Forces Between Dislocations

A dislocation senses a force due to the presence of a stress field around it The presence of more dislocations, each with its own smrounding stress field, means that all these forces influence one another. Consider the presence of two dislocations without the application of an external stress. The total force is assumed to be the sum of the individual forces. In Sect. 3.3.7, it was indicated that a dislocation must have certain strain energy and, indeed, a strain field exists at the site of each dislocation, affecting its motion. 

Given two parallel dislocations with the same sign at sites 1 (the origin) and 2 (as shown schematically in Fig. 3.48a), the stress fields at these screw dislocations have a radial symmetry and their Bmgers vectors are parallel to the z axis. Due to their radial symmetry, their stress is  

There is only one force, because of the radial symmetry, Fr, which is a radial component. The first stress field is acting on dislocation 2 at a distance, r, from the first dislocation and, therefore, this force is  

These two same-signed, parallel dislocations repel each other, depending inversely on the distance, r, (how far apart they are from each other) and, therefore, they stay relatively distant on the same or on near slip planes. 

In Cartesian coordinates, two forces exist, those of and Fy, expressed as  

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The physical interpretation of this result is that it is the force per unit length of dislocation acting on a segment by virtue of the presence of the stress field a. This expression is the famed Peach-Koehler formula and will be seen to yield a host of interesting insights into the interactions between dislocations. 

Fig. 8.23. Interaction force between two perpendicular screw dislocations (adapted from Hartley and Hirth (1965)).

In principle, eqn (11.59) is all that we need to go about computing the interaction force between an obstacle and a dislocation. However, we will find it convenient to rewrite this equation in a more transparent form for the present discussion. Using the principle of virtual work, we may rewrite the interaction energy as... 

With this result in hand, the interaction force between the dislocation and the... 

In 1930 Fritz London demonstrated that he could account for a weak attractive force between any two molecules, whether polar or nonpolar. He postulated that the electron distribution in molecules is not fixed electrons are in continuous motion, relative to the nucleus. So, for a short time a nonpolar molecule could experience an instantaneous dipole, a short-lived polarity caused by a temporary dislocation of the electron cloud. These temporary dipoles could interact with other temporary dipoles, just as permanent dipoles interact in polar molecules. We now call these intermolecular forces London forces. 

Nevertheless, there is little doubt that dislocations play an important part in many solid-state decomposition reactions. Faraday was the first to notice a relationship between dislocations and reaction when he described the spontaneous dehydration of sodium carbonate crystals when scratched with a pin as a curious illustration of the influence of mechanical forces over chemical affinity ... 

Pande and Suenaga [ ] have recently claimed that grain boundary flux pinning is caused by the elastic interaction between the dislocations constituting the grain boundaries and the fluxoids. The interactions between dislocations and fluxoids have long been the subject of studies. The two modes of interaction are (1) the first-order, or volume difference, effect, and (2) the second-order, or shear modulus difference, effect. The former usually dominates The Peach-Koehler equation [ ] can be used to calculate the interaction force between the stress field of the fluxoid lattice (a calculation of which has recently become available [ " ]) and the strain field of the dislocations. In the experiments of this study, the calculation of fpL... 

Equations (3.66-3.72) are based on the expressions in Sect. 3.3.9, specifically Sects. 3.3.9.1 and 3.3.9.2 for screw and edge dislocations, respectively. In Fig. 3.48a and b, the Burgers vectors are parallel to the z and x axes of the screw and edge dislocations, respectively. As mentioned in Sect. 3.3.11.1, there is a repulsive force between same-signed dislocations, an attractive force between unlike dislocations. When y = 0, Eq. (3.69) reduces to... 

Fig. 17. Map of the interaction between dislocations in MgO as function of angles that define the orientation relative to the intersection of sUp planes. Gray shades display interaction force (white attraction, black repulsion), (a) Lomer lock, (b) Hirth lock [443].

An image dislocation, located at —z with the same in-plane Burgers vector, cancels the in-plane displacements at z = 0. The vertical repulsive force between... 

Elastic constants are fundamentetl physical constants that are measures of the interatomic forces in materials, and are often used for the estimation of an interatomic potential that is applied in a computer simulation. They give information about the stiffness of the material and are used for understanding of mechanical properties. For example, the properties of dislocations like Peierls stress, self-energy, interaction between dislocations, etc., are explained by elastic theory. The Peierls stress rp is given by the following equation (Peieris, 1940 Nabarro, 1947) ... 

Figure 8. Interaction force between parallel dislocation lines with Burgers vector 4 [110] for NijAl. The radial component F tangential component Fg, and radial component for an isotropic material F, (iso) are normalized by (2-Krb ), where r is the distance between dislocations and b is the magnitude of the Burgers vector. The orientation of the dislocation lines varies from the direction of the Burgers vector, [110] (Reproduced by permission of Pergamon Press from Yoo, 1987a)...

In materials of nominal purity small holes and cracks are formed when necking starts. These holes and cracks are produced during deformation at inclusions and, possibly, by dislocation interactions. In cylindrical specimens the formation of the neck is accompanied by the setting up of a triaxial stress system in the neck the forces between adjacent transverse sections of the neck have trajectories that follow the profile of the neck and will, therefore, have components normal to the specimen axis. This triaxial stress system may be considered as a hydrostatic stress plus a longitudinal stress. The former does not produce plastic deformation and so the material is effectively hardened. Any holes that are formed in the neck are able to grow transversely more rapidly than they can axially and so are able to coalesce, leading to an internal fracture surface at the centre of the minimum cross-section. The formation of this internal surface is followed by shearing on a surface of maximum shear stress. This surface is conical and the result is the so-called cup and cone fracture, typical of the fracture of many metallic materials at room temperature. 

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