Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.9 Energetics of Dislocations in Elastic Solids

 

Dislocations play a crucial role in determining the response of crystalline materials to stress.  For example, plastic flow in ductile metals occurs as a result of dislocation motion; dislocation emission from a crack tip can determine whether a material is ductile or brittle; and stress induced dislocation nucleation plays a critical role in semiconductor devices.

 

Dislocations tend to move through a crystal in response to stress.  The goal of this and following sections is to derive some results that can be used to predict this motion.

 

 

5.9.1 Classical solution for potential energy of an isolated dislocation loop in an infinite solid

 

In this section, we show that the energy of an isolated dislocation loop with burgers vector b in an infinite solid can be calculated using the following expressions:

$\begin{array}{l}{V}^{D\infty }=\frac{E}{32\pi (1+\nu )}{\displaystyle \underset{C}{\oint }{\displaystyle \underset{C}{\oint }\frac{{\partial }^{2}R(x-\xi )}{\partial x_p\partial x_p}{b}_i{b}_j{\tau }_i(x){\tau }_j(\xi )d{s}_x}}d{s}_{\xi }\\ +\frac{E}{16\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\oint }{\displaystyle \underset{C}{\oint }{\in }_{ikl}{\in }_{jmn}{b}_k{b}_m\frac{{\partial }^{2}R(x-\xi )}{\partial x_i\partial x_j}{\tau }_l(x){\tau }_n(\xi )d{s}_{x}d{s}_{\xi }}}\end{array}$

Here, $R(x-\xi )=\sqrt{\left(x_k-{\xi }_k\right)\left(x_k-{\xi }_k\right)}$ and the integral is taken around the dislocation line twice.  In the first integral, $\xi$ is held fixed, and $x$ varies with position around the dislocation line; then $\xi$ is varied for the second line integral.

 

Difficulties with evaluating the potential energy in the classical solution: In practice, this is a purely formal result $–$ in the classical solution, the energy of a dislocation is always infinite.   You can see this clearly using the solution for a straight dislocation in an infinite solid given in Section 5.3.4.  Recall that the stress state for a pure edge dislocation, with line direction parallel to the $e_3$ axis and burgers vector $b=b_1{e}_1+b_2{e}_2$ at the origin of an infinite solid is given (in polar coordinates) by 

${\sigma }_{rr}={\sigma }_{\theta \theta }=-\frac{E(b_1\sin \theta -b_2\cos \theta )}{4\pi (1-{\nu }^2)r}{\sigma }_{r\theta }=\frac{E(b_1\cos \theta +b_2\sin \theta )}{4\pi (1-{\nu }^2)r}$

The strain energy density distribution around the dislocation follows as

$U=\frac{1+\nu }{2E}\left\{\left({\sigma }_{rr}^2+{\sigma }_{\theta \theta }^2+2{\sigma }_{r\theta }^2\right)-\nu {\left({\sigma }_{rr}+{\sigma }_{\theta \theta }\right)}^2\right\}$

We can use this to calculate the total strain energy in an annular region around the dislocation, with inner radius a, and outer radius b.   The result is

$V={\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}U(r,\theta )rd\theta dr}=\frac{E(b_1^2+b_2^2)}{8\pi (1-{\nu }^2)}\log \left(\frac{b}{a}\right)}$

Taking $a\to 0$ gives an infinite energy, because the strain energy density varies as $1/r^2$ near the dislocation core.

 

Various ways to avoid this problem have been proposed.   The simplest approach is to neglect the strain energy in a tubular region with small radius $r_0\approx \left|b\right|/4$ surrounding the dislocation, on the grounds that the elastic solution does not accurately characterize the atomic-scale deformation near the dislocation core. This works for straight dislocations, but is not easy to apply to 3D dislocation loops.   A more satisfactory approach is described in the next section.

 

Application to a circular prismatic dislocation loop As an example, we attempt to apply the general formula to calculate the energy of a circular dislocation loop, with radius a, which lies in the $(x_1,x_2)$ plane, and has a burgers vector $b=b{e}_3$ that is perpendicular to the plane of the loop.  For this case, the contour integral for the potential energy reduces to

$V^{D\infty }=\frac{Ea{b}^2}{8(1-{\nu }^2)}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\frac{1+\cos \theta }{2\sqrt{2(1-\cos \theta )}}d\theta }$

(To see this, note that the result of the integral with respect to x must be independent of $\xi$ by symmetry). As expected, the integral is divergent.  In the classical theory, the energy of the loop is estimated by truncating the integral around the singularity, so that

$V^{D\infty }=\frac{Ea{b}^2}{8(1-{\nu }^2)}{\displaystyle \underset{r/a}{\overset{2\pi -r/a}{\int }}\frac{1+\cos \theta }{2\sqrt{2(1-\cos \theta )}}d\theta }\approx \frac{Ea{b}^2}{4(1-{\nu }^2)}\left\{\log \left(\frac{4a}{r}\right)-1\right\}+{\rm O}{(\rho /a)}^2$

where r is a small cut-off distance.  This is somewhat similar to the core cutoff radius $r_0$, but the relationship between $r_0$ and r is not clear.

