蜂窝材料弹性波频散关系分析及带隙特性微结构设计
发布时间：2018-08-05 09:54
【摘要】：蜂窝材料是一种周期性的二维有序多孔材料,具有优良的吸声、隔振、减振、波导等功能,在航空、航天、航海、轻工业、建筑等领域和声功能器件行业中得到广泛的应用。因此,近二十年来,有关蜂窝材料弹性波传播问题的研究受到学术界越来越多的关注。对于该问题的研究,根据是否考虑蜂窝材料单胞拓扑构型及尺寸的影响,可以分为等效连续介质的方法和离散化的晶格动力学方法。基于这两种分析建模的思想,本文分别对蜂窝夹层结构和二维周期蜂窝材料的波传播问题展开研究。内容主要包括:(1)基于Timoshenko梁模型及单胞应变能等效原理,研究棱镜型夹层结构的蜂窝夹芯均匀化问题。选取蜂窝夹芯材料单胞为研究对象,将其胞壁近似为Timoshenko梁模型,通过连续介质小变形理论,建立宏观应变与胞壁应变之间的关系,进而将代表体单元的应变能密度表述成宏观应变的函数,然后对宏观应变求二阶导便可得到夹芯层的等效弹性常数。通过有限元计算等效模型与实际模型的结构响应和前五阶振动频率,并将Timoshenko等效模型的结果与Euler等效模型(令剪切因子为零,Timoshenko等效模型即可退化为传统的Euler等效模型)、实际模型的结果进行对比,验证本文方法的有效性。同时进一步分析蜂窝夹芯材料的胞壁长细比(或相对密度)对等效弹性常数的影响,结果表明Timoshenko等效模型较Euler等效模型能更加准确地预测蜂窝材料的面内等效弹性常数。该均匀化方法的研究与探讨,为轻质夹层结构波传播问题(计算模型的建立)提供理论依据。(2)扩展的Wittrick-Williams算法和精细积分方法的结合,在处理各种边值问题时具有明显的优势,本文将该方法推广至蜂窝夹层圆柱壳弹性波传播问题的研究中。基于虚位移原理和勒让德变换,建立柱坐标系下的混合变量状态空间方程,进而采用分段平均假设、扩展的Wittrick-Williams算法和精细积分方法,得到夹层圆柱壳中简谐弹性波传播的频散关系,与多项式方法的计算结果进行比较,验证该方法在研究波传播问题中的有效性。采用(1)中应变能等效的均匀化方法,将正方形、菱形、三角形-6型和三角形-8型等四种构型的夹芯层等效为均匀的正交各向异性材料,并分析拓扑构型、相对密度及夹层柱壳的内外径边界条件对夹层圆柱壳波传播特性的影响。(3)从材料的单胞微结构角度出发,对二维星形蜂窝结构的等效弹性常数及波传播特性进行研究,并对比分析等效泊松比与带隙特性的变化关系。采用卡氏第二定理导出单胞的横向及纵向位移,根据应变、杨氏模量及泊松比的定义,得到星形蜂窝材料的等效杨氏模量和等效泊松比的解析表达式。应用Bloch定理,将无限大周期结构的波传播分析简化为代表体单元(单胞)波动特性的研究,基于有限元分析得到单胞的动力刚度矩阵,进而利用变分原理推导得到频域下的控制方程,将波传播问题转化为本征值问题,采用扩展的Wittrick-Williams算法对该本征值问题进行求解。通过对不同几何参数下星形蜂窝材料的等效弹性常数及波传播带隙的分析,发现当等效泊松比为负值时,蜂窝材料的等效杨氏模量明显提升,即结构整体强度得到增强;同时在保证带隙宽度几乎不变的情况下,可以通过调节结构的几何尺寸,达到降低带隙的目的。因此,在星形蜂窝材料的力学优化设计中,泊松比是一个重要的参数。(4)将变截面设计的思想引入到传统蜂窝材料中,开展带隙的可设计性研究。采用(3)中波传播特性的分析方法,研究单胞夹角、胞壁长细比和变截面系数对二维变截面六边形蜂窝结构带隙特性和方向特性的影响。通过分析得到:这三个几何参数对结构的带隙特性有重要的影响,尤其是变截面系数的引入,使蜂窝材料较传统等截面的周期材料拥有更多、更宽的全带隙;而对于方向特性而言,结构夹角的影响较另外两个因素更为显著。综上所述,变截面系数可以作为带隙特性研究中一个重要的设计参数,这对工程中的滤波隔振等实际应用具有很重要的指导意义。(5)基于单胞拓扑构型的对称性,对二维周期结构波传播问题中不可约布里渊区的适用性进行探讨。以正方形晶格(具有四重旋转对称性和四重轴对称性)为研究对象,分别考虑两种类型的单胞构型:第一种是指与正方形晶格具有完全相同对称性的基元单胞(类型Ⅰ),如正方形和内凹正方形;第二种是指对称性低于正方形晶格对称性的基元单胞(类型Ⅱ),如方形zigzag和四手性(只具有四重旋转对称性)。研究分析这两种类型蜂窝材料的前八阶频率极值的大小及其出现的位置,结果表明:对于类型Ⅰ的单胞构型,频率极值均出现在不可约布里渊区的边界;对于类型Ⅱ的单胞构型,其相平面不再具有轴对称性,部分频率极值偏离不可约布里渊区的边界,此时必须对整个第一布里渊区的频散关系进行计算,以获取正确的带隙特性。因此,在研究周期结构带隙特性时,不仅需要考虑晶格的对称性,还必须充分考虑基元单胞的对称性。
[Abstract]:Honeycomb material is a periodic two-dimensional ordered porous material, with excellent sound absorption, vibration isolation, vibration damping, waveguide and other functions. It has been widely used in the field of aviation, aerospace, navigation, light industry, building and other fields. Therefore, the research on the elastic wave propagation of honeycomb materials has been studied in the past twenty years. More attention should be paid to the problem. Based on the consideration of the effect of the single cell topology and size of honeycomb material, it can be divided into the equivalent continuous medium method and the discrete lattice dynamics method. Based on the two analytical modeling ideas, the wave propagation of the honeycomb sandwich structure and the two-dimensional periodic honeycomb material is discussed in this paper. The main contents are as follows: (1) based on the Timoshenko beam model and the equivalent principle of the single cell strain energy, the homogenization of the honeycomb sandwich is studied. The cell wall of the honeycomb sandwich material is selected as the research object, and the cell wall is approximated to the Timoshenko beam model, and the macroscopic strain and cell are established through the theory of small deformation of continuous medium. The relationship between the wall strain and the strain energy density of the representative element is expressed as the function of the macroscopic strain. Then the equivalent elastic constant of the sandwich layer can be obtained by the two order stool of the macroscopic strain. The structural response of the equivalent model and the actual model and the first five order vibration frequencies are calculated by the finite element method, and the results of the Timoshenko equivalent model are obtained. With the Euler equivalent model (the shear factor is zero, the Timoshenko equivalent model can degenerate into the traditional Euler equivalent model). The results of the actual model are compared to verify the effectiveness of the method. At the same time, the effect of the cell wall length ratio (or relative density) on the equivalent elastic constants of the honeycomb sandwich materials is further analyzed, and the results show that Timoshenk O equivalent model can predict the equivalent elastic constant of honeycomb material more accurately than the Euler equivalent model. The research and discussion of this homogenization method provides a theoretical