ABSTRACTS
(In Order Presented)
Go to section:
Introductory
Lecture:
WORK HARDENING AND DISLOCATION PATTERNING:
INTRODUCTORY REMARKS
F. R. N. Nabarro
The ultimate aim of the theory is to predict
the behaviour of a complex mechanical structure from atomic parameters
alone. This involves a hierarchy of techniques which are advancing
in parallel: electron theory, molecular dynamics, dislocation patterning,
macroscopic stress analysis. We attempt an order-of-magnitude estimate
of the time scales involved.
At a more fundamental level, we enquire what determines
the sequence of dislocation patterns which form as work hardening proceeds,
e.g. Stages I, II, III, IV, V for a single crystal of copper. Why
do we not have similitude, so that, as work hardening proceeds, the dislocation
configuration remains self-similar, with all linear dimensions l inversely
proportional to the applied stress? Why does the configuration which
in the course of one stage is slowly shrinking give way rather suddenly
to the different configuration of the succeeding stage?
There are two basic possibilities:
One is that the configuration which has the lowest
energy at a certain stress or dislocation density is not the one of lowest
energy at a higher stress or dislocation density. While it is mathematically
convenient to consider changes of configuration at constant dislocation
density, the physical reality is that the changes occur at constant flow
stress. The change of energy arising from a change of configuration
at constant dislocation density involves three factors. First, there
may be a change from a high proportion of edge dislocations of high elastic
energy to a high proportion of screw dislocations with a low elastic energy.
Since the elastic energy increases with the stress screening length and
so with decreasing dislocation density, the change of energy per unit length
of dislocation when the configuration changes is, paradoxically, more important
at low dislocation densities. Secondly, the ratio of the screening
length to the mean separation of dislocations will be different in different
configurations. If there is no change in the edge/screw ratio, the
change of energy per unit length of dislocation when the configuration
changes is independent of the dislocation density. This is easily
seen: the force on the unit length of dislocation as it moves from
one configuration to another is inversely proportional to the scale of
the pattern, and distance moved is proportional to this scale. Thirdly,
the core energy per unit length is independent of dislocation density,
while the elastic energy per unit length decreases with increasing density.
A primitive model of a change of configuration involved by this effect
under conditions of constant flow stress is presented.
The other possibility is that the change of configuration
is determined by other factors. One is kinematic. The postulated
uniform shrinkage of the pattern is difficult to achieve by glide alone,
while a sequence of patterns may be easier. When gliding dislocations
pass through the densely tangled cell walls considered by Mughrabi, point
defects may be generated in sufficient numbers to cause edge dipoles to
annihilate by climb, so leading to the transition from this pattern to
one free of redundant dislocations. In an f.c.c. lattice, screw dislocations
are dissociated. The non-glide component of the shear stress on the
glide plane alters the separation of the partials. Where the internal
stress, which increases with the dislocation density, reduces this separation,
it assists cross slip, which can then spread along the whole length of
the dislocation segment.
I. Experiments
/ Observations:
DISLOCATION PATTERNING AND CELL FORMATION
IN FCC METALS: OFTEN OVERLOOKED FACTS
A. S. Argon
Massachusetts Institute of Technology
Cambridge MA 02139
Dislocation patterning occurs in somewhat different
forms throughout monotonic plastic flow in most metals of high symmetry,
in ranges where the lattice resistance to glide is a negligible fraction
of the overall glide resistance. It has been studied most widely
in Cu single crystals. It manifests itself, starting in Stage I by
the formation of kink walls, transforming into quasi-periodically placed
open braids during Stage II, and develops into well formed closed cells
upon the onset of dynamic recovery in Stage III, whereupon the cells undergo
a remarkable self similar reduction in size, inversely proportional to
the increasing plastic resistance. The dislocation content in the
braids and cell walls in overwhelmingly redundant.
From the earliest stages of deformation in Stage
I, initiation of dislocation aggregation requires anchoring sites.
These form where primary glide dislocations encounter and react with grown-in
dislocations to form sessile segments that serve to nucleate clusters.
In Stage II the dislocations in braids are anchored down by readily observable
short Lomer-Cottrell segments, the production of which is monotonic and
kinematically related to the secondary slip process associated with primary
shear strain. The result is the characteristic linear hardening of Stage
II. In Stage III the production of sessile segments with strain is
counteracted by dynamic recovery that must result in a systematic thermally
assisted rate of dismantling of sessile segments by encounters with the
mobile "flux carriers," resulting in spikes of "glide collapse" of substantial
densities of unpinned redundant density. While it is most likely
that the process of dislocation loss in dynamic recovery is similarly jerky
as that of static thermal recovery, its mechanistic details remain largely
unclear, but almost certainly is not keyed to cross-slip which is prevalent
even in Stage I.
Thus, the central question in patterning as well
as in strain hardening itself is less one of understanding the interactions
of the clustered dislocations with the mobile flux, but more one of understanding
why the clusters are where they are.
MECHANICAL BEHAVIOR OF Ta AND Ta-W ALLOYS
AND CORRELATED DISLOCATION STRUCTURES
D. H. Lassila
Lawrence Livermore National Laboratory
Livermore, CA 94550
C. L. Briant and C. Bull
Brown University, Division of Engineering
Providence, RI 02912
A wide range of mechanical testing
was performed on numerous annealed tantalum and tantalum-tungsten alloy
plate materials. These tests included; dynamic tensile testing, uniaxial
testing in compression of strain rates of 0.001/s and 3000/s. In
general the Ta-W alloys exhibited greater work-hardening behavior.
