lecture 2 fundamentals of discrete element modeling anthony d. rosato granular science laboratory me...
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Lecture 2 Fundamentals of Discrete Element
Modeling
Anthony D. RosatoGranular Science Laboratory
ME DepartmentNew Jersey Institute of Technology
Newark, NJ, USA
Lecture 2: Fundamentals of Discrete Element Modeling
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Presentation OutlinePresentation Outline
Part 1: Background & Motivation
Part 2: Particle Collision Models
Part 3: Wave Propagation Ideas
Part 4: Elastic Impact of Spheres
Lecture 2: Fundamentals of Discrete Element Modeling
Granular Science Lab - NJIT
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Discrete Element Modeling is an outgrowth of molecular dynamics simulations used in computational statistical physics.
However, the discrete element method was independently developed by P. Cundall in the 1970’s.
B. J. Alder and T. E. Wainwright, “Statistical mechanical theory of transport properties,” Proceedings of the International Union of Pure and Applied Physics, Brussels, 1956.
W. T. Ashurst and W. G. Hoover, “Argon shear viscosity via a Lennard-Jones potential with equilibrium and nonequilibrium molecular dynamics,” Phys. Rev. Lett. 31, 206 (1973).
P. A. Cundall, Symposium of the International Society of Rock Mechanics,” Nancy, France (1971).P. A. Cundall, “Computer model for rock-mass behavior using interactive graphics for input and output of geometrical data,” U.S. Army Corps. Of Engineers, Technical Report No. MRD, 1974, p. 2074.
Lecture 2: Fundamentals of Discrete Element Modeling
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Approximate the collisional interactions between particles using idealized force models that dissipate energy.
Integrate system equations of motion Determine individual particle positions and velocities.
Compute relevant transport quantities, bulk properties and analyze evolving microstructure.
Basic Idea of DEM Modeling
Lecture 2: Fundamentals of Discrete Element Modeling
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Initial positions, orientations and velocities
Update particle link-list(find new or broken contacts)
Calculate the force and torque on each particle
Integrate the equation of motion to calculate the newpositions, velocities and orientation
Accumulate statistics to calculate transport properties
Time increment: t = t + dt
LOOP
“F = ma”
“T = I”
Basic Flow Chart
- for spheres -
Lecture 2: Fundamentals of Discrete Element Modeling
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Equations of Motion for the Particle - Integration
Integration time step t approximated from normal force model by dividing twice the time spent in unloading period during a particle collision into n steps
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et m K
n
tvtF
txPosition, velocity and force at time t
tω
tθ
tT
Angular position, velocity and Torque at time t
0 ,4
0 ,
0 ,2
oo21
211
2/121
tm
tt
ttm
ttt
ttt
Fvv
vxx
Fvv
0 ,4
0 ,
0 ,2
oo21
211
2/121
tm
tt
ttm
ttt
ttt
Tωω
ωθθ
Tωω
- for spheres -
Lecture 2: Fundamentals of Discrete Element Modeling
Granular Science Lab - NJIT
See Appendix L2-C for derivation
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M o d e lM o d e l
Seek qualitative / quantitative agreement with physical experiments.
Critical Goal
Once validation has been established, the simulations provides a means of carrying out an ‘experiment’ on the computer.
Lecture 2: Fundamentals of Discrete Element Modeling
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In contrast to the energy conservation of molecular systems-energy dissipation is a critical characteristic of granular systems, and consequently, it is necessary to employ realistic approximations to model energy loss in colliding particles.
For this purpose, there are essentially two basic approaches.
By considering particles to be infinitely stiff, “hard particle” models assume instantaneous, binary collisions governed by a collision operator, which is a function of particle properties (i.e., friction, normal and tangential restitution coefficients) and the pre- and postcollisional velocities and spins. Such an approach is appropriate in collision-dominated systems, where continuous and/or multiple contacts are not characteristic.
‘Hard’ Particle Models
‘Soft’ Particle Models
The interaction is a function of an allowed overlap between colliding particles that is intended to model the plastic deformation* at the contact.
*0. R. Walton, K. A. Hagen, and J. M. Cooper, “Modeling of inelastic frictional, contact forces in flowing granuIar assemblies,” 16th Znternational Congress of Theoretical and Applied Mechanics, Lyngby, Denmark, 1984.
Lecture 2: Fundamentals of Discrete Element Modeling
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Elementary Hard Particle Model
When particles collide, some of the kinetic energy is lost – that is, a portion of the initial kinetic energy goes into deforming the objects.
Thus, a ball that is dropped and hits the ground will not rebound to the same height from which is was released.
