Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation

Nanoindentation experiments have shown that microstructural inhomogeneities across the surface of gold thin films lead to position-dependent nanoindentation behavior [Phys. Rev. B (2002), to be submitted]. The rationale for such behavior was based on the availability of dislocation sources at the grain boundary for initiating plasticity. In order to verify or refute this theory, a computational approach has been pursued. Here, a simulation study of the initial stages of indentation using the embedded atom method (EAM) is presented. First, the principles of the EAM are given, and a comparison is made between atomistic simulations and continuum models for elastic deformation. Then, the mechanism of dislocation nucleation in single crystalline gold is analyzed, and the effects of elastic anisotropy are considered. Finally, a systematic study of the indentation response in the proximity of a high angle, high sigma (low symmetry) grain boundary is presented; indentation behavior is simulated for varying indenter positions relative to the boundary. The results indicate that high angle grain boundaries are a ready source of dislocations in indentation-induced deformation.     

A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

Two-dimensional dislocation dynamics (2D-DD) simulations under fully periodic boundary conditions are employed to study the relation between microstructure and strength of a material. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a preexisting dislocation population, dislocation sources, and grain boundaries. The mechanical response of such a material is tested by uniaxially loading it up to a certain stress and allowing it to relax until the strain rate falls below a threshold. The total plastic strain obtained for a certain stress level yields the quasi-static stress-strain curve of the material. Besides assuming Frank-Read-like dislocation sources, we also investigate the influence of a pre-existing dislocation density on the flow stress of the model material. Our results show that - despite its inherent simplifications - the 2D-DD model yields material behavior that is consistent with the classical theories of Taylor and Hall-Petch. Consequently, if set up in a proper way, these models are suited to study plastic deformation of polycrystalline materials.     

On dislocation models of grain boundaries

Dislocation models of grain boundaries was suggested by Bragg (Proc Phys Soc 52:54–55, 1940) and Burgers (Proc Phys Soc 52:23–33, 1940). The first quantitative study of these models was given by Read and Shockley (Phys Rev 78(3):275–289, 1950). They obtained a formula for the dependence of the grain boundary energy on the misorientation of the neighboring grains, which became a cornerstone of the grain boundary theory. The Read–Shockley formula was based on a proposition that the grain boundary energy is the sum of energies of the two sets of dislocations that come from the two neighboring grains. This proposition was proved under an assumption on a quite special geometry of the slip planes. This paper aims to show that the assumption is not necessary and the proposition holds for arbitrary geometry of slip planes. Another goal of this paper is to provide all basic formulas of the theory: though the dislocation model of grain boundaries is considered in all treatises on dislocation theory, a complete analysis, including the relations for lattice rotations and displacements, has not been given. This analysis shows, in particular, that continuum theory does not yield the proper relations for the lattice misorientations, and these relations must be introduced by an independent ansatz.     

Three-dimensional formulation of dislocation climb

We derive a Green's function formulation for the climb of curved dislocations and multiple dislocations in three-dimensions. In this new dislocation climb formulation, the dislocation climb velocity is determined from the Peach–Koehler force on dislocations through vacancy diffusion in a non-local manner. The long-range contribution to the dislocation climb velocity is associated with vacancy diffusion rather than from the climb component of the well-known, long-range elastic effects captured in the Peach–Koehler force. Both long-range effects are important in determining the climb velocity of dislocations. Analytical and numerical examples show that the widely used local climb formula, based on straight infinite dislocations, is not generally applicable, except for a small set of special cases. We also present a numerical discretization method of this Green's function formulation appropriate for implementation in discrete dislocation dynamics (DDD) simulations. In DDD implementations, the long-range Peach–Koehler force is calculated as is commonly done, then a linear system is solved for the climb velocity using these forces. This is also done within the same order of computational cost as existing discrete dislocation dynamics methods.     

