Dislocations and Plastic Deformation

A dislocation is a line defect characterized by a Burgers vector: the lattice-translation mismatch obtained from a circuit around the defect. The vector has units of length. An edge dislocation has Burgers vector perpendicular to its line direction; a screw dislocation has them parallel. A curved dislocation can have mixed character along its length. These descriptions classify lattice geometry, not a stream of extra atoms travelling with the observed macroscopic strain. Continuum elasticity describes the surrounding stress field but cannot resolve the atomic core accurately.

Glide moves a dislocation on a slip plane and produces a permanent relative translation after the defect has passed through the crystal. Climb requires transport of vacancies or atoms and introduces diffusion and temperature sensitivity. Many different microscopic mechanisms can therefore contribute to the same measured plastic strain. A permanent deformation is not automatically a crack, and recoverable elastic distortion still surrounds the moving defects. Separating the elastic and plastic contributions is essential when interpreting unloading: only the elastic part is removed immediately in an ideal quasistatic unload.

A crystal does not respond to the magnitude of an applied tensile stress alone. For a chosen slip system, the relevant driving stress is the shear traction resolved along its slip direction. Under uniaxial stress, two directional projections give the Schmid factor. One angle is between the loading axis and the slip-plane normal; the other is between the loading axis and the slip direction. Those directions have a geometric relationship, so their cosines cannot be independently set to arbitrary values for one physical slip system.

Obstacles and lattice resistance oppose motion. Solute atoms, grain boundaries, precipitates, and other dislocations can change the stress needed for appreciable glide. This is why processing and microstructure affect strength even when the basic elastic stiffness changes little. In a polycrystal, neighbouring grains constrain one another, and a single resolved-shear calculation does not predict the whole specimen response. The simplified relation is useful for understanding orientation, provided the uniaxial loading and one-slip-system assumptions are stated rather than silently extended to every stress tensor.

A moving line sweeps out slip area. Add the swept areas from the mobile defects in a representative volume and multiply by the Burgers-vector magnitude to obtain plastic shear. Differentiating gives the Orowan relation for a single slip system. Mobile dislocation density is total mobile line length divided by volume, so its units are inverse square metres. Multiply that by Burgers-vector length and average glide speed: the resulting units are inverse seconds, exactly those of a dimensionless shear strain rate.

The relation is kinematic; it does not supply a velocity law or an evolution law for density. Increasing mobile density while holding speed fixed raises the rate, but increasing total stored density can hinder motion and cause work hardening. Total and mobile density are therefore not interchangeable. In the vectors lab, first predict the x projection of a fixed-magnitude vector at 0° and 90°, then test it. This illustrates one directional projection with dimensionless quantities. It does not calculate a complete Schmid factor, a dislocation stress field, defect mobility, or a plastic flow history.