A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes

Abstract We present a self-consistent formulation of 3-D parametric dislocation dynamics (PDD) with the boundary element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. The error in the image force on a screw dislocation approaching a free surface is shown to increase as the dislocation approaches the surface, but is nevertheless controllable. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000 × 2000 (in units of lattice parameter), and 64 quadrature points. The Eshelby twist angle in a finite-length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Finally, large scale 3-D simulation results of single slip behavior in cylindrical microcrystals are presented. Plastic flow characteristics and the stress–strain behavior of cylindrical microcrystals under compression are shown to be in agreement with experimental observations. It is shown that the mean length of dislocations trapped at the surface is the dominant factor in determining the size effects on hardening of single crystals. The influence of surface image fields on the flow stress is finally explored. It is shown that the flow stress is reduced by as much as 20% for small single crystals of size less than 0.15 μ m .

[1]  J. D. Eshelby.  Screw Dislocations in Thin Rods , 1953 .

[2]  M Verdieryk.  Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications , 1998 .

[3]  Marc Fivel,et al.  Implementing image stresses in a 3D dislocation simulation , 1996 .

[4]  A. Needleman,et al.  Plasticity size effects in tension and compression of single crystals , 2005 .

[5]  Michael D. Uchic,et al.  Contribution to size effect of yield strength from the stochastics of dislocation source lengths in finite samples , 2007 .

[6]  Carlos Alberto Brebbia,et al.  The Boundary Element Method in Engineering Practice , 1984 .

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

[8]  D. Dimiduk,et al.  Sample Dimensions Influence Strength and Crystal Plasticity , 2004, Science.

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

[10]  Julia R. Greer,et al.  Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients , 2005 .

[11]  Ladislas P. Kubin,et al.  Dislocation Microstructures and Plastic Flow: A 3D Simulation , 1992 .

[12]  H. Espinosa,et al.  Dislocation escape-related size effects in single-crystal micropillars under uniaxial compression , 2007 .

[13]  Michael D. Uchic,et al.  Size-affected single-slip behavior of pure nickel microcrystals , 2005 .

[14]  R. V. Kuktat,et al.  Three-dimensional numerical simulation of interacting dislocations in a strained epitaxial surface layer , 1998 .

[15]  N. Ghoniem,et al.  The Influence of Crystal Surfaces on Dislocation Interactions in Mesoscopic Plasticity: A Combined Dislocation Dynamics- Finite Element Approach , 2002 .

[16]  Akiyuki Takahashi,et al.  A computational method for dislocation-precipitate interaction , 2008 .

[17]  van der Erik Giessen,et al.  Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics , 2002 .

[18]  A. Benzerga,et al.  Scale dependence of mechanical properties of single crystals under uniform deformation , 2006 .

[19]  C. A. Volkert,et al.  Size effects in the deformation of sub-micron Au columns , 2006 .

[20]  J. Lothe,et al.  Elastic Field and Self-Force of Dislocations Emerging at the Free Surfaces of an Anisotropic Halfspace , 1982 .

[21]  Jens Lothe John Price Hirth,et al.  Theory of Dislocations , 1968 .

[22]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[23]  R. Pascual,et al.  Low amplitude fatigue of copper single crystals—I. The role of the surface in fatigue failure , 1983 .

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

[25]  H. Zbib,et al.  The treatment of traction-free boundary condition in three-dimensional dislocation dynamics using generalized image stress analysis , 2001 .

[26]  J. Willis,et al.  A line-integral representation for the stresses due to an arbitrary dislocation in an isotropic half-space , 1994 .

[27]  Michael D. Uchic,et al.  A methodology to investigate size scale effects in crystalline plasticity using uniaxial compression testing , 2005 .

[28]  van der Erik Giessen,et al.  Discrete dislocation plasticity: a simple planar model , 1995 .

[29]  Peter Gumbsch,et al.  Dislocation sources in discrete dislocation simulations of thin-film plasticity and the Hall-Petch relation , 2001 .

[30]  Oliver Kraft,et al.  Interface controlled plasticity in metals: dispersion hardening and thin film deformation , 2001 .

[31]  H. Mughrabi.  Introduction to the viewpoint set on : surface effects in cyclic deformation and fatigue , 1992 .

[32]  Nasr M. Ghoniem,et al.  Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation , 2000 .