ANALYSIS OF THE EFFECT OF THE THERMAL AXIAL FORCE ON THE THERMOELASTIC DAMPING IN BI-LAYERED MICRO BEAMS
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摘要: 近年来, 已有不少关于复合材料层合梁/板谐振器热弹性阻尼的研究论文发表. 然而, 在这些研究中热轴力(热薄膜力)对热弹性阻尼的贡献都被忽略了. 众所周知, 若梁/板的材料性质分布关于几何中面不对称则其物理中面将偏离几何中面. 于是, 热−弹耦合振动引起的变温场不但会形成热弯矩, 还会产生热轴力(热薄膜力), 二者将共同产生热弹性阻尼. 文章基于Euler-Bernoulli 梁理论和经典热传导理论, 建立了双层矩形截面微梁热−弹耦合振动数学模型, 精确考虑了热轴力对微结构内能耗的贡献. 然后, 采用解析方法求得了用变形几何量表示的热轴力和热弯矩, 进而求得了系统自由振动的复频率以及用逆品质因数表示的热弹性阻尼解析解. 作为数值算例, 选取由均匀的银(Ag)和氮化硅(Si3N4)分层组成的双层微梁, 通过大量的数值实验定量分析了分层体积分数变化对热弹性阻尼的影响规律, 考察了热轴力对热弹性阻尼的影响程度. 结果表明, 忽略热轴力将会低估层合梁谐振器的热弹性阻尼. 在金属银的体积分数为70% (氮化硅体积分数为30%)时, 忽略热轴力后对热弹性阻尼的低估最大可达16.3%.Abstract: In recent years, there have been many publications on the TED of composite laminated beam/plate resonators. However, the contribution of the thermal axial force (or thermal membrane force) on the thermoelastic damping (TED) in the resonators were neglected in all those investigations. It is well known that if the distribution of material properties along the thickness is asymmetric about the geometric midplane of a beam/plate resonator then the physical neutral surface will deviate from it. As a result, the temperature field in the resonator arising in the thermoelastic coupling vibration will produce both the thermal bending moment and the thermal axial/membrane force each of them will produce the TED. In this paper, based on the Euler-Bernoulli beam theory and the classical heat conduction theory, mathematical formulations for the thermo-elastically coupled free vibration of bi-layered laminated micro beams with rectangular cross sections are established in which the contribution of the thermal axial force on the internal energy dissipation is considered accurately. Then, analytical solutions of the thermal axial force and bending moment are found in terms of the geometric quantities representing the beam deformation. Furthermore, the complex frequency of the system and the TED represented by the inverse quality factor are obtained. As an example of numerical analysis, a bi-layered micro beam with homogenous layers of silver (Ag) and silicon nitride (Si3N4) is selected to quantitatively examine the effects of the thermal axial force on the TED by changing the volume fractions of the laminas and the total thickness of the beam. The numerical results show that neglection of the thermal axial force will underestimate the TED in the bi-layered beam resonators. Especially, for the resonator with the volume fraction of the silver at 70% (that of the silicon nitride at 30%), the TED will be underestimated about 16.3% if the thermal axial force is neglected.
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引 言
热弹性阻尼(thermoelastic damping, TED) 是谐振器在一个振动周期内由热弹性耦合变形引起的内部耗能, 它不能通过外部条件的改善而消除[1-2]. 随着谐振尺寸达到微/纳尺度, TED 将会变得更加显著. 在通过优化设计最大限度地消除外部阻尼的情况下, TED 将会决定谐振器品质因子的上限. 因此, 精确地分析和预测 TED 对高品质微/纳谐振器的研究和设计具有重要意义.
随着微/纳机电系统 (MEMS/NEMS)科技以及材料科学技术的发展, 除了传统的均匀各向同性材料谐振器, 复合材料谐振器也已得到广泛应用. 例如, 在陶瓷基底上铺设金属或压电层来增强谐振器的功能. 因此, 在理论上精确地分析和预测层热弹性阻尼对复合材料谐振器的优化设计就显得十分必要. 自从 Bishop等[3-4] 首先采用能量法研究多层微/纳结构的 TED 以来, 已有大量的文献研究了双层和多层微/纳梁板的TED. 鉴于本文的研究主题和篇幅所限, 下面只介绍关于复合材料微/纳梁式谐振的TED研究现状[3-16].
Bishop等[3-4]首次将Zener[1] 关于均匀材料微/纳梁谐振器TED分析的能量法推广应用到了具有完善和非完善界面复合材料多层微/纳梁/板谐振器的TED研究中, 并通过计算一个振动周期内由不可逆传热产生的总热量与总弹性势能之比给出了层合结构的逆品质因子解析解, 具体分析了对称铺设三层矩形截面微梁的TED. 接着, 已有不少作者采用能量法分别研究了双层和三层微梁中的TED. 其中, Vengllatore [5]和Prabhakar等[6] 分别预测了对称铺设三层和双层微梁中的TED, 通过数值结果分析了具有不同分层厚度(体积分数)的微梁中的TED随着等温固有频率的变化规律. Vahdat 等[7] 基于包含一个松弛参数的广义二维热传导方程采用复频率法分析了上下表面粘贴压电层的三层微梁的TED, 数值结果表明可以通过改变外加电压来调整谐振器的TED以及临界厚度. Zamanian 等[8]分析了表层不完全覆盖的双层微梁在静电场作用下的热弹性耦合自由振动响应, 其中考虑了静电场的吸入电压(pull-in voltage)产生的静态非线性弯曲变形以及轴线伸长对层合微梁中TED的影响. Nourmohammadi 等[9]在对双层微梁的TED分析中发现以SiO2/Si分层组成的微梁中的TED随着等温频率的增加会出现双峰值现象. Zuo 等[10] 推导出了非对称铺设三层微梁中TED的解析解, 并在SiO2/Si/Zn三层微梁的TED随频率变化曲线中也发现了多峰值现象. Yang 等[11-13]进一步基于二维热传导方程分别研究了具有顶层完全覆盖[11]和部分覆盖[12-13]的双层微梁中的TED, 数值结果中进一步分析了SiO2/Si 微梁TED的双峰值现象.
