Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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170 5 Dislocation Kinetics, Work-Hardening, and Recovery

The ﬁrst factor is the number of atomic sites per unit length of the tube around

the dislocation. Fraction c

of these sites is occupied by vacancies. The third

term is the jump frequency of the vacancies having a diﬀusion coeﬃcient D

across the tube surface. The factor 12/3 considers the coordination number

and the fact that only one third of the jumps are forward jumps. Finally, the

bracketed term is the fractional diﬀerence between the inward and outward

ﬂuxes. μ

is the chemical potential of the vacancies at the dislocation. The

product of the diﬀusivity of vacancies and their concentration is the self-

diﬀusion coeﬃcient via vacancies D

= c

. To obtain the total ﬂux, i.e.,

the number of vacancies emitted or absorbed per unit time, J has to be

multiplied by the length of the dislocation

Φ =2πr

loop

J =

(2πr

loop

)(2πr

)



exp

− 1



In its area, the prismatic dislocation loop contains n = πr

loop

a/Ω extra

atoms or vacancies. Thus, the shrinkage rate is given by

Φ =

∂n

∂t

= −

2πr

loop

∂r

loop

∂t

resulting in

−

∂r

loop

∂t

2πr

loop

Φ ≈

8πr



exp

− 1



. (5.4)

Hirth and Lothe [130] treat the problem by assuming that point sources

are located on the dislocation at a distance x

and that the ﬂux is controlled

by diﬀusion into a shell having a diameter much larger than the loop radius.

The shrinkage rate of the loop is then

−

∂r

loop

∂t

2πD

b ln(8r

loop

)



exp

− 1



. (5.5)

The driving force of the climb is the line tension Γ of the dislocation loop.

With (4.82) and (4.84), the chemical potential becomes

ΩΓ

loop

The prismatic loop is everywhere of edge character so that sin β =1and

the line tension equals the line energy E

. For the dislocation loop, the outer

cut-oﬀ radius is equal to some fraction of the loop diameter, or R = αr

loop

with α near 1. Thus, with (3.14)

μbΩ

4π(1 − ν)r

loop

αr

loop

. (5.6)

This has to be inserted into (5.4) or (5.5). For small forces, that is, 0 <

/(kT)  1, the exponential exp(μ

/(kT)) can be replaced by 1+μ

/(kT).

5.2 Work-Hardening and Recovery 171

Equation (5.5) has the advantage that then the logarithmic terms cancel if

the inner cut-oﬀ radius of the dislocation is set to r

= αx

/8, yielding

−

∂r

loop

∂t

μbΩD

2(1 − ν)kT r

loop

. (5.7)

This equation can easily be integrated. For larger stresses, (5.4) or (5.5)

with (5.6) have to be integrated numerically. As mentioned earlier, the obser-

vation of loop shrinkage during annealing opens the possibility of measuring

the self-diﬀusion coeﬃcient via the velocity of climb. This method can be used

at temperatures of deformation tests which are considerably lower than those

of conventional diﬀusion experiments. Besides, in more-component materials

like ionic crystals or intermetallic alloys, the climb rate is controlled by the

diﬀusion of the slower diﬀusing component. In the loop-annealing studies,

this rate is measured directly. In [295], the method is extended to loops of

dissociated dislocations of nonpure prismatic character.

Dislocation annihilation can be introduced into the kinetic equation of the

dislocation density in a way similar to immobilization. Again, two dislocations

are involved in the process leading to second-order kinetics. In contrast to

immobilization, the process coordinate is now the time t, and the rate depends

on the temperature so that the kinetic equation may read

d = wτ

∗

ds − q

dt =



wτ

∗

v

− q



dt, (5.8)

with v

being the dislocation velocity. The temperature depending annihila-

tion rate coeﬃcient can be written as

q = q

exp (−ΔG

ann

/kT ) . (5.9)

ΔG

ann

is the activation energy of annihilation by climb. After the discussion

above, it should equal the activation energy of self-diﬀusion. Kinetic equations

of the dislocation density similar to (5.8) and (5.9) are part of a system of con-

stitutive equations describing the plastic behavior of materials. Such a set of

equations will be discussed in Sect. 5.2.4. Kinetic equations with annihilation

by climb can be applied to deformation at high temperatures. An example is

given in Video 9.15 showing the dynamic formation and annihilation of dis-

locations during the deformation of an FeAl single crystal. Dislocation length

disappears also by shrinkage of relatively large loops.