Extensive transmission electron microscopy study of annealed and deformed
materials indicated that the annealed samples had a significant dislocation
density, and at a given value of plastic strain this density increased
with increasing strain rate, increasing tungsten concentration, and decreasing
test temperature. The effect of W alloying additions on stress-strain
response and dislocation character is discussed.
INSTANTANEOUS DISLOCATION VELOCITIES AND
MOBILE DISLOCATIONS
J. M. Galligan and T. J. McKrell
University of Connecticut
Institute of Materials Science
Metallurgy and Materials Engineering
97 North Eagleville Road
Storrs, CT 06269-3136
Deformation involves the motion and multiplication of
dislocations. Understanding of this process requires instantaneous measurements
of how fast dislocations move and how many are moving at a given strain,
for a given deformation temperature and a given applied strain rate. Below
we describe methods of measuring instantaneously these variables; these
methods exploit the electronic properties of metals.
Measurements of the instantaneous dislocation velocity
are attained by utilizing the variation of the electron-dislocation interaction
with a magnetic field.(1) Spatially varying a magnetic field relative to
the slip vector of a dislocation results in a change in the electron-dislocation
interaction.(2) This is observed(3) in the tensile stress necessary to
deform a crystal; with this variation in stress occurring at an angle given
by (=Vi/Ve where ( is the angle that the magnetic field vector makes with
respect to the slip system, Vi is the instantaneous dislocation velocity
and Ve the Fermi velocity of the electrons. Measurement of ( gives Vi,
for a known value of Ve. These measurements, carried out in Zn and Fe single
crystals, will be discussed in terms of their variation with temperature,
strain, strain rate and magnetic field.
Measurement of the mobile dislocation density, again
an instantaneous measurement, is carried out as follows: When dislocations
move in the mixed state of a type II superconductor this results in a change
of magnetic flux, (f; this change in flux is proportional to the mobile
dislocation density.(4) By measuring (f as a function of various deformation
parameters, the relative mobile dislocation density is obtained as a function
of deformation.
More recently we have measured the "noise" associated
with the flux change attendant the movement of dislocations; these measurements
show that correlated dislocation motion occurs during deformation.(5) The
measurements, carried out as a function of plastic strain, demonstrate
how these correlations vary as a crystal starts to deform and as a function
of strain.
References:
1) a) T. Holstein, Phys. Rev., [2], 151, 187 (1966).
b) C. Ya Kravchenko, Sov. Phys. Solid State, 8,
740 (1966).
2) A. M. Grishin, E. A. Kaner and E. P. Feldman, Zh. Fur Theor. Exp.
Phys., 70, 1445 (1976).
3) C. S. Kim, T. J. Garosshen and J. M. Galligan, Scr. Met., 23, 1591,
1959 (1989).
4) C. S. Pang and J. M. Galligan, Phys. Rev. Lett., 43, 1595 (1979).
5) Ji-Fu Kung, H. Bao and J. M. Galligan, Scr. Mat., 34, 479 (1996).
SCALING OF DISLOCATION STRUCTURE EVOLUTION
D. A. Hughes
Center for Materials and Engineering Sciences
Sandia National Laboratories, Livermore, CA 94550 USA
An experimental approach is used to identify dislocation
microstructure relationships that reflect the evolution of deformation
structures with increasing strain but do not vary with material, deformation
and material properties. This experimental approach is proposed due to
the complexity and size of the evolving dislocation population during deformation.
Transmission electron microscopy is used to quantify large areas of the
deformation microstructure arising from grain subdivision by dislocation
boundaries. A scaling hypothesis is used in conjunction with the experimental
results to aid in the identification of these microstructural relationships.
This identification is intended to provide a window on the origins of the
dislocation structures and simplify microstructure model development.
In this context, the misorientation angle across
dislocation boundaries is presented as a microstructural parameter to describe
the plastic deformation. The misorientation angle is both related to important
macroscopic properties including strength and recrystallization and to
the fundamental process of deformation. Recently, it was reported that
the evolution of the misorientation angles associated with one type of
dislocation boundary, incidental dislocation boundaries (IDBs) in cold
rolled high purity Al is consistent with a dynamic scaling hypothesis.
Note that IDB is a more general term for what are typically called dislocation
cell boundaries. A second type of dislocation boundary, a geometrically
necessary boundary (GNB) is also a part of the deformation microstructure
and has its own scaling behavior.
To test the universality and evolutionary limits
of this scaling hypothesis, a new series of measurements is reported. This
series was chosen to investigate key factors that make large differences
in the motion of individual dislocations and in macroscopic mechanical
properties as well as span a broad class of fcc metals/alloys. A wide variation
in these factors is made within the limits for the formation of a dislocation
cell structure. In order to form dislocation cell structures, dislocations
must have some three-dimensional mobility. The chosen factors include stacking
fault energy (SFE), the presence of solute atoms, temperature, strain rate,
deformation mode and strain. Note that SFE, solute atoms, temperature and
strain rate all alter the dislocation dynamics. Deformation mode leads
to differences in slip system activity and hence to differences in the
Burgers vector population of the dislocations. Strain changes the evolutionary
stage of the microstructure. It is not expected a priori that the misorientation
angles should behave in a similar fashion for all of these different factors.
Surprisingly, the distributions for the small to large strain regimes for
aluminum, 304L stainless steel, nickel and copper (taken from the literature)
appear to be identical. Hence the distributions may be ``universal.'' These
results have large implications in the development of dislocation based
deformation models.
This work was supported in part by the U.S. DOE
Office of Basic Energy Sciences,
Division of Materials Sciences under contract no. DE-AC04-94AL85000.