 

A Circular glide loop, which has burgers vector b (with magnitude b) in the plane of the loop, has energy

$V^{D\infty }=\approx \frac{Ea{b}^2(2-\nu )}{4(1-{\nu }^2)}\left\{\log \left(\frac{4a}{r}\right)-2\right\}+{\rm O}{(\rho /a)}^2$

 

Derivation of the solution for the energy of a 3D dislocation loop

1.      Let $[u_i(\xi ),{\epsilon }_{pq}(\xi ),{\sigma }_{pq}(\xi )]$ denote the displacement, strain and stress induced by the dislocation loop.   The total potential energy of the solid can be calculated by integrating the strain energy density over the volume of the solid

$V^{D\infty }={\displaystyle \underset{V}{\int }\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}dV}$

2.      The potential energy can also be expressed in terms of the displacement field in the solid, as

$V^{D\infty }={\displaystyle \underset{V}{\int }\frac{1}{2}{\sigma }_{ij}\frac{\partial u_i}{\partial x_j}dV}=\frac{1}{2}{\displaystyle \underset{V}{\int }\frac{\partial ({\sigma }_{ij}{u}_i)}{\partial x_j}-u_i\frac{\partial ({\sigma }_{ij})}{\partial x_j}dV}=\frac{1}{2}{\displaystyle \underset{V}{\int }\frac{\partial ({\sigma }_{ij}{u}_i)}{\partial x_j}dV}$

where we have used the symmetry of ${\sigma }_{ij}$ and recalled that the stress field satisfies the equilibrium equation $\partial {\sigma }_{ij}/\partial x_j=0$

3.      Applying the divergence theorem, and taking into account the discontinuity in $u_i$ across S,

$V^{D\infty }=\frac{1}{2}{\displaystyle \underset{S}{\int }{\sigma }_{ij}(-m_j)u_i^{+}dA}+\frac{1}{2}{\displaystyle \underset{S}{\int }{\sigma }_{ij}{m}_j{u}_i^{-}dA}=\frac{1}{2}{\displaystyle \underset{S}{\int }{\sigma }_{ij}{m}_j{b}_{i}dA}$

4.      Next, we substitute the expression given in Section 5.8.4 for ${\sigma }_{ij}$ and reverse the order of integration

$\begin{array}{l}{V}^{D\infty }=\frac{E}{32\pi (1+\nu )}{\displaystyle \underset{C}{\int }{\displaystyle \underset{S}{\int }\left(\left[{\in }_{imp}{b}_m{\tau }_q+{\in }_{imq}{b}_m{\tau }_p\right]\frac{{\partial }^{3}R(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right)m_p{b}_{q}d{A}_x}}d{s}_{\xi }\\ +\frac{E}{16\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\int }{\displaystyle \underset{S}{\int }\left(b_m{\in }_{imk}{\tau }_k\left[\frac{{\partial }^{3}R(x-\xi )}{\partial x_i\partial x_p\partial x_q}-{\delta }_{pq}\frac{{\partial }^{3}R(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right]\right)}}{m}_p{b}_{q}d{A}_{x}d{s}_{\xi }\end{array}$

5.      Finally, the surface integrals in this expression can be transformed into a contour integral around C by means of Stoke’s theorem

${\displaystyle \underset{S}{\int }{\in }_{npj}\frac{\partial F_j}{\partial x_p}{m}_{n}d{A}_x={\displaystyle \underset{C}{\int }{F}_j{\tau }_{j}d{s}_x}}$

After some tedious index manipulations, this gives the required result.

 

 

 

5.9.2 Non-singular dislocation theory

 

The infinite potential energy associated with the classical description of a dislocation is unphysical, and highly unsatisfactory.  A straightforward approach to avoiding this difficulty was proposed by Cai et al, Journal of the Mechanics and Physics of Solids, 54, 561-587, (2006).

 

In the classical solution, the dislocation core is localized at a single point in space, which leads to an infinite energy.  In practice, dislocation cores are distributed over a small, but finite, area as indicated in the figure. The TEM micrograph is from Tillmann et al Microsc. Microanal. 10, 185–198, 2004.

