# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 1 Introduction  
Metamaterials are composed of periodically arranged artificial cells, which can effectively improve the coupling between unit cells and waves, thus controlling and dissipating low-frequency energy [1]. In vibration and noise reduction [2], energy recovery [3], and fluctuation regulation [4], metamaterials show a broad prospect. In recent years, designing novel lightweight unit cells to achieve low-frequency broadband is one of the current research hotspots of metamaterials in engineering applications [5]. Acoustic black hole (ABH) refers to the use of power-law variation in geometric thickness or material parameters of thin-walled structures, which can achieve the convergence effect on elastic waves and reduce the wave speed to 0 when the geometric thickness of the tip is 0 [6, 7, 8]. [PERSON] applied the ABH structure to beams and found that the ABH thickness variation can gather the elastic wave energy at the tip and achieve the vibration suppression of elastic waves [9, 10]. [PERSON] investigated the attenuation of elastic fluctuations in ABH plate structures and achieved a significant reduction in vibration amplitude by increasing the power exponent and adding a small amount of damping material [11]. [PERSON] embedded the ABH structure into a thin-walled structure and observed the attenuation of broadband vibrations in the high-frequency range above the cut-off frequency [12]. Conventional ABH structures can only achieve vibration attenuation at high frequencies above the cut-off frequency. [PERSON] arranged the ABH structures periodically to form a one-dimensional ABH metamaterial, and based on the wavelet-decomposed energy method, he demonstrated the existence of a local resonance bandgap in the ABH metamaterial structure above the cut-off frequency caused by the ABH effect [13]. Subsequently, [PERSON] proposed a double-leafABH metamaterial to increase structural stiffness, and coupling of the local resonance and the Bragg scattering effect achieved the broadening of the bandgap at mid-frequency and high-frequency [14]. To further reduce the bandgap of the ABH metamaterial structure, [PERSON] proposed a composite double-leaf ABH metamaterial by introducing ethylene vinyl acetate material into the ABH unit-cell uniform beams, which opens up the bandgap in the low-frequency range of 179-317 Hz [15]. [PERSON] applied multi-materials to ABH metamaterial unit cells to open a band gap with a bandwidth of 96 Hz in a low-frequency range [16]. [PERSON] attached a periodic local resonator to the ABH plate to construct ABH metamaterials, which can generate a bandgap of 48-52 Hz in the low-frequency range by controlling resonators [17]. Although ABH metamaterials achieve a low-frequency bandgap below the cut-off frequency, the bandgap width is narrow. [PERSON] achieved low-frequency bandgap broadening by embedding multiple ABHs in a two-dimensional ABH metamaterial plate, with an in-bandgap attenuation effect of up to 50 dB [18]. [PERSON] attached several different ABH dynamic vibration absorbers to the Euler-Bernoulli beam, which achieved the broadband suppression of low-frequency vibration by combining with the mistuning effect, with an attenuation bandwidth percentage of 0.587 below 1000 Hz [19]. However, by attaching a number of ABHs to the structure, the width of the attenuation band and attenuation depends on the number of ABH structures. [PERSON] designed an ABH metamaterial based on 3D printing with simultaneous double power-law variations in geometric thickness and material to widen the low-frequency bandgap to 287 Hz [20]. He proposed to embed ABH cells into intercalated beams to form multiphase metamaterials, which form a wide frequency band gap below 400 Hz [21]. While the use of multi-material ABH metamaterials can widen the low-frequency bandgap, the increase in the number of combined interfaces makes it difficult to process and limits its application.