众所周知, 在层合微梁的物理中面偏离几何中面的情况下(典型的如由分层材料性质不同的双层微梁), 微梁在振动过程中将会产生拉完耦合变形. 于是, 由热−弹耦合振动导致的变温场在横截面内不仅会形成热弯矩, 而且还会产生热轴力. 然而, 在上述关于双层梁和不对称铺设的三层微梁的TED预测中[6, 8-13]热轴力全被忽略了, 其中只考虑了热弯矩的耗能效应. 由于未计热轴力, 上述研究中都采用物理中面法将轴向位移用挠度来表示, 从而在数学上消去了拉−弯耦合, 简化了问题的数学分析和求解. 然而, 忽略热轴力对层合微梁TED的预测精度的影响至今没有任何的定量分析和讨论.
不同于材料性质横向阶梯型变化的层合微梁, 功能梯度材料微梁的物性参数是沿着高度连续变化的. 例如典型的金属−陶瓷组分的矩形截面功能梯度梁, 其材料性质可设计为从上表面的纯陶瓷沿厚度连续变化为下表面的纯金属. 因此, 功能梯度材料微梁的热−弹耦合横向振动伴随拉−弯耦合变形. 文献[14-16]基于Euler-Bernoulli梁理论和经典的准一维热传导理论研究功能梯度材料微梁的热−弹耦合自由振动响应. 采用分层均匀化方法获得了变系数热传导方程的半析解. 进而通过求解结构自由振动的复特征值问题, 获得了用逆品质因子表示的TED解析解, 精确地考虑了热轴力对TED的贡献[15-16]. Zhang等[16]还基于修正的偶应力理论分析了尺度效应对TED的影响.
综上所述, 热轴力对层合微/纳梁谐振器热弹性阻尼的影响依然是复合材料微结构热−弹耦合振动响应研究的新课题. 本文基于Euler-Bernoulli 梁理论和准一维热传导理论研究双层微梁谐振器的热弹耦合振动响应, 精确考虑热轴力, 建立热−弹耦合兼拉−弯耦合变形的复特征值问题的数学模型, 寻求温度场、位移场、复频率以及逆品质因子的解析解, 通过数值算例定量分析热轴力对TED的影响程度和规律, 给出更加精确的TED预测.
1. 问题的数学模型
1.1 运动方程
考虑矩形截面双层微梁, 长为$ l $、宽为$ b $、高为$ h = {h_1} + {h_2} $, 其中 $ {h_1} $和$ {h_2} $分别为分层厚度 (如图1所示). 两个分层分别由两种材料性质不同的均匀各向同性材料组成. $ (x,y,z) $分别表示长度、宽度和高度方向的直角坐标, $ z = 0 $ 为几何中面.
基于Euler-Bernoulli 梁理论, 位移场可表示为
$$ u(x,z,t) = {u_0} - z\frac{{\partial w}}{{\partial x}} \text{, } w(x,z,t) = {w_0}(x,t) $$ (1) 其中$ {u_0}(x,t) $和$ {w_0}(x,t) $分别为几何中面上点的轴向和横向位移; $ u $和$ w $ 分别为梁内任意一点的位移分量; $ t $ 为时间坐标.
小振幅振动下的应变与位移的关系为
$$ {\varepsilon _x} = \frac{{\partial {u_0}}}{{\partial x}} - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} $$ (2) 由胡克定律给出各分层的正应力
$$\left.\begin{split} & {\sigma _x}^{(j)} = {E_j}\left( {\frac{{\partial {u_0}}}{{\partial x}} - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} - {\alpha _j}{\theta _j}} \right) \\ &\qquad ({z_j} < z < {z_{j + 1}},\;\;j = 1,2) \end{split} \right\}$$ (3) 其中$ {E_j} $和$ {\alpha _j} $ (j = 1, 2) 分别为分层的材料弹性模量和热膨胀系数; $ {\theta _j}(x,z,t) = {T_j}(x,z,t) - {T_0} $ 为热弹性耦合振动产生的变温场, $ {T_0} $和$ {T_j} $分别为初始温度和瞬态温度场.
忽略轴向惯性力, 层合梁自由振动微分方程为
$$\qquad\qquad\qquad \frac{{\partial {F_N}}}{{\partial x}} = 0 $$ (4) $$\qquad\qquad\qquad \frac{{{\partial ^2}M}}{{\partial {x^2}}} = {I_{{\text{eq}}}}\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} $$ (5) 其中, $ {I_{{\text{eq}}}} $ 为等效惯性参数; $ {F_N} $和 $ M $分别为轴力和弯矩, 可表示为
$$ {F_N} = \int_A {{\sigma _x}{\text{d}}} A = {S_0}\frac{{\partial {u_0}}}{{\partial x}} - {S_1}\frac{{{\partial ^2}{w_0}}}{{\partial x}} - {N_T} \tag{6a}$$ $$ M = \int_A {{\sigma _x}z{\text{d}}} A = {S_1}\frac{{\partial {u_0}}}{{\partial x}} - {S_2}\frac{{{\partial ^2}{w_0}}}{{\partial x}} - {M_T} \tag{6b}$$ 其中$ {N_T} $为热轴力, $ {M_T} $为热弯矩; $ {S_0} $, $ {S_1} $和$ {S_2} $ 分别为拉伸、拉−弯耦合和弯曲刚度. 惯性参数、刚度参数以及热轴力和热弯矩分别由下式计算
$$ {I_{{\text{eq}}}} = b\sum\limits_{i = 2}^2 {{\rho _i}} ({z_{i + 1}} - {z_i}) ,\quad {S_0} = b\sum\limits_{i = 2}^2 {{E_i}} ({z_{i + 1}} - {z_i}) \tag{7a}$$ $$ {S_1} = \frac{b}{2}\sum\limits_{i = 1}^2 {{E_i}} (z_{i + 1}^2 - z_i^2) ,\quad {S_2} = \frac{b}{3}\sum\limits_{i = 1}^2 {{E_i}} (z_{i + 1}^3 - z_i^3) \tag{7b}$$ $$ {N_T} = b\sum\limits_{i = 1}^2 {{E_i}} {\alpha _i}\int_{{z_i}}^{{z_{i + 1}}} {{\theta _i}{\text{d}}z} ,\; {M_T} = b\sum\limits_{i = 1}^2 {{E_i}} {\alpha _i}\int_{{z_i}}^{{z_{i + 1}}} {{\theta _i}z{\text{d}}z} \tag{7c}$$ 其中, $ {z_1} = - h/2 $, $ {z_2} = {h_1} - h/2 $, $ {z_3} = h/2 $.