IN SITU ULTRA-SMALL-ANGLE X-RAY
SCATTERING MEASUREMENTS OF DISLOCATION STRUCTURE EVOLUTION
L. E. Levine, G. G. Long, Robb Thomson (Emeritus)
Materials Science and Engineering Laboratory
National Institute of Standards and Technology
Gaithersburg, MD 20899
Most experimental measurements of dislocation structures
have been restricted to ex situ studies of surfaces or thin cross-sections.
Although such measurements have provided considerable information, they
also suffer from limitations. The extent of these limitations depends upon
the specific material system, but they include the large size of the 3D
dislocation structures relative to the sample thickness; an inability to
follow the rapid, inhomogeneous nature of the evolution process;
and the sensitivity of dislocation structures to destructive sample preparation
techniques. Thus, in situ measurements of dislocation structure
evolution would have great advantages over ex situ measurements.
However, since the formation of dislocation cells is a bulk process and
the cells usually extend over several micrometers in size, a minimum sample
thickness of approximately 50 micrometers is required for in situ
experiments. This requirement places a difficult restriction
on experiments. Small-angle scattering (SAS) techniques can provide quantitative
microstructural information from thick specimens; thus, the potential applicability
of SAS techniques for dislocation structure studies has long been recognized.
Over the past 47 years, many attempts have been
made to use SAS to study the microstructure of cold-worked metals.
These attempts met with little success due to the low scattering contrast
of dislocations, the strong angular dependence of the scattering, and problems
associated with avoiding other, much higher contrast, processes such as
accidental Bragg diffraction. In addition, the range of scattering
angle where the dislocation walls are visible is outside the range of most
SAS experiments.
Over the past year, we have developed a comprehensive
analytical theory of small-angle scattering from dislocation structures
and used this theory to design new experiments at NIST's materials science
beam line at the National Synchrotron Light Source. In situ
ultra-small-angle X-ray scattering experiments have been conducted on several
samples and a high sensitivity to the developing dislocation microstructure
has been demonstrated. These measurements were conducted on single-crystal,
high-purity aluminum samples deformed uniaxially in situ, reaching
strains up to 0.22; initial sample thicknesses ranged from 0.17 - 0.21
mm. All of the predictions of the theory have been confirmed, allowing
quantitative results to be obtained from the data. The experiments
have successfully probed positional correlations between dislocations,
measured the changing `diffuse width' of the walls, detected the presence
of dislocation dipoles in uniaxially deformed samples, examined the inhomogeneity
of the microstructure, and allowed measurements of dislocation structure
relaxation during room temperature annealing.
II. Atomic
Scale:
ATOMISTIC SIMULATIONS FOR MULTISCALE MODELING
IN BCC METALS*
John A. Moriarty, Wei Xu, Per Söderlind, Lin H.
Yang, James F. Belak, and Jing Zhu
Lawrence Livermore National Laboratory
Livermore, CA 94551
Multiscale modeling of plastic flow and other mechanical
properties in bcc transition metals requires an accurate atomistic description
of deformation and defect energetics as input into larger length scale
simulations such as 3D dislocation dynamics at the microscale. We
are using state-of-the-art electronic-structure and interatomic-potential
methods to study a wide range of fundamental properties of prototype bcc
metals, such as Mo and Ta, including elastic moduli, ideal shear strength,
the atomic structure and energetics of vacancies, dislocations, and grain
boundaries, and dislocation-dislocation interactions. Special emphasis
is being given to the case of tantalum, where mechanical behavior at both
ambient and extreme conditions is of interest. A comprehensive set
of ab initio electronic-structure calculations have been performed in the
0-10 Mbar pressure range in Ta [1] and used together with rigorous generalized
pseudopotential theory (GPT) to develop corresponding model-GPT (MGPT)
multi-ion interatomic potentials [2] suitable for realistic atomistic simulations.
Many-body angular forces, which are accounted for in the MGPT through explicit
three- and four-ion potentials, are generally important to the structural
and mechanical properties of bcc transition metals. In this regard,
selected grain boundary structures are being studied for comparison with
concurrent HREM measurements, as an additional validation test of the MGPT
potentials. With regard to dislocations, our initial studies on Mo
[3] have now been extended to Ta and generalized in scope. We have
now investigated the core structure, the gamma surfaces, and the Peierls
stress and related energetic barriers associated with the motion of <111>
screw dislocations on the primary {110} and {112} slip planes in both metals.
In addition, we are calculating kink-pair formation, migration, and activation
energies, including both their stress and orientation dependence.
These latter quantities control the low-temperature plasticity in bcc metals
and are essential input for microscale dislocation-dynamics simulations.
We are also studying dislocation-dislocation interactions in an attempt
to accurately model junction formation and breaking, which are fundamental
to the description of strain hardening at the microscale. In the
future, we hope to perform dynamic simulations of dislocation mobility
and dislocation-dislocation interactions using accelerated molecular dynamics
schemes on massively parallel computers. Advanced parallel versions
of our static dislocation simulation codes are also being developed.
[1] P. Söderlind and J. A. Moriarty, Phys. Rev. B (1998, in press).
[2] J. A. Moriarty, Phys. Rev. B 49, 12431 (1994) and 42, 1609 (1990).
[3] W. Xu and J. A. Moriarty, Phys. Rev. B 54, 6941 (1996) and Comput.
Mater. Sci. 9, 348 (1998).
*This work has been performed under the auspices of the U.S. Department
of Energy by the Lawrence Livermore National Laboratory under contract
number W-7405-ENG-48.
ATOMISTIC STUDIES OF DEFORMATION AND FRACTURE
MECHANISMS IN INTERMETALLICS
Diana Farkas
Department of Materials Science and Engineering
Virginia Polytechnic Institute
Blacksburg, VA 24061
These studies use interatomic
potentials and molecular level simulations to investigate the detailed
atomistic nature of the deformation and fracture mechanisms in intermetallic
alloys. The many body interatomic potentials (of the embedded atom
type) used for the calculations are obtained from the perfect lattice experimental
properties and from first principle quantum mechanical calculations.
