Test Case: Frank-Read Source

Frank-Read source is a type of dislocation multiplication mechanism. Consider a segment whose ends are pinned(corresponding to nodes in a network, precipitates, or sites where the dislocation leaves the glide plane). Under a certain applied stress the segment bows out by glide. As bow-out proceeds, the radius of curvature of the line decreases and the line-tension forces tending to restore the line to a straight configuration increase. For stress less than a critical value, a metastable equilibrium configuration is attained, in which the line-tension force balances that caused by the applied stress. For the large bow-out case, following equilibrium condition holds:

$b\mathbf{\sigma} = \frac{\mathbb{S}}{r}=\frac{\mu b^2}{4\pi r\left( 1-\nu\right)} \left \{ \left [ 1- \frac{\nu}{2} \left ( 3-4cos^2\beta \right ) \right ] \mathrm{ln} \frac{L}{\rho} -1 + \frac{\nu}{2} \right \}$

where r is the radius of curvature