Force-field-induced energy-based design method for arbitrary prescribed modes in elastic metamaterials

Elastic metamaterials possess flexible regulatory capabilities of elastodynamic field information and energy through engineering and tailoring wave amplitudes, phase, and polarization vectors. However, due to the lack of general wave quantities and dynamic mode characterization methods, it is difficult to describe and design customized elastic dispersions with prescribed eigenmodes of interest, especially under large wave vectors or high frequencies. To tackle this challenge, we propose a systematic design method based on force-field-induced energy to inversely customize arbitrary prescribed eigenmodes at required frequencies for both small and large wave vectors. We build up a dynamic mode characterization theory based on energy, which contributes to portraying eigenmode response behavior under external excitations. It theoretically reveals the distribution features of the energy, induced by external excitations, in wave vector-frequency (k-ω) domain for the solid media. A systematic inverse-design method, using responsive energy maximization, is proposed to tailor-make eigenmodes and dispersions under arbitrarily prescribed k-ω conditions. Then, a series of periodic porous structures are optimized to support orthotropic/anisotropic longitudinal, transversal and rotational modes at different k-ω points, alongside customized dispersion. Meanwhile, an inverse strategy fusing longitudinal and transversal modes is forged and used to realize broadband fluid-like mode in porous microstructure with an effective refractive index, in which a strongly suppressed transversal mode in the extremely low-frequency region of the dispersion and a single broadband longitudinal mode are supported. In addition, through inversely designing local vibration modes at three k-ω points simultaneously, a dispersion passband supporting negative group velocity is generated within an expected frequency range. Meanwhile, entire dispersion curves satisfying the prescribed k-ω relationship and supporting prescribed modes are customized. The wave behaviors of the optimized metamaterials are elucidated by phonon-band-structure experiments as well as numerical simulations. The established approach provides a universal design paradigm of wave modes that promises to pave the route for engineering extreme dispersion and functionalities.