Inverse Design of Nonlinear Phononic Metamaterials for Nonlinear Ultrasonic Wave Cloaking

The interaction of monochromatic ultrasonic waves with initial damages in solids—such as dislocation substructures, grain boundary plasticity, micro-voids, and cracks—generates higher harmonics, as explored by Cantrell (2004). Tang, Achenbach, Wang, and Kube (2012-17) extended this by studying how longitudinal waves scatter from single nonlinear circular inclusions, giving rise to higher harmonics in both longitudinal and transverse components.

Building on this, our focus shifts to the more complex scenario involving multiple nonlinear inclusions of varied shapes and sizes, where the challenge of harmonic scattering increases. To tackle this, we employ finite element methods and construct a parametric model for nonlinear inclusion clusters. Inclusions are designed and organized within a linear elastic matrix, with parameters such as size, shape, and orientation carefully controlled.

Our work centers on optimizing this configuration using gradient-free algorithms to reduce higher harmonic amplitudes detected at the receiver's periphery. This process, framed as a constrained optimization problem, leads to an achievement in nonlinear cloaking. The outcome is a metamaterial that can "cloak" nonlinear ultrasonic waves, rendering them undetectable. Furthermore, we've engineered a single nonlinear inclusion that replicates the cloaking effect of multiple inclusion clusters, marking a significant step in the evolution of nonlinear mechanical metamaterials.