As a first approach, this energy loss is modeled through a
“coefficient of restitution”, denoted by e.
e = 1 No energy loss (perfectly elastic)
e = 0 Complete energy loss (plastic)
0 < e < 1 A portion of the incident energy is loss
Lecture 2: Fundamentals of Discrete Element Modeling
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The coefficient of restitution is related to the velocities of the two particles by the relation
Consider a particle A of mass mA that has a velocity vA ,and collides with a stationary particle B of mass mB.
What is the kinetic energy lost KE during the impact?
It can be shown that …
)1( )(2
22 emm
mmKE A
BA
BA
v
BA
ABevv
vv
A BVA
VB
Before collision
A BBV
AV
After collision
Lecture 2: Fundamentals of Discrete Element Modeling
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See Appendix L2-A
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A ball is dropped from a height h = 36 inches onto a smooth floor. Knowing that the height of the first bounce is h1 = 32”, determine the coefficient of restitution e and the expected height h2 after the second bounce and after n bounces.
e..
e 913913
vv
v
v
Smooth floor
ho
d1 d2
h1 h2
vft/sec 9.132
2
1 2 oo ghmmgh vv
Conservation of Energy
oo h
hem ghegh(
mmghm 12
12
12 )2
2
2
1v
9428.036
322 ee
It can be shown that "44.2842 ehh o and in general, after k bounces,
ok
k heh 2
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Three Parameter Hard Sphere Collision Model (Walton)
L. Labous, et al. (1997), “Measurement of collisional properties of spheres using high-speed video analysis,” PRE 56(6), 5717-5727.
S.F. Foerster, et al.(1994), “Measurements of the collision properties of small spheres,” Phys. Fluids 6 (3), 1108-1115.
O. R. Walton (1992), in Particulate Two Phase Flow (ed. M. C. Roco), pp. 884-907.
The model mimics the phenomenon explained by R. Mindlin* in his solution for the collision of two elastic spheres – where there is a partial retrieval of elastic energy stored as tangential deformation of the spheres in the contact region.
In Walton’s model, the collision is ‘instantaneous’, and the post-collisional velocities are determined by their pre-collisional values and 3 phenomenological material parameters, i.e.,
e Coefficient of normal restitution Coefficient of tangential restitution Coefficient of sliding friction
*R. D. Mindlin (1949), J. Appl. Mech 16, 249; R. D. Mindlin & H. Deresiewicz (1953), J. Appl. Mech. 21, 237
Comment: This model was motivated by the experiments of Maw Barber & Fawcett and by Savage and Lun’s ‘beta’ parameter in their kinetic theory for particles with rotational degrees of freedom and friction
Lecture 2: Fundamentals of Discrete Element Modeling
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sv cv
1m
2m
nvP
Consider 2 homogeneous spheres of masses 21 , mm
moments of inertia about their centers 21 , II and
diameters 21 , dd
21 , rr Locations of the particle centers at the instant of
collision with respect to an inertial reference frame
21
21 , ,vv Translational and rotational velocities of
the particles immediately before collision P
Impulse that sphere 2 exerts on sphere 1
21
21rr
rrn
Unit vector in the direction of the line of centers - from sphere 1 to sphere 2
sn vvcot Angle of incidence
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Apply conservation of linear and angular momentum to determine the velocities after collision (denoted by a ‘prime’)
222111 vvvv
mmP
222
211
1
1 22
d
I
d
IPn
(1)
(2)
Relative velocity of spheres at contact point (sliding velocity) immediately before collision
ndd
2
21
121 22
vvvc (3)
nncn
vv Normal component of relative sliding velocity immediately before collision
nnccncs vvvvv Tangential component of relative sliding
velocity immediately before collision(4)
Lecture 2: Fundamentals of Discrete Element Modeling
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ss vv t Tangent direction
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nen 2121 vvvv Definition of the normal restitution coefficient e
ss vv Definition of the tangential restitution coefficient
models the collisional energy loss due to the tangential compliance of the particle surface. It reduces the magnitude of the relative surface velocity after collision.
*Maw et al. showed that is a function of the angle of incidence , such that [-1, 1].
* N. Maw, J. Barber, and J. N. Fawcett, Wear 38, 101 (1976).
By using definition (5b) and the conservation of linear momentum (1), it can be shown that
the normal component of P
is given by nv
rn meP 1 where
2121 mmmmmr is the reduced mass.
It now remains to find the tangential component of the momentum change.
(5a)
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(5b)
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Walton’s assumption: Collision is either sliding or rolling throughout the contact
Sliding Contact tPP nt
o
o is a measure of the friction at the particle surface
(6)
It can be shown that for collisions that are almost head-on, i.e., is close to , (6) yields -values larger than 1. This is physically impossible since it means that there is an increase in the energy.
Therefore, an apriori known value of is assumed to be known, 1o
such that whenever the collision yields o the collision is in rolling contact.