Mixed potentials are created to describe the experimental thermodynamics
of the system. The structure of lattice defects are primarily responsible
for the observed mechanical behavior of these materials.
These interatomic potentials
were used to study the local atomistic configuration around dislocation
cores in intermetallic alloys, and their response to applied stress.
The cores were studied as they react to applied stress and start to move.
Large-scale simulations of cracked crystals allow the study of the fracture
processes in these alloys. Dislocation emission from the crack tip
and/or crack propagtion are observed, and correlated with the ductile/brittle
response.
THEORY/SIMULATION STUDIES OF DISLOCATION
MOTION AND INSTABILITIES
Robin L. B. Selinger
Department of Physics
Catholic University
Washington, DC 20064-0001
When modeled as a simple damped elastic string, a mobile
dislocation can be morphologically unstable under a positive gradient in
the driving force, that is, a driving force that increases in the direction
of dislocation motion. This fundamental instability plays a role in dislocation
pair annihilation (in bulk systems or in thin films) and in annihilation
at a free surface or at a crack tip. Closely related line defects
such as vortices in magnets are expected to have the same instability.
We report on our ongoing simulation studies of magnetic vortices to test
the quantitative predictions of the theory, and describe (so far unsuccessful)
efforts to test the theory in atomistic simulations of screw islocations
in Al, where the dynamics are complicated by a dislocation's anisotropic
line tension and by cross-slip..
Our analysis of the instability points to a characteristic
length scale and we speculate that it might play a role in work hardening.
We also draw a parallel between work-hardening and the formation of the
twist grainboundary phase in smectic liquid crystals, an equilibrium phase
in which a layered system forms a regular array of twist grain boundariescontaining
regularly spaced screw dislocations. We speculate that theselection
of cell size in work hardening of metals may arise through the emergence
of a similar but metastable phase.
ATOMIC MODELLING OF CRITICAL PROCESSES
IN THE NONLINEAR CRACK TIP ZONE OF CRYSTALLINE SOLIDS
R. G. Hoagland
School of Mech. and Matl's Eng'g
Washington State University
Pullman, WA 99164-2920
Because of the analytical intractabilities posed by
the nonlinear properties of the cores of crack tips and dislocations, elasticity
offers little help in describing the details of the energetics of crack
extension and dislocation emission from crack tips. However, this problem
is ideally suited to exploration by atomistic modelling. Atomistic modelling
offers a method for studying some of the key features of crack tip behavior
and the factors that distinguish a brittle material from an intrinsically
tough material. In this talk, we describe some of the nonlinear properties
within crack tip and dislocation cores that derive from such calculations.
We also examine the energetics of crack tip evolution and show that the
Peierls-like criterion posed by Rice et. al.[1,2] This criterion invokes
a critical emission force which then implies a critical stress intensity
to cause emission. By systematically changing the orientation of the crystal
relative to the crack coordinates the behaviors of atomic models of EAM
nickel are observed to change from emission to crack extension in a way
that is quantitatively consistent with the combined predictions of the
Rice et al. And Griffith criteria based on a constant emission force and
relevant surface enthalpies. Furthermore, using the value of emission force
that best represents single crystal results, the predictions are also found
to agree favorably with the behavior of a double-ended crack on a grain
boundary where one end emits dislocations while the other extends in a
brittle manner. Finally, we explore the emission of trailing dislocations
and find that the second and third dislocations require somewhat higher
stress intensities to emit but, because the K needed to extend the crack
is also increased, the emission of the first dislocation remains the critical
step. If time permits, the origins of the critical emission force will
be discussed and it will be shown that it may not, in general be adequately
defined solely in terms of the unstable stacking energy.
1. Rice, J. R. J. Mech. Phys. Solids, 40, (1992) p. 239.
2. Rice, J. R., Beltz, G. E., and Sun, Y., in Topics in Fracture and
Fatigue, ed. by Argon (Springer) (1992) p. 1.
This work was supported by the Div. of Materials Sci., Office of Basic
Energy Sciences, U. S. Dept of Energy through Grant DE-FG-6-87ER45287.
ATOMISTIC SIMULATIONS OF DISLOCATION INTERSECTION
PROCESS
- A STEP TO BRIDGE APPROACHES ACROSS
LENGTH SCALE
S.J. Zhou
Applied Theoretical & Computational Physics Division
Los Alamos National Laboratory, Los Alamos, NM 87545
Deformation of metals and intermetallics is closely
related to dislocations and their interactions. One of important
questions is how a dislocation moves through a dislocation forest, which
is one of theimportant mechanisms restricting the dislocation motion and
contributing to work hardening. In this talk, we will report our
study on the perpendicular intersection process of extended dislocation
in single-crystal copper with 3D molecular dynamics simulations with up
to 50 million atoms at very low temperature [1]:
The repulsive intersection process, which involves
three of the four possible {111} glide planes in the face-centered cubic
lattice, begins with junction formation, followed by unzipping, partial
dislocation bowing, cutting, and finally unit jog formation. The
critical stress estimated from the measured breaking angle, 70 degree,
is in good agreement with the value measured from MD simulations. We also
find that the critical stress for the attractive intersection is not very
different from that for the repulsive intersection. The mechanism of vacancy
formation due to the nonconservative motion of the jogged dissociated dislocation
will be discussed.
[1] S.J. Zhou, D.L. Preston, P.S. Lomdahl, and D.M. Beazley,
"Large-scale molecular dynamics simulations of dislocation
intersection in copper", Science, Vol. 279, Page 1525, 6 March
1998.
