Abstract
We establish non-Hermitian topological mechanics in one dimensional (1D) and two dimensional (2D) lattices consisting of mass points connected by meta-beams that lead to odd elasticity. Extended from the "non-Hermitian skin effect" in 1D systems, we demonstrate this effect in 2D lattices in which bulk elastic waves exponentially localize in both lattice directions. We clarify a proper definition of Berry phase in non-Hermitian systems, with which we characterize the lattice topology and show the emergence of topological modes on lattice boundaries. The eigenfrequencies of topological modes are complex due to the breaking of $\mathcal{PT}$-symmetry and the excitations could exponentially grow in time in the damped regime. Besides the bulk modes, additional localized modes arise in the bulk band and they are easily affected by perturbations. These distinguishing features may manifest themselves in various active materials and biological systems.
Highlights
Recent years have witnessed advances in applying the notion of “topological protection” to mechanical systems which have led to the blossom of the new field “topological mechanics” [1,2,3,4,5,6]
Besides the bulk modes which are extended in space, we observe additional modes whose eigenvalues are not separated from the bulk band, and they are localized on the lattice boundaries
We extend the notion of “non-Hermitian topological theory” to mechanical systems
Summary
Recent years have witnessed advances in applying the notion of “topological protection” to mechanical systems which have led to the blossom of the new field “topological mechanics” [1,2,3,4,5,6]. We study non-Hermitian topological lattices composed of mass points and metabeams that lead to odd elasticity. Odd elasticity offers unconventional elastostatics and dynamics [92] which are absent in passive solids, such as horizontal deflection and wave propagation in the overdamped regime It elucidates nonreciprocal linear responses of active materials. Major efforts of non-Hermitian mechanics have been limited to one-dimensional (1D) parity-time (PT ) symmetric systems [102,103] whose eigenvalues are real and the eigenmode amplitudes do not grow in time Our research lifts this PT symmetry by allowing the eigenvalues to be complex, meaning that in the dissipationless limit the lattice is unstable against infinitesimal stimulations. The unit cells of all three lattices are subjected to fixed boundary conditions and are composed of two mass particles labeled A and B with mass m which are connected by odd-elastic metabeams.
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