Kosevich A.M. The crystal lattice

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324

14 Dislocation Dynamics

One of the methods of solving the system of equations (14.1.5) consists in replac-

ing an inﬁnitely small displacement of the dislocation line element by an inﬁnitely

small dislocation loop. This method is only fruitful when the time dependence of the

deformation ﬁeld is studied.

We differentiate the equation of elastic medium motion with respect to time us-

ing (14.1.2). We then obtain a dynamic equation of elasticity theory for the velocity

vector v:

∂

∂t

− λ

iklm

∇

= λ

iklm

∇

. (14.1.12)

The role of the force density in this equation is played by th e vector f

= λ

iklm

∇

The solution to (15,1.12) can be written as

(r, t)=







−∞



(r − r



, t −t



) f



, t



) ,

where G

(r, t) is the Green tensor of the elasticity theory dynamic equation. This

formula solves completely the problem of ﬁnding the displacement velocity and de-

termines the time dependence of the displacements.

We recall that the Green dynamic tensor for an isotropic medium is written explic-

itly as

(r, t)=

4πρr





t −



−n



t −





t(δ

−3n

)

4πρr





t −



− Θ



t −



where s

and s

are the longitudinal and transverse sound velocities; r = nr, Θ(x) is

the Heaviside discontinuity function.

The present Green tensor resembles the Green function for an electromagnetic ﬁeld

in the medium. The equations for the dynamics of an elastic ﬁeld with dislocations

are not very different from the dynamic equations for an electromagnetic ﬁeld with

charges and currents. These are the differential equations in partial derivatives of the

type of wave equations with sources. Since nonstationary (accelerated) motion of

the sources generates a radiation ﬁeld, the accelerated motion of dislocations induces

radiation of elastic (sound) waves. The description of acoustic radiation of disloca-

tions can be made by a standard scheme developed in ﬁeld theory, but is somewhat

complicated by the tensor character of the deformation ﬁelds.

14.2 Dislocations as Plasticity Carriers

325

14.2

Dislocations as Plasticity Carriers

Dislocation motion is accompanied (apart from changes of elastic deformation) by

changes in the crystal not associated directly with stresses, i. e., plastic deformation

(Fig. 14.1). By a plastic deformation we understand a residual crystal deformation

that does not vanish after the process that generates it is over.

Fig. 14.1 Scheme of a residual atomic rearrangement resulting from the

motion of an edge dislocation.

Let an edge dislocation pass through a crystal from left to right. As a result, part of

the crystal above the glide plane is displaced by one lattice period . Since the lattice

at any point inside the specimen is regular after the dislocation has moved through

it, the crystal remains un stressed. Unlike the elastic deformation associated directly

with the thermal state of a solid, plastic deformation is a function of the process. (In

considering stationary dislocations the question of distinguishing between elastic and

plastic deformation does not arise. We are interested in stresses that are independent

of the previous crystal history.)

It is clear that the dislocation displacement is inevitably related to the occurrence of

plastic deformation: dislocations are elementary plasticity carriers.

For a quan titative description of dislocation elasticity, it is necessary to return to

(14.1.2). We introduce the vector of the total geometric displacement of the points of

the medium u measured from their position befo r e the deformation starts. The time

derivative of the vector u determines the displacement velocity of the medium element

v = ∂u/∂t. Therefore,

∂w

∂t

= ∇

, (14.2.1)

where w

is the total distortion tensor, w

= ∇

Using (14.2.1) we rewrite (14.1.2) as

∂

∂t

−u

)=−j

The difference w

−u

determines the part of the total distortion tensor that is not

associated with elastic stresses and is generally called the plastic distortion of a solid.

We denote this quantity by u

and obtain

∂u

∂t

= −j

. (14.2.2)

326

14 Dislocation Dynamics

Thus, the variation in the plastic distortion tensor at a certain point of the medium

for small time δt is equal to

δu

= −j

δt.

A similar relation for the plastic strain tensor ε

has the form

δε

= −

+ j

)δt. (14.2.3)

The relation between dislocation ﬂow density and plastic deformation velocity, i. e.,

the relations equivalent to (14.2.2) or (14.2.3) were indicated by Kroener and Rider

(1956).

If, during the dislocation motion, the elements of the medium move without break-

ing the continuity, then j

= 0. It then follows from (14.2.2) that ε

= 0. Thus,

we come to the assertion, known in plasticity theory, that a pur ely plastic deformation

taking place without breaking the medium continuity does not result in a hydrostatic

compression (which should be connected with internal stresses).

