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A Discrete Element Method model for frictional fibers
J´erˆome Crassous
Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35000 Rennes, France∗
(Dated: April 26, 2022)
We present a Discrete Element Method algorithm for the simulation of elastic fibers in frictional
contacts. The fibers are modeled as chains of cylindrical segments connected to each other by
springs taking into account elongation, bending and torsion forces. The frictional contacts between
the cylinders are modeled using a Cundall and Strack model routinely used in granular material
simulations. The physical scales for simulations, the determination and the tracking of contacts,
and the algorithm are discussed. Tests on different situations involving few or many contact points
are presented and compared to experiments or to theoretical predictions.
I. INTRODUCTION
The use of natural or artificial fibers allows to design
materials with original mechanical properties. At the
nanometric or micrometric scales, carbon nanotubes [1]
or polymer fibers [2] can be assembled into threads or
networks. At the micrometer and millimeter scales, the
frictional forces act with the elasticity of the fibers to pro-
duce a wide variety of materials. The fibers can just be
deposited without any special preparation to form highly
elastic media [3] such as cushions or non-woven fabrics [4].
Textile fibers can be twisted to produce yarns [5–7],
which are then assembled into cords [8], woven [9] or knit-
ted fabrics [9, 10]. Cyclic mechanical stresses can form
very compact natural structures [11], and birds also as-
semble fibers to build their nests [12, 13]. The contacts
between fibers play a fundamental role in describing the
physics of knots, which is a subtle competition between
tension and friction [14, 15], as well as eventual bending
of the fibers [16–19].
Several approaches have been proposed to numerically
simulate these structures. One approach is to use finite
element algorithms to discretize the fibers [20]. This ap-
proach allows a complete solution of the elasticity equa-
tions in complex geometries such as nodes [18], but is
only possible for systems with small numbers of contacts.
Another approach is to model the fibers as connected
spheres [21] or sphero-cylinders [22] and to use the dis-
crete element method algorithm widely used for the study
of granular materials. However, the periodic variations of
diameter of such fiber may induce very specific physical
properties as interlocked granular chains stiffening [23].
More realistic approaches are the simulations of
fibers as discrete [24] or continuous [25, 26] cylindri-
cal elastic chains of circular cross-sections. The non-
interpenetration condition between fibers and surfaces,
or between fibers, is then treated as constraints on the
displacements. The introduction of frictional tangential
forces in such model has been proposed using methods for
finding forces that match the Coulomb conditions [25–
27]. In those algorithms, the fibers are moved in order
∗jerome.crassous@univ-rennes1.fr
to find the positions of the surfaces that match the non-
penetration of fibers, with forces verifying the Coulomb
condition. Those positions are found using an iterative
procedure with proper regularization of Coulomb law to
ensure the convergence towards one solution verifying the
force balance. In the case where many frictional contacts
are present, the problem becomes hyperstatic, and the
solution is expected not to be unique. This is a well
known situation in granular material [28] simulations,
and the solutions selected by iterative algorithms are not
well controlled [28], and presumably depend on the algo-
rithm itself. Those drawbacks are of course of minimal
importance in situations where the indeterminacy in con-
tact forces is absent (hypo- or iso-static problem) such as
in knots with few contacts [29], or if qualitative simula-
tions are needed as in computer graphic community [30].
In explicit methods, the forces are obtained directly from
the kinematic of the body in contacts. The selection of
one solution of the Coulomb friction forces among many
ones is then ensured by the dynamics of the system. In
counterpart, explicit algorithm are usually slower.
We describe in this manuscript a model of discrete
elastic rod where the contact forces are calculated ex-
plicitly. The basis of the algorithm is the Discrete Ele-
ment Method which is popularly used for studying static
and dynamics of granular material. The specificity arises
from particles which are connected cylindrical particles.
In the section II, we first describe the mechanical model
of our fibers, including internal elastic forces and con-
tact forces. The numerical resolution is then detailed in
section III, where we insist on points that are specific
compared to standard DEM simulations, i.e. the numer-
ical scales that are used, the integration of displacement,
and the search of neighbors. In section IV, we illustrate
this algorithm on various situations including static and
dynamics, with few and many contacts.
arXiv:2204.11542v1 [cond-mat.soft] 25 Apr 2022
2
FIG. 1. (a) Ensemble of connected point forming the skeleton
of the fiber. (b) Cylinders and spheres forming the shell of
the fibers.
