Simulation for the Oblique Impact of a Lattice System

Kuninaka, Hiroto; Hayakawa, Hisao

doi:10.1143/jpsj.72.1655

Cited by 9 publications

(26 citation statements)

References 32 publications

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“…We obtain Poisson's ratio ν = (7.50±0.11)×10 −2 and Young's modulus E = (9.54 ± 0.231) × 10 3 mc 2 /R 2 , respectively [10].…”

mentioning

confidence: 82%

“…We place 800 mass points at random in a disk with the radius R and 4000 mass points at random in a wall for the disk and the wall, respectively. We connect each mass point with its neighbor mass points by the Delaunay triangulation algorithm [11], and undeformed nonlinear springs are placed on all the connections.Each mass point i on the lower half boundary of the disk feels the force,is the distance between i-th surface mass point of the disk and the nearest surface spring of the wall, a = 300/R, V 0 = amc 2 R/2, m is the mass of each mass point, c is the one-dimensional velocity of sound, and n (i) s is the unit vector normal to the connection between two surface mass points of the wall [10]. We should note that the strong repulsion F(l (i) s ) is introduced to inhibit the penetration of the disk to the surface of the wall [5].…”

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confidence: 99%

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Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact

Kuninaka

Hayakawa

2004

Phys. Rev. Lett.

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The coefficient of normal restitution in an oblique impact is theoretically studied. Using a twodimensional lattice model for an elastic disk and an elastic wall, we demonstrate that the coefficient of normal restitution can exceed unity and has a peak against the incident angle in our simulation. We also explain this behavior based upon a phenomenological theory.The coefficient of normal restitution e is introduced to determine the normal component of the post-collisional velocity in the collision of two materials. The coefficient e is defined bywhere v(τ ) is the relative velocity of the centers of mass of two colliding materials at time τ measured from the initial contact, τ c is the duration of a collision, and n is the unit vector normal to the contact plane. Though some text books of elementary physics state that e is a material constant, many experiments and simulations show that e decreases with increasing impact velocity [1]. The dependence of e on the low impact velocity is theoretically treated by the quasi-static theory [2,3,4]. We also recognize that e can be less than unity for the normal impact without the introduction of any explicit dissipation, because the macroscopic inelasticities originate in the transfer of the energy from the translational mode to the internal modes such as the vibrations [3,5,6]. While e has been believed to be less than unity in most situations, it is recently reported that e can exceed unity in oblique impacts [7,8,9]. In particular, Louge and Adams [9] observed oblique impacts of a hard aluminum oxide sphere on a thick elastoplastic polycarbonate plate and found that e grows monotonically with the magnitude of the tangent of the incident angle γ. In their experiment, Young's modulus of the plate is 100 times smaller than that of the aluminum oxide sphere. They also suggested that e can exceed unity for the most oblique impacts.In this letter, we demonstrate that our twodimensional simulation of the oblique impact based on Hamilton's equation has yielded an increasing e with tan γ and e exceeds unity at the critical incident angle. Finally, we explain our results by our phenomenological theory.Let us introduce our numerical model [10]. Our model consists of an elastic disk and an elastic wall (Fig. 1). The width and the height of the wall are 8R and 2R, respectively, where R is the mean radius of the undeformed disk. The both side ends and the bottom of the wall are fixed. We place 800 mass points at random in a disk with the radius R and 4000 mass points at random in a wall for the disk and the wall, respectively. We connect each mass point with its neighbor mass points by the Delaunay triangulation algorithm [11], and undeformed nonlinear springs are placed on all the connections.Each mass point i on the lower half boundary of the disk feels the force,is the distance between i-th surface mass point of the disk and the nearest surface spring of the wall, a = 300/R, V 0 = amc 2 R/2, m is the mass of each mass point, c is the one-dimensional velocity of sound, and n (i) ...

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“…We obtain Poisson's ratio ν = (7.50±0.11)×10 −2 and Young's modulus E = (9.54 ± 0.231) × 10 3 mc 2 /R 2 , respectively [10].…”

mentioning

confidence: 82%

mentioning

confidence: 99%

Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact

Kuninaka

Hayakawa

2004

Phys. Rev. Lett.

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“…(42), (43), and (44) into Eqs. (26), (27), and (28), we obtain a set of hydrodynamic equations for the deviation fields ρ, w, and θ. To avoid to use time dependent coefficients, we non-dimensionalize the set of equations by using…”

Section: Hydrodynamic Equationsmentioning

confidence: 99%

Long-time tails in freely cooling granular gases

Hayakawa

Otsuki

2007

Phys. Rev. E

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The long-time behavior of the current autocorrelation functions for the velocity, the shear stress, and the heat flux is investigated in freely cooling granular gases. It is found that the correlation functions for the velocity and the shear stress have the long-time tails obeying tau(-d/2), while the correlation function for heat flux decays as tau(-(d+2)/2) exp(-zeta*tau) with the dimensionless cooling rate zeta*, the spatial dimension d, and the scaled time tau in terms of the collision frequency. The result of our numerical simulation of the freely cooling granular gases is consistent with the theoretical prediction.

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“…It is basically the same as our previous model which obeys Hamilton equation. 11,14 The disk consists of 1099 mass points while the wall consists of 1269 mass points. The two corner points of the bottom of the wall are fixed.…”

Section: Modelmentioning

confidence: 99%

“…(i) s ) = aV 0 exp(−al(i) s )n (i) s , where l (i)s is the distance between i-th surface mass point of the disk and the nearest surface spring of the wall, a = 500/R, V 0 = amc 2 R/2, m is the mass of each mass point i, c is the one-dimensional speed of sound, and n (i) s is the unit vector normal to the connection between two surface mass points of the wall 11,14. Thus, the dynamical equation of motion for each mass point i of the lower half…”

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confidence: 99%