Effective medium theory for elastic metamaterials in thin elastic plates

An effective medium theory for resonant and nonresonant metamaterials for flexural waves in thin plates is presented. The theory provides closed-form expressions for the effective mass density, rigidity, and Poisson's ratio of arrangements of isotropic scatterers in thin plates, valid for low frequencies and moderate filling fractions. It is found that the effective Young's modulus and Poisson's ratio are induced by a combination of the monopolar and quadrupolar scattering coefficient, as it happens for bulk elastic waves, while the effective mass density is induced by the monopolar coefficient, contrarily as it happens for bulk elastic waves, where the effective mass density is induced by the dipolar coefficient. It is shown that resonant positive or negative effective elastic parameters are possible, being therefore the monopolar resonance responsible for creating an effective medium with negative mass density. Several examples are given for both nonresonant and resonant effective parameters and the results are numerically verified by multiple scattering theory and finite element analysis.

Figure 1

Schematic view of the homogenization process based on the effective $T$-matrix formalism. An incident field ${\psi }_0$ arrives to a circular cluster of $N$ scatterers. It is expected that in the low frequency the cluster behaves like an effective circular scatterer of radius $R_{\mathrm{eff}}$ and parameters ${\rho }_{\mathrm{eff}},D_{\mathrm{eff}}$, and ${\nu }_{\mathrm{eff}}$. The parameters are obtained by comparing the low-frequency limit of the scattering properties, defined by means of the cluster and effective $T$ matrices.

Figure 2

Continuous lines: effective phase velocity for circular holes embedded in an elastic plate of different Poisson's ratio. Dashed lines: results from an approximated model (see text for further details).

Figure 3

Effective mass density (a), rigidity (b), Poisson's ratio (c), and phase velocity (d) as a function of the filling fraction for circular inclusions in aluminium. Results are shown for holes (blue continuous line), lead inclusions (green dashed line), and rubber inclusions (red dashed-dotted line).

Figure 4

Real part of the frequency-dependent parameters ${\Gamma }_0,{\Gamma }_1$, and ${\Gamma }_2$ for rubber inclusions in an aluminium matrix. Only the ${\Gamma }_0$ and ${\Gamma }_1$ parameters present a resonance, while the ${\Gamma }_2$ parameter is nearly constant along all the frequency range.

Figure 5

Frequency-dependent effective mass density (a), rigidity (b), and Poisson's ratio (c) for the system described in Fig. 4. The monopolar resonance creates a resonant behavior in the three parameters, although only for the mass density is divergent.

Figure 6

Far-field amplitude in the backward (upper panel) and forward (lower panel) directions as a function of frequency for a circular cluster of 151 holes of radius $R_a=0.1a$ embedded in a triangular lattice of lattice constant $a$ (dashed green line) compared with that of the effective homogeneous scatterer obtained with the presented homogenization theory.

Figure 7

Same system of Fig. 6 but being the radius of the holes $R_a=0.3a$.

Figure 8

Same system of Fig. 6 but being the radius of the holes $R_a=0.45a$.

Figure 9

Relative total scattering cross section of the cluster of holes and the corresponding effective parameters. Results are shown for three filling fractions $f=0.04$ (blue continuous line), $f=0.33$ (green dashed line), and $0.74$ (red dashed-dotted line). As can be seen, the differences are higher for higher-filling fractions and frequency, but always smaller than 0.1.

Figure 10

Same system of Fig. 6 but with lead inclusions in the aluminium matrix, being the radius of the inclusions $R_a=0.4a$. Even being a situation of high-filling fraction, the far-field amplitude in the back and forward directions is very similar.

Figure 11

Relative total scattering cross section of the cluster of lead inclusions and the corresponding effective parameters. Results are shown for three filling fractions $f=0.04$ (blue continuous line), $f=0.33$ (green dashed line), and $0.74$ (red dashed-dotted line). As can be seen, the differences are higher for higher-filling fractions and frequency, showing that for the high-filling fraction the differences are very important for $a/\lambda >0.1$.

Figure 12

Far-field amplitude in the back (upper panel) and forward (lower panel) directions for a cluster of 151 rubber inclusions in an aluminium plate (red dashed lines), for a homogeneous scatterer with frequency-independent parameters (blue continuous line), and with a metamaterial with frequency-dependent parameters (green dotted line). The radius of the inclusions is $R_a=0.3a$, being the effective parameters of the cluster those depicted in Fig. 5. It is seen how the frequency-dependent theory predicts better the behavior of the cluster, although some corrections must be added near the homogenization limit.

Figure 13

Relative total scattering cross section for different filling fractions for the cluster of rubber inclusions in an aluminium plate compared with that of the frequency-dependent effective scatterer. It is seen how the differences are important for high frequencies and filling fractions, being therefore necessary the inclusion of the multiple scattering corrections. However, the behavior of the cluster is properly predicted for low-filling fraction and even near the resonance for the filling fraction $f=0.33$.

Figure 14

(a) Multiple scattering simulation of a plane wave interacting with circular cluster of holes in aluminium. (b) Scattering of the same wave by a single scatterer with the corresponding effective parameters. (c), (d) Field profiles along the $y$ axis for $x=0$ and along the $x$ axis for $y=0$, respectively, for the cluster (blue dots) and the effective scatterer (green line).

Figure 15

(a) Multiple scattering simulation of a plane wave interacting with circular cluster of lead inclusions in aluminium. (b) Scattering of the same wave by a single scatterer with the corresponding effective parameters. (c) and (d) Field profiles along the $y$ axis for $x=0$ and along the $x$ axis for $y=0$, respectively, for the cluster (blue dots) and the effective scatterer (green line).

Figure 16

(a) Multiple scattering simulation of a plane wave interacting with circular cluster of rubber inclusions in aluminium. (b) Scattering of the same wave by a single scatterer with the corresponding frequency-dependent effective parameters. (c), (d) Field profiles along the $y$ axis for $x=0$ and along the $x$ axis for $y=0$, respectively, for the cluster (blue dots) and the effective scatterer (green line).

Figure 17

Red dots: three-dimensional band structure of a square array of holes of radius $R_a=0.3a$ (upper panel) and $R_a=0.45a$ (lower panel) in an aluminium plate of thickness $h_b=0.1a$ computed by FEM. Blue line: linear dispersion relation obtained with the effective medium theory of this work.

Figure 18

Red dots: three-dimensional band structure of a triangular array of lead inclusions of radius $R_a=0.3a$ (upper panel) and $R_a=0.45a$ (lower panel) in an aluminium plate of thickness $h_b=0.1a$ computed by FEM. Blue line: linear dispersion relation obtained with the effective medium theory of this work.

Figure 19

Red dots: three-dimensional band structure of a triangular array of rubber inclusions of radius $R_a=0.3a$ (upper panel) and $R_a=0.45a$ (lower panel) in an aluminium plate of thickness $h_b=0.1a$ computed by FEM. Blue line: linear dispersion relation obtained with the effective medium theory of this work.