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

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.


Introduction
A dislocation is a long linear defect in crystalline solids, originating from an abrupt local change in the arrangement of atoms. Dislocations are of two types: edge and screw. An edge dislocation is formed when a single extra monoatomic halfplane is inserted midway through the complete crystalline lattice of a metal or alloy, which distorts the nearby planes of atoms. This extra monoatomic surface imposes a substantial lattice Head [19] in the past; however, both the results we obtained and the method we used are different from those given earlier, as detailed at the end of §2 in the present article.

Stress field due to real and imaginary edge dislocations
We consider an edge dislocation near the flat traction-free surface of an isotropic semi-infinite elastic medium. Figure 2 illustrates the configuration of the system embedded in the right-handed Cartesian coordinate system; the z-axis extends from the back to the front, perpendicular to the plane of the paper. The region of x ≤ 0 (shaded in figure 2) is occupied by the semi-infinite elastic medium, and the y-z plane serves as the free surface boundary to the motion of the dislocation. The dislocation is straight and parallel to the z-axis, positioned at (x, y) = (−d, 0), and is at a distance d from the free surface. As shown in the left side of figure 2, the Burgers vector of the edge dislocation (labelled by 'real dislocation' in the panel) is assumed to be tilted by an angle u ðreÞ with respect to the x-axis; the superscript 're' denotes a real dislocation that exists within the actual elastic medium.
If the elastic medium were infinitely large in the three-dimensional space, this edge dislocation would produce a stress field over the entire space. The resulting stress components, denoted by s ðreÞ xx ðx, yÞ, s ðreÞ yy ðx, yÞ and t ðreÞ xy ðx, yÞ, can be derived using coordinate transformation (see appendices A and B). We can prove that all three components obey the following unique function: with different definitions of α c and α s . In equation (2.1), G is a material-dependent constant defined by Nevertheless, the stress field given by equation (2.1) will not be realized within the semi-infinite system, because the components s ðreÞ xx and t ðreÞ xy do not vanish at x = 0. In other words, the stress field realized in the present system is modified to satisfy the condition that the flat surface at x = 0 should be free from any traction forces. The image force method is a theoretical approach for determining the stress field in a semi-infinite system by superposing the stress field produced by a virtual additional dislocation with a reversed sign onto the stress field produced by the real dislocation within the medium. In the present system, this is partly achieved by virtually introducing an image edge dislocation with a negative sign at (x, y) = (d, 0), as shown in the right side of figure 2. The tilt angle of this image dislocation is u ðimÞ . If the image dislocation is solely present at (x, y) = (d, 0) in an infinitely large elastic medium, it produces a stress field denoted by s ðimÞ xx ðx, yÞ, s ðimÞ yy ðx, yÞ and royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 220151 t ðimÞ xy ðx, yÞ, which are also expressed in equation (2.1); however, the following three modifications are required: This indicates that an additional stress field should be superimposed to cancel the component expressed by equation (2.10). This additional stress field can be identified using a method based on Airy's stress function, as explained in the next section.

