Anomalous Stress Profile in a Sheared Granular Column

Vishwajeet Mehandia,* Kamala Jyotsna Gutam,† and Prabhu R. Nott

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India(Received 28 December 2011; revised manuscript received 9 July 2012; published 19 September 2012)

We present measurements of the stress as a function of vertical position in a column of granular

material sheared in a cylindrical Couette device. All three components of the stress tensor on the outer

cylinder were measured as a function of distance from the free surface at shear rates low enough that the

material was in the dense, slow flow regime. We find that the stress profile differs fundamentally from that

of fluids, from the predictions of plasticity theories, and from intuitive expectation. We argue that the

anomalous stress profile is due to an anisotropic fabric caused by the combined action of gravity and shear.

DOI: 10.1103/PhysRevLett.109.128002 PACS numbers: 45.70.�n, 83.80.Fg, 47.57.Gc, 81.05.Rm

The stress in a column of granular material confined byvertical walls has been of interest since the mid-19thcentury, when food grains began to be stored in tall silos.In modern times, silos are used for the storage of a varietyof granular materials. It was realized quite early [1] that,unlike in liquid columns, the normal stress at the base of astatic granular column does not increase linearly with thehead of material. Apart from its importance in the design ofsilos, the stress in sheared granular columns is also ofinterest from the standpoint of rheometry—rheologicalproperties of fluids are often measured in a cylindricalCouette device (Fig. 1), which in essence shears a verticalcolumn of the sample. The flow in this device falls in aclass called viscometric flows, wherein the directions ofthe velocity and velocity gradient are everywhere orthogo-nal. For such flows, stress measurements may be readilyrelated to the rheological properties of the fluid. In granularmaterials, however, the relation between the stress and thekinematics is more complex. A better understanding of theresponse of the material in such simple flows will help inthe development of better rheological models for granularmaterials.

In a static column of an incompressible fluid, the effectof gravity is solely to increase the pressure linearly withthe vertical distance from the free surface z as a result ofthe hydrostatic balance. The only effect of shearing is theexertion of a shear stress on the walls if the fluid isNewtonian, and normal stress differences may be exhibitedin addition if the fluid is non-Newtonian; however, none ofthese quantities will vary with z. For a granular column, thevariation of the stress with z is qualitatively different. In astatic column, Coulomb friction causes the walls to impartan upward vertical shear stress on the column; this resultsin all components of the stress asymptotically reachingfinite values, the length scale of variation being a multipleof the lateral dimension of the column. This behavior wasfirst explained by Janssen [2] using a simple analysis,which was later refined by others [3]. The stress profileduring flow has been measured in the draining of bins [4],wherein the gravity direction lies in the plane of shear

(velocity-gradient plane) and are found to be in qualitativeagreement with the Janssen solution. In this Letter, weconsider a viscometric flow wherein the gravity directionis orthogonal to the plane of shear; previous measurementsfor this flow [5] suffered from limitations and inaccuracieswhich are discussed later in the Letter.We report experimental measurements of the stress pro-

file in a granular material sheared in a cylindrical Couettedevice. The novel features of our experiment are that allthree components (one normal and two shear) of the stressacting on the outer cylinder were measured as a function ofthe depth z. The results of our experiments are intriguing:the stress profile differs fundamentally from that of asheared fluid column and a static granular column. Thenormal stress at the outer cylinder rises much faster with zthan the Janssen and hydrostatic profiles. Secondly, thevertical shear stress �rz changes sign when the column issheared. We argue that our results point to an anisotropicfabric that results from a combination of gravity and shear.The experiments were conducted in a custom-built cy-

lindrical Couette apparatus [Fig. 1(a)]. The outer cylinder(radius Ro ¼ 7:5 cm, length 30 cm) was kept stationary,and the inner cylinder (radius Ri) was rotated at a constantangular speed � ¼ 0:34 s�1. Inner cylinders of radiiRi ¼ 3:5, 4.5, 5, and 6 cm were used to vary the Couettegap W � Ro � Ri. A vertical slot of width 3 cm was cutalong the length of the outer cylinder, into which a slider ofthe same width was inserted; the slider could be movedvertically using a rack-and-pinion arrangement. A multi-axis force sensor (ATI Industrial Automation) capable ofmeasuring the three orthogonal components of the tractionwas used to measure the stress on the outer cylinder [insetof Fig. 1(a)]. With this assembly, we achieved a measure-ment resolution of 15 Pa. The space between the inner andouter cylinders was filled with glass beads of mean diame-ter dp ¼ 0:83 mm. The base of the Couette cell was a

porous, rigid diffuser plate through which air could bepumped.Each experimental run started by placing the sensor at

the lowest point of measurement z ¼ H (� 8 cm above

PRL 109, 128002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 SEPTEMBER 2012

