C. Wassgren, Purdue University
DEM Modeling: Lecture 02
Introduction to the Hard-Particle Algorithm
Collision Model
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Introduction
• Hard particle simulations are deterministic - given the
same initial conditions the final results will be the same
– the state of every particle in the system and all particle
interactions are determined using physical laws
• The term “hard-particle” refers to the fact that particles
are considered to be perfectly rigid
– ⇒ particles do not deform during a collision
– ⇒ collisions occur instantaneously
– ⇒ only binary (two particle) collisions occur
• Used most often for modeling dilute, energetic granular
flows
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Collision Dynamics
ω1
m1, r1, I1
m2, r2, I2
J
ω2
n̂
Assuming spheres
⇒ I = 2/5mr2
x1
y
x2
x
z
J=
∫
tcollision
Fdt
≡
≡
≡
≡
≡
≡
position
angular velocity
mass
radius
moment of inertia
impulse of momentum
acting on particle 1 due
to collision with particle 2
n̂ ≡ unit vector pointing from the
center of particle 1 to the center
of particle 2
x
ω
m
r
I
J
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Collision Dynamics…
• Unit normal vector for the contact (assuming spheres)
nˆ =
x 2 − x1
x 2 − x1
• Velocity of particle 2 relative to particle 1 at the point of
contact
xɺ 1,C = xɺ 1 + ω1 × r1nˆ
xɺ 2,C = xɺ 2 + ω 2 × ( −r2 nˆ )
∆xɺ C = xɺ 2,C − xɺ 1,C = ( xɺ 2 − xɺ 1 ) − ( ω 2 × r2 nˆ + ω1 × r1nˆ )
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Collision Dynamics…
• Unit tangential vector for the contact
– perpendicular to unit normal and points in the direction of the
relative contact velocity just prior to the collision
∆xɺ C− = ( ∆xɺ C− ⋅ nˆ ) nˆ + ( ∆xɺ C− ⋅ sˆ ) sˆ
sˆ =
∆xɺ C− − ( ∆xɺ C− ⋅ nˆ ) nˆ
∆xɺ C− − ( ∆xɺ C− ⋅ nˆ ) nˆ
• Normal and tangential components of the impact velocity
nɺ = ∆xɺ C ⋅ nˆ
sɺ = ∆xɺ C ⋅ sˆ
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Collision Dynamics…
• Conservation of linear momentum for each particle
m1 ( xɺ − xɺ
)=J
( xɺ − xɺ ) = −J
m2
+
1
−
1
+
2
−
2
xɺ 1+ = xɺ 1− +
J
m1
xɺ +2 = xɺ 2− −
J
m2
⇒
• Conservation of angular momentum for each particle
I1 ( ω − ω
I2
) = r nˆ × J
( ω − ω ) = ( −r nˆ ) × ( −J )
+
1
−
1
+
2
−
2
1
2
ω1+ = ω1− +
⇒
r1
( nˆ × J )
I1
r2
ω = ω + ( nˆ × J )
I2
+
2
−
2
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Collision Dynamics…
• Consider the change in the relative contact speed after and before
the collision
∆xɺ C+ − ∆xɺ C− = ( xɺ 2+ − xɺ 2− ) − ( xɺ 1+ − xɺ 1− ) − r2 ( ω 2+ − ω 2− ) × nˆ − r1 ( ω1+ − ω1− ) × nˆ
r12
J
J r22
=−
−
− ( nˆ × J ) × nˆ − ( nˆ × J ) × nˆ
m2 m1 I 2
I1
 1
 r12 r22 
1 
= − +
 J −  +  ( nˆ × J ) × nˆ 
 m1 m2 
 I1 I 2 
• Make use of a vector identity
nˆ ⋅ nˆ ) J − ( J ⋅ nˆ ) nˆ = J − ( J ⋅ nˆ ) nˆ
( nˆ × J ) × nˆ = (
=1
• Also use an effective mass
1
m
1
=
′
m1
+ 1
m2
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Collision Dynamics…
• Simplify
 r12 r22 
1
∆xɺ − ∆xɺ = − J −  +   J − ( J ⋅ nˆ ) nˆ 
m′
 I1 I 2 
 1 r12 r22 
 r12 r22 
= −  + +  J +  +  ( J ⋅ nˆ ) nˆ
 m′ I1 I 2 
 I1 I 2 
+
C
−
C
• Write the normal and tangential components of the
change in the relative impact speed
2
2
2
2




r
r
r
r
1
1
+
−
1
2
1
2
ˆ
ˆ
ˆ
ɺ
ɺ
∆
x
−
∆
x
⋅
n
=
−
+
+
J
⋅
n
+
+
J
⋅
n
=
−
) 
)
( J ⋅ nˆ )
( C C)

