Stress Fields and Energies of Dislocation

Stress Field Around Dislocations Dislocations are defects; hence, they introduce stresses and strains in the surrounding lattice of a material. The mathematical treatment of these stresses and strains can be substantially simplified if the medium is considered to be isotropic and continuous. Under conditions of isotropy, a dislocation is completely described by the line and Burgers vectors.

With this in mind, and considering the simplest situation, dislocations are assumed to be straight, infinitely long lines. Figure 14-1 shows a hollow cylinder sectioned along the longitudinal direction. This is an idealization of the strains around an edge dislocation.

Figure Simple model for edge dislocation. The deformation fields can be obtained by cutting a slit longitudinally along a thick-walled cylinder and displacing the surface by b perpendicular to the dislocation line. Edge Dislocation

Figure 14-2b. Deformation of a circle containing an edge dislocation. The unstrained circle is shown by a dashed line. The solid line represents the circle after the dislocation has been introduced.

The cylinder, with external radius R, was longitudinally and transversally displaced by the Burgers vector b, which is perpendicular to the cylinder axis in the representation of an edge dislocation. An internal hole with radius r o is made through the center. This is done to simplify the mathematical treatment.

In a continuous medium, the stresses on the center would build up and become infinite in the absence of a hole; in real dislocations the crystalline lattice is periodic, and this does not occur. In mechanics terminology, this is called singularity. A “singularity” is a spike, or a single event. For instance, the Kilimanjaro is a singularity in the African plans. Therefore, we “drill out” the central core, which is a way of reconciling the continuous-medium hypothesis with the periodic nature of the structure.

To analyze the stresses around a dislocation, we use the formal theory of elasticity. For that, one has to use the relationships between stresses and strains (constitutive relationships), the equilibrium equations, the compatibility equations, and the boundary conditions. Hence, the problem is somewhat elaborate.

(14.1) (14.2) (14.3) Stress Field Due to Edge Dislocations where (14.4)

(14.5) (14.6) (14.7)

The largest normal stress is along the x-axis. –This is compressive--- above slip plane. – tensile below slip plane.  xy shear stress is maximum in the slip plane, i.e. when y=0 The stress field can also be written in Polar Coordinates, and this is given as: (14.8) (14.9)

Figure 14-2a. Simple model for screw dislocation. The deformation field can be obtained by cutting a slit longitudinally along a thick-walled cylinder and displacing a surface by b parallel to the dislocation line. Screw Dislocation

Stress Field Due to Screw Dislocations This has complete cylindrical symmetry The non zero components are: In Cartesian coordinate, the stress field matrix is given as: (14.10) (14.11) (14.12)

There are no extra half plane of atoms. Therefore, there are no compressive or tensile normal stresses. The stress field of the screw dislocation can also be expressed in Polar-coordinate system as: (14.13)

Strain Energy The elastic deformation energy of a dislocation can be found by integrating the elastic deformation energy over the whole volume of the deformed crystal. The deformation energy is given for (a) Edge Dislocation (14.14) (14.15)

(b) Screw Dislocation Note that: for both edge and screw dislocations If we add the core energy (r o ~ b), the total Energy will be given by: (14.16) (14.18) (14.17)

For an annealed crystal: r 1 ~ cm, b ~ 2 x10 -8 cm Therefore, Strain energy of dislocation ~ 8eV for each atom plane threaded by the dislocation. Core energy ~0.5eV per atom plane Free energy of crystal increases by introducing a dislocation. (14.19)

Forces on Dislocations When a sufficiently high stress is applied to a crystal: Dislocation move Produce plastic deformation Slip (glide) Climb (high Temperatures)

When dislocations move it responds as though it experiences a force equal to the work done divided by the distance it moves The force is regarded as a glide force if no climb is involved.

b dl ds Figure Force acting on a dislocation line.

The crystal planes above & below the slip plane will be displaced relative to each other by b Average shear displacement = where, A is the area of the slip plane The external force on the area is Therefore, work done when the elements of slip occur is: (14-20) (14-21)

The glide force F on a unit length of dislocation is defined as the work done when unit length of dislocation moves unit distance. Therefore, Shear stress in the glide plane resolved in the direction of b (14.22) (14.23)

Line Tension: In addition to the force due to an externally applied stress, a dislocation has a line tension, T which is defined as the energy per unit length = force tending to straighten the line The is analogous to the surface tension of a soap bubble or a liquid. Consider the curved dislocation. The line tension will produce forces tending to straighten the line & so reduce the total energy of the line. (Energy)

Figure Forces on a curved dislocation line.  

The direction of the net force is perpendicular to the dislocation and towards the center of curvature For small, F ~ 2T But dislocation segment Radius of curvature (14.24)

The line will only remain curved if there is a shear stress which produces a force on the dislocation line in the opposite sense. [recall equation 14.23] equation & gives (14.25) (14.26)

Recall Stress required to bend a dislocation to a radius R (14.27) (14.28)

A more general form of eqn is given as where, t is the dislocation line vector Expanding equation gives: (14.29) (14.30)

Note eqns , & are the same A particular direct applicant of these is in the understanding of the Frank-Read dislocation multiplication source (14.31)

Forces on dislocations can be due to other dislocations, precipitates, point defects, thermal gradients, second- phases, etc. (14.32) (14.33)