Fluid-structure interactions of bristled wings: the trade-off between weight and drag.

The smallest flying insects often have bristled wings resembling feathers or combs. We combined experiments and three-dimensional numerical simulations to investigate the trade-off between wing weight and drag generation. In experiments of bristled strips, a reduced physical model of the bristled wing, we found that the elasto-viscous number indicates when reconfiguration occurs in the bristles. Analysis of existing biological data suggested that bristled wings of miniature insects lie below the reconfiguration threshold, thus avoiding drag reduction. Numerical simulations of bristled strips showed that there exist optimal numbers of bristles that maximize the weighted drag when the additional volume due to the bristles is taken into account. We found a scaling relationship between the rescaled optimal numbers and the dimensionless bristle length. This result agrees qualitatively with and provides an upper bound for the bristled wing morphological data analysed in this study.

Magneto-active elastic shells with tunable buckling strength.

Shell buckling is central in many biological structures and advanced functional materials, even if, traditionally, this elastic instability has been regarded as a catastrophic phenomenon to be avoided for engineering structures. Either way, predicting critical buckling conditions remains a long-standing challenge. The subcritical nature of shell buckling imparts extreme sensitivity to material and geometric imperfections. Consequently, measured critical loads are inevitably lower than classic theoretical predictions. Here, we present a robust mechanism to dynamically tune the buckling strength of shells, exploiting the coupling between mechanics and magnetism. Our experiments on pressurized spherical shells made of a hard-magnetic elastomer demonstrate the tunability of their buckling pressure via magnetic actuation. We develop a theoretical model for thin magnetic elastic shells, which rationalizes the underlying mechanism, in excellent agreement with experiments. A dimensionless magneto-elastic buckling number is recognized as the key governing parameter, combining the geometric, mechanical, and magnetic properties of the system.

Nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature.

In this work, we consider the stability of a spherical shell under combined loading from a uniform external pressure and a homogenous natural curvature. Nonmechanical stimuli, such as one that tends to modify the rest curvature of an elastic body, are prevalent in a wide range of natural and engineered systems, and may occur due to thermal expansion, changes in pH, differential swelling, and differential growth. Here we investigate how the presence of both an evolving natural curvature and an external pressure modifies the stability of a complete spherical shell. We show that due to a mechanical analogy between pressure and curvature, positive natural curvatures can severely destabilize a thin shell, while negative natural curvatures can strengthen the shell against buckling, providing the possibility to design shells that buckle at or above the theoretical limit for pressure alone, i.e., a strengthening factor. These results extend directly from the classical analysis of the stability of shells under pressure, and highlight the important role that nonmechanical stimuli can have on modifying the membrane state of stress in a thin shell.

Evolution of critical buckling conditions in imperfect bilayer shells through residual swelling.

We propose and investigate a minimal mechanism that makes use of differential swelling to modify the critical buckling conditions of elastic bilayer shells, as measured by the knockdown factor. Our shells contain an engineered defect at the north pole and are made of two layers of different crosslinked polymers that exchange free molecular chains. Depending on the size of the defect and the extent of swelling, we can observe either a decreasing or increasing knockdown factor. FEM simulations are performed using a reduced model for the swelling process to aid us in rationalizing the underlying mechanism, providing a qualitative agreement with experiments. We believe that the working principle of our mechanism can be extended to bimetallic shells undergoing variations in temperature and to shells made of pH-responsive gels, where the change in knockdown factor could be changed dynamically.

Buckling of geometrically confined shells.

We study the periodic buckling patterns that emerge when elastic shells are subjected to geometric confinement. Residual swelling provides access to range of shapes (saddles, rolled sheets, cylinders, and spherical sections) which vary in their extrinsic and intrinsic curvatures. Our experimental and numerical data show that when these moderately thick structures are radially confined, a single geometric parameter - the ratio of the total shell radius to the amount of unconstrained material - predicts the number of lobes formed. We present a model that interprets this scaling as the competition between radial and circumferential bending. Next, we show that reducing the transverse confinement of saddles causes the lobe number to decrease with a similar scaling analysis. Hence, one geometric parameter captures the wave number through a wide range of radial and transverse confinement, connecting the shell shape to the shape of the boundary that confines it. We expect these results to be relevant for an expanse of shell shapes, and thus applicable to the design of shape-shifting materials and the swelling and growth of soft structures.

