Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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150 4 Dislocation Motion

τ =

4πc

ωβ

ωb

As a consequence, the induced Snoek eﬀect shows a behavior similar to that

of the Cottrell eﬀect with a positive strain rate sensitivity at low velocities, and

a negative one at high velocities. In both eﬀects, the drag stress is proportional

to the solute concentration, in contrast to the action of the solutes as localized

obstacles where the ﬂow stress contribution is proportional to the square root

of the concentration of defects. The temperature dependence of the Cottrell

eﬀect is given by the diﬀusion coeﬃcient of the solutes, which mostly includes

the activation energy of the formation of vacancies in addition to the activation

energy of migration, as most diﬀusion mechanisms are vacancy mechanisms.

The activation energy of the Snoek eﬀect is only the migration energy.

4.12 Dynamic Laws of Dislocation Mobility

In many materials, dislocation velocities v

have been measured as a function

of the applied (eﬀective) stress τ

∗

and temperature T , mainly by selective etch-

ing described in Sect. 2.2. The stress dependence of screw and edge dislocations

in LiF single crystals is demonstrated in Fig. 4.41 as a double-logarithmic plot

taken from the pioneering work of Johnston and Gilman [19]. The dislocation

velocities increase strongly with increasing stress. The velocities of edge dis-

locations are higher than those of screws. At high velocities, the slope of the

curves decreases in agreement with the asymptotic behavior near the veloc-

ity of sound. The curves in Fig. 4.41 are typical also for other materials in

Fig. 4.41. Velocity of edge (squares) and screw dislocations (circles)inanas-grown

LiF single crystal as a function of stress. Regression curves according to (4.104). Data

from Johnston and Gilman [19]

4.12 Dynamic Laws of Dislocation Mobility 151

Fig. 4.42. Shear stress dependence of the velocity of screw (full symbols)and

◦

dislocations (open symbols) in high-purity silicon crystals grown by the ﬂoat

zone technique. The dislocation motion was monitored by X-ray topography. Tem-

peratures are indicated on the left side. Data from Imai and Sumino [262]. For

comparison, single data from in situ straining experiments in an HVEM from [163]

shown by large open squares. These data will be discussed in Sect. 6.2. Temperatures

are indicated on the right side

which the dislocation mobility is not controlled by the double-kink mecha-

nism. Characteristic curves for materials with the double-kink mechanism are

presented in Fig. 4.42. Here, dislocations of 60

◦

character move faster than

screw dislocations. For these materials, the high-speed range has not been

accessible. Some further examples will be presented in Part II. For incorpo-

ration into other theories, it is useful to describe the relation v

(τ

∗

,T)over

a wide range of the parameters by phenomenological laws. In a variety of rel-

evant deformation processes, the dislocation motion is supported by thermal

activation so that the Arrhenius equation (4.7) with the Gibbs free energy of

activation from (4.3) represents the natural, physics-based law

= v

exp



−

ΔG(τ

∗

)



= v

exp



−

ΔF (τ

∗

) − V (τ

∗

) τ

∗



. (4.101)

The activation volume is then given by

V = kT

∂ ln v

∂τ

∗

. (4.102)

The disadvantage of (4.101) consists in the fact that the activation free

energy ΔF and the activation volume V are, in general, still functions of τ

∗

.If

a particular process is suggested in the range where diﬀusion can be neglected,

the respective stress dependence of ΔG can be inserted, for example, (4.62)

for the combination of the Fleischer potential and Friedel statistics. Kocks

et al. [9] propose a general formula

152 4 Dislocation Motion

ΔG =ΔG



1 −



∗





where p primarily describes the tail of the interaction proﬁle, and q, the top.

The possible ranges are 0 <p≤ 1and1≤ q ≤ 2. The respective shapes

of the potentials are given in [9]. For practical use, this formula contains

too many variables so that the exponents have to be chosen for theoretical

reasons. If the dislocation velocities are analyzed over a wide range of the

temperature, it has to be considered that ΔG

and τ

are proportional to the

shear modulus, which depends on the temperature. Therefore, these quantities

should be multiplied by a factor μ(T )/μ

,withμ

being the shear modulus at

zero temperature. Respective analyses of experimental data are discussed, for

instance in Sect. 7.2.5, Fig. 7.21, for a simple box potential, and in Sect. 7.3.5

for a phenomenological potential suggested in [9]. A further disadvantage of

the Arrhenius relation (4.101) is the incorrect limiting behavior for both very

low and very high velocities. The correct behavior of v

→ 0forτ

∗

→ 0is

obtained by including back jumps with a changed sign of the work term Vτ

∗

which corresponds to the exchange of the exponential function by two times

the hyperbolic sine function. At high stresses where the dislocation velocities

are reduced by damping, (4.101) can no longer be used.

