李杨^*,?, 秦庆华^?, 张亮亮^*, 高阳^,*,2)*中国农业大学理学院, 北京 100083
^?澳大利亚国立大学工程与计算机科学学院, 澳大利亚堪培拉 2601

EXACT THERMO-ELECTRO-ELASTIC SOLUTION OF FUNCTIONALLY GRADED MULTILAYERED ONE-DIMENSIONAL QUASICRYSTAL CYLINDRICAL SHELLS ¹⁾

Li Yang^*,?, Qin Qinghua^?, Zhang Liangliang^*, Gao Yang^,*,2)*College of Science, China Agricultural University, Beijing 100083, China
^?College of Engineering and Computer Science, Australian National University, Canberra 2601, Australia

通讯作者: 2)高阳, 教授, 主要研究方向: 智能材料与结构力学. E-mail:ygao@cau.edu.cn

收稿日期:2020-04-18接受日期:2020-04-18网络出版日期:2020-09-18

基金资助:

1)国家自然科学基金.11972365 国家自然科学基金.51704015 辽宁省教育厅科学研究经费.L2019006

Received:2020-04-18Accepted:2020-04-18Online:2020-09-18
作者简介 About authors


摘要
两种或多种不同性质材料组成的层状结构可以满足工业发展的需求. 然而, 材料属性在接触面的突变问题, 容易导致层间界面处产生应力集中、裂纹以及分层等问题. 功能梯度材料利用连续变化的组分梯度来代替突变界面, 可以消除界面处的物理性能突变, 提高结构的粘结强度. 本文以一维准晶功能梯度层合圆柱壳为研究对象, 利用类Stroh公式和传递矩阵方法, 建立了材料参数沿径向呈现幂函数变化的层合圆柱壳模型, 获得了简支边界条件对应的一维准晶功能梯度层合圆柱壳的热电弹性精确解. 数值算例中讨论了层合圆柱壳内外表面承受温度载荷时, 功能梯度指数因子对温度场、电场、声子场和相位子场的影响, 尤其是对层合圆柱壳内外表面的影响. 结果表明, 指数因子改变了材料参数的空间分布情况, 进而对温度场、电场、声子场和相位子场都有影响; 增加功能梯度指数因子, 可减小温度载荷引起的内表面变形, 进而提升结构强度. 本文得到的结果可以为功能梯度准晶层合圆柱壳的设计和制造提供可靠的理论依据.
关键词： 功能梯度材料;准晶;层合圆柱壳;热电弹耦合;精确解

Abstract
Layered structures made of two and more materials with different properties can meet the needs of industrial development. However, the abrupt change of material properties at the interface of the laminated structures can easily cause some interface problems, such as stress concentration, interface cracks, and interface delamination phenomena. Functionally graded materials refer to utilize a continuously changing component gradient instead of the original sudden change interface, which can eliminate or weaken the abrupt change of the physical properties and then increase the bonding strength of the layered structures. In this paper, the research object is the functionally graded multilayered one-dimensional quasicrystal cylindrical shells. By virtue of the pseudo-Stroh formalism and the propagator matrix method, we establish the layered one-dimensional quasicrystal cylindrical shells model with the material parameters following the power-law type distribution along its radius direction, and obtain the exact thermo-electro-elastic solution of the functionally graded layered one-dimensional quasicrystal cylindrical shells with simply supported boundary condition. Numerical examples are carried out to investigate the influences of the exponential factor on temperature, electric, phason and phonon fields of the functionally graded layered one-dimensional quasicrystal cylindrical shells subjected to both inner and outer surfaces temperature variations, especially the effects on physical quantities at the inner and outer surfaces of the layered one-dimensional quasicrystal cylindrical shells. The obtained results indicate that: the exponential factor can change the distribution characteristic of material parameters, which can cause a significant influence on the physical quantities in the temperature, electric, phason, and phonon fields; by increasing the exponential factor, the deformation at the internal surface induced by temperature stimuli is reduced and the strength of the layered one-dimensional quasicrystal cylindrical shells is improved. The results obtained in this paper can provide a reliable theoretical basis for the design and manufacture of functionally graded layered one-dimensional quasicrystal cylindrical shells.
Keywords：functionally graded materials;quasicrystals;layered cylindrical shells;thermo-electro-elastic coupling effect;exact solution


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本文引用格式
李杨, 秦庆华, 张亮亮, 高阳. 一维准晶功能梯度层合圆柱壳热电弹性精确解 ¹⁾. 力学学报[J], 2020, 52(5): 1286-1294 DOI:10.6052/0459-1879-20-126
Li Yang, Qin Qinghua, Zhang Liangliang, Gao Yang. EXACT THERMO-ELECTRO-ELASTIC SOLUTION OF FUNCTIONALLY GRADED MULTILAYERED ONE-DIMENSIONAL QUASICRYSTAL CYLINDRICAL SHELLS ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(5): 1286-1294 DOI:10.6052/0459-1879-20-126

引言

层状结构因其优异的性能被广泛应用于航空、航天、土木和机械等领域^[1]. 常规的层状结构是由均质材料层组合而成, 以期获得更好的机械和热力学性能. 但是, 这种层状结构界面处材料参数的突然变化会产生较大的层间应力, 从而导致应力集中、裂纹和分层等问题. 为了克服这些不利影响, 科学家们提出功能梯度材料的概念, 即利用连续变化的组分梯度来代替突变界面, 进而消除界面处的物理性能突变, 达到提高结构强度和优化结构性能的目的^[2-3]. 鉴于功能梯度材料具有组成结构连续变化和可设计性的特点, 越来越多的科研人员对功能梯度材料表现出极大的兴趣. 仲政等^[2]对功能梯度材料与结构若干力学问题的最新研究进展进行了综述, 并对非均匀介质力学研究进行了展望. 柯燎亮和汪越胜^[3]结合功能梯度材料接触力学的若干基本问题, 综述了相关理论的研究进展. 郑保敬等^[4]提出一种模型降阶方法用于分析非均质材料结构在复杂载荷作用下的动态响应. 杨健鹏和王惠明^[5]研究了功能梯度球形水凝胶在一定条件下的非均匀大变形溶胀行为. Yang等^[6]利用复变函数方法研究了含有功能梯度层加固椭圆孔的无限大板的应力集中问题, 并获得了其通解. 基于广义England方法, Yang等^[7]研究了功能梯度圆板在中心承受集中力作用下的轴对称弯曲问题. Pan和Han^[8]提出类Stroh理论用于研究功能梯度磁电弹层合板的弯曲问题, 获得了材料常数沿厚度方向呈现指数函数分布的精确解.

