Realization of active metamaterials with odd micropolar elasticity

Abstract

Materials made from active, living, or robotic components can display emergent properties arising from local sensing and computation. Here, we realize a freestanding active metabeam with piezoelectric elements and electronic feed-forward control that gives rise to an odd micropolar elasticity absent in energy-conserving media. The non-reciprocal odd modulus enables bending and shearing cycles that convert electrical energy into mechanical work, and vice versa. The sign of this elastic modulus is linked to a non-Hermitian topological index that determines the localization of vibrational modes to sample boundaries. At finite frequency, we can also tune the phase angle of the active modulus to produce a direction-dependent bending modulus and control non-Hermitian vibrational properties. Our continuum approach, built on symmetries and conservation laws, could be exploited to design others systems such as synthetic biofilaments and membranes with feed-forward control loops.

Fig. 1. Design and mechanics of an odd micropolar metabeam.

a A single unit cell featuring three piezoelectric patches mounted on a beam: one that acts as a sensor, and two that act as actuators. b A segment of the full metabeam. c Each unit cell has an electronic loop. The voltage V_s induced by the central piezoelectric is fed into a transfer function H(ω) = V_a(ω)/V_s(ω) that sends opposing voltages V_a and −V_a to the piezoelectric actuators. d A photograph of the metabeam (horizontal) with the electronic circuits in the foreground. We note that the mechanical forces from the attached wires are negligible. The wires act only as sources of energy and computation, but not of linear or angular momentum. e The motion of the metabeam can be described by two independent fields, φ and h, which parameterize the angular and vertical displacements of the metabeam. Notice that under a reflection about the z-axis, we have φ → −φ and h → h. f When the beam bends, the center piezoelectric is stretched. g The antisymmetric electronic actuation then gives rise to a shearing stress proportional to the modulus P.

Fig. 2. Quasistatic deformation cycles with odd micropolar elasticity.

a The state of the unit cell is tracked in the space of shear and bend. When a quasistatic closed path is traced out in this space, the unit cell performs work per unit volume that is proportional to the area times the modulus P. The z-displacement is provided in arbitrary units. b We numerically compute the work done for a clockwise (top) and a counterclockwise (bottom) path. The solid lines are predictions from the continuum theory, and the black dots result from finite element simulations of the unit cell. In the simulations, maximum amplitudes of bending and shearing are b_max ≈ 10⁻¹ m⁻¹ and s_max ≈ 10⁻², respectively. See Supplementary Note 1 for further details on the simulation.

Fig. 3. Non-Hermitian skin effect via the odd micropolar elasticity.

a The vibrational spectrum for the flexural mode of a metabeam with periodic boundary conditions and odd micropolar modulus P = 3Π. The black line results from the continuum theory given by Eq. (28). The data points are obtained via fully piezoelectrically coupled simulations in COMSOL with the hue indicating the wavenumber kL, where L is the unit cell length. For the full spectrum plotted as a function of k in the continuum theory and in the numerics, see Fig. 7 and S2, respectively. The inset compares the continuum theory and simulations for small wavenumbers $\ll 1/\sqrt{l_1{l}_2}$. b The inverse penetration depth κ for real ω in a medium with open boundary conditions. The points are the results of COMSOL simulations, the black lines are Eq. (29), and the dark lines are the result of the transfer matrix method, see Supplementary Note 3. c The localized states are connected to a topological index ν(ω). The periodic boundary spectrum for P > 0, P = 0, and P < 0 are represented schematically by the solid lines. The arrows indicate the direction of increasing k. For a given frequency ω, the winding number ν(ω) of the periodic boundary spectrum indicates the presence of a localized mode. d The localization of eigenmode at the value of ω denoted by the star in (c) is schematically illustrated.

Fig. 4. Pseudo-Hermitian dynamics.

a The spectrum is shown for the metamaterial with arg(P) = π/2. We note that the reality of the frequencies is maintained, while the modulus P breaks the k → −k symmetry. L is the unit cell length. b Transverse displacement wave fields for the waves traveling in different directions. The left and right traveling modes are excited at equal frequencies, but have differing wavenumbers due to the odd micropolarity. The red arrows indicate the direction of travel of the wave, and H is the transfer function such that P = ΠH.

Fig. 5. Discrete model for odd micropolar beam.

a A discrete model of a Timeoshenko beam consists of a central mass m with moment of inertial J, a Hookean spring of spring constant k_μ and a torsional spring of spring constant κ, Band lattice spacing L. b The unit cell is described by the height of the mass h_i and the angle φ_i of the black connecting rod. c, d The odd micropolar beam has an internal feedback P that senses the angle change of the torsional spring and actuates additional tension in the Hookean spring. The control loop is unidirectional: stretching or compressing the Hookean spring does not affect the torsional spring.

Fig. 6. Experimental demonstration of skin modes and odd micropolar moduli.

a Experimental schematic. Flexural waves are generated in the active metabeam from either the right or left side using piezoelectric actuators (yellow), see “Methods”. A scanning laser Doppler vibrometer (SLDV) measures the transverse velocity of the surface of the active metabeam. b, c Unidirectional amplification of waves. A metamaterial consisting of 9 unit cells is actuated from either the right (blue) or left (red) with a 2 kHz tone burst signal (gray). The output velocity is normalized by the maximum velocity observed when the experiment is performed with no active feedback. d Observation of the non-Hermitian skin effect. Experiments are performed between 1.5 kHz and 4 kHz for right to left (blue) and left to right (red) traveling waves. A 2D FFT shows the intensity of the observed spectrum. The intensity is normalized by its maximum value. e The inverse decay length. In d, e the solid theoretical curves are based on the transfer matrix method. In d the gray dashed curves are theoretical predictions with no activity. f A plot of arg(P) as a function of frequency. At ω = ω₀ (= 3 kHz), arg(P) = −π/2, indicating that the system is pseudo-Hermitian and accordingly we observe κ = 0 at ω = ω₀.

Fig. 7. Non-Hermitian band topology via odd micropolar elasticity.

a The spectrum for P = 0 features a pair of bending dominated bands (black) and shear dominated bands (gray) separated by a band gap ω₁. (b-d) The spectrum is shown in the complex ω plane for P = 0, P > 0, and P < 0. The thick black lines represent the bending dominated band, while the thick gray lines represent the shear dominated bands, both with k ∈ [−R, R] for a finite R. The thin black lines represent the analytical continuation of the spectrum for k = Re^iφ for φ ∈ [0, π]. The arrows indicate the direction of increasing k. The numbers indicate the value of ν(ω) for ω in the corresponding colored regions of the complex plane. This number corresponds to the number of times that the spectrum winds around a given region. e For a semi-infinite system with a free boundary, the winding number of $\tilde{\nu }$ = 1 ($\tilde{\nu }$  = 3) for our continuum theory indicates a mode localized to the right (left) boundary. The wave forms schematically depict the localization with A(x) representing amplitude. For a calculation of the precise eigenvectors, see Supplementary Note 1.