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Chapter 5
Analytical
techniques and solutions for linear elastic solids
5.9 Energetics of Dislocations in Elastic
Solids
Dislocations
play a crucial role in determining the response of crystalline materials to
stress. For example, plastic flow in
ductile metals occurs as a result of dislocation motion; dislocation emission
from a crack tip can determine whether a material is ductile or brittle; and
stress induced dislocation nucleation plays a critical role in semiconductor
devices.
Dislocations
tend to move through a crystal in response to stress. The goal of this and following sections is
to derive some results that can be used to predict this motion.
5.9.1
Classical solution for potential energy of an isolated dislocation loop in an
infinite solid
In
this section, we show that the energy of an isolated dislocation loop with
burgers vector b in an infinite
solid can be calculated using the following expressions:
Here,
 and the integral is taken around the
dislocation line twice. In the first
integral,  is held fixed, and  varies with position around the dislocation
line; then  is varied for the second line integral.
Difficulties with evaluating the
potential energy in the classical solution: In practice, this is a purely formal result  in the classical solution, the energy of a
dislocation is always infinite.  You
can see this clearly using the solution for a straight dislocation in an
infinite solid given in Section 5.3.4.Â
Recall that the stress state for a pure edge dislocation, with line
direction parallel to the  axis and burgers vector  at the origin of an infinite solid is given
(in polar coordinates) byÂ
The
strain energy density distribution around the dislocation follows as
We
can use this to calculate the total strain energy in an annular region around
the dislocation, with inner radius a,
and outer radius b.  The result is
Taking
 gives an infinite energy, because the strain
energy density varies as  near the dislocation core.
Various
ways to avoid this problem have been proposed.  The simplest approach is to neglect the
strain energy in a tubular region with small radius  surrounding the dislocation, on the grounds
that the elastic solution does not accurately characterize the atomic-scale
deformation near the dislocation core. This works for straight dislocations,
but is not easy to apply to 3D dislocation loops.  A more satisfactory approach is described
in the next section.
Application to a circular prismatic
dislocation loop As an example, we
attempt to apply the general formula to calculate the energy of a circular
dislocation loop, with radius a,
which lies in the  plane, and has a burgers vector  that is perpendicular to the plane of the
loop. For this case, the contour
integral for the potential energy reduces to
(To
see this, note that the result of the integral with respect to x must be independent of  by symmetry). As expected, the integral is
divergent. In the classical theory,
the energy of the loop is estimated by truncating the integral around the
singularity, so that
where
r is a small cut-off distance. This is somewhat similar to the core cutoff
radius ,
but the relationship between  and r
is not clear.
A Circular glide loop, which has burgers vector b (with magnitude b) in the plane of the loop, has energy
Derivation of the solution for the
energy of a 3D dislocation loop
1. Let  denote the displacement, strain and stress
induced by the dislocation loop.  The
total potential energy of the solid can be calculated by integrating the
strain energy density over the volume of the solid
2. The potential energy can also be expressed in terms
of the displacement field in the solid, as
where we have used the symmetry of  and recalled that the stress field satisfies
the equilibrium equation
3. Applying the divergence theorem, and taking into
account the discontinuity in  across S,
4. Next, we substitute the expression given in Section
5.8.4 for  and reverse the order of integration
5.
Finally, the
surface integrals in this expression can be transformed into a contour
integral around C by means of Stoke’s theorem
After some tedious index manipulations, this gives the required result.
5.9.2 Non-singular dislocation theory
The
infinite potential energy associated with the classical description of a
dislocation is unphysical, and highly unsatisfactory. A straightforward approach to avoiding this
difficulty was proposed by Cai et al, Journal of the Mechanics and Physics of
Solids, 54, 561-587, (2006).
In
the classical solution, the dislocation core is localized at a single point
in space, which leads to an infinite energy.Â
In practice, dislocation cores are distributed over a small, but
finite, area as indicated in the figure. The TEM micrograph is from Tillmann
et al Microsc. Microanal. 10, 185–198, 2004.
