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

304 13 Elastic Field of Dislocations in a Crystal

A screw dislocation generates around itself a pure shear strain field due to the isotropic medium. To show the role of anisotropy in the elastic field variation we show the distribution of dilatation deformation (the distribution of ε kk = div u) around the screw dislocation parallel to [111] in a cubic crystal. In Fig. 13.2, the levels of constant div u value forming a rosette that exhibits the anisotropy of a α-Fe monocrystal are given.

Calculation of elastic fields around a straight edge dislocation is much more complicated (in contrast to a screw dislocation, a purely edge dislocation can be curvilinear if the dislocation line is in a plane perpendicular to the vector b). Let the z- axis be directed along the dislocation axis and the x-axis along the Burgers vector: bx = b, by = bz = 0. It follows from the symmetry of the problem when ε zz = 0 that the deformation vector lies in the plane xOy and is independent of z.

Since the solution to this seemingly simple elasticity theory problem is cumbersome, we do not repeat it here in detail.

We write down the final results for the displacement field2

where ν is Poisson’s ratio.

The first term in the expression for ux is written in a form analogous to (13.3.2). Thus, the condition (13.1.1) is satisfied.

The stress tensor calculated on the basis of (13.3.5) has the Cartesian components

or in polar coordinates (the corresponding coordinates r, ϕ in the plane xOy):

with M = G/2π(1 − ν).

It follows from the condition ε zz = 0 that σzz = ν(σxx + σyy) = ν(σrr + σϕ ϕ ). Therefore, the mean hydrostatic pressure generated by the edge dislocation in an isotropic medium is

2) The solution to this problem can be found in the literature on dislocation theory.

13.3 Fields and the Interaction of Straight Dislocations 305

In an isotropic medium the levels of constant pressure p0 are at the same time the levels of constant dilatation ε0 ≡ ε kk = −p0/K (K is the compression modulus). Therefore, the distribution of an elastic dilatation of the deformation field of an edge dislocation is characterized by the levels

The curves (13.3.9) represent a family of circles where centers are on the Oy-axis and pass through the coordinate origin (the dislocation axis).

In a crystal the dilatation distribution around the edge dislocation has levels that are different from circles (13.3.9), but preserving the same character on the whole (Fig. 13.3).

Fig. 13.3 The dilatation field of an edge dislocation in plane (110): (a) the crystal α-Fe; (b) the crystal Li calculated by Chou (1965).

The expressions for the elastic fields of straight dislocations allow us to describe the interaction of parallel dislocations easily.

To determine the force acting from one dislocation on another and parallel to it, we use (13.2.11), (13.2.12) and the expression for the stress tensor of an elastic field around the second dislocation. As seen from Fig. 13.2, the elastic field around a screw dislocation in a crystal differs considerably from that in an isotropic medium. Therefore, a detailed description of the interaction of screw dislocations in the isotropic model is meaningless, although it is justified for edge dislocations, and therefore we shall now describe the interaction of edge dislocations in an isotropic medium.

If a dislocation coincides with the z-axis, it acts on a second dislocation running through the point (x, y) in the (xOy) plane with a force whose projection on the slip

306 13 Elastic Field of Dislocations in a Crystal

Fig. 13.4 Stable equilibrium configurations of a system of two parallel edge dislocations: (a) of a single sign, (b) of opposite signs.

plane is

Just this projection is of great interest, since in the slip plane only migration of the dislocation may have the character of a mechanical motion. The vanishing of this force projection corresponds to the equilibrium configuration of two dislocations relative to gliding. It follows from (13.3.10) that Fx vanishes at ϕ = π/2 and ϕ = π/4. It is easy to verify that the first of these positions (Fig. 13.4a) corresponds to the stable equilibrium condition of two dislocations of the same sign (b1b2 > 0), and the second one (Fig. 13.4b) to that of two dislocations of different sign (b1b2 < 0). Obviously, the first configuration (Fig. 13.4a) remains in stable equilibrium if the number of dislocations is greater than two. This is the reason for the dislocation wall (Section 10.5, Figs 10.8) to be an equilibrium and stable structure.