5.2 Work-Hardening and Recovery

In the preceding sections, the processes of the generation of dislocations and

their immobilization and annihilation were described, leading to an evolution

of the density of mobile and stored dislocations. This section discusses the

dislocation structures forming during deformation, their internal stress ﬁelds,

and their inﬂuence on the dislocation motion.

172 5 Dislocation Kinetics, Work-Hardening, and Recovery

Fig. 5.12. Motion of a probe dislocation S through a random arrangement of parallel

dislocations of opposite signs

5.2.1 Work-Hardening Models

In the early stages of plastic deformation of materials generating the new

dislocations by multiplication, the dislocation density is not very high, and

the dislocations are distributed in the developing slip bands quite uniformly.

The mobile dislocations have to move in the ﬁeld of the long-range inter-

nal stresses of all the other dislocations as sketched in Fig. 5.12 and ﬁrst

treated by Taylor [5]. Parallel dislocations of both signs are considered to be

distributed randomly. The mutual elastic interactions between two disloca-

tions passing each other on parallel planes of distance y

were discussed in

Sect. 3.2.4. According to this, the bypassing stress is

τ =

μb

4πy

for screw dislocations, and

τ =

μb

8π(1 − ν)y

for edge dislocations. The average distance between the dislocations in the

random array is given approximately by 1/

√



,where

is the density of

parallel dislocations. It may be assumed that

¯y

2π

√



(5.10)

is a suitable eﬀective distance between the slip planes to represent the bypass-

ing stress in the random array of dislocations. Thus, the latter can be

described by

= π

μb

2π

√



= α

μb

2π

√



. (5.11)

The factor α

for parallel dislocations assumes values between 1 and 2π,

depending on the details of the particular model. A frequently used value

is α

= π corresponding to screw dislocations and the estimate in (5.10). The

bypassing stress was designated τ

to characterize it as some average internal

5.2 Work-Hardening and Recovery 173

Fig. 5.13. Cutting of dislocations of slip system A with the projection of the Burgers

vector b

through dislocations of the oblique system B with Burgers vector b

during

in situ deformation of an MgO single crystal. Micrographs from the work in [281]

stress. Its meaning will further be discussed in the next section. The important

properties are its proportionality to the shear modulus and the square root of

the dislocation density. The increase of the internal stress component due to

the increasing dislocation density is termed work-hardening or strain harden-

ing, and Taylor hardening in the special case of randomly distributed parallel

dislocations.

In general, slip is not restricted to a single slip system as considered in

the Taylor hardening model. In multiple slip, dislocations of the primary slip

system have to intersect dislocations of the other (secondary) slip systems.

These processes have been described in Sect. 4.8. Figure 5.13 shows an example

of dislocations of a slip band A in an MgO single crystal which intersect an

oblique slip band B. The dislocations of the band B act as forest dislocations

between which the gliding dislocations A have to bulge. In the empty region

behind band B, the dislocations can freely expand like the loop indicated by

an arrow. Interactions between dislocations of diﬀerent slip systems include

also the formation of junctions as discussed in Sect. 4.8. According to [296],

so-called collinear interactions form strong pinning agents. In this interaction,

two dislocations AA



and BB



in Fig. 5.14 of the same Burgers vector glide

on non-coplanar planes, for example, on the slip and cross slip planes in an

f.c.c. crystal. When they meet at C, parts of the dislocations between D and E

can annihilate. In elastic equilibrium, the dashed segments form right angles

with the intersection line between the slip planes at the transition points D

and E. In the case of junctions, this angle is smaller than 90

◦

so that the

re-mobilization stress is higher for the collinear interaction.

