# Dispersion of elastic and acoustic waves in multilayer structures involving poroelastic materials

Mathieu Maréchal, Jean-Philippe Groby, Olivier Dazel, Vicent Romero-García

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Mathieu Maréchal, Jean-Philippe Groby, Olivier Dazel, Vicent Romero-García. Dispersion of elastic and acoustic waves in multilayer structures involving poroelastic materials. Resonance -Journées Jeunes Chercheurs/Chercheuses en Acoustique Vibrations et Bruit (JJCAB), Jul 2023, Toulouse, France. hal-04178076

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Submitted on 7 Aug 2023

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Distributed under a Creative Commons Attribution 4.0 International License
# Dispersion of elastic and acoustic waves in multilayer structures involving poroelastic materials

Mathieu Maréchal

Jean-Philippe Groby, Olivier Dazel & Vicent Romero-García

## Numerical computation of dispersion relations

* Interest in the modal behavior of various structures: metaporous surfaces/interfaces and composite multilayer

* Influence of the losses in poroelastic materials (PEM): viscoelastic (from the elastic part), the viscothermal (from the fluid part) and leakage (in the structure and from the structure to the surrounding medium)

* Investigation of these mechanisms using $(\tilde{k}, \omega)$ dispersion relations: complex wavenumber and real frequency

* Need of robust, versatile and easy to use numerical methods to compute the dispersion relations in these different configurations

## Spectral collocation method

* Decomposition of fields as polynomials

Basis functions:
Chebyshev
polynomials

* Chebyshev nodes: roots of the polynomials

* Residue at the nodes $r(x_m) = 0$

* Quasi-guided wave ansatz $u(x) = u(x_2)e^{ik_1x_1}e^{-i\omega t}$

## Root-finding with Müller method

* Secant-based method with quadratic approximation

* Search of roots from an analytical description of the

system

$$ \det(M(\tilde{k}, \omega)) = 0 \qquad f(\tilde{k}) $$

* Sensitivity to initial values and conditioning of M

* Slow and computationally expensive method

* No certainty of finding all roots at a given $\omega$

<figure><img src="image_2.png" /></figure>

## 1 - Discretization across the layers of the structure

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## 3 - Solution of the problem

$$ (k_1^2 K_2 + k_1 K_1 + K_0 + ik_2^{(f)} K_1') S = 0 \qquad S = (A_{0-}, s_0, \dots, s_n, \dots, s_N, A_{0+}) $$

Algebraic
manipulations
$
\xrightarrow{\text{to linearize the problem}}
$

eigenvalues:
wavenumbers
$(A - \gamma B) S' = 0$ eigenvectors:
mode shapes

## 2 - Construction of the multilayer numerical scheme

<figure><img src="image_4.png" /></figure>

## Solution of the eigenvalue problem ($A - \gamma B) S' = 0$

* Linearization: change of variable for $k_1$ and $k_2$ to go from a nonlinear problem to a polynomial problem

* Recovery of the physical fields mode shapes at no additional computational cost.

## Dispersion of waves in a bilayer aluminium-melamin structure

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

## Overview of the SCM

* Numerical computation of dispersion relations for general multilayer structures

* Computationally very efficient and guarantees to yield all solutions of the system

* Further developments: experimental validation and study of similar structures with embedded periodic inclusions

## Poroelastic materials (PEM)

* Biphasic material: fluid and solid phases

* Full Biot model accounting for the skeleton motion

* Formulation $(u^s, p)$ used to express the motion equations, involving 3 physical fields in the PEM

## Results

* Aluminium plate: $h_2 = 1$ mm, $M_2 = 5$

* Melamine foam: $h_1 = 5$ cm, $M_1 = 17$

* Symmetry in the results, with $\tilde{k}_1 = -\tilde{k}_1^*$

* A0 mode with a cutoff frequency from the rigid-backing (0)

* Large attenuation for most branches: fluid-coupling and losses in the material

* Aluminium plate modes not visible in this frequency range: large thickness difference

## References

* D. Kiefer et al., J. Acoust. Soc. Am., 145, 3341 (2019)

* L. Trefethen, Spectral methods in MATLAB, SIAM (2000)

* N. Atalla et al., J. Acoust. Soc. Am. 104.3 (1998)

* A. Hood, Localizing the eigenvalues of matrix-valued functions, Cornell University (2017)

<figure><img src="image_8.png" /></figure>