A reprogrammable mechanical metamaterial with stable memory.

Metamaterials are designed to realize exotic physical properties through the geometric arrangement of their underlying structural layout1,2. Traditional mechanical metamaterials achieve functionalities such as a target Poisson's ratio3 or shape transformation4-6 through unit-cell optimization7-9, often with spatial heterogeneity10-12. These functionalities are programmed into the layout of the metamaterial in a way that cannot be altered. Although recent efforts have produced means of tuning such properties post-fabrication13-19, they have not demonstrated mechanical reprogrammability analogous to that of digital devices, such as hard disk drives, in which each unit can be written to or read from in real time as required. Here we overcome this challenge by using a design framework for a tileable mechanical metamaterial with stable memory at the unit-cell level. Our design comprises an array of physical binary elements (m-bits), analogous to digital bits, with clearly delineated writing and reading phases. Each m-bit can be independently and reversibly switched between two stable states (acting as memory) using magnetic actuation to move between the equilibria of a bistable shell20-25. Under deformation, each state is associated with a distinctly different mechanical response that is fully elastic and can be reversibly cycled until the system is reprogrammed. Encoding a set of binary instructions onto the tiled array yields markedly different mechanical properties; specifically, the stiffness and strength can be made to range over an order of magnitude. We expect that the stable memory and on-demand reprogrammability of mechanical properties in this design paradigm will facilitate the development of advanced forms of mechanical metamaterials.

Mechanics of two filaments in tight orthogonal contact.

Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple, yet canonical, problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (µCT) and mechanical testing experiments are in excellent agreement with finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibit strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contact-pressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the µCT and FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small, but nonzero, tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures.

Regulation of the swimming kinematics of lampreys Petromyzon marinus across changes in viscosity.

The bodies of most swimming fishes are very flexible and deform as result of both external fluid dynamic forces and internal musculoskeletal forces. If fluid forces change, the body motion will also change unless the fish senses the change and alters its muscle activity to compensate. Lampreys and other fishes have mechanosensory cells in their spinal cords that allow them to sense how their body is bending. We hypothesized that lampreys (Petromyzon marinus) actively regulate body curvature to maintain a fairly constant swimming waveform even as swimming speed and fluid dynamic forces change. To test this hypothesis, we measured the steady swimming kinematics of lampreys swimming in normal water, and water in which the viscosity was increased by 10 or 20 times by adding methylcellulose. Increasing the viscosity over this range increases the drag coefficient, potentially increasing fluid forces up to 40%. Previous computational results suggested that if lampreys did not compensate for these forces, the swimming speed would drop by about 52%, the amplitude would drop by 39%, and posterior body curvature would increase by about 31%, while tail beat frequency would remain the same. Five juvenile sea lampreys were filmed swimming through still water, and midlines were digitized using standard techniques. Although swimming speed dropped by 44% from 1× to 10× viscosity, amplitude only decreased by 4%, and curvature increased by 7%, a much smaller change than the amount we estimated if there was no compensation. To examine the waveform overall, we performed a complex orthogonal decomposition and found that the first mode of the swimming waveform (the primary swimming pattern) did not change substantially, even at 20× viscosity. Thus, it appears that lampreys are compensating, at least partially, for the changes in viscosity, which in turn suggests that sensory feedback is involved in regulating the body waveform.

Snap buckling of bistable beams under combined mechanical and magnetic loading.

We investigate the mechanics of bistable, hard-magnetic, elastic beams, combining experiments, finite-element modelling (FEM) and a reduced-order theory. The beam is made of a hard magneto-rheological elastomer, comprising two segments with antiparallel magnetization along the centreline, and is set into a bistable curved configuration by imposing an end-to-end shortening. Reversible snapping is possible between these two stable states. First, we experimentally characterize the critical field strength for the onset of snapping, at different levels of end-to-end shortening. Second, we perform three-dimensional FEM simulations using the Riks method to analyse high-order deformation modes during snapping. Third, we develop a reduced-order centreline-based beam theory to rationalize the observed magneto-elastic response. The theory and simulations are validated against experiments, with an excellent quantitative agreement. Finally, we consider the case of combined magnetic loading and poking force, examining how the applied field affects the bistability and quantifying the maximum load-bearing capacity. Our work provides a set of predictive tools for the rational design of one-dimensional, bistable, magneto-elastic structural elements. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Probabilistic buckling of imperfect hemispherical shells containing a distribution of defects.

