Summary The developing vertebrate gut tube forms a reproducible looped pattern as it grows into the body cavity. Here we use developmental experiments to eliminate alternative models and show that gut looping morphogenesis is driven by the homogeneous and isotropic forces that arise from the relative growth between the gut tube and the anchoring dorsal mesenteric sheet, tissues that grow at different rates. A simple physical mimic, using a differentially strained composite of a pliable rubber tube and a soft latex sheet is consistent with this mechanism and produces similar patterns. We devise a mathematical theory and a computational model for the number, size and shape of intestinal loops based solely on the measurable geometry, elasticity and relative growth of the tissues. The predictions of our theory are quantitatively consistent with observations of intestinal loops at different stages of development in the chick embryo. Our model also accounts for the qualitative and quantitative variation in the distinct gut looping patterns seen in a variety of species including the quail, finch and mouse illuminating how the simple macroscopic mechanics of differential growth drives the morphology of the developing gut.

We study the role of connectivity on the linear and nonlinear elastic behavior of amorphous systems using a two-dimensional random network of harmonic springs as a model system. A natural characterization of these systems arises in terms of the network coordination relative to that of an isostatic network δz; a floppy network has δz < 0, while a stiff network has δz > 0. Under the influence of an externally applied load we observe that the response of both floppy and rigid network are controlled by the same critical point, corresponding to the onset of rigidity. We use numerical simulations to compute the exponents which characterize the shear modulus, the amplitude of nonaffine displacements, and the network stiffening as a function of δz, derive these theoretically and make predictions for the mechanical response of glasses and fibrous networks. The mechanics of crystalline solids is a fairly well understood subject owing to the simplicity of the underlying lattice which is periodic. In contrast, an understanding of the mechanics of amorphous solids is complicated by the presence of quenched disorder, often on multiple scales. Two structural properties affecting the elasticity of disordered solids are their coordination, and the presence of different types of interactions between the constituents of vastly dissimilar strengths. In the case of weaklycoordinated covalent glass such as amorphous selenium, the backbone is floppy, i.e. it is continuously deformable with almost no energy cost, but weak interactions such as van der Waals are responsible for the non-vanishing elastic moduli. On the other hand, highly-coordinated covalent glasses such as silica, or amorphous particle assemblies where the main interaction is radial, such as emulsions, metallic glasses or granular matter, the backbone is stiff. In foams and fibrous networks which are made of low-dimensional structures such as filaments and membranes, there is a wide separation of energetic scales between stretching and bending modes. This leads to a range of curious mechanical responses in these materials including strongly non-affine deformations [1,2,3,4,5,6] and elastic moduli that can be quite sensitive to the applied stress [1,7]. Despite several theoretical advances [8,9,10,11,12], a unified descriptions of these behaviors remains to be given. Here we study the mechanical response of simple floppy and rigid systems as the coordination is continuously varied and propose such a unifying approach.We start by recalling Maxwell's criterion for rigidity in a central force network [13] by considering a set of N points in d dimensions, subject to N c constraints in the form of bonds that connect these points. This network has N d − N c effective degrees of freedom (ignoring the d(d + 1)/2 rigid motions of the entire system), and an average coordination number z = 2N c /N . The system is said to be isostatic when the system is just rigid, i.e. the number of constraints and the number of degrees of freedom are just balanced, so that N d = N c , and z = 2d. W...

Origami structures are mechanical metamaterials with properties that arise almost exclusively from the geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we characterize the geometry and planar and nonplanar effective elastic response of a simple periodically folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and out-of-plane Poisson's ratios are equal in magnitude, but opposite in sign, independent of material properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we solve the inverse design problem of determining the geometric parameters for the optimal geometric and mechanical response of these extreme structures.