 

This effect can be modeled approximately by using the classical solution to construct a distributed dislocation core.  To this end, we suppose that the burgers vector of the dislocation can be represented by a distribution $b\beta (x)$, which must be chosen to satisfy

$b(x)=b{\displaystyle \underset{V}{\int }\beta (\xi -x)d{V}_{\xi }}$

where the volume integral extends over the entire infinite solid.  In principle, $\beta (x)$ could be constructed to give an accurate description of the atomic-scale strain field in the immediate neighborhood of the dislocation core, but this is difficult to do, and is not the main intent of the theory.  Instead, $\beta (x)$ is selected to make the expressions for the energy and stress field of the dislocation as simple as possible.  It is particularly convenient to choose $\beta (x)$ to satisfy

$R_{\rho }(x-\xi )={\displaystyle \underset{V}{\int }{\displaystyle \underset{V}{\int }R(y-z)\beta (z-\xi )\beta (y-x)d{V}_{z}d{V}_y}}R(x)=\sqrt{x_k{x}_k}{R}_{\rho }(x)=\sqrt{x_k{x}_k+{\rho }^2}$

where $\rho$ is a small characteristic length, comparable to the dimensions of the dislocation core. The required distribution cannot be calculated exactly, but is closely approximated by

$\beta (x)\approx \frac{15}{8\pi }\left\{\frac{1-m}{{\rho }_1^3{\left(x_k{x}_k/{\rho }_1^2+1\right)}^{7/2}}+\frac{m}{{\rho }_2^3{\left(x_k{x}_k/{\rho }_2^2+1\right)}^{7/2}}\right\}$

with ${\rho }_1=0.9038\rho$, ${\rho }_2=0.5451\rho$, $m=0.6575$.  The distribution can also be shown to satisfy

${\displaystyle \underset{V}{\int }\beta (x-\xi )\beta (\xi )dV}=\frac{15}{8\pi }\frac{{\rho }^3}{{\left(x_k{x}_k+{\rho }^2\right)}^{7/2}}$

 

 Nonsingular energy: The expression for the energy of a dislocation loop then reduces to

$\begin{array}{l}{V}^{D\infty }=\frac{E}{32\pi (1+\nu )}{\displaystyle \underset{C}{\oint }{\displaystyle \underset{C}{\oint }\frac{{\partial }^2{R}_{\rho }}{\partial x_p\partial x_p}{b}_i{b}_j{\tau }_i(x){\tau }_j(\xi )d{s}_x}}d{s}_{\xi }\\ +\frac{E}{16\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\oint }{\displaystyle \underset{C}{\oint }{\in }_{ikl}{\in }_{jmn}{b}_k{b}_m\frac{{\partial }^2{R}_{\rho }}{\partial x_i\partial x_j}{\tau }_l(x){\tau }_n(\xi )d{s}_{x}d{s}_{\xi }}}\end{array}$

This is virtually identical to the classical singular solution, except that the derivatives of $R_{\rho }(x)=\sqrt{x_k{x}_k+{\rho }^2}$ are bounded everywhere, so the integral is finite.

 

 Nonsingular Stress: The stress due to the dislocation loop can be expressed in terms of a function ${{\rm B}}_{\rho }(x)$, defined as

${{\rm B}}_{\rho }(x)={\displaystyle \underset{V}{\int }R(y)\beta (y-x)d{V}_y}$

This function cannot be calculated exactly, but can be estimated using the approximation to $\beta (x)$ as

${{\rm B}}_{\rho }(x)=(1-m)\sqrt{x_k{x}_k+{\rho }_1^2}+m\sqrt{x_k{x}_k+{\rho }_2^2}$

The stress field then becomes

$\begin{array}{l}{\sigma }_{pq}(x)=\frac{E}{16\pi (1+\nu )}{\displaystyle \underset{C}{\int }\left(\left[{\in }_{imp}{b}_m{\tau }_q+{\in }_{imq}{b}_m{\tau }_p\right]\frac{{\partial }^3{{\rm B}}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right)}d{s}_{\xi }\\ +\frac{E}{8\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\int }\left(b_m{\in }_{imk}{\tau }_k\left[\frac{{\partial }^3{{\rm B}}_{\rho }(x-\xi )}{\partial x_i\partial x_p\partial x_q}-{\delta }_{pq}\frac{{\partial }^3{{\rm B}}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right]\right)}d{s}_{\xi }\end{array}$

Alternatively, one may calculate exactly a modified stress measure, defined as

${\sigma }_{pq}^{(\rho )}(x)={\displaystyle \underset{V}{\int }{\sigma }_{pq}(\xi )\beta (\xi -x)d{V}_{\xi }}$

This stress measure is particularly convenient for calculating the force tending to make a dislocation move, as shown in a subsequent section.  In addition, ${\sigma }_{pq}^{(\rho )}(x)\approx {\sigma }_{pq}(x)$ except very close to the core of a dislocation.   It is straightforward to show that

$\begin{array}{l}{\sigma }_{pq}^{(\rho )}(x)=\frac{E}{16\pi (1+\nu )}{\displaystyle \underset{C}{\int }\left(\left[{\in }_{imp}{b}_m{\tau }_q+{\in }_{imq}{b}_m{\tau }_p\right]\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right)}d{s}_{\xi }\\ +\frac{E}{8\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\int }\left(b_m{\in }_{imk}{\tau }_k\left[\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_p\partial x_q}-{\delta }_{pq}\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right]\right)}d{s}_{\xi }\end{array}$

where $R_{\rho }(x)=\sqrt{x_k{x}_k+{\rho }^2}$, as before. 

