A non-singular continuum theory of dislocations. (English) Zbl 1120.74329

Summary: We develop a non-singular, self-consistent framework for computing the stress field and the total elastic energy of a general dislocation microstructure. The expressions are self-consistent in that the driving force defined as the negative derivative of the total energy with respect to the dislocation position, is equal to the force produced by stress, through the Peach-Koehler formula. The singularity intrinsic to the classical continuum theory is removed here by spreading the Burgers vector isotropically about every point on the dislocation line using a spreading function characterized by a single parameter \(a\), the spreading radius. A particular form of the spreading function chosen here leads to simple analytic formulations for stress produced by straight dislocation segments, segment self and interaction energies, and forces on the segments. For any value \(a > 0\), the total energy and the stress remain finite everywhere, including on the dislocation lines themselves. Furthermore, the well-known singular expressions are recovered for \(a = 0\). The value of the spreading radius a can be selected for numerical convenience, to reduce the stiffness of the dislocation equations of motion. Alternatively, a can be chosen to match the atomistic and continuum energies of dislocation configurations.


74A60 Micromechanical theories
Full Text: DOI


[1] Blin, J., Acta metall., 3, 199, (1995)
[2] Brown, L.M., The self-stress of dislocations and the shape of extended nodes. philos. mag., 10, 441, (1964) · Zbl 0125.25701
[3] Bulatov, V.V., Cai, W., Fier, J., Hiratani, M., Pierce, T., Tang, M., Rhee, M., Yates, K., Arsenlis, A., 2004. Scalable line dynamics of ParaDiS. SuperComputing 19. \(<\)http://www.sc-conference.org/sc2004/schedule/pdfs/pap206.pdf>.
[4] Cai, W., 2001. Atomistic and mesoscale modeling of dislocation mobility. Ph.D. Thesis, Massachusetts Institute of Technology.
[5] Cai, W., Arsenlis, A., 2005. MATLAB codes that implement the non-singular stress, energy and force expressions. Available from: \(<\)http://micro.stanford.edu/\(\sim\)caiwei/Forum/2005-04-18-NSDD/>.
[6] Cai, W.; Bulatov, V.V.; Chang, J.; Li, J.; Yip, S., Anisotropic elastic interactions of a periodic dislocation array, Phys. rev. lett., 86, 5727, (2001)
[7] Cai, W.; Bulatov, V.V.; Chang, J.; Li, J.; Yip, S., Periodic image effects in dislocation modelling, Philos. mag. A, 83, 539, (2003)
[8] Cai, W.; Bulatov, V.V.; Pierce, T.G.; Hiratani, M.; Rhee, M.; Bartelt, M.; Tang, M., Massively-parallel dislocation dynamics simulations, (), 1
[9] Cai, W.; Bulatov, V.V.; Chang, J.; Li, J.; Yip, S., Dislocation core effects on mobility, (), 1
[10] Devincre, B.; Kubin, L.P., Mesoscopic simulations of dislocations and plasticity, Mater. sci. eng. A, 8, 234-236, (1997)
[11] de Wit, R., Solid state phys., 10, 249, (1960)
[12] de Wit, R., Some relations for straight dislocations, Phys. status solidi, 20, 567, (1967)
[13] de Wit, R., The self-energy of dislocation configurations made up of straight segments, Phys. status solidi, 20, 575, (1967)
[14] Fedelich, B., The glide force on a dislocation in finite elasticity, J. mech. phys. solids, 52, 215, (2004) · Zbl 1074.74011
[15] Gavazza, S.D.; Barnett, D.M., The self-force on a planar dislocation loop in an anisotropic linear-elastic medium, J. mech. phys. solids, 24, 171, (1976) · Zbl 0368.73094
[16] Ghoniem, N.M.; Sun, L.Z., Fast-sum method for the elastic field of three-dimensional dislocation ensembles, Phys. rev. B, 60, 128, (1999)
[17] Gutkin, M.Y.; Aifantis, E.C., Screw dislocation in gradient elasticity, Scr. mater., 35, 1353-1358, (1996)
[18] Gutkin, M.Y.; Aifantis, E.C., Edge dislocation in gradient elasticity, Scr. mater., 36, 129-135, (1997)
[19] Hirth, J.P.; Lothe, J., Theory of dislocations, (1982), Wiley New York
[20] LeSar, R., Ambiguities in the calculation of dislocation self energies, Phys. status solidi B, 241, 2875, (2004)
[21] Lothe, J., 1992. In: Indenbohm, V.L., Lothe, J. (Eds.), Elastic Strain Fields and Dislocation Mobility. North-Holland, Amsterdam, p. 182.
[22] Lothe, J.; Hirth, J.P., Dislocation core parameters, Phys. status solidi B, 242, 836, (2005)
[23] Mura, T., Micromechanics of defects in solids, (1982), Kluwer Dordrecht
[24] Nabarro, F.R.N., Dislocations in a simple cubic lattice, Proc. phys. soc., 59, 256, (1947)
[25] Peierls, R.E., The size of a dislocation, Proc. phys. soc., 52, 34, (1940)
[26] Schwarz, K.W., Simulation of dislocations on the mesoscopic scale. I. methods and examples, J. appl. phys., 85, 108, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.