Force Schemes in Simulations of Granular Materials
J. Shäfer, S. Dippel, D. Wolf
To cite this version:
J. Shäfer, S. Dippel, D. Wolf. Force Schemes in Simulations of Granular Materials. Journal de
Physique I, EDP Sciences, 1996, 6 (1), pp.5-20. <10.1051/jp1:1996129>. <jpa-00247176>
HAL Id: jpa-00247176
https://hal.archives-ouvertes.fr/jpa-00247176
Submitted on 1 Jan 1996
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
J.
Phys.
I
France
Force
(1996)
6
Schemes
Schifer(*),
J.
Simulations
in
S.
JANUARY1996,
5-20
Dippel and
3
July1995,
Forschungszentrum
received
final
in
5
Materials
Wolf
D. E.
H6chstleistungsrechenzentrum,
(Received
Granular
of
PAGE
form
and
Jiilich,
accepted
D-52425
12
Jfilich,
Germany
1995)
October
of granular flow, one widely used technique is classical
In
simulations
Abstract.
computer
Molecular
soft-sphere
Dynamics, where the equations of motion of the particles are numerically
integrated. This requires specification of the forces acting between grains. In this paper, we systematically study the properties of the force laws most commonly used and compare
them with
experiments on the impact of spheres. We point out possible problems and give criteria
recent
Finally, two generic problems of soft-sphere
for the right choice of
simulations
parameters.
are
discussed.
Computer modeling and
PACS.
07.05Tp
PACS.
46.30Pa
Friction,
PACS.
83.70Fn
Granular
1.
Introduction
Flows
of
hardness,
mechanical
contacts,
and
tribology.
solids.
granular
importance,
simulation.
adherence,
wear,
materials
ubiquitous
are
properties (among which
sticking [3], density waves [4, 5]
in
their
flowing to
Computer
have
turned
simulations
granular flow, especially valuable as
difficulties
far, and experimental
are
out
are
"silo
and
be
to
there
industry and nature.
Despite their technological
from
size segregation [1,2], sudden
transitions
is
no
a
music"
[6])
far
are
powerful tool to
generally accepted
from
well
understood.
investigate the physics of
theory of granular flow so
A very
simulation
popular
scheme
is
an
adaptation of the classical
Molecular
Dynamics technique. It consists of integrating
Newton's
equations of motion for a system of "soft" grains starting from a given initial configuration.
This requires giving an explicit expression for the forces that act
In principle,
between
grains.
should provide such
the
of
touching
bodies
mechanics
but
problem
contact
two
expressions,
considerable.
general conditions is very complicated (see, e.g., [7] and references therein). )(any more
strongly simplified force schemes have thus been suggested and employed in simulations
or
(e.g., [8-20] ), often without a thorough discussion of their properties.
three-fold:
of
The
of the
firstly, to give a brief
theoretical
present
account
paper
is
aim
considerations
results
and
experimental
concerning the free impact of spheres; secondly, to
critically compare the properties of existing force schemes on these grounds, and to give hints
their
and thirdly, to discuss
problems that can
in soft
correct
on
use;
some
generic
occur
sphere simulations
independently of the force laws used. ~ie perform example
simulations
to
under
less
illustrate
the properties of
algorithm [21]. Although
(*) Permanent
Germanyj e-mail:
©
Les
(ditions
address:
FB
the
force
it is in
10,
laws
Theoretical
j.schaefer%kfa-juebch.de
de
Physique
1996
with
principle
a
constant-timestep
desirable
Physics,
to
Gerhard
work
with
Mercator
fifth
order
predictor-corrector
units, they
non-dimensional
University,
D-47048
Duisburg,
JOURNAL
PHYSIQUE
DE
I
N°1
~
~
v~- v~
w~
Grain
I
Ri
~
R
~
r~
2
~
2
2
Fig.
Definition
1.
quantities
of the
used
description
for
impact.
of the
always practical; for example, non-linear force laws do not
discussed
later.
Therefore, we decided to use SI units, and
cellulose
mimic a specific granular material, namely the
acetate
are
will
to
not
define
a
be
tuned
all
by Drake [22-24]
Thus, we are able
and
Foerster
al. [25], of
simulation
et
radius
R
3
=
as
such
as
parameters
spheres used
and
mm
unique timescale,
mass
m
experiments
1.48 x 10~~ kg.
data [25].
in
=
directly to experimental
compare
our
of
ourselves
non-cohesive,
spherical grains which are
restricted
We limit
dry,
to the
to
case
(two translational, one rotational), as would be the case m a twothree degrees of freedom
between two grains of radii R~, positions r~,
dimensional
velocities
setup. A general contact
ii
sketched
deformation
of
and
angular
velocities
is
The
the
1,
in
Figure
grains
2)
I.
w~
v~,
overlap"
parametrized
by
the
"virtual
(,
is
to
results
=
j
Two
unit
shear
vectors
and
n
s
are
(0, Ri
max
=
used
+
R2
)r2
decompose
to
ri
)).
forces
the
and
velocities
normal
into
and
components:
n
=
s
Thus, the
normal
relative
velocity
~n
~~
"
=
~~
~~
)r2
ri
=
(n~,
n~
(ny, -n~).
"
and
~n
relative
(V2
VI
n
(v2
vi
s
velocity
shear
+ WI
RI
+
~~
are
given by
w2R2
with
velocity component ~~ is equal to zero at the beginning of a contact, the impact
normal, otherwise it is shearing or obliq~te. We begin our discussion in Section 2
or
on
normal impacts introducing the normal force Fn, and proceed to oblique impacts and the
shear
force F~ in
If the
is
general,
extremes
are
3.
colliding spheres undergo a
perfectly inelastic and perfectly
two
of
transformation
heat)
Section
Impacts
Normal
2.
In
shear
head
of
[7] plastic
kinetic
energy
deformation,
into
other
deformation
elastic.
forms
viscoelasticity
which
will be
Possible
of
of the
energy
somewhere
mechanisms
which
material,
and
for
ultimately
also
elastic
between
the
dissipation ii-e-,
transform
waves
into
excited
FORCE
N°1
by
that
energy
one
-~l/~l
en
the following,
(final) quantities.
