Fig. 2: Blueprint of metamaterial supporting anomalous frozen evanescent phonons.

This nonlocal mechanical metamaterial composed of a single constituent polymer material allows obtaining frozen evanescent Bloch modes with large characteristic exponential decay length \(l\) (cf. Fig. 1b). a, b Two different views onto a single unit cell. The geometrical parameters are defined. Two of the yellow cylinders are rendered semi-transparent to indicate the height of blue helices. The colors are for illustration only. c Beam composed of a one-dimensional periodic arrangement of this unit cell along the \(z\)-direction with period \({a}_{z}\). Adjacent yellow plates are connected by four blue helices (“springs”), fixed to the plates by yellow cylinders. The handedness of the springs alternates, such that the overall structure has two mirror planes, making it achiral. The strength of this nearest-neighbor interaction can be tailored by the radius \({R}_{1}\). The yellow plates are additionally connected to their \(N\)-th neighbors by the red rods with radius \({R}_{N}\). This radius determines the strength of the \(N\)-th nearest neighbor interactions. The example shown refers to \(N=3\). We will discuss \(N=2,\,3,\,4\) with different geometrical parameters. The geometric parameters for \(N=2,\,3,\) and \(4\) are chosen as \(2{R}_{1}/{a}_{z}=0.10\), \(2{R}_{N}/{a}_{z}=0.156\), \(2{R}_{1}/{a}_{z}=0.10\), \(2{R}_{N}/{a}_{z}=0.16\), and \(2{R}_{1}/{a}_{z}=0.072\), \(2{R}_{N}/{a}_{z}=0.10\), respectively. All other geometrical parameters are fixed: \({a}_{z}=100\,{{\rm{\mu m}}}\), \(2{R}_{1}/{a}_{z}=0.10\), \(2{R}_{N}/{a}_{z}=0.16\), \(w/{a}_{z}=2.0,\,h/{a}_{z}=0.34\), \({h}_{1}/{a}_{z}=0.16\), \({h}_{2}/{a}_{z}=0.34\), \({h}_{3}/{a}_{z}=0.50\), \({D}_{1}/{a}_{z}=0.30\), \({D}_{2}/{a}_{z}=0.60\), \({L}_{1}/{a}_{z}=0.57\), \({L}_{2}/{a}_{z}=0.30\), and \({L}_{3}/{a}_{z}=0.70\). For the material parameters of the constituent polymer, we choose mass density \(\rho=1190\,{{\rm{kg}}}/{{{\rm{m}}}}^{3}\), Young’s modulus \(E=4.19\,{{\rm{GPa}}}\), and Poisson’s ratio \(v=0.3\).