How Sandcastles Fall

4 Pages·0.12 MB·English

by  Thomas C. Halsey

How Sandcastles Fall
Thomas C. Halsey and Alex J. Levine
Exxon Research and Engineering, Route 22 East, Annandale, N.J. 08801
(February 1, 2008)
Capillary forces significantly affect the stability of sandpiles. We analyze the stability of sandpiles
with such forces, and find that the critical angle is unchanged in the limit of an infinitely large
8 system; however, this angle is increased for finite-sized systems. The failure occurs in the bulk of
9 the sandpile rather than at the surface. This is related to a standard result in soil mechanics. The
9 increase in the critical angle is determined by the surface roughness of the particles, and exhibits
1 three regimes as a function of the added-fluidvolume. Our theory is in qualitative agreement with

n therecentexperimentalresultsofHornbakeret al.,althoughnotwith theinterpretation theymake
a of theseresults.
J

0 81.05.Rm, 68.45.Gd, 91.50.Jc
2
The continuum mechanics of most materials was es- τ >kσ, (1)

] tablished in the 19th century; however, the mechanics
ft of granular materials is still largely mysterious [1]. The whereτ isthetangentialstressacrosssomeplaneinterior
o studyofgranularmediaisalsomotivatedbytheubiquity tothesandpile,σ isthenormalcompressivestressacross
s of this form of matter in a variety of industrial contexts, thatplane,andkistheinternalfrictioncoefficient. Fora
.
at as well as in geophysical ones. givenstressstatestabilityrequiresthattherebenoplane
m While mostrecentattentionhasfocussedondrygran- forwhichthe ratioofτ toσ exceedsk. Todeterminethe
- ularmedia,arecentexperimentalstudybyHornbakeret plane on which this ratio is maximized we turn to the
d al. has opened the relatively unexplored subject of “hu- Mohr construction [4].
n mid” granular media, in which small amounts of added The Mohr circle provides a geometric construction to
o
fluid generate, through capillarity, adhesive forces be- transform under rotations a two dimensional symmetric
c
[ tween the grains [2]. Somewhat whimsically, these au- tensor such as the stress tensor. Suppose the two prin-
thors argue that their work is appropriate for the un- cipal stresses at a point are σ1 and σ2. Then a circle
1
derstanding of sandcastles; we actually take this point drawn in the σ τ plane through the points (σ1,0) and
v −
4 seriously,becauseadhesiveforcesandotherliquideffects (σ2,0), symmetrically about the σ axis, gives the nor-
0 are extremely important in geophysical applications, of mal and tangential stresses on any plane. To find these
2 which sandcastles are an unusual example. stressesfora physicalsystemrotatedbyanangleφ from
1 InthisLetter,wepresentatheoryofthestabilityofhu- the principalsystemonelooksatthe pointsonthe circle
0
mid sandpiles, based upon a continuum analysis of their at an angle 2φ from the horizontal. These two opposite
8
9 statics. While previous work on the statics and dynam- points give the normaland tangentialstresses acrossthe
/ ics of dry sandpiles concentrated on the behavior of the physicalplanes of the rotatedsystem. Thus the stability
t
a pile’s surface [3], we find that the addition of small ad- criterion for a dry sandpile is that no Mohr circle may
m hesive forcesbetween the grainscauses the site of failure extend above the line τ =kσ.
