dislocation plasticity

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Dislocation Plasticity: Overview
1. DISLOCATIONS AND PLASTIC DEFORMATION
An arbitrary deformation of a material can always be described as the sum of a change in
volume and a change in shape at constant volume (shear). Assuming constant structure,
the change in volume is recovered when the load is removed, since the atoms can simply
relax back to their equilibrium sizes. The change in shape, on the other hand, may or may
not be recovered, since the atoms can relax into new positions that are configurationally
identical to the original ones, but displaced from them. The part of the shear that is
recovered is elastic, the part that remains is plastic. Plastic deformation is a permanent
change in shape through shear.
There are three generic ways in which the shape of a crystal can change at constant volume.
First, individual atoms can move so that the crystal becomes longer in one or more of its
dimensions and correspondingly shorter in the others. In a crystal individual atoms move
by diffusion, and this process is known as diffusional creep. Second, all of the atoms in
the crystal, or some subvolume of it, can move simultaneously to accomplish the shear.
Something of this sort happens in mechanical twinning, in which one part of the crystal is
uniformly sheared, but remains atomically matched to the remainder along a common
twinning plane.
Third, and most commonly, planes of atoms can slip over one another like cards in a deck,
leading to an overall shear that is localized within specific atom planes. It is always
energetically favorable to accomplish this slip a little at a time, as one would move a large
rug across a floor. And it is usually favorable to slip in increments that correspond to a
lattice displacement, so that the area of the plane that has slipped maintains a perfect
crystallographic match with the plane beneath it. In this case the boundary of the slipped
area is a linear defect, called a dislocation.
1.1 Concept of a slip dislocation
The concept of a dislocation in a solid was developed mathematically by Volterra in the
early 20th century (Volterra, 1907). However, the mechanistic connection between
dislocations and plastic deformation was not clearly recognized until the 1930's, when
Orowan, Taylor and Polanyi published almost simultaneous papers describing the essential
mechanisms of dislocation plasticity (Orowon, 1934; Polanyi, 1934, Taylor, 1934).
The Volterra dislocation can be created as illustrated in Fig. 1. Let a solid body be cut over
the plane indicated in the figure, and let the material above the cut be displaced with respect
to that below it by the vector, b. Then let the lips of the cut be welded back together so that
the cut becomes invisible. The only remnant of the operation is the linear distortion at the
edge of the slipped region. This linear defect is called a dislocation, and the vector slip, b,
J.W. Morris, Jr.: Overview of Dislocation Plasticity
on the plane it bounds is called the Burgers vector of the dislocation . If the solid is
crystalline, the slipped faces of the cut can only be welded to leave no trace if the Burgers
vector, b, is a lattice vector of the crystal. In this case the dislocation is called a perfect
dislocation. Fig. 1b shows an example of a perfect dislocation in a simple cubic crystal.
“
b
(a)
Fig. 1:
(b)
(a) Method of creating a Volterra dislocation.
dislocation in a simple cubic lattice.
(b) An edge
In the example shown in Fig. 1 the dislocation is a straight line perpendicular to the slip, b.
Such a dislocation is called an edge dislocation since it can be visualized as the edge of an
extra half-plane of atoms in the crystal (Fig. 1b). In general, however, the planar region of
slip can have an arbitrary shape with the consequence that the dislocation, which is its
boundary, can be curved or looped as shown in Fig. 2.
b
edge screw
Fig. 2:
A dislocation loop in a crystal.
There are at least three useful ways to visualize a "dislocation". First, as illustrated in Figs.
1 and 2, a dislocation is the linear boundary of a planar region that has experienced a slip,
b. Several important geometric properties of dislocations follow immediately from this
fact. Among them, a dislocation cannot begin or end inside a material; it must either
intersect a free surface, close on itself, or end at a junction form which other dislocation
emanate. If a dislocation lies between regions that have slipped, respectively, by b1 and
b2, its Burgers' vector is the vector sum: b = b1 - b2. If a single dislocation, with Burgers'
vector, b, divides into two dislocations (b1, b2) at a node, then b = b1 + b2.
Second, a dislocation can be regarded as a one-dimensional defect that exists independent
of the slip that created it. This viewpoint has the advantage of objectivity, since a perfect
dislocation carries no record of how it was created. The edge dislocation drawn in Fig. 1b,
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
for example, could have been formed by a vector slip, b, on the half-plane to the left of the
dislocation line, by a slip, -b, on the plane to the right, or by a sequence of slips that have
either net result. As a one-dimensional defect the dislocation is characterized, locally, by
two vectors, l , a unit vector tangent to the dislocation line that defines the orientation of the
dislocation, and the Burgers vector, b, which defines the strength of the slip it carries.
Both l and b are ambiguous as to sign. It is conventional to remove the ambiguity by
constructing a Burgers circuit around the dislocation. Choose a closed circuit that can be
drawn in a perfect crystal by taking sequential steps from atom to atom, as illustrated for a
{100} plane in a simple cubic crystal in Fig. 3a. Now draw that same circuit in a
dislocated crystal, as in Fig. 3b. If the circuit encloses a dislocation, it will not close. If
the direction of the dislocation line is chosen so that the circuit is clockwise (right-handed
screw), the Burgers vector, b, is the vector displacement of the end point of the circuit from
the start (Fig. 3b). It measures the net displacement experienced by an imaginary observer
who completes a circuit around the dislocation that would be closed in a perfect crystal.
b
(a)
Fig. 3:
(b)
A Burgers circuit closes in a {100} plane of a cubic crystal, but
fails to close by the Burgers vector, b, when the same circuit
encloses an edge dislocation.
The dislocation is an edge dislocation when b is perpendicular to l, as in Fig. 3b. It is
called a screw dislocation when b is parallel to l, since an imaginary observer who
followed a Burgers circuit around it would advance along its length by the vector b per
circuit, as if he were following the thread of a right-hand screw. Dislocations intermediate
between the edge and screw configurations are called mixed dislocations, and are often
characterized by the angle between b and l. A curved dislocation like that shown in Fig. 2
is mixed over most of its length, becoming edge (screw) only when the dislocation lies
perpendicular (parallel) to b.
A third way to visualize a dislocation is to model the slipped region (Fig. 2) as a thin elastic
inclusion with a thickness, h, equal to the interplanar distance perpendicular to the slip
plane. If the slip is b, the strain in the inclusion is the simple shear
γoij
=
1
nb
h i j
(1)
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
where γij denotes a shear displacement of the ith coordinate face in the jth direction, ni is the
ith component of the normal to the slip plane and bj is the jth component of the Burgers'
vector. While this may appear to be a needlessly complicated description, it is, in fact, a
very useful model for treating the energies and associated strains of dislocations in real
materials. Since elastic strains are additive a single dislocation strains the crystal by the
amount
γpij
=
Vp o
γ
V ij
=
A
nb
V i j
(2)
where Vp = Ah is the volume of the equivalent inclusion created by slip over the area, A. It
has recently been recognized (Jin, Artemev and Khachaturyan, 2001) that the inclusion
model makes it possible to compute the elastic energy of an arbitrary distribution of
dislocations with methods that are straightforward adaptations of established theories
(Khachaturyan, 1983). The formalism is a bit too elaborate for review here, but is
undergoing rapid development.
1.2 Dislocations and shear strain
Let a dislocation move so that it sweeps out the incremental area, ∂A. It follows from eq. 2
that the associated strain increment to the shear strain is
δγpij
=
nibj
V
δA =
nibj⌠
⌡ δxn dL = L nibj<∂xn>
V L
V
(3)
where <δxn> is the average normal displacement of the dislocation line and L is its total
length. More generally, let the dislocations within a solid be set in motion by an applied
shear stress. Let the index å denote a particular slip set, that is, a particular combination of
slip plane (n) and Burgers' vector (b). If vå is the average normal velocity of dislocations
in the slip set, å, then the shear strain rate is
• ©ij = ∑ ®å(nibj)vå
(4)
å
where ®å = Lå/V, is the total line length per unit volume of the dislocations in the åth slip
set. We are often interested in the simple shear, ©, produced almost entirely by dislocations
from a particular slip set. In this case we have the simpler and familiar relation
•© = ®bv
(5)
where ® is the density of mobile dislocations from the active slip set, b is their Burgers'
vector and v is their average normal velocity.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
1.3 The energy of a perfect dislocation
In the usual case the energy of a dislocation is the elastic energy stored by the distortion of
interatomic bonds around the defect. As illustrated in Fig. 1b, the distortion is pronounced
in the immediate vicinity of the dislocation line, but decays rapidly with distance. It is
convenient to separate the strain field of the dislocation into two parts: a narrow, cylindrical
core of radius, r0, that includes the severely distorted material immediately around the
dislocation line, and a long-range field in which the strain is small enough to be treated by
the methods of linear elasticity. Unfortunately, both are difficult to calculate with
precision. The stress in the elastic field decreases only as 1/r, where r is the radial distance
from the core, with the consequence that the elastic energy diverges. This result is
unphysical, since elastic fields in real materials are eventually canceled by the fields of other
defects or terminated at free surfaces. However, it requires that we select a finite cut-off
radius, R, for the outer boundary of the elastic field. The elastic energy per unit length of a
straight dislocation (line tension) is, then,
Γ=
E
L

=

Gb2  R 
ln
4π  r0
screw
Gb2  R 
ln
4π(1-ˆ)  r0
edge
Γesin2œ + Γscos2œ
mixed
(6)
where G is the shear modulus and ˆ is Poisson's ratio. The angle, œ, of the mixed
dislocation is the angle between b and l, and Γe and Γs are, respectively, the line tensions of
edge and screw dislocations with be = b sin(œ), bs = bcos(œ)). The cut-off radius, R, is
usually taken to be the mean spacing between dislocations, on the grounds that a
dislocation will tend to minimize its elastic energy by surrounding itself with dislocations of
opposite sign.
