MORRIS JR., J. W. Overview of Dislocation Plasticity.

8/20/2019 MORRIS JR., J. W. Overview of Dislocation Plasticity.

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Dislocation Plasticity: Overviewislocation Plasticity Overview

1. D

ISLOCATIONS AND PLASTIC DEFORMATION

An arbitrary deformation of a material can always be described as the sum of a change involume 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 simplyrelax back to their equilibrium sizes. The change in shape, on the other hand, may or maynot be recovered, since the atoms can relax into new positions that are configurationallyidentical to the original ones, but displaced from them. The part of the shear that isrecovered is elastic , the part that remains is plastic . Plastic deformation is a permanentchange 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 itsdimensions and correspondingly shorter in the others. In a crystal individual atoms moveby diffusion, and this process is known as diffusional creep . Second, all of the atoms inthe 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 isuniformly sheared, but remains atomically matched to the remainder along a commontwinning 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 largerug across a floor. And it is usually favorable to slip in increments that correspond to alattice displacement, so that the area of the plane that has slipped maintains a perfectcrystallographic match with the plane beneath it. In this case the boundary of the slippedarea 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 theearly 20th century (Volterra, 1907). However, the mechanistic connection betweendislocations and plastic deformation was not clearly recognized until the 1930's, when

Orowan, Taylor and Polanyi published almost simultaneous papers describing the essentialmechanisms 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 overthe plane indicated in the figure, and let the material above the cut be displaced with respectto that below it by the vector, b. Then let the lips of the cut be welded back together so thatthe cut becomes invisible. The only remnant of the operation is the linear distortion at theedge of the slipped region. This linear defect is called a dislocation, and the vector slip,

b

,


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J.W. Morris, Jr.: Overview of Dislocation Plasticity

on the plane it bounds is called the Burgers vector of the dislocation . If the solid iscrystalline, the slipped faces of the cut can only be welded to leave no trace if the Burgersvector, 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.

(a) (b)

Fig. 1: (a) Method of creating a Volterra dislocation. (b) An edgedislocation 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 anextra 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 itsboundary, can be curved or looped as shown in Fig. 2.

screwedge

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 thisfact. 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 dislocationemanate. If a dislocation lies between regions that have slipped, respectively, byb

1

andb

2

, its Burgers' vector is the vector sum: b = b1

-b2

. If a single dislocation, with Burgers'vector, b, divides into two dislocations (b

1

, b2

) at a node, then b = b1

+ b2

.

Second, a dislocation can be regarded as a one-dimensional defect that exists independentof the slip that created it. This viewpoint has the advantage of objectivity, since a perfectdislocation carries no record of how it was created. The edge dislocation drawn in Fig. 1b,


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for example, could have been formed by a vector slip,b

, on the half-plane to the left of thedislocation line, by a slip, -

b

, on the plane to the right, or by a sequence of slips that haveeither net result. As a one-dimensional defect the dislocation is characterized, locally, bytwo vectors, l

, a unit vector tangent to the dislocation line that defines the orientation of thedislocation, and the Burgers vector, b, which defines the strength of the slip it carries.

Bothl

andb

are ambiguous as to sign. It is conventional to remove the ambiguity byconstructing a Burgers circuit around the dislocation. Choose a closed circuit that can bedrawn 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 adislocated 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-handedscrew), the Burgers vector,

b

, is the vector displacement of the end point of the circuit fromthe start (Fig. 3b). It measures the net displacement experienced by an imaginary observerwho completes a circuit around the dislocation that would be closed in a perfect crystal.

(a) (b)

Fig. 3: A Burgers circuit closes in a {100} plane of a cubic crystal, butfails to close by the Burgers vector,

b

, when the same circuitencloses an edge dislocation.

The dislocation is an edge dislocation when

b

is perpendicular tol

, as in Fig. 3b. It iscalled a screw dislocation whenb is parallel to l, since an imaginary observer whofollowed a Burgers circuit around it would advance along its length by the vector b percircuit, as if he were following the thread of a right-hand screw. Dislocations intermediatebetween the edge and screw configurations are called mixed dislocations , and are oftencharacterized by the angle between

b

andl

. A curved dislocation like that shown in Fig. 2is 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 elasticinclusion with a thickness, h, equal to the interplanar distance perpendicular to the slipplane. If the slip is

b

, the strain in the inclusion is the simple shear

!oij =1h nib j (1)


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where !ij denotes a shear displacement of the ith coordinate face in the jth direction, ni is theith component of the normal to the slip plane and b j is the jth component of the Burgers'vector. While this may appear to be a needlessly complicated description, it is, in fact, avery 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 theamount

!pij =VpV

!oij =AV

nib j (2)

where Vp = Ah is the volume of the equivalent inclusion created by slip over the area, A. Ithas recently been recognized (Jin, Artemev and Khachaturyan, 2001) that the inclusionmodel 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. 2that the associated strain increment to the shear strain is

