Refraction of shear zones in granular materials

0.25 MB·English

by  Tamas Unger

journal ref: Phys.Rev.Lett., 98, 018301 (2007)
Refraction of shear zones in granular materials
Tama´s Unger
Research group “Theory of Condensed Matter” of the Hungarian Academy of Sciences,
Institute of Physics, Budapest University of Technology and Economics,H-1111 Budapest, Hungary and
Department of Physics, University Duisburg-Essen, D-47048 Duisburg, Germany
(Dated: February 3, 2008)
Westudystrainlocalizationinslowshearflowfocusingonlayeredgranularmaterials. Aheretofore
7 unknown effect is presented here. We show that shear zones are refracted at material interfaces in
0 analogywithrefractionoflightbeamsinoptics. Thisphenomenoncanbeobtainedasaconsequence
0
ofarecentvariationalmodelofshearzones. Thepredictionsofthemodelaretestedandconfirmed
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by3Ddiscreteelementsimulations. WefoundthatshearzonesfollowSnell’slawoflightrefraction.

n
a PACSnumbers: 47.57.Gc, 45.70.-n,83.80.Fg,83.50.Ax,42.25.Gy
J

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The motion of granularmaterials,such as sand, is dif- effects ofgravityandcurvedshearingareabsent. There-

] ficulttopredict,astheybehaveneitherlikeelasticsolids fore a long cylindrical container is considered, which is
t
f norlike normalfluids. When they yieldunder stress and cut along its axis into two halves (Fig. 1a). (In com-
o start flowing, the material consists of large, almost solid puter simulations periodic boundary conditions in the
s
parts, and the relative motion is confined to narrow re- axial direction are applied.) The container is completely
.
at gions between them, called shear zones or shear bands filled with two granular materials having different fric-
m [1,2,3]. Shearzonesrepresentmaterialfailureandthere- tional properties. They are separated by a planar inter-
fore play a crucial role in geophysics (geological faults), face parallel to the axis, but at an angle to the cut of
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d in engineering (building foundations) and in industrial the container. The system is sheared quasi-statically [3]
n processes. It is an important problem of these fields to by moving the two halfs of the cylinder wall slowly par-
o
predict where the failure takes place. This is a difficult allel to the axis in opposite directions. This creates a
c
task for those shear zones which arise in the bulk of the shear zone starting and ending where the container wall
[
material far from the confining walls [2, 4, 5, 6, 7, 8]. is cut. In order to find the shape of the zone in between

