Shells and Plates


From viral capsids to architectural domes, slender shells are ubiquitous in natural and engineered structures. Their Mechanics is tightly intertwined with the underlying geometry and the onset for instability is a classic problem in structural Mechanics. At the flexLab, we have revived this old field by an innovative experimental approach that focuses on the postbuckling regime, where strong geometric nonlinearities and energy focusing can emerge. Central to this effort, we have devised a robust, versatile,and precise fabrication mechanism to rapid prototype thin shells in a lab setting. We are also interested in problems involving the large deformation of elastic plates, for example when loaded under aerodynamic or hydrodynamic conditions.

Topics that we have investigated on plates and shells include: predicting knock-down factor in the buckling of pressurized shells, periodic patterning of shell-substrate systems, form-finding in gridhells, stress-focusing in buckled shells and tearing of thin sheets. A more detailed account of these examples and other problems is provided below.


Form finding in elastic gridshells

with: Changyeob Baek, Andrew O. Sageman-Furnas, and Mohammad K. Jawed

 



Elastic gridshells comprise an initially planar network of elastic rods that are actuated into a shell-like structure by loading their extremities. The resulting actuated form derives from the elastic buckling of the rods subjected to inextensibility. We study elastic gridshells with a focus on the rational design of the final shapes. Our precision desktop experiments exhibit complex geometries, even from seemingly simple initial configurations and actuation processes. The numerical simulations capture this nonintuitive behavior with excellent quantitative agreement, allowing for an exploration of parameter space that reveals multistable states. We then turn to the theory of smooth Chebyshev nets to address the inverse design of hemispherical elastic gridshells. The results suggest that rod inextensibility, not elastic response, dictates the zeroth-order shape of an actuated elastic gridshell. As it turns out, this is the shape of a common household strainer. Therefore, the geometry of Chebyshev nets can be further used to understand elastic gridshells. In particular, we introduce a way to quantify the intrinsic shape of the empty, but enclosed regions, which we then use to rationalize the nonlocal deformation of elastic gridshells to point loading. This justifies the observed difficulty in form finding. Nevertheless, we close with an exploration of concatenating multiple elastic gridshell building blocks.

We have also studied what sets the rigidity of elastic gridshells under point load indentation, finding scaling law in terms of the dimension of the structure and the number of the rods it contains, as well as the geometric and material properties of the individual rods. Our proposed empirical relation for the rigidity also points to the underlying nonlocal nature of the mechanical response of gridshells, in contrast to the local response of isotropic continuum shells. We further assess this nonlocality by quantifying the resulting radial displacement field as well as inspecting the effect of the location of the indentation point on the rigidity

Publications:
• C. Baek, A.O. Sageman-Furnas, M.K. Jawed, and P.M. Reis, “Form finding in elastic gridshells” PNAS, 115, 75-80 (2018). [html, pdf].

• C. Baek and P.M. Reis, “Rigidity of hemispherical elastic gridshells under point load indentation” Journal of the Mechanics and Physics of Solids, 124, 411-426 (2019). [html, pdf].


Buckling patterns in biaxially pre-stretched bilayer shells

with: Rashed Al-Rashed, Francisco López Jiménez, and Joel Marthelot

 



We introduce a new experimental system to study the effects of pre-stretch on the buckling patterns that emerge from the biaxial compression of elastomeric bilayer shells. Upon fabrication of the samples, releasing the pre-stretch in the substrate through deflation places the outer film in a state of biaxial compression and yields a variety of buckling patterns. We systematically explore the parameter space by varying the pre-stretch of the substrate and the ratio between the stiffness of the substrate and film. The phase diagram of the system exhibits a variety of buckling patterns: from the classic periodic wrinkle to creases, folds, and high aspect ratio ridges. Our system is capable of readily transitioning between these buckling patterns, a first for biaxial systems. We focus on the wrinkle to ridge transition. In the latter, we find that pre-stretch plays an essential role and that the ridge geometry (width, height) remains nearly constant throughout their formation process. For the localized ridged patterns, we find that the propagation of the ridge tip depends strongly on both strain and stiffness ratio, in a way that is akin to hierarchical fracture.