According to Johnson [7],
and
little
The
elasticity of the
Here
SIMULATIONS
IN
but
latter
always present
they make the noise
carry
so
are
Phenomenologically,
them
of
dissipation.
generally
neglect
source
can
as
restitution
impact is described by the coefficient of normal
en,
impact.
the
SCHEMES
in
the
superscript
the
necessary
lo, il.
E
=
the
refers
pre-collisional (initial)
to
and
f to
post-
collisional
yield velocity causing plastic
critical
deformation
is
given by
~3 y5
107tj
G3
~),~>~
elf
Ii)
off
(RiR2 /(Ri +R2) are the reduced mass and radius,
(mim2) /(mi +m2) and R~n
yield strength of the softer of the spheres, and E~n is related to Young's modulus E and
v])/E2. If ~[ < ~~,~j~,
ii VI) /Ei + ii
Poisson's
ratio v of both spheres through I/E~,
deformation
plastic
during the impact, and all energy loss should be due to
occur
no
may
viscoelasticity. If ~[ > ~~,~,~, on the other hand, the energy loss due to plastic deformation
dominate
the energy loss due to viscoelasticity.
have a critical yield
Most
materials
must
over
velocity that is quite small (e.g. ~~,~,~ m 0.14 m/s for cellulose
acetate, ~~,~,~ m 0.02 m/s for
where
m~a
=
=
Y is the
=
steel).
For the
deformation,
of plastic
case
increasing
impact velocity. For the
obtain
coefficient
close
a
to
en)
ii
one,
of
normal
c~
al. [28],
Kuwabara
and
class of different
materials
compatible
is
hand,
spheres
which
with
show
cases
m/s < ~[
Modelling a
0.29
<
and
of
sort
some
l.2
slight
a
of en
narrow
(Foerster
~yn is
(, the
is
a
force
leads
that
dissipation.
damping
advantage
that
the
calculation
of
~[
on
for
et al.
and
the
of en
range
e,i
en
[27], Bridges
others
spheres made of a large
increasing ~[ which
al. [29] for
measured
[26j
Kono
increasing ~[; for
~N.ith
with
impact
of
0.87
m
in
the
other
acetate
velocities
used,
velocity
range
a
inelastic
to
The
requires
collisions
simplest force
with
least
at
two
properties
desired
the
terms:
the
is
repulsion
damped
force
deformation
the
and
m/s).
Fn
where
Kuwabara
Goldsmith
decrease
monotonic
dependence
rather
was
oscillator
harmonic
show
systematic
no
both
en
decreases
Experimental studies by
[26] and Sondergaard et
Kono
all
also
that
~[~~~~ with
to fall off like
~[~~~~ power law over several orders of magnitude of ~[. On
by Drake and Shreve [22] and
Foerster
cellulose
et al. [25] on
a
measurements
in
case
restitution
~[~~~
et
simple theory [7] predicts en
of purely
viscoelastic
losses,
a
and km
constant
=
is
-kn(
related
to
important
quantities.
en
For
instance,
'~"
=exp(-
stiffness
the
grain. This model (also
analytic solution (with initial
of the
its
(2)
'fn(,
referred
conditions
the
to
of
as
((0)
coefficient
spring
a
linear
=
of
0
elongation
whose
spring-dashpot) has
~[) allows
and ((0)
=
normal
restitution
tn),
is
(3)
2111~ff
~~~~~
~
i~~i
~"
denotes
the
duration
of the
collision.
~~~
~2~ii~
~
The
fmax
maximum
<
~[t»/~,
overlap during
~~~
a
collision
is
is)
PHYSIQUE
DE
JOURNAL
I
N°1
1.0
~
~
~
°
<
o
°'~
o
~
o
o
n
~
(
~
8
o
o
6
n-
o
o
o
~
~
°
T
©
~
c
o~~
~
o
n
0.8
-o
°
°
i
~
~
°
u
°
0.7
°
n
~
a
0.6
A
o.5
~~
jm/Sj
~~
Fig.
Dependence
2.
normal
lo)
forces.
Kuwabara-Kono
of
coefficient
(9), IA)
force
normal
of
spring-dashpot
Linear
restitution
(2), In)
Hertz
force
(10).
Walton-Braun
e~
impact
the
on
law
linear
with
velocity v[
for
various
damping (8), IO)
Hertz-
I). An accurate
(reproducing the
where the equality holds for elastic grains (en
simulation
analytic en with relative
of
the
order
10~~)
requires
At m tn /100.
time
constant
step
a
errors
In principle, force (2) has no free
since km and ~yn can be set adjusting en and
parameters,
exhibited
experimental values
by a given
material
in a velocity range
tn to the corresponding
=
for
relevant
simulations.
the
=
order
In
formulate
to
theory of elastic
spheres:
a
behaviour
its
=
refined
more
force
stiffness
with
(6),
en
used
0.87
=
in
works
numerous
and
tn
I
=
x
10~~
s
=
for the
Hertz
case
Drake's
~fi)
properties
and
the
to
radii
no
~~~
of
=
~~~
~[
the
spheres, km
9.0
acetate
longer independent of ~( [32]
cellulose
tn is
time
of
(6)
elastic
the
to
collision
3.21
of the
results
force
the
-kn(~/~
For
the
tn
been
=
connected
4/3/$E~n.
=
for
(2), one can use the
following repulsive
than
predicts
Fn
Here, in is a
non-linear
spheres through In
10~ Nm~~/~. Note that
has
it
illustrated
is
kg/s).
2.06
[7, 31, 32], which
contact
advantages,
of its
Because
[9-12, IS,16,30]. In Figure 2,
(w km
7.32 x 10~ N/m, ~yn
x
(7)
km
This
about
In
force
Hertz
for
obtain
to
in
an
on
no
which
a
maximum
fashion
However,
as
m
For
low
impact
10~~
x
s,
and
we
a
-kn(~/~
leads
force
the
have
we
set
viscous
a
of the
time
simulation
to
impact velocity
maximum
tn /100.
damping term
At
numerical
during the
=
added
was
to
the
(8)
~yn(.