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

Shocks and slip systems: Predictions from a mesoscale theory of continuum dislocation dynamics

Exploring a recently developed mesoscale continuum theory of dislocation dynamics, we derive three predictions about plasticity and grain boundary formation in crystals. (1) There is a residual stress jump across grain boundaries and plasticity-induced cell walls as they form, which self-consistently acts to attract neighboring dislocations; residual stress in this theory appears as a remnant of the driving force behind wall formation under both polygonization and plastic deformation. We derive the predicted asymptotic late-time dynamics of the grain-boundary formation process. (2) During grain boundary formation at high temperatures, there is a predicted cusp in the elastic energy density. (3) In early stages of plasticity, when only one type of dislocation is active (single-slip), cell walls do not form in the theory; instead we predict the formation of a hitherto unrecognized jump singularity in the dislocation density.     

Dislocation nucleation in Σ3 asymmetric tilt grain boundaries

Atomistic simulations were used to investigate dislocation nucleation from Σ3 asymmetric (inclined) tilt grain boundaries under uniaxial tension applied perpendicular to the boundary. Molecular dynamics was employed based on embedded atom method potentials for Cu and Al at 10 K and 300 K. Results include the grain boundary structure and energy, along with mechanical properties and mechanisms associated with dislocation nucleation from these Σ3 boundaries. The stress and work required for dislocation nucleation were calculated along with elastic stiffness of the bicrystal configurations, exploring the change in response as a function of inclination angle. Analyses of dislocation nucleation mechanisms for asymmetric Σ3 boundaries in Cu show that dislocation nucleation is preceded by dislocation dissociation from the boundary. Then, dislocations preferentially nucleate in only one crystal on the maximum Schmid factor slip plane(s) for that crystal. However, this crystal is not simply predicted based on either the Schmid or non-Schmid factors. The synthesis of these results provides a better understanding of the dislocation nucleation process in these faceted, dissociated grain boundaries.     

Dislocation nucleation from bicrystal interfaces and grain boundary ledges: Relationship to nanocrystalline deformation

Molecular dynamics simulations are used to evaluate the primary interface dislocation sources and to estimate both the free enthalpy of activation and the critical emission stress associated with the interfacial dislocation emission mechanism. Simulations are performed on copper to study tensile failure of a planar Σ5 {2 1 0} 53.1° interface and an interface with the same misorientation that contains a ledge. Simulations reveal that grain boundary ledges are more favorable as dislocation sources than planar regions of the interface and that their role is not limited to that of simple dislocation donors. The parameters extracted from the simulations are utilized in a two-phase composite mesoscopic model for nanocrystalline deformation that includes the effects of both dislocation emission and dislocation absorption mechanisms. A self-consistent approach based on the Eshelby solution for grains as ellipsoidal inclusions is augmented by introduction of stress concentration in the constitutive law of the matrix phase to account for more realistic grain boundary effects. Model simulations suggest that stress concentration is required in the standard continuum theory to activate the coupled grain boundary dislocation emission and absorption mechanisms when activation energy of the dislocation source is determined from atomistic calculation on grain boundaries without consideration of impurities or other extrinsic defects.     

Size effects and dislocation patterning in two-dimensional bending

We perform atomistic Monte Carlo simulations of bending a Lennard-Jones single crystal in two dimensions. Dislocations nucleate only at the free surface as there are no sources in the interior of the sample. When dislocations reach sufficient density, they spontaneously coalesce to nucleate grain boundaries, and the resulting microstructure depends strongly on the initial crystal orientation of the sample. In initial yield, we find a reverse size effect, in which larger samples show a higher scaled bending moment than smaller samples for a given strain and strain rate. This effect is associated with source-limited plasticity and high strain rate relative to dislocation mobility, and the size effect in initial yield disappears when we scale the data to account for strain rate effects. Once dislocations coalesce to form grain boundaries, the size effect reverses and we find that smaller crystals support a higher scaled bending moment than larger crystals. This finding is in qualitative agreement with experimental results. Finally, we observe an instability at the compressed crystal surface that suggests a novel mechanism for the formation of a hillock structure. The hillock is formed when a high angle grain boundary, after absorbing additional dislocations, becomes unstable and folds to form a new crystal grain that protrudes from the free surface.     

