Granular impact drag force and its material-dependent scaling - - PowerPoint PPT Presentation

Granular impact drag force and its material-dependent scaling

Hiroaki Katsuragi & Douglas J. Durian Nagoya University University of Pennsylvania

• Fundamental process from planetary scale to our everyday
• Fundamental physics of granular matter
• Solid - fluid - solid transition
• Response to disturbance

Impact!

Contradictory previous works?

Uehara et al, PRL 2003, Ambroso et al, PRE 2004 de Bruyn et al, CJP 2004 Ciamarra et al, PRL 2004 Lohse et al, Nature 2004

F ∼ |z|α|v|

4−2α 3

F ∼ F0 + C|v| F ∼ |v0|

F ∼ |z|

d ∼ H1/3 d = α + cv0 tstop = const. d ∼ m (v0 = 0)

Experimental Apparatus

sand light line scan camera trigger electromagnet PC metal tip transparent stripe z sieve wind box ←N2

• Sand is fluidized

before each impact.

• Free fall is triggered

by an electromagnet holder.

• Dropping transparent

stripe is captured by a line-scan camera.

Raw data

z t

10 ms 1mm impact cessation z(t), v(t), a(t)

impact & stop time

• 80
• 60
• 40
• 20

v [cm/s]

0.06 0.04 0.02

t [s]

• 4
• 2

2 0.065 0.060

• 56
• 55
• 54
• 0.001

0.001

−g

vr 2vr

v = H(t−t0){v0+g(t−t0)}+H(t0−t){v0+(g−v2

0/d1)(t−t0)+([v3 0−2gd1v2 0]/d2 1)(t−t0)2}

• Low speed impact

takes longer time.

• Velocity is NOT

linear function of time.

• Acceleration shows

discontinuity at the stopping point.

a(t) = dv dt

Various drop heights :

(1” steel ball & glass beads)

• 8
• 6
• 4
• 2

2

z [cm]

• 400
• 300
• 200
• 100

v [cm/s]

h=83 cm h=50 h=30 h=15 h=7.0 h=3.8 h=2.2 h=1.6 h=1.1 h=0.52 h=0.28 h=0.18 h=0 101 102 103 104

a + g [cm/s2]

0.10 0.08 0.06 0.04 0.02 t [s]

z(t) = t v(t′)dt′ v(t)

Empirical laws & our new data

Constant stop time Coulomb friction

8 6 4 2

d [cm]

• 400
• 300
• 200
• 100

v0 [cm/s]

d = d0+|v0|

1 10

d [cm]

1 10 100

H [cm]

d= (d0

2H)1/3

d>H

0.10 0.08 0.06 0.04 0.02

tstop [s]

• 400
• 300
• 200
• 100

v0 [cm/s]

2.0x103 1.5 1.0 0.5

a + g [cm/s2]

• 2.0
• 1.5
• 1.0
• 0.5

z [cm]

v0 = 0

d/d0 = (H/d0)1/3 d = d0 + α|v0|

The data are completely consistent with empirical laws.

What is the stopping force?

Stopping force model

: independent of depth z : independent of velocity v

a+g v

f(zi)/m{

v2/d1

ΣF(= ma) = −mg + f(z) + mv2 d1 a + g = f(z) m + v2 d1 d1 f(z)

Inertial drag

a + g = (1/d1)v2 + f(zi)/m

zi = {0, −1, −2, −3, −4 [cm]}

20x103 15 10 5

a+g [cm/s2]

• 400
• 300
• 200
• 100

v [cm/s]

zi=-4 cm zi=-3 cm zi=-2 cm zi=-1 cm zi= 0 cm

20x103 15 10 5

a+g [cm/s2]

160x103 120 80 40

v2 [cm2/s2]

Friction force

8000 6000 4000 2000

• 2000

a + g - v2/d1 [cm/s2]

