Elastic hyperbolic strip lattices

Abstract

We investigate the dynamic properties of elastic lattices defined by tessellations of a hyperbolic strip domain. These strip lattices are generated by a conformal map of tessellations of the hyperbolic disk. Their vibrational modes are organized into three distinct classes: boundary-localized, interior-localized, and global. This mode classification is governed by a localization index quantifying the spatial localization of each mode along the strip's width. We show that, like hyperbolic lattices in the disk, hyperbolic strip lattices exhibit dynamic spectra populated primarily by localized modes. This finding is supported by numerical studies of the dynamics of a strip lattice whose hyperbolically distributed sites are coupled by structural beams. The integrated density of states computation for boundary, interior, and global modes reveals the predominance of localized modes and allows for the identification of spectral bands dominated by particular mode classes. This analysis informs time domain simulations of the lattice response to bandlimited inputs dominated by each mode class. The results illustrate distinctive wave propagation behavior when the excited frequency band is dominated by boundary-localized, interior-localized, or global modes. We experimentally confirm these observations in the frequency and time domains. In the frequency domain, the measured response confirms that the spectral neighborhoods of each excitation are indeed populated by the numerically predicted mode class. We further show that the time-averaged responses are consistent with simulations. Through this work, elastic hyperbolic strips emerge as a new class of lattices with characteristic truss-core architectures and novel nodal arrangements. The considered configuration shows promising capabilities to confine and guide elastic waves along varying spatial regions depending on the excited frequency content.