Two-dimensional Discrete Dislocation Plasticity Research Articles

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.

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.

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.

Climb-enabled discrete dislocation plasticity

A small strain two-dimensional discrete dislocation plasticity framework coupled to vacancy diffusion is developed wherein the motion of edge dislocations is by a combination of glide and climb. The dislocations are modelled as line defects in a linear elastic medium and the mechanical boundary value problem is solved by the superposition of the infinite medium elastic fields of the dislocations and a complimentary non-singular solution that enforces the boundary conditions. Similarly, the climbing dislocations are modelled as line sources/sinks of vacancies and the vacancy diffusion boundary value problem is also solved by a superposition of the fields of the line sources/sinks in an infinite medium and a complementary non-singular solution that enforces the boundary conditions. The vacancy concentration field along with the stress field provides the climb rate of the dislocations. Other short-range interactions of the dislocations are incorporated via a set of constitutive rules. We first employ this formulation to investigate the climb of a single edge dislocation in an infinite medium and illustrate the existence of diffusion-limited and sink-limited climb regimes. Next, results are presented for the pure bending and uniaxial tension of single crystals oriented for single slip. These calculations show that plasticity size effects are reduced when dislocation climb is permitted. Finally, we contrast predictions of this coupled framework with an ad hoc model in which dislocation climb is modelled by a drag-type relation based on a quasi steady-state solution.

A dynamic discrete dislocation plasticity method for the simulation of plastic relaxation under shock loading

In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (10 6 s −1 or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of time-dependent plasticity during shock loading is outlined here and will be presented in detail elsewhere.

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.

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.

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.

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.

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.

Discrete dislocation analyses of circular nanoindentation and its size dependence in polycrystals

Nanoindentation has been extensively used to measure the mechanical behavior of materials at the micro- and nanoscale. However, the material response of nano/microindentation and its intrinsic mechanisms are very complicated, especially when considering heterogeneous polycrystalline materials. In this contribution, nanoindentation in polycrystals, performed with a circular indenter, is studied by numerical modeling based on the two-dimensional discrete dislocation plasticity of Van der Giessen and Needleman [Van der Giessen E, Needleman A. Model Simul Mater Sci Eng 1995;3:689]. The dependence of indentation hardness is investigated with respect to four typical characteristic dimensions: indenter radius, grain size, indentation depth, and the distance between the grain boundary and the indenter. Results show that these characteristic dimensions have considerable influence on the nanoindentation hardness. Further investigation shows that their influence takes effect mainly in two ways, i.e. via strain hardening and the indentation size effect. Although both effects are size dependent, their underlying mechanisms are clearly different. For the present polycrystal case, the strain hardening effect is mainly associated with the constraints of grain boundaries and dislocation obstacles to the dislocation glide, while the indentation size effect is related to the average strain gradient beneath the circular indenter.

Discrete dislocation plasticity analysis of single crystalline thin beam under combined cyclic tension and bending

The cyclic plastic response of a single crystalline thin beam subject to combined cyclic tension and bending is analyzed using two-dimensional discrete dislocation plasticity. In this contribution, special attention is paid to the difference in the inherent mechanism of the size effect for different cyclic loads. Results show that the cyclic plastic response has a strong size effect for both cyclic pure tension–compression and pure bending. However, the inherent mechanisms are different. The dislocation starvation mechanism dominates the cyclic tension–compression while the geometrically necessary dislocation dominates the cyclic pure bending. When the combined cyclic tension and bending are applied to the thin beam, the cyclic moment–rotation response shows strong size effect while the stress–strain response shows weak or even no size effect. In addition, it is also found that the cyclic loading paths have considerable influences on the shape of the cyclic stress–strain loops.

Discrete dislocation analysis of the wedge indentation of polycrystals

This paper reports a study of the indentation of a model polycrystal using two-dimensional discrete dislocation plasticity. The polycrystal consists of square grains having the same orientation. Grain boundaries are modelled as being impenetrable to dislocations. Every grain has three slip systems, with a random distribution of initial sources and obstacles, and edge dislocations that glide in a drag-controlled manner. The indenter is wedge shaped, so that the indentation depth is the only geometrical length scale. The microstructural length scale on which we focus attention is the grain size, which is varied from 0.625 to 5 μm. While the predicted uniaxial yield strength of the polycrystals follows the Hall–Petch relation, this grain size dependence couples to the dependence on indentation depth. Polycrystals with a sufficiently large grain size exhibit the same “smaller is harder” dependence on indentation depth as single crystals, but an inverse indentation depth dependence occurs for fine-grained materials. For sufficiently deep indentation, the predicted nominal hardness is found to scale with grain size d according to H N = H N ∞ ( 1 + d * / d ) 1 / 2 , where H N ∞ is the single-crystal nominal hardness and d * is a material length scale.

Two hardening mechanisms in single crystal thin films studied by discrete dislocation plasticity

Two-dimensional discrete dislocation plasticity simulations of the evolution of thermal stress in single crystal thin films on a rigid substrate are used to study size effects. The relation between the residual stress and the dislocation structure in the films after cooling is analyzed using dislocation dynamics. A boundary layer characterized by a high stress gradient and a high dislocation density is found close to the impenetrable film-substrate interface. There is a material-dependent threshold film thickness above which the dislocation density together with the boundary layer thickness and stress state are independent of film thickness. In such films the stress outside the boundary layer is on average very low, so that the film-thickness-independent boundary layer is responsible for the size effect. A larger size effect is found for films thinner than the threshold thickness. The origin of this size effect stems from nucleation activity being hindered by the geometrical constraint of the small film thickness, so that by decreasing film thickness, the dislocation density decreases while the stress in the film increases. The size dependence is only described by a Hall–Petch type relation for films thicker than the threshold value.