Sharkskin-Inspired Magnetoactive Reconfigurable Acoustic Metamaterials

Most of the existing acoustic metamaterials rely on architected structures with fixed configurations, and thus, their properties cannot be modulated once the structures are fabricated. Emerging active acoustic metamaterials highlight a promising opportunity to on-demand switch property states; however, they typically require tethered loads, such as mechanical compression or pneumatic actuation. Using untethered physical stimuli to actively switch property states of acoustic metamaterials remains largely unexplored. Here, inspired by the sharkskin denticles, we present a class of active acoustic metamaterials whose configurations can be on-demand switched via untethered magnetic fields, thus enabling active switching of acoustic transmission, wave guiding, logic operation, and reciprocity. The key mechanism relies on magnetically deformable Mie resonator pillar (MRP) arrays that can be tuned between vertical and bent states corresponding to the acoustic forbidding and conducting, respectively. The MRPs are made of a magnetoactive elastomer and feature wavy air channels to enable an artificial Mie resonance within a designed frequency regime. The Mie resonance induces an acoustic bandgap, which is closed when pillars are selectively bent by a sufficiently large magnetic field. These magnetoactive MRPs are further harnessed to design stimuli-controlled reconfigurable acoustic switches, logic gates, and diodes. Capable of creating the first generation of untethered-stimuli-induced active acoustic metadevices, the present paradigm may find broad engineering applications, ranging from noise control and audio modulation to sonic camouflage.


Modeling of the Mie resonator pillar (MRP)
Analytical modeling. Here, we estimate the resonance of the MRP based on a homogenization procedure presented in Reference 7 . For the MRP in Fig. S4a, the medium can be simplified into six channels (Fig. S4b), which are shorter than a wavelength 8 . Such a model can then be homogenized and equivalently represented as three two-dimensional regions with different effective properties as seen in Fig. S4c. The pressure fields in the different regions, denoted here as I, II and III, are expressed mathematically by a combination of Bessel function and Hankel function as 7,9 : where /2 is the total radius of the individual resonator, is the inner radius of the resonator housing the air core, and is the outer radius of a uniform equivalent layer with being the filling ratio of the air channels. In our case, the filling ratio is approximately equal to 0.36. Note here that all the radii are measured from the center of the resonator, i.e. 0. To satisfy the continuity at the interfaces and , the pressure field and are matched at the boundaries 10 , which gives two sets of conditions: where , denote the density of regions I, III and II, respectively. Consequently, one can get the following equations, for the th term of the summation: which can be cast into a compact matrix form: where the coefficients through are now normalized by the coefficient . We assume that the effective uniform medium density has the same bulk modulus as the background medium, but different density defined as / 10 . Of interest here is the determination of scattering coefficient . Using the scheme presented in reference 9 for calculating via a physically revealing model, the scattering coefficient is found by: where is the imaginary unit and and are defined as the following determinants: where is the Bessel function of the second kind. Following the assumptions ≪ 1 and ≪ 1, the first monopole frequency is found by setting 0 7 . If the diameter of the MRP is 1.5 cm, the Mie resonance is estimated as ~8.9 kHz, which is found from the frequency response of | | where it approaches 1, signaling the occurrence of a scattering resonance ( Fig.   S4d) 7 . The magnitude | | and angle of are depicted in Figs. S4d, e. This result is close to the experimentally observed and numerically simulated 9.1 kHz (Fig. 2). Hz. This theoretically predicted effect of the geometrical inconsistency also agrees well with the numerical simulations shown in Fig. S2c. 6

