ABSTRACTS
(In Order Presented)
 

 
Go to section:
Introductory Lecture
I. Experiments / Observations
II. Atomic Scale
III. Dislocation Dynamics Simulations
IV. Multiscale
V. Statistical / Reaction Diffusion
 


 

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.