The formula for the contribution τ

forest

of the intersection of a dislocation

forest of density 

to the ﬂow stress (4.73) is exactly equal to (5.11) with

another interaction constant α

. On a local scale, the square root dependence

was illustrated above in Fig. 4.32. Thus, this dependence of the internal stress

contribution to the ﬂow stress on the dislocation density is of very general

174 5 Dislocation Kinetics, Work-Hardening, and Recovery

Fig. 5.14. Collinear interaction after [296] between two dislocations of the same

Burgers vector on non-coplanar slip planes. Dashed lines show segments after the

reaction. The lines between D and E are annihilated

character. In this respect, the total dislocation density is the most important

parameter to describe a dislocation structure. The square root dependence of

the internal stress component on the dislocation density has been proved by

many experiments (see the review in [297]) and by three-dimensional com-

puter simulations of the evolution of the dislocation structure during plastic

deformation [298], reviewed in [299]. Recently, the results of three-dimensional

simulations of the development of dislocation structures and the mean free

path of dislocations are incorporated into macroscopic hardening models on

the basis of Taylor hardening predicting the orientation dependence of the

stress-strain curves of f.c.c. metals [300]. A typical experimental value is the

interaction constant in f.c.c. Cu of α =0.66 π for the total dislocation density,

and of α =0.52 π for dislocations breaking through the forest in cell walls (see

the next section) [301]. Equation (5.11) with a suitable constant α is a further

equation in the set of constitutive equations describing plastic deformation.

However, as (5.11) is explained using very diﬀerent models, the numerical

factor α

does not represent a universal constant. This is shown by a study

of deformed NaCl single crystals [302] presented in Fig. 5.15. The dislocation

density was varied either by work-hardening up to diﬀerent plastic strains in

a material of constant composition (squares), or by a change in composition

but with measurements at a constant (low) plastic strain (triangles). The

dislocation density was determined by electron microscopy of replicas of etched

cross-sectional faces. As the ﬁgure shows, the square root dependence of the

ﬂow stress on the dislocation density is observed in both sets of measurements.

The slope of the curves, however, is most diﬀerent.

The random distribution of dislocations assumed in the Taylor harden-

ing model is a structure of intermediate dislocation strain energy where the

outer cut-oﬀ radius R in (3.15) equals the mutual distance between the dis-

locations or

√

ρ. Such a uniform distribution of dislocations may occur in

the early stages of deformation. With increasing strain, the dislocation struc-

ture becomes heterogeneous with regions of low dislocation density and others

5.2 Work-Hardening and Recovery 175

Fig. 5.15. Dependence of the ﬂow stress of NaCl single crystals on the square

root of the dislocation density. Variation of the stress by strain hardening with

plastic strains between 1 and 12% in a material with 26 ppm of divalent cationic

impurities (Ca

)(squares) and by changing the impurity concentration between

1.7 and 256 ppm at a constant strain of 1% (triangles). Data from [302]

of high density. Such structures have been called cell structures. Very regular

heterogeneous structures designated as persistent slip bands (PSBs) form dur-

ing cyclic deformation (fatigue) as shown in Fig. 5.16. If the dislocations in

the dense regions form dipolar structures, the long-range stresses cancel and

the outer cut-oﬀ radius can be set to the distance between the dislocations

in the dipoles. As this distance is considerably smaller than the average dis-

tance between the dislocations, the total strain energy is lower than that for a

uniform distribution. Such low energy dislocation structures (LEDs) without

long-range stress ﬁelds were the basis of work-hardening theories represented

mainly by the work of Kuhlmann-Wilsdorf (e.g., [86, 303]).

On the other hand, dislocations can also form structures with an energy

higher than that for the uniform distribution in the Taylor hardening model.

This happens when dislocations of the same sign, for example, those generated

by a localized Frank-Read source, pile up against some obstacle to slip. In

this case, the stress ﬁelds of the individual dislocations superimpose to form

far-reaching stress ﬁelds. The obstacles to glide may be sessile products of

certain dislocation reactions, which are not discussed here. In particular, they

are grain and phase boundaries. A pile-up of N = 5 edge dislocations with

positions x

... against a boundary is shown in Fig. 5.17. The repulsive

interaction energy between the dislocations solely depends on the relative

spacings,

= W (x

− x

, ..., x

− x

) .

