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Asymptotic Analysis for Plane Stress Problems
Journal of Elasticity ( IF 2 ) Pub Date : 2021-03-17 , DOI: 10.1007/s10659-021-09821-z
Dan Givoli

In this classroom note, the old and well-known plane-stress elastic class of problems is revisited, using an analysis technique which is different than that commonly found in the literature, and with a pedagogical benefit. An asymptotic analysis is applied to problems of thin linear elastic plates, made of a homogeneous and rather general anisotropic material, under the plane stress assumption. It is assumed that there are no body forces, that the boundary conditions are uniform over the thickness, and that the material (hence also the solution) is symmetric about the middle plane. The small parameter in this analysis is \(\epsilon =t/D\) where \(t\) is the (uniform) thickness of the plate and \(D\) is a measure of its overall size. The goal of this analysis is to show how the three-dimensional (3D) problem of this type is reduced asymptotically to a sequence of essentially two-dimensional (2D) problems for a small \(\epsilon \). As expected, the leading problem in this sequence is shown to be the classical plane-stress problem. The solutions of the higher-order problems are corrections to the plane-stress solution. The analysis also shows that all six 3D compatibility equations are satisfied as \(\epsilon \) goes to zero, and that the error incurred by the plane stress assumption is \(O(\epsilon ^{2})\). For the special case of an isotropic in-the-plane material, the second-order solution is shown to be the exact solution of the 3D problem, up to an \(O(\epsilon ^{2})\) error in the close vicinity of the edge (which agrees with a well-known result for an isotropic material).



中文翻译:

平面应力问题的渐近分析

在本课堂笔记中,使用一种不同于文献中常见的分析技术,重新探讨了古老而著名的平面应力弹性问题类别,并具有教学上的优势。在平面应力假设下,将渐近分析应用于由均匀且相当普遍的各向异性材料制成的线性弹性薄板的问题。假定没有体力,边界条件在整个厚度上是均匀的,并且材料(因此也是解)关于中间平面对称。此分析中的小参数为\(\ epsilon = t / D \),其中\(t \)是平板的(均匀)厚度,\(D \)是衡量其总体规模的指标。该分析的目的是说明对于一个小\(\ epsilon \),如何将这种类型的三维(3D)问题渐近地简化为一系列基本二维(2D)问题。不出所料,此序列中的主要问题被证明是经典的平面应力问题。高阶问题的解决方案是对平面应力解决方案的更正。分析还表明,随着\(\ epsilon \)趋于零,所有六个3D兼容性方程均得到满足,并且平面应力假设引起的误差为\(O(\ epsilon ^ {2})\)。对于各向同性面内材料的特殊情况,表明二阶解是3D问题的精确解,直至边缘附近的\(O(\ epsilon ^ {2})\)误差(与各向同性材料的众所周知结果相符)。

更新日期:2021-03-18
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