Geometrically necessary dislocations

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 equations 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 ${\displaystyle {\frac {\partial W}{\partial k_{ij}}}={\frac {1}{2}}\alpha _{ij}+{\frac {1}{2}}{\frac {\partial \alpha _{kl}}{\partial k_{ij}}}k_{kl}={\frac {1}{2}}\alpha _{ij}+{\frac {1}{2}}k_{ji}-{\frac {1}{2}}\delta _{ij}k_{kk}=\alpha _{ij}}$.

The uniaxial tensile test has largely been performed to obtain the stress-strain relations and related mechanical properties of bulk specimens. However, there is an extra storage of defects associated with non-uniform plastic deformation in geometrically necessary dislocations, and ordinary macroscopic test alone, e.g. uniaxial tensile test, is not enough to capture the effects of such defects, e.g. plastic strain gradient. Besides, geometrically necessary dislocations are in the micron scale, where a normal bending test performed at millimeter-scale fails to detect these dislocations.[5]

Only after the invention of spatially and angularly resolved methods to measure lattice distortion via electron backscattered diffraction by Adams et al.[6] in 1997, experimental measurements of geometrically necessary dislocations became possible. For example, Sun et al.[7] in 2000 studied the pattern of lattice curvature near the interface of deformed aluminum bicrystals using diffraction-based orientation imaging microscopy. Thus the observation of geometrically necessary dislocations was realized using the curvature data.

But due to experimental limitations, the density of geometrically necessary dislocation for a general deformation state was hard to measure until a lower bound method was introduced by Kysar et al.[8] at 2010. They studied wedge indentation with a 90 degree included angle into a single nickel crystal (and later the included angles of 60 degree and 120 degree were also available by Dahlberg et al.[9]). By comparing the orientation of the crystal lattice in the after-deformed configuration to the undeformed homogeneous sample, they were able to determine the in-plane lattice rotation and found it an order of magnitude larger than the out-of-plane lattice rotations, thus demonstrating the plane strain assumption.

The Nye dislocation density tensor[4] has only two non-zero components due to two-dimensional deformation state and they can be derived from the lattice rotation measurements. Since the linear relationship between two Nye tensor components and densities of geometrically necessary dislocations is usually under-determined, the total density of geometrically necessary dislocations is minimized subject to this relationship. This lower bound solution represents the minimum geometrically necessary dislocation density in the deformed crystal consistent with the measured lattice geometry. And in regions where only one or two effective slip systems are known to be active, the lower bound solution reduces to the exact solution for geometrically necessary dislocation densities.

Because ${\displaystyle \rho _{g}}$ is in addition to the density of statistically stored dislocations ${\displaystyle \rho _{s}}$, the increase in dislocation density due to accommodated polycrystals leads to a grain size effect during strain hardening; that is, polycrystals of finer grain size will tend to work-harden more rapidly.[2]

Geometrically necessary dislocations can provide strengthening, where two mechanisms exists in different cases. The first mechanism provides macroscopic isotropic hardening via local dislocation interaction, e.g. jog formation when an existing geometrically necessary dislocation is cut through by a moving dislocation. The second mechanism is kinematic hardening via the accumulation of long range back stresses.[10]

Geometrically necessary dislocations can lower their free energy by stacking one atop another (see Peach-Koehler formula for dislocation-dislocation stresses) and form low-angle tilt boundaries. This movement often requires the dislocations to climb to different glide planes, so an annealing at elevated temperature is often necessary. The result is an arc that transforms from being continuously bent to discretely bent with kinks at the low-angle tilt boundaries.[1]