Energy Absorption and Recovery in Tapered Granular Chains: Small Chains and Low Tapering
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Energy Absorption and Recovery in Tapered Granular Chains: Small Chains and Low Tapering
Jan Pfannes, Surajit Sen Department of Physics, State University of New York, Buffalo, New York 14260, USA Soumya Chakravarti Department of Physics, California State Polytechnic University, Pomona, California 91768, USA Farhat I. Surve Department of Physics, State University of New York, Buffalo, New York 14260, USA and Nowrosjee Wadia College, Pune, India
Abstract Shock absorption traditionally exploits visco-elastic devices, fluid viscous dampers and friction dampers, or "soft" materials. Recent work on impulse propagation in granular assemblies suggests that it may be possible to absorb shock waves using a different variety of shock absorbers and thus unlock the possibility of building structures that are significantly more shock absorbent than is currently possible, an issue of relevance to concerns of security. These new shock absorbers are composed of granular materials and exploit the nonlinear energy propagation properties in assemblies of granular materials. They have the potential to partially recover the energy of the absorbed shock waves for useful purposes, a property that might allow one to design devices to convert the incident energy to other useful forms. The basic physics of the “tapered chain shock absorber” is briefly discussed in this work. Introduction When elastic grains are squeezed against one another they repel according to the Hertz law [1,2]. Let us consider two equal sized elastic spheres, each of diameter d and assume that the distance between their centers upon squeezing is r (usually referred to as overlap). The repulsive force between the spheres is then given by F ∝ δ 3/2, where δ ≡ d – r. More generally, the index of the force law depends upon the geometry of the interface between two elastic objects in 3D but always exceeds unity. If one considers a linear alignment of elastic spheres that are in mutual contact (but are not precompressed) and initiates an impulse at one end of the chain of spheres, the impulse travels through the chain as a tight energy bundle or as a solitary wave [3-12]. Propagation of solitary waves in chains of elastic beads has been extensively discussed in the literature. Restitution and polydispersity can attenuate the amplitude of this solitary wave but does not affect its width [13]. When one considers a linear alignment of spheres that are barely in contact and where the grains progressively shrink in radius, impulse propagation behavior changes
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dramatically. If we assume that an impulse is incident on the largest sphere at one extremity of an alignment of elastic spheres in a tapered chain, momentum conservation demands that the smaller sphere end up moving at a higher velocity than the larger sphere and carry less kinetic energy. The upshot of momentum conservation and nonlinearity implies that the propagating shock wave, which is an energy bundle, breaks up into myriads of smaller energy bundles. The degree of tapering, as well as the number
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