"!pij =ni b j

V "A =

ni b j

V#$

L "xn dL =

LV

nib j (3)

where


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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 pronouncedin 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, cylindricalcore of radius, r0, that includes the severely distorted material immediately around thedislocation line, and a long-range field in which the strain is small enough to be treated bythe methods of linear elasticity. Unfortunately, both are difficult to calculate withprecision. The stress in the elastic field decreases only as 1/r, where r is the radial distancefrom the core, with the consequence that the elastic energy diverges. This result isunphysical, since elastic fields in real materials are eventually canceled by the fields of otherdefects 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 astraight dislocation (line tension) is, then,

& =EL

=

'()(* Gb24 π ln+,- ./0Rr0 screw

Gb2

4 π(1-ˆ)ln+,-

./0R

r0edge

&esin2œ + &scos2œ mixed

(6)

where G is the shear modulus and ˆ is Poisson's ratio. The angle, œ, of the mixeddislocation is the angle between b andl, and &e and&s 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 adislocation will tend to minimize its elastic energy by surrounding itself with dislocations of opposite sign.

To complete the calculation of the line tension, & , of a straight dislocation we need theenergy 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 thelong-range elastic field, and can be roughly accounted for by setting the core cut-off radiusat r0 = b.

The computation of the line tension is further complicated when the dislocation is curved orthe specific arrangement of dislocations is considered. In lieu of an accurate calculation, itis often useful to use the qualitative relation

& «12 Gb2 (7)

which is a reasonable approximation for typical metals and alloys.


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1.4 Partial dislocations

While most crystal dislocations are total dislocations when viewed from sufficiently faraway, 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 astacking 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 arethe 1

2 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-typeatom sites (Fig. 8), the element of slip carries an atom from one B-site to another. It iseasiest to accomplish the slip in two sequential steps. The B atoms are first slipped into Cpositions, then moved from C back to B again. This slip can be accomplished by thesequential passage of two dislocations: b

1

(=16 [–1–12]) and b2 (=16 [–211]). However, b1

and b2 are not lattice vectors; they are examples of partial dislocations . Their sum is thetotal dislocation , b.

B

C

A

A A

AA

A A

B

B

C C

1

2

1

2

A

A

A

A

A

A

B

B

B

B

B

B

(a) (b)

Fig. 4: The slip of close-packed planes in fcc: (a) in stacking of close-packed {111} planes, (b) in fcc unit cell. The dashed arrowshows slip by a total dislocation, b = 12 [–101]. The solid arrowsshow successive slip by partial dislocations. The shaded atom isthe intermediate, C-site position.

Splitting the total dislocation, b, into the partials, b1

andb2

, not only facilitates slip, butalso lowers the line energy, since |b|2 > |b

1

|2 + |b2

|2. However, the separation of the two

partials, b1 andb2, also creates a stacking fault with a positive energy, ßs, per unit area, asillustrated in Fig. 5. Minimization of the total energy dictates the separation between thepartials, which is of the order of 5-500‹ in typical fcc crystals.

Similar considerations apply to hcp and diamond cubic crystals. Total dislocations in theclose-packed planes tend to divide into partials separated by ribbons of stacking fault. Inbcc crystals, however, the stacking fault energies are very high and separated partialscannot ordinarily be resolved. Nonetheless, incipient decomposition into partials


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apparently occurs in some cases, producing a complex structure along the dislocation core.The most important example is the 12 screw dislocation in bcc. This dislocation,which dominates low-temperature plasticity in many bcc alloys, is believed to have acomplex, triangular splitting in the core. The triangular core structure not only restricts itsmobility, 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).

12

Fig. 5: The total dislocation, b, divided into partial dislocations, b1

andb

2

separated by a stacking fault. 1.5 Ordered structures

Compounds and ordered structures have several atoms per unit cell, and, therefore, havevery large lattice vectors. As a consequence, dislocations in ordered structures are almostalways split into two or more partials. In many cases, the ordered structure is achieved byordering species on the sites of a simple fcc, bcc, NaCl or diamond cubic parent lattice. Inthis case, while the unit cell may be large, the partial dislocations are the usual dislocationsthat 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 adisplacement of the type 1

2 in the third plane.


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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 theantiphase boundary that is created by an interplanar slip by the vector b

= 12 , whichis a common Burgers vector for dislocations in bcc. The original structure can restored by

a second slip by an identical amount. As a consequence,1

2 dislocations in thisstructure tend to be paired, with an antiphase boundary between them. The perfectdislocation 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 metalthe 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 materialsthat are subjected to high temperature for any period of time.

On the other hand, the energy necessary to create dense distributions of dislocations isreadily available from the elastic energy that is stored under moderate loads. A solid that issheared elastically by © has a stored elastic energy per unit volume of 12 G©

2. If thisenergy were used to create dislocations with the line tension given by eq. 7 the resultingdislocation 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 dislocationdensities can be achieved; densities of the order of 1016/m2 are observed in cold-workedmetals, 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 theminimum value of b is the nearest neighbor distance. Hence the common dislocations infcc have

b

=12 while the common dislocations in bcc haveb =12 . In

ordered solids or compounds with multiatom unit cells the minimum Burgers' vectors arelarge 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 andpoor ductility.

