Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results
Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results
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DOI：
10.1016/j.ijsolstr.2011.03.006
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发表时间：
2011-06-15
影响因子：
3.6
通讯作者：
Aifantis, Elias C.
中科院分区：
文献类型：
--
作者：
Askes, Harm;Aifantis, Elias C.
在本文中，我们讨论了梯度弹性的各种形式及其在静态和动态应用中的性能。梯度弹性理论提供了经典弹性方程的扩展，以及应变、应力和/或加速度的附加高阶空间导数。我们关注梯度弹性理论的通用类别，其中高阶项是相应低阶项的拉普拉斯算子。制定梯度弹性理论的挑战之一是将附加本构参数的数量保持在最低限度。我们首先讨论一般 Mindlin 理论，该理论最一般的形式有 903 个本构弹性参数，但被 Mindlin 简化为三个独立的材料长度尺度。通常可以进行进一步的简化。特别是，Aifantis 理论在静力学中只有一个附加参数，开辟了解析和数值求解过程的全新领域。我们还讨论了如何将其扩展到动力学。给出了长度尺度识别和量化程序的概述。讨论了最常用的梯度弹性版本的有限元实现以及变分一致的边界条件。提供了可以使用简单的线性有限元形状函数实现的特定梯度弹性格式的详细信息。新的数值结果表明消除了静力学和动力学中的奇异性，以及梯度弹性预测的与尺寸相关的机械响应。 (C) 2011 Elsevier Ltd. 保留所有权利。
In this paper, we discuss various formats of gradient elasticity and their performance in static and dynamic applications. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order spatial derivatives of strains, stresses and/or accelerations. We focus on the versatile class of gradient elasticity theories whereby the higher-order terms are the Laplacian of the corresponding lower-order terms. One of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum. We start with discussing the general Mindlin theory, that in its most general form has 903 constitutive elastic parameters but which were reduced by Mindlin to three independent material length scales. Further simplifications are often possible. In particular, the Aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures. We also address how this can be extended to dynamics. An overview of length scale identification and quantification procedures is given. Finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions. Details are provided for particular formats of gradient elasticity that can be implemented with simple, linear finite element shape functions. New numerical results show the removal of singularities in statics and dynamics, as well as the size-dependent mechanical response predicted by gradient elasticity. (C) 2011 Elsevier Ltd. All rights reserved.