1. Theoretical shear strength of a solid
shear stress displacement relation
Slip in a perfect, defect free lattice x1
Elastic shear stress-strain relation a
b/2 b
For small x/b
This is 100 – 1000 times larger than the measured
shear strength of solids.

2. Self - interstitial Perfect Crystal Vacancy
Dislocation Mechanics
Defects in Solids
Point Defects; these are thermodynamic in the sense that T and ΔGf define the concentration.
Self - interstitial
Perfect Crystal
Vacancy
For Tmp ~ 103 K , xv ~ 10 -5

3. Line defects - Dislocations
Crystal analogue
Movement of an “edge” dislocation

4. edge dislocation

5. Points in direction of extra half plane Useful rule:
edge dislocation
Def: Burgers vector; b
The “strength” of an edge
Points in direction of extra half plane
Useful rule:
Take the same path around the dislocation in the imperfect crystal. If the path encircles a dislocation (the area encircled by the path should be large enough so that it avoids the heavily distorted area around the supposed dislocation), then it will no longer close. In this case, the Burgers vector points from the start of the path to the end of the path (hence "start-finish").
SF/RH
convention
In a perfect crystal trace out a right-handed path around the line. In the perfect crystal this path should be constructed so that it closes. "Right-handed path" means you point the thumb on your right hand in the line direction , then the path is traced following the sense of your curved fingers.

6. b b Continuum analogue of an edge dislocation
plane-strain displacement of
an edge dislocation.
A cut is made along the cylinder axis
and the surfaces are displaced by b.

7. screw dislocation
Right-handed screw
Left-handed-handed screw

8. Continuum analogue of an screw dislocation
Right-handed screw dislocation along the axis of
a cylinder of radius R and length L. A Burgers circuit
operation shows the dislocation has a positive b.

9. mixed dislocation
where
and is a unit vector normal to the slip plane

10. “node” Conservation of b Formal def. of b
Dislocation must end at a free surface, dislocation node (intersection with other dislocations), grain boundary, or other type of defect (but not a point defect, for instance). A dislocation can not end in the middle of a perfect region of crystal.
Formal def. of b
Reversing the sense of the
dislocation line by reversing
the direction of causes
to reverse direction.

11. Stress fields of straight dislocations in isotropic solids
Screw dislocation
The dislocation is “made” by making a cut
defined by and sliding the bottom
of the cut surface over the top by an amount
b in the +z direction. The displacement is
discontinuous as the cut surface and is
The diagram also shows that the displacement uz increases uniformly with θ to give the discontinuity;
Right-handed screw dislocation along the axis of
a cylinder of radius R and length L. A Burgers circuit
operation shows the dislocation has a positive b.

12. Note that the displacement is only a function
of x and y. Using the definition of strains in terms
of displacements;
we can immediately obtain the strains and then using
the linear elastic isotropic constitutive equations obtain
the stresses.
Rectangular coordinates
Cylindrical polar coordinates

13. Edge dislocation The displacement field and strains of the
edge dislocation can not be determined by
inspection as was the case for the screw.
Instead, we must solve the bi-harmonic
equation. It turns out the appropriate Airy
stress function takes the form,
Using the relation between the stress function
and the stresses we obtain;
An edge dislocation with indicated plain strain displacements.
Rectangular coordinates
Cylindrical polar coordinates

14. Mixed dislocation The stress components for edge dislocations include
while those for a screw dislocation are
Since a mixed dislocation has both screw and edge character it will include all the stress
components from both edge and screw. The only issue is that for the edge component
of the mixed dislocation in place of b in the stress field equations for an edge dislocation
you must use only the edge component of b, bsinθ. Likewise, for the screw component
of the mixed dislocation in place of b in the stress field equations for a screw dislocation
you must use only the screw component of b, bcosθ.

15. Strain energy and self- energy of straight dislocations in isotropic media
Screw dislocation
It’s easiest to work this out for the stress filed in cylindrical polar coordinates since we have only one stress
The strain energy is
The strain energy per unit length of dislocation line is
ro is called the “core” region of the dislocation and corresponds to a region around the
dislocation where the displacements are so large that linear elasticity breaks down.

16. The size of this core region depends on crystal structure
The size of this core region depends on crystal structure. For example dislocations in
fcc solids have a bit larger core regions than that in bcc solids. Generally one can write
, where  is 0.5 – 2 for metals, ~ 2 for an ionic solid such as NaCl and ~1.2 for
covalent crystals.
The self energy of a dislocation includes the elastic energy per unit length plus a term
related to the core energy of the dislocation. For screw dislocations this additional term is
so, we can write the self energy of the dislocation line per unit length as
Edge dislocation
The strain energy is calculated for the edge dislocation the same way as it was
for the screw except we have more terms!

