Discrete Dislocation Plasticity Research Articles

Plane-strain discrete dislocation plasticity with climb-assisted glide motion of dislocations

A small-strain two-dimensional discrete dislocation plasticity (DDP) framework is developed wherein dislocation motion is caused by climb-assisted glide. The climb motion of the dislocations is assumed to be governed by a drag-type relation similar to the glide-only motion of dislocations: such a relation is valid when vacancy kinetics is either diffusion limited or sink limited. The DDP framework is employed to predict the effect of dislocation climb on the uniaxial tensile and pure bending response of single crystals. Under uniaxial tensile loading conditions, the ability of dislocations to bypass obstacles by climb results in a reduced dislocation density over a wide range of specimen sizes in the climb-assisted glide case compared to when dislocation motion is only by glide. A consequence is that, at least in a single slip situation, size effects due to dislocation starvation are reduced. By contrast, under pure bending loading conditions, the dislocation density is unaffected by dislocation climb as geometrically necessary dislocations (GNDs) dominate. However, climb enables the dislocations to arrange themselves into lower energy configurations which significantly reduces the predicted bending size effect as well as the amount of reverse plasticity observed during unloading. The results indicate that the intrinsic plasticity material length scale associated with GNDs is strongly affected by thermally activated processes and will be a function of temperature.

Discrete dislocation modeling of fracture in plastically anisotropic metals

The intrinsic lattice resistance to dislocation motion, or Peierls stress, depends on the core structure of the dislocation and is one essential feature controlling plastic anisotropy in materials such as HCP Zn, Mg, and Ti. Here, we implement an anisotropic Peierls model as a friction stress within a 2d discrete dislocation (DD) plasticity model and investigate the role of plastic anisotropy on the crack tip stress fields, crack growth, toughening, and microcracking. First, tension tests for a pure single crystal with no obstacles to dislocation motion are carried out to capture the general flow behavior in pure HCP-like materials having slip on basal and pyramidal planes. Then Mode-I crack growth in such a single crystal of the HCP material is analyzed using the 2d-DD model. Results show that the fracture toughness scales inversely with the tensile yield stress, largely independent of the plastic anisotropy, so that increasing Peierls stress on the pyramidal planes gives decreasing resistance to crack growth, consistent with recent experiments on Zn. Analyzing the results within the framework of Stress Gradient Plasticity concepts shows that the equilibrium dislocation dipole spacing serves as an internal material length scale for controlling fracture toughness. Furthermore, the fracture toughness of materials with flow stress controlled by a Peierls stress (this work) and of materials with flow stress controlled by dislocation obstacles (prior literature) is unified through the Stress Gradient Plasticity concept. Finally, the DD simulations show that local stress concentrations exist sporadically along the pyramidal plane(s) that emanate from the current crack tip, suggesting an origin for experimentally observed basal-plane microcracking near the tip of large cracks.

A discrete dislocation plasticity analysis of a single-crystal semi-infinite medium indented by a rigid surface exhibiting multi-scale roughness

Discrete dislocation plasticity was used to analyse plane-strain indentation of a single-crystal elastic–plastic semi-infinite medium by a rigid surface exhibiting multi-scale roughness, characterised by self-affine (fractal) behaviour. Constitutive rules of dislocation emission, glide and annihilation were used to model short-range dislocation interactions. Dislocation multiplication and the development of subsurface shear stresses due to asperity microcontacts forming between a single-crystal medium and a rough surface were examined in terms of surface roughness and topography (fractal) parameters, slip-plane direction and spacing, dislocation source density, and contact load (surface interference). The effect of multi-scale interactions between asperity microcontacts on plasticity is elucidated in light of results showing the evolution of dislocation structures. Numerical solutions yield insight into plastic flow of crystalline materials in normal contact with surfaces exhibiting multi-scale roughness.

