Transmission of a screw dislocation across a coherent, non-slipping interface

Current research on nanocrystalline metals and nanoscale multilayer thin films suggests extraordinary plastic strength is due to confinement of slip to individual grains or layers. To assess the magnitude of confinement, a Peierls model of slip transmission of a screw dislocation across a coherent, non-slipping interface is presented. The results reflect that large interfacial barriers to transmission are generated by rapid fluctuations in dislocation line energy near the interface due to elastic modulus mismatch, stacking fault energy mismatch, and antiphase boundary energy for transmission into an ordered phase. Coherency stress is predicted to dramatically alter the dislocation core configuration and impart additional strength regardless of the sign. Contributions to strength are not additive due to nonlinear coupling via the dislocation core configuration. The predicted barrier strength for a coherent (0 0 1) Cu/Ni interface is comparable to atomistic (EAM) results but larger than estimates from hardness data.     

Atomistically informed dislocation dynamics in fcc crystals

We develop a nodal dislocation dynamics (DD) model to simulate plastic processes in fcc crystals. The model explicitly accounts for all slip systems and Burgers vectors observed in fcc systems, including stacking faults and partial dislocations. We derive simple conservation rules that describe all partial dislocation interactions rigorously and allow us to model and quantify cross-slip processes, the structure and strength of dislocation junctions, and the formation of fcc-specific structures such as stacking fault tetrahedra. The DD framework is built upon isotropic non-singular linear elasticity and supports itself on information transmitted from the atomistic scale. In this fashion, connection between the meso and micro scales is attained self-consistently, with all material parameters fitted to atomistic data. We perform a series of targeted simulations to demonstrate the capabilities of the model, including dislocation reactions and dissociations and dislocation junction strength. Additionally we map the four-dimensional stress space relevant for cross-slip and relate our findings to the plastic behavior of monocrystalline fcc metals.     

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.     

On dislocation patterning: Multiple slip effects in the rate equation approach

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development of various types of dislocation microstructures (dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc.), depending on the externally applied loading and the intrinsic lattice constraints. The term “dislocation patterning” was introduced over 20 years ago by the third author and a corresponding “gradient dislocation dynamics” framework was suggested to describe such phenomena. In the W–A model proposed at that time by the last two authors, it was shown how coupled nonlinear evolution equations of the reaction-diffusion type for the forest (immobile) and gliding (mobile) dislocation densities can generate dislocation microstructures which correspond to walls perpendicular to the slip direction for Cu-crystals oriented for single slip under cyclic loading conditions. This model is adapted to the multiple slip case here. Weakly nonlinear analysis predicts that dislocation patterns should correspond to domains of walls perpendicular to each slip direction and separated by domain walls in the same orientations. This result is confirmed by numerical analysis and experimental observations. The present model generalizes the original W–A model to the case of multiple slip and considers also explicitly gradient effects by allowing for non-uniform dislocation velocities and internal stress effects.     

The role of partial mediated slip during quasi-static deformation of 3D nanocrystalline metals

We present dislocation simulations involving the collective behavior of partials and extended full dislocations in nanocrystalline materials. While atomistic simulations have shown the importance of including partial dislocations in high strain rate simulations, the behavior of partial dislocations in complex geometries with lower strain rates has not been explored. To account for the dissociation of dislocations into partials we include the full representation of the gamma surface for two materials: Ni and Al. During loading, dislocation loops are emitted from grain boundaries and expand into the grain interiors to carry the strain. In agreement with high strain rate simulations we find that Al has a higher density of extended full dislocations with smaller stacking fault widths than Ni. We also observe that configurations with smaller average grain size have a higher density of partial dislocations, but contrary to simplified analytical models we do not find a critical grain size below which there is only partial dislocation-mediated deformation. Our results show that the density of partial dislocations is stable in agreement with in situ X-ray experiments that show no increase of the stacking fault density in deformed nanocrystalline Ni (Budrovic et al., 2004). Furthermore, the ratio between partial and extended full dislocation contribution to strain varies with the amount of deformation. The contribution of extended full dislocations to strain grows beyond the contribution of partial dislocations as the deformation proceeds, suggesting that there is no well-defined transition from full dislocation- to partial dislocation-mediated plasticity based uniquely on the grain size.     