basis for the wave propagation problem of light sandwich structures (2) the combination of extended Wittrick-Williams method and fine integration method to deal with various boundary values. The problem has obvious advantages. In this paper, this method is extended to the study of elastic wave propagation in a honeycomb sandwich cylindrical shell. Based on the principle of virtual displacement and Legendre transformation, the state space equation of mixed variables in the cylindrical coordinate system is established, and then the piecewise mean hypothesis, the extended Wittrick-Williams algorithm and the fine integration method are obtained. The frequency dispersion relation of simple harmonic wave propagation in a sandwich cylindrical shell is compared with the calculation results of polynomial method to verify the effectiveness of the method in the study of wave propagation. Using the homogenization method of the equivalent strain energy in (1), the sandwich layer of four configurations, such as square, diamond, triangle -6 and triangle -8, is equivalent to uniform. The influence of the topological configuration, the relative density and the inner and outer diameter boundary conditions of the sandwich cylindrical shell on the wave propagation characteristics of a sandwich cylindrical shell is analyzed. (3) the equivalent elastic constants and wave propagation characteristics of a two-dimensional star honeycomb structure are studied from the single cell microstructure angle of the material, and the equivalent Poisson ratio and band are compared and analyzed. The transverse and longitudinal displacement of the single cell is derived by the Carson's second theorem. According to the definition of strain, Young's modulus and Poisson's ratio, the analytic expression of the equivalent Young's modulus and the equivalent Poisson's ratio of the star honeycomb material is obtained. The Bloch theorem is used to simplify the wave propagation analysis of the infinite periodic structure to the representative unit. Based on the finite element analysis, the dynamic stiffness matrix of the single cell is obtained by the finite element analysis. Then the control equation in the frequency domain is derived by the variational principle. The wave propagation problem is converted to the eigenvalue problem. The extended Wittrick-Williams algorithm is used to solve the eigenvalue problem. When the equivalent Poisson's ratio is negative, the equivalent Young's modulus of the honeycomb material increases obviously when the equivalent Poisson's ratio is negative, that is, when the width of the band gap is almost invariable, the aim of reducing the band gap can be achieved through the geometric size of the joint structure. Therefore, the Poisson's ratio is an important parameter in the mechanical optimization design of the star honeycomb material. (4) the idea of the variable section design is introduced into the traditional honeycomb material and the design of the band gap is studied. The angle of single cell, the length of the cell wall and the variable cross section coefficient are studied by the analysis method of the wave propagation characteristics in (3) six. The effect of the band gap and direction characteristics of the beehive structure is analyzed. The three geometric parameters have an important influence on the band gap characteristics of the structure, especially the variable section coefficient, which makes the honeycomb material have more and wider full band gap than the traditional equivalent section of the periodic material; and for the direction characteristics, the angle of the structure is reflected. The response of the other two factors is more significant. To sum up, the variable section coefficient can be used as an important design parameter in the study of band gap characteristics, which is of great guiding significance for the practical application of filtering and vibration isolation in engineering. (5) based on the symmetry of the single cell topology, the irreducible Brillouin in the problem of two-dimensional periodic structure wave propagation is made. The applicability of the region is discussed. Taking the square lattice (with four heavy rotation symmetry and four heavy axisymmetric symmetry) as the research object, two types of single cell configurations are considered respectively: the first is the basic unit cell (type I), which has the same symmetry as the square lattice, such as the square and the concave square; the second means