Plastic flattening of a sinusoidal metal surface: A discrete dislocation plasticity study

The plastic flattening of a sinusoidal metal surface is studied by performing plane strain dislocation dynamics simulations. Plasticity arises from the collective motion of discrete dislocations of edge character. Their dynamics is incorporated through constitutive rules for nucleation, glide, pinning and annihilation. By analyzing surfaces with constant amplitude we found that the mean contact pressure is inversely proportional to the wavelength. For small wavelengths, due to interaction between plastic zones of neighboring contacts, the mean contact pressure can reach values that are about 1/10 of the theoretical strength of the material, thus significantly higher than what is predicted by simulations that do not account for size dependent plasticity. Surfaces with the same amplitude to period ratio have a size dependent response, such that if we interpret each period of the sinusoidal wave as the asperity of a rough surface, smaller asperities are harder to be flattened than large ones. The difference between the limiting situations of sticking and frictionless contacts is found to be negligible.

Tensile response of passivated films with climb-assisted dislocation glide

The tensile response of single crystal films passivated on two sides is analysed using climb enabled discrete dislocation plasticity. Plastic deformation is modelled through the motion of edge dislocations in an elastic solid with a lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation incorporated through a set of constitutive rules. The dislocation motion in the films is by glide-only or by climb-assisted glide whereas in the surface passivation layers dislocation motion occurs by glide-only and penalized by a friction stress. For realistic values of the friction stress, the size dependence of the flow strength of the oxidised films was mainly a geometrical effect resulting from the fact that the ratio of the oxide layer thickness to film thickness increases with decreasing film thickness. However, if the passivation layer was modelled as impenetrable, i.e. an infinite friction stress, the plastic hardening rate of the films increases with decreasing film thickness even for geometrically self-similar specimens. This size dependence is an intrinsic material size effect that occurs because the dislocation pile-up lengths become on the order of the film thickness. Counter-intuitively, the films have a higher flow strength when dislocation motion is driven by climb-assisted glide compared to the case when dislocation motion is glide-only. This occurs because dislocation climb breaks up the dislocation pile-ups that aid dislocations to penetrate the passivation layers. The results also show that the Bauschinger effect in passivated thin films is stronger when dislocation motion is climb-assisted compared to films wherein dislocation motion is by glide-only.

NM-JP-7 Homogenization analysis of plastic size effects using discrete dislocation plasticity approach

Homogenization analysis of lamellar composites consisting of elastic and elastic-plastic layers is performed using a discrete dislocation plasticity approach.Plastic size effects are caused by dislocation pile-ups at interfaces.It is shown that the strain hardening in elastic-plastic layers arises depending on the thickness of elastic layers as well as the gap between slip planes in adjacent elastic-plastic layers.In a case where the thickness of elastic layers is several dozen nm,the strain hardening in elastic-plastic layers reduces as the gap of the slip planes decreases.It is found that this effect is attributed to the long range stress due to the pile-ups in adjacent elastic-plastic layers.

527 離散転位塑性法を用いたすべり系間相互作用のミクロ・マクロ解析(OS8-1 ナノスケールフロンティアにおける材料と構造の新たな強度信頼評価手法の開拓1,OS8 ナノスケールフロンティアにおける材料と構造の新たな強度信頼評価手法の開拓)

527 離散転位塑性法を用いたすべり系間相互作用のミクロ・マクロ解析(OS8-1 ナノスケールフロンティアにおける材料と構造の新たな強度信頼評価手法の開拓1,OS8 ナノスケールフロンティアにおける材料と構造の新たな強度信頼評価手法の開拓)

Multiscale modeling of dislocation/grain-boundary interactions: III. 60° dislocations impinging on Σ3, Σ9 and Σ11 tilt boundaries in Al

