Buckling of geometrically confined shells Lucia Stein-Montalvo, Paul Costa, Matteo Pezzulla...
The geometry and topology of thin structures dictates how they deform – it is easy to role a sheet of paper into a cylinder, but impossible to wrap it around a sphere without crumpling it, as this requires you to stretch it. A comparison of the energies for stretching and bending suggests that, if possible, a thin sheet will deform in a manner that avoid stretching as much as possible. This suggests that if we can actively tune a structure’s local geometry by prescribing regions that stretch and shrink we will be able to programmatically shape and morph 3D structures on command. One pathway to actively morphing materials is by selectively swelling portions of a structure. This is analogous to the continuous shape change during the growth and decay of biological structures. Structures morph to accommodate an in flux of new material, either growing from an external nutrient source, or swelling from the absorption of water. Some of the most dramatic growth–induced deformations occur with slender structures, such as growing leaves, wrinkling skin, and the writhing of tendril–bearing climbers. We have demonstrated the controlled morphing of sheets into shells with a process known as residual swelling, which results in growth-like deformations.
Swelling is a robust approach to structural change as it occurs naturally in humid environments and can easily be adapted into industrial design. Small volumes of fluid that interact favorably with a material can induce large, dramatic, and geometrically nonlinear deformations. If these fluids are precisely dispersed and polymerizable, they can control the morphology of a structure across many length scales. Consider the coupling of elastocapillarity and swelling. For example, when you put a straw into a liquid, the liquid rises via capillary action – surface tension draws the fluid up while gravity pulls it down. The smaller the straw diameter, the higher the fluid rises. If the walls of that straw are flexible, the fluid rises higher still as surface tension pulling on the walls is strong enough to bend them closer together. This is known as elastocapillarity, and it is what you see when bristles of a paintbrush or wet hairs clump together. Now, if the material is flexible and absorbent, like a sponge, the fluid will swell the walls, causing them to curl apart when wetted. So there are two competing effects – surface tension pulling the flexible objects together, and swelling curling them apart. In the image on the cover, two flexible and absorbent silicone rubber fibers are dipped into a bath of silicone oil. Initially, a balance between elasticity and capillarity pulls the fibers together, and then the swelling of the fluid into the material slowly curls the fibers apart. Eventually, the swelling-induced bending peeling them off the surface of the fluid bath, and a fluid droplet moves upward. The addition of swelling to the problem of elastocapillarity may bring new insights to the swelling and drying of many soft, porous, engineered materials, such as textiles and paper, as well the study of swellable biological structures, such as hair, certain types of plants, and other soft tissues. It is likely that swelling occurs in various elastocapillary environments, and to date it’s effect has been ignored. For instance, elastocapillary coalescence is a common cause of failure during the photolithography of high aspect ratio pillars – as scientists begin printing and replicating soft structures, swelling could add significant complications. While similar large, curling deformations may not always be observed, confinement could lead to unexpected residual stresses, localized deformations, and delamination – these types of things are common culprits for the failure of soft materials. Finally, elastocapillary swelling could lead to the design of new types of soft actuators involving liquid transport and shape changes, thereby bridging the worlds of capillary origami and swelling-induced shape-changing structures.
While a structure’s intrinsic geometry dictates the shape it would ideally like to adopt, its surroundings provide additional constraints. For example, a structure embedded and deforming in soft and fragile matter, such as tissues and granular media, requires considering the interplay between the deforming structure and its surrounding media. If we can actively control and direct these a flexible elastica, we can create advanced, autonomous structures capable of “swimming” around obstacles in various media. The structure may be an investigative drill penetrating an oily shale field with rock outcroppings, or a surgical probe passing around delicate neural tissue towards an elusive tumor in the brain.
