We formulate a discrete-to-continuum approach to the dispersion relation of one-dimensional lattice metamaterials. With reference to a generic lattice structure that can be described as a biatomic mass-spring chain, we formulate a higher order gradient continuum theory of the competent dynamical problem using a homogenization approach. The proposed theory allows us to obtain an analytic description of the bandgap-type response of the homogenized chain, and to estimate the frequency boundary region that is affected by the full transmission of mechanical waves. Numerical applications of the proposed discrete-to-continuum approach are given with reference to tensegrity metamaterials, which exhibit a prestress-tunable bandgap response over a wide range of frequencies.
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