Geometrically necessary dislocations

As straining progresses, the dislocation density increases and the dislocation mobility decreases during plastic flow. There are different ways through which dislocations can accumulate. Many of the dislocations are accumulated by multiplication, where dislocations encounters each other by chance. Dislocations stored in such progresses are called statistically stored dislocations, with corresponding density ${\displaystyle \rho _{s}}$.^[2] In other words, they are dislocations evolved from random trapping processes during plastic deformation.^[3]

In addition to statistically stored dislocation, geometrically necessary dislocations are accumulated in strain gradient fields caused by geometrical constraints of the crystal lattice. In this case, the plastic deformation is accompanied by internal plastic strain gradients. The theory of geometrically necessary dislocations was first introduced by Nye^[4] in 1953. Since geometrically necessary dislocations are present in addition to statistically stored dislocations, the total density is the accumulation of two densities, e.g. ${\displaystyle \rho _{s}+\rho _{g}}$, where ${\displaystyle \rho _{g}}$ is the density of geometrically necessary dislocations.

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The plastic bending of a single crystal can be used to illustrate the concept of geometrically necessary dislocation, where the slip planes and crystal orientations are parallel to the direction of bending. The perfect (non-deformed) crystal has a length ${\displaystyle l}$ and thickness ${\displaystyle t}$. When the crystal bar is bent to a radius of curvature ${\displaystyle r}$, a strain gradient forms where a tensile strain occurs in the upper portion of the crystal bar, increasing the length of upper surface from ${\displaystyle l}$ to ${\displaystyle l+dl}$. Here ${\displaystyle dl}$ is positive and its magnitude is assumed to be ${\displaystyle t\theta /2}$. Similarly, the length of the opposite inner surface is decreased from ${\displaystyle l}$ to ${\displaystyle l-dl}$ due to the compression strain caused by bending. Thus, the strain gradient is the strain difference between the outer and inner crystal surfaces divided by the distance over which the gradient exists

${\displaystyle strain\ gradient=2{\frac {dl/l}{t}}=2{\frac {t\theta /2l}{t}}={\frac {\theta }{l}}}$. Since ${\displaystyle l=r\theta }$, ${\displaystyle strain\ gradient={\frac {1}{r}}}$.

The surface length divided by the interatomic spacing is the number of crystal planes on this surface. The interatomic spacing ${\displaystyle b}$ is equal to the magnitude of Burgers vector ${\displaystyle b}$. Thus the numbers of crystal planes on the outer (tension) surface and inner (compression) surface are ${\displaystyle (l+dl)/b}$ and ${\displaystyle (l-dl)/b}$, respectively. Therefore, the concept of geometrically necessary dislocations is introduced that the same sign edge dislocations compensate the difference in the number of atomic planes between surfaces. The density of geometrically necessary dislocations ${\displaystyle \rho _{g}}$ is this difference divided by the crystal surface area

${\displaystyle \rho _{g}={\frac {(l+dl)/b-(l-dl)/b}{lt}}=2{\frac {dl}{ltb}}={\frac {1}{rb}}={\frac {strain\ gradient}{b}}}$.

More precisely, the orientation of the slip plane and direction with respect to the bending should be considered when calculating the density of geometrically necessary dislocations. In a special case when the slip plane normals are parallel to the bending axis and the slip directions are perpendicular to this axis, ordinary dislocation glide instead of geometrically necessary dislocation occurs during bending process. Thus, a constant of order unity ${\displaystyle \alpha }$ is included in the expression for the density of geometrically necessary dislocations

${\displaystyle \rho _{g}=\alpha {\frac {strain\ gradient}{b}}}$.

Between the adjacent grains of a polycrystalline material, geometrically necessary dislocations can provide displacement compatibility by accommodating each crystal's strain gradient. Empirically, it can be inferred that such dislocations regions exist because crystallites in a polycrystalline material do not have voids or overlapping segments between them. In such a system, the density of geometrically necessary dislocations can be estimated by considering an average grain. Overlap between two adjacent grains is proportional to ${\displaystyle {\overline {\varepsilon }}d}$ where ${\displaystyle {\overline {\varepsilon }}}$ is average strain and ${\displaystyle d}$ is the diameter of the grain. The displacement ${\displaystyle dl}$ is proportional to ${\displaystyle {\overline {\varepsilon }}}$ multiplied by the gage length, which is taken as ${\displaystyle d}$ for a polycrystal. This divided by the Burgers vector, b, yields the number of dislocations, and dividing by the area (${\displaystyle \cong d^{2}}$) yields the density