2. DISLOCATION MOTION

2.1 Glide and climb

Except in the special case of a screw dislocation, the vectors b andl define a plane, whichis called the glide plane of the dislocation. A dislocation that moves in its glide plane cando so stepwise by simply breaking and reconfiguring the bonds immediately around itsline, 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 (orvacancies) 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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(a) (b) (c)

Fig. 7: Glide and climb of an edge dislocation. Only a single bond mustbe 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 andl are parallel, a straight screwdislocation can glide in any plane. Note, however, that any bending of the dislocationintroduces 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 appliedstress in an infinitesimal displacement of the dislocation. Let the dislocation line bedisplaced by the vector, ∂x, so that it sweeps out the area ∂A. The mechanical work done is

∂W =#$V ßij∂‰ ij dV = ßij!

oij ∂Vp = ßijb jni∂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∂xkßijb j =∂xkekimßijb jlm = f k∂xk (9)

hence the effective force on the dislocation, f, is

f

= b ^ ß x l (f k = ekimßijb jlm) (10)

It follows immediately from eq. (8) that the glide force per unit length on an edge or mixeddislocation is

f = †ben

(11)

where †b =b ^ ß ^ n

(=ßijb jni) is the shear stress on the glide plane resolved in the directionof

b

, ande

n

is a unit vector perpendicular to the dislocation line. The force impelling glideis a shear stress that acts perpendicular to the dislocation line.


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The force driving the climb of an edge dislocation can be found from eq. (10). Choose acoordinate system in which the dislocation line lies in the direction e

1

while the edgecomponent of the Burgers' vector, b

e

, 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, whichpoints along the normal to the glide plane. To interpret this force physically, note that ß22acts to stretch the crystal in the direction of b. If the dislocation climbs down (Fig. 7) theextra 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 bothl

andb

alonge

1

. Then

f =e2

ß31b - e3ß21b (13)

A screw dislocation can glide in any plane that containsb

. The glide force in a particularplane is (†b

e

n

), as in eq. (11), where † is the shear stress on that plane resolved alongb

.

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 criticalvalue, 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, †ccontrols 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 aclose-packed direction within the crystal. A finite shear stress is needed to move thedislocation from this energy well to another. This stress, the Peierls-Nabarro stress , setsthe minimum of †c in the low-temperature limit. Its value was estimated by Nabarro (1947)to be

†P ~2G1-ˆ

exp+,-

./0-

4 π b

«2G1-ˆ

exp+,-

./0-

2 πhb(1-ˆ)

(14)

where is the effective width of the dislocation in the glide plane. Eq. 14 pertains to anedge dislocation; the same equation without the factor (1-ˆ) applies to a screw. Criticismsand modifications of eq. (14) have been suggested by a number of authors over the years


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(for example, Huntington, 1955, Joos and Duesbery, 1997, Nabarro, 1997), but itsqualitative features survive: †P increases with the shear modulus, G, and decreasesexponentially with the interplanar spacing, (h/b). Moreover, the simple Nabarro formula isin rough agreement with the available experimental data (Nabarro, 1997).

From eq. (14) the dimensionless stress, †/G, depends almost entirely on the crystalstructure, through the minimum value of the factor (h/b). Predicted values of (†/G) areabout 10-5 for fcc crystals, 10-2-10-3 for bcc, hcp and NaCl, and 10-1 for oxides,∫-ZnSand diamond cubic materials. It follows that dislocation glide should be relatively easy inthe common metals, alloys and simple ionic materials, as is observed, but may be muchmore difficult in materials with diamond-like structures. In fact, at shear stresses near0.1G crystal lattices themselves become unstable with respect to spontaneous shear, anddislocation glide may no longer be the preferred mechanism of deformation (Morris, et al.,2000). Si, for example, deforms by a spontaneous structural transformation rather thandislocation glide in indentation hardness tests at low temperature.

The simple estimate of the Peierls-Nabarro stress is inaccurate for dislocations withcomplex structures. Screw dislocations in bcc are a particular example because of theircomplex core structures (as discussed above). The core asymmetry of these dislocationshas 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 byadopting a configuration in which segments along close-packed directions are joined byshort 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 normalmotion 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, sokink 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 infcc and hcp metals, so dislocations in these structures glide at stresses well below †P atmoderate temperatures (Hirth and Lothe, 1982). Double-kink nucleation is more difficult


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in bcc metals and in ionic and covalent crystals, and may be an important barrier to glide atlow temperature.

2.3.4 Slip systems

Taken together, the concepts of line tension and Peierls-Nabarro stress suggest that thedislocations that dominate deformation at low temperature will have the minimum possibleBurgers vectors and will lie in glide planes with maximum interplanar separation (h/b). Itfollows that crystalline solids have a strong tendency to slip in particular directions onspecific crystallographic planes, both determined by the crystal structure. The combinationof slip plane and slip direction is called the slip system. The members of a slip system arethe slip sets (nb) defined in eq. (1).

The common slip systems in fcc and hcp crystals combine close-packed directions withclose-packed planes: {111} in fcc (and diamond cubic), {0001} in hcp. Themost common prismatic slip system in hcp is {1–101}. The slip direction in bcc

crystals is almost always along , but several slip planes compete, including {110},{112} and, less commonly, {123}. The variety of available slip planes in bcc has theconsequence that slip sometime occurs in pencil glide , slip in a direction onapparently random slip planes. Slip in ionic and covalent crystals may be complicated bythe need to preserve bond or charge configurations, which may affect the operative slipsystems.

