Test Case: Frank-Read Source: Difference between revisions
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<math>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 \}</math> |
<math>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 \}</math> |
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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 b the equation above with r=L/2. For the typical case that <math>L=10^3 \rho</math> |
where r is the radius of the loop. The radius of the curvature is a minimum when <math>r=L/2</math>. Hence the maximum stress for which local equilibrium is possible is given b the equation above with <math>r=L/2</math>. For the typical case that <math>L=10^3 \rho</math> |
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Revision as of 08:48, 19 December 2007
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:
![{\displaystyle 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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69872050f9083d6a830312ee03f4d7b888bd3468)
where r is the radius of the loop. The radius of the curvature is a minimum when 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 r=L/2} . Hence the maximum stress for which local equilibrium is possible is given b the equation above with 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 r=L/2} . For the typical case that 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 L=10^3 \rho}