Session Q52: Focus Session: Extreme Mechanics - Shells & Snapping

Mechanics and Dynamics of a Snapping Arch

Snap-buckling of geometric arches and thin spherical shells occur in a variety of different geometric situations, exhibiting a highly nonlinear response dictated by the geometry and material properties of the system. As this elastic instability often precedes the catastrophic failure of a mechanical system, significant work has focused on the stability criteria for such structures. In order to properly understand the biomechanics of plants that rely on this instability, and in addition use snap-buckling in the design of advanced materials, it is necessary to also develop a fundamental understanding of the timescale and post-buckling response of a snapping structure. Currently, a fundamental understanding of the osmotically-induced snap-buckling phenomena is lacking. In this presentation, we examine the osmotic swelling of a bistable arch to identify the stability criteria, relevant snap-through timescale, and the impact of geometric confinement on snap-through symmetry and damping.   

Elastocapillary-driven snap-through instability

The snap-through instability, which is present in a wide range of systems ranging from carnivorous plants to MEMS, is a well-known phenomenon in solid mechanics : when a buckled elastic beam is subjected to a transverse force, above a critical load value the buckling mode is switched. Here, we revisit this phenomenon by studying snap-through under capillary forces. In our experiment, a droplet (which replaces the usual dry load) is deposited on a buckled thin strip, clamped horizontally at both ends. In this setup both the weight of the drop and capillary forces jointly act toward the instability. The possibility of reverse elastocapillary snap-through, where the droplet is put under the beam, is then tested and successfully observed, showing the predominance of capillary forces at small enough scales.   

Fast Motion of Plants through mechanical instability: Mechanics without Muscles

Plants are not well known for fast motions, yet some plants such as the Venus flytrap can move in a fraction of a second to capture insects, even though they do not have nerves or muscles. This type of rapid motion has intrigued scientists for centuries. Darwin did a first systematic study on the trap closure mechanism, and considered the plant as ``one of the most wonderful in the world". Thereafter, several physical mechanisms have been proposed, such as the rapid loss of turgor pressure, an irreversible acid-induced wall loosening mechanism, and the snap-through model by mechanical instability, but with no unanimous agreement among researchers. Here we propose a coupled mechanical bistable mechanism that explains the rapid closure of the Venus flytrap in a comprehensive manner, consistent with a series of experimental observations. Such bistabile behaviors are theoretically modeled and validated with table-top experiments. Based on the principles learnt from the Venus flytrap, we are also able to manufacture a preliminary ``flytrap robot''. Hence, it is promising to design smart bio-mimetic materials and devices with snapping mechanisms as sensors, actuators, artificial muscles and biomedical devices.   

Strange instabilities of simple elastic structures

A class of simple elastic structures is shown exhibiting bifurcation and instability under tensile dead loading, multiple bifurcations, and softening/hardening behaviour in the postcritical regime. These structures evidences new and unexpected behaviours which are theoretically predicted and experimentally verified. These nonlinear behaviours can be exploited in the design of flexible mechanics devices and open new perspective in the control of vibrations.   

Materials with Tunable Behavior due to Constrained Instabilities: Performance and Stability Analysis

Combining several materials into a composite permits the creation of new materials with overall properties tunable via a careful choice of the constituent materials with favorable specifics. The probably simplest example is a particle-matrix system, in which particles of one material enhance the mechanical behavior of the matrix material. Recent advances have confirmed that the overall performance of such a composite (e.g., its viscoelastic properties) can be dramatically altered, and stiffness and damping can be tuned to an extreme if one allows for temporarily negative elastic moduli in the inclusion. Such incremental negative moduli imply instability; e.g. a free-standing body of negative stiffness is thermodynamically unstable. However, through its geometric constraint a matrix phase can stabilize the otherwise unstable state of the inclusion phase, thus rendering the overall composite stable. In this contribution, we show, based on dynamic stability analyses, that the matrix constraint does indeed allow for the existence and use of negative moduli, and that this effect can be utilized to design novel composites of superior performance. Approaches to stabilize the negative-stiffness effect will be discussed as well as the performance of such composites.   

Snap-Through of Graphene: An Elasto-Capillary Perspective

Understanding the interaction between graphene flakes and various substrates is of crucial importance for nanoelectromechanical systems (NEMS) applications, among others. The `snap-through' instability of graphene flakes placed onto corrugated substrates has recently received much attention as a potential assay for the study of this interaction. A sharp transition has been found in the morphology of the graphene between a) closely adhering to the corrugations of the substrate and b) lying almost completely flat on top. Which of these morphologies is observed depends on the geometry of the substrate and the mechanical properties of the flake. In this talk we shall focus on understanding the nature of this transition and, in particular, the 'sharpness' of the transition. We investigate how the location of snap through and its sharpness might be used to yield estimates of adhesion strength and friction with the substrate.   