=
this
choice
[13,17,33]:
studies
contrary to
increases,
where the Hertz
velocities,
simulations,
our
some
Taguchi [34] pointed out,
impact velocity
In
The
expected
velocities
Hertz-type force,
Fn
the
collisions.
to
relative
accuracy.
tn m 1.88
dissipative
ad hoc
timescale
intrinsic
the
numerical
m/s,
100
order
is
depend
must
satisfactory
ensure
of
there
that
means
At
step
to
collisions
experimental
results
for
elastic
that
evidence:
contacts
become
ii
should
en
be
more
elastic
as
~(~~~~ [35].
regained, force
c~
FORCE
N°1
(8) produces a
behaviour using
~yn
of
=
that
so
Kuwabara
and
Kono
material
the
and
[26]
be
to
In
where
the
identical
is
m
Figure 2,
In
zero.
illustrate
we
covered by
range
this
Foerster's
of the
elastic.
-(n(~/~
approach
Hertz
§n(~/~(,
theory
Hertz
in
This
and in is
force
(9)
leads
connected
to
a
experimental
190 kg m~~/~ s~~.
deformation
radii
of
ii
results:
spheres
of the
normal
restitution
en)
presented by
was
~(~~~
c~
Walton
and
ki and k2, for the loading
constants,
spring
the
to
coefficient
with
agreement
=
unloading part
the original
They derive
extend
[36]
of
=
viscosity.
increasing ~(,
al.
et
instead
[26, 35]. In Figure 2, we illustrate this using fin
An approach guided by the picture of plastic
different
Braun [37]. They
that there are
assume
and
9
velocity
0.87 in the
Brilliantov
from
of bulk
with
decreases
In
the
to
coefficients
two
that
en
en
viscoelastic
Fn
and
approaches
that
restitution
kg/s,
0.35
SIMULATIONS
IN
[25].
measurements
assuming
coefficient
SCHEMES
contact:
kit
~
k2 If
~
j
> 0
(
< 0
pleading)
(Unloading)
,
(0)
~~~~
,
to
where
the
is
circumstances,
k2 a function
made
be
As
a
(omitted
the
or
remark,
model,
under
abscissa
fij.
en
=
the given
By making
=
behaviour
want
we
this
"
function
this
the
intersects
In
force F~'*~ achieved
during loading, k2
ki + sfj-~, en
(sv[(m~n/ki)~/~ + l)~~/~.
of the impact velocity ~(, en
maximum
illustrate
we
final
the unloading
curve
plastic deformation.
( where
permanent
of the
decreasing
a
Figure 2,
of
value
ki
with
comment
to
on
7.32
"
the
10~ N/m and
x
of the
use
s
=
reduced
2
mass
can
In
10~ m~~.
x
in
m~a
the
forces
studies, it is understood as an additional prefactor for the
some
damping term [10, II,13,14,17,18, 30], leading to a coefficient of restitution
decreases
that
with
Other
authors
prefactor
both
the
elastic
the
dissipative
it
and
put
term
increasing
as
m~n.
in
[IS,16,34], such that the coefficient of restitution
becomes
independent of m,.,. If neither of the
function of m~n.
Unfortunately,
two
terms is given a prefactor m~~~ [8, 9,12], en is an
increasing
research
dependence
of
knowledge
there is no systematic
experimental
the
to
mass
our
on
en
from
by
work
dating
cited
Goldsmith
which
indicates
slight
decrease
1864,
except
[27],
some
a
of en with
result
This
increasing m~~~ for spheres of equal size.
suggests that the first of the
possibilities described
desirable
above might be most
suited.
It is definitely most
that
more
experimental
work be done in
order to settle
However, when the granular flow
this
point.
without large mass differences, putting m~~~ into the forces merely
consists of particles
amounts
stiffness and for damping
to redefining the
constant.
here for
3.
Oblique
We
now
of the
force
simplicity).
Impacts
to
impacts
or
shear
force
Coulomb
laws
and ~ld
that
means
~1~
are
the
for the
Contacts
where ~( # 0, such that there
for short.
In general, the
turn
force,
by the
F~
Here
Frictional
and
F~
ill)
In
of friction,
~1~Fn
for
static
=
~1dFn
for
dynamic
case
of
of static
static
a
non-vanishing
shear
force is
tangential
connected
to
component
the
normal
namely
<
coefficients
is
and
friction
(~~
friction
=
III
0),
(~~ #
0).
dynamic friction, respectively. The "<"
compensates exactly the (unknown)
friction, F~
(12)
sign
in
external
JOURNAL
10
PHYSIQUE
DE
N°1
I
If Fj~~ > ~1sFn, one
force Fz' applied to the contact, so that ~~
maintained.
the
0 is
enters
dynamic friction regime and equation (12) applies. Normally, ~ls > ~ld and both are about 0.5.
of two colliding
bodies, a local version of equations ill and (12)
In the
contact
convex
area
along the contact. For elastically similar bodies, the
relates the normal and the shear
stresses
=
shear
the
stresses
results
of the
of
do
influence
not
theory
Hertz
contact
can
where
area,
the
normal
still
be
the
normal
distribution
stress
for the
used
stresses
the
over
contact
normal
stresses.
Then,
small
because
the
are
in
area
the
strains
[7],
so
that
regions
small, one
outer
are
general expect the condition for dynamic friction to be fulfilled, whereas in the central
and
friction
This
strains
regions, where large normal
static
stresses
present,
are
may
occur.
leads to the development of an ann~tl~ts of microslip surrounding an
inner region of sticking
friction
Because
the
laws are strongly nonlinear, the size and form of
the
contact
area.
in
annulus of microslip depend on the loading-unloading history of the contact, making the
the
deformation
and friction forces, in a given
prediction of tangential
situation, complicated [38].
discussed
friction
forces
Mindlin
and
Deresiewicz
the tangential
between
elastic
two
[39] have
loading-unloading
and assuming the Hertz
spheres for the case of several
distinct
histories
theory to hold. Maw et al. [40, 41] and Walton [42] performed such an analysis for the case of
the oblique impact of spheres. An interesting result is that due to the ability of the sticking
"tangential" kinetic energy, there may be a reversion of tangential
and
store
restore
contact
to
for discs [41] and
fact
experimentally
confirmed
velocity ~~ under
certain
circumstances,
a
in
must
spheres [25, 2 7].
following, we implemented
varying obliqueness. We
directly compare the results to the experimental results of [25]. The linear spring-dashpot force
(2) was used as normal force in all cases, because it is simple, robust and, most importantely,
absolute
makes the
results of oblique impacts only dependent on the obliqueness, not on the
value of the impact velocity ~[. We adopt the values en
7.32 x 10° N/m,
0.87 (~ km
simulations
otherwise.
