1.Key Laboratory of Grain Information Processing and Control (Henan University of Technology), Ministry of Education, Zhengzhou 450001, China 2.Henan Provincial Key Laboratory of Grain Photoelectric Detection and Control, Henan University of Technology, Zhengzhou 450001, China 3.College of Information Science and Engineering, Henan University of Technology, Zhengzhou 450001, China 4.School of Electronic and Information Engineering, Key Laboratory for Physical Electronics and Devices of the Ministry of Education, Xi’an Jiaotong University, Xi’an 710049, China
Abstract: Pentamode metamaterial (PM) is a kind of artificial microstructure extremum material with solid morphology and fluid properties proposed by Milton and Cherkaey. By decoupling the compression and the shear waves, the periodic structure is difficult to be compressed, but the shear deformation occurs easily. Theoretically, acoustic metamaterials consisting of such periodic arrangement of structural units can achieve complete matching with water. Therefore, the characteristics of adjustable modulus anisotropy, small stuffing rate and broadband endow the PMs with excellent acoustic control ability, which has attracted more attention of researchers. In this paper, the narrow-diameter intersection point P (0.25a, 0.25a, 0.25a) of an isotropic three-dimensional PM selected as the reference point in four different directions (X-axis, Y-axis, Z-axis and body diagonal). When the P-point moves, the farther the P-point is, the greater the degree of anisotropy is. The introduction of anisotropy will cause the structural bifurcation of the three-dimensional PM to change structural parameters, and the structural parameters are important factors affecting the band characteristics of the three-dimensional PM of Bragg scattering. In order to study the influence of anisotropy on the band structure and pentamode properties of three-dimensional asymmetric double-cone PMs, we use the finite element simulation software COMSOL to calculate the primitive-cell of three-dimensional anisotropic PMs under Bloch boundary conditions. By adjusting the position of P point, four different types of three-dimensional anisotropic asymmetric double-cone PMs are constructed. Since the anisotropy changes in different directions have different effects on the parameters of the asymmetric double-cone structure, the band characteristics and the pentamode characteristics will also receive different degrees of influence. In this paper, the relationship between the degree of anisotropy and the band gap characteristics, single-mode region and figure of merit (FOM) are given, and the result can provide guidance for the design of asymmetric double-cone PM acoustic device. Compared with the isotropic double-cone PMs, the relative bandwidth of the first band gap of the anisotropic double-cone PMs can be broadened to 123%, and the FOM can be increased to 6.9 times. Due to the introduction of anisotropy, Due to the introduction of anisotropy, the structure of three-dimensional asymmetric double-cone PMs are more complex, the demand for sample fabrication is further improved, and the stability of PMs also reduced. Therefore, PMs with high stability and easy to be fabricated still needs further research and exploration. Keywords:pentamode metamaterial/ anisotropy/ phononic band gap/ broadband
式中${C_{11}}$, ${C_{12}}$和${C_{44}}$为三维五模超材料弹性矩阵的3个独立弹性系数, $\upsilon _{110}^{T, xy}$, $\upsilon _{110}^{L, xy}$和$\upsilon _{110}^{T, z}$分别为压缩波与剪切波沿[110] ($\varGamma K$)方向的相速度, $\rho = f{\rho _0}$是三维五模超材料的等效密度, $f$和${\rho _0}$分别为三维五模超材料的填充率与组成材料的质量密度. 因此, 图2中三维五模超材料的品质因数约为251. 3.数值仿真结果及讨论基于有限元方法, 对于四种类型的三维各向异性非对称双锥五模超材料的原胞进行能带结构数值计算, 并对其第一带隙、单模区域、品质因数的变化规律进行了系统研究, 结果如图3—图7所示. 图 3 模型1的第一带隙与单模区域的(a)上下界频率; (b)相对带宽 Figure3. (a) The upper and lower edges and (b) relative bandwidth of the first phononic band gaps and single mode area of model 1.
图 7 各向异性对非对称双锥五模超材料品质因数的影响 Figure7. The influence of anisotropy on the figure of merit of asymmetric double-cone pentamode materials.
