【摘要】：蜂窩材料是一種周期性的二維有序多孔材料,具有優(yōu)良的吸聲、隔振、減振、波導(dǎo)等功能,在航空、航天、航海、輕工業(yè)、建筑等領(lǐng)域和聲功能器件行業(yè)中得到廣泛的應(yīng)用。因此,近二十年來(lái),有關(guān)蜂窩材料彈性波傳播問(wèn)題的研究受到學(xué)術(shù)界越來(lái)越多的關(guān)注。對(duì)于該問(wèn)題的研究,根據(jù)是否考慮蜂窩材料單胞拓?fù)錁?gòu)型及尺寸的影響,可以分為等效連續(xù)介質(zhì)的方法和離散化的晶格動(dòng)力學(xué)方法�；谶@兩種分析建模的思想,本文分別對(duì)蜂窩夾層結(jié)構(gòu)和二維周期蜂窩材料的波傳播問(wèn)題展開(kāi)研究。內(nèi)容主要包括:(1)基于Timoshenko梁模型及單胞應(yīng)變能等效原理,研究棱鏡型夾層結(jié)構(gòu)的蜂窩夾芯均勻化問(wèn)題。選取蜂窩夾芯材料單胞為研究對(duì)象,將其胞壁近似為T(mén)imoshenko梁模型,通過(guò)連續(xù)介質(zhì)小變形理論,建立宏觀應(yīng)變與胞壁應(yīng)變之間的關(guān)系,進(jìn)而將代表體單元的應(yīng)變能密度表述成宏觀應(yīng)變的函數(shù),然后對(duì)宏觀應(yīng)變求二階導(dǎo)便可得到夾芯層的等效彈性常數(shù)。通過(guò)有限元計(jì)算等效模型與實(shí)際模型的結(jié)構(gòu)響應(yīng)和前五階振動(dòng)頻率,并將Timoshenko等效模型的結(jié)果與Euler等效模型(令剪切因子為零,Timoshenko等效模型即可退化為傳統(tǒng)的Euler等效模型)、實(shí)際模型的結(jié)果進(jìn)行對(duì)比,驗(yàn)證本文方法的有效性。同時(shí)進(jìn)一步分析蜂窩夾芯材料的胞壁長(zhǎng)細(xì)比(或相對(duì)密度)對(duì)等效彈性常數(shù)的影響,結(jié)果表明Timoshenko等效模型較Euler等效模型能更加準(zhǔn)確地預(yù)測(cè)蜂窩材料的面內(nèi)等效彈性常數(shù)。該均勻化方法的研究與探討,為輕質(zhì)夾層結(jié)構(gòu)波傳播問(wèn)題(計(jì)算模型的建立)提供理論依據(jù)。(2)擴(kuò)展的Wittrick-Williams算法和精細(xì)積分方法的結(jié)合,在處理各種邊值問(wèn)題時(shí)具有明顯的優(yōu)勢(shì),本文將該方法推廣至蜂窩夾層圓柱殼彈性波傳播問(wèn)題的研究中。基于虛位移原理和勒讓德變換,建立柱坐標(biāo)系下的混合變量狀態(tài)空間方程,進(jìn)而采用分段平均假設(shè)、擴(kuò)展的Wittrick-Williams算法和精細(xì)積分方法,得到夾層圓柱殼中簡(jiǎn)諧彈性波傳播的頻散關(guān)系,與多項(xiàng)式方法的計(jì)算結(jié)果進(jìn)行比較,驗(yàn)證該方法在研究波傳播問(wèn)題中的有效性。采用(1)中應(yīng)變能等效的均勻化方法,將正方形、菱形、三角形-6型和三角形-8型等四種構(gòu)型的夾芯層等效為均勻的正交各向異性材料,并分析拓?fù)錁?gòu)型、相對(duì)密度及夾層柱殼的內(nèi)外徑邊界條件對(duì)夾層圓柱殼波傳播特性的影響。(3)從材料的單胞微結(jié)構(gòu)角度出發(fā),對(duì)二維星形蜂窩結(jié)構(gòu)的等效彈性常數(shù)及波傳播特性進(jìn)行研究,并對(duì)比分析等效泊松比與帶隙特性的變化關(guān)系。采用卡氏第二定理導(dǎo)出單胞的橫向及縱向位移,根據(jù)應(yīng)變、楊氏模量及泊松比的定義,得到星形蜂窩材料的等效楊氏模量和等效泊松比的解析表達(dá)式。