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A Numerical Scheme for Generalized Peierls-Nabarro Model of Dislocations Based on the Fast Multipole Method and Iterative Grid Redistribution

Part of: Equilibrium (steady-state) problems Integral equations, integral transforms Partial differential equations, boundary value problems Generalities, axiomatics, foundations of continuum mechanics of solids

Published online by Cambridge University Press:  23 November 2015

Aiyu Zhu ,
Congming Jin ,
Degang Zhao ,
Yang Xiang  and
Jingfang Huang
Aiyu Zhu
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Congming Jin
Affiliation:
Department of Mathematics, Zhejiang Sci-Tech University, Zhejiang, China
Degang Zhao
Affiliation:
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
Yang Xiang*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Jingfang Huang
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA.
*
*Corresponding author. Email addresses:maayzhu@ust.hk (A. Zhu), jincm@lsec.cc.ac.cn (C. Jin), dgzhao@hust.edu.cn (D. Zhao), maxiang@ust.hk (Y. Xiang), huang@email.unc.edu (J. Huang)
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Abstract

Dislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

Keywords

DislocationelasticityPeierls-Nabarro modeliterative grid redistributionfast multipole method

MSC classification

Secondary: 65R20: Integral equations 65N50: Mesh generation and refinement 74A50: Structured surfaces and interfaces, coexistent phases 74G65: Energy minimization
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1282 - 1312
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Hirth, J.P., Lothe, J., Theory of Dislocations, 2nd ed., John Wiley, New York, 1982.Google Scholar
[2]Peierls, R.E., The size of a dislocation, Proc. Phys. Soc. 52 (1940) 34–37.CrossRefGoogle Scholar
[3]Nabarro, F.R.N., Dislocations in a simple cubic lattice, Proc. Phys. Soc. 59 (1947) 256–272.CrossRefGoogle Scholar
[4]Vitek, V., Intrinsic stacking faults in body-centered cubic crystals, Phil. Mag. 18 (1968) 773–786.Google Scholar
[5]Kaxiras, E. and Duesbery, M. S., Free energies of generalized stacking faults in Si and implications for the brittle-ductile transition, Phys. Rev. Lett. 70 (1993) 3752–3755.CrossRefGoogle Scholar
[6]Payne, A. P., Nix, W. D., Lairson, B. M., and Clemens, B. M., Strain relaxation in ultrathin films: A modified theory of misfit-dislocation energetics, Phys. Rev. B 47 (1993), 13730–13736.Google Scholar
[7]Beltz, G. E. and Freund, L. B., Analysis of the strained-layer critical thickness concept based on a Peierls-Nabarro model of a threading dislocation, Phil. Mag. A 69 (1994) 183–202.CrossRefGoogle Scholar
[8]Schoeck, G., The generaliezd Peierls-Nabarro model, Phil. Mag. A 69 (1994) 1085–1095.CrossRefGoogle Scholar
[9]Joos, B. and Duesbery, M. S., The Peierls stress of dislocations: An analytic formula, Phys. Rev. Lett. 78 (1997) 266–269.CrossRefGoogle Scholar
[10]Bulatov, V. V. and Kaxiras, E., Semi-discrete variational Peierls framework for dislocation core properties, Phys. Rev. Lett. 78 (1997) 4221–4223.Google Scholar
[11]Hartford, J., von Sydow, B., Wahnstrom, G. and Lundqvist, B. I., Peierls barriers and stresses for edge dislocations in Pd and Al calculated from first principles, Phys. Rev. B 58 (1998) 2487–2496.Google Scholar
[12]Mryasov, O. N., Gornostyrev, Y. N. and Freeman, A. J., Generalized stacking-fault energetics and dislocation properties: Compact versus spread unit-dislocation structures in TiAl and CuAu, Phys. Rev. B 58 (1998) 11927–11932.Google Scholar
[13]Movchan, A. B., Bullough, R. and Willis, J. R., Stability of a dislocation: Discrete model, Eur. J. Appl. Math. 9 (1998) 373–396.Google Scholar
[14]Schoeck, G., The Peierls energy revisited, Phil. Mag. A 79 (1999) 2629–2636.CrossRefGoogle Scholar
[15]Lu, G., Kioussis, N., Bulatov, V. V. and Kaxiras, E., Generalized-stacking-fault energy surface and dislocation properties of aluminum, Phys. Rev. B 62 (2000) 3099–3108.Google Scholar
[16]Xu, G. and Argon, A. S., Homogeneous nucleation of dislocation loops under stress in perfect crystals, Phil. Mag. Lett. 80 (2000) 605–611.CrossRefGoogle Scholar
[17]Lu, G., Bulatov, V. V. and Kioussis, N., Dislocation constriction and cross-slip: an ab initio study, Phys. Rev. B 66, (2002) 144103.Google Scholar