II. MECHANICAL MODEL OF FIBERS IN
CONTACT
A. Description of the fiber
Following [24], we model a fiber as an ensemble of N
connected points (see figure 1(a)). Let ri, with 0 ≤i≤
N−1 be the position of the point, and ei= (ri+1 −
ri)/kri+1 −rik, with 0 ≤i≤N−2 the unit vector joining
two successive points. We note li=kri+1 −rik. The
segment joining two successive points is the generatrix of
a cylinder of circular basis of diameter d. In addition,
each point riis the center of a sphere of diameter d.
So each fiber is a set of Nspheres connected by N−1
cylindrical segments. A mass m0is assigned to each point
of the string.
B. Internal elastic forces
The internal elastic forces that we consider in the fol-
lowing are elongation and flexion forces. The elonga-
tional forces are modeled using springs of stiffness k0
with dashpots of damping λ. The equilibrium length of
the spring is l0, and the elongation force exerted by point
i+ 1 on the mass located at riis:
f(e)
i+1;i=k0(li−l0) + λ˙
liei(1)
Each point iis submitted to forces from points i−1 and
i+ 1 so that f(e)
i=f(e)
i+1;i+f(e)
i−1;i, excepted the first i= 0
and last i=N−1 points. The flexion forces acting on
the point iis obtained from the elastic bending energy
E(b)=Rs(B/2) κ2ds with Bthe bending stiffness of the
fiber, and κthe curvature. The bending energy of the
FIG. 2. Contact between two cylinders.
discrete fiber is:
E(b)=B l0
2
i=N−2
X
i=1
κ2
i(2)
where κiis the curvature at node i, and the summation
is extended to all nodes except ending ones. Writing
the curvatures κias function of nodes positions ri, the
flexion force f(b)
i=−(∂E(b)/∂ri) acting on nodes iis (see
Appendix A):
f(b)
i=−B
l3
0ri−2−4ri−1+ 6ri−4ri+1 +ri+2(3)
for (N−3) ≥i≥2. Expressions of the forces f(b)
ifor
i < 2 and i > (N−3) are given in Appendix A. The
calculation supposes that the fibers are weakly extended
and bent (see Appendix A).
The internal elastic torque is obtained from the twist-
ing energy of a weakly bent and stretched discrete
fiber [24]:
E(t)=C
2l0
i=N−2
X
i=0
(θi+1 −θi)2(4)
with Cthe torsion modulus of the fiber. The elastic
moment along axis of cylinder is:
m(t)
i=−∂E(t)
∂θi=C
l0θi+1 −2θi+θi−1(5)
for N−2>i>0.
C. Contact forces
Contact between fibers may occur between segments
of cylinders or spheres belonging to identical or differ-
ent fibers. The figure 2 shows the contact between
two sections of cylinders. The contact point C(t) is lo-
cated on the segment with ending points H(1)
iand H(2)
i
3
on axis cylinders which minimizes the distance between
axis. This segment is unique if the axis are not paral-
lel. The determination of this segment will be detailed
in section III C. Let d(t) = H(1)
iH(2)
ithis minimal dis-
tance, and n(t) the normal unitary vector. We note
δ(t)=(d1+d2)/2−d(t) the interpenetration of the two
cylinders, and C(t) = H(1)
i+ (r1−δ/2) nthe contact
point. We use the Cundall-Strack model for the contact
force [31]. The normal contact force exerted by cylinder
1 on cylinder 2 is modeled as a spring-dashpot system:
f(c)
n=−knδ+λn˙
δn(6)
with knthe contact stiffness between the cylinders, and
λnthe contact damping. This contact law is a simpli-
fied version of the elastic contact force between 2 cylin-
ders [32] which varies non-linearly with the interpenetra-
tion f(c)
n∼δ3/2, and which depends on the angle between
cylinder axis. The tangential contact force is a Coulomb-
Force:
f(c)
t=−Minktut;µknδut
ut
(7)
where ktis the tangential stiffness, utthe tangential
displacement, and µthe microscopic friction coefficient.
The tangential displacement utis initialized to 0 when
the contact is first formed. The tangential displacement
vector first is integrated as:
ut(t) = ut(t−dt)+(v(2) −v(1) )dt (8)
with v(1) = (1 −s(1)
i)˙
r(1)
i+s(1)
i˙
r(1)
i+1 +ωiei×H(1)
iC(and
similar for v(2)) is the velocity of the point of the cylinder
(1) coinciding with the contact point C. The normal
component (ut·n)nis then removed. Finally, if ktut>
µknδ, then the tangential displacement is renormalized
such that ut=µknδ/kt.
