Dislocations stress field

One consequence of Equation 7.23 is that edge dislocations attract impurity atoms. Impurities (and intrinsic defects) are nearly always the wrong size for the site on which they lie. An oversized atom will be attracted to the tensile portion of the edge dislocation stress field while an undersized atom will be attracted to the compressive region. In other words, essentially all impurities will be attracted to edge dislocations. This is good when the impurities are detrimental but can be harmful when the impurities are intended dopant atoms. 

Next, let us compile some quantitative relations which concern the stress field and the energy of dislocations. Using elastic continuum theory and disregarding the dislocation core, the elastic energy, diS, of a screw dislocation per unit length for isotropic crystals is found to be... 

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... 

Let us derive the force F which is exerted by an externally applied stress field a (or rather o) on a unit length segment of a dislocation. If this segment is differentially displaced by d/, the (surface) force is a-dA (cL4 = s-dr), and by this displacement the shift, b, of atoms on opposite sides of cL4 extracts an amount of work... 

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)]... 

Dislocations move when they are exposed to a stress field. At stresses lower than the critical shear stress, the conservative motion is quasi-viscous and is based on thermal activation that overcomes the obstacles which tend to pin the individual dislocations. At very high stresses, > t7crit, the dislocation velocity is limited by the (transverse) sound velocity. Damping processes are collisions with lattice phonons. 

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. 

This chapter is concerned with the influence of mechanical stress upon the chemical processes in solids. The most important properties to consider are elasticity and plasticity. We wish, for example, to understand how reaction kinetics and transport in crystalline systems respond to homogeneous or inhomogeneous elastic and plastic deformations [A.P. Chupakhin, et al. (1987)]. An example of such a process influenced by stress is the photoisomerization of a [Co(NH3)5N02]C12 crystal set under a (uniaxial) chemical load [E.V. Boldyreva, A. A. Sidelnikov (1987)]. The kinetics of the isomerization of the N02 group is noticeably different when the crystal is not stressed. An example of the influence of an inhomogeneous stress field on transport is the redistribution of solute atoms or point defects around dislocations created by plastic deformation. 

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. 

Let us return to the reduction of shear stress at the crack tip due to the emission of dislocations. Figure 14-9 illustrates a possible stress reduction mechanism. It can be seen that the tip of a crack is no longer atomically sharp after a dislocation has been emitted. It is the interaction of the external stress field with that of the newly formed dislocations which creates the local stress responsible for further crack growth. Thus, the plastic deformation normally impedes embrittlement because the dislocations screen the crack from the external stress. Theoretical calculations are difficult because the lattice distortions of both tension and shear near the crack tip are large so that nonlinear behavior is expected. In addition, surface effects have to be included. 

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... 

In a more general stress field, the force (which is always perpendicular to the dislocation line) can have a component in the glide plane of the dislocation as well as a component normal to the glide plane. In such a case, the overall force will tend to produce both glide and climb. However, if the temperature is low enough that no significant diffusion is possible, only glide will occur. 

Use Eq. 11.12 to show that a dislocation in a crystal possessing a uniform nonequilibrium concentration of point defects and a uniform stress field will... 

Heterophase Interfaces. In certain cases, sharp heterophase interfaces are able to move in military fashion by the glissile motion of line defects possessing dislocation character. Interfaces of this type occur in martensitic displacive transformations, which are described in Chapter 24. The interface between the parent phase and the newly formed martensitic phase is a semicoherent interface that has no long-range stress field. The array of interfacial dislocations can move in glissile fashion and shuffle atoms across the interface. This advancing interface will transform... 

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 force per unit length, /, exerted on the dislocation by the stress field in the climb direction normal to the glide plane is then... 

Coherent Nucleation. The elastic interaction between the strain field of the nucleus and the stress field in the matrix due to the dislocation provides the main catalyzing force for heterogeneous nucleation of coherent precipitates on dislocations. This elastic interaction is absent for incoherent precipitates. 

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