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Some interface theories and Hall-Petch relationship.

Hiroaki Yokoyama1, Hiroyuki Nagahama1

  • 1Department of Earth Science, Tohoku University, Sendai, Japan.

Royal Society Open Science
|May 15, 2025
PubMed
Summary

This study explains how increased interface density, like deformation twins, strengthens polycrystalline materials. It mathematically links interface density to flow stress, providing a theoretical basis for empirical observations.

Keywords:
Hall–Petch relationshipdeformation twindislocation densityrank-1 connectionstresssurface dislocation

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Area of Science:

  • Materials Science
  • Solid Mechanics
  • Crystallography

Background:

  • Polycrystalline materials strengthen with increased interface density.
  • The relationship between deformation-twin density and flow stress is empirically observed but lacks theoretical understanding.

Purpose of the Study:

  • To theoretically explain the correlation between interface density and flow stress in polycrystalline materials.
  • To establish a mathematical framework linking deformation-twin density to flow stress.

Main Methods:

  • Utilizing interface theories to analyze the relationship between interface density and flow stress.
  • Establishing macroscopic and mathematical equivalence between surface dislocation density and dislocation density.
  • Applying surface dislocation theory and other interface theories to various boundary types.

Main Results:

  • Derived the relationship between deformation-twin density and flow stress from dislocation density correspondence.
  • Demonstrated mathematical equivalence between surface dislocation theory and other interface theories (rank-1 connection, Hadamard jump condition, 0-lattice theory).
  • Explained Hall-Petch and inverse Hall-Petch relationships through the lens of rank-1 connection.

Conclusions:

  • Provided a theoretical foundation for the observed strengthening effect of interfaces in polycrystalline materials.
  • Unified various interface theories under a common mathematical framework.
  • Offered a new perspective on understanding strengthening mechanisms like the Hall-Petch relationship.