2.06 kg Is) and ~1= 0.25 for all
except where explicitly stated
~yn
for
the
obliqueness
of
There
the
the
angle
impact,
impact
measures
are
various
e-gillustrate
To
and
them
the
carried
properties of the
out
force
shear
discussed
laws
binary
of free
simulations
test
in
the
with
impacts
=
=
=
~
~~~~]
=
(
~~ .
~
arCCOS
,
or
the
In
all
We
dimensionless
are
impacts
going
dimensionless
tangential velocity,
initial
shown
in
the
following,
the
quantities
two
to
measure
final tangential velocity
as
a
~f
as
a
function
of
#' and the
measured
in
the
coefficient
of
tangential
coefficient
center-of-mass
as
function
~~s
of total
system
particle spins
initial
f
of the
that
#'
tan ~.
namely the
=
restitution
fi
a
function
~l/~i
=
impact
so
obliqueness,
/ ~n
restitution
e~
zero,
are
~f/~>
=
of
siniJ.
Plotting #~
~ers~ts
#',
the
SCHEMES
FORCE
N°1
SIMULATIONS
IN
11
la)
16)
3.0
3.0
2.
2.0
/~
°
9'
o/f~
°
O$~
~
O
1'
O
~.°
l'
_-~
O-O
'~~~~~~~~~~~
aaoo°
OOOO°
~'~0.0
2.0
1-o
3.0
~0.0
4.0
2.0
0
/
ll'~
o
o
2.0
/
/
o
l.0
0
)/
%fi'
~
°~
O-o
o-o
2.0
0
~a%a°
~0.0
4.0
3.0
2.0
0
~~~
~
%
l-o
°
~'°
o,
o
o
f~
°
°
°'°
g
-10
a
1-o
o-O
3.0
2.0
0
4.0
o-O
0
2.0
i'~
3.
~i~-~i' plots for
denote
(14) with
d) linear
k$/kn
theirs
we
(16)
spring
off for
read
any
in
Figure
3.
role for
decisive
On
the
the
to
zero.
force
that
(13)
The
circles
denote
grains,
0
e~
(reversal
#~ vs. #'
directly
to
we
are
going
compare
our
the other hand, the form of e as a function of impact obliqueness plays
dissipation of granular temperature in the granular system. In Figure 4,
acetate
Fs
this
(2).
resulting curve.
non-frictional
For perfectly
(frictional
losses)
I
-I
<
< e~ <
or
even
elasticity). Foerster et al. [25] used plots
spheres;
cannot
provide
is
discontinuous
=
-/1.
reversal
at
~~
the
following.
the
Coulomb
of
results
to
plot e(sin ~) for the force laws presented in
simplest shear force [10,30] just applies
Note
force
=
The
Obviously,
normal
with
The
[25].
=
#~ from
cellulose
their
combined
speres
=
be
characterize
to
a
tangential
4.0
dotted, dashed and
dot-dashed
lines
friction
law (13) with y
friction
law
0.15, 0.25, 0.35; b)
viscous
of la) and 16) according to (15) with ~fs
combination
1, 3, 20 kg/sj
with ks/kn
tangential spring (20) with
1, 2/7, 1/5; e) variable
model (22) with (o "10~~ (only dotted line).
equal to one; in practice, e~
tangential velocity due to tangential
would
of
tangential forces
impact of acetate
1, 2/3, 1/3j f) stick-slip
be
can
the
3.0
i"
various
for
data
respectively: a) Coulomb
kg/s; c)
~fs =1, 3, 20
=
°
°
o
/
O-o
°~
~'°
o
experimental
4.0
~~~
2.0
the
3.0
3.o
3.o
Fig.
~
~
/
,
~0.0
/
o
o
l'
aaoa°
~~-
4.0
3.0
2.0
~~
3.0
(d)
(c)
3.0
~9-
o
_~~-~~~~
o
~~~~~~/
O-O
o
/
+
law of
dynamic friction, thus
jfnj signjiJs).
of
=
j13)
tangential
velocity;
When
0
0.
~~
-
it
can
only
slow
~~
down
(rolling regime), numerically
JOURNAL
12
PHYSIQUE
DE
I
N°1
(b)
(a)
i-o
i-o
0.9
0.9
~~
0.8
©
~"~
/~
~<
0.8
fi
/
~
~
0.7
0.7
~~0.0
0.2
0.4
0.6
~~0.0
1-O
0.8
0.2
0.4
(cl
0.8
,j~
~
0.6
0.8
0
0.6
0.8
0
0.9
~/
~
0.8
/~
~,,,~
~
0.8
~,
~'
0.7
0.7
~'~0.0
0A
0.2
0.6
~'~0.0
1-O
0.8
0.2
0A
le)
(f)
i o
i o
0.9
0.9
/~
~h~,~~
0.8
w
0
(d)
0.9
w
0.6
/
0.8
~~~~
0.7
0.7
0.6
0.6
O-o
0.2
0.6
0.4
Sin
Fig. 4.
Dependence
corresponding
parameters
gets F~ jumping
one
has
effect
no
amplitude
the
on
of
0A
Sin
linestyles
between
final
0.2
O-O
0
coefficient
the
and
1-o
0.8
as
positive
velocity ~(,
of
total
in
Figure
the
e
on
the
offset
impact
sin
~.
Forces,
3.
negative
and
as
restitution
0
values
time
average
end of the
of F~
instead
of F~
vanishes
0.
=
as
it
However, this
should, and the
#~ vs. ~i~ as obtained with
contact.
distinguishes a regime of ~~ where,
during the impact ~~ was slowed down to zero (rolling) and a regime where v) was too large, so
that a finite ~( resulted.
Both regimes are
characterized
by constant slope, and the transition
between
them is governed by the value of /~: the higher ~1, the longer the rolling regime is
sustained.