(
(
m′
 m′ I1 I 2 
 I1 I 2 
2
2


r
r
1
+
−
1
2
( ∆xɺ C − ∆xɺ C ) ⋅ sˆ = −  m′ + I + I  ( J ⋅ sˆ )
1
2 

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Collision Dynamics…
• Now consider the normal component of the impulse
J ⋅ nˆ = −m′ ( ∆xɺ C+ − ∆xɺ C− ) ⋅ nˆ = −m′ ( ∆xɺ C+ ⋅ nˆ ) − ( ∆xɺ C− ⋅ nˆ ) 
• Make use of the definition of the normal coefficient of
restitution, εN
 ∆xɺ C+ ⋅ nˆ 
ε N ≡ −  −  ⇒ ( ∆xɺ C+ ⋅ nˆ ) = −ε N ( ∆xɺ C− ⋅ nˆ )
 ∆xɺ C ⋅ nˆ 
(Note: 0 ≤ εN ≤ 1 with εN = 0 being completely inelastic and
εN = 1 being perfectly elastic.
J ⋅ nˆ = m′ (1 + ε N ) ( ∆xɺ C− ⋅ nˆ )
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Collision Dynamics…
• Consider the tangential component of the impulse
 1 r12 r22 
J ⋅ sˆ = −  + + 
 m′ I1 I 2 
−1
−1
 1 r12 r22 
+
−
( ∆xɺ C − ∆xɺ C ) ⋅ sˆ = −  m′ + I + I  ( ∆xɺ C+ ⋅ sˆ ) − ( ∆xɺ C− ⋅ sˆ )

1
2 
• Make use of the definition of the tangential coefficient of
restitution, εS
 ∆xɺ C+ ⋅ sˆ 
ε S ≡ −  −  ⇒ ( ∆xɺ C+ ⋅ sˆ ) = −ε S ( ∆xɺ C− ⋅ sˆ )
 ∆xɺ C ⋅ sˆ 
(Note: -1 ≤ εS ≤ 1 with εS = -1 being frictionless, εS = 0 resulting
in no-slip and εS = 1 being perfectly elastic.
 1 r12 r22 
J ⋅ sˆ =  + + 
 m′ I
I 2 
1

−1
(1 + ε S ) ( ∆xɺ C− ⋅ sˆ )
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Collision Dynamics…
• Re-write the collision impulse
J = ( J ⋅ nˆ ) nˆ + ( J ⋅ sˆ ) sˆ
 1 r
r 
−
′
ˆ
ˆ
ɺ
= m (1 + ε N ) ( ∆xC ⋅ n ) n +  + + 
 m′ I1 I 2 
2
1
2
2
−1
(1 + ε S ) ( ∆xɺ C− ⋅ sˆ ) sˆ
• Note that since we’re assuming spheres, I = 2/5mr2
J = m′ (1 + ε N ) ( ∆xɺ C− ⋅ nˆ ) nˆ + 72 m′ (1 + ε S ) ( ∆xɺ C− ⋅ sˆ ) sˆ
C. Wassgren, Purdue University
normal coefficient of restitution, εN
Collision Model Parameters
Goldsmith, W., 1960, Impact: The Theory and Physical Behaviour of Colliding Solids, Dover.
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Contact Model Parameters…
Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press.
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Contact Model Parameters…
normal and oblique
impacts give similar
normal coefficients
of restitution
Gorham, D.A. and Kharaz, A.H., 2000, “The measurement of particle rebound characteristics,”
Powder Technology, Vol. 112, pp. 193 – 202.
C. Wassgren, Purdue University
Contact Model Parameters…
Kruggel-Emden, H., Simsek, E., Rickelt, S., Wirtz, S., and Scherer, V., 2007, “Review and
extension of normal force models,” Powder Technology, Vol. 171, pp. 157 – 173.
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Contact Model Parameters…
Walton, O.R., 1993, “Numerical
simulation of inelastic, frictional
particle-particle interactions,” in
Particulate Two-Phase Flow,
Chap. 25, M.C.Roco ed.,
Butterworth-Heinemann.
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Collision Model Parameters…
• (1 – εN) ~ Vimpact1/5 as εN → 1
in the visco-elastic regime
– Kuwabara and Kono (1987)
– Vimpact < ~1 m/s
•
εN ~ Vimpact-1/4 in the elastoplastic regime
– Johnson (1985)
– Vimpact > ~1 m/s
Labous, L., Rosato, A.D., and Dave, R.N., 1997, “Measurements of
collisional properties of spheres using high-speed video analysis,”
Physical Review E, Vol. 56, pp. 5717-5725.
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Elastic-Perfectly Plastic Collisions
• Plastic yield starts to occur at
2
Vyield
– e.g. two 3 mm diameter stainless steel (grade
316) spheres (ρ = 8030 kg/m3, E = 193 GPa, n =
0.35, Y = 310 MPa) ⇒ Vyield ≈ 4 mm/s
– plastic yield is common
R ′3Y 5
≈ 107
m′E ′4
• Assuming fully plastic loading and elastic unloading:
3 Y  2 
≈   
8  E′  

