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Multi-layer structures are ubiquitous in constructing metamaterial devices torealise various frontier applications including super-resolution imaging andinvisibility cloaking. In this paper, we develop a general mathematicalframework for studying elastostatics within multi-layer material structures in$\mathbb{R}^d$, $d=2,3$. The multi-layer structure is formed by concentricballs and each layer is filled by either a regular elastic material or anelastic metamaterial. The number of layers can be arbitrary and the materialparameters in each layer may be different from one another. In practice, themulti-layer structure can serve as the building block for various materialdevices. Considering the impingement of an incident field on the multi-layerstructure, we first derive the exact perturbed field in terms of an elasticmomentum matrix, whose dimension is the same as the number of layers. By highlyintricate and delicate analysis, we derive a comprehensive study of thespectral properties of the elastic momentum matrix. This enables us toestablishe a handy algebraic framework for studying polariton resonancesassociated with multi-layer metamaterial structures, which forms thefundamental basis for many metamaterial applications.

Elastostatics with multi-layer metamaterial structures and an algebraic framework for polariton resonances