We have also studied what sets the rigidity of elastic gridshells under point load indentation, finding scaling law in terms of the dimension of the structure and the number of the rods it contains, as well as the geometric and material properties of the individual rods. Our proposed empirical relation for the rigidity also points to the underlying nonlocal nature of the mechanical response of gridshells, in contrast to the local response of isotropic continuum shells. We further assess this nonlocality by quantifying the resulting radial displacement field as well as inspecting the effect of the location of the indentation point on the rigidity

Publications:
• Rashed Al-Rashed, Francisco López Jiménez, Joel Marthelot and Pedro M. Reis, “Buckling patterns in biaxially pre-stretched bilayer shells: wrinkles, creases, folds and fracture-like ridges” Soft Matter, 13, 7969-7978 (2017). [html, pdf].


Reversible patterning of spherical shells through constrained buckling

with: Joel Marthelot, Pierre-Thomas Brun, and Francisco López Jiménez

 




Recent advances in active soft structures envision the large deformations resulting from mechanical instabilities as routes for functional shape-morphing. Numerous such examples exist for filamentary and plate systems. However, examples with double-curved shells are rarer, with progress hampered by challenges in fabrication and the complexities involved in analyzing their underlying geometrical nonlinearities. We show that on-demand patterning of hemispherical shells can be achieved through constrained buckling. Their post-buckling response is stabilized by an inner rigid mandrel. Through a combination of experiments, simulations and scaling analyses, our investigation focuses on the nucleation and evolution of the buckling patterns into a reticulated network of sharp ridges. The geometry of the system, namely the shell radius and the gap between the shell and the mandrel, is found to be the primary ingredient to set the surface morphology. This prominence of geometry suggests a robust, scalable, and tunable mechanism for reversible shape-morphing of elastic shells.

Videos that summarize this project can be found in the following links [Video1] [Video2].

Publications:
• J. Marthelot, P.-T. Brun, F. López Jiménez and P.M. Reis, “Reversible patterning of spherical shells through constrained buckling” Phys. Rev. Materials, 1, 025601 (2017). [html, pdf].


Imperfection sensitivity: Critical buckling pressure of precisely imperfect shells

with: Anna Lee, Francisco López Jiménez, Joel Marthelot, and John W. Hutchinson





We study the effect of a dimplelike geometric imperfection on the critical buckling load of spherical elastic shells under pressure loading. This is a canonical problem in Structural Mechanics that has been longstanding for more than 50 years. Our investigation combines precision experiments, finite element modeling, and numerical solutions of a reduced shell theory, all of which are found to be in excellent quantitative agreement. In the experiments, the geometry and magnitude of the defect can be designed and precisely fabricated through a customizable rapid prototyping technique. Our primary focus is on predictively describing the imperfection sensitivity of the shell to provide a quantitative relation between its knockdown factor and the amplitude of the defect. In addition, we find that the buckling pressure becomes independent of the amplitude of the defect beyond a critical value. The level and onset of this plateau are quantified systematically and found to be affected by a single geometric parameter that depends on both the radius-to-thickness ratio of the shell and the angular width of the defect.

To the best of our knowledge, this is the first time that experimental results on the knockdown factors of imperfect spherical shells have been accurately predicted, through both finite element modeling and shell theory solutions.

A video that summarizes this project can be found in the following link [Video].

This project was done in collaboration with John Hutchinson (Harvard University).

