Table 1 Coefficients for the evolution equations.

\({g}_{sour}^{\alpha }\)

\({b}^{\alpha }\varphi {\sum }_{\beta }\sqrt{{\rho }_{im}^{\beta }}\)

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

\({g}_{immob-}^{\alpha }\)

\(\frac{{l}_{c}{f}_{o}}{\sqrt{{\rho }_{im}^{\alpha }}}\sum_{\beta }\sqrt{{a}_{\alpha \beta }}{\rho }_{im}^{\beta }\)

\({g}_{mnter+}^{\alpha }\)

\(\frac{{l}_{c}{f}_{o}}{{\dot{\gamma }}^{\alpha }{\rho }_{m}^{\alpha }}\sum_{\beta ,\gamma }{n}_{\alpha }^{\beta \gamma }\sqrt{{a}_{\beta \gamma }}\left(\frac{{\rho }_{m}^{\gamma }{\dot{\gamma }}^{\beta }}{{b}^{\beta }}+\frac{{\rho }_{m}^{\beta }{\dot{\gamma }}^{\gamma }}{{b}^{\gamma }}\right)\)

\({g}_{immob+}^{\alpha }\)

\(\frac{{l}_{c}{f}_{o}}{{\dot{\gamma }}^{\alpha }\sqrt{{\rho }_{im}^{\alpha }}}\sum_{\beta }{n}_{\alpha }^{\beta \gamma }\sqrt{{a}_{\beta \gamma }}{\rho }_{im}^{\gamma }{\dot{\gamma }}^{\beta }\)

\({g}_{recov}^{\alpha }\)

\(\frac{{l}_{c}{f}_{o}}{{\dot{\gamma }}^{\alpha }\sqrt{{\rho }_{im}^{\alpha }}}\left(\sum_{\beta }\sqrt{{a}_{\beta \gamma }}\frac{{\dot{\gamma }}^{\beta }}{{b}^{\beta }}\right)\text{exp}\left(\frac{-{H}_{0}+{H}_{0}\sqrt{{\rho }_{im}^{\alpha }/{\rho }_{s}}}{kT}\right)\)

Recovery for annihilation of dislocation-densities