The corresponding e(sin~) diagram is displayed in Figure 4a. For normal
impacts
(sin ~
0), e
contribution
of the
impacts one sees a growing
en; for increasingly oblique
force to the total
shear
dissipation, and grazing impacts approach e
I, as physical
intuition
(13)
of the
shown
is
in
=
jumping
Figure 3a
goes
for
to
~1
at
zero
=
the
0.15, 0.25,0.35.
One
=
=
suggests.
Some
authors
[12, IS, 20, 33,35]
use
a
viscous
F~
where
~y~
celeration
is
shear
a
is
a
linear
damping
function
constant
of the
=
force of the
form
(14)
-'f~~~,
physical
velocity, one
without
initial
friction
interpretation.
obtains
a
Because
constant
e~
here
>
0 for
the
all
de#'
FORCE
N°1
(Fig. 3b)
rolling case is never
go smoothly to one
e
(14) is not governed by the normal
Thus, in the limits of nearly normal
IN
reached.
On
the
does
tution
SCHEMES
the
grazing
for
not
SIMULATIONS
hence
does
nearly grazing
and
hand, the
(Fig. 4b).
impacts
Fn and
force
other
13
coefficient
of
total
This is due to the
vanish for grazing
not
force
impacts,
resti-
fact
that
impacts.
(14) yields unphysical
results.
discontinuity
The
(13)
in
be
can
Fs
Here,
~y~
considered
be
may
properties
collisional
for
to
with
(14) [II,13,17,19,43-45],
(IS)
rnirlll~/s~sl,IJLF«I) Sigrll~s).
=
to be
not
by combining (13)
avoided
a
technical
should
which
parameter
substantially
With
increasing
differ
from
those
of
have
force
value
a
high enough
#' for
vs.
(13). #~
approaches the
behaviour
is shown in Figure 3c.
~y~, force (IS)
(13). The same trend is observed for e(sin~) (Fig. 4c). Apparently, a small ~y~
changes mainly the properties of "moderately oblique" impacts, making them more elastic.
Considering that the form of e(sin~) is decisive for the dissipation of granular temperature in
extended
granular system, it seems very important that ~y~ is given a high enough value.
an
A simpler possibility is to use (13) instead of (IS), avoiding a
physical
parameter of unclear
force
this
of
force
interpretation.
presented so far do not account for tangential elasticity. Therefore, none
negative
contrary to the experimental
a
e~ in any part of the #~ vs. #' diagram,
data.
There is a further
disadvantage to them which is important in static or quasi-static
systems: a pile made of particles which interact through the force laws (13)-(15) is not stable.
Stability would require that finite shear forces act between particles also at ~~
0 in order to
withstand
gravitational force components in shear direction of the
contacts.
Tangential elasticity was first introduced by Cundall and Strack [8] and used by many others
[9, 14, 16, 33, 46] writing
All the
of them
force
schemes
shows
=
F~
where k~
took
some
is
place
that
tangential
since
the
min()ks(),
=
stiffness
time
( denotes
and
to, when
the
(lt)
It is
essentially
individual
half
the
values
period
of
ratio
of the
tangential
was
(16)
displacement in the tangential
first established,
e.
direction
/~ ~slt')dt"
determines
k~ /kn that
stiffnesses.
This
be
can
the
II?)
results
oblique
an
considering
of
understood
impact
that
k~
and
the
not
determines
a
[47],
oscillation
(fill
-i/2
~
t~
the
contact
=
sign((),
)~1Fn))
=
~
+
m~ii
mR~ II))
(18)
oscillation
of inertia). Thus
just like km determines a half period of normal
in II is the
moment
the phase of the tangential
oscillation
when the
determined by
at the
moment
contact
ceases, is
fi$ (this proportionality is valid strictly only for en I). For uniform
the ratio t~ /tn c~
spheres,
=
1
and
value
force
=
2 IS
mR~,
(19)
2/7. The
periods of tangential and normal
oscillation
equal when k~ /kn
are
~( additionally depends on ~(. In Figure 3d, we present ~fi~ vs. #' as obtained
2/7, one obtains a regime of
(16) for k~/kn
1, 2/7, 1IS. With k~/kn
the
=
of
negative
=
e~
for
small
to
=
intermediate
#',
caused
by
the
restoring
of
tangential
kinetic
actual
using
constant
energy
JOURNAL
14
by
tangential spring.
the
regime of sliding
a
higher #',
For
region
The
friction.
PHYSIQUE
DE
the
Coulomb
where
reversal
N°1
I
condition
of
(16),
governs
and
tangential velocity
initial
enters
one
observed
is
from
normal
impacts to ~~ m 1.6 (~ m 58°), in good agreement with the experimental
by Foerster et al. [25]. For k~/kn smaller or higher than 2/7, the extension of the
subtleties
arise.
For k~/kn
I, there is a slight rise in
(e~ < 0) region is unchanged, but new
behaviour
has been predicted theoretically by lilaw
~~ before it falls below the abscissa; this
observed in Foerster's
data. For k~ /kn
elastic spheres, but is not
I IS.
et al. [40] for perfectly
match the experimental
values quite well. The
behaviour
the
simulation
results
porresponding
much of a difference
which
of e(sin~) is shown in Figure 4d. Here, it does not
to make
seem
extends
results
=
=
k~/kn
of
value
chosen.
is
leads to very
realistic impact
behaviour
used by Walton and Braun
was
force and
patterned after Mindlin's
results for
normal
constant
varying
tangential force [39], assuming that in each time step, the normal force changes only by a
influence
force
that will not significantly
the tangential force.
It
introduces
small
amount
a
dependent k~, such that for each time step
Another
model
Their
[37].
that
scheme
is
Fs~fl+ks'l(~l'l'
where
refers
prime
a
respective
the
to
values
previous
the
in
~
if v~ in
initial
denotes
during impact
k~. In
by kj
stiffness
such
that
(20)
they
with
to
seem
Brilliantov
(thus,
=
the
surfaces
et al.
