Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES http://www.jomms.org Founded by Charles R. Steele and Marie-Louise Steele EDITORS CHARLES R. STEELE Stanford University, U.S.A. DAVIDE BIGONI University of Trento, Italy IWONA JASIUK University of Illinois at Urbana-Champaign, U.S.A. YASUHIDE SHINDO Tohoku University, Japan EDITORIAL BOARD H. D. BUI École Polytechnique, France J. P. CARTER University of Sydney, Australia R. M. CHRISTENSEN Stanford University, U.S.A. G. M. L. GLADWELL University of Waterloo, Canada D. H. HODGES Georgia Institute of Technology, U.S.A. J. HUTCHINSON Harvard University, U.S.A. C. HWU National Cheng Kung University, R.O. China B. L. KARIHALOO University of Wales, U.K. Y. Y. KIM Seoul National University, Republic of Korea Z. MROZ Academy of Science, Poland D. PAMPLONA Universidade Católica do Rio de Janeiro, Brazil M. B. RUBIN Technion, Haifa, Israel A. N. SHUPIKOV Ukrainian Academy of Sciences, Ukraine T. TARNAI University Budapest, Hungary F. Y. M. WAN University of California, Irvine, U.S.A. P. WRIGGERS Universität Hannover, Germany W. YANG Tsinghua University, P.R. China F. ZIEGLER Technische Universität Wien, Austria PRODUCTION PAULO NEY DE SOUZA Production Manager SHEILA NEWBERY Senior Production Editor SILVIO LEVY Scientific Editor Cover design: Alex Scorpan Cover photo: Wikimedia Commons See inside back cover or http://www.jomms.org for submission guidelines. JoMMS (ISSN 1559-3959) is published in 10 issues a year. The subscription price for 2010 is US $500/year for the electronic version, and $660/year (+$60 shipping outside the US) for print and electronic. Subscriptions, requests for back issues, and changes of address should be sent to Mathematical Sciences Publishers, Department of Mathematics, University of California, Berkeley, CA 94720–3840. JoMMS peer-review and production is managed by EditFLOW from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers http://www.mathscipub.org A NON-PROFIT CORPORATION Typeset in LATEX ©Copyright 2010. Journal of Mechanics of Materials and Structures. All rights reserved.
JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 5, No. 5, 2010 AXIAL COMPRESSION OF HOLLOW ELASTIC SPHERES ROBERT SHORTER, JOHN D. SMITH, VINCENT A. COVENEY AND JAMES J. C. BUSFIELD When thin-walled hollow elastic spheres are compressed between two parallel rigid surfaces, there is an initial flattening of the sphere in the contact regions, followed by a snap-through buckling of the flattened surface. As the compression increases the sphere undergoes further buckling modes as a number of ridges and folds are formed. This elastic buckling deformation is investigated using a finite element analysis (FEA) technique. It is shown that the ratio of displacement at buckling to wall thickness depends weakly not only on Poisson’s ratio, ν, but also on the ratio of the geometric wall thickness, h, to sphere radius, R. This approach is validated by comparison with experimental compression results on microspheres of approximately 40 µm in diameter to table tennis balls with a diameter of 40 mm. The analysis shows that a simple axial compression of a thin-walled hollow sphere can be used to measure both the average wall thickness of the sphere, from the deformation at the buckling snap-through, and the modulus from the force at this point. This provides a good technique to fully characterise the geometry and the elastic behaviour of thin-walled spheres of any size. Introduction Hollow thin-walled spheres are used for a variety of applications ranging from the recreational (table tennis balls, tennis balls, footballs), the industrial (lightweight and syntactic foams), through to the medical (the use of ultrasonic contrast agents to enhance ultrasonic imaging). In many of these applications the mechanical properties of the spherical shell play an important role, yet reliable measurements of those properties are often difficult to obtain. For example, with sintered foam structures made from hollow spheres, such as that proposed in [Taguchi and Karushige 2007; Peng et al. 2000], it is important that the mechanical properties of the individual spheres are known. With polymer spheres, even if samples of the polymer constituent are available, the manufacturing method may influence the final mechanical properties. This is often the case for example with polymers made by blow or injection moulding where an in-situ measurement may be desirable. The situation is even starker for microscopic fillers: for example, with ultrasonic contrast agents where experimental evidence indicates that the elasticity and thickness of the shell are important components to the overall dynamics [Leong-Poi et al. 2002; Ketterling et al. 2007]. Such shells often have a diameter of a few µm and are constructed of proteins and lipids. Another type of hollow sphere, commercially available as Expancel � microspheres, are encountered in industrial applications as either a way of reducing weight or to act as a blowing agent. These range in diameter from 20 to 55 µm with shell thicknesses of approximately 0.1 µm. In [Trivett et al. 2006], when the behaviour of these microspheres was studied in castor oil, the Young’s modulus of the shell was estimated to be ∼3 GPa from the sound speed of the Keywords: compression, buckling, instability, hollow spheres, finite element analysis, 693
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