Collective Behavior Of Dislocations Research Articles

The mechanism for annealing-induced ductile to brittle transition in a high-temperature titanium alloy and its mitigation

By investigating the relation between microstructure evolution and tensile responses of a BTi6431S alloy at room temperature and 700 °C, this study finds an annealing-induced ductile-to-brittle transition phenomenon in the near-α titanium alloy. The high strength of the as-received BTi6431S alloy is rooted in the high dislocation density and collective dislocation behavior in the α phase. However, after annealing at 650 °C, the alloy exhibits brittle fracture at room temperature. Using multiscale characterization methods, we attribute this phenomenon to the presence of brittle precipitates and the change of dislocation structures near the surface of BTi6431S titanium alloy. Silicides are generated in the vicinity of the surface region due to the relatively high dislocation density on the surface after the rolling process. The room temperature ductility can be restored by removing the near-surface area. When the alloy is tested at 700 °C, most of the near-surface brittle silicides and dislocations can be eliminated, because the deformation is mainly affected by dislocation behavior and dynamic recrystallization at high temperature. The findings in this study can advance the understanding of alloying effects on the mechanical behavior of high-temperature titanium alloy. The results suggest that inhibiting silicide formation and controlling the environmental temperature during service are potential approaches for maintaining the mechanical performance of this near-α titanium alloy.

Complex patterning in jerky flow from time series analysis and numerical simulation

The paper is a tribute to Ladislas P Kubin’s long-standing work on the collective behavior of dislocations in jerky flow. In a first part, it reviews his contributions to the statistical, dynamical and multifractal analyses carried out on stress-time series recorded from both single crystals and polycrystalline samples of dilute alloys subjected to tensile tests at constant strain rate. Various spatio-temporal dynamical regimes were found as the applied strain rate was varied. Type C static bands were associated with quasi-random collective behavior, the hopping type B and propagating type A bands could be shown to correspond to chaotic and self-organized critical dynamics, respectively. The crossover between the A and B regimes was characterized by a large spread in the multifractal spectrum of stress drops, associated with heterogeneity of the dynamics. In a second part, the paper reviews the nonlocal models Ladislas inspired to interpret these results from numerical solutions of the boundary value problem, on the basis of dynamic strain aging, the incompatibility stresses associated with dislocations, their plastic relaxation and the spatial couplings they inherently involve. Eventual developments of this research, rooted in the same ideas, on the statistical and multifractal analyses of the accompanying acoustic emission are reviewed and discussed in terms of the synchronization of small-scale plastic events.

Situating the Vector Density Approach Among Contemporary Continuum Theories of Dislocation Dynamics

Abstract For the past century, dislocations have been understood to be the carriers of plastic deformation in crystalline solids. However, their collective behavior is still poorly understood. Progress in understanding the collective behavior of dislocations has primarily come in one of two modes: the simulation of systems of interacting discrete dislocations and the treatment of density measures of varying complexity that are considered as continuum fields. A summary of contemporary models of continuum dislocation dynamics is presented. These include, in order of complexity, the two-dimensional statistical theory of dislocations, the field dislocation mechanics treating the total Kröner–Nye tensor, vector density approaches that treat geometrically necessary dislocations on each slip system of a crystal, and high-order theories that examine the effect of dislocation curvature and distribution over orientation. Each of theories contain common themes, including statistical closure of the kinetic dislocation transport equations and treatment of dislocation reactions such as junction formation. An emphasis is placed on how these common themes rely on closure relations obtained by analysis of discrete dislocation dynamics experiments. The outlook of these various continuum theories of dislocation motion is then discussed.

Plasticity without phenomenology: A first step

A novel, concurrent multiscale approach to meso/macroscale plasticity is demonstrated. It utilizes a carefully designed coupling of a partial differential equation (pde) based theory of dislocation mediated crystal plasticity with time-averaged inputs from microscopic Dislocation Dynamics (DD), adapting a state-of-the-art mathematical coarse-graining scheme. The stress-strain response of mesoscopic samples at realistic, slow, loading rates up to appreciable values of strain is obtained, with significant speed-up in compute time compared to conventional DD. Effects of crystal orientation, loading rate, and the ratio of the initial mobile to sessile dislocation density on the macroscopic response, for both load and displacement controlled simulations are demonstrated. These results are obtained without using any phenomenological constitutive assumption, except for thermal activation which is not a part of microscopic DD. The results also demonstrate the effect of the internal stresses on the collective behavior of dislocations, manifesting, in a set of examples, as a Stage I to Stage II hardening transition.