2.4 Dislocation climb: jogs

Dislocation climb is an important mechanism of deformation at higher temperatures, wherethe 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 orvacancies that are required for climb are most easily added at kink-like featuresperpendicular 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 thedislocation line. A single atom or vacancy added to an edge dislocation effectively creates apair of jogs. The second source is the intersection of dislocations. If dislocations withBurgers' vectors

b

1

andb

2

pass through one another, each will create a jog on the otherequal in length and direction to its own Burgers vector (Friedel, 1964). In both cases theshort 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 theparent line requires climb of the jog segment. For these reasons jogs are almost alwaysimpediments to dislocation glide and are often sessile segments that pin the dislocationlocally.


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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 screworientation 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 glideplane, as illustrated in Fig. 9. It may then slip back onto a plane that is parallel to, butdisplaced from the original glide plane. Cross-slip is a common mechanism for multiplyingthe number of active slip planes, and for by-passing microstructural barriers during plasticdeformation.

Note that the dislocation segments that bridge the parallel glide planes in Fig. 9 lie in adifferent 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 parallelglide plane.

When the dislocation is split into partials, as it commonly is in fcc and dc structures, thecross-slip process becomes more complicated, and often requires some thermal activation.The interaction between cross-slipped partials may also create sessile segments in thebridging plane that act as strong pinning points.

2.6 Dislocation multiplication

The density of dislocations (®) increases rapidly during plastic deformation. This happensfor two principal reasons. First, dislocations naturally become longer as loops expand andsegments extend to avoid microstructural barriers. Since the dislocation density is the linelength per unit volume, these natural processes increase ®.

Second, new dislocations are continually created by a variety of mechanisms. A commonmechanism that serves as the prototype case is the Frank-Read source that is illustrated inFig. 10. Let a dislocation be firmly pinned at two points. These may, for example, beprecipitate particles, or the bridging segments in a cross-slipped configuration like thatillustrated in Fig. 9. If the shear stress on the dislocation is † and the line tension isconstant, the dislocation bows out between the pinning points in a circular arc of radius


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r =&

†b «

Gb2†

(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, theyannihilate, 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. Thesource continues to operate, generating new dislocation loops, so long as the local stressremains above †c.

L

Fig. 10: Diagram of a Frank-Read source.

There are several common variants on the classic Frank-Read source, and several otherkinds of sources, most of which involve irregularities at free surfaces, grain boundaries ormisfitting inclusion particles. As a consequence, dislocation multiplication is relativelysimple in metals and alloys, and is only difficult in materials in which the line tension ishigh or the mobility is restricted.

2.7 Dislocation dislocation interactions

To understand how dislocations interact with one another, we note that a dislocation is bothan elastic and a crystallographic defect. Consider, for example, the edge dislocation shownin 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 thedislocation line. The elastic distortion near the dislocation is indicated in the figure. Ineffect, there are too many atoms in the region just above the dislocation line, which sets itin 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 understoodqualitatively from the simple principal that elastic fields superimpose, and attract if theyrelax 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 onadjacent planes they form a dipole pair, as in Fig. 11c.


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Edge dislocations also interact to form vertical arrays. If like dislocations approach oneanother on adjacent planes that are separated in the vertical direction there is an attractiveinteraction between them, since the tensile field of one partially cancels the compressivefield 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, iscalled a low-angle tilt boundary because lattice planes that cross the boundary are tilted bythe 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 dislocations attract and annihilate (b).Unlike dislocations on nearby planes trap one another to formdislocation dipoles (c).

Dislocations pile-ups result when like dislocations attempt to glide on a plane that isblocked by some obstacle, most commonly a grain boundary. The lead dislocation isblocked, and the trailing dislocations pile up against it. An important effect of the pile-up isto magnify and concentrate the shear stress, †. If there are n dislocations in a pile-upcreated 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 inmechanical equilibrium under the applied stress and the stresses due to one another. Theirequilibrium 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)+,-

./0L

b +,-

./0 †

G (18)


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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 thepile-up is penetrated or fractured when the effective stress reaches a critical value, †c. Theapplied stress, †, at yield or fracture then varies with (L/b)-1/2, where L is the maximumpile-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, edgedislocations 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 glideplane is preserved on the far side.

A further set of important crystallographic interactions is illustrated in Fig. 12. Whendislocation 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 forit 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 withBurgers' vector

b

=b

1

+b

2

, 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. THE YIELD STRENGTH

The stress required to initiate plastic deformation in a solid that is pulled in tension is calledits yield strength. The yield strength is the usual measure of the useful structural strengthof a ductile metal or alloy. As we shall discuss below, the yield strength depends on thetemperature and strain rate at which the test is done. However, for structural metals testedat low to moderate strain rate near room temperature, which is the case of greatestengineering interest, the yield strength depends primarily on the microstructure. Thestrength is controlled metallurgically by modifying the microstructure to influence themobility of dislocations.