The contact force f(c)=f(c)
t+f(c)
nis then applied to
the two point masses of fiber (2) ending the segments:
f(c)
i= (1 −s(2)
i)f(c)(9a)
f(c)
i+1 =s(2)
if(c)(9b)
The opposite force is similarly applied to fiber (1). This
decomposition ensures that the resultant and the com-
ponents of the moment perpendicular to the axis are the
same for the contact force and the two points forces. The
component m(c)
iof the moment of the contact force along
the axis is:
m(c)
i= (riC×f(c))·ei(10)
If the contact between two fibers involve one cylindrical
segment of the fiber and one sphere, or two spheres, the
contact point is calculated accordingly to the type of the
surfaces in contact.
D. Miscellaneous forces.
In addition misc extra forces may be added. A global
viscous damping force f(v)
i=−λv˙
rimay be added. It
is useful to damp transverse motion of fibers. Indeed,
our mechanical model of fiber does not include any dis-
sipation for motion perpendicular to fiber axis if there
is no contacts. Volumetric forces such as gravity forces
f(g)
i=m0gwith gthe gravity field may be also added.
Other external forces may applied to fibers such pre-
tension at ends of fibers.
III. NUMERICAL IMPLEMENTATION
A. Integration of equation of motions
The dynamical equations of motions writes as:
M0˙
vi=f(e)
i+f(b)
i+f(c)
i+f(v)
i+f(g)
i(11a)
Jz˙ωi=m(t)
i+m(c)
i(11b)
The second equation described the rotation of cylinder
segment around its axis. We did not consider in (11b)
any elastic torque due to torsion of the fiber, and the
fiber is free to rotate around node ri. The dynamical
equations are integrated using a standard second-order
Verlet algorithm [33].
B. Physical parameters for simulations
1. Physical scales
We first define mass, length and stiffness scale for the
simulation. The mass scale m0is the point mass of nodes,
and the length scale l0is the equilibrium length of each
segment, and the stiffness scale k0is the elongation stiff-
ness of spring. If all the fibers do not have identical phys-
ical properties, those scales are chosen from the fibers of
smallest radius. For every physical quantities x, with a
physical scale x0, we note the non-dimensional quantity
as x∗=x/x0.
The time scale is t0= (m0/k0)1/2. For fibers of di-
ameters r=d/2 made of an elastic material (Young
modulus E, Poisson coefficient ν) of density ρ, we have
k0=Eπr2/l0,m0=ρπr2l0, and then t0=l0(ρ/E)1/2.
The time scale t0is then the time of propagation of com-
pression waves through one segment of the fiber. The
force scale f0=k0l0=Eπr2is the force that extend a
hypothetical perfectly elastic fiber by 100%.
2. Elastic forces and damping
When submitted to a traction force f, the relative ex-
pansion of the fibers is f /f0=f∗. It follows that if we
4
want to stay in the limit of small extension, we should
keep f∗1. In practice the simulations are done with
f∗∼10−5−10−3. It should be noted that if f∗is too
small, the propagation of transverse waves is very slow
when no bending forces are present. Indeed, the veloc-
ity of transverse wave vtin a string of linear density ρl
under a tension fis vt= (f/ρl)1/2. With ρl=m0/l0,
we have ρ∗
l= 1, and the non-dimensional speed of trans-
verse wave is v∗
t= (f∗/ρ∗
l)1/2= (f∗)1/2when no bending
stiffness are present.
The non-dimensional bending stiffness is B∗=B/k0l3
0.
For an elastic fiber as consider in III B1, we have B=
Eπr4/4, and then B∗= (r∗)2/4. Similarly, the non-
dimensional torsional modulus is C∗=C/k0l3
0. For an
elastic fiber of radius r, we have C=Eπr4/2(1 + ν), and
then C∗= (r∗)2/2(1 + ν).
The longitudinal damping λis chosen to avoid com-
pression waves that travel continuously through the
fibers, needing very long time to return to equilibrium.
We take λ∼(k0m0)1/2, and then λ∗∼1 for this.