If a dislocation pair is formed by two dislocations of opposite sign (Fig. 13.4b) separated by a small distance (e. g., dislocations are positioned in the neighboring slip planes) such a pair is called a dislocation dipole. This system is stable enough to occur as a single dislocation entity.

Edge dislocations that are in the same glide plane (whose trace is the Ox-axis) interact in a very simple way

The formula (13.3.11) is also valid in an anisotropic medium when the constant M is connected in a different way with the elastic moduli, but remains positive.

We consider a set of a large number of identical edge dislocations placed parallel to one another in the same slip plane. The interaction of each pair of them will then be determined by the force (13.3.11).

We choose the x-axis along the Burgers vector and assume that the dislocations are distributed on the part (a1 , a2) of the x-axis. For definiteness, we can assume that at the point x = a2 there is an obstacle holding the dislocations and the whole

308 13 Elastic Field of Dislocations in a Crystal

The upper integration limit (13.3.15) is determined by the quantity R that is of the order of the dislocation length or crystal dimension. Then,

Comparing (13.3.16), (13.3.7), we see that (13.3.17) determines the main part of the dislocation energy under the condition log R/r0 1. The formula (13.3.17) determines the dislocation energy with logarithmic accuracy.

We note that the scalar field vortex energy is exactly equal to (13.3.17). For an edge dislocation the integral (13.3.14) equals

It is natural that the energy of unit length of an edge dislocation has the same order of magnitude as the screw dislocation. Moreover, it can be shown (although it is quite obvious) that for any weakly distorted dislocation line (with radius of curvature R b) its energy per unit length will have the order of magnitude

However, it should be noted that the theoretical “large parameter” log(R/r0 ) is, in fact, small. Indeed, taking the relation R/r0 105 − 106 (and it is often less), we obtain that ln(R/r0 ) is not different from 4π. Therefore, for rough estimates the energy per unit length of an isolated dislocation can be taken approximately equal to

U≡ Gb2. Such an estimate also remains valid for an anisotropic medium. Analyzing straight dislocation fields, we considered a screw and an edge disloca-

tion separately. This is so because in an isotropic medium any straight dislocation with tangent vector τ can be represented as consisting of two independent dislocations: a purely screw dislocation, the displacements around which are parallel to τ, and a purely edge dislocation, the displacements around which are perpendicular to τ. It is interesting to know whether the displacement field of a straight dislocation in an anisotropic medium can be divided into a screw part where all displacements are parallel to the dislocation line and an edge part where all displacements are perpendicular to this line. It turns out that this division is possible if the dislocation line is a two-fold symmetry axis or if the dislocation line is perpendicular to the crystal symmetry plane.

13.4 The Peierls Model 309

13.4

The Peierls Model

For a quantitative study of the role of crystal lattice discreteness in a dislocation deformation, Peierls (1940) suggested a semi-microscopic model that on the one hand, takes into account the crystal translation symmetry, and on the other hand allows one to obtain a direct limiting transition to the elasticity theory results.

We return to (13.3.12) and recall the geometrical meaning of the quantity ρ(x). We assume that the shift generated by dislocations along the slip plane is absent at x = ∞. By the definition of an edge dislocation the integral

x

equals the atomic layer displacement at the point x above the slip plane relative to that under this plane. It follows from (13.4.1) that

Thus, knowing the relative crystal displacements on both sides of the slip plane we can, using (13.4.2), find a corresponding dislocation density.

