A Pragmatic Design of Elastic Metamaterial with Extreme Anisotropic Stiffness

Traditional elastic metamaterials encounter difficulties when trying to independently adjust the operating frequency in two perpendicular directions. In this research, we present a practical active elastic metamaterial with the capability to achieve extreme stiffness anisotropy. Our approach involves incorporating piezoelectric patches, coupled with adjustable conductance, into the microstructure unit cell to control the effective elastic stiffness in both primary directions. We conduct simulations to manipulate wave propagation within such metamaterials and conduct experimental investigations to validate the design. The obtained results closely align with mathematical analysis and numerical predictions, demonstrating the effectiveness of our proposed metamaterial. This novel approach offers a versatile and innovative design methodology for elastic metamaterials.


Instruction
Super-resolution imaging (SR) refers to a collection of techniques aimed at enhancing the resolution of imaging systems [1][2][3] [4].Successful demonstrations of subwavelength imaging using metamaterials have been achieved, both in the realm of electromagnetic waves and acoustic waves.Furthermore, researchers have extended their efforts to explore total transmission capabilities in these contexts [2][5] [6].However, only a few groups are conducting the research on hyper-lens for elastic waves [7][8] [9].Nevertheless, recent studies conducted by these groups have demonstrated the feasibility of designing elastic metamaterials (EMMs) with anisotropic mass density or anisotropic stiffness through the appropriate utilization of local resonant microstructures [9] [10].Oh et al. introduced an elastic hyperbolic metamaterial lens, demonstrating total transmission subwavelength imaging through experimental observation of the wave field inside the metamaterial lens [7].To extend the operational frequency range of the hyper-lens, Shen et al. devised a broadband single-phase hyperlens capable of supporting multipolar resonances at various deep-subwavelength scales.They further demonstrated the super-resolution imaging capabilities of the lens specifically for longitudinal waves [11].Piezoelectric materials have gained significant attention in active vibration control applications in recent years [12][13][14] [15].Due to their promising active capabilities, piezoelectric shunts have gained extensive adoption in the field of mitigating flexible vibrations in structures [16] [17].In recent times, the utilization of negative capacitance shunted piezoelectric patches has emerged as a notable approach in the development of active and/or adaptive electromagnetic metamaterials (EMM).This strategy effectively addresses the limitation of working frequency ranges commonly associated with passive EMM.Furthermore, negative capacitance shunted piezoelectric patches offer significant advantages over their passive counterparts [18][19] [20][21].

Microstructure design of the active EMMs
In this article, the first physical implementation of an active EMM plate capable of independently manipulating elastic moduli along two principal in-plane directions ( and ) is presented, as depicted in Figure 1(b).To achieve tunable working frequencies in both orthogonal directions, we propose four identical piezoelectric patches with independent external circuits in the EMM unit cell, as shown in Figure 1(a).The square unit cell's in-plane lattice constant is denoted by a, and the plate's thickness is denoted by .The length and width of the piezoelectric patches are denoted by  and , respectively, while  represents the length of the central square piece's sides.The capacitance of the piezoelectric patch is denoted by  in the unit cell.The yellow arrow highlights the in-plane polarization direction of the piezoelectric patches, as shown in Figure 1(a).The material properties and geometric parameters are listed in Table 1.Each piezoelectric patch's external circuit is illustrated in Figure 1(c), and it consists of a variable conductor, a resistor, and the piezoelectric patch itself connected in series.

