Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact

Abstract

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) ...

Simulation of cohesive head-on collisions of thermally activated nanoclusters

“…(13), (14), and (15). To draw Fig.9(b), we adopt V * = 0.018 ǫ/m which is slightly larger than the calculated value V * = 0.015 ǫ/m by Eq.…”

“…[7,12,13] Although it is believed that the restitution coefficient for head-on collisions is smaller than unity, the restitution coefficient can be larger than unity in oblique collisions. [14,15,16] For example, Louge and Adams observed such an anomalous impact in which the restitution coefficient is larger than unity in oblique collisions of a hard aluminum oxide sphere onto a thick elastoplastic polycarbonate plate in which the restitution coefficient increases monotonically with the increase of the magnitude of the tangent of the angle of incidence. [14] They explained that this phenomena can be attributed to the change in rebound angle resulting from the local deformation of the contact area between the sphere and the plate, which causes the increase in the normal component of the rebound velocity against the collision plane.…”

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