The fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.

在这项工作中解决了半个多世纪未解决的 Mindlin 型不相容各向同性应变梯度弹性理论中位错的基本问题。Mindlin 类型的不相容应变梯度弹性是 Mindlin 相容应变梯度弹性的推广，包括塑性场，以这种方式为研究位错等缺陷提供了适当的本征应变框架。推导出螺型位错和刃型位错的位移场、弹性变形、柯西应力、塑性变形和位错密度的精确解析解。对于位错场的数值分析，已经使用了从 ab initio DFT 计算中获取的弹性常数和梯度弹性常数。位移、弹性变形、螺旋位错和刃位错的塑性变形和柯西应力场是非奇异的、有限的和光滑的。螺旋位错的位错场取决于一个特征长度，而刃位错的位错场取决于多达三个特征长度。对于螺型位错，在 Mindlin 型不相容应变梯度弹性中获得的位错场与在简化的不相容应变梯度弹性中对应的位错场一致。在边缘位错的情况下，在 Mindlin 类型的不相容应变梯度弹性中获得的位错场比在简化的不相容应变梯度弹性中的相应位错场更真实。其中，在 Mindlin 类型的不相容各向同性应变梯度弹性中获得的边缘位错的柯西应力在位错核心区域看起来比在简化的不相容应变梯度弹性中获得的柯西应力更物理，并且与计算的边缘位错的应力场非常一致在原子模拟中。此外，研究表明，由于其在 Mindlin 型不相容各向同性应变梯度弹性中固有的不对称性，刃型位错的位错核的形状比在简化的不相容应变梯度弹性中具有圆柱对称性的位错核具有更现实的不对称形式。 .