- to move from the surface into the bulk of the sandpile, a Consider a semi-infinite dry sandpile, whose surface
d
fact well-known in soil mechanics. Even though the fail- is oriented at an angle θ to the horizontal. We choose
n
o ure of the sandpile at the critical angle occurs at some anx z coordinatesystem,inwhichz givesthedistance
−
c finite depth, in the limit of infinite system size, the crit- fromthesurfaceofthepile(z >0down),andxgivesthe
v: ical angle is actually unchanged by the adhesion. For distance parallel to the surface. Then the stress tensor
i finite systems the angle of repose is increased from the σij satisfies the static equations
X
infinite-system/non-adhesivecriticalangle. Byanalyzing
r the cohesive effect of small amounts of wetting fluid, we ∂zσzz +∂xσxz =ρgcosθ, (2)
a
find that this increase in the critical angle as a function ∂zσzx+∂xσxx =ρgsinθ, (3)
oftheadded-fluidvolumeexhibitsarangeoverwhichthe
dependence is linear,in agreementwith the principalre- where ρ is the density of the sandpile. To solve these
sult of Hornbaker et al. However, we disagree with the equations, we first restrict ourselves to solutions which
suggestionoftheseauthorsthatmostofthewettingfluid are functions of z alone–in a semi-infinite system, any
will be found outside the particle contact zones. x-dependence of the solutions would be liable to gen-
To determine the stability ofa sandpile, we must have erate arbitrarily large stresses near the surface, which
a criterion for local failure of the sandpile. For a non- would cause the system to buckle. The most general
adhesive (dry) sandpile, a simple phenomenological cri- z-dependent solution, which also satisfies the boundary
terion for failure is that condition that the surface is stress-free, is
1
σ =ρgzcosθ, (4) Performing a calculation similar to the one above, we
zz
find that the failure criterion, which is now an explicit
σ =ρgzsinθ, (5)
xz
function of depth, is
σ =C(z), (6)
xx
sA −1
k =tanθ 1+ . (11)
The well-known stress indeterminacy in granular media ρgzcosθ
impliesthatC(z)isanunknownfunction. Wecanfixthis (cid:0) (cid:1)
function, however, by finding the form of C(z) that al- Note that since the sandpile will only be in a state of
lowsustomaximizethecriticalangle. Theanglethereby incipientfailurebelowsomefixeddepth,thestresstensor
obtained will certainly be an upper bound on the true retainssomeindeterminacyabovethatdepth. OurMohr
critical angle; furthermore, if the sandpile is free to ad- analysis, now local, applies only at the incipent failure
just its undetermined stress C(z) within some range, we depth and does not determine the global stress state of
expectthatthisupperboundwillbeidenticaltothecrit- the sand pile.
ical angle. Thus, while it is impossible in general to fix The criterion thus derived, Eq. (11), is most stringent
C(z)withoutsomeconstitutiveorclosurerelation,wedo as z , in which case the dry sandpile result θc =
−1→ ∞
believe that it is possible to fix it at the critical angle. tan k is recovered. On the other hand, for a sandpile
Clearly the function C(z) that will maximize the crit- offixeddepthD,thefailuremustoccuratmostatdepth
ical angle will have the form C(z) = C′z. To apply the D. Thus the critical angle will be the solution of the
geometric Mohr construction, we first determine the di- equation
ameter and the position of the center of the Mohr cir- sA −1
k =tanθ (D) 1+ , (12)
cle given by the difference and sum of eigenvalues of the c
ρgDcosθ (D)
c
stress tensor respectively. From Eqs. (4-6) we find that
(cid:0) (cid:1)
these eigenvalues are giving an critical angle θc(D) that decreases monotoni-
cally with D. Thus finite humid sandpiles have a depth-
1+c (1+c)2 dependent critical angle, unlike dry sandpiles, which is
2
σ1,2 =zρgcosθ 2 ±r 4 +tan θ−c!, (7) a well-known result in soil mechanics [5]. In addition,
we see that humid(i.e. cohesive)sandpiles failat depth,
where c = C′/(ρgcosθ). Note that both the radius and whereas from statics alone we were unable to determine
center position of the Mohr circle depend linearly onthe the failure depth of dry sandpiles. In the case of small
depth z below the surface. The maximum value of τ/σ adhesion stress sA/ρgD 1, we can write
≪
as a function of c is
ks
A −1
tanθ k+ sec tan (k) . (13)
c
τ (1+c)2 ≈ ρgD
= 1. (8)
σ max s4(c tan2θ) − If, as we are assuming, the a(cid:2)dhesion a(cid:3)rises from cap-
(cid:12) −
(cid:12) illary forces, we must still connect the adhesive stress to
Now this quan(cid:12)tity must be <k, so to find the maximum
the amount of fluid present. We suppose that the sand
critical angle, we wish to minimize the right-hand side
is composed of macroscopically spherical grains (radius
of Eq. (8) with respect to c. We then find that at this
R) whose surface roughness may be characterizedas fol-
minimum value of c,
lows: the spatial correlation of fluctuations in local sur-
τ face height saturates at height l at a lateral distance d
R
=tanθ, (9)
σ max that is much smaller than the particle radius, d R.
≪
(cid:12) Since the particles are macroscopically spherical, we re-
(cid:12) −1
sothatthecriticalang(cid:12)leisθc =tan k,whichisaclassic quire that l R (see Fig. 1).