Effect of hydrogen on the collective behavior of dislocations in the case of nanoindentation

Most of the studies reported treats the effect of hydrogen on single dislocation line, while models that describe the collective interaction are missing. In this study, hydrogen-induced softening of metallic materials is studied from a perspective of collective behavior of dislocations. Building on the evolution of dislocation density, a hydrogen-informed expanding cavity model is developed for the first time to predict the dynamic evolution of load-displacement curve obtained from nanoindentation tests. Large-scale molecular dynamics simulations on the mechanical behavior of fcc Ni with and without hydrogen (H) charged are performed to calibrate the proposed continuum model. The results show that the H-induced decrease of indentation force is due to that the energy barrier for dislocation nucleation is lowered by the solute drag of the H atmosphere formed around dislocations. Envisioned as a complex non-equilibrium process, it is found that the power-law exponent of the self-organized criticality of dislocations increases due to the insertion of H atoms. Analysis also indicates that H can reduce the probability of dislocation pile-up, thus promote the delivery of dislocations to the surface of specimens during nanoindentation.

The strength and dislocation microstructure evolution in superalloy microcrystals

In this work, the evolution of the dislocations microstructure in single crystal two-phase superalloy microcrystals under monotonic loading has been studied using the three-dimensional discrete dislocation dynamics (DDD) method. The DDD framework has been extended to properly handle the collective behavior of dislocations and their interactions with large collections of arbitrary shaped precipitates. Few constraints are imposed on the initial distribution of the dislocations or the precipitates, and the extended DDD framework can support experimentally-obtained precipitate geometries. Full tracking of the creation and destruction of anti-phase boundaries (APB) is accounted for. The effects of the precipitate volume fraction, APB energy, precipitate size, and crystal size on the deformation of superalloy microcrystals have been quantified. Correlations between the precipitate microstructure and the dominant deformation features, such as dislocation looping versus precipitate shearing, are also discussed. It is shown that the mechanical strength is independent of the crystal size, increases linearly with increasing the volume fraction, follows a near square-root relationship with the APB energy and an inverse square-root relationship with the precipitate size. Finally, the flow strength in simulations having initial dislocation pair sources show a flow strength that is about one half of that predicted from simulations starting with single dislocation sources. The method developed can be used, with minimal extensions, to simulate dislocation microstructure evolution in general multiphase materials.

Atomistically-informed crystal plasticity in MgO polycrystals under pressure

This paper addresses multi-scale modeling of a special kind of pressure-dependent plasticity. In MgO, which is a major constituent of the Earth's lower mantle, the relative activity of the 12〈110〉{110}and12〈110〉{100} slip modes depends on the level of hydrostatic pressure. The influence of pressure on both the dislocation core structures and the collective behavior of dislocations may be computed based on atomistic modeling and dislocation dynamics simulations. In the present study, results from such lower-scale simulations are used to determine the parameters of a crystal plasticity model that is suitable in order to probe the large-scale mechanical response of MgO polycrystals at pressures up to 100 GPa. The model is assessed based on experimental compression tests performed at different pressure levels. It turns out that the responses of single crystals and polycrystals are fundamentally different. Moreover, due to the large anisotropy of individual crystals, the outcome of polycrystalline simulations is found to depend strongly on the modeling assumption made about grain interactions. Crystal plasticity based finite element modeling provides the best predictions of the texture development when comparing to recent high pressure experiments.

Collective behaviour of dislocations in a finite medium

We derive the grand-canonical partition function of straight and parallel dislocation lines without making a priori assumptions on the temperature regime. Such a systematic derivation for dislocations has, to the best of our knowledge, not been carried out before, and several conflicting assumptions on the free energy of dislocations have been made in the literature. Dislocations have gained interest as they are the carriers of plastic deformation in crystalline materials and solid polymers, and they constitute a prototype system for two-dimensional Coulomb particles. Our microscopic starting level is the description of dislocations as used in the discrete dislocation dynamics (DDD) framework. The macroscopic level of interest is characterized by the temperature, the boundary deformation and the dislocation density profile. By integrating over state space, we obtain a field theoretic partition function, which is a functional integral of the Boltzmann weight over an auxiliary field. The Hamiltonian consists of a term quadratic in the field and an exponential of this field. The partition function is strongly non-local, and reduces in special cases to the sine–Gordon model. Moreover, we determine implicit expressions for the response functions and the dominant scaling regime for metals, namely the low-temperature regime.