Continuum dynamics of the formation,migration and dissociation of self-locked dislocation structures on parallel slip planes

In continuum models of dislocations, proper formulations of short-range elastic interactions of dislocations are crucial for capturing various types of dislocation patterns formed in crystalline materials. In this article, the continuum dynamics of straight dislocations distributed on two parallel slip planes is modelled through upscaling the underlying discrete dislocation dynamics. Two continuum velocity field quantities are introduced to facilitate the discrete-to-continuum transition. The first one is the local migration velocity of dislocation ensembles which is found fully independent of the short-range dislocation correlations. The second one is the decoupling velocity of dislocation pairs controlled by a threshold stress value, which is proposed to be the effective flow stress for single slip systems. Compared to the almost ubiquitously adopted Taylor relationship, the derived flow stress formula exhibits two features that are more consistent with the underlying discrete dislocation dynamics: (i) the flow stress increases with the in-plane component of the dislocation density only up to a certain value, hence the derived formula admits a minimum inter-dislocation distance within slip planes; (ii) the flow stress smoothly transits to zero when all dislocations become geometrically necessary dislocations. A regime under which inhomogeneities in dislocation density grow is identified, and is further validated through comparison with discrete dislocation dynamical simulation results. Based on the findings in this article and in our previous works, a general strategy for incorporating short-range dislocation correlations into continuum models of dislocations is proposed.     

Studying the effect of grain boundaries in dislocation density based crystal-plasticity finite element simulations

A dislocation density based constitutive model for the face centered cubic crystal structure has been implemented into a crystal-plasticity finite element framework and extended to consider the mechanical interaction between mobile dislocations and grain boundaries by the authors [Ma, A., Roters, F., Raabe, D., 2006a. A dislocation density based constitutive model for crystal-plasticity FEM including geometrically necessary dislocations. Acta Materialia 54, 2169–2179; Ma, A., Roters, F., Raabe, D., 2006b. On the consideration of interactions between dislocations and grain boundaries in crystal-plasticity finite element modeling – theory, experiments, and simulations. Acta Materialia 54, 2181–2194]. The approach to model the grain boundary resistance against slip is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of internal interfaces. This energy barrier is derived from the assumption of thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems. In this study we focus on the influence of the one remaining model parameter which can be used to scale the obstacle strength of the grain boundary.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations.Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.     

Defect redistribution within a continuum grain boundary plasticity model

The mechanical response of polycrystalline metals is significantly affected by the behaviour of grain boundaries, in particular when these interfaces constitute a relatively large fraction of the material volume. One of the current challenges in the modelling of grain boundaries at a continuum (polycrystalline) scale is the incorporation of the many different interaction mechanisms between dislocations and grain boundaries, as identified from fine-scale experiments and simulations. In this paper, the objective is to develop a model that accounts for the redistribution of the defects along the grain boundary in the context of gradient crystal plasticity. The proposed model incorporates the nonlocal relaxation of the grain boundary net defect density. A numerical study on a bicrystal specimen in simple shear is carried out, showing that the spreading of the defect content has a clear influence on the macroscopic response, as well as on the microscopic fields. This work provides a basis that enables a more thorough analysis of the plasticity of polycrystalline metals at the continuum level, where the plasticity at grain boundaries matters.     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

EBSD-based continuum dislocation microscopy

Recent advances in high-resolution electron backscatter diffraction (EBSD)-based microscopy are applied to the characterization of elastic fields and incompatibility structures near the grain boundaries (GBs) in polycrystals. Two main recoveries are reported here: surface geometrically necessary dislocation (density) tensors, as described by Kröner, and the elastic fields near cracks (unconsolidated portions of interface) in loaded samples. Context for the application of these recoveries is described, using Green’s function solutions for combined heterogeneity and dislocation. Featured recoveries required the cross-correlation based determination of the elastic distortion tensor, aided by application of the simulated pattern method, and determination of the absolute pattern center utilizing the expected pattern properties in a spherical Kikuchi reference frame. High-resolution data obtained along an ultrasonically consolidated nickel boundary of varying amalgamation indicates that the imposed traction free boundary condition at free surfaces is well observed in the data structure. Further, high-resolution data acquired near a single grain boundary in well-annealed, low content steel suggests that it may be possible to measure the intrinsic elastic properties of GBs.     

