motons are governed by the mechancs of rgd bodes wth complant contacts: partcle motons produce contact deformatons; contact deformatons produce contact forces; and the contact forces on each partcle must be n equlbrum. Each of these three relatons ntroduces a length scale. Partcles p and q are n contact at the pont c (Fg. 2). The contact forces f on a partcle must be n equlbrum: f pq = b, r pq f pq = m, (1) q q 3

4 Partcle p Motons u p, θ p x p r pq l pq Partcle q Motons u q, θ q r qp x q FIG. 2. Two partcles n contact. where the sums are over all partcles q that are n contact wth p, and b and m are the body force and moment on p. The radal vectors r pq are drected from a sngle reference pont on p to ts contact wth q. Equaton (1 2 ) ncludes an ntrnsc length the radus r pq. We wll later test the scalng of shear band phenomena by runnng smulatons wth entrely contrved mechancal rad αr pq nstead of the real rad r pq n the moment equlbrum equaton [A 2 ][δf]. The ncrement of a contact force df pq depends upon the contact deformaton, perhaps n the form df pq = F pq ( δu pq, def, f pq) δu pq, def. (2) We have excluded vscous effects n ths form (see Pöschel et al. 21), along wth any effect of the contact hstory, but we allow the ncremental response to depend on the current contact force f pq, as would apply wth frctonal contacts. The consttutve form (2) s ncrementally non-lnear, as would be expected for frctonal contacts. The choce of a contact law F pq ( ) wll affect materal scalng, and n Secton 3, we gve the results of numercal, DEM smulatons n whch a smple, lnear contact law s altered by selectng dfferent contact frcton coeffcents. We then determne the effect, f any, on shear band thckness. The contact deformaton δu pq, def depends upon the partcle motons, δu pq, def = du q du p + (dθ q r qp dθ p r pq ), (3) where du p and du q are the ncremental dsplacements of the two partcles, and dθ p and dθ q are ther ncremental rotatons. Equaton (3) contans the components r pq of partcle rad. We consder these rad as beng mechancal rad, snce they are assocated wth the knematcs of partcle nteracton. We wll later test the scalng of shear band phenomena by runnng smulatons wth contrved rad βr pq n place of the real rad r pq. Equatons (1), (2), and (3) can be gathered nto a matrx stffness equaton for all N partcles of an assembly: [ ] [ H ] du 6N 6N = [ c ] dθ 6N 1 (4) 6N 1 where [H] s the stffness matrx, [c] s the forcng vector, and vector [du/dθ] contans the dsplacements and rotatons of all N partcles. By consderng the ncremental form of Eq. (1), we can expand Eq. (4) as follows: ( [ ] [ ] [ ] [ ] [ ] ) [ ] A1 + A2 A3 + A2 F([δu def ], [f]) [ B ] du = [ c ]. (5) dθ 4

