Floppy modes and nonaffine deformations in random fiber networks.
ABSTRACT We study the elasticity of random fiber networks. Starting from a microscopic picture of the nonaffine deformation fields, we calculate the macroscopic elastic moduli both in a scaling theory and a self-consistent effective medium theory. By relating nonaffinity to the low-energy excitations of the network ("floppy modes"), we achieve a detailed characterization of the nonaffine deformations present in fibrous networks.
- SourceAvailable from: asu.edu[show abstract] [hide abstract]
ABSTRACT: The percolation of rigidity in 2D central-force networks with no special symmetries (generic networks) has been studied using a new combinatorial algorithm. We count the exact number of floppy modes, uniquely decompose the network into rigid clusters, and determine all overconstrained regions. With this information we have found that, for the generic triangular lattice with random bond dilution, the transition from rigid to floppy occurs at pcen=0.6602±0.0003 and the critical exponents include ν=1.21±0.06 and β=0.18±0.02.Physical Review Letters 12/1995; 75(22):4051-4054. · 7.94 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Rigidity percolation is analyzed in two-dimensional random networks of stiff fibers. As fibers are randomly added to the system there exists a density threshold q=q(min) above which a rigid stress-bearing percolation cluster appears. This threshold is found to be above the connectivity percolation threshold q=q(c) such that q(min)=(1.1698+/-0.0004)q(c). The transition is found to be continuous, and in the universality class of the two-dimensional central-force rigidity percolation on lattices. At percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters, whose number diverges in the limit of infinite system size, when a critical bond is removed. The scaling with system size of the average size of these clusters was found to give a new scaling exponent delta=1.61+/-0.04.Physical Review E 01/2002; 64(6 Pt 2):066117. · 2.31 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Glasses are rigid, but flow when the temperature is increased. Similarly, granular materials are rigid, but become unjammed and flow if sufficient shear stress is applied. The rigid and flowing phases are strikingly different, yet measurements reveal that the structures of glass and liquid are virtually indistinguishable. It is therefore natural to ask whether there is a structural signature of the jammed granular state that distinguishes it from its flowing counterpart. Here we find evidence for such a signature, by measuring the contact-force distribution between particles during shearing. Because the forces are sensitive to minute variations in particle position, the distribution of forces can serve as a microscope with which to observe correlations in the positions of nearest neighbours. We find a qualitative change in the force distribution at the onset of jamming. If, as has been proposed, the jamming and glass transitions are related, our observation of a structural signature associated with jamming hints at the existence of a similar structural difference at the glass transition--presumably too subtle for conventional scattering techniques to uncover. Our measurements also provide a determination of a granular temperature that is the counterpart in granular systems to the glass-transition temperature in liquids.Nature 07/2005; 435(7045):1075-8. · 38.60 Impact Factor
arXiv:cond-mat/0603697v3 [cond-mat.soft] 12 Sep 2006
Floppy modes and non-affine deformations in random fiber networks
Claus Heussinger and Erwin Frey
Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,
Department of Physics, Ludwig-Maximilians-Universit¨ at M¨ unchen,
Theresienstrasse 37, D-80333 M¨ unchen, Germany
(Dated: February 6, 2008)
We study the elasticity of random fiber networks. Starting from a microscopic picture of the non-
affine deformation fields we calculate the macroscopic elastic moduli both in a scaling theory and
a self-consistent effective medium theory. By relating non-affinity to the low-energy excitations of
the network (“floppy-modes”) we achieve a detailed characterization of the non-affine deformations
present in fibrous networks.
PACS numbers: 62.25.+g, 87.16.KA, 81.05.Lg
Materials as different as granular matter, colloidal sus-
pensions or lithospheric block systems share the com-
mon property that they may exist in a highly fragile
state [1, 2]. While in principle able to withstand static
shear stresses, small changes in the loading conditions
may lead to large scale structural rearrangements or even
to the complete fluidization of the material [2, 3, 4]. To
understand the extraordinary mechanical properties of
these systems new concepts have to be developed that go
beyond the application of classical elasticity theory and
that sufficiently reflect the presence of the microstruc-
ture . One example is the “stress-only” approach to
the elasticity of granular materials , where the elimina-
tion of the kinematic degrees of freedom accounts for the
infinite stiffness of the grains. This seems to capture the
inhomogeneous distribution of stresses in the sample and
their concentration along the so called force-chains .