2 Nature often stratifies granular media, i.e. different we apply the principle of minimum dissipation. This,
v
materialsare depositedin distinct layers. We study here in zero width approximation [7], leads to the following
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the question what effect such inhomogeneities have on variational problem (explained below):
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0 shear zones. Based on a recent theory and computer
0 simulations we showthat shearzonesare refractedwhen vpµeffdS =min. (1)
Z
1 they pass through the interface between different granu-
6 lar layers. Moreover, the angle of refraction obeys a law Herethelocalrateofenergydissipationisintegratedover
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analogous to Snell’s law of light refraction which reveals the whole surface of the shear zone. The shear zone is
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at an unexpected analogy between granular media and ge- regarded as being infinitely thin, outside the shear zone
m ometric optics. The effect of refraction presented here no deformation and no dissipation takes place. The lo-
influences the position of shear zones in various kinds cal dissipation rate per unit area vpµeff is obtained by
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d of materials including powders, sand, soil and rock lay- the sliding velocity v between the two sides of the shear
n ers providing implications on the behavior of geological zone times the shear stress pµeff. The parameters p and
o faults [9, 10]. µeff denote the overall pressure in the system and the
c
Recentlythe variationalprincipleofminimumdissipa- coarse grained effective friction coefficient [3]. The ef-
:
v tion was successfully applied [7, 8] to describe the non- fective friction has different values for the two materials
Xi trivial shape of the shear zone in a homogeneous granu- while v and p are taken constant for our setup [19]. The
lar material which is sheared in a modified Couette cell surface that minimizes the total rate of dissipation pro-
r
a [4, 5, 6]. Minimum rate of energy dissipation is a widely vides the shape of the shear zone. This means that the
appliedselectionprinciple[11]todeterminewhichsteady system yields along the surface where it has the least
states are realized in nature. When applying this prin- resistance against the external shear.
ciple to the cylindrical geometry of the modified Cou- The system is translation invariant along the axis,
ette cell[7],notonlythenon-trivialpositionofthe shear therefore the above surface integral is reduced to a line
zone could be calculated without any fitting parameter, integral. We have to find the path (cross section of the
but new, closed shear zones were predicted which were shear zone) which connects two fixed points and along
discovered also in experiments [12] and simulations [6]. which the integral of µeff is minimal. This problem has
In the following,a so far unknown but measurable effect exactly the form of Fermat’s principle of geometric op-
will be derived from the same principle. It concerns the ticswheretheeffectivefrictionplaystheroleoftheindex
shape of shear zones in layered granular materials. of refraction. It is known that Fermat’s principle of the
This phenomenon can be understood best, if the side shortest travel time leads to refraction of light beams
2
thereforeonecanexpectasimilarphenomenonofrefrac-
tion also for shear zones.
In order to check this prediction we carried out com-
puter simulations of 105 frictional and hard spherical
beads (Fig. 1b). Based on a standard discrete element
method(DEM)[13,14]themotionofeachbeadisdeter-
mined by Newton’s equation. This grain-level dynamics
leads to a collective steady state flow where we measure
the coarse-grained velocity field. The coarse-grained ve-
locities are parallelto the cylinder axis. The local strain
rate, which is calculated from the velocity field, corre-
sponds to pure shear deformation (without volumetric
strain). The spatial distribution of the magnitude of the
shear rate is shown in Fig. 1c. As predicted, the arising
shearzonedepartsfromthecutplaneanditsdirectionis
changed significantly at the material interface (Fig. 1b,
c). Note, that the shear zone we obtained is quite wide
relative to the system size. The relative width becomes
smaller for larger systems [4] which, however, would de-
mand also a computational effort out of reach. Still, the
effect of refraction is clearly shown.
The analogy with light refraction provides quantita-
tive predictions that can be measured in numerical and
realexperiments. Fermat’sprincipleindicates thatshear
zones have to follow Snell’s law of light refraction which
will be tested here by DEM simulations. In order to do
so we characterizethe behavior of the shear zone by two
anglesφ1 andφ2 whichareanalogoustotheincidentand
refractive angles used in geometric optics. These are de-
fined with the help of the shear deformation (Fig. 1c)
as follows: first, the refraction point is identified as the
point along the interface where the local shear rate has
its maximum. Then the refraction point is connected
to the starting and ending points of the shear zone by
straight lines. Thus the direction of the shear zone on
eachsideoftheinterfaceisdetermined. Thesedirections
α
provide the angles φ1 and φ2 with respect to the normal
of the interface. Another angle, which is an important
input parameter of the shear test, is the tilt angle of the
interface α between the normal of the cut plane and the
interface (Fig. 1c). For various values of α we measured
φ1 and φ2. If the interface is perpendicular to the cut
plane (α = 0) no refraction was observed, however, as
α is increased the refraction of the shear zone becomes
more and more pronounced.
If we apply Snells law to our case it indicates that,
no matter how the angles φ1 and φ2 are changed, the