Airy's stress function approach
The linear elasticity theory states that any in-plane strain problem in two dimensions can be reduced to a partial differential equation with a single unknown, ϕ(x, y) [25]: Once the solution of ϕ in a particular domain of interest is obtained under the given boundary conditions, the stress components within the domain can be derived through partial differentiation of ϕ(x, y) (see equation (2.12)). Thus, our immediate task is to derive such a solution of equation (2.11), denoted by f ðexÞ ðx, yÞ, that cancels the shear components given by equation (2.10); the three components of the stress field derived from the solution are  Figure 2. Illustration of the image force method. A negative dislocation with tilt angle u ðimÞ is virtually introduced at the opposite side of the free surface with respect to the positive dislocation tilted by u ðreÞ that exists inside the elastic medium. The distance d from the free surface to the dislocations is the same for both the real (left) and image (right) dislocations.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 220151 Accordingly, each component of the actual stress field generated within the present semi-infinite system can be expressed as the sum of three components (real, image and excess), as follows: with j = re, im, ex. Furthermore, since the boundary condition at the free surface requires σ xx (x, y) and τ xy (x, y) to vanish at x = 0, the two components s ðexÞ xx ðx, yÞ and t ðexÞ xy ðx, yÞ, derived from the solution f ðexÞ ðx, yÞ must satisfy the following relations: and t ðexÞ xy ð0, yÞ ¼ À2G We hypothesize that such a solution, f ðexÞ ðx, yÞ, that satisfies both equations (2.16) and (2.17) can be obtained using variable separation: f ðexÞ ðx, yÞ ¼ jðxÞhðyÞ: ð2:18Þ Substituting it to equation (2.11), we have  Because the stress field produced within the elastic medium far from the free surface should converge to zero, ξ(x) must vanish at the limit of x → −∞, which implies that c 5 = c 6 = 0. In addition, because the traction-free condition at the surface, s ðexÞ xx ð0, yÞ ¼ 0, is satisfied only if ξ(x)∂ 2 η( y)/∂y 2 = 0 at x = 0, ξ(x) mush vanish at x = 0, which implies that c 3 = 0. As a consequence, the solution of f ðexÞ ðx, yÞ that satisfies the boundary conditions reads where we explicitly expressed that a 1 and a 2 depend on k.  with γ c and γ s defined as below: in the case of s ðexÞ xx ðx, yÞ, we have g c ¼ À6dxðx À dÞ 2 y þ 2dxy 3 and g s ¼ xðx þ dÞðx À dÞ 3 À 6dxðx À dÞy 2 À xy 4 :
The complete stress distribution within the semi-infinite medium is determined by superposing the stress fields of the real edge dislocation at (−d, 0) and the image dislocation at (d, 0), and the stress field derived from Airy's stress function f ðexÞ ðx, yÞ, as shown in equations (2.13)-(2.15).
As briefly mentioned in the Introduction, we are aware that the stress components under similar conditions to those dealt with in the present work have been considered earlier by Head [19] using a different approach. However, Head's solution does not satisfy the equilibrium conditions expressed by @s xx @x þ @t xy @y ¼ 0 and @t xy @x þ @s yy @y ¼ 0: ð2:40Þ The conditions of equation (2.40) ensure that the forces acting on the inside of the system are balanced with each other and thus the system is stationary. We believe, therefore, that it remains to be debated whether the Head's solution that does not satisfy the equilibrium condition is consistent with the stress field in actual elastic media. On the other hand, that our solutions of the stress components satisfy the equilibrium condition can be proved straightforwardly by substituting them into equation (2.40). Figure 3 shows the spatial distributions of the three stress components produced by the real edge dislocation at (x, y) = (−1, 0). The tilt angle θ of the Burgers vector with the x-axis is θ = 0 in (a)-(c), θ = π/4 in (d)-( f ), and θ = π/2 in (g)-(i). In all the plots, G=b and b are taken as the unit of stress and the length scale, respectively. In many cases, μ of a metal reaches in the order of tens of gigapascals, and b is a few angstroms. Assuming Poisson ratio ν to be 0.3, therefore, G=b is estimated to be several gigapascals.

Total stress distribution
All the plots show a clear deviation in the stress field from the field generated in an infinitely large system with no surface boundary; in the latter, a vertical or lateral symmetry around the core of dislocation should be observed based on the function form given by equations (A 3)-(A 5). In particular, the σ yy distribution close to the free surface exhibits a significant deviation from the symmetric distribution observed in an infinite system; the figure shows that because of the region having a non-zero stress component expanding significantly toward the edge of the sample, a large tensile or compressive force along the y-direction acts on the free surface. The region that deviates considerably from the stress distribution in such an infinite system extends from the centre of the dislocation to a point several times the magnitude of the Burgers vector.
Notably, the physical quantity measured in an actual experiment is often strain rather than stress. By using the analytical solution for the stress distribution derived herein, the strain distribution can be easily obtained from the relational expression below: where E is Young's modulus of the material considered.

