Meta-dissipation: A framework for quantifying energy dissipation in dissipative discrete periodic metamaterials

Quantifying energy dissipation in dissipative periodic discrete metamaterials remains a challenge, as traditional interpretations of metadamping, based on summing modal damping ratios, do not consistently reflect actual time-domain energy loss in multi-modal systems. The study addresses this critical gap by introducing a framework grounded in a consistent definition of the unit cell for discrete dissipative systems. A single-term exponential energy decay coefficient, obtained through least squares minimization of the transient response, is proposed as a robust descriptor of system-level dissipation. The coefficient approximately captures energy decay behavior in both phononic crystals and acoustic metamaterials, offering a more physically meaningful measure than conventional damping metrics. The proposed meta-dissipation framework establishes a critical link between dispersion-based analysis and transient energy dissipation, enabling more effective design and optimization of metamaterials for vibration attenuation across a wide range of engineering applications.