 

Nonsingular energy of circular dislocation loops.  It is straightforward to calculate the energy of a circular dislocation loop.  Cai et al Journal of the Mechanics and Physics of Solids, 54, 561-587, (2006) give:

 Prismatic Loop: (b perpendicular to loop) $V^{D\infty }=\frac{Ea{b}^2}{4(1-{\nu }^2)}\left\{\log \left(\frac{8a}{\rho }\right)-1\right\}+{\rm O}{(\rho /a)}^2$

 Glide Loop: (b in the plane of the loop): $V^{D\infty }\approx \frac{Ea{b}^2}{8(1+\nu )}\left\{\frac{2-\nu }{1-\nu }\left[\log \left(\frac{8a}{r}\right)-2\right]-\frac{1}{2}\right\}+{\rm O}{(\rho /a)}^2$

 

 

 

 

5.9.3 Energy of a dislocation loop in a stressed, finite elastic solid

 

The figure shows a dislocation loop in an elastic solid.  Assume that:

1.      The solid is an isotropic, homogeneous, linear elastic material with Young’s modulus E and Poisson’s ratio $\nu$

2.      The solid contains a dislocation, which is characterized by the loop C, and the burgers vector b for the dislocation, following the conventions described in the preceding section.  As before, we can imagine creating the dislocation loop by cutting the crystal over some surface S, and displacing the two material surfaces adjacent to the cut by the burgers vector.  The figure shows the dislocation loop to be completely contained within the solid, but this is not necessary $–$ the surface S could intersect the exterior boundary of the solid, in which case the dislocation line C would start and end on the solid’s surface.

3.      Part of the boundary of the solid ${\partial }_{1}R$ is subjected to a prescribed displacement, while the remainder ${\partial }_{2}R$ is subjected to a prescribed traction.  Note that there is some ambiguity in specifying the prescribed displacement.  In some problems, the solid contains a dislocation before it is loaded: if so, displacements are measured relative to the solid with traction free boundary, but containing a dislocation.  In other problems, the dislocation may be nucleated during deformation.  In this case, displacements are measured with respect to the initial, stress free and undislocated solid.  In the discussion to follow, we consider only the latter case.

 

 

To express the potential energy in a useful form, it is helpful to define several measures of stress and strain in the solid, as follows:

1.      The actual fields in the loaded solid containing the dislocation will be denoted by $[u_i,{\epsilon }_{ij},{\sigma }_{ij}]$.  Note that $u_i$ is measured with respect to a stress-free solid, which contains no dislocations.  The displacement is discontinuous across S.

2.      The fields induced by the applied loading in an un-dislocated solid will be denoted by $[u_i^{*},{\epsilon }_{ij}^{*},{\sigma }_{ij}^{*}]$.  The displacement field $u_i^{*}$ is continuous.

3.      The fields in a solid containing a dislocation, but with ${\partial }_{2}R$ traction free, and with zero displacement on ${\partial }_{1}R$ will be denoted by $[u_i^D,{\epsilon }_{ij}^D,{\sigma }_{ij}^D]$

4.      The fields in an infinite solid containing a dislocation with line C and burgers vector b will be denoted by $[u_i^{D\infty },{\epsilon }_{ij}^{D\infty },{\sigma }_{ij}^{D\infty }]$.  If the dislocation line terminates on the solid’s surface, any convenient procedure can be used to close the loop when deriving the infinte solid solution, but the fields will depend on this choice.

5.      The difference between the fields for a dislocation in a bounded solid and the solution for a dislocation in an infinite solid will be denoted by $[u_i^I=u_i^D-u_i^{D\infty },{\epsilon }_{ij}^I={\epsilon }_{ij}^D-{\epsilon }_{ij}^{D\infty },{\sigma }_{ij}^I={\sigma }_{ij}^D-{\sigma }_{ij}^{D\infty }]$

 

The potential energy of the solid can be expressed as

$V=V^D+V^{D*}+V^{*}$

where

 $V^D={\displaystyle \underset{S}{\int }\frac{1}{2}{\sigma }_{ij}^D{b}_i{m}_{j}dA}$ is the strain energy of the dislocation itself

 $V^{D*}={\displaystyle \underset{S}{\int }{\sigma }_{ij}^{*}{b}_i{m}_{j}dA}$ is the work done to introduce the dislocation into the externally applied stress

 $V^{*}={\displaystyle \underset{R}{\int }\frac{1}{2}{\sigma }_{ij}^{*}{\epsilon }_{ij}^{*}dV}-{\displaystyle \underset{\partial 2R}{\int }{t}_i{u}_i^{*}dA}$ is the potential energy of the applied loads

 

The strain energy of the dislocation can also be expressed as a sum of two terms:

$V^D=V^{D\infty }+V^I$

where

 $V^{D\infty }$ is the energy of a dislocation with line C in an infinite solid, which can be calculated using the expressions in 5.9.2.

