Abstract
The strain and stress fields of a rectangular dislocation loop in an isotropic solid that is a semi-infinite medium (half medium) are developed here for a Volterra-type dislocation. Specifically, the loop is parallel to the free surface of the solid. The elastic fields of the dislocation loop are developed by integrating the displacement equation of infinitesimals dislocation loops over a finite rectangular loop area below the free surface. The strains and stress then follow from the small strain tensor and Hooke’s law for isotropic materials, respectively. In this paper, analytical verification and numerical verification for the elastic fields are both demonstrated. Equilibrium equations and strain compatibility equations are applied in the verification. Also, a comparison with a newly-developed numerical method for dislocations near a free surface is performed as well. The developed solution is a function of the loop depth beneath the surface and can be used as a fundamental solution to solve elasticity, plasticity or dislocation problems.
Highlights
The problem of finding analytical solutions for the elastic fields of dislocations in different material types, material geometry and sizes has occupied researchers for tens of years
In [1] [2], to treat the unphysical stress traction brought on the free surface by a screw dislocation line whose fundamental solution is that of a screw dislocation in an infinite medium, an image screw dislocation with opposite Burgers vector is utilized
Elastic field solutions for dislocation problems as presented are beneficial for several reasons: 1) They serve as fundamental solutions, similar to a Green’s function, for other elasticity, plasticity or dislocation problems, and 2) They serve as verification problems for numerical methods like the collocation-point method or different dislocation dynamic simulation codes
Summary
The problem of finding analytical solutions for the elastic fields of dislocations in different material types, material geometry and sizes has occupied researchers for tens of years. Maurissen and Capella [23] [24] derived the field stress correction terms of a dislocation segment parallel and perpendicular to a free surface in a semi-infinite elastic medium. In [1] [2], to treat the unphysical stress traction brought on the free surface by a screw dislocation line (parallel to the surface) whose fundamental solution is that of a screw dislocation in an infinite medium, an image screw dislocation with opposite Burgers vector is utilized. Several papers utilized the “collocation point” numerical method to solve the problem of dislocation near a flat free surface These collocation point methods enforce zero traction on a select number of surface points and not infinite number of them as in analytical methods. Elastic field solutions for dislocation problems as presented are beneficial for several reasons: 1) They serve as fundamental solutions, similar to a Green’s function, for other elasticity, plasticity or dislocation problems (e.g. for disclination problems, fracture problems, or general eigenstrain problems [32] [33]), and 2) They serve as verification problems for numerical methods like the collocation-point method or different dislocation dynamic simulation codes
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