0031-9007=12=109(12)=128002(5) 128002-1 � 2012 American Physical Society

the base). The Couette gap was filled with the glass beadsand the column aerated for a brief period at an air velocitybelow that at minimum fluidization. Aeration was used toobtain a reasonably uniform packing and to erase themicrostructural history of filling. Shearing was then com-menced, with the nominal shear rate _� � Ri�=W smallenough that the material was in the dense, slow flow regime[6]. A statistically steady state was reached after shearingfor about 20 minutes, after which the forces recorded bythe sensor at a frequency of 1 Hz were averaged for 5 min.The slider was then moved up to the next position, thematerial sheared for a few minutes to reach a steady state,and the forces averaged; this procedure was repeated untilthe free surface (z ¼ 0) was reached. As the flow is axi-symmetric, we report the stress in the cylindrical coordi-nate frame (�, r, z), the � direction coinciding with thedirection of rotation. The ‘‘as poured’’ packing fraction inthe bed is estimated to be 0.62, but it is likely to be slightlylower after aeration and shear.

The kinematics of viscometric flows of dense granularmaterials has been investigated extensively in earlier stud-ies [7]; their common finding is that the velocity decaysroughly exponentially from the moving boundary. Theshear layer thickness �, related to the decay length ofthe velocity, is a function of the roughness of the grainsand the walls, and is a weak function of W [8,9]. Forthe range of W explored in this study, � is estimated tovary between 10 and 13dp [8]. The radial variation of the

azimuthal velocity v� at the free surface is given in theSupplemental Material (SM) [10] for W ¼ 18dp.

The stress profiles in sheared and static columns arecompared in Figs. 1(b) and 1(c); the data are reported interms of the scaled vertical distance � � z=W, and the

scaled stress ��ij � �ij=ð�0gWÞ, where �0 ¼ 1250 kg=m3

is the nominal bulk density of the granular material (theactual density is significantly higher). In a static column,the asymptotic saturation of the stress components to con-stant values is evident [see inset in Fig. 1(c) for a magnifiedview]. The nonmonotonic variation at large z is probablydue to proximity to the base of the inner cylinder. Thevertical shear stress �rz is positive, so the reaction from thewall is an upward traction on the granular column; asalready noted, this is the cause of the asymptotic saturationof the stress with z. The dashed lines fitted to the staticstress profiles are the Janssen solution for a granular ma-terial in a bin of arbitrary cross section [3,4],

h ��rri ¼ K

M½1� e�M��; h ��rzi ¼ h ��rri tan�; (1)

whereM � ðPW=AÞK tan�, K is the Janssen constant, � isthe angle of wall friction, P and A are the perimeter lengthand area of the cross section, and the angle brackets signifyaveraging over the perimeter. Thus, the magnitude of thestress components saturate within a length proportional toW; our data agree well with this relatively crude model, asin previous studies [3,11]. The stress profile in a shearedcolumn is strikingly different: the normal stress �rr ap-pears to rise exponentially with�, and the axial shear stress�rz has changed sign. The latter implies that rather thancounteract the weight of the granular column, the wallimposes an additional downward traction on it. The pro-files of �rr in repeated runs almost coincide, suggestinghigh reproducibility, but small run-to-run variations areseen in �rz. The profile of the azimuthal shear stress �r�

is similar to that of �rr [Fig. 2(c)].

60Static

Sheared 1

Janssen

Sheared 2

50

40

30

20

10

00(a)

(b) (c)

2 4 6 8 10 12η

0

00

1

2

3

3 6 9 12–7

–5

–3

–1

0

1

2 4 6 8 10 12η

FIG. 1 (color online). (a) Schematic figure of our cylindrical Couette apparatus. The inset shows the sensor assembly. The forcesensor is fastened to a rigid aluminum transmitter on one side and anchored to the slider on the other. The sensor-transmitter assemblyis inserted through a circular aperture (radius 1 cm) in the slider. The surfaces of the slider and transmitter were machined to the samecurvature as the outer cylinder and mounted flush. A clearance of� 100 �m was maintained between the transmitter and the aperture.(b), (c) Profiles of �rr and �rz for static and sheared columns of glass beads. The dashed lines are the Janssen [Eq. (1)] and modifiedJanssen [Eq. (2)] solutions with the following (fitted) parameter values: static—� ¼ 10�, K ¼ 0:82; sheared—� ¼ 5:8�, K ¼ 1:1. Theinset in (c) is an expanded plot of the static profiles (�rr solid circles, �rz open circles).