R
result[5]. This simple model, by itself, does notindicate ≪
We can characterize the surface roughness of two par-
at which depth failure initiates–presumably a dynamical
ticles in contact by considering the function δ(x) which
model would resolve this ambiguity [6].
gives the average distance between the two particles a
Now consider a sandpile in which a normal adhesive
lateraldistance xfromanasperityatwhichthe twopar-
stress s is exerted across every plane, in addition to
A ticles are in contact. We write δ(x) in the form [7]
whateverotherstressesmayexistduetothebodyforces.
This stress introduces a normal force between pairs of δ(x)=l f(x/d) , (14)
R
contiguousparticleswhichallowsthesandpiletosupport
afiniteshearstress,eveninthelimitofzeroappliedcom- where f(w) is a scaling function with the limits
pressivestress. The maximumsupportedshearstress,in wχ w 0,
this case, is ks and we therefore replace the dry sand f(w) → (15)
A ∼ 1 w .
failure criterion, Eq. (1), by n →∞
The roughness exponent, χ, of the surface satisfies 0 <
τ >k(σ+s ) . (10) χ 1.
A
≤
2
2−χ
ΓV1 V 2+χ
fA ∼ lR2 (cid:18)V1(cid:19) , for V <V1, (18)
where V1 =lRd2.
R For a rough surface where χ =1, Eq. (18) shows that
the force depends on the cube root of the fluid volume.
d
This is identical to the dependence of the cohesive force
between a cone and plate on the volume of the liquid
bridge connecting them [2]. It is to be expected that the
l cone-and-plate model will reproduce the cohesive force
R
near a single asperity.
Roughness Regime– For larger values of V, the fluid
will occupy a statistically rough region, which is still
small enough that the macroscopic curvature of the par-
l R
R ticlesplaysnorole–however,thefluidoccupiesmorethan
the areaarounda single asperity. This regimeoccurs for
V1 <V <V2, where V2 =lR2R. The pressure is
Γ
P , (19)
∼−l
R
and the force will be
FIG. 1. The contact zone between two rough particles of
radius R. The scale of height deviations from themean is lR ΓV
and the height fluctuations are correlated over a distance d. fA ∼ l2 , for V1 <V <V2 , (20)
R
Thelateralsize ofthecontactzoneinwhichthemacroscopic
curvatureof the particles is not apparent is √lRR. In this roughness regime, the cohesive force is linear in
∼
the volume of the added fluid, reproducing the linear
dependence found by Hornbaker et al.
Note that this form can only be valid for x < √l R.
R Spherical Regime– When the lateral extent of the
For larger values of x, the macroscopic curvature of the
fluid contact exceeds d, then the wetting region will be
particles will determine the local distance between them
determinedbythemacroscopiccurvatureoftheparticles,
(see Fig. 1).
andthesurfaceroughnesswillnolongerplayasignificant
For suchparticles,there are three regimesfor the cap-
role. In this case the pressure is given by [8]
illary force exerted by a wetting fluid as a function ofV,
the total amount of fluid present per particle contact. Γ
P = , (21)
Asperity Regime– For the smallest values of V, the − V/2πR
capillaryforce is dominatedby the accumulationof fluid
around a single or a small number of asperities at which and the force by p
two neighboring particles are in contact. This will hold
until the lateral extent of the fluid-filled region exceeds fA =2πΓR, for V >V2 , (22)
d, determining a maximum contact fluid volume for this
which is independent of the volume of the liquid bridge
regime, V1 lRd2. joining the two grains. Thus the linear increase of the
∼
We can write the adhesion force f as
A cohesive force with fluid volume saturates for volumes
ΓA V >V2 =lR2R (see Fig. 2).
f = , (16)
A If the fluid wets the surface of the particles, then in
r
additiontothe fluidinthe contactregion,therewillalso
whereΓ isthe surfacetensionofthe fluid,r isthe radius
bealayeroffluidofthicknesstonthesurfaceofthepar-
of curvature of the meniscus of the fluid layer connect-
ticles. Typically this film will be no thicker than a few
ing the two grains near the asperity and A V/δ is the
∼ monolayers; hence there are a complicated set of forces
area of the contact patch. Γ/r is the pressure reduc-
− between this film and the surface. To simplify, we con-
tion due to the capillary meniscus. Because r will be
sideronlytheVanderWaalsforces,whichforathickness
approximately the distance between the particles at the
t generate a “disjoining pressure” P given by [9]
d
meniscus, we find
2H
− χ P = , (23)
Γ V 2+χ d t3
P , (17)
∼−lR (cid:18)V1(cid:19) where H is the Hamaker constant (H < 0 for a wetting
and the adhesion force is fluid). To determine the thickness of the wetting region,
3
I II III
Now let us consider the experiment of Hornbaker et
f
A al. They measured the critical angle for a medium com-
posed of radius 4 10−2cm polystyrene spheres with
×
small amounts of added oil using the draining crater
method [2]. They found a linear increase in the criti-
cal angle measured as a function of the volume of added
oil. They claimed that the failure of their systems was
at the surface, and concluded that they could account
for their results by assuming that 99.9%of the fluid was
outsideofthe contactzonesbetweenthe particles. Their
particles had a surface roughness on the order of 1µm.