It is clear that (14.2.2), (14.2.3) are valid for any mechanism responsible for disloca-

tion migration, in particular, their nonconservative motion (climb). In the last case, as

shown above, the crystal volume changes locally, but the general scheme of plasticity

is not broken.

To evaluate the role of dislocation climb it is exp edient to imagine a semi-

microscopic scheme for this process. In Fig. 14.2, the edge of an extra atomic

half-plane coincident with the edge dislocation axis is shown. When the interstitial

atom gets attached to the edge of a half-plane inserted in a crystal, it becomes a “reg-

ular” atom , resulting in a protuberance of atomic dimensions on the dislocation line.

The dislocation line itself is determined up to atomic dimensions, so that its po sition

can only be affected by the “condensation” of a macroscopic number of interstitial

atoms. The change in position of an extra half-plane edge under the condensation of a

great number of interstitial atoms (I), when an extended row of extra atoms is formed

on the dislocation, is shown in the middle of Fig. 14.2. The appearance of an extra

row of atoms results in the displacement of a corresponding part of the dislocation line

by the value a in a direction perpend icular to the slip plane. The dislocatio n appears

to go down by one atomic layer; it goes to the next slip plane.

The condensation of vacancies (V) on the edge dislocation leads to similar results,

the difference being that the dislocation displacement is oppositely directed – the dis-

location line goe s up to a higher slip plane.

Both the glide and climb occur under the action of elastic stresses. The glide, when

viewed as a fast mechanical motion, differs from climb in its threshold character: it

only begins when the stress exceeds a deﬁn ite value (the initiation stress). But in a

number of crystals (e. g., in many metals and quantum crystals) comparatively small

stresses are needed to start dislocation motion. If the crystal has a regular structure

the dislocations at the ﬁrst stage of their motion glide easily and often travel large

14.3 Energy and Effective Mass of a Moving Dislocation

327

Fig. 14.2 Climb of an edge dislocation via elongation of an extra atomic

half-plane under the condensation of interstitial atoms (I) or shortening

of an extra half-plane under the condensation of vacancies (V).

distances in their slip planes. However, the process o f easy g liding is short since

obstacles decelerating the dislocation motion arise in a crystal.

The obstacles decelerating the dislocation are different. These may be defects such

as impurities, ﬁxing the dislocation line at some points only. But sometimes the dislo-

cation stops gliding over an extended part of its line. This occurs when the dislocation

is retard ed by clusters of imp urities or macroscopic heteroge neous inclusions. Ob-

stacles of any kind retard the dislocation motion. As a result, the velocity of crystal

plastic deformation is determined by factors such as climb and glide, and whether the

dislocation can overcome the obstacles.

14.3

Energy and Effective Mass of a Moving Dislocation

The system of equations (14.1.5) determines the elastic distortion tensor u

and the

vector v

of the medium displacement velocity using a given distribution of disloca-

tions and ﬂows. The tensors α

and j

and, hence, the dislocation motion are assumed

to be given. For a system of equations to be completely closed and to determine a self-

consistent evolution of dislocations and elastic ﬁeld, it is necessary to take into account

the changes in the dislocation d ensity and their ﬂow under the action of elastic ﬁelds.

In other words, it is necessary to know the equation of dislocation motion.

Since separate dislocation loops represent lines of elastic ﬁeld singularities, the

equation for dislocation motion is an equation for the motion of an elastic ﬁeld singu-

larity. The physical idea of obtaining the equation of motion of a ﬁeld singularity (in

the given case, a dislocation) is well known.

A dislocation is a source of internal stress, which creates stress and strain ﬁelds

in a crystal free of external loads. A deﬁnite elastic energy that can be regarded as

the dislocation energy is associated with this ﬁeld. When the dislocation moves, the

elastic ﬁeld induced by it also moves, but it always has inertia, because the dynamic

328

14 Dislocation Dynamics

elastic ﬁeld energy is different from the static ﬁeld energy. The inertia of the elastic

ﬁeld of the dislocation can be interpreted as the inertia of the dislocation, which can

be described by an effective mass. With such an approach, the energy and the mass of

the dislocation and, hence, the equa tion of th e dislocation motion are of a pure ﬁeld

origin.

