A non-singular continuum theory of dislocations

Abstract 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.

[1]  Nasr M. Ghoniem,et al.  Fast-sum method for the elastic field of three-dimensional dislocation ensembles , 1999 .

[2]  K. Schwarz,et al.  Simulation of dislocations on the mesoscopic scale. I. Methods and examples , 1999 .

[3]  Hussein M. Zbib,et al.  On plastic deformation and the dynamics of 3D dislocations , 1998 .

[4]  J Li,et al.  Anisotropic elastic interactions of a periodic dislocation array. , 2001, Physical review letters.

[5]  J. Bogdanoff,et al.  On the Theory of Dislocations , 1950 .

[6]  K. Schwarz,et al.  Simulation of dislocations on the mesoscopic scale. II. Application to strained-layer relaxation , 1999 .

[7]  J. Lothe,et al.  Elastic Strain Fields and Dislocation Mobility , 1992 .

[8]  Wei Cai,et al.  Atomistic and mesoscale modeling of dislocation mobility , 2001 .

[9]  F. Nabarro.  Dislocations in a simple cubic lattice , 1947 .

[10]  Elias C. Aifantis,et al.  Edge dislocation in gradient elasticity , 1997 .

[11]  L. M. Brown.  The self-stress of dislocations and the shape of extended nodes , 1964 .

[12]  B. Fedelich.  The glide force on a dislocation in finite elasticity , 2004 .

[13]  Sidney Yip,et al.  Periodic image effects in dislocation modelling , 2003 .

[14]  R. de Wit,et al.  Some Relations for Straight Dislocations , 1967 .

[15]  Ladislas P. Kubin,et al.  Mesoscopic simulations of dislocations and plasticity , 1997 .

[16]  Sidney Yip,et al.  Molecular dynamics simulations of motion of edge and screw dislocations in a metal , 2002 .

[17]  F. Nabarro,et al.  Dislocations in solids , 1979 .

[18]  David M. Barnett,et al.  The self-force on a planar dislocation loop in an anisotropic linear-elastic medium , 1976 .

[19]  J. Hirth,et al.  Dislocation core parameters , 2005 .

[20]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[21]  R. Wiť.  The Self‐Energy of Dislocation Configurations Made up of Straight Segments , 1967 .

[22]  D. Barnett.  The singular nature of the self-stress field of a plane dislocation loop in an anisotropic elastic meidum , 1976 .

[23]  R. Peierls.  The size of a dislocation , 1940 .

[24]  G. Gladwell,et al.  Solid mechanics and its applications , 1990 .

[25]  R. Wiť.  The Continuum Theory of Stationary Dislocations , 1960 .

[26]  Benoit Devincre,et al.  Three dimensional stress field expressions for straight dislocation segments , 1995 .

[27]  Sidney Yip,et al.  Chapter 64 – Dislocation Core Effects on Mobility , 2004 .

[28]  Wei Cai,et al.  Scalable Line Dynamics in ParaDiS , 2004, Proceedings of the ACM/IEEE SC2004 Conference.

[29]  W. Cai,et al.  Massively-Parallel Dislocation Dynamics Simulations , 2004 .

[30]  Maurice de Koning,et al.  Anomalous dislocation multiplication in FCC metals. , 2003, Physical review letters.