Two-dimensional Discrete Dislocation Research Articles

Construction of a Two-Dimensional Discrete Dislocation Model to Describe the Plastic Deformation Process of a Single Crystal

Construction of a Two-Dimensional Discrete Dislocation Model to Describe the Plastic Deformation Process of a Single Crystal

A discrete dislocation dynamics framework for modeling polycrystal plasticity with hardening

A two-dimensional discrete dislocation dynamics (DDD) framework is extended to model plastic deformation in polycrystalline materials having realistic grain morphology in finite geometries. The formulation includes constitutive rules to represent intra-grain dislocation mechanisms as well as dislocation/slip transmission across grain boundaries. Rules for intra-granular interactions mimic key three-dimensional dislocation mechanisms, such as line tension and dynamic junction and source formation, in addition to other short-range dislocation interactions. Various effects of idealized three-dimensional dislocation mechanisms on the mechanical response in tension are explored. The scaling of flow strength with grain size is investigated with and without slip transmission at grain boundaries. To aid in the analysis, spatial mappings of relevant stress components and geometrically necessary dislocation density are produced, which bring out competing dislocation-based strengthening mechanisms leading to the observed grain size effect in polycrystals.

Effect of hydrogen on the dislocation evolution in an aluminum alloy under cyclic loading by two-dimensional discrete dislocation dynamics

The aim of this research is to reveal the effect of hydrogen on fatigue crack initiation in an aluminum alloy. First, the effect of hydrogen on the dislocation movement, that is, the hydrogen-enhanced localized plasticity (HELP), was modeled. Two mechanisms of HELP were modeled: reduction of the dislocation interaction stress and an increase of the dislocation velocity by hydrogen. Next, simulations of the early stage of crack initiation were performed by discrete dislocation dynamics with the HELP model. The simulation results indicated that HELP accelerates fatigue crack initiation and the increase of the dislocation velocity by hydrogen is the main mechanism of HELP.

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.

Dislocation avalanches from strain-controlled loading: A discrete dislocation dynamics study.

We study strain-controlled plastic deformation of crystalline solids via two-dimensional discrete dislocation dynamics simulations. To this end, we characterize the average stress-strain curves as well as the statistical properties of strain bursts and the related stress drops as a function of the imposed strain rate and the stiffness of the specimen-machine system. The dislocation system exhibits strain-rate sensitivity such that a larger imposed strain rate results in a higher average stress at a given strain. In the limit of small strain rate and driving spring stiffness, the sizes and durations of the dislocation avalanches are power law distributed up to a cutoff scale, and exhibit temporally asymmetric average shapes. We discuss the dependence of the results on the driving parameters and compare our results to those from previous simulations where quasistatic stress-controlled loading was used.

On the origin of microstructural discontinuities in sliding contacts: A discrete dislocation plasticity analysis

Two-dimensional discrete dislocation plasticity (DDP) calculations that simulate single crystal films bonded to a rigid substrate under sliding by a rigid sinusoid-shaped asperity are performed with various contact sizes. The contact between the thin film and the asperity is established by a preceding indentation and modelled using a cohesive zone method (CZM), whose behavior is governed by a traction-displacement relation. The emergence of microstructural changes observed in sliding tests has been interpreted as a localized lattice rotation band produced by the activity of dislocations underneath the contact. The depth of the lattice rotation band is predicted to be well commensurate with that observed in the corresponding tests. Furthermore, the dimension and magnitude of the lattice rotation band have been linked to the sliding distance and contact size. This research reveals the underpinning mechanisms for the microstructural changes observed in sliding tests by explicitly modelling the dislocation patterns and highly localized plastic deformation of materials under various indentation and sliding scenarios.

Effective mobility of BCC dislocations in two-dimensional discrete dislocation plasticity

