Discrete Dislocation Plasticity Research Articles

On the interaction of grain-scale and hydride-scale stresses in hydrogen enriched zirconium alloy nuclear cladding via combined discrete dislocation plasticity and crystal plasticity finite element modelling

The interaction of Zircaloy fuel cladding components with coolant water in a nuclear reactor leads to embrittlement and potentially delayed hydride cracking (DHC). We explore rate controlling mechanisms for the detrimental DHC process via Discrete Dislocation Plasticity (DDP) modelling of an intragranular δ-hydride, informed by Crystal Plasticity Finite Element (CPFE) analysis of a notched Zircaloy-4 (Zr-4) polycrystal. It is believed that nano-hydride plasticity occurs under a background (polycrystalline) stress state that depends on the grain-scale stress re-distribution associated with plastic deformation at a notch. We find that depending on grain size the background stresses can enhance plasticity during hydride growth (cooling), enhancing the residual hydrostatic stresses on hydride dissolution (heating), which encourages local hydrogen accumulation and re-precipitation. This ‘memory effect’ can be enhanced further by obstacles preventing dislocations from gliding backwards and annihilating during dissolution, highlighting that the discrete nature of plasticity can play important role in the DHC process. Our analysis provides a stepping-stone to modelling interacting nano-hydrides and irradiation effects for supporting the design of better nuclear materials.

A Perspective on Plasticity, Dissipation and the Second Law of Thermodynamics

Abstract The requirement of a non-negative dissipation rate for all possible deformation histories is generally imposed on plastic constitutive relations. This is a constraint analogous to the Coleman–Noll [Coleman, B. D., and Noll, W., 1964, “The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity,” Arch. Ration. Mech. Anal., 13, pp. 167–178. 10.1007/BF01262690] postulate that the Clausius–Duhem inequality needs to be satisfied for all possible deformation histories. The physical basis for the Clausius–Duhem inequality is as a statistical limit for a large number of discrete events for a long time and is not a fundamental physical requirement for small systems for a short time. The relation between the requirement of a non-negative dissipation rate and the Clausius–Duhem inequality is considered. The consequences of imposing a non-negative dissipation rate for all possible deformation histories are illustrated for: (i) a single crystal plasticity framework that accounts for elastic lattice curvature changes as well as elastic lattice straining and (ii) for discrete defect theories of plasticity, with attention specifically on discrete dislocation plasticity for crystalline solids and discrete shear transformation zone (STZ) plasticity for amorphous solids. Possible less restrictive conditions on the evolution of dissipation in plasticity formulations are considered as are implications for stability. The focus is on open questions and issues.

On the role of plastic relaxation in stress assisted grain boundary oxidation

The influence of plasticity on the high-temperature stress-assisted grain boundary oxidation of nickel-based superalloys used in applications such as turbine rotor discs is investigated using the method of discrete dislocation plasticity (DDP). The misfit stress fields of nib-shaped intrusions are captured by a continuous distribution of edge dislocations whose extra half planes represent the volumetric misfit of the oxide, which is implemented in a planar formulation of DDP by invoking the linear superposition principle. DDP simulations show that stresses generated by an intrusion several microns or more in size are large enough to generate dislocation pileups with associated stresses at the intrusion interface on the order of 1 GPa, which in turn lead to localized growth and morphology change of the intrusion by stress-assisted diffusion. This morphology change relaxes the compression stress inside the intrusion near the base, and therefore increases the fracture resistances of the intrusion. The effects of applied loading and background plasticity on the growth rate of the intrusion in defect-free and prestrained samples are predicted. It is found that applied tensile stress generally increases grain boundary oxidation, while in prestrained samples the enhancement of the intrusion growth rate by the applied load is insignificant due to dislocation pile-ups ahead of the oxidation process.

