Anomalous Collisions of Elastic Vector Solitons in Mechanical Metamaterials

Phase transitions in 2D multistable mechanical metamaterials via collisions of soliton-like pulses

In recent years, mechanical metamaterials have been developed that support the propagation of an intriguing variety of nonlinear waves, including transition waves and vector solitons (solitons with coupling between multiple degrees of freedom). Here we report observations of phase transitions in 2D multistable mechanical metamaterials that are initiated by collisions of soliton-like pulses in the metamaterial. Analogous to first-order phase transitions in crystalline solids, we observe that the multistable metamaterials support phase transitions if the new phase meets or exceeds a critical nucleus size. If this criterion is met, the new phase subsequently propagates in the form of transition waves, converting the rest of the metamaterial to the new phase. More interestingly, we numerically show, using an experimentally validated model, that the critical nucleus can be formed via collisions of soliton-like pulses. Moreover, the rich direction-dependent behavior of the nonlinear pulses enables control of the location of nucleation and the spatio-temporal shape of the growing phase.

Emerging topics in nanophononics and elastic, acoustic, and mechanical metamaterials: an overview

This broad review summarizes recent advances and "hot" research topics in nanophononics and elastic, acoustic, and mechanical metamaterials based on results presented by the authors at the EUROMECH 610 Colloquium held on April 25-27, 2022 in Benicássim, Spain. The key goal of the colloquium was to highlight important developments in these areas, particularly new results that emerged during the last two years. This work thus presents a "snapshot" of the state-of-the-art of different nanophononics- and metamaterial-related topics rather than a historical view on these subjects, in contrast to a conventional review article. The introduction of basic definitions for each topic is followed by an outline of design strategies for the media under consideration, recently developed analysis and implementation techniques, and discussions of current challenges and promising applications. This review, while not comprehensive, will be helpful especially for early-career researchers, among others, as it offers a broad view of the current state-of-the-art and highlights some unique and flourishing research in the mentioned fields, providing insight into multiple exciting research directions.

Discrete breathers in a mechanical metamaterial

We consider a previously experimentally realized discrete model that describes a mechanical metamaterial consisting of a chain of pairs of rigid units connected by flexible hinges. Upon analyzing the linear band structure of the model, we identify parameter regimes in which this system may possess discrete breather solutions with frequencies inside the gap between optical and acoustic dispersion bands. We compute numerically exact solutions of this type for several different parameter regimes and investigate their properties and stability. Our findings demonstrate that upon appropriate parameter tuning within experimentally tractable ranges, the system exhibits a plethora of discrete breathers, with multiple branches of solutions that feature period-doubling and symmetry-breaking bifurcations, in addition to other mechanisms of stability change such as saddle-center and Hamiltonian Hopf bifurcations. The relevant stability analysis is corroborated by direct numerical computations examining the dynamical properties of the system and paving the way for potential further experimental exploration of this rich nonlinear dynamical lattice setting.

Conformal elasticity of mechanism-based metamaterials

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied-conformal-maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

Characterization, stability, and application of domain walls in flexible mechanical metamaterials

Domain walls, commonly occurring at the interface of different phases in solid-state materials, have recently been harnessed at the structural scale to enable additional modes of functionality. Here, we combine experimental, numerical, and theoretical tools to investigate the domain walls emerging upon uniaxial compression in a mechanical metamaterial based on the rotating-squares mechanism. We first show that these interfaces can be generated and controlled by carefully arranging a few phase-inducing defects. We establish an analytical model to capture the evolution of the domain walls as a function of the applied deformation. We then employ this model as a guideline to realize interfaces of complex shape. Finally, we show that the engineered domain walls modify the global response of the metamaterial and can be effectively exploited to tune its stiffness as well as to guide the propagation of elastic waves.

Rarefactive lattice solitary waves with high-energy sonic limit

We compute rarefactive solitary wave solutions in a nonlinear lattice with nearest-neighbor interaction forces that are sublinear near the undeformed state. This setting includes bistable bonds governed by a double-well potential. In contrast to the prototypical Korteweg-de Vries-type delocalization, the obtained solutions feature a nontrivial sonic limit (Chapman-Jouguet regime) with nonzero energy and algebraic decay at infinity. In the bistable case the waves are strongly localized and have high energy over the entire velocity range. Direct numerical simulations suggest stability of the computed solitary waves. We consider several quasicontinuum models that mimic some features of the obtained solutions, including the nontrivial nature of the sonic limit, but fail to accurately approximate their core structure for all velocities in the bistable regime.

Magnetic Solitons in a Spin-1 Bose-Einstein Condensate

Vector solitons are a type of solitary or nonspreading wave packet occurring in a nonlinear medium composed of multiple components. As such, a variety of synthetic systems can be constructed to explore their properties, from nonlinear optics to ultracold atoms, and even in metamaterials. Bose-Einstein condensates have a rich panoply of internal hyperfine levels, or spin components, which make them a unique platform for exploring these solitary waves. However, existing experimental work has focused largely on binary systems confined to the Manakov limit of the nonlinear equations governing the soliton behavior, where quantum magnetism plays no role. Here we observe, using a "magnetic shadowing" technique, a new type of soliton in a spinor Bose-Einstein condensate, one that exists only when the underlying interactions are antiferromagnetic and which is deeply embedded within a full spin-1 quantum system. Our approach opens up a vista for future studies of "solitonic matter" whereby multiple solitons interact with one another at deterministic locations.

Solitary waves in a nonintegrable chain with double-well potentials

We study solitary waves in a one-dimensional lattice of identical masses that are connected in series by nonlinear springs. The potential of each spring is nonconvex, where two disjoint convex regions, phase I and phase II, are separated by a concave, spinodal region. Consequently, the force-strain relation of the spring is nonmonotonous, which gives rise to a bistable behavior. Based on analytical treatment, with some approximations, combined with extensive numerical simulations, we are able to reveal important insights. For example, we find that the solitary-wave solution is indifferent to the energy barrier that separates the two energy wells associated with phase I and phase II, and that the shape of the wave can be described by means of merely two scalar properties of the potential of the springs, namely, the ratio of stiffness in phase II and phase I, and the ratio between the Maxwell's force and corresponding transition strain. The latter ratio provides a useful measure for the significance of the spinodal region. Linear stability of the solitary-wave solution is studied analytically using the Vakhitov-Kolokolov criterion applied to the approximate solutions obtained in the first part. These results are validated by numerical simulations. We find that the solitary-wave solution is stable provided that its velocity is higher than some critical value. It is shown that, practically, the solitary waves are stable for almost the entire range of possible wave velocities. This is also manifested in the interaction between two solitary waves or between a solitary wave and a wall (rigid boundary). Such interaction results in a minor change of height and shape of the solitary wave along with the formation of a trail of small undulations that follow the wave, as expected in a nonintegrable system. Even after a significant number of interactions the changes in the wave height and shape are minor, suggesting that the bistable chain may be a useful platform for delivering information over long distances, even concurrently with additional information (other solitary waves) passing through the chain.