An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods
An objective 3D large deformation finite element formulation for geometrically exact curved Kirchhoff rods
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DOI：
10.1016/j.cma.2014.05.017
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发表时间：
2014-08-15
影响因子：
7.2
通讯作者：
Wall, Wolfgang A.
中科院分区：
文献类型：
--
作者：
Meier, Christoph;Popp, Alexander;Wall, Wolfgang A.
本文的目的是根据Kirchhoff细杆理论建立一种新的梁的有限元列式，包括轴向拉伸、扭转和弯曲的变形状态。所提出的公式考虑了三维问题中的大变形，该问题由具有任意弯曲的初始几何形状和任意横截面形状的细长棱柱梁组成。与几何精确的Reissner理论一样，所导出的变形测量在几何上是精确的，因为对于任何构型和任意大的平移、旋转和应变，它们与平衡方程和梁的几何形状的强弱形式是一致的。为了满足强意义上剪切应变消失的Kirchhoff约束，并保持空间离散问题的客观性，将一种新的正交内插策略应用于表示横截面方向的三维场。由相应的弱形式产生的连续性要求通过三次Hermite插值来满足，从而导致梁中心线的C-1连续表示。通过适当的数值算例验证了所发展的梁单元的客观性、路径无关性、一致性和精度等基本性质。(C)2014爱思唯尔B.V.保留所有权利。
The objective of this work is the development of a new finite element formulation for beams according to the Kirchhoff theory of thin rods, which includes the deformation states of axial tension, torsion and bending. The proposed formulation accounts for large deformations in three-dimensional problem settings consisting of slender, prismatic beams with arbitrarily curved initial geometries and arbitrary cross-section shapes. Like in geometrically exact Reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations, rotations and strains. A novel orthogonal interpolation strategy is applied to the triad field representing the cross-section orientation in order to fulfill the Kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem. The continuity requirements resulting from the corresponding weak form are fulfilled by a cubic Hermite interpolation, thus leading to a C-1-continuous representation of the beam centerline. Fundamental properties such as objectivity, path-independence, consistency and accuracy of the developed beam element are verified by means of suitable numerical examples. (C) 2014 Elsevier B.V. All rights reserved.