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October  2018, 23(8): 2989-3021. doi: 10.3934/dcdsb.2017183

Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation

Yang Xiang 1, and  Xiaodong Yan 2,

1. 

Department of Mathematics, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong
2. 

Department of Mathematics, University of Connecticut 341 Mansfield Road, Storrs, CT 06269, USA

Received  June 2016 Revised  June 2017 Published  October 2018 Early access  July 2017

Fund Project: The first author is partially supported by the Hong Kong Research Grants Council General Research Fund 606313. The second author thanks HKUST for hospitality.

Low angle grain boundaries can be modeled as arrays of line defects (dislocations) in crystalline materials. The classical continuum models for energetics and dynamics of curved grain boundaries are mainly based on those with equilibrium dislocation structures without the long-range elastic interaction, leading to a capillary force proportional to the local curvature of the grain boundary. The new continuum model recently derived by Zhu and Xiang (J. Mech. Phys. Solids, 69,175-194,2014) incorporates both the long-range dislocation interaction energy and the local dislocation line energy, and enables the study of low angle grain boundaries with non-equilibrium dislocation structures that involves the long-range elastic interaction. Using this new energy formulation, we show that the orthogonal network of two arrays of screw dislocations on a planar twist low angle grain boundary is always stable subject to both perturbations of the constituent dislocations within the grain boundary and the perturbations of the grain boundary itself.

Keywords: Dislocation, grain boundary, continuum model, calculus of variations, stability.
Mathematics Subject Classification: 93D20, 49K20, 74G65.
Citation: Yang Xiang, Xiaodong Yan. Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2989-3021. doi: 10.3934/dcdsb.2017183
References:
[1]

A. Arsenlis, W. Cai, M. Tang, M. Rhee, T. Oppelstrup, G. Hommes, T. G. Pierce and V. V. Bulatov, Enabling strain hardening simulations with dislocation dynamics, Modell. Simul. Mater. Sci. Eng., 15 (2007), 553-595.  doi: 10.1088/0965-0393/15/6/001.  Google Scholar

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[3]

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[4]

F. C. Frank, The resultant content of dislocations in an arbitrary intercrystalline boundary, in Symposium on the Plastic Deformation of Crystalline Solids, Office of Naval Research, Pittsburgh, 1950,150-154 Google Scholar

[5]

N. M. Ghoniem, S. H. Tong and L. Z. Sun, Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation, Phys. Rev. B, 61 (2000), 913-927.  doi: 10.1103/PhysRevB.61.913.  Google Scholar

[6]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, American Mathematical Society, 1999.  Google Scholar

[7]

C. Herring, Surface tension as a motivation for sintering, Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, (1999), 143-179.  doi: 10.1007/978-3-642-59938-5_2.  Google Scholar

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J. P. Hirth and J. Lothe, Theory of Dislocations 2nd edition, John Wiley, New York, 1982. Google Scholar

[9]

L. P. Kubin, G. Canova, M. Condat and B. Devincre, Dislocation microstructures and plastic flow: A 3d simulation, Solid State Phenomena, 23/24 (1992), 455-472.  doi: 10.4028/www.scientific.net/SSP.23-24.455.  Google Scholar

[10]

A. T. Lim, D. J. Srolovitz and M. Haataja, Low-angle grain boundary migration in the presence of extrinsic dislocations, Acta Mater., 57 (2009), 5013-5022.  doi: 10.1016/j.actamat.2009.07.003.  Google Scholar

[11]

A. T. Lim, M. Haataja, W. Cai and D. J. Srolovitz, Stress-driven migration of simple low-angle mixed grain boundaries, Acta Mater., 60 (2012), 1395-1407.  doi: 10.1016/j.actamat.2011.11.032.  Google Scholar

[12]

A. A. Pihlaja, Modeling Grain Boundary Structures Using Energy Minimization Ph. D. thesis, New York University, 2000.  Google Scholar

[13]

S. S. Quek, Y. Xiang and D. J. Srolovitz, Loss of interface coherency around a misfitting spherical inclusion, Acta Mater., 59 (2011), 5398-5410.  doi: 10.1016/j.actamat.2011.05.012.  Google Scholar

[14]

W. Read and W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev., 78 (1950), 275-289.  doi: 10.1103/PhysRev.78.275.  Google Scholar

[15]

Strichartz, Improved sobolev inequalities, Trans. Amer. Math. Soc., 279 (1983), 397-407.  doi: 10.1090/S0002-9947-1983-0704623-6.  Google Scholar

[16]

A. P. Sutton and R. W. Balluffi, Interfaces in Crystalline Materials Clarendon Press, Oxford, 1995. Google Scholar

[17]

A. P. Sutton and V. Vitek, On the structure of tilt grain boundaries in cubic metals Ⅰ. Symmetical tilt boundaries, Philos. Trans. Roy. Soc. Lond. A, 309 (1983), 1-36.  doi: 10.1098/rsta.1983.0020.  Google Scholar

[18]

Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424.   Google Scholar

[19]

Y. Xiang, L. T. Cheng, D. J. Srolovitz and W. E, A level set method for dislocation dynamics, Acta Mater., 51 (2003), 5499-5518.  doi: 10.1016/S1359-6454(03)00415-4.  Google Scholar

[20]

L. C. Zhang, Y. J. Gu and Y. Xiang, Energy of low angle grain boundaries based on continuum dislocation structure, Acta Mater., 126 (2017), 11-24.  doi: 10.1016/j.actamat.2016.12.035.  Google Scholar

[21]

X. H. Zhu and Y. Xiang, Stabilizing force on perturbed grainboundaries using dislocation model, Scripta Mater., 64 (2011), 5-8.  doi: 10.1016/j.scriptamat.2010.08.050.  Google Scholar

