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Field Dislocation Mechanics

The objective of this work is to numerically study dislocation interaction with mesoscale microstructures and, together with input from experimental observations, develop constitutive models suitable for use in continuum plasticity theories.

Background

The Field Dislocation Mechanics (FDM) theory developed by Acharya§ is used for this purpose.  The motivation behind this theory is that dislocations create a stress field that, along with the stress field due to applied boundary conditions, drives their motion.  The governing equations are
  1. Incompatibility equation, which relates dislocation density with lattice distortion
  2. Equilibrium equation, which is used to obtain the displacement and stress fields
  3. Dislocation velocity equation, which is specified as a function of stress through a constitutive statement.
  4. Dislocation evolution equation, which determines the motion of dislocations
  5. Slip evolution equation, which describes plastic strain generation due to slip
This system is solved numerically using the Finite Element Method (FEM): Least-squares FEM for the incompatibility equation, Galerkin FEM for the equilibrium equation, and Galerkin-Least-Squares FEM for the dislocation evolution equation.  The other equations are ordinary differential equations that are integrated explicitly.

A 3-D parallel code based on the Message Passing Interface (MPI) standard and the PetSc solver library is being developed to implement this FDM theory.

Some Results

A range of problems have been solved both to validate the code and explore the FDM theory.

1.  Single Edge Dislocation

The stress field of a single edge dislocation is determined using the imcompatibility and equilibrium equations.   The edge dislocation is  represented by dislocation density over a small region. Figures (1) and (2) show the mesh and the stress field respectively, while Figure (3) compares the normal stress along the x-axis with the analytical solution.


Figure 1: Single Edge Dislocation (Mesh)

Figure 2:  Single Edge Dislocation (Stress Field)

Figure 3:  Single Edge Dislocation (Comparison with Analytical Solution)

2.  Zero Stress Equivalent Distribution

The stress field of a distribution of dislocations called a Zero Stress Equivalent (ZSE) arrangement is obtained using the incompatibilty and equilibrium equations.  ZSE distributions do not result in long range stress fields.  The mesh and the stress distribution are shown in Figures (4) and (5).   Screw dislocations on the front and rear faces merge into edge dislocations on the other four faces to form a grid.  It is clear that the stress is zero away from the grid.

Figure 4:  Finite Element Mesh for the ZSE Distribution



Figure 5:   Stress Field for the ZSE Distribution

3.  Dislocation Reaction

The dislocation evolution equation is used to model interaction between dislocations, with dislocation velocity specified as a function of the sign of dislocation density.  Each dislocation loop in Figure (6) is formed by positive edge density merging into  positive screw density, negative edge density, and negative screw density.  The two loops expand, the positive and negative edge densities of the two loops meet and annihilate each other, and a single loop results.  The evolution equation is capable of modeling dislocation annihilation, which is a short range interaction.



Figure 6:  Dislocation Loop Interaction

4.  Simple Shear

A prismatic domain containing positive and negative edge dislocation densities is subjected to simple shear.  The dislocation velocity is specified as a function of the stress and sign of dislocation density.  The dislocation density and shear stress evolution are shown in Figures (7) and (8).  Note the formation of the slipsteps where the dislocation exit the domain.  Figure (9) shows the evolution of the volume-averaged plastic strain.



Figure 7:  Dislocation Evolution Under Applied Shear



Figure 8:  Shear Stress Evolution Under Applied Shear


Figure 9:  Evolution of Plastic Shear Strain Under Applied Shear

Ongoing Work

  • Study dislocation interaction with grain boundaries
  • Dynamics of dislocation-solute interaction (Portevin-Le Chatelier effect)
  • Extension of FDM to include crystal plasticity for statistically-stored dislocations

§ Acharya A., "A model of crystal plasticity based on the theory of continuously distributed dislocations," JMPS, 49, 2001, 761-784.