A phenomenological dynamic law, which approximates the Arrhenius rela-

tion but shows correct behavior at low velocities, is the power law similar to

(4.10)

= v

exp



−

ΔG



∗



. (4.103)

For a ﬁxed temperature, the Boltzmann factor with the activation energy can

be included into v

so that there are only two parameters, the pre-factor v

and the exponent m. Except for high velocities, this relation gives a good ﬁt

to experimental data, thus being frequently used. In Fig. 4.41, the low-velocity

range exhibits a relatively great slope of m =dlnv

/dlnτ ≈ 20, typical of

the obstacle mechanisms. The slope in the high-velocity range is m ≈ 1.8,

close to m = 1 characteristic of the viscous range. In Fig. 4.42, the slope m is

close to unity as expected for the Peierls mechanism.

Gilman [263,264] has suggested an equation of the form

= v

exp



−

ΔG



exp (−D/τ

∗

) (4.104)

to interpret the measurements in [19]. Again, the Boltzmann factor can be

included in v

for a ﬁxed temperature. This relation describes the correct

limiting behavior at both low and high velocities and ﬁts the measured veloc-

ities over a range of more than ten orders of magnitude as the regression

lines in Fig. 4.41 demonstrate. This equation has little physical justiﬁcation

as the processes are very diﬀerent in the diﬀerent velocity ranges. In spite of

its simplicity, it is rarely used in further evaluations.

4.12 Dynamic Laws of Dislocation Mobility 153

To determine the parameters of the phenomenological laws, dislocation

velocities have to be plotted vs. the stress at constant temperature in suitable

coordinates to obtain straight lines. Plots of ln v

vs 1/T at a ﬁxed stress,

so-called Arrhenius plots, yield the activation energy. Many data on the dislo-

cation mobility are concluded indirectly from macroscopic deformation tests

described in Sect. 2.1 using the formalism of thermal activation of Sect. 4.1.

The dislocation velocity is related to the plastic strain rate ˙γ by the Orowan

relation (3.5) via the mobile dislocation density 

. In the special tests of

determining the dislocation velocity, the dislocation density is mostly low so

that τ

in (4.1) is also low, and τ

∗

occurring in the dislocation mobility laws

is equal to the applied stress τ. This does not hold for the macroscopic defor-

mation experiments. Therefore, incremental tests are performed with sudden

changes of the deformation parameters: the strain rate by strain rate cycling

(SRC) or stress relaxation tests, and the temperature by temperature change

tests. It is assumed and mostly fulﬁlled only approximately that then 

and

remain constant so that

∂ ln (v

)=∂ ln ( ˙γ/˙γ

)and∂τ

∗

= ∂τ.

This problem will further be discussed in Sect. 5.2.2. Changes in 

and τ

after the sudden change of the deformation parameters will result in transient

eﬀects. Some of them will be described in Part II. An example of determining

the parameters of a dislocation mobility law will be given in Sect. 7.2.

The parameters of the Arrhenius law (4.101) may help identifying the

processes controlling the dislocation mobility. Table 4.1 lists the main mech-

anisms discussed in this chapter, giving characteristic values of the activation

parameters.

Table 4.1. Characteristic values of activation parameters of dislocation mobility

Mechanism Act. vol. (V/b

) m Activation energy Comments

Peierls mechanism 1–20 1–3 Related to line

energy

V only weakly

dependent on

stress

Loc. obstacles,

Friedel case

100–10

30–10

Fraction to several

V approx.

∝ c

−1/2

Loc. obst., Mott-

Labusch case

10–10

Mostly <1eV

Dislocation

intersection

>10

for

<10

−2

Climb ≈1 ≈1Energyof

self-diﬀusion

Jog dragging 10–10

>1Energyof

self-diﬀusion minus

work term

154 4 Dislocation Motion

Most activation energies depend on the type of binding and the elastic

constants. They can therefore vary between a few tenths of an electron-volt

up to several electron-volt. It should be kept in mind that for processes taking

place within time intervals of seconds up to some days, reasonable rates, for

example, in (4.6), are necessary. For atomic jumps in 100 s, for instance, with

an attempt frequency of ν

=10

−1

, the Gibbs free energy of the process

should be ΔG ≈ 35 kT or approximately 0.9 eV at room temperature. Thus, at

low temperatures the Gibbs free energies of activation will amount to fractions

of an electron-volt and to several ones at high temperatures. However, the

activation energies at zero stress are higher than ΔG by the work term Vτ

∗

which is large at low temperatures, and small at high ones.