[36]
experimental
formulated
values
force
a
tangential force)
apparent
yield
local
the
if
is
stress
a
F~
and
[J~j
surface
saw-tooth
tangential
to
10~~
used in
(o "10~~
m
means
the
=
truncation
integer
initially equal
is
in the
normal
value
instantaneous
kn(I v) Ill
v/2). The Poisson
2/3. Figure 3e and Figure 4e
=1, 2/3, 1/3. The results look
the
direction.
tangential
ratio
bit
stiffness
v
vs.
quite
similar
is
better
for
k) /kn
the
-~1Fn
certain
of force
of z, (o
is
a
using
(16), though
2/3.
threshold.
length
of
normal
obtained
With
their
transmission
contacting
model, they
(o~
(o
value
occurs
materials.
most
assuming that the tangential
moment
by microscopic asperities on the
a
the
to
mediated
exceeds
the
evaluation
related
1/3 for
e(sin~)
those
=
in
to
set
inevitably
that
of Fn
k)
#~ and
to
and
to 0
force
of order
is
#~
show
m
derive
where
F]
change
for by using the
theory [39], the initial
match
which
The
accounted
elastic
kj/kn
kj /kn
opposite
stiffness.
direction.
its
reverses
is
Mindlin's
tangential
initial
the
~~
(~~~
l/3
F)
~1Fn +
S
whenever
direction
~j
i
if ~~ in
Here, k)
step and
~~~
ks
of F~
time
~i)
~~
~0
120)
(22)
connected
to
material
constants
expressed in terms
microscopic
Force (22) is a
parameters.
function in ( with period (o. Clearly, (o should be much
smaller
than (~, the final
displacement immediately before the contact
ends. (~ is in a range from
about 10~~
roughness,
and
(from nearly
simulations.
our
m
(changing (o
~1
is
stiffness
to nearly grazing) for the
impact angles and normal
properties of force (22) are shown m Figure 3f and Figure 4f for
10~~ m or10~~ m does not visibly alter the curve). One obtains
normal
The
to
of
SCHEMES
FORCE
N°1
SIMULATIONS
IN
15
3.0
%1'
~Q
°
O/,~
°
'/
II
/
/~i
,</
l-Q
~9-
,/
p/O
O/~'
/
~
,/
II
__---~~'
Q-Q
O
coop*
~°
1'
/
'
'/
-1.0
I-O
O-O
2.0
3.0
4.0
if'
combination
of
Hertz-Kuwabara-Kono
Fig. 5.
normal
force (9) with
linear
tangential
~i~ vs. ~i' for
elasticity (16) for v]~,
0.1 m Is (dotted line), 1.0 m/s (dashed line), 10.0 m Is (dot-dashed line). The
circles
denote the experimental
values for the impact of
spheres [25].
acetate
=
of
reversion
is
smaller
e
seems
sin ~
to
with
be
(16)
forces
overestimated
and
with
for
(20).
regard
small
but
the
hand, the
other
forces
the
to
angles,
impact
On the
(16)
of total
(20) especially
and
region
of this
extension
coefficient
restitution
the
in
region
o.7.
m
Finally, let
from
tangential velocity
initial
than
(2)
normal
discuss
us
applied.
are
force indirectly
changes
the
coefficient
The
through F~
collisional
in
behaviour
when
normal
force
different
laws
the
tangential
restitution
on
e~ and #~ vs. ~' depend
f(Fn) and through tn, the normal oscillation half period:
of
=
~(
~[
=
/~~
A~
(23)
F~ dt.
c~
force (2) by other
changes in the behaviour of
normal forces,
some
expected. We discuss those changes for the dynamic friction force
(13) and the elastic tangential force (16).
force 11 3) is not likely to be very
The
non-elastic
sensitive to the normal force chosen, because
either to rolling or slipping. Of course, choosing a non-linear
normal force,
it essentially leads
velocity-dependent, and the same must be true for A~. In practice, however, the
in becomes
velocity dependence in (7) is so weak that for a large range of ~[, the changes in #~ vs. #~ are
small to be represented
here.
too
tangential force (16), the situation is different. Here, as pointed out before,
For the
elastic
the impact
results
depend on the ratio of the half period of tangential
oscillation t~ and the
collision
normal
In order to get meaningful results, k~ must
be set such
that for
time tn.
particular impact velocity, t~/tn
I. According to (18) and (19), this leads to the
one
c m
replaces
Therefore, if one
oblique impacts
must
be
=
condition
~
k~
Figure 5,
In
combination
put k~
the
=
9.43
different
show
we
with
x
the
2/7
~
m~~~
(24)
tnc
#~ vs. #' for impacts obeying the
Hertz-Kuwabara-Kono
tangential force (16) with varying impact velocities ~]~,
lo~ N/m
impact
=
m
velocities
order
to
achieve
affect
and
tn
t~
/tn
=
therefore
I for ~)~,
the
=
l
collision
m
Is.
It is
behaviour.
force
)v]
clearly
=
On
(9)
v)).
seen
the
in
We
how
other
JOURNAL
16
PHYSIQUE
DE
I
N°1
.o
o-g
~~~-
~,,
'~,
0.8
w
~
/
',
--,__
~~-,
i
If
~,~j~_
'
_,/,
~'~'~
0.7
0.6
O-O
0A
0.2
0.6
sin
Fig.
Dependence
6.
forces
and
of the
parameters
coefficient
Figure
as
of
total
k~
constant
selected
is
smaller
times
have
constants
Since
with
than
restitution
coefficient
the
whereas
of km,
total
restitution
tangential dissipation, the normal force
especially for small impact angles.
curve,
dependent
for
true
by
non-elastic
Generic
The
and
and
feature
describe
First
consider
apart
and
all
grains
the
whole
that
a
away
grain
to
catch
wall
impact
on
a
a
up
wall)
hand,
limiting
damped
the
contact
the
In
smaller
m
the
with
large number
considerably
mutual
[35],
discussed
of binary
force
tat
other,
£~
~N
l
at
oscillators
the
case
same
rather
to
case
between
coefficient
to/tn
lead
to
e(sin ~)
as
in
the
and
e(sin ~)
of the
a
velocityThis
is
influenced
is
Figure
5.
to
equations (4)
as
of
I
IN,
the
N
/tn
the
to
»
wall,
a
until
wait
coefficient of
restitution
~f
N~~
I
/tn,
where
when
the
I, the
grains.
restitution
so
"
is
total
/~o
is
the
coefficient
restitution
impact, en.
a binary
hitting the wall leads
The
with
total
the
needed
time
for
suddenly stopped (e.g. by
hitting the wall leads to a
for
interact
grains.
to
latter
column's
The
column's
grains
distinct
hit
column
the
i=I
of it
the
and
<
time,
f
~
this
measure
i
on
front
in
grain
than
normal
according
that
is
Let
and
~~
~;
depends
e,~,
limiting
the
that
how
parameters
models
ao~is.
common
each
and
~
collisions
than
spring
velocity-dependent.
