A non-singular theory of dislocations in anisotropic crystals
A non-singular theory of dislocations in anisotropic crystals
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DOI：
10.1016/j.ijplas.2017.10.003
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发表时间：
2017-06
期刊：
影响因子：
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通讯作者：
G. Po;M. Lazar;N. Admal;N. Ghoniem
中科院分区：
文献类型：
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作者：
G. Po;M. Lazar;N. Admal;N. Ghoniem
我们发展了三维位错环的非奇异理论在一个特定版本的Mindlin的各向异性梯度弹性与多达六个长度尺度参数。该理论是作为经典各向异性理论在线性化不相容弹性框架下的推广而系统发展起来的。各向异性位错理论的所有关键方程，包括孤立立体角的Burgers位移方程、Peach-Koehler应力方程、弹性变形的Mura-Willis方程和Peach-Koehler力方程，均以线积分的形式导出。在不使用应力函数的情况下，直接得到了两个位错环之间相互作用能的双线积分表达式。结果表明，所有的弹性场都是非奇异的，并且在远离位错核的几个特征长度处收敛到经典的弹性场。在实际应用中，将经典（奇异）各向异性格林张量替换为Lazar和Po （2015b）导出的非奇异各向异性格林张量，可以从经典场中得到非奇异场。弹性解适用于任意各向异性介质。除了经典的各向异性弹性常数外，非奇异格林张量还依赖于描述弱非定域性的二阶对称长度尺度张量，其结构取决于晶体对称性的特定类别。由该张量定义的各向异性亥姆霍兹算子允许一个格林函数作为Burgers向量密度的扩展函数。因此，伯格斯矢量密度在不同的晶体结构中分布不同。基于经典弹性常数和梯度弹性常数的独立原子计算，提出了确定长度尺度参数张量的两种方法。各向异性非奇异理论在没有拟合参数的情况下与分子静力学很好地吻合，并且与奇异理论不同，应力分量的符号在接近核时不显示反转。与各向同性溶液相比，边位错和螺位错单位长度能量密度的差异更为明显。
We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.