Energy Transfer and Nonlinear Waves in the Models of Locally Resonant Granular Acoustic Metamaterials

Nowadays, the design of new metamaterials and the study of their properties is a common task, since the structures obtained with their help have wide areas of application. In this work, a model of a granular locally resonant metamaterial on a substrate is considered. The interaction between granules is determined by the hertzian contact potential with the power of nonlinearity p=3/2. In each granule has additional vibrational degree of freedom with a certain mass and linear interaction. In our case, the interaction with the substrate will also be considered linear. In such systems nonlinear waves named breathers exist, which are the main carrier of energy. A breather is a localized nonlinear wave with an internal degree of freedom. Here it is nessesary to note that the granular chain of the mass-in-mass type without a support has been well studied. The existence of stable breathers has been mathematically rigorously proven. However, when a system with a substrate is considered, breathers become long-lived as they radiate energy into an oscillating tail. To study the energy transfer and propagation of breathers in a granular metamaterial, a transition is made from the original system of equations to the asymptotic P-Schrödinger equation, which describes the envelope-functions of the variables describing the original system. Conditions for the propagation of breathers in a locally resonant granular chain without loss of energy into the oscillating tail were also obtained.