The interactions between 60° dislocation pile-ups with grain boundaries (GBs) are studied using multiscale modeling. Careful quantitative analyses of complex processes associated with 60° dislocation absorption and transmission phenomena at Σ3, Σ9 and Σ11 symmetric tilt boundaries in Al are interpreted in terms of a set of modified Lee–Robertson–Birnbaum (MLRB) criteria. Our results and the MLRB criteria (i) explain experimental observations, (ii) rationalize new mechanisms such as deformation twinning and formation of extended stacking faults, (iii) show that reactions can be controlled more strongly by the leading partial of an incoming dislocation rather than the full Burgers vector and (iv) demonstrate that non-Schmid stresses, e.g. shear and compressive stresses along the GB, GB dislocation processes and step-height changes on the GB all influence the critical nucleation stress, but to differing degrees among different tilt boundaries. The MLRB criteria do not capture the effects of local GB structure that can also influence behavior. Quantitative metrics based on the MLRB criteria are formulated, using the simulation results, for various absorption and transmission phenomena. These metrics can be used as input into mesoscale models such as discrete dislocation plasticity, so that atomic-scale observations can inform higher-scale predictions plasticity.

Plastic size effect analysis of lamellar composites using a discrete dislocation plasticity approach

Plastic size effect analysis of lamellar composites consisting of elastic and elastic–plastic layers is performed using a discrete dislocation plasticity approach, which is based on applying periodic homogenization to the superposition method for discrete dislocation plasticity. In this approach, the decomposition of displacements into macro and perturbed components circumvents the calculation of superposing displacement fields induced by dislocations in an infinitely homogeneous medium, resulting in two periodic boundary value problems specialized for the analysis of representative volume elements. The present approach is verified by analyzing a model lamellar composite that includes edge dislocations fixed at interfaces. The plastic size effects due to dislocation pile-ups at interfaces are also analyzed. The analysis shows that, strain hardening in elastic–plastic layers arises depending on two factors, namely the thickness and stiffness of elastic layers; and the gap between slip planes in adjacent elastic–plastic layers. In the case where the thickness of elastic layers is several dozen nm, strain hardening in elastic–plastic layers is restrained as the gap of the slip planes decreases. This particular effect is attributed to the long range stress due to pile-ups in adjacent elastic–plastic layers.

A Discrete Dislocation Plasticity Analysis of a Single-Crystal Half-Space Indented by a Rigid Cylinder

Elastic-plastic indentation of a single-crystal half-space by a rigid cylinder was analyzed by discrete dislocation plasticity. Short-range dislocation interactions were modeled by a set of constitutive rules of dislocation emission, glide, pinning (by obstacles), and annihilation. The occurrence of the first dislocation dipole, multiplication of dislocations, and evolution of subsurface stress field were examined in terms of contact load, dislocation source density, slip-plane distance and orientation angle, and indenter radius. In the presence of defects (dislocation sources), the critical load for dislocation initiation is less than that of a defect-free medium and depends on dislocation source density, slip-plane distance, and indenter radius. The critical indenter radius resulting in deformation under the theoretical material strength is determined from numerical results, and the role of dislocation obstacles is interpreted in terms of their spatial density. Simulations provide insight into yielding and plastic deformation of indented single-crystal materials, and establish a basis for developing coarse-grained plasticity models of localized contact deformation in polycrystalline solids.

OS1513 多結晶体モデルの離散転位塑性シミュレーション(OS15-3 ひずみ速度効果,OS-15 微視構造を有する材料の変形と破壊)

Work hardening character of polycrystalline material is studied by using a discrete dislocation plasticity. Grain boundaries are modeled as barriers. Grain size dependences of yield stress and work hardening factor are investigated for model of different grain size, and the result is compared with that of single crystal model. Dislocation pile-up is observed and it is shown that the work hardening property is more significant in fine grain model.

Plane-strain discrete dislocation plasticity incorporating anisotropic elasticity

The increasing application of plane-strain testing at the (sub-) micron length scale of materials that comprise elastically anisotropic cubic crystals has motivated the development of an anisotropic two-dimensional discrete dislocation plasticity (2D DDP) method. The method relies on the observation that plane-strain plastic deformation of cubic crystals is possible in specific orientations when described in terms of edge dislocations on three effective slip systems. The displacement and stress fields of such dislocations in an unbounded anisotropic crystal are recapitulated, and we propose modified constitutive rules for the discrete dislocation dynamics of anisotropic single crystals. Subsequently, to handle polycrystalline problems, we follow an idea of O’Day and Curtin (J. Appl. Mech. 71 (2004) 805–815) and treat each grain as a plastic domain, and adopt superposition to determine the overall response. This method allows for a computationally efficient analysis of micro-scale size effects. As an application, we study freestanding thin copper films under plane-strain tension. First, the computational framework is validated for the special case of isotropic thin films modeled by means of a standard 2D DDP method. Next, predictions of size dependent plastic behavior in anisotropic columnar-grained thin films with varying thickness/grain size are presented and compared with the isotropic results.