准晶由以色列科学家Shechtman教授首次在急冷的Al-Mn 合金电子衍射图形中发现^[9]. 准晶是一种不同于晶体和非晶体的新型固体材料, 具有长程有序的原子排列, 但不具备平移对称性^[10]. 基于Landau的元激发唯象理论, 准晶中存在两个低能元激发: 声子和相位子^[11-12]. 相位子场的引入, 使准晶表现出不同于晶体和非晶体的多种性能, 如: 高强度、高硬度、耐磨性、耐腐蚀、低摩擦系数和低导热率等^[13-14]. 由于准晶表现出来的优异性能, 其主要用作表面涂层、薄膜以及复合材料增强相^[15]. 随着准晶应用的推广, 准晶材料的性能研究受到许多****的关注. Maugin^[16]将提出的准晶弹性方程扩展到准晶热弹性问题中. Li等^[17]利用广义势能理论, 研究了二维六方准晶中的三维热弹性平面裂纹问题, 并获得了解析解. Guo等^[18]基于Stroh理论, 研究了二维十次热弹性准晶的缺陷问题. Li等^[19-20]分别给出了考虑热效应的一维六方压电准晶中平面裂纹问题的理论解和数值解.

层合圆柱壳具有重量轻、强度高等特点, 是工程中常用的结构元件之一. 目前, 关于准晶层状结构的动态和静态问题研究多数是基于矩形层合板开展的^[21-23]. 此外, 同时考虑热电弹耦合效应和材料不均匀性的准晶层合圆柱壳的研究开展的很少. 因此, 本文采用类Stroh理论和传递矩阵方法, 获得了一维准晶功能梯度层合圆柱壳的热电弹性精确解, 讨论了功能梯度指数因子对层合圆柱壳物理量的影响, 以期为准晶功能梯度层合圆柱壳的多场耦合效应及非均匀性分析提供可靠的参考依据.

1 基本方程

图1是柱坐标系($r$, $\theta$, $z)$中一维准晶功能梯度层合圆柱壳的示意图, 准周期方向和极化方向均沿着$r $轴, 其外表面半径为$r_{b}$, 内表面半径为$r_{a}$, 沿$z$向长度远远大于层合圆柱壳半径, 层合圆柱壳的角度跨度为$\theta_{0}$, 第$j$层的内外半径分别为$r_{j}$和$r_{j+1}$.

图1

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图1一维准晶功能梯度层合圆柱壳的示意图

Fig.1Schematic of functionally graded layered 1Dquasicrystal cylindrical shells



对于考虑热-电-弹耦合的一维正交准晶材料, 本构方程为^[24]

(1) $ \begin{eqnarray} \label{eq1} \left. \begin{array}{ll} \sigma _{rr} =C_{33} \varepsilon _{rr} +C_{13} \varepsilon _{\theta \theta } +C_{23} \varepsilon _{zz} +R_3 w_{rr} -\\ \qquad e_{33} E_r -\beta _3 T \\ \sigma _{\theta \theta } =C_{13} \varepsilon _{rr} +C_{11} \varepsilon _{\theta \theta } +C_{12} \varepsilon _{zz} +R_1 w_{rr} -\\ \qquad e_{31} E_r -\beta _1 T \\ \sigma _{zz} =C_{23} \varepsilon _{rr} +C_{12} \varepsilon _{\theta \theta } +C_{22} \varepsilon _{zz} +R_2 w_{rr} -\\ \qquad e_{32} E_r -\beta _1 T \\ \sigma _{\theta z} =\sigma _{z\theta } =2C_{66} \varepsilon _{\theta z} \\ \sigma _{rz} =\sigma _{zr} =2C_{44} \varepsilon _{zr} +R_5 w_{rz} -e_{24} E_z \\ \sigma _{r\theta } =\sigma _{\theta r} =2C_{55} \varepsilon _{r\theta } +R_6 w_{r\theta } -e_{15} E_\theta \\ H_{rr} =R_3 \varepsilon _{rr} +R_1 \varepsilon _{\theta \theta } +R_2 \varepsilon _{zz} +K_3 w_{rr} -d_{33} E_r \\ H_{rz} =2R_5 \varepsilon _{zr} +K_2 w_{rz} -d_{24} E_z \\ H_{r\theta } =2R_6 \varepsilon _{r\theta } +K_1 w_{r\theta } -d_{15} E_\theta \\ D_r =e_{33} \varepsilon _{rr} +e_{31} \varepsilon _{\theta \theta } +e_{32} \varepsilon _{zz} +d_{33} w_{rr} +\\ \qquad \xi _{33} E_r +p_3 T \\ D_\theta =2e_{15} \varepsilon _{r\theta } +d_{15} w_{r\theta } +\xi _{11} E_\theta \\ D_z =2e_{24} \varepsilon _{zr} +d_{24} w_{rz} +\xi _{11} E_z \\ \end{array} \right\} \end{eqnarray}$
式中, $\sigma _{kl} $为声子场应力; $\varepsilon _{kl} $为声子场应变; $H_{il} $为相位子场应力; $w_{il} $为相位子场应变; $D_k $为电位移; $E_k $为电场强度; $T$为温度变化; $C_{kl} $, $R_{kl} $, $K_{kl} $分别为声子场弹性常数, 声子场-相位子场耦合弹性常数和相位子场弹性常数; $e_{kl} $, $d_{kl} $, $\xi _{ll} $分别为声子场压电系数, 相位子场压电系数和介电系数; $p_3 $和$\beta _k $分别为热电系数和导热系数.

