Graded anisotropic metamaterials for elastic wave mode conversion

Abstract

Efficient transmission of elastic waves across interfaces is central to several applications, including medical imaging, seismic isolation, and transducer design. Interfaces with abrupt changes in the material properties significantly impede wave transmission, leading to reflections. This limitation, known as impedance mismatch, becomes even more prominent for mode conversion between different wave types due to polarization mismatch. In this study, we investigate a mechanism employing two-dimensional functionally graded anisotropic metamaterials to facilitate longitudinal--shear mode conversion as waves propagate from a stiff to a compliant medium. By embedding density and anisotropic shape gradients within the functionally graded metamaterial, polarization-induced impedance mismatch is mitigated and efficient mode conversion is enabled. We use unit cell dispersion analysis to tailor the frequency range for mode conversion through gradation in the dispersion behavior and coupling between modes. Using frequency-domain finite element analysis, we demonstrate broadband mode conversion across interfaces with large stiffness contrast operating in the 1--10 kHz range. We then experimentally validate and quantify mode conversion through full-field velocity measurements on an additively manufactured specimen. We further apply the methodology to design a device capable of converting radial--tangential wave modes.

Figures (10)

• Figure 1: (A) A semi-infinite medium made of two isotropic materials with a sharp interface between a stiff and soft material. At steady state, a longitudinal periodic excitation (at the left) results in only a smaller amplitude longitudinal response (at the right) due to impedance mismatch. (B) A functionally graded design is sandwiched between two isotropic materials in the semi-infinite medium. The graded design enhances conversion by coupling deformation modes. The shape gradients in unit cells (8 to 20) allow the wave mode to convert from longitudinal to a hybrid shear-dominant mode. The density gradients in unit cells (1 to 8 and 20 to 27) help gradually reduce the wave speedwhile retaining the mode shape. (C) Unit cells in the graded design domain numbered from the left end to the right end.
• Figure 2: (A) Dispersion band diagram for waves propagating along $x_1$ axis (Normalized wavenumber ($\frac{\gamma a}{\pi}$) -- frequency $(\omega)$ plots). Only, the first two modes of a subset of unit cells are shown. The first mode is purely shear, and the second mode is purely longitudinal for the unit cell 1. Rest of the unit cells exhibit mode hybridization. (B-C) Mode shapes of the same subset of unit cells are plotted at a fixed wavenumber $\gamma = 0.167\frac{\pi}{a}$. The color contour shows normalized values of displacement magnitude. (D) Plot of the fill fraction of the stiff material with unit cell number, showing density gradients between unit cells 1 to 8 and 20 to 27. (E-F) Overly of dispersion band diagrams for all the unit cells in the functionally graded structure for modes 1 and 2 respectively.
• Figure 3: Comparison of transmission spectrum for three designs under longitudinal excitation. (A-C) Contours of horizontal velocity component $V_1$ for symmetric, asymmetric, and graded designs at three different frequencies (2000 Hz, 4000 Hz, 8000 Hz). (D-F) Plots of the variation of the transmitted signal evaluated at the output with the frequency of excitation (normalized with the input). (G-I) Plots of the variation of the polarization evaluated at the output with the frequency of excitation.
• Figure 4: Comparison of deformation shapes for the three designs under shear excitation at 4000 Hz, showing a mixed deformation mode in the softer region of the graded design. Contours display the horizontal velocity component ($V_1$).
• Figure 5: Plot of magnitude of velocity components at selected frequencies, measured experimentally. Mode shapes are symmetric at frequencies near 2400 Hz and 8000 Hz, indicating no conversion. Mode shapes are asymmetric near 4400 Hz, indicating a shear deformation mode for longitudinal excitation.