Discrete dislocation analysis of the mechanical response of silicon carbide reinforced aluminum nanocomposites

Metal matrix composites reinforced with ceramic particles exhibit improved thermo-mechanical properties compared to the host metal. In recent years, experiments have shown that reducing the size of the particles to the nanoscale dramatically increases the mechanical strength of these composites even at low particle volume fractions. However, numerical simulations using classical plasticity laws are unable to capture these trends correctly since these classical frameworks are length-scale independent. In this paper, the mechanical response of aluminum matrix reinforced with nanosized silicon carbide is analyzed using plane strain, discrete dislocation plasticity. In the simulations, plasticity arises from the collective motion of dislocations within an elastic medium. Constitutive rules are prescribed for nucleation, motion and annihilation of the dislocations. Calibration of various parameters or quantities which affect these processes is performed. The numerical results show improvements in the mechanical strength of the nanocomposite material with increasing particle volume fraction and decreasing particle size.

Cylindrical nano-indentation on metal film/elastic substrate system with discrete dislocation plasticity analysis: A simple model for nano-indentation size effect

The cylindrical nano-indentation on metal film/elastic substrate is computationally studied using two-dimensional discrete dislocation plasticity combined with the commercial software ANSYS®, with a focus on the storage volume for geometrically necessary dislocations (GNDs) inside the films and the nano-indentation size effect (NISE). Our calculations show that almost all GNDs are stored in a rectangular area determined by the film thickness and the actual contact width. The variations of indentation contact width with indentation depth for various film thicknesses and indenter radii are fitted by an exponential relation, and then the GND density underneath the indenter is estimated. Based on the Taylor dislocation model and Tabor formula, a simple model for the dependence of the nano-indentation hardness of the film/substrate system on the indentation depth, the indenter radius and the film thickness is established, showing a good agreement with the present numerical results.

A hybrid multiscale computational framework of crystal plasticity at submicron scales

Mechanical behavior for crystals at submicron-to-nanometer scales shows many atypical plastic phenomena. In this paper, a new hybrid multiscale simulation method named DD-FEM is developed by coupling the 3D discrete dislocation dynamics (DDD) simulation with the finite element method (FEM) under elasto-viscoplasticity continuum model. In this methodology, the discrete dislocation plasticity in a finite crystal is solved under a completed continuum mechanics framework. Our approach is competent at dealing with the large and non-linear deformation by communicating deformation information between DDD and FEM. As an application, the deformation of micropillar under uniaxial compression has been predicted. Our results show that the source-truncation hardening and mobile dislocation exhaustion hardening are the main mechanisms for flow intermittency, and stress heterogeneity plays a key role in the observed strain hardening. The effects of contact boundary condition between the indenter and the upper surface of the specimen, as well as the taper of the pillar are also investigated.

Modelling dislocation transmission across tilt grain boundaries in 2D

A line tension approach is reviewed to model slip transfer across tilt grain boundaries (GBs). It is implemented as a constitutive rule in the already existing two-dimensional discrete dislocation plasticity model of polycrystalline thin films. GBs are treated as obstacles to dislocation motion but with a finite strength. Dislocation transmission across a GB is potentially allowed, provided that three known geometrical criteria are satisfied. If allowed, the motion of the dislocation across the GB is enabled by dynamically creating a GB source near the GB, the stress on which must exceed the boundary strength. To separate transmission effects from other discrete dislocation features of the plastic deformation of thin films, comparisons are carried out by modelling GBs as impenetrable barriers to dislocation motion. It is shown that slip transfer tends to make the film softer if the density of bulk sources is sufficiently low. Since the position of the boundary source lies within the high-stress region of dislocation pile-ups, once activated the transmission process becomes independent of the source strength. In addition, the Bauschinger effect (BE) in thin films has been studied and a decrease of the BE is seen when dislocation transmission across GBs is enabled.