Electromechanical phase transition in dielectric elastomers under uniaxial tension and electrical voltage

Subject to forces and voltage, a dielectric elastomer may undergo electromechanical phase transition. A phase diagram is constructed for an ideal dielectric elastomer membrane under uniaxial force and voltage, reminiscent of the phase diagram for liquid-vapor transition of a pure substance. We identify a critical point for the electromechanical phase transition. Two states of deformation (thick and thin) may coexist during the phase transition, with the mismatch in lateral stretch accommodated by wrinkling of the membrane in the thin state. The processes of electromechanical phase transition under various conditions are discussed. A reversible cycle is suggested for electromechanical energy conversion using the dielectric elastomer membrane, analogous to the classical Carnot cycle for a heat engine. The amount of energy conversion, however, is limited by failure of the dielectric elastomer due to electrical breakdown. With a particular combination of material properties, the electromechanical energy conversion can be significantly extended by taking advantage of the phase transition without electrical breakdown.   

Soft Dielectrics: Heterogeneity and Instabilities

Dielectric Elastomers are capable of large deformations in response to electrical stimuli. Heterogeneous soft dielectrics with proper microstructures demonstrate much stronger electromechanical coupling than their homogeneous constituents. In turn, the heterogeneity is an origin for instability developments leading to drastic change in the composite microstructure. In this talk, the electromechanical instabilities are considered. Stability of anisotropic soft dielectrics is analyzed. Ways to achieve giant deformations and manipulating extreme material properties are discussed. 1. S. Rudykh and G. deBotton, ``Instabilities of Hyperelastic Fiber Composites: Micromechanical Versus Numerical Analyses.'' Journal of Elasticity, 2011. http://dx.doi.org/2010.1007/s10659-011-9313-x 2. S. Rudykh, K. Bhattacharya and G. deBotton, ``Snap-through actuation of thick-wall electroactive balloons.'' International Journal of Non-Linear Mechanics, 2011. http://dx.doi.org/10.1016/j.ijnonlinmec.2011.05.006 3. S. Rudykh and G. deBotton, ``Stability of Anisotropic Electroactive Polymers with Application to Layered Media.'' Zeitschrift f\"ur angewandte Mathematik und Physik, 2011. http://dx.doi.org/10.1007/s00033-011-0136-1 4. S. Rudykh, A. Lewinstein, G. Uner and G. deBotton, ``Giant Enhancement of the Electromechanical Coupling in Soft Heterogeneous Dielectrics.'' 2011 http://arxiv.org/abs/1105.4217v1   

Modeling of dielectric elastomeric materials: theory, finite element simulation, and applications

Elastomeric materials that undergo large deformations in response to an electric field have garnered attention in recent years. Applications of these dielectric elastomeric materials include actuators capable of converting electrical energy to mechanical work and energy harvesting devices that convert mechanical energy into electrical energy. Furthermore, dielectric elastomers exhibit interesting instabilities, especially under constrained geometries, opening the door for possible applications in active surfaces. Interest has increased in the mechanics community concerning the formulation of a finite-deformation constitutive theory for an electro-mechanically-coupled material. While the details of the formulation of such a theory are beginning to come into focus in the literature, numerical techniques for solving these equations are in their infancy. In this work, we have developed a finite-element-based numerical simulation capability for dielectric elastomers. This talk will highlight the application of our numerical simulation capability to dielectric elastomeric actuators, energy harvesting devices, as well instabilities of small dielectric elastomeric structures on a constraining substrate.   

Giant linear voltage-induced deformation of a dielectric elastomer actuator

For dielectric elastomers, one of the most conspicuous attributes is large deformation of actuation induced by voltage. However, electromechanical instability may limit their deformation. In this seminar, I will illustrate how dielectric elastomers survive or eliminate electromechanical instability, through mechanical designs. For example, I will analyze a dielectric elastomer with a ``pure shear'' boundary condition. The membrane is first prestretched along the transverse direction, and then fixed by a rigid bar. As a result, the stretch in transverse direction is fixed, and the membrane can only be actuated along the vertical direction. The theory shows that the actuator can avert electromechanical instability, and achieve a giant linear deformation of actuation. The experiments confirm the theoretical predictions. For SEBS material, the linear strain of actuation can be 80{\%}. For VHB material, the linear strain of actuation can be 300{\%}. The actuator shows advantages compared to the classic designs (say, tube and circular actuators), and can be used as artificial muscles in soft robots.   

