Quadrupole higher-order topological phases in static mechanical metamaterials

Higher-order topological insulators (HOTIs), an extended concept of traditional first-order topological insulators, are capable of hosting multidimensional topological states and provide an unprecedented platform for robust wave manipulation and energy enhancement. HOTIs have recently been extended to mechanical systems. However, most studies focus primarily on dynamic wave motion and signal transmission. In this paper, we show a realization of quadrupole HOTIs using static Rayleigh deformation modes in three-dimensional layered mechanical metamaterials. The quadrupole topological phase, usually shown in dynamic systems as charge accumulation or energy concentration at corners, is present in the mechanical metamaterial through its patterned static deformation. It is demonstrated analytically that for the topologically protected corner modes, an externally applied corner point load can be fully confined within a localized region for any layer, thereby constructing a deformation isolation device capable of facilitating deformation shielding for a macroscopic bulk region. The obtained results shed light on the exploration of higher-order topological states with a static system and the control of bulk and boundary static deformations in various dimensions.