1
εN
1
2
2
m′Vimpact
3
YR ′




− 18
Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press.
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Stick-Slip Contacts
• In a real contact, the contact area has regions that “stick”
toward the center of the contact area and regions that
“slip” at the edges
pressure
stick zone
slip occurs if:
τ > µp
position
contact area
slip zone
• For sufficiently small pressures, the entire contact area
may slip
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Stick-Slip Contacts…
• Assume that the contact has two tangential
coefficients of restitution
– a constant tangential coefficient of restitution in the
stick-slip (SS) regime given by εSSS
– a variable tangential coefficient of restitution in the
pure slip (PS) regime denoted by εSPS
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Stick-Slip Contacts…
• In the stick-slip regime:
( ∆xɺ ⋅ sˆ ) = −ε ( ∆xɺ
+
c
S
−
c
∆xɺ ⋅ sˆ )
∆xɺ ⋅ sˆ )
(
(
⋅ sˆ ) ⇒
= −ε
⇒ψ
( ∆xɺ ⋅ nˆ )
( ∆xɺ ⋅ nˆ )
+
c
−
c
−
c
S
−
c
∴ψ 2SS = −ε SSSψ 1
( J ⋅ sˆ )
SS
= 72 m′ (1 + ε SSS )( ∆xɺ C− ⋅ sˆ )
2
= −ε Sψ 1
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Stick-Slip Contacts…
• In the pure-slip regime:
( J ⋅ sˆ )
= 72 m′ (1 + ε SPS )( ∆xɺ C− ⋅ sˆ ) = sign ( ∆xɺ c− ⋅ sˆ ) µ J ⋅ nˆ
PS
– the magnitude of the tangential momentum impulse is
limited by sliding friction
( J ⋅ sˆ )
PS
≤ ( J ⋅ sˆ )
µ ≤ m′
2
7
(
SS
1 + ε SSS
(
)
(
)( ∆xɺ ⋅ sˆ )
⇒ sign ∆xɺ c− ⋅ sˆ µ J ⋅ nˆ ≤ 72 m′ 1 + ε SSS
)
−
C
∆xɺ C− ⋅ sˆ 2
∆xɺ C− ⋅ sˆ
SS
= 7 m′ 1 + ε S
ˆ
J ⋅n
m′ (1 + ε N ) ∆xɺ C− ⋅ nˆ
(
)
∆xɺ C− ⋅ sˆ
7  1+ ε N
slipping occurs if:
= ψ1 ≥ µ 
−
2  1 + ε SSS
∆xɺ C ⋅ nˆ
(



)
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Stick-Slip Contacts…
• In the pure-slip regime…
( J ⋅ sˆ )
2
7
PS
= 72 m′ (1 + ε SPS )( ∆xɺ C− ⋅ sˆ ) = sign ( ∆xɺ c− ⋅ sˆ ) µ J ⋅ nˆ
m′ (1 + ε SPS )( ∆xɺ C− ⋅ sˆ ) = sign ( ∆xɺ c− ⋅ sˆ ) µ m′ (1 + ε N ) ∆xɺ C− ⋅ nˆ
ε
ψ 2PS
=
PS
S
∆xɺ C− ⋅ nˆ
7
= µ (1 + ε N )
−1
−
2
∆xɺ C ⋅ sˆ
−ε SPSψ 1
⇒ ψ 2PS
7