由式(4)可知轴力为常数. 考虑到对于小振幅自由振动的微梁不考虑轴向惯性力, 且有一端为轴向可移约束, 则有$ {F_N}(l,t) = 0 $(或$ {F_N}(0,t) = 0 $), 由此可断定$ {F_N}(x,t) = 0 $ (0 < x < l). 于是由式(6a)可得
$$ \frac{{\partial {u_0}}}{{\partial x}} = {z_0}\frac{{{\partial ^2}{w_0}}}{{\partial x}} + \frac{{{N_T}}}{{{S_0}}} $$ (8) 其中$ {z_0} = {S_1}/{S_0} $为物理中面的位置. 在已有文献[6, 8-13]中, 上式中的热轴力$ {N_T} $却被不加说明地忽略了. 即在式(8)中令$ {N_T} = 0 $, 并将其代入式(2)后可得正应变下近似表示
$$ {\varepsilon _x} = ({z_0} - z)\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} $$ (9) 从而消去了拉−弯耦合, 简化了温度场的求解[6, 8-13].
将式(6b)代入式(5)并利用式(8), 得到只用挠度表示的运动微分方程
$$ {\bar S_2}\frac{{{\partial ^4}w}}{{\partial {x^4}}} + \frac{{{\partial ^2}{M_T}}}{{\partial {x^2}}} - {z_0}\frac{{{\partial ^2}{N_T}}}{{\partial {x^2}}} + {I_{{\text{eq}}}}\frac{{{\partial ^2}w}}{{\partial {t^2}}} = 0 $$ (10) 其中$ {\bar S_2} = {S_2} - {z_0}{S_1} $为层合梁的等效抗弯刚度. 方程(10)中包含了热弯矩和热轴力, 二者都由热−弹耦合的变温场确定.
1.2 热传导方程
忽略温度梯度在轴向的变化, 微梁的傅里叶热传导方程可分别在两个分层给出[1, 2, 5-11]
$$ {\kappa _j}\frac{{{\partial ^2}{\theta _j}}}{{\partial {z^2}}} = {\rho _j}{C_j}\frac{{\partial {\theta _j}}}{{\partial t}} + \frac{{{\alpha _j}{E_j}{T_0}}}{{1 - 2{\nu _j}}}\frac{{\partial \varepsilon _{ii}^{(j)}}}{{\partial t}} \;\; ( j = 1,2 ) $$ (11) 其中 $ {\kappa _j} $ 为热传导系数; $ {C_j} $ 为比热; $ \varepsilon _{ii}^{(j)} $是第j层的体积应变, 具体表示为[2]
$$ \varepsilon _{ii}^{(j)} = (1 - 2{\nu _j})\left( {\frac{{\partial {u_0}}}{{\partial x}} - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}} \right) + 2(1 + {\nu _j}){\alpha _j}{\theta _j} $$ (12) 将式(12)代入式(11), 相比单位1忽略高阶微量$ {E_j}{\alpha _j}^2{T_0}/({\rho _j}{C_j}) $, 得到用位移分量和变温场表示的热−弹耦合的热传导方程
$$ \frac{{{\partial ^2}{\theta _j}}}{{\partial {z^2}}} - \frac{{{\rho _j}{C_j}}}{{{\kappa _j}}}\frac{{\partial {\theta _j}}}{{\partial t}} = \frac{{{\alpha _j}{E_j}{T_0}}}{{{\kappa _j}}}\frac{\partial }{{\partial t}}\left( {\frac{{\partial {u_0}}}{{\partial x}} - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}} \right) $$ (13) 2. 问题的求解
假设谐振器的位移场和温度场的自由振动响应为
$$ ({u_0},{w_0}) = (\bar u(x),\bar w(x)){{\rm{e}}^{{\text{i}}\omega t}} \tag{14a}$$ $$ ({\theta _1},{\theta _2}) = ({\bar \theta _1}(x,z),{\bar \theta _2}(x,z)){{\rm{e}}^{{\text{i}}\omega t}} \tag{14b}$$ $$ ({M_T},{N_T}) = ({\bar M_T}(x,z),{\bar N_T}(x,z)){{\rm{e}}^{{\text{i}}\omega t}} \tag{14c}$$ 其中$ \omega $为固有频率, $ {\text{i}} = \sqrt { - 1} $ ; $ \bar u(x) $, $ \bar w(x) $, $ {\bar \theta _1}(x,z) $, $ {\bar \theta _2}(x,z) $, $ {\bar N_T}(x) $和 $ {\bar M_T}(x) $分别为位移、变温场、热轴力和热弯矩的振幅. 将式(14)代入式(10)和式(13)得到特征方程
$$ {\bar S_2}\frac{{{{\text{d}}^4}\bar w}}{{{\text{d}}{x^4}}} + \frac{{{{\text{d}}^2}{{\bar M}_T}}}{{{\text{d}}{x^2}}} - {z_0}\frac{{{{\text{d}}^2}{{\bar N}_T}}}{{{\text{d}}{x^2}}} - {I_{{\rm{eq}}}}{\omega ^2}\bar w = 0 $$ (15) $$ \frac{{{\partial ^2}{{\bar \theta }_j}}}{{\partial {z^2}}} + p_j^2{\bar \theta _j} = - p_j^2{q_j}\left( {\frac{{{\text{d}}\bar u}}{{{\text{d}}x}} - z\frac{{{{\text{d}}^2}\bar w}}{{{\text{d}}{x^2}}}} \right)\;\; ( j = 1,\;2 ) $$ (16) 其中
$$ {p_j} = \sqrt { - \frac{{{\text{i}}\omega }}{{{\chi _j}}}} ,\quad {q_j} = \frac{{\varDelta _{Ej}^{}}}{{{\alpha _j}}} $$ (17) 式中, $ {\chi _j} = {\kappa _j}/({\rho _j}{C_j}) $为材料的热扩散系数, $\varDelta _{Ej}^{} = {E_j}\alpha _j^2{T_0}/ ({\rho _j}{C_j})$为弹性模量的松弛强度.