Discrete dislocation dynamics simulations of [formula omitted]-type prismatic loops in zirconium

Neutron irradiation of zirconium alloys in light water nuclear reactors generates nano-scale defects in the form of vacancy and interstitial 〈a〉-type prismatic loops which lie in prismatic planes of the sample. The dynamics of idealised conservative square 〈a〉-type prismatic loops have been investigated for a range of loop lengths in the framework of linear isotropic elasticity. Three-dimensional dislocation dynamics (DD) simulations of a dislocation-loop interaction have been performed to investigate the dislocation-loop interaction mechanism. For this purpose, a mobility law developed for hexagonally close-packed materials has been implemented and described in detail. Analytical and numerical calculations have been performed to obtain expressions for the restoring force and angular stability of prismatic loops. These analyses have been used to inform a 2.5D discrete dislocation plasticity (DDP) model in order to emulate realistic prismatic loop physics and improve irradiation hardening simulations. From the 2.5D DDP prismatic loop analyses, it has been observed that the stable angle of smaller sized loops is less sensitive to external stresses compared to that of larger loops, which may have implications for the mechanisms of irradiation hardening. Furthermore, initial 2.5D single-slip simulations predict that prismatic loops cause significantly elevated flow stress that increases with increasing loop density in accord with experimental observations, and that the restraining effect of the out-of-plane loop segments (the restoring force) plays an important role in the strengthening caused by loops.

Slip intermittency and dwell fatigue in titanium alloys: a discrete dislocation plasticity analysis

Slip intermittency and stress oscillations in titanium alloy Ti–7Al–O that were observed using in-situ far-field high energy X-ray diffraction microscopy (ff-HEDM) are investigated using a discrete dislocation plasticity (DDP) model. The mechanistic foundation of slip intermittency and stress oscillations are shown to be dislocation escape from obstacles during stress holds, governed by a thermal activation constitutive law. The stress drop events due to <a>-basal slip are larger in magnitude than those along <a>-prism, which is a consequence of their differing rate sensitivities, previously found from micropillar testing. It is suggested that interstitial oxygen suppresses stress oscillations by inhibiting the thermal activation process. Understanding of these mechanisms is of benefit to the design and safety assessment of jet engine titanium alloys subjected to dwell fatigue.

Discrete defect plasticity and implications for dissipation

Inelastic deformation of solids is almost always, if not always, associated with the evolution of discrete defects such as dislocations, vacancies, twins and shear transformation zones. The focus here is on discrete defects that can be modeled appropriately by continuum mechanics, but where the discreteness of the carriers of plastic deformation plays a significant role. The formulations are restricted to small deformation kinematics and the defects considered, dislocations and discrete shear transformation zones (STZs), are described by their linear elastic fields. In discrete defect plasticity both the stress–strain response and the partitioning between defect energy storage and defect dissipation are outcomes of an initial/boundary value problem solution. For such defects an explicit expression for the dissipation rate is presented and, because there is a length scale associated with the discrete defects, the stress–strain response and the evolution of the dissipation rate can be size dependent. Discrete dislocation plasticity modeling results are reviewed that illustrate the implications of defect dissipation evolution for friction, fracture, fatigue crack growth and thermal softening. Examples are also given of the consequences in constrained shear of three modes of the evolution of discrete defects for the size dependence of the stress–strain response and the associated dissipation. For a purely mechanical formulation the Clausius–Duhem inequality specializes to the requirement that the dissipation rate is non-negative. Requiring a non-negative dissipation rate for all points of a body and for all time imposes restrictions on kinetic relations for the evolution of discrete defects. In statistical mechanics, the Clausius–Duhem inequality can be violated for sufficiently small regions for a sufficiently short time. Explicit kinetic relations for discrete dislocation plasticity dissipation and for discrete STZ plasticity dissipation identify conditions that can lead to a negative dissipation rate. In continuum mechanics, at least in some circumstances, satisfaction of the Clausius–Duhem inequality can be regarded as a stability requirement. Nevertheless, simple one-dimensional continuum calculations illustrate that there can be a negative dissipation rate over a small region and for a short time period with overall stability maintained. Implications for discrete defect plasticity modeling are briefly discussed.

