Stress Fields and Energies of Dislocation

1
Stress Fields and Energies of Dislocation
2
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.

3

• 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.

4
Edge Dislocation

• Figure 14-1. 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.

5
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.
6

• 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 ro is made through
  the center.
• This is done to simplify the mathematical
  treatment.

7

• 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.

8

• 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.

9
Stress Field Due to Edge Dislocations
(14.1)
(14.2)
(14.3)
where
(14.4)
10
(14.5)
(14.6)
(14.7)
11

• 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 y0
• The stress field can also be written in Polar
  Coordinates, and this is given as

(14.8)
(14.9)
12
Screw Dislocation

• 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.

13
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)
14

• 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)
15
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)
16
(b) Screw Dislocation
(14.16)

• Note that
• for both edge and screw dislocations
• If we add the core energy (ro b), the total
  Energy will be given by

(14.17)
(14.18)
17

• For an annealed crystal r1 10-5cm, 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)
18
Forces on Dislocations

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

19

• 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.

20
Figure 14-3. Force acting on a dislocation line.
21

• 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)
22

• 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)
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)
24
?
?
Figure 14-4. Forces on a curved dislocation line.
25

• 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)
26

• 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 14.25 14.26 gives

(14.25)
(14.26)
27

• Recall
• Stress required to bend a dislocation to a radius
  R

(14.27)
(14.28)
28

• A more general form of eqn 14.23 is given as
• where, t is the dislocation line vector
• Expanding equation 14.29 gives

(14.29)
(14.30)
29

• Note eqns. 14.26, 14.29 14.31 are the same
• A particular direct applicant of these is in the
  understanding of the Frank-Read dislocation
  multiplication source

(14.31)
30

• Forces on dislocations can be due to other
  dislocations, precipitates, point defects,
  thermal gradients, second-phases, etc.

(14.32)
(14.33)