The above defines a critical value of the incidence angle o above which (6) is no longer valid. In this situation, the is set to its limiting value o
Lecture 2: Fundamentals of Discrete Element Modeling
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In summary, the three model parameters which characterize the collisional energy losses, and which must be determined experimentally, are
e Normal coefficient of restitution
o Tangential restitution of coefficient
Coefficient of friction
After some work, the equations (1-6) can be combined to yield the following
(rolling) for ,
(sliding) for ,111
oo
o1
os
neKvv
dmd
IK diameter of spheres homogenousfor
5
242
(7)
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1
1
oRolling
Energy limit
Sliding (slope = 1)
s
neKvv
11 1
Graphical representation of the Walton 3-Parameter Model (7)
From a single collision experiment between two spheres, one can determine e and either or o.
The factor determining which of the latter two can be found is the ratio sn vv
Perhaps the first experiments on real spheres to determine the collision parameters were done by Foester et al. and reported inS.F. Foerster, et al.(1994), “Measurements of the collision properties of small spheres,” Phys. Fluids 6 (3), 1108-1115.
Lecture 2: Fundamentals of Discrete Element Modeling
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S. F. Foerester, M. Y. Louge, H. Chang, K. Allia, Physics of Fluids 6(3), 1108 – 1115, 1994.
Lecture 2: Fundamentals of Discrete Element Modeling
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Lecture 2: Fundamentals of Discrete Element Modeling
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Comprehensive Data can be found on the website of Prof. Michel Louge, Mechanical & Aerospace Engineering, Cornell University, Ithaca, NY, USA.
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The behavior of real colliding particles is a great deal more complex than the model just explained because of the interplay of translational and rotational motion. The behavior of a “superball” demonstrates this ….
The ball reverses spin upon rebounding from the bottom of the table top.
Laboratory experiments to measure particle collision properties
Lecture 2: Fundamentals of Discrete Element Modeling
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Results from experiments with 2.54 cm nylon spheres
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Soft Particle Models
Linear Spring – Dashpot Model for Two Interacting Spheres
x1
x2 KD
Fig. 1: Sphere interaction modeled as a spring and dashpot.
Consider a sphere of mass m1 and diameter d1 whose collisional
interaction with another sphere (mass m2, diameter d2) is
idealized by a dashpot and spring as depicted in Fig. 1. Define the penetration distance x as:
21212
21
1 21
22ddxx
dx
dxx
Hence the interpenetration velocity and acceleration is given by
21
21
xxx
xxx
(1)
(2)
The material presented in this section was taken from S. Luding, Models and Simulations of Granular
Materials, Ph.D. Dissertation, 1994.
Lecture 2: Fundamentals of Discrete Element Modeling
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For each sphere, the equation of motion is written, and the resulting two equations can be combined to obtain the governing differential equation for the penetration x,
0 r rx D / m x K / m x (3)
Dimensions m ~ FT2/LK = ~ F/LK/m ~ 1/T2
D ~ FT/LD/m ~ 1/T________________L = unit of lengthF = unit of forceT = unit of time
Equation (4) is solved for the penetration x in the usual fashion to yield,
(4)
21
21
mm
mmmr
is the reduced mass. where
teAteAtx tt sincos 21
2o 2
r r
K D,
m mwhere22
o and
The natural frequency is
22
22of
Thus, the collision time tc can be approximated as 22
o2
1
ftc
(5b) (5a)
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are applied to (5), we obtaino0v
tx 0
0
txAs an example, if initial conditions and
tetx t
sinvo
ttetx t
cossinvo
(6)
Contact between the spheres is broken when ttx 0
Hence in this case, the contact time is exactly 22
o
ct
Normal Coefficient of Restitution
/
021
21
00v
v-
vv
vve
x
txt cc
t
tt c
/
ov etx cwhere
Therefore, 22
o
e(7)
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Maximum penetration xmax occurs when dx/dt = 0. Differentiation of the first of equation (6) yields
omax arcsin
1arctan
1t from which
arcsinexpv
o
omaxx (8)
Case of low dissipation:
22 1 o r r o o K m D m
Therefore, 22 o
maxctt
Time for maximum penetration is approximately ½ the collision time
From (8), it follows that o
o
oo
omax
v
2exp
v
x (9)
To numerical example
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Numerical example of maximum penetration length for steel spheres
d = 1.5 mm
= 0.28
m = 1.38 x 10-5 kg
E = 2.06 x 1011 N/m2
kg 1069.02
5m
mr
02 20 xxx
redm
K2
0red
n
m
D
2
9.0
sec. 106.4 6
22o
ct
c
e
t
sec 108.6
sec 1029.2
5o
-14
arcsinexpv
o
omaxx For Vo = 1 m/s, %093.0max
d
x
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Nonlinear Spring – Dashpot Model for Two Interacting Spheres
rij
Particle i Particle j
rij
x
½ (di+dj)
ijij xxr
ij
ijij
r
rn
Consider two uniform spheres of diameters d, where the vector from the center of sphere i to j is
and the unit normal is
.
ijijji ddx rn 2
1The penetration distance is jiij vvv and the relative velocity of the spheres is
Note that when x < 0, the spheres are not in contact with each other.
