Abstract
The shape of a mechanically equilibrated dislocation line is of considerable interest in the study of plastic deformation of metals and alloys. A general numerical method for finding such configurations in arbitrary stress fields has been developed. Analogous to the finite-element method (FEM), a general dislocation line is approximated by a series of straight segments (elements) bounded by nodes. The equilibrium configuration is found by minimizing the system energy with respect to nodal positions using a Newton-Raphson procedure. This approach, termed the finite-segment method (FSM), confers several advantages relative to segment-based, explicit formulations. The utility, generality, and robustness of the FSM is demonstrated by analyzing the Orowan bypass mechanism and a model of dislocation generation and equilibration at misfitting particles. Energy differences from previous analytical methods based on simple loop shapes are significant, up to 80 pct. Explicit expressions for the coordinate transformations, energies, and forces required for numerical implementation are presented.
Similar content being viewed by others
References
G. DeWit and J.S. Koehler: Phys. Rev., 1959, vol. 116, p. 1113.
J.P. Hirth and J. Lothe: Theory of Dislocations, 2nd ed., John Wiley & Sons, Inc., New York, NY, 1982.
R.O. Scattergood and D.J. Bacon: Phil. Mag., 1975, vol. 31, pp. 179–98.
L.M. Brown: Phil. Mag., 1964, vol. 10, pp. 441–66.
M.O. Peach and J.S. Koehler: Phys. Rev., 1950, vol. 80, p. 436.
M.F. Ashby and L. Johnson: Phil. Mag., 1969, vol. 18, pp. 1009–22.
X.J. Xin, R.H. Wagoner, and G.S. Daehn: Acta Mater., 1998, vol. 46, pp. 6131–44.
D.J. Bacon: Phys. Status Solidi, 1967, vol. 23, p. 527.
A.J.E. Foreman: Phil. Mag., 1967, vol. 15, pp. 1011–21.
A.J.E. Foreman, P.B. Hirsch, and F.J. Humphreys: Proc. Conf. on Fundamental Aspects of Dislocation Theory, National Bureau of Standards Publication, National Bureau of Standards, Gaithersburg, MD, 1970, vol. 317, p. 1083.
D.J. Bacon, U.F. Kocks, and R.J. Scatergood: Phil. Mag., 1973, vol. 28, pp. 1241–63.
L.A. Gore: Ph.D. Thesis, Stanford University, Stanford, CA, 1981.
D.T. Knight and B. Burton: Phil. Mag. A, 1989, vol. 59, pp. 1027–44.
D.T. Knight and B. Burton: Scripta Metall., 1989, vol. 23, pp. 429–34.
M.S. Duesbery and K. Sadananda: Phil. Mag. A, 1991, vol. 63, pp. 535–58.
K.W. Schwarz and J. Tersoff: Appl. Phys. Lett., 1996, vol. 69, pp. 1220–22.
K.W. Schwarz: Appl. Phys. Lett., 1997, vol. 78, pp. 4785–88.
K.W. Schwarz and F.K. LeGoues: Appl. Phys. Lett., 1997, vol. 78, pp. 1877–80.
H.M. Zbib, M.R. Rhee, and J.P. Hirth: Int. J. Mech. Sci., 1998, vol. 40, pp. 113–27.
L.P. Kubin, G. Canova, M. Conat, B. Devincre, V. Pontikis, and Y. Brechet: Solid State Phenomena, 1992, vols. 23–24, p. 455.
B. Devincre and M. Condat: Acta Metall. Mater., 1992, vol. 40, pp. 2629–37.
A. Moulin, M. Condat, and L.P. Kubin: Acta Metall. Mater., 1997, vol. 45, pp. 2339–48.
B.A. Bilby, R. Bullough, and E. Smith: Proc. R. Soc., 1955, vol. A231, p. 263.
Y.T. Keum, E. Nakamachi, R.H. Wagoner, and J.K. Lee: Int. J. Num. Meth. Eng., 1990, vol. 30, pp. 1471–1502.
D. Zhou and R.H. Wagoner: Int. J. Mech. Sci., 1997, vol. 39, pp. 1363–84.
D.J. Bacon, D.M. Barnett, and R.O. Scattergood: Progr. Mater. Sci., 1979, vol. 23, p. 51.
M.F. Ashby, S.H. Gelles, and L.E. Tanner: Phil. Mag., 1969, vol. 18, p. 757.
Ching-Yao Huang and Glenn S. Daehn: Acta Mater., 1996, vol. 44, p. 1035.
Ching-Yao Huang and Glenn S. Daehn: Acta Mater., 1997, vol. 45, p. 4283.
Ching-Yao Huang and Glenn S. Daehn: Net Shape Processing of Powder Materials, S. Krishnaswami, R.M. McMeeking, and J.R.L. Trasorras, eds., ASME, Fairfield, NJ, 1995, AMD-vol. 216, p. 69.
J.R. Rice and R. Thomson: Phil. Mag., 1974, vol. 29, p. 73.
S.J. Zhou and R. Thomson: J. Mater. Res., 1991, vol. 6, p. 639.
X.J. Xin, R.H. Wagoner, and G.S. Daehn: Scripta Mater., 1998, vol. 39, pp. 397–407.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Xin, X.J., Wagoner, R.H. & Daehn, G.S. A general numerical method to solve for dislocation configurations. Metall Mater Trans A 30, 2073–2087 (1999). https://doi.org/10.1007/s11661-999-0018-8
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11661-999-0018-8