Layer-wise dynamic stiffness formulation for free vibration analysis of multilayered composite structures

纤维增强先进复合材料正在迅速取代传统的各向同性材料，特别是在航空工业中。过去，复合材料的使用主要局限于民用客机的二级结构，但由于其高比强度，更重要的是，其方向性和定制能力，它们现在正在向一级结构迈进。然而，存在与复合材料的动态行为相关的潜在问题，这些问题在各向同性材料中是闻所未闻的。因此，克服这些问题的需要需要开发新的方法。本计画提出一种新的分层动态刚度法（LW DSM）来分析厚层复合材料结构的自由振动。目前复合材料的建模工具主要是基于经典层合理论（CLT），该理论将非均匀层合板视为具有等效性能的等效正交各向异性均匀层。将每层的刚度特性相加以获得全局刚度。显然，这种简单的方法是有缺陷的，因为它违反了层间界面处的一致性和平衡条件，并且可能导致较大的误差。当寻求结构的宏观行为时，用CLT获得的精度水平可能是可以接受的，而结构的宽厚比大于100，但对于主要结构，要承受大载荷，因此要厚，误差会高得多。最近的研究表明，对于某些结构的基本固有频率的误差可以超过30%。在航空航天工业中，安全系数通常较低，以实现更轻的质量，30%的误差是不可接受的，并且可能导致实验室或飞行测试期间的结构故障。为此，将开发改进的建模技术。新的建模技术确实可能需要计算，但另一方面，它们提供了急需的准确性。在这一领域的一个重要的最新发展是所谓的逐层技术，其中每个单层作为一个单独的板建模，通过使用适当的位移假设和组装程序。使用FEM的LW模型的主要缺点是，它导致大量的未知数，这取决于层数。例如，20层4节点有限板单元具有252个自由度。LW有限元模型需要大量的有限元单元，自由度过多，使得传统的LW理论在实际应用中变得不可行，而将LW理论与动态刚度法（DSM）相结合将是一个重大的突破，使LW理论的应用成为现实。对于板组件的自由振动，申请人已经开发了基于经典板理论和一阶剪切变形理论的动态刚度（DS）单元，其显示出比常规有限元巨大的优越性。DSM与FEM形成鲜明对比的潜在优势之一是，一个单一的单元就足以在精确意义上对具有均匀几何形状的结构的任何部分进行建模，而不会损失任何精度，从而大大减少了问题的未知数。这只有在DSM中才有可能，因为不是离散结构，而是以封闭形式求解运动微分方程，并且将解推广到开发元素属性，然后可以旋转，偏移，组装以模拟复杂结构，如机翼盒。由于DSM可以在研究真实的复合材料结构时应用LW理论，本课题的目的是利用LW理论研制一种高精度的DS元件。这一建议是一个突破，克服了可能是最大的绊脚石，准确预测复合材料飞机结构的动态行为厚层压板。

Fibre reinforced advanced composite materials are rapidly replacing conventional isotropic materials, particularly in aircraft industry. In the past, the use of composite materials was mostly confined to secondary structures of civil airliner, but given their high specific strength and more importantly, their directional properties and hence ability to be tailored, they are now making headway to primary structures. However, there are potential problems associated with the dynamic behaviour of composites that are unheard of in isotropic materials. Thus, the need to overcome these problems requires the development of a new method. This project proposes a novel Layer-Wise Dynamic Stiffness Method (LW DSM) for free vibration analysis of thick composite structures. Current modelling tools for composites are primarily based on classical lamination theory (CLT) which considers an inhomogeneous laminate as an equivalent orthotropic homogeneous layer with equivalent properties. The stiffness properties of each layer are summed to obtain the global stiffness. Clearly, such a simple approach is flawed because it violates congruency and equilibrium conditions at the interfaces between layers and it may lead to large errors. The level of accuracy obtained by CLT is probably acceptable when macro behaviour of a structure with width over thickness ratio >100 is sought, but for primary structures that are intended to carry large loads and are thus thicker, the error will be much higher. Recent research has shown that for some structures the error can be over 30% on the fundamental natural frequency. In aerospace industry where safety factors are generally low to achieve a lighter mass, an error of 30 % is unacceptable and it may lead to structural failures during laboratory or flight tests. For this reason, improved modelling techniques are to be developed. Novel modelling techniques could indeed, be computationally demanding but, on the other hand, they deliver the much needed accuracy. One important recent development in this area is the so-called Layer-Wise technique in which each single layer is modelled as an individual plate by using appropriate displacement assumptions and assembly procedure. The main drawback of the LW model using FEM is that it leads to a large number of unknowns which depends on the number of layers. For example, a 20-layer 4-noded finite plate element has 252 DOF. Clearly, lots of finite elements are required to model a structure, and thus the number of DOF for a LW FE model becomes excessive, making the use of conventional LW theory practically impossible.A major break-through would be to use LW theory in conjunction with the dynamic stiffness method (DSM) to make this application realistically possible. For free vibration of plate assemblies, Dynamic stiffness (DS) elements based on classical plate theory and first order shear deformation theory have already been developed by the applicants showing huge superiority over conventional finite elements. One of the potential benefits of the DSM in sharp contrast to FEM, is that one single element is enough to model any part of the structure with uniform geometry in an exact sense without losing any accuracy, and thus reducing the number of unknowns of the problem drastically. This is possible only in DSM because instead of discretising the structure, the differential equation of motion is solved in closed-form and the solution is generalised to develop element properties which can then be rotated, offset, assembled to model a complex structure such as wing boxes.As the DSM can make the application of a LW theory feasible when investigating real composite structures, the aim of the project is to develop an accurate high precision DS element using LW theory. The proposal epitomises a break-through to overcome probably the biggest stumbling block in accurately predicting the dynamic behaviour of composite aircraft structures made of thick laminates.