To complete the calculation of the line tension, Γ, of a straight dislocation we need the
energy of the dislocation core. The accurate calculation of the core energy requires ab initio
methods at the atomic level (Blase, et al., 2000), and has only recently become possible.
However, the available models suggest that the core energy is small compared to that of the
long-range elastic field, and can be roughly accounted for by setting the core cut-off radius
at r0 = b.
The computation of the line tension is further complicated when the dislocation is curved or
the specific arrangement of dislocations is considered. In lieu of an accurate calculation, it
is often useful to use the qualitative relation
Γ«
1
2
Gb2
(7)
which is a reasonable approximation for typical metals and alloys.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
1.4 Partial dislocations
While most crystal dislocations are total dislocations when viewed from sufficiently far
away, it is not uncommon to find them dissociated locally into a configuration that can be
described as two parallel partial dislocations connected by a planar defect that is called a
stacking fault in the crystal. The prototypic example is found in FCC crystals.
The common dislocation in the fcc structure is the dislocation that causes close-packed
{111} planes to slip over one another. The Burgers vectors that accomplish unit slip are
the 12 <110> vectors that connect atoms to their nearest neighbors in the {111} planes.
Assuming that the glide plane of the dislocation lies between (111) planes of A and B-type
atom sites (Fig. 8), the element of slip carries an atom from one B-site to another. It is
easiest to accomplish the slip in two sequential steps. The B atoms are first slipped into C
positions, then moved from C back to B again. This slip can be accomplished by the
sequential passage of two dislocations: b1 (= 16 [–11– 2]) and b2 (= 16 [–211]). However, b1
and b2 are not lattice vectors; they are examples of partial dislocations. Their sum is the
total dislocation, b.
A
A
b2
A
b1
B
B
B
A
C
C
b
A
A
A
A
C
A
b1
Fig. 4:
b
A
A
(a)
B
b2
B
B
A
B
B
A
B
(b)
The slip of close-packed planes in fcc: (a) in stacking of closepacked {111} planes, (b) in fcc unit1 cell. The dashed arrow
shows slip by a total dislocation, b = 2 [–101]. The solid arrows
show successive slip by partial dislocations. The shaded atom is
the intermediate, C-site position.
Splitting the total dislocation, b, into the partials, b1 and b2, not only facilitates slip, but
also lowers the line energy, since |b|2 > |b1|2 + |b2|2. However, the separation of the two
partials, b1 and b2, also creates a stacking fault with a positive energy, ßs, per unit area, as
illustrated in Fig. 5. Minimization of the total energy dictates the separation between the
partials, which is of the order of 5-500‹ in typical fcc crystals.
Similar considerations apply to hcp and diamond cubic crystals. Total dislocations in the
close-packed planes tend to divide into partials separated by ribbons of stacking fault. In
bcc crystals, however, the stacking fault energies are very high and separated partials
cannot ordinarily be resolved. Nonetheless, incipient decomposition into partials
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
apparently occurs in some cases, producing a complex structure along the dislocation core.
The most important example is the 12 <111> screw dislocation in bcc. This dislocation,
which dominates low-temperature plasticity in many bcc alloys, is believed to have a
complex, triangular splitting in the core. The triangular core structure not only restricts its
mobility, but also causes a pronounced asymmetry in its behavior under load. The
dislocation tends to glide on {112} planes, but moves much more easily in one direction
(the twinning direction) than in its opposite (Vitek, 1974; Hirth and Lothe, 1982).
b2
b1
“
b
Fig. 5:
The total dislocation, b, divided into partial dislocations, b1 and
b2 separated by a stacking fault.
1.5 Ordered structures
Compounds and ordered structures have several atoms per unit cell, and, therefore, have
very large lattice vectors. As a consequence, dislocations in ordered structures are almost
always split into two or more partials. In many cases, the ordered structure is achieved by
ordering species on the sites of a simple fcc, bcc, NaCl or diamond cubic parent lattice. In
this case, while the unit cell may be large, the partial dislocations are the usual dislocations
that appear in the fcc or bcc parent lattice, and the faults between them are often antiphase
boundaries that involve a discontinuity in the state of order rather than in the structure.
(a)
Fig. 6:
(b)
(a) The CsCl structure drawn as a stacking of {110} planes. (b)
An antiphase boundary1 in the CsCl structure made by a
displacement of the type 2 <110> in the third plane.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
A simple illustrative example is provided by ∫-brass, the low-temperature modification of
CuZn. The Cu and Zn atoms are ordered into a CsCl configuration, as illustrated in Fig.
6a, which is a simple substitutional ordering of a bcc solid solution. Fig. 6b shows the
antiphase boundary that is created by an interplanar slip by the vector b = 12 <110>, which
is a common Burgers vector for dislocations in bcc. The original structure can restored by
a second slip by an identical amount. As a consequence, 12 <110> dislocations in this
structure tend to be paired, with an antiphase boundary between them. The perfect
dislocation in the parent bcc structure is a partial dislocation in the ordered structure.
1.6 The line tension and the dislocation density
It is important to note that dislocations are non-equilibrium defects. Even in a simple metal
the line tension of a dislocation (eq. 7) is large enough that the equilibrium concentration of
dislocations is almost zero. It follows that dislocations will tend to anneal out of materials
that are subjected to high temperature for any period of time.
On the other hand, the energy necessary to create dense distributions of dislocations is
readily available from the elastic energy that is stored under moderate loads. A solid that is
sheared elastically by © has a stored elastic energy per unit volume of 12 G ©2. If this
energy were used to create dislocations with the line tension given by eq. 7 the resulting
dislocation density would be ® « (©/b)2, or about 1013/m2 for a typical metal strained to © =
0.001. This is well above the dislocation density that is typical of an annealed metal («
1010/m2). If the elastic shear is maintained during deformation much higher dislocation
densities can be achieved; densities of the order of 1016/m2 are observed in cold-worked
metals, and are produced by the multiplication processes discussed below.
The dislocations that are most likely to appear are those that have minimum energy, or, by
eq. 7, minimum values of b. For a perfect dislocation in a simple Bravais lattice the
minimum value of b is the nearest neighbor distance. Hence the common dislocations in
fcc have b = 12 <110> while the common dislocations in bcc have b = 12 <111>. In
ordered solids or compounds with multiatom unit cells the minimum Burgers' vectors are
large and, even after splitting into partials to minimize energy, are high-energy defects.
This is a major reason why ordered compounds tend to have low dislocation densities and
poor ductility.
2. DISLOCATION MOTION
2.1 Glide and climb
Except in the special case of a screw dislocation, the vectors b and l define a plane, which
is called the glide plane of the dislocation. A dislocation that moves in its glide plane can
do so stepwise by simply breaking and reconfiguring the bonds immediately around its
line, as illustrated in Fig. 7(a-b). This conservative motion is called glide. Motion out of
the glide plane, on the other hand, requires the addition or subtraction of atoms (or
vacancies) along the dislocation line (Fig. 7c). Since climb requires interatomic diffusion,
glide is the dominant mechanism of motion at low temperature.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
(a)
Fig. 7:
(b)
(c)
Glide and climb of an edge dislocation. Only a single bond must
be broken per plane for each increment of glide. An atoms
(vacancy) must be added per plane for each increment of
downward (upward) climb.
