Envelope soliton in locally resonant piezoelectric metamaterials with nonlinear shunt circuits

Electromechanical metamaterials, comprising bimorph piezoelectric layers sandwiching an elastic substrate and connected to inductive shunt circuits, are well known for their locally resonant bandgaps. Beyond linear resonant circuits, synthetic impedance circuits with digital control enable precise tuning capabilities of resonant frequencies and introducing intentionally designed nonlinearities. In this work, we investigate nonlinear wave propagation in a piezoelectric metamaterial beam with Duffing-type nonlinear shunts, characterized by both linear and cubic nonlinear inductances. After homogenizing the model, we employ the method of multiple scales to study the nonlinear wave propagation in the limit of weak nonlinearity and long wavelength approximation. Our analysis reveals that the wave evolution adheres to the nonlinear Schrödinger equation that supports shape-preserving solitary waves where the dispersion is balanced with nonlinearity. In addition, the significant tunability of the nonlinear shunt circuit enables us to manipulate the dispersion curves and bandgap frequencies, facilitating the generation of solitons through either softening or stiffening nonlinearity. The analytical findings are validated by numerical studies using a nonlinear finite element method. These results may open an avenue of the nonlinear design of electromechanical metamaterial beams with various applications such as targeted energy transfer, enhanced sensing, and physical intelligence.