PRL 109, 128002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 SEPTEMBER 2012

128002-2

Two previous studies have measured the stress in cylin-drical Couette flow [5]. In the first, the torque on the innercylinder was determined for only two fill heightsH, yet theauthors infer that it varies quadratically with H; with a fewassumptions, they arrive at the unconvincing conclusionthat the normal stresses are equal to �gz, in accord with thehydrostatic balance. In their second paper, �rr is measuredas a function of z with a sensor that appears to intrude intothe Couette gap, presumably disturbing the flow, and �zz ismeasured at the base. However, here the normal stressesare found to be 3 tan��gz, where � is angle of internalfriction of the material. Neither study gives an estimate ofthe measurement accuracy. In our study, considerable carewas taken to minimize experimental artifacts and errors.Our data show clearly that �rr grows roughly exponen-tially with z and that �rz changes sign upon shearing.

The sharp rise in the magnitude of all components of thestress during shear can be understood from the Janssenanalysis itself, if we reverse the direction of �rz at thewalls. The modified Janssen solution is

�� rr ¼ K

M½eM� � 1�; ��rz ¼ � ��rr tan�: (2)

The data in Figs. 1(b) and 1(c) agree well with the pre-dicted exponential rise of the stress. However, the values of� and K for the fit differ significantly from those used forthe static stress. In addition, it is evident from Fig. 2, whichshows the stress profile for different Couette gaps W, thatthat their values also depend on W, which is inconsistentwith the assumptions of the model. Most importantly, theJanssen analysis does not explain why �rz switches signupon shearing. Hence, the Janssen solution (1) and itsmodified form (2) for sheared columns are, at best, roughphenomenological models.

Though the stress was measured at the outer cylinder,the data for different inner cylinder radii allow limitedanalysis of their radial variation. The � component of the

momentum balance, with the assumptions of axisymmetryand symmetry of the stress tensor yields �r� ¼ðRi=rÞ2�r�ðRiÞ. With the additional assumption of thefriction boundary condition �r� ¼ �rr tan� at r ¼ Ri, weget �r�=�rr ¼ ðRi=rÞ2½�rrðRiÞ=�rr� tan�. If �rrðRiÞ=�rrðRoÞ is independent of Ri, the ratio of shear and normalstress at the outer cylinder must vary as ðRo=RiÞ�2. Thedata in the inset of Fig. 2(c) shows that it broadly decreaseswith Ro=Ri, in agreement with the above analysis, but therange of Ro=Ri is too small to validate the functionaldependence.The observed stress profile for a sheared column is

anomalous as it is counter to intuitive expectation, previousmeasurements, and the predictions of continuum plasticitytheories. The theories assume isotropy of the granular ma-terial, and most assume coaxiality of the principal direc-tions of the stress and deformation rate, but a few allowdeviation from coaxiality. For viscometric flows, all thesetheories predict coaxiality, equality of all the normalstresses, and zero shear stress in the vorticity direction[4,5,9]. For the problem at hand, this implies �rr ¼ �zz ¼�gz, and �rz ¼ 0. Needless to say, plasticity theories maynot hold if the condition for plastic yield is not satisfied. Thestate of the material outside the shear layer—whether thedeformation is elastic or plastic—is not well understood. Ifit is in an elastic state, �rz on the outer cylinder depends onthe preparation history, but in our experiments it is positivewhen the column is static; upon shearing,�rzmust vanish inthe shear layer, as mentioned above. Hence, its value in theelastic zone layer may decrease, but it would continue to bepositive [11]. Hence, a negative �rz is not predicted byisotropic plasticity and elasticity theories.The important question then is what causes the anoma-

lous response? One possibility is that the flow is not trulyviscometric as a result of the rigid base—this is readilytested by determining the stress profile for different fillheights. It is clear from Fig. 2 that the profiles for two fill