Publications:
• J. Marthelot, F. López Jiménez, A. Lee, J.W. Hutchinson and P.M. Reis, “Buckling of a pressurized hemispherical shell subjected to a probing force” J. Appl. Mech., 84, 121005 (2017). [html, pdf]
• F. López Jiménez, J. Marthelot, A. Lee, J.W. Hutchinson and P.M. Reis, “Technical brief: Knockdown factor for the buckling of spherical shells containing large-amplitude geometric defects” J. Appl. Mech., 84, 034501 (2017). [html, pdf]
• A. Lee, F. López Jiménez, J. Marthelot, J.W. Hutchinson and P.M. Reis “The geometric role of precisely engineered imperfections on the critical buckling load of spherical elastic shells” J. Appl. Mech., 83(11), 111005 (2016). [html, pdf]


Rapid fabrication of slender elastic shells

with: Anna Lee, Pierre-Thomas Brun, Joel Marthelot, Gioele Balestra, and François Gallaire




Various manufacturing techniques exist to produce double-curvature shells, including injection, rotational and blow molding, as well as dip coating. However, these industrial processes are typically geared for mass production and are not directly applicable to laboratory research settings, where adaptable, inexpensive and predictable prototyping tools are desirable.

In this project we have studied the rapid fabrication of hemispherical elastic shells by coating a curved surface with a polymer solution that yields a nearly uniform shell, upon polymerization of the resulting thin film. We experimentally characterize how the curing of the polymer affects its drainage dynamics and eventually selects the shell thickness. The coating process is then rationalized through a theoretical analysis that predicts the final thickness, in quantitative agreement with experiments and numerical simulations of the lubrication flow field. This robust fabrication framework should be invaluable for future studies on the mechanics of thin elastic shells and their intrinsic geometric nonlinearities.

A video that summarizes this project can be found in the following link [Video].

This project was done in collaboration with the group of François Gallaire at EPFL (Switerzland).

Publications:
• A. Lee, P. -T. Brun, J. Marthelot, G. Balestra, F. Gallaire and P. M. Reis, “Fabrication of slender elastic shells by the coating of curved surfaces” Nature Communications, 7, 11155 (2016). [html, pdf]

Press coverage:
• Jennifer Chu, “New theory, inspired by chocolate coatings, predicts thickness of thin shells” MIT News 4/5/2016. Featured video [here].
Spotlight on MIT’s Homepage. 5/4/2016


Mechanics of thin elastic shells: Geometry-Induced Rigidity and Localization

with: Arnaud Lazarus, Bastiaan Florijn, Amin Ajdari and Ashkan Vaziri

 


If one compresses an eggshell along its major axis, the shell is strikingly rigid and it is extremely challenging to break it with our bare hands. Conversely, if the eggshell is compressed along its equator, the resulting deflections are larger and, past a critical load one is typically able to fracture it. We have rationalized this difference in the rigidity of an eggshell depending on the shell-load orientation to be due to the local geometry near the points of indentation.

We have introduced a predictive framework for the rigidity of thin elastic shells which can also account for the situation when the shell is over-pressurized. Our concept of Geometry-Induced Rigidity can be used in reverse, as a precision non-destructive tool, to measure parameters of a shell (e.g. thickness) upon knowing the geometry of the underlying surface and the local mechanical response. The scale-invariance of Geometry-Induced Rigidity suggests that our framework should find uses across length scales: from the mechanical testing of viral capsids through Atomic Force Microscopy, to ocular tonometry procedures or in the design of architectural shells. All this work was inspired by the remarkable physics of an elegant eggshell!




More recently, we have been studying the emergence and evolution of point and linear-like loci of localization on thin shells indented well into the nonlinear regime. For large enough indentation, sharp points of localized curvature form, which we refer to as ‘s-cones’ (for shell-cones), in contrast with their developable cousins in plates, ‘d-cones’. Through experiments and FEM, e have found that the shape of the indenter has a significant effect on the mechanical response and that there is a qualitative different between sharp and blunt indenters. Given the importance of geometry and the scale-invariance of this problem, our results should find uses at the microscale, e.g. for AFM, where it is crucial to understand how the curvature of the tip, relative to the object being indented, affects the mechanical response.

