  Nonlinear acoustic metamaterial (NAM) initiates new fields for controlling
elastic waves. In this work, the flexural wave propagation in the half-infinite
NAM beam consisting of periodic Duffing resonators is reported by considering
the third harmonic generation (THG). Different analytical methods are proposed
to describe the wave propagation in the equivalent homogenous medium. Then
their effectiveness and accuracy are demonstrated in comparison with the finite
element methods. We unveil analytically and numerically extensive physical
properties of the strongly nonlinear AM, including the nonlinear resonance in a
cell, the effective density, nonlinear locally resonant (NLR) bandgap,
propagations and couplings of the fundamental and the third harmonics. These
characteristics are highly interrelated, which facilitates the prediction of
functionalities. In the near field, the identical bifurcation frequency of
these features acts as the start frequency of the NLR bandgap for fundamental
waves, whose width is narrower for a stronger nonlinearity. While in the far
field, the NLR bandgap characterizes a distance-amplitude-dependent behavior
leading to a self-adaptive bandwidth. Moreover, the transmission in the
passband of the infinite NAM is different from the chaotic band effect of
finite NAMs, and it is influenced by the shifted NLR gap. Our work will promote
future studies and constructions of NAMs with novel properties.
