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Figure from:A non-singular continuum theory of dislocations

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Fig. 8. The radial component of the self-force per unit length on a glide loop with radius R = 10 and regularization parameter a = 0.1 (uw = 1, v = 0.3). Angle ¢ specifies the orientation of a point on the loop with respect to x-axis. The thin solid line: f?(@), the force predicted by the present model. The dashed line: f'(@), the force predicted by the Gavazza-Barnett model. The thick solid line: f2>2(@), the force difference due to core energy differences predicted by Eq. (56). The circles: the prediction of our model plus the core energy contribution, ie., f'(¢) + f2,d(¢).

Figure 8 The radial component of the self-force per unit length on a glide loop with radius R = 10 and regularization parameter a = 0.1 (uw = 1, v = 0.3). Angle ¢ specifies the orientation of a point on the loop with respect to x-axis. The thin solid line: f?(@), the force predicted by the present model. The dashed line: f'(@), the force predicted by the Gavazza-Barnett model. The thick solid line: f2>2(@), the force difference due to core energy differences predicted by Eq. (56). The circles: the prediction of our model plus the core energy contribution, ie., f'(¢) + f2,d(¢).

Related Figures (13)

Fig. 1. To remove singularities from the elastic energy of a dislocation loop, one convention is to ignore the interaction between differential segments whose distance from each other is less than p. This removes part of the interaction energy between two neighboring segments 7 and j sharing a node. Unfortunately, this “distance cut-off” convention has not been enforced consistently owing to the lack of analytic expressions for the interaction energy between two hinged segments with “excluded” ength (Hirth and Lothe, 1982).
Fig. 1. To remove singularities from the elastic energy of a dislocation loop, one convention is to ignore the interaction between differential segments whose distance from each other is less than p. This removes part of the interaction energy between two neighboring segments 7 and j sharing a node. Unfortunately, this “distance cut-off” convention has not been enforced consistently owing to the lack of analytic expressions for the interaction energy between two hinged segments with “excluded” ength (Hirth and Lothe, 1982).
Fig. 2. (a) Because the stress field is anti-symmetric around an infinite straight dislocation, the average of stress taken at two points at a distances +p from the line is zero. Consequently, the straight dislocation produces no self-force in Brown’s approach. Here € is the line direction, n is the glide plane normal, m is a unit vector perpendicular to € and n, and b is the Burgers vector. (b) P is a point on a dislocation loop on a plane with normal vector n. € is the local tangent direction shown here at an angle a with respect to a datum. m is orthogonal to both n and €. In (Brown 1964), the self force f at point P is computed by applying the Peach-Koehler formula on the averaged stress taken at two points P + pm.
Fig. 2. (a) Because the stress field is anti-symmetric around an infinite straight dislocation, the average of stress taken at two points at a distances +p from the line is zero. Consequently, the straight dislocation produces no self-force in Brown’s approach. Here € is the line direction, n is the glide plane normal, m is a unit vector perpendicular to € and n, and b is the Burgers vector. (b) P is a point on a dislocation loop on a plane with normal vector n. € is the local tangent direction shown here at an angle a with respect to a datum. m is orthogonal to both n and €. In (Brown 1964), the self force f at point P is computed by applying the Peach-Koehler formula on the averaged stress taken at two points P + pm.
Again, as a — 0, the classical singular solution for the stress field about an infinite straight edge dislocation is recovered. For an edge dislocation along z axis, with b, = b, b, = b, = 0, the non-singular solution is
Again, as a — 0, the classical singular solution for the stress field about an infinite straight edge dislocation is recovered. For an edge dislocation along z axis, with b, = b, b, = b, = 0, the non-singular solution is
Fig. 4. The Burgers vector is conserved at every point in the dislocation network. This holds true both for “discretization” nodes (on the left) and for “physical” nodes, e.g. the nodes connecting three or more dislocation together (on the right). Following the convention that the line direction € always flows out of the node, the Burgers vector conservation can be written as, }>;b; = 0 for every node P in the dislocation network.
Fig. 4. The Burgers vector is conserved at every point in the dislocation network. This holds true both for “discretization” nodes (on the left) and for “physical” nodes, e.g. the nodes connecting three or more dislocation together (on the right). Following the convention that the line direction € always flows out of the node, the Burgers vector conservation can be written as, }>;b; = 0 for every node P in the dislocation network.
Fig. 5. Computing driving force on node rg. The contribution from segment rj-rg is the integral pe f(z)o(z)dz, where f(z) = z is the weight function. area due to a virtual displacement of node at ro, ie.,
Fig. 5. Computing driving force on node rg. The contribution from segment rj-rg is the integral pe f(z)o(z)dz, where f(z) = z is the weight function. area due to a virtual displacement of node at ro, ie.,
Fig. 6. (a) A circular glide loop (dashed line) in the z-y plane with Burgers vector b = [100] is discretized into N = 12 segments (solid line) connecting N nodes (small open circles). (b) The radial component on nodal force shown here as a function of node (angular) position along the loop. The straight lines are obtained by stress integration over the segments and circles are obtained by numerical differentiation. For every point on the loop, ¢ marks its orientation angle with respect to x axis.
Fig. 6. (a) A circular glide loop (dashed line) in the z-y plane with Burgers vector b = [100] is discretized into N = 12 segments (solid line) connecting N nodes (small open circles). (b) The radial component on nodal force shown here as a function of node (angular) position along the loop. The straight lines are obtained by stress integration over the segments and circles are obtained by numerical differentiation. For every point on the loop, ¢ marks its orientation angle with respect to x axis.
Fig. 7. (a) Elastic energy WS" computed for N = 12, 100, 200, 400, 500, 600 (circle), where N is the number of straight segments representing a glide dislocation loop of radius R. The dashed line is the analytic solution Eq. (40) (neglecting the O(a?/R?) terms). (b) Total radial force summed over all nodes, compared to the analytic solution for {gb = OR given in Eq. (41)(dashed line). (c) (Crosses) The relative deviation of WS! computed for N = 12,100,200, 400,500,600, from the converged numerical value (NV = 1200). (Circles) The relative deviation of f§) as a function of the number of nodes JN.
Fig. 7. (a) Elastic energy WS" computed for N = 12, 100, 200, 400, 500, 600 (circle), where N is the number of straight segments representing a glide dislocation loop of radius R. The dashed line is the analytic solution Eq. (40) (neglecting the O(a?/R?) terms). (b) Total radial force summed over all nodes, compared to the analytic solution for {gb = OR given in Eq. (41)(dashed line). (c) (Crosses) The relative deviation of WS! computed for N = 12,100,200, 400,500,600, from the converged numerical value (NV = 1200). (Circles) The relative deviation of f§) as a function of the number of nodes JN.
Fig. B.1. A coordinate system for deriving the stress field of a straight dislocation segment.
Fig. B.1. A coordinate system for deriving the stress field of a straight dislocation segment.
the stress components are,
the stress components are,