首先, 可求得热传导方程(16)的通解
$$ {{\bar \theta }_j} = {A_j}\sin ({p_j}z )+ {B_j}\cos ({p_j}z) - {q_j}\left( {\frac{{{\text{d}}\bar u}}{{{\text{d}}x}} - z\frac{{{{\text{d}}^2}\bar w}}{{{\text{d}}{x^2}}}} \right) \;\; ( j = 1,\;\;2 ) $$ (18) 其中$ {A_j} $和$ {B_j} $是运动学参数$ {\text{d}}\bar u/{\text{d}}x $和$ {{\text{d}}^2}\bar w/{\text{d}}{x^2} $的表达式, 形式上可表示为
$$\left.\begin{array}{l} A_j=A_{1 j} \dfrac{\mathrm{d} \bar{u}}{\mathrm{d} x}+A_{2 j} \dfrac{\mathrm{d}^2 \bar{w}}{\mathrm{d} x^2}\;\;(j=1,2) \\ B_j=B_{1 j} \dfrac{\mathrm{d} \bar{u}}{\mathrm{d} x}+B_{2 j} \dfrac{\mathrm{d}^2 \bar{w}}{\mathrm{d} x^2}\;\;(j=1,2) \end{array}\right\}$$ (19) 其中$ {A_{kj}} $和$ {B_{kj}} $$ (k,j = 1,2) $ 是与微梁的几何尺寸、材料性质以及固有频率有关的常数.
假设上下表面为绝热, 则变温场的边界条件和界面处的连续性条件分别为
$$ {\left. {\frac{{\partial {{\bar \theta }_1}}}{{\partial z}}} \right|_{z = {z_1}}} = 0 ,\quad {\left. {\frac{{\partial {{\bar \theta }_2}}}{{\partial z}}} \right|_{z = {z_3}}} = 0 $$ (20) $$ {\left. {{{\bar \theta }_1}} \right|_{z = {z_2}}} = {\left. {{{\bar \theta }_2}} \right|_{z = {z_2}}} ,\quad {\kappa _1}{\left. {\frac{{\partial {{\bar \theta }_1}}}{{\partial z}}} \right|_{z = {z_2}}} = {\kappa _2}{\left. {\frac{{\partial {{\bar \theta }_2}}}{{\partial z}}} \right|_{z = {z_2}}} $$ (21) 将式(18)和式(19) 代入式(20)和式(21), 利用$ {\text{d}}\bar u/{\text{d}}x $和$ {{\text{d}}^2}\bar w/{\text{d}}{x^2} $的任意性可得关于式(19)中8个待定系数的代数方程组, 由此容易求得这8个常数. 然后将式(19)代入式(18), 最终可得分层的变温场解析解
$$ {\bar \theta _j} = {\bar A_j}\frac{{{\text{d}}\bar u}}{{{\text{d}}x}} + {\bar B_j}\frac{{{{\text{d}}^2}\bar w}}{{{\text{d}}{x^2}}}\;\;(j = 1,2 )$$ (22) 其中
$$\left.\begin{array}{l} \bar{A}_j=A_{1 j} \sin( p_j z)+B_{1 j} \cos (p_j z)-q_j\\ \bar{B}_j=A_{2 j} \sin (p_j z)+B_{2 j} \cos( p_j z)+q_j z \end{array}\right\}$$ (23) 进一步可将式(14)和式(22)代入式(7c)得到用位移量表示的热轴力和热弯矩的振幅
$$ {\bar N_T} = {\beta _u}\frac{{{\text{d}}\bar u}}{{{\text{d}}x}} + {\beta _w}\frac{{{{\text{d}}^{\text{2}}}\bar w}}{{{\text{d}}{x^2}}} \tag{24a}$$ $$ {\bar M_T} = {\gamma _u}\frac{{{\text{d}}\bar u}}{{{\text{d}}x}} + {\gamma _w}\frac{{{{\text{d}}^{\text{2}}}\bar w}}{{{\text{d}}{x^2}}} \tag{24b}$$ 其中
$$\left.\begin{array}{l} \beta_u=b E_1 \alpha_1 \int_{z_1}^{z_2} \bar{A}_1 \mathrm{~d} z+b E_2 \alpha_2 \int_{z_2}^{z_3} \bar{A}_2 \mathrm{~d} z \\ \beta_w=b E_1 \alpha_1 \int_{z_1}^{z_2} \bar{B}_1 \mathrm{~d} z+b E_2 \alpha_2 \int_{z_2}^{z_3} \bar{B}_2 \mathrm{~d} z \\ \gamma_u=b E_1 \alpha_1 \int_{z_1}^{z_2} z \bar{A}_1 \mathrm{~d} z+b E_2 \alpha_2 \int_{z_2}^{z_3} z \bar{A}_2 \mathrm{~d} z \\ \gamma_w=b E_1 \alpha_1 \int_{z_1}^{z_2} z \bar{B}_1 \mathrm{~d} z+b E_2 \alpha_2 \int_{z_2}^{z_3} z \bar{B}_2 \mathrm{~d} z \end{array}\right\}$$ (25) 显然, 如果要忽略热轴力, 则只需要在式(24)中令$ {\beta _u} = {\beta _w} = 0 $.
