# Dynamic cloaking in elastic thin plates using architected metamaterials

Léo Pradier, Baptiste Chomette, Arnaud Hubert, Manuel Collet

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Léo Pradier, Baptiste Chomette, Arnaud Hubert, Manuel Collet. Dynamic cloaking in elastic thin plates using architected metamaterials. Active and Passive Smart Structures and Integrated Systems XIX, Mar 2025, Vancouver, Canada. pp.8, 10.1117/12.3051957. hal-05073297

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Submitted on 19 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
Dynamic cloaking in elastic thin plates using
architected metamaterials

Léo Pradier<sup>a</sup>, Baptiste Chomette<sup>a</sup>, Arnaud Hubert<sup>b</sup>, and Manuel Collet<sup>a</sup>

<sup>a</sup>Ecole Centrale de Lyon, CNRS, ENTPE, LTDS, UMR5513, 69130
Ecully, France

<sup>b</sup>Université de Technologie de Compiègne, Alliance Sorbonne
Université, Laboratoire Roberval, Centre de recherche de Royallieu,
CS 60319, 60203 Compiègne, cedex, France

May 19, 2025

**Abstract**

We investigate the case of cloaking metamaterials for Kirchhoff-Love plates. We propose a design methodology still based on the geometrical transformation method initially introduced by Pendry [1], no longer with the aim of transforming the governing equation of Kirchhoff-Love plates but with the aim of generating curved waveguides forcing bending waves to propagate in the direction advocated by the preliminary idea of cloaking. Numerical results show that the resulting metamaterial is capable of making a hole in a plate invisible over a wide frequency band.

# 1 Introduction

Cloaking was introduced by Pendry [1], Leonhardt [2] and implemented by Schurig [3] in 2006 for the case of electromagnetic microwaves. The key idea was the introduction of a transformation of the space coordinates associating any element whose coordinates belong to the domain to be cloaked with new coordinates outside this domain. By performing this change of coordinates in Maxwell's equations governing the propagation of electromagnetic waves, Pendry shows that Maxwell's equations are form invariant. The coordinates transformation therefore maps Maxwell's equations to Maxwell's equations generalised to the case of propagation in an inhomogeneous and anisotropic medium which can be achieved by using metamaterials. For the more complex case of elastodynamics, it was shown by Milton et al. [4] that, in general, the governing equations of elasticity are not form invariant
under coordinates transformations. This means that the method described above is inapplicable. Cloaking as originally defined aims to bend the trajectories of a wave, within a medium of finite size, forcing the wave to "take an extended path" around an object. Consequently, we will use geometric transformation as a guideline to generate an architected metamaterial that forces the waves to propagate around a hole. We begin by mathematically defining the ray equations obtained from classical cloaking theory, and the resulting phase velocity field to be matched along these rays. We will then show that it is possible to approximate this desired behaviour using a set of curved beams with variable out-of-plane thicknesses. Finite element (FE) simulations will support our modelling and a performance indicator will be introduced to quantify the effectiveness of the proposed design as a function of frequency.

## 2 Cloaking design methodology

As specified by Pendry [1], cloaking allows a wave to bypass a zone of space. This is particularly visible when the ray tracing is performed inside the cloaking layer. It is also known in the literature that the coordinate transformation induces a non-constant phase velocity in the cloaking layer, which is logical since a wave bypassing a zone will travel a greater distance than a wave travelling straight through. Our approach is to physically recreate this propagation path using curved waveguides with space-dependent properties in order to recover the phase velocity advocated. We suppose that this waveguide is governed by the Euler-Bernoulli thin beam equation.

$$ \frac{\partial^2}{\partial s^2} \left( EI(s) \frac{\partial^2 w}{\partial s^2} \right) + \rho A(s) \frac{\partial^2 w}{\partial t^2} = 0 \quad (1) $$

$w(s,t) \in \mathbb{R}$, $E$, $I$, $\rho$, and $A$ are respectively the out-of-plane displacement, Young's modulus, second moment of area, mass density, and cross-sectional area. A variable thickness $h_{\text{beam}}$ is assumed here in order to define the variation in phase velocity along the curvilinear abscissa $s$. This dependency appears in (1) through the coefficients $I = lh_{\text{beam}}^3(s)/12$ and $A = lh_{\text{beam}}(s)$, where $l$ is the width of the beam. Note that this model does not take into account the coupling between bending and torsion due to the curvature of the beam [5, 6]. Assuming the spatial and temporal parts of $w$ are separable (see details in Graff's book [7]), one can derive the dispersion relation (i.e. the dependence between the angular frequency $\omega$ and the wavenumber $k$) and the beam phase velocity.