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3.1 The yield strength of a single crystal

While we measure yield strength in tension, plastic deformation is ordinarily controlled bydislocations 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, †, thatexceeds the critical resolved shear stress for glide in at least one slip system.

œ

ß

ƒ

Fig. 13: A uniaxial tension, ß, produces a resolved shear stress, †, alongthe Burgers vector, b, of a dislocation that lies in a plane whosenormal is tilted by œ from the tensile axis.

Assume a cylindrical tensile specimen, as illustrated in Fig. 13. A dislocation with Burgersvector,

b

, lies in a plane whose normal makes the angle, œ, with the tensile axis. Thedirection of

b

makes the angle, ƒ, with the tensile axis. The shear stress on the planeresolved 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 criticalresolved shear stress for the sip set

nb

. It follows that the tensile yield strength is

ßy = min+,-

./0 †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 materialproperty (for a given slip set), the tensile yield strength is not. It varies with the orientationof the crystal. Only its minimum value is a material property, and is realized when the mostfavorable slip system (minimum †c) has œ = ƒ = 45º. In general, ßy ≥ 2†c.


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Second, the preferred slip system may change with the orientation of the crystal. This doesnot happen in fcc metals. The preferred slip system in fcc is {111} and the anglebetween {111} planes is sufficiently small that there is always a {111} set availablefor glide at stresses not too far above the minimum yield stress. But it does happen inmaterials 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 importanceas structural materials (e.g., iron and steel). The asymmetric core structures of dislocationsin bcc (discussed above) and the complexity of the possible dislocation-dislocationinteractions 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 exhibitanomalous slip , in which the active slip set is not the one that experiences the highest shearstress (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 theHCP 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 minimumwhen 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 isambiguous. 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 weakestelement of the microstructure. In the ideal case, this happens when the applied stress is justsufficient to move the most favorably oriented dislocations in the polygranular body, thatis, when ß = 2†c. Most polygranular materials yield at even smaller stresses, since theyhave residual internal stresses that add to the applied stress, and heterogeneities that causelocal stress concentrations. The earliest incidents of local yielding ordinarily do notpropagate, since the grains around the yielded grain are unlikely to have equally favorableslip systems. Nonetheless, local yielding produces a net plastic strain of the overall sampleand 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 noticeablyfrom its initial, linear slope. Eventually, the stress becomes sufficient to cause generalyielding, 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 sizeseparately below), a polygranular material will fracture along its boundaries unless itsgrains deform together. This requires that the typical grain have enough active slip systemsto accomplish an arbitrary change of shape. It can be shown that at least five independent


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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 fiveindependent slip systems, the yield stress must exceed its minimum value, 2†c, by a factorknown as the Taylor factor . For a cubic crystal, the Taylor factor is about 1.5, so thestress 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 isused as the criterion for the onset of plastic deformation, the material property that is mostimportant is the critical resolved shear stress, †c. The microstructural control of yieldstrength is accomplished by manipulating the microstructure to adjust †c.

ß

‰

0 2

ß

y

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 strengthas 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% offsetload as the nominal yield stress has the dual advantages that it is relatively easy to measurein practice, and, for most materials, corresponds fairly well to the stress required forgeneral yielding.

4. MICROSTRUCTURAL CONTROL OF THE YIELD STRENGTH

The yield strength is controlled by adjusting the critical resolved shear stress, †c, fordislocation 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. Theinherent value of †c is thePeierls-Nabarro stress that was discussed in sec. 2.3.2. Thecritical resolved shear stress is increased by placing microstructural obstacles in the plane of the dislocation that make it difficult to move.


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4.1 Obstacle hardening

The microstructural obstacles that inhibit dislocation slip through the grain interiors may besolute atoms, forest dislocations that thread through the slip plane, or small second-phaseprecipitates. 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 aspoint barriers in the slip plane. Their effect is captured in a few, simple constitutiverelations that are widely applicable.

Let a dislocation move over its slip plane under the action of a stress, †, that is significantlylarger than the Peierls-Nabarro stress, †p. In this case, the atomic structure of the slip planeis relatively unimportant, and the dislocation behaves roughly like a flexible, extensiblestring with a constant line tension & (« Gb2/2). When the dislocation encounters an arrayof obstacles that oppose its motion, it presses against them to create local configurationslike that shown in Fig. 15. The dislocation bows out between adjacent obstacles in acircular arc of dimensionless radius

R* =RLs

=&

†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* issignificantly greater than 1/2, there will always be at least one configuration of obstacles inthe plane that the dislocation cannot penetrate unless it passes through the obstaclesthemselves.

s

¥

†b

b

Fig. 15: A dislocation, modeled as a flexible string, pressing againstobstacles 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 = 2&cos(¥/2) (23)

where ¥ is the angle between the arms of the dislocation at the obstacle. Let Fc be the forcerequired 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 =Fc2&

(24)


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If we define the dimensionless stress as

†* =1

2R* =

†bLs2&

« †Ls

Gb =

†GbÔn

(25)

then it can be shown (Friedel, 1964, Hanson and Morris, 1975a, Labusch, 1977, Altintasand Morris, 1986) that the critical resolved shear stress, †*c , is a function of the obstaclestrength, ∫c, and the geometry of the obstacle distribution. In particular, the criticalresolved 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 fordislocation 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 obstacleconcentration. The equation holds reasonably well for hardening by solute atoms (in thelimit 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. Forexample, 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. Thecritical 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 andMorris, 1975b; Glazer and Morris, 1987),

(†*c )2 =%

i

(†*i )2xi (27)

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

†i (28)

This superposition is only strictly accurate when the strengths are so different thatsuccessively weaker mechanisms are so much weaker that their effect can be modeled as auniform friction stress.