The head dislocation is in elastic equilibrium with the repulsive force of

the obstacle and with the interaction forces of all the other dislocations. In

equilibrium, the force on all other dislocations is zero

τb =

∂W

∂x

∂W

∂x

= ···=

∂W

∂x

176 5 Dislocation Kinetics, Work-Hardening, and Recovery

1 µm

Fig. 5.16. Persistent slip band (PSB) structure formed during cyclic deformation of

an Ni single crystal unloaded after the compression cycle followed by in situ tensile

deformation in an HVEM. Arrows: dislocations with strong curvature near dense

regions. Micrograph from the work in [304]

boundary

Fig. 5.17. Pile-up of ﬁve edge dislocations against a boundary

The equilibrium positions can be found by numerically solving the set of

algebraic equations of the interaction forces. The force exerted on the head

dislocation 1 by the dislocations 2 to N is given by

−

∂W

∂x

∂W

∂x

∂W

∂x

+ ···+

∂W

∂x

=(N − 1) τb .

As the external stress acts on the head dislocation, too, the total force on it

per unit length is

head

= τNb. (5.12)

Thus, the force acting on the head dislocation is the same as that on a single

dislocation of an N -fold Burgers vector. This rule holds for pile-ups of all

types of dislocations in homogeneous stress ﬁelds. The elastic far-ﬁeld of the

5.2 Work-Hardening and Recovery 177

pile-up converges for distances greater than the length of the pile-up against

that of the single dislocation of N-fold Burgers vector. The stress at a point A

with coordinates r, φ in front of the head dislocation and close to it (as shown

in Fig. 5.17) can be written as

pileup

(r, φ)=f(φ)



τ, (5.13)

where f(φ) is a function of φ,andL is the length of the pile-up. Remarkable

is the weak square root decrease of the stress with increasing radius r.This

near-ﬁeld resembles that of a crack.

The establishment of long-range stress ﬁelds by dislocation arrangements

like pile-ups was the basis of work-hardening theories put forward by Seeger

and coworkers (e.g., [87]). These theories are controversial to those of the

formation of low-energy dislocation structures by Kuhlmann-Wilsdorf, men-

tioned earlier. In any case, pile-ups form in materials with grain or phase

boundaries when dislocations emitted from localized sources queue in front

of the grain or phase boundaries. The strong stress concentrations ahead of

the pile-ups may then initiate slip in the neighboring grains or lead to the

formation of cracks. Pile-ups of a few dislocations are regularly observed

in connection with localized Frank–Read sources as the video sequences

Videos 8.8 or 9.16 illustrate. The fresh dislocations pile up in front of the

source. Their back-stress blocks the source until the head dislocation breaks

through its obstacle, a boundary, or simply a dense region of dislocations.

A more realistic approach to the heterogeneous dislocation structures

formed by deformation and their internal stresses is found in the composite

model established by Mughrabi (e.g., [305]). Similar ideas were also developed

by Holste and coworkers [306]. In the composite model, a deformed crystal

with cell or wall structures is treated as a material composite consisting of

regions of high and low dislocation densities as outlined in Fig. 5.18. According

to the diﬀerent dislocation densities, the local ﬂow stresses are diﬀerent, too.

They are denoted τ

in the hard walls of high dislocation density, and τ

in the

soft regions. Because of the necessary compatibility, the hard and soft regions

have to be sheared in parallel so that the total shear strains, that is, elastic

plus plastic ones, are equal. Thus, under low applied stress τ, both phases

deform elastically. When the stress reaches τ

, the soft regions start to deform

plastically but the hard regions still resume elastic deformation. Only when

the total shear strain reaches γ

= τ

/μ,whereμ is the shear modulus, both

phases deform plastically. The ﬂow stress of the composite is then given by

τ = x

+ x

where x

and x

are the area fractions of the hard and soft regions with

+ x

= 1. It can easily be shown [305] that τ

>τ and τ

<τ and that

(τ

− τ)+x

(τ

− τ)=0.