The screw dislocation is a special case. Since b and l are parallel, a straight screw
dislocation can glide in any plane. Note, however, that any bending of the dislocation
introduces some edge component and establishes a glide plane.
2.2 The force on a dislocation
The resolved force on a dislocation can be computed from the work done by the applied
stress in an infinitesimal displacement of the dislocation. Let the dislocation line be
displaced by the vector, ∂x, so that it sweeps out the area ∂A. The mechanical work done is
⌡ ßij∂‰ij dV = ßijγo ∂Vp = ßijbjni∂A
∂W = ⌠
ij
V
(8)
where we have used eqs. (2) and (3) and use the summation convention (indices are
summed 1 to 3 if repeated) . Since the vector, n, is the normal to the slip plane, n∂A = l x
∂x, or in Cartesian tensor notation, ni∂A = eimklm∂xk, where eimn is the permutation tensor
(= 1 (-1) for imn = 123 (213) or their cyclic permutations, 0 otherwise). It follows that
∂W = eimklm∂xkßijbj = ∂xkekimßijbjlm = fk∂xk
(9)
hence the effective force on the dislocation, f, is
f = b ^ ß x l
(fk = ekimßijbjlm)
(10)
It follows immediately from eq. (8) that the glide force per unit length on an edge or mixed
dislocation is
f = †ben
(11)
where †b = b ^ ß ^ n (=ßijbjni) is the shear stress on the glide plane resolved in the direction
of b, and en is a unit vector perpendicular to the dislocation line. The force impelling glide
is a shear stress that acts perpendicular to the dislocation line.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
The force driving the climb of an edge dislocation can be found from eq. (10). Choose a
coordinate system in which the dislocation line lies in the direction e1 while the edge
component of the Burgers' vector, be, lies along e2. Then, from equation (10),
f = ß32be2 - ß22be3
(12)
The first term reproduces eq. (11). The second term is the climb force, -ß22b, which
points along the normal to the glide plane. To interpret this force physically, note that ß22
acts to stretch the crystal in the direction of b. If the dislocation climbs down (Fig. 7) the
extra half-plane of atoms is extended, and the crystal stretches along b, hence the sign of
the force is negative. If the dislocation climbs up, the crystal contracts along b.
The force on a screw dislocation can be found by orienting both l and b along e1. Then
f = e2ß31b - e3ß21b
(13)
A screw dislocation can glide in any plane that contains b. The glide force in a particular
plane is (†ben), as in eq. (11), where † is the shear stress on that plane resolved along b.
2.3 Dislocation glide
2.3.1 The critical resolved shear stress
Dislocation glide is driven by the shear stress on the glide plane, resolved in the direction of
b. Since there is always some resistance to glide, the shear stress must reach a critical
value, the critical resolved shear stress, †c, before glide can occur. When the yield strength
of a material is governed by dislocation plasticity, as it ordinarily is in crystalline solids, †c
controls its value.
2.3.2 The Peierls-Nabarro stress
The minimum value of †c applies to an isolated dislocation in an otherwise perfect crystal.
At low temperature the dislocation can minimize its core energy by aligning itself along a
close-packed direction within the crystal. A finite shear stress is needed to move the
dislocation from this energy well to another. This stress, the Peierls-Nabarro stress, sets
the minimum of †c in the low-temperature limit. Its value was estimated by Nabarro (1947)
to be
†P ~
 4π 
2G
exp  - 
1-ˆ
b
«
 2πh 
2G
exp  - 1-ˆ
b(1-ˆ)
(14)
where
is the effective width of the dislocation in the glide plane. Eq. 14 pertains to an
edge dislocation; the same equation without the factor (1-ˆ) applies to a screw. Criticisms
and modifications of eq. (14) have been suggested by a number of authors over the years
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
(for example, Huntington, 1955, Joos and Duesbery, 1997, Nabarro, 1997), but its
qualitative features survive: †P increases with the shear modulus, G, and decreases
exponentially with the interplanar spacing, (h/b). Moreover, the simple Nabarro formula is
in rough agreement with the available experimental data (Nabarro, 1997).
From eq. (14) the dimensionless stress, †/G, depends almost entirely on the crystal
structure, through the minimum value of the factor (h/b). Predicted values of (†/G) are
about 10-5 for fcc crystals, 10-2-10-3 for bcc, hcp and NaCl, and 10-1 for oxides, ∫-ZnS
and diamond cubic materials. It follows that dislocation glide should be relatively easy in
the common metals, alloys and simple ionic materials, as is observed, but may be much
more difficult in materials with diamond-like structures. In fact, at shear stresses near
0.1G crystal lattices themselves become unstable with respect to spontaneous shear, and
dislocation glide may no longer be the preferred mechanism of deformation (Morris, et al.,
2000). Si, for example, deforms by a spontaneous structural transformation rather than
dislocation glide in indentation hardness tests at low temperature.
The simple estimate of the Peierls-Nabarro stress is inaccurate for dislocations with
complex structures. Screw dislocations in bcc are a particular example because of their
complex core structures (as discussed above). The core asymmetry of these dislocations
has the consequence that †P depends on the direction of glide.
2.3.3 Kinks
A dislocation that does not lie along a close-packed direction can minimize its energy by
adopting a configuration in which segments along close-packed directions are joined by
short kinks, and will tend toward such a configuration in the low-temperature limit (Fig.
8). The kinks in such a line are much more mobile than the segments themselves. They
ordinarily move under stresses well below †P .
Fig. 8:
A kinked dislocation. Lateral motion of kinks causes normal
motion of the dislocation line, as illustrated at right. A double
kink is shown at left.
However, single kinks eventually annihilate at free surfaces or dislocation junctions, so
kink migration does not provide a viable mechanism of plasticity unless paired, doublekinks can form spontaneously along the line. Double-kink nucleation is particularly easy in
fcc and hcp metals, so dislocations in these structures glide at stresses well below †P at
moderate temperatures (Hirth and Lothe, 1982). Double-kink nucleation is more difficult
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
in bcc metals and in ionic and covalent crystals, and may be an important barrier to glide at
low temperature.
2.3.4 Slip systems
Taken together, the concepts of line tension and Peierls-Nabarro stress suggest that the
dislocations that dominate deformation at low temperature will have the minimum possible
Burgers vectors and will lie in glide planes with maximum interplanar separation (h/b). It
follows that crystalline solids have a strong tendency to slip in particular directions on
specific crystallographic planes, both determined by the crystal structure. The combination
of slip plane and slip direction is called the slip system. The members of a slip system are
the slip sets (nb) defined in eq. (1).
The common slip systems in fcc and hcp crystals combine close-packed directions with
close-packed planes: <110>{111} in fcc (and diamond cubic), <11–20>{0001} in hcp. The
most common prismatic slip system in hcp is <11–20>{1–101}. The slip direction in bcc
crystals is almost always along <111>, but several slip planes compete, including {110},
{112} and, less commonly, {123}. The variety of available slip planes in bcc has the
consequence that slip sometime occurs in pencil glide, slip in a <111> direction on
apparently random slip planes. Slip in ionic and covalent crystals may be complicated by
the need to preserve bond or charge configurations, which may affect the operative slip
systems.
2.4 Dislocation climb: jogs
Dislocation climb is an important mechanism of deformation at higher temperatures, where
the rate of self-diffusion is appreciable. Its principal role is providing a mechanism for
dislocations to by-pass microstructural barriers by changing slip planes. The atoms or
vacancies that are required for climb are most easily added at kink-like features
perpendicular to the glide plane that are called jogs.
There are two basic sources of jogs. The first is the addition of atoms or vacancies to the
dislocation line. A single atom or vacancy added to an edge dislocation effectively creates a
pair of jogs. The second source is the intersection of dislocations. If dislocations with
Burgers' vectors b1 and b2 pass through one another, each will create a jog on the other
equal in length and direction to its own Burgers vector (Friedel, 1964). In both cases the
short segments of the jogs have the Burgers' vector, b, of the parent line.
The short segment of a jog lies in a different glide plane from the parent line and will,
therefore, experience a different stress. Often, the configuration is such that glide of the
parent line requires climb of the jog segment. For these reasons jogs are almost always
impediments to dislocation glide and are often sessile segments that pin the dislocation
locally.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
2.5 Cross-slip
There is a second mechanism, cross-slip, by which a dislocation can change its glide plane.
Let a dislocation loop expand in its glide plane. The segments of the loop that are in screw
orientation can glide in any plane that contains the Burgers' vector, b. It is, therefore,
possible for a segment of the loop to slip onto a plane that is angled to the primary glide
plane, as illustrated in Fig. 9. It may then slip back onto a plane that is parallel to, but
displaced from the original glide plane. Cross-slip is a common mechanism for multiplying
the number of active slip planes, and for by-passing microstructural barriers during plastic
deformation.