00

20

40

6018

30

36

48

W/dp

–8

–6

–4

–2

0 10

8

6

4

2

02 4 6 8 10 12

(a) (b) (c)

η0 2 4 6 8 10 12

1.2 1.6 20

0.1

0.2

Ro/Ri

η0 2 4 6 8 10 12

η

FIG. 2 (color online). The profiles of (a) ��rr, (b) ��rz, and (c) ��rz for different Couette gaps. The (red) filled and open circles are forthe two fill heightsH ¼ 7:3W and 12W, respectively. The inset in (c) shows the stress ratio �r�=�rr as a function of the ratio of radii ofthe outer to inner cylinders for three vertical positions: z=dp ¼ 36 (circles), 72 (stars), and 144 (squares).

PRL 109, 128002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 SEPTEMBER 2012

128002-3

heights for W ¼ 18dp (filled and open circles) virtually

coincide, indicating the base has little or no influence [12].In other words, the stress is determined solely by shear andgravity.

Another hypothesis for the anomalous stress is that thegranular material is in a state of dilation during shear. It iswell known that dilation accompanies deformation ingranular materials. As the cylinders and base are rigid,dilation can be accommodated only by a rise of the freesurface, i.e., by an upward motion of the granularmaterial—this would result in a negative �rz, consistentwith our observation. However, in standard shear tests, a‘‘critical state’’ of isochoric deformation is reached at astrain of Oð1Þ [4]—this implies that �rz should asymptoti-cally vanish. It is possible that the material near the outerwall has not deformed enough to reach a critical state and istherefore in a state of incipient dilation. However, thishypothesis too is not supported by our observations:Firstly, we find that the free surface drops after commence-ment of shear, indicating that material compacts [13].Secondly, from the velocity profile v�ðrÞ (see SM [10])for W ¼ 18dp, we estimate the strain rate at a distance of

dp from the outer cylinder to be � 1:5% of the nominal

strain rate _�. Though this is small, over the duration of theexperiment it yields a strain of Oð10Þ, which is clearlylarge enough for dilation to be fully achieved. Though thestrain near the outer cylinder for larger Couette gaps ismuch smaller, the qualitative features of the stress profiledo not depend on the gap—the profiles for narrow and widegaps in Fig. 2 are of similar shape. These arguments lead usto conclude that incipient dilation is not the cause of thestress anomaly. Nevertheless, a direct measurement of thedensity near the walls is desirable for a confirmation.

The above arguments suggest that the most plausiblehypothesis for the anomalous stress is that it is a result of

anisotropic rheology. Indeed, it is well known that grainsform an anisotropic and inhomogeneous fabric in a staticpile [14] and during shear [15]; it is reasonable to expectthat this will lead to anisotropic rheology. This hypothesisis corroborated by Fig. 3, where �rz is shown as a functionof the strain � � _�t, from the unsheared state (� ¼ 0) till asteady sheared state is reached; it is clear that the steadystate is reached only after a strain of �300, and the strainrequired for reversal of �rz is�50 (inset of Fig. 3). This isin contrast to the strain of Oð1Þ required to reach a steadystate in standard shear tests, presumably because theyrequire small rearrangements of the grains. Thus, apartfrom negating dilation as the cause for the anomalousstress, Fig. 3 indicates that the development of the fabricin the cylindrical Couette geometry requires cooperativerearrangement of particles over a length scale large com-pared to the grain size. Clearly gravity is essential for thisbehavior as the up-down symmetry in its absence causes�rz to vanish. These observations lead us to propose thatthe anomalous stress is a result of the formation of ananisotropic fabric due to the combined effects of gravityand shear. In standard shear tests, where the direction ofgravity falls within the shear plane, fabric anisotropy de-velops more rapidly [16] and does not appear to have astrong rheological signature. The cylindrical Couette is oneof a few devices where gravity is perpendicular to the planeof shear, and as seen in our experiments, the effect ofanisotropy is striking. To our knowledge, the fabric hasnot been studied in flows where gravity is orthogonal to theplane of shear; our results suggest that this would be aworthwhile endeavor.Theories for anisotropic fluids that introduce a fabric

tensor in the constitutive relation have been proposed [17],having the general form

�� ¼ F ðD;AÞ; (3a)