In the derivation that follows, we consider only a normal force
model. ije kx nF 1
Elastic force: Damping force: ijijijd xD nnvF
the linear model is recovered When 0
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The equations of motion are:
221
111
xmxDxkx
xmxDxkx
(10)
Multiplication the first of these by m2 and the second by m1 followed by a subtraction of the resulting equations from each other yields desired model for the evolution of the penetration x, i.e.,
01 xxm
Dx
m
kx
rr (11)
21
21
mm
mmmr
is the reduced mass. where
By using definitions (5a), the equation can be written as
02 12 xxxx o (12)
The solution of equation (12) is the penetration x between two spheres, whose behavior is governed by an elastic and dissipative normal contact force models.
Lecture 2: Fundamentals of Discrete Element Modeling
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2o 2
r r
K D,
m m22
o and
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Equation (12) can be re-expressed in terms of the Hertzian* contact force between two elastic spheres of radii r1 and r2 with elastic moduli E1, E2 and Poisson’s ratios 1 and 2.
3/ 21 2 1 22 21 1 2 2 1 2
4
3 (1 ) (1 )
H
E E r rF x
E E r r
*S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd edition, pg. 412, McGraw Hill, 1970.
When the spheres have the same elastic properties (E, ) and diameter d=2r, the above relation reduces to 3 2
2
3(1 )
H
EF d x
Upon equating FH to the assumed form for the elastic interaction force
1F ne ijkx one finds
2, where
3(1 )
E
k E d E
Following S. Luding**, assume the forms
**S. Luding, “Models and Simulations of Granular Materials, Ph.D. Dissertation, 1994.
1k Ed 1D d and
Observe that when = ½, Hertzian contact is recovered.
Lecture 2: Fundamentals of Discrete Element Modeling
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2/3 xkF HH or
Details
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A substitution of the latter two forms into equation (11) yields the re-expressed model for the penetration x, i.e.,
0
r r
d x Ed xx x x
m d m d
(13)
Maximum Penetration Depth, Contact Time and Normal Restitution Coefficient
To determine the maximum overlap or depth of penetration of the two spheres xmax, assume that all of the incident kinetic energy is stored as elastic energy.
Hence, if vo is the incident relative velocity, then, 21 v2 r o elasm E
Eelas is approximated for the case of weak or no dissipation for which 0
x
Multiply (13) by x and integrate the resulting equation.
(14)
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222
1
2
10 x
dt
d
d
xdExm
dt
d
d
xxdxExxm rr
2 2
2max1
011
00
2
max
maxmax xdEdxxdEdtxx
d
xdExmdE
x
xx
relas
A substitution of (15) into (14) yields the maximum penetration,
(15)
)2(12)2(1
1
)2(1
max )(v2
1
o
r
dE
mx (16)
The contact time tc can be estimated from energy conservation (i.e., little or no dissipation)
xdd
xdEeFdxFxmd e
r
where,02
2
2/
0
)2/()2/()2/(1
1
)2/(1cos2v
2
2/12 d
dE
mt o
rc
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This latter expression is simplified using the beta function* to produce the result…
G
dE
mt o
rc
)2/()2/(1
1
)2/(1
v2
1 (17a)
244
2
1
21
G
(17b)
Reduction to Hertzian model when = ½: 5/1v~ oct(18)
*Abramowitz & Stegun, Handbook of Mathematical Functions, pg. 258, Dover, 1972.
Normal Coefficient of Restitution
The coefficient of restitution can be determined from the energy dissipated in a collision max xFE ddis
ood xdDxF vv 1 where
)22()24()2()1(
11 v~
o
rdis
dE
mdE (19)
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energy ofn dissipatio smallfor 2
111
oo
E
EEE dis
dis
After substituting Edis into the above expression, one obtains )2()2(v2
11~ o
For Hertzian contact, = ½ and = 0 corresponding to a constant damping force
From this, one can find the approximate energy loss in a collision
)2()2(ov ~ Loss
In this case, the restitution coefficient o
5/1o vas 0 v
2
11~
This is physically unrealistic behavior as it is known from experiments* and finite element calculations** that the restitution coefficient decreases as a function of the incident velocity.