Note that the dislocation segments that bridge the parallel glide planes in Fig. 9 lie in a
different crystallographic plane and, hence, experience a different glide force. These crossslipped segments often act as pinning points along the dislocation line.
b
Fig. 9:
Double cross-slip of a dislocation allows it to move onto a parallel
glide plane.
When the dislocation is split into partials, as it commonly is in fcc and dc structures, the
cross-slip process becomes more complicated, and often requires some thermal activation.
The interaction between cross-slipped partials may also create sessile segments in the
bridging plane that act as strong pinning points.
2.6 Dislocation multiplication
The density of dislocations (®) increases rapidly during plastic deformation. This happens
for two principal reasons. First, dislocations naturally become longer as loops expand and
segments extend to avoid microstructural barriers. Since the dislocation density is the line
length per unit volume, these natural processes increase ®.
Second, new dislocations are continually created by a variety of mechanisms. A common
mechanism that serves as the prototype case is the Frank-Read source that is illustrated in
Fig. 10. Let a dislocation be firmly pinned at two points. These may, for example, be
precipitate particles, or the bridging segments in a cross-slipped configuration like that
illustrated in Fig. 9. If the shear stress on the dislocation is † and the line tension is
constant, the dislocation bows out between the pinning points in a circular arc of radius
page 13
J.W. Morris, Jr.: Overview of Dislocation Plasticity
r=
Γ
†b
«
Gb
2†
(15)
If the stress, † , is greater than †c = Gb/L, where L is the spacing between pinning points,
then r < L/2. In this case the dislocation penetrates between the pinning points and spirals
around them as shown in the figure. When the two arms of the dislocation meet, they
annihilate, creating a dislocation loop that expands out into the crystal and a pinched-off
segment between the two pinning points that will spiral out to repeat the process. The
source continues to operate, generating new dislocation loops, so long as the local stress
remains above †c.
L
Fig. 10:
Diagram of a Frank-Read source.
There are several common variants on the classic Frank-Read source, and several other
kinds of sources, most of which involve irregularities at free surfaces, grain boundaries or
misfitting inclusion particles. As a consequence, dislocation multiplication is relatively
simple in metals and alloys, and is only difficult in materials in which the line tension is
high or the mobility is restricted.
2.7 Dislocation-dislocation interactions
To understand how dislocations interact with one another, we note that a dislocation is both
an elastic and a crystallographic defect. Consider, for example, the edge dislocation shown
in Fig. 1. The crystallographic discontinuity at the defect is set by the Burgers' vector, b,
and appears physically in the form of the extra half-plane of atoms that terminates at the
dislocation line. The elastic distortion near the dislocation is indicated in the figure. In
effect, there are too many atoms in the region just above the dislocation line, which sets it
in compression, and too few atoms in the region below, which causes it to be in tension.
2.7.1 Elastic interactions
The elastic interaction between two dislocations is relatively long-range, and is difficult to
calculate in general. However, many of its qualitative features can be understood
qualitatively from the simple principal that elastic fields superimpose, and attract if they
relax one another.
The interactions between dislocations in the same plane are illustrated in Fig. 11(a,b). If
the dislocations have the same sign, they repel one another. If they have opposite signs,
they attract and annihilate. If dislocations of opposite sign approach one another on
adjacent planes they form a dipole pair, as in Fig. 11c.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
Edge dislocations also interact to form vertical arrays. If like dislocations approach one
another on adjacent planes that are separated in the vertical direction there is an attractive
interaction between them, since the tensile field of one partially cancels the compressive
field of the other. As a consequence, edge dislocations of like sign tend to gather into
vertically stacked arrays. A vertical array of edge dislocations with Burgers' vector, b, is
called a low-angle tilt boundary because lattice planes that cross the boundary are tilted by
the angle
œ=
b
∂
(16)
as they pass through, where ∂ is the separation between dislocations.
compression
compression
tension
tension
(a)
(b)
(c)
Fig. 11: Interaction of edge dislocations in the same plane. Like dislocations repel (a), unlike dislocations attract and annihilate (b).
Unlike dislocations on nearby planes trap one another to form
dislocation dipoles (c).
Dislocations pile-ups result when like dislocations attempt to glide on a plane that is
blocked by some obstacle, most commonly a grain boundary. The lead dislocation is
blocked, and the trailing dislocations pile up against it. An important effect of the pile-up is
to magnify and concentrate the shear stress, †. If there are n dislocations in a pile-up
created by the external stress, †, the effective shear stress at the head of the pile-up is
†e = n†
(17)
and can be very high if many dislocations participate. The dislocations in a pile-up are in
mechanical equilibrium under the applied stress and the stresses due to one another. Their
equilibrium spacing can, therefore, be calculated (Chou and Li, 1969; Hirth and Lothe,
1982). The expected number of edge dislocations in a pile-up of length L under stress † is
 L  † 
n = π(1-ˆ)   
b G
(18)
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
It follows that the effective stress at the tip of a pile-up of fixed length, L, increases as †2,
and can be many times large than †. It is often the case that the barrier at the head of the
pile-up is penetrated or fractured when the effective stress reaches a critical value, †c. The
applied stress, †, at yield or fracture then varies with (L/b)-1/2, where L is the maximum
pile-up length. When L is the grain size, d, as it often is in metals and alloys, the yield or
fracture stress varies as d-1/2, in agreement with the empirical Hall-Petch relation.
2.7.2 Crystallographic interactions
Many important dislocation interactions are crystallographic. For example, edge
dislocations of opposite sign annihilate when they meet on the same slip plane (Fig. 11b)
and a dislocation cannot simply move through a grain or phase boundary unless its glide
plane is preserved on the far side.
A further set of important crystallographic interactions is illustrated in Fig. 12. When
dislocation that lie on different planes meet, they may pass through one another, as
illustrated in Fig. 12a. If they do, each leaves a jog on the other equal to its own Burgers'
vector. The jog will ordinarily exert a drag on the dislocation, making it more difficult for
it to continue glide. The second possibility, illustrated in Fig. 12b, is the combination of
the two dislocations along part of their length to form two nodes joined by a segment with
Burgers' vector b = b1 + b2, the sum of the Burgers' vectors of the interacting dislocations.
This union of dislocations only occurs when fairly stringent conditions are satisfied
(Friedel, 1964), but then creates what is often a strong barrier to further dislocation motion.
b2
b1
b2
b1 + b2
b1
(a)
Fig. 12:
b1
b2
(b)
Crystallographic interactions when dislocations cross.
3. THE YIELD STRENGTH
The stress required to initiate plastic deformation in a solid that is pulled in tension is called
its yield strength. The yield strength is the usual measure of the useful structural strength
of a ductile metal or alloy. As we shall discuss below, the yield strength depends on the
temperature and strain rate at which the test is done. However, for structural metals tested
at low to moderate strain rate near room temperature, which is the case of greatest
engineering interest, the yield strength depends primarily on the microstructure. The
strength is controlled metallurgically by modifying the microstructure to influence the
mobility of dislocations.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
3.1 The yield strength of a single crystal
While we measure yield strength in tension, plastic deformation is ordinarily controlled by
dislocations that are driven by shear. To initiate plastic deformation of a single crystal the
applied tensile stress, ß, must be large enough to produce a resolved shear stress, †, that
exceeds the critical resolved shear stress for glide in at least one slip system.
œ
ƒ
ß
†
b
Fig. 13:
A uniaxial tension, ß, produces a resolved shear stress, †, along
the Burgers vector, b, of a dislocation that lies in a plane whose
normal is tilted by œ from the tensile axis.
Assume a cylindrical tensile specimen, as illustrated in Fig. 13. A dislocation with Burgers
vector, b, lies in a plane whose normal makes the angle, œ, with the tensile axis. The
direction of b makes the angle, ƒ, with the tensile axis. The shear stress on the plane
resolved in the direction of b is
† = Ft/A' = ßcos(œ)cos(ƒ)
(19)
To move the dislocation and shear the crystal we must have † ≥ †c, where †c is the critical
resolved shear stress for the sip set nb. It follows that the tensile yield strength is

ßy = min

†c
cos(œ)cos(ƒ)
(20)
ßy is the value of ß that produces † ≥ †c for the most favorable slip set in the crystal.