FIG.1: (coloronline)a,Schematicviewshowingtheshearcell ratio of sinφ1 and sinφ2 has to be independent of the
andtherefraction oftheshearzone. Densegranularmaterial tilt angle α. Furthermore it is expected that this ratio
is sheared between two half-cylinders. The shear direction is can be expressed by the effective frictions:
indicatedbythearrows. b,Asnapshotofthesimulation. The
upperlight-gray(yellow)materialandlowerdark-gray(blue) sinφ1 µeff2
= (2)
material have different frictional properties (blue beads have sinφ2 µeff1
stronger friction). If the velocity of any bead drops below
half of the external shear velocity, it is overpainted by dark- First we report the influence of α observed in simula-
gray (red). Despite the fluctuations of single bead-velocities, tions. We used materials that contained hard spheres of
a dark (red) zone appears in thebulk that separates the two radii distributed uniformly between Rmin and 1.3Rmin.
moving parts of the system. c, The spatial distribution of
This polydispersity was needed to avoid shear induced
the shear rate which shows the structure of the shear zone.
crystallization [15]. In order to control the frictional
Lightercolorsrepresentstrongerdeformations(orangemeans
the largest shear rate, followed by red and blue, while black
means almost no deformation). The straight line indicates
thematerial interface.
3
80o 80o 0.4
A A
60o B 60o B
C C
φ140o φ240o 0.35
20o 20o
(a) (b)
0o 0o 0.3
10o 20o 30oα40o 50o 60o 10o 20o 30oα40o 50o 60o effµ
0.25
3.5
φφ) / sin()12 2 .35 CAB ::: µµµµµ111ee===ffff(( 00000.....55001)) ,,,// µµµµµee222ffff===((00 000..10...555)) 0.2
sin( 0.15
n : 2 0 0.5 1µ 1.5 2
o
acti 1.5
efr FIG. 3: Effective friction versus microscopic friction. The
of r 1 open circles correspond to the materials used in systems A,
x
e B and C.
nd 0.5
el. i (c)
r 0
10o 20o 30o 40o 50o 60o rial parameter, the effective friction coefficient, can be
tilt angle of the interface α
defined which turns out to be independent of the prepa-
ration history, the actual shear rate and the confining
FIG.2: (coloronline) φ1 (a),φ2 (b)andtherelativeindexof pressure [3, 17]. The effective friction µeff is given by
refraction (c) measured for shear zones in computer simula-
tions. Thehorizontal axisshow thetilt angle of theinterface the shear stress divided by the normal stress, both mea-
between the two materials. Data points denoted by triangle, sured in the plane along which the shear deformation
squareand circle stand for systemsA,B andC, respectively. takes place.
The pairs of microscopic friction that are used in these sys- The quantity µeff is not a microscopic input parame-
tems are indicated in the figure. The straight lines show the
ter of the computer simulation, thus the question arises
theoretically predicted values (the upper line for system A what values µeff has for the materials that appeared in
and B, thelower onefor system C).
systems A, B and C. This can be deduced from simu-
lations which are independent of the previous refraction
tests: we put the same materials into a 3D rectangular
propertiesofthe materialswe variedthe valueofthe mi- boxunderplaneshear[3]measurethecomponentsofthe
croscopicfrictionµwhichisthefrictioncoefficientatthe stress tensor [18] and calculate the value of µeff. In this
particle-particle contacts. It is important to note that way we achieve a calibration curve that provides a one
the microscopic friction µ and the effective friction µeff to one correspondencebetweenmicroscopicandeffective
are not the same although they are closely related as we friction, see Fig. 3.
will see later. We simulated three systems called A, B
With that we arrived to a point where we can make
andC. Thetotalnumberofparticleswas100000forsys-
quantitativestatementsabouttheextentoftherefraction
temAand50000forsystemsB andC. Foreachcasethe
based on Snell’s law. E.g. for system A, with help of
shear cell was filled with two materials with microscopic the prescribed values of µ1 and µ2, the ratio µeff2/µeff1
frictions µ1 and µ2. For systems A and B µ1 = 0 and canbecalculatedandcomparedtothedatasinφ1/sinφ2
µ2 =0.5 while for system C µ1 = 0.1 and µ2 =0.5. For recordedin the refraction test. This is done in Fig. 2 for
allsystemstheradiusofthecontainerwasapproximately
systems A, B and C. A surprisingly good agreement is
65Rmin. found between the predicted and measured values which
The influence of the tilt angle of the interface is pre- holds for all systems and for various tilt angles of the
sentedinFig.2. Thecomputersimulationsshowthatφ1 interface. This is our main result.
and φ2 vary strongly but sinφ1/sinφ2 remains constant Dealingwithshearlocalizationinlayeredgranularma-
for a wide range of the angle α. This ratio seems to de- terials we investigated the predictions of a recent varia-
pend only on the materials in which the shear zone was tional model and compared them to computer simula-
created, in full agreement with the theoretical consider- tions. The results of the present work convey two mes-
ations. sages. First, we gave verification of the variational ap-
Next we deal with the role of the effective friction. It proach to granular shear flows. We tested the model in
is knownthatif adensenoncohesivegranularmaterialis a new situation and, according to the numerical data, it
shearedthen after a shorttransient [16] it reaches a well gaveanexcellentdescriptionofthe behavioroftheshear
definedresistanceagainstsheardeformationprovidedthe zones. Second, an interesting analogy between granular
flow is slow enough (quasistatic shear). Thus a mate- flow and geometric optics is revealed. We showed that
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shear zones are refracted at material interfaces similarly support and help. Support by grant OTKA T049403
to light beams. The phenomenon presented here should andbytheG.I.F.researchgrantI-795-166.10/2003isac-
be accessible by experiments. knowledged.
We are grateful to J. Kert´esz and D.E. Wolf for their
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