忽略体力、自由电荷以及内部热源, 静态平衡方程可写为

(2) $ \begin{eqnarray} \label{eq2} \left. \begin{array}{ll} \sigma _{rr,r} +{\sigma _{r\theta ,\theta } } / r+\sigma _{rz,z} +{\left( {\sigma _{rr} -\sigma _{\theta \theta } } \right)} / r=0 \\ \sigma _{r\theta ,r} +{\sigma _{\theta \theta ,\theta } } / r+\sigma _{\theta z,z} +{2\sigma _{r\theta } } / r=0 \\ \sigma _{rz,r} +{\sigma _{\theta z,\theta } } / r+\sigma _{zz,z} +{\sigma _{rz} } / r=0 \\ H_{rr,r} +{H_{r\theta ,\theta } } / r+H_{rz,z} +{H_{rr} } / r=0 \\ D_{r,r} +{D_{\theta ,\theta } } / r+D_{z,z} +{D_r } / r=0 \\ q_{r,r} +{q_{\theta ,\theta } } / r+q_{z,z} +{q_r } / r=0 \\ \end{array} \right\} \end{eqnarray}$
式中, $q_r $, $q_\theta $, $q_z $为热流. 热流与温度的关系, 可以表示为^[25]

(3) $ \begin{eqnarray} \label{eq3} q_r =-k_{11} T_{,r},\ \ q_\theta =-k_{22} T_{,\theta} / r,\ \ q_z =-k_{33} T_{,z} \end{eqnarray}$
式中, $k_{ll} $为热传导系数. 几何方程为

(4) $ \begin{eqnarray} \label{eq4} \left. \begin{array}{ll} \varepsilon _{rr} =u_{r,r} , \ \ \varepsilon _{\theta \theta } ={\left( {u_{\theta,\theta } +u_r } \right)} / r, \ \ \varepsilon _{zz} =u_{z,z}\\ \varepsilon _{\theta z} =\varepsilon _{z\theta } =0.5\left( {u_{\theta ,z} +{u_{z,\theta } } / r} \right) \\ \varepsilon _{rz} =\varepsilon _{zr} =0.5\left( {u_{z,r} +u_{r,z} } \right)\\ \varepsilon _{\theta r} =\varepsilon _{r\theta } =0.5\left( {{u_{r,\theta }} / r+u_{\theta ,r} -{u_\theta } / r} \right) \\ w_{rr} =w_{r,r} , \ \ w_{r\theta } =w_{r,\theta}/r, \ \ w_{rz} =w_{r,z} \\ E_r =-\phi _{,r}, \ \ E_\theta =-\phi _{,\theta } / r, \ \ E_z =-\phi _{,z} \\ \end{array} \right\} \end{eqnarray}$
式中, $u_r $, $u_\theta $, $u_z $为声子场位移; $w_r $为相位子场位移; $\phi$为电势. 考虑层合圆柱壳两侧边简支的边界条件^[24], 即

(5) $ \begin{eqnarray} \label{eq5} \theta =0,\ \ \theta _0;\ \ \ u_r =w_r =\phi =T=0 \end{eqnarray}$
对于一维准晶功能梯度材料, 假设其材料参数沿着径向呈现幂函数分布, 即

(6) $ \begin{eqnarray} \label{eq6} \left. \begin{array}{ll} C_{kl} (r)=C_{kl}^0 (r/r_j )^\alpha , \ \ R_{kl} (r)=R_{kl}^0 (r/r_j )^\alpha \\ K_{kl} (r)=K_{kl}^0 (r/r_j )^\alpha , \ \ e_{kl} (r)=e_{kl}^0 (r/r_j )^\alpha \\ d_{kl} (r)=d_{kl}^0 (r/r_j )^\alpha , \ \ \xi _{ll} (r)=\xi _{ll}^0 (r/r_j )^\alpha \ \\ \beta _k (r)=\beta _k^0 (r/r_j )^\alpha , \ \ k_{ll} (r)=k_{ll}^0 (r/r_j )^\alpha \\ p_3 (r)=p_3^0 (r/r_j )^\alpha \\ \end{array} \right\} \end{eqnarray}$
式中, $\alpha $为功能梯度指数因子, 表示材料参数在半径$r$方向的梯度分布程度, 后文中$\alpha $均表示功能梯度指数因子; 上标带"0"的物理量表示均质材料对应的材料参数.

2 单层圆柱壳的热电弹性精确解

2.1 温度场的精确解

满足两侧边简支边界条件的温度函数可假设为

(7) $ \begin{eqnarray} \label{eq7} T=\bar{{T}}\left( r \right)\sin \left( {p\theta } \right)=f\rho ^\eta \sin \left( {p\theta } \right) \end{eqnarray}$
式中, $\rho = r/r_j $, $p = m\pi/\theta _0 $, $\bar{{T}}\left( r \right)$为与圆柱壳角度$\theta $无关的温度, 文中其他顶部带横线的物理量均表示独立于圆柱壳角度的物理量.

