Non-conservative Motion Research Articles

ON THE INFLUENCE OF NON-EQUILIBRIUM VACANCIES ON THE CHARACTERISTICS OF STRAIN INDUCED BROKEN DISLOCATION BOUNDARIES

The influence of material supersaturation with deformation vacancies on the characteristics of broken dislocation boundaries that appear near wedge disclinations as a result of accommodative plastic deformation is considered. An analysis is made for the elastic and osmotic forces acting on the dislocations of the broken boundary. Using discrete dislocation dynamics method, computer simulation of the nonconservative motion (climb) of dislocations in the boundary plane has been carried out. The broken boundary formed as a result of the splitting of the original disclination into two partial with equal strengths and the boundary with an inhomogeneous distribution of dislocations (with a linearly decreasing misorientation) were considered as initial ones. It is shown that in the case of broken dislocation boundary formed near a positive wedge disclination, dislocation climb leads to its unlimited growth through the model grain with a simultaneous decrease of misorientation. In the case of a boundary formed near a negative disclination, the osmotic forces acting on the dislocations of the broken boundary can, at sufficiently large super saturations of the material with nonequilibrium vacancies, lead to its significant compression with a simultaneous increase of the density of the Burgers vector. In this case, the choice of the initial configuration of the broken boundary has practically no effect on the characteristics of the equilibrium distribution of dislocations. The density distributions of the Burgers vector along the broken boundary are calculated for the case of small and large supersaturations of the material with nonequilibrium vacancies. The critical value of supersaturation above which the boundary compression occurs is calculated. The boundary compression leads to a significant increase in tensile stresses in its vicinity. In this case, the maximum tensile stresses in the plane coinciding with the plane of the boundary are reached at the boundary break point. The results obtained may be of interest in the analysis of possible mechanisms for the initiation of cracks and pores during ductile fracture of metals.

Immobilization of partial dislocations bounding double Shockley stacking faults in 4H-SiC observed by in situ synchrotron X-ray topography

The expansion of double Shockley stacking faults (DSFs) in an n-type 4H-SiC substrate with a nitrogen concentration of 3.9 × 1019 cm−3 was investigated using in situ synchrotron X-ray topography. DSF expansion was observed to be suppressed and immobilized above 1590 K, along with the partial dislocation (PD) shape being changed from a straight to zig-zag configuration. For a different heating process (higher heating rate), the PDs could continue to expand, even above 1590 K. Ex situ topography experiments revealed that the DSFs close to the specimen surface expanded widely, although those expanding toward the specimen interior became immobile. One possible mechanism for this immobilization was proposed, where the core structural changes from a Si-core to the C-core by non-conservative motion induced by the interaction between the PDs and point defects (C interstitials).

Incorporating point defect generation due to jog formation into the vector density-based continuum dislocation dynamics approach

During plastic deformation of crystalline materials, point defects such as vacancies and interstitials are generated by jogs on moving dislocations. A detailed model for jog formation and transport during plastic deformation was developed within the vector density-based continuum dislocation dynamics framework (Lin and El-Azab, 2020; Xia and El-Azab, 2015). As a part of this model, point defect generation associated with jog transport was formulated in terms of the volume change due to the non-conservative motion of jogs. Balance equations for the vacancies and interstitials including their rate of generation due to jog transport were also formulated. A two-way coupling between point defects and dislocation dynamics was then completed by including the stress contributed by the eigen-strain of point defects. A jog drag stress was further introduced into the mobility law of dislocations to account for the energy dissipation during point defects generation. A number of test problems and a fully coupled simulation of dislocation dynamics and point defects generation and diffusion were performed. The results show that there is an asymmetry of vacancy and interstitial generation due to the different formation energies of the two types of defects. The results also show that a higher hardening rate and a higher dislocation density are obtained when the point defect generation mechanism is coupled to dislocation dynamics.

