In this paper, an approach to model thin composite plates reinforced with curvilinear fibres is presented and applied to analyse modes of vibration. Particular attention is given to plates with non-standard geometries, which are less commonly addressed in studies on this topic. Aiming to achieve accuracy with a small number of degrees-of-freedom, the model is based on Kirchhoff’s plate theory, combined with an hp-version finite element method. Assembling p-version Kirchhoff plate elements, while ensuring continuity, presents a significant challenge. Elastic connections are introduced to address this issue. Additionally, elastic boundaries are also considered to impose the boundary conditions. Regarding the reinforcing fibres, cubic polynomial splines are employed to represent the path of the fibres, which also adds to the proposed model generality. To discretise the displacement field of the plate, three sets of interpolation functions are investigated. The convergence properties of the model, and the effects of the intervening features, are analysed based on hp-refinement. The proposed approach is shown to require fewer degrees-of-freedom to effectively analyse irregular-shaped plates, when compared to the more commonly used h-version finite elements. Moreover, the capability of cubic polynomial splines to represent fibre paths is validated. The paper concludes with modal analysis of a composite plate with a complex shape to verify tailoring abilities of reinforcing curvilinear fibres.

中文翻译：

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用于曲线纤维增强层压板振动分析的 hp 有限元


在本文中，提出了一种用曲线纤维增强的薄复合板建模的方法，并将其应用于分析振动模式。特别关注具有非标准几何形状的板，这些在有关该主题的研究中不太常见。该模型以 Kirchhoff 板理论为基础，结合 hp 版有限元方法，以小自由度实现精度为目标。在确保连续性的同时，组装 p 型 Kirchhoff 板元件是一项重大挑战。为了解决这个问题，引入了弹性连接。此外，弹性边界也被考虑用于施加边界条件。关于增强纤维，采用三次多项式样条来表示纤维的路径，这也增加了所提出的模型的通用性。为了离散板的位移场，研究了三组插值函数。模型的收敛特性以及干预特征的影响基于 hp 细化进行分析。与更常用的 h 型有限元相比，所提出的方法需要更少的自由度来有效分析不规则形状的板。此外，三次多项式样条表示纤维路径的能力也得到了验证。本文最后对具有复杂形状的复合板进行了模态分析，以验证增强曲线纤维的剪裁能力。