【作者】 吴振；

【导师】 陈万吉；

【作者基本信息】 大连理工大学 ， 固体力学， 2007， 博士

【摘要】 本文旨在客观地评价位移基层合板理论，并且提出应用范围广范且准确高效的层合板理论。基于这一目的，本文用解析解对十几种有代表性的层合板理论进行数值比较。结果表明：对于层数较少的层合板，整体-局部1，2-3高阶理论GLHT-30（3表示整体面内位移沿板厚度方向展开三次多项式，0表示横向位移沿厚向为常数）能够通过本构方程直接准确计算横向剪切应力不需要任何后处理。基于整体-局部1，2-3高阶理论，本文建立了几种层合板单元并分析了层合板弯曲、动力和屈曲问题。进一步研究表明，整体-局部1，2-3高阶理论不能处理多层板弯曲，面内和任意温度分布载荷作用的层合板问题。从提高效率和扩展应用范围考虑，本文基于整体-局部1，2-3高阶理论建立了几种增强整体-局部高阶理论并系统地研究了复合材料层合板有限元方法。具体工作为：1．建立几种整体-局部高阶理论。●为了提高计算效率，本文建立了Reddy型整体-局部高阶理论。Reddy型整体-局部理论能够预先满足面内位移和横向剪切应力层间连续，而且独立变量个数比整体-局部1，2-3高阶理论少了4个。此理论已被用于分析层数较少层合板和功能梯度板弯曲问题。●通过提高整体-局部1，2-3高阶理论整体面内位移沿板厚度方向展开多项式的阶数，本文提出整体-局部高阶理论GLHT-m0（m为整体面内位移沿厚度方向展开多项式的阶数，m取不同的值可以得到不同的理论），基于此理论研究了高阶剪切变形对多于五层层合板弯曲、动力及屈曲响应的影响，并且给出结论即对于多于五层层合板问题应采用五阶整体-局部理论（m=5）。●对于热膨胀问题，横法向应变扮演重要角色。为了处理热膨胀问题，本文把整体-局部1，2-3高阶理论的横向位移w沿厚向展开二次多项式建立了考虑横法向应变的整体-局部高阶理论GLHT-32（3表示整体面内位移沿厚度方向展开三次多项式，2表示横向位移沿厚向展开二次多项式）。数值试验表明，GLHT-32确实能够处理热膨胀问题。●本文进一步提出了能够处理在任意温度载荷作用下任意铺设层合板问题的整体-局部高阶理论GLHT-52。此工作把整体-局部高阶理论应用范围扩展到任意温度载荷问题和角铺设层合板问题。●复合材料层合板自由边问题是典型三维问题，现有等效单层板理论不能处理此类问题，不得不采用三维有限元和Layerwise理论（效率低）。通过提高整体面内位移和横向位移沿厚度方向展开多项式的阶数，本文建立了增强型整体-局部高阶理论GLHT-mn（m，n分别表示整体面内位移和横向位移沿厚度方向展开多项式的阶数，m，n取不同的值可以得到不同的理论）并应用此理论分析了层合板自由边问题。数值试验表明，GLHT-99能够准确分析多种自由边问题。本文增强型整体-局部高阶理论属于等效单层板理论但可以准确分析自由边问题，直接验证了等效单层板理论有能力分析层合板自由边问题。●本文建立了整体-局部高阶层合壳理论GLHST-52并给出了简单解析解。当前的层合壳理论特点是未知变量个数独立于层合壳层数，并且应用本构方程可以直接准确地计算横向剪切应力。数值比较表明，当前理论计算的结果精度明显高于同类层合壳理论。2．基于建立的整体-局部高阶理论，本文系统地研究了复合材料层合板有限元方法。●应用精化元法分别提出了精化四边形单元RQLP13和精化三角形单元RTLP23；基于满足单元间C¹连续的离散Kirchhoff薄板单元，建立了三角形层合板单元TLP13和四边形单元QLP19。整体-局部高阶理论的应变项中出现了横向位移w的一阶导数和二阶导数，构造有限元时应使用同时满足C⁰和C¹连续的横向位移函数，被称为C^0-1连续，是一种新提法。为此，本文用同样节点参数分别构造满足C⁰和C¹连续的两套横向位移函数。本文有限元法列式简单，便于实施，计算过程中无须特殊的数值技巧，求解各种算例效果很好，在规则和不规则网格下，它们都能保持良好的性态。●基于整体-局部1，2-3高阶理论推导了动力及稳定问题的有限元，把修改几何刚度阵的方法推广到复合材料层合板稳定问题。用本文方法分析了软核夹层板和变厚度层合板动力及屈曲问题，结果表明：一阶理论甚至整体型高阶理论不适于预测软核夹层板和变厚度层合板静动力及屈曲响应，本文有限元能够给出准确结果。●组合整体-局部1，2-3高阶理论和Layerwise理论，本文建立了混合模型并应用此模型分析压电层合板问题，研究沿厚度方向电位移选取对结果的影响。在混合模型中整体-局部1，2-3高阶理论模拟压电板的结构变形，Layerwise理论模拟电位移沿厚度方向的变化。该模型突破了以往用位移基高阶理论求解层合压电板必需使用后处理方法计算横向剪切应力的限制。●基于整体-局部1，2-3理论推导了几何非线性有限元并且分析了层合板几何非线性问题。结果表明：对于层合板几何非线性分析，面内位移的二次项对横向剪切应力确实影响较大。更多还原