[22]

X. H. Zhu and Y. Xiang, Continuum frmework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries, J. Mech. Phys. Solid, 69 (2014), 175-194.  doi: 10.1016/j.jmps.2014.05.005.  Google Scholar

show all references

References:
[1]

A. Arsenlis, W. Cai, M. Tang, M. Rhee, T. Oppelstrup, G. Hommes, T. G. Pierce and V. V. Bulatov, Enabling strain hardening simulations with dislocation dynamics, Modell. Simul. Mater. Sci. Eng., 15 (2007), 553-595.  doi: 10.1088/0965-0393/15/6/001.  Google Scholar

[2]

Á. Bényi and T. Oh, The Sobolev inequality on the torus revisited, Publicationes mathematicae, 83 (2013), 359-374.  doi: 10.5486/PMD.2013.5529.  Google Scholar

[3]

B. A. Bilby, Bristol Conference Report on Defects in Crystalline Materials Physical Society, London, 1955,123. Google Scholar

[4]

F. C. Frank, The resultant content of dislocations in an arbitrary intercrystalline boundary, in Symposium on the Plastic Deformation of Crystalline Solids, Office of Naval Research, Pittsburgh, 1950,150-154 Google Scholar

[5]

N. M. Ghoniem, S. H. Tong and L. Z. Sun, Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation, Phys. Rev. B, 61 (2000), 913-927.  doi: 10.1103/PhysRevB.61.913.  Google Scholar

[6]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, American Mathematical Society, 1999.  Google Scholar

[7]

C. Herring, Surface tension as a motivation for sintering, Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, (1999), 143-179.  doi: 10.1007/978-3-642-59938-5_2.  Google Scholar

[8]

J. P. Hirth and J. Lothe, Theory of Dislocations 2nd edition, John Wiley, New York, 1982. Google Scholar

[9]

L. P. Kubin, G. Canova, M. Condat and B. Devincre, Dislocation microstructures and plastic flow: A 3d simulation, Solid State Phenomena, 23/24 (1992), 455-472.  doi: 10.4028/www.scientific.net/SSP.23-24.455.  Google Scholar

[10]

A. T. Lim, D. J. Srolovitz and M. Haataja, Low-angle grain boundary migration in the presence of extrinsic dislocations, Acta Mater., 57 (2009), 5013-5022.  doi: 10.1016/j.actamat.2009.07.003.  Google Scholar

[11]

A. T. Lim, M. Haataja, W. Cai and D. J. Srolovitz, Stress-driven migration of simple low-angle mixed grain boundaries, Acta Mater., 60 (2012), 1395-1407.  doi: 10.1016/j.actamat.2011.11.032.  Google Scholar

[12]

A. A. Pihlaja, Modeling Grain Boundary Structures Using Energy Minimization Ph. D. thesis, New York University, 2000.  Google Scholar

[13]

S. S. Quek, Y. Xiang and D. J. Srolovitz, Loss of interface coherency around a misfitting spherical inclusion, Acta Mater., 59 (2011), 5398-5410.  doi: 10.1016/j.actamat.2011.05.012.  Google Scholar

[14]

W. Read and W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev., 78 (1950), 275-289.  doi: 10.1103/PhysRev.78.275.  Google Scholar

[15]

Strichartz, Improved sobolev inequalities, Trans. Amer. Math. Soc., 279 (1983), 397-407.  doi: 10.1090/S0002-9947-1983-0704623-6.  Google Scholar

[16]

A. P. Sutton and R. W. Balluffi, Interfaces in Crystalline Materials Clarendon Press, Oxford, 1995. Google Scholar

[17]

A. P. Sutton and V. Vitek, On the structure of tilt grain boundaries in cubic metals Ⅰ. Symmetical tilt boundaries, Philos. Trans. Roy. Soc. Lond. A, 309 (1983), 1-36.  doi: 10.1098/rsta.1983.0020.  Google Scholar

[18]

Y. Xiang, Modeling dislocations at different scales, Commun. Comput. Phys., 1 (2006), 383-424.   Google Scholar

[19]

Y. Xiang, L. T. Cheng, D. J. Srolovitz and W. E, A level set method for dislocation dynamics, Acta Mater., 51 (2003), 5499-5518.  doi: 10.1016/S1359-6454(03)00415-4.  Google Scholar

[20]

L. C. Zhang, Y. J. Gu and Y. Xiang, Energy of low angle grain boundaries based on continuum dislocation structure, Acta Mater., 126 (2017), 11-24.  doi: 10.1016/j.actamat.2016.12.035.  Google Scholar

[21]

X. H. Zhu and Y. Xiang, Stabilizing force on perturbed grainboundaries using dislocation model, Scripta Mater., 64 (2011), 5-8.  doi: 10.1016/j.scriptamat.2010.08.050.  Google Scholar

[22]

X. H. Zhu and Y. Xiang, Continuum frmework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries, J. Mech. Phys. Solid, 69 (2014), 175-194.  doi: 10.1016/j.jmps.2014.05.005.  Google Scholar

Figure 1.  (a) Schematic illustration of a twist boundary. (b) Dislocation network consisting of two orthogonally intersecting screw dislocation arrays on a low angle twist boundary. For this twist boundary, across each horizontal dislocation (with Burgers vector $\mathbf b_1$) in the upper grain, the atoms above it shift to the left by $b_1/2$ and those below it shift to the right by $b_1/2$ with respect to the lower grain; across each vertical dislocation (with Burgers vector $\mathbf b_2$) in the upper grain, the atoms on its left shift downward by $b_2/2$ and those on its right shift upward by $b_2/2$ with respect to the lower grain. The net effect is making the upper grain rotate in the counterclockwise direction with respect to the lower grain
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