3. Contact force
The value of the contact stiffness is fixed from a lin-
earization of the Hertzian contact between two elastic
cylinders. If two cylinders of radius r, with perpendicu-
lar axis are in contact, the problem is equivalent to the
the contact between a sphere of radius rand a plane,
and the normal force is fn= (4/3) Eeff r1/2δ3/2, with
Eeff =E/(1 −ν2), νbeing the Poisson ratio of the mate-
rial. For doing the linearization, we arbitrary set that the
elastic energy of the Hertzian contact ∼Eeff r1/2δ5/2
is equal elastic energy knδ2/2 of the spring for a normal
force fwhich is of the order or the traction force that we
applied on the fibers. Dropping numerical factor of order
1, we obtain kn=E2/3f1/3r1/3. The non-dimensional
stiffness may then be obtain as:
k∗
n=(f∗)1/3
r∗(12)
where we again dropped constant term. f∗is the typical
non-dimensional force (i.e. the non-dimensional traction
applied to the fibers). This value of k∗
nis a reasonable
choice for modeling contact, but evidently different values
may be set. In practice, since the tension is of order
f∗∼10−5−10−3, and typical radius are r∗∼10−1,
we have k∗
n∼1. For sake of simplicity, the tangential
stiffness is taken as k∗
t=k∗
n.
Some damping of the normal force λnmay be intro-
duced. We took λ∗
n∼1 for rapid relaxation of oscillating
motion of contact.
4. Time scale for simulation
The time step dt for simulation is chosen such that
the dynamic of length relaxation and of contact es-
FIG. 3. (a) 2D view of two sets composed of one sphere and
one cylinder. (b) Motions of an external Cext and internal
Cint contact points at the junction between two cylinders.
tablishment is correctly described. The length of seg-
ment relaxes on a time scale ∼(m0/k0)1/2=t0,
whereas the time scale for a contact to establish is
∼(m0/kn)1/2=t0(k0/kn)1/2. The time step is cho-
sen as dt =Mint0;t0(k0/kn)1/2/10, leading to:
dt∗=1
10 Min1; (k∗
n)−1/2(13)
such that both relaxat ions occur on at least 10 time
steps. In practice, since k∗
n∼1, we take dt∗= 0.1. For
a given set of parameters, it is checked that results are
unchanged if time steps are divided by a factor 2.
C. Computation of contact points
The Discrete Element Method is mainly used in assem-
blies of spherical particles. Due to the anisotropic shape
of the segments, our algorithm for the determination of
the contact points has some particularities compared to
sphere-sphere contact that we discuss in this section.
1. Distance between fibers
The distance between fibers is calculated in the follow-
ing way. We first consider a segment as a set composed
of a sphere and a part of cylinder as shown on figure 3(a).
We first calculate the distance between the two parts of
cylinders following the method described in Appendix
VI B. If contact does not occur along two cylinders, con-
tact between spheres and cylinders are searched, and fi-
nally between the two spheres. The hull of the fiber is
therefore composed of the external surface of the cylin-
ders and of the spheres as shown ib figure 3(b). The
starting and the ending of fibers are finished by spheres.
2. Integration of displacement of contact point
The contact point is followed continuously during the
motion of the fibers. This may be done easily as long as
5
FIG. 4. (a-b): Two possible choices for the size of the cells
into which collisions between fibers mat be searched: (a) size
of the cell scales as the length of segments; (b) size of the cell
scales as the radius of segments. (c) Simplification arising
from the fact that the segments belonging to each fibers are
connected.
the contact point between one segment and one fiber is
unique as in example the contact point Cext of 3(b). In
this case, the displacement of the contact point is con-
tinuously integrated along the motion. In some case, two
contact points may exist simultaneously as the two points
as in example the contact points C(a)
int and C(b)
int of 3(b).
In this case, because Cint is discontinuous, the tangen-
tial displacement is artificially reset to 0 when the contact
flips from one cylinder to the other one. Since the fibers
are weakly bend with rl0, we expect that the num-
ber of such contacts are very small compared to the total
number of contact, producing very negligible errors. A
possible refinement may be to interpolate the two contact
points as a single one, allowing a continuous integration
of displacement.
3. Neighbor search method
The search for contacts between discrete objects can
significantly increase the computation time of DEM al-
gorithms. In our case, the algorithm for measuring the
distance between cylinders is slightly more complex than
for spheres, further increasing the computation time of
collisions. Several strategies are possible to significantly
improve the computation time of the collisions. They are
based on the use of neighbor list (Verlet list) or on the
partition of the system in boxes (Linked Cell Method).