the low symmetry. The basic unit cell (type II) of square lattice symmetry, such as square zigzag and four chiral (only four heavy rotation symmetry). The size and location of the first eight order frequency extremes of these two types of honeycomb materials are studied and analyzed. The results show that the frequency extremes of the type I single cell configuration appear in the irreducible Brillouin region. For the single cell configuration of type II, the phase plane is no longer axisymmetric and some of the frequency extremes deviate from the boundary of the irreducible Brillouin region. At this time, the dispersion relation of the whole first Brillouin region must be calculated to obtain the correct band gap characteristics. Therefore, in the study of the band gap characteristics of the periodic structure, the lattice is not only needed to consider the lattice. Symmetry of the unit cell must also be fully considered.
【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O347.41
[Abstract]:Honeycomb material is a periodic two-dimensional ordered porous material, with excellent sound absorption, vibration isolation, vibration damping, waveguide and other functions. It has been widely used in the field of aviation, aerospace, navigation, light industry, building and other fields. Therefore, the research on the elastic wave propagation of honeycomb materials has been studied in the past twenty years. More attention should be paid to the problem. Based on the consideration of the effect of the single cell topology and size of honeycomb material, it can be divided into the equivalent continuous medium method and the discrete lattice dynamics method. Based on the two analytical modeling ideas, the wave propagation of the honeycomb sandwich structure and the two-dimensional periodic honeycomb material is discussed in this paper. The main contents are as follows: (1) based on the Timoshenko beam model and the equivalent principle of the single cell strain energy, the homogenization of the honeycomb sandwich is studied. The cell wall of the honeycomb sandwich material is selected as the research object, and the cell wall is approximated to the Timoshenko beam model, and the macroscopic strain and cell are established through the theory of small deformation of continuous medium. The relationship between the wall strain and the strain energy density of the representative element is expressed as the function of the macroscopic strain. Then the equivalent elastic constant of the sandwich layer can be obtained by the two order stool of the macroscopic strain. The structural response of the equivalent model and the actual model and the first five order vibration frequencies are calculated by the finite element method, and the results of the Timoshenko equivalent model are obtained. With the Euler equivalent model (the shear factor is zero, the Timoshenko equivalent model can degenerate into the traditional Euler equivalent model). The results of the actual model are compared to verify the effectiveness of the method. At the same time, the effect of the cell wall length ratio (or relative density) on the equivalent elastic constants of the honeycomb sandwich materials is further analyzed, and the results show that Timoshenk O equivalent model can predict the equivalent elastic constant of honeycomb material more accurately than the Euler equivalent model. The research and discussion of this homogenization method provides a theoretical basis for the wave propagation problem of light sandwich structures (2) the combination of extended Wittrick-Williams method and fine integration method to deal with various boundary values. The problem has obvious advantages. In this paper, this method is extended to the study of elastic wave propagation in a honeycomb sandwich cylindrical shell. Based on the principle of virtual displacement and Legendre transformation, the state space equation of mixed variables in the cylindrical coordinate system is established, and then the piecewise mean hypothesis, the extended Wittrick-Williams algorithm and the fine integration method are obtained. The frequency dispersion relation of simple harmonic wave propagation in a sandwich cylindrical shell is compared with the calculation results of polynomial method to verify the effectiveness of the method in the study of wave propagation. Using the homogenization method of the equivalent strain energy in (1), the sandwich layer of four configurations, such as square, diamond, triangle -6 and