Geometry-induced rigidity in pressurized elastic shells

We study the indentation of pressurized thin elastic shells, with positive Gauss curvature. In our precision desktop-scale experiments, the geometry of the shells and their material properties are custom-controlled using rapid prototyping and digital fabrication techniques. The mechanical response is quantified through load-displacement compression tests and the differential pressure is set by a syringe-pump system under feedback control. Focus is given to the linear regime of the response towards quantifying the geometry-induced rigidity of pressurized shells with different shapes. We find that this effective stiffness is proportional to the local mean curvature in the neighborhood of the locus of indentation. Combining classic theory of shells with recent developments by D. Vella et al. (2011), we rationalize the dependence of the geometry-induced rigidity on: i) the mean curvature at the point of indentation, ii) the material properties of the shell and iii) the in-out differential pressure. The proposed predictive framework is in excellent agreement with our experiments, over a wide range of control parameters. The prominence of geometry in this class of problems points to the relevance and applicability of our results over a wide range of lengthscales.   

S-cones in thin shells under indentation

We perform a hybrid experimental and numerical investigation of the localization of deformation in indented thin spherical elastic shells. Past the initial linear response, an inverted cap develops as a Pogorelov circular ridge. For further indentation, this ridge looses axis-symmetry and sharp points of localized curvature form. We refer to these localized objects as \emph{s-cones} (for shell-cones), in contrast with their developable cousins in plates (d-cones). We quantify the effect of systematically varying the indenter's radius of curvature (from point to plate load) on the formation and evolution of s-cones. In our precision desktop-scale experiments we use rapid prototyped elastomeric shells and rigid indenters of various shape. The mechanical response is measured through load-displacement compression tests and the deformation process is further characterized through digital imaging. In parallel, the experimental results are contrasted against nonlinear Finite Element simulations. Merging these two complementary approaches allows us to gain further physical insight towards rationalizing this geometrically nonlinear process.   

Folding and buckling pathways in spherical shells with soft spots

Thin elastic spherical shells subject to an external pressure undergo a buckling transition when the pressure reaches a critical value. Past the buckling instability, the shell typically takes on a shape with one or more inversions that focus the elastic deformation energy within narrow circular regions on the sphere. These inversions are associated with large volume changes and hysteresis, and their location is highly sensitive to very slight imperfections in the sphere. Recently, it has been demonstrated [1] that natural pollen grains have evolved soft sectors in their hard outer walls which guide them toward particular folding pathways when their internal volume is reduced due to dessication, thus avoiding sudden and uncontrolled changes in shape. Motivated by these results, we study the effect of circular soft spots on the buckling of otherwise uniform spherical shells. Through a combination of scaling arguments and numerical simulations, we demonstrate that the shell can be tuned to follow distinct buckling pathways by varying the size and stiffness of the soft spot. [1] E. Katifori \textit{et al}, \textit{Proc. Natl. Acad. Sci. USA} \textbf{107}, 7635 (2010)   

Buckliballs: Buckling-Induced Pattern Transformation of Structured Elastic Shells

We present a class of continuum shell structures, the buckliball, which, undergo a structural transformation induced by buckling under pressure loading. The geometry of the buckliball comprises a spherical shell patterned with a regular array of circular voids. Moreover, we show that the buckling-induced pattern transformation is possible only with five specific hole arrangements. These voids are covered with a thin membrane, thereby making the ball air tight. Beyond a critical internal pressure, the thin ligaments between the voids buckle leading to a cooperative buckling cascade of the skeleton of the ball. Both precision desktop-scale experiments and finite element simulations are used to explore the underlying mechanics in detail and proof of concept of the proposed structures. We find excellent qualitative and quantitative agreement between experiments and simulations. This pattern transformation induced by a mechanical instability opens the possibility for reversible encapsulation, over a wide range of length scales.   

Wrinkling of a collapsing viscous bubble

Thin-sheets of sufficiently viscous liquid can behave similar to elastic sheets and buckle under certain external forces. A classic example is the ``parachute instability'' for which a ruptured viscous bubble wrinkles as it relaxes, with the explanation for the wrinkles being based on the liquid film falling under its own weight. In this talk we revisit the viscous bubble-bursting experiments and demonstrate that gravity is responsible for neither the collapse nor the resulting wrinkling instability. Using a combination of experiments and theory, we highlight the importance of capillary forces and elucidate their role in the wrinkling instability.