  $V^I={\displaystyle \underset{S}{\int }{\sigma }_{ij}^I{b}_i{m}_{j}dA}$ is the change in potential energy due to the presence of boundaries in the solid. 

 

If the classical approach is used to represent the burgers vector of the dislocation, the energy is infinite, because $V^{D\infty }$ contains a contribution from the singular dislocation core.  The remaining terms are all bounded.  The simplest way to avoid this unsatisfactory behavior is to estimate $V^{D\infty }$ using the non-singular dislocation theory presented in 5.9.2, but use the classical expressions for all the remaining terms.   This is not completely consistent, because in a rigorous non-singular dislocation theory all the terms should be computed by taking a convolution integral with the burgers vector distribution.   However, provided the solid is large compared with the dislocation core, the error in the approximate result is negligible.

 

Derivation

1. The potential energy of the solid is given by the usual expression $V={\displaystyle \underset{R}{\int }\frac{1}{2}{\sigma }_{ij}{\epsilon }_{ij}dV}-{\displaystyle \underset{\partial 2R}{\int }{t}_i{u}_{i}dA}$
2. The total stress consists of the dislocation fields, together with the externally applied fields, so that

$V={\displaystyle \underset{R}{\int }\frac{1}{2}\left[{\sigma }_{ij}^D+{\sigma }_{ij}^{*}\right]\left[{\epsilon }_{ij}^D+{\epsilon }_{ij}^{*}\right]dV}-{\displaystyle \underset{\partial 2R}{\int }{t}_i\left(u_i^D+u_i^{*}\right)dA}$

3. This expression can be re-written as

$V={\displaystyle \underset{R}{\int }\left(\frac{1}{2}{\sigma }_{ij}^D\frac{\partial u_i^D}{\partial x_j}+{\sigma }_{ij}^{*}\frac{\partial u_i^D}{\partial x_j}+\frac{1}{2}{\sigma }_{ij}^{*}{\epsilon }_{ij}^{*}\right)dV}-{\displaystyle \underset{\partial 2R}{\int }{t}_i\left(u_i^D+u_i^{*}\right)dA}$

To see this, note that ${\sigma }_{ij}^D{\epsilon }_{ij}^D={\sigma }_{ij}^D(\partial u_i^D/\partial x_j)$ from the symmetry of ${\sigma }_{ij}^D$ and the strain-displacement relations, and that ${\sigma }_{ij}^D{\epsilon }_{ij}^{*}=C_{ijkl}{\epsilon }_{kl}^D{\epsilon }_{ij}^{*}={\sigma }_{ij}^{*}{\epsilon }_{ij}^D$ because of the symmetry of the elasticity tensor $C_{ijkl}$.

4. The terms involving $(\partial u_i^D/\partial x_j)$ can now be integrated by parts, by writing, for example

${\sigma }_{ij}^D(\partial u_i^D/\partial x_j)=\partial ({\sigma }_{ij}^D{u}_i^D)/\partial x_j-u_i(\partial {\sigma }_{ij}^D/\partial x_j)=\partial ({\sigma }_{ij}^D{u}_i^D)/\partial x_j$

because ${\sigma }_{ij}^D$ is an equilibrium stress field.  Using this result, applying the divergence theorem, and taking into account the discontinuity in the displacement field across S gives

${\displaystyle \underset{R}{\int }\frac{1}{2}{\sigma }_{ij}^D\frac{\partial u_i^D}{\partial x_j}dV={\displaystyle \underset{S+}{\int }\frac{1}{2}{\sigma }_{ij}^D{u}_i^{D+}(-m_j)dA}+{\displaystyle \underset{S-}{\int }\frac{1}{2}{\sigma }_{ij}^D{u}_i^{D-}{m}_{j}dA}+{\displaystyle \underset{\partial R}{\int }\frac{1}{2}{\sigma }_{ij}^D{u}_i^D{n}_{j}dA}}={\displaystyle \underset{S}{\int }\frac{1}{2}{\sigma }_{ij}^D{b}_i{m}_{j}dA}$

where we have noted that ${\sigma }_{ij}^D{n}_j{u}_i^D=0$ on the exterior boundary of the solid, and that $u_i^{-}-u_i^{+}=b_i$.  A similar procedure gives

${\displaystyle \underset{R}{\int }{\sigma }_{ij}^{*}\frac{\partial u_i^D}{\partial x_j}dV={\displaystyle \underset{S}{\int }{\sigma }_{ij}^{*}{b}_i{m}_{j}dA}+{\displaystyle \underset{\partial R}{\int }{\sigma }_{ij}^{*}{n}_j{u}_i^{D}dA}}$

Finally, substituting this result back into the expression for V and noting ${\sigma }_{ij}^{*}{n}_j=t_i$ on ${\partial }_{2}R$ and $u_i^D=0$ on ${\partial }_{1}R$ gives the required result.