Avalanches and rate effects in strain-controlled discrete dislocation plasticity of Al single crystals

Three-dimensional discrete dislocation dynamics simulations are used to study strain-controlled plastic deformation of face-centered cubic aluminium single crystals. After describing the rate and size dependence of the average stress-strain curves, we study the power-law distributed strain bursts and the average avalanche shapes, and find a universal power-law exponent $\tau \approx 1.0$ for all imposed strain rates and system sizes, characterizing both the event sizes and their durations. We discuss the dependence of our results on loading rate and compare these with previous studies of strain-controlled two-dimensional systems of discrete dislocations as well as of quasistatic stress-controlled loading of aluminium single crystals.

Bayesian optimization of discrete dislocation plasticity of two-dimensional precipitation-hardened crystals

Discovering relationships between materials' microstructures and mechanical properties is a key goal of materials science. Here, we outline a strategy exploiting Bayesian optimization to efficiently search the multidimensional space of microstructures, defined here by the size distribution of precipitates (fixed impurities or inclusions acting as obstacles for dislocation motion) within a simple two-dimensional discrete dislocation dynamics model. The aim is to design a microstructure optimizing a given mechanical property, e.g., maximizing the expected value of shear stress for a given strain. The problem of finding the optimal discretized shape for a distribution involves a norm constraint, and we find that sampling the space of possible solutions should be done in a specific way in order to avoid convergence problems. To this end, we propose a general mathematical approach that can be used to generate trial solutions uniformly at random while enforcing an Euclidean norm constraint. Both equality and inequality constraints are considered. A simple technique can then be used to convert between Euclidean and other Lebesgue $p$-norm (the 1-norm in particular) constrained representations. Considering different dislocation-precipitate interaction potentials, we demonstrate the convergence of the algorithm to the optimal solution and discuss its possible extensions to the more complex and realistic case of three-dimensional dislocation systems where also the optimization of precipitate shapes could be considered.

Discrete dislocation modelling of δ hydrides in Zr: towards an understanding of the importance of interfacial stresses for crack initiation

We carried out discrete dislocation plasticity (DDP) simulations in the infinite-matrix approximation of δ Zr micro-hydrides – including their elastic field - first precipitating in Zr and then subjected to an external stress. We calculate and present the dislocation density, maximum principal shear and stresses, configurational energy density and plastic displacement. It was found that even hydrides as thin as 20 nm have a large impact on dislocation density, plastic deformation and stress build-up, and that this effect increases for increasing hydride thickness. It also appears that the critical location for crack nucleation is at the Zr/hydride interface. In fact, slip bands that develop on one side of the interface, and are stopped by it, are accompanied by the formation of high stresses. The observation that the interfacial stress is the only way in which hydrides are represented in the simulations, with no point obstacles or change in the slip systems at the interface, combined with the observed strong influence of the hydrides on the results, suggests a possible mechanistic explanation for the embrittlement caused by the precipitation of intragranular hydrides. The mechanism is rooted in the volumetric expansion upon precipitation and naturally includes a thickness dependence that is also experimentally observed.

Dislocation modelling of the plastic relaxation and thermal ratchetting induced by zirconium hydride precipitation

The precipitation of hydrides in zirconium alloys is accompanied by a significant and anisotropic volumetric expansion. Previous literature quantified the misfit both theoretically and experimentally, but these values differ greatly; the experimental values are consistently lower. One possibility is that the experimental measurements include the effect of dislocations generated by the hydride, which relax the transformation stresses. To test this hypothesis, it is important to determine the stress field of a hydride and its associated dislocations, combined. A simple planar dislocation model was developed of the hydride—dislocation ensemble in α-Zr. By capturing details of the dislocation structures given in the literature, it is shown in this study that including the interfacial dislocations largely reconciles the predicted and experimental values. Discrete dislocation plasticity is then used to model the diffuse plastic relaxation associated with hydride formation. The effects of plastic relaxation on the equilibrium hydrogen profile, hence the implications for subsequent hydride precipitation, are discussed. In particular, precipitation–dissolution cycles were simulated to calculate the magnitude of the residual hydrostatic tension, which is argued to be the primary cause of the “memory effect” for the re-precipitation of both γ and δ hydrides.