This material is based upon work supported by the National Science Foundation under Grant No. 1454153. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Not long ago, the loss of structural stability through buckling generally referred to failure and disaster. It was a phenomenon to be designed around, and rarely did it provide functionality. The increasing focus on soft materials, from rubbers and gels to biological tissues, encouraged scientists to revisit the role of elastic instabilities in the world around us and inspired their utilization in advanced materials. Now the field of elastic instabilities, or extreme mechanics, brings together the disciplines of physics, mechanics, mathematics, biology, and materials science to extend our understanding of structural instabilities for both form and function. Our research examines the fundamental mechanics and dynamics of wrinkling, crumpling, and snapping of soft or slender structures.
For example, we’ve used the voltage-induced buckling deformation of thin films within microfluidic channels to control and direct fluid flow. The simple and robust design we present can have multiple internal and external actuators, such as mechanical and electrical stimuli, to move fluids within three-dimensional, hierarchical structures.
Snapping Shells – When a structure “snaps” to an alternate shape – like the inversion of an umbrella on a windy day – its structural and material integrity are often irreversibly lost. Many soft structures, however, are able to reversibly change between two stable configurations, presenting a fascinating opportunity to design dynamic, adaptable engineering structures across a multitude of length scales. The snap-through elastic instability enables large and fast deformations, as a structure switches between two stable states once a critical criterion is met. Snapping provides advanced functionality in nature with the rapid leaf closure of the Venus flytrap and the waterwheel plant, and has been employed with great amusement in the ‘jumping disc’ and ‘popper’ toys that jump with an audible pop. In order to design engineering systems that use instabilities as a feature rather than a fault, we aim to understand what dictates the dynamics of the snap-through instability, and to provide a means for snap as a mechanism for energy harvesting.
This material is based upon work supported by the National Science Foundation under Grant No. 1505125. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Toys have long captured and stimulated the imaginations of scientists and children alike. The snap bracelet, Slinky, rattleback, and tippe top often challenge our understanding of angular momentum, friction, rigid body contact kinematics, and other nonintuitive “elementary” mechanics. Online videos of toys – like the “levitating Slinky” – have helped bring concepts of mechanics and dynamics to the attention of millions of people through the internet. Toys overlap with disciplines of classical mechanics, dynamical systems, soft matter physics, mechanical engineering, robotics, and applied mathematics.
Slinky Mechanics – The floppy nature of a tumbling Slinky has captivated children and adults alike for over half a century. Highly flexible, the spring will walk down stairs, turn over in your hands, and – much to the chagrin of children everywhere – become easily entangled and permanently deformed. The Slinky can be used as an educational tool for demonstrating standing waves, and a structural inspiration due to its ability to extend many times beyond its initial length without imparting plastic strain on the material. Engineers have scaled the iconic spring up to the macroscale as a pedestrian bridge, and down to the nanoscale for use as conducting wires within flexible electronic devices, while animators have simulated its movements in a major motion picture. Yet, perhaps the most recognizable and remarkable features of a Slinky are simply its ability to splay its helical coils into an arch, and to tumble over itself down a steep incline. We have studied the mechanics of this soft, helical spring, and developed a model to describe is static shapes and unstable states.
