Gilewski Wojciech, Pełczyński Jan
Faculty of Civil Engineering, Warsaw University of Technology, 00-637 Warsaw, Poland.
Materials (Basel). 2020 Feb 10;13(3):803. doi: 10.3390/ma13030803.
A four-noded finite element of a moderately thick plate made of functionally graded material (FGM) is presented. The base element is rectangular and can be extended to any shape using a transformation based on NURBS functions. The proposed 2D shape functions are consistent with the physical interpretation and describe the states of element displacement caused by unit displacements of nodes. These functions depend on the FGM's material parameters and are called material-oriented. The shape function matrix is based on a superposition displacement field of two plate strips with 1D exact shape functions. A characteristic feature of the proposed formulation is full coupling of the membrane and bending states in the plate. The analytical form of the stiffness matrix and the nodal load vector was obtained, which leads to the numerical efficiency of the formulation. The element has been incorporated into Abaqus software with the use of Maple program. The finite element shows good convergence properties for different FGM models in the transverse direction to the middle plane of the plate. During derivation of the 2D plate element the formally exact 1D finite element for transverse nonhomogeneous FGM plate strip was developed.
提出了一种由功能梯度材料（FGM）制成的中等厚度板的四节点有限元。基本单元为矩形，可通过基于非均匀有理B样条（NURBS）函数的变换扩展为任意形状。所提出的二维形状函数与物理解释一致，描述了节点单位位移引起的单元位移状态。这些函数取决于功能梯度材料的材料参数，被称为面向材料的函数。形状函数矩阵基于具有一维精确形状函数的两个板条的叠加位移场。所提出公式的一个特征是板中膜状态和弯曲状态的完全耦合。获得了刚度矩阵和节点载荷向量的解析形式，这导致了该公式的数值效率。该单元已通过Maple程序并入Abaqus软件。对于不同的功能梯度材料模型，该有限元在垂直于板中面的方向上显示出良好的收敛特性。在二维板单元的推导过程中，开发了用于横向非均匀功能梯度材料板条的形式上精确的一维有限元。