${\displaystyle \rho _{g}\cong {\frac {\overline {\varepsilon }}{bd}}}$

which, with further geometrical considerations, can be refined to

${\displaystyle \rho _{g}={\frac {\overline {\varepsilon }}{4bd}}}$.^[2]

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Nye has introduced a set of tensor (so-called Nye's tensor) to calculate the geometrically necessary dislocation density.^[4]

For a three dimension dislocations in a crystal, considering a region where the effects of dislocations is averaged (i.e. the crystal is large enough). The dislocations can be determined by Burgers vectors. If a Burgers circuit of the unit area normal to the unit vector ${\displaystyle l_{j}}$ has a Burgers vector ${\displaystyle B_{i}}$

${\displaystyle B_{i}=\alpha _{ij}l_{j}}$ (${\displaystyle i,j=1,2,3}$)

where the coefficient ${\displaystyle \alpha _{ij}}$ is Nye's tensor relating the unit vector ${\displaystyle l_{j}}$ and Burgers vector ${\displaystyle B_{i}}$. This second-rank tensor determines the dislocation state of a special region.

Assume ${\displaystyle B_{i}=b_{i}(nr_{j}l_{j})}$, where ${\displaystyle r}$ is the unit vector parallel to the dislocations and ${\displaystyle b}$ is the Burgers vector, n is the number of dislocations crossing unit area normal to ${\displaystyle r}$. Thus, ${\displaystyle \alpha _{ij}=nb_{i}r_{j}}$. The total ${\displaystyle \alpha _{ij}}$ is the sum of all different values of ${\displaystyle nb_{i}r_{j}}$. Assume a second-rank tensor ${\displaystyle k_{ij}}$ to describe the curvature of the lattice, ${\displaystyle d\phi _{i}=k_{ij}dx_{j}}$, where ${\displaystyle d\phi _{i}}$ is the small lattice rotations about the three axes and ${\displaystyle dx_{j}}$ is the displacement vector. It can be proved that ${\displaystyle k_{ij}=\alpha _{ji}-{\tfrac {1}{2}}\delta _{ij}\alpha _{kk}}$where ${\displaystyle \delta _{ij}=1}$ for ${\displaystyle i=j}$, and ${\displaystyle \delta _{ij}=0}$ for ${\displaystyle i\neq j}$.

The equation of equilibrium yields ${\displaystyle {\frac {\partial \alpha _{ij}}{\partial x_{j}}}=0}$. Since ${\displaystyle k_{ij}={\frac {\partial \phi {i}}{\partial x_{j}}}}$, thus ${\displaystyle {\frac {\partial k_{ij}}{\partial x_{k}}}={\partial ^{2} \over \partial x_{j}\partial x_{k}}\phi _{i}={\frac {\partial k_{ik}}{\partial x_{j}}}}$. By substituting ${\displaystyle \alpha }$ for ${\displaystyle k}$, ${\displaystyle {\frac {\partial \alpha _{ji}}{\partial x_{k}}}-{\frac {\partial \alpha _{ki}}{\partial x_{j}}}={\frac {1}{2}}(\delta _{ij}{\frac {\partial \alpha _{ll}}{\partial x_{k}}}-\delta _{ik}{\frac {\partial \alpha _{ll}}{\partial x_{j}}})}$. Due to the zero solution for equations with ${\displaystyle j=k}$ are zero and the symmetry of ${\displaystyle j}$ and ${\displaystyle k}$, only nine independent equations remain of all twenty-seven possible permutations of ${\displaystyle i,j,k}$. The Nye's tensor ${\displaystyle \alpha _{ij}}$ can be determined by these nine differential equations.

Thus the dislocation potential can be written as ${\displaystyle W={\tfrac {1}{2}}\alpha _{ij}k_{ij}}$, where ${\displaystyle {\frac {\partial W}{\partial k_{ij}}}={\frac {1}{2}}\alpha _{ij}+{\frac {1}{2}}{\frac {\partial \alpha _{kl}}{\partial k_{ij}}}k_{kl}={\frac {1}{2}}\alpha _{ij}+{\frac {1}{2}}k_{ji}-{\frac {1}{2}}\delta _{ij}k_{kk}=\alpha _{ij}}$.