Mechanical metamaterials are engineered materials whose structures give them novel mechanical properties, including negative Poisson's ratios, negative compressibilities and phononic bandgaps. Of particular interest are systems near the point of mechanical instability, which recently have been shown to distribute force and motion in robust ways determined by a nontrivial topological state. Here we discuss the classification of and propose a design principle for mechanical metamaterials that can be easily and reversibly transformed between states with dramatically different mechanical and acoustic properties via a soft strain. Remarkably, despite the low energetic cost of this transition, quantities such as the edge stiffness and speed of sound can change by orders of magnitude. We show that the existence and form of a soft deformation directly determines floppy edge modes and phonon dispersion. Finally, we generalize the soft strain to generate domain structures that allow further tuning of the material.

We show that two-dimensional mechanical lattices can generically display topologically protected bulk zero-energy phonon modes at isolated points in the Brillouin zone, analogs of massless fermion modes of Weyl semimetals. We focus on deformed square lattices as the simplest Maxwell lattices, characterized by equal numbers of constraints and degrees of freedom, with this property. The Weyl points appear at the origin of the Brillouin zone along directions with vanishing sound speed and move away to the zone edge (or return to the origin) where they annihilate. Our results suggest a design strategy for topological metamaterials with bulk low-frequency acoustic modes and elastic instabilities at a particular, tunable finite wave vector. DOI: 10.1103/PhysRevLett.116.135503 The topological properties of the energy operator and associated functions in wave vector (momentum) space can determine important properties of physical systems [1][2][3]. In quantum condensed matter systems, topological invariants guarantee the existence and robustness of electronic states at free surfaces and domain walls in polyacetylene [4,5], quantum Hall systems [6,7], and topological insulators [8-13] whose bulk electronic spectra are fully gapped (i.e., conduction and valence bands separated by a gap at all wave numbers). More recently, topological phononic and photonic states have been identified in suitably engineered classical materials as well [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], provided that the band structure of the corresponding wavelike excitations has a nontrivial topology.A special class of topological mechanical states occurs in Maxwell lattices, periodic structures in which the number of constraints equals the number of degrees of freedom in each unit cell [34]. In these mechanical frames, zero-energy modes and states of self-stress (SSSs) are the analogs of particles and holes in electronic topological materials [16]. A zero-energy (frequency) mode is the linearization of a mechanism, a motion of the system in which no elastic components are stretched [19,20]. States of self-stress on the other hand guide the focusing of an applied stress and can be exploited to selectively pattern buckling or failure [18]. Such mechanical states can be topologically protected in Maxwell lattices, such as the distorted kagome lattices of Ref. [16], in which no zero modes exist in the bulk phonon spectra (except those required by translational invariance at wave vector k ¼ 0). These lattices are the analog of a fully gapped electronic material. They are characterized by a topological polarization equal to a lattice vector R T (which can be zero) that, along with a local polarization R L , determines the number of zero modes localized at free surfaces, interior domain walls separating different polarizations, and dislocations [17]. Because R T only changes upon closing the bulk phonon gap, these modes are robust against disorder or imperfections.In this Letter we demonstrate how to create topolo...

The exceptional mechanical properties of graphene have made it attractive for nanomechanical devices and functional composite materials [1]. Two key aspects of graphene's mechanical behavior are its elastic and adhesive properties. These are generally determined in separate experiments, and it is moreover typically difficult to extract parameters for adhesion. In addition, the mechanical interplay between graphene and other elastic materials has not been well studied. Here, we demonstrate a technique for studying both the elastic and adhesive properties of few-layer graphene (FLG) by placing it on deformable, micro-corrugated substrates. By measuring deformations of the composite graphene-substrate structures, and developing a related linear elasticity theory, we are able to extract information about graphene's bending rigidity, adhesion, critical stress for interlayer sliding, and sample-dependent tension. The results are relevant to graphene-based mechanical and electronic devices, and to the use of graphene in composite [1], flexible [2], and strainengineered [3] materials.The elastic properties of graphene have previously been measured using nano-indentation [4] and pressurization [5] techniques, and Young's modulus E was found to be exceptionally high, ∼ 1 TPa. Graphene's van der Waals adhesion to surfaces has been examined theoretically [6], and local adhesion to nanoparticles has been studied [7]. Substrate interactions, due to surface adhesion, highly modify graphene's doping [8,9] and carrier mobility. In addition, adhesion to sidewalls in suspended graphene introduces strain and modifies mechanical behavior [10,11]. Here, we explore both elasticity and adhesion, which are evident in the interaction between microscale-corrugated elastic substrates and graphene samples of thickness ranging from 1 to 17 atomic layers. By using an atomic force microscope (AFM) to determine surface adhesion and deformations, we find that the FLG can fully adhere to the patterned substrate, and that thicker samples flatten the corrugated substrate more than thinner samples do. By developing a linear elasticity theory to model the flattening and adhesion as a function of layer thickness, we are able to extract estimates of, or bounds on, various fundamental and sample-dependent properties of the system. Sample substrates were prepared by casting a 3 mm thick layer of polydimethylsiloxane (PDMS)-which cures into a flexible, rubbery material-onto the exposed surface of a writable compact disc. This resulted in ap- proximately sinusoidal corrugations on the PDMS, having a wavelength of 1.5 µm and a depth of 200 nm (see Fig. 1a). Graphene was then deposited onto the PDMS via mechanical exfoliation [12]. Candidate samples were first located using optical microscopy, then imaged on an Asylum Research MFP-3D AFM. Figure 1a shows a topographic image of FLG on the PDMS; it is evident from the image that the graphene conforms to the corrugations, as illustrated in Fig. 1b.In order to fit the experimental data to a theoretical model, it...