The buckling of spherical shells is plagued by a strong sensitivity to imperfections. Traditionally, imperfect shells tend to be characterized empirically by the knockdown factor, the ratio between the measured buckling strength and the corresponding classic prediction for a perfect shell. Recently, it has been demonstrated that the knockdown factor of a shell containing a single imperfection can be predicted when there is detailed a priori knowledge of the defect geometry. Still, addressing the analogous problem for a shell containing many defects remains an open question. Here, we use finite element simulations, which we validate against precision experiments, to investigate hemispherical shells containing a well-defined distribution of imperfections. Our goal is to characterize the resulting knockdown factor statistics. First, we study the buckling of shells containing only two defects, uncovering non-trivial regimes of interactions that echo existing findings for cylindrical shells. Then, we construct statistical ensembles of imperfect shells, whose defect amplitudes are sampled from a lognormal distribution. We find that a 3-parameter Weibull distribution is an excellent description for the measured statistics of knockdown factors, suggesting that shell buckling can be regarded as an extreme-value statistics phenomenon. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Smooth Triaxial Weaving with Naturally Curved Ribbons.

Triaxial weaving is a handicraft technique that has long been used to create curved structures using initially straight and flat ribbons. Weavers typically introduce discrete topological defects to produce nonzero Gaussian curvature, albeit with faceted surfaces. We demonstrate that, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously, which is not feasible using traditional techniques. Further, we reveal that the shape of the physical unit cells is dictated solely by the in-plane geometry of the ribbons, not elasticity. Finally, we leverage the geometry-driven nature of triaxial weaving to design a set of ribbon profiles to weave smooth spherical, ellipsoidal, and toroidal structures.

Bending Response of a Book with Internal Friction.

We study the bending of a booklike system, comprising a stack of elastic plates coupled through friction. The behavior of this layered system is rich and nontrivial, with a nonadditive enhancement of the apparent stiffness and a significant hysteretic response. A dimension reduction procedure is employed to develop a centerline-based theory describing the stack as a nonlinear planar rod with internal shear. We consider the coupling between the nonlinear geometry and the elasticity of the stacked plates, treating the interlayer friction perturbatively. This model yields predictions for the stack's mechanical response in three-point bending that are in excellent agreement with our experiments. Remarkably, we find that the energy dissipated during deformation can be rationalized over 3 orders of magnitude, including the regimes of a thick stack with large deflection. This robust dissipative mechanism could be harnessed to design new classes of low-cost and efficient damping devices.

Form finding in elastic gridshells.

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.

Printing on liquid elastomers.

In recent years the research community has paid significant attention to geometrically engineered materials. These materials derive their unique properties from their structure rather than their chemistry alone. Despite their success in the laboratory, the assembly of such soft functional materials remains an outstanding challenge. Here, we propose a robust fluid-mediated route for the rapid fabrication of soft elastomers architected with liquid inclusions. Our approach consists of depositing water drops at the surface of an immiscible liquid elastomer bath. As the elastomer cures, the drops are encapsulated in the polymer and impart shape and function to the newly formed elastic matrix. Using the framework of fluid mechanics, we show how this type of composite material can be tailored.

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.

Wrinkling crystallography on spherical surfaces.

