C. Wassgren, Purdue University 1. DEM Modeling: Lecture 07 Normal Contact Force Models. Part II

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1 C. Wassgren, Purdue University 1 DEM Modeling: Lecture 07 s. Part II

2 C. Wassgren, Purdue University 2 Excluded volume error Effects of Soft Springs consider the solid fraction in a compressed system containing particles with stiff vs. soft springs stiff springs solid fraction < 1 (as expected) soft springs solid fraction > 1! (not realistic)

3 C. Wassgren, Purdue University 3 Effects of Soft Springs Excluded volume error Ketterhagen et al. (2005) found that sheared systems with soft springs produce stresses similar to those that would be generated if smaller particles were used

4 C. Wassgren, Purdue University 4 Effects of Soft Springs 2D shear simulations: as spring stiffness at large solid fraction, stresses at small solid fraction, stresses From Ketterhagen et al. (2005)

5 C. Wassgren, Purdue University 5 Effects of Soft Springs Detachment effect (Luding et al.,1994) the effective coefficient of restitution of a system of particles depends upon the ratio of the time between collisions, t 0 = s 0 /v 0 to the duration of an impact, T, t t 0 0 T 1 ε N ε T N,eff N,eff ε 1 spring stiffnesses that are too soft (i.e. have too large of a T) may have a much smaller degree of energy dissipation than expected N s 0 v 0

6 C. Wassgren, Purdue University 6 Effects of Soft Springs Brake efficiency failure (Schäfer and Wolf, 1995) θ R λ v if contact is stiff, then t T and v v contact if contact is soft, then tcontact λ and v v 1 v λ = 2R cosθ T λ 2R k = τ π vcrit cos θ k v π crit for contacts with v > v crit, the braking efficiency decreases with increasing impact speed; particle rebound response is significantly altered

7 C. Wassgren, Purdue University 7 Effects of Soft Springs From Campbell (2002) Stress scaling varies depending on the dimensionless stiffness and solid fraction elastic-quasi-static: τ = f ( ν, µ, ε ) kd inertial-collisional: τ = f,, 2 2 ρd γ elastic-inertial: ( ν µ ε ) τ τ k or = f,, 2 2 ν µ ε 3 2 ρd γ kd ρd γ

8 C. Wassgren, Purdue University 8 Hysteretic Linear Spring First proposed by Walton and Braun (1986) Widely used R i ˆn R j F i k ˆ Ln max = k ( ) ˆ U n < < 0 0 res res res max k L loading spring stiffness k U unloading spring stiffness (k U k L ) res residual overlap maximum overlap during contact max k L, k U energy dissipation due to spring force hysteresis contact force is continuous energy dissipation is position dependent relatively simple model to implement, but need to retain history of max

9 C. Wassgren, Purdue University 9 The hysteretic linear spring model mimics elastic-perfectly plastic, quasi-static FEM model results. The hysteretic stiffnesses (k L, k U ) model the strain hardening of the material due to plastic deformation. The residual overlap ( res ) represents the permanent plastic deformation of the contact. From: Walton, O.R. (1993)

10 C. Wassgren, Purdue University 10 A simple example: F B A B k L k L ( ) = k max U max res B C k U res = max 1 k k L U A,D k L k U C res max C D energy lost in contact Note: After full unloading (after reaching pt. D), the residual overlap is forgotten for the contact. The plastic deformation for the contact only exists during the contact.