We discuss here the problem arising when using strongly
anisotropic objects. In linked cell method, the particles
are assigned in cells, and the list of particles is each cell is
updated periodically. The collisions are searched only for
particles within the same or the neighboring cells. This
strategy is very effective for approximately monodisperse
spheres. In the case of polydisperse spheres, the size of
the cell must be a multiple of the size of the largest parti-
cles, so that the number of particles per box increases. As
a consequence, the computation time grows rapidly with
the polydispersity as shown by Luding et al [34]. The
problem is very similar for strongly anistropic particles
such fibers, or segments of fibers. The figure 4(a) shows
an assembly of fibers with segments of size l0. If collisions
between segments are searched within one or neighboring
cells, the size of the cell should be ∼2l0. For segments
of section ∼4r2, the number of segments in each cell is
∼2 (l0/r)2for dense 3Dsystem. Since l0/r 1, sorting
particles in cell of size ∼l0is not efficient. A more conve-
nient way to define cell may be considered. It consists, as
shown 4(b), of replacing segments by fictitious spheres
of radius rinside each segment of length l0, and to con-
sider cells of size ∼4r. In this case for a system of Nf
fibers of Nsegments each, the total numbers of fictitious
spheres is ∼NfN(l0/2r). However those two methods do
not use the fact that different segments of one fiber are
linked together. Taking advantage of this knowledge may
significantly speed up the search of neighboring. The fig-
ure 4(c) shows two fibers, and we search contact between
segment iof fiber 1, with fiber 2, by increasing j. For a
segment j, we calculate the distance d(i, j). If this dis-
tance is larger that 2rthere is no contact, and we are
sure that there is no contact between the two fibers for
|j0−j| ≤ d(i, j)−2r. So the next segment where we need
to search contact verifies j0> j +d(i, j)−2r.
The optimal strategy to find contacts is expected de-
pendent on the type of fiber under studies. In case of
fibers with numerous segments, taking advantage of the
constraint that the segment are linked as depicted in 4(c)
is presumably the better. At the opposite, in the case of
an assembly of very short fibers, such as a packing of
one-segment needles, use of celld as 4(b) should be pre-
ferred. The further study of such optimization is outside
the scope of this study.
IV. ILLUSTRATION EXPERIMENTS.
The program has been tested on various simple geome-
tries in order to check the consistency with the theory, to
verify the numerical stability of the algorithm, and test
the numerical precision. Those configurations were the
rolling or sliding of a cylinder on a inclined plane, the ve-
locity of transverse waves of a string, the static flexion of
a fiber loaded at extremity by a point force, the catenary
shape of a massive string under gravity. We present in
the following four more complex situations. If not other-
wise specified, the simulation parameters are: time step
dt∗= 0.1, internal damping λ∗= 2.8, contact stiffness
k∗
n=k∗
t= 1, contact damping λ∗
n= 1, global viscous
damping λ∗
v= 0.001, inertia momentum J∗=m∗r∗2/2
(homogeneous cylinder).
6
FIG. 5. Tension in a rolled string around a cylinder: T∗is
the tension in the string, and θis the rolling angle.Circles are
symbol, plain line is an exponential fit. See for simulations
parameters Inset: Schematic of the experiment.
A. Static without flexion : capstan
We simulate the tension along a string which is rolled
up around a cylinder. For this, we prepare a infinitely
flexible spring (B∗= 0, N= 200, r∗= 0.1) which makes
5 turns around a cylinder (R∗= 5). The cylinder had a
huge mass and moment of inertia to prevent any motion.
The friction coefficient is µ= 0.2. We first apply an
equal tension T∗
1=T∗
2= 0.01, with opposite directions,
to the two ends of the string. We let the system to reach
equilibrium. Then, we slowly decreases T∗
2while keeping
T∗
1= 0.01. For a threshold value of T∗
2, the sliding of
the string occurs. We measure the tension in the string
using (1) at the onset of sliding. The fig.5 shows the
decrease of the tension T∗along the string as a function
of θ= (s∗−s∗
0)/(R∗+r∗), with s∗the abscissa along the
curve, and s∗
0the abscissa of first contact contact point.
The solution of capstan problem with a finite thickness
rod predicts that [35]: T∗/T ∗
1= exp(−µ θ), which is the
observed behavior on figure 5. The measured decay is
µ= 0.198 in agreement with the imposed value µ= 0.2.
B. Static with flexion : elastic knots
We consider the mechanical response of an elastic rod
with an open knot. An elastic fiber of length L, with
a circular section of radius r, and bending modulus Bis
bent in an open trefoil knot (31). We then apply a tension
Tto the ends of the fibers. This experimental situation
has been addressed by Audoly et al. [16, 17]. When the
tension is weak, the loop radius Ris very large compared
to r. In this limit, authors found analytical solutions for
the shape of the knot, either in the frictionless case, but
also for weak friction µ1. This knot has been simu-
lated very recently by Choi et al. using an discrete rod
FIG. 6. (a) Snapshot of a (31) knot. For seek of clarity,
illustration is made with r∗= 0.2. (b) Tension as a function
of ε=pr/R for frictionless and frictional strings. Symbols
are numerical data, and lines are theoretical expressions given
by equation (14).
model with an implicit solver for the contact force [29].
