Cloaking In-Plane Elastic Waves with Swiss Rolls

Abstract

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 Pendry's transform. Unlike the hitherto known situations, the present geometric transform starts from a Willis medium and further assumes that displacement fields u in original medium and u in transformed medium remain unaffected (u = 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 120(25), 254301 (2018)], notwithstanding potential chiral elastic effects [Frenzel et al., Science 358(6366), 1072 (2017)], and thus mitigates roles of Willis and Cosserat media in the achieved elastodynamic cloaking.

On Near-cloaking for Linear Elasticity

“…1. The spatially varying components of the elasticity tensor \BbbC \prime coss (r \prime ) of a cylindrical Cosserat cloak from (54) for r \prime \in [1,2] and \varepsi = 0.2 for a bulk medium with Lam\' e parameters \lambda 0 = 1.5 \times 10 11 N.m - 2 and \mu 0 = 7.5 \times 10 10 N.m - 2 . One notes the substantial minor symmetry breaking for r \prime = 1.…”

“…This is indeed the proposal of Greenleaf et al [35] (see also the recent work of Ghosh and Tarikere [33]) in applying periodic homogenization arguments to design approximate multilayered cloaks. We consider an alternation of concentric layers of isotropic elasticity tensors \BbbC (1) and \BbbC (2) such that the effective elasticity tensor for this arrangement equates to the symmetrized Cosserat tensor \BbbC s coss . As hinted above, we will be considering periodic arrangement in radial direction (essentially, a onedimensional periodic homogenization problem).…”

“…Let us denote by H -1 2 \scrR (\partial\Omega ; \BbbR d ) the subspace of all those elements in H -1 2 (\partial\Omega ; \BbbR d ) which satisfy the compatibility condition (2). It is a classical matter in this setting to show that for any given g \in H -1 2 \scrR (\partial\Omega ; \BbbR d ), there exists a unique solution u \in W(\Omega ; \BbbR d ) to the system (1)---we refer the reader to [30,15].…”

Pulse Dynamics of Flexural Waves in Transformed Plates

“…When one moves to the area of elastic waves, however, the elasticity equations are in general not form‐invariant under a general coordinate transformation, [ 7 ] except in the framework of Cauchy elasticity [ 8–10 ] and in the framework of Willis materials. [ 11,12 ] Consequently, if cloaking exists for such a class of waves, it would be of a different nature than its acoustic [ 13 ] and electromagnetic counterparts. [ 9 ] Researchers resort to studying the special case of flexural waves in thin plates, which are described by the the fourth order Kirchhof—Love equation.…”

Physical Realization of Elastic Cloaking with a Polar Material

“…At first sight, this should mean that cloaking in solids for full elasticity [28][29][30] is more accessible than in fluids for acoustics [31][32][33] since anisotropic solids are potentially easier to fabricate than "anisotropic fluids." Nonetheless, a closer look reveals that elastic cloaks, in general, further require their materials to be polar, i.e., to exhibit asymmetric stresses [34][35][36][37][38][39]. Polarity turns out to be necessary if shear and hydrostatic stresses are coupled as is typically the case in elasticity [36].…”