Influence of varying nitrogen on creep deformation behaviour of 316LN austenitic stainless steel in the framework of the state-variable approach

Abstract Primary and steady-state creep deformation of 316LN stainless steel with four different nitrogen weight percent (0.07, 0.11, 0.14, and 0.22% N) have been examined in the framework of internal stress based kinetic creep law. The interrelationship between activation area or activation volume and effective stress has been employed in the present model, rather than the applied stress dependency on these thermodynamic quantities. The model predicted the decrease in obstacle spacing and the increase in internal stress with nitrogen at all the applied stress levels. Across the nitrogen content, the evaluated activation volume ranging from 400 to 900 b3 suggested the dominance of forest dislocation intersections coupled with the non-conservative motion of jogged dislocations during steady-state creep. The rate constant parameter of the model associated with the dynamic recovery exhibited a decreasing trend with nitrogen. The variations in the strength enhanced activation volume and length of the dislocation segment against the applied stress/yield strength exhibited the unified power law relationships, irrespective of nitrogen content. The model appropriately predicted the creep strain/rate vs. time curves and steady-state creep rate vs. internal stress and effective stress quantities. By incorporating the damage term into the creep rate equation, the developed model is extended for the prediction of entire creep strain-time curves.

Is crystal plasticity non-conservative? Lessons from large deformation continuum dislocation theory

Crystal plasticity for large plastic deformations is usually modeled as volume conserving. However, there is experimental evidence that huge amounts of vacancies are produced during plastic deformation. The only regularly considered mechanism of non-conservative dislocation motion is climb of edge dislocations, but this is supposed to be enabled by high vacancy concentrations rather than producing them. In the current contribution we show that large deformation kinematics applied to evolving fields of geometrically necessary dislocations accounts for kinking and jogging of dislocations when cutting through one another. Therefore, dislocation fields in multiple slip situations will not be confined to slip planes and their motion will necessarily involve non-conservative motion of jogs. Because inelastic volume changes in crystals are only possible in the presence of point defects, we find that large deformation continuum dislocation theory must be coupled to an evolving vacancy concentration. We suggest to treat vacancy diffusion as an independent inelastic deformation mechanism yielding a three-way multiplicative decomposition of the deformation gradient. Small numerical examples are used to illustrate the kinematics of emerging and moving kinks and jogs, and the ensuing production of vacancy concentration. Continuum thermodynamics is used to derive coupled evolution equations for geometrically necessary dislocation density vectors and vacancy concentration. We conclude that large deformation crystal plasticity as an averaged theory of dislocations may not be conservative and that the usual kinematic assumption Lp=∑sγ˙sms⊗ns does not hold in multiple slip deformation.

Compressive creep behavior of a γ-TiAl based Ti–45Al–8Nb–2Cr-0.2B alloy: The role of β(B2)-phase and concurrent phase transformations

The primary and steady-state compressive creep deformation behavior of an as-cast high Nb containing γ-TiA1 alloy having remnant β(B2)-phase has been studied over the temperature range of 750–850 °C at constant initial applied stress levels between 75-200 MPa. The creep curves derived over the entire creep testing conditions show a prominent primary creep regime which is attributed to the presence of dislocation debris of the coarse γ-grains at the colony boundaries. The stress and temperature dependence of the steady-state creep rate follows the Norton-Bailey power law and the corresponding average value of the stress exponent and apparent activation energy are estimated to be 3.6 and 375 kJ/mol, respectively. The average value of the stress exponent in conjunction with the dislocation substructure of the crept samples suggests that the kinetics of creep deformation within the studied experimental conditions is controlled by the non-conservative motion (climb) of dislocations. Further misorientation analyses of the phase-resolved EBSD maps imply that for most of the conditions creep strain is carried by the γ-TiAl phase. Consequently, contrary to the commonly believed idea, the β(B2)-phase does not appear to deteriorate the creep resistance of the present alloy below suitable combinations of temperature and stress. Beyond these conditions, the concurrent deformation of γ-TiAl and β(B2)-phases is found to enhance the steady-state creep rate. In addition, characteristics of the dynamic transformation of β(B2) phase into γ and α2 phases during creep and its effect on the steady-state creep rate are further considered. Finally, the application potential of the present alloy is compared through a composite plot of Larson-Miller parameter with several other potential alloys reported in the literature.