5 TABLE 1. Test condtons for DEM smulatons. Statc Knematc Test length factor length factor Frcton No. α β µ Although t s dffcult to entrely separate geometrc and mechancal effects, we would say that the stffnesses [A 1 ] 6N 6N and [A 3 ] 3M 6N are geometrc n orgn: they nclude the products of the partcle rad, the surface curvatures of the partcles at ther M contacts ponts, and the cumulatve contact forces. Matrces [A 2 ] 6N 3M, [F] 3M 3M, and [B] 3M 6N are mechancal n orgn: [A 2 ] s the statcs matrx; [F] s the contact stffness matrx; and [B] s the assembly knematcs matrx. In the next secton we use smulatons to explore the effect of alterng the length scales wthn the mechancal stffness, whle leavng the geometrc stffness unchanged. 3 SIMULATIONS Several DEM shearng smulatons were conducted wth altered mechancal rad and wth dfferent coeffcents of contact frcton, and we compare the thcknesses of the shear bands that appeared n these smulatons. The mechancal rad were altered wth fve combnatons of the factors α and β: the statc factor α was multpled by the rad that appear n the moment equlbrum Eq. (1 2 ), and the knematc factor β was multpled by the rad wthn the knematc Eq. (3). Fve combnatons of factors α and β were appled n the fve smulatons, but wthn each smulaton, the same par of values was used wth all partcles (Tests 1 5, Table 1). Contact detecton was based entrely upon the actual, geometrc rad, so that a mechancal orgn of materal scalng could be dstngushed from a geometrc orgn. That s, the contact detecton process assured that partcles would roll across ther geometrc surfaces, and that contact formaton and dsengagement would also conform to the actual geometrc shapes. Another three tests were conducted wth the same α = β = 1 but wth dfferent contact frcton coeffcents µ (Tests 1, 6, and 7, Table 1). The purpose of these three tests was to dstngush any consttutve-mechancal orgn of the materal scale. Other than modfyng the mechancal rad, the Dscrete Element Method (DEM) was mplemented n a conventonal manner. The DEM algorthm uses dynamc relaxaton to resolve the equlbrum, knematc, and consttutve equatons, rather than the matrx approach outlned n Secton 2.2. The DEM algorthm s smply an effcent approach for the solvng the same set of (ncrementally non-lnear) equatons. The meanng of the mechancal scalng factors may be more clearly understood n the context of the DEM algorthm, partcularly when appled to crcular dsks. Wth an α =, no partcle rotatons wll occur, snce any moment mbalances that would be produced by tangental forces are nullfed by the α and wll not mpel partcle rotatons. Wth a β =, partcle rotatons produce no tangental contact forces. The rectangular assembly contaned 4,5 unbonded crcular dsks of multple dameters. The dsk szes were randomly dstrbuted over a farly narrow range of between.56d and 1.7D, where D s the mean partcle dameter. The assembly was created by slowly and 5

6 Rough, rgd boundary D 85 D Perodc boundary Vertcal poston, x2/d Horzontal movement, u 1 /D Vertcal poston, x2/d Horzontal movement, u 1 /D (a) Assembly proportons (b) Shearng movements (c) Movement detal FIG. 3. The assembly of 4,5 crcular dsks and the observed shearng dsplacements. sotropcally compactng a sparse arrangement of partcles wthn a set of perodc boundares that surrounded the assembly. Durng compacton, the frcton between partcles was dsallowed (although frcton was later restored for the shearng tests). Ths technque produced a materal that was dense, random, and sotropc, at least when vewed at a macro-scale. The average ntal vod rato was.173 (sold fracton of.8525), the ntal average coordnaton number was 3.93, and the average overlap between neghborng partcles was about of D. Contact stffness was n the form of normal and tangental sprngs of equal stffness. After compacton, the perodc boundares were removed from the top and bottom of the assembly and were replaced wth rough rgd platens. These platens were smply thn layers of tghtly ntermeshed partcles that were placed by shftng a (perodc) subset of partcles onto the top and bottom of the assembly. The fnal assembly was 84D wde and 432D tall (Fg. 3a). The assembly was horzontally sheared n all of the tests. Vertcal dlaton was freely allowed by mantanng a constant vertcal stress throughout the shearng process, but the assembly wdth was mantaned constant. These condtons are smlar to those employed by Cundall (1989) and Matsushma et al. (23). Shear bands developed n all of the smulatons, regardless of the choces of α, β, and µ. An example s shown n Fgs. 3b c, whch plot the horzontal partcle movements u p 1 of all 4,5 partcles as a functon of ther vertcal postons x p 2 (α = β = 1, µ =.5). The plots show the horzontal dsplacements that had occurred between the shearng strans of 9% and 1%. All movements and postons are expressed n a dmensonless form by dvdng by the mean (geometrc) dameter D. Plots of the shearng stress are shown n Fg. 4 for each of the frcton coeffcents (µ =.25,.5, and 1., α = β = 1). Although the peak strength ncreases wth an ncreasng frcton coeffcent, the resdual strength s ndependent of µ. The prmary queston s whether a mechancal scalng of rad by the factors α and β wll 6