【Abstract】 Aim of this thesis is to propose more accurate and efficient laminated plate theories afterobjectively estimating previous the displacement-based laminated plate theories. To pursuethis goal, this thesis firstly compared several theories by using analytical methods. Numericalresults show that 1,2-3 global-local higher-order theory is able to accurately predict transverseshear stresses without any smoothing methods as the number of layers of laminated plates isless than five. Based on this theory, several laminated plate elements have been constructed.Moreover, these elements were used to analyze bending, vibration and buckling of laminatedplates. By further research on 1,2-3 global-local higher-order theory, it is found that thistheory will encounter some accuracy difficulties to analyze bending of multilayered plates andthermal expansion of laminated plates. In view of computational efficiency and range ofapplicability, this thesis proposes several enhanced global-local higher-order theories andstudies detailedly laminated plate elements. Main work includes the following aspect:1. Proposed several global-local higher-order theories.●To improve computional efficiency, Reddy-type global-local higher-order theory ispresented. This theory can a priori satisfy continuity of in-plane displacements andtransverse shear stresses at interfaces. Moreover, compared to 1,2-3 global-localhigher-order theory, number of independent variables is reduced to 7. This theory hasbeen used to analyze bending of lamianted plates with few layers and functionally gradedplates.●By increasing the number of orders of the global in-plane displacement component in1,2-3 global-local higher-order theory, the global-local higher-order theory GLHT-m0 ispresented. Using this theory, effects of higher-order global-local shear deformations onbending, vibration and buckling of multilayered plates have been studied and thecorresponding conclusions are drawn: considering computational efficiency and accuracyof the results, the fifth-order global-local theory （m=5） should be applied to predict thestatic, the dynamic and the buckling response of multilayered plates.●For thermal expansion problems of laminated plates, transverse normal strain plays animportant role. To analyze such problems, global-local higher-order theory GLHT-32 isproposed by considering transverse normal strain of 1,2-3 global-local higher-ordertheory. Numerical results show that GLHT-32 is suitable for predicting thermal responseof laminated plates under uniform temperature. ●This thesis further developes the global-local higher-order theory GLHT-52 which canaccuratelly predict thermal response of angle-ply laminated plates subjected to arbitrarytemperature loads. This work extends the range of applicability of the global-localhigher-order theory.●It is well known that the free-edge problem is a typical three-dimensional problem.Three-dimensional finite elements and layerwise theories have to be used to analyzefree-edge effects because previous equivalent single layer plate theories can not analyzeso complicated problem. By increasing the number of order of in-plane and transversedisplacements, an enhanced global-local higher-order theory GLHT-mn has beendeveloped to analyze free-edge problems. Numerical results show that GLHT-99 which isa typical equivalent single layer plate theory is able to predict accurately static responseof the free-edge effects. Thereby, this work directly proves that equivalent single layerplate theory can analyze free-edge problems of laminated plates.●Global-local higher-order laminated shell theory GLHST-52 as well as correspondinganalytical results have been given in this thesis. Unknown variables of this theory areindependent of number of layers and transverse shear stresses can be directly computedfrom constitutive equations. Numerical results show that present shell theory is moreaccurate than other shell models.2. Based on global-local higher-order theory, this thesis detailedly studies laminatedplate elements.●Based on the refined nonconforming element method, a refined four-node quadrilateralplate element RQLP13 and a refined three-node triangular element RTLP23 are presented.Subsequently, a triangular plate element TLP13 and a quadrilateral plate element QLP19are also proposed by using the discrete Kirchhoff thin plate bending element satisfying C¹continuity on the element boundary. Global-local higher-order theory possesses first andsecond derivatives of transverse displacement w in the strain components. Thustransverse displacement function satisfying C⁰ as well as C¹ continuity on the elementboundary, which is named as C^0-1 continuity, should be employed. This thesiscircumvents the requirement of C^0-1 continuity by using C⁰ continuity displacementfunction and C¹ continuity displacement function, respectively. Finite element modelsconstructed in this thesis are simple and are convenient to use, which are suitable for allkinds of cases without any numerical technique. Moreover, they exhibit goodperformance for both regular and irregular meshes.●Based on 1,2-3 global-local higher order theory, laminated plate elements for vibration and buckling analysis are given. Furthermore, the modified geometric stiffness matrixmethod is extended to buckling problems of laminated composite plate. By using ourapproaches, dynamic and buckling problems on soft-core sandwich plates as well aslaminated composite plates with different thickness and materials at each ply have beenanalyzed. Numerical results show that first order theories even global higher-ordertheories are inadequate to predict dynamic and buckling response of so special structureswhereas 1,2-3 global-local higher order theory is still suitable for analyzing dynamic andbuckling problems of laminated composite plates with variational thickness and materialsat each layer and soft-core sandwich plates.●Combining 1,2-3 global-local higher order theory and layerwise theory, the mixed modelis developed in this thesis. This model is used to analyze bending problems of laminatedpiezoelectric plates. Moreover, effects of electric displacement on static response oflaminated piezoelectric plates are also studied. The concept of mixed model is that themechanical component is modeled by 1,2-3 global-local higher-order theory whereas theelectric field is modeled with layerwise theory. Present mixed model is able to accuratelypredict transverse shear stresses of laminated piezoelectric plate without any smoothingmethods.●Geometrically nonlinear finite element based on 1,2-3 global-local higher-order theoryhas been constructed, which is used for geometrically nonlinear analysis of laminatedplates. Numerical results show that for geometrically nonlinear analysis, effect of thesecond-order components in in-plane displacement on transverse shear stresses is actuallysignificant.更多还原