On the origin of plasticity-induced microstructure change under sliding contacts

Discrete dislocation plasticity (DDP) calculations are carried out to investigate the response of a single crystal contacted by a rigid sinusoidal asperity under sliding loading conditions to look for causes of microstructure change in the dislocation structure. The mechanistic driver is identified as the development of lattice rotations and stored energy in the subsurface, which can be quantitatively correlated to recent tribological experimental observations. Maps of surface slip initiation and substrate permanent deformation obtained from DDP calculations for varying contact size and normal load suggest ways of optimally tailoring the interface and microstructural material properties for various frictional loads.

Machine learning reveals strain-rate-dependent predictability of discrete dislocation plasticity

Predicting the behavior of complex systems is one of the main goals of science. An important example is plastic deformation of micron-scale crystals, a process mediated by collective dynamics of dislocations, manifested as broadly distributed strain bursts and significant sample-to-sample variations in the response to applied loading. Here, by combining large-scale discrete dislocation dynamics simulations and machine learning, we study the problem of predicting the fluctuating stress-strain curves of individual small single crystals subject to strain-controlled loading using features of the initial dislocation configurations as input. Our results reveal an intriguing rate dependence of deformation predictability: For small strains predictability improves with increasing strain rate, while for larger strains the predictability vs strain rate relation becomes nonmonotonic. We show that for small strains the rate dependence of deformation predictability can be captured by considering the fraction of dislocations moving against the direction imposed by the external stress, serving as a measure of strain-rate-dependent complexity of the dislocation dynamics. The nonmonotonic predictability vs strain rate relation for large strains is argued to be related to a transition from fluctuating to smooth plastic flow when strain rate is increased.

Evolution of partial dislocation slip–mediated deformation twins in single crystals: a discrete dislocation plasticity model and an analytical approach

Deformation twinning is an essential plastic deformation mechanism that realizes the trade–off between strength and ductility. Twins nucleate and grow by the coordinated slip of partial dislocations on consecutive {111}–type slip planes. The route of twin growth is responsible for the evolution of twin morphology, which affects the non–coplanar dislocation slips by adjusting the twinning–associated mean free path. Incorporating such twinning mechanisms is critical for the accurate modelling and simulation of deformation behavior. In this study, a discrete dislocation plasticity (DDP) model was developed by integrating the source introduction methods and source activation criteria of Shockley partial dislocation. In the model, two twin nucleation mechanisms, i.e., the internal source and surface source, were considered concurrently, and the additional effect of stacking fault energy on the motion of partial dislocations was introduced. The evolution of partial dislocation slip–mediated deformation twins in micron–sized pillars of twinning–induced plasticity steel under uniaxial compression was investigated. The predicted twin morphologies and stress–strain curves from DDP simulation both agreed well with the experimental results, highlighting the inherent characteristics of partial–dislocation–based twinning behavior. The simulation results showed that the formation of nanometer–sized sharp twin tips was caused by the strong interaction between the front and rear dislocations on adjacent slip planes. In addition, a novel analytical model verified with the DDP simulation was proposed by considering the kinetics of the newly formed twin embryos. The competition between new twin activation and near–twin merging in determining the evolution of twin thickness was analyzed using the analytical model. The dependence of flow stress and twin morphology on the density and distribution of internal sources was demonstrated by considering the new twinning route. This research thus advances the understanding of partial dislocation slip–mediated twinning mechanisms.

