Elastic energy and polarization transport through spatial modulation

Thouless pumping, a specific type of quantum Hall effect, enables topological transport of energy through internal pathways by modulating adiabatically the Hamiltonian of a system. This intriguing phenomenon has been mostly observed in discrete waveguide systems. In this study, we propose a similar phenomenon for a continuous in-plane elastic system and explore its topological properties, including vibrational spectra and localized modes. This pumping is achieved by directly incorporating spatial modulation on material elasticity. We illustrate that a given polarization of elastic waves can be transported and converted along customized paths through phase modulation of elastic tensor. This transport is topologically protected, allowing precise and robust control over elastic wave propagation. To actualize this phenomenon, a family of lattice microstructures, termed as pentamode materials, is specifically engineered to accommodate the distribution of elastic tensor. The topological properties of the modulated lattice are found to agree well with the continuum model. The approach offers an alternative and promising strategy for effectively manipulating elastic waves, paving the way for various applications in elastic waveguiding and wave-based technologies.