A discrete mechanics approach to dislocation dynamics in BCC crystals

A discrete mechanics approach to modeling the dynamics of dislocations in BCC single crystals is presented. Ideas are borrowed from discrete differential calculus and algebraic topology and suitably adapted to crystal lattices. In particular, the extension of a crystal lattice to a CW complex allows for convenient manipulation of forms and fields defined over the crystal. Dislocations are treated within the theory as energy-minimizing structures that lead to locally lattice-invariant but globally incompatible eigendeformations. The discrete nature of the theory eliminates the need for regularization of the core singularity and inherently allows for dislocation reactions and complicated topological transitions. The quantization of slip to integer multiples of the Burgers’ vector leads to a large integer optimization problem. A novel approach to solving this NP-hard problem based on considerations of metastability is proposed. A numerical example that applies the method to study the emanation of dislocation loops from a point source of dilatation in a large BCC crystal is presented. The structure and energetics of BCC screw dislocation cores, as obtained via the present formulation, are also considered and shown to be in good agreement with available atomistic studies. The method thus provides a realistic avenue for mesoscale simulations of dislocation based crystal plasticity with fully atomistic resolution.     

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.     

Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires

Superelastic deformation of thin Ni-Ti wires containing various nanograined microstructures was investigated by tensile cyclic loading with in situ evaluation of electric resistivity. Defects created by the superelastic cycling in these wires were analyzed by transmission electron microscopy. The role of dislocation slip in superelastic deformation is discussed. Ni-Ti wires having finest microstructures (grain diameter <100 nm) are highly resistant against dislocation slip, while those with fully recrystallized microstructure and grain size exceeding 200 nm are prone to dislocation slip. The density of the observed dislocation defects increases significantly with increasing grain size. The upper plateau stress of the superelastic stress-strain curves is largely grain size independent from 10 up to 1000 nm. It is hence claimed that the Hall-Petch relationship fails for the stress-induced martensitic transformation in this grain size range. It is proposed that dislocation slip taking place during superelastic cycling is responsible for the accumulated irreversible strains, cyclic instability and degradation of functional properties. No residual martensite phase was found in the microstructures of superelastically cycled wires by TEM and results of the in situ electric resistance measurements during straining also indirectly suggest that none or very little martensite phase remains in the studied cycled superelastic wires after unloading. The accumulation of dislocation defects, however, does not prevent the superelasticity. It only affects the shape of the stress-strain response, makes it unstable upon cycling and changes the deformation mode from localized to homogeneous. The activity of dislocation slip during superelastic deformation of Ni-Ti increases with increasing test temperature and ultimately destroys the superelasticity as the plateau stress approaches the yield stress for slip. Deformation twins in the austenite phase ({1 1 4} compound twins) were frequently found in cycled wires having largest grain size. It is proposed that they formed in the highly deformed B19′ martensite phase during forward loading and are retained in austenite after unloading. Such twinning would represent an additional deformation mechanism of Ni-Ti yielding residual irrecoverable strains.     

Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics

A computational method (CADD) is presented whereby a continuum region containing dislocation defects is coupled to a fully atomistic region. The model is related to previous hybrid models in which continuum finite elements are coupled to a fully atomistic region, with two key advantages: the ability to accomodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. The resulting CADD model allows for the study of 2d problems involving large numbers of defects where the system size is too big for fully atomistic simulation, and improves upon existing discrete dislocation techniques by preserving accurate atomistic details of dislocation nucleation and other atomic scale phenomena. Applications to nanoindentation, atomic scale void growth under tensile stress, and fracture are used to validate and demonstrate the capabilities of the model.     