III. Dislocation
Dynamics Simulations:
MODELS FOR LONG/SHORT RANGE INTERACTIONS
AND CROSS SLIP IN 3D DISLOCATION SIMULATION OF BCC SINGLE CRYSTALS
H. M. Zbib, M. Rhee and J. P. Hirth
School of Mechanical and Materials Engineering
Pullman, WA 99164-2920
and
H. Huang and T. de la Rubia
Lawrence Livermore National Laboratory
P.O. Box 808, L-268, Livermore, CA 90224
Models and rules for short range interactions, cross
slip and long range interactions of dislocation segments for implementation
into a 3D dislocation dynamics (3DD) model are developed. Dislocation curves
of arbitrary shapes are discretized into sets of straight segments of mixed-dislocations.
Long range interactions are evaluated explicitly based on results from
the theory of dislocations. Models for short range interactions, including,
annihilation, formation of jogs, junctions, and dipoles are developed on
the basis of a "critical-force" criterion that captures the effect of the
local fields from surrounding dislocations. Also a model for the
cross slip mechanism is developed and coupled with a Monte-Carlo type analysis
to simulate the development of double cross-slip and composite slip. The
model is then used to simulate stage I (easy glide) stress-strain behavior
in bcc single crystals, illustrating the feasibility of the 3DD model in
predicting macroscopic properties such as flow stress and hardening, and
their dependence on microscopic parameters such as dislocation mobility,
dislocation structure, and pinning points.
PLASTIC DEFORMATION IN BCC (Ta) SINGLE
CRYSTALS: 3D DISLOCATION DYNAMICS SIMULATIONS
M. Tang
LLNL, P.O.Box 808, L-407, Livermore, CA 94550
B. Devincre and L. Kubin
LEM, CNRS-ONERA (OM), 29 Av. de la Division
Leclerc, BP 72, 92322 Chatillon Cedex, France
A 3D dislocation dynamics method
is developed for bcc single crystals to study the plastic deformation at
low temperatures. The method consists of using discretized screw and edge
segments and discretized time step. It's based on a method initially developed
for fcc crystals[1]. And is adapted and incorporated features characteristic
of bcc crystals[2]. The method focuses on identifying the key mechanisms
and critical parameters that control the plastic deformation. For bcc single
crystals, the screw dislocations have low mobility and control the deformation.
They move by thermally activated double kink mechanism, where the activation
enthalpy is an intrinsic function of the local resolved shear stress. The
yielding behavior depends strongly on both temperature and strain
rate. For strain hardening, junctions are considered as the main
mechanism. Since the dislocation microstructure in bcc crystals at low
temperatures mainly consists of long screw dislocations, the formation
of junctions not only modify the dislocation length, but also provide locally
additional friction stress against the dislocation motion.
Single crystal Tantalum
is used as a prototype. We will show results of yield stresses as a function
of both temperature and strain rates. The results will be compared directly
with experimental data. We will also show 2D modeling of dislocation cutting
through forest obtacles and to study the effect of forest density on the
strain hardening. Also, preliminary results of strain hardening obtained
from 3D simulations will be shown. Issues will be addressed regarding linking
with atomistic simulations and as a supplementary tool for experimental
measurements of single dislocation activation properties.
[1] L. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis and
Y. Brechet, Solid State Phenomena 23-24, 455 (1992).
[2] M. Tang, L. Kubin, and G. Canova, Acta Mater. , in press (1998).
RAPID EVALUATION OF ELASTIC INTERACTIONS
AMONG DISLOCATIONS
Yuhong Fu and Gregory J. Rodin
Texas Institute for Computational and Applied
Mathematics
The University of Texas at Austin, Austin,
TX 78712
e-mail: yuhong@ticam.utexas.edu
email: gjr@ticam.utexas.edu
This talk is concerned with fast
summation methods for dislocation dynamics. For large-scale simulations,
such methods are indispensable since they allow one co compute the long-range
elastic interactions in O(N) instead of O(N2) operations, where N is the
number of the dislocation segments. Specific topics include:
Introduction to fast summation methods.
Fast summation methods for dislocation dynamics in
isotropic media.
Performance and reliability of fast summation methods.
Representations of fundamental solutions in anisotropic
media.
HARDENING AND PLASTIC INSTABILITIES OF
IRRADIATED MATERIALS
Nasr M. Ghoniem
Mechanical and Aerospace Engineering Department
University of California, Los Angeles (UCLA)
Los Angeles, California 90095-1600
Tel.: (310) 825-4866
Fax: (310) 206-4830
e-mail: ghoniem@ucla.edu
Bachu Singh
Materials Department
RISOE Danish National Laboratory
DK-4000 Roskilde, Denmark
e-mail: bachu.singh@risoe.dk
Irradiated structural materials
exhibit unique deformation characteristics as a result of the extreme non-equilibrium
conditions created by the irradiation field. High-energy particle
irradiation introduces damage in the form of small vacancy and interstitial
loops and stacking fault tetrahedra. These features of the microstructure
lead to substantial modifications in the way dislocation loops are generated
and mobilized under further plastic deformation conditions. In most
alloys, extreme radiation hardening is observed. Moreover, it is
also show that irradiated materials undergo plastic instabilities when
subjected to critical values of an externally applied stress. A preliminary
3-D Dislocation Dynamics model for the critical stress required to unlock
immobile dislocation from small interstitial loops is presented.
Under irradiation, small interstitial clusters migrate rapidly to existing
dislocations, leading to their decoration and immobilization. The
conditions required for the decoration of these dislocations will be identified.