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The approximations used in the point obstacle model are substantial. In real crystals thedislocation line tension is neither constant nor isotropic, the obstacles are finite and interferewith 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 oftenquantitatively 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 localdistortion 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 isalso 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 theslip 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 thesolution 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 åsis a constant that includes both the obstacle strength, ∫c, and the Taylor factor that relatesßy to †c. When the solute concentration becomes appreciable, the strain fields of theindividual solute atoms overlap so they no longer behave like discrete obstacles. In thisregime 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. Itfollows that interstitial solutes strengthen an alloy much more effectively than substitutionalones. 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 andequiaxed. Nonetheless, solution hardening is widely used in structural alloys with fccstructures, including Al alloys and austenitic steels.

The diffusional mobility of solute atoms may also affect the strength, particularly when thesolute is a mobile interstitial or when the test temperature is relatively high. The reason isthat solute atoms are attracted to dislocations, and diffuse so that they accumulate there,forming solute atmospheres . The most important engineering consequence of impurity


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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 temperaturesnear room temperature, and migrate to form atmospheres around dislocations. Since thereis 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, asillustrated 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

y

yield

point

yield point

elong tion

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 theinteractions between a gliding dislocation and the other dislocations that cut through itsglide 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 reasonableapproximation, 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 numberof dislocation that intersect a unit area of the glide plane, is one-half of the volumetricdislocation density, ®, the total length of dislocation line per unit volume. The yieldstrength of a material increases with its dislocation density according to the relation

ß = ß0 + ådGb ® (30)


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where åd ordinarily lies in the range 0.3-0.9. Eq. (30) has the form suggested by theobstacle model, and is reasonably well obeyed by structural metals and alloys.

Fig. 17: A mobile dislocation that is resisted by forest dislocations in itsglide 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 dislocationsand lower its strength. Dislocations are non-equilibrium defects, and heat treatmentdecreases their density by either of two mechanisms. The first is recovery . If a materialcontaining a high density of dislocations is annealed at a temperature high enough to permitdislocation climb, dislocations migrate and interact, both with one another and with freesurfaces. Some of the dislocations are annihilated, others are gathered into stable, planarconfigurations, such as low-angle grain boundaries (called subgrain boundaries ). The neteffect 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 isheated to a temperature above its recrystallization temperature , then new, defect-free grainsnucleate and grow at the expense of the old, producing a microstructure that is relativelyfree of dislocations. While heat treatments can decrease the dislocation density, they do noteliminate it entirely. A typical structural alloy that has been recrystallized and annealed hasa dislocation density in the order of 1010/m2.

2. Mechanical deformation . The dislocation density, ®, increases with plastic strain, andthe material work hardens according to eq. (30). When the mechanical deformation is doneat high temperature, as it is during the hot deformation that is used to roll metal ingots intoplates or sheets, work hardening is counterbalanced by recrystallization and recovery. To

achieve a high residual dislocation density it is necessary to deform at relatively lowtemperature. Metal products that are strengthened in this way are said to be cold-worked orcold-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 inthe annealed condition.

3. Transformation strengthening . Transformation strengthening is possible in materialsthat undergo martensitic transformations on cooling. The martensitic transformation


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changes the structure by shearing the parent lattice (fcc in the case of structural steel) intothe product (bcc in steel). If the steel is properly alloyed, for example, by increasing the Nicontent and decreasing the carbon, the mechanism of the martensitic transformation can beadjusted so that the transformation strains are accommodated by dislocations (the product iscalled 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. Suchprecipitates are normally introduced by aging a slightly supersaturated material at relativelylow temperature, so the precipitates nucleate primarily in the grain interiors. The volumefraction of the precipitates is determined by the phase diagram, and is, hence, fixed by thecomposition 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 particlesconsume 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 temperatureat given composition increases the volume fraction of precipitates with the consequence thatthe strength increases to a higher value. However, the material hardens more slowly dueits the lower diffusivity.

ß

y

t m

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 precipitatedensity. The former reason is more common. Most hardening precipitates are coherentwith the matrix, and lattice dislocations cut through them when they are small. As theygrow, their strength (∫c) increases, but their mean separation (Ls) also increases since theirvolume fraction is nearly constant. In the early stages of growth, the increase in obstaclestrength outweighs the increase in separation, and †c increases. However, there is an upperlimit 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 theobstacle, but wraps around it, and the obstacle strength is independent of precipitate size.


