Precipitation Hardening - Governing Equations

Governing Equations

There are two equations to describe the two mechanisms for precipitation hardening:

Dislocations cutting through particles:

where is material strength, is the second phase particle radius, is the surface energy, is the magnitude of the Burgers vector, and is the spacing between pinning points. This governing equation shows that the strength is proportional to, the radius of the precipitate particles. This means that it is easier for dislocations to cut through a material with smaller second phase particles (small r). As the size of the second phase particles increases, the particles impede dislocation movement and it becomes increasingly difficult for the particles to cut through the material. In other words, the strength of a material increases with increasing r.

Dislocations bowing around particle:

where is the material strength, is the shear modulus, is the magnitude of the Burgers vector, is the distance between pinning points, and is the second phase particle radius. This governing equation shows that for dislocation bowing the strength is inversely proportional to the second phase particle radius r. Dislocation bowing, also called Orowan strengthening, is more likely to occur when there are large particles present in the material.

These governing equations show that the precipitation hardening mechanism depends on the size of the precipitate particles. At small r, cutting will dominate, while at large r, bowing will dominate.

Looking at the plot of both equations, it is clear that there is a critical radius at which max strengthening occurs. This critical radius is typically 5-30 nm.