A non-singular theory of dislocations in anisotropic crystals

Abstract

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 nonsingular, 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.

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

“…">1.The proposed material model is thermodynamically consistent, the related set of deformation variables conforms to that of a material with microstructure advanced by []. Also, in analogy with anisotropic strain gradient elasticity, it can be cast as a model with separable anisotropy featured by an internal length described by a two‐rank tensor …”

“…In this case, the usual concept of internal length parameter (ℓ 2 ) is replaced by the internal length tensor governing the one‐to‐one correspondence between the so‐called dual gradient directions . The latter class of materials is of particular interest for engineering applications …”

“…[51] The latter class of materials is of particular interest for engineering applications. [23,31,44]…”

“…Also, in analogy with anisotropic strain gradient elasticity, it can be cast as a model with separable anisotropy featured by an internal length described by a two-rank tensor. [31,32,44,51] 2. The variational principles here presented lead to a set of six extra boundary conditions of kinematic Dirichlet type on the whole boundary surface.…”

Singularity‐free dislocation continuum theory for anisotropic crystals

“…In this work, a non-singular theory of three-dimensional dislocations in a particular version of Mindlin's anisotropic strain gradient elasticity of form II [5] with up to six length scale parameters is presented [3,4,6]. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of incompatible elasticity.…”