應(yīng)用Bloch定理,將無(wú)限大周期結(jié)構(gòu)的波傳播分析簡(jiǎn)化為代表體單元(單胞)波動(dòng)特性的研究,基于有限元分析得到單胞的動(dòng)力剛度矩陣,進(jìn)而利用變分原理推導(dǎo)得到頻域下的控制方程,將波傳播問(wèn)題轉(zhuǎn)化為本征值問(wèn)題,采用擴(kuò)展的Wittrick-Williams算法對(duì)該本征值問(wèn)題進(jìn)行求解。通過(guò)對(duì)不同幾何參數(shù)下星形蜂窩材料的等效彈性常數(shù)及波傳播帶隙的分析,發(fā)現(xiàn)當(dāng)?shù)刃Р此杀葹樨?fù)值時(shí),蜂窩材料的等效楊氏模量明顯提升,即結(jié)構(gòu)整體強(qiáng)度得到增強(qiáng);同時(shí)在保證帶隙寬度幾乎不變的情況下,可以通過(guò)調(diào)節(jié)結(jié)構(gòu)的幾何尺寸,達(dá)到降低帶隙的目的。因此,在星形蜂窩材料的力學(xué)優(yōu)化設(shè)計(jì)中,泊松比是一個(gè)重要的參數(shù)。(4)將變截面設(shè)計(jì)的思想引入到傳統(tǒng)蜂窩材料中,開(kāi)展帶隙的可設(shè)計(jì)性研究。采用(3)中波傳播特性的分析方法,研究單胞夾角、胞壁長(zhǎng)細(xì)比和變截面系數(shù)對(duì)二維變截面六邊形蜂窩結(jié)構(gòu)帶隙特性和方向特性的影響。通過(guò)分析得到:這三個(gè)幾何參數(shù)對(duì)結(jié)構(gòu)的帶隙特性有重要的影響,尤其是變截面系數(shù)的引入,使蜂窩材料較傳統(tǒng)等截面的周期材料擁有更多、更寬的全帶隙;而對(duì)于方向特性而言,結(jié)構(gòu)夾角的影響較另外兩個(gè)因素更為顯著。綜上所述,變截面系數(shù)可以作為帶隙特性研究中一個(gè)重要的設(shè)計(jì)參數(shù),這對(duì)工程中的濾波隔振等實(shí)際應(yīng)用具有很重要的指導(dǎo)意義。(5)基于單胞拓?fù)錁?gòu)型的對(duì)稱(chēng)性,對(duì)二維周期結(jié)構(gòu)波傳播問(wèn)題中不可約布里淵區(qū)的適用性進(jìn)行探討。以正方形晶格(具有四重旋轉(zhuǎn)對(duì)稱(chēng)性和四重軸對(duì)稱(chēng)性)為研究對(duì)象,分別考慮兩種類(lèi)型的單胞構(gòu)型:第一種是指與正方形晶格具有完全相同對(duì)稱(chēng)性的基元單胞(類(lèi)型Ⅰ),如正方形和內(nèi)凹正方形;第二種是指對(duì)稱(chēng)性低于正方形晶格對(duì)稱(chēng)性的基元單胞(類(lèi)型Ⅱ),如方形zigzag和四手性(只具有四重旋轉(zhuǎn)對(duì)稱(chēng)性)。研究分析這兩種類(lèi)型蜂窩材料的前八階頻率極值的大小及其出現(xiàn)的位置,結(jié)果表明:對(duì)于類(lèi)型Ⅰ的單胞構(gòu)型,頻率極值均出現(xiàn)在不可約布里淵區(qū)的邊界;對(duì)于類(lèi)型Ⅱ的單胞構(gòu)型,其相平面不再具有軸對(duì)稱(chēng)性,部分頻率極值偏離不可約布里淵區(qū)的邊界,此時(shí)必須對(duì)整個(gè)第一布里淵區(qū)的頻散關(guān)系進(jìn)行計(jì)算,以獲取正確的帶隙特性。因此,在研究周期結(jié)構(gòu)帶隙特性時(shí),不僅需要考慮晶格的對(duì)稱(chēng)性,還必須充分考慮基元單胞的對(duì)稱(chēng)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.
【學(xué)位授予單位】：西北工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O347.41

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