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

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