Mixed finite element formulations of strain-gradient elasticity problems
Mixed finite element formulations of strain-gradient elasticity problems
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DOI：
10.1016/s0045-7825(01)00353-x
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发表时间：
2002-01-01
影响因子：
7.2
通讯作者：
Aravas, N
中科院分区：
文献类型：
--
作者：
Amanatidou, E;Aravas, N
内在或材料长度尺度的理论在尺寸依赖现象的建模中找到了应用。在弹性力学中，长度尺度通过弹性应变能函数进入本构方程，在这种情况下，它不仅取决于应变张量，而且取决于旋转和应变张量的梯度。本文详细介绍了Mindlin等人在60年代提出的应变梯度弹性理论。在这些理论中，当问题用位移形式表示时，控制偏微分方程是四阶的。如果使用传统的有限元数值求解此类问题，则要求C-1位移连续。另一种“混合”的有限元公式的开发，其中的位移和位移梯度被用作独立的未知数和它们的关系是强制执行的“积分意义”。一个变分公式，可用于线性和非线性应变梯度弹性理论。所得到的有限元只需要C-0连续性，并且公式化简单。所提出的技术被应用到一些问题，并与现有的精确解进行比较。(C)2002 Elsevier Science B. V.保留所有权利。
Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C-1 displacement continuity is required. An alternative "mixed" finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an "integral-sense". A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C-0 continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made. (C) 2002 Elsevier Science B.V. All rights reserved.