 

 

 

 

5.9.4 Energy of two interacting dislocation loops

 

Consider two dislocation loops in an infinite elastic solid, as shown in the figure.  Assume that

1. The solid is an isotropic, homogeneous, linear elastic material with Young’s modulus E and Poisson’s ratio $\nu$
2. The dislocations can be characterized by surfaces, contours and burger’s vectors $[S_1,C_1,b^{(1)}]$ and $[S_2,C_2,b^{(2)}]$.

 

The total potential energy of the solid can be calculated from the following expressions

$V=V^{D1}+V^{D1D2}+V^{D2}$

where

 $V^{D1}$ and $V^{D2}$ are the energies of the two dislocation loops in isolation, which can be computed from the formulas in 5.9.1 (or 5.9.2 if you need a non-singular expression)

 $V^{D1D2}$ is an `interaction energy,’ which can be thought of as the work done to introduce dislocation 2 into the stress field associated with dislocation 1 (or vice-versa).  The interaction energy is given by

$\begin{array}{l}{V}^{D1D2}=\frac{E}{16\pi (1+\nu )}{\displaystyle \underset{C2}{\oint }{\displaystyle \underset{C1}{\oint }\frac{{\partial }^{2}R}{\partial x_p\partial x_p}{b}_i^{(1)}{b}_j^{(2)}{\tau }_i^{(1)}(x){\tau }_j^{(2)}(\xi )d{s}_x}}d{s}_{\xi }\\ -\frac{E}{16\pi (1+\nu )}{\displaystyle \underset{C2}{\oint }{\displaystyle \underset{C1}{\oint }{\in }_{ijq}{\in }_{mnq}{b}_i^{(1)}{b}_j^{(2)}\frac{{\partial }^{2}R}{\partial x_k\partial x_k}{\tau }_m^{(1)}(x){\tau }_n^{(2)}(\xi )d{s}_{x}d{s}_{\xi }}}\\ +\frac{E}{8\pi (1-{\nu }^2)}{\displaystyle \underset{C2}{\oint }{\displaystyle \underset{C1}{\oint }{\in }_{ikl}{\in }_{jmn}{b}_k^{(1)}{b}_m^{(2)}\frac{{\partial }^{2}R}{\partial x_i\partial x_j}{\tau }_l^{(1)}(x){\tau }_n^{(2)}(\xi )d{s}_{x}d{s}_{\xi }}}\end{array}$

Although this integral is bounded (provided the dislocation lines only meet at discrete points), it is sometimes convenient to replace $R(x)=\sqrt{x_k{x}_k}$ by $R_{\rho }(x)=\sqrt{x_k{x}_k+{\rho }^2}$ for a non-singular treatment of dislocations.

 

HEALTH WARNING: Notice that the expression for the interaction energy is very similar to the formula for the self-energy of a dislocation loop, except that (i) it contains an extra term (which vanishes if $b^{(1)}=b^{(2)}$ ), and (ii) the integrals in the interaction energy are twice those in the self-energy.  The latter is an endless source of confusion.

 

Derivation: We can regard the two interacting dislocations as a special case of a dislocation loop subjected to an applied stress:  one dislocation generates the `applied stress,’ which influences the second dislocation.  The total potential energy follows as $V=V^D+V^{D*}+V^{*}$, where $V^D=V^{D1}$, $V^{*}=V^{D2}$ are the potential energies of the two isolated dislocations, and $V^{D*}=V^{D1D2}$ is the interaction energy.  We have that

$V^{D1D2}={\displaystyle \underset{S1}{\int }{\sigma }_{ij}^{D2}{b}_i^{(1)}{m}_j^{(1)}dA}$

where ${\sigma }_{ij}^{D2}$ is the stress induced by dislocation 2.  We can express this stress in terms of a line integral around dislocation 2.  Finally, the surface integral over S1 can reduced to a contour integral around dislocation 1 by applying Stokes theorem.

 

 

 

5.9.5 Driving force for dislocation motion $–$ The Peach-Koehler formula

 

If a dislocation is subjected to stress, it tends to move through the crystal.  This motion is the mechanism for plastic flow in a crystalline solid, as discussed in Section 3.7.12.

 

The tendency of a dislocation to move can be quantified by a force.  This force needs to be interpreted carefully: it is not a mechanical force (in the sense of Newtonian mechancics) that induces motion of a material particle, but rather a generalized force (in the sense of Lagrangean mechanics) that causes a rearrangement of atoms around the dislocation core.   It is sometimes known as a `configurational force’

 

The generalized force for dislocation motion is defined as follows.

