Application of a continuum theory to vertical vibrations of a layer of granular material

Hill, James M.; Spencer, A. J. M.; McCue, Scott W.

doi:10.1016/j.ijengsci.2009.05.012

Cited by 2 publications

(20 citation statements)

References 33 publications

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“…We comment that this system is identical in form to two corresponding systems examined in Hill et al [12] and for which the full solution is derived in detail. Accordingly, we do not repeat these and state only the following final results:…”

Section: Polynomial Solutionsmentioning

confidence: 75%

“…where l 2 = À c 1 =c 2 . Precisely, as described in Hill et al [12], this equation has a solution in terms of Bessel functions of order one, and it may be verified that the following expressions,…”

Section: Polynomial Solutionsmentioning

confidence: 89%

“…Following Hill et al [12], we now consider small disturbances of the above motion of the form, u(r, z, t) = e u 1 (r, z, t), w(r, z, t) = da=dt + e w 1 (r, z, t), c(r, z, t) = e c 1 (r, z, t), q(r, z, t) = q 0 (z, t) + e q 1 (r, z, t), ð12Þ…”

Section: Perturbed Solutionmentioning

confidence: 99%

“…Again, on following Hill et al [12], we seek separable solutions of ( 17), (18), and (21) of the form,…”

Section: Perturbed Solutionmentioning

confidence: 99%

“…As detailed in Hill et al [12], equation (30) 1 can be readily transformed to the standard form of Mathieu's equation (see, for example, Abramowitz and Stegun [32]). Solutions of Mathieu's equation exhibit a wide range of behaviour, including stable, unstable, and periodic solutions, all of which are consistent with the observed behaviour of vertically vibrating granular layers.…”

Section: Time-dependencementioning

confidence: 99%

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Axially symmetric continuum mechanical model for the oscillon

Hill

Spencer

McCue

2022

Mathematics and Mechanics of Solids

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The oscillon is a highly localized dynamical phenomenon occurring in a thin horizontal layer of granular material, which rests on a rigid plate oscillating in the vertical direction. The geometry is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. The oscillon moves from “peak” to “crater” and “crater” to “peak” such that the time from “peak” to “peak” or “crater” to “crater” is twice the period of the oscillating plate. The physics of granular phenomena is not properly understood, and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behaviour. Here for a free-flowing (cohesion-less) granular material, under axially symmetric conditions, we present a partial continuum mechanical analysis assuming the Coulomb–Mohr yield function and non-dilatant double-shearing theory. We examine small perturbations superimposed upon a purely vertical vibration, and make the assumption that throughout the motion, the lower surface of the layer remains in contact with the rigid metal plate. We show how the temporal dependence, which decouples from the spatial structure, is governed by Mathieu’s equation for the physically relevant case of the rigid plate oscillating sinusoidally, and therefore stability is determined by certain key parameters. We explore a variety of possible forms for spatial dependence. The present axially symmetric analysis complements that presented by the authors for plane strain conditions, and we find, quite remarkably, that apart from constants, both flows are governed by similar fourth-order systems of ordinary differential equations. This means that for both plane strain and axially symmetry, analogous pattern forming conditions can operate.

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Section: Polynomial Solutionsmentioning

confidence: 75%

Section: Polynomial Solutionsmentioning

confidence: 89%

Section: Perturbed Solutionmentioning

confidence: 99%

“…Again, on following Hill et al [12], we seek separable solutions of ( 17), (18), and (21) of the form,…”

Section: Perturbed Solutionmentioning

confidence: 99%

Section: Time-dependencementioning

confidence: 99%

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Axially symmetric continuum mechanical model for the oscillon

Hill

Spencer

McCue

2022

Mathematics and Mechanics of Solids

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Preface of MMS special issue honoring the memory and legacy of Professor AJM Spencer, FRS

Soldatos¹

2022

Mathematics and Mechanics of Solids

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When he was appointed Professor of Theoretical Mechanics, back in 1966, Professor Anthony James Merrill Spencer [1] (Elected FRS in 1987) delivered an inaugural lecture, entitled ''Mechanics, Mathematics and Materials.'' Despite the substantial progress that continuum mechanics has enjoyed since that time, the lecture remains surprisingly contemporary. Its contents continue to resonate today to the extent that its reading leaves the strong impression that if published in the open literature, the lecture might be both revealing and inspirational to yet another generation of researchers. These impulses led me to make a request and obtain publication permission from both Tony Spencer's family and the George Green Library of the University of Nottingham.I next approached a small number of colleagues and proposed them to compose a relevant research article that could accompany and support the lecture's publication. The proposed enterprise was embraced with overwhelming support. It was destined to escalate into a relevant journal special issue that was naturally directed toward and finally associated with the journal ''Mathematics and Mechanics of Solids'' (MMS).Formal submission invitations were sent out at the very beginning of August 2021, after a proper consultation with potential contributors settled an indicative submission deadline in the middle of March 2022. The received contributions went through the standard review process of this journal and the whole process progressed smoothly. I thankfully express my sincere gratitude and appreciation to every author and every anonymous reviewer involved in the composition of this volume, as well as to the MMS Editor in Chief, Professor David Steigmann, for his sterling work, support, and pleasant editorial collaboration. Special thanks are also due to Mr Milad Shirani, Department of Mechanical Engineering, University of California at Berkeley, for the substantial technical assistance he offered in the pre-publication preparation of Professor Spencer's inaugural lecture.The lecture is naturally the leading article of this MMS special edition. It is followed by a comprehensive supporting article which, in brief, marks the significance that Professor Tony Spencer gave to Mathematics and its interaction with Continuum Mechanics, describes the state of the art at the time of the lecture, provides a fair account of the development that have followed since in both subjects, and acknowledges the marked successes of their interaction as well as the room that remains available for many more. Tony Spencer is remarkably one of the co-authors of the third article, while the fourth contribution contains a photograph of handwritten text, penciled by him more than half a century ago. Every single one of the eighteen remaining papers bears clear marks and signs of Professor Spencer's influential publications, and lifelong commitment and contribution to theoretical solid mechanics.A detailed list of Tony Spencer's publications needs not be presented here. This is available in the liter...

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Application of a continuum theory to vertical vibrations of a layer of granular material

Cited by 2 publications

References 33 publications

Axially symmetric continuum mechanical model for the oscillon

Axially symmetric continuum mechanical model for the oscillon

Preface of MMS special issue honoring the memory and legacy of Professor AJM Spencer, FRS

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