• 8
• 6
• 4
• 2

z [cm]

data g{1+[3(z/d0)2-1]exp(-2|z|/d1)} (k/m)|z|

f(z)/m = a + g − (1/d1)v2 f(z) m = k m|z|

Unified Force law

ΣF v z ΣF = −mg + k|z| + mv2 d1

gravitational force depth proportional frictional drag (velocity independent) velocity dependent inertial drag (depth independent)

• H. Katsuragi & D.J. Durian (2007)

Solving equation

Ke = 1 2mv2 dKe dz = −mg + 2 d1 Ke − k|z|

linearized!

v = −

• v2

0e− 2|z|

d1 − kd1|z|

m + (1 − e− 2|z|

d1 )

• gd1 + kd2

1

2m 1/2

dKe dz = mv dv dz = mdv dt

Clark & Behringer, EPL (2013) Katsuragi & Durian, PRE (2013)

v(z) & force model

• 400
• 300
• 200
• 100

v [cm/s]

• 8
• 6
• 4
• 2

z [cm] v = −

• v2

0e− 2|z|

d1 − kd1|z|

m + (1 − e− 2|z|

d1 )

• gd1 + kd2

1

2m 1/2

d1 = 8.7 cm k/m = 1040 s−2

Universality

For steel ball vs glass beads:

How can we predict these values for other material impacts?

Two parameters and determine the dynamics. d1 k/m d1 = 8.7 cm k/m = 1040 s−2

Expectation for inertial drag

ρp : density of projectile ρg : density of granular media Dp : diameter of projectile

(momentum transfer)

α = 3/2 (ball), 4/π (cylinder) A : impact area (ratio of area and volme factor)

m d1 v2 ∼ ρgAv2 d1 ∼ α−1 ρp ρg Dp

Expectation for friction force

(hydrostatic pressure + Coulomb friction)

ρp : density of projectile ρg : density of granular media µ = tan θr : friction coefficient Dp : diameter of projectile α = 3/2 (ball), 4/π (cylinder) A : impact area (ratio of area and volme factor)

k|z| ∼ µgρg|z|A k m ∼ αµg ρg ρp 1 Dp

Various projectiles and sand

glass beads beach sand rice sugar

1” Tungsten, steel,polymer, wood 2” - 1/8” steel 1/2” - 1/4” diameter 2” - 6” length aluminum

• 400
• 200

v [cm/s]

• 8
• 6
• 4
• 2

z [cm]

• 400
• 200

v [cm/s]

• 400
• 200

v [cm/s]

• 400
• 200

v [cm/s] (a) (b) (c) (d) Wood Delrin PTFE Steel

Limited fitting of v(z)

Good fitting for all impacts Bad fitting for all shallow impacts by fixed and

d1 k/m

v = −

• v2

0e− 2|z|

d1 − kd1|z|

m + (1 − e− 2|z|

d1 )

• gd1 + kd2

1

2m 1/2

Scaling of parameters

Inertial parameter: Friction parameter:

1 10

kDp'/mg

1 10 100

ρp [g/cm3]

• 1/2
• 1

0.1 1 10 100

d1/Dp' 1

(a) (b)

kDp mg ∼ ρp

1/2

d1 Dp

0 ∼ ρp

Dp

0 = 3π/8Dp for cylinder (shape factor)

Internal friction dependence

Inertial parameter Friction parameter

10

(kDp'/mg)(p/g)

1/2

1

µ 1

0.1 1

(d1/Dp')(g/p)

• 1

(a) (b)

kDp mg = 12µ ✓ρg ρp ◆1/2 d1 Dp

0 = 0.25

µ ρg ρp

d1 = 1/(1 + 2.2µ)

UNIFIED FORCE LAW

ΣF = −mg + k|z| + mv2 d1

final form of the drag force:

d1 Dp = 0.25 µ ρp ρg k m Dp g = 12µ ρp ρg 1/2

Scaling by material properties:

• H. Katsuragi & D.J. Durian, Phys. Rev. E 87, 052208 (2013)