利用式(14a)则式(8)可改写为
$$ \frac{{{\text{d}}\bar u}}{{{\text{d}}x}} = {z_0}\frac{{{{\text{d}}^2}\bar w}}{{{\text{d}}{x^2}}} + \frac{1}{{{S_0}}}{\bar N_T} $$ (26) 进一步将式(24a)代入式(26)得到
$$ \frac{{{\text{d}}\bar u}}{{{\text{d}}x}} = \eta \frac{{{{\text{d}}^2}\bar w}}{{{\text{d}}{x^2}}} $$ (27) 其中$ \eta = ({S_1} + {\beta _w})/({S_0} - {\beta _u}) $. 再将式(27)代入式(24)可得
$$\left.\begin{array}{l} \bar{N}_T=\left(\eta \beta_u+\beta_w\right) \dfrac{\mathrm{d}^2 \bar{w}}{\mathrm{~d} x^2}\\ \bar{M}_T=\left(\eta \gamma_u+\gamma_w\right) \dfrac{\mathrm{d}^2 \bar{w}}{\mathrm{~d} x^2} \end{array}\right\}$$ (28) 最后, 将式(28)代入式(15), 得到只用挠度的振幅表示的结构振动方程
$$ \frac{{{{\text{d}}^4}\bar w}}{{{\text{d}}{x^4}}} - \frac{{{I_{{\text{eq}}}}{\omega ^2}}}{{(1 + \psi ){{\bar S}_2}}}\bar w = 0 $$ (29) 其中
$$ \psi = {\psi _{{N_T}}} + {\psi _{{M_T}}} $$ (30) 这里$ {\psi _N}_{_T} = - {z_0}(\eta {\beta _u} + {\beta _w})/{\bar S_2} $, $ {\psi _{{M_T}}} = (\eta {\gamma _u} + {\gamma _w})/{\bar S_2} $ 均为复频率$ \omega $的复函数, 分别反映热轴力和热弯矩引起的内耗能效应.
如果令$ \psi = 0 $, 则方程(29)退化为层合微梁无阻尼自由振动的控制方程
$$ \frac{{{{\text{d}}^4}{{\bar w}_0}}}{{{\text{d}}{x^4}}} - \frac{{{I_{{\text{eq}}}}}}{{{{\bar S}_2}}}\omega _0^2{\bar w_0} = 0 $$ (31) 其中$ {\omega _0} $和$ {\bar w_0} $分别为无阻尼(等温)微梁的固有频率和振幅. 根据梁振动理论无阻尼固有频率可表示为[12-15]
$$ {\omega _0} = \frac{{\varOmega _0^*}}{{{l^2}}}\sqrt {\frac{{{{\bar S}_2}}}{{{I_{{\text{eq}}}}}}} $$ (32) 其中$\varOmega _0^*$ 为均匀材料梁的无量纲频率参数, 它与梁的端部支承条件有关.
利用方程(29)和式(31)之间的相似性可得特征值之间的关系
$$ \omega = {\omega _0}\sqrt {1 + \psi (\omega )} $$ (33) 然而, 由式(17)、式(23) ~ 式(25)可知方程(33)是关于复频率$ \omega $的超越方程. 为了简化计算, 采用文献[2, 14-17]中的近似计算方法, 在式(33)中令$ \psi (\omega ) = \psi ({\omega _0}) $即可得到复频率$ \omega $的解析解. 然后, 利用复频率法[2, 12-19]可得用逆品质因子表示的双层微梁的热弹性组尼
$$ {Q^{ - 1}} = 2\left| {\frac{{{{\rm{Im}}} (\omega )}}{{{{\rm{Re}}} (\omega )}}} \right| $$ (34) 其中${{\rm{Re}}} (\omega )$和$ {\text{Im}}(\omega ) $分别为复频率的实部和虚部. 若在式(30)中令$ {\psi _u} = 0 $, 则式(34)退化为忽略热轴力时的TED解答[5, 7, 9].
3. 数值结果与讨论
本节通过数值实验定量分析热轴力对TED的影响. 作为数值算例, 选取由第一层氮化硅(Si3N4) 和第2层银(Ag)组成的层合微梁. 陶瓷和金属分层所占的体积分数分别为$ {H_1} = {h_1}/h $和$ {H_2} = {h_2}/h $. 在表1中列出了参考温度$ {T_0} = 300\;{\text{K}} $的条件下分层材料的物性参数. 设双层(Ag/Si3N4)微梁的支承为两端夹紧 (clamped-clamped, 或C-C). 于是微分方程(29)和式(31)的边界条件可记为
表 1 分层材料的物性参数 ($ {{\boldsymbol{T}}_{\boldsymbol{0}}} = {\boldsymbol{300}}\;{\bf{K}} $)Table 1. Material properties of the laminas (${{{T}}_{{0}}} = {{300}}\;{\rm{K}}$)Materials $E / {\text{GPa} }$ $\rho /({\text{kg} }\cdot{ {\text{m} }^{ - 3} })$ $\kappa / ({\text{W} }\cdot{ {\text{m} }^{ - 1} }\cdot{ {\text{K} }^{ - 1} })$ $\alpha /{10^{ - 6}} \;{{\text{K} }^{ - 1} }$ $ C $ Si3N4 250 3200 8 3.0 937.5 Ag 76 10500 430 18.0 286 $$\left.\begin{array}{l} \bar{w}(0)=\bar{w}^{\prime}(0)=0, \quad \bar{w}(0)=\bar{w}^{\prime}(0)=0\\ \bar{w}(l)=\bar{w}^{\prime}(l)=0, \quad \bar{w}(l)=\bar{w}^{\prime}(l)=0 \end{array}\right\}$$ (35) 其中“$ (·)'\; $”表示关于坐标x的导数. 对应上述边界条件的前3阶的无量纲固有频率参数分别为$\varOmega _0^* = 22.373, 61.670,\;\;120.90$.