To perform simulations of the motion of many dislocations, the dynamic

laws have to be inserted into the computer codes. This can be done by deﬁning

discretization nodes along the dislocation lines and physical nodes where the

dislocations branch. Then, the simulation consists in integrating the nodal

equations of motion [265].

Dislocation Kinetics, Work-Hardening,

and Recovery

In the preceding chapter, the motion of individual dislocations was described

as a function of the locally acting stress. During plastic deformation, however,

the dislocations have to move in an evolving dislocation structure. The main

eﬀect of this structure is a spatially and temporarily varying ﬁeld of inter-

nal stresses, which are superimposed onto the external stress. The eﬀective

stress τ

∗

controlling the velocity of the moving dislocations is inﬂuenced by

these internal stress ﬁelds. The most important parameter determining the

strength of internal stresses is the total dislocation density. The next section

on dislocation kinetics will describe the processes of the evolution of the dis-

location density and structure. The formation of the dislocation substructure

with an increasing dislocation density results in a hardening of the crystals.

This inﬂuence of the internal stress ﬁelds on the dislocation dynamics and

the ﬂow stress will be treated in Sect. 5.2. Finally, Sect. 5.3 will discuss plastic

instabilities.

5.1 Dislocation Kinet ics

In most crystalline materials, the dislocation density increases drastically

during plastic deformation leading to work-hardening. This process may be

described by an evolution law of the dislocation density containing a rate of

dislocation generation minus rates of dislocation immobilization and annihi-

lation. The dependencies of the dislocation density itself on the experimental

parameters like stress, strain, and temperature have been extensively studied

for numerous materials. However, the processes of dislocation generation and

recovery are much less understood, at least on a quantitative level, although

the basic models are as old as about 50 years. The present section describes

these models and summarizes the information on dislocation kinetics obtained

from in situ straining experiments in an HVEM.

156 5 Dislocation Kinetics, Work-Hardening, and Recovery

5.1.1 Models of Dislocation Generation

As described in Sect. 3.2.2, the energy to form even small dislocation loops is

very high so that the homogeneous nucleation of dislocations requires stresses

of about one tenth of the shear modulus. The generation of dislocations during

plastic deformation has to take place at far lower stresses. An exception is the

concentrated loading of the crystals using micro-indentation, where the acting

force during elastic loading suddenly decreases down to a lower level. During

this pop-in eﬀect, dislocations are nucleated at the high stresses under the

indenter tip (e.g., [266]). Most materials, especially those for structural appli-

cations, contain a certain dislocation density originating from the production

process. These grown-in dislocations form a three-dimensional network with

certain distributions of the segment lengths between the nodes [267–269]. The

ﬁrst mobile dislocations are formed by elongation, that is, bow-out, of such

existing dislocation segments. For this process, a segment of a certain length

has to lie on a slip plane, and the stress has to be high enough to fulﬁl the

same criterion as that for dislocation sources described below. If the grown-in

dislocations are pinned by impurities, an additional stress component is nec-

essary to unpin the dislocations. The dislocation density further grows when

the moving dislocations increase their length, mainly by one of the following

mechanisms.

The best-known model of dislocation generation is the Frank–Read source

[116] mentioned in Sect. 3.2.7, which may be characterized as a localized

source. A dislocation segment lying on a slip plane is pinned at its ends.

The pinning may be caused by some extrinsic pinning agents A and B like

precipitates as shown in Fig. 5.1a, when the dislocation changes onto another

plane where it is not mobile (b), or by nodes of the dislocation network (c).

The case of Fig. 5.1b is sketched again in Fig. 5.2. Under the applied stress, the

mobile segment moves through diﬀerent stages marked a to d. At stage d, the

arcs marked by arrows have opposite signs of the line direction. When they

approach each other, they can annihilate and the original source dislocation

e is restored. This localized source can operate repeatedly to emit a greater

number of dislocations.

In the cases of Fig. 5.1a, b, only one dislocation is involved so that all

dislocations are generated on the same slip plane leading to planar slip. In

Fig. 5.1c, where the dislocation nodes act as pinning agents, three cases are

possible depending on whether the Burgers vectors b

and b

of node A

(a) (c)(b)

Fig. 5.1. Diﬀerent types of pinning agents for localized dislocation sources

5.1 Dislocation Kinetics 157

Fig. 5.2. Localized (Frank–Read) source for the generation of many dislocations on

a single slip plane

and the Burgers vectors b

and b

have components perpendicular to the

glide plane of the source dislocation with Burgers vector b

,ornot.

1. These Burgers vectors do not have a component perpendicular to the

slip plane. Then, the dislocations are all emitted on the same plane as in

Fig. 5.1b.

