Dislocation-density kinematics: a simple evolution equation for dislocation density involving movement and tilting of dislocations


In this paper, a simple evolution equation for dislocation densities moving on a slip plane is proven. This equation gives the time evolution of dislocation density at a general field point on the slip plane, due to the approach of new dislocations and tilting of dislocations already at the field point. This equation is fully consistent with Acharya’s evolution equation and Hochrainer et al.’s “continuous dislocation dynamics” (CDD) theory. However, it is shown that the variable of dislocation curvature in CDD is unnecessary if one considers one-dimensional flux divergence along the dislocation velocity direction.

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  1. 1.

    A. Acharya: A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids 49, 761 (2001).

    Article  Google Scholar 

  2. 2.

    A. El-Azab: Statistical mechanics treatment of the evolution of dislocation distributions in single crystals. Phys. Rev. B 61, 11956 (2000).

    CAS  Article  Google Scholar 

  3. 3.

    R. Sedláček, J. Kratochvil and E. Werner: The importance of being curved: bowing dislocations in a continuum description. Philos. Mag. 83, 3735 (2003).

    Article  Google Scholar 

  4. 4.

    T. Hochrainer, M. Zaiser and P. Gumbsch: A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation. Philos. Mag. 87, 1261 (2007).

    CAS  Article  Google Scholar 

  5. 5.

    J.F. Nye: Some geometrical relations in dislocated crystals. Acta Metal. 1, 153 (1953).

    CAS  Article  Google Scholar 

  6. 6.

    E. Kröner: Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, Berlin, 1958).

    Google Scholar 

  7. 7.

    H.S. Leung and A.H.W. Ngan: Dislocation-density function dynamics—an all-dislocation, full-dynamics approach for modeling intensive dislocation structures. J. Mech. Phys. Solids 91, 172 (2016).

    Article  Google Scholar 

  8. 8.

    A. Acharya and A. Roy: Size effects and idealized dislocation microstructure at small scales: Predictions of a phenomenological mode of mesoscopic field dislocation mechanics: Part I. J. Mech. Phys. Solids 54, 1687 (2006).

    Article  Google Scholar 

  9. 9.

    V. Taupin, S. Varadhan, C. Fressengeas and A.J. Beaudoin: Directionality of yield point in strain-aged steels: The role of polar dislocations. Acta Mater. 56, 3002 (2008).

    CAS  Article  Google Scholar 

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The work described in this paper is supported by Kingboard Endowed Professorship in Materials Engineering and Seed Fund for Basic Research (Project code: 201411159129) at the University of Hong Kong. This Research Letter was invited by Editor of MRS Communications, based on an invited presentation at 2017 MRS Spring Meeting.

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Correspondence to A. H. W. Ngan.

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Ngan, A.H.W. Dislocation-density kinematics: a simple evolution equation for dislocation density involving movement and tilting of dislocations. MRS Communications 7, 583–590 (2017). https://doi.org/10.1557/mrc.2017.66

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