Two-dimensional discrete dislocation plasticity (2D-DDP) has shown to be a powerful tool for studying micro-plasticity problems such as size effects in single crystals, fracture of bimaterial interfaces, delamination of thin films, fatigue crack growth etc. The power of 2D-DDP lies in the application of edge dislocation dipoles as the vehicle for plastic slip: the loss of accuracy in the description of dislocation structures is counter-balanced by its simplicity and the possibility to reach larger plastic strains. The constitutive rules for dislocation evolution in 2D-DDP used so far are tuned to FCC crystals and need to be modified to be used for BCC materials. One of the key challenges in extending the method to BCC materials is that, contrary to FCC, the mobilities of edge and screw dislocations in BCC crystals differ vastly from each other, so that the screw mobility will be rate limiting the plastic slip. Thus, a method is required to map the edge and screw mobilities of dislocation loops into an effective mobility to be used in 2D. To do so, we here propose a 3D-to-2D procedure that is based on the notion of conservation of in-plane plastic strain rate. The consequence of this approach is that the effective 2D mobility for FCC crystals is not simply equal to the uniform mobility of a dislocation loop, as has been assumed by all 2D models to date, but also on the size of edge dipoles. In order to assess the consequences of this departure from the current literature, we considered a few key problems involving plasticity size effects and crack growth, and compared the predictions assuming constant mobility versus the proposed effective mobility. After observing that, overall, the predictions do not deviate substantially, we proceed with application of the 3D-to-2D procedure to compute the effective 2D mobility for BCC materials based on their edge and screw mobilities. The validation of the approach is done by comparison of the predicted rate sensitivity of polycrystalline iron with the experimental rate sensitivity at room temperature, which are found to be in fairly good agreement.

Theoretical framework for predicting solute concentrations and solute-induced stresses in finite volumes with arbitrary elastic fields

A theoretical model for computing the interstitial solute concentration and the interstitial solute-induced stress field in a three-dimensional finite medium with any arbitrary elastic fields was developed. This model can be directly incorporated into two-dimensional or three-dimensional discrete dislocation dynamics simulations, continuum dislocation dynamics simulations, or crystal plasticity simulations. Using this model, it is shown that a nano-hydride can form in the tensile region below a dissociated edge dislocation at hydrogen concentration as low as χ0=5×10−5, and its formation induces a localized hydrogen elastic shielding effect that leads to a lower stacking fault width for the edge dislocation. Additionally, the model also predicts the segregation of hydrogen at Σ109(13 7 0)/33.4∘ symmetric tilt grain boundary dislocations. This segregation strongly alters the magnitude of the shear stresses at the grain boundary, which can subsequently alter dislocation-grain boundary interactions and dislocation slip transmissions across the grain boundary. Moreover, the model also predicts that the hydrogen concentration at a mode-I central crack tip increases with increasing external loading, higher intrinsic hydrogen concentration, and/or larger crack lengths. Finally, linearized approximate closed-form solutions for the solute concentration and the interstitial solute-induced stress field were also developed. These approximate solutions can effectively reduce the computation cost to assess the concentration and stress field in the presence of solutes. These approximate solutions are also shown to be a good approximation when the positions of interest are several nanometers away (i.e. long-ranged elastic interactions) from stress singularities (e.g. dislocation core and crack tip), for low solute concentrations, and/or at high temperatures.

Elucidating the dual role of grain boundaries as dislocation sources and obstacles and its impact on toughness and brittle-to-ductile transition

In this paper, we resolve the role of grain boundaries on toughness and the brittle-to-ductile transition. On the one hand, grain boundaries are obstacles for dislocation glide. On the other hand, the intersection points of grain boundaries with the crack front are assumed to be preferred dislocation nucleation sites. Here, we will show that the single contributions of grain boundaries (obstacles vs. source) on toughness and the brittle-to-ductile transition are contradicting, and we will weight the single contributions by performing carefully designed numerical experiments by means of two-dimensional discrete dislocation dynamics modelling. In our parameter studies, we vary the following parameters: (i) the mean free path for dislocation glide, δ, combined with (ii) the (obstacle) force of the grain boundary, ϕ, and (iii) the dislocation source spacing along the crack front, λ. Our results show that for materials or microstructures for which the mean distance of the intersection points of grain boundaries with the crack front is the relevant measure for λ, a decrease of grain size results in an increase of toughness. The positive impact of grain boundaries outweighs the negative consequences of dislocation blocking. Furthermore, our results explain the evolving anisotropy of toughness in cold-worked metals and give further insight into the question of why the grain-size-dependent fracture toughness passes through a minimum (and the brittle-to-ductile transition temperature passes through a maximum) at an intermediate grain size. Finally, a relation of the grain-size-dependence of fracture toughness in the form of K(dδ, dλ) = KIC + kdδ0.5/dλ is deduced.

