Nonreciprocal phenomenon in nonlinear elastic wave metamaterials with continuous properties

The nonreciprocal phenomenon appears in nonlinear acoustic metamaterial which are composed of a linear phononic crystal and a nonlinear layer. Based on the nonlinear characteristics, the acoustic waves can pass in one direction and are prohibited in reverse. Naturally, people will wonder whether the nonreciprocal phenomenon can be found in nonlinear elastic wave metamaterial. Because of the coupling of the multi-displacements, this problem becomes more complicated but interesting for waves in solid. Motivated by this enlightenment, we will discuss the band gap and transmission properties of elastic waves in both linear and nonlinear periodic systems. The fundamental and double frequency waves can be generated by the interaction between SH waves and the nonlinearity of materials. According to continuity conditions of incident SH wave, the transfer matrices are derived. The stop band and transmission coefficient are obtained by Bloch's law. This research is expected to be helpful to develop new elastic wave metamaterial devices.