Geometrically necessary dislocations

[2] They are in contrast to statistically stored dislocations, with statistics of equal positive and negative signs, which arise during plastic flow from multiplication processes like the Frank-Read source.

[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.

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.

, a strain gradient forms where a tensile strain occurs in the upper portion of the crystal bar, increasing the length of upper surface from

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.

{\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.

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.

This second-rank tensor determines the dislocation state of a special region.

is the Burgers vector, n is the number of dislocations crossing unit area normal to

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.

[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.

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.

, 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 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.