Strain / Work Hardening

Increase in strength and hardness of a metal by plastic deformation i.e., cold work known as Work hardening or strain hardening

In general, an anneales crystal have a dislocation density of \( 10^{-8} \) m^-2 , which can be incease up to \( 10^{-10} - 10^{12} \) m^-2 for a moderate cold worked material and \( 10^{-14} - 10^{16} \) m^-2 for a heavy cold worked material. During cold wroking when a material is strained beyond its yield point. An increasing stress is required to produce additional plastic deformation and the metal apparently becomes stronger and more difficult to deform. This happens due to interaction between the dislocation which restrict theer motion and hence requires more stess to move same dislocation. In a simple way dislocations themselves are obstacles to dislocation motion.. And these obstacles can be soft or hard depending on type of interation.
A typical curve showing effect of strain hardening on yield strength and ductility. On work hardening materials strength or yield strength and hardness increase, hence %elongation falls.

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Types of interactions that takes place during dislocation motion:

A dislocation can restrict another dislocation motion by following ways...

1. Stress field :
   A dislocation develops a steess field around it. And hardening can be arise from the interaction of the stress fields aroung a dislocations moving on parallel slip planes


2. Dislocation Nodes :
   In general, a dislocation line can not terminate inside a crystal. The exception is at the nodes, where 2 or more dislocation lines meet. These nodes can be generated when an dislocation spit into two dislocation or when two dislocation mearge to one. These nodes are immobile and act as a pinning point for the dislocation, which restrict the motion of dislocation and dislocation bows at these points . Hence requires more stress for further deformation.


3. Formation of immobile jogs and kinks :
   When a dislocation intersect with another dislocation they form kinks and jogs. And for some special cases these jogs and kinks can be immobile hence disloction can not glide/move in the same plane.

These obscles can act alone or as a conbinations to restrict the the motion of dislocation.

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Effect of dislocation density on yield strength :

As we have seen earlier, on cold working the dislocation density arises and hence increasing the strength of the materials. This can be formulated as - \[ \boxed{ \tau = \tau_{0} + A\sqrt{\rho} } \] Here,
\( \tau \) : Stress required to move a dislocaion in a matrix of dislocation density \( \rho \).
\( \tau_{0} \) : Stress required to move a dislocaion in the same matrix with zero dislocation density
A : constant
\( \rho \) : Dislocation density