将式(7)代入式(3), 然后代入式(2), 得

(8) $ \begin{eqnarray} \label{eq8} \left( {k_{11}^0 \eta ^2+k_{11}^0 \alpha \eta -k_{22}^0 p^2} \right)f=0 \end{eqnarray}$
式中, 特征值$\eta $可求解为

(9) $ \begin{eqnarray} \label{eq9} \eta _1 =\frac{-\alpha +\sqrt {\alpha ^2+4k_{22}^0 p^2/k_{11}^0 }}{2}, \ \ \eta _2 =-\eta _1 -\alpha \end{eqnarray}$
将式(8)所示特征关系改写成

(10) $ \begin{eqnarray} \label{eq10} \left[ {{\begin{array}{cc} 0 & {{-1} / {k_{11}^0 }} \\ {-p^2k_{22}^0 } & {-\alpha } \\ \end{array} }} \right]\left[ {{\begin{array}{c} f \\ g \\ \end{array} }} \right]=\eta \left[ {{\begin{array}{c} f \\ g \\ \end{array} }} \right] \end{eqnarray}$
式中, $\left[ {f,g} \right]^{\rm T}$为特征向量. 为了令特征向量和特征值对应, 对特征向量加下标, 即$\eta _1 $对应$\left[ {f_1 ,g_1 } \right]^{\rm T}$, $\eta _2 $对应$\left[ {f_2 ,g_2 } \right]^{\rm T}$, 式中上标"T"表示矩阵的转置.

利用特征值和特征向量, 可得到一维准晶功能梯度单层圆柱壳温度场的精确解

(11) $ \begin{eqnarray} \label{eq11} \left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r\bar{{q}}_r } \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} 1 & 0 \\ 0 & {\rho ^\alpha } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\rho ^{\eta _1 }} & 0 \\ 0 & {\rho ^{\eta _2 }} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\chi _1 } \\ {\chi _2 } \\ \end{array} }} \right] \end{eqnarray}$
式中, $\chi _1 $和$\chi _2 $为待定未知量.

2.2 电弹耦合场的精确解

对于简支层合圆柱壳, 广义位移矢量可表示为

(12) $ \begin{eqnarray} \label{eq12} && { u}=\left[ {{\begin{array}{*{20}c} {u_r } \\ {u_\theta } \\ {u_z } \\ {w_r } \\ \phi \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} {\bar{{u}}_r \sin \left( {p\theta } \right)} \\ {\bar{{u}}_\theta \cos \left( {p\theta } \right)} \\ {\bar{{u}}_z \cos \left( {p\theta } \right)} \\ {\bar{{w}}_r \sin \left( {p\theta } \right)} \\ {\bar{{\phi }}\sin \left( {p\theta } \right)} \\ \end{array} }} \right]= \rho ^s\left[ {{\begin{array}{*{20}c} {a_1 \sin \left( {p\theta } \right)} \\ {a_2 \cos \left( {p\theta } \right)} \\ {a_3 \cos \left( {p\theta } \right)} \\ {a_4 \sin \left( {p\theta } \right)} \\ {a_5 \sin \left( {p\theta } \right)} \\ \end{array} }} \right]+ \\ &&\qquad f\rho ^{1+\eta }\left[ {{\begin{array}{*{20}c} {c_1 \sin \left( {p\theta } \right)} \\ {c_2 \cos \left( {p\theta } \right)} \\ {c_3 \cos \left( {p\theta } \right)} \\ {c_4 \sin \left( {p\theta } \right)} \\ {c_5 \sin \left( {p\theta } \right)} \\ \end{array} }} \right] \end{eqnarray}$
将式(12)代入式(4), 然后代入式(1), 得到径向的广义应力矢量

(13) $ \begin{eqnarray} \label{eq13} && r{ t}=r\left[ {{\begin{array}{*{20}c} {\sigma _{rr} } \\ {\sigma _{\theta r} } \\ {\sigma _{zr} } \\ {H_{rr} } \\ {D_r } \\ \end{array} }} \right]=r\left[ {{\begin{array}{*{20}c} {\bar{{\sigma }}_{rr} \sin \left( {p\theta } \right)} \\ {\bar{{\sigma }}_{\theta r} \cos \left( {p\theta } \right)} \\ {\bar{{\sigma }}_{zr} \cos \left( {p\theta } \right)} \\ {\bar{{H}}_{rr} \sin \left( {p\theta } \right)} \\ {\bar{{D}}_r \sin \left( {p\theta } \right)} \\ \end{array} }} \right] =\rho ^{s+\alpha }\left[ {{\begin{array}{*{20}c} {b_1 \sin \left( {p\theta } \right)} \\ {b_2 \cos \left( {p\theta } \right)} \\ {b_3 \cos \left( {p\theta } \right)} \\ {b_4 \sin \left( {p\theta } \right)} \\ {b_5 \sin \left( {p\theta } \right)} \\ \end{array} }} \right]+ \\ &&\qquad f\rho ^{1+\eta +\alpha }\left[ {{\begin{array}{*{20}c} {d_1 \sin \left( {p\theta } \right)} \\ {d_2 \cos \left( {p\theta } \right)} \\ {d_3 \cos \left( {p\theta } \right)} \\ {d_4 \sin \left( {p\theta } \right)} \\ {d_5 \sin \left( {p\theta } \right)} \\ \end{array} }} \right] \end{eqnarray}$
引入4个矢量

(14) $ \begin{eqnarray} \label{eq14} \left. \begin{array}{ll} { a}=\left[ {a_1 ,a_2 ,a_3 ,a_4 ,a_5 } \right]^{\rm T}, { b}=\left[ {b_1 ,b_2 ,b_3 ,b_4 ,b_5 } \right]^{\rm T} \\ { c}=\left[ {c_1 ,c_2 ,c_3 ,c_4 ,c_5 } \right]^{\rm T}, { d}=\left[ {d_1 ,d_2 ,d_3 ,d_4 ,d_5 } \right]^{\rm T} \\ \end{array} \right\} \end{eqnarray}$
式(14)所示矢量存在如下关系

