Topology optimization of hard-magnetic soft phononic structures for wide magnetically tunable band gaps

Abstract Hard-magnetic soft materials, which exhibit finite deformation under magnetic loading, have emerged as a promising class of soft active materials for the development of phononic structures with tunable elastic wave band gap characteristics. In this paper, we present a gradient-based topology optimization framework for designing the hard-magnetic soft materials-based two-phased phononic structures with wide and magnetically tunable anti-plane shear wave band gaps. The incompressible Gent hyperelastic material model, along with the ideal hard-magnetic soft material model, is used to characterize the constitutive behavior of the hard-magnetic soft phononic structure phases. To extract the dispersion curves, an in-house finite element model in conjunction with Bloch's theorem is employed. The {method of moving asymptotes} is used to iteratively update the design variables and obtain the optimal distribution of the hard-magnetic soft phases within the phononic structure unit cell. Analytical sensitivity analysis is performed to evaluate the gradient of the band gap maximization function with respect to each one of the design variables. Numerical results show that the optimized phononic structures exhibit a wide band gap width in comparison to a standard hard-magnetic soft phononic structure with a central circular inclusion, demonstrating the effectiveness of the proposed numerical framework. The numerical framework presented in this study, along with the derived conclusions, can serve as a valuable guide for the design and development of futuristic tunable wave manipulators.