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Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory
European Journal of Mechanics - A/Solids ( IF 4.1 ) Pub Date : 2021-01-19 , DOI: 10.1016/j.euromechsol.2021.104221
Jalal Torabi , Jarkko Niiranen , Reza Ansari

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.



中文翻译:

应变梯度弹性范围内的非线性有限元分析:Reissner-Mindlin板理论与三维理论

通过采用某种非标准的有限元方法,研究了Mindlin的应变梯度弹性理论(SGT)中的非线性板弯曲。主要目标是在SGT中比较几何非线性三维(3D)理论和几何非线性Reissner-Mindlin板理论(即一阶剪切变形板理论(FSDT))提供的弯曲结果。对于3D理论,采用非线性格林-拉格朗日应变关系,而FSDT采用vonKármán非线性应变。对于这两个模型,导出了能量泛函的矩阵向量形式。为了执行相应的有限元离散化,C1个-连续的四节点四面体固体单元和准-C1个-连续的6节点三角形板单元分别用于3D模型和板模型。将原始问题量的一阶导数用作附加节点值,以响应类别的连续性要求C1个。针对两种不同的板几何形状,给出了突出显示3D模型与FSDT模型之间差异的各种计算结果:带圆孔的矩形板和椭圆形板。

更新日期:2021-01-29
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