Two finite element discretizations for gradient elasticity
Two finite element discretizations for gradient elasticity
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DOI：
10.1061/(asce)0733-9399(2009)135:3(203
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发表时间：
2009-03
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通讯作者：
A. Zervos;Stefanos-Aldo Papanicolopulos;I. Vardoulakis
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作者：
A. Zervos;Stefanos-Aldo Papanicolopulos;I. Vardoulakis
我们提出并比较了两种不同的数值求解梯度弹性边值问题的方法。第一种方法基于使用位移公式的有限元离散化，其中需要保证应变连续性的元素（即 C1 插值）。两个这样的元素被呈现并显示为收敛：具有直边的三角形和等参四边形。第二种方法基于 Mindlin 弹性与微观结构的有限元离散，梯度弹性是其中的一个特例。提出了两个等参元素，一个三角形和一个四边形，对位移和微变形场进行插值。结果表明，通过适当选择材料参数，它们可以为梯度弹性边值问题提供近似解。使用这两种方法来解决基准问题，以评估它们在准确性、简单性和计算性方面的相对优点和缺点。
We present and compare two different methods for numerically solving boundary value problems of gradient elasticity. The first method is based on a finite-element discretization using the displacement formulation, where elements that guarantee continuity of strains (i.e., C1 interpolation) are needed. Two such elements are presented and shown to converge: a triangle with straight edges and an isoparametric quadrilateral. The second method is based on a finite-element discretization of Mindlin’s elasticity with microstructure, of which gradient elasticity is a special case. Two isoparametric elements are presented, a triangle and a quadrilateral, interpolating the displacement and microdeformation fields. It is shown that, using an appropriate selection of material parameters, they can provide approximate solutions to boundary value problems of gradient elasticity. Benchmark problems are solved using both methods, to assess their relative merits and shortcomings in terms of accuracy, simplicity and computa...