We present the results of an experimental investigation on the crystallography of the dimpled patterns obtained through wrinkling of a curved elastic system. Our macroscopic samples comprise a thin hemispherical shell bound to an equally curved compliant substrate. Under compression, a crystalline pattern of dimples self-organizes on the surface of the shell. Stresses are relaxed by both out-of-surface buckling and the emergence of defects in the quasi-hexagonal pattern. Three-dimensional scanning is used to digitize the topography. Regarding the dimples as point-like packing units produces spherical Voronoi tessellations with cells that are polydisperse and distorted, away from their regular shapes. We analyze the structure of crystalline defects, as a function of system size. Disclinations are observed and, above a threshold value, dislocations proliferate rapidly with system size. Our samples exhibit striking similarities with other curved crystals of charged particles and colloids. Differences are also found and attributed to the far-from-equilibrium nature of our patterns due to the random and initially frozen material imperfections which act as nucleation points, the presence of a physical boundary which represents an additional source of stress, and the inability of dimples to rearrange during crystallization. Even if we do not have access to the exact form of the interdimple interaction, our experiments suggest a broader generality of previous results of curved crystallography and their robustness on the details of the interaction potential. Furthermore, our findings open the door to future studies on curved crystals far from equilibrium.

Buckling patterns in biaxially pre-stretched bilayer shells: wrinkles, creases, folds and fracture-like ridges.

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.

Coiling of elastic rods on rigid substrates.

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study.

Curvature-induced symmetry breaking determines elastic surface patterns.

Symmetry-breaking transitions associated with the buckling and folding of curved multilayered surfaces-which are common to a wide range of systems and processes such as embryogenesis, tissue differentiation and structure formation in heterogeneous thin films or on planetary surfaces-have been characterized experimentally. Yet owing to the nonlinearity of the underlying stretching and bending forces, the transitions cannot be reliably predicted by current theoretical models. Here, we report a generalized Swift-Hohenberg theory that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. By testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions for the critical curves separating labyrinth, hybrid and hexagonal phases. Furthermore, a comparison to earlier experiments suggests that the theory is universally applicable to macroscopic and microscopic systems. Our approach builds on general differential-geometry principles and can thus be extended to arbitrarily shaped surfaces.

Curvature-Controlled Defect Localization in Elastic Surface Crystals.

We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal, and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling, or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test bed for exploring the effects of geometrically induced forces on assemblies of topological charges.

Deformation of a soft helical filament in an axial flow at low Reynolds number.

We perform a numerical investigation of the deformation of a rotating helical filament subjected to an axial flow, under low Reynolds number conditions, motivated by the propulsion of bacteria using helical flagella. Given its slenderness, the helical rod is intrinsically soft and deforms due to the interplay between elastic forces and hydrodynamic loading. We make use of a previously developed and experimentally validated computational tool framework that models the elasticity of the filament using the discrete elastic rod method and the fluid forces are treated using Lighthill's slender body theory. Under axial flow, and in the absence of rotation, the initially helical rod is extended. Above a critical flow speed its configuration comprises a straight portion connected to a localized helix near the free end. When the rod is also rotated about its helical axis, propulsion is only possible in a finite range of angular velocity, with an upper bound that is limited by buckling of the soft helix arising due to viscous stresses. A systematic exploration of the parameter space allows us to quantify regimes for successful propulsion for a number of specific bacteria.

Buckling-induced encapsulation of structured elastic shells under pressure.

We introduce a class of continuum shell structures, the Buckliball, which undergoes 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. In order for the pattern transformation to be induced by buckling, the possible number and arrangement of these voids are found to be restricted to 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 ligament buckling 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. We use a combination of precision desktop-scale experiments, finite element simulations, and scaling analyses to explore the underlying mechanics of these foldable structures, finding excellent qualitative and quantitative agreement. Given that this folding mechanism is induced by a mechanical instability, our Buckliball opens the possibility for reversible encapsulation, over a wide range of length scales.

Buckling of an elastic rod embedded on an elastomeric matrix: planar vs. non-planar configurations.

We investigate the buckling of a slender rod embedded in a soft elastomeric matrix through a combination of experiments, numerics and theory. Depending on the control parameters, both planar wavy (2D) or non-planar coiled (3D) configurations are observed in the post-buckling regime. Our analytical and numerical results indicate that the rod buckles into 2D configurations when the compression forces associated to the two lowest critical modes are well separated. In contrast, 3D coiled configurations occur when the two buckling modes are triggered at onset, nearly simultaneously. We show that the separation between these two lowest critical forces can be controlled by tuning the ratio between the stiffness of the matrix and the bending stiffness of the rod, thereby allowing for specific buckling configurations to be target by design.

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.