In jammed systems of soft spheres, on the other hand,
fragility has recently been shown to directly affect the
deformation response of the system. While it may in-
duce anomalous deformation fields that strongly deviate
from the expectations of homogeneous elasticity (“non-
affine” deformations) , it may also lead to a prolifer-
ation of low-frequency vibrational states far beyond the
usual Debye-behaviour of ordinary solids . It has been
argued that these low-energy vibrations derive from a set
of zero-frequency modes (floppy modes) that are present
just below the jamming threshold  and relate to the
ability of the structure to internally rearrange without
any change in its potential energy. This concept of floppy
modes has also been used in connection with elastic per-
colation networks where the fragile state is reached by
diluting a certain fraction of nearest-neighbour contacts.
In these systems constraint-counting arguments may be
used to determine the percolation transition at which the
system looses its rigidity .
Here our focus is on a particular class of heterogeneous
networks composed of crosslinked fibers. These systems
have recently been suggested as model systems for study-
ing the mechanical properties of paper sheets  or bio-
logical networks of semiflexible polymers [13, 14]. While
these networks are known to have a rigidity percolation
transition at low densities [15, 16], we show here that
even networks in the high-density regime in many ways
resemble the behaviour of fragile matter, despite the fact
that they are far away from the percolation threshold.
We identify the relevant floppy modes and highlight their
importance for understanding the macroscopic elasticity
of the network.In particular, we will be able to ex-
plain the occurence of an anomalous intermediate scaling
regime observed in recent simulations [16, 17, 18]. In this
regime the shear modulus was found to depend on den-
sity (measured relative to the percolation threshold) as
G ∼ δρµwith a fractional exponent as large as µ ≈ 6.67
. Also, highly non-affine deformations [17, 19] as well
as inhomogeneous distribution of stresses in the network
have been found. Heuristic non-affinity measures have
been devised [17, 19], however, little is known about the
actual nature of the deformations present.
expression “non-affine” is exclusively used to signal the
absence of conventional homogeneous elasticity, scarce
positive characterization of non-affine deformations has
been achieved up to now . This Letter tries to fill this
gap by characterizing in detail the non-affine deformation
field present in fibrous networks. By relating non-affinity
to the floppy modes of the structure we can, starting from
a microscopic picture, calculate the macroscopic elastic
moduli both in a scaling theory and a self-consistent effec-
tive medium approximation. In analogy with the affine
theory of rubber elasticity for flexible polymer gels, our
approach might very well serve as a second paradigm to
understand the elasticity of microstructured materials.
Due to the proximity to the fragile state, it might also be
of relevance to force transmission in granular media and
to the phenomenon of jamming.
The two-dimensional fiber network we consider is de-
fined by randomly placing N elastic fibers of length lf
on a plane of area A = L2such that both position and
orientation are uniformly distributed. We consider the
fiber-fiber intersections to be perfectly rigid, but freely
rotatable cross-links that do not allow for relative sliding
of the filaments. The randomness entails a distribution
of angles θǫ[0,π] between two intersecting filaments
while distances between neighbouring intersections, the
segment lengths ls, follow an exponential distribution 
P(ls) = ?ls?−1e−ls/?ls?. (2)
The mean segment length ?ls? is inversely related to the
line density ρ = Nlf/A as ?ls? = π/2ρ. The segments
are modeled as classical beams with cross-section ra-
dius r and bending rigidity κ. Loaded along their axis
(“stretching”) such slender rods have a rather high stiff-
ness k?(ls) = 4κ/lsr2, while they are much softer with re-
spect to transverse deformations k⊥(ls) = 3κ/l3
ing”). Numerical simulations for the effective shear mod-
ulus G of this network have identified a cross-over scaling
scenario characterized by a length scale ξ = lf(δρlf)−ν
and ν ≈ 2.84  that mediates the transition between
two drastically different elastic regimes. For fiber radius
r ≫ ξ the system is in an affine regime where the elastic
response is mainly dominated by stretching deformations
homogeneously distributed throughout the sample. The
modulus in this regime is simply proportional to the typ-
ical stretching stiffness, Gaff∝ k?(?ls?) and independent
of the fiber length lf. This is in marked contrast to the
second regime at r ≪ ξ. There, only non-affine bending
deformations are excited and the modulus shows a strong
dependence on fiber length Gna ∝ k⊥(?ls?)(lf/?ls?)µ−3.