11 C. Wassgren, Purdue University 11 A more complex example: F F k L D k L A B: B C: C B D: D E: E D F: F G H: initial loading partial unloading reloading beyond max,b partial unloading beyond res,d reloading beyond max,d full unloading B k U k U k L C k U A,H E G res,b max,b res,d res,f max,d max,f

12 C. Wassgren, Purdue University 12 For a two particle contact (derivation is left as an exercise): = ɺ max 0 k L R i m i ˆn m j R j ε = N k k L U k L, k U T π = ε N + k 2 L ( 1) ɺ 0 relative impact speed m effective mass (= (m -1 i + m -1 j ) -1 ) ε N normal coefficient of restitution Τ contact duration 2 ( ) = ɺ ε res 0 1 N kl

13 C. Wassgren, Purdue University dimensionless overlap or velocity, * or dot* εn = thick lines: * thin lines: dot* εn = 0.1 dimensionless time, t* εn = 0.5 εn = 0.5 εn = 0.9 εn = 0.9 m k L, k U * ɺ 0 ɺ ɺ * ɺ * t t 0 k k

14 C. Wassgren, Purdue University 14 dimensionless overlap acceleration, ddot* εn = 0.1 εn = 0.5 εn = 0.9 m k L, k U ɺɺ ɺɺ * ɺ * t t 0 * ɺ 0 k k k -1.0 dimensionless overlap *

15 C. Wassgren, Purdue University 15 Some observations the contact force is continuous at the start and end of the contact the contact force is always repulsive the loading portion of the contact is the same regardless of the coefficient of restitution energy dissipation is due to the hysteresis in the overlap

16 C. Wassgren, Purdue University 16 Some observations coefficient of restitution is independent of impact speed (in real collisions ε N as ɺ 0 ) contact duration as k L, m, and ε N coefficient of restitution dependence is opposite that for the damped linear spring model contact duration is independent of impact speed (in real collisions, contact duration as impact speed ) maximum overlap as ɺ 0, m, k L larger overlaps make the geometrically rigid particle assumption less accurate and can cause modeling errors due to excluded volume effects unlike the damped linear spring model, coefficient of restitution does not play a role

17 C. Wassgren, Purdue University 17 The unloading spring stiffness (k U ) may be found from the loading spring stiffness (k L ) and the coefficient of restitution (ε N ): k Method for determining the loading spring stiffness is not widely agreed upon consider three methods, all of which set particular contact parameters equal to those found using a Hertzian contact model maximum overlap contact duration maximum strain energy ε = N k L U

18 C. Wassgren, Purdue University 18 equivalent maximum overlap, max HLS model: = ɺ max 0 k L Hertzian spring model: 15 2 = ɺ 1 16R 2 E max k ɺ L,overlap 0 ( ) RE 1 5

19 C. Wassgren, Purdue University 19 equivalent contact duration, T HLS model: T π = ε N + k 2 L ( 1) Hertzian spring model: T RE ɺ L,duration 0 ( ) 2 ɺ 2 5 RE ( εn ) 12 2 k =

20 C. Wassgren, Purdue University 20 maximum strain energy, SE max HLS model: SE max = 1 2 L k 2 max Hertzian spring model: SE max = 2 5 Hz k 5 2 max where khz = R E 15 2 = ɺ 1 16R 2 E max k ɺ L,SE 0 ( ) RE 1 5

21 C. Wassgren, Purdue University 21 stiffness ratio koverlap/kduration kse/kduration kse/koverlap kl,overlap klse, = = k k ε L,duration L,duration N ( + 1) normal coefficient of restitution, ε N For example: two 3.18 mm diam. soda lime glass spheres (ρ = 2500 kg/m 3, ν = 0.22, E = 71 GPa) impacting at 1 m/s (ε N = 0.97; Foerster et al., 1994): max overlap/eff. diameter = 0.2% k overlap = 2.02 MN/m contact duration = 9.51 µs k duration = 2.23 MN/m max strain energy = 10.5 µj k SE = 2.02 MN/m

22 C. Wassgren, Purdue University 22 With a constant k U, the coefficient of restitution remains constant regardless of impact speed A variable coefficient of restitution may be modeled using a variable unloading stiffness (Walton and Braun, 1986). F F max,2 F max,1 k = k + SF where S is a constant and F max is the U k L k U1 L k U2 max maximum force achieved before unloading. The constant S can be determined empirically from experimental data. (S ~ m -1 ). ε = T N 1 1+ S ɺ 0 π = ε N + k 2 L k ( 1) L 1 k 1 S = ɺ m 0 L 1 2 ε N