为了定量地显示热轴力对层合微梁TED的影响规律, 同时也为其他研究者提供便于进行比较的数据, 首先在表2中给出了长度$ l = $300 μm、具有不同厚度($ h $)和银层体积分数($ {H_2} $)的双层微梁的TED ($ {Q^{ - 1}} $)值. 其中, 对应一个$ {H_2} $值, 前两行数据分别为忽略和考虑热轴力时的TED, 第3行数据为二者之间的相对误差. 从表2中结果可见, 总体上热轴力使得热弹性阻尼增大. 而且在两种材料的体积分数相近时, 热轴力对TED 的贡献显著增大, 忽略热轴力后导致的最大相对误差可达到16.3%. 随着金属银的体积分数的增大, 层合梁TED的最大值不断增大. 同样几何尺寸的纯银梁的TED的最大值为$1.215 \times {10^{-3}}$, 远大于纯氮化硅梁的TED最大值$1.112 \times {10^{-4}}$. 随着分层体积分数的改变, 层合梁的TED将在上述两个数值之间变化.
表 2 两端夹紧双层微梁的热弹性阻尼($ {{\boldsymbol{Q}}^{ - {\boldsymbol{1}}}} \times {{\boldsymbol{10}}^{\boldsymbol{5}}} $)随总厚度$ {\boldsymbol{h}} $和体积分数$ {{\boldsymbol{H}}_{\boldsymbol{2}}} $的变化(${\boldsymbol{ l}} = $300 μm)Table 2. TED ($ {Q^{ - 1}} \times {10^5} $) in clamped-clamped (C-C) bilayer micro beam for some specified values of h and $ {H_2} $ ($ l = $300 μm)$ {H_2} $ h/μm 2 4 6 8 10 12 14 16 18 20 0.2 6.8936 28.058 16.719 9.3470 6.1540 4.4410 3.4126 2.7479 2.2949 1.9747 7.1491$ 29.130 17.490 9.8975 6.5836 4.7884 3.7009 2.9920 2.5052 2.1583 3.71& 3.82 4.61 5.89 6.98 7.82 8.4 8.9 9.2 9.30 0.3 6.8414* 33.915 24.220 13.557 8.8933 6.4921 5.0929 4.2370 3.7024 3.3697 7.3044$ 36.243 26.070 14.799 9.8228 7.2268 5.6937 4.7408 4.1335 3.7442 6.77& 6.86 7.64 9.16 10.5 11.3 11.8 11.9 11.6 11.1 0.4 5.7672* 33.747 30.799 17.770 11.718 8.8199 7.2869 6.4861 6.1237 6.0181 6.3419$ 37.130 34.056 19.900 13.267 10.030 8.2718 7.3099 6.8261 6.6248 9.96& 10.0 10.6 12.0 13.2 13.7 13.5 12.7 11.5 10.1 0.5 4.1236* 27.473 34.242 21.511 14.586 11.555 10.337 10.051 10.267 10.668 4.6595$ 31.051 38.792 24.552 16.756 13.234 11.700 11.189 11.231 11.487 13.0& 13.0 13.3 14.1 14.9 14.5 13.2 11.3 9.3826 7.67 0.6 2.4685* 18.042 32.221 24.492 17.713 15.150 14.836 15.644 16.820 17.759 2.8490$ 20.824 37.191 28.269 20.369 17.190 16.484 17.011 17.959 18.701 15.4& 15.4 15.4 15.4 15.0 13.5 11.1 8.73 6.77 5.30 0.7 1.2081* 9.3489 23.578 25.967 21.751 20.438 21.745 24.142 26.345 27.364 1.4047$ 10.869 27.374 29.957 24.697 22.661 23.526 25.596 27.523 28.300 16.3& 16.3 16.1 15.4 13.6 10.9 8.19 6.02 4.47 3.42 0.8 0.4869* 3.8690 12.208 22.187 26.807 29.020 32.703 37.056 39.974 40.215 0.5509$ 4.3772 13.792 24.907 29.595 31.253 34.437 38.424 41.042 41.020 13.2& 13.1 13.0 12.3 10.4 7.69 5.3 3.7 2.67 2.00 注：*忽略$ {N_T} $; $考虑$ {N_T} $; &相对误差$ = 100{\text{%}} \times [{({Q^{ - 1}})^\$ } - {({Q^{ - 1}})^*}]/{({Q^{ - 1}})^*} $
Note: *$ {N_T} $is ignored; $$ {N_T} $is considered; &the relative error$ = 100{\text{%}} \times [{({Q^{ - 1}})^\$ } - {({Q^{ - 1}})^*}]/{({Q^{ - 1}})^*} $为了更加清晰地反映具有不同分层体积分数的层合微梁的TED随总厚度的变化规律, 在图2中分别绘出了给定金属银的体积分数$ {H_2} $不同值时层合微梁的$ {Q^{ - 1}} $值随厚度h连续变化的曲线. 其中红色实线和蓝色点画线分别代表忽略和考虑热轴力时的TED曲线. 相比表2, 图2更加直观地反映了层合梁的TED随厚度和分层体积分数的变化规律. 从图中可见, 当$ {H_2} < 0.5 $时, 曲线的形态与均匀(单层)微梁的$ {Q^{ - 1}} $ ~ $ h $曲线类似. 在$ {H_2} = 0.6 $, $ 0.7 $时曲线具有明显的双峰值. 与表2中的数据所反映的变化规律相同, 当分层的体积分数相近时热轴力对TED的影响显著. 另外, 从图2中可以看出曲线的峰值($ Q_{\max }^{ - 1} $)对应的厚度(称临界厚度$ {h_{{\text{cr}}}} $)随着金属银层的体积分数的增加而增大. 为便于研究者进行数据比较, 在表3中列出了图2中各曲线(蓝色点画线)的热弹性阻尼最大值和相应的临界厚度. 由此可见, 随着银层体积分数的增大临界厚度单调增加.
