Dispersion relation of viscoelastic phononic crystals
Periodic phononic crystals (PnCs) in the form of metallic layered show unique wave characteristics typically at high frequency ranges (e.g., $MHz$ or higher). But, unwanted vibrations and noises relevant to human surroundings are characterized by sonic frequency ranges (e.g., a few kHz or lower) which require low impedance polymeric materials as PnC constituents. Generally, polymeric composites are analyzed as linear elastic or hyperelastic materials without considering the viscoelastic damping behavior. We have studied the wave propagation at arbitrary angles in the sagittal plane of viscoelastic multilayered composites.
The fundamental of wave motion in phononic crystals is derived from Bloch theorem. The theorem explains that a plane wave eik3.a3 propagating through the crystal would produce periodic boundary conditions (e.g, displacement, stress) between n-th and (n+1)-th unit cells along the periodic direction x3. From continuum mechanics, we also obtain continuous displacement and stress boundary conditions at the interfaces between similar unit cells. A combination of the Bloch periodic condition and mechanics boundary conditions provide an eigenvalue formulation. We use Cayley-Hamilton theorem to obtain a fourth-order polynomial solution for the eigenvalue equation. The coefficients are simplified using composite periodicity to obtain the complex valued dispersion relation of viscoelastic multilayered PnCs.
Furthermore, when a viscoelastic medium is attached to an elastic layer, the wave attenuation in a viscoelastic layer occurs only in the direction perpendicular the layers, i.e., zero k1-component of kI. The propagation and the attenuation characteristics of sagittal plane waves can be formally presented in a 3-D plot consisting of ω over the k1R– k3R plane and ω over the k1R– k3I plane, respectively.
Spatial aliasing solution of numerical dispersion relations
Dispersion analysis of laminated phononic crystals (LPnCs) requires numerical methods for intricate applications. FE based methods are predominantly used to incorporate the effects of material and geometric nonlinearity in dispersion relations. However, dispersion analysis of LPnCs by FE methods suffers from fictitious dispersion modes. Because, one dimensional periodicity of LPnCs are modeled using 2-D elements by applying a redundant Bloch periodicity in non-periodic direction. Our study solves the fictitious mode issue by proving that spatial aliasing arising form artificial periodicity is the source of the problem.
In order to solve the complex eigenvalue problem of Bloch periodicity, FE methods decompose the real (RE) and imaginary (IM) parts using two identical structures. We apply the coupled periodic relations for dependent (d) and independent (i) nodes of two structures using ABAQUS subroutine MPC. We use a rectangular unit cell a1✕ a3 to model the LPnCs where periodicity only occurs in the a3 direction. New fictitious modes appear at πa1 spacing in reciprocal space based on the nonperiodic length a1 .
For an inclined wavevector k* in real space, FE analysis shows actual and fictitious dispersion modes shown by blue and red lines. These fictitious modes don’t appear in analytical approach. We use the sagittal plane wave equations to calculate dispersion relations of aliasing wavevectors. The isolated dispersion relations of real and aliasing wavevectors are presented in a 3-D format for inclined wavevector. Superposition of all these modes recreates the spectrally distorted FE result which proves that the fictitious modes are result of spatial aliasing.
Amplitude-dependent wave in periodic viscoelastic composites
Due to soft, viscoelastic, and damping behavior of polymers, high impact forces cause nonlinear wave motion in the polymeric PnCs. A suitable experimental method to study the effect of high amplitude and rate dependent spectral response in flexible PnCs is unavailable. We have conducted experimental and computational mechanics study to developed a hybrid Split Hopkinson Pressure Bar (SHPB) to determine impact dependent wave transmission of polymeric PnCs.
We first obtain the small amplitude excitation behavior of a multilayered polymer-metal PnC as a reference using an electrodynamic shaker. The input vibration and output response are respectively measured by force transducer and accelerometer. The transmission coefficient Ct(f) is determined for sonic frequency (f) range. The impact induced phononic behavior is determined by the hybrid SHPB where a metal input bar and polymeric output bar are used for detectable signal-to-noise ratios. We convert the strain signals of output and input bars respectively to spectral accelerations and force values so that we can calculate a comparable Ct(f).
A series of computational simulations are conducted by FEA to explicitly determine the change of wave transmission as a function of impact. The rate-dependent large deformation of silicone rubber of PnCs are modeled by a hyper-viscoelastic material. The hyperelastic and viscoelastic behavior are fitted with material experiment data using Yeoh and Maxwell models, respectively. The The FE analysis For the time-domain simulations of our crystal, the SHPB components (i.e., striker, input bar, output bar) are also modeled by appropriate elastic and viscoelastic materials.
Small amplitude excitation of viscoelastic PnC by Vibration shaker indicates two wave attention frequencies 2 and 8 kHz. In case of the impact tests by SHBP we use different lengths and speeds of striker bar to gradually increase the impulse. Here impulse is used as control variable which represents the force over time of an impact. Wave propagation results for the SHPB has indicated appearance of two new attenuation frequencies at higher impulse loadings. To clearly detect the change of transmission coefficient we need high speed impact force of small strikers. So, we conduct FE simulation using a parametric script to gradually increase the impulse. The simulation results are compiled in a contour plot which confirms formation of low transmission frequency zones at 5.5 kHz and 12 kHz for impulse greater than 15 N.s. The experimentally guided numerical study have demonstrated evolution of nonlinear wave motion of viscoelastic phononic crystals.
Data-driven graded kirigami for programmed strain distribution
The emerging technologies of wearable electronics, biomedical sensors, and soft robotics constitute the broad domain of stretchable electronics. These devices are developed by integrating rigid microchips in soft elastomeric substrates. However, stretchability of these devices are low due to premature failure at the soft-rigid interface between elastomers and microchips. The compliance mismatch of such interface cause high stress concentration which accelerates mechanical failure and electronic malfunction. Commonly adopted approaches of placing rigid electronics on stiff polymeric island or multilayered substrate has found limited success because the soft-rigid interface still remains in the device.
We have developed a programmable stiffness-graded kirigami composites as a gradual strain transition platform for soft-rigid interfaces. Stiff polymeric films with kirigami beam architectures is embedded in soft elastomers to tune the stiffness from rigid electronics to soft elastomer. Consequently, the graded kirigami composite achieves a gradual strain gradient between rigid zone strain εr and soft elastomer strain εs. Kirigami geometry is designed by coupling finite element analysis (FEA) with experimental digital image correlation (DIC) to achieve diverse strain profiles for flexible interfaces. Architected kirigami patterns provide versatile design space for parametric study using beam geometry, spacing, and arrangement. We have utilized the mechanics of stiffness-strain relation to develop an inverse design method. For a target strain profile, kirigami patterns are placed in sequence of segments so that the various linear and nonlinear strain distribution can be programmed. Such predictive mechanical performance make the graded kirigami composites suitable for robust soft-rigid interfaces and defect-tolerant stretchable devices and robots.