23 C. Wassgren, Purdue University 23 From Walton and Braun (1986)

24 C. Wassgren, Purdue University 24 Some observations coefficient of restitution as impact speed matches experimental values reasonably well contact duration as impact speed very poor match to experimental data For example: two 3.18 mm diam. spheres soda lime glass: ρ = 2500 kg/m 3, ν = 0.22, E = 71 GPa match k dur and S at 1 m/s (ε N = 0.97) k duration = 2.23 MN/m S = 2.04*10 4 m -1 contact duration [µ s] WB Hertz error 60% 40% 20% 0% -20% -40% -60% % error from Hertz impact speed [m/s] -80%

25 C. Wassgren, Purdue University 25 Damped Hertzian Spring Taguchi (1992); Ristow (1992); Pöschel (1993); Lee and Hermann (1993); Zhou et al. (1999) F i ( k ) Hz ν = + ɺ n 3 2 ˆ R i ˆn R j k Hz Hertzian spring stiffness = 4/3R 1/2 E ν damping coefficient k Hz ν spring force is consistent with Hertz model dashpot added (in an ad hoc fashion) to provide energy dissipation contact force is discontinuous at start/end of contact due to damping force energy dissipation is velocity dependent simple model to implement

26 C. Wassgren, Purdue University 26 Damped Hertzian Spring For a two particle contact (derivation is left as an exercise): R i m i ˆn k Hz ν m j R j ɺ 0 relative impact speed m effective mass (= (m -1 i + m -1 j ) -1 ) ν damping coefficient k Hz Hertz spring stiffness = 4/3R 1/2 E ɺɺ * * * * + ν ɺ = ν * ( k t t ɺ k 0 * Hz 0 0 ɺ ν 0 Hz * k Hz ɺ 2 0m ɺ ɺ* = ɺ 2 5 * m ɺɺ ɺɺ ɺ 3 0k Hz 2 5 )2 5

27 C. Wassgren, Purdue University 27 Damped Hertzian Spring dimensionless overlap, * ν = 0.00 ν = 0.25 ν = 0.50 ν = 0.75 ν = * ν * k ɺ Hz 0 * k Hz ν t t ɺ ( ɺ 0 khz ) dimensionless time, t * ɺ * * m ɺ = ɺɺ ɺɺ ɺ 3 ɺ 0 0k Hz dimensionless overlap speed, dot* m k Hz ν ν* = 0.00 ε N = 1.00 ν* = 0.25 ε N = 0.62 ν* = 0.50 ε N = 0.37 ν* = 0.75 ε N = 0.17 ν* = 1.00 ε N = dimensionless time, t * ν = 0.00 ν = 0.25 ν = 0.50 ν = 0.75 ν = 1.00

28 C. Wassgren, Purdue University 28 Damped Hertzian Spring dimensionless maximum overlap, max* * * Hz 0 * ( ɺ 0 khz ) ν k ɺ k Hz t t 0 ν m ɺ ɺ * * m ɺ = ɺɺ ɺɺ ɺ 3 ɺ 0 0k Hz ν * dimensionless contact time, Tcontact* coefficient of restitution, ε N m k Hz ν ν *

29 C. Wassgren, Purdue University 29 Damped Hertzian Spring 0.4 m k Hz dimensionless acceleration, ddot* ν = 0.00 ν = 0.25 ν = 0.50 ν = 0.75 ν = 1.00 dimensionless overlap, * ν ν ( ɺ 0 khz ) ν ν* = 0.00 ε N = 1.00 ν* = 0.25 ε N = 0.62 ν* = 0.50 ε N = 0.37 ν* = 0.75 ε N = 0.17 ν* = 1.00 ε N = 0.04 * * Hz k ɺ t t m * k ɺ Hz * ɺ = ɺ2 0 ɺ 0 * m ɺɺ ɺɺ ɺ 3 0k Hz