Effects of Grain Size, Orientation, and Source Density on Dislocation Configurational Energy Density

The effects of grain size, source density, and misorientations on the dislocation configurational energy area density are investigated using two-dimensional discrete dislocation plasticity. Grain boundaries are modeled as impenetrable to dislocations. The considered grain size ranges from $$ 0.4\;\upmu{\text{m}}^{2} $$ to $$ 8.0\;\upmu{\text{m}}^{2} $$ . The configurational energy area density displays a strong size dependence, similar to the stress response. Two sets of materials are considered, with low and high source/obstacle density. The high-source-density specimens exhibit negative configurational energy, implying that the dislocation structure is more stable than for isolated dislocations . The contribution of misorientation to the configurational energy density is analyzed using specimens with a single orientation or a checkerboard arrangement. The configurational energy density is found not only to depend on the dislocation spacing but also to be related to the local stress states. Low source densities lead to higher (positive) configurational energy densities.

Universality Class of Nanocrystal Plasticity: Localization and Self-Organization in Discrete Dislocation Dynamics.

The universality class of the avalanche behavior in plastically deforming crystalline and amorphous systems has been commonly discussed, despite the fact that the microscopic defect character in each of these systems is different. In contrast to amorphous systems, crystalline flow stress increases dramatically at high strains and/or loading rates. We perform simulations of a two-dimensional discrete dislocation dynamics model that minimally captures the phenomenology of nanocrystalline deformation. In the context of this model, we demonstrate that a classic rate dependence of dislocation plasticity at large rates (>10^{3}/s) fundamentally controls the system's statistical character as it competes with dislocation nucleation: At large rates, the behavior is statistically dominated by long-range correlations of "dragged" mobile dislocations. At small rates, plasticity localization dominates in small volumes and a spatial integration of avalanche behavior takes place.

Mimicking complex dislocation dynamics by interaction networks

Two-dimensional discrete dislocation models exhibit complex dynamics in relaxation and under external loading. This is manifested both in the time-dependent velocities of individual dislocations and in the ensemble response, the strain rate. Here we study how well this complexity may be reproduced using so-called Interaction Networks, an Artificial Intelligence method for learning the dynamics of complex interacting systems. We test how to learn such networks using creep data, and show results on reproducing individual and collective dislocation velocities. The quality of reproducing the interaction kernel is discussed.

Discrete dislocation dynamics simulations of nanoindentation with pre-stress: Hardness and statistics of abrupt plastic events

Nanoindentation of crystalline materials has been thought as a primarily surface-driven technique that is not able to probe bulk mechanical properties directly, such as material yield strength or bulk plastic flow rates. We elucidate this question through extensive discrete dislocation plasticity simulations of nanoindentation on a single crystal. We consider the competition between nanoindentation and tensile loading (pre-stress) towards crystal plasticity. For this purpose, we study a two-dimensional discrete dislocation model where indentation is performed by using cylindrical indentation with varying radius under both displacement and load control. We focus on the behavior of the hardness and pop-in event statistics during nanoindentation under various pre-stress levels and we correlate them to the spatially correlated dislocation microstructure behavior. At small indentation depths (relative to other microstructural or tip length scales), we find that the hardness is inversely dependent on the plastic strain/dislocation density induced by the applied tensile pre-stress; consequently, we argue that small-depth indentation may be useful for identifying bulk plastic yielding behavior prior to indentation. In contrast, for larger indentation depths, pre-stress has no effect on hardness. However, effect of pre-stress can be revealed through plastic events statistics, both in load and displacement controlled protocols. Moreover, post-indentation surface morphology clearly shows the effect of the pre-stress.

Stress-Driven Migration of Low-Angle Grain Boundaries Near Crack Tips in Nanocomposites Containing Incoherent Nanoinclusions

Abstract Theoretical model describing stress-driven migration of low-angle grain boundaries (GBs) in the vicinity of growing crack in metal matrix nanocomposites with reinforcing (metallic or ceramic) incoherent nanoinclusions is proposed. Using two-dimensional discrete dislocation dynamics approach profiles of migrating GBs are analytically calculated and critical stress for transition into unstable migration mode is found. It is shown that the presence of crack always promotes stress-driven migration and thus grain growth.

Dislocation evolution during plastic deformation: Equations vs. discrete dislocation dynamics study

Equations for dislocation evolution bridge the gap between dislocation properties and continuum descriptions of plastic behavior of crystalline materials. Computer simulations can help us verify these evolution equations and find their fitting parameters. In this paper, we employ discrete dislocation dynamics to establish a continuum-based model for the evolution of the dislocation structure in polycrystalline thin films. Expressions are developed for the density of activated dislocation sources, as well as dislocation nucleation and annihilation rates. We demonstrate how size effect naturally enters the evolution equation. Good agreement between the simulation and the model results is obtained. The current approach is based on a two-dimensional discrete dislocation dynamics model but can be extended to three-dimensional models.