Accordingly, to evaluate the dislocation dynamics, it is necessary to know the dislo-

cation e nergy. To calculate the latter it is necessary to express the elastic energy of the

ﬁeld caused by a moving dislocation loop (or a system of loops) through the instanta-

neous coordinates an d velocities of d islocation line elements. In accordance with the

electromagnetic ﬁeld theory of moving charges, this procedure is generally possible

using an approximation that is qua dratic in velocities (Section 3.6). Since the struc-

ture of the dynamical equations of an elastic ﬁeld with dislocations is qualitatively the

same as that o f Maxwell’s eq uations, the difﬁculties in carrying out the a bove p ro-

cedure may be due to additional calculations only. The latter are rather cumbersome

(even in an isotropic approximation) for the general case of a dislocation loop moving

arbitrarily in a crystal. Taking this into account, we try, by analyzing a simple example

and its almost obvious generalizations, to explain the general features of deriving the

equation for dislocation motion.

We consider th e uniform motion of a linear screw dislocation in an isotropic

medium. We choose the z-axis along the dislocation line and the x-axis parallel to

the direction of the d islocation velocity V. As in the static case (Section 3.14), the

elastic ﬁeld of a screw dislocation is completely described by a single nonzero com-

ponent of the displacement vector u

= u(x, y, t) and the equations of the elastic

medium motion reduce to the 2D wave equation

∂

∂t

= s



∂

∂y

∂

∂y



, s

. (14.3.1)

The solution to (14.3.1) should have a standard dislocation singularity on the line

x = Vt, y = 0, and, thus it is convenient by changing the variables

ξ =

x −Vt



1 − β

, β

ρV

to write it as

∆u

= 0, ∆ ≡

∂

∂ξ

∂

∂y

, (14.3.2)

where u

(ξ, y)=u(x, y, t).

Equation (14.3.2) coincides with (13.3.1) and its solution has a d islocation singular-

ity at the point ξ = y = 0. Thus the function u

(ξ, y) is identical to the corresponding

static solution (13.3.2) if the following replacement x → ξ is made.

14.3 Energy and Effective Mass of a Moving Dislocation

329

The total ﬁeld energy per the dislocation unit length is

E =







∂u

∂t



+ G



+ U





dx dy





1 + β

1 − β



∂u

∂ξ





∂u

∂y





dx dy



1 − β





1 + β

1 − β



∂u

∂ξ





∂u

∂y





dξ dy.

(14.3.3)

Using polar coordinates in the plane ξOy we obtain instead of (14.3.3) the following

expression:

E =



1 − β





1 + β

1 − β

sin

ϕ + cos



ϕz

rdrdϕ

πG



1 − β



ϕz

rdr.

Finally, we substitute the expression for ε

ϕz

that follows from (13.3.3):

E =



2π



πG



1 − β



4π

log(R/r

)



1 − β

, (14.3.4)

where the parameters R and r

are chosen as in the static case.

Expressing in (14.3.4) the shear modulus G through the transverse sound wave

velocity (G = ρs

), we arrive at the following formula for the energy per unit length

of a screw dislocation

E =

∗



1 − v

, m

∗

ρb

4π

log

. (14.3.5)

In spite of the “relativistic” form of (14.3.5) this formula is valid only for small dis-

location velocities, assuming V  s. Such a “pseudodepletion” of (14.3.5) results

from the fact that the dislocation energy has the ﬁeld as or igin and involves, in par-

ticular, the interaction energy of different elements of the same dislocation line. To

take into account the interaction of distant dislocation loop elements in the case of

nonstationary motion the elastic dislocation ﬁeld at a certain time should be expressed

through its coordinates and velocities at the same time. In the general case, because

the elastic waves are retarded, this expression does not exist. As mentioned above for

a nonstationary motion this is possible within the approximation quadratic in V/s.

Thus an expression for E, valid for any time depend e nce of the screw dislocation

velocity V will be obtained using the condition V  s and replacing (14.3.5) by

E = m

∗

. (14.3.6)

330

14 Dislocation Dynamics

The ﬁrst term in (14.3.6) coincides with the eigenenergy per unit length o f the dislo-

cation at rest (13.3.17), the second is assumed to be its kinetic energy. Therefore, m

∗

can be called the unit length effective mass of a screw dislocation. As in the case of a

rest energy, our assumption is justiﬁed, i. e., when a real dislocation moves in a crystal,

some atoms in the vicinity of the dislocation axis at a distance of the order of r

from

it also start moving. This generates an additional dislocation ine rtia due to the usual

atomic mass. The order of magnitude of the atomic mass inside tube of radius r

∼ b

per unit length of the dislocation can be estimated as ρr

∼ ρb

. Comparing this

estimate with (14.3.5) for m

∗

, the dislocation mass can be regarded with logarithmic

accuracy as the ﬁeld mass.

