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Journal of Elasticity

Erschienen in:

13.09.2021

Effective Model for Elastic Waves in a Substrate Supporting an Array of Plates/Beams with Flexural and Longitudinal Resonances

verfasst von: Jean-Jacques Marigo, Kim Pham, Agnès Maurel, Sébastien Guenneau

Erschienen in: Journal of Elasticity | Ausgabe 1/2021

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Abstract

In a previous study (Marigo et al. in J. Mech. Phys. Solids 143:104029, 2020) we have studied the effect of a periodic array of subwavelength plates or beams over a semi-infinite elastic ground on the propagation of waves hitting the interface. The study was restricted to the low frequency regime where only flexural resonances take place. Here, we present a generalization to higher frequencies which allows us to account for both flexural and longitudinal resonances and to evaluate their interplay. An effective model is obtained using asymptotic analysis and homogenization techniques, which can be expressed in terms of the ground alone with an effective dynamic (frequency-dependent) boundary conditions of the Robin’s type. For an in-plane wave at oblique incidence, the scattered displacement fields and the reflection coefficients are obtained in closed forms and their effectiveness to reproduce the actual scattering is inspected by comparison with direct numerics in a two-dimensional setting.

Vorheriger Artikel A Constrained Cosserat Shell Model up to Order : Modelling, Existence of Minimizers, Relations to Classical Shell Models and Scaling Invariance of the Bending Tensor

Nächster Artikel Representation of Solutions of Lamé–Navier System

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The dispersion relation of the flexural resonances reads $D(\kappa )=(1+\cos \kappa h_{\text{\tiny b}}\text{ch}\kappa h_{\text{\tiny b}})=0$ resulting in $\kappa h_{\text{\tiny b}}=1.875$, 4.694, 7.855, 10.995, ⋯; the approximation $\kappa h_{\text{\tiny b}}\sim \frac{\pi }{2}+n\pi $ provides $\kappa h_{\text{\tiny b}}=$ 1.571, 4.712, 7.854, 10.995 ⋯ and the agreement gets better as $n$ increases.

The parameter ${\mathcal{K}}$ can be written ${\mathcal{K}}=\frac{K}{\kappa }= \frac{h_{\text{\tiny F}}}{h_{\text{\tiny L}}}$; at prescribed frequency, it measures the relative wavelengths associated with flexural and longitudinal motions.

At the occurrence of the first longitudinal resonance, $f_{\text{\tiny L}}$ diverges in (44) resulting in $u_{z}(x,0)=0$ in (43); this was not captured in the model of [1] as $f_{\text{\tiny L}}=\varphi \frac{\rho _{\text{\tiny b}}}{\rho _{\text{\tiny s}}}k_{\text{\tiny T}}h_{\text{\tiny b}}$ was obtained as an approximation of $f_{\text{\tiny L}}$ in (43) for $Kh_{\text{\tiny b}}\ll 1$.

In Fig. 7, spectra are reported against $\theta _{\text{\tiny L}}\in (0,90^{\circ })$ in their lower-parts. As $\theta _{\text{\tiny L}}=90^{\circ }$, $\theta _{\text{\tiny T}}=\theta _{c}$ with $\theta _{c}=\text{asin}\sqrt{ \frac{\mu _{\text{\tiny s}}}{\lambda _{\text{\tiny s}}+2\mu _{\text{\tiny s}}}} \simeq 37.8^{\circ }$. Increasing further the incidence of the transverse wave above $\theta _{c}$, the longitudinal waves become evanescent and no mode conversion occurs (hence $|R_{\text{\tiny TT}}|=1$ and Fig. 7 shows the real part). Note that the expressions of the reflection coefficients in (50) remain valid, with $\sin \theta _{\text{\tiny L}}=\frac{\beta }{k_{\text{\tiny L}}}>1$, hence $\alpha _{\text{\tiny L}}$ becomes purely imaginary.

It is worth noting that the discrepancy between our model (dotted black lines) and that neglecting flexural motions (dashed-dotted green lines) is more prominent for the absolute values reported in Fig. 9 than for the real parts in Figs. 7 and 8. This is due to the fact that imaginary parts of the reflexion coefficients have more pronounced variations around resonances (they cancel for the substrate on its own, (50)).

Expressions of the reflection coefficients (53) apply at longitudinal resonances, $|f_{\text{\tiny L}}|\to \infty $. Symmetrically, at flexural resonances, $|f_{\text{\tiny F}}|\to \infty $, (50) simplify to

$$ \displaystyle R_{\text{\tiny TT}}= \frac{ \cos \theta _{\text{\tiny T}}-i\xi ^{-1}f_{\text{\tiny L}}\cos (\theta _{\text{\tiny L}}+\theta _{\text{\tiny T}})}{\cos \theta _{\text{\tiny T}}-i\xi ^{-1}f_{\text{\tiny L}}\cos (\theta _{\text{\tiny L}}-\theta _{\text{\tiny T}})}, \quad \displaystyle R_{\text{\tiny LT}}= \frac{-i\sin 2\theta _{\text{\tiny T}}f_{\text{\tiny L}}}{\cos \theta _{\text{\tiny T}}-i\xi ^{-1}f_{\text{\tiny L}}\cos (\theta _{\text{\tiny L}}-\theta _{\text{\tiny T}})}, $$

hence strictly, we do not have $|R_{\text{\tiny TT}}|=1$ and $|R_{\text{\tiny LT}}=0$ at flexural resonances. This is less visible in Fig. 9 as most of these resonances take place as $|f_{\text{\tiny F}}|\ll 1$ (far from a longitudinal resonance).

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Titel: Effective Model for Elastic Waves in a Substrate Supporting an Array of Plates/Beams with Flexural and Longitudinal Resonances
verfasst von: Jean-Jacques Marigo
Kim Pham
Agnès Maurel
Sébastien Guenneau
Publikationsdatum: 13.09.2021
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 1/2021
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-021-09854-4

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