1
ov
* W. Goldsmith, Impact, E. Arnold Publishers (1960).** O. R. Walton, “Numerical Simulation of Inelastic, Frictional Particle-Particle Interactions,” in Particulate Two-Phase Flows (Ed. M.C. Roco), Buterworth-Heinemann, Boston (1993).
xDxxDFd
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Lecture 2: Fundamentals of Discrete Element Modeling
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J.C. Mccrae & W. A. Gray, “Significance of the properties of materials in the packing of realspherical particles,” Brit. J. Appl. Phys. 12 [4], 164-171 (1964).
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A further problem with using linear damping and Hertzian contact is a discontinuous force-displacement curve.
dampingelas FFF
F
x
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Selection of a ‘Proper’ Value of
t
K
m
DAt
m
K
m
DAex
rmrrr
tmD
r2
1
2
2
22
1
2
2
1
)2
(
4exp
4exp
Critical Damping: KmDm
K
m
Drcrit
rr
2 4 2
2
(from O.R. Walton, “Force Models for Particle-Dynamics Simulations of Granular Materials”, in Mobile Particulate Systems, ed. E. Guazzell & L. Oger, NATO AS1 Series E: Applied Sciences, Vol. 287, Kluwer Academic Publishers, (1995), Chpt. 20, Pg. 367)
We want Deff to scale with the slope of the Hertzian spring force in order to maintain a fixed fraction of critical damping for each incremental step along the force-displacement curve. Recall the solution to the linear spring-dashpot model 0 KxxDxmr
Recall the general non-linear model 01 xxm
Dx
m
Kx
rr
Define an ‘effective damping coefficient as .effDDx
whose solution is given by
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For Hertzian contact, = ½ and = 1/4, corresponding to 4/1xxDxxDFd 2/3kxFelas and
.02/34/1 kxxDxxmr the model becomes
kmxD
kxdx
dmdxdFmmKD
r/
crit
elascrit
6
222
41
2/12/32/1
1/4 criteff DDxD
In this case, 1v2
11~ )2()2(
o
Thus, the model produces a constant restitution coefficient (in contrast to the physically unrealistic behavior that occurs with linear damping and Hertzian contact), which is acceptable for incident (relative) collision velocities that are less than a critical value (which depends on the particle material).
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Two-Dimensional Soft Particle ModelsInclude the effects of friction and particle rotation
Idealized contact between an elastic disk and a planar surface:
1. No sliding ( = )
2. Stress in the ‘normal’ direction is independent of the tangential force. (This is the usual assumption in most theoretical & numerical treatments of contacts for the local stress field.
m Mass of disk
Io = mR2/2 Moment of inertiay
x
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y
x KxKy
The model consists of two springs
Kx Normal spring stiffness
Ky Tangential spring stiffness
Assume that the spring contact forces are establish as soon as the disk encounters the planar surface.
x Tangential displacement of the disk’s center of gravity measured from its initial contact point
y Normal displacement of the disk’s center of gravity from its initial contact point
Orientation of the disk relative to when the contact is first established
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Equations of Motion
RxRKRFTRxKFyKF xxxxyy
(3)
(2) 1
(1) 1
oo
θ
RxKI
R
I
N
yKmm
Fy
RxKmm
Fx
x
yy
xx
Upon adding equations (1) and (3), we obtain
Rx
I
mR
m
KRx x
o
21 (4)
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The objective is to design the system so that the spring in the normal direction will return to its equilibrium position at the same time as the tangential spring. In order to do this, the period of oscillation for the simple harmonic motion described by equations (2) and (4) must be the same. Hence,
o
21
I
mRKK xy (5)
Tuning the Springs
xy
xy
KKmRI
KKmRI
72 52 :Spheres
3 2 :Disks
2o
2o
Mindlin’s theory for real elastic materials (R.D. Mindlin, J. Appl. Mech. 16, 259 (1949)) indicates that
13
2
y
x
K
K
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Implications
• For most real contacts, the frequency of rotational motion is higher than that of the normal motion
• If the contacting disk or spheres is initially in rotation only slightly, it could experience a reversal in the direction of rotation during the time it is in contact with the surface.
• (cf. Walton’s 3-parameter hard sphere model)
(3)
(2) 1
(1) 1
oo
θ
RxKI
R
I
N
yKmm
Fy
RxKmm
Fx
x
yy
xx
The solution of the system will be discussed in the next slides.
For x < 0:
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0
0
0
0
00
0
o2
o
y
x
IKRIRK
mK
mRKmK
y
x
xx
y
xx
:0When y
:0When y
0
0
o
I
gym
xm
Superball Motion
How to set the parameters so that the disk jumps back and forth without deviation ?