An atomistically validated continuum model for strain relaxation and misfit dislocation formation

In this paper, molecular dynamics (MD) calculations have been used to examine the physics behind continuum models of misfit dislocation formation and to assess the limitations and consequences of approximations made within these models. Without compromising the physics of misfit dislocations below a surface, our MD calculations consider arrays of dislocation dipoles constituting a mirror imaged “surface”. This allows use of periodic boundary conditions to create a direct correspondence between atomistic and continuum representations of dislocations, which would be difficult to achieve with free surfaces. Additionally, by using long-time averages of system properties, we have essentially reduced the errors of atomistic simulations of large systems to “zero”. This enables us to deterministically compare atomistic and continuum calculations. Our work results in a robust approach that uses atomistic simulation to accurately calculate dislocation core radius and energy without the continuum boundary conditions typically assumed in the past, and the novel insight that continuum misfit dislocation models can be inaccurate when incorrect definitions of dislocation spacing and Burgers vector in lattice-mismatched systems are used. We show that when these insights are properly incorporated into the continuum model, the resulting energy density expression of the lattice-mismatched systems is essentially indistinguishable from the MD results.     

A theory for grain boundaries with strain-gradient plasticity

In this work, the effect of the material microstructural interface between two materials (i.e., grain boundary in polycrystalls) is adopted into a thermodynamic-based higher order strain gradient plasticity framework. The developed grain boundary flow rule accounts for the energy storage at the grain boundary due to the dislocation pile up as well as energy dissipation caused by the dislocation transfer through the grain boundary. The theory is developed based on the decomposition of the thermodynamic conjugate forces into energetic and dissipative counterparts which provides the constitutive equations to have both energetic and dissipative gradient length scales for the grain and grain boundary. The numerical solution for the proposed framework is also presented here within the finite element context. The material parameters of the gradient framework are also calibrated using an extensive set of micro-scale experimental measurements of thin metal films over a wide range of size and temperature of the samples.     

Constitutive analysis of finite deformation field dislocation mechanics

Driving forces for dislocation motion and nucleation in finite-deformation field dislocation mechanics are derived. The former establishes a rigorous analog of the Peach-Koehler force of classical elastic dislocation theory in a nonlinear, nonequilibrium field-theoretic context; the latter is a prediction of the theory. The structure of the stress response and permanent distortion are also derived. Sufficient boundary and initial conditions are indicated, and invariance under superposed rigid motions is discussed. Hyperelasticity and finite-deformation elastic theory of dislocations are shown to be special cases of the framework. Owing to the nonlocal nature of the theory, the results as well as the methods used to derive them appear to be novel.     

Mechanical response of multicrystalline thin films in mesoscale field dislocation mechanics

A continuum model of plasticity, Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM), is used to study the effect of surface passivation, grain orientation, grain boundary constraints, and film thickness on the mechanical response of multicrystalline thin films. The numerical experiments presented in this paper show that a surface passivation layer on thin films introduces thickness dependence of the mechanical response. However, the effect of passivation decreases in films with impenetrable grain boundaries. The orientation of individual grains of the multicrystal also has a significant effect on the mechanical response. Our results are in qualitative agreement with experimental observations. A primary contribution of this work is the implementation of a jump condition that enables the modeling of important limits of grain boundary constraints to plastic flow, independent of ad-hoc constitutive assumptions and interface conditions.