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1. Static elastic cloaking, low-frequency elastic wave transparency and neutral inclusions

   https://doi.org/10.1098/rspa.2019.0725  

   Norris, Andrew N.; Parnell, William J. (August 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   New connections between static elastic cloaking, low-frequency elastic wave scattering and neutral inclusions (NIs) are established in the context of two-dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low-frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core–shell composite cylinder as a neutral elastic inclusion. Two types of NI are distinguished, weak and strong with the former equivalent to low-frequency transparency and the classical Christensen and Lo generalized self-consistent result for in-plane shear from 1979. Our introduction of the strong NI is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed ‘unphysical’ materials. The relationships between low-frequency transparency, static cloaking and NIs provide the material designer with options for achieving elastic cloaking in the quasi-static limit. 

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2. Bifurcations of an elastic disc coated with an elastic inextensible rod

   https://doi.org/10.1098/rspa.2023.0491  

   Gaibotti, M; Mogilevskaya, S G; Piccolroaz, A; Bigoni, D (January 2024, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   An analytical solution is derived for the bifurcations of an elastic disc that is constrained on the boundary with an isoperimetric Cosserat coating. The latter is treated as an elastic circular rod, either perfectly or partially bonded (with a slip interface in the latter case) and is subjected to three different types of uniformly distributed radial loads (including hydrostatic pressure). The proposed solution technique employs complex potentials to treat the disc’s interior and incremental Lagrangian equations to describe the prestressed elastic rod modelling the coating. The bifurcations of the disc occur with modes characterized by different circumferential wavenumbers, ranging between ovalization and high-order waviness, as a function of the ratio between the elastic stiffness of the disc and the bending stiffness of its coating. The presented results find applications in various fields, such as coated fibres, mechanical rollers, and the growth and morphogenesis of plants and fruits. 

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3. Elastic and thermodynamic consequences of lipid membrane asymmetry

   https://doi.org/10.1042/ETLS20220084  

   Foley, Samuel L.; Varma, Malavika; Hossein, Amirali; Deserno, Markus (March 2023, Emerging Topics in Life Sciences)

   Many cellular lipid bilayers consist of leaflets that differ in their lipid composition — a non-equilibrium state actively maintained by cellular sorting processes that counter passive lipid flip-flop. While this lipidomic aspect of membrane asymmetry has been known for half a century, its elastic and thermodynamic ramifications have garnered attention only fairly recently. Notably, the torque arising when lipids of different spontaneous curvature reside in the two leaflets can be counterbalanced by a difference in lateral mechanical stress between them. Such membranes can be essentially flat in their relaxed state, despite being compositionally strongly asymmetric, but they harbor a surprisingly large but macroscopically invisible differential stress. This hidden stress can affect a wide range of other membrane properties, such as the resistance to bending, the nature of phase transitions in its leaflets, and the distribution of flippable species, most notably sterols. In this short note we offer a concise overview of our recently proposed basic framework for capturing the interplay between curvature, lateral stress, leaflet phase behavior, and cholesterol distribution in generally asymmetric membranes, and how its implied signatures might be used to learn more about the hidden but physically consequential differential stress. 

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4. Corner singularities and shape of stretched elastic sheets

   https://doi.org/10.1103/PhysRevE.98.043003  

   Chopin, Julien; Panaitescu, Andreea; Kudrolli, Arshad (October 2018, Physical Review E)
5. Elastic solids with strain-gradient elastic boundary surfaces

   https://doi.org/10.1007/s10659-024-10073-w  

   Rodriguez, C (June 2024, Journal of Elasticity)

   Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen’s extension of Noll’s theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips. 

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