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.