Curvature-Induced Instabilities of Shells.

Induced by proteins within the cell membrane or by differential growth, heating, or swelling, spontaneous curvatures can drastically affect the morphology of thin bodies and induce mechanical instabilities. Yet, the interaction of spontaneous curvature and geometric frustration in curved shells remains poorly understood. Via a combination of precision experiments on elastomeric spherical shells, simulations, and theory, we show how a spontaneous curvature induces a rotational symmetry-breaking buckling as well as a snapping instability reminiscent of the Venus fly trap closure mechanism. The instabilities, and their dependence on geometry, are rationalized by reducing the spontaneous curvature to an effective mechanical load. This formulation reveals a combined pressurelike term in the bulk and a torquelike term in the boundary, allowing scaling predictions for the instabilities that are in excellent agreement with experiments and simulations. Moreover, the effective pressure analogy suggests a curvature-induced subcritical buckling in closed shells. We determine the critical buckling curvature via a linear stability analysis that accounts for the combination of residual membrane and bending stresses. The prominent role of geometry in our findings suggests the applicability of the results over a wide range of scales.

Curvature-driven morphing of non-Euclidean shells.

We investigate how thin structures change their shape in response to non-mechanical stimuli that can be interpreted as variations in the structure's natural curvature. Starting from the theory of non-Euclidean plates and shells, we derive an effective model that reduces a three-dimensional stimulus to the natural fundamental forms of the mid-surface of the structure, incorporating expansion, or growth, in the thickness. Then, we apply the model to a variety of thin bodies, from flat plates to spherical shells, obtaining excellent agreement between theory and numerics. We show how cylinders and cones can either bend more or unroll, and eventually snap and rotate. We also study the nearly isometric deformations of a spherical shell and describe how this shape change is ruled by the geometry of a spindle. As the derived results stem from a purely geometrical model, they are general and scalable.

Geometry and mechanics of thin growing bilayers.

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness that describes a sheet in terms of both thickness and plate shape. We find that the mean curvature of the isometric state is three fourths the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.

Morphing of geometric composites via residual swelling.

Understanding and controlling the shape of thin, soft objects has been the focus of significant research efforts among physicists, biologists, and engineers in the last decade. These studies aim to utilize advanced materials in novel, adaptive ways such as fabricating smart actuators or mimicking living tissues. Here, we present the controlled growth-like morphing of 2D sheets into 3D shapes by preparing geometric composite structures that deform by residual swelling. The morphing of these geometric composites is dictated by both swelling and geometry, with diffusion controlling the swelling-induced actuation, and geometric confinement dictating the structure's deformed shape. Building on a simple mechanical analog, we present an analytical model that quantitatively describes how the Gaussian and mean curvatures of a thin disk are affected by the interplay among geometry, mechanics, and swelling. This model is in excellent agreement with our experiments and numerics. We show that the dynamics of residual swelling is dictated by a competition between two characteristic diffusive length scales governed by geometry. Our results provide the first 2D analog of Timoshenko's classical formula for the thermal bending of bimetallic beams - our generalization explains how the Gaussian curvature of a 2D geometric composite is affected by geometry and elasticity. The understanding conferred by these results suggests that the controlled shaping of geometric composites may provide a simple complement to traditional manufacturing techniques.

Anisotropic swelling of thin gel sheets.

We describe the anisotropic swelling within the Flory-Rehner thermodynamic model through an extension of the elastic component of the free-energy, which takes into account the oriented hampering of the swelling-induced deformations due to the presence of stiffer fibers. We also characterize the homogeneous free-swelling solutions of the corresponding anisotropic stress-diffusion problem, and discuss an asymptotic approximation of the key equations, which allows us to explicitly derive the anisotropic solution of the problem. We propose a proof-of-concept of our model, realizing thin bilayered gel sheets with layers having different anisotropic structures. In particular, for seedpod-like sheets, we observe and quantitatively measure the helicoid versus ribbon transition determined by the aspect ratio of the composite sheet.