Table 1.

Coefficients for the evolution equations.

$g$	Expression	General coefficient description
$g_{\mathit{sour}}^{\alpha }$	$b^{\alpha }\phi {\sum }_{\beta }\sqrt{{\rho }_{\mathit{im}}^{\beta }}$	Dislocation-density source term
$g_{mnter-}^{\alpha }$	$l_c{f}_o{\sum }_{\beta }\sqrt{a_{\alpha \beta }}\left(\frac{{\rho }_m^{\beta }}{{\rho }_m^{\alpha }{b}^{\alpha }},+,\frac{{\dot{\gamma }}_m^{\beta }}{{\dot{\gamma }}_m^{\alpha }{b}^{\beta }}\right)$	Mobile and immobile dislocation density interactions
$g_{immob-}^{\alpha }$	$\frac{l_c{f}_o}{\sqrt{{\rho }_{\mathit{im}}^{\alpha }}}{\sum }_{\beta }\sqrt{a_{\alpha \beta }}{\rho }_{\mathit{im}}^{\beta }$
$g_{mnter+}^{\alpha }$	$\frac{l_c{f}_o}{{\dot{\gamma }}^{\alpha }{\rho }_m^{\alpha }}{\sum }_{\beta ,\gamma }{n}_{\alpha }^{\beta \gamma }\sqrt{a_{\beta \gamma }}\left(\frac{{\rho }_m^{\gamma }{\dot{\gamma }}^{\beta }}{b^{\beta }},+,\frac{{\rho }_m^{\beta }{\dot{\gamma }}^{\gamma }}{b^{\gamma }}\right)$
$g_{immob+}^{\alpha }$	$\frac{l_c{f}_o}{{\dot{\gamma }}^{\alpha }\sqrt{{\rho }_{\mathit{im}}^{\alpha }}}{\sum }_{\beta }{n}_{\alpha }^{\beta \gamma }\sqrt{a_{\beta \gamma }}{\rho }_{\mathit{im}}^{\gamma }{\dot{\gamma }}^{\beta }$
$g_{\mathit{recov}}^{\alpha }$	$\frac{l_c{f}_o}{{\dot{\gamma }}^{\alpha }\sqrt{{\rho }_{\mathit{im}}^{\alpha }}}\left({\sum }_{\beta },\sqrt{a_{\beta \gamma }},\frac{{\dot{\gamma }}^{\beta }}{b^{\beta }}\right)\text{exp}\left(\frac{-H_0+H_0\sqrt{{\rho }_{\mathit{im}}^{\alpha }/{\rho }_s}}{\mathit{kT}}\right)$	Recovery for annihilation of dislocation-densities