The inertial p roperties o f an arbitrary weakly deformed dislocation loop are charac-

terized by some tensor of line density of the effective mass m

∗

dependent on a point

on the dislocation line. It can be concluded that at a point where the radius of curva-

ture of the dislo cation loop R

curv

 b, an estimate of the order of magnitude of the

effective mass is the same as that of the dislocation rest energy (13.3.18):

∗

∼

ρb

4π

log

. (14.3.7)

In the case o f translational motion of the linear dislocation, R

curv

is the dislocation

length. If the dislocatio n vibrates, R

curv

equals the wavelength of the dislocation

bending vibrations.

A very important physical conclusion follows from the previous comments con-

cerning the estimation o f the parameter R

curv

. By writing the dislocation energy in the

form of (14.3.6), we introduced the effective mass of unit length of the dislocation, but

characterized in fact the motion of the dislocation. This mass is not a local character-

istic of the dislocation. We recall that with the retardation of electromagnetic waves in

a 2D electron crystal taken into account, the mass of a vibrating atom has transformed

into a nonlocal characteristic of the inertial properties of the crystal. Analogously, the

inertial properties of a dislocation loop should be characterized by a nonlocal mass

density. This m eans that the energy of a moving dislocation loop can be written as

E = E



(l, l



(l)V



) dl dl



, (14.3.8)

where E

is the quasi-static dislocation energy, which is dependent only on the instan-

taneous form and th e instantaneous positio n of the dislocation and p lays the role of a

rest energy. The second term in (14.3.8) should b e considered as the kinetic energy of

the dislocation. V(l) is the velocity of a dislocation line element dl , and the double

integration is over the length of the entire dislocation loop. Then µ

(l, l



) plays the

role of a nonlocal density of the effective mass of the dislocation.

Even in an isotropic medium, the effective mass density of the dislocation is

anisotropic. In view of the above expression for the effective mass of a screw dislo-

cation (14.3.5), it is easy to understand the general form of the tensor function of two

14.4 Equation for Dislocation Motion

331

variables µ

(l, l



) for a dislocation moving in its slip plane in an isotropic medium.

This is, as follows (Kosevich, 1962):

(l, l



ρb



ττ



− τ







1 +





sin



K(l, l



) ,

b cos θ = nb , τ = τ(l) , τ



= τ(l



) ,

(14.3.9)

where n is the unit vector in the direction of a straight line linking two points l and l



on the dislocation: n =[r(l) −r(l



)] /(r − r



). The scalar function of two variables

K(l, l



) resembles the formula for the self-induction coefﬁcient of a linear conductor

coincident with the dislocation loop. It is a purely geometrical characteristic of the

dislocation loop D, independent of the crystal properties and even of those physical

properties of the medium that are related with this line. This function c haracterizes the

self-action of any linear singularity of the classical ﬁeld in a 3D space if the dynamic

ﬁeld equations are differential equations in partial derivatives, such as wave equations

with sources. Its general form is

K(l, l



4π



γ(ξ)γ(ξ



) d

ξ d



|r(l, ξ) −r(l



, ξ



, (14.3.10)

where ξ is the two-dimensional radius vector measured from the dislocation axis in a

plane perpendicular to it; γ(ξ) is some smooth function localized in the vicinity of the

dislocation axis and describing the Burgers vector “distribution” in the cross section

of the dislocation core:

= τ

γ(ξ) ,



γ(ξ) d

ξ = 1. (14.3.11)

The necessity of introducing such an arbitrary function is “payment” for changing to

a continuum description of the dislocation as a speciﬁc crystal d efect. If we replace

the function γ(ξ) by the delta-function γ(ξ)=δ(ξ), a two-fold integral (14.3.10)

will become meaningless. Thus, by introducing this function we avoid a nonphysical

divergence in elasticity theory. Since the dependence on the dislocation-core radius is

involved in the eigenenergy and the effective mass logarithmically, then arbitrariness

in choosing the function γ(ξ) does not affect the results. Generally, this function is

assumed to be constant and nonzero inside a cylinder of radius r

To obtain the expression (14.3.10) by direct calculation, it is necessary to calcu-

late the self-induction coefﬁcient of a linear conductor and also to solve problems on

dislocation vortices in a scalar crystal model.

14.4

Equation for Dislocation Motion

Knowing the effective mass of the dislocation makes it possible to write its equation

of motion. The equatio n of motion of an element of a dislocation loop located at the

332

14 Dislocation Dynamics

point l on the d islocation line can be written similarly to Newton’s secon d law:



(l, l



)dl



= F

(l)+e

ikm

(l)σ



(l)b

+ S

(l, V ), (14.4.1)

where W (l)=∂V/∂t is the dislocation acceleration; F

is the force of a quasi-static

self-action of a distorted dislocation, generated by the energy density (13.3.18). In a

line tension approximation used with certain reservations the order of magnitude of

this force is evaluated when R  b as

, T

∼ GB

. (14.4.2)

The last term on the r.h.s. of (14.4.1) is the retardation force experienced by the

dislocation in a real crystal (this has not been discussed yet). We distinguish two

physically different parts in the value of this force.

