The Stress Field of a Dislocation

Because dislocations distort the crystal lattice, an elastic stress field forms around the dislocation line. This will now be shown using the example of an 

As the local strain at the core of the dislocation in the direction of its line vector, 33(2 1 = X2= 0), must be the same as everywhere else in the material, the stress state is a state of plane strain with 33 = 0. Using Hooke s law, we thus find 

These equations have been derived using continuum mechanics. They are thus only valid a few atomic distances away from the core of the dislocation too close to it the assumptions of continuum mechanics do not hold. 

The energy per unit length T has the unit of a force. Analogous to a taut string, T can be considered as a force stretching the dislocation line. Therefore, T is often called the line tension of the dislocation. Frequently, its value is of the order of 10 N. 

It has been mentioned above that it is simpler to consider screw dislocations, since they possess a cylindrical symmetry and only shear stress acts when displacement occurs in the z direction. Elastic distortion occurs with the introduction of a dislocation into an isotropic material. 

Here is another case of a cylindrical ring cut from good material surrounding a screw dislocation, as shown in Fig. 3.42a. Recall that dislocations are defects in 

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  

Once again, the magnitude of the elastic stress is directly proportional to the shear modulus, G, and the Burgers vector, b, and inversely proportional with increasing distance from the dislocation, given as  

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/ 

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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... 

For example, the stress field of a dislocation wiU likely influence the structural configuration of intrinsic and extrinsic defects localised in or near the core region of the dislocation. 

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)]... 

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... 

In order to obtain the stress field of a screw dislocation in an isotropic solid, we can define the displacement field as... 

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... 

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. 

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. 

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... 

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. 

Fig. 6.28. Part (a) shows a schematic illustration of a quantum wire in a V-groove formed by patterning a 100 surface of a cubic crystal substrate, with the glide dislocation on a 111 crystallographic plane within the structure. The conditions for formation of such a dislocation are considered in terms of a superposition of the stress fields of the configurations depicted in parts (b) and (c). The superposition scheme is described in the text.

The determination of a dilatation line stress field is detailed elsewhere (Chateau et al., 1999). The expressions are presented in the same form as by Lin and Thomson (1986), to allow the same use as for dislocations (complex potentials, interaction forces, boundary conditions). Expressed in the complex plane, the stress field at a position z of the dilatation line associated with... 

In order to obtain the stress field of an edge dislocation in an isotropic solid, we can use the equations of plane strain, discussed in detail in Appendix E. The geometry of Fig. 10.1 makes it clear that a single infinite edge dislocation in an isotropic solid satisfies the conditions of plane strain, with the strain along the axis of the dislocation vanishing identically. The stress components for plane strain are given in terms of the Airy stress function, A(r, 0), by Eq. (E.49). We define the function... 

The elastic stress field around a dislocation affects the elastic energy of a dislocation and the interaction between parallel dislocations. The elastic energy per unit length of dislocation between two cylindrical surfaces of radius Cq and R is given by ... 

The long-range stress field about a dislocation loop also cancels and thus the strain energy of a curved dislocation is approximately Gb /4n) ln(p/ro), where p is the radius of curvature. " ... 

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