Videos of S-cones of a thin shell under indentation: [Experiments, FEM Simulations]

Publications:
• A. Lazarus, H. C. B. Florijn, and P. M. Reis “Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells” Phys. Rev. Lett, 109 144301 (2012) [html, pdf] (Cover [pdf], Editor’s Suggestion and Physics Focus).
• A. Nasto, A. Ajdari, A. Lazarus, A. Vaziri, and P.M. Reis, “Localization of deformation in thin shells under indentation” Soft Matter 9, 6796 (2013). [html, pdf]. (Special Themed Issue on “Emerging Investigators in Soft Matter”).
• A. Nasto and P.M. Reis “Localized Structures in Indented Shells: a Numerical Investigation” J. App. Mech., 81 121008 (2014). [html, pdf].

Press coverage:
• Don Monroe “Connecting a Thin-Shell’s Stiffness with Its Geometry” Physics 5, 110 (2012). [html] • Jennifer Ouellette, “Cracking Eggs 101” Slate, September 2012, 12th. [html] • Jennifer Ouellette “Walking on Eggshells: Anatomy of a Science Story“Scientific American, September, 12th, 2012. [html] • Mike Lucibella “Sharper Curve, Stronger Egg” Physics Central, September, 7th, 2012. [html] • David Larousserie “Pourquoi l’ouef a-t-il la tête dure?“, Le Monde, September 24th, 2012. [html]


The Buckliball and Buckligami: buckling-induced encapsulation and soft Actuation

with: Jongmin Shim, Elizabeth Chen, Claude Perdigou and Katia Bertoldi




We introduce a class of continuum shell structures, the Buckliball, which undergoes folding induced by buckling under pressure loading. The geometry of the Buckliball consists of a spherical shell patterned with a regular array of circular voids. Topological constraints set that the possible number and arrangement of these voids are found to be restricted to five and only five specific configurations. Below a critical internal pressure, the narrow ligaments between the voids buckle, leading to a cooperative buckling cascade of the skeleton of the ball. This leads to closure of the voids and a reduction of the total volume of the shell by up to 54\%, while remaining spherical, thereby opening the possibility of encapsulation. Mechanical instabilities, which are often associated with failure in engineering, are here turned into an asset for functionality.

Video of the Buckliball in action: [Movie]




In a separate study, we have introduced a new class of soft actuators based on the auxetic behavior of patterned cylindrical shells containing a layout of voids that can be designed to reversibly achieve flexural or twisting motion. Depressurizing our samples allows for tunable and controllable motion. Given that the deformation is primarily governed by the geometry of the design, coupled to the buckling of the thin ligaments of the pattern, the resulting modes of actuation should be readily scalable.

Publications:
• J. Shim, C. Perdigou, E.R. Chen, K. Bertoldi and P.M. Reis , “Buckling induced encapsulation of structured elastic shells under pressure” Proc. Natl. Acad. Sci. U.S.A. 109, 16 (2012). [html, pdf] (Supplementary Information pdf).
• A. Lazarus and P.M. Reis “Soft Actuation of Structured Cylinders through Auxetic Behavior” Adv. Eng. Mater. 17, 815-820 (2015) [html, pdf].

Press coverage:
• “Buckle In‘” MIT News, March 26th, 2012.
• Kim Krieger, “Extreme mechanics: Buckling down“, Nature 488, 146 (2012). [html, pdf].


Wrinklons as Building-blocks in Wrinkling Cascades:
From Curtains to Graphene Sheets




We show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles. From simple geometry arguments and energy scalings, we develop a formalism based on wrinklons (the transition zones in the merging of two wrinkles) as building-blocks of the global pattern. Contrary to the case of crumpled paper where elastic energy is focused, this transition is described as smooth in agreement with a recent numerical work by B. Davidovich et al. This formalism is validated through experiments from hundreds of nm for graphene sheets to meters for ordinary curtains, which shows the universality of our description. We finally describe the effect of an external tension to the distribution of the wrinkles.