The ﬁrst part of the force S is composed of forces that arise because the crystal

structure is discrete and the dislocation core is constructed of atoms. Among these

forces there is a static component (the Peierls force) and a component dependent on

the dislocation motion. The last force is related to lattice distortions moving together

with the dislocation on its axis and the induced reconstruction of the dislocation core

in motion . The dislocation -core reconstru ction involves, in particular, the formation

and migration (along the dislocation) of so-called steps (kinks) that link parts of the

same dislocation positioned in the neighboring “valleys” of the Peierls potential relief.

A similar defect (kink) was observed by us in studying solutions to the sine-Gordon

equation that, as a good model, can be applied to the kink dynamics on the dislocation.

In a number of cases the kink kinetics is the main mechan ism of dislocation motion.

The second part of the force S is due to various dynamic mechanisms responsible

for the energy dissipation of a moving dislocation. First, these are the microscopic

processes o f interaction between the dislocation and phonons and other e lementary

crystal excitations. Then, there are the macroscopic processes of energy losses in the

dynamic elastic dislocation ﬁeld connected with the dispersion of the elastic moduli

of a real crystal. Both processes are greatly dependent on the defect structure of a real

crystal.

As well as with the equation of motion of a dislocation element (14.4.1) we can

consider the equation of motion of the entire dislocation loop:



(l)W

(l) dl = e

ikm





dl +



(l) dl , (14.4.3)

where



(l, l



) dl



. (14.4.4)

When the integration is over the whole loop, the ﬁrst term on the r.h.s. of (14.4.1)

is omitted, as the total force of the dislocation static self-action is zero.

14.4 Equation for Dislocation Motion

333

It follows from (14.4.3) that just m

(l) can be considered as the value of the effec-

tive mass of a unit dislocation for the motion of the entire dislocation loop. But the

mass per unit length introduced in this way is not a local property of a given point on

the dislocation loop and is dependent on the dimensions and form of the entire loop.

Using (14.4.4) and the deﬁnition of the tensor µ

it is easy to obtain the estimate

(14.3.7) for the effective mass per unit length of the dislocation.

The given estimate for m

makes it possible to justify our assumption of a purely

ﬁeld origin of the dislocation mass. When a real dislocation moves in a crystal, some

of the atoms in the vicinity of the dislocation a xis (at distances of the order of r

)

also start moving. This leads to the appearance of an additional dislocation energy

associated with the ordinary mass of these atoms. The order of magnitude of atomic

masses inside a tube of radius r

∼ b per unit length of the dislocation can be estimated

as ρr

∼ ρb

. Comparing this estimate with (14.3.7), we see that for log R

curv



1, taking into account the mass of atoms moving near the dislocation axis practically

does not affect the dislocation energy and, up to a logarithmic accuracy, the dislocation

mass can be assumed to be the ﬁeld mass.

For a linear dislocation, when the vectors τ and b remain unchanged along the

dislocation line, the tensor m

ρb

4π

(δ

− τ

)



1 +





sin



cos

θ =(τb)

(14.4.5)

where R

is the dislocation length (R

 r

). In conclusion, we note that the inertia

term in the equation of motion is essential only for nonstationary motion of the dislo-

cation when its acceleration is too great. If the dislocation acceleration is small, the

action of retarding forces involving dissipative forces will be dominant. Their values

and the dependence on the dislocation velocity mainly determine the character of an

almost stationary motion of the dislocation.

The e quation of dislocation mo tion is often app lied in formulating a string model

where the dislocation line is considered as a heavy string possessing a certain tension

and lying in a “corrugated” surface. The corrugated surface relief is described by the

Peierls potential and the valleys on it correspond to the potential minima o n the slip

plane that are occupied by a straight dislocation in equilibrium (Fig. 14.3).

Let the Ox-axis b e directed along the equilib rium position of a straight disloca-

tion and the transverse dislocation displacement η be along the Oy-axis. In the string

model, this displacement as a function of x and t is described by the following equa-

tion of motion:

∂

∂t

− T

∂

∂x

+ bσ

sin 2π

= bσ, (14.4.6)

where m is the mass p er unit length o f the dislocation; T

is the line tension; σ

the Peierls stress; σ is the corresponding component of the stress tensor caused by

external loads.