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 ρ s {\displaystyle \rho _{s}} . 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. ρ s + ρ g {\displaystyle \rho _{s}+\rho _{g}} , where ρ g {\displaystyle \rho _{g}} 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 l {\displaystyle l} and thickness t {\displaystyle t} . When the crystal bar is bent to a radius of curvature r {\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 l {\displaystyle l} to l + d l {\displaystyle l+dl} . Here d l {\displaystyle dl} is positive and its magnitude is assumed to be t θ / 2 {\displaystyle t\theta /2} . Similarly, the length of the opposite inner surface is decreased from l {\displaystyle l} to l − d l {\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
s t r a i n g r a d i e n t = 2 d l / l t = 2 t θ / 2 l t = θ l {\displaystyle strain\ gradient=2{\frac {dl/l}{t}}=2{\frac {t\theta /2l}{t}}={\frac {\theta }{l}}} . Since l = r θ {\displaystyle l=r\theta } , s t r a i n g r a d i e n t = 1 r {\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 b {\displaystyle b} is equal to the magnitude of Burgers vector b {\displaystyle b} . Thus the numbers of crystal planes on the outer (tension) surface and inner (compression) surface are ( l + d l ) / b {\displaystyle (l+dl)/b} and ( l − d l ) / b {\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 ρ g {\displaystyle \rho _{g}} is this difference divided by the crystal surface area
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