2. If the dislocations with Burgers vectors b

and b

have equal compo-

nents perpendicular to the plane of the source, each subsequent dislocation

loop enters a new parallel plane at the distance of the normal component

of the Burgers vectors. Such a source is called a spatial source [87].

3. If the normal components are not equal, the source assumes a spiral char-

acter. These so-called pole mechanisms are important for the creation of

twins. The diﬀerent types of sources are reviewed in more detail in [270].

The dislocations emitted from the source cause a back stress on the source

dislocation. The dynamics of the emission of dislocations into an otherwise

undisturbed crystal was modeled in [271] by the motion of straight screw

dislocations emitted from the source in one direction. The dependence of

the velocity of the ith dislocation on the eﬀective stress is described by an

exponential law like the Arrhenius equation (4.101), in the form of

= A exp (Bτ

∗

) ,

where A and B are constants. The eﬀective stress on the ith dislocation is

given by the applied stress τ plus the sum of the internal stresses of all the

other dislocations

∗

= τ + τ

The shear stress exerted by the jth dislocation on the probe dislocation i is

given by (3.10) with y = 0 (all dislocations move on the same plane) and the

mutual distance x

− x

μb

2π

− x

158 5 Dislocation Kinetics, Work-Hardening, and Recovery

or that of all dislocations j

μb

2π



j=0,i=j

− x

μb

2π

The index j = 0 represents the source. Thus, the velocity of the ith dislocation

= A exp (Bτ)exp



Bμbξ

2π



or in the abbreviated form

= v

exp (αξ

) ,

where α = Bμb/(2π)andv

is the velocity of an isolated dislocation under

the uniform stress τ. The latter equation forms a set of diﬀerential equations

which were solved numerically. The dislocation source was represented by a

source dislocation ﬁxed at x = 0. After reaching an eﬀective stress higher

than the initiation stress τ

∗

= τ

at a site x ≥ 0, the source dislocation was

mobilized with a new source dislocation being introduced.

The above study yielded the following main results:

• After initial transients, all dislocations reach approximately the same con-

stant velocity, as demonstrated in the displacement vs. time plot of Fig. 5.3

for four selected dislocations. Thus, during their steady state motion, all

dislocations experience the same eﬀective stress τ

∗

. In other terms

∗

= τ +Δτ,

with a ﬁxed Δτ.

Fig. 5.3. Collective movement of dislocations emitted from a source. Displacement

vs. time plot for dislocations number 1, 3, 6, and 12. Parameters: ∂ ln v

/∂(τ

∗

/μ)=

μB = 1000, applied stress τ/μ =5× 10

−3

,sourcestressτ

/μ =2× 10

−3

.Velocity

of an individual dislocation at the same stress v

=1μms

−1

, velocity of leading

dislocation v

=9.1 μms

−1

. Data from [271]

5.1 Dislocation Kinetics 159

Fig. 5.4. Linear relation between the number N of dislocations emitted from a

source and the distance of the leading dislocation to the source L. Parameters:

normalized activation volume parameter μB = 500 (circles), 750 (triangles), 1,000

(squares), applied stress τ/μ =5× 10

−3

,sourcestressτ

/μ =2× 10

−3

.Datafrom

[271]

• The velocity of the leading dislocation v

, and accordingly also that of all

the others, is higher than the velocity of an individual dislocation v

under

the same applied stress τ.Thus,Δτ>0with

=exp(BΔτ) . (5.1)

• As demonstrated in Fig. 5.4, the number of emitted dislocations is pro-

portional to the distance of the ﬁrst dislocation from the source, that

is,

dL(t)

πΔτ

μb

This relation is independent of the normalized activation volume parame-

ter ∂ ln v

/∂(τ

∗

/μ)=μB = μV/(kT).

• The positions of the dislocations resemble those in a pile-up (see Sect. 5.2)

in front of the source dislocation.

The described model of a Frank–Read source corresponds to the prop-

erties of slip localized in a narrow slip band propagating into a crystal of

low dislocation density. Such a situation holds for some measurements of the

velocity–stress relation on dislocations generated at a scratch. The develop-

ment of a constant velocity of the leading dislocation, that is, of the head of

the slip band, is a prerequisite to such kind of measurements. On the other

hand, the velocity of the head dislocation is much higher than that of the

individual dislocations, thus making the interpretation diﬃcult. According to

(5.1), the velocity ratio depends on the stress diﬀerence Δτ, which, in turn,

depends on the applied stress τ and weakly on the source initiation stress τ

Velocity ratios were found up to 200. In another study [272], the behavior of a

double-ended array was analyzed, that is, of a source that emits dislocations