(15) $ \begin{eqnarray} \label{eq15} \left. \begin{array}{ll} { b}=\left( {-{ P}^{\rm T}+s{ T}} \right){ a} \\ { d} \ \ =\left[ {-{ P}^{\rm T}+\left( {1+\eta } \right){ T}} \right]{ c}-{ \beta }_2 \\ \end{array} \right\} \end{eqnarray}$
利用基本方程和式(12), 推导出满足一维热-电-弹性准晶功能梯度层合圆柱壳的类Stroh公式^[26]

(16) $ \begin{eqnarray} \label{eq16} \left. \begin{array}{ll} \left[ {{ Q}-\alpha { P}^T+s\left( {{ P}-{ P}^T+\alpha { T}} \right)+s^2{ T}} \right]{ a}={\bf 0} \\ \left[{ Q}-\alpha { P}^T+\left( {1+\eta } \right)\left( {{ P}-{ P}^T+\alpha { T}} \right)+ \\ \qquad \left( {1+\eta } \right)^2{ T} \right]{ c}={ \beta }_1 +\left( {1+\eta +\alpha } \right){ \beta }_2 \\ \end{array} \right\} \end{eqnarray}$
式(15)和式(16)中矩阵分别为

(17) $ \begin{eqnarray} \label{eq17} \left. \begin{array}{ll} { Q}=\left[ {{\begin{array}{c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c} {-\left( {C_{55}^0 p^2+C_{11}^0 } \right)} & {\left( {C_{55}^0 +C_{11}^0 } \right)p} & 0 & {-R_6^0 p^2} & {-e_{15}^0 p^2} \\ {\left( {C_{55}^0 +C_{11}^0 } \right)p} & {-\left( {C_{11}^0 p^2+C_{55}^0 } \right)} & 0 & {R_6^0 p} & {e_{15}^0 p} \\ 0 & 0 & {-C_{66}^0 p^2} & 0 & 0 \\ {-R_6^0 p^2} & {R_6^0 p} & 0 & {-K_1^0 p^2} & {-d_{15}^0 p^2} \\ {-e_{15}^0 p^2} & {e_{15}^0 p} & 0 & {-d_{15}^0 p^2} & {\xi _{11}^0 p^2} \\ \end{array} }} \right], \ \ { P}=\left[ {{\begin{array}{c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c} {-C_{13}^0 } & {-C_{55}^0 p} & 0 & {-R_1^0 } & {-e_{31}^0 } \\ {C_{13}^0 p} & {C_{55}^0 } & 0 & {R_1^0 p} & {e_{31}^0 p} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & {-R_6^0 p} & 0 & 0 & 0 \\ 0 & {-e_{15}^0 p} & 0 & 0 & 0 \\ \end{array} }} \right]\\[10mm] { T}=\left[ {{\begin{array}{{c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c@{\ \ \ }c}} {C_{33}^0 } & 0 & 0 & {R_3^0 } & {e_{33}^0 } \\ 0 & {C_{55}^0 } & 0 & 0 & 0 \\ 0 & 0 & {C_{44}^0 } & 0 & 0 \\ {R_3^0 } & 0 & 0 & {K_3^0 } & {d_{33}^0 } \\ {e_{33}^0 } & 0 & 0 & {d_{33}^0 } & {-\xi _{33}^0 } \\ \end{array} }} \right], \ \ { \beta }_1 =r_j \left[ {{\begin{array}{*{20}c} {-\beta _1^0 } \\ {\beta _1^0 p} \\ 0 \\ 0 \\ 0 \\ \end{array} }} \right], \ \ { \beta }_2 =r_j \left[ {{\begin{array}{*{20}c} {\beta _3^0 } \\ 0 \\ 0 \\ 0 \\ {-p_3^0 } \\ \end{array} }} \right] \\ \end{array} \right\} \end{eqnarray}$
为了便于求解, 将式(16)转换为对应的标准特征关系

(18) $ \begin{eqnarray} \label{eq18} { N}\left[ {\begin{array}{l} { a} \\ { b} \\ \end{array}} \right]=s\left[ {\begin{array}{l} { a} \\ { b} \\ \end{array}} \right], \ \ { N}\left[ {\begin{array}{l} { c} \\ { d} \\ \end{array}} \right]=\left( {1+\eta } \right)\left[ {\begin{array}{l} { c} \\ { d} \\ \end{array}} \right]+{ \beta } \end{eqnarray}$
式中, 矩阵${ N}$和${ \beta }$分别为

(19) $ \begin{eqnarray} \label{eq19} \begin{array}{l} { N}=\left[ {{\begin{array}{*{20}c} {{ T}^{-1}{ P}^{\rm T}} & {{ T}^{-1}} \\ {-{ Q}-{ PT}^{-1}{ P}^{\rm T}} & {-{ PT}^{-1}-\alpha { I}} \\ \end{array} }} \right] \\ { \beta }=-\left[ {{\begin{array}{*{20}c} {\bf 0} & {{ T}^{-1}} \\ { I} & {-{ PT}^{-1}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{ \beta }_1 } \\ {{ \beta }_2 } \\ \end{array} }} \right] \\ \end{array} \end{eqnarray}$
求解式(18), 可得到电-弹耦合场对应的特征值$s$和特征向量$[{ a}\ \ { b}]^{\rm T}$, $[{ c} \ \ { d}]^{\rm T}$. 因此, 一维准晶功能梯度单层圆柱壳电-弹耦合场的精确解为