30 C. Wassgren, Purdue University 30 Damped Hertzian Spring Some observations the contact force is discontinuous at the start and end of the contact due to the viscous damping force (real contact forces are continuous) the contact force is cohesive toward the end of the impact (real contact forces are always repulsive for cohesionless systems) ε N as impact speed (just the opposite in real collisions) ν * as dot ε N 0 as dot 0! contact duration as impact speed (consistent with real data)

31 C. Wassgren, Purdue University 31 References Buchholtz, V. and Pöschel, T., 1994, Numerical investigations of the evolution of sandpiles, Physica A, Vol. 202, Nos. 3-4, pp Campbell, C.S., 2002, Granular shear flows at the elastic limit, Journal of Fluid Mechanics, Vol. 465, pp Corkum, B.T. and Ting, J.M., 1986, The discrete element method in geotechnical engineering, Publication 86-11, Department of Civil Engineering, University of Toronto, ISBN Cundall, P.A. and Strack, O.D.L., 1979, A discrete numerical model for granular assemblies, Géotechnique, Vol. 29, No. 1, pp Dury, C.M. and Ristow, G.H., 1997, Radial segregation in a two-dimensional rotating drum, Journal de Physique I, Vol. 7, No. 5, pp Foerster, S.F., Louge, M.Y., Chang, H., and Allis, K., 1994, Measurements of the collision properties of small spheres, Physics of Fluids, Vol. 6, No. 3, pp Goddard, J.D., 1990, Nonlinear elasticity and pressure-dependent wave speeds in granular media, Proceedings of the Royal Society London A: Mathematical and Physical Sciences, Vol. 430, No. 1878, pp Goldsmith, W., 1960, Impact: The Theory and Physical Behaviour of Colliding Solids, Dover. Hertz, H., 1882, Über die Berührung fester elastischer Körper, J. reine und angewandte Mathematik, Vol. 92, pp Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press. Ketterhagen, W.R., Curtis, J.S., and Wassgren, C.R., 2005, Stress results from two-dimensional granular shear flow simulations, Physical Review E, Vol. 71, Art Kruggel-Emden, H., Simsek, E., Rickelt, S., Wirtz, S., and Scherer, V., 2007, Review and extension of normal force models for the Discrete Element Method, Powder Technology, Vol. 171, pp Kuwabara, G. and Kono, K., 1987, Restitution coefficient in a collision between two spheres, Jpn. J.Appl. Phys., Vol. 26, pp Labous, L., Rosato, A.D., Dave, R.N., 1997, Measurement of collisional properties of spheres using high-speed video analysis, Physical Review E, Vol. 56, pp Lan, Y. and Rosato, A.D., 1995, Macroscopic behavior of vibrating beds of smooth inelastic spheres, Physics of Fluids, Vol. 7, No. 8, pp Lan, Y. and Rosato, A.D., 1997, Convection related phenomena in granular dynamics simulations of vibrated beds, Physics of Fluids, Vol. 9, No 12, pp Luding, S., Clément, E., Blumen, A., Rajchenback, J., and Duran, J., 1994, Anomalous energy dissipation in molecular-dynamics simulation of grains: The detachment effect, Physical Review E, Vol. 50, No. 5, pp Mullier, M., Tüzün, U., and Walton, O.R., 1991, A single-particle friction cell for measuring contact frictional properties of granular materials, Powder Technology, Vol. 65, pp Schäfer, J. and Wolf, D.E., 1995, Bistability in simulated granular flow along corrugated walls, Physical Review E, Vol. 51, No. 6, pp Stevens, A.B. and Hrenya, C.M., 2005, Comparison of soft-sphere models to measurements of collision properties during normal impacts, Powder Technology, Vol. 154, pp Walton, O.R., 1993, Numerical simulation of inelastic, frictional particle-particle interactions, in Particulate Two-Phase Flow, M.C. Roco, ed., Chap. 25, pp , Butterworth-Heinemann.