Theoretical analysis of magnetically-induced buckling
Following the previously reported work on magnetically-induced buckling of tilted beam 11 , we here develop an analytical model for the magnetically-induced bending of the pillar. The magnetically-induced bending can be modeled as a tilted pillar under a magnetic field with an angle α to the pillar (Fig. S7). We construct two coordinate systems: global Cartesian coordinate (x,y) and local curvilinear coordinate (s,θ) shown in Fig. S7. The free energy of the magnetic-field deformed pillar can be written as [11][12][13][14][15][16] ⋅ (S16) where s is the curvilinear coordinate along the beam, θ is the angle between the tangent line and the horizontal axis, E is Young's modulus of the magnetoactive elastomer, H is the pillar length, I is the second moment of the cross-section area, A is the cross-section area, B is the applied magnetic field vector, and M is the magnetization vector.
We first assume that the pillar aspect ratio (H/D) is relatively large, and the pillar can be considered as a slender structure. The variation of Eq. S16 leads to a governing equation of the magnetically-induced bending written as 11-16 is the magnetic permittivity of the vacuum, and α is the initial tilted angle of the beam. ∆χ is the effective magnetic susceptibility difference between the axial and orthogonal direction and can be estimated as , where is the magnetic susceptibility of the elastomer.
At the critical point of the buckling, the characteristic length should scale with the beam length H. Therefore, the critical magnetic field of the buckling should follow a scaling law as 7 ∝ (S18) Once the applied magnetic field is larger than the critical magnetic field for the pillar buckling, the pillar will be bent and pinched on the substrate.
According to Eq. S18, the critical magnetic field is affected by the Young's modulus, magnetic susceptibility, cross-section geometry, and the length of the pillar. Besides, we notice that the critical magnetic field should also be affected by the tilting angle of the magnetic field.
The effect of the tilting angle should be reflected by the pre-factor of the scaling law, . In addition, Eq. S18 is derived based on a slender assumption. If the slender condition is relaxed to extended to relatively small pillar aspect ratio (e.g., H/D=2.83 in this work), the contribution of the geometrical factor H/D should be also reflected by the pre-factor . Therefore, the critical magnetic field of the buckling can be written as , where the pre-factor is a function of tilting angle and pillar aspect ratio H/D.
To validate the theoretical model in Eq. S19, we employ Mie resonator pillars (MRPs) with varied concentration of the ferromagnetic iron particle within the magnetoactive elastomer but maintain the geometry of the pillar and tilting angle 4 ⁄ . The variation of the iron concentration changes the Young's modulus and magnetic susceptibility of the magnetoactive elastomer. We measure the material and geometrical parameters of the MRPs and show them in Table S2. Experiments verify that the critical magnetic field indeed follows a linear relationship with (Fig. 2n). 8

Design principle of the reconfigurable acoustic logic gates
The design of the switchable logic gate is based on two mechanisms: (1) The transmission ratios of the MRP arrays decreases with the row number of the array.
For example, the acoustic transmission through one row of MRP array is around 0.35 at 8760 Hz; thus, the acoustic transmission through two rows of MRP array is around 0.35 2 =0.12 at 8760 Hz (Figs. 4c, h, m, and S14). If two inputs are applied, the output transmission of one-row MRP array is 0.35+0.35=0.7 (above 0.5), but the output transmission of two-row MRP array is only 0.12+0.12=0.24 (below 0.5). We here denote that the normalized pressure equal to or larger than 0.5 as "1" and otherwise as "0". Then, these two types of MRP array show different output states: the former outputs "1" but the latter outputs "0".
(2) The magnetic field can be used to bend a row of MRP array to leave only one-row MRP array. On-off switching the magnetic field can one-demand switch between one-row array and two-row array.
According to the design principle, to enable switching among NOT, AND, and OR operators, the acoustic transmission of one-row MRP array can be around 0.25-0.5. The corresponding frequency range is slightly below or above the Mie resonance frequency range, for example, 8700-8830 Hz and 8930-9070 Hz (according to data shown in Fig. S14a). To verify this working frequency range for the switching of three logic operators, we carry out numerical simulations that show the frequencies through 8700-8830 Hz and 8930-9070 Hz work well for the switching of three logic operators (Figs. S14 and S15). This point is also confirmed by experiments at 8700 and 9050 Hz which are located at two frequency branches, respectively (Fig.   S16). 9