Michael Moshe

Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses

Although Nature has always been a common source of inspiration in the development of artificial materials, only recently has the ability of man-made materials to produce complex three-dimensional (3D) structures from two-dimensional sheets been explored. Here we present a new approach to the self-shaping of soft matter that mimics fibrous plant tissues by exploiting small-scale variations in the internal stresses to form three-dimensional morphologies. We design single-layer hydrogel sheets with chemically distinct, fibre-like regions that exhibit differential shrinkage and elastic moduli under the application of external stimulus. Using a planar-to-helical three-dimensional shape transformation as an example, we explore the relation between the internal architecture of the sheets and their transition to cylindrical and conical helices with specific structural characteristics. The ability to engineer multiple three-dimensional shape transformations determined by small-scale patterns in a hydrogel sheet represents a promising step in the development of programmable soft matter.

Shape selection in chiral ribbons: from seed pods to supramolecular assemblies

We provide a geometric-mechanical model for calculating equilibrium configurations of chemical systems that self-assemble into chiral ribbon structures. The model is based on incompatible elasticity and uses dimensionless parameters to determine the equilibrium configurations. As such, it provides universal curves for the shape and energy of self-assembled ribbons. We provide quantitative predictions for the twisted-to-helical transition, which was observed experimentally in many systems, and demonstrate it with synthetic ribbons made of responsive gels. In addition, we predict the bi-stability of wide ribbons and also show how geometrical frustration can cause arrest of ribbon widening. Finally, we show that the models predictions provide explanations for experimental observations in different chemical systems.

Metric Description of Singular Defects in Isotropic Materials

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.

Pattern selection and multiscale behaviour in metrically discontinuous non-Euclidean plates

We study equilibrium configurations of non-Euclidean plates, in which the reference metric is uniaxially periodic. This work is motivated by recent experiments on thin sheets of composite thermally responsive gels (Wu et al 2013 Nature Commun. 4). Such sheets bend perpendicularly to the periodic axis in order to alleviate the metric discrepancy. For abruptly varying metrics, we identify multiple scaling regimes with different power law dependences of the elastic energy and the axial curvature κ on the sheets thickness h. In the h → 0 limit the equilibrium configuration tends to an isometric embedding of the reference metric, and . Two intermediate asymptotic regimes emerge in between the buckling threshold and the h → 0 limit, in which the energy scales either like h4/5 or like h2/3. We believe that this system exemplifies a much more general phenomenon, in which the thickness of the sheet induces a cutoff length scale below which finer structures of the metric cannot be observed. When the reference metric consists of several separated length scales, a decrease of the sheets thickness results in a sequence of conformational changes, as finer properties of the reference metric are revealed.

Geometry and mechanics of two-dimensional defects in amorphous materials

Significance Modeling defects, or localized strain carriers, are a central challenge in the formulation of elasto-plastic theory of amorphous solids. Whereas in crystalline solids defects are identified as local deviations from the crystal order, it is not clear how, or even if, equivalent intrinsic entities can be defined in amorphous solids. This work presents a new way of defining and describing localized intrinsic geometrical defects in amorphous solids and for computing the stresses within defected bodies. The methods and results that are presented here can be integrated into phenomenological theories of plasticity and can be applied to biomechanical problems that involve strain localization. We study the geometry of defects in amorphous materials and their elastic interactions. Defects are defined and characterized by deviations of the material’s intrinsic metric from a Euclidian metric. This characterization makes possible the identification of localized defects in amorphous materials, the formulation of a corresponding elastic problem, and its solution in various cases of physical interest. We present a multipole expansion that covers a large family of localized 2D defects. The dipole term, which represents a dislocation, is studied analytically and experimentally. Quadrupoles and higher multipoles correspond to fundamental strain-carrying entities. The interactions between those entities, as well as their interaction with external stress fields, are fundamental to the inelastic behavior of solids. We develop analytical tools to study those interactions. The model, methods, and results presented in this work are all relevant to the study of systems that involve a distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.

Microalloying and the mechanical properties of amorphous solids

The mechanical properties of amorphous solids like metallic glasses can be dramatically changed by adding small concentrations (as low as 0.1%) of foreign elements. The glass-forming-ability, the ductility, the yield stress and the elastic moduli can all be greatly effected. This paper presents theoretical considerations with the aim of explaining the magnitude of these changes in light of the small concentrations involved. The theory is built around the experimental evidence that the microalloying elements organise around them a neighbourhood that differs from both the crystalline and the glassy phases of the material in the absence of the additional elements. These regions act as isotropic defects that in unstressed systems modify the shear moduli. When strained, these defects interact with the incipient plastic responses which are quadrupolar in nature. It will be shown that this interaction interferes with the creation of system-spanning shear bands and increases the yield strain. We offer experimentally testable estimates of the lengths of nano-shear bands in the presence of the additional elements.

Elastic interactions between two-dimensional geometric defects.

In this paper, we introduce a methodology applicable to a wide range of localized two-dimensional sources of stress. This methodology is based on a geometric formulation of elasticity. Localized sources of stress are viewed as singular defects-point charges of the curvature associated with a reference metric. The stress field in the presence of defects can be solved using a scalar stress function that generalizes the classical Airy stress function to the case of materials with nontrivial geometry. This approach allows the calculation of interaction energies between various types of defects. We apply our methodology to two physical systems: shear-induced failure of amorphous materials and the mechanical interaction between contracting cells.