Calculations of the critical value of the applied stress, which is necessary
to free deformation-type dislocations from the immobilizing action of radiation-induced
defects, will be presented. Trapping and conversion of mobile irradiation-induced
defect clusters is shown to be critical to the initiation of deformation
instabilities in irradiated materials. Once the instability is initiated,
plastic deformation is confined to very narrow dislocation channels, while
the majority of the irradiated volume appears to be totally un-deformed.
A RULE-BASED DISLOCATION SIMULATION IN
SILICON ?
K. W. Schwarz
IBM Research, Yorktown Heights, NY
Simulations of large numbers of interacting dislocations
promise to be computationally very demanding. A particularly vexing
issue is how to deal with the occasional strong local interactions which
occur whenever two dislocation cores approach each other closely.
It is not computationally efficient to resolve each such interaction as
it occurs. This has led to the idea of a "rule-based" approach, in which
the program is given a predetermined set of rules to let it decide on the
outcome of each strong local interaction without doing detailed calculations.
We have explored the usefulness of this approach
for the relatively simple problem of two Frank-Read sources on intersecting
glide planes in a strained SiGe layer. Surprisingly, we find that
for this system the outcome of even simple close encounters can depend
sensitively on the local stress environment, and on the dislocation configurations.
A fully resolved calculation of the interacting Frank-Read sources is then
performed, following each close encounter down to the atomic-scale limit.
The outcome of certain kinds of encounters are easy to predict, and these
can be replaced by rules. Most close encounters, however, turn out
to be essentially unpredictable. For this physical system, therefore,
a rule-based approach is found to be untenable.
These results do not necessarily invalidate the
rule-based approach in general, in that the system investigated may be
a particularly difficult candidate for a rule-based approach. The
nonuniform stress field makes interactions more complicated; the geometry
is such as to lead to a concentration of dislocations in one region; and
the scales are small (layer thickness is 100 nm). Thus it may be
more constructive to view our results as indicating how a rule-based simulation
might run into trouble. From this perspective, one can draw two morals
from the numerical results. First, individual dislocation interactions
can be surprisingly sensitive to configurational and environmental details:
Any rules to be used must be verified and connected to the atomic scale
by doing fully resolved elastic calculations. Secondly, rule-based
calculations can be expected to break down when they produce dislocation
structures which are so dense that several dislocations interact strongly
at the same time. Fully resolved calculations will be needed to sort
out such situations. The applicability of these ideas to the simulations
currently being developed promises to be an especially interesting area
of investigation.
IV. Multiscale:
LARGE COUPLING OF MESOSCOPIC AND MACROSCOPIC
SIMULATIONS OF PLASTIC DEFORMATION
C. Lemarchand*, B. Devincre**, L.P. Kubin**,
and J.L. Chaboche*
** Laboratoire d'Etude des Microstructures,
CNRS-ONERA, BP 72, 92322 Chatillon Cedex, France
* D'epartement M'ecanique du Solide et de l'Endommagement,
ONERA, BP 72, 92322 Chatillon Cedex, France
In order to describe the plastic behavior of crystalline materials containing
structural heterogeneities and/or under complex loadings, a new non local
method that incorporates both dislocation properties and a rigorous treatment
of boundary value problems is proposed. This new computer model involves
a coupling of two different types of three-dimensional simulations: a Finite
Elements (FE) code and a Dislocation Dynamics Simulation (DDS) at the mesoscopic
scale. The simulation method is presented and illustrated by validation
tests perfomed on the plastic deformation of copper single crystals at
small strains.
THEORY AND EXAMPLES OF BOUNDARY VALUE
PROBLEMS WITH DISCRETE DISLOCATIONS
H.H.M. Cleveringa, E. Van der Giessen
Delft University of Technology, Koiter Institute Delft,
Mekelweg 2, 2628 CD Delft, The Netherlands
A. Needleman
Brown University, Division of Engineering,
Providence, RI 02912, USA
We describe our recent work on the simulation of plastic
flow by the collective motion of large numbers of discrete dislocations.
The simulations are carried out using a general framework for boundary
value problems involving many dislocations. Although valid for full three-dimensional
problems, attention is confined to a small strain, two-dimensional formulation.
The dislocations are modeled as line defects in an isotropic linear elastic
solid. The key idea in the approach is to write the stresses, strains and
displacements as superpositions of two fields: (i) fields due to the discrete
dislocations and (ii) complimentary (or image) fields that enforce the
boundary conditions. The first part of the decomposition is conveniently
taken to be an appropriate analytical solution. This can be the well-known
one for dislocations in an infinite solid, but may also be any other convenient
solution (e.g. in a half space). The formulation for the complimentary
fields leads to a linear elastic boundary value problem which is solved
by the finite element method. Hence, the long range interactions between
dislocations are accounted for through the continuum elasticity fields.
Short range interactions, i.e. at an atomic scale,
are incorporated through a set of constitutive rules. Dislocation motion
is taken to be governed by a linear drag relation, which relates the dislocation
velocity to the (Peach-Koehler) force on the dislocation. New dislocation
pairs are generated by simulating Frank-Read sources; annihilation of dislocations
occur when two opposite dislocations come within a critical distance, and
dislocations can get locked at obstacles in the material. Some of the applications
to be discussed here are for multiple slip systems (in two dimensions).
Interactions between dislocations near intersections of slip planes are
considered to take place solely in accordance with the above-mentioned
rules; no additional {\it ad hoc} rules are introduced for intersections.
This presentation will present some salient features
of the simulations using this approach. The applications to be discussed
include: (a) (reverse) shear of a composite material with a single slip
system; (b) uniaxial tension of strip with traction-free sides with multiple
slip; (c) bending of the same strip; (d) dislocation plasticity near a
(propagating) crack tip. In all cases, the plastic stress-strain response
and the evolution of the dislocation structure are outcomes of the boundary
value problem solution.