triangle -8, is equivalent to uniform. The influence of the topological configuration, the relative density and the inner and outer diameter boundary conditions of the sandwich cylindrical shell on the wave propagation characteristics of a sandwich cylindrical shell is analyzed. (3) the equivalent elastic constants and wave propagation characteristics of a two-dimensional star honeycomb structure are studied from the single cell microstructure angle of the material, and the equivalent Poisson ratio and band are compared and analyzed. The transverse and longitudinal displacement of the single cell is derived by the Carson's second theorem. According to the definition of strain, Young's modulus and Poisson's ratio, the analytic expression of the equivalent Young's modulus and the equivalent Poisson's ratio of the star honeycomb material is obtained. The Bloch theorem is used to simplify the wave propagation analysis of the infinite periodic structure to the representative unit. Based on the finite element analysis, the dynamic stiffness matrix of the single cell is obtained by the finite element analysis. Then the control equation in the frequency domain is derived by the variational principle. The wave propagation problem is converted to the eigenvalue problem. The extended Wittrick-Williams algorithm is used to solve the eigenvalue problem. When the equivalent Poisson's ratio is negative, the equivalent Young's modulus of the honeycomb material increases obviously when the equivalent Poisson's ratio is negative, that is, when the width of the band gap is almost invariable, the aim of reducing the band gap can be achieved through the geometric size of the joint structure. Therefore, the Poisson's ratio is an important parameter in the mechanical optimization design of the star honeycomb material. (4) the idea of the variable section design is introduced into the traditional honeycomb material and the design of the band gap is studied. The angle of single cell, the length of the cell wall and the variable cross section coefficient are studied by the analysis method of the wave propagation characteristics in (3) six. The effect of the band gap and direction characteristics of the beehive structure is analyzed. The three geometric parameters have an important influence on the band gap characteristics of the structure, especially the variable section coefficient, which makes the honeycomb material have more and wider full band gap than the traditional equivalent section of the periodic material; and for the direction characteristics, the angle of the structure is reflected. The response of the other two factors is more significant. To sum up, the variable section coefficient can be used as an important design parameter in the study of band gap characteristics, which is of great guiding significance for the practical application of filtering and vibration isolation in engineering. (5) based on the symmetry of the single cell topology, the irreducible Brillouin in the problem of two-dimensional periodic structure wave propagation is made. The applicability of the region is discussed. Taking the square lattice (with four heavy rotation symmetry and four heavy axisymmetric symmetry) as the research object, two types of single cell configurations are considered respectively: the first is the basic unit cell (type I), which has the same symmetry as the square lattice, such as the square and the concave square; the second means the low symmetry. The basic unit cell (type II) of square lattice symmetry, such as square zigzag and four chiral (only four heavy rotation symmetry). The size and location of the first eight order frequency extremes of these two types of honeycomb materials are studied and analyzed. The results show that the frequency extremes of the type I single cell configuration appear in the irreducible Brillouin region. For the single cell configuration of type II, the phase plane is no longer axisymmetric and some of the frequency extremes deviate from the boundary of the irreducible Brillouin region. At this time, the dispersion relation of the whole first Brillouin region must be calculated to obtain the correct band gap characteristics. Therefore, in the study of the band gap characteristics of the periodic structure, the lattice is not only needed to consider the lattice. Symmetry of the unit cell must also be fully considered.
【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O347.41
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