|       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. 
  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) by   
 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. 
  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   
     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.   Nonsingular energy of circular
  dislocation loops.  It is straightforward to calculate the
  energy of a circular dislocation loop. 
  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. 
  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. 
  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. 
  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      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. 
  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:  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. 
  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.ÂSuppose 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   .   
   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 expression   
 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. 
      is a force generated by stress
       associated with the solid’s boundaries. 
       These are generally non-singular. 
       This force is often referred to as the `image force’   is the force exerted on the dislocation
       by externally applied loading. 
       This, too, is generally nonsingular.     Derivation:  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.             |