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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 asthe 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 theobstacles became impenetrable. However, in real solids precipitates have a distribution of sizes, and each size generates obstacle of several different strengths since it penetratesseveral 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 becomeimpenetrable (Glazer and Morris, 1986).

(a) (b)

Fig. 19: The interaction of a dislocation with impenetrable obstacles. (a)The arms of the dislocation wrap around the obstacle and attractone another. (b) The arms intersect and annihilate, producing apropagating dislocation, and leaving dislocation loops around theobstacles.

Some materials form very hard precipitates, which are uncuttable even when their size isvery 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 isaged. The reason is that such precipitates are ordinarily so difficult to form that theirvolume fraction is low when their size is small. In these materials the increase in hardeningwith 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 todislocations. Since the orientations of the active slip planes change abruptly at grainboundaries, slip must be transmitted indirectly from grain to grain. When a dislocationimpinges on a grain boundary its stress field produces shear stresses on the potentiallyactive slip planes of the adjacent grain. These add to the applied load and help to propagateplastic deformation by the motion of independent dislocations in the adjacent grain.


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Large grains are particularly efficient at transmitting strain to their neighbors. When a largegrain slips, a number of dislocations glide along the preferred plane (or along closelyspaced, parallel planes) and pile up against the grain boundary, as illustrated in Fig. 20. Aswe discussed in section 2.7.1, the stress at the head of such a dislocation pile-up ismagnified; 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 themore easily strain is transmitted across the boundary. The consequence is that the yieldstrength of a material decreases with its grain size.

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 itsgrain 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 andthe characteristics of the microstructure. The Hall-Petch relation is very well obeyed by avariety 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, therelation breaks down for the finest grain sizes where alternate deformation mechanismssuch as grain boundary sliding dominate.

5. WORK HARDENING

When a material deforms plastically it also hardens. The yield strength increases with thestrain, a phenomenon known as work hardening . The basic mechanism of work hardeningwas described above, when we considered how dislocations harden materials. As strainbuilds up inside the material, dislocations slip, intersect and interact with one another, asillustrated schematically in Fig. 21. These interactions cause an increase in the dislocationdensity (total dislocation length per unit volume) that is monotonic in the strain. In apolygranular material the dislocation density builds for two reasons: the multiplication of dislocations within grains and necessary development of a dislocation density along thegrain boundaries to match plastic strain across the boundary and keep adjacent grains


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together. Following Ashby (1966) these are usually classified as statistically stored andgeometrically necessary dislocations , respectively.

To a good first approximation the yield strength increases with the dislocation densityaccording 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 bestachieved in shear tests. The associated work hardening rate , œs, is, then

œs =d†d©

= -åGb

2 ® +,-

./0d®

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 dividedinto five separate stages (Argon, 1996).

Stage 1 is only observed when the crystal is oriented foreasy glide , with only one slip setactivated. The work hardening rate, œ1, is low, of the order of 10--4-10-3G. The primarymechanism of work hardening is the interaction of dislocations on neighboring slip planesthrough, 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 andwith one another, multiplying dislocations that act as obstacles to further glide. The workhardening 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 thatassociation and annihilation reactions between dislocations produce a significant rate of recovery. Along with the recovery, the dislocation distribution begins to develop a well-defined cell structure , in which roughly rectangular regions that are relatively free of


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dislocations are defined by relatively diffuse walls made up of dense tangles of dislocations. Since recovery decreases the dislocation density, the work hardening ratedecreases 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 3usually begins relatively early in the deformation process, and lasts until the flow stressapproaches values of the order of 5x10-3G.

Stage 4 evolves from stage 3 and is marked by the sharpening of dislocation cell walls intowell-defined subgrain boundaries. The sharp cell boundaries resist transmission of dislocations, and large elastic strains develop within the cells that also impede dislocationmotion (Mughrabi, et al,. 1986). The work hardening rate, œ4, is low, of the order of 10-4G, 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 5the dislocation and recovery processes are in balance and hardening terminates.

Fig. 22: The variation of the work hardening rate with the stress foraluminum alloys.

The work hardening behavior of a polygranular solid that is deformed in tension isordinarily much simpler, and can often be represented by two straight lines, as shown inFig. 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 localyielding. The effective work-hardening rate, œ = dß/d‰, is equal to Young's modulusduring the elastic portion of the stress-strain curve, and decreases rapidly as an increasingfraction of the volume of the material yields and contributes to plastic deformation. Aftergeneral yielding the deformation of the crystal is dominated by the overall dislocationdensity, which develops as expected for stage 3 hardening. A specimen in tension losesstability with respect to necking and fracture when the work hardening rate falls to the valueset by the Considere criterion:


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œ =dßd‰

= ß (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 understrain 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 motionduring 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 istaken, for example, by Kocks, Argon and Ashby (1975) and by Argon (1996) (theauthor'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 becauseit is simpler to do so, not because it is more accurate. We shall, therefore, finish thissummary with a brief discussion of rate and temperature effects, including the phenomenonof 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 inFig. 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 themelting point. Ambient temperature is in the intermediate temperature regime for Al and itsalloys, is slightly into the low-temperature regime for typical structural steels, and is in thehigh-temperature regime for low-melting metals like Pb.