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图 2 具有不同体积分数$ {H_2} $的双层微梁的TED ($ {Q^{ - 1}} \times {10^4} $)随固有频率$ {\omega _0} $的变化曲线 ($ l = $300 μm, 一阶模态)Figure 2. Continuously variation of the TED ($ {Q^{ - 1}} \times {10^4} $) with the frequency $ {\omega _0} $ of a bi-layered micro beam for some specified values of$ {H_2} $ ($ l = $300 μm, 1st mode)表 3 对应于不同体积分数$ {{\boldsymbol{H}}_{\boldsymbol{2}}} $的TED最大值$ {\boldsymbol{Q}}_{{\bf{max}} }^{ - {\boldsymbol{1}}} \times {{\boldsymbol{10}}^{\boldsymbol{5}}} $和临界厚度$ {{\boldsymbol{h}}_{{\bf{cr}}}} $ ($ {\boldsymbol{l }}=$300 μm)Table 3. The maximum, $ Q_{\max }^{ - 1}\times {{{10}}^{{5}}} $ and the critical thickness,$ {h_{{\text{cr}}}} $for different values of $ {H_2} $ ($ l = $300 μm)$ {H_2} $ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 $ Q_{\max }^{ - 1} $ 11.12 19.33 28.06 34.91 38.04 36.80 32.22 31.06 40.46 62.77 12.54 $ {h_{{\text{cr}}}}/$μm 3.47 3.74 4.02 4.34 4.74 5.25 5.98 7.21 19.06 18.34 19.43 图3进一步展示了给定总厚度的双层微梁的TED随体积分数$ {H_2} $连续变化的特性曲线. 由此可见, 大约在$ 0.3 < {H_2} < 0.8 $的区间, 热轴力对热弹性阻尼的影响变得显著. 图4给出了表2中定义的相对误差随金属银的体积分数连续变化的曲线. 清晰地显示了最大误差对应的分层体积分数值($ {H_2} $).
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图 3 具有不同厚度的双层微板的TED ($ {Q^{ - 1}} $)随银层的体积分数($ {H_2} $)连续变化的特性曲线($ l = $300 μm)Figure 3. Curves of the TED ($ {Q^{ - 1}} $) in bilayer micro beam varying continuously with the volume fraction of the silver layer ($ {H_2} $) ($ l = $300 μm)![]()
图 4 考虑和忽略热轴力时的热弹性阻尼之间的相对误差随$ {H_2} $的变化曲线($l = 300{\text{ μm}}$)Figure 4. Curves of the relative error between the TEDs with considering and neglecting the thermal axial force versus$ {H_2} $($l = 300{\text{ μm}}$)由式(30)可知, 热轴力产生的热弹性阻尼取决于复函数$ {\psi _{{N_T}}}({\omega _0}) $的虚部. 图5绘出了具有不同厚度的${{\rm{Im}}} ({\psi _{{N_T}}})$随体积分数$ {H_2} $的连续变化曲线. 结果再次表明当两个分层的体积分数相近时阻尼效应显著.
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图 5 热轴力产生的阻尼函数${{\rm{Im}}} ({\psi _{{N_T}}})$随银层体积分数(H2)以及厚度(h)的变化Figure 5. Variation of damping function ${{\rm{Im}}} ({\psi _{{N_T}}})$ produced by thermal axial force with H2 and h热轴力是由物理中面与几何中面的偏离引起的. 最后, 图6绘出了物理中面的位置($ {\zeta _0} = {z_0}/h $)与分层体积分数$ {H_2} $的关系曲线. 显然, 在两个分层的体积分数相近的区间物理中面偏离几何中面显著, 热弹性阻尼相对误差的最大值正是出现在该区间. 这是因为在此种情况下层合梁的材料性质在横向的非均匀程度最强, 从而拉−弯耦合变形最显著, 由此产生的热轴力也显著.
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图 6 物理中面的位置($ {\zeta _0} = {z_0}/h $) 随银层体积分数($ {H_2} $)的变化Figure 6. Position of the physical neutral surface ($ {\zeta _0} = {z_0}/h $) changing with the volume fraction of the silver layer ($ {H_2} $)4. 结论
首次定量地分析了热轴力对双层微梁热弹性阻尼的影响程度和规律. 基于Euler-Bernoulli 梁理论和准一维热传导理论, 建立了包含了热轴力的双层微梁热−弹耦合自由振动微分方程. 给出了系统振动的复频率以及用逆品质因子表示的热弹性阻尼解析解. 以银(Ag)和氮化硅(Si3N4)分层组成的双层微梁为例, 分别计算了考虑和忽略热轴力后的热弹性阻尼以及二者之间的相对误差, 详细地定量分析了热弹性阻尼随分层体积分数和梁的总厚度的变化以及热轴力对热弹性阻尼的影响规律. 本文得到主要结论如下.
(1) 忽略热轴力将会低估双层微梁的热弹性阻尼. 随着两分层体积分数接近, 热轴力对热弹性阻尼的影响变得显著. 在金属银的体积分数为70% (氮化硅的体积分数为30%)时, 如果忽略热轴力, 则对热弹性阻尼低估比例将达到16.3%.
(2) 由于热轴力是由物理中面偏离几何中面引起的, 在此种情况下层合梁的材料性质在横向的非均匀程度最强, 从而拉−弯耦合变形最显著, 由此产生的热轴力也最大. 因此, 在物理中面的位置偏离几何中面显著时, 忽略热轴力将会严重低估双层微梁的热弹性阻尼.
(3) 当Si3N4/Ag双层微梁的分层体积分数相近时, TED随总厚度的变化曲线存在双峰值.
