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

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

Contradictory previous works? Uehara et al, PRL 2003, de Bruyn et al, CJP 2004 Ciamarra et al, PRL 2004 Lohse et al, Nature 2004 Ambroso et al, PRE 2004 d ∼ m t stop = const. d = α + cv 0 d ∼ H 1 / 3 ( v 0 = 0) 4 − 2 α F ∼ | z | α | v | F ∼ F 0 + C | v | F ∼ | v 0 | F ∼ | z | 3

sieve ←N 2 light line scan camera trigger electromagnet PC metal tip transparent stripe z 0 sand wind box Experimental Apparatus • 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 10 ms t 1mm impact cessation z ( t ) , v ( t ) , a ( t )

impact & stop time -80 -4 -2 -60 2v r 0 v r − g v [cm/s] 2 -40 0.060 0.065 -56 -20 -55 -54 -0.001 0 0.001 0 0 0.02 0.04 0.06 t [s] v = H ( t − t 0 ) { v 0 + g ( t − t 0 ) } + H ( t 0 − t ) { v 0 +( g − v 2 0 /d 1)( t − t 0 )+([ v 3 0 − 2 gd 1 v 2 0 ] /d 2 1 )( t − t 0 ) 2 }

2 � t v ( t ′ ) dt ′ z ( t ) = Various drop heights : 0 0 -2 z [cm] -4 • Low speed impact -6 takes longer time. -8 -400 h=83 cm h=50 • Velocity is NOT v ( t ) h=30 -300 h=15 h=7.0 linear function of v [cm/s] h=3.8 h=2.2 -200 time. h=1.6 h=1.1 h=0.52 h=0.28 -100 h=0.18 • Acceleration shows h=0 0 discontinuity at the 10 4 stopping point. a + g [cm/s 2 ] 10 3 a ( t ) = dv 10 2 (1” steel ball & glass beads) dt 10 1 0 0.02 0.04 0.06 0.08 0.10 t [s]

Empirical laws & our new data d/d 0 = ( H/d 0 ) 1 / 3 d = d 0 + α | v 0 | 10 8 d>H 6 d [cm] d [cm] 4 2 H) 1/3 d = d 0 + � |v 0 | d= (d 0 2 0 1 0 -100 -200 -300 -400 1 10 100 v 0 [cm/s] H [cm] Constant stop time Coulomb friction 2.0x10 3 0.10 v 0 = 0 0.08 1.5 a + g [cm/s 2 ] t stop [s] 0.06 1.0 0.04 0.5 0.02 0 0 0 -0.5 -1.0 -1.5 -2.0 0 -100 -200 -300 -400 z [cm] v 0 [cm/s]

The data are completely consistent with empirical laws. What is the stopping force?

Stopping force model Σ F (= ma ) = − mg + f ( z ) + mv 2 d 1 + v 2 a + g = f ( z ) a+g v 2 /d 1 m d 1 f ( z i ) /m { v : independent of depth z d 1 : independent of velocity v f ( z )

Inertial drag a + g = (1 /d 1 ) v 2 + f ( z i ) /m z i = { 0 , − 1 , − 2 , − 3 , − 4 [cm] } 20x10 3 20x10 3 a +g [cm/s 2 ] 15 15 10 5 a +g [cm/s 2 ] 0 160x10 3 0 40 80 120 10 v 2 [cm 2 /s 2 ] z i =-4 cm z i =-3 cm 5 z i =-2 cm z i =-1 cm z i = 0 cm 0 0 -100 -200 -300 -400 v [cm/s]

Friction force f ( z ) /m = a + g − (1 /d 1 ) v 2 8000 f ( z ) = k m | z | a + g - v 2 /d 1 [cm/s 2 ] 6000 m 4000 2000 data g{1+[3(z/d 0 ) 2 -1]exp(-2|z|/d 1 )} 0 (k/m)|z| -2000 0 -2 -4 -6 -8 z [cm]

