变分渐近精细模型及在功能梯度压电层合板壳多场耦合分析中的应用

With the increase of the service environments and accuracy requirements for the functionally graded piezoelectric material (FGPM), the in-depth study on the structural behavior of FGPM sturctures has important theoretical significance and application values. The linear distribution assumptions of field variables (electric potential, temperature and displacement) made by the traditional plates and shells theories result in the serious distribution phenomenons of "self-locking" along the thickness direction. In order to overcome these serious defects, the 3D mechanic-electro-thermo coupling problem is rigidly decoupled into a 2D non-linear plate and shell analysis alone the mid-plane and a warping deformation analysis alone the thickness direction by taking into account the geometric nonlinearity, elastic-piezoelectric-thermoelectric coupling and the properties coeffectives related to the temperature. A refined model without any assumptions is built by taking advantage of the small parameters inherent in the structures to asymptotically correct the leading items in the series 2D energy equations. Then, based on this refined model, the nonlinear dynamic equations of FGPM laminated plates and shells under multi-field coupling are established and soluted. The criterion to determine the stable equilibrium paths before and after the mode jumping is confirmed. Finally, the results of theoretical analysis are verified and evaluated by experimental study. The research results will undoubtedly reveal the basic laws of mechanics-electro-thermo coupling effects, and provide the theoretical basis for the structure optimization, intelligent control and application.

随着功能梯度压电材料(FGPM)服役环境和精度要求的不断提高，多场耦合下对FGPM结构性能进行深入研究具有深远的理论意义和应用价值。为克服传统板壳理论所做的场变量(电、热、位移)沿厚向线性分布假设导致各种"自锁"等严重失真现象，项目在综合考虑几何非线性、力-电-热耦合和物性系数与温度相关性的基础上，利用变分渐近法巧妙地将FGPM层合板壳的3维力-电-热耦合问题解耦为沿中面2维非线性板壳分析和沿厚向翘曲变形分析；利用板壳固有小参数渐近修正系列2维能量方程中的主导变分项，从而构建无需任何场变量假设的精细模型；基于构建的模型探讨多场耦合下FGPM板壳非线性屈曲运动控制方程的建立和求解方法, 确定模态跃迁前后稳定平衡路径的判定准则，并通过实验研究对理论分析结果进行验证和评估。以上研究成果无疑会揭示FGPM层合板壳多场耦合作用的基本规律，并为FGPM层合结构优化设计、智能控制和推广应用提供理论依据。

功能梯度压电材料(FGPM)具有组分材料体积分数从一侧表面向另一侧表面平滑过渡的特性，可降低压电材料在极端环境下灾难性断裂的风险。随着FGPM服役环境和精度要求的不断提高，多场耦合下对FGPM 板壳结构性能进行深入研究具有深远的理论意义和应用价值。研究内容主要分以下几个部分：.1..基于变分渐近均匀化方法建立压电材料机-电耦合细观力学模型。从非均质连续体控制机-电耦合问题的总能量泛函入手，采用场变量波动函数代替位移函数作为未知量，利用细宏观尺度比为小参数对该未知量进行渐近扩展，从而将细观均匀化及局部涨落问题转换为约束条件下泛函最小化—取驻值求解波动函数问题； 将该方法与有限元方法相结合推导出离散形式泛函的最小化求解过程。通过与细观力学有限元法的结果和试验值对比分析表明：在保证计算精度不变的前提下，构建的模型可大大提高计算效率，并可考虑纤维复杂截面形状和排列方式对压电材料宏观性能的影响。.2..基于变分渐近法构建了各层材料系数为常数或连续变化的FGPM 层合板广义Reissner-Mindlin 模型。 利用结构固有小参数，采用变分渐近法将原三维非线性模型渐近降维为系列二维模型，可准确捕捉大位移小应变产生的所有几何非线性效应；推导了层间界面上的连续性条件，并求解了相应的解析解。通过正弦型热机荷载下FGPM 层合板柱形弯曲算例表明：重构的三维位移和应力沿厚向分布满足板顶、底面自由边界和层间连续条件，计算精度可与三维精确解相媲美；面内应力沿厚向变化是非线性的，与均质材料面内应力线性变化完全不同；采用功能梯度面层替代均质面层可消除界面应力的不一致，减小面层开裂的可能性。.3..基于渐近修正几何非线性理论，探讨多场耦合下 FGPM 层合板非线性屈曲控制方程的建立和求解方法, 确定模态跃迁前后稳定平衡路径的判定准则和平衡构型附近的线性振动。研究表明：构建模型与FEA在主后屈曲区域的线性屈曲荷载计算结果有很好的吻合性；FEA在靠近二次分岔点时失去收敛性，而模型能深入后屈曲区域，准确捕捉不同板型和边界条件的模态跃迁现象；通过沿稳定主后屈曲和跃迁平衡路径计算得到层合板的固有频率，发现不同板型和边界条件下存在振型交换或“模态跃迁”现象。