Stress fields of dislocations

An internal (residual) strain field must exist around a dislocation since atoms associated with the dislocation are displaced from their equilibrium positions. The atoms near the dislocation line have substantial displacements from their normal position but outside this region the stress field will be linearly elastic. The 

The stress field around an edge dislocation is more complex, with both shear and dilatational stresses. For example, from Fig. 6.9, one expects o-, to be compressive in the region above the slip plane due to the insertion of the extra half-plane. 

This is compensated by a tensile value of cr, below the slip plane. The stress components around an edge dislocation are given by 

Performing the integration and assuming the logarithmic term can be represented by a geometric coefficient a, one obtains an energy per unit length of dislocation 

---

TWo remarks, however, seem appropriate. 1) If the distance, a, between individual dislocations is very small on an atomic scale, diffusion coefficients obtained from macroscopic experiments can not be used in Eqn. (14.29) (as explained in Sections.1.3). 2) Since diffusional transport takes place in the stress field of dislocations, in principle, fluxes in the form of Eqn. (14.18) should be used. This, however, would complicate the formal treatment appreciably. In the zeroth order approach, one therefore neglects the influence of the stress gradient, which can partly be justified by the symmetry of the transport problem. 

When two dislocations get close together, much of their stress fields cancel out, especially if they are arrayed in a tilt-type grain boundary, or if they are the two ends of a small loop. Therefore if their strain energy fields were important in dissolution, isolated dislocations would etch more rapidly than those in boundaries. In fact they etch at almost identical rates (7), so again it may be concluded that the stress fields of dislocations have little effect cp dissolution at moderate undersaturations. 

An important result of applying Eq. 12.4 is that the force acting on dislocation 1 due to the stress field of dislocation 2 depends on bi b2 if bi b2 is positive, then the dislocations repel one another if it is negative, then the interaction is attractive (but be careful about the line direction). 

Mechanisms involving the nucleation of fresh dislocations in the stress field of dislocations of another type can also be considered but will not be discussed here. 

Since screw and edge components of a mixed dislocation have no common stress components, one can add the respective strain energies in order to obtain the line energy of a mixed dislocation. The strain and stress fields of a screw dislocation (in direction 5) are respectively... 

One may conclude from Eqn. (3.6) that an (arbitrary) stress a exerts both a glide force and a climb force on edge dislocations, but no climb force on screw dislocations (s 6 F=0). Equation (3.6) can also be used to calculate the interaction between two dislocations, that is, the force which the stress field of one dislocation exerts on the unit length of another dislocation at a given coordinate. For parallel dislocations, this force can be written as [J. P. Hirth, J. Lothe (1982)]... 

Although this line of reasoning shows one of the principal features of heterogeneous nucleation, the real situation of nucleation near a dislocation line is much more complex [S. Q. Xiao, P. Haasen (1989)]. The stress field of the dislocation changes the composition of both the matrix and the precipitate, which in turn influences both yp and Agp. In view of this fact, one has to determine whether nucleation near the dislocation occurs before or after the Cottrell atmosphere around the dislocation had sufficient time to form. 

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

Show that the forces exerted on interstitial atoms by the stress field of an edge dislocation are tangent to the dashed circles in the directions of the arrows shown in Fig. 3.8. 

The diffusion of interstitial atoms in the stress field of a dislocation was considered in Section 3.5.2. Interstitials diffuse about and eventually form an... 

An approximate model for the rate of boundary motion can be developed if it is assumed that the rate of dislocation climb is diffusion limited [2], Neglecting any effects of the dislocation motion and the local stress fields of the dislocations on... 

The stress fields of edge dislocations interact with other edge dislocations. The systems energy is lowered if they are aligned so that the compressive field of one... 

The dislocation method of stress analysis is also useful for determining craze stress fields in anisotropic (e.g., oriented) polymers . All one needs here is the stress field of a single dislocation in a single crystal with the same symmetry as the oriented polymer (the text by Hirth and Lothe provides a number of simple cases plus copious references to more complete treatments in the literature) the craze stress field can be generated by superposition of the stress fields of an array of these dislocations of density a(x). Dislocations may also be used to represent the self-stress fields of curvilinear crazes (produced by craze growth in a non-homogeneous stress field for example). Such a method has been developed by Mills... 

Another manifestation of the strain in the films is the presence of the half-loop dislocations extending out from the open tubes in the GaN films grown on the porous substrates, as discussed in the previous section. Regarding the origin of these half-loops, it is easy to see that open tubes (or voids) in a strained film will act as stress concentrators since the normal component of the stress is necessarily zero at the tube wall, the material near the wall will be displaced relative to its position in the absence of the void, and the tangential in-plane component of the stress is thereby increased. In other words, during growth, the shear stress field of the GaN film will be locally concentrated around these open tubes in the film. The open tubes provide a free surface where these half-loops can nucleate due to the increased stress. 

Here (T33 and ajj are the stresses at the origin, where the first dislocation is situated, due to the presence of the other dislocation at (x3,Zo). This is known as the Peach-Koehler force on a dislocation arising from the stress field of the other. This can be generalized to mean that under application of a stress a dislocation experiences a force Ft whose exact relationship is given by the above expressions. 

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... 

The stress field of a screw dislocation is pure shear. As indicated earlier, high strains exist in the core region and, therefore, Hooke s Law of elasticity does not apply and so will not be considered. The dislocation line is parallel to the z axis there are no displacements in the x and y directions and the other stress components are zero ... 

The stress field of a screw dislocation is manifested by two active stresses Tez. acting in radial planes parallel to the z axis and acting in planes normal to the z axis. The stress has a long range, because it is inversely proportional to r and perpendicular to the radius. To get a feeling for this, take a distance of 10" b for 1/... 

If two identically oriented edge dislocations are parallel and lie almost on top of each other, the tensile stress field of one overlaps with the compressive stress field of the other. This is energetically favourable, so the dislocations attract and, in the ideal case, finally stop if one is exactly on top of the other. If several dislocations are arranged in this way, the crystal regions on both sides of the dislocation lines are tilted (figure 6.18). This is called a low-angle grain boundary. 

Grain boundaries are barriers for the movement of dislocations. As the crystal orientation in the neighbouring grain is different, a dislocation cannot simply enter it. The stress field of the dislocation may initiate dislocation movement in the neighbouring grain, but if the slip systems are less favourably oriented there, a larger stress is needed to move dislocations than in the first grain. 

---