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 the loop. The radius of the curvature is a minimum when $r=L/2$. Hence the maximum stress for which local equilibrium is possible is given by the equation above with $r=L/2$. For the typical case that $L=10^{3}\rho$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu=0.33} , the critical stress for a dislocation initially pure edge and pure screw, respectively, is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma ^*=0.5\mu b/L} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma ^*=1.5\mu b/L} .

When the net local resolved shear stress( the applied stress plus the internal stresses) exceeds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^*} , the loop has no stable equilibrium configuration but passes through the successive positions. Provided that the expanding loop neither jogs out of the original glide plane because of intersections with other dislocations nor is obstructed from rotating about the pinning point, it will annihilate over a portion of its length, creating a complete closed loop and restoring the original configuration. A sequence of loops then continues to form from the source until sufficient internal stresses are generated for the net resolved shear stress at the source to drop below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^*} .

DDLab has included a Frank-Read Source input file in its latest version 2007-12-28.

To compare the analytic result with DDLab simulation, we need to use the relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = rc/2 } . According to the non-singular continuum theory of dislocation (JMPS 54, 561-587, 2006), the energy of a circular prismatic dislocation loop is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = 2 \pi R \frac{\mu b^2}{4 \pi (1-\nu)} \left( \ln \frac{8 R}{r_c} - 1 \right) + O\left(\frac{a^2}{R^2}\right) }

According to Hirth and Lothe, the energy of the same loop is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = 2 \pi R \frac{\mu b^2}{4 \pi (1-\nu)} \left( \ln \frac{4 R}{\rho} - 1 \right) + O\left(\frac{a^2}{R^2}\right) }

Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = rc_2} Reference: J. P. Hirth and J. Lothe, Theory of Dislocations", 2nd ed. (Wiley, New York, 1982)