Publications:
• H. Vandeparre, M. Pineirua, F. Brau, B. Roman, J. Bico, C. Gay, W. Bao, C.N. Lau, P.M. Reis and P. Damman “Wrinkling Hierarchy in Constrained Thin Sheets from Suspended Graphene to Curtains” Phys. Rev. Lett, 106 224301 (2011) [html, pdf]. Cover Story, [pdf].

Press coverage:
• “A hierarchy of wrinkles” Physics Synopses, June 2, 2011.
• “Introducing the ‘wrinklon’” Physicsworld, Jun 20, 2011.


Rolling of Flexible Ribbons

with: Pascal Raux, John Bush and Christophe Clanet




Galileo’s study of rigid spheres rolling down an inclined ramp is often considered as the starting point of modern physics, since it involves both theory and experiment. In this study we consider a variant of Galileo’s problem in which the ramp is rigid but the rolling body, an elastic cylindrical shell, is thin, flexible and therefore deformable. Particular attention is given to characterizing the steady shapes that arise in static and dynamic rolling configurations. In both cases, above a critical value of the forcing (either gravitational or centrifugal), the ribbon assumes a two-lobed peanut shape. Our theoretical model allows us to rationalize the observed shapes through consideration of the ribbon’s bending and stretching in response to the applied forcing. This dynamical elastic problem presents some common features with the rolling of a liquid drop on a hydrophobic surface or a lubricated ramp.

Publications:
• P.S. Raux, P.M. Reis, J.W.M. Bush, and C. Clanet, “Rolling ribbons” Phys. Rev. Lett. 105 044301 (2010) [html, pdf]. (selected as “Editor Suggestion“).

Press coverage:
• The Physics of a Rolling Rubber Band, Science Now, 28 July (2010).
• Rolling ribbons get the bends, Physics Today, 22 July (2010).
Why a rolling rubber band squashes, Physics Synopsis, July 23 (2010).
Galileo revisited: How ribbons roll, MITNews, September 3 (2010).


Delamination of thin films from an elastic substrate

with: Dominic Vella, Benoit Roman, José Bico and Arezki Boudaoud




The wrinkling and delamination of stiff thin films adhered to a polymer substrate have important applications in `flexible electronics’. The resulting periodic structures, when used for circuitry, have remarkable mechanical properties since stretching or twisting of the substrate is mostly accommodated through bending of the film, which minimizes fatigue or fracture. To date, applications in this context have used substrate patterning to create an anisotropic substrate-film adhesion energy, thereby producing a controlled array of delamination `blisters’. However, even in the absence of such patterning, blisters appear spontaneously, with a characteristic size. Here, we perform well-controlled experiments at macroscopic scales to study what sets the dimensions of these blisters in terms of the material properties and explain our results using a combination of scaling and analytical methods. As well as pointing to a novel method for determining the interfacial toughness our analysis suggests a number of design guidelines for the thin films used in flexible electronic applications. Crucially, we show that to avoid the possibility that delamination may cause fatigue damage, the thin film thickness must be greater than a critical value, which we determine. [Video here]

Publications:
• D. Vella, J. Bico, A. Boudadoud, B. Roman and P.M. Reis, “Delamination of thin elastic sheets adhered to an elastic substrate”, Proc. Natl. Acad. Sci. U.S.A. 106, 10901 (2009) [html, pdf].

Press coverage:
MIT Press release.
• “A curvy, stretchy future for electronics”, John A. Rogers and Yonggang Huang, Commentry, Proc. Natl. Acad. Sci. U.S.A. 106, 10875 (2009) [html, pdf].
• “High Flex: failing stickers lead to research that could improve stretchable electronicsTechnology Review, September/October 2009.
• Online press coverage of our article on delamination blisters [A22]: Nanowerk, Science Daily, EU-Cordis, Inside Engineer, ZDNet, Gizmag, Physorg, Frost & Sullivan.
• Small-scale thin film experiments can provide models for large-scale engineering applications. CEE Newsletter “On Balance” (October 2009) [pdf].