ß

y

m

t

h

m

v

o

h

m

e

t

h

m

d

o

Fig. 23: Typical variation of yield strength with homologous temperature.


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The rapid strength increase at low temperature is due to the strong temperature dependenceof hardening by obstacles that can be cut or passed by thermal activation. Theseparticularly include the weak, short-range obstacles created by isolated solute atoms. Asthe temperature increases, the increased amplitude of atomic thermal vibrations produces aneffective 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 kinknucleation is also important as a mechanism for increasing dislocation mobility. The higherPeierls stress, greater interstitial lattice strain, and need for thermally activated kinkformation to sustain low-temperature deformation have the consequence that the increase instrength at low temperature is much more pronounced in typical bcc metals than in typicalfcc metal.

The low-temperature strengthening mechanisms are ineffective at temperatures above 1/4 to1/3 of the melting point. At intermediate temperature the yield strength is a relatively weakfunction of temperature. The strength in this region tends to be controlled by dislocation-dislocation and dislocation-precipitate interactions. These present obstacles that have

relatively large effective sizes; they spread over at least several atom spacings in the slipplane, and are not easily passed by thermal vibrations of the dislocation.

At temperature above 0.5Tm the yield strength begins to drop dramatically. The principalreason is the increasing rate of solid state diffusion, which affects both the dislocations andthe microstructural barriers. The microstructural effect is most important. Dislocationconfigurations recover by climb and recombination, precipitates coarsen and overage, andgrains grow; virtually all of the available microstructural barriers become ineffective. At thesame time, the high diffusivity makes it possible for dislocations to climb at an appreciablerate, so the obstacles that remain are more easily passed. The only practical mechanism forhardening 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 theydo not coarsen at a rapid rate, and relatively large, so that they are not easily passed bydislocation climb.

In assessing the strengths of the common metals and alloys, it is useful to keep in mind theprinciple 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 Mohave 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 thesample is strained. However, the magnitude of the rate effect varies dramatically with thematerial and with the temperature at which the test is done. The reason is that the commonrate effects have the same source as the temperature effects. If deformation is thermally


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activated it becomes easier to accomplish as the rate of deformation is lower; more time isprovided for thermal activation. To a good first approximation, increasing the strain rate isequivalent 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 strainrate when they are tested at intermediate temperature. As might be expected, the strengthsof structural steels are sensitive to strain rate, while those of Al and its alloys are morenearly 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 singlevariable, the Zener-Holloman parameter , Tln(•©), where the strain rate, •©, is defined in anappropriate dimensionless form.

7. HIGH TEMPERATURE CREEP

It has been known since the original work of Gibbs (1878) that a material that is subject toa shear stress is unstable with respect to any deformation that will relieve that stress andreturn it to a state of hydrostatic pressure. It follows that a material that is subjected to aconstant shear load will gradually deform in a direction that attempts to relax the load, aprocess known as creep , while a material that is given a shear strain will gradually deformto 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 rateat temperatures above about 0.5 Tm, where the rate of diffusion becomes appreciable.

‰

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 athigh temperature under constant load. The creep response is conveniently divided intothree regimes, called primary, steady-state, and tertiary creep. During primary creep thestrain rate decreases with strain (that is, the materials hardens), reaching a steady-statevalue that is preserved for some period of time, then monotonically increases until thematerial fails. Primary creep is ordinarily not extensive, and the creep behavior at small to


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moderate strains is well characterized by the steady-state creep rate. The steady state creeprate 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 stressexponent, 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 stressexponent, n, varies from 1 to 7 or more, and may have different values in different regimesof the applied stress.

While the precise mechanisms that govern dislocation-dominated high-temperature creepremain uncertain, their common feature is dislocation climb. A material can only support astatic load if the dislocations within it are effectively pinned at microstructural barriers. Tomove, they must be liberated, which requires dislocation climb onto slip planes that arefree. 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 fairlystraightforward interpretation. If we assume one dominant slip set, the shear strain rate isgiven by eq. 5,

•© = ®bv (5)

Both ® and v are functions of †. In the usual case, the dislocation density depends on themagnitude 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 oddpowers. 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 †, acharacteristic of Harper-Dorn creep (Wang, 1996). Larger stresses commonly produce adislocation density proportional to †2 and power-law creep with n = 3, while still higherstresses 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 bythe more complex eq. 4, which is further complicated by the fact that slip planes havedifferent orientations in adjacent grains. It is not surprising that the "best-fit" stressexponent is often a non-integral number.

Of course, creep is not always dominated by dislocations. At low stresses and hightemperatures, 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.


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In this case, n « 2, Q is the activation energy for grain boundary diffusion, and the materialmay exhibit superplastic deformation in which it sustains very large deformations beforefracture (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 rateof stress relaxation is governed by the steady-state creep rate. The residual stress in a partstrained in shear is † = G©. Stress relaxation under steady-state conditions is, hence,governed by the differential equation

d†dt

= Gd©dt

= GA†nexp+,-

./0-

QkT

(38)

The higher the strain rate at given stress, the higher the rate of stress relaxation.

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MORRIS JR., J. W. Overview of Dislocation Plasticity.

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