A Deterministic Limit for Acoustic Metamaterials: Deriving the Critical Bandgap Radius via Phononic Scattering

The prediction of structural failure and acoustic transmission in phononic crystals and acoustic metamaterials under high-amplitude shockwaves relies heavily on linear Bragg scattering models and weakly nonlinear perturbation methods. While these frameworks effectively estimate bandgap frequencies in the low-amplitude linear regime, they fail to deterministically define the exact spatial boundary where localized anharmonic yielding triggers catastrophic bandgap collapse. This paper introduces a strict continuum framework for elastodynamics. By modeling the periodic phononic lattice as a dynamic kinematic balance between the spatial capacity for wave attenuation (acoustic scattering) and the localized rate of anharmonic yielding (nonlinear forcing), we derive a universal critical bandgap radius (Rbandgap). We demonstrate that bandgap collapse is not a statistical breakdown of wave mechanics, but an exact deterministic limit where localized kinetic injection strictly overpowers the advective destructive-interference capacity of the surrounding microstructure. We propose a blueprint for Absolute Sonic Cloaking, guaranteeing unbreakable acoustic attenuation for defense and aerospace applications.