Tunable elastic modulus of the EMMs
First, let us consider a thin, shunted piezoelectric patches parallelly connected to negative capacitances   = Cn * .According to Hagood and Flotow, the effective young's modulus of the shunted patch become independent of frequency ω and can be expressed as: where   is the Young's modulus of the piezoelectric material when the shunting network is in a short circuit configuration,  11 denotes the electromechanical coupling coefficient constant, and C is the electrical capacitance of the piezoelectric material under constant stress or no stress.To validate the wave manipulation ability of the EMM, the longitudinal wave fields within a 2-D array of the proposed EMM unit are plotted in Fig. 2(a).When short circuit boundary conditions are applied to all the piezoelectric patches, clear longitudinal wave field only appears along the -axis and -axis direction, as shown in Fig. 2(a).While longitudinal wave propagating along the  axis is forbidden at 40000, when negative resonance ( = −1.35) is applied to selected horizontal piezoelectric patches, as shown in Fig. 2(b).When the elastic moduli of the 15 unit cells reach extreme high value, they will vibrate left and right with respect to the plate as one object, and then the shear wave mode will be generated to up and down, as in Fig. 2(d).

Experimental Validation
In this section, experimental validation is conducted to evaluate our numerical simulations in section 3, as it is shown in Figure 3.A two-dimensional elastic metamaterial plate was fabricated through cutting in a 20cm×20cm×5mm aluminum plate.The piezoelectric patch center mass piece is assembled and insulated using super glue.The external circuit for each piezoelectric patch is connected according to Figure 1c.All the parameters of the EMMS plate and the external circuit are same to table 1.All the piezoelectric patches are carefully wired to the external circuits, the insulation and connection of the all the electric components are checked individually before the test.A circular actuator powered by a signal generator is used to generate longitudinal waves propagating in all directions in the metamaterial plate.To be consistent with the numerical simulation, the actuator is placed at the center of the metamaterial plate.Continuous sinusoidal waves are used in this test.Three piezoelectric sensors are used to collect At first, we set the conductance of conductor of the horizontal components (x direction) and vertical components (y direction) and to Cn=-1.357 and Cn=-1.38 respectively, the stimulation frequency was sweeping from 0Hz to 50000Hz.The normalized frequency response (H1/H2) was plotted in Figure 4.According to the simulation in previous sections, the extreme stiffness of the plate at x and y direction should be archived at 10000Hz and 40000Hz respectively, which is consistent with the experimental results.In addition, we fixed the stimulation frequency at 30000Hz and sweep the conductance Cn of the horizontal components and from -1.35to -1.39.Both Figure 4(a) and Figure 4(b) shows that the stiffness of the proposed metamaterial plate depends on the wave frequency and conductance of the shunted circuits.The propagation of longitudinal wave can be manipulated by changing the conductance of the conductor in the eternal circuit.All these results are consistent with our initial predictions in section 3. The error is around 5% for both case which is due to the imperfection of the manufacture and setups.We also notice that the noise also affects the results due to the reflection and scattering of longitudinal waves in the plate due to the size of the plate.

Conclusion
The paper presents a novel active EMM with tunable anisotropic effective stiffness by integrating smart components into the microstructure.By adjusting the shunted negative capacitance value, the effective mass density can be tailored over a wide frequency range.The effective medium theory yields high anisotropy in the effective elastic stiffness, which can even have opposite signs.A metamaterial plate was manufactured and tested by sweeping both wave frequency and conductance of conductors in the external circuits, and the experimental data agrees well with the simulation results.The results of this study demonstrate that our proposed method can accurately and efficiently predict the output elastic waves of composite bars based on their respective architectures and input waves.This valuable insight into the behavior of composite materials under different loading conditions could lead to the optimization of their design and performance.

Figure 1
Figure 1 (a) Unit cell design of the proposed active EMM.Aluminum is in blue color, and piezoelectric patch is in gray.The direction of polarization of the piezoelectric patch is indicated by yellow arrow.(b) EMM array composed of periodic unit cells.(C) External electric circuit for each piezoelectric patch.

Figure 2
Figure 2 (a) Effective stiffness of piezoelectric patch with different inductance, (b) Electric field of the piezoelectric patches at resonance frequency.(a ) (b)

Table 1
Geometric and material parameters of the unit cell of the active EMM 31 -6.55 C/m 2  33 23.3 C/m 2