Problem (a) demonstrates the evolution of geometrically
necessary and statistical dislocations around particles. Also, it is used
to address the issue of how the limit of continuum slip is approached.
Problem (c) illustrates how geometrically necessary dislocations are being
generated during bending, whereas those under uniaxial tension in problem
(b) are essentially statistically stored. The difference in the macroscopic
hardening is emphasized. The last problem is motivated by the observation
that continuum plasticity theory can give a useful description of a large
part of the plastic zone around a crack tip, but at sufficiently small
distances from the tip, the elastic interaction between the dislocations
and the crack control the crack-tip state and the tendency for propagation.
DISLOCATION MOBILITY IN Si FROM ATOMIC
CORE TO MICRON SCALE
Vasily V. Bulatov
Massachusetts Insitute of Technology
It is commonly accepted that dislocation motion in Si
is controlled by the kink mechanisms. Based on this notion, the well
known kink diffusion model leads to a simple rate equation describing
temperature and stress dependence of dislocation mobility. However,
both experimental data and atomistic simulations suggest that this simple
approach does not fully account for the realistic complexity of dislocation
motion, at the core level and on a larger, mesoscopic scale. We propose
a quantitative theory of conservative dislocation motion in Si that bridges
these distinct length and time scales. The theory contains no adjustable
parameters and combines accurate atomistic calculations of the core energetics
and mechanisms with kinetic Monte Carlo simulations of micron
scale dislocation behavior.
SUBGRAIN FORMATION AND WORK HARDENING
W. Blum*, J. Kratochivil** and
R. Sedlacek*
*Institut fur Werkstoffwissenschaften, Lehrstuhl
I,
Universitat Erlangen-Nurnberg
Martensstrasse 5, D-91058 Erlangen
Germany
**Faculty of Civil Engineering, Department
of Physics
Czech Technical University Prague
Thakurova 7, CZ-166 29 Prague,
Czech Republic
Subgrain formation is a fundamental
process of dislocation patterning not only in pure crystalline materials,
but also in solid solutions and particle hardened alloys. Subgrains
or misoriented dislocation cells can be found in plastically deformed metals
on very different length scales, from sub-_m subgrains in cold rolled Al
foils or martensitic steels to mm subgrains in Al deformed near the melting
point.
The process of work hardening
is clearly correlated with the process of subgrain formation. The
decrease of the deformation rate during a test at a constant applied stress
corresponds to the increase of the materials volume filled with subgrains.
This relation is explained by the building up of internal stresses in the
subgrain structure and is described by the composite model. The backward
stresses in the subgrains can be correlated with the subrain size.
They were explained tentatively by the inhomogeneity of plastic deformation
due to the bowing-out of glide dislocations in the subgrain interiors.
The forward stresses at the subgrain boundaries are then a necessary consequence
of the stress equilibrium. The existence of the internal stresses
can be demonstrated experimentally e.g. by analysis of x-ray line profiles.
The question concerning the
origin of the subgrain formation remains to be answered. The basic
reason for the subgrain formation is the inhomogeneity of plastic deformation.
The subgrains form, as the plastic deformation and lattice rotation of
the different volume elements proceeds in an organized way so that the
incompatibility of material deformation which would be set up by the deformation
of a single volume element is eliminated by the deformation and rotation
of the neighboring volume elements. Subgrains are the volume elements
rigidly rotated relative to each other and separated by dislocation boundaries.
It has been suggested that this inhomogeneity of plastic deformation is
caused by an instability of a homogeneous plastic flow which is of non-linear
continuum mechanics origin. In addition to macroscopic instabilities
of plastic deformation in the form of necking or shear band formation,
there is an instability in the form of internal bending. This instability
is caused by an anisotropy of plastic flow provided either by deformation
on primary and secondary slip systems or by glide and climb of dislocations.
The stability behaviour is additionally modified by the geometrical softening
or hardening. The internal bending leads to the build up of misorientations
between neighboring volume elements. The corresponding pattern of
dislocations which are necessary to accommodate the lattice misorientations
are interpreted as the beginning of subgrain formation.
V. Statistical
/ Reaction Diffusion:
COOPERATIVE HOMOGENEOUS DISLOCATION GENERATION
AT FINITE TEMPERATURES: THEORY AND MONTE CARLO SIMULATION
V. Vitek and M. Khantha
Department of Materials Science and Engineering, University of
Pennsylvania, Philadelphia, PA 19104
We have recently proposed that sudden homogeneous
nucleation of dislocations leading to extensive plasticity may occur at
certain temperatures in loaded solids [1, 2]. It was suggested that
this mechanism is central to the phenomenon of the brittle-to-ductile transition
(BDT) in all crystalline materials [3]. This kind of dislocation
generation is a collective phenomenon, involving simultaneous formation
of a large number of interacting dislocation loops, analogous to the Kosterlitz-Thouless
[4] dislocation-mediated transition to the hexatic phase in two dimensions
(2D) that occurs near melting.
The analysis of this mechanism was first developed
for the 2D case, when generation of dislocation dipoles takes place [1,
2]. In this presentation we outline the theory in three dimensions
(3D), when many dislocation loops are generated, and discuss its application
to the BDT. We then return to a two dimensional case and present
Monte Carlo simulations of the generation of interacting dislocation dipoles
in a 2D loaded crystal.
Dislocation loops of atomic sizes have finite formation
energies in the range 1.0-2.0eV in most materials and, therefore, the density
of loops of radius r at a temperature T is proportional to exp[-H(r)/kBT],
where H(r) is the formation enthalpy of the loop and kB the Boltzmann constant.