Using renormalization-group language the parameters r
and lfmay be viewed as scaling fields (measured in units
of the “lattice-constant” ?ls?). The stretching dominated
regime may then be characterized by an (affine) fixed-
point at lf → ∞ and finite radius r ?= 0. On the other
hand, the (non-affine) fixed-point of the bending domi-
nated regime is obtained by first letting r → 0 and then
performing lf→ ∞. This suggests that the elastic prop-
erties in the latter regime may be analysed at vanishing
radius r = 0, that is by putting the system on the stable
manifold of the fixed point.
In the following we will exploit this limit to calculate
the modulus Gnain the non-affine regime. Central to the
analysis is the recognition that in this limit the ratio of
bending to stretching stiffness k⊥/k?∝ r2tends to zero
and bending deformations become increasingly soft. We
thus obtain the much simpler problem of a central-force
network. However, as only two fibers may intersect at a
cross-link the coordination is z < 4  and rigid regions
may not percolate through the system [22, 23].
implies that on a macroscopic level, the elastic moduli
will be zero, while microscopically displacements can be
chosen such that segment lengths need not be changed.
These are the floppy modes of the structure that entail
the fragility of the network in the bending dominated
FIG. 1: Construction of a floppy mode by axial displacement
δz of the primary fiber (drawn horizontally) and subsequent
transverse deflection ¯ y = −δz cotθ of the crosslinks to restore
the segment lengths on the secondary fibers (dashed lines,
possible to first order in δz). Initial cross-link positions are
marked as black squares, final configurations as red circles.
regime. It has been argued that a critical coordination
of zc= 4 is necessary to give the network rigidity .
This value defines the “isostatic” point, which in our net-
work corresponds to taking the limit lf → ∞. Thus,
we arrive at the conclusion that isostaticity and the on-
set of rigidity seem to be intimately connected to the
fixed point governing the non-affine regime. While it is
usually not possible to deduce the specific form of the
floppy modes, the fibrous architecture allows for their
straightforward construction. In a first step (see Fig.1)
we perform an arbitrary axial displacement δz of a given
(primary) fiber as a whole. This, of course, will also af-
fect the crossing (secondary) fibers such that the lengths
of interconnecting segments change. In a second step,
therefore, one has to account for the length constraints
on these segments by introducing cross-link deflections
¯ yi= −δz cotθitransverse to the contour of the primary
fiber. As a result all segment lengths remain unchanged
to first order in δz . The construction is therefore
suitable to describe the linear response properties of the
network, while at the same time it offers an explanation
for the stiffening behaviour found in fully nonlinear sim-
ulations [19, 24]. Any finite strain necessarily leads to
the energetically more expensive stretching of bonds and
therefore to an increase of the modulus.
The identified modes take the form of localized exci-
tations that affect only single filaments and their imme-
diate surroundings. By superposition we may therefore
construct a displacement field that allows the calcula-
tion of macroscopic quantities like the elastic moduli. To
achieve this we need to know the typical magnitude of
displacements δz of a given fiber relative to its surround-
ings, the crossing secondary fibers. Since δz is defined
on the scale of the complete fiber we do not expect any
dependence on average segment length ?ls?, such that
δz ∝ lf remains as the only conceivable possibility. Al-
ternatively, one may obtain the same result by assuming
that the individual fiber centers follow the macroscopic
strain field in an affine way. Then, relative displacements
of centers of neighbouring fibers would be proportional
to their typical distance. This is of the order filament
length lf and again δz ∝ lf. Note, however, that the as-
sumption of affine displacement of the fiber centers can-
not be literally true for fibers intersecting at very small
angles θ → 0. To avoid a diverging transverse deflection
¯ yi= −δz cotθ → ∞ the two fibers will most likely not
experience any relative motion at all and δz → 0. Truly
affine displacements can therefore only be established on
scales larger than the filament length. It should also be
clear, that the assumption of affine displacements of the
fiber centers is different from the usual approach of as-
signing affine deformations on the scale of the single seg-
ment. The latter would lead to deformations δaff ∝ ls,
proportional to the length ls of the segment. Instead,
axial displacements of the fiber as a whole are, by con-
struction of the floppy mode, directly translated into non-
affine deformations δna∝ lf, which do not depend on the
length of the segment.
Restoring the radius r to its finite value, the floppy
modes acquire energy and lead to bending of the fibers.
