Efi Efrati

Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics

The connection between a surfaces metric and its Gaussian curvature (Gauss theorem) provides the base for a shaping principle of locally growing or shrinking elastic sheets. We constructed thin gel sheets that undergo laterally nonuniform shrinkage. This differential shrinkage prescribes non-Euclidean metrics on the sheets. To minimize their elastic energy, the free sheets form three-dimensional structures that follow the imposed metric. We show how both large-scale buckling and multiscale wrinkling structures appeared, depending on the nature of possible embeddings of the prescribed metrics. We further suggest guidelines for how to generate each type of feature.

Geometry and Mechanics in the Opening of Chiral Seed Pods

Two joined latex strips show complex twisting behavior similar to that of seed pods. We studied the mechanical process of seed pods opening in Bauhinia variegate and found a chirality-creating mechanism, which turns an initially flat pod valve into a helix. We studied configurations of strips cut from pod valve tissue and from composite elastic materials that mimic its structure. The experiments reveal various helical configurations with sharp morphological transitions between them. Using the mathematical framework of “incompatible elasticity,” we modeled the pod as a thin strip with a flat intrinsic metric and a saddle-like intrinsic curvature. Our theoretical analysis quantitatively predicts all observed configurations, thus linking the pod’s microscopic structure and macroscopic conformation. We suggest that this type of incompatible strip is likely to play a role in the self-assembly of chiral macromolecules and could be used for the engineering of synthetic self-shaping devices.

Elastic theory of unconstrained non-Euclidean plates

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

The mechanics of non-Euclidean plates

Non-Euclidean plates are plates (“stacks” of identical surfaces) whose two-dimensional intrinsic geometry is not Euclidean, i.e. cannot be realized in a flat configuration. They can be generated via different mechanisms, such as plastic deformation, natural growth or differential swelling. In recent years there has been a concurrent theoretical and experimental progress in describing and fabricating non-Euclidean plates (NEP). In particular, an effective plate theory was derived and experimental methods for a controlled fabrication of responsive NEP were developed. In this paper we review theoretical and experimental works that focus on shape selection in NEP and provide an overview of this new field. We made an effort to focus on the governing principles, rather than on details and to relate the main observations to known mechanical behavior of ordinary plates. We also point out to open questions in the field and to its applicative potential.

The metric description of elasticity in residually stressed soft materials

Living tissue, polymeric sheets and environmentally responsive gel are often described as elastic media. However, when plants grow, plastic sheets deform irreversibly and hydrogels swell differentially the different material elements within an object change their rest lengths often resulting in objects that possess no stress-free configuration making the standard elastic description inappropriate. In this paper we review an elastic framework based on Riemannian geometry devised to describe such objects lacking a stress-free configuration. In this framework the growth or irreversible deformation are associated with the change of a reference Riemannian metric that prescribes local distances within the body, and the elastic problem is one of optimal embedding. We discuss and resolve points of controversy regarding the Riemannian metric formulation. We give examples for dimensionally reduced theories, such as plates and shells theories, which arise naturally and discuss the relation between geometric frustration and residual stress.

Why Do Sleeping Nematodes Adopt a Hockey-Stick-Like Posture?

A characteristic posture is considered one of the behavioral hallmarks of sleep, and typically includes functional features such as support for the limbs and shielding of sensory organs. The nematode C. elegans exhibits a sleep-like state during a stage termed lethargus, which precedes ecdysis at the transition between larval stages. A hockey-stick-like posture is commonly observed during lethargus. What might its function be? It was previously noted that during lethargus, C. elegans nematodes abruptly rotate about their longitudinal axis. Plausibly, these “flips” facilitate ecdysis by assisting the disassociation of the old cuticle from the new one. We found that body-posture during lethargus was established using a stereotypical motor program and that body bends during lethargus quiescence were actively maintained. Moreover, flips occurred almost exclusively when the animals exhibited a single body bend, preferentially in the anterior or mid section of the body. We describe a simple biomechanical model that imposes the observed lengths of the longitudinally directed body-wall muscles on an otherwise passive elastic rod. We show that this minimal model is sufficient for generating a rotation about the anterior-posterior body axis. Our analysis suggests that posture during lethargus quiescence may serve a developmental role in facilitating flips and that the control of body wall muscles in anterior and posterior body regions are distinct.

Orientation-Dependent Handedness and Chiral Design

Chirality occupies a central role in fields ranging from biological self-assembly to the design of optical metamaterials. The definition of chirality, as given by Lord Kelvin, associates chirality with the lack of mirror symmetry: the inability to superpose an object on its mirror image. While this definition has guided the classification of chiral objects for over a century, the quantification of handed phenomena based on this definition has proven elusive, if not impossible, as manifest in the paradox of chiral connectedness. In this work, we put forward a quantification scheme in which the handedness of an object depends on the direction in which it is viewed. While consistent with familiar chiral notions, such as the right-hand rule, this framework allows objects to be simultaneously right and left handed. We demonstrate this orientation dependence in three different systems - a biomimetic elastic bilayer, a chiral propeller, and optical metamaterial - and find quantitative agreement with chirality pseudotensors whose form we explicitly compute. The use of this approach resolves the existing paradoxes and naturally enables the design of handed metamaterials from symmetry principles.

Hydrodynamic Singularities and Clustering in a Freely Cooling Inelastic Gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite-time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.

Strong Resistance to Bending Observed for Nanoparticle Membranes

We demonstrate how gold nanoparticle monolayers can be curled up into hollow scrolls that make it possible to extract both bending and stretching moduli from indentation by atomic force microscopy. We find a bending modulus that is 2 orders of magnitude larger than predicted by standard continuum elasticity, an enhancement we associate with nonlocal microstructural constraints. This finding opens up new opportunities for independent control of resistance to bending and stretching at the nanoscale.

Three-dimensional geometry of the Heineke-Mikulicz strictureplasty.

Background:The objective of this study was to assess the regional geometry of the Heineke–Mikulicz (HM) strictureplasty. The HM intestinal strictureplasty is commonly performed for the treatment of stricturing Crohns disease of the small intestine. This procedure shifts relatively normal proximal and distal tissue to the point of narrowing and thus increases the luminal diameter. The overall effect on the regional geometry of the HM strictureplasty, however, has not been previously described in detail. Methods:HM strictureplasties were created in latex tubing and cast with an epoxy resin. The resultant casts of the lumens were then imaged using computed tomography. Using 3-dimensional vascular reconstruction software, the cross-sectional areas were determined and the surface geometry was examined. Results:The HM strictureplasty, while increasing the lumen at the point of the stricture, also results in a counterproductive luminal narrowing proximal and distal to the strictureplasty. Within the model used, cross-sectional area was diminished 25% to 50% below baseline. This effect is enhanced when 2 strictureplasties are placed in close proximity to each other. Conclusions:The HM strictureplasty results in alterations in the regional geometry that may result in a compromise of the lumen proximal and distal to the location of the strictureplasty. When 2 HM strictureplasties are created in close proximity to each other, care should be undertaken to assure that the lumen of the intervening segment is adequate.