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Superball Motion Continued
Find the initial conditions so that the direction of the surface velocity is reversed after the collision
oo 1
RvRvVV
V
Vxxss
s
s
Conservation of momentum oxxx vvmdtF
xFjj xx FFRR
and
Conservation of angular momentum
ooo kk IIdtRFx o
o R
IdtFx
By equating the above two relations, one obtains oo
o
R
Ivvm xx
Spin reversal also requires that o , ,oo
yyxx vvvv
A substitution into * gives the desired condition
*
mR
IVx o
o
o
Lecture 2: Fundamentals of Discrete Element Modeling
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Lecture 2: Fundamentals of Discrete Element Modeling
Implementation of Superball Model
Superball.Exe
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Copyrighted application by Y. Chung
52
Walton and Braun Soft Particle ModelsWalton, O. R., Numerical simulation of inelastic, frictional particle-particle interactions, in M. C. Roco, ed., Particulate Two-Phase Flow, Butterworths, Boston, 1992, pp. 884-911.
fc
d
b
a
K1
K2
Fn
Normal Interaction
Model produces a constant restitution coefficient
1 2e K K
Linear Loading – Unloading ModelF1 = K1F2 = K2
Initial Loading: a b with slope K1
Unloading initiated at b, follows b c with slope K2
Reloading from c follows path c b d
Subsequent unloading from d follows path d f c a
K1 estimate
Lecture 2: Fundamentals of Discrete Element Modeling
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See Appendix L2-B
54
Tangential Interaction
O. R. Walton, R. L. Braun, J. Rheol. 30(5), 949-980, 1986.
KT = Tangential stiffness of a contact, which decreases with tangential displacement. Full sliding occurs when KT = 0
* 0 initially when relative tangential slip reverses direction
TT
T = total tangential forceKo = initial tangential stiffness ~ K1
= coefficient of sliding friction
decreasingfor 1
increasingfor 1
3/1
*
*
3/1
*
*
TTN
TTK
TTN
TTK
K
o
o
T
At each new time step, compute TnewsKTT Toldnew
s = Relative surface displacement between contacting particles
Full sliding of the contact occurs at the friction limit when KT becomes zero.
Based on. R. D. Mindlin, H. Deresiewicz, “Elastic spheres in contacts under varying oblique forces,” J. Appl. Mech. 20, 327 (1953).
Lecture 2: Fundamentals of Discrete Element Modeling
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Notes taken from O. R. Walton, “Force Models for Particle-Dynamics Simulations of Granular Materials,” in Mobile Particulate Systems (eds. E. Guazzelli & L. Oger), NATO ASI Series (Series E: Applied Sciences), Vol. 287, pp. 370 (1994).
Lecture 2: Fundamentals of Discrete Element Modeling
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Consider a 1-D “Chain of Spheres”
Spheres of mass m and diameter d are held in contact by externally applied boundary load. Assume that there is a small disturbance so that we can model contacts as linear springs of stiffness K.
1iu iu 1iu
Let x be the equilibrium spacing between the point masses, s.t. x = dLet ui-1, ui, ui+1 be displacements from equilibrium of 3 adjacent masses.
Fi Force acting on ith particle
ii uuK 1 1 ii uuK
11 iiiii uuKuuKF
Newton’s 2nd Law:
2112
112
2x
uuuxKuuuKum iii
iiii
Wave speed mKdmKxc
Lecture 2: Fundamentals of Discrete Element Modeling
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Wave Speed for Rigid Particles Not in Contact
Systems which are sufficiently energetic so that time between collisions >> tc Then, if this time scale separation is large enough, system can be modeled as rigid spheres which experience instantaneous, rigid-body collisions.
“Information” is propagated through the motion of particles.
s
doV
Equally spaced chain of spheres
Particle strikes one end of chain at velocity Vo.
velocity V0.
Lecture 2: Fundamentals of Discrete Element Modeling
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Suppose that collisions are perfectly elastic, i.e., e = 1. Then kinetic energy is preserved, and hence, at any instant, there is only one particle moving at velocity Vo.
collision after 2 particle ofVelocity 0 2 particle of velocity Initial
collisionafter 1 particle ofVelocity 1 particle of velocity Initial
2o2
1o1
VV
VV
VVVVVmmVmV 1o121o2o1
12o1
o1
21 1 VVVV
VVe
o121 ,0 VVV
travels that waveDistance sd
s)(d V
sΔ distance the travel tofor wave Time
o
fraction void theis Δ
Δ where,
Δvoid
void
oo s
sd f
f
VV
s
sd c
csf 0), that meaning( ,0 If void
Lecture 2: Fundamentals of Discrete Element Modeling
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Elastic Particle – Finite Contact Time
Account for finite contact time of elastic particles in estimating wave speed.Consider a 1-D chain of elastic spheres equally spaced.
What is the response of the chain to a single particle hitting chain at one end?
Factors which contribute to wave velocity in non-contacting granular assemblies:
Velocity of the particles in the system.Duration of contacts in collisions.
In order to approximate effect of finite contact time, assume that during contact period, each particle has an average velocity of ½ of the incident colliding particles velocity.