The effect of strain rate asymmetry on the Bauschinger effect: A discrete dislocation plasticity analysis

A two-dimensional discrete dislocation plasticity model was used to examine the role of strain rate asymmetry on the Bauschinger effect (BE). A simple single crystal single slip plane model was used first to investigate stress–strain response within high strain rate regimes where the rate sensitivity is mainly governed by dislocation nucleation and the free-flight of dislocations. Both the yield strength and the BE of the specimen showed a strong correlation to the forward and reverse deformation rate. Simulations were also carried out on a single crystal with multiple slip systems. The effect of source and obstacle density on the BE and its rate dependence was investigated. We find that the rate dependence of the BE is enhanced by increasing the source density in obstacle-free specimens and is reduced by obstacles when they are present. At low strain rates, where thermally-activated dislocation escape from obstacles becomes the predominant rate-controlling mechanism, the BE of single crystals is only affected by strain rates if the activation energy is low. Grain boundaries in polycrystalline specimens and obstacles with high activation energies act as barriers to dislocation motion, suggesting that the rate dependence under low strain rate regimes is enhanced by decreasing the grain size or reducing the activation energy associated with dislocation escape from obstacles. The effect mechanisms of strain rate asymmetry on the BE can be fundamentally crucial for multiple industrial applications, including lifetime assessments of aero-engine disks, zirconium cladding of nuclear reactor and the spring-back prediction of metal forming.

Effect of shape on void growth: A coupled Extended Finite Element Method (XFEM) and Discrete Dislocation Plasticity (DDP) study

Voids are one of the many material defects present at the microscopic length scale. They are primarily responsible for the formation of cracks and hence contribute to ductile fracture. Circular voids tend to deform into elliptical voids just before their coalescence to form cracks. The principle aim of this study is to investigate the effect of void shape on the micro-mechanism of void growth by using Discrete Dislocation Plasticity simulations. For voided crystals, conventional DDP produces a continuous slip step throughout the material even if a dislocation escapes from a non-convex domain. To overcome this issue, the Extended Finite Element Method (XFEM) is used here to incorporate the displacement discontinuity. Different aspect ratios of elliptical voids are considered under uniaxial and biaxial deformation boundary conditions. The results suggest that voids having the largest surface area tend to have maximum growth rate as compared to void with lower surface area, i.e. “larger is faster”. Under biaxial loading, a higher magnitude of strain hardening, and void growth rate are observed as compared to uniaxial loading. The results also suggest that the orientation of slip planes as well as voids, affect the overall plastic behavior of the voided-ductile material. Furthermore, circular void tends to induce minimum growth rate but have the maximum strain hardening effect as compared to other void shapes under both loading conditions. The results of this study provide a deeper understanding of ductile fracture with applications in manufacturing industry, aerospace industry and in the design of nano/micro-electromechanical devices i.e. NEMS/MEMS.

Effect of dislocation core fields on discrete dislocation plasticity

Discrete dislocation plasticity is a modeling technique that treats plasticity as the collective motion of dislocations. The dislocations are described through their elastic Volterra fields, outside of a cylindrical core region, with a few Burgers vectors of diameter. The contribution of the core fields to the dislocation dynamics is neglected, because it is assumed that their range is too short to be of influence. The aim of this work is to assess the validity of this assumption.In recent ab-initio studies it has been demonstrated that the dislocation core fields are significant up to a distance of ten Burgers vector from the dislocation line. This is a longer range influence than expected and can give rise to changes in the evolving dislocation structure and in the overall response of a plastically deforming body. It is indeed experimentally observed that dislocations pile up against strong interfaces, and that the spacing between dislocations at the front of these pile-ups can be less than ten Burgers vectors.In this work, 2-D discrete dislocation plasticity simulations are performed to investigate the effect of core fields on edge dislocation interactions. The results of the simulations, which include core fields for the first time, show indeed that dislocations that are very closely spaced experience additional glide or climb due to core fields. The effect is however negligible when compared to glide and climb due to Volterra fields or due to the external load.