6 shows
and
the
behaviour
forces
also
Figure
forces
both
the
normal
both
Simulations
i=I
et al.
of
on
(2),
force
I.
m
c
function
becomes
alike.
same
normal
influence
non-linear
forces
along the
~o
the
achieve
column,
Luding
same
decreases
with increasing
stiffness.
We
tn that
connected
duration of contacts.
to the finite
column of N grains of same
and material, all a distance
size
a
~
As
the
duration
finite
are
velocity
from
linear
to
e(sin~)
soft-sphere
all
one-dimensional
a
having
move
by
have
effects
two
strong
the
using
the
direct
a
a
Soft-Sphere
shared
collisions
normal
of en
with
With
tangential
elastic
of
sin ~ for
results
is
has
restitution,
normal
Problems
common
(7),
now
of
velocity dependence
the
4.
so
coefficient
offset
impact
of
those
magnitude
order of
same
of
the
on
e
Figure 3d (obtained with force (2)
the tangential
as long as
spring
with force (9), the
numerical
value of k~ is nearly
that
Note
care.
that
be of the
to
1-O
5.
hand, the results of Figure 5 look very similar to
and varying k~ /kn) and also fit the
experimental
1000
0.8
0
wall
restitution
as
a
On
all
chain
thus
the
is
other
grains into
of coupled
coefficient
is
close
SCHEMES
FORCE
N°1
SIMULATIONS
IN
17
~
~lel
Fig.
Sketch
7.
illustrate
to
brake
the
failure
effect.
disperses.
detachment
Therefore, this regime was called the
regime [35]. This example illustrates that it is very important to choose realistic collision times,
realistic
stiffnesses, especially when dense systems are
considered.
Note also that in the
i-e"inelastic collapse"
observed by
McNamara
and Young in ID granular gases [48]
instantaneous
of vital
importance.
contacts
are
The second effect is related to
described by Schifer
times in oblique impact and was
contact
Consider
and Wolf [30].
other
with
relative
velocity ~'
two grains hitting each
)v]~,) under
of the
duration
impact angle ~ as sketched in Figure 7. We ask for the
contact, t~~,,,.
an
0), t~~,,~
Of course, for normal
impacts (sin ~
tn, but for an oblique impact this is not
necessarily so. One can distinguish two regimes: a hard-sphere regime, where the stiffness of
the
tn still holds, and a soft-sphere regime, where the stiffness
grains is so high that t,~,,~
unhinderedly. In this case, t~~,,,
each other
~/~~,
is so low that the grains essentially
cross
to
this
in
one
the
case,
column
=
=
=
=
=
where
in
~
Figure
we
7.
a
of the
Cartesian
chord
initial
between
coordinate
with
system
and
the
final
contact
z-axis
m
the
p$in.t
as
direction
drawn
of v]~~,
~[ ~[ on the initial velocity ~' in the two limiting
dependence of A~~
hard-sphere limit, the collision follows approximately the law of reflection, such
~~. In the soft-sphere limit, one has
the
=
the
In
A~~
length
is the
Introducing
ask for
now
regimes.
that
2Rcos~
=
c~
~~~
~
mn< l/~(t)
l/~ t~,i~.
dt
~
during the impact, ll~, is independent of ~~ m this regime, while t~~,,~ c~ 1/~',
1/~~. The braking efficiency goes down with increasing ~~; therefore, this regime
has been called the brake fail~tre regime. It is clear that in a given impact it depends on ~',
for the
between
the regimes
estimate
transition
km, and ~ which of the regimes applies. An
~fi
~/~'; solving for u~ gives the critical velocity for
provided by equating tn m
is
The
so
mean
that
~
A~~
force
c~
=
brake
failure
~~
~w
cos
=
~
fi.
(25)
~
0.999).
and ~
Figure 8 for the cellulose
87.4° (sin ~
acetate
parameters
Figure 8, the two regimes are clearly seen. The dotted
corresponds
curve
takes place at about
usual tn
between
the
I x 10~~ s; here, the
transition
to the
regimes
m/s,
good
predicted
by equation (25), ~w m 27 m/s.
in
agreement with the value
~w m 35
Increasing tn by a factor of 10 Ii-e-, decreasing km by a factor of100) leads to a decrease of
factor
of10 (dashed line in Fig. 8).
The corresponding ~w m 3.5 m/s is a speed
~~~ by a
failure
typically obtained in experiments or simulations.
The possible
of brake
consequences
for
simulated
granular flow have been discussed by Schifer and Wolf [30].
This is
In
the
illustrated
in
log-log plot
=
of
=
=
DE
JOURNAL
18
PHYSIQUE
I
N°1
o-i
0.001
0. I
I e+01
I
e+02
1e+03
v'
Fig.
t~
=
Braking
8.
I
x
10~~
s,
efficiency
dashed
line:
as
tn
I
of
function
a
=
x
10~~
initial
relative
velocity
for
~
87.4°.
=
Dotted
line:
s.
Conclusion
5.
specific and generic properties of force schemes used in the Molecular Dygranular materials.
Some of the forces
exhibit a quite
realistic
behaviour
namics
Which of the force
schemes is the most
normal
and oblique impacts,
others less so.
in
approbe
priate for simulations
answered
in general; this depends very much on the specific
cannot
geometry, particle density, mean flow velocity, etc. In any case, we strongly recommend testing
force schemes, and paying special
influence of the corresponding
attention
various
out
to the
free
the
flow
Complementary
discussion
for free impact
properties.
parameters
to
our
on
ii. e. applying to rapid granular flow), Sadd et al. [49] have conducted an investigation on the
laws for the case of dense
properties of several
contact
systems with long-lasting
contacts.
approach to granular dynamics is its versatility
While
the general merit of the soft-sphere
and ability to
simulate
also very dense and /or quasi-static
inaccessible
systems la regime
to
hard-sphere simulations), there are also some generic problems which we shortly described in
effect and brake failure are especially predominant
the preceding section.
