Dislocations in the theory of gradient elasticity
Dislocations in the theory of gradient elasticity
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DOI：
10.1016/s1359-6462(98)00424-2
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发表时间：
1999-02-05
影响因子：
6
通讯作者：
Aifantis, EC
中科院分区：
文献类型：
--
作者：
Gutkin, MY;Aifantis, EC
本文是我们以前研究梯度弹性理论中位错[1，2]和向错[3]的工作的进一步发展。主要结果是消除了缺陷线处的应变奇异性，类似于裂纹问题的类似工作[4-10]。值得注意的是，以前的连续介质模型，这类缺陷已考虑到偶应力或非局部性（见[3]的审查），不免除在应变场的奇异性，即使其中一些[11-13]声称消除应力奇异性。本文提出的模型消除了应变场和应力场的奇异性。使用线性弹性理论的简单梯度修正，形式为[4] trI 2 c 2 trI 2，（1）其中和是Lamé常数，是应力和应变张量，I是单位张量，2表示Laplacian，c 0是梯度系数，文[4-10]的作者证明了从裂纹尖端应变场的解中消除经典奇异性。受这些结果的鼓舞，我们采用[1，2]同样的梯度理论来考虑位错。特别地，四个位错组态，即螺型位错，刃型位错，和这样的位错的偶极子已被考虑。已经表明，在螺位错的情况下[1]，弹性应变在位错线处为零，并在距离c处达到最大值（bz/10 c）。值得注意的是，对于原子晶格，梯度系数c可以估计为ca/4 [4]，其中a是晶格常数。在Burgers矢量bz取为等于a的情况下，得出上述最大值被估计为12%。在刃型位错的情况下，我们曾错误地指出，只有位错线处的弹性膨胀严格等于零，而弹性应变的其它分量在离位错线很小的区域r 10 3 c（原子晶格为10 3 c）内保持奇异。然而，对极限的更仔细的评估（如r 3 0）表明，所有应变分量在位错线处等于零，在位错核（r 4c）内达到最大值（3-14）%。
The present paper is a further development of our previous works [1–3] dealing with dislocations [1, 2] and disclinations [3] in a special theory of gradient elasticity. The main result there was the elimination of strain singularities at the defect line, in analogy to similar work for crack problems [4–10]. It is worth noting, that previous continuum models for such kind of defects which have taken into account couple stresses or non-locality (see [3] for a review), do not dispense with the singularity in the strain field, even though some of them [11–13] claim elimination of stress singularity. The model proposed here results into elimination of the singularity in both the strain and stress fields. Using a simple gradient modification of the linear theory of elasticity in the form [4] trI 2 c 2 trI 2,(1) where and are the Lamé constants, and are the stress and strain tensors, I is the unit tensor, 2 denotes Laplacian and c 0 is the gradient coefficient, the authors of [4–10] have demonstrated the elimination of classical singularity from the solution for the strain field at the crack tip. Encouraged by these results, we employed [1, 2] the same gradient theory to consider dislocations. In particular, four dislocation configurations ie a screw dislocation, an edge dislocation, and the dipoles of such dislocations have been considered. It has been shown that in the case of screw dislocation [1], the elastic strain is zero at the dislocation line and achieves a maximum value (bz/10c), at a distance c from it. It is worth noting that for an atomic lattice, the gradient coefficient c can be estimated [4] as ca/4, where a is the lattice constant. With the Burgers vector bz taken to be equal to a, it follows that the aforementioned maximum value is estimated as 12%. In the case of edge dislocations [2], we have erroneously reported that only the elastic dilatation is strictly equal to zero at the dislocation line, while the other components of the elastic strain remain singular within an extremely small region r 10 3 c (10 3Å for an atomic lattice) from the dislocation line. More careful evaluation of the limits (as r 3 0), however, shows that all strain components are equal to zero at the dislocation line achieving maximum values (3–14)% within the dislocation core (r 4c).