Jan 05, 2017

The stress of dislocation is dependent on the shear modulus, G, the magnitude of the Burgers vector, b, and the dislocation density, {\displaystyle \rho _{\perp }}

where {\displaystyle \tau _{0}} is the intrinsic strength of the material with low dislocation density and {\displaystyle \alpha }is a correction factor specific to the material.

As shown in Figure 1 and the equation above, work hardening has a half root dependency on the number of dislocations. The material exhibits high strength if there are either high levels of dislocations (greater than 1014 dislocations per m2) or no dislocations. A moderate number of dislocations (between 107 and 109 dislocations per m2) typically results in low strength.

For an extreme example, in a tensile test a bar of steel is strained to just before the distance at which it usually fractures. The load is released smoothly and the material relieves some of its strain by decreasing in length. The decrease in length is called the elastic recovery, and the end result is a work-hardened steel bar. The fraction of length recovered (length recovered/original length) is equal to the yield-stress divided by the modulus of elasticity. (Here we discuss true stress in order to account for the drastic decrease in diameter in this tensile test.) The length recovered after removing a load from a material just before it breaks is equal to the length recovered after removing a load just before it enters plastic deformation.

The work-hardened steel bar has a large enough number of dislocations that the strain field interaction prevents all plastic deformation. Subsequent deformation requires a stress that varies linearly with the strain observed, the slope of the graph of stress vs. strain is the modulus of elasticity, as usual.

The work-hardened steel bar fractures when the applied stress exceeds the usual fracture stress and the strain exceeds usual fracture strain. This may be considered to be the elastic limit and the yield stress is now equal to the fracture toughness, which is of course, much higher than a non-work-hardened steel yield stress.

The amount of plastic deformation possible is zero, which is obviously less than the amount of plastic deformation possible for a non-work-hardened material. Thus, the ductility of the cold-worked bar is reduced.

Substantial and prolonged cavitation can also produce strain hardening.

Additionally, jewelers will construct structurally sound rings and other wearable objects (especially those worn on the hands) that require much more durability (than earrings for example) by utilizing a material's ability to be work hardened. While casting rings is done for a number of economical reasons (saving a great deal of time and cost of labor), a master jeweler may utilize the ability of a material to be work hardened and apply some combination of cold forming techniques during the production of a piece.

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