Higher order interfacial effects for elastic waves in one dimensional phononic crystals via the Lagrange-Hamilton's principle

•New transmission conditions modeling a thin elastic layer are proposed.•They are obtained using asymptotic analysis and the Lagrange-Hamilton's principle.•They unify and extend four classical models (perfect, mass, spring, spring-mass).•The thickness, inertia, elasticity and anisotropy of the layer are accounted for.•Their effect on the band structure of 1d Bloch-Floquet waves is discussed.

This work proposes new transmission conditions at the interfaces between the layers of a three-dimensional composite structures. The proposed transmission conditions are obtained by applying the asymptotic expansion technique in the framework of Lagrange-Hamilton's principle. The proposed conditions take into account interfacial effects of higher order, thus representing an extension of the classical zero-thickness interface models. In particular, the (small) thickness of the interface together with its inertia, stiffness and anisotropy are accounted for. The effect of the transmission conditions on the band structure of Bloch-Floquet waves propagating in a one dimensional phononic crystal is discussed based on numerical results.