(20) $ \begin{eqnarray} \label{eq20} && \left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r{\bar{{ t}}}} \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} { I} & {\bf 0} \\ {\bf 0} & {\rho ^\alpha { I}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{ A}_1 } & {{ A}_2 } \\ {{ B}_1 } & {{ B}_2 } \\ \end{array} }} \right]\left\langle {\rho ^{s\ast }} \right\rangle \left[ {\begin{array}{l} { K}_1 \\ { K}_2 \\ \end{array}} \right]+ \left[ {{\begin{array}{*{20}c} {\rho { I}} & {\bf 0} \\ {\bf 0} & {\rho ^{1+\alpha }{ I}} \\ \end{array} }} \right]\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} {{ c}_1 } & {{ c}_2 } \\ {{ d}_1 } & {{ d}_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\rho ^{\eta _1 }} & 0 \\ 0 & {\rho ^{\eta _2 }} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & 0 \\ 0 & {f_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\chi _1 } \\ {\chi _2 } \\ \end{array} }} \right] \end{eqnarray}$
式中

(21) $ \begin{eqnarray} \label{eq21} \left. \begin{array}{ll} \left[ \begin{array}{*{20}c} {{\bar{{ u}}}} & {r{\bar{{ t}}}} \\ \end{array} \right]^{\rm T}=\left[ \bar{{u}}_r ,\ \ \bar{{u}}_\theta ,\ \ \bar{{u}}_z ,\ \ \bar{{w}}_r , \ \ \bar{{\phi }}, \ \ r\bar{{\sigma }}_{rr} , \ \ r\bar{{\sigma }}_{\theta r} , \\\qquad r\bar{{\sigma }}_{zr} ,\ \ r\bar{{H}}_{rr} , \ \ r\bar{{D}}_r \right]^{\rm T} \\ { A}_1 =\left[ {{ a}_1 , \ \ { a}_2 , \ \ { a}_3 , \ \ { a}_4 , \ \ { a}_5 } \right]\\ { A}_2 =\left[ { \ \ \ { a}_6 , \ \ { a}_7 , \ \ { a}_8 , \ \ { a}_9 , \ \ { a}_{10} } \right] \\ { B}_1 =\left[ {{ b}_1 , \ \ { b}_2 , \ \ { b}_3 , \ \ { b}_4 , \ \ { b}_5 } \right]\\{ B}_2 =\left[ {{ b}_6 , \ \ { b}_7 , \ \ { b}_8 , \ \ { b}_9 , \ \ { b}_{10} } \right] \\ \left\langle {\rho ^{s^\ast }} \right\rangle = \mbox{diag}\left[ \rho ^{s_1 }, \ \ \rho ^{s_2 }, \ \ \rho ^{s_3 }, \ \ \rho ^{s_4 },\ \ \rho ^{s_5 }, \ \ \rho ^{-s_1 -\alpha }, \\\qquad \rho ^{-s_2 -\alpha }, \ \ \rho ^{-s_3 -\alpha }, \ \ \rho ^{-s_4 -\alpha },\ \ \rho ^{-s_5 -\alpha } \right] \\ \end{array} \right\} \end{eqnarray}$

3 层合圆柱壳的热电弹性精确解

本节引入传递矩阵方法^[26], 用于处理层状结构问题. 假设层间界面为完美连接, 首先处理温度场的传递问题, 然后求解电-弹耦合场多层结构问题, 最后将温度场、电场和弹性场的结果合并, 得到一维准晶功能梯度层合圆柱壳的热-电-弹性精确解.

考虑温度场, 对于层合圆柱壳第$j$层, 利用式(11), 有

(22) $ \begin{eqnarray} \label{eq22} && \left[ {{\begin{array}{*{20}c} {\chi _1 } \\ {\chi _2 } \\ \end{array} }} \right]_j =\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]^{-1}\left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r_j \bar{{q}}_r } \\ \end{array} }} \right]_{r_j } =\left[ {{\begin{array}{*{20}c} {(r_{j+1} /r_j )^{-\eta _1 }} & 0 \\ 0 & {(r_{j+1} /r_j )^{-\eta _2 }} \\ \end{array} }} \right]\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]^{-1}\left[ {{\begin{array}{*{20}c} 1 & 0 \\ 0 & {(r_{j+1} /r_j )^{-\alpha }} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r_{j+1} \bar{{q}}_r } \\ \end{array} }} \right]_{r_{j+1} } \end{eqnarray}$
利用式(22), 圆柱壳$j$层任意半径处的物理量为

(23) $ \begin{eqnarray} \label{eq23} \left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r\bar{{q}}_r } \\ \end{array} }} \right]={ T}_j (r/r_j )\left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r_j \bar{{q}}_r } \\ \end{array} }} \right]_{r_j } \end{eqnarray}$
式中, 温度场的传递矩阵${ T}_j $为

(24) $ \begin{eqnarray} \label{eq24} && { T}_j (r/r_j )=\left[ {{\begin{array}{*{20}c} 1 & 0 \\ 0 & {(r/r_j )^\alpha } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} {(r/r_j )^{\eta _1 }} & 0 \\ 0 & {(r/r_j )^{\eta _2 }} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]^{-1} \end{eqnarray}$
考虑电-弹耦合场, 对于圆柱壳$j$层, 借助式(20), 可以得到

