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:

Failed to parse (syntax error): {\displaystyle b \mathbf{\sigma} = \frac{\mu b^2}{4\pi r \left \( 1-\nu \right \) } \left \{ \left \[ 1-\frac{\nu}{2} \left \( 3-4cos^2 \beta \right \) \right \] \ln\frac{L}{\rho}-1+\frac{nu]{2} \right \} }