Two important results follow immediately from eq. (20). First, while †c is a material
property (for a given slip set), the tensile yield strength is not. It varies with the orientation
of the crystal. Only its minimum value is a material property, and is realized when the most
favorable slip system (minimum †c) has œ = ƒ = 45º. In general, ßy ≥ 2†c.
page 17
J.W. Morris, Jr.: Overview of Dislocation Plasticity
Second, the preferred slip system may change with the orientation of the crystal. This does
not happen in fcc metals. The preferred slip system in fcc is {111}<110> and the angle
between {111} planes is sufficiently small that there is always a {111}<110> set available
for glide at stresses not too far above the minimum yield stress. But it does happen in
materials with other crystal structures, such as bcc and hcp metals, where several slip
systems appear.
The complex behavior of the bcc metals is particularly relevant because of their importance
as structural materials (e.g., iron and steel). The asymmetric core structures of dislocations
in bcc (discussed above) and the complexity of the possible dislocation-dislocation
interactions has the consequence that †c may vary significantly from one slip set to another,
even when both are members of the same favorable slip system. Bcc crystals often exhibit
anomalous slip, in which the active slip set is not the one that experiences the highest shear
stress (Christian, 1983).
Non-cubic crystals are, ordinarily, strongly anisotropic in their yield behavior. The hcp
metals are classic examples. The only close-packed plane in HCP is the basal plane of the
HCP cell, and dislocation glide is ordinarily much easier on this plane than on the prismatic
planes (those angled to the basal plane). The yield strength has a pronounced minimum
when the basal plane is « 45º from the tensile axis.
3.2 The yield strength of a polygranular material
In polygranular materials the process of yielding is complex and the yield strength itself is
ambiguous. Most polygranular materials exhibit two distinct kinds of yielding behavior:
local yielding, and general yielding.
Local yielding occurs when the applied stress triggers local dislocation glide in the weakest
element of the microstructure. In the ideal case, this happens when the applied stress is just
sufficient to move the most favorably oriented dislocations in the polygranular body, that
is, when ß = 2†c. Most polygranular materials yield at even smaller stresses, since they
have residual internal stresses that add to the applied stress, and heterogeneities that cause
local stress concentrations. The earliest incidents of local yielding ordinarily do not
propagate, since the grains around the yielded grain are unlikely to have equally favorable
slip systems. Nonetheless, local yielding produces a net plastic strain of the overall sample
and causes the stress-strain curve to deviate from linearity.
As the stress is raised beyond that required to cause local yielding, an increasing volume of
the specimen is plastically deformed, and the stress-strain curve deviates more noticeably
from its initial, linear slope. Eventually, the stress becomes sufficient to cause general
yielding, in which the whole specimen behaves as an essentially plastic body.
Even when the grain size is relatively large (we shall discuss the influence of grain size
separately below), a polygranular material will fracture along its boundaries unless its
grains deform together. This requires that the typical grain have enough active slip systems
to accomplish an arbitrary change of shape. It can be shown that at least five independent
page 18
J.W. Morris, Jr.: Overview of Dislocation Plasticity
slip systems are required (the symmetric strain tensor has six independent elements; one of
these governs the change in volume, five govern the change in shape). To activate five
independent slip systems, the yield stress must exceed its minimum value, 2†c, by a factor
known as the Taylor factor. For a cubic crystal, the Taylor factor is about 1.5, so the
stress required for general yielding, which is often used as the theoretical definition of the
yield strength, is
ßy ~ 3†c
(21)
Note two features of yielding in polycrystals. First, whether local or general yielding is
used as the criterion for the onset of plastic deformation, the material property that is most
important is the critical resolved shear stress, †c. The microstructural control of yield
strength is accomplished by manipulating the microstructure to adjust †c.
ßy
ß
}
0.2%
Fig. 14:
‰
The method of measuring the 0.2% offset yield strength.
Second, the yield of a typical polycrystal is gradual rather than abrupt. The yield strength
is, therefore, largely a matter of definition. The usual practice is to define the yield strength
as the 0.2% offset load, that is, the stress required to accomplish a plastic strain of 0.2%.
The method of taking the measurement is illustrated in Fig. 14. The use of the 0.2% offset
load as the nominal yield stress has the dual advantages that it is relatively easy to measure
in practice, and, for most materials, corresponds fairly well to the stress required for
general yielding.
4. MICROSTRUCTURAL CONTROL OF THE YIELD STRENGTH
The yield strength is controlled by adjusting the critical resolved shear stress, †c, for
dislocation glide within a grain or by changing the grain size to inhibit the transmission of
strain from one grain to another. We first consider the mechanisms that influence †c. The
inherent value of †c is the Peierls-Nabarro stress that was discussed in sec. 2.3.2. The
critical resolved shear stress is increased by placing microstructural obstacles in the plane of
the dislocation that make it difficult to move.
page 19
J.W. Morris, Jr.: Overview of Dislocation Plasticity
4.1 Obstacle hardening
The microstructural obstacles that inhibit dislocation slip through the grain interiors may be
solute atoms, forest dislocations that thread through the slip plane, or small second-phase
precipitates. When the obstacles are widely spaced their elastic fields do not overlap
strongly and they act as independent barriers. In this case the obstacles can be modeled as
point barriers in the slip plane. Their effect is captured in a few, simple constitutive
relations that are widely applicable.
Let a dislocation move over its slip plane under the action of a stress, †, that is significantly
larger than the Peierls-Nabarro stress, †p. In this case, the atomic structure of the slip plane
is relatively unimportant, and the dislocation behaves roughly like a flexible, extensible
string with a constant line tension Γ (« Gb2/2). When the dislocation encounters an array
of obstacles that oppose its motion, it presses against them to create local configurations
like that shown in Fig. 15. The dislocation bows out between adjacent obstacles in a
circular arc of dimensionless radius
R* =
R
Ls
=
Γ
†bLs
(22)
Where Ls is the mean spacing between obstacles (= n-1/2, where n is the number of
obstacles per unit area). If the obstacles are distributed uniformly over the plane and R* is
significantly greater than 1/2, there will always be at least one configuration of obstacles in
the plane that the dislocation cannot penetrate unless it passes through the obstacles
themselves.
¥
†b
Ls
R
†b
Fig. 15: A dislocation, modeled as a flexible string, pressing against
obstacles that are separated by the distance, Ls.
The force that the dislocation exerts on the obstacles is due to its line tension, and equal to
F = 2Γcos(¥/2)
(23)
where ¥ is the angle between the arms of the dislocation at the obstacle. Let Fc be the force
required for the dislocation to pass the obstacle by cutting through it or wrapping around it.
The obstacle is passed when ¥ falls to ¥c, where
cos(¥c/2) = ∫c =
Fc
2Γ
page 20
(24)
J.W. Morris, Jr.: Overview of Dislocation Plasticity
If we define the dimensionless stress as
†* =
1
2R*
=
†bLs †Ls
†
«
=
2Γ
Gb GbÔn
(25)
then it can be shown (Friedel, 1964, Hanson and Morris, 1975a, Labusch, 1977, Altintas
and Morris, 1986) that the critical resolved shear stress, †*c , is a function of the obstacle
strength, ∫c, and the geometry of the obstacle distribution. In particular, the critical
resolved shear stress for random and square arrays of obstacles is
†*c

=

Q∫c3/2
random
∫c
square
(26)
where Q is a factor of about 0.9. Eq. 26 predicts that the critical resolved shear stress for
dislocation glide through a random array of obstacles increases with the shear modulus,
with the 3/2 power of the obstacle strength, and with the square root of the obstacle
concentration. The equation holds reasonably well for hardening by solute atoms (in the
limit of low concentration), in which case ∫c is in the range 0.01-0.05, for hardening by
"forest" dislocations that thread through the glide plane, with ∫c in the range 0.1 to 0.3,
and for hardening by small precipitate particles, with ∫c in the range 0.5-0.8.
Quite often a solid is hardened by obstacles of several different effective strengths. For
example, a solution-hardened material may also contain a significant density of
dislocations, and hardening precipitates of finite size cut through several glide planes,
placing obstacles of different sizes (hence, different effective strengths) on each. The
critical resolved shear stress for a dislocation gliding through a plane that contains a mixture
of obstacles can be found to a good approximation from the geometric sum (Hanson and
Morris, 1975b; Glazer and Morris, 1987),
(†*c )2 = ∑ (†*i) 2xi
(27)
i
where the sum is over distinct obstacle types, xi is the fraction of obstacles of type i, and †*i
is the value the critical resolved shear stress would have if all obstacles were of type i.