We simulate numerically such knot by considering a
flexible spring (r∗= 0.1, B∗= (r∗)2/4=2.5 10−3,N=
500, λ∗
v= 4.10−4) as shown on figure 6(a). We first knot
the fiber by setting µ= 0 and applying a tension ±T∗ez
at ends. After this preparation stage, we set µto its
actual value, and we increase or decrease T∗depending
if we tighten or lossen the knot. When the knot begins
to move, we measure the radius of curvature of the loop
as R=<kdti/dsk−1>, where dti/ds =ei+1 −eiis the
derivative of the tangent vector, and the average < > is
over all segments in the loop which are at a distance of
at least one segment from any contact point. Following
[16, 17], we introduce ε=pr/R. The figure 6(b) shows
the tension T∗as a function of εfor frictionless (µ= 0),
and frictional (µ= 0.1) loosening and opening knots.
The analytical solutions in the limits ε1 and µ1
are [16, 17]:
T r2
B=ε4
2±µσε (14)
where the sign ±depends if the knot is tightened (+)
or loosened (−), and σis a numerical constant which is
σ'0.492 for trefoil knot. As shown on figure 6(b), the
numerical data agrees correctly with the analytical one.
In the frictionless case, we may observe deviations from
7
FIG. 7. (a) Experimental snapshots of the impact of a
metallic chain on a fixed perpendicular cylinder of radius
R= 10 mm. The perimeter of the cylinder is underlined
in red. Time t= 0 is defined as the first contact time. Chain
length L= 190 mm, mass m= 8.5g. (b) Simulated impact.
See text for physical parameters of the simulation.
the scaling T∼ε4when ε&0.15. Two possible sources
of deviations may be identified. First, the equation (14)
is obtained in the limit ε1, and deviations may arise
from high order εterms in equation (14). Second, for
ε&0.15, we have R∗=R/l0=r∗/ε2∼4, so that the
discretization of loop may then be an issue. The discret
nature of the rod may be clearly identified on numerical
data form µ= 0.1 loosening, where some steps in εare
visible. For the frictional case, the model (14) slightly
underestimates the role of friction compared to numerical
simulations. It may be due to some departure from the
hypothesis µ1 which is used to obtain (14).
C. Impact: falling chain.
We consider the dynamics of the impact of a metallic
chain on a cylindrical obstacle. We restrict this anal-
ysis to a qualitative analysis. A metallic chain (length
L= 190 mm, mass m= 8.5g) is held at its extremities
by hands. The chain is released and its fall is recorded
with a fast camera operating at 200 f ps. The figure 7(a)
show some snapshots of the impact. The chain is simu-
lated as a infinitely flexible spring B∗= 0. We set the
length scale to l0= 2 mm, and N= 95, so that L=N l0.
The choice of the time scales may be done in the follow-
ing way. We want to simulate a non-extensible chain, so
we need that the non-dimensional typical force is << 1.
The gravity force is Fg=Nm0g, with m0the mass scale
of one segment, and gthe gravity. The non-dimensional
gravity force is then F∗
g=Fg/k0l0=Ng/l0t−2
0=Ng∗.
We take g∗= 4.9 10−5so that N g∗'5.10−31. This
sets the time scale t0= 0.1ms. It should be noted that
in the limit of a non-extensible chain, the mass scale does
not need to be specified. Other parameters are dt∗= 0.1,
R∗= 5, r∗= 0.1, µ= 0.1, k∗
n=k∗
t= 1. Figure 7(b)
shows the results of the simulations which qualitatively
agree with the experiments. We may remark that the
behaviour of the experimental chain is not symmetric in
compression and in extension (nearly infinite stiffness in
extension, zero stiffness in compression), whereas the nu-
merical chain is symmetric (same stiffness in compression
and in extension). However, in impact experiment, the
chain is always in tension, and the lack of symmetry does
not have importance.
D. Multiple fibers: a yarn model.
In a recent study, Seguin et al. considered the situation
of a staple yarn made of twisted totally flexible fibers [7].