1. Consider an elastic solid, which contains a dislocation loop. The loop is characterized by a curve C, the tangent vector $\tau$, and the burgers vector b. As usual, we can imagine creating the dislocation loop by cutting the crystal over some surface S that is bounded by C, and displacing the two material surfaces adjacent to the cut by the burgers vector. 
2. Suppose that the dislocation moves, so that a point at $x(s)$ on C advances to a new position $x(s)+\delta a(s)n(s)$, where n(s) is a unit vector normal to C, as shown in the figure (in the figure, the dislocation moves ina single plane, but this is not necessary).
3. As the dislocation moves, the potential energy of the solid changes by an amount $\delta V$.  This change of energy provides the driving force for dislocation motion.
4. The driving force is defined as a vector function of arc-length around the dislocation $F(s)$, whose direction is perpendicular to C, and which satisfies

$-\delta V={\displaystyle \underset{C}{\int }F(s)\cdot n(s)\delta a(s)ds}$

for all possible choices of $n(s)$ and $\delta a(s)$ (the change in energy is negative because the displacement is in the same direction as the force).

 

The Peach-Koehler formula states that the driving force for dislocation motion can be computed from the following formula

$F_i(s)={\in }_{ijk}{\sigma }_{jm}(s)b_m{\tau }_k(s)$

where ${\sigma }_{ij}$ is the total stress acting on the dislocation at a point s along the curve C (the stress includes contributions from the dislocation itself, as well as stresses generated by external loading on the solid).

 

The Peach-Koehler equation is meaningless without further discussion, because the classical solution predicts that the stress acting on the dislocation line is infinite.   To avoid this, we need to partition the stress according to its various origins, as described in Section 5.9.3.

 

 

1. We assume that the dislocation loop lies within an elastic solid, which is subjected to some external loading.  The external fields subject part of the boundary of the solid ${\partial }_{1}R$ to a prescribed displacement; and the remainder of the boundary ${\partial }_{2}R$ to a prescribed traction.

 

2. The actual fields in the loaded solid containing the dislocation will be denoted by $[u_i,{\epsilon }_{ij},{\sigma }_{ij}]$. 

 

3. The fields induced by the applied loading in an un-dislocated solid will be denoted by $[u_i^{*},{\epsilon }_{ij}^{*},{\sigma }_{ij}^{*}]$.

 

4. The fields in a solid containing a dislocation, but with ${\partial }_{2}R$ traction free, and with zero displacement on ${\partial }_{1}R$ will be denoted by $[u_i^D,{\epsilon }_{ij}^D,{\sigma }_{ij}^D]$

 

5. The fields in an infinite solid containing a dislocation with line C and burgers vector b will be denoted by $[u_i^{D\infty },{\epsilon }_{ij}^{D\infty },{\sigma }_{ij}^{D\infty }]$.  If the dislocation line terminates on the solid’s surface, any convenient procedure can be used to close the loop when deriving the infinte solid solution, but the fields will depend on this choice.

 

6. The difference between the fields for a dislocation in a bounded solid and the solution for a dislocation in an infinite solid will be denoted by $[u_i^I=u_i^D-u_i^{D\infty },{\epsilon }_{ij}^I={\epsilon }_{ij}^D-{\epsilon }_{ij}^{D\infty },{\sigma }_{ij}^I={\sigma }_{ij}^D-{\sigma }_{ij}^{D\infty }]$

 

The Peach-Koehler force can then be divided into contributions from three sources:

$F_i(s)=F_i^{D\infty }(s)+F_i^I(s)+F_i^{*}(s)$

where

1. $F_i^{D\infty }(s)={\in }_{ijk}{\sigma }_{jm}^{D\infty }(s)b_m{\tau }_k(s)$ is the `self-force’ of the dislocation, i.e. the force exerted by the stresses generated by the dislocation itself.   This force always acts so as to reduce the length of the dislocation line.  In the classical solution, this force is infinite.  The procedure described in Section  5.9.2 can be used to remove the singularity $–$ in this case the stress in the Peach-Koehler formula should be calculated using the expression 

$\begin{array}{l}{\sigma }_{pq}^{D\infty }(x)=\frac{E}{16\pi (1+\nu )}{\displaystyle \underset{C}{\int }\left(\left[{\in }_{imp}{b}_m{\tau }_q+{\in }_{imq}{b}_m{\tau }_p\right]\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right)}d{s}_{\xi }\\ +\frac{E}{8\pi (1-{\nu }^2)}{\displaystyle \underset{C}{\int }\left(b_m{\in }_{imk}{\tau }_k\left[\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_p\partial x_q}-{\delta }_{pq}\frac{{\partial }^3{R}_{\rho }(x-\xi )}{\partial x_i\partial x_j\partial x_j}\right]\right)}d{s}_{\xi }\end{array}$

where $R_{\rho }(x)=\sqrt{x_k{x}_k+{\rho }^2}$.   Note that, if the dislocation remains straight, the total length of the dislocation line does not change as the dislocation moves.  In this case, the self-force is zero.  In 2D descriptions of dislocation motion, therefore, the core singularity has no effect $–$ this is why it has been possible to live with the classical dislocation fields for so long.

2. $F_i^I(s)={\in }_{ijk}{\sigma }_{jm}^I(s)b_m{\tau }_k(s)$ is a force generated by stress associated with the solid’s boundaries.  These are generally non-singular.  This force is often referred to as the `image force’
3. $F_i^{*}(s)={\in }_{ijk}{\sigma }_{jm}^{*}(s)b_m{\tau }_k(s)$ is the force exerted on the dislocation by externally applied loading.  This, too, is generally nonsingular.