When the obstacles are very different in strength it is common to estimate †c from the linear
superposition (Argon, 1996),
†c = ∑ †i
(28)
i
This superposition is only strictly accurate when the strengths are so different that
successively weaker mechanisms are so much weaker that their effect can be modeled as a
uniform friction stress.
page 21
J.W. Morris, Jr.: Overview of Dislocation Plasticity
The approximations used in the point obstacle model are substantial. In real crystals the
dislocation line tension is neither constant nor isotropic, the obstacles are finite and interfere
with one another, and the distribution is never fully random. The consequences of some of
the shortcomings have been discussed by Ardell (1985) and by Kocks, et al. (1975),
among others. However, the constitutive equations that emerge from the model, eqs. (26)
and (27), are qualitatively applicable to several hardening mechanisms and are often
quantitatively reasonable as well.
The important specific microstructural hardening species include solute atoms, dislocations,
precipitates and grain boundaries. We discuss these in turn.
4.2 Solution hardening
Solute atoms never "fit" quite properly in the parent lattice, so there is always some local
distortion of the lattice in the vicinity of the solute (the misfit defect). Moreover, the
bonding around the solute is never quite the same as that in the parent lattice, so there is
also some difference in the local value of the elastic constants near the solute (the modulus
defect). The result is that a dilute distribution of solute atoms acts as a distribution of
obstacles of the type considered in the previous section.
While the obstacle strength of solute atoms (∫c) is relatively small, the areal density in the
slip plane is relatively large, even when the solute concentration is much less than 1%.
Solution hardening is an effective hardening mechanism that is widely used. When the
solution is dilute, the yield strength is given by an equation of the form
ßy = ß0 + åsGb c
(29)
where ß0 is the yield strength of a solute-free material with the same microstructure and ås
is a constant that includes both the obstacle strength, ∫ c, and the Taylor factor that relates
ßy to †c. When the solute concentration becomes appreciable, the strain fields of the
individual solute atoms overlap so they no longer behave like discrete obstacles. In this
regime the strength varies with concentration roughly as c2/3 (Labusch, 1970).
The strength (∫c) of the solute defect is primarily due to its misfit in the parent lattice. It
follows that interstitial solutes strengthen an alloy much more effectively than substitutional
ones. Interstitial solution hardening is more pronounced in bcc crystals, where the
interstitial sites are small and asymmetric, than in fcc crystals, where they are larger and
equiaxed. Nonetheless, solution hardening is widely used in structural alloys with fcc
structures, including Al alloys and austenitic steels.
The diffusional mobility of solute atoms may also affect the strength, particularly when the
solute is a mobile interstitial or when the test temperature is relatively high. The reason is
that solute atoms are attracted to dislocations, and diffuse so that they accumulate there,
forming solute atmospheres. The most important engineering consequence of impurity
page 22
J.W. Morris, Jr.: Overview of Dislocation Plasticity
atmospheres is the yield point observed in the room-temperature stress-strain curves of
high-carbon steels. Interstitial carbon atoms have moderate mobility even at temperatures
near room temperature, and migrate to form atmospheres around dislocations. Since there
is a significant binding energy between the carbon atoms and the dislocation, the dislocation cannot move until the resolved shear stress is sufficient to literally rip it away from its
atmosphere. Plastic strain initiates at a sharp yield point in the stress-strain curve, as
illustrated in Fig. 16.
Immediately following yielding a material that exhibits a yield point experiences yield point
elongation, a plastic elongation at a lower value of the stress. If the stress is controlled,
and increased until the sample yields, the yield point elongation may occur rapidly and dramatically, and appear in the form of discrete bands of deformation across the sample.
ßy
yield
point
}
ß
yield point
elongation
‰
Fig. 16:
The stress-strain curve of a material that exhibits a yield point.
4.3 Dislocation hardening
From the perspective of strength, the most important dislocation interactions are the
interactions between a gliding dislocation and the other dislocations that cut through its
glide plane (Fig. 17). The dislocations that intersect the plane are called forest dislocations,
and they provide obstacles to the motion of the gliding dislocation that, to a reasonable
approximation, can be treated as point obstacles in the glide plane.
The dislocation-dislocation interaction is much stronger than the dislocation-solute interaction; the forest dislocations act as point barriers that have strengths (∫c) that typically lie in
the range 0.1-0.3. If the dislocations are randomly oriented, their density, n, the number
of dislocation that intersect a unit area of the glide plane, is one-half of the volumetric
dislocation density, ®, the total length of dislocation line per unit volume. The yield
strength of a material increases with its dislocation density according to the relation
ß = ß0 + ådGb ®
(30)
page 23
J.W. Morris, Jr.: Overview of Dislocation Plasticity
where åd ordinarily lies in the range 0.3-0.9. Eq. (30) has the form suggested by the
obstacle model, and is reasonably well obeyed by structural metals and alloys.
Fig. 17:
A mobile dislocation that is resisted by forest dislocations in its
glide plane.
There are three common methods for controlling the dislocation density in structural materials: heat treatment, mechanical deformation and martensitic phase transformations.
1. Heat treatment. A material is annealed at elevated temperature to remove dislocations
and lower its strength. Dislocations are non-equilibrium defects, and heat treatment
decreases their density by either of two mechanisms. The first is recovery. If a material
containing a high density of dislocations is annealed at a temperature high enough to permit
dislocation climb, dislocations migrate and interact, both with one another and with free
surfaces. Some of the dislocations are annihilated, others are gathered into stable, planar
configurations, such as low-angle grain boundaries (called subgrain boundaries). The net
effect is to leave the bulk of the volume relatively free of dislocations. The second mechanism is recrystallization. If the dislocation density is high enough, and the material is
heated to a temperature above its recrystallization temperature, then new, defect-free grains
nucleate and grow at the expense of the old, producing a microstructure that is relatively
free of dislocations. While heat treatments can decrease the dislocation density, they do not
eliminate it entirely. A typical structural alloy that has been recrystallized and annealed has
a dislocation density in the order of 1010/m2.
2. Mechanical deformation. The dislocation density, ®, increases with plastic strain, and
the material work hardens according to eq. (30). When the mechanical deformation is done
at high temperature, as it is during the hot deformation that is used to roll metal ingots into
plates or sheets, work hardening is counterbalanced by recrystallization and recovery. To
achieve a high residual dislocation density it is necessary to deform at relatively low
temperature. Metal products that are strengthened in this way are said to be cold-worked or
cold-rolled. A severely cold-worked metal has a dislocation density of 1014-1016/m2,
producing a dislocation hardening that can be two orders of magnitude greater than that in
the annealed condition.
3. Transformation strengthening. Transformation strengthening is possible in materials
that undergo martensitic transformations on cooling. The martensitic transformation
page 24
J.W. Morris, Jr.: Overview of Dislocation Plasticity
changes the structure by shearing the parent lattice (fcc in the case of structural steel) into
the product (bcc in steel). If the steel is properly alloyed, for example, by increasing the Ni
content and decreasing the carbon, the mechanism of the martensitic transformation can be
adjusted so that the transformation strains are accommodated by dislocations (the product is
called dislocated martensite). Martensitic steels of this kind combine very high strength
with reasonably good toughness is the as-quenched condition.
4.4 Precipitation hardening
The final type of hardening obstacle is a small precipitate in the interior of the grain. Such
precipitates are normally introduced by aging a slightly supersaturated material at relatively
low temperature, so the precipitates nucleate primarily in the grain interiors. The volume
fraction of the precipitates is determined by the phase diagram, and is, hence, fixed by the
composition and temperature. The size of the precipitates then depends on the aging time.
The precipitates form as very small particles, and coarsen with time as the larger particles
consume the smaller ones to decrease the total interface area.
The yield strength of a precipitation-hardened material varies with the aging time and temperature as illustrated in Fig. 18. The strength increases to a maximal value, the peak
hardness, then decreases on further aging, or overaging. Lowering the aging temperature
at given composition increases the volume fraction of precipitates with the consequence that
the strength increases to a higher value. However, the material hardens more slowly due
its the lower diffusivity.
lower
temperature
ßy
time
Fig. 18:
The variation of yield strength with aging time for a precipitationhardened material.
The hardness peaks for one of two reasons: the precipitate strength or the precipitate
density. The former reason is more common. Most hardening precipitates are coherent
with the matrix, and lattice dislocations cut through them when they are small. As they
grow, their strength (∫c) increases, but their mean separation (Ls) also increases since their
volume fraction is nearly constant. In the early stages of growth, the increase in obstacle
strength outweighs the increase in separation, and †c increases. However, there is an upper
limit to the strength of the obstacles. As they grow they eventually become impenetrable,
as illustrated in Fig. 19. When this happens the dislocation does not cut through the
obstacle, but wraps around it, and the obstacle strength is independent of precipitate size.
page 25
J.W. Morris, Jr.: Overview of Dislocation Plasticity
Because of the attraction between the arms of the dislocation, the strength, ∫c, is about 0.8,
less than the ideal value of 1.0 (Bacon et al., 1973).
