Maximization of operating frequency ranges of hyperbolic elastic metamaterials by topology optimization

Abstract

Hyperbolic elastic metamaterials developed for sub-wavelength resolution allow wave propagation in the radial direction but prohibit wave propagation in the circumferential direction. Recently, a two-dimensional elastic metamaterial truly exhibiting the hyperbolic behavior has been realized and also experimented but there is a practically important design issue that its operating frequency range should be widened. Motivated by this need, the present investigation aims to set up a topology optimization formulation to maximize the operating frequency range. Because different wave physics are involved along the circumferential and radial directions, the topology optimization requires the extraction of the key physical phenomena along the two different directions. In doing so, the wave physics occurring in the hyperbolic elastic metamaterial is analyzed by using equivalent discrete models and the findings from the analysis are used to set up a topology optimization problem. The topology optimization that maximizes the operating frequency range of the hyperbolic elastic metamaterial is newly formulated by using the finite element method. After the metamaterial configuration maximizing the frequency range is found, the mechanics hidden in the optimized configuration is explained in some details by using analytic mass-spring model.

Appendix. Validation of the mass-spring system based approaches

In Appendix, the wave dispersion relations in (3) and (17), which are based on the mass-spring systems shown in Figs. 3 and 4, will be validated. To this end, the equivalent mass and spring coefficients of the unit cell are first calculated analytically. Then the dispersion curves based on the mass-spring model are compared with those of the finite element analysis incorporating a full continuum model of the unit cell.

Since the operating frequencies are relatively low for which the unit cell size is much smaller than the wavelength, the equivalent mass coefficients (m ₁, m ₂, m ₃ and m ₄) can be calculated directly from the density and volume of each segment shown in Fig. 13a. Considering the wave motion in Segment S ₁ shown in Fig. 13b, the spring coefficients a ₁ and a ₃ can be evaluated as

Fig. 13

(Color Online) a The unit cell considered in this work, the detailed geometric parameters to calculate the equivalent mass and spring coefficients for b Segment S1 and c Segment S2

$$ {a}_1=\frac{E{h}_{S_1}{t}_{S_1}}{l_{S_1}\left(1-{v}^2\right)},{a}_3=\frac{E{l}_{S_1}{t}_{S_1}}{h_{S_1}}. $$

(47)

On the other hand, due to the complicated configuration of Segment S ₂, the spring coefficients a ₂ and a ₄ are calculated from the energy method. From the total energy of Segment S ₂, the spring coefficient a ₂ is calculated as (Oh et al. 2014)

$$ \begin{array}{c}\hfill \frac{1}{a_2}=\left[\frac{2}{3}{\beta}_{{\mathrm{S}}_2}^3+{\beta}_{{\mathrm{S}}_2}^2{\gamma}_{{\mathrm{S}}_2}+\Big[\frac{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\beta}_{{\mathrm{S}}_2}^2}{2}\right.\hfill \\ {}\hfill \left.\kern1.75em +\frac{\left\{{\left({\gamma}_{{\mathrm{S}}_2}+{\alpha}_{{\mathrm{S}}_2}\right)}^2-{\alpha}_{{\mathrm{S}}_2}^2\right\}{\beta}_{{\mathrm{S}}_2}}{2}\Big]A+\left({\beta}_{{\mathrm{S}}_2}^2+{\beta}_{{\mathrm{S}}_2}{\gamma}_{{\mathrm{S}}_2}\right)B\right]/E{I}_{{\mathrm{S}}_2}\hfill \end{array}, $$

(48)

where ${I}_{{\mathrm{S}}_2}$ is the moment of inertia of Segment S ₂ defined as ${I}_{{\mathrm{S}}_2}={l}_{{\mathrm{S}}_2}{t}_{{\mathrm{S}}_2}^3/12$. One can show that A and B can be calculated from the boundary condition as

$$ \begin{array}{l}\begin{array}{c}\hfill \left[\begin{array}{cc}\hfill \frac{{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right)}^3}{3}+\left\{{\left({\gamma}_{{\mathrm{S}}_2}+{\alpha}_{{\mathrm{S}}_2}\right)}^2+{\alpha}_{{\mathrm{S}}_2}^2\right\}{\beta}_{{\mathrm{S}}_2}\hfill & \hfill \frac{{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right)}^2}{2}+\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\beta}_{{\mathrm{S}}_2}\hfill \\ {}\hfill \frac{{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right)}^2}{2}+\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\beta}_{{\mathrm{S}}_2}\hfill & \hfill 2{\alpha}_{{\mathrm{S}}_2}+2{\beta}_{{\mathrm{S}}_2}+{\gamma}_{{\mathrm{S}}_2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill A\hfill \\ {}\hfill B\hfill \end{array}\right]\hfill \\ {}\hfill = \left[\begin{array}{c}\hfill -\frac{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\beta}_{{\mathrm{S}}_2}^2}{2}-\frac{\left\{{\left({\gamma}_{{\mathrm{S}}_2}+{\alpha}_{{\mathrm{S}}_2}\right)}^2-{\alpha}_{{\mathrm{S}}_2}^2\right\}{\beta}_{{\mathrm{S}}_2}}{2}\hfill \\ {}\hfill -{\beta}_{{\mathrm{S}}_2}^2-{\beta}_{{\mathrm{S}}_2}{\gamma}_{{\mathrm{S}}_2}\hfill \end{array}\right]\hfill \end{array}.\\ {}\kern7em \end{array} $$

