Wave amplitude decay driven by anharmonic potential in nonlinear mass-in-mass systems

Periodic or quasi-periodic arrangements of artificial structures can be used to design a class of materials, i.e., metamaterials, with intriguing
properties. Recently, it has been proposed to use periodic systems with internal resonances for the attenuation of acoustic/seismic waves.
However, large input displacements due to seismic waves can drive the working point of these systems in a nonlinear regime. Here, we have
studied the nonlinear dynamics of periodic chain of mass-in-mass systems, which can be used to model composite foundations, where the
external spring is characterized by an anharmonic potential. The main finding of this work is the identification of two attenuation
mechanisms, one is characterized by an exponential amplitude decay and is observed in the strongly anharmonic regime, whereas the other
has a linear decay pattern and characterizes the weak anharmonic dynamics. This result has a direct impact in the design of low frequency
seismic metamaterials.