Once the precipitate strength has maximized, the yield strength decreases monotonically as
the precipitate separation increases on further aging. If all of the obstacles introduced by
the precipitates were identical, the peak strength would coincide with the point at which the
obstacles became impenetrable. However, in real solids precipitates have a distribution of
sizes, and each size generates obstacle of several different strengths since it penetrates
several glide planes. A detailed analysis of the hardening of Al by coherent precipitates of
∂'-Al3Li suggests that the strength peaks when the strongest of the obstacles first become
impenetrable (Glazer and Morris, 1986).
†b
†b
†b
†b
b
b
(a)
(b)
Fig. 19: The interaction of a dislocation with impenetrable obstacles. (a)
The arms of the dislocation wrap around the obstacle and attract
one another. (b) The arms intersect and annihilate, producing a
propagating dislocation, and leaving dislocation loops around the
obstacles.
Some materials form very hard precipitates, which are uncuttable even when their size is
very small. An example is the (Si,Ge) precipitate in Al (Hornbogan, et al. 1992).
However, even in this case the hardness tends to increase to a maximum as the alloy is
aged. The reason is that such precipitates are ordinarily so difficult to form that their
volume fraction is low when their size is small. In these materials the increase in hardening
with aging time is due to an increase in the volume fraction of the precipitate phase (Mitlin,
et al., 2001).
4.5 Grain refinement
One of the simplest and most useful ways to strengthen a structural metal is by refining its
grain size. Grain boundaries are discontinuities in the crystal structure that act as barriers to
dislocations. Since the orientations of the active slip planes change abruptly at grain
boundaries, slip must be transmitted indirectly from grain to grain. When a dislocation
impinges on a grain boundary its stress field produces shear stresses on the potentially
active slip planes of the adjacent grain. These add to the applied load and help to propagate
plastic deformation by the motion of independent dislocations in the adjacent grain.
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J.W. Morris, Jr.: Overview of Dislocation Plasticity
Large grains are particularly efficient at transmitting strain to their neighbors. When a large
grain slips, a number of dislocations glide along the preferred plane (or along closely
spaced, parallel planes) and pile up against the grain boundary, as illustrated in Fig. 20. As
we discussed in section 2.7.1, the stress at the head of such a dislocation pile-up is
magnified; if the resolved shear stress is †, the stress at the head of a pile-up of n
dislocations is n†. The larger the grain the more easily extensive pile-ups develop, and the
more easily strain is transmitted across the boundary. The consequence is that the yield
strength of a material decreases with its grain size.
d
Fig. 20:
A dislocation pile-up in a grain that has yielded.
As discussed following eq. (18) above, the yield strength of a typical metal varies with its
grain size according to the Hall-Petch relation:
ßy = ß0 +
K
(31)
d
where d is the mean grain size and K is a constant whose value depends on the material and
the characteristics of the microstructure. The Hall-Petch relation is very well obeyed by a
variety of structural alloys to sub-micron grain sizes, including both bcc (å-Fe) and fcc
(Cu, ©-Fe) materials (Jang and Koch, 1990, Kimura and Takaki, 1995). However, the
relation breaks down for the finest grain sizes where alternate deformation mechanisms
such as grain boundary sliding dominate.
5. WORK HARDENING
When a material deforms plastically it also hardens. The yield strength increases with the
strain, a phenomenon known as work hardening. The basic mechanism of work hardening
was described above, when we considered how dislocations harden materials. As strain
builds up inside the material, dislocations slip, intersect and interact with one another, as
illustrated schematically in Fig. 21. These interactions cause an increase in the dislocation
density (total dislocation length per unit volume) that is monotonic in the strain. In a
polygranular material the dislocation density builds for two reasons: the multiplication of
dislocations within grains and necessary development of a dislocation density along the
grain boundaries to match plastic strain across the boundary and keep adjacent grains
page 27
J.W. Morris, Jr.: Overview of Dislocation Plasticity
together. Following Ashby (1966) these are usually classified as statistically stored and
geometrically necessary dislocations, respectively.
To a good first approximation the yield strength increases with the dislocation density
according to eq. (30),
† = †0 + åGb ®
(32)
where å ordinarily lies in the range 0.1-0.3 and we have written the equation in shear,
since the very large strains that are used in fundamental studies of work hardening are best
achieved in shear tests. The associated work hardening rate, œs, is, then
œs =
d†
d©
=-
åGb  d®
 
2 ® d©
(33)
and depends on the rate at which dislocations multiply with the strain.
Fig. 21:
Dense dislocation network in a severely strained material.
Work hardening is simplest in a single crystal, but even in that case is customarily divided
into five separate stages (Argon, 1996).
Stage 1 is only observed when the crystal is oriented for easy glide, with only one slip set
activated. The work hardening rate, œ1, is low, of the order of 10--4-10-3G. The primary
mechanism of work hardening is the interaction of dislocations on neighboring slip planes
through, for example, formation of dislocation dipoles.
Stage 2 evolves from stage 1 when the shear stress is sufficient to operate secondary slip
sets. Dislocations gliding in these slip sets interact with those in the primary slip set and
with one another, multiplying dislocations that act as obstacles to further glide. The work
hardening rate, œ2, is relative constant in this stage and has a value of the order 10-2G .
Stage 3 evolves from stage 2 when the dislocation density becomes high enough that
association and annihilation reactions between dislocations produce a significant rate of
recovery. Along with the recovery, the dislocation distribution begins to develop a welldefined cell structure, in which roughly rectangular regions that are relatively free of
page 28
J.W. Morris, Jr.: Overview of Dislocation Plasticity
dislocations are defined by relatively diffuse walls made up of dense tangles of
dislocations. Since recovery decreases the dislocation density, the work hardening rate
decreases with the stress, often according to the simple linear relation
œ 3 = œ2 - k†
(34)
where k is a dimensionless constant of order 1 (Mecking and Kocks, 1981). Stage 3
usually begins relatively early in the deformation process, and lasts until the flow stress
approaches values of the order of 5x10-3G .
Stage 4 evolves from stage 3 and is marked by the sharpening of dislocation cell walls into
well-defined subgrain boundaries. The sharp cell boundaries resist transmission of
dislocations, and large elastic strains develop within the cells that also impede dislocation
motion (Mughrabi, et al,. 1986). The work hardening rate, œ 4, is low, of the order of 104G, and only weakly dependent on the stress.
Stage 5 is the termination of work hardening at saturation at the end of stage 4. In stage 5
the dislocation and recovery processes are in balance and hardening terminates.
œ
ß
Fig. 22: The variation of the work hardening rate with the stress for
aluminum alloys.
The work hardening behavior of a polygranular solid that is deformed in tension is
ordinarily much simpler, and can often be represented by two straight lines, as shown in
Fig. 22. Both fcc metals, including many Al alloys, and bcc metals, including many steels
show this simple behavior. The rapid initial decrease in work hardening is due to local
yielding. The effective work-hardening rate, œ = dß/d‰, is equal to Young's modulus
during the elastic portion of the stress-strain curve, and decreases rapidly as an increasing
fraction of the volume of the material yields and contributes to plastic deformation. After
general yielding the deformation of the crystal is dominated by the overall dislocation
density, which develops as expected for stage 3 hardening. A specimen in tension loses
stability with respect to necking and fracture when the work hardening rate falls to the value
set by the Considere criterion:
page 29
J.W. Morris, Jr.: Overview of Dislocation Plasticity
œ=
dß
=ß
d‰
(35)
and, therefore, usually fails before the fourth stage of work hardening is reached.
6. THE INFLUENCE OF TEMPERATURE AND STRAIN RATE
The mechanical tests that probe the nature of dislocation plasticity are ordinarily done under
strain control at given strain rate and temperature. While we have discussed the behavior of
dislocations as if it were quasi-static, in truth dislocations are in thermally activated motion
during most of these tests. There is some merit in developing the whole theory of
dislocation plasticity in terms of the dynamics of dislocation motion, an approach that is
taken, for example, by Kocks, Argon and Ashby (1975) and by Argon (1996) (the
author's analysis of thermally activated glide in the point-obstacle approximation is given in
Klahn and Morris, 1973). The present paper has adopted a quasi-static viewpoint because
it is simpler to do so, not because it is more accurate. We shall, therefore, finish this
summary with a brief discussion of rate and temperature effects, including the phenomenon
of high-temperature creep in which they are most clearly revealed.
