Tabor, who was a physicist and engineer, first proposed this idea i...
The angles that the sheets make as they approach the contact region...
Here's an example of the Chinese finger trap - the only way to rele...
In order to better understand how to derive the differential equati... F
riction has a long scientiﬁc pedigree. Leonardo da Vinci
established that the friction force, or traction, T applied
when an object begins to slide is given by the simple for-
mula T = µN, where N is the load, or normal force, and
µ is the coecient of friction. Leonardo had suggested
that µ =
1
4 always, but Guillaume Amontons and
Charles Augustin Coulomb later showed that µ is not univer-
sal; rather, it depends on the nature of the sliding object and
the surface on which it slides. In total, the classic Amontons–
Coulomb laws state that the friction force during sliding is in-
dependent of both the area of contact and the sliding velocity
and that it is proportional to the load.
The independence of T from the contact area is hardly intu-
itive; indeed, it was not fully explained until the mid-20th cen-
tury, in work by David Tabor and Frank Philip Bowden. More
recently, theoretical and computational progress, in conjunc-
tion with technological advances such as the atomic force mi-
croscope, surface forces apparatus, and quartz crystal microbal-
ance, have renewed interest in friction and improved scientists’
understanding. Many systems, we now know, exhibit behav-
ior much more complex than envisioned in the Amontons–
Coulomb laws—particularly at or near the nanoscale. Multi-
walled nanotubes furnish an example. In stunning experiments
(see the ﬁnal two additional resources), inner and outer tubes
were slid along each other to study true molecular friction.
Those investigations and others have shown area dependence
and vanishing friction. Nevertheless, the classic laws apply in
many cases.
A simple, striking experiment
Take two phone books, interleave their sheets, and a"empt to
separate the books by pulling on their spines. You cannot do
it. The accumulated friction between the pages is so great that,
as ﬁgure 1 shows, even the weight of a car cannot pull the
phone books apart.
Less spectacular than li#ing a car, but more instructive, is a
pair of experiments you can try with two perfect-bound note-
books (that is, having rigid spines) whose sheets can be easily
removed. Interleave the sheets and you will ﬁnd that the trac-
tion force needed to separate the two books is immense. Next,
remove every other sheet in each notebook and repeat the
experiment. In that case you will have no trouble separating
the notebooks.
If you want to go one step further, set up a simple traction-
measuring device such as a spring balance. If your device is
sensitive enough, you will ﬁnd that when alternating sheets are
removed from each notebook and the remaining sheets are all
parallel to each other, the traction force scales linearly with the
number of sheets. The friction at each interface is small, so it is
easy to separate the books. In contrast, when sheets are not re-
moved, they fan out as they approach the overlap region. As a
result, the traction force is greater than in the sheets-removed
case, and it increases dramatically with the number of sheets.
In fact, our research team—which includes Héctor Alarcón,
Christophe Poulard, Jean-Francis Bloch, and Élie Raphaël—
showed that in some cases a 10-fold increase in the number of
sheets results in a traction force that is increased by more than
four orders of magnitude.
The enigma of the interleaved phone books, though o#en
retical a"ention, a circumstance that delighted, perplexed, and,
at times, frustrated us. As we will now discuss, the secret to the
ampliﬁcation of friction lies in the angles the sheets make as
they approach the overlap region.
The Hercules number
In our laboratory experiments, we prepared booklets from
identical sheets of paper having a width w = 12 cm, length
L = 25 cm, and thickness ε = 0.01 mm. The total number 2M of
74 PHYSICS TODAY
|
JUNE 2016
Kari Dalnoki-Veress is a
professor in the department
of physics and astronomy
at McMaster University in
Thomas Salez is a CNRS
research associate at ESPCI
ParisTech in Paris. Frédéric Restagno is a CNRS research associate
at Université Paris–Sud in Orsay, France.
QUICK STUDY
Why can’t you separate interleaved books?
Kari Dalnoki-Veress, Thomas Salez, and Frédéric Restagno
Because of the way interpenetrating sheets deform, the harder you pull, the greater the friction
force resisting separation.