Constitutive modelling of transient and steady state creep behaviour of type 316LN austenitic stainless steel

In the present study, sine hyperbolic creep rate law coupled with the evolution of internal stress with time has been employed for modelling transient and steady state creep deformation behaviours of type 316LN austenitic stainless steel (SS). The model predicts the creep strain-time and creep rate-time data appropriately for the stress levels ranging from 120 to 225 MPa at 923 K, with good agreement between experimental and predicted steady state creep rates. At all the stress levels, the predicted internal stress increases with time during transient creep, and it approaches saturation at steady state creep. The predicted increase in internal stress was interrelated to the increase in dislocation density and a decrease in the mean free path during creep deformation. At steady state condition, a decrease in activation volume with an increase in applied stress levels has been observed for the steel. The estimated value of activation volume for the steel demonstrates the dominance of intersection of forest dislocations and non-conservative motion of jogs during transient creep in type 316LN SS. The applicability of the present model has been demonstrated towards the prediction of stress-relaxation behaviour of type 316LN SS based on the power law interrelation between internal stress and applied stress.

Fundamental insights into the mass transfer via full dislocation loops due to alternative surface cuts

Mass transfer during the formation and emission of a dislocation loop without non-conservative motions can be very tricky. We combine the molecular statics, nonsingular dislocation theory, finite element (FE) method and molecular dynamics (MD) approach to offer new insights. It is proved both theoretically and numerically that the formation and the dynamic emission of a full shear loop can grow a nanovoid. For the first time, a novel approach is proposed to create a shear loop via a hemispherical surface cut by the atomistic simulation. Such a formation process can produce a similar shear loop to that via a flat surface cut, and yet lead to void growth. We incorporate the non-singular dislocation theory into a FE model to calculate the potential energy regarding different surface cuts. Among possible cuts to create a shear loop, the potential energy would favour the one that grows a void under triaxial tension. We show that the arbitrariness regarding the surface cut of dislocation segments should also be recognized during the dynamic emission process. Under triaxial tension, we confirm the mass transfer during the cross-slipping of V-shape shear loops prior to the completion of prismatic loops. By comparison, the uniaxial tension case shows that a V-shape shear loop is not necessarily the precursor of a prismatic loop. Hence, its mass transfer mechanism should be treated differently.

Strengthening mechanisms of different oxide particles in 9Cr ODS steel at high temperatures

The influence of different oxide particles on deformation mechanisms of 9Cr ODS steels at temperatures 400–600 °C is tested. Two ODS variants with Al2O3 and Y2O3 dispersed particles respectively and one variant without particles have been prepared by powder metallurgy. The deformation mechanisms are investigated by the tensile and stress-relaxation test at temperatures 400 °C and 600 °C. The analysis is based on a comparison of athermal and thermal stress components of the flow stress, apparent activation volume and a strain rate hardening. Microstructural parameters are evaluated upon SEM and TEM observations. The contributions of different strengthening mechanisms are assessed using the Taylor, Hall-Petch, dispersed barrier and Orowan models. Alumina particles provide the highest strengthening which can be explained by the stress decrease around the particles induced by a high difference between Young's moduli of alumina and ferritic matrix. This stress field can trap the dislocation in the vicinity of particles. Due to this stress decrease, alumina particles provide strengthening even at 600 °C in contrast to yttria particles that lose their effect due to their overcoming by a non-conservative motion of dislocations.

Improvement of field method and its application to integrating motion equation in Riemann-Cartan space

Like the Hamilton-Jacobi method, the Vujanović field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanović field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanović field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanović field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m (m n) first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution (the complete solution is a special case of m=n) of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanović field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanović field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subjected to an active force, its motion equation in its Riemann-Cartan configuration space can be obtained by a first order nonlinear nonholonomic mapping. Since the motion equations in Riemann-Cartan configuration space contain quasi-speeds, they are often considered to be difficult to solve directly. In this paper we give a briefing of how to construct the motion equations of a first order linear nonholonomic constraint system in its Riemann-Cartan configuration space, and how to obtain the first integrals of the motion equations in the Riemann-Cartan configuration space by the improved field method. This is an effective method to study some nonholonomic nonconservative motions.

Mechanisms of high-speed deformation of polycrystalline copper

The mechanisms of high-speed deformation of ultrafine-grained copper produced during severe plastic deformation by equal-channel angular pressing were analyzed using numerical modeling in comparison with those in the case of coarse-crystalline copper. The activity of annihilation processes during nonconservative motion and double cross slip of dislocations was estimated. Their effect on the macroscopic behavior of samples is shown.