Peach-Koehler force
We now consider the Peach-Koehler force exerted on the real-edge dislocation at (−d, 0). The x-and y-components of the force, f x and f y , can be written as [ Substituting these values into equations (3.4) and (3.5), we obtain a highly concise expression: Since the sign of f x is always positive, the force acting on the edge dislocation is always directed toward the free surface. Notably, when considering the motion of edge dislocations, the component of the force acting on the dislocations that is parallel to the slip plane is often important. In such a case, it is necessary to only decompose the attraction force into parallel (f glide ) and normal (f climb ) components in the gliding  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 220151 direction (i.e. b); then, we have f glide = f x cos θ and f climb = f x sin θ. In actual crystalline samples, edge dislocations start to move along the slip plane only when the force f glide exceeds various frictional forces. An important implication of the concise expression, equation (3.9), is that the force exerted on the edge dislocation near the free surface is always oriented normal to the surface, as consistent with the earlier theoretical work [29,30]. This observation may seem to be in contrast to our finding that the stress distribution created by the edge dislocation changes differently depending on the change in the direction of the Burgers vector, while the same conclusion can be obtained by an energetics argument on the near-surface edge dislocation [11]. It should also be mentioned that under certain conditions, the direction of the attraction force does not have to be perpendicular to the free surface if it is uneven rather than flat [31].
Another interesting finding is that the magnitude of the attraction force is independent of the tilt angle θ of the Burgers vector. It depends only on the distance d from the surface; specifically, it is inversely proportional to this distance. The θ-independence of the attraction force may also seem counterintuitive, given that the stress distribution is strongly dependent on θ, while it has been proved exactly through our formulation. Special attention should be paid to the fact that the magnitude of the action force, Gb=ð2dÞ, is identical to that of the force generated when two edge dislocations with Burgers vectors having opposite signs are located on a straight line parallel to the Burgers vector and separated by a distance of 2d (i.e. when u ðreÞ ¼ u ðimÞ ¼ 0 in figure 2). That is, the attraction force from the free surface acting on the edge dislocations in the semi-infinite elastic medium is determined only by the contribution from the image dislocation; the stress distribution created by the Airy function offers no contribution. This phenomenon is known to occur in a system with θ = 0; however, we found that it also holds true for any choice of θ.

Summary
Herein, we derived an analytical solution for the stress field distribution around edge dislocations positioned near the slanted free surface of a semi-infinite elastic medium. The explicit function forms of the stress components were derived using the image force method and Airy's function method. The findings revealed significant variations in the stress distribution in response to changes in the direction of the Burgers vector of the dislocation. By contrast, the stress field, however, the attraction force exerted by the free surface on the edge dislocation is independent of the direction of the Burgers vector; it follows a universal function that is inversely proportional to the distance between the dislocation and the free surface.
Data accessibility. All data and models generated or used during the study appear in the submitted article.

Appendix C. Residue theorem
This appendix explains how to compute the integrations given by equations (2.32) and (2.33) for obtaining the coefficients ka 1 (k) and ka 2 (k) that are important for determining the stress components derived from the field f ðexÞ ðx, yÞ. The computation is based on the residue theorem, which is a powerful tool for evaluating the line integrals of analytic functions over closed curves.
Assume a complex plane associated with the complex-valued variable z (figure 5). C 1 denotes the line segment [−R, R] on the real axis (oriented to the right), and C 2 represents the semicircular arc of radius R on the upper half-plane centred at the origin, in the counterclockwise direction. Connecting them, we obtain a closed curve, designated as C(= C 1 + C 2 ).
Next, we consider the contour integral Þ C f ðzÞdz with respect to the complex-valued function f (z) defined by f ðzÞ ¼ ðz 2 À d 2 Þðz sin u À d cos uÞ ðz 2 þ d 2 Þ 2 e ikz ðk . 0Þ: ðC 1Þ It follows from equation (C 1) that f (z) has two singular points at z = id and z = −id, both of which are second order. Of these, only the former is included inside C.  Figure 5. Semi-circular integral path on the complex plane, which allows us to calculate the integrals in equations (2.32) and (2.33). The integrands have two points of singularity, z = id and z = −id, among which only the former point is surrounded by the integral path.