Dynamics of threading dislocations in porous heteroepitaxial GaN films

Behavior of threading dislocations in porous heteroepitaxial gallium nitride (GaN) films has been studied using computer simulation by the two-dimensional discrete dislocation dynamics approach. A computational scheme, where pores are modeled as cross sections of cylindrical cavities, elastically interacting with unidirectional parallel edge dislocations, which imitate threading dislocations, is used. Time dependences of coordinates and velocities of each dislocation from dislocation ensembles under investigation are obtained. Visualization of current structure of dislocation ensemble is performed in the form of a location map of dislocations at any time. It has been shown that the density of appearing dislocation structures significantly depends on the ratio of area of a pore cross section to area of the simulation region. In particular, increasing the portion of pores surface on the layer surface up to 2% should lead to about a 1.5-times decrease of the final density of threading dislocations, and increase of this portion up to 15% should lead to approximately a 4.5-times decrease of it.

Statistical model of rough surface contact accounting for size-dependent plasticity and asperity interaction

The work by Greenwood and Williamson (GW) has initiated a simple but effective method of contact mechanics: statistical modeling based on the mechanical response of a single asperity. Two main assumptions of the original GW model are that the asperity response is purely elastic and that there is no interaction between asperities. However, as asperities lie on a continuous substrate, the deformation of one asperity will change the height of all other asperities through deformation of the substrate and will thus influence subsequent contact evolution. Moreover, a high asperity contact pressure will result in plasticity, which below tens of microns is size dependent, with smaller being harder. In this paper, the asperity interaction effect is taken into account through substrate deformation, while a size-dependent plasticity model is adopted for individual asperities. The intrinsic length in the strain gradient plasticity (SGP) theory is obtained by fitting to two-dimensional discrete dislocation plasticity simulations of the flattening of a single asperity. By utilizing the single asperity response in three dimensions and taking asperity interaction into account, a statistical calculation of rough surface contact is performed. The effectiveness of the statistical model is addressed by comparison with full-detail finite element simulations of rough surface contact using SGP. Throughout the paper, our focus is on the difference of contact predictions based on size-dependent plasticity as compared to conventional size-independent plasticity.

Dislocation dynamic simulation of dislocation pattern evolution in the early fatigue stages of aluminium single crystal

The two-dimensional discrete dislocation dynamics simulations under fully periodic boundary conditions has been employed to study the dislocation pattern evolution in the early stages of fatigue in aluminium single crystal. Long-range force among of dislocations is solved by line elasticity model, and short-range force is obtained by constitutive equations of dislocation nucleation, slip, pileup and annihilate. Dislocation movement mechanisms of single-slip-oriented and multi-slip systems are simulated and the evolution process of fatigue pattern is revealed. The result shows that dislocation quantity and microstructure strongly depend on external load and internal configuration. The dislocation pattern of single-slip-oriented generate matrix wall, and positive dislocations are vertical alignment and negative dislocations are at the angle of 45° with slip oriented in the initial stage. For multi-slip system, maze structure of dislocations is produced during dislocation multiplication, which eventually transforms into persistent slip band. The result is consistent with the existing experiment.

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.

Collective dynamics of dislocations interacting with mobile solute atoms

We study the effect of diffusing solute atoms on the collective dynamics of dislocations in plastically deforming crystals, by simulating a two-dimensional discrete dislocation dynamics model with solute atoms included. We employ various protocols to apply the external stress, including constant, oscillatory and quasistatically increasing stress, and study the resulting dynamics for various values of the solute mobility, temperature, and interaction strength with the dislocations. The values of these parameters dictate if Cottrell clouds are formed around the dislocations, and whether the dislocations are able to drag them along as they move. The relevant solute-induced processes include a temporally increasing average Cottrell cloud size due to cloud merging during the evolution of the dislocation structures subject to constant stresses, and a crossover between a solute-free ‘phase’ and a regime where solute drag is important for cyclic stresses, controlled by the solute mobility and temperature. Statistics of deformation bursts under quasistatic loading exhibit atypical scaling where the average burst size is directly proportional to its duration, and are also affected by solute-induced strain hardening in the high-stress regime.