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，特别是在飞机工业中。过去，复合材料的使用大多局限于民用客机的二次结构，但由于其较高的比强度，更重要的是其方向性和可裁剪能力，现在它们正在向一次结构发展。然而，与复合材料的动态行为相关的潜在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一种新的方法。本课题提出了一种新的层状动力刚度法(LW-DSM)，用于分析复合材料厚板结构的自由振动。目前的复合材料建模工具主要基于经典层合板理论，该理论将非均匀层合板视为具有相同性能的等效正交各向异性均质层。将每一层的刚度属性相加，得到整体刚度。显然，这种简单的方法是有缺陷的，因为它违反了层与层之间界面的一致性和平衡条件，可能会导致较大的误差。当寻求宽厚比为100的结构的宏观性能时，CLT获得的精度水平可能是可以接受的，但对于要承载较大荷载的初级结构，因此厚度较大，误差将大得多。最近的研究表明，对于某些结构，其基频误差可达30%以上。在航天工业中，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，它可能会导致在实验室或飞行试验中的结构故障。为此，需要开发改进的建模技术。新的建模技术确实可能在计算上要求很高，但另一方面，它们提供了亟需的精度。这一领域最近的一项重要发展是所谓的层智能技术，在这种技术中，通过使用适当的位移假设和组装程序，将每一单层模拟为单独的板。LW有限元模型的主要缺点是产生了大量的未知量，而未知量的大小与层数有关。例如，20层4节点有限板单元有252个自由度。显然，建立结构模型需要大量的有限元单元，因此LW有限元模型的自由度变得过多，使得使用传统的LW理论实际上是不可能的。一个主要的突破将是将LW理论与动力刚度法(DSM)相结合，使这一应用成为可能。对于板组件的自由振动，基于经典板理论和一阶剪切变形理论的动力刚度元已经由申请者发展起来，显示出比传统有限元的巨大优势。与有限元形成鲜明对比的是，DSM的一个潜在优点是，一个单元就足以精确地以均匀的几何形状模拟结构的任何部分，而不会损失任何精度，从而极大地减少了问题的未知数。这只在DSM中是可能的，因为不是离散结构，而是以闭合形式求解运动微分方程，并且该解被推广以发展单元属性，然后可以旋转、偏移、组装成诸如翼盒之类的复杂结构的模型。由于DSM可以使LW理论在研究真实复合材料结构时的应用是可行的，因此该项目的目标是利用LW理论开发精确的高精度DS单元。这项提议是一个突破的缩影，目的是克服在准确预测厚层压板复合材料飞机结构的动态性能方面可能是最大的绊脚石。

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.