(25) $ \begin{eqnarray} \label{eq25} && \left[ {\begin{array}{l} { k}_1 \\ { k}_2 \\ \end{array}} \right]_j =\left[ {{\begin{array}{*{20}c} {{ a}_1 } & {{ a}_2 } \\ {{ b}_1 } & {{ b}_2 } \\ \end{array} }} \right]^{-1}\left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r_j {\bar{{ t}}}} \\ \end{array} }} \right]_{r_j } -\left[ {{\begin{array}{*{20}c} {{ a}_1 } & {{ a}_2 } \\ {{ b}_1 } & {{ b}_2 } \\ \end{array} }} \right]^{-1}\left[ {{\begin{array}{*{20}c} {{ c}_1 } & {{ c}_2 } \\ {{ d}_1 } & {{ d}_2 } \\ \end{array} }} \right]\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} {f_1 } & 0 \\ 0 & {f_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\chi _1 } \\ {\chi _2 } \\ \end{array} }} \right]_j =\left\langle {(r_{j+1} /r_j )^{-s\ast }} \right\rangle \left[ {{\begin{array}{*{20}c} {{ a}_1 } & {{ a}_2 } \\ {{ b}_1 } & {{ b}_2 } \\ \end{array} }} \right]^{-1}\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} { i} & {\bf 0} \\ {\bf 0} & {(r_{j+1} /r_j )^{-\alpha }{ i}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r_{j+1} {\bar{{ t}}}} \\ \end{array} }} \right]_{r_{j+1} } -\left\langle {(r_{j+1} /r_j )^{-s\ast }} \right\rangle\cdot \\&&\qquad\left[ {{\begin{array}{*{20}c} {{ a}_1 } & {{ a}_2 } \\ {{ b}_1 } & {{ b}_2 } \\ \end{array} }} \right]^{-1} \left[ {{\begin{array}{*{20}c} { i} & {\bf 0} \\ {\bf 0} & {(r_{j+1} /r_j )^{-\alpha }{ i}} \\ \end{array} }} \right]\cdot \\&&\qquad\left[ {{\begin{array}{*{20}c} {(r_{j+1} /r_j ){ i}} & {\bf 0} \\ {\bf 0} & {(r_{j+1} /r_j )^{1+\alpha }{ i}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{ c}_1 } & {{ c}_2 } \\ {{ d}_1 } & {{ d}_2 } \\ \end{array} }} \right]\cdot \\&&\qquad \left[ {{\begin{array}{*{20}c} {(r_{j+1} /r_j )^{\eta _1 }} & 0 \\ 0 & {(r_{j+1} /r_j )^{\eta _2 }} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & 0 \\ 0 & {f_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {\chi _1 } \\ {\chi _2 } \\ \end{array} }} \right]_j \end{eqnarray}$
利用式(25), 层合圆柱壳$j$层任意半径处的物理量可以表示为

(26) $ \begin{eqnarray} \label{eq26} \left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r{\bar{{ t}}}} \\ \end{array} }} \right]={ E}_j (r/r_j )\left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r_j {\bar{{ t}}}} \\ \end{array} }} \right]_{r_j } +{ S}_j (r/r_j )\left[ {{\begin{array}{*{20}c} {\bar{{T}}} \\ {r_j \bar{{q}}_r } \\ \end{array} }} \right]_{r_j } \end{eqnarray}$
式中, 矩阵${ E}_j $和${ S}_j $分别为

(27) $ \begin{eqnarray} \label{eq27} \left. \begin{array}{ll} { E}_j (r/r_j )=\left[ {{\begin{array}{*{20}c} { I} & {\bf 0} \\ {\bf 0} & {(r/r_j )^\alpha { I}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{ A}_1 } & {{ A}_2 } \\ {{ B}_1 } & {{ B}_2 } \\ \end{array} }} \right]\left\langle {(r/r_j )^{s\ast }} \right\rangle\cdot\\\qquad \left[ {{\begin{array}{*{20}c} {{ A}_1 } & {{ A}_2 } \\ {{ B}_1 } & {{ B}_2 } \\ \end{array} }} \right]^{-1} \\ { S}_j (r/r_j )=\left[ {{\begin{array}{*{20}c} {(r/r_j ){ I}} & {\bf 0} \\ {\bf 0} & {(r/r_j )^{1+\alpha }{ I}} \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {{ c}_1 } & {{ c}_2 } \\ {{ d}_1 } & {{ d}_2 } \\ \end{array} }} \right]\\ \qquad\left[ {{\begin{array}{*{20}c} {(r/r_j )^{\eta _1 }} & 0 \\ 0 & {(r/r_j )^{\eta _2 }} \\ \end{array} }} \right] \left[ {{\begin{array}{*{20}c} {f_1 } & 0 \\ 0 & {f_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]^{-1}-\\ \qquad{ E}_j (r/r_j )\left[ {{\begin{array}{*{20}c} {{ c}_1 } & {{ c}_2 } \\ {{ d}_1 } & {{ d}_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & 0 \\ 0 & {f_2 } \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} {f_1 } & {f_2 } \\ {g_1 } & {g_2 } \\ \end{array} }} \right]^{-1} \\ \end{array} \right\} \end{eqnarray}$
将式(23)和式(26)中精确解合并, 同时重复利用传递矩阵, 得到一维准晶功能梯度层合圆柱壳的热电弹性精确解

(28) $ \begin{eqnarray} \label{eq28} &&\left[ \begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r_b {\bar{{ t}}}} \\ {\bar{{T}}} \\ {r_b \bar{{q}}_r } \\ \end{array} \right]_{r_b } ={ G}_N (\delta _N ){ G}_{N-1} (\delta _{N-1} )\cdot \cdot \cdot { G}_2 (\delta _2 ){ G}_1 (\delta _1 )\cdot \\&&\qquad\left[ {{\begin{array}{*{20}c} {{\bar{{ u}}}} \\ {r_a {\bar{{ t}}}} \\ {\bar{{T}}} \\ {r_a \bar{{q}}_r } \\ \end{array} }} \right]_{r_a } \end{eqnarray}$
式中, $\delta _j =r_{j+1} /r_j $, 传递矩阵${ G}_j (\delta _j )$为

(29) $ { G}_j (\delta _j )=\left[ {{\begin{array}{c@{\quad }c} {{ E}_j (\delta _j )} & {{ S}_j (\delta _j )} \\ {\bf 0} & {{ T}_j (\delta _j )} \\ \end{array} }} \right]$

4 数值算例

本节主要讨论功能梯度指数因子对温度场、电场、声子场和相位子场的影响. 考虑3层功能梯度圆柱壳, 第一层和第三层为功能梯度准晶材料Al-Ni-Co, 第二层为均质压电材料BaTiO$_{3}$, 且每层厚度相等, 材料常数见表1^[27-29]. 计算中为避免产生奇异矩阵, 晶体的相位子场弹性常数按照10$^{-8}$倍的准晶相位子场弹性常数选取^[30], 同时晶体中声子场-相位子场耦合弹性常数为0.