Unified Force law Σ F = − mg + k | z | + mv 2 d 1 velocity dependent inertial drag gravitational force (depth independent) depth proportional frictional drag (velocity independent) Σ F v z H. Katsuragi & D.J. Durian (2007)

Solving equation dK e = mv dv dz = mdv K e = 1 2 mv 2 dz dt dK e = − mg + 2 K e − k | z | linearized! dz d 1 Clark & Behringer, EPL (2013) �� 1 / 2 gd 1 + kd 2 � � d 1 − kd 1 | z | 0 e − 2 | z | + (1 − e − 2 | z | v 2 1 d 1 ) v = − m 2 m Katsuragi & Durian, PRE (2013)

v(z) & force model �� 1 / 2 gd 1 + kd 2 � d 1 − kd 1 | z | � 0 e − 2 | z | + (1 − e − 2 | z | v 2 1 d 1 ) v = − m 2 m d 1 = 8 . 7 cm k/m = 1040 s − 2 -400 -300 v [cm/s] -200 -100 0 0 -2 -4 -6 -8 z [cm]

Universality Two parameters and determine the dynamics. k/m d 1 For steel ball vs glass beads: d 1 = 8 . 7 cm k/m = 1040 s − 2 How can we predict these values for other material impacts?

Expectation for inertial drag m v 2 ∼ ρ g Av 2 d 1 (momentum transfer) d 1 ∼ α − 1 ρ p D p ρ g ρ p : density of projectile A : impact area ρ g : density of granular media α = 3 / 2 (ball), 4 / π (cylinder) D p : diameter of projectile (ratio of area and volme factor)

Expectation for friction force k | z | ∼ µg ρ g | z | A (hydrostatic pressure + Coulomb friction) k 1 m ∼ α µg ρ g D p ρ p ρ p : density of projectile A : impact area ρ g : density of granular media µ = tan θ r : friction coe ffi cient D p : diameter of projectile α = 3 / 2 (ball), 4 / π (cylinder) (ratio of area and volme factor)

Various projectiles and sand 1” Tungsten, steel,polymer, wood 2” - 1/8” steel 1/2” - 1/4” diameter 2” - 6” length aluminum glass beads beach sand rice sugar

Wood (a) Limited fitting of v(z) -400 v [cm/s] -200 by fixed and k/m d 1 0 �� 1 / 2 gd 1 + kd 2 � d 1 − kd 1 | z | � Delrin (b) 0 e − 2 | z | + (1 − e − 2 | z | 1 v 2 d 1 ) -400 v = − m 2 m v [cm/s] -200 Bad fitting for all shallow impacts 0 PTFE (c) -400 v [cm/s] -200 0 Steel (d) -400 v [cm/s] Good fitting for all impacts -200 0 0 -2 -4 -6 -8 z [cm]

Scaling of parameters 100 Inertial parameter: 1 10 d 1 d 1 /D p ' 0 ∼ ρ p D p 1 (a) 0.1 10 Friction parameter: (b) kD p '/mg 0 kD p � 1 / 2 ∼ ρ p -1/2 mg 1 -1 1 10 100 ρ p [g/cm 3 ] 0 = 3 π / 8 D p for cylinder (shape factor) D p

Internal friction dependence Inertial parameter -1 (d 1 /D p ')( � g / � p ) 1 d 1 0 = 0 . 25 ρ g µ D p ρ p (a) 0.1 d 1 = 1 / (1 + 2 . 2 µ ) (b) Friction parameter 1/2 (kD p '/mg)( � p / � g ) 1 10 ◆ 1 / 2 0 ✓ ρ g kD p = 12 µ mg ρ p 1 µ

UNIFIED FORCE LAW final form of the drag force: Σ F = − mg + k | z | + mv 2 d 1 Scaling by material properties: d 1 = 0 . 25 ρ p D p µ ρ g � 1 / 2 � ρ p k D p = 12 µ m g ρ g H. Katsuragi & D.J. Durian, Phys. Rev. E 87, 052208 (2013)