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Register a new userFinite indentation of highly curved elastic shells
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Experimentally measuring the elastic properties of thin biological surfaces is non-trivial, particularly when they are curved. One technique that may be used is the indentation of a thin sheet of material by a rigid indenter, whilst measuring the applied force and displacement. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also theoretically possible to determine the elastic properties by comparing the resulting force-displacement curves with a mathematical model. Existing mathematical studies generally assume that the elastic surface is initially flat, which is often not the case for biological membranes. We previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness) which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here we introduce the effect of bending stiffness, ensuring that energy is required to change the shell shape without stretching, and find that commonly neglected terms in the shell equilibrium equation must be included. The theory presented here allows for the estimation of shape- and size-independent elastic properties of highly curved surfaces via indentation experiments, and is particularly relevant for biological surfaces.
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Surface tension is a prominent factor for the deformation of solids at micro-/nano-scale. This paper investigates the effects of surface tension on the two-dimensional contact problems of an elastic layer bonded to the rigid substrate. Under the plane strain assumption, the elastic field induced by a uniformly distributed pressure within a finite width is formulated by applying the Fourier integral transform, and the limiting process leading to the solutions for a line force brings the requisite surface Greens function. For the indentation of an elastic layer by a rigid cylinder, the corresponding singular integral equation is derived, and subsequently solved by using an effective numerical method based on Gauss-Chebyshev quadrature formula. It is found from the theoretical and numerical results that the existence of surface tension strongly enhances the hardness of the elastic layer and significantly affects the distribution of contact pressure, when the size of contact region is comparable to the elastocapillary length. In addition, an approximated relationship between external load and half-width of contact is generalized in an explicit and concise form, which is useful and convenient for practical applications.
Morphogenetic dynamics of tissue sheets require coordinated cell shape changes regulated by global patterning of mechanical forces. Inspired by such biological phenomena, we propose a minimal mechanochemical model based on the notion that cell shape changes are induced by diffusible biomolecules that influence tissue contractility in a concentration-dependent manner -- and whose concentration is in turn affected by the macroscopic tissue shape. We perform computational simulations of thin shell elastic dynamics to reveal propagating chemical and three-dimensional deformation patterns arising due to a sequence of buckling instabilities. Depending on the concentration threshold that actuates cell shape change, we find qualitatively different patterns. The mechanochemically coupled patterning dynamics are distinct from those driven by purely mechanical or purely chemical factors. Using numerical simulations and theoretical arguments, we analyze the elastic instabilities that result from our model and provide simple scaling laws to identify wrinkling morphologies.
Origami and crumpling are two extreme tools to shrink a 3-D shell. In the shrink/expand process, the former is reversible due to its topological mechanism, while the latter is irreversible because of its random-generated creases. We observe a morphological transition between origami and crumple states in a twisted cylindrical shell. By studying the regularity of crease pattern, acoustic emission and energetics from experiments and simulations, we develop a model to explain this transition from frustration of geometry that causes breaking of rotational symmetry. In contrast to solving von Karman-Donnell equations numerically, our model allows derivations of analytic formula that successfully describe the origami state. When generalized to truncated cones and polygonal cylinders, we explain why multiple and/or reversed crumple-origami transitions can occur.
Thin elastic membranes form complex wrinkle patterns when put on substrates of different shapes. Such patterns continue to receive attention across science and engineering. This is due, in part, to the promise of lithography-free micropatterning, but also to the observation that similar patterns arise in biological systems from growth. The challenge is to explain the patterns in any given setup, even when they fail to be robust. Building on the theoretical foundation of [Tobasco, to appear in Arch. Ration. Mech. Anal., arXiv:1906.02153], we derive a complete and simple rule set for wrinkles in the model system of a curved shell on a liquid bath. Our rules apply to shells whose initial Gaussian curvatures are of one sign, such as cutouts of saddles and spheres. They predict the surprising coexistence of orderly wrinkles alongside disordered regions where the response appears stochastic, which we confirm in experiment and simulation. They also unveil the role of the shells medial axis, a distinguished locus of points that we show is a basic driver in pattern selection. Finally, they explain how the sign of the shells initial curvature dictates the presence or lack of disorder. Armed with our simple rules, and the methodology underlying them, one can anticipate the creation of designer wrinkle patterns.
In this paper we study the elastic response of synthetic hydrogels to an applied shear stress. The hydrogels studied here have previously been shown to mimic the behaviour of biopolymer networks when they are sufficiently far above the gel point. We show that near the gel point they exhibit an elastic response that is consistent with the predicted critical behaviour of networks near or below the isostatic point of marginal stability. This point separates rigid and floppy states, distinguished by the presence or absence of finite linear elastic moduli. Recent theoretical work has also focused on the response of such networks to finite or large deformations, both near and below the isostatic point. Despite this interest, experimental evidence for the existence of criticality in such networks has been lacking. Using computer simulations, we identify critical signatures in the mechanical response of sub-isostatic networks as a function of applied shear stress. We also present experimental evidence consistent with these predictions. Furthermore, our results show the existence of two distinct critical regimes, one of which arises from the nonlinear stretch response of semi-flexible polymers..
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