PHYSICAL REVIEW E 96, 042902 (2017) Energy decay in a tapped granular column: Can a one-dimensional toy model provide insight into fully three-dimensional systems? C. R. K. Windows-Yule, 1, 2, 3 D. L. Blackmore, 4 and A. D. Rosato 5 1 Multiscale Mechanics (MSM), MESA+, CTW, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands 2 Institute for Multiscale Simulation, Engineering of Advanced Materials, Friedrich-Alexander Universität Erlangen-Nürnberg, Schloßplatz 4, 91054 Erlangen, Germany 3 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom 4 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA 5 Granular Science Laboratory, Mechanial and Industrial Engineering Department, New Jersey Institute of Technology, Newark, New Jersey 07102, USA (Received 5 January 2017; revised manuscript received 19 August 2017; published 5 October 2017) The decay of energy within particulate media subjected to an impulse is an issue of significant scientific interest, but also one with numerous important practical applications. In this paper, we study the dynamics of a granular system exposed to energetic impulses in the form of discrete taps from a solid surface. By considering a one-dimensional toy system, we develop a simple theory, which successfully describes the energy decay within the system following exposure to an impulse. We then extend this theory so as to make it applicable also to more realistic, three-dimensional granular systems, assessing the validity of the model through direct comparison with discrete particle method simulations. The theoretical form presented possesses several notable consequences; in particular, it is demonstrated that for suitably large systems, effects due to the bounding walls may be entirely neglected. We also establish the existence of a threshold system size above which a granular bed may be considered fully three dimensional. DOI: 10.1103/PhysRevE.96.042902 I. INTRODUCTION Granular and particulate media are ubiquitous both in industry and our everyday lives, representing—aside from water—the most widely handled commodity on Earth [1]. Despite this fact, the dynamical behaviors of these materials remain incompletely understood [1–3] and thus, in many cases, impossible to accurately predict. This unpredictability carries many negative consequences, both in nature and in numerous diverse industries [4–7]. While we are currently witnessing a continual improvement in the speed and power of simulational techniques for modeling granular systems [8] as well as in the capabilities of experi- mental techniques, we are still a long way from being able to reliably analyze the behaviors of large, three-dimensional systems with precision. However, within the limitations of contemporary technology, we can successfully model small, often one-dimensional or quasi-two-dimensional toy systems to a high degree of precision, producing simulations and models that accurately and quantitatively represent observa- tions from similar experimental systems [9–11]. Nonetheless, such systems are typically beset by a recurring question: can the behaviors of these simple models provide gainful insight into the behavior of more complex, larger, and/or fully three-dimensional systems more relevant to the majority of real-world processes? In this paper, we consider a very simple toy model of a granular system: a one-dimensional column of dissipative particles subjected to a single discrete excitation event, or tap. The use of this system ensures simplicity not only in terms of geometry, but also in terms of the manner of excitation. Tapped granular systems are of direct relevance to a number of contemporary applications, such as the creation of granular dampers used, for example, in the construction, aerospace and even medical sectors [12–15], and the compaction of particulate media [16]. Perhaps more importantly, however, these systems provide fundamental insight into the behaviors of excited granular media as a whole, in particular vibrated and vibrofluidized beds. In the present work, we focus specifically on the dissipation of energy within the tapped granular systems studied. It is the dissipative nature of granular materials that predominantly separates them from classical solids, liquids, and gases, of which our understanding is far superior. It is hoped that by gaining an improved, predictive knowledge of said energy dissipation within the fundamental canonical systems studied here, we may take a vital early step towards an improved understanding of excited granular media as a whole. Due to their value as a means through which to gain insight into the fundamental physics of particulate systems, one-dimensional granular columns have been widely studied. Falcon et al. studied the collision of a granular column with a static wall [9]. The paper focused predominantly on the force exerted upon the wall by the falling column, making the surprising observation that the maximal force experienced by the wall remained constant irrespective of the number of particles forming the column. However, more relevantly to the current work, Falcon et al. also provided insight into the dissipative behaviors of the column, which we touch upon in subsequent sections. Louge [17] utilized a simple granular column, impacted from above by a single particle, in order to better understand the invariance of particle packing fraction with depth in granular flows along inclined planes, which he hypothesized could be considered as a series of individual columns. Other authors have, in place of experimental systems and discrete particle simulations, utilized one-dimensional lattice models to explore the fundamental behaviors of various important processes within granular systems. The work of Brey 2470-0045/2017/96(4)/042902(11) 042902-1 ©2017 American Physical Society