17. 1-n. The core-cutoff size is the same as that discussed for the screw.
Substituting for the stresses and integrating, we obtain
This result is identical in form to that obtained for the screw except for the factor of
1-n. The core-cutoff size is the same as that discussed for the screw.
The core energy of the edge is
so the self energy of the edge dislocation
can be written as,

18. Mixed dislocation
The strain energy of a mixed dislocation can be written as
where is
In a similar fashion the self energy of a mixed dislocation can be written as

19. Forces on dislocations
We can expect a gradient force to act on a dislocation if the mechanical energy
of the system is a function of the position of the dislocation:
To a dislocation we can expect that it is immaterial whether the stresses acting
on it are internal or external in origin.
We want to find a relation between an external or internal stress and it’s action
on the dislocation.

20. An interesting and often not appreciated result is that since the strain energy of
a body is independent of the position of the dislocation, the presence of the
dislocation does not alter the elastic constants of the material.
Consider an unstrained body with no dislocation present. Apply and external force
per unit area over the surface of the body.
causes an elastic displacement and the strain energy
in the body is just the work done by the force,

21. Now form a dislocation by cutting and applying to the cut faces a force causing
Also, there is work done by as is being applied. The total strain energy is

22. Now we can do exactly the same thing in reverse
Now we can do exactly the same thing in reverse. First form the dislocation
by applying resulting in and then apply
Alternatively we could remove and then in the manner of a
thermodynamic cycle.
Since does not act on
Also, is single valued whereas is equal and opposite on the two sides of the
cut surface; the work gained on one face of the cut is exactly balanced by the
work lost on the other face,

23. So, therefore is independent of the position of the dislocation.
This means that the elastic constants are insensitive to the presence or
location of stationary dislocations. *This is for an “infinite” solid as the
presence or effect of surfaces is not taken into account.
Also, this means that the work done by over S and A as the dislocation is
formed must sum to zero.
The “force” on the dislocation will be due entirely to the external work that is
done when it moves,

24. and
This may be considered as the force per unit length acting on a dislocation line
where is the stress in the slip plane in the slip direction.

25. S glide S climb S Edge dislocation Screw dislocation edge & mixed
only one glide plane
(mixed dislocations also)
any crystallographically allowable glide plane
climb S
edge & mixed
dislocations
screw dislocations don’t climb (unless they have a small edge component)

26. Figure showing edge dislocation climb

27. Jogged edge dislocation indicating annihilation and removal of atomic planes.

28. Climb: important in high temperature creep
(helps in dislocation annihilation and circumventing obstructions)
dislocations are important sinks and sources for vacancies
adsorbing
vacancies
emitting
vacancies
dislocation emitting or absorbing vacancies?
dislocation emitting or absorbing vacancies?

29. Mixed & edge dislocations have only one glide plane and can’t cross slip
Screw dislocations
can cross-slip
Cross slip is important as a means of dislocations 1) rearranging themselves, 2) getting around obstacles and 2) annihilating (dynamic recovery) -S +S

30. Generalized PK Force: Exampley
Mixed dislocation with a tangent vector in the negative
z-direction and a Burgers vector x z
force for glide
force for climb

31. Osmotic force
Suppose that there is a supersaturation or deficit of vacancies: The crystal has N vacancies instead of N0, the equilibrium number for a given temperature and pressure. In this case the excess free energy per vacancy is
A dislocation which is at least partly edge can climb to create or destroy vacancies, moving the crystal back toward equilibrium and thereby lowering the free energy of the crystal. Suppose the dislocation absorbs dN vacancies by climbing a distance dx, r b is
the
edge
component of b
where e is
the
volume
per vacanc y

32. Osmotic force (continued)
The change in free energy of the crystal is
The osmotic force, per unit length, is v v v v

33. Line Tension of a dislocation
The line tension T can be approximated as l l r
The maximum stress occurs at r=l/2

34. Dislocation Sources in Crystals
The Frank-Read source

35. Dislocation point defect interactions
The elastic interaction energy between a point defect and a dislocation
results from the interaction of the volume change associated with the defect
and the hydrostatic stress field of the dislocation. The interaction is in the
form of a pDV work term:
The volume change associated with a point defect is
The mean stress of an edge dislocation is

36. Dislocation point defect interactions
The interaction energy is
For r ~ b the interaction energy is maximum at
negative misfit r
positive misfit

37. Strain caused by a glide dislocation moving through a crystalx t h l
When a glide dislocation moves
a distance x on its slip plane in a
crystal of length l, height h, and
thickness t the plastic shear strain
that results is
When there are N dislocations
parallel to the t direction and they
all move an average distance
the total strain in the crystal is
Defining
as the dislocation
density,

38. Image forces on dislocations
A free surface or interface separating materials with “different” elastic constants
will exert a “force” on a dislocation. This is a gradient force in the sense that
the dislocation can lower its energy by moving either toward or away from the
interface. y A x z B r
A screw dislocation A is near a free surface. B is the “image” of A and is of
opposite sign placed symmetrically wrt A so that the free surface boundary
conditions are satisfied;

39. Image forces on dislocations
The stress field of the screw dislocation is
at y = 0,
For x = r at the free surface
Application of the PK equation for the force that B exerts on A yields

40. Image forces on dislocations
This is the equivalent of a gradient force. Consider the elastic energy
per unit length of a screw dislocation;
The gradient force is given by