On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.

[1]  A. Needlemana,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2002 .

[2]  Amit Acharya,et al.  Grain-size effect in viscoplastic polycrystals at moderate strains , 2000 .

[3]  Magdalena Ortiz,et al.  A micromechanical model of hardening, rate sensitivity and thermal softening in BCC single crystals , 2001, cond-mat/0103284.

[4]  J. Nye.  Some geometrical relations in dislocated crystals , 1953 .

[5]  L. Anand,et al.  Polycrystalline plasticity and the evolution of crystallographic texture in FCC metals , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[6]  Meijie Tang,et al.  Simulations on the growth of dislocation density during Stage 0 deformation in BCC metals , 2003 .

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

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

[9]  Y. Milman,et al.  Microindentations on W and Mo oriented single crystals: An STM study , 1993 .

[10]  Michael Zaiser,et al.  Spatial Correlations and Higher-Order Gradient Terms in a Continuum Description of Dislocation Dynamics , 2003 .

[11]  Elias C. Aifantis,et al.  On the formation and stability of dislocation patterns—III: Three-dimensional considerations , 1985 .

[12]  Huajian Gao,et al.  Indentation size effects in crystalline materials: A law for strain gradient plasticity , 1998 .

[13]  Morton E. Gurtin,et al.  A comparison of nonlocal continuum and discrete dislocation plasticity predictions , 2003 .

[14]  E. B. Marin,et al.  On modelling the elasto-viscoplastic response of metals using polycrystal plasticity , 1998 .

[15]  M. Ashby.  The deformation of plastically non-homogeneous materials , 1970 .

[16]  Paul Steinmann,et al.  Views on multiplicative elastoplasticity and the continuum theory of dislocations , 1996 .

[17]  Morton E. Gurtin,et al.  A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations , 2002 .

[18]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[19]  S. Nemat-Nasser,et al.  A constitutive model for fcc crystals with application to polycrystalline OFHC copper , 1998 .

[20]  R. J.,et al.  I Strain Localization in Ductile Single Crystals , 1977 .

[21]  James R. Rice,et al.  Strain localization in ductile single crystals , 1977 .

[22]  R. Bullough,et al.  Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[23]  Maurice de Koning,et al.  Modelling grain-boundary resistance in intergranular dislocation slip transmission , 2002 .

[24]  D. Parks,et al.  Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density , 1999 .

[25]  Norman A. Fleck,et al.  The role of strain gradients in the grain size effect for polycrystals , 1996 .

[26]  E. Aifantis,et al.  PLASTIC INSTABILITIES, DISLOCATION PATTERNS AND NONEQUILIBRIUM PHENOMENA. , 1988 .

[27]  D. A. Hughes,et al.  Scaling of the spacing of deformation induced dislocation boundaries , 2000 .

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

[29]  M. Ashby,et al.  Strain gradient plasticity: Theory and experiment , 1994 .

[30]  Lucia Nicola,et al.  Discrete dislocation analysis of size effects in thin films , 2003 .

[31]  Michael Ortiz,et al.  Computational modelling of single crystals , 1993 .

[32]  J. Stolken,et al.  Orientation imaging microscopy investigation of the compression deformation of a [011] ta single crystal , 1999 .

[33]  Huajian Gao,et al.  Mechanism-based strain gradient plasticity— I. Theory , 1999 .

[34]  Discrete Dislocation Plasticity , 2003 .

[35]  W. King,et al.  Observations of lattice curvature near the interface of a deformed aluminium bicrystal , 2000 .

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

[37]  Y. Estrin,et al.  The analysis of shear banding with a dislocation based gradient plasticity model , 2000 .

[38]  N. Fleck,et al.  Strain gradient plasticity , 1997 .

[39]  Anthony G. Evans,et al.  A microbend test method for measuring the plasticity length scale , 1998 .

[40]  Alberto M. Cuitiño,et al.  Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations , 2001 .

[41]  D. Clarke,et al.  Size dependent hardness of silver single crystals , 1995 .

[42]  Amit Acharya,et al.  Lattice incompatibility and a gradient theory of crystal plasticity , 2000 .

[43]  Michael Ortiz,et al.  Plastic yielding as a phase transition , 1999 .

[44]  Norman A. Fleck,et al.  Strain gradient crystal plasticity: size-dependentdeformation of bicrystals , 1999 .

[45]  E. Aifantis,et al.  Dislocation patterning in fatigued metals as a result of dynamical instabilities , 1985 .

[46]  Amit Acharya,et al.  A model of crystal plasticity based on the theory of continuously distributed dislocations , 2001 .

[47]  U. F. Kocks,et al.  A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable , 1988 .

[48]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[49]  M. Gurtin,et al.  On the characterization of geometrically necessary dislocations in finite plasticity , 2001 .

[50]  Paul Steinmann,et al.  On the continuum formulation of higher gradient plasticity for single and polycrystals , 2000 .