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Register a new userStatic elastic cloaking, low frequency elastic wave transparency and neutral inclusions
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New connections between static elastic cloaking, low frequency elastic wave scattering and neutral inclusions 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 neutral inclusion are distinguished, textit{weak} and textit{strong} with the former equivalent to low frequency transparency {and the classical Christensen and Lo generalised self-consistent result for in-plane shear from 1979. Our introduction of the textit{strong neutral inclusion} 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 neutral inclusions provide the material designer with options for achieving elastic cloaking in the quasi-static limit.
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A theory is presented showing that cloaking of objects from antiplane elastic waves can be achieved by elastic pre-stress of a neo-Hookean nonlinear elastic material. This approach would appear to eliminate the requirement of metamaterials with inhomogeneous anisotropic shear moduli and density. Waves in the pre-stressed medium are bent around the cloaked region by inducing inhomogeneous stress fields via pre-stress. The equation governing antiplane waves in the pre-stressed medium is equivalent to the antiplane equation in an unstressed medium with inhomogeneous and anisotropic shear modulus and isotropic scalar mass density. Note however that these properties are induced naturally by the pre-stress. Since the magnitude of pre-stress can be altered at will, this enables objects of varying size and shape to be cloaked by placing them inside the fluid-filled deformed cavity region.
We propose a design of cylindrical elastic cloak for coupled in-plane shear waves consisting of concentric layers of sub-wavelength resonant stress-free inclusions shaped as swiss-rolls. The scaling factor between inclusions sizes is according to Pendrys transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields ${bf u}$ in original medium and ${bf u}$ in transformed medium remain unaffected (${bf u}={bf u}$), and this breaks the minor-symmetries of the rank-4 and rank-3 tensors in the Willis equation that describes the transformed effective medium. We achieve some cloaking for a shear polarized source at specific, resonant sub-wavelength, frequencies, when it is located near a clamped obstacle surrounded by the structured cloak. Such an effective medium allows for strong Willis coupling [Quan et al., Physical Review Letters {bf 120}(25), 254301 (2018)], notwithstanding potential chiral elastic effects [Frenzel et al., Science {bf 358}(6366), 1072 (2017)], and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.
The propagation of acoustic or elastic waves in artificial crystals, including the case of phononic and sonic crystals, is inherently anisotropic. As is known from the theory of periodic composites, anisotropy is directly dictated by the space group of the unit cell of the crystal and the rank of the elastic tensor. Here, we examine effective velocities in the long wavelength limit of periodic acoustic and elastic composites as a function of the direction of propagation. We derive explicit and efficient formulas for estimating the effective velocity surfaces, based on second-order perturbation theory, generalizing the Christofell equation for elastic waves in solids. We identify strongly anisotropic sonic crystals for scalar acoustic waves and strongly anisotropic phononic crystals for vector elastic waves. Furthermore, we observe that under specific conditions, quasi-longitudinal waves can be made much slower than shear waves propagating in the same direction.
The resonance modes and the related effects to the transmission of elastic waves in a two dimensional phononic crystal formed by periodic arrangements of a two blocks unit cell in one direction are studied. The unit cell consists of two asymmetric elliptic cylinders coated with silicon rubber and embedded in a rigid matrix. The modes are obtained by the semi-analytic method in the least square collocation scheme and confirmed by the finite element method simulations. Two resonance modes, corresponding to the vibration of the cylinder along the long and short axes, give rise to resonance reflections of elastic waves. One mode in between the two modes, related to the opposite vibration of the two cylinders in the unit cell in the direction along the layer, results in the total transmission of elastic waves due to zero effective mass density at the frequency. The resonance frequency of this new mode changes continuously with the orientation angle of the elliptic resonator.
We design a two-dimensional ultra-thin elastic metasurface consisting of steel cores coated with elliptical rubbers embedded in epoxy matrix, capable of manipulating bulk elastic wave modes for reflected waves. The energy exchanges between the longitudinal and transverse modes are completely controlled by the inclined angle of rubber. One elastic mode can totally convert into another by the ultra-thin elastic metasurface. The conversion mechanism based on the non-degenerate dipolar resonance is a general method and easily extended to three-dimensional or mechanical systems. A mass-spring model is proposed and well describe the conversion properties. We further demonstrate that high conversion rates (more than 95%) can be achieved steadily for one elastic metasurface working on almost all different solid backgrounds. It will bring wide potential applications in elastic devices.
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