This
effect can be modeled approximately by using the classical solution to
construct a distributed dislocation
core. To this end, we suppose that the
burgers vector of the dislocation can be represented by a distribution ,
which must be chosen to satisfy
where
the volume integral extends over the entire infinite solid. In principle,  could be constructed to give an accurate
description of the atomic-scale strain field in the immediate neighborhood of
the dislocation core, but this is difficult to do, and is not the main intent
of the theory. Instead,  is selected to make the expressions for the
energy and stress field of the dislocation as simple as possible. It is particularly convenient to choose  to satisfy
where  is a small characteristic length, comparable
to the dimensions of the dislocation core. The required distribution cannot
be calculated exactly, but is closely approximated by
with ,
,
. The distribution can also be shown to
satisfy
 Â Nonsingular
energy: The expression for the energy
of a dislocation loop then reduces to
This
is virtually identical to the classical singular solution, except that the
derivatives of  are bounded everywhere, so the integral is
finite.
 Nonsingular
Stress: The stress due to the
dislocation loop can be expressed in terms of a function ,
defined as
This
function cannot be calculated exactly, but can be estimated using the
approximation to  as
The
stress field then becomes
Alternatively,
one may calculate exactly a modified stress measure, defined as
This
stress measure is particularly convenient for calculating the force tending
to make a dislocation move, as shown in a subsequent section. In addition,  except very close to the core of a
dislocation.  It is straightforward to
show that
where
,
as before.Â
Nonsingular energy of circular
dislocation loops. It is straightforward to calculate the
energy of a circular dislocation loop.Â
Cai et al Journal of the
Mechanics and Physics of Solids, 54,
561-587, (2006) give:
 Prismatic Loop: (b
perpendicular to loop)
 Glide Loop: (b in
the plane of the loop):
5.9.3
Energy of a dislocation loop in a stressed, finite elastic solid
The
figure shows a dislocation loop in an elastic solid. Assume that:
1. The solid is an isotropic, homogeneous, linear
elastic material with Young’s modulus E
and Poisson’s ratio
2. The solid contains a dislocation, which is
characterized by the loop C, and
the burgers vector b for the
dislocation, following the conventions described in the preceding
section. As before, we can imagine
creating the dislocation loop by cutting the crystal over some surface S, and displacing the two material
surfaces adjacent to the cut by the burgers vector. The figure shows the dislocation loop to be
completely contained within the solid, but this is not necessary  the surface S could intersect the exterior boundary of the solid, in which
case the dislocation line C would
start and end on the solid’s surface.
3. Part of the boundary of the solid  is subjected to a prescribed displacement,
while the remainder  is subjected to a prescribed traction. Note that there is some ambiguity in
specifying the prescribed displacement.Â
In some problems, the solid contains a dislocation before it is
loaded: if so, displacements are measured relative to the solid with traction
free boundary, but containing a dislocation.Â
In other problems, the dislocation may be nucleated during
deformation. In this case,
displacements are measured with respect to the initial, stress free and
undislocated solid. In the discussion
to follow, we consider only the latter case.
To
express the potential energy in a useful form, it is helpful to define
several measures of stress and strain in the solid, as follows:
1. The actual fields in the loaded solid containing the
dislocation will be denoted by . Note that  is measured with respect to a stress-free
solid, which contains no dislocations.Â
The displacement is discontinuous across S.
2. The fields induced by the applied loading in an
un-dislocated solid will be denoted by . The displacement field  is continuous.
3. The fields in a solid containing a dislocation, but
with  traction free, and with zero displacement on
 will be denoted by
4. The fields in an infinite
solid containing a dislocation with line C and burgers vector b will
be denoted by . If the dislocation line terminates on the
solid’s surface, any convenient procedure can be used to close the loop when
deriving the infinte solid solution, but the fields will depend on this
choice.