Bauschinger effect in unpassivated freestanding thin films

Two-dimensional (2D) discrete dislocation plasticity simulations are carried out to investigate the Bauschinger effect (BE) in freestanding thin films. The BE in plastic flow of polycrystalline materials is generally understood to be caused by inhomogeneous deformation during loading, leading to residual stress upon unloading. This inhomogeneity can be caused by dislocation pile-ups, variations in texture, grain orientations, and grain size. To study the BE, columnar-grained films as well as films with multiple grains across the thickness are considered. The film is modeled in a 2D framework by a unit cell consisting of an array of grains with different orientation. In order to capture the interaction among grains, we motivate and explore the use of an affine deformation assumption on the grain level to mimic the three-dimensional geometry in this framework. It is shown that the dispersion of grain size in a film together with the size-dependence of yield strength leads to significant BEs in bare films. Quantitative comparison of simulations with experimental data is provided.

Compliant interfaces: A mechanism for relaxation of dislocation pile-ups in a sheared single crystal

Discrete dislocation plasticity models and strain-gradient plasticity theories are used to investigate the role of interfaces in the elastic–plastic response of a sheared single crystal. The upper and lower faces of a single crystal are bonded to rigid adherends via interfaces of finite thickness. The sandwich system is subjected to simple shear, and the effect of thickness of crystal layer and of interfaces upon the overall response are explored. When the interface has a modulus less than that of the bulk material, both the predicted plastic size effect and the Bauschinger effect are considerably reduced. This is due to the relaxation of the dislocation stress field by the relatively compliant surface layer. On the other hand, when the interface has a modulus equal to that of the bulk material a strong size effect in hardening as well as a significant reverse plasticity are observed in small specimens. These effects are attributed to the energy stored in the elastic fields of the geometrically necessary dislocations (GNDs).

Distribution of dislocation source length and the size dependent yield strength in freestanding thin films

A method is proposed to estimate the size-dependent yield strength of columnar-grained freestanding thin films. The estimate relies on assuming a distribution of the size of Frank–Read sources, which is then translated into a log-normal distribution of the source strength, depending on film thickness, grain size and theoretical strength of the material, augmented with a single fit parameter. Two-dimensional discrete dislocation plasticity (DDP) simulations are carried out for two sets of Cu films and the fit parameter is determined from independent experiments. Subsequent DDP predictions of the stress–strain curves in comparison with the corresponding experimental data show excellent agreement of initial yield strength and hardening rate for films of varying film thickness and grain size. Having thus demonstrated the power of the proposed strength distribution, it is shown that the mode of this distribution governs the most effective source strength. This is then used to suggest a method to estimate the yield strength of thin films as a function of film thickness and grain size. Simple maps are presented that are in very good agreement with recent experimental results for Cu thin films.

Circular nano-indentation in particle-reinforced metal matrix composites: Simply uniformly distributed particles lead to complex nano-indentation response

Abstract The indentation response of aluminum–matrix SiC particle-reinforced metal matrix composites (PR-MMCs) impressed by a circular indenter has been numerically analyzed under the two-dimensional discrete dislocation plasticity combined with the ANSYS® finite element software. Dependence of the nominal hardness H N on multiple factors, including the particle size, the particle area fraction, the matrix thickness between the indenter and the particle, the indentation location as well as the particle shape, is addressed in detail. Herein, we show that the nano-indentation response of the PR-MMCs is strongly associated with the particle area fraction of PR-MMCs as well as the blocking of dislocation free gliding within the matrix. At a high particle area fraction of 25%, H N increases with the decrease of the particle size due to the increasing blocking effect of particles on the dislocation gliding. However, at a low particle area fraction of 9%, a reverse dependence of H N could be observed, which is attributed to the formation of long free-slip bands in the matrix.