Dislocation Plasticity: Overview

1. D

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,

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) (b)

Fig. 1: (a) Method of creating a Volterra dislocation. (b) An edge

dislocation in a simple cubic lattice.

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

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,

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) (b)

Fig. 3: 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

γo_ij = 1

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 γp_ij = Vp V γ o ij = A V nibj (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

δγp_ij = n_Vibj δA = n_Vibj⌡⌠_L δxn dL = L_V nibj<∂xn> (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

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.

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 =









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)

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 [–1–12]) 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. B C A A A A A A A B B C C b b1 b₂ b b1 b₂ A A A A A A B B B B B B (a) (b)

Fig. 4: The slip of packed planes in fcc: (a) in stacking of

close-packed {111} planes, (b) in fcc unit cell. The dashed arrow

shows slip by a total dislocation, b = 1₂ [–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

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).

“ b

b1

b2

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) (b)

Fig. 6: (a) The CsCl structure drawn as a stacking of {110} planes. (b)

An antiphase boundary in the CsCl structure made by a

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. D

M

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.

(a) (b) (c)

Fig. 7: 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

∂W = ⌡⌠

V

o

ij  ∂Vp = ßijbjni∂A (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

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 ~ 2G_1-ˆ exp _     - 4π b « 2G 1-ˆ exp      -  2πh 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

(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,

double-kinks 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

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.

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 cross-slipped 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

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.

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

tension

compression

tension

(a) (b) (c)

Fig. 11: Interaction of edge dislocations in the same plane. Like dislo-cations repel (a), unlike dislodislo-cations 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

n = π(1-ˆ)_L_ b      † G (18)

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 b₁ b₁ b₂ b₂ b₁+b₂ (a) (b)

Fig. 12: Crystallographic interactions when dislocations cross.

3. T

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.

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

ß

†

b ƒ

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

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

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.

ß

‰

}

0.2%

ßy

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. M

C

Y

S

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.

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. R Ls ¥ †b †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

If we define the dimensionless stress as †* = 1 2R* = †bLs 2Γ « †Ls 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

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

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.

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,

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 disloca-tion 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 dra-matically, and appear in the form of discrete bands of deformation across the sample.

ß

‰

}

ß

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

interac-tion; 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

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 materi-als: 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 mecha-nism 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

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

tem-perature 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.

ß_y

time

lower temperature

Fig. 18: The variation of yield strength with aging time for a

precipitation-hardened 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.

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.

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

d (31)

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. W

H

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

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, œ_s, is, then

œs = d†_d© = - åGb 2 ®       d® 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

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

œ3 = 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, œ4, is low, of the order of 10

-4_{G, 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.

ß œ

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:

œ = dß

d‰ = ß (35)

and, therefore, usually fails before the fourth stage of work hardening is reached.

6. T

I

T

S

R

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

If the yield strength of a typical material is plotted as a function of its homologous

tempera-ture (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.

ßy

T/Tm

thermal

activation

athermal barriers _{thermal degradation}