6.1 The variation of yield strength with temperature
thermal
degradation
ßy
athermal
barriers
thermal
activation
If the yield strength of a typical material is plotted as a function of its homologous temperature (T/Tm, where Tm is the melting temperature), the result is a curve that resembles that in
Fig. 23. The yield strength is relatively insensitive to the temperature over a range of
intermediate values of the homologous temperature, but increases dramatically as the
temperature is lowered toward zero, and decreases dramatically as it is raised to near the
melting point. Ambient temperature is in the intermediate temperature regime for Al and its
alloys, is slightly into the low-temperature regime for typical structural steels, and is in the
high-temperature regime for low-melting metals like Pb.
T/T m
Fig. 23:
Typical variation of yield strength with homologous temperature.
page 30
J.W. Morris, Jr.: Overview of Dislocation Plasticity
The rapid strength increase at low temperature is due to the strong temperature dependence
of hardening by obstacles that can be cut or passed by thermal activation. These
particularly include the weak, short-range obstacles created by isolated solute atoms. As
the temperature increases, the increased amplitude of atomic thermal vibrations produces an
effective vibration of the dislocation line, which permits it to cut through obstacles that
could not be bypassed by the stress alone. In bcc metals thermally activated kink
nucleation is also important as a mechanism for increasing dislocation mobility. The higher
Peierls stress, greater interstitial lattice strain, and need for thermally activated kink
formation to sustain low-temperature deformation have the consequence that the increase in
strength at low temperature is much more pronounced in typical bcc metals than in typical
fcc metal.
The low-temperature strengthening mechanisms are ineffective at temperatures above 1/4 to
1/3 of the melting point. At intermediate temperature the yield strength is a relatively weak
function of temperature. The strength in this region tends to be controlled by dislocationdislocation and dislocation-precipitate interactions. These present obstacles that have
relatively large effective sizes; they spread over at least several atom spacings in the slip
plane, and are not easily passed by thermal vibrations of the dislocation.
At temperature above 0.5Tm the yield strength begins to drop dramatically. The principal
reason is the increasing rate of solid state diffusion, which affects both the dislocations and
the microstructural barriers. The microstructural effect is most important. Dislocation
configurations recover by climb and recombination, precipitates coarsen and overage, and
grains grow; virtually all of the available microstructural barriers become ineffective. At the
same time, the high diffusivity makes it possible for dislocations to climb at an appreciable
rate, so the obstacles that remain are more easily passed. The only practical mechanism for
hardening metals that are to be used at a significant fraction of their melting points is
precipitation hardening by precipitates that are thermodynamically very stable, so that they
do not coarsen at a rapid rate, and relatively large, so that they are not easily passed by
dislocation climb.
In assessing the strengths of the common metals and alloys, it is useful to keep in mind the
principle that the homologous temperature (T/Tm), rather than the actual temperature,
governs the strength. Materials with low melting points, like Pb and eutectic Pb-Sn alloys,
are very soft at room temperature. The reason is not so much their low inherent strength
(though they do have relatively low shear moduli) but their high homologous temperature,
which invalidates conventional deformation processes. Similarly, materials like pure Mo
have relatively high room temperature strength, largely because of their high melting
points.
6.2 The influence of strain rate on strength
The yield strength of a ductile metal or alloy always increases with the rate at which the
sample is strained. However, the magnitude of the rate effect varies dramatically with the
material and with the temperature at which the test is done. The reason is that the common
rate effects have the same source as the temperature effects. If deformation is thermally
page 31
J.W. Morris, Jr.: Overview of Dislocation Plasticity
activated it becomes easier to accomplish as the rate of deformation is lower; more time is
provided for thermal activation. To a good first approximation, increasing the strain rate is
equivalent to decreasing the temperature.
It follows that materials exhibit pronounced strain rate effects when they are tested in either
the high-temperature or the low-temperature regimes, and are relatively insensitive to strain
rate when they are tested at intermediate temperature. As might be expected, the strengths
of structural steels are sensitive to strain rate, while those of Al and its alloys are more
nearly constant, even when the strain rate is changed by several orders of magnitude.
Thermal and rate effects in dislocation plasticity are often gathered together in a single
variable, the Zener-Holloman parameter, Tln(•©), where the strain rate, •©, is defined in an
appropriate dimensionless form.
7. HIGH TEMPERATURE CREEP
It has been known since the original work of Gibbs (1878) that a material that is subject to
a shear stress is unstable with respect to any deformation that will relieve that stress and
return it to a state of hydrostatic pressure. It follows that a material that is subjected to a
constant shear load will gradually deform in a direction that attempts to relax the load, a
process known as creep, while a material that is given a shear strain will gradually deform
to relax the stress associated with that strain, a process known as stress relaxation. Both of
these processes are usually governed by dislocation motion, and occur at a significant rate
at temperatures above about 0.5 Tm, where the rate of diffusion becomes appreciable.
‰
I
II
III
t
Fig. 24: The variation of strain with time in normal creep, showing I:
primary creep, II: steady-state creep, III: tertiary creep to rupture.
Fig. 24 is a standard creep curve that illustrates the variation of plastic strain with time at
high temperature under constant load. The creep response is conveniently divided into
three regimes, called primary, steady-state, and tertiary creep. During primary creep the
strain rate decreases with strain (that is, the materials hardens), reaching a steady-state
value that is preserved for some period of time, then monotonically increases until the
material fails. Primary creep is ordinarily not extensive, and the creep behavior at small to
page 32
J.W. Morris, Jr.: Overview of Dislocation Plasticity
moderate strains is well characterized by the steady-state creep rate. The steady state creep
rate is often well-represented by a semi-empirical constitutive equation of the form
•© = A†nexp[-Q/kT]
(36)
The Dorn equation, where •© is the shear strain rate, † is the shear stress, n is the stress
exponent, and Q is the activation energy (Bird, et. al, 1969; Sherby and Burke, 1968).
The activation energy, Q, is ordinarily the activation energy for self-diffusion. The stress
exponent, n, varies from 1 to 7 or more, and may have different values in different regimes
of the applied stress.
While the precise mechanisms that govern dislocation-dominated high-temperature creep
remain uncertain, their common feature is dislocation climb. A material can only support a
static load if the dislocations within it are effectively pinned at microstructural barriers. To
move, they must be liberated, which requires dislocation climb onto slip planes that are
free. Climb requires the diffusion of atoms or vacancies, a process that is governed by the
activation energy for self-diffusion.
The observation of several distinct values for the stress exponent, n, also has a fairly
straightforward interpretation. If we assume one dominant slip set, the shear strain rate is
given by eq. 5,
•© = ®bv
(5)
Both ® and v are functions of †. In the usual case, the dislocation density depends on the
magnitude of †, but not on its sign, while both ©• and v change sign if the stress is reversed.
It follows that ® should be an even function while v should be an odd function; a series
expansion for ® contains even powers of † while a series expansion for v involves odd
powers. Hence
•© = (®0 + a†2 + ...)b(å† + ∫†3 + ...)
= ®0bå† + (®0∫+åå)†3 + a∫†5 + ...
(37)
When the dislocation density is constant and the stress is small, •© is proportional to †, a
characteristic of Harper-Dorn creep (Wang, 1996). Larger stresses commonly produce a
dislocation density proportional to †2 and power-law creep with n = 3, while still higher
stresses raise the exponent to values of 5 or larger. In polygranular materials at moderate to
high stress, more than one slip set will ordinarily be active, and eq. 5 should be replaced by
the more complex eq. 4, which is further complicated by the fact that slip planes have
different orientations in adjacent grains. It is not surprising that the "best-fit" stress
exponent is often a non-integral number.
Of course, creep is not always dominated by dislocations. At low stresses and high
temperatures, lattice diffusion is often the dominant mechanism, particularly in non-metals.
When the stress is low and the grain size is small, grain boundary sliding may dominate.
page 33
J.W. Morris, Jr.: Overview of Dislocation Plasticity
In this case, n « 2, Q is the activation energy for grain boundary diffusion, and the material
may exhibit superplastic deformation in which it sustains very large deformations before
fracture (Langdon, 1981)
The final creep phenomenon we shall discuss is stress relaxation, in which the elastic stress
imposed by a fixed strain is gradually relaxed by creep. Neglecting primary creep, the rate
of stress relaxation is governed by the steady-state creep rate. The residual stress in a part
strained in shear is † = G©. Stress relaxation under steady-state conditions is, hence,
governed by the differential equation
d†
dt
=G
d©
dt
 Q
= GA†nexp - 
kT
(38)
The higher the strain rate at given stress, the higher the rate of stress relaxation.
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