On bandgap sensitivity to three-to-one internal resonances between acoustic and optical waves in metamaterials

Abstract

We investigate the nonlinear equations governing wave propagation across a metamaterial consisting of a cellular periodic structure hosting resonators with linear and cubic springs. The resulting system of two coupled equations with cubic nonlinearity is Hamiltonian, with the origin being an elliptic equilibrium characterized by two distinct linear frequencies associated with the acoustic and optical wave modes. Understanding this complex wave propagation problem requires the explicit derivation of analytical formulas for the nonlinear frequencies as functions of the relevant physical parameters. In the small wave amplitude regime and away from internal resonances, by using Hamiltonian Perturbation Theory, we obtain the first-order nonlinear correction to the linear frequencies. Additionally, we analytically estimate the threshold for the applicability of the perturbative solution, identifying the maximum admissible amplitude at which the obtained formulas remain valid. This threshold decreases significantly in the presence of internal resonances, particularly when the ratio between the optical and acoustic frequencies is close to 3 due to a 3:1 resonance which affects the correction. Furthermore, we analytically evaluate the remainder after the first step of the perturbation scheme. The specific application deals with the wave propagation equations obtained for metamaterial honeycombs with periodically distributed nonlinear one-dof resonators. Our analysis provides quantitative insights into how internal resonances affect the bandgap. Specifically, we identify a 3:1 resonant curve in the parameter space represented by the resonator modal mass and stiffness. We also identify a region away from the above resonant curve where the nonlinear bandgap for softening resonators springs is significantly larger than the linear counterpart.

Figures (13)

• Figure 1: Schematic view of a 2D metamaterial with periodically distributed resonators, proposed in SW23jsv.
• Figure 2: The irreducible Brillouin triangle $\triangle:=\bf{\Gamma}\overset{{\triangle}}{\bf{X}} \bf{M}$ with $\mathbf{\Gamma}=(0,0)$, $\mathbf{X}=(\frac{4}{3}\pi,0)$, $\mathbf{M}=(\pi,\frac{\pi}{\sqrt{3}})$.
• Figure 3: The solid red curve $\mathfrak R$ and the dashed red curve are the two ${\bf X}$-resonant curves in the $(\tilde{M},\tilde{K})$-plane. The light yellow region within the reference rectangle $[0.05, 0.3]\times[1, 20]$ indicates the "good" nonresonant set where the representation formula \ref{['formulaSL1']} can be applied. In contrast, the light purple region shows the "bad" set of resonant parameters, where a "resonant" representation formula is necessary.
• Figure 4: Contour plot of the maximum percentage difference between nonlinear and linear bandgap in the $(\tilde{M},\tilde{K})$ plane. The ${\bf X}$-resonant curve $\mathfrak R$ is plotted in red. Here, $N_3=- 10^4$ (softening spring). In this softening case, the majority of parameter pairs above $\mathfrak R$ result in an increase in the bandgap width, while those below $\mathfrak R$ either show a decrease or, at most, a very slight increase. The region where the increase is most pronounced is contained in the set of "good" parameters highlighted in light yellow in Figure \ref{['buoni']}.
• Figure 5: (Left) The reference rectangle $[0.05,0.3]\times[1,20]$. For every pair $(\tilde{M},\tilde{K})$ in the blue region, there exist two resonant curves intersecting $\triangle$, while for pairs in the green region, there exists only one resonant curve intersecting $\triangle$. The red curve that separates the two regions is exactly the ${\bf X}$-resonant curve $\mathfrak R$. (Right) Resonant curves in the $(\tilde{k}_1,\tilde{k}_2)$-plane for different values of the pairs $(\tilde{M},\tilde{K})$ and their intersection with the boundary of the Brillouin triangle $\triangle$. The six blue curves correspond to three different points $(\tilde{M},\tilde{K})$ in the blue region of the rectangle. Specifically, when $(\tilde{M},\tilde{K})=(0.146,2)=A$ the corresponding two blue curves, the dotted-dashed ones, have 4 intersections. When $(\tilde{M},\tilde{K})=(0.146,3.6)=B$ the corresponding two blue curves, the solid ones, have 6 intersections. When $(\tilde{M},\tilde{K})=(0.146,5)=C$ the corresponding two blue curves, the dashed ones, have 4 intersections. The red curves, corresponding to $(\tilde{M},\tilde{K})=(0.146,5.73)=D\in\mathfrak R$, have 3 intersections. Finally the green curves, corresponding to $(\tilde{M},\tilde{K})=(0.146,10.79)=E$, have only 2 intersections.

Theorems & Definitions (10)

• Proposition 1
• Theorem 1: Birkhoff Normal Form
• Proposition 2: Quantitative estimates on nonresonant BNF
• Corollary 1
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Proposition 3
• Remark 5