FIGURE 1. THE FRICTION FORCE generated by the two interleaved
phone books on the left is enough to support a car, as illustrated in
the scene at right from the show On nest pas que des cobayes!, aired
in France on 2 May 2014. France 5/2P2L, used with permission.) JUNE 2016
|
PHYSICS TODAY 75
sheets ranged from 24 to 200. A tensile tester measured the total
traction force as the notebooks were pulled vertically at a con-
stant velocity, typically 1 mm per minute. We stopped pulling
intermi"ently so that we could probe the coecient of friction
at various distances d separating the clamp of one book from
the overlap region. Figure 2a shows the geometry of the inter-
leaved books and some of the important parameters used in
our analysis. Figure 2b presents our measurements. It clearly
shows the nonlinear variation of T with both the number of
sheets and the separation distance.
The mechanism leading to ampliﬁed friction can be under-
stood with the help of ﬁgure 2a. The index n labels individual
sheets in a booklet, and the nth sheet makes an angle θ
n
as it
goes from the clamp at the end of the notebook to the overlap
region. Therefore, a component of the local traction T
n
exerted
on a sheet by an external operator at point A results in a local
normal force T
n
tanθ
n
exerted on the stack of sheets below it at
point B. The result is a self-ampliﬁed friction force that resists
the traction: The harder the operator pulls, the greater the local
Summing over all the local tractions yields the total trac-
tion T. To proceed analytically, we use a continuum descrip-
tion corresponding to a large M. Introducing the quantities
z = n/M, T(z)=T
n
, and the dimensionless ampliﬁcation param-
eter α =2µεM
2
/d yields the ordinary dierential equation
T(z)+2αzT(z)=0, where the prime denotes dierentiation with
respect to z. To integrate and solve that equation, one needs a
boundary condition T
for the traction on the outermost sheet
of the stack. The source of that bounding traction could be fric-
tion introduced by the elasticity of the paper as it bends or by
any tiny adhesion. Upon integration, we obtain the total traction
force T =2MT
·(π/4α)
1/2
· e
α
·erf(
α), where “erf denotes the
error function.
Note that as α approaches zero, which corresponds to par-
allel sheets and small angles, T =2MT
. That makes sense: As
the angles tend to zero, the friction force is due to all the inde-
pendent sheets, each experiencing local friction T
. In general,
however, the total traction force depends nearly exponentially
on a single quantity—the dimensionless ampliﬁcation param-
eter α, which has been given the lovely name “Hercules num-
ber.” And as the inset to ﬁgure 2b shows, when the appropri-
ately normalized traction is plo"ed against α, the results of all
our experiments collapse onto a single master curve.
Not just a compelling muse
We began our phone-book study because we wanted to solve
an intriguing popular mystery. But the eect we have studied
can be seen in many other systems in which a pulling force re-
sults in an orthogonal load that enhances friction. An example
is a toy known as the Chinese ﬁnger trap. Fingers inserted
into the trap—a braided tube—are dicult to extract because
tension tightens the tube. Surgeons use so-called ﬁnger-trap
sutures based on that principle. Likewise, a ship can be moored
to a capstan because the rope can tighten around the capstan
and thereby increase friction. The friction-amplifying mecha-
nism also recalls catch bonds, biomolecular links that strengthen
with tensile stress and facilitate cell adhesion. We hope that our
study, in addition to elucidating the long-standing and fun
phone-book puzzle, will help to clarify the mechanical behav-
ior of more complex interleaved systems involving textiles,
biological entities, and nanoscale mechanical devices.
/watch?v=AX_lCOjLCTo.
‣ Q. J. Wang, Y.-W. Chung, eds., Encyclopedia of Tribology,
Springer (2013).
‣ A. A. Pitenis, D. Dowson, W. G. Sawyer, “Leonardo da Vinci’s
friction experiments: An old story acknowledged and re-
peated,” Tribol. Le%. 56, 509 (2014).
‣ H. Alarcón et al., “Self-ampliﬁcation of solid friction in
interleaved assemblies,” Phys. Rev. Le%. 116, 015502 (2016).
‣ J. Cumings, A. Ze"l, “Low-friction nanoscale linear bearing
realized from multiwall carbon nanotubes,” Science 289, 602
(2000).
‣ A. Niguès et al., “Ultrahigh interlayer friction in multiwalled
boron nitride nanotubes,” Nat. Mater. 13, 688 (2014).