We now formulate a semi-microscopic model of an edge dislocation. Let nondeformed crystal atoms form a simple tetragonal lattice with the constant b along the x-axis and the constant a along the y-axis (Fig. 13.5a). We consider two parts of this crystal, one of which (A) contains an extra atomic plane compared to the other part (B). We form a single crystal of these two parts, making the upper and lower parts come closer at an interatomic distance a and matching the left and right edges of these parts. To make the latter, with fixed right edges of parts A and B, it is necessary to perform a relative shift along the x-axis by the value b of their left edges. After joining the two parts of a crystal, the atoms form a single lattice with an edge dislocation (Fig. 13.5b).

We denote by u(x) the relative displacement of the atoms on two sides of the slip plane in the direction of the x-axis. At an infinitely large distance from the dislocation, the lattice should be ideal. This means that

The function u(x) should have a form qualitatively similar to the plot in Fig. 5.1. We set u(x0 ) = b/2, i. e., assume the dislocation center to be at the point x = x0 = X. For an exact definition of the function u(x) the upper and lower parts of a crystal can be regarded as two continuous elastic solids. However, the tangential stress in the slip plane (in a plane where elastic half-spaces link) that prevents the deformation is treated as a periodic function of the local relative displacement u(x) with period b.

310 13 Elastic Field of Dislocations in a Crystal

Fig. 13.5 A scheme of the formation of a monocrystal with an edge dislocation: (a) crystal part A contains an extra atomic plane. (b) Parts A and B are combined in a monocrystal with a dislocation.

Peierls chooses a simple periodic dependence, namely

where the coefficient µ1 has the order of magnitude of crystal elastic moduli. The procedure determining it will be indicated below.

The local stresses (13.4.4) should be coincident with the macroscopic stresses generated by an inhomogeneous distribution of the displacement u(x) along the slip plane. We have already seen that an inhomogeneous relative displacement u(x) is equivalent to a set of edge dislocations with the density (13.4.2) and, hence, it causes the shear stresses (13.3.12). Substituting (13.4.4) into (13.3.12) and equating the result to σxyP , we get a singular integral equation for u(x):

−∞

The solutions to (13.4.5) should have the property (13.4.3).

First, we shall find the relation between the coefficient µ1 and the crystal elastic moduli. It is clear that far from the dislocation core (x b), where the displacements u(x) are small, the solution to (13.4.5) should coincide with the known solution of elasticity theory if we take uyx = u/a. It follows from (13.4.5) that for x → ∞ when

−∞

We now calculate the derivative ∂ux /∂y using (13.3.5) for x → ∞ and compare it with (13.4.6). We get µ1 = bG/πa(3 − 2ν). Thus, the Peierls equation takes the

13.4 The Peierls Model 311

−∞

A very important property of (13.4.7) that distinguishes it from the Frenkel–Kontorova equation for dislocations in a 1D crystal is the very slow variation of the quantity u(x) with distance x. While in a 1D crystal the decay of u(x) as x → ∞ is exponential, the solution to (13.4.7) decreases as a hyperbolic law (13.4.6). The solution corresponding to the conditions (13.4.3) is

The quantity l may conventionally be called a dislocation half-width. This notation is tentative, because the displacement u(x) varies more slowly in space, which leads to the dislocation width being dependent on a specific form of the periodic function in (13.4.7). In the isotropic approximation the dislocation half-width depending on Poisson’s ratio can range from l = (3/4a) (for ν = 0) to l = a (for ν = 1/2).

The Peierls model allows one to estimate the critical stress necessary to displace the dislocation in the slip plane. This was made by Nabarro (1947) using the following considerations. The stress σxyP corresponds to the following potential energy per interval b of the x-axis

We now take into account the discrete structure of a slip plane and associate each atomic layer with the coordinate

choosing the layer numeration origin in the vicinity of the dislocation center. If we now substitute a discrete coordinate (13.4.11) into (13.4.10), then we obtain the n-th atomic layer energy in a lattice distorted by the presence of a dislocation. Hence, the dislocation energy related to local stresses along the slip plane is equal to

Using the explicit form of (13.4.8) for the function u(x), it is easy to obtain