Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

324 14 Dislocation Dynamics

One of the methods of solving the system of equations (14.1.5) consists in replacing 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 using (14.1.2). We then obtain a dynamic equation of elasticity theory for the velocity vector v:

The role of the force density in this equation is played by the vector fiD = λiklm k jim. The solution to (15,1.12) can be written as

t

vi(r, t) = dV dt Gik(r − r , t − t ) fkD (r , t ) ,

−∞

where Gik(r, t) is the Green tensor of the elasticity theory dynamic equation. This formula solves completely the problem of finding the displacement velocity and determines the time dependence of the displacements.

We recall that the Green dynamic tensor for an isotropic medium is written explicitly as

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 dislocations 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 unstressed. 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 quantitative 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 before the deformation starts. The time derivative of the vector u determines the displacement velocity of the medium element v = ∂u/∂t. Therefore,

where wik is the total distortion tensor, wik = iuk. Using (14.2.1) we rewrite (14.1.2) as

∂

∂t (wik − uik) = −jik.

The difference wik − uik 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.

pl

We denote this quantity by uik and obtain

pl

∂uik = −jik. (14.2.2)

∂t

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

δuplik = −jikδt.

pl

A similar relation for the plastic strain tensor ε ik has the form

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 breaking the continuity, then jkk = 0. It then follows from (14.2.2) that εplkk = 0. Thus, we come to the assertion, known in plasticity theory, that a purely 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 dislocation 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 expedient to imagine a semimicroscopic 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 “regular” 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 position 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 perpendicular to the slip plane. The dislocation 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 dislocation line goes 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 definite 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 of easy gliding 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 dislocation stops gliding over an extended part of its line. This occurs when the dislocation is retarded by clusters of impurities or macroscopic heterogeneous inclusions. Obstacles 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 uik and the vector vα of the medium displacement velocity using a given distribution of dislocations and flows. The tensors αik and jik and, hence, the dislocation motion are assumed to be given. For a system of equations to be completely closed and to determine a selfconsistent evolution of dislocations and elastic field, it is necessary to take into account the changes in the dislocation density 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 singularity. 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 equation of the dislocation motion are of a pure field origin.

Accordingly, to evaluate the dislocation dynamics, it is necessary to know the dislocation energy. 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 instantaneous coordinates and velocities of dislocation line elements. In accordance with the electromagnetic field theory of moving charges, this procedure is generally possible using an approximation that is quadratic in velocities (Section 3.6). Since the structure of the dynamical equations of an elastic field with dislocations is qualitatively the same as that of Maxwell’s equations, the difficulties in carrying out the above procedure 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 the 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 dislocation velocity V . As in the static case (Section 3.14), the elastic field of a screw dislocation is completely described by a single nonzero component of the displacement vector uz = u(x, y, t) and the equations of the elastic medium motion reduce to the 2D wave equation

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

where u0 (ξ , y) = u(x, y, t).

Equation (14.3.2) coincides with (13.3.1) and its solution has a dislocation singularity at the point ξ = y = 0. Thus the function u0 (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

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

where the parameters R and r0 are chosen as in the static case.

Expressing in (14.3.4) the shear modulus G through the transverse sound wave velocity (G = ρs2), we arrive at the following formula for the energy per unit length

In spite of the “relativistic” form of (14.3.5) this formula is valid only for small dislocation velocities, assuming V s. Such a “pseudodepletion” of (14.3.5) results from the fact that the dislocation energy has the field as origin and involves, in particular, 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 dependence of the screw dislocation velocity V will be obtained using the condition V s and replacing (14.3.5) by

330 14 Dislocation Dynamics

The first term in (14.3.6) coincides with the eigenenergy per unit length of the dislocation 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 r0 from it also start moving. This generates an additional dislocation inertia due to the usual atomic mass. The order of magnitude of the atomic mass inside tube of radius r0 b per unit length of the dislocation can be estimated as ρr02 ρb2. Comparing this estimate with (14.3.5) for m , the dislocation mass can be regarded with logarithmic accuracy as the field mass.

The inertial properties of an arbitrary weakly deformed dislocation loop are characterized by some tensor of line density of the effective mass mik dependent on a point on the dislocation line. It can be concluded that at a point where the radius of curvature of the dislocation loop Rcurv 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):

In the case of translational motion of the linear dislocation, Rcurv is the dislocation length. If the dislocation vibrates, Rcurv equals the wavelength of the dislocation bending vibrations.

A very important physical conclusion follows from the previous comments concerning the estimation of the parameter Rcurv. 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 characteristic 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 means that the energy of a moving dislocation loop can be written as

where E0 is the quasi-static dislocation energy, which is dependent only on the instantaneous form and the instantaneous position of the dislocation and plays the role of a rest energy. The second term in (14.3.8) should be 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 µik(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 dislocation (14.3.5), it is easy to understand the general form of the tensor function of two

332 14 Dislocation Dynamics

point l on the dislocation line can be written similarly to Newton’s second law:

where W (l) = ∂V /∂t is the dislocation acceleration; F0 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

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 reconstruction 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 mechanism 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 of interaction between the dislocation and phonons and other elementary 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:

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 mik(l) can be considered as the value of the effective 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 µik it is easy to obtain the estimate (14.3.7) for the effective mass per unit length of the dislocation.

The given estimate for mik 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 axis (at distances of the order of r0 ) 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 r0 b per unit length of the dislocation can be estimated as ρr02 ρb2. Comparing this estimate with (14.3.7), we see that for log Rcurv/r0 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 mik is

b2 cos2 θ = (τb)2 ,

where Rm is the dislocation length (Rm r0). In conclusion, we note that the inertia term in the equation of motion is essential only for nonstationary motion of the dislocation 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 equation of dislocation motion is often applied 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 on the slip plane that are occupied by a straight dislocation in equilibrium (Fig. 14.3).

Let the Ox-axis be directed along the equilibrium position of a straight dislocation 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 equation of motion:

where m is the mass per unit length of the dislocation; TD is the line tension; σp is the Peierls stress; σ is the corresponding component of the stress tensor caused by external loads.