A segment of length ls will then typically store the en-
ergy wb(ls) ≃ κδ2
segment length distribution Eq.(2) one may calculate the
average bending energy ?Wb?, stored in a fiber consisting
of n ≃ ρlf segments,
s. By averaging over the
?Wb? ≃ ρlf
We assume the integral to be regularized by a lower cut-
off length lmin, that we now determine in a self-consistent
manner. Physically, lmincorresponds to the shortest seg-
ments along the fiber that contribute to the elastic en-
ergy. Even though we know (see Eq.(2)) that arbitrar-
ily short segments do exist, their high bending stiffness
k⊥(ls) ∝ l−3
makes their deformation increasingly ex-
pensive. Segments with length ls < lmin will therefore
be able to relax from their floppy mode deformation δna,
thereby reducing their bending energy from wb(lmin) to
nearly zero. However, due to the length constraints this
relaxation necessarily leads to the movement of an en-
tire secondary fiber and to the excitation of a floppy
mode there. By balancing wb(lmin) = ?Wb? this gives
lmin ≃ 1/ρ2lf and for the average bending energy of a
single fiber ?Wb? ≃ κ/lf(ρlf)6. This implies for the mod-
ulus Gna≃ ρ/lf?Wb? ∝ ρ7, which compares well with the
simulation result of µ = 6.67. What is more, by equat-
ing the energy ?Wb? with ?Ws? ≃ κlfr−2valid in the
affine stretching regime, one can also infer the crossover
exponent ν = 3.
In summary, we have succeeded in explaining the elas-
ticity of the bending dominated regime starting from the
microscopic picture of the floppy modes that character-
ize directly the deformation field deep inside the non-
affine regime. Alternatively, one might try to understand
the emergent non-affinity in a perturbative approach that
considers deviations from an affine reference state. Such
a line of reasoning has recently been suggested in ,
where non-affine boundary layers, growing from the fila-
ment ends, are assumed to perturb the perfect affine or-
der. However, comparing with their simulation data the
authors could not confirm the scaling picture unambigu-
ously and acknowledged the need for further numerical
as well as improved theoretical work . Thus, non-
affine elasticity in fibrous networks appears to be intrin-
sically a non-perturbative strong-coupling phenomenon
for which the floppy mode picture provides the correct
low-energy excitations. As we will explicitly show next,
one particular strength of our approach is that the scal-
ing picture can readily be extended to a full theory that
self-consistently calculates the modulus in a non-affine
effective medium theory.
To set up the theory we consider a single filament to-
gether with its cross-links that provide the coupling to
the medium. The energy of this assembly consists of two
parts. First, the bending energy of the primary fiber
dz , (4)
due to a transverse deflection y(z). A second “stretch-
ing” energy contribution arises whenever a cross-link
deflection yi = y(zi) differs from its prescribed value
¯ yi = −δz cotθi and may be written in the form of an
harmonic confining potential Ws(yi) =1
acts individually on each of the n ≃ ρlf cross-links. It
allows the filament to reduce its own energy at the cost
of deforming the elastic matrix it is imbedded into. Per-
forming a configurational average ?.? over cross-link posi-
tions ziand orientations θiwe obtain the average elastic
energy stored in a single fiber as
2ki(yi− ¯ yi)2that
2(yi− ¯ yi)2
To solve the model we further need to specify the stiff-
ness ki= k(θi) of the medium that relates to the relax-
ation mode of a cross-link on the primary filament from
its floppy mode deflection. As we have argued above, any
relaxation of this kind must act as axial displacement on
a secondary fiber, thus exciting a new floppy mode there.
The energy scale associated with this is ?W? such that
we can write
k(θi) = 2?W?sin2(θi)
where the angular dependence derives from the projec-
tion onto the axis of the secondary filament. Eqs.(5) and
n = 20
FIG. 2: Graphical solution of Eqs.(5) and (6) for various num-
bers n of cross-links obtained by calculating the intersection
between the left side of the equation ?W?lhs (bisecting line,
dashed curve) with the right side ?W?rhs (full curves). The
different curves for a given n correspond to ensembles of vary-
ing size. They seem to diverge in the limit ?W?rhs≫ ?W?lhs.
In fact, there (and only there) the averaging procedure is ill
defined . Inset: Resulting dependence of ?W? on n.