On the Origin of Plastic Deformation and Surface Evolution in Nano-Fretting: A Discrete Dislocation Plasticity Analysis

Discrete dislocation plasticity (DDP) calculations were carried out to investigate a single-crystal response when subjected to nano-fretting loading conditions in its interaction with a rigid sinusoidal asperity. The effects of the contact size and preceding indentation on the surface stress and profile evolution due to nano-fretting were extensively investigated, with the aim to unravel the deformation mechanisms governing the response of materials subjected to nano-motion. The mechanistic drivers for the material’s permanent deformations and surface modifications were shown to be the dislocations’ collective motion and piling up underneath the contact. The analysis of surface and subsurface stresses and the profile evolution during sliding provides useful insight into damage and failure mechanisms of crystalline materials subject to nano-fretting; this can lead to improved strategies for the optimisation of material properties for better surface resistance under micro- and nano-scale contacts.

Plastic relaxation and solute segregation to β-Nb second phase particles in Zr-Nb alloys: A discrete dislocation plasticity study

There is clear evidence in the literature that iron segregates to the interface of second phase particles (SPPs) in unirradiated Zr-Nb alloys, and that it does not do so in the presence of radiation damage. In this work, a discrete dislocation plasticity model is developed that takes into account the long-range stress field of the SPP interface. A simple analytical model is also outlined, providing an upper bound for estimating the amount of interstitial segregation. The model provides a possible mechanism to explain both the iron segregation to coherent SPPs and its subsequent loss after irradiation. Qualitatively, the model proved to be insensitive to variations of all geometrical and computational parameters, allowing for general conclusions to be drawn. The model suggests that the segregation originates from a tensile field of order 1 GPa induced by the dislocations generated during the plastic relaxation around the SPP. This leads to the six-fold increase in the iron concentration observed in experiments. In the model, the loss of SPP/matrix coherency after irradiation causes the dislocations to drift away from the interface, and the iron concentration is homogenised accordingly. The hydrogen concentration was also predicted and found to be about 50% higher than in the bulk zirconium matrix at room temperature. The computational framework is built to be fast, making possible a statistical analysis on over five hundred simulations for improved reliability of the predictions.

Structural transition and ductility enhancement of a tungsten heavy alloy under high pressure

High pressure structural transformation of 95 W-3.5Ni-1.5Fe alloy was investigated by in-situ high pressure synchrotron radiate-on X-ray diffraction technique. Starting sample was compressed to 52.1 GPa in a diamond anvil cell under hydrostatic condition at ambient temperature. Experimental results demonstrated that the extreme compressive stress led to a set of reversible structural evolution. The grain size, atomic positions, bond distance and unit cell volume exhibited remarkable dependence on high pressure. The equation of state (EOS) together with bulk modulus K0 = 409(11) GPa and K0’ = 6.4(0.8) GPa of tungsten phase and K0 = 221(9) GPa and K0′=7.2(0.8) GPa of matrix phase were obtained based on the unit cell volume data varies with pressure for the first time. In addition, the structure responses of tungsten heavy alloy under high pressure and high temperature were revealed by large volume cubic press. Results indicated that the thermo-mechanical coupling effect facilitated the dynamic recrystallization process. Three-dimensional discrete dislocation plasticity simulations were proposed to illustrate the formation and evolution of screw dislocation in body-center cubic under various hydrostatic pressures. Results demonstrated that the extreme compression conditions fueled the nucleation and propagation of screw dislocations, which can be looked upon as the significant contributors to dislocation density increasing and ductility strengthening.

Avalanche correlations and stress-strain curves in discrete dislocation plasticity

The sequence of deformation bursts during plastic deformation exhibits scale-free features. In addition to the burst or avalanche sizes and the rate of avalanches the process is characterized by correlations in the series which become manifest in the resulting shape of the stress-strain curve. We analyze such features of plastic deformation with two-dimensional and three-dimensional simulations of discrete dislocation dynamics models and we show that only with severe plastic deformation the ensuing memory effects become negligible. The role of past deformation history and dislocation pinning by disorder are studied. In general, the correlations have the effect of reducing the scatter of the individual stress-strain curves around the mean one.