Introducing the stress distribution near grain boundaries to improve the di
slocation pileup model for the Halt-Petch (H-P) relation, the continuous di
stribution of dislocations in the pileup could be solved by means of Tscheb
ysheff polynomials for the Hilbert transformation. An analytical formula of
the stress intensity factor for the dislocation pileup is obtained. The re
verse H-P relation may be explained by the modified dislocation-pileup-mode
l.