(6) represent a closed set of equations to calculate the
configurationally averaged deformation energy ?W? as a
function of the number of cross-links n. In implement-
ing this scheme we have generated ensembles of filaments
with a distribution of cross-linking angles as given by
Eq.(1) and segment-lengths according to Eq.(2). Note,
that there is no free parameter in this calculation. The
equations are solved graphically in Fig.2 by plotting both
sides of Eq.(5) as a function of ?W?. The point of inter-
section, which solves the equation, is shown in the inset
as a function of the number of cross-links n. For the
same parameter window as used in the network simula-
tions , it yields the scaling behaviour of ?W? ∝ n5.75.
This implies for the modulus the exponent of µ = 6.75,
which improves upon the simple scaling picture presented
above and provides a very accurate calculation of the
scaling exponent µ.
In conclusion, we have succeeded in deriving the
macroscopic elasticity of random fibrous networks start-
ing from a microscopic description of the displacement
field in a manner that does not rely on the notion of affine
deformations. We have given a floppy mode construction
that may be applied to any two or three-dimensional net-
work with fibrous architecture, for example paper or bi-
ological networks of semiflexible filaments. It may also
be shown to be relevant to systems where the constraint
of straight fibers is relaxed . The unusually strong
density dependence of the modulus found here is a con-
sequence of the exponential segment length distribution
(2) and the presence of the length-scale lmin. While iden-
tification of the floppy modes has been recognized to be
highly important for a description of force transmission
in granular media or the jamming transition in colloidal
systems, one can rarely give the exact form of these zero-
energy excitations. On the contrary, we have achieved an
explicit construction of the floppy modes that can be put
in the form of localized elementary excitations (“floppi-
ons”) affecting only single filaments and their immediate
It is a pleasure to acknowledge fruitful discussions with
David Nelson and Mikko Alava.
 Soft and Fragile Matter, ed. by M. E. Cates and M. R.
Evans (Institute of Physics, London, 2000).
 A. Soloviev and A. Ismail-Zadeh, in Nonlinear Dynamics
of the Lithosphere and Earthquake Prediction, ed. by V. I.
Keilis-Borok and A. Soloviev (Springer, Berlin, 2003).
 M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P.
Claudin, Phys. Rev. Lett. 81, 1841 (1998).
 E. I. Corwin, H. M. Jaeger, and S. Nagel, Nature 435,
 S. Alexander, Phys. Rep. 296, 65 (1998)
 J. P. Wittmer, P. Claudin, M. E. Cates, and J.-P.
Bouchaud, Nature 382, 336 (1996).
 J.-P. Bouchaud, P. Claudin, D. Levine, and M. Otto, Eur.
Phys. J. E 4, 451 (2001).
 A. Tanguy, J. P. Wittmer, F. Leonforte, and J.-L. Barrat,
Phys. Rev. B 66, 174205 (2002).
 L. E. Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett.
95, 098301 (2005).
 M. Wyart, S. R. Nagel, and T. A. Witten, Europhys.
Lett. 72, 486 (2005), Ann. Phys. Fr. 30, 1 (2005)
 D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. 75,
 M. Alava and K. Niskanen, Rep. Prog. Phys. 69, 669
 A. Bausch and K. Kroy, Nature Physics 2, 231 (2006).
 C. Heussinger and E. Frey, Phys. Rev. Lett. 96, 017802
 M. Latva-Kokko and J. Timonen, Phys. Rev. E 64,
 J. Wilhelm and E. Frey, Phys. Rev. Lett. 91, 108103
 D. A. Head, A. J. Levine, and F. C. MacKintosh, Phys.
Rev. Lett. 91, 108102 (2003).
 D. A. Head, A. J. Levine, and F. C. MacKintosh, Phys.
Rev. E 68, 061907 (2003).
 P. R. Onck, T. Koeman, T. van Dillen, and E. van der
Giessen, Phys Rev Lett 95, 178102 (2005).
 B. A. DiDonna and T. C. Lubensky, Phys. Rev. E 72,
 O. Kallmes and H. Corte, Tappi 43, 737 (1960).
 M. Kellom¨ aki, J.˚ Astr¨ om, and J. Timonen, Phys. Rev.
Lett. 77, 2730 (1996).
 J. C. Maxwell, Philos. Mag. 27, 294 (1864).
 C. Heussinger and E. Frey, in preparation.
 Due to the finite fiber length there are also two- and
three-fold coordinated cross-links.
 A similar construction holds in 3d, where, in addition,
each cross-link acquires a floppy “out-of-plane” degree of