A� ¼ H ðD;AÞ; (3b)

where D and A are the rate of deformation and fabrictensors, F and H are tensor functions that have termslinear and quadratic in their arguments, and � represents aframe-invariant time derivative. A simplified version ofthis model has shown good agreement with discrete ele-ment method simulations of oscillatory shear [16]. In lightof our observations, the above constitutive relation must bealtered to include gravity by extending Eq. (3b) to the formH ðD;A; gÞ, such that the result is a frame invariant tensor.In summary, we have shown that the stress profile in a

granular material in a cylindrical Couette device differsradically from that of a fluid, the predictions of isotropicplasticity theories, and intuitive expectation. We argue thatthe the anomalous stress is caused by an anisotropic fabric,which evolves slowly when gravity is perpendicular to theshear plane, and propose that plasticity theories must beextended to incorporate the fabric in the constitutive

0–12

–9

–6

–3

0

3

62

0

–20 100 200

150 300 450 600

FIG. 3 (color online). The vertical shear stress as a function ofstrain, from the static state till a steady sheared state is reached.The data are for W ¼ 18dp at � ¼ 12.

PRL 109, 128002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 SEPTEMBER 2012

128002-4

relation for the stress. Further studies are required, such asthe measurement of the stress at the inner cylinder andevaluation of the fabric by discrete element method simu-lations, to verify this hypothesis.

We acknowledge useful discussions with K. Kesava Raoand Tejas Murthy, and thank Tejas Murthy for help withimage analysis. Funding from the Department of Scienceand Technology, India is acknowledged.

*Present address: ISIS, University of Strasbourg, France†Present address: Dr. Reddy’s Laboratories, Qutubullapur,Andhra Pradesh 500072, India

[1] I. Roberts, Proc. R. Soc. London 36, 225 (1883).[2] H. A. Janssen, Z. Ver. Dtsch. Ing. 39, 1045 (1895) [M.

Sperl, Granular Matter 8, 59 (2006)].[3] D.M. Walker, Chem. Eng. Sci. 21, 975 (1966); S. C.

Cowin, J. Appl. Mech. 44, 409 (1977).[4] K. K. Rao and P. R. Nott, An Introduction to Granular

Flow (Cambridge University Press, New York, 2008).[5] G. Tardos, M. Khan, and D. Schaeffer, Phys. Fluids 10,

335 (1998); G. Tardos, S. McNamara, and I. Talu, PowderTechnol. 131, 23 (2003).

[6] The Savage number, defined as � _�2d2p=�rr, where � is thebulk density and signifying the ratio of the collisional tototal normal stress, was never greater than 10�7.

[7] U. Tuzun and R.M. Nedderman, Chem. Eng. Sci. 40, 337(1985); W. Losert, L. Bocquet, T. C. Lubensky, and J. P.Gollub, Phys. Rev. Lett. 85, 1428 (2000); D.M. Mueth,

G. F. Debregeas, G. S. Karczmar, P. J. Eng, S. R. Nagel,and H.M. Jaeger, Nature (London) 406, 385 (2000).

[8] K. S. Ananda, S. Moka, and P. R. Nott, J. Fluid Mech. 610,69 (2008).

[9] L. S. Mohan, P. R. Nott, and K.K. Rao, J. Fluid Mech. 457,377 (2002).

[10] See Supplemental Material at http://link.aps.org/supple-mental/10.1103/PhysRevLett.109.128002 for a profile ofthe azimuthal velocity for a Couette gap of 18dp andexplanatory text.

[11] G. Ovarlez, C. Fond, and E. Clement, Phys. Rev. E 67,060302 (2003).

[12] The base does influence the stress in a region close to it, asindeed one expects, but our data are for distances greaterthan 8 cm from the base.

[13] This is most likely because the initial filling and aerationleaves the material in a relatively loose state, whichcompacts when sheared in accord with the critical statetheory.

[14] P. Dantu, in Proceedings of the 4th InternationalConference on Soil Mechanics and FoundationsEngineering (Butterworths, London, 1957),pp. 144–148; L. Vanel, D. Howell, D. Clark, R. P.Behringer, and E. Clement, Phys. Rev. E 60, R5040(1999).

[15] D. Howell, R. P. Behringer, and C. Veje, Phys. Rev. Lett.82, 5241 (1999).

[16] J. Sun and S. Sundaresan, J. Fluid Mech. 682, 590(2011).

[17] G. L. Hand, J. Fluid Mech. 13, 33 (1962); J. D. Goddard,J. Fluid Mech. 568, 1 (2006).

PRL 109, 128002 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 SEPTEMBER 2012

128002-5