(49)

In the same manner, the spring coefficient a ₄ is calculated as (Oh 2014)

$$ \begin{array}{l}\begin{array}{c}\hfill \frac{1}{a_4}=\left[\frac{\left(4{\alpha}_{{\mathrm{S}}_2}+3{\gamma}_{{\mathrm{S}}_2}\right){\alpha}_{{\mathrm{S}}_2}^2}{12}+\frac{\left({\alpha}_{{\mathrm{S}}_2}+{\beta}_{{\mathrm{S}}_2}\right){\gamma}_{{\mathrm{S}}_2}^2}{16}+\frac{\gamma_{{\mathrm{S}}_2}^3}{160}\right.\hfill \\ {}\hfill +\left[-\frac{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\alpha}_{{\mathrm{S}}_2}{\beta}_{{\mathrm{S}}_2}}{2{\gamma}_{{\mathrm{S}}_2}}-\frac{\beta_{{\mathrm{S}}_2}^2}{4}\right]C\ \left.+\left[\frac{\alpha_{{\mathrm{S}}_2}^2}{\gamma_{{\mathrm{S}}_2}}+\frac{\alpha_{{\mathrm{S}}_2}+{\beta}_{{\mathrm{S}}_2}}{2}+\frac{\gamma_{{\mathrm{S}}_2}}{12}\right]D\right]/E{I}_{{\mathrm{S}}_2}\hfill \\ {}\hfill +\frac{1}{G{A}_{{\mathrm{S}}_2}\kappa}\left({\alpha}_{{\mathrm{S}}_2}+\frac{\gamma_{{\mathrm{S}}_2}}{6}\right)\hfill \end{array},\\ {}\kern1.5em \\ {}\kern1.5em \end{array} $$

(50)

where κ is the shear-correction factor (κ = 5/6) and C and D can be calculated from the following equation;

$$ \begin{array}{l}\begin{array}{c}\hfill \left[\begin{array}{cc}\hfill \frac{\beta_{{\mathrm{S}}_2}^3}{3}+{\beta}_{{\mathrm{S}}_2}^2{\alpha}_{{\mathrm{S}}_2}+\frac{E{I}_{{\mathrm{S}}_2}}{G{A}_{{\mathrm{S}}_2}\kappa }{\beta}_{{\mathrm{S}}_2}\hfill & \hfill -{\alpha}_{{\mathrm{S}}_2}{\beta}_{{\mathrm{S}}_2}-\frac{\beta_{{\mathrm{S}}_2}^2}{2}\hfill \\ {}\hfill -{\alpha}_{{\mathrm{S}}_2}{\beta}_{{\mathrm{S}}_2}-\frac{\beta_{{\mathrm{S}}_2}^2}{2}\hfill & \hfill {\alpha}_{{\mathrm{S}}_2}+{\beta}_{{\mathrm{S}}_2}+\frac{\gamma_{{\mathrm{S}}_2}}{2}\hfill \end{array}\right]\left[\begin{array}{c}\hfill C\hfill \\ {}\hfill D\hfill \end{array}\right]\hfill \\ {}\hfill =\left[\begin{array}{c}\hfill \frac{\left({\gamma}_{{\mathrm{S}}_2}+2{\alpha}_{{\mathrm{S}}_2}\right){\alpha}_{{\mathrm{S}}_2}{\beta}_{{\mathrm{S}}_2}{\gamma}_{{\mathrm{S}}_2}}{4}+\frac{\gamma_{{\mathrm{S}}_2}^2{\beta}_{{\mathrm{S}}_2}^2}{16}\hfill \\ {}\hfill -\frac{\alpha_{{\mathrm{S}}_2}^2{\gamma}_{{\mathrm{S}}_2}}{4}-\frac{\left({\alpha}_{{\mathrm{S}}_2}+{\beta}_{{\mathrm{S}}_2}\right){\gamma}_{{\mathrm{S}}_2}^3}{16}-\frac{\gamma_{{\mathrm{S}}_2}^3}{24}\hfill \end{array}\right]\hfill \end{array}.\\ {}\kern12.5em \end{array} $$

(51)

For more details about the calculation of mass and spring coefficients, see Oh (2014) and Oh et al. (2014).

By using the equivalent mass and spring coefficients, the wave dispersion curves are calculated from (3, 17) and from the finite element analysis method, as in Figs. 14 and 15. Here, the metamaterials having the width of 9.8, 14.4 and 19.2 mm are considered. Good agreements between two dispersion curves can be observed in the Figs. 14 and 15, which demonstrate a good correlation in mechanics between the mass-spring systems and actual unit cell structures. Also, considering the definition of the shear resonance frequency ${\omega}_{shear}=\sqrt{2{a}_4/{m}_4}$, it can be seen that the internal resonance frequency ω _shear can be calculated by using Segment S₂ only, as employed in the present multi-model based analysis.

Fig. 14

(Color Online) The θ -directional wave dispersion curves calculated by analytic mass-spring system and numerical simulations of actual unit cell. The width of the metamaterials are a 9.8, b 14.4 and c 19.2 mm, respectively

Fig. 15

(Color Online) The r-directional wave dispersion curves calculated by analytic mass-spring system and numerical simulations of actual unit cell. The width of the metamaterials are a 9.8, b 14.4 and c 19.2 mm, respectively