We find that the increase in the critical angle is linear
with the fluid volume in the roughnessregime, up to the
V V V saturation result Eq. (27). We expect the Hamaker con-
1 2 stant for a wetting fluid to be negative, and of the order
FIG. 2. The behavior of the adhesive force between two of magnitude of H 10−20erg, so no more than a few
rough, “spherical” particles. The three regimes of the force ∼
monolayersoffluidshouldbepresentontheparticlesur-
vs. volume of the wetting layer are I – Asperity Regime, II
faces. Furthermore, in the linear regime, all fluid added
–RoughnessRegime,andIII–SphericalRegime. Theinsets
to the system will enter the particle contacts. Thus we
show the extentof thewetting region typical of each regime.
disagree with the claim that the overwhelming majority
ofthe fluid inthis case willbe outside the contactzones.
wemustsetthisdisjoiningpressureequaltothepressure Finally, we disagree with the interpretation of their ex-
inside the contact regions. If the radius of curvature of periment, according to which surface failure is the most
the contact meniscus is r, then relevant failure mode– the failure plane in cohesive ma-
terials should be at depth.
1
2Hr 3 We are grateful to P. Schiffer for providing us with
t= . (24)
− Γ copies of Ref. [2] before publication and to D. Erta¸s for
(cid:18) (cid:19)
many useful discussions.
Intheasperityregime,twillincreasewiththemeniscus
radius of curvature. In the roughnessregime, however,r
saturates at a value l , and t will be constant. In this
R
∼
regimeanyaddedfluidentersthecontactregion. Finally,
in the spherical regime, t will again increase.
Now consider a system with a volume V of liquid per
l
[1] H.M. Jaeger, S.R. Nagel, and R.P. Behringer, Rev. Mod.
particle. Ifwesupposethatthespheresareclose-packed,
Phys. 68, 1259 (1996).
theneachspherehas12neighbors,sothereisanaverage
[2] D.J.Hornbaker,R.Albert,I.Albert,A.-L.Barab´asi, and
of 6 contacts per sphere, each with a fluid volume of
P. Schiffer,Nature 387, 765 (1997), and unpublished.
V =V/6. The averagenumber ofcontacts per unit area
l [3] J.-P. Bouchaud, M. E. Cates, J. R. Prakash, and S. F.
will be (3φV/πR2), where φV is the volume fraction of Edwards, J. Phys. I,4, 1383 (1994).
the particles. Thus for a close-packed lattice, for which [4] A.C. Ugaral and S.K. Fenster, Advanced Strength and
φV =√2π/6, the adhesive stress will be approximately Applied Elasticity (Prentice-Hall, Englewood Cliffs, NJ,
1987) Chapter 1.
fA [5] R.M. Nedderman, Statics and Kinematics of Granular
s , (25)
A ≈ √2R2 Materials (Cambridge University Press, New York, 1992)
Chapter 3.
and we can now substitute into Eq. (13) to obtain the [6] H.M. Jaeger and S.R.Nagel, Science 255, 1523 (1992).
dependence of the critical angle upon fluid volume. For [7] A.-L. Barab´asi and H.E. Stanley, Fractal Concepts in
a sandpile of fixed depth D, this is linear in V, Surface Growth (Cambridge University Press, New York,
1995) Chapter 2.
kΓV −1 [8] F.P.BowdenandD.Tabor,The Friction and Lubrication
tanθ k+ sec tan (k) , (26)
c ≈ √2l2R2ρgD of Solids, (Oxford UniversityPress, 1986), Chapter 15.
R
(cid:2) (cid:3) [9] S.A. Safran, Statistical Thermodynamics of Surfaces, In-
up to a saturation result determined by (see Fig. 2) terfaces, and Membranes (Addison - Wesley Publishing
Co., New York,1994) Chapter 5.
√2πkΓ
−1
tanθ k+ sec tan (k) . (27)
c
≈ RρgD
(cid:2) (cid:3)
4