7 Shear stress, τ/po Frcton µ =.25 µ =.5 µ = Shearng stran, γ.8.1 FIG. 4. Evoluton of shear stress for three frcton coeffcents. Vertcal poston, x2/d Scalng α =.5, β = 2 α = 2, β =.5 α = 1, β = 1 α =.5, β =.5 α = 2, β = 2 Vertcal poston, x2/d Frcton coeffcent µ =.25 µ =.5 µ = Shear stran Shear stran.25.3 (a) Combnatons of mechancal scalng, α and β (b) Values of frcton coeffcent µ FIG. 5. Shear stran profles wth shear bands for varous combnatons of α, β, and µ. affect shear band thckness. The factors α and β have no effect on thckness. Fgure 5a shows the smoothed profles of shearng stran wthn the shear bands for fve smulatons havng dfferent combnatons of α and β. The stran profles have been centered at md-thckness of the bands, even though shear bands appeared at dfferent heghts n the fve tests. The fve bands share the same thckness and almost dentcal stran profles. The frcton coeffcent µ also has not effect on shear band thckness, as s shown n Fg. 5b. 4 CONCLUSION The smulaton results were surprsng. We had expected some effect of mechancal scalng on shear band thckness, but the thcknesses were the same for all values of mechancal scalng, α and β, and for all frcton coeffcents µ. Shear band thckness seems to have a geometrc 7

8 orgn and to depend upon the geometrc szes, sze dstrbuton, and shape of the partcles, although ths concluson wll need to be confrmed wth addtonal tests. We have already noted that regular packngs of equal-sze spheres and dsks exhbt shear bands that are much thnner than those of the current study (Babc et al. 199; O Sullvan and Bray 23; Bag 23). We plan to conduct 2D tests wth dfferent dstrbutons of dsk szes (smlar to the physcal experments of Vggan et al. 21) and wth oval shapes havng dfferent aspect ratos. REFERENCES Babc, M., Shen, H. H., and Shen, H. T. (199). The stress tensor n granular shear flows of unform, deformable dsks at hgh solds concentratons. J. Flud Mech., ASME, 219(1), Bag, K. (23). From order to chaos: the mechancal behavor of regular and rregular assembles. Quas-Statc Deformatons of Partculate Materals, K. Bag, ed., Proc. of the QuaDPM 3 Workshop, Aug , Budapest, Hungary Cundall, P. (1989). Numercal experments on localzaton n frctonal materals. Ingeneur- Archv, 59(2), Matsushma, T., Saomoto, H., Tsubokawa, Y., and Yamada, Y. (23). Observaton of gran rotaton nsde granular assembly durng shear deformaton. Sols and Found., 43(4), O Sullvan, C. and Bray, J. D. (23). Evoluton of localzaton n dealzed granular materals. Quas-Statc Deformatons of Partculate Materals, K. Bag, ed., Proc. of the QuaDPM 3 Workshop, Aug , Budapest, Hungary Pöschel, T., Salueña, C., and Schwager, T. (21). Scalng propertes of granular materals. Contnuous and Dscontnuous Modellng of Cohesve-Frctonal Materals, P. A. Vermeer, S. Debels, W. Ehlers, H. J. Herrmann, S. Ludng, and E. Ramm, eds., Sprnger, Berln, Vggan, G., Küntz, M., and Desrues, J. (21). An expermental nvestgaton of the relatonshps between gran sze dstrbuton and shear bandng n sand. Contnuous and Dscontnuous Modellng of Cohesve-Frctonal Materals, P. A. Vermeer, S. Debels, W. Ehlers, H. J. Herrmann, S. Ludng, and E. Ramm, eds., Sprnger, Berln, Yoshda, T. and Tatsuoka, F. (1997). Deformaton property of shear band n sand subjected to plane stran compresson and ts relaton to partcle characterstcs. Proc. 14th Int. Conf. Sol Mech. and Found. Engrg., Hamburg, Vol. 1,