We present in this section some numerical details about
this simulation. The yarn is made of an assembly of Nf
identical fibers of Nsegments initially parallel to an axis
z(see figure 8(a). Their positions (xi;yi) in the plane
perpendicular to the zaxis, with 1 ≤i≤Nfare the
positions of a packing of disks in 2D obtained from a
separate simulation.
In a first phase of the simulation the fibers are twisted.
The fiber are submitted to a tension T∗= 10−4along z,
applied at both ends. A torque C∗ezis applied to both
ends of the assembly of fibers. For this, each fiber iwith
1≤i≤Nfis submitted at both ends j= 0 and j=N
to an external shear force:
τi(j) = ±C∗
Pir2
i(j)ez×ri(j)(15)
where sign is −for j= 0, and + for j=Nends. The
torque is gradually increased until it reaches its target
value and the shear forces are updated at each time step.
Under the action of this torque, the fibers twist and be-
comes approximately helicoidal as shown on figure 8(b).
During this preparation, the friction coefficient is set to a
low value µ= 0.05. This is important in order to obtain
a regular pitch along the thread. Indeed, since the yarn
is twisted by the application of torques at ends, the pres-
ence of an important friction between fibers has the effect
of concentrating the twist near ends, with a central zone
of low twist. This behavior is also observed experimen-
tally [7] if the twist is not homogenized along the yarn.
8
FIG. 8. (a) Assembly of of initially straight fibers. (b) Thread
of fiber after a torque is applied at ends. (c) Separation of
the fibers due to applied forces. (d) Force ratio necessary to
separate the slivers as a function of the twist angles. Line is for
guidelines. (e) Same data as (d) plotted as a function of H=
µθ2R/L. Dotted line is 0.75 µθ2R/L. Simulation parameters
are Nf= 20, N= 30, r∗= 0.1, B∗= 0. For clarity, the
sub-figures (a-c) are enlarged by a factor 6 perpendicularly to
z-axis.
The duration of this preparation stage is t∗= 5.105, and
the total twist θis measured at the end of this phase.
In a second phase the fibers are separated. The fric-
tion is first set at its target value. The fibers are ran-
domly partitioned is two set -up- and -down-. The ten-
sion of the up-fibers are multiplied by a factor f > 1
at the up-extremities: Tup(j= 0) = T∗and Tup (j=
N) = f T ∗. Symmetrically, Tdown(j= 0) = f T ∗and
Tdown(j=N) = T∗. The factor is f= 1 at the begin-
ning of the separating stage and is increased at a fixed
rate (∆f/∆t∗) = 2.10−6. During this phase, the torque
is kept constant. The difference between the average po-
sitions of the -up and -down fibers is measured. This dif-
ference stays constant, until a threshold value of fwhere
the two slivers of fiber separates (see figure 8(c)). A me-
chanical model of this problem developed in [7], show
that the force necessary to separate the two slivers is
ln(1 + f)'0.75 µθ2R/L which is the behavior that is
observed on figure 8(e).
V. CONCLUSION
We have described a discrete element mechanics algo-
rithm for the simulation of flexible and frictional fibers.
This algorithm is similar to the DEM type algorithms
widely used for the study of granular materials. The dif-
ference arises from the type of surfaces in contact (cylin-
ders and not spheres) and from the elastic forces between
the cylinders which are connected to form a fiber. The
algorithm has been tested on various configurations that
can be compared to experiments or to theoretical models.
The assumptions and approximations used to design
this algorithm are quite limited. The low bending as-
sumption is not very compelling for many applications,
but could eventually be minimized by a finer discretiza-
tion of the fiber. The simplification of the Hertzian elastic
contact law between the cylindrical segments by a linear
spring has probably a very small impact on the model-
ing of real systems. An extension to non-linear contact
laws should not be a problem. Finally, the discretization
of the fiber generates a discontinuity of the displacement
for some contact points at the passage between successive
segments of a fiber. A priori the number of such jumps is
negligible compared to the total number of contacts for
thin and weakly bent fibers, and this should not be an
issue for simulations of real systems.
The main difference between this algorithm and those
previously described to simulate elastic fibers lies in the
level of simplification of the mechanical problem. Simula-
tions of fibers with finite element algorithms are certainly
of high accuracy but can only simulate small systems.
Implicit algorithms are probably faster, but the indermi-
nation of forces in multi-contact cases is not resolved by
the dynamics of the system. The use of a DEM algo-
rithm is a compromise that allows to consider relatively
complex assemblies of fibers and that correctly handles
the multiplicity of equilibrium solutions.
