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
- Incompatibility equation, which relates dislocation
density with lattice distortion
- Equilibrium equation, which is used to obtain the
displacement and stress fields
- Dislocation velocity equation, which is specified as
a function of stress through a constitutive statement.
- Dislocation evolution equation, which determines the
motion of dislocations
- 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.
|