PT
FIGURE 2. WHEN SUITABLY SCALED, the friction of interleaved
books obeys a universal law. (a) As this schematic shows, when
interleaved books are separated by a distance d, the nth sheet
makes an angle θ
n
as it goes from the overlap region to a binding
clamp. Here, T
n
denotes the local traction force on the nth sheet.
(b) Total traction force T is measured as a function of distance d.
From bottom to top, the number 2M of sheets in each book is 24,
30, 46, 54, 100 (two data sets), 150, and 200. As the inset shows,
when the total traction is appropriately scaled and plotted as a
function of the Hercules number α =2μεM
2
/d, all the curves from
the unscaled-traction plot coincide. The solid black line corresponds
to the expression for T given in the text.
Tabor, who was a physicist and engineer, first proposed this idea in 1950, based on his observations of the behavior of friction in a number of different systems. He observed that the force of friction acting on an object is proportional to the normal force (the force that a surface exerts on an object), but is independent of the surface area of the object. Bowden, who was a physicist, later confirmed Tabor's observations through experiments in which he measured the force of friction acting on objects of different sizes and shapes. He found that the force of friction was always proportional to the normal force, but did not depend on the surface area of the object. Here's an example of the Chinese finger trap - the only way to release the fingers is to push the ends toward the middle, which enlarges the openings and reduces the traction. ![](https://i.makeagif.com/media/3-03-2019/vhBaV6.gif) The angles that the sheets make as they approach the contact region are crucial for the geometrical amplification of friction in interleaved books. The traction forces create loads perpendicular to the paper-paper interfaces through these angles, resulting in large self-created friction forces. This can be verified by creating an interleaved-book system with $θ_n = 0$ by removing alternating sheets in two notepads. In this case, the books can be easily pulled apart, which is consistent with the theory presented. In order to better understand how to derive the differential equation for this problem it's useful to analyze the following diagram. ![](https://i.imgur.com/2db83HI.png) The first thing to note is that the problem is symmetric with respect to the central line of the book. We start by indexing each sheet of a given book by $n$ ($n=1$, middle of the book to $n = M$ at one extremity). We also define $H_n \equiv h_n/d = n/d$, where $h_n$ is the shift in position of the $n^{th}$ sheet in the contact zone with respect to its position of clamping. ![](https://i.imgur.com/HsWnXbJ.png) As a consequence, we can build a triangle with angle $\theta_n$ and sides $1$ and $h_n/d$ such that the nth sheet satisfies $\sin \theta_n = H_n/\sqrt{1 + H^2_n}$ and $\cos \theta_n = 1/\sqrt{1 + H^2_n}$. According to Amontons-Coulomb law, this leads to a self-induced inner friction force that resists the traction when an operator pulls on an object. At the onset of sliding, the change in traction with respect to the index n can be calculated using these principles. $$T_n − T_{n+1} = 4 \mu H_n T_n$$ where $\mu$ is the coefficient of kinetic friction, and the two factors of 2 come from the identical contributions of the two books and the two pages of one sheet. The boundary condition is given by $T_M = T^{*}$, where $T^{*}$ is the unknown traction exerted on the outer sheet, and the total traction force is defined by $\Gamma = 2\sum_{k=1}^M T_k$. We introduce the variable $z= n/M$ and the dimensionless amplification parameter $\alpha = 2 \mu M^2/d$. In order to use a continuous description, we replace $T_n$ with $T(z)$, since M is much greater than 1. This allows us to obtain the following ordinary differential equation: $$T'(z)+2\alpha z T(z) = 0$$ When the boundary condition $T(1) = T^{*}$ is applied, the local traction force $T(z)$ can be found by integration: $T(z) = T^{*}e^{\alpha(1 − z^2)}$. Using this result, we can calculate the total traction force $\Gamma = 2M \int^1_0 dz T(z)$ $$\frac{\Gamma}{2MT^{*}}=\sqrt{\frac{\pi}{4\alpha}}e^{\alpha}erf({\sqrt{\alpha}})$$ Two intersting things about this last expresion: it tends to 1 as alpha approaches 0, meaning that it represents the geometrical amplification of friction compared to a simple linear collection of 2M independent flat sheets with local friction $T^{*}$. Secondly, it depends solely and almost exponentially on the amplification parameter alpha, which is therefore the central dimensionless parameter of this study.