Analytic formulation of elastic field around edge dislocation adjacent to slanted free surface

Abstract

Explicit and tractable formulation of the internal stress field around edge dislocations is indispensable for considering the mechanics of fine crystalline solids, because the motion of edge dislocations in a slanted direction with respect to the free surface often plays a vital role in the plastic deformation of the solids under loading. In this study, we formulated an analytical solution for the stress distribution that occurs around edge dislocations embedded in a semi-infinite elastic medium. This formulation is based on the image force method and the Airy stress function method; it describes the variation in the stress distribution with changes in the slanted angle between the traction-free flat surface of the medium and the Burgers vector of the edge dislocation. Furthermore, our analytical solution shows that the attractive force acting on the edge dislocation due to the presence of the free surface is always perpendicular to the surface, regardless of the relative angle of the Burgers vector with the surface.

Screw dislocations in a holed wedge

“…The trend for the location of “c” is expected to be controlled by forces on § 0 from S1, S2, and the hole surface. Also from literature, the force which a free surface exerts on § 0 is always along a perpendicular direction to the free surface (i.e., toward the shortest distance from the surface) [18]. The movement of “c” measured from the hole surface as a function of wedge angle is plotted in Figure 9(a) in normalized units of D / b (see Figure 9(c) for definition of D ).…”

“…An M × 1 vector B contains the unknown Burgers vectors b j . Also, the tractions T k!i z from equation ( 18) must be found for M collocation points to create the M × 1 forcing vector F. The elements of F are from equation (18):…”

Spot–Ladder Selection of Dislocation Patterns in Metal Fatigue

“…Nevertheless, qualitative agreement with experimental results alone is not sufficient for the model to be accepted as reliable; quantitative agreement between theory and experiment also needs to be scrutinized using realistic parameter values [53]. In addition, when considering a realistic metallic specimen of a finite size, it is necessary to consider the attractive force exerted by the outer surface of the specimen on the dislocations inside [54,55]. To accomplish these tasks, numerical simulations based on coupled differential equations will be discussed in future work.…”

Nonsingular Stress Distribution of Edge Dislocations near Zero-Traction Boundary

“…Otherwise, the nucleation and mobility of dislocations near the tip of a crack have a strong impact on the fracture mechanics of the material [ 7 ]. Furthermore, an in-depth understanding of dislocation dynamics and dislocation-induced elastic fields near traction-free boundaries [ 8 , 9 , 10 ] and grain boundaries [ 9 , 11 , 12 , 13 , 14 ], as well as the assessment of the boundary conditions used in the existing formulations [ 15 ] are critical to predicting the performance of advanced materials: they include gold nanowires [ 16 , 17 ], barium titanium nanomaterials [ 18 ], functionally graded materials [ 19 ], and other metallic and semiconductor materials [ 20 , 21 , 22 , 23 ].…”