The potential applications of this algorithm are ob-
viously multiple. The study of complex knots between
fibers of ropes, with or without bending energy is possi-
ble. The mechanical response of fiber clusters in nests,
cushions, or in rigid needle stacks are also possible. For
these studies, the contact search should be optimized ac-
cording to the aspect ratio of the fibers and the geometry
of the packing. The simulation of knitted or woven fab-
rics can also be considered. For this, large systems can
be simulated, but the introduction of periodic boundary
conditions should be more suitable. Finally, systems mix-
ing fibers and grains for the study of soils reinforced by
fibers or roots are also possible applications of this work.
ACKNOWLEDGMENTS
The author would like to thank Antoine Seguin and
Sean McNamara for discussions and careful reading of
the manuscript, and Laurent Courbin for his help in ex-
periments on falling chains.
9
VI. APPENDICES
A. Flexion forces
The curvature κiat a node N−2≥i≥1 is first
expressed as a function of the positions of nodes ri−1,
riand ri+1. The radius Ri= 1/κiof the circle join-
ing those three points may be expressed as a function of
the surface Siand the perimeter piof the triangle with
vertices (ri−1,ri,ri+1) using the Heron formula. After
elementary calculus, we obtain:
κ2
i= 4 l2
i−1l2
i−(li−1·li)2
l2
i−1l2
i(li−1+li)2(16)
where we noted li=ri−ri−1. The bending energy is
E(b)=B l0
2
i=N−2
X
i=1
κ2
i(17)
The flexion force is then:
f(b)
i=B l0
2
∂
∂ri
i0=N−2
X
i0=1
κ2
i0(18)
First, we notice that for weakly bend fibers li'li−1,
and for weakly extended fibers li'l0. Then, the denom-
inator of (16) is '4l6
0:
∂κ2
i0
∂ri
'1
l6
0
∂
∂ril2
i0−1l2
i0−(li0−1·li0)2(19)
Using li=ri−ri−1, we obtain:
∂κ2
i−1
∂ri
'1
l4
02(li−1−li−2)(20a)
∂κ2
i
∂ri
'1
l4
0−4(li−li−1)(20b)
∂κ2
i+1
∂ri
'1
l4
02(li+1 −li)(20c)
and (∂κ2
j/∂ri) = 0 if |i−j|>1. We obtain finally:
f(b)
i=B
l3
0−li−2+ 3li−1−3li+li+1(21)
=B
l3
0ri−2−4ri−1+ 6ri−4ri+1 +ri+2(22)
for N−3≤i≤2. Expressions of the forces for i < 2,
and i>N−3 are obtained by noticing that summation
FIG. 9. Distance between two points located on two segments
of line.
in (18) is for i0= 1 to i0=N−2.
f(b)
0=−B
l3
0r0−2r1+r2(23a)
f(b)
1=−B
l3
0−2r0+ 5r1−4r2+r3(23b)
f(b)
N−2=−B
l3
0rN−4−4rN−3+ 5rN−2−2rN−1(23c)
f(b)
N−1=−B
l3
0rN−3−2rN−2+rN−1(23d)
B. Distance
We consider two segments 1 and 2 whose axis are
drawn on figure 9. On each axis are located at abscissa
s= 0 a sphere of rayon r, and a segment of cylinder of
radius rfor 0 ≤s≤1. The distance between two points
at abscissa s1and s2is:
d2(s1, s2)=(a+s1l1+s2l2)2(24)
The distance is minimal for s∗
1and s∗
2which verify:
∂d2(s1, s2)
∂s1(s∗
1, s∗
2) = ∂d2(s1, s2)
∂s1(s∗
1, s∗
2) = 0 (25)
Equation 25 is solved to obtain (s∗
1, s∗
2), and the mini-
mal distance d(s∗
1, s∗
2) is obtained. If d(s∗
1, s∗
2)<2r, with
0≤s∗
1≤1 and 0 ≤s∗
2≤1, the contact is found between
the two cylinders.
It not, the contact is checked between the sphere lo-
cated at s1= 0 and the cylinder 2. For this the minimal
distance is obtained for s∗
2verifying:
∂d2(0, s2)
∂s2(0, s∗
2) = 0 (26)
.
Equation 26 is solved to obtain s∗
2, and the minimal
distance d(0, s∗
2) is obtained. If d(0, s∗
2)<2r, with 0 ≤
10
s∗
2≤1, the contact is found between the sphere (1) and
the cylinder (2).
The contact between sphere (2) and cylinder (1) is
searched in a similar way. If not, we check for a con-
tact between the two spheres.
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