Stochastic dispersion behavior and optimal design of locally resonant metamaterial nanobeams using nonlocal strain gradient theory

This study examines the stochastic response of a metamaterial (MM) nanobeam, focusing on bandgap formation and analyzed using machine learning. The nanobeam is modeled as an infinitely long Euler–Bernoulli beam with two length scale parameters: the nonlocal and strain gradient parameter. Periodically distributed linear resonators along its length introduce periodicity. The deterministic analysis is conducted by estimating bandgap edge frequencies using the dispersion of elastic waves in a representative unit cell. The impact of uncertainties on wave propagation behavior indicate that geometric properties predominantly influence variability in frequency response, followed by material properties, affecting the location and width of the bandgap. Scale dependent parameters, however, have a negligible effect. A Gaussian process (GP) surrogate model is employed to efficiently capture the stochastic behavior of the nanobeam. To highlight the utility of machine learning in computationally intensive tasks, a multi-objective optimization problem is formulated to tailor the bandgap features of the nanobeam. The offline-trained GP model yields a Pareto front of design configurations closely aligned with actual simulations, eliminating the need for repeated analyses during optimization. This surrogate based optimizer efficiently facilitates reverse engineering of MM designs for user defined wave dispersion characteristics, showcasing its potential for large scale optimization. Importantly, the stochastic dispersion framework grounded in nonlocal strain gradient theory can be directly applied to other periodic MM nanostructures. By varying unit cell configurations and materials within the same computational pipeline, new insights into bandgap emergence across applications ranging from phononic waveguides, nanoscale acoustic devices to structure–property relationships in next-generation MMs can be rapidly obtained.