Both the
detachment
stiffness kn of the grains is lower than in real systems (or, equivalently, when the
when the
therefore be taken in
normal
collision
should
simulations
time tn is too large). Special care
to
We
have
discussed
simulation
rule
out
the
of
presence
of any of the
two
effects.
Acknowledgments
We thank M. Louge for providing
fully acknowledge support by the
577/1-1.
us
with
the
Deutsche
original experimental
Forschungsgemeinschaft
data
from [25]. We
through
grant
grate-
No.
Wo
SCHEMES
FORCE
N°1
SIMULATIONS
IN
19
References
iii
[2]
[3]
Williams J-C-, Powder
Technol. 15 (1976) 245.
Knight J-B-, Jaeger H-M- and Nagel S-R-, Phys. Rev. Lett.
Savage S-B- and Sayed M., J. Fl~tid. Mech. 142 (1984) 391.
[4]
Schick
[5]
Baxter
A-A-,
Verveen
and
K-L-
Nat~tre
Behringer R-P-, Fagert
G-W-,
251
T.
(1974)
and
70
(1993)
3728.
599.
G-A-, Phys.
Johnson
Rev.
Lett.
62
(1989)
Eds
(Else-
50
(1994)
2825.
[6]
[7]
Tejchman
J.
and
K-L-,
Johnson
[8]
Cundall
P-A-
and
[9]
Walton
O-R-,
in:
Amsterdam,
vier,
[10]
[iii
[12]
[13]
[14]
[15]
[16]
ii?]
[I8j
[19]
[20]
Gudehus
Contact
Technol. 76 (1993) 201.
(Univ. Press, Cambridge, 1989).
O-D-L-, Gdotechniq~te 29 (1979) 47.
G.,
Powder
Mechanics
Strack
of
hlechanics
Granular
Media, J. T.
Jenkins
and
M.
Satake,
1983).
P.K. and
Werner B-T-,
Powder
Technol. 48 (1986) 239.
Thompson P-A- and Grest G-S-, Phys. Rev. Lett. 67 (1991) 1751.
Gallas J-A-C-,
Herrmann
H-J- and
Sokolowski S., Physica A 189 (1992) 437.
Ristow G-H-, J. Phys. I France 2 (1992) 649.
Technol. 71 (1992) 239.
Tsuji Y., Tanaka T. and Ishida T., Powder
Taguchi Y.-h., Phys. Rev. Lett. 69 (1992) 1367.
Zhang Y. and Campbell C-S-; J. Fl~tid Mech. 237 (1992) 541.
P6schel T., J. Phys. II France 3 (1993) 27.
Lee J. and
Herrmann
H-J-, J. Phys. A 26 (1993) 373.
llelin S., Phys. Rev. E 49 (1994) 2353.
Luding S., ClAment E., Blumen A., Rajchenbach J. and Duran J., Phys. Rev. °E
Haff
R1762.
[21]
Allen
M-P-
Tildesley D-J-, Computer
and
Simulation
of
Liquids (Clarendon Press, Oxford,
1987).
[22] Drake T-G- and Shreve R-L-, J. Rheol. 30 (1986) 981.
[23] Drake T-G-, J. Geophysical Research 95 (1990) 8681.
[24j Drake T-G-, J. Fl~tid Mech. 225 (1991)121.
[25] Foerster S-F-, Louge M-Y-, Chang H. and Allia K., Phys. Fluids 6 (1994) l108.
1<uwabara
G. and I<ono Il., Jap. J. Appl. Phys. 26 (1987) 1230.
[26]
Goldsmith
W., Impact (Edward Arnold Publ., London, 1960).
[27]
[28] Bridges F-G-, Hatzes A. and Lin D-N-C-. Nat~tre 309 (1984) 333.
[29j Sondergaard R., Chaney K. and Brennen C-E-, J. Appl. Mech. 57 (1990) 694.
[30] Schifer J. and Wolf D-E-, Phys. Rev. E 51 (1995) 6154.
H., J. fir die
[31j
Hertz
[32]
Landau
L-D.
and
reine
Lifschitz
angew.
u.
E-M-,
Math.
Lehrbuch
92
(1882)
der
Verlag, Berlin, 1989).
[33] Lee J., Phys. Rev. E 49 (1994) 281.
[34] Taguchi Y.-h., J. Phys. II France 2 (1992) 2103.
[35] Luding S., ClAment E. and Blumen A., Rajchenbach
(1994)
[36]
[37]
[38]
[39]
[40]
Brilliantov
J.
Duran
and
N-V-, Spahn F., Hertzsch J-M- and P6schel T.,
and Braun It-L-, J. Rheoi. 30 (1986) 949.
O-It-
Adams
M-J-
and
Briscoe
Mehta, Ed. (Springer,
Mindlin
Maw
Physik,
Vol.
VII
J., Phys.
(Akademie-
Rev.
E 50
4113.
Walton
A.
136.
Theoretischen
N.,
It-D-
and
Barber
B-J-,
New
Deresiewicz
J-R-
and
m:
Granular
Matter:
York, 1994).
H., J. Appl. Mech.
Fawcett
J-N-,
Wear
38
preprint
An
20 (1953)
(1976) 101.
(1995).
Interdisciplinary
327.
Approach,
JOURNAL
20
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
DE
PHYSIQUE
I
N°1
N., Barber J-It- and Fawcett J-N-, J. Lubr. Technol. 103 (1981) 74.
K., J. Mech. Phys. Solids 26 (1978) 139.
P0schel T., J. Phys. I France 4 (1994) 499.
P6schel T. and
Herrmann
H-J-, Europhys. Lett. 29 (1995) 123.
Melin S., Inn. J. Mod. Phys. C4 (1993) l103.
Itistow G-H- and
Herrmann
H-J-, Phys. Rev. E 50 (1994) Its.
Systems, E. Guazzelli and L. Oger, Eds. (Kluwer,
Walton O-It-, in:
Mobile
Particulate
Dordrecht, 1995).
McNamara
S. and Young W-R-, Phys. Fluids A 4 (1992) 496.
Sadd M-H-, Tai Q. and Shukla A., Inn. J.
Non-Linear
Mechanics
28 (1993) 251.
Maw
Walton