COARSE-GRAINED ATOMISTIC MODELING AND SIMULATION OF INELASTIC MATERIAL BEHAVIOR

This paper presents a new methodology for coarse-grained atomistic simulation of inelastic material behavior including phase transformations in ceramics and dislocation mediated plasticity in metals. The methodology combines an atomistic formulation of balance equations and a modified finite element method. With significantly fewer degrees of freedom than those of a fully atomistic model and without additional constitutive rules but the interatomic force field, the new coarse-grained (CG) method is shown to be feasible in predicting the nonlinear constitutive responses of materials and also reproducing atomic-scale phenomena such as phase transformations (diamond →β-Sn) in silicon and dislocation nucleation and migration, formation of dislocation loops and stacking faults ribbons in single crystal nickel. Direct comparisons between CG and the corresponding full molecular dynamics (MD) simulations show that the present methodology is efficient and promising in modeling and simulation of inelastic material behavior without losing the essential atomistic features. The potential applications and the limitations of the CG method are also discussed.     

An atomistically-informed dislocation dynamics model for the plastic anisotropy and tension-compression asymmetry of BCC metals

Atomistic simulations have shown that a screw dislocation in body-centered cubic (BCC) metals has a complex non-planar atomic core structure. The configuration of this core controls their motion and is affected not only by the usual resolved shear stress on the dislocation, but also by non-driving stress components. Consequences of the latter are referred to as non-Schmid effects. These atomic and micro-scale effects are the reason slip characteristics in deforming single and polycrystalline BCC metals are extremely sensitive to the direction and sense of the applied load. In this paper, we develop a three-dimensional discrete dislocation dynamics (DD) simulation model to understand the relationship between individual dislocation glide behavior and macro-scale plastic slip behavior in single crystal BCC Ta. For the first time, it is shown that non-Schmid effects on screw dislocations of both {110} and {112} slip systems must be implemented into the DD models in order to predict the strong plastic anisotropy and tension-compression asymmetry experimentally observed in the stress-strain curves of single crystal Ta. Incorporation of fundamental atomistic information is critical for developing a physics-based, predictive meso-scale DD simulation tool that can connect length/time scales and investigate the underlying mechanisms governing the deformation of BCC metals.     

Coarse-grained atomistic simulation of dislocations

This paper presents a new methodology for coarse-grained atomistic simulation of dislocation dynamics. The methodology combines an atomistic formulation of balance equations and a modified finite element method employing rhombohedral-shaped 3D solid elements suitable for fcc crystals. With significantly less degrees of freedom than that of a fully atomistic model and without additional constitutive rules to govern dislocation activities, this new coarse-graining (CG) method is shown to be able to reproduce key phenomena of dislocation dynamics for fcc crystals, including dislocation nucleation and migration, formation of stacking faults and Lomer-Cottrell locks, and splitting of stacking faults, all comparable with fully resolved molecular dynamics simulations. Using a uniform coarse mesh, the CG method is then applied to simulate an initially dislocation-free submicron-sized thin Cu sheet. The results show that the CG simulation has captured the nucleation and migration of large number of dislocations, formation of multiple stacking fault ribbons, and the occurrence of complex dislocation phenomena such as dislocation annihilation, cutting, and passing through the stacking faults. The distinctions of this method from existing coarse-graining or multiscale methods and its potential applications and limitations are also discussed.     

The key role of dislocation dissociation in the plastic behaviour of single crystal nickel-based superalloy with low stacking fault energy: Three-dimensional discrete dislocation dynamics modelling