(31.) D.P. Holmes, J.H. Lee, H.S. Park, and M. Pezzulla, “The nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature,” (2018). [arXiv] [Mathematica]
(30.) D.P. Holmes, “Elasticity and Stability of Shape Changing Structures,” (2018). [arXiv]
(29.) L. Stein-Montalvo, P. Costa, M. Pezzulla, D.P. Holmes, “Buckling of geometrically confined shells,” Soft Matter, 15(6), 1215-1222, (2019). [PDF] [arXiv]
Special Issue: Emerging Investigators
(26.) S. Wei, H. Shao, X. Jiang, D.P. Holmes, and T.K. Ghosh, “Bioinspired Electrically Activated Soft Bistable Actuators,” Advanced Functional Materials, 1802999, (2018). [Link]
(24.) D.J. Schunter Jr., M. Brandenbourger, S. Perriseau, and D.P. Holmes, “Elastogranular Mechanics: Buckling, Jamming, and Structure Formation,” Physical Review Letters, 120, 078002, (2018). [PDF] [arXiv]
(22.) M.A. Dias, M.P. McCarron, D. Rayneau-Kirkhope, P.Z. Hanakata, D.K. Campbell, H.S. Park, and D.P. Holmes, “Kirigami Actuators,” Soft Matter, 13, 9087-9802, (2017). [PDF] [arXiv]
Back Cover: [Link]
(22.) A.R. Mojdehi, D.P. Holmes, and D.A. Dillard, “Revisiting the Generalized Scaling Law for Adhesion: Role of Compliance and Extension to Progressive Failure,” Soft Matter, 13, 7529-7536, (2017). [PDF]
(20.) B. Tavakol, D.P. Holmes, G. Froehlicher, and H.A. Stone, “Extended Lubrication Theory: Estimation of Fluid Flow in Channels with Variable Geometry,” Proceedings of the Royal Society A, 474, 0234, (2017).[PDF] [arXiv]
(19.) A.R. Mojdehi, D.P. Holmes, and D.A. Dillard, “Friction of extensible strips: An extended shear lag model with experimental evaluation,” International Journal of Solids and Structures, (2017). [PDF]
(17.) A.R. Mojdehi, B. Tavakol, W. Royston, D.A. Dillard, D.P. Holmes, “Buckling of elastic beams embedded in granular media,” Extreme Mechanics Letters, 9, 237-244, (2016). [PDF]
(12.) R.H. Plaut, A.D. Borum, D.P. Holmes, and D.A. Dillard, “Falling vertical chain of oscillators, including collisions, damping, and pretensioning,” Journal of Sound and Vibration, 349, 195-205, (2015). [PDF]
(11.) D.P. Holmes, A.D. Borum, B.F. Moore III, R.H. Plaut, and D.A. Dillard, “Equilibria and Instabilities of a Slinky: Discrete Model”, International Journal of Nonlinear Mechanics, 65, 236-244, (2014). [PDF] [arXiv]
(10.) B. Tavakol, M. Bozlar, C. Punckt, D.P. Holmes, G. Froehlicher, H.A. Stone, I.A. Aksay, and D.P. Holmes, “Buckling of Dielectric Elastomeric Plates for Soft, Electrically Active Microfluidic Pumps “, Soft Matter, 10(27), 4789-4794, (2014). [PDF]
(8.) A. Pandey and D.P. Holmes, “Swelling-Induced Deformations: A Materials-Defined Transition from Macroscopic to Microscopic Deformations,” Soft Matter, 9, 5524, (2013). [PDF]
(7.) D.P. Holmes, B. Tavakol, G. Froehlicher, and H.A. Stone, “Control and Manipulation of Microfluidic Fluid Flow via Elastic Deformations”, Soft Matter, 9, 7049, (2013). [PDF]
Special Issue: Emerging Investigators [PDF]
(6.) D.P. Holmes, “Elastic Instabilities for Form and Function”, iMechanica, (2012). [Link]
(5.) D.P. Holmes, M. Roché, T. Sinha, and H.A. Stone, “Bending and Twisting of Soft Materials by Non-Homogenous Swelling,” Soft Matter, 7, 5188, (2011). [PDF]
(4.) M. Staykova, D.P. Holmes, C. Read, and H.A. Stone, Proc. Natl. Acad. Sci, 108(22), 9084-9088, (2011). [PDF]
(3.) D.P. Holmes and A.J. Crosby, “Draping Films: A Wrinkle to Fold Transition,” Physical Review Letters, 105, 038303, (2010). [PDF]
(2.) D.P. Holmes, M. Ursiny, and A.J. Crosby, “Crumpled Surface Structures,” Soft Matter, 4, 82, (2008). [PDF]
(1.) D.P. Holmes and A.J. Crosby, “Snapping Surfaces,” Advanced Materials, 19, 3589, (2007). [PDF]