5. The difference between the fields for a dislocation
in a bounded solid and the solution for a dislocation in an infinite solid
will be denoted by
Â
The potential energy of the
solid can be expressed as
where
  is the strain energy of the dislocation
itself
  is the work done to introduce the
dislocation into the externally applied stress
  is the potential energy of the applied loads
The strain energy of the
dislocation can also be expressed as a sum of two terms:
where
  is the energy of a dislocation with line C in an infinite solid, which can be
calculated using the expressions in 5.9.2.
   is the change in potential energy due to the
presence of boundaries in the solid.Â
If
the classical approach is used to represent the burgers vector of the
dislocation, the energy is infinite, because  contains a contribution from the singular
dislocation core. The remaining terms
are all bounded. The simplest way to
avoid this unsatisfactory behavior is to estimate  using the non-singular dislocation theory
presented in 5.9.2, but use the classical expressions for all the remaining
terms.  This is not completely
consistent, because in a rigorous non-singular dislocation theory all the
terms should be computed by taking a convolution integral with the burgers
vector distribution.  However,
provided the solid is large compared with the dislocation core, the error in
the approximate result is negligible.
Derivation
- The potential energy of the solid is given by
the usual expression
- The total stress consists of the dislocation
fields, together with the externally applied fields, so that
- This expression can be re-written as
To see this, note that  from the symmetry of  and the strain-displacement relations, and
that  because of the symmetry of the elasticity
tensor .
- The terms involving  can now be integrated by parts, by
writing, for example
because  is an equilibrium stress field. Using this result, applying the divergence
theorem, and taking into account the discontinuity in the displacement field
across S gives
where we have noted that  on the exterior boundary of the solid, and
that . A similar procedure gives
Finally,
substituting this result back into the expression for V and noting  on  and  on  gives the required result.
5.9.4
Energy of two interacting dislocation loops
Consider two dislocation
loops in an infinite elastic solid, as shown in the figure. Assume that
- The solid is an
isotropic, homogeneous, linear elastic material with Young’s modulus E and Poisson’s ratio
- The dislocations can
be characterized by surfaces, contours and burger’s vectors  and .
The total potential energy
of the solid can be calculated from the following expressions
where
  and  are the energies of the two dislocation
loops in isolation, which can be computed from the formulas in 5.9.1 (or 5.9.2
if you need a non-singular expression)
  is an `interaction energy,’ which can be
thought of as the work done to introduce dislocation 2 into the stress field
associated with dislocation 1 (or vice-versa). The interaction energy is given by
Although
this integral is bounded (provided the dislocation lines only meet at
discrete points), it is sometimes convenient to replace  by  for a non-singular treatment of
dislocations.
HEALTH WARNING: Notice that the expression for the interaction
energy is very similar to the formula for the self-energy of a dislocation
loop, except that (i) it contains an extra term (which vanishes if  ), and (ii) the integrals in the interaction
energy are twice those in the self-energy.Â
The latter is an endless source of confusion.
Derivation: We can regard the two interacting dislocations as a
special case of a dislocation loop subjected to an applied stress:Â one dislocation generates the `applied
stress,’ which influences the second dislocation. The total potential energy follows as ,
where ,
 are the
potential energies of the two isolated dislocations, and  is the interaction energy. We have that
where
 is the stress induced by dislocation 2. We can express this stress in terms of a
line integral around dislocation 2.Â
Finally, the surface integral over S1 can reduced to a contour
integral around dislocation 1 by applying Stokes theorem.
5.9.5
Driving force for dislocation motion  The Peach-Koehler formula
If
a dislocation is subjected to stress, it tends to move through the
crystal. This motion is the mechanism
for plastic flow in a crystalline solid, as discussed in Section 3.7.12.
The
tendency of a dislocation to move can be quantified by a force. This force needs to be interpreted
carefully: it is not a mechanical force (in the sense of Newtonian
mechancics) that induces motion of a material particle, but rather a generalized force (in the sense of
Lagrangean mechanics) that causes a rearrangement of atoms around the
dislocation core.  It is sometimes
known as a `configurational force’
The generalized force for
dislocation motion is defined as follows.