Plasticity of bcc micropillars controlled by competition between dislocation multiplication and depletion

Recent micropillar experiments have shown strong size effects at small pillar diameters. This “smaller is stronger” phenomenon is widely believed to involve dislocation motion, which can be studied using dislocation dynamics (DD) simulations. In the present paper, we use a three-dimensional DD model to study the collective dislocation behavior in body-centered cubic micropillars under compression. Following the molecular dynamics (MD) simulations of Weinberger and Cai, we consider a surface-controlled cross-slip process, involving image forces and non-planar core structures, that leads to multiplication without the presence of artificial dislocation sources or pinning points. The simulations exhibit size effects and effects of initial dislocation density and strain rate on strength, which appear to be in good agreement with recent experimental results and with a simple dislocation kinetics model described here. In addition, at the high strain rates considered, plasticity is governed mainly by the kinetics of dislocation motion, not their elastic interactions.

Scale transitions in crystal plasticity by dislocation dynamics simulations

This article shows, on the base of recent results, that dislocation dynamics simulations provide a unique opportunity for establishing scale transitions in crystal plasticity. In the present study, transitions are performed between elementary dislocation mechanisms, collective and intermittent slip events at the mesoscale and the mechanical response of bulk materials with low lattice resistance. In fcc crystals, the insight that is obtained provides guidelines for the modeling of collective dislocation behavior. It also allows incorporating more physical input into phenomenological models for strain hardening and improving their predictive ability.

Atomic and dislocation dynamics simulations of plastic deformation in reactor pressure vessel steel

The collective behavior of dislocations in reactor pressure vessel (RPV) steel involves dislocation properties on different phenomenological scales. In the multiscale approach, adopted in this work, we use atomic simulations to provide input data for larger scale simulations. We show in this paper how first-principles calculations can be used to describe the Peierls potential of screw dislocations, allowing for the validation of the empirical interatomic potential used in molecular dynamics simulations. The latter are used to compute the velocity of dislocations as a function of the applied stress and the temperature. The mobility laws obtained in this way are employed in dislocation dynamics simulations in order to predict properties of plastic flow, namely dislocation–dislocation interactions and dislocation interactions with carbides at low and high temperature.

Effects of pre-strain on the compressive stress–strain response of Mo-alloy single-crystal micropillars

A NiAl–Mo eutectic was directionally solidified to produce composites with well-aligned single-crystal Mo-alloy fibers embedded in a NiAl matrix. They were pre-strained by compressing along the fiber axis and then the matrix was etched away to expose free-standing micropillars having different sizes (360–1400 nm) and different amounts of pre-strain (0–11%). Compression testing of the pillars revealed a variety of behaviors. At one extreme were the as-grown pillars (0% pre-strain) which behaved like dislocation-free materials, with yield stresses approaching the theoretical strength, independent of pillar size. At the other extreme were pillars pre-strained 11% which behaved like the bulk, with reproducible stress–strain curves, relatively low yield strengths, stable work-hardening and no size dependence. At intermediate pre-strains (4–8%), the stress–strain curves were stochastic and exhibited considerable scatter in strength. This scatter decreased with increasing pre-strain and pillar size, suggesting a transition from discrete to collective dislocation behavior.

Nanoindentation-Induced Collective Dislocation Behavior and Nanoplasticity

The present paper summarizes the crystallographic dependence of the displacement burst behavior observed in nanoindentation using two single crystalline aluminum (Al) materials and copper (Cu) with three kinds of surface indices, namely (001), (110) and (111). From the critical indent load at the first burst, the critical resolved shear stresses (CRSSs) of the collective dislocation nucleation were estimated in reference to molecular dynamics (MD) simulations. These are almost one-tenth of the shear modulus, which are close to the ideal values. We explain the nanoplastic mechanics by a comprehensive energy balance model to describe the linear relation between the indent load and the burst width of the first displacement burst and by the nucleation model consisting of three-dimensional discrete dislocations to evaluate the number of dislocations nucleating. The distance between the emitted dislocation loops of Al is found to be fairly large. Thus, Al is expected to exhibit a less tangled network of dislocations just below the indentation than Cu, which has a lower stacking fault energy.

Dislocation motion in high strain-rate deformation

We present a systematic investigation of dislocation motion, dislocation interactions, and the collective behaviour of dislocations in high strain-rate deformation. Based on results from three-dimensional dislocation dynamics simulations, we find that employing the accurate, full-dynamics, equation of motion (i.e. that includes inertial effects) significantly changes the predictions of microstructural evolution and the macroscopic response compared to the commonly used overdamped equation of motion (i.e. with no inertial effects), especially at high strain rates (103–106 s−1). While we find that inertial effects cannot be neglected, the net velocities are not high enough that ‘relativistic’ effects are important. We also present results on the effects of high strain rates on single-crystal deformation, which show good agreement with experimental trends, including increased hardening with increasing strain rate.