 

 

Derivation:  The following expression for the total energy of a dislocation in an elastic solid was derived in Section 5.9.3.

$V=V^D+V^{D*}+V^{*}$

where

 $V^D={\displaystyle \underset{S}{\int }\frac{1}{2}{\sigma }_{ij}^D{b}_i{m}_{j}dA}$ is the strain energy of the dislocation itself

 $V^{D*}={\displaystyle \underset{S}{\int }{\sigma }_{ij}^{*}{b}_i{m}_{j}dA}$ is the work done to introduce the dislocation into the externally applied stress

 $V^{*}={\displaystyle \underset{R}{\int }\frac{1}{2}{\sigma }_{ij}^{*}{\epsilon }_{ij}^{*}dV}-{\displaystyle \underset{\partial 2R}{\int }{t}_i{u}_i^{*}dA}$ is the potential energy of the applied loads

We wish to calculate the change in potential energy resulting from a small change in area $\delta S$ as the dislocation line advances by a small distance $\delta a(s)n(s)$.  We consider each term in the potential energy

1. The last term is independent of S, and therefore $\delta V^{*}=0$.

2. The change in $V^{D*}$ follows as  $\delta V^{D*}={\displaystyle \underset{\delta S}{\int }{\sigma }_{ij}^{*}{b}_i{m}_{j}dA}$, where $\delta S$ is the increment in area swept by the dislocation.  Note that an area element swept by the advancing dislocation line can be expressed as $mdA=\delta an\times \tau ds$, so we can write

$\delta V^{D*}={\displaystyle \underset{C}{\int }{\sigma }_{ij}^{*}{b}_i{\in }_{jkl}{n}_k{\tau }_l\delta a(s)ds}$

3. The change in $V^D$ can be written as

$\delta V^D={\displaystyle \underset{S}{\int }\frac{1}{2}\delta {\sigma }_{ij}^D{b}_i{m}_{j}dA}+{\displaystyle \underset{\delta S}{\int }\frac{1}{2}{\sigma }_{ij}^D{b}_i{m}_{j}dA}$

To calculate the change in stress $\delta {\sigma }_{ij}$ arising from the motion of the dislocation line, recall that the displacement and stress due to the dislocation loop can be calculated from the expression

$u_k(x)={\displaystyle \underset{S}{\int }{m}_i{\Sigma }_{ij}^{(k)}(x-\xi )b_{j}d{A}_{\xi }}{\sigma }_{pq}^D=C_{pqkl}\frac{\partial u_k}{\partial x_l}=C_{pqkl}{\displaystyle \underset{S}{\int }{m}_i\frac{\partial }{\partial x_l}{\Sigma }_{ij}^{(k)}(x-\xi )b_{j}d{A}_{\xi }}$

where ${\Sigma }_{ij}^{(k)}$ is the stress due to a point force in the (bounded) elastic solid.  The change in stress therefore follows as

$\delta {\sigma }_{pq}=C_{pqkl}{\displaystyle \underset{C}{\int }\frac{\partial }{\partial x_l}{\Sigma }_{ij}^{(k)}(x-\xi )b_j{\in }_{jkl}{n}_k{\tau }_l\delta a(s)d{s}_{\xi }}$

This shows that

$\delta V^D=\frac{1}{2}{\displaystyle \underset{S}{\int }{C}_{pqkl}{\displaystyle \underset{C}{\int }\frac{\partial }{\partial x_l}{\Sigma }_{ij}^{(k)}(x-\xi )b_j{\in }_{jkl}{n}_k{\tau }_l\delta a(s)d{s}_{\xi }}dA}+{\displaystyle \underset{\delta S}{\int }\frac{1}{2}{\sigma }_{ij}^D{b}_i{m}_{j}dA}$

Reversing the order of integration in the first integral and using the expression for ${\sigma }_{pq}$ then gives

$\delta V^D={\displaystyle \underset{\delta S}{\int }{\sigma }_{ij}^D{b}_i{m}_{j}dA}={\displaystyle \underset{C}{\int }{\sigma }_{ij}^D{b}_i{\in }_{jkl}{n}_k{\tau }_l\delta a(s)ds}$

4. Finally, combining the results of (3) and (4) and noting that ${\in }_{ijk}=-{\in }_{jik}$ then gives

$-\delta V={\displaystyle \underset{C}{\int }{\in }_{kjl}\left[{\sigma }_{ij}^D+{\sigma }_{ij}^{*}\right]b_i{\tau }_l{n}_k\delta a(s)ds}={\displaystyle \underset{C}{\int }F(s)\cdot n(s)\delta a(s)ds}$

This has to hold for all possible $\delta a(s)$, which shows that $F_i(s)={\in }_{ijk}{\sigma }_{jm}(s)b_m{\tau }_k(s)$ as required.