$$ v_{\phi, \text{beam}} = \frac{\omega}{k} = \sqrt[4]{\frac{\omega^2 Eh_{\text{beam}}^2(s)}{12\rho}} \quad (2) $$

From (2), one can show that $h_{\text{beam}}$ must match:
$$h_{\text{beam}}(s) = \sqrt{\frac{12v_{\text{ext}}^4(s)\rho}{\omega^2 E}} \qquad (3)$$

Where $v_{\text{ext}}$ is the phase velocity to be imposed in order to maintain the phase between the wave propagating in the cloak and the wave propagating in its vicinity.

## 3 Numerical results

We now apply the presented method to the case of a real structure composed of several curved beams. Such a structure could be manufactured using additive manufacturing. The chosen mechanical properties for numerical simulations, are those of PPS 25% GF, which is a printable material, with high stiffness and low damping (dynamic damping is neglected in this study). All the geometric and material properties are listed in the table 1. Finite element simulations in the time domain are proposed using COMSOL Multiphysics. A square plate with free edge conditions is excited on its left side by a Gaussian pulse (force boundary condition) with a centre frequency of 6000 Hz. The resulting wave propagation, for both an uncloaked hole and a cloaked hole are depicted in figure 1. It appears that the defined cloak is effective in cancelling out the scattering created by the hole for this frequency, both upstream and downstream of the cloak.

## 4 Conclusion

Since the literature shows that it is not possible to transform the Kirchhoff-Love equation in such a way as to satisfy the exact cloaking behaviour, we present here a problem simplification to a unidirectional case showing the possibility of near cloaking. By using an effective medium through Euler-Bernoulli beam theory, we define a thickness variation in the cloak layer that reproduces a desired velocity field. We can then design a cloaking layer composed of several curved beams surrounding a hole. Numerical simulations of cloaked and uncloaked voids demonstrate the effectiveness of

<table><caption>Table 1: Parameters of the numerical model</caption><tr><td>Cloak inner radius, <i>a</i> [m]</td><td>0.05</td></tr><tr><td>Cloak outer radius, <i>b</i> [m]</td><td>0.1</td></tr><tr><td>Host plate length [m]</td><td>0.5</td></tr><tr><td>Host plate thickness, <i>h</i> [mm]</td><td>3</td></tr><tr><td>Young's modulus, <i>E</i> [GPa]</td><td>11</td></tr><tr><td>Poisson's ratio, <i>ν</i> [1]</td><td>0.35</td></tr><tr><td>Mass density, <i>ρ</i> [kg/m<sup>3</sup>]</td><td>1650</td></tr></table>
<figure><img src="image_2.png" /><figcaption>Figure 1: Time-domain simulation results for an uncloaked hole (right) and a cloaked hole (left) at $t = 0.44, 0.82$ and $1$ ms.</figcaption></figure>

the cloak for a Gaussian pulse excitation with a centre frequency of 6000 Hz. The resulting design has the advantage of being easily made using additive manufacturing technologies. Consequently, the manufacturing and experimental validation of the proposed cloak will be considered in a near future.

## Acknowledgments

This research was funded, in whole or in part, by l'Agence Nationale de la Recherche (ANR), project ANR-21-CE46-001-05. For the purpose of open access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
References

[1] J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling Electromagnetic Fields," *Science*, vol. 312, pp. 1780-1782, June 2006.

[2] U. Leonhardt, "Optical Conformal Mapping," *Science*, vol. 312, no. 5781, pp. 1777-1780, 2006. -eprint: https://www.science.org/doi/pdf/10.1126/science.1126493.

[3] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," *Science*, vol. 314, no. 5801, pp. 977-980, 2006. Publisher: American Association for the Advancement of Science.

[4] G. W. Milton, M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," *New J. Phys.*, vol. 8, pp. 248-248, Oct. 2006.

[5] S.-K. Lee, B. Mace, and M. Brennan, "Wave propagation, reflection and transmission in curved beams," *Journal of Sound and Vibration*, vol. 306, pp. 636-656, Oct. 2007.

[6] B. K. Lee, S. J. Oh, J. M. Mo, and T. E. Lee, "Out-of-plane free vibrations of curved beams with variable curvature," *Journal of Sound and Vibration*, vol. 318, pp. 227-246, Nov. 2008.

[7] K. Graff, *Wave Motion in Elastic Solids*. Oxford engineering science series, Ohio State University Press, 1975.