178 5 Dislocation Kinetics, Work-Hardening, and Recovery

slip planes

Fig. 5.18. Composite model of a sheared structure of hard and soft regions after

Mughrabi [305]. Dislocations in the soft regions are not shown. s stored dislocations,

m misﬁt dislocations

The local diﬀerences in the acting stress can be understood as the con-

sequence of long-range internal stresses. With no external stress acting, the

hard dislocation walls consist of statistically stored dislocations, mainly in

low-energy dipole conﬁgurations. These are the dislocations in regions s with

thin symbols in Fig. 5.18. When the deformation starts in the soft cell inte-

riors, the dislocations are blocked at the hard cell walls and form the misﬁt

dislocations in regions m marked by bold symbols. These geometrically nec-

essary dislocations (GNDs) accommodate the elastic mismatch between the

hard and soft regions. They are the sources of the long-range internal stresses

necessary for the simultaneous compatible deformation of the whole crystal, as

proved experimentally in [307]. If the stress is high enough for the deformation

of the hard walls, the dislocation ﬂux becomes equal in the walls and the soft

cell interiors [308]. Dislocations in cell walls formed during multiple slip do not

represent GNDs. These walls may be called incidental dislocation boundaries,

whereas the kink walls and dislocation sheets forming during deformation in

work-hardening stage II consist of the GNDs.

In the channels between persistent slip bands, the local stresses were mea-

sured from the curvature of dislocations, which were pinned under load by

neutron irradiation [309]. As described in Sect. 3.2.7, there is the problem of

the adequate values of the line tension. In [309], the line tension was scaled so

that the average of the local stresses equals the macroscopic external stress.

In this frame, the study shows how the local stress varies across the channels

in the persistent slip band structure. The stress at the center of the channels

is approximately half the external stress and near the walls it is about three

times of it. Strongly curved segments near the walls are also marked by arrows

in Fig. 5.16. The variation of the local stresses in a PSB structure was also

estimated from dislocation curvatures during in situ straining experiments on

Ni single crystals [304]. However, the line tension data used there have to be

corrected (see the discussion in Sect. 5.2.3).

The composite model certainly represents a bridge between the low-energy

dislocation structure and the pile-up theories of work-hardening. In this model,

5.2 Work-Hardening and Recovery 179

0.5 µm

Fig. 5.19. Dislocations and dislocation debris in an NiAl single crystal during in

situ deformation in an HVEM. Image normal [110], g =(

110), b projection of [010]

Burgers vector. From the work in [287]

long-range internal stresses result from the inhomogeneity of the dislocation

density, without the necessity of evoking the existence of pile-ups with very

high stress concentrations.

The hardening models discussed so far have all been concerned with long-

range internal stresses. An exception is the production of so-called dislocation

“debris” during deformation. The debris consists of small dislocation dipoles

produced by double-cross slip events of low cross slip height, which do not meet

the criteria for dislocation multiplication, as described above in Sects. 5.1.1

and 5.1.2 (Figs. 5.5 and 5.8). The idea of dislocation dipoles acting as harden-

ing agents was introduced by Gilman [310]. Figure 5.19 exhibits a dislocation

structure with many short dipoles during in situ deformation of the intermetal-

lic alloy NiAl. Arrows mark two dislocations that are just trailing dislocation

dipoles.

Dislocation debris occurs only in materials that generate the new disloca-

tions by the double-cross slip mechanism. Within the single specimens, the

density of the debris is frequently not uniform. A certain part of the ﬂow stress

is necessary to produce the debris. It can be estimated from the Frank–Read

criterion for fully bowing dislocation segments (3.47), with L being the dis-

tance between jogs trailing dipoles. It may be concluded from Fig. 5.19 that

the distances between other pinning agents along the dislocations (the cusps

in the dislocations) are much shorter than those between the trailing jogs, so

that the ﬂow stress contribution from generating the debris is mostly small.

Existing debris can contribute to the ﬂow stress by elastic interactions with