∆xɺ C− ⋅ nˆ
= −  µ (1 + ε N )
− 1ψ 1
−
∆xɺ C ⋅ sˆ
 2

ψ 2PS = ψ 1 − 72 µ (1 + ε N ) sign (ψ 1 )
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Collision Model Parameters…
SS
 ∆xɺ c− ⋅ sˆ 
ψ 1 = −  − 
 ∆xɺ c ⋅ nˆ 
dimensionless pre-collision
tangential speed
 ∆xɺ c+ ⋅ sˆ 
ψ 2 = −  − 
 ∆xɺ c ⋅ nˆ 
dimensionless post-collision
tangential speed
PS
ψ 2SS = −ε SSSψ 1
ψ 2PS = ψ 1 − 72 µ (1 + ε N ) sign (ψ 1 )
Foerster, S.F., Louge, M.Y., Chang, H., and Allia, K., 1994,
“Measurements of the collision properties of small spheres,”
Physics of Fluids, Vol. 6, pp. 1108-1115.
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tangential coefficient of restitution, εS
Collision Model Parameters…
PS
SS
= 1/ψ1
Labous, L., Rosato, A.D., and Dave, R.N., 1997, “Measurements of collisional properties of
spheres using high-speed video analysis,” Physical Review E, Vol. 56, pp. 5717-5725.
C. Wassgren, Purdue University
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Measurements Methods
Foerster et al. (1994)
• High speed photography of marked, colliding
particles.
• Typically use “large” (> ~1 mm) particles for easy
visualization.
Labous et al. (1997)
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Summary
• Three parameter hard particle collision model
x 2 − x1
x 2 − x1
1
1
1
=
+
m′ m1 m2
nˆ =
xɺ 1,C = xɺ 1 + ω1 × r1nˆ
xɺ 2,C = xɺ 2 + ω 2 × ( − r2 nˆ )
sˆ =
ε
PS
S
∆xɺ C = xɺ 2,C − xɺ 1,C
∆xɺ C− − ( ∆xɺ C− ⋅ nˆ ) nˆ
∆xɺ C− − ( ∆xɺ C− ⋅ nˆ ) nˆ
∆xɺ C− ⋅ nˆ
7
= µ (1 + ε N )
−1
−
2
∆xɺ C ⋅ sˆ
ε S = min ( ε SSS , ε SPS )
J = m′ (1 + ε N ) ( ∆xɺ C− ⋅ nˆ ) nˆ + 72 m′ (1 + ε S ) ( ∆xɺ C− ⋅ sˆ ) sˆ
xɺ 1+ = xɺ 1− +
J
m1
xɺ 2+ = xɺ 2− −
J
m2
ω1+ = ω1− +
r1
( nˆ × J )
I1
ω 2+ = ω 2− +
r2
( nˆ × J )
I2
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Some Additional Comments
• The three parameter hard-particle model
(εN, εSSS, and µ) is a “lumped parameter”
model since the details of what occurs
during the collision are not described.
• The collision parameters εN, εSSS, and µ
are not properties of a single material, but
are the properties of the two interacting
materials.
• Difficult to find and measure these model
parameters.
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References
Foerster, S.F., Louge, M.Y., Chang, H., and Allia, K., 1994, “Measurements of the
collision properties of small spheres,” Physics of Fluids, Vol. 6, pp. 1108-1115.
Goldsmith, W., 1960, Impact: The Theory and Physical Behaviour of Colliding Solids,
Dover.
Gorham, D.A. and Kharaz, A.H., 2000, “The measurement of particle rebound
characteristics,” Powder Technology, Vol. 112, pp. 193 – 202.
Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press.
Kruggel-Emden, H., Simsek, E., Rickelt, S., Wirtz, S., and Scherer, V., 2007, “Review
and extension of normal force models,” Powder Technology, Vol. 171, pp. 157 – 173.
Kuwabara, G. and Kono, K., 1987, “Restitution Coefficient in a Collision between Two
Spheres,” Japanese Journal of Applied Physics, Vol. 26, pp. 1230 – 1233.
Labous, L., Rosato, A.D., and Dave, R.N., 1997, “Measurements of collisional properties
of spheres using high-speed video analysis,” Physical Review E, Vol. 56, pp. 57175725.
Maw, N., Barber, J.R., and Fawcett, J.N., 1976, “The oblique impact of elastic spheres,”
Wear, Vol. 38, pp. 101 – 114.
Maw, N., Barber, J.R., and Fawcett, J.N., 1981, “The role of elastic tangential compliance
in oblique impact,” Journal of Lubrication Technology, Vol. 103, pp. 74 – 80.
Walton, O.R., 1993, “Numerical simulation of inelastic, frictional particle-particle
interactions,” in Particulate Two-Phase Flow, Chap. 25, M.C. Roco ed., ButterworthHeinemann.