[18]Movchan, A. B., Bullough, R. and Willis, J. R., Two-dimensional lattice models of the Peierls type, Phil. Mag. 83 (2003) 569–587.Google Scholar
[19]Schoeck, G. and Krystian, M., The peierls energy and kink energy in fcc metals, Phil. Mag. 85 (2005) 949–966.Google Scholar
[20]Xiang, Y., Modeling dislocations at different scales, Commun. Comput. Phys. 1 (2006) 383–424.Google Scholar
[21]Xiang, Y., Wei, H., Ming, P. B. and W. E., , A generalized Peierls-Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu, Acta Mater. 56 (2008) 1447–1460.Google Scholar
[22]Wei, H., Xiang, Y. and Ming, P. B., A generalized Peierls-Nabarro model for curved dislocations using discrete Fourier transform, Commun. Comput. Phys. 4 (2008) 275–293.Google Scholar
[23]Wei, H. and Xiang, Y., A generalized Peierls-Nabarro model for kinked dislocations Phil. Mag. 89 (2009) 2333–2354.Google Scholar
[24]Dai, S. Y., Xiang, Y., and Zhang, T. Y., A continuum model for core relaxation of incoherent twin boundaries based on the Peierls-Nabarro framework, Scripta Mater. 64 (2011), 438–441.Google Scholar
[25]Schoeck, G., The core structure and Peierls potential of dislocations in Al, Mater. Sci. Eng. A 558 (2012), 162–169.CrossRefGoogle Scholar
[26]Kamimura, Y., Edagawa, K., and Takeuchi, S., Experimental evaluation of the Peierls stresses in a variety of crystals and their relation to the crystal structure, Acta Mater. 61 (2013), 294–309.Google Scholar
[27]Dai, S. Y., Xiang, Y., and Srolovitz, D. J., Structure and energy of (111) low angle twist boundaries in Al, Cu and Ni, Acta Mater. 61 (2013), 1327–1337.Google Scholar
[28]Dai, S. Y., Xiang, Y., and Srolovitz, D. J., Atomistic, generalized Peierls-Nabarro, and analytical models for (111) twist boundaries in Al, Cu and Ni for all twist angles, Acta Mater., 69 (2014), 162–174.Google Scholar
[29]Shen, C., Li, J., and Wang, Y., Microscopic phase field model for dislocation core and grain boundary, preprint, 2013.Google Scholar
[30]Xu, G. and Ortiz, M., A variational boundary integral method for the analysis of 3D cracks of arbitrary geometry modeled as continuous distributions of dislocation loops, Int.J. Numer. Meth. Engng. 36 (1993) 3675–3701.Google Scholar
[31]Nye, J. F., Some geometrical relations in dislocated crystals, Acta Metall. 1 (1953), 153–162.Google Scholar
[32]Zimmerman, J. A., Gao, H., and Abraham, F. F., Generalized stacking fault energies for embedded atom FCC metals, Modell. Simul. Mater. Sci. Eng. 8 (2000), 103–115.Google Scholar
[33]Greengard, L. and Rokhlin, V., A new version of the Fast Multipole Method for the Laplace equation in three dimensions, Acta Numerica 6 (1997), 229–269.Google Scholar
[34]Zhao, D. G., Huang, J. F., and Xiang, Y., A new version fast multipole method for evaluating the stress field of dislocation ensembles, Modelling Simul. Mater. Sci. Eng. 18 (2010), 045006.Google Scholar
[35]Ren, W. and Wang, X. P., An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys. 159 (2000) 246–273.Google Scholar
[36]Winslow, A., Numerical solution of the quasi-linear Poisson equation, J. Comput. Phys. 1 (1966), 149–172.Google Scholar
[37]Van der Giessen, E. and Needleman, A., Discrete dislocation plasticity: a simple planar model, Model. Simul. Mater. Sci. Eng. 3 (1995), 689–735.CrossRefGoogle Scholar
[38]Quek, S. S., Xiang, Y., Zhang, Y. W., Srolovitz, D. J., and Lu, C., Level set simulation of dislocation dynamics in thin films, Acta Mater. 54 (2006), 2371–2381.CrossRefGoogle Scholar
[39]Zhao, D. G., Huang, J. F., and Xiang, Y., Fast multipole accelerated boundary integral equation method for evaluating the stress field associated with dislocations in a finite medium, Commun. Comput. Phys. 12 (2012), 226–246.Google Scholar
[40]Freund, L. B. and Barnett, D. M., A two-dimensional analysis of surface deformation due to dip-slip faulting, Bull. Seism. Soc. Am. 66 (1976), 667–675.Google Scholar
[41]Balluffi, R. W., Introduction to Elasticity Theory for Crystal Defects, Cambridge University Press, Cambridge, 2012.Google Scholar
[42]Lothe, J., The image force on dislocations at free surfaces – Comments on the concept of line tension. In: Fundamental aspects of dislocation theory, vol. 1. National Bureau of Standards (US) Special Publication 317, 1970. pp. 11–22.Google Scholar
[43]Lardner, R. W., Mathematical Theory of Dislocations and Fracture, University of Toronto Press, Toronto, 1974.Google Scholar