Statistically based continuum model of misoriented dislocation cell structure formation

The formation of misoriented dislocation cells in the course of plastic deformation is explained within the framework of continuum mechanics as a result of the trend to reduce the energetically costly hardening in multislip by locally decreasing the number of active slip systems by local lattice rotations. A model of an infinite crystal deformed in plane strain by symmetric double slip, where the plastic strain is carried by straight, parallel, edge dislocations, is considered. The nonlocal constitutive equations of the model are derived from statistical mechanics description of the collective behavior of dislocations. The finite cell size as well as the orientation of the cell boundaries result from the competition between two tendencies: the internal and dissipative energy tend to decrease the cell size, whereas the short-range dislocation interactions oppose this tendency.

A combined dislocation—cohesive zone model for fracture in a confined ductile layer

The collective dislocation behavior near a crack tip in a ductile layer sandwiched between two brittle solids is analyzed via two-dimensional dislocation dynamics (DD) simulations that incorporate a cohesive zone (CZ) model. The cohesive crack tip is treated as part of a much larger finite crack confined in the ductile layer. The underlying boundary value problem is formulated with a set of boundary integral equations and numerically evaluated with a collocation method. The fracture energy of the layered composite material is shown to be strongly correlated with the layer thickness and is directly influenced by the cohesive strength of the ductile layer (Hsia KJ et al. (1994) J Mech Phys Solids 6 877–896).

Statistical aspects of discrete dislocation plasticity

Abstract Discrete dislocation calculations suggest that statistical effects are important in the modeling of crystal plasticity. These include: (i) effects of source limited plasticity; (ii) statistical variations of dislocation sources and obstacles in small volumes; and (iii) the sensitivity of the discrete dislocation predictions to small perturbations. Implications of these for phenomenological constitutive models and for statistical theories of the collective behavior of dislocations are discussed.

Incorporating the influence of the temperature into a mesoscopic continuum description of dislocation systems

Understanding the collective behaviour of dislocations is a key issue of recent research activity on dislocation theory. Around the micron scale a continuum description operating with continuous fields (like stored and geometrically necessary dislocation densities) seems to be an efficient approach for describing phenomena like pattern formation or size effects. The aim of the present paper is to incorporate the influence of temperature and dislocation climb into the time evolution equations of the different dislocation densities. For a system of straight dislocations a Fokker–Planck type equation is derived. By performing a coarse graining procedure it is shown that thermal noise and climb lead to the appearance of additional gradient-like terms.

Size effects in single crystal thin films: nonlocal crystal plasticity simulations

Stress relaxation in single crystalline thin films on substrates subjected to thermal loading is studied using a recently proposed nonlocal continuum crystal plasticity theory. The theory is founded on a statistical-mechanics description of the collective behaviour of dislocations in multiple slip, which is coupled to a small-strain continuum crystal plasticity description. The theory is inherently nonlocal with the length scale being determined by the evolving dislocation density. Symmetric double slip is considered with the film being in plane strain. The predicted stress versus temperature response and the evolution of the dislocation structure are analyzed for different orientations and film thicknesses. The effect of film size is associated with the formation of a boundary layer of dislocations at the film-substrate interface which does not scale with the film thickness. The width of the boundary layer itself is shown to be dependent on the slip system orientation. The results are consistent with those of recent discrete dislocation simulations. C 2005 Elsevier SAS. All rights reserved.

518 離散転位法と境界要素法のマルチスケールモデリングによるプリズマティック転位の運動(GS-1 局所構造・き裂,研究発表講演)

Collective dislocation emissions and their dynamic behaviors are observed in the defect-free region of the crystalline materials. In order to describe the nano-plastic deformation in this situation, it is necessary to develop the mesoscopic modeling which deals with both the dislocation nucleation and the collective dislocation behavior as the result of long/short-range interaction. Discrete Dislocation Mechanics (DD) is the most suitable methodology to treat dislocation motion directly. In this paper, we use Molecular Dynamics (MD) to obtain some criteria of dislocation emissions and apply DD combined by Boundary Element Method (BEM) to describe the dislocation motions in the semi-infinite media with traction-free surface. The developed scheme is applied to the inclusion problem embedded into the ductile matrix, where we have captured the dislocation emission and the short-ranged behavior by the full MD atomistic simulation. Stress field of the DD-BEM problem can be obtained by the superposition of the stress field generated by dislocation segments and the free surface image stress calculated by BEM.