Intro talk: Emergence of Willis constitutive coupling in elastodynamic heterogeneous media

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• William Parnell (University of Manchester)
• Tuesday 21 March 2023, 09:00-10:00
• Seminar Room 1, Newton Institute.

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MWSW02 - Theory of wave scattering in complex and random media

Co-authors – Philip A Cotterill, David Nigro, Marie Touboul

Abstract: We will begin this talk with an introduction to the formalism of scattering and multiple scattering in linear elastic media. We work entirely in two dimensions, describing scattering by a circular, traction-free void in an isotropic elastic medium and subsequently show that as the medium becomes highly incompressible (e.g. rubber-like), a low-frequency resonance arises; the Giant Monopole Resonance (GMR). This can be allied with the “rigid-in-soft” (RSR) resonance that has been employed in metamaterials science since the early 2000s, for the basis of numerous acoustic and elastodynamic metamaterials. We employ the multiple scattering formalism to illustrate how the resonators described above behave when they are in close proximity, so that multiple scattering effects cannot be neglected. We associate the terms “co-void” and “co-dipole” resonators with pairs of such scatterers in the case of the GMR and RSR respectively. We make the following conclusions: 1.       Co-void resonators allow a lower frequency resonance compared to a single GMR of the same cross-sectional area, but the resonant strength is largely unaffected. 2.       Co-dipole resonators allow a much stronger resonance than an isolated RSR , but the resonant frequency is largely unaffected. These resonators are thus suggested as the basis for future elastodynamic metamaterials. Following this we move onto the consideration of dynamic constitutive coupling associated with heterogeneous/inhomogeneous elastic materials. It is now well established that a general and inclusive dynamic constitutive law in acoustics and elasticity is that which involves Willis coupling, embedded in the Milton-Brianne-Willis equations. Its analogue in electrodynamics is bianisotropy. We will describe the emergence of this coupling phenomena from the starting point of the form-invariance of the governing equations to spatial transformations, and from material symmetry. Finally, we link the first two parts of the talk with an example of an inhomogeneous slab consisting of arrays of GMRs embedded in an elastic medium. We present results for the case of normal incidence where Willis coupling is required for the concept of dynamic homogenization to be valid. Of specific interest is the case where we allow the slab medium to be lossy and where we employ asymmetry in the arrays. This leads to reciprocal, direction-dependent absorption. We close with discussion of oblique incidence, with specific reference to which components of the coupling tensor can be discerned from reflection and transmission coefficients.

This talk is part of the Isaac Newton Institute Seminar Series series.

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