Nonlinear Inclusions Invisible to Longitudinal Ultrasonic Waves

Interaction of monochromatic ultrasonic waves with early-stage damages in solids such as dislocation substructures, micro-voids, and micro-cracks generate higher harmonics, Cantrell (2004). Damages are accumulated locally as a cluster of nonlinear inclusions. Theoretical solutions for the interaction of the longitudinal wave with a single quadratic and cubically nonlinear circular inclusion are obtained by Tang et al. (2012), Wang & Achenbach (2017-18), and Kube (2017-18) to demonstrate harmonic scattering, the presence of higher harmonics of the scattered longitudinal and transverse waves, and their sensitivity towards the size and shape of the inclusion. Harmonic scattering from multiple nonlinear inclusions of various sizes and shapes arranged in a cluster is of critical importance, and obtaining theoretical solutions is challenging. Finite element studies help us understand harmonic scattering from such complex clusters of nonlinear inclusions. In this study, a parametric cluster of nonlinear inclusions is modeled by defining design parameters that control the elliptical inclusions size, shape, and angle of rotation and the relative positions of the inclusions embedded in the linear elastic material. Inclusions are modeled as Murnaghan hyperelastic material by keeping linear impedance as of linear matrix material. Design parameters are optimized using the Nelder-Mead gradient-free algorithm to reduce amplitudes of higher harmonics measured over the edge of the receiving end by solving a time-dependent constrained optimization problem and validated using an optimal cluster configuration. Such inversely designed cluster of nonlinear inclusions acts as nonlinear metamaterials that are invisible to longitudinal elastic waves despite their presence.