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图 2 具有不同体积分数$ {H_2} $的双层微梁的TED ($ {Q^{ - 1}} \times {10^4} $)随固有频率$ {\omega _0} $的变化曲线 ($ l = $300 μm, 一阶模态)
Figure 2. Continuously variation of the TED ($ {Q^{ - 1}} \times {10^4} $) with the frequency $ {\omega _0} $ of a bi-layered micro beam for some specified values of$ {H_2} $ ($ l = $300 μm, 1st mode)
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图 3 具有不同厚度的双层微板的TED ($ {Q^{ - 1}} $)随银层的体积分数($ {H_2} $)连续变化的特性曲线($ l = $300 μm)
Figure 3. Curves of the TED ($ {Q^{ - 1}} $) in bilayer micro beam varying continuously with the volume fraction of the silver layer ($ {H_2} $) ($ l = $300 μm)
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图 4 考虑和忽略热轴力时的热弹性阻尼之间的相对误差随$ {H_2} $的变化曲线($l = 300{\text{ μm}}$)
Figure 4. Curves of the relative error between the TEDs with considering and neglecting the thermal axial force versus$ {H_2} $($l = 300{\text{ μm}}$)
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图 5 热轴力产生的阻尼函数${{\rm{Im}}} ({\psi _{{N_T}}})$随银层体积分数(H2)以及厚度(h)的变化
Figure 5. Variation of damping function ${{\rm{Im}}} ({\psi _{{N_T}}})$ produced by thermal axial force with H2 and h
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图 6 物理中面的位置($ {\zeta _0} = {z_0}/h $) 随银层体积分数($ {H_2} $)的变化
Figure 6. Position of the physical neutral surface ($ {\zeta _0} = {z_0}/h $) changing with the volume fraction of the silver layer ($ {H_2} $)
表 1 分层材料的物性参数 ($ {{\boldsymbol{T}}_{\boldsymbol{0}}} = {\boldsymbol{300}}\;{\bf{K}} $)
Table 1 Material properties of the laminas (${{{T}}_{{0}}} = {{300}}\;{\rm{K}}$)
Materials $E / {\text{GPa} }$ $\rho /({\text{kg} }\cdot{ {\text{m} }^{ - 3} })$ $\kappa / ({\text{W} }\cdot{ {\text{m} }^{ - 1} }\cdot{ {\text{K} }^{ - 1} })$ $\alpha /{10^{ - 6}} \;{{\text{K} }^{ - 1} }$ $ C $ Si3N4 250 3200 8 3.0 937.5 Ag 76 10500 430 18.0 286 
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表 2 两端夹紧双层微梁的热弹性阻尼($ {{\boldsymbol{Q}}^{ - {\boldsymbol{1}}}} \times {{\boldsymbol{10}}^{\boldsymbol{5}}} $)随总厚度$ {\boldsymbol{h}} $和体积分数$ {{\boldsymbol{H}}_{\boldsymbol{2}}} $的变化(${\boldsymbol{ l}} = $300 μm)
Table 2 TED ($ {Q^{ - 1}} \times {10^5} $) in clamped-clamped (C-C) bilayer micro beam for some specified values of h and $ {H_2} $ ($ l = $300 μm)
$ {H_2} $ h/μm 2 4 6 8 10 12 14 16 18 20 0.2 6.8936 28.058 16.719 9.3470 6.1540 4.4410 3.4126 2.7479 2.2949 1.9747 7.1491$ 29.130 17.490 9.8975 6.5836 4.7884 3.7009 2.9920 2.5052 2.1583 3.71& 3.82 4.61 5.89 6.98 7.82 8.4 8.9 9.2 9.30 0.3 6.8414* 33.915 24.220 13.557 8.8933 6.4921 5.0929 4.2370 3.7024 3.3697 7.3044$ 36.243 26.070 14.799 9.8228 7.2268 5.6937 4.7408 4.1335 3.7442 6.77& 6.86 7.64 9.16 10.5 11.3 11.8 11.9 11.6 11.1 0.4 5.7672* 33.747 30.799 17.770 11.718 8.8199 7.2869 6.4861 6.1237 6.0181 6.3419$ 37.130 34.056 19.900 13.267 10.030 8.2718 7.3099 6.8261 6.6248 9.96& 10.0 10.6 12.0 13.2 13.7 13.5 12.7 11.5 10.1 0.5 4.1236* 27.473 34.242 21.511 14.586 11.555 10.337 10.051 10.267 10.668 4.6595$ 31.051 38.792 24.552 16.756 13.234 11.700 11.189 11.231 11.487 13.0& 13.0 13.3 14.1 14.9 14.5 13.2 11.3 9.3826 7.67 0.6 2.4685* 18.042 32.221 24.492 17.713 15.150 14.836 15.644 16.820 17.759 2.8490$ 20.824 37.191 28.269 20.369 17.190 16.484 17.011 17.959 18.701 15.4& 15.4 15.4 15.4 15.0 13.5 11.1 8.73 6.77 5.30 0.7 1.2081* 9.3489 23.578 25.967 21.751 20.438 21.745 24.142 26.345 27.364 1.4047$ 10.869 27.374 29.957 24.697 22.661 23.526 25.596 27.523 28.300 16.3& 16.3 16.1 15.4 13.6 10.9 8.19 6.02 4.47 3.42 0.8 0.4869* 3.8690 12.208 22.187 26.807 29.020 32.703 37.056 39.974 40.215 0.5509$ 4.3772 13.792 24.907 29.595 31.253 34.437 38.424 41.042 41.020 13.2& 13.1 13.0 12.3 10.4 7.69 5.3 3.7 2.67 2.00 注：*忽略$ {N_T} $; $考虑$ {N_T} $; &相对误差$ = 100{\text{%}} \times [{({Q^{ - 1}})^\$ } - {({Q^{ - 1}})^*}]/{({Q^{ - 1}})^*} $
Note: *$ {N_T} $is ignored; $$ {N_T} $is considered; &the relative error$ = 100{\text{%}} \times [{({Q^{ - 1}})^\$ } - {({Q^{ - 1}})^*}]/{({Q^{ - 1}})^*} $
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表 3 对应于不同体积分数$ {{\boldsymbol{H}}_{\boldsymbol{2}}} $的TED最大值$ {\boldsymbol{Q}}_{{\bf{max}} }^{ - {\boldsymbol{1}}} \times {{\boldsymbol{10}}^{\boldsymbol{5}}} $和临界厚度$ {{\boldsymbol{h}}_{{\bf{cr}}}} $ ($ {\boldsymbol{l }}=$300 μm)
Table 3 The maximum, $ Q_{\max }^{ - 1}\times {{{10}}^{{5}}} $ and the critical thickness,$ {h_{{\text{cr}}}} $for different values of $ {H_2} $ ($ l = $300 μm)
$ {H_2} $ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 $ Q_{\max }^{ - 1} $ 11.12 19.33 28.06 34.91 38.04 36.80 32.22 31.06 40.46 62.77 12.54 $ {h_{{\text{cr}}}}/$μm 3.47 3.74 4.02 4.34 4.74 5.25 5.98 7.21 19.06 18.34 19.43
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