13 research outputs found
Mesoscale Theory of Grains and Cells: Polycrystals & Plasticity
Solids with spatial variations in the crystalline axes naturally evolve into cells or grains separated by sharp walls. Such variations are mathematically described using the Nye dislocation density tensor. At high temperatures, polycrystalline grains form from the melt and coarsen with time; the dislocations can both climb and glide. At low temperatures under shear the dislocations (which allow only glide) form into cell structures. While both the microscopic laws of dislocation motion and the macroscopic laws of coarsening and plastic deformation are well studied, we hitherto have had no simple, continuum explanation for the evolution of dislocations into sharp walls. We present here a mesoscale theory of dislocation motion. It provides a quantitative description of deformation and rotation, grounded in a microscopic order parameter field exhibiting the topologically conserved quantities. The
current of the Nye dislocation density tensor is derived from downhill motion driven by the microscopic Peach--Koehler forces between dislocations, making use of a simple closure approximation. The resulting theory is shown to form sharp dislocation walls in finite time mathematically described by Burgers equation---similar to those seen in the theory of sonic booms and traffic jams. Our finite-difference methods use special upwind and Fourier techniques for dealing with the shock formation. The outcomes of our simulations in one and two dimensions are in good agreement with experiment and other discrete dislocation simulations. These results provide fundamental insights into the basic phenomena of plastic deformation in crystals, and offers predictions for residual stress, cell-structure refinement, and the distinguish features of different hardening stages in plastic deformation.ITR/ASP ACI0085969
DMR-021847
Study of size effects in thin films by means of a crystal plasticity theory based on DiFT
In a recent publication, we derived the mesoscale continuum theory of
plasticity for multiple-slip systems of parallel edge dislocations, motivated
by the statistical-based nonlocal continuum crystal plasticity theory for
single-glide due to Yefimov et al. (2004b). In this dislocation field theory
(DiFT) the transport equations for both the total dislocation densities and
geometrically necessary dislocation densities on each slip system were obtained
from the Peach-Koehler interactions through both single and pair dislocation
correlations. The effect of pair correlation interactions manifested itself in
the form of a back stress in addition to the external shear and the
self-consistent internal stress. We here present the study of size effects in
single crystalline thin films with symmetric double slip using the novel
continuum theory. Two boundary value problems are analyzed: (1) stress
relaxation in thin films on substrates subject to thermal loading, and (2)
simple shear in constrained films. In these problems, earlier discrete
dislocation simulations had shown that size effects are born out of layers of
dislocations developing near constrained interfaces. These boundary layers
depend on slip orientations and applied loading but are insensitive to the film
thickness. We investigate stress response to changes in controlled parameters
in both problems. Comparisons with previous discrete dislocation simulations
are discussed.Comment: 20 pages, 11 figure
Growth instability due to lattice-induced topological currents in limited mobility epitaxial growth models
The energetically driven Ehrlich-Schwoebel (ES) barrier had been generally
accepted as the primary cause of the growth instability in the form of
quasi-regular mound-like structures observed on the surface of thin film grown
via molecular beam epitaxy (MBE) technique. Recently the second mechanism of
mound formation was proposed in terms of a topologically induced flux of
particles originating from the line tension of the step edges which form the
contour lines around a mound. Through large-scale simulations of MBE growth on
a variety of crystalline lattice planes using limited mobility, solid-on-solid
models introduced by Wolf-Villain and Das Sarma-Tamborenea in 2+1 dimensions,
we propose yet another type of topological uphill particle current which is
unique to some lattice, and has hitherto been overlooked in the literature.
Without ES barrier, our simulations produce spectacular mounds very similar, in
some cases, to what have been observed in many recent MBE experiments. On a
lattice where these currents cease to exist, the surface appears to be
scale-invariant, statistically rough as predicted by the conventional continuum
growth equation.Comment: 10 pages, 12 figure
Mesoscale theory of grains and cells: crystal plasticity and coarsening
Solids with spatial variations in the crystalline axes naturally evolve into
cells or grains separated by sharp walls. Such variations are mathematically
described using the Nye dislocation density tensor. At high temperatures,
polycrystalline grains form from the melt and coarsen with time: the
dislocations can both climb and glide. At low temperatures under shear the
dislocations (which allow only glide) form into cell structures. While both the
microscopic laws of dislocation motion and the macroscopic laws of coarsening
and plastic deformation are well studied, we hitherto have had no simple,
continuum explanation for the evolution of dislocations into sharp walls. We
present here a mesoscale theory of dislocation motion. It provides a
quantitative description of deformation and rotation, grounded in a microscopic
order parameter field exhibiting the topologically conserved quantities. The
topological current of the Nye dislocation density tensor is derived from a
microscopic theory of glide driven by Peach-Koehler forces between dislocations
using a simple closure approximation. The resulting theory is shown to form
sharp dislocation walls in finite time, both with and without dislocation
climb.Comment: 5 pages, 3 figure
Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor
We derive general relations between grain boundaries, rotational
deformations, and stress-free states for the mesoscale continuum Nye
dislocation density tensor. Dislocations generally are associated with
long-range stress fields. We provide the general form for dislocation density
fields whose stress fields vanish. We explain that a grain boundary (a
dislocation wall satisfying Frank's formula) has vanishing stress in the
continuum limit. We show that the general stress-free state can be written
explicitly as a (perhaps continuous) superposition of flat Frank walls. We show
that the stress-free states are also naturally interpreted as configurations
generated by a general spatially-dependent rotational deformation. Finally, we
propose a least-squares definition for the spatially-dependent rotation field
of a general (stressful) dislocation density field.Comment: 9 pages, 3 figure
Statistical approach to dislocation dynamics: From dislocation correlations to a multiple-slip continuum plasticity theory
Due to recent successes of a statistical-based nonlocal continuum crystal
plasticity theory for single-glide in explaining various aspects such as
dislocation patterning and size-dependent plasticity, several attempts have
been made to extend the theory to describe crystals with multiple slip systems
using ad-hoc assumptions. We present here a mesoscale continuum theory of
plasticity for multiple slip systems of parallel edge dislocations. We begin by
constructing the Bogolyubov-Born-Green-Yvon-Kirkwood (BBGYK) integral equations
relating different orders of dislocation correlation functions in a grand
canonical ensemble. Approximate pair correlation functions are obtained for
single-slip systems with two types of dislocations and, subsequently, for
general multiple-slip systems of both charges. The effect of the correlations
manifests itself in the form of an entropic force in addition to the external
stress and the self-consistent internal stress. Comparisons with a previous
multiple-slip theory based on phenomenological considerations shall be
discussed.Comment: 12 pages, 3 figure