High temperature deformation mechanisms in the 14% Cr ODS alloy

The stress relaxation behavior of the ODM401 ODS steel is tested at the temperature range 400–800°C in order to investigate the deformation mechanisms. The repeated relaxation tests are performed on the tensile specimens within the plastic strain range 10−4–10−1. The deformation mechanisms are analyzed in terms of an activation volume, dislocation density and velocity, a thermal stress component and a strain rate hardening. The deformation is driven by the dislocation glide and climb. Dispersed particle-dislocation interactions is a dominant effect at temperatures 400–600°C. Short-range interactions between dislocations and a lattice are most significant at 600°C. The non-conservative motion of dislocations becomes active at temperatures above 600°C. The majority of the mobile dislocations is not trapped and is able to move freely in the microstructure.

Density of dislocations in CdHgTe heteroepitaxial structures on GaAs(013) and Si(013) substrates

Epitaxial layers of Cd x Hg1–x Te (MCT) on GaAs(013) and Si(013) substrates were grown by molecular beam epitaxy. The introduction of ZnTe and CdTe intermediate layers into the structures made it possible to retain the orientation close to that of the substrate in MCT epitaxial layers despite the large mismatch between the lattice parameters. The structures were investigated using X-ray diffraction and transmission electron microscopy. The dislocation families predominantly removing the mismatch between the lattice parameters were found. Transmission electron microscopy revealed Γ-shaped misfit dislocations (MDs), which facilitated the annihilation of threading dislocations. The angles of rotation of the lattice due to the formation of networks of misfit dislocations were measured. It was shown that the density of threading dislocations in the active region of photodiodes is primarily determined by the network of misfit dislocations formed in the MCT/CdTe heterojunction. A decrease in the density of threading dislocations in the MCT film was achieved by cyclic annealing under conditions of the maximally facilitated nonconservative motion of dislocations. The dislocation density was determined from the etch pits.

Elastoplastic behavior of copper upon high-strain-rate deformation

The deformation behavior of copper under conditions of high-strain-rate deformation has been investigated based on the model of elastoplastic medium with allowance for the kinetics of plastic deformation. Data have been obtained on the evolution of the dislocation subsystem, namely, on the average dislocation density, density of mobile dislocations, velocity of dislocation slip, concentration of deformation-induced vacancies, and density of twins. The coefficient of the annihilation of screw dislocations has been estimated depending on pressure and temperature. It has been shown that severe shear stresses that arise upon high-strain-rate deformation can lead to a significant increase in the concentration of vacancies. The time of the dislocation annihilation upon their nonconservative motion has been estimated. It has been shown that this time is much greater than the time of the deformation process in the samples, which makes it possible to exclude the annihilation of dislocations upon their nonconservative motion from the active mechanisms of deformation.

High-temperature discrete dislocation plasticity

A framework for solving problems of dislocation-mediated plasticity coupled with point-defect diffusion is presented. The dislocations are modeled as line singularities embedded in a linear elastic medium while the point defects are represented by a concentration field as in continuum diffusion theory. Plastic flow arises due to the collective motion of a large number of dislocations. Both conservative (glide) and nonconservative (diffusion-mediated climb) motions are accounted for. Time scale separation is contingent upon the existence of quasi-equilibrium dislocation configurations. A variational principle is used to derive the coupled governing equations for point-defect diffusion and dislocation climb. Superposition is used to obtain the mechanical fields in terms of the infinite-medium discrete dislocation fields and an image field that enforces the boundary conditions while the point-defect concentration is obtained by solving the stress-dependent diffusion equations on the same finite-element grid. Core-level boundary conditions for the concentration field are avoided by invoking an approximate, yet robust kinetic law. Aspects of the formulation are general but its implementation in a simple plane strain model enables the modeling of high-temperature phenomena such as creep, recovery and relaxation in crystalline materials. With emphasis laid on lattice vacancies, the creep response of planar single crystals in simple tension emerges as a natural outcome in the simulations. A large number of boundary-value problem solutions are obtained which depict transitions from diffusional to power-law creep, in keeping with long-standing phenomenological theories of creep. In addition, some unique experimental aspects of creep in small scale specimens are also reproduced in the simulations.