2 mecontact ti during traveledDistance oo Vc
considered are collisions isolated with casesonly that so that Assume cs
ooo contactsbetween flight of Time
VV
s
V
s ccg
oy at velocit contactsbetween moves particle a Distance Vs c
Lecture 2: Fundamentals of Discrete Element Modeling
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ccds in time travelswave'' Distance
2s
2-s Speedn Propagatio
o
o
c
oo
cVcgc
VsdVsdVsdc
2s o
o
cV
Vsdc
Compare the speed to the model for which the contact time was not included, i.e.,
oΔ
Vs
sd c
Observe that when the contact time is included in the rough model, the wave speed is slower because of the “Voc/2” term in the denominator.
Lecture 2: Fundamentals of Discrete Element Modeling
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Material taken from K. Johnson, Contact Mechanics, Chapter 11, Cambridge University Press (1985).
Lecture 2: Fundamentals of Discrete Element Modeling
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1G
2G
1zV
2zV
11 yx RV
22 yx RV
1y
2y
The classical Hertz theory for the impact of frictionless, elastic spheres assumes (1) that the deformation is restricted to the vicinity of the contact area and is given by a static theory and (2) elastic wave motion is ignored.
Collinear Impact of Spheres spheres of Masses , 21 mm
O
Collision occurs at point O
collision of at time centers of line along spheres of Velocities , 21 zz VV
Assume colinear impact so that 02121 xxyy VV
zDisplacement amount by which the sphere centers approach each other due to elastic deformation.
12 zzz VV
dt
d
tP Force acting between the spheres during their contact.
dtzdV
mdt
zdVmtP 2
21
1
(a)
(b)
Lecture 2: Fundamentals of Discrete Element Modeling
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Multiply the 1st by m2 and the second by m1 and then subtract the resulting equations.
1221
21
21
2121
1122
zz VVdt
dmmP
mm
mm-
dtzdV
mmtPm
dtzdV
mmtPm
dt
d z
Therefore, the equation relating P to the evolution of z is given by z
dt
dP
mm
mm
2
2
11
21
Next, assume that P and are related through Hertz’s static, elastic contact law.
21*
2
22
1
21
*
3/1
2**
2
111 and
111 where
16
9
RRR
EEEER
P
(c)
Lecture 2: Fundamentals of Discrete Element Modeling
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Solve the latter equation for P in terms of
2/32/3**2/3221
212
21
21
21 )()(3
4
)1()1(
3
4zz KER
EE
EE
RR
RRP
and substitute the result into equation (c) to obtain 2/32
2)( z
zr K
dt
dm
where
21
21
mm
mmmr
To solve this equation, multiply by sides by z and integrate. This yields,
elocityapproach v relative theis where5
2
2
1012
2/52
2
tzzzz
r
zz VVV
m
K
dt
dV
At the maximum compression, * and 0 zzz dtd and hence,
(d)
(e)5/2
**
2*
16
15
ER
Vm zrz
5/22*
4
5
K
Vm zrz or
Lecture 2: Fundamentals of Discrete Element Modeling
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Time versus Compression Curve
5
2
2
1 2/32
2z
r
zz m
K
dt
dV
Multiply this equation by *z and simplify to yield:
)()(1 *
2/12/5*** zzzzz
z
z
z dV
dt
d
)()(1 *
2/12/5**
zzzzz
z dV
t
The above was numerically integrated by Deresiewicz+ to produce a t vs z curve, which, by using P(t) = Kz
3/2, yields a force versus time curve.
+ H. Deresiewicz, “A note on Hertz impact”, Acta Mechanica 6, 110 ( 1968)
(f)
Lecture 2: Fundamentals of Discrete Element Modeling
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Contact time Tc in terms of the relative incident velocity Vz
ncompressio maximum of at time Load
occursn compressio maximumat which Time *
*
P
t
Equation (d):
2/3
**2/3
2/3**
)(
)(
z
z
z
z
P
P
KP
KP*tt
1 2
*
The total contact time is 2t* because the spheres are perfectly elastic and frictionless. Hence there is no loss of energy (i.e., reversible).
94.2)()(122*1
0
*2/12/5*
**
z
zzzzz
z
zc V
dV
tT
(g)
Lecture 2: Fundamentals of Discrete Element Modeling
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By substituting equation (e), rewritten as
51
2**
51*
)(256
225
z
rzz VER
mV
into (g), we obtain the result that
51
51
2** )(87.2
z
rc V
ER
mT
The question arises under what conditions is the use of Hertz’s theory in the above analysis valid? This is the topic of discussion in Chapter 11, Sections 1-3 of Johnson’s book.
(h)
Lecture 2: Fundamentals of Discrete Element Modeling
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Section 11.1, Johnson’s book
When is the quasi-static analysis for the contact model valid?