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Long leaves in terrestrial plants and their submarine counterparts, algal blades, have a typical, saddle-like midsurface and rippled edges. To understand the origin of these morphologies, we dissect leaves and differentially stretch foam ribbons to show that these shapes arise from a simple cause, the elastic relaxation via bending that follows either differential growth (in leaves) or differential stretching past the yield point (in ribbons). We quantify these different modalities in terms of a mathematical model for the shape of an initially flat elastic sheet with lateral gradients in longitudinal growth. By using a combination of scaling concepts, stability analysis, and numerical simulations, we map out the shape space for these growing ribbons and find that as the relative growth strain is increased, a long flat lamina deforms to a saddle shape and/or develops undulations that may lead to strongly localized ripples as the growth strain is localized to the edge of the leaf. Our theory delineates the geometric and growth control parameters that determine the shape space of finite laminae and thus allows for a comparative study of elongated leaf morphology.growing surfaces | edge actuation | leaves | buckling | rippling L aminae, or leaf-like structures, are thin, i.e., they have one dimension much smaller than the other two. They arise in biology in a variety of situations, ranging from the gracefully undulating submarine avascular algal blades (1) to the saddle-shaped, coiled or edge-rippled leaves of many terrestrial plants (2). The variety of their planforms and three-dimensional shapes reflects both their growth history and their mechanical properties and poses many physico-chemical questions that may be broadly classified into two kinds: (i) How does inhomogeneous growth at the molecular and cellular level lead to the observed complex shapes at the mesoscopic/macroscopic level? and (ii) how does the resulting mesoscopic shape influence the underlying molecular growth processes? At the molecular level, mutants responsible for differential cell proliferation (3) lead to a range of leaf shapes. At the macroscopic level, stresses induced by external loads lead to phenotypic plasticity in algal blades that switch between long, narrow, blade-like shapes in rapid flow to broader undulating shapes in slow flow (1). Understanding the origin of these morphological variants requires a mathematical theory that accounts for the process by which shape is generated by inhomogeneous growth in a tissue. Recent work has focused on some of these questions by highlighting the self-similar structures that form near the edge because of variations in a prescribed intrinsic metric of a surface (4, 5), and also on the case of a circular disk with edge-localized growth (6-8), but does not consider the subtle role of the boundary conditions at the free edge, the effect of the finite width of a leaf, or the phase space of different shapes that quantify the diversity in leaf morphology.Motivated by our experimental observations of ...

Many species of macroalgae have flat, strap-like blades in habitats exposed to rapidly flowing water, but have wide, ruffled "undulate" blades at protected sites. We used the giant bull kelp, Nereocystis luetkeana, to investigate how these ecomorphological differences are produced. The undulate blades of N. luetkeana from sites with low flow remain spread out and flutter erratically in moving water, thereby not only enhancing interception of light, but also increasing drag. In contrast, strap-like blades of kelp from habitats with rapid flow collapse into streamlined bundles and flutter at low amplitude in flowing water, thus reducing both drag and interception of light. Transplant experiments in the field revealed that shape of the blade in N. luetkeana is a plastic trait. Laboratory experiments in which growing blades from different sites were subjected to tensile forces that mimicked the hydrodynamic drag experienced by blades in different flow regimes showed that change in shape is induced by mechanical stress. During growth experiments in the field and laboratory, we mapped the spatial distribution of growth in both undulate and strap-like blades to determine how these different morphologies were produced. The highest growth rates occur near the proximal ends of N. luetkeana blades of both morphologies, but the rates of transverse growth of narrow, strap-like blades are lower than those of wide, undulate blades. If rates of longitudinal growth at the edges of a blade exceed the rate of longitudinal growth along the midline of the blade, ruffles along the edges of the blade are produced by elastic buckling. In contrast, flat blades are produced when rates of longitudinal growth are similar across the width of a blade. Because ruffles are the result of elastic buckling, a compliant undulate N. luetkeana blade can easily be pushed into different configurations (e.g., the wavelengths of the ruffles along the edges of the blade can change, and the whole blade can twist into left- and right-handed helicoidal shapes), which may enhance movements of the blade in flowing water that reduce self-shading and increase mass exchange along blade surfaces.

As the thinnest atomic membrane, graphene presents an opportunity to combine geometry, elasticity, and electronics at the limits of their validity. We describe the transport and electronic structure in the neighborhood of conical singularities, the elementary excitations of the ubiquitous wrinkled and crumpled graphene. We use a combination of atomistic mechanical simulations, analytical geometry, and transport calculations in curved graphene, and exact diagonalization of the electronic spectrum to calculate the effects of geometry on electronic structure, transport, and mobility in suspended samples, and how the geometry-generated pseudomagnetic and pseudoelectric fields might disrupt Landau quantization.