超材料是一种具有超常物理特性的人工复合材料，它能够打破某些自然规律的限制，实现传统材料不能实现的超常特性，为材料设计提供了崭新的理念，具有重要的历史意义。在声学领域中，由于质量密度定律的限制，低频、宽频声波控制一直是个颇具挑战的难题。采用传统材料进行低频声波控制时，往往需要结构的尺度非常大，不利于工程实际应用。局域共振声学超材料打破了质量密度定律的限制，采用小尺度结构单元就可以实现长波长控制，但是由于共振机制所限，其只能在共振频率处实现低频声波的控制，存在频带较窄的问题，为了实现宽频声波控制，往往需要将具有不同共振频率的谐振单元进行耦合，增加了结构的复杂性。与局域共振声学超材料不同，空间折叠声学超材料通过延长声波传输路径使结构具有超常声学特性，采用单一的结构单元就可以产生多重振动模态，为采用较为简单的结构单元实现声波的低频、宽频控制提供了新方法。但目前在二维空间折叠结构中还存在频带较窄的问题，尤其是1000Hz以下的宽频声波控制能力较差；三维空间折叠结构主要集中于结构的隔声特性研究，负折射等超常声学特性研究较少，并且采用传统的“之”形通道进行结构设计，所设计的结构较为复杂，不利于向三维空间延伸，给实际应用带来了困难。针对空间折叠声学超材料存在的这些问题，我们开展了以下研究：

（1）将曲线通道引入到二维自相似分形结构的设计中来，设计了不同阶数的二维自相似分形结构，通过对其带隙特性和传输特性进行研究，我们发现新设计的自相似分形结构具有较长的声通道，较高的折射率和较好的低频、宽频隔声特性。然后，以二维自相似分形结构为基础，进行旋转耦合，设计制备了具有Mie共振特性的二维自相似分形结构，通过对其声学特性进行分析，我们发现二维Mie共振分形结构不仅具有较好的鲁棒性，可以实现低频、宽频隔声，还具有较好的近零密度特性，可以实现声波的超常传输。

（2）将顺时针和逆时针相结合的盘绕方式引入到空间折叠结构的单元设计中，设计了不同折叠次数的反螺旋结构，并对其带隙特性和传输特性开展了研究，发现我们新设计的反螺旋结构可以产生多条全方向带隙，在100Hz至1000Hz范围内能够更好的实现低频、宽频声波控制。

（3）我们首先对Menger分形结构开展了研究，与之前报道的局域共振型和空间折叠型三维声学超材料相比，其不仅结构简单、对称性较好，易于向三维空间进行延伸，而且还具有负折射、声聚焦和声隧穿等超常声学特性。然后，将Menger分形结构与传统的ZigZag结构进行耦合，设计制备了具有结构形式简单易于向三维空间延伸的三维空间折叠型声学超材料，它可以产生负折射和声隧穿等超常声学现象，为制备结构简单的具有超常声学特性的三维空间折叠声学超材料提供了一种新的设计方法，具有较为光明的应用前景。

在本论文中，我们设计了二维自相似分形结构和反螺旋结构，提高了二维空间折叠声学超材料的低频、宽频隔声特性，设计了结构简单、易于向三维空间延伸的三维空间折叠超材料，实现了双负特性。这些研究丰富了空间折叠声学超材料的设计方式，提高了结构的声学特性，具有光明的应用前景。

Metamaterials are artificial composite materials, can break the limitations of some natural laws and achieve extraordinary physical properties, which provide new concepts to design materials and have historical significance. In acoustics, achieving the control of low frequency and wide frequency acoustic waves is always a challenging problem, due to the limitation of mass density law. When using ordinary materials to control low frequency acoustic waves, the scale of structure is very large, which is not conducive to the practical application of engineering. Locally resonant acoustic metamaterials break the limitation of mass law and achieve the control of long waves with a small scale unit, however, because of the limitation of locally resonant mechanism, it only can control the acoustic waves in the range of resonant frequency, which caused the narrower bandgaps problems. In order to realize the broadband acoustic waves control, it need to couple resonant units with different resonant frequency together, which caused the structure complexed. Different from the locally resonant metamaterials, space-coiling structure with extreme acoustic properties can produce multiple vibration modes with a single structural unit, which provides a new way to control the low frequency and wide frequency acoustic waves with a simpler structural unit. However, there is still a narrower bandgaps in two-dimensional space-coiling structure, especially the poor ability of controlling acoustic waves under 1000Hz. Three-dimensional space-coiling structure reported to date are mainly focused on the sound insulation property, there have been no reports on 3D space-coiling fractal metamaterials with a double negative property. Furthermore, the structure of the 3D space-coiling fractal metamaterials reported are complex, which is difficult to extend into the 3D space and make the construction of these metamaterials difficult. To solve these problems in space-coiling structure, we carried out the following studies:

(1) We carried out a new design of self-similar fractal metamaterials by adopting the curve channels and designed different order fractal structures. Through the study of bandgap property and transmission property, we found that the new designed self-similar fractal structure has a longer channel, high refractive index, better low frequency and broadband acoustic properties. Then, based on the new designed two-dimensional self-similar fractal structure and through the rotating coupling, we construct the Mie resonance self-similar fractal structure, we also analysis their acoustic properties, and found that the two-dimensional Mie resonance fractal structure not only has good robustness, but also has good nearly zero density properties, which can achieve the extreme transmission of the sound waves.

(2) We introduce the combination of clockwise and anticlockwise winding into the unit design of space-coiling structure, and design the anti-spiral structure in which the sound wave can alternately propagate in clockwise and anticlockwise way. Through the study of bandgap property and transmission property, we found that the newly designed anti-spiral structure can generate multiple all-directional band gaps, which can better realize the control of low-frequency and wide-frequency sound waves in the range of 100Hz to 1000Hz.

(3) Firstly, we carried out the study of Menger fractal structure, and found that the Menger fractal structure not only has simple structure units, good symmetry and can extend into 3D space easily, but also has extreme acoustic properties, can achieve negative refraction, acoustic focusing and sound tunneling. Then, we constructed a three-dimensional fractal acoustic metamaterial using the combination of zigzag channels and the Menger fractal structures that had high structural symmetry and could extend into 3D space easily, it can also achieve the extreme acoustic properties such as negative refraction and sound tunneling, and would be promising to construct the 3D double negative structure and control acoustic waves on a subwavelength scale within a single unit.

In this text, we constructed two-dimensional self-similar fractal structures and anti-spiral structures, which can effectively improve the low-frequency and broadband frequency properties of two-dimensional space-coiling fractal structure. A three-dimensional space-coiling metamaterial is designed, which not only has high structural symmetry and could extend into 3D space easily, but also can achieve double negative property. These studies enrich the design of space-coiling acoustic metamaterials, improve the acoustic properties of space-coiling structures, and are promising for acoustic applications.