Geometrically necessary dislocations
Geometrically necessary dislocations are like-signed dislocations needed to accommodate for plastic bending in a crystalline material. They are present when a material's plastic deformation is accompanied by internal plastic strain gradients. 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. Dislocations in crystalline materials. Statistically stored 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 formula_1. In other words, they are dislocations evolved from random trapping processes during plastic deformation. Geometrically necessary dislocations. 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 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. formula_2, where formula_3 is the density of geometrically necessary dislocations. Concept. Single crystal. 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 formula_4 and thickness formula_5. When the crystal bar is bended to a radius of curvature formula_6, a strain gradient forms where a tensile strain occurs in the upper portion of the crystal bar, increasing  the length of upper surface from formula_4 to formula_8. Here formula_9 is positive and its magnitude is assumed to be formula_10. Similarly, the length of the opposite inner surface is decreased from formula_4 to formula_12 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 formula_13. Since formula_14, formula_15. The surface length divided by the interatomic spacing is the number of crystal planes on this surface. The interatomic spacing formula_16 is equal to the magnitude of Burgers vector  formula_16. Thus the numbers of crystal planes on the outer (tension) surface and inner (compression) surface are formula_18 and formula_19, 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 formula_3 is this difference divided by the crystal surface area formula_21. 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 formula_22 is included in the expression for the density of geometrically necessary dislocations formula_23. Polycrystalline material. 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 formula_24 where formula_25 is average strain and formula_26 is the diameter of the grain. The displacement formula_9 is proportional to formula_25 multiplied by the gage length, which is taken as formula_26 for a polycrystal. This divided by the Burgers vector, "b", yields the number of dislocations, and dividing by the area (formula_30) yields the density formula_31 which, with further geometrical considerations, can be refined to formula_32. Nye's tensor. Nye has introduced a set of tensor (so-called Nye 's tensor) to calculate the geometrically necessary dislocation density. 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 formula_33 has a Burgers vector formula_34 formula_35 (formula_36) where the coefficient formula_37 is Nye's tensor relating the unit vector formula_33 and Burgers vector formula_34. This second-rank tensor determines the dislocation state of a special region. Assume formula_40, where formula_6 is the unit vector parallel  to the dislocations and formula_16 is the Burgers vector, n is the number of dislocations crossing unit area normal to formula_6. Thus, formula_44. The total formula_37 is the sum of all different values of formula_46. Assume a second-rank tensor formula_47 to describe the curvature of the lattice, formula_48, where formula_49 is the small lattice rotations about the three axes and formula_50 is the displacement vector. It can be proved that formula_51where formula_52 for formula_53, and formula_54 for formula_55. The equations of equilibrium yields formula_56. Since formula_57, thus formula_58. By substituting formula_22 for formula_60, formula_61. Due to the zero solution  for equations with formula_62 are zero and the symmetry of formula_63 and formula_60, only nine independent equations remain of all twenty-seven possible permutations of formula_65. The Nye's tensor formula_37 can be determined by these nine differential equations. Thus the dislocation potential can be written as formula_67, where formula_68. Measurement. 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. Only after the invention of spatially and angularly resolved methods to measure lattice distortion via electron backscattered diffraction by Adams et al. in 1997, experimental measurements of geometrically necessary dislocations became possible. For example, Sun et al. 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. 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.). 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 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. Application. Because formula_3 is in addition to the density of statistically stored dislocations formula_1, 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. 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. 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.