Table 1
表1
表1材料常数
Table 1Material Constants tabular

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层合圆柱壳的几何参数为: 内表面半径$r_{a}= 4$ mm, 外表面半径$r_{b }= 10$ mm, 角度跨度$\theta_{0 }= 1$ rad. 假设在功能梯度层合圆柱壳的内外表面同时施加温度载荷, 外表面载荷为$T(r_b ) = 1K$, 内表面载荷为$T(r_a )=0.5K$. 令$m = 1$, 指数因子分别为$\alpha = -5$, 0, 5. 当$\alpha = 0$时, 功能梯度材料退化为均质材料. 数值算例中给出的物理量均独立于圆柱壳角度$\theta $.

图2给出指数因子$\alpha $对温度$\bar{{T}}$的影响, 图3为电势$\bar{{\phi }}$和电位移$\bar{{D}}_r $随着指数因子$\alpha $的变化曲线. 从图2可以看到, $\bar{{T}}$在功能梯度层合圆柱壳的内表面为0.5, 外表面为1.0. 同时, 图3(b)显示电位移$\bar{{D}}_r $在功能梯度层合圆柱壳的内外表面均为0. 上述物理量在圆柱壳内外表面的数值满足加载的边界条件. 文中考虑了温度场对弹性场、电场的单向耦合问题, 根据温度场传递矩阵关系, 图2所示$\bar{{T}}$与$\alpha $和特征值$\eta $有关. 同时考虑层合圆柱壳内外表面加载的温度边界条件. 因此, 不同$\alpha $对应的$\bar{{T}}$曲线出现交叉现象. 此外, $\alpha = -5$对应的温度$\bar{{T}}$在界面处更光滑, 是因为不同层间的热传导系数差异性减小导致的. 在图3(a)中, 最外层的$\bar{{\phi }}$随着$\alpha $的增大略有增大. 图3(b)所示中间层的$\bar{{D}}_r $数值随着$\alpha $的减小而增大, 并且数值远大于外层和内层, 这主要是因为晶体的热电系数大于准晶材料的热电系数导致.

图2

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图2指数因子对温度的影响

Fig.2Influence of exponential factor on temperature

图3

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图3指数因子对电场的影响

Fig.3Influence of exponential factor on electric field



图4为不同指数因子$\alpha $对应的声子场和相位子场应力沿半径方向的变化曲线. 图4(a)为在圆柱壳界面处连续的声子场应力$\bar{{\sigma }}_{rr} $. 和$\alpha =-5$对应的$\bar{{\sigma }}_{rr} $相比, $\alpha = 5$对应的$\bar{{\sigma }}_{rr} $最大值更大. 此外, $\alpha = 5$对应的$\bar{{\sigma }}_{rr} $在靠近内表面处数值最小, 在靠近外表面处数值最大. 从图4(b)可以看到, 相位子场应力$\bar{{H}}_{rr} $在中间层为0, 因为晶体中不存在相位子场. 准晶层中的$\bar{{H}}_{rr} $随着$\alpha $的增大而增大.

图4

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图4指数因子对声子场和相位子场应力的影响

Fig.4Influence of exponential factor on phonon and phason stresses



图5给出指数因子$\alpha $对声子场和相位子场位移的影响. 图5(a)显示声子场位移$\bar{{u}}_\theta $在界面处是连续的. 文中功能梯度材料参数遵循幂函数变化规律, 当指数因子$\alpha >0$时, 功能梯度材料性质呈现渐变硬化趋势, $\alpha <0$ 则呈现渐变软化特性. 所以层合圆柱壳内表面$\bar{{u}}_\theta $随着$\alpha $增大而变小. 同时, $\bar{{u}}_\theta$的大小还与层合圆柱壳半径有关. 因此, 不同的指数因子对应的$\bar{{u}}_\theta $在最外层出现交点. 此外, 和$\alpha =-5$对应的$\bar{{u}}_\theta $相比, $\alpha = 5$对应的$\bar{{u}}_\theta $在界面处曲线更光滑. 在图5(b)中, 和$\alpha$对$\bar{{u}}_\theta $的影响相比, $\alpha $对内外表面处的相位子场位移$\bar{{w}}_r $的影响较小.

图5

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图5指数因子对声子场和相位子场位移的影响

Fig.5Influence of exponential factor on phonon and phason displacements

5 结论

本文利用类Stroh理论和传递矩阵方法, 研究了一维准晶功能梯度层合圆柱壳的热电弹性耦合问题, 基于材料参数沿径向呈现幂函数分布特点, 得到了简支边界条件下的一维准晶功能梯度层合圆柱壳的热电弹性精确解. 数值算例中讨论了功能梯度指数因子对温度场、电场、声子场和相位子场的影响. 所得结论如下:

(1) 指数因子的变化影响层合圆柱壳的材料性质分布情况, 进而对温度、电势、电位移、声子场和相位子场应力和位移产生较大的影响.

(2) 随着指数因子的增大, 径向声子场应力增大, 且最大值出现在层合圆柱壳外层. 外层中较大的应力易于被外层凸面分散, 从而提升层合圆柱壳的承载能力.

(3) 随着指数因子的增大, 层合圆柱壳外表面周向声子场位移略有增大, 内表面周向声子场位移减小. 此外, 周向位移在界面处更为光滑, 可有效减缓层合圆柱壳在界面处出现开裂和分层等现象.

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