Static topological mechanics: from space-time duality to localized deformations

Manipulating localized static deformations in materials that are otherwise sensitive to stochastic imperfections and impurities is a long-standing pursuit in solid mechanics. Developments in topological physics have unveiled an unprecedented paradigm for steering robust, localized transport of mass and energy and have been extended to classical dynamic mechanical systems, known as topological mechanics. Recently, Wang, Zhou, and Chen (2023) showed that phonons are not the only elementary excitations in topological mechanics; topology is also inherent in static load-induced mechanical deformations of lattice materials and elastic continua. In these quasistatic and frequency-irrelevant systems, topologically nontrivial modes manifest as ordered static deformations localized at boundaries or interfaces, providing a new method to rationally regulate localized deformations so that they are robust against structural disorders and defects. In this review, we introduce the fundamental concepts of this topology, referred to as static topological mechanics, emphasizing the space-time duality and the imaginary time transformation between static and associated wave dynamic systems. We outline several archetypal topological states reconstructed in static mechanical systems, including topological zero modes, multipole higher-order topologies, and non-Hermitian topologies. In parallel with their dynamic counterparts, robust static topologies can be harnessed to customize localized deformations, such as constructing multidirectional stress guides in lattice materials. This review concludes by envisioning future challenges and opportunities in static topological mechanics.