Tailoring dispersion and evanescent modes in multimodal nonlocal lattices using positive-only interactions

TL;DR

This work tackles dispersion control in uniform nonlocal lattices under the constraint of positive, passive stiffness. It introduces a parametric interpolation framework that fixes a small set of target frequency–wavenumber points, solving for nonlocal stiffness parameters $β_p$ to tailor local dispersion features without reconstructing an entire curve. The method is demonstrated on both spring and Euler–Bernoulli beam lattices, enabling roton-like extrema, tunable group-velocity dispersion, and actively designed evanescent decay within band gaps, all while maintaining real, positive-only stiffnesses. This approach provides a physically consistent and computationally efficient pathway for advanced phononic design with potential benefits for vibration isolation and wave control in metamaterials.

Abstract

Metamaterials derive their unconventional properties from engineered microstructures, with periodic lattices providing a versatile framework for modeling wave propagation. Dispersion relations, obtained from Bloch-Floquet theory, govern how waves propagate, attenuate, or localize within such systems. Extending interactions beyond nearest neighbors, through nonlocality, substantially enriches the design space of band diagrams, enabling phenomena such as negative or zero group velocities, roton-like extrema, and band-gap localization. However, existing approaches to dispersion tailoring often rely on analytical formulations or Fourier-based identifications, which become impractical for complex coupling mechanisms and offer limited control over physical constraints such as stiffness positivity. This work introduces a general interpolation-based framework for customizing dispersion relations in uniform nonlocal lattices. Rather than reconstructing full dispersion curves, the method enforces prescribed frequency-wavenumber points as interpolation constraints, enabling localized and tunable control of wave behavior. The formulation is applied to both spring- and beam-interaction lattices, and demonstrated on an Euler-Bernoulli beam model with adjustable nonlocal couplings. Through systematic parameter tuning, the framework enables the creation of rotons, the adjustment of group-velocity dispersion, and the design of evanescent waves with controlled exponential decay within band gaps, all while ensuring real, positive-only stiffness parameters and passive mechanical behavior. Altogether, this parametric interpolation strategy provides a physically consistent and computationally efficient route for engineering advanced phononic functionalities in periodic nonlocal systems.

Figures (6)

• Figure 1: Monoatomic chain with non-local interactions up to order $P=2$. Local interactions are shown in black, while higher-order couplings are highlighted in blue.
• Figure 2: Examples of dispersion curves for the Euler-Bernoulli beam lattice. In the first two cases, a single branch is interpolated, while in the third case both branches are simultaneously constrained (see Table \ref{['table:dispersion_beam']}).
• Figure 3: (\ref{['fig:roton_disp02']}) Example of a roton created at the frequency $\omega_{+}=8$. Three interpolation points are selected to enforce a vanishing central finite difference at $\omega_{+}$ (see Table \ref{['table:Rotons']}). (\ref{['fig:roton_disp01']}) Model exhibiting two propagating solutions for the same frequency $\omega_{-}=6$, obtained by prescribing two interpolation points at $(\kappa_{1}=1, \omega_{-})$ and $(\kappa_{2}=\pi, \omega_{-})$. (\ref{['fig:roton_transt']}) Transient response associated with (\ref{['fig:roton_disp01']}): the excitation at $\omega_{-}=6$ splits into two wave packets with distinct group velocities.
• Figure 4: (\ref{['fig:GVD_01']}) Dispersion curves of three parametrized models, each tailored in a frequency range around $\omega_0 = 12.0$. (\ref{['fig:GVD_02']}) Zoom on the customized region. Three interpolation points are selected to simultaneously control the slope and curvature of the dispersion relation near $\omega_0 = 12.0$ (see Table \ref{['table:GVD']}). For the three models, the group velocity is fixed at $c_0 = -2$, while the group velocity dispersion (GVD) takes values $0$, $1.25$, and $2.5$, respectively. (\ref{['fig:GVD_03']}) Transient wave-packet analysis. The excitation imposed on node $x_0$ follows Eq. \ref{['eq:wave_packet']}. The corresponding spectral content, shown in green in (\ref{['fig:GVD_02']}), is centered around $\omega_0$ and lies within the parametrized region of the dispersion curves.
• Figure 5: Two uniform nonlocal beam-lattice models exhibiting the same band gap $[6.0, 12.0]$. The colormap represents the modulus of $\lambda$, providing a complementary visualization of $\Im(\kappa)$. An evanescent wave solution at $\omega = 6.5$ is parametrized with different decay rates: $\kappa_I = -0.2$ in the first case and $\kappa_I = -0.7$ in the second. Transient analysis confirms that the spatial attenuation follows approximately $e^{\kappa_I}$. The excitation applied to the reference node $x_0$ follows Eq. \ref{['eq:wave_packet']}. The parameters used are listed in Table \ref{['table:evanescent_wave']}.