Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial

Concepts from quantum topological states of matter have been extensively utilized in the past decade in creating mechanical metamaterials with topologically protected features, such as one-way edge states and topologically polarized elasticity. Maxwell lattices represent a class of topological mechanical metamaterials that exhibit distinct robust mechanical properties at edges/interfaces when they are topologically polarized. Realizing topological phase transitions in these m...

Based on the Maxwell-Beatty reciprocity theorem, static non-reciprocity has been realized by using nonlinearity, but this non-reciprocity has strict restrictions on input amplitude and structure size (number of units). Here, we propose a robotic metamaterial with two components of displacement and rotation, which uses active control to add external forces on the units to break reciprocity at the level of the interactions between the units. We show analytically and simulativel...

Recent progress in topological mechanics have revealed a family of Maxwell lattices that exhibit topologically protected floppy edge modes. These modes lead to a strongly asymmetric elastic wave response. In this paper, we show how topological Maxwell lattices can be used to realize non-reciprocal transmission of elastic waves. Our design leverages the asymmetry associated with the availability of topological floppy edge modes and the geometric nonlinearity built in the mecha...

Topological mechanical metamaterials have demonstrated exotic and robust mechanical properties which led to promising engineering applications. One of such properties is the focusing of stress at the interface connecting domains of topological Maxwell lattices of opposite topological polarizations, which protects the bulk of the material against fracturing. Here we generalize this theory to non-ideal Maxwell lattices, incorporate real material features that leads to interacti...

Topological metamaterials have invaded the mechanical world, demonstrating acoustic cloaking and waveguiding at finite frequencies and variable, tunable elastic response at zero frequency. Zero frequency topological states have previously relied on the Maxwell condition, namely that the system has equal numbers of degrees of freedom and constraints. Here, we show that otherwise rigid periodic mechanical structures are described by a map with a nontrivial topological degree (a...

Topological edge modes are excitations that are localized at the materials' edges and yet are characterized by a topological invariant defined in the bulk. Such bulk-edge correspondence has enabled the creation of robust electronic, electromagnetic and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter and geophysical flows. Recently, the advent of non-Hermitian topological systems---wherein energy is not conserved-...

The motive of this work is to understand the complex spatial characteristics of finite-amplitude elastic wave propagation in periodic structures and leverage the unique opportunities offered by nonlinearity to activate complementary functionalities and design adaptive spatial wave manipulators. The underlying assumption is that the magnitude of wave propagation is small with respect to the length scale of the structure under consideration, albeit large enough to elicit the ef...

In this paper we study Maxwell lattices with non-rectilinear constraints, where the elastic energy is determined by the collective motion of three or more particles, in contrast to a rectilinear spring whose elastic energy only relies on the displacement of two particles attached to the two ends of the spring. Utilizing polygon-shaped constraints, we found that the Maxwell counting argument and the topological construction, based on the compatibility matrix and the equilibriu...

We introduce a method to design topological mechanical metamaterials that are not constrained by Newtonian dynamics. The unit cells in a mechanical lattice are subjected to active feedback forces that are processed through autonomous controllers, pre-programmed to generate the desired local response in real-time. As an example, we focus on the quantum Haldane model, which is a two-band system with directional complex coupling terms, violating Newton's third law. We demonstrat...

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 t...