Unveiling Extreme Anisotropy in Elastic Structured Media

Periodic structures can be engineered to exhibit unique properties observed at symmetry points, such as zero group velocity, Dirac cones, and saddle points; identifying these and the nature of the associated modes from a direct reading of the dispersion surfaces is not straightforward, especially in three dimensions or at high frequencies when several dispersion surfaces fold back in the Brillouin zone. A recently proposed asymptotic high-frequency homogenization theory is applied to a challenging time-domain experiment with elastic waves in a pinned metallic plate. The prediction of a narrow high-frequency spectral region where the effective medium tensor dramatically switches from positive definite to indefinite is confirmed experimentally; a small frequency shift of the pulse carrier results in two distinct types of highly anisotropic modes. The underlying effective equation mirrors this behavior with a change in form from elliptic to hyperbolic exemplifying the high degree of wave control available and the importance of a simple and effective predictive model.

Figure 1

Asymptotic prediction of the local mode structure and respective dispersion surfaces: Effective medium tensor components govern the parabolic ($|T_{11}|\gg |T_{22}|$), hyperbolic ($T_{11}{T}_{22}<0$), or elliptic ($T_{11}{T}_{22}>0$) character of Eq. (1) and capture the extreme anisotropic features of waves propagating within a doubly periodic medium shown in (a),(c),(e). The corresponding local dispersion surfaces are shown in (b),(d),(f). The algebraic criteria for critical points straightforwardly extend to $n$-dimensional periodic systems and reveal effective elliptic ($T_{11}{T}_{22},\dots ,T_{nn}>0$) or hyperbolic ($T_{11}{T}_{22},\dots ,T_{nn}<0$) features without resorting to visualization of hypersurfaces.

Figure 2

Experimental apparatus for imaging pinned elastic plate vibrations. The heterodyne laser interferometer (Thalès SH-140) probes the out-of-plane vibration at the surface of a $10\times 10{\mathrm{cm}}^2$ large, 0.5-mm-thick Duralumin plate clamped on a square periodic lattice by 3-mm-diameter columns screwed between the thin plate and a metallic base (see inset). 3-mm-thick adhesive rubber is sandwiched between the plate’s edges and a square metallic frame attached to the base to reduce unwanted elastic edge reflections.

Figure 3

HFH asymptotics: (a)–(c) The flexural wave field predicted for a point frequency source placed centrally in the array. (a) 94-kHz isotropic response as predicted by effective medium tensor components $T_{11}=T_{22}=1025$; (b) 122-kHz hyperbolic behavior following from $T_{11}=-180.17$ and $T_{22}=311.74$; (c) 138 kHz creating a $+$ shape with $T_{11}=-4189.10$ and $T_{22}=19.51$. (d) Bloch dispersion diagram using the irreducible Brillouin zone $\mathrm{\Gamma }XM$ (shown in inset) with frequencies used in (a)–(c) as red dots. (e)–(g) Log-log validates the HFH asymptotics in the neighborhood of the frequencies used in (a)–(c). Circles are HFH asymptotics of the dispersion curves for the $X\mathrm{\Gamma }$ path; dashed curves are from finite element numerics for the KL model, and the vertical scale is the difference of frequency from that at the band edge.

Figure 4

Snapshots of flexural wave field evolution from a pulse excitation at the center of the system with central frequency $\mathrm{\Omega }$ and bandwidth $\mathrm{\Delta }\mathrm{\Omega }$. (a),(c),(e) Experiment at $\mathrm{\Omega }=90$, 105, 128 kHz and $\mathrm{\Delta }\mathrm{\Omega }=10\mathrm{kHz}$; see, also, the video in Ref. [30]. We observe wavelengths at 90 kHz: 9 mm, at 105 kHz: 27.9 mm, and at 128 kHz: 6.8 mm, which are at least 10 times the scanning resolution. (b),(d),(f) Numerical simulations, FDTD solving the full 3D Navier system, at $\mathrm{\Omega }=95$, 110, 119 kHz with $\mathrm{\Delta }\mathrm{\Omega }=1\mathrm{kHz}$. The minor difference in frequency is attributed to finite boundary effects in the experiments.