*If the contact time is much larger than the time required for the wave to traverse a rod many times.
A. E. Love, Treatise on the Mathematical Theory of Elasticity, 4 th ed., Cambridge University Press (1952).
Love suggested that the same criterion * for rods is also true for spheres. Let R be the radius of the sphere and let
*o
44
4 diameters two travel to wavesallongitudinfor required time
E
RR/C
R t sphc
(i)
Recall equation (h) for the contact time. 51
51
2** )(87.2
z
rc V
ER
mT
Lecture 2: Fundamentals of Discrete Element Modeling
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514o
c )(
85.5
cV
RT
z
If the two colliding spheres are the same size (R1 = R2 = R), then R* = R/2. Substitute this into
(h), as well as the mass in terms of the density , and 2o
* cE to yield
c-sphc tTLet Upon substituting Tc into this definition, one obtains
51
o
~
c
Vz
1According to Love’s criterion,
in order to the quasi-static analysis to hold for colliding spheres
Lecture 2: Fundamentals of Discrete Element Modeling
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Lecture 2: Fundamentals of Discrete Element Modeling
Granular Science Lab - NJIT
End of Lecture 2
71
Lecture 2: Fundamentals of Discrete Element Modeling
Granular Science Lab - NJIT
Appendix L2-A: Coefficient of Restitution
Am v1( )A Am v Pdt
1( ) ( ) ( )A A Am v Pdt m v 11111111111111
1( )B Bm v Pdt Bm v
1( ) ( ) ( )B B Bm v Pdt m v 11111111111111
Deformation Phase
2( )B Bm vRdtBm v
2( )A Am vAm v Rdt
2( ) ( ) ( )B B Bm v Rdt m v 11111111111111
2( ) ( ) ( )A A Am v Rdt m v 11111111111111
Restitution Phase
2
1
( )
( )A
A
Rdt v v
v vPdt
2
1
( )
( )B
B
Rdt v v
v vPdt
2 2
1 1
( ) ( )
( ) ( )B A
A B
v ve
v v
72
Lecture 2: Fundamentals of Discrete Element Modeling
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Appendix L2-B: Derivation of Normal Restitution Coefficient for Walton Soft-Sphere Model
Consider the impact of two spheres approaching along the line joining the centers of the spheres. The normal interaction force F1 begins to act and change the velocities of the spheres after their initial contact. Suppose v1 and v2 are the value of these velocities, m1 and m2 sphere masses.
121 21
1 Fdt
dvmF
dt
dvm
Relative velocity of approach 21 vv
overlap nonzero from reloadingor g, Unloadin),(
loading ,
22
11
oKF
KF
o - overlap value where the unloading curve goes to zero along the slope K2.
(2)
(1)
(3)
21
211 mm
mmK
Combine (1) – (3) (4)
21
21Let mm
mm
73
Lecture 2: Fundamentals of Discrete Element Modeling
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Appendix L2-B: Derivation of Normal Restitution Coefficient for Walton Soft-Sphere Model
Multiply both sides of (4) by
dK
d 2
1 12 (5)
Let be the velocity of approach (immediately before collision when = 0)
av
Let be the maximum velocity – when . mv 0
Integrate (5) (6)
Now consider the unloading period when the spheres are moving apart. The equations of motion are
2221 21 Fdt
dvmF
dt
dvm (7)
Relative velocity of approach 21 vv
Relative approach velocity )( and o21 22 KFvv
1K
v ma
74
Lecture 2: Fundamentals of Discrete Element Modeling
Granular Science Lab - NJIT
)( )( 2
1oo
2 2
dK
d
Now follow the same procedure used to derive (6)
Appendix L2-B: Derivation of Normal Restitution Coefficient for Walton Soft-Sphere Model
)( ])[( 2
)( 2
1o
0
α0
2 222
mso
v Kvd
Kd
m
s
A substitution of (6) and (8) into the definition of the restitution coefficient yields,
(8)
1
2 )( o
K
K
v
ve
m
m
a
s
Since
(9)
,2
1
12
22oK
K
KF
KF
m
m
2
1 K
Ke (10)
75
Lecture 2: Fundamentals of Discrete Element Modeling
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Appendix L2-C: Derivation of the Integration Time Step for Walton’s Soft-Sphere Model
The time step is found from the unloading period.
2o
2o
α0
2 )()( ])[( 2
)( 2
1 222
moK
dt
dαd
Kzd
m
22o
2o
o
2 2
)()(
)( Timen Restitutio
oK
μd
K
μt
m
mR
For a monodisperse system, m1 = m2 = m, so that = m/2. Use this result and (10) to obtain
12 22
22 K
me
K
mtR
We estimate the time step by simply doubling the
restitution period, and then divide this result by ‘n’.
12
K
m
n
et