When the applied stress is well below the ideal shear strength of the material,
the activation energy needed for the expansion of a single dislocation
loop to the critical radius is so high that generation of individual dislocations
is practically impossible. An alternative is the cooperative process
facilitated by two distinct factors. First, the exponential increase
of the density of loops with temperature and the second, and more important,
the effective lowering of the total self-energy of the loops as more sub-critical
loops form in the solid. This effect can be described either by a
"mean-field" theory or using a "direct" approach which is employed in the
simulation. In the "mean-field" treatment, the energy of a dislocation
loop is indirectly influenced by the induced rearrangement of the existing
loops. There is a net plastic strain associated with such rearrangement
which leads to an effective decrease of the moduli that relate stresses
and total strains in the medium. The self energy of the newly formed
'test' loop is then proportional to these lower 'effective moduli'.
When the free energy of the test loop becomes negative the density of loops
in the medium increases dramatically.
In order to investigate this process by a "direct"
approach, we have carried out Monte Carlo simulations of the formation
of dislocation dipoles in 2D. In this simulation, smallest dislocation
dipoles are spontaneously nucleated on a planar triangular lattice at finite
temperatures assuming a constant applied stress. The dipoles are
then allowed to annihilate, expand and contract in subsequent steps of
the Monte Carlo procedure. The energy of many interacting dislocation
dipoles is calculated by taking into account all the pair-wise interactions
between dislocations forming the dipoles. These simulations clearly
demonstrate the occurrence of the collective nucleation of dipoles and
the decrease of the transition temperature with increasing applied stress.
Acknowledgments: This research was supported by the U.S. Air Force Office
of Scientific Research Grant F49620-98-1-0245.
1. M. Khantha, D. P. Pope and V. Vitek,
Phys. Rev. Lett. 73, 684 (1994).
2. M. Khantha and V. Vitek, Acta Mater.
45, 4675 (1997).
3. M. Khantha, D. P. Pope and V. Vitek,
Acta Mater. 45, 4687 (1997).
4. J. M. Kosterlitz and D. J. Thouless,
J. Phys. C : Solid State Phys. 6, 1181 (1973).
KINETIC MONTE CARLO SIMULATIONS OF DISLOCATIONS
DYNAMICS
D. C. Chrzan
Department of Materials Science and Mineral Engineering
University of California, Berkeley, CA
and
Division of Materials Science
Lawrence Berkeley National Laboratory
Karin Lin
Department of Physics
University of California, Berkeley, CA
and
Division of Materials Science
Lawrence Berkeley National Laboratory
A kinetic Monte Carlo simulation of dislocation dynamics
is developed and studied. The simulations are based on isotropic
elasticity theory, and are meant to describe the motion of dislocations
in materials with a large Peierls barrier.
The simulations are used to study the dynamics of
dislocations moving via double-kink nucleation, and lateral kink motion,
but without the mean-field approximations inherent in traditional formulations.
In the low stress limit, the dynamics of the dislocations are consistent
with existing theories.
An analogy with rudimentary models of crystal growth
suggests that the dislocations might display kinetic roughening.
The kinetic roughening of the dislocations is studied through a dynamic
scaling analysis of the dislocation width. The simulated dislocations
do, indeed, display kinetic roughening, and the scaling exponents are those
derived from the Kardar-Parisi-Zhang equation describing the growth of
thin films. The implications of this scaling for modeling the dynamics
of dislocations is discussed.
THEORY OF STRAIN PERCOLATION IN METALS
Robb Thomson (Emeritus) and Lyle E. Levine
Materials Science and Engineering Laboratory
NIST, Gaithersburg, MD 20899
Once information about the cell structure in a deformed
metal has been obtained by any means, it is necessary to describe how additional
dislocations percolate through the structure in order to obtain the stress/strain
relations. We will present such a theory.
It is assumed that the walls which inhibit the transmission
of strain from one cell to the next are described by a stochastic variable
which reflects the statistical variation from cell wall to cell wall, and
that the strain propagation is given by the following linear law,
s* = sa,
where s* is the strain transmitted to an initially unstrained
cell through the wall separating it from a strained cell with strain, s.
The transmission through the wall is characterized by the amplification
factor, a. The linearity in s follows from the fact that the strain
burst is a dislocation pileup. Since the force on the front of a pileup
is proportional to the number of dislocations in the pileup, and the propagation
of strain from one cell to another is proportional to the force on the
front dislocation, the propagation must be linear in s.
We assume the amplification factor is composed of
two terms,
a = P1 * Z + P2 * exp(-Z / L),
where P1, P2 and L are parameters, and 0 <
Z < 1 is a random variable. The two terms in the equation reflect
our opinion that the wall action is bi-modal. In the first, the walls serve
as ordinary sources for dislocations, which respond to the pileup in the
normal way. In the second, we presume that in relatively rare cases, a
wall will contain a weak lock which unzips under the influence of the pileup,
so that a large increment of strain is propagated to the previously unstrained
cell.
We develop both 1D models for the percolation, which
are analytic in character, and 2D models which must be explored numerically.
We show that at the percolation limit, the model exhibits a critical surface
in the space of the (internal) parameters, and that as strain progresses,
the system clings to this critical surface. The universality class of this
percolation problem appears to be the same as for ordinary 2D percolation.
Thus, the deforming metal is a self organizing critical system.
Finally, we show that a differential form of the
stress-strain relation can be written on the critical surface which contains
three terms. Two of these are physical evolution laws associated with the
physics of the dislocations interacting with walls in a representative
dislocation cell, and which are expressed in terms of internal variables
of the system. The third term is a (universal) function of the critical
surface.