对于模型1, 当P点沿X轴方向移动时, 其第一带隙与单模区域随移动距离的变化如图3所示. 当${O_x}P/a = 0.25$时, 五模超材料的原胞为各向同性的三维结构, 这时P点的各向异性程度最小; 当${O_x}P/a = 0.05$或0.45时, 各向异性程度最大. 随着各向异性程度的增加, 第一带隙的下界频率从8.845 kHz上升到11.753 kHz, 对应的上界频率从9.538 kHz上升到14.846 kHz, 其相对带宽从0.148增加到0.233, 大约增加了57.4%. 单模区域的下界频率在0.457与0.601 kHz之间变化, 对应的上界频率在1.938与2.333 kHz之间变化, 其相对带宽从1.344减小到1.053, 大约减小了21.6%. 对于模型2, 当P点沿Y轴方向移动时, 各向异性五模超材料的第一带隙与单模区域随移动距离的变化如图4所示. 随着各向异性程度的增加, 第一带隙的下界频率在8.017—11.27 kHz之间变化, 对应的上界频率在8.421—14.852 kHz之间变化, 其相对带宽从0.147增加到0.328, 大约增加了123%. 单模区域的下界频率从0.457 kHz上升到0.632 kHz, 对应的上界频率从2.333 kHz减小到1.448 kHz, 其相对带宽从1.344减小到0.881, 大约减小了34.4%. 单模区域相对带宽在各向异性程度最大处明显下降的主要原因是由于第一带隙相对带宽增大, 单模区域的频率范围受到压缩. 图 4 模型2的第一带隙与单模区域的(a)上下界频率; (b)相对带宽 Figure4. (a) The upper and lower edges and (b) relative bandwidth of the first phononic band gaps and single mode area of model 2.
对于模型3, 当P点沿Z轴方向移动时, 各向异性五模超材料的第一带隙与单模区域随移动距离的变化如图5所示. 随着各向异性程度的增加, 第一带隙的下界频率从8.839 kHz上升到11.764 kHz, 对应的上界频率从9.527 kHz上升到14.847 kHz, 其相对带宽从0.148增加到0.236, 大约增加了59.4%. 单模区域的下界频率在0.438—0.562 kHz之间变化, 对应的上界频率在1.941—2.333 kHz之间变化, 其相对带宽从1.344减小到1.101, 大约减小了18.1%. 图 5 模型3的第一带隙与单模区域的(a)上下界频率; (b)相对带宽 Figure5. (a) The upper and lower edges and (b) relative bandwidth of the first phononic band gaps and single mode area of model 3.
对于模型4, 当P点沿对角线方向移动时, 各向异性五模超材料的第一带隙与单模区域随移动距离的变化如图6所示. 随着各向异性程度的增加, 第一带隙的下界频率在4.701—10.034 kHz之间变化, 对应的上界频率在4.992—17.794 kHz之间变化, 其相对带宽从0.147增加到0.256, 约增加了74.1%. 单模区域的下界频率在0.322—0.923 kHz之间变化, 对应的上界频率在1.217—2.333 kHz之间变化, 其相对带宽从1.344减小到0.449, 大约减小了66.6%. 在${O_{\rm{s}}}P/\sqrt 3 a = 0.05$处, 第一带隙的拓宽对单模区域的上界产生了较强的影响, 使单模区域的上界频率下降, 同时其下界频率上升, 从而单模区域相对带宽在此处下降比较明显. 图 6 模型4的第一带隙与单模区域的(a)上下界频率; (b)相对带宽 Figure6. (a) The upper and lower edges and (b) relative bandwidth of the first phononic band gaps and single mode area of model 4.
图 1 (a)各向同性五模超材料晶胞与(b)?(e)各向异性五模超材料晶胞结构示意图
图 2 P点沿空间对角线方向偏移0.25倍对角线长度时的能带结构
图 3 模型1的第一带隙与单模区域的(a)上下界频率; (b)相对带宽
图 7 各向异性对非对称双锥五模超材料品质因数的影响
图 4 模型2的第一带隙与单模区域的(a)上下界频率; (b)相对带宽
图 5 模型3的第一带隙与单模区域的(a)上下界频率; (b)相对带宽
图 6 模型4的第一带隙与单模区域的(a)上下界频率; (b)相对带宽