- Consider an elastic
solid, which contains a dislocation loop. The loop is characterized by a
curve C, the tangent vector ,
and the burgers vector b. As
usual, we can imagine creating the dislocation loop by cutting the
crystal over some surface S that
is bounded by C, and
displacing the two material surfaces adjacent to the cut by the burgers
vector.Â
- Suppose that the
dislocation moves, so that a point at  on C
advances to a new position ,
where n(s) is a unit vector
normal to C, as shown in the
figure (in the figure, the dislocation moves ina single plane, but this
is not necessary).
- As the dislocation moves, the potential energy
of the solid changes by an amount . This change of energy provides the
driving force for dislocation motion.
- The driving force is
defined as a vector function of arc-length around the dislocation ,
whose direction is perpendicular to C,
and which satisfies
for all possible choices of  and  (the change in energy is negative because
the displacement is in the same direction as the force).
The Peach-Koehler formula
states that the driving force for dislocation motion can be computed from the
following formula
where  is the total
stress acting on the dislocation at a point s along the curve C (the stress includes contributions from the dislocation itself,
as well as stresses generated by external loading on the solid).
The
Peach-Koehler equation is meaningless without further discussion, because the
classical solution predicts that the stress acting on the dislocation line is
infinite.  To avoid this, we need to partition the
stress according to its various origins, as described in Section 5.9.3.
- We assume that the
dislocation loop lies within an elastic solid, which is subjected to
some external loading. The
external fields subject part of the boundary of the solid  to a prescribed displacement; and the
remainder of the boundary  to a prescribed traction.
- The actual fields in
the loaded solid containing the dislocation will be denoted by .Â
- The fields induced
by the applied loading in an un-dislocated solid will be denoted by .
- The fields in a
solid containing a dislocation, but with  traction free, and with zero
displacement on  will be denoted by
- The fields in an infinite solid containing a
dislocation with line C and
burgers vector b will be
denoted by . If the dislocation line terminates on
the solid’s surface, any convenient procedure can be used to close the
loop when deriving the infinte solid solution, but the fields will
depend on this choice.
- The difference
between the fields for a dislocation in a bounded solid and the solution
for a dislocation in an infinite solid will be denoted by
The Peach-Koehler force can
then be divided into contributions from three sources:
where
-  is the `self-force’ of the dislocation,
i.e. the force exerted by the stresses generated by the dislocation
itself.  This force always acts
so as to reduce the length of the dislocation line. In the classical solution, this force
is infinite. The procedure
described in Section 5.9.2 can be
used to remove the singularity  in this case the stress in the
Peach-Koehler formula should be calculated using the expressionÂ
where .  Note that, if the dislocation remains straight, the total length of the
dislocation line does not change as the dislocation moves. In this case, the self-force is zero. In 2D descriptions of dislocation motion,
therefore, the core singularity has no effect  this is why it has been possible to live
with the classical dislocation fields for so long.
- Â is a force generated by stress
associated with the solid’s boundaries.Â
These are generally non-singular.Â
This force is often referred to as the `image force’
- Â is the force exerted on the dislocation
by externally applied loading.Â
This, too, is generally nonsingular.
Derivation:Â The
following expression for the total energy of a dislocation in an elastic
solid was derived in Section 5.9.3.
where
  is the strain energy of the dislocation
itself
  is the work done to introduce the
dislocation into the externally applied stress
  is the potential energy of the applied loads
We
wish to calculate the change in potential energy resulting from a small
change in area  as the dislocation line advances by a small
distance . We consider each term in the potential
energy
- The last term is independent of S, and therefore .
- The change in  follows as ,
where  is the increment in area swept by the
dislocation. Note that an area
element swept by the advancing dislocation line can be expressed as ,
so we can write
- The change in  can be written as
To
calculate the change in stress  arising from the motion of the dislocation
line, recall that the displacement and stress due to the dislocation loop can
be calculated from the expression
where
 is the stress due to a point force in the
(bounded) elastic solid. The change in
stress therefore follows as
This
shows that
Reversing
the order of integration in the first integral and using the expression for  then gives
- Finally, combining the results of (3) and (4)
and noting that  then gives
This
has to hold for all possible ,
which shows that  as required.
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