Statistics of dislocation pinning at localized obstacles

Pinning of dislocations at nanosized obstacles like precipitates, voids, and bubbles is a crucial mechanism in the context of phenomena like hardening and creep. The interaction between such an obstacle and a dislocation is often studied at fundamental level by means of analytical tools, atomistic simulations, and finite element methods. Nevertheless, the information extracted from such studies cannot be utilized to its maximum extent on account of insufficient information about the underlying statistics of this process comprising a large number of dislocations and obstacles in a system. Here, we propose a new statistical approach, where the statistics of pinning of dislocations by idealized spherical obstacles is explored by taking into account the generalized size-distribution of the obstacles along with the dislocation density within a three-dimensional framework. Starting with a minimal set of material parameters, the framework employs the method of geometrical statistics with a few simple assumptions compatible with the real physical scenario. The application of this approach, in combination with the knowledge of fundamental dislocation-obstacle interactions, has successfully been demonstrated for dislocation pinning at nanovoids in neutron irradiated type 316-stainless steel in regard to the non-conservative motion of dislocations. An interesting phenomenon of transition from rare pinning to multiple pinning regimes with increasing irradiation temperature is revealed.

Inflating Graphene with Atomic Scale Blisters

Using 80 kV electron beam irradiation we have created graphene blister defects of additional carbon atoms incorporated into a graphene lattice. These structures are the antithesis of the vacancy defect with blister defects observed to contain up to six additional carbon atoms. We present aberration-corrected transmission electron microscopy data demonstrating the formation of a blister from an existing divacancy, together with further examples that undergo reconfiguration and annihilation under the electron beam. The relative stability of the observed variations of blister are discussed and considered in the context of previous calculations. It is shown that the blister defect is seldom found in isolation and is more commonly coupled with dislocations where it can act as an intermediate state, permitting dislocation core climb without the atom ejection from the graphene lattice required for nonconservative motion.

Creep behaviour of a new air-hardenable intermetallic Ti–46Al–8Ta alloy

Creep behaviour of a new cast air-hardenable intermetallic Ti–46Al–8Ta (at.%) alloy was investigated. Constant load tensile creep tests were performed at initial applied stresses ranging from 200 to 400 MPa in the temperature range from 973 to 1073 K. The minimum creep rate is found to depend strongly on the applied stress and temperature. The power law stress exponent of the minimum creep rate is n = 5.8 and the apparent activation energy for creep is calculated to be Q a = (382.9 ± 14.5) kJ/mol. The kinetics of creep deformation of the specimens tested to a minimum creep rate (creep deformation about 2%) is suggested to be controlled by non-conservative motion of dislocations in the γ(TiAl) matrix. Besides dislocation mechanisms, deformation twinning contributes significantly to overall measured strains in the specimens tested to fracture. The initial γ(TiAl) + α 2(Ti 3Al) microstructure of the creep specimens is unstable and transforms to the γ + α 2 + τ type during creep. The particles of the τ phase are preferentially formed along the grain and lamellar colony boundaries.

Erratum: First-order nonconservative motion of optically trapped nonspherical particles [Phys. Rev. E82, 031141 (2010)

Received 27 October 2010DOI:https://doi.org/10.1103/PhysRevE.82.059903©2010 American Physical Society

First-order nonconservative motion of optically trapped nonspherical particles

It is well known that optical force fields are not conservative. This has important consequences for the thermal motion of optically trapped dielectric spheres. In particular, the spheres do not reach thermodynamic equilibrium. Instead, a steady state is achieved in which the stochastic trajectory contains an underlying deterministic bias toward cyclic motion, and the energy of the sphere deviates from that implied by the equipartition theorem. Such effects are second order and only observed at low trap powers when the sphere is able to explore regions of the trap beyond the linear regime. Analogous effects may be expected for particles of less than spherical symmetry. However, in this case the effects are first order and depend on the linear term in the optical force field. As such they are not suppressed by increases in beam power, although the frequency and amplitude of the cyclic motion will be affected by it. In this paper, we present an analysis of the first-order nonconservative behavior of nonspherical particles in optical traps. The analysis is supported by optical force calculations and brownian dynamics simulations of dielectric microrods held vertically in gaussian optical traps.