To model the deformation of single crystal nickel based superalloys (SCNBS) with low stacking fault energy (SFE), three-dimensional discrete dislocation dynamics (3D-DDD) is extended by incorporating dislocation dissociation mechanism. The present 3D-DDD simulations show that, consistent with the existing TEM observation, the leading partial can enter the matrix channel efficiently while the trailing partial can hardly glide into it when the dislocation dissociation is taken into account. To determine whether the dislocation dissociation can occur or not, a critical percolation stress (CPS) based criterion is suggested. According to this CPS criterion, for SCNBS there exists a critical matrix channel width. When the channel width is lower than this critical value, the dislocation tends to dissociate into an extended configuration and vice versa. To clarify the influence of dislocation dissociation on CPS, the classical Orowan formula is improved by incorporating the SFE. Moreover, the present 3D-DDD simulations also show that the yielding stress of SCNBSs with low SFE may be overestimated up to 30% if the dislocation dissociation is ignored. With dislocation dissociation being considered, the size effect due to the width of γ matrix channel and the length of γ′ precipitates on the stress–strain responses of SCNBS can be enhanced remarkably. In addition, due to the strong constraint effect by the two-phase microstructure in SCNBS, the configuration of formed junctions is quite different from that in single phase crystals such as Cu. The present results not only provide clear understanding of the two-phase microstructure levelled microplastic mechanisms in SCNBSs with low SFE, but also help to develop new continuum-levelled constitutive laws for SCNBSs.     

Multiscale diffusion method for simulations of long-time defect evolution with application to dislocation climb

In many problems of interest to materials scientists and engineers, the evolution of crystalline extended defects (dislocations, cracks, grain boundaries, interfaces, voids, precipitates) is controlled by the flow of point defects (interstitial/substitutional atoms and/or vacancies) through the crystal into the extended defect. Precise modeling of this behavior requires fully atomistic methods in and around the extended defect, but the flow of point defects entering the defect region can be treated by coarse-grained methods. Here, a multiscale algorithm is presented to provide this coupling. Specifically, direct accelerated molecular dynamics (AMD) of extended defect evolution is coupled to a diffusing point defect concentration field that captures the long spatial and temporal scales of point defect motion in the presence of the internal stress fields generated by the evolving defect. The algorithm is applied to study vacancy absorption into an edge dislocation in aluminum where vacancy accumulation in the core leads to nucleation of a double-jog that then operates as a sink for additional vacancies; this corresponds to the initial stages of dislocation climb modeled with explicit atomistic resolution. The method is general and so can be applied to many other problems associated with nucleation, growth, and reaction due to accumulation of point defects in crystalline materials.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

Modeling finite deformations in trigonal ceramic crystals with lattice defects

A model is developed for thermomechanical behavior of defective, low-symmetry ceramic crystals such as α$\alpha$-corundum. Kinematics resolved are nonlinear elastic deformation, thermal expansion, dislocation glide, mechanical twinning, and residual lattice strains associated with eigenstress fields of defects such as dislocations and stacking faults. Multiscale concepts are applied to describe effects of twinning on effective thermoelastic properties. Glide and twinning are thermodynamically irreversible, while free energy accumulates with geometrically necessary dislocations associated with strain and rotation gradients, statistically stored dislocations, and twin boundaries. The model is applied to describe single crystals of corundum. Hardening behaviors of glide and twin systems from the total density of dislocations accumulated during basal slip are quantified for pure and doped corundum crystals. Residual lattice expansion is predicted from nonlinear elasticity and dislocation line and stacking fault energies.     

Dislocation pile-ups in bicrystals within continuum dislocation theory

Within continuum dislocation theory the plastic deformation of bicrystals under a mixed deformation of plane constrained uniaxial extension and shear is investigated with regard to the nucleation of dislocations and the dislocation pile-up near the phase boundaries of a model bicrystal with one active slip system within each single crystal. For plane uniaxial extension, we present a closed-form analytical solution for the evolution of the plastic distortion and of the dislocation network in the case of symmetric slip planes (i.e. for twins), which exhibits an energetic as well as a dissipative threshold for the dislocation nucleation. The general solution for non-symmetric slip systems is obtained numerically. For a combined deformation of extension and shear, we analyze the possibility of linearly superposing results obtained for both loading cases independently. All solutions presented in this paper also display the Bauschinger effect of translational work hardening and a size effect typical to problems of crystal plasticity.