Non-equilibrium grain boundary structure and inelastic deformation using atomistic simulations

Grain boundary influence on material properties becomes increasingly significant as grain size is reduced to the nanoscale. Nanostructured materials produced by severe plastic deformation techniques often contain a higher percentage of high-angle grain boundaries in a non-equilibrium or energetically metastable state. Differences in the mechanical behavior and observed deformation mechanisms are common due to deviations in grain boundary structure. Fundamental interfacial attributes such as atomic mobility and energy are affected due to a higher non-equilibrium state, which in turn affects deformation response. In this research, atomistic simulations employing a biased Monte Carlo method are used to approximate representative non-equilibrium bicrystalline grain boundaries based on an embedded atom method potential, leveraging the concept of excess free volume. An advantage of this approach is that non-equilibrium boundaries can be instantiated without the need of simulating numerous defect/grain boundary interactions. Differences in grain boundary structure and deformation response are investigated as a function of non-equilibrium state using Molecular Dynamics. A detailed comparison between copper and aluminum bicrystals is provided with regard to boundary strength, observed deformation mechanisms, and stress-assisted free volume evolution during both tensile and shear simulations.     

On the nonlocal nature of dislocation nucleation during nanoindentation

Using static atomistic simulations, we study the full details of the mechanism by which dislocations homogeneously nucleate beneath the surface of a initially defect-free crystal during indentation. The mechanism involves the collective motion of a finite disk of atoms over two adjacent slip planes, the diameter of which depends on the indenter size. The nucleation mechanism highlights the need for nonlocal considerations in the development of a nucleation criterion. We review three nucleation criteria from the literature, each of which is based on purely local measures of the state of stress, and show that none are sufficiently general to predict nucleation in realistic atomic systems. We then propose a criterion based on an eigenmode analysis of the atomic-scale acoustic tensor. We demonstrate the accuracy of the criterion, which also works in the presence of existing topological defects like free surfaces or dislocation cores. The dependence of the size of the nucleated disk on the indenter radius leads to a self-similar nucleation process and virtually no indentation size effect (ISE), suggesting that homogeneous nucleation is only possible for very small indenters.     

Plasticity of metal wires in torsion: Molecular dynamics and dislocation dynamics simulations

The orientation dependent plasticity in metal nanowires is investigated using molecular dynamics and dislocation dynamics simulations. Molecular dynamics simulations show that the orientation of single crystal metal wires controls the mechanisms of plastic deformation. For wires oriented along , dislocations nucleate along the axis of the wire, making the deformation homogeneous. These wires also maintain most of their strength after yield. In contrast, wires oriented along and directions deform through the formation of twist boundaries and tend not to recover when high angle twist boundaries are formed. The stability of the dislocation structures observed in molecular dynamics simulations are investigated using analytical and dislocation dynamics models.     

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.     

Size effect in the initiation of plasticity for ceramics in nanoindentation

The indentation size effect has been observed for many years and is usually associated with increasing hardness as the depth of indentation is reduced for pyramidal indenters. The indentation size effect for spherical indenters has recently been associated with an increase in the yield stress of metals proportional to the inverse cube root of indenter radius [Spary, I.J., Bushby, A.J., Jennett, N.M., 2006. On the indentation size effect in spherical indentation. Philos. Mag. 86 (33), 5581-5593]. Here we investigate ceramic materials where the yield point is high enough to be easily distinguished in nanoindentation tests. A robust method for determining the yield point from a nanoindentation test with spherical indenters is presented. The results for a range of ceramics confirm that the increase in yield pressure is directly proportional to the inverse cube root of indenter radius. Furthermore, the yield pressure is also shown to be proportional to the inverse square root of the contact radius. Revisiting data in the literature shows that this inverse square root relationship is also true for pyramidal indenters. This implies that the indentation size effect is driven by the contact area rather than by the depth of indentation or by the indenter radius.     

Surface versus bulk nucleation of dislocations during contact

The indentation of single crystals by a periodic array of flat rigid contacts is analyzed using discrete dislocation plasticity. Plane strain analyses are carried out with the dislocations all of edge character and modeled as line singularities in a linear elastic solid. The limiting cases of frictionless and perfectly sticking contacts are considered. The effects of contact size, dislocation source density, and dislocation obstacle density and strength on the evolution of the mean indentation pressure are explored, but the main focus is on contrasting the response of crystals having dislocation sources on the surface with that of crystals having dislocation sources in the bulk. When there are only bulk sources, the mean contact pressure for sufficiently large contacts is independent of the friction condition, whereas for sufficiently small contact sizes, there is a significant dependence on the friction condition. When there are only surface dislocation sources the mean contact pressure increases much more rapidly with indentation depth than when bulk sources are present and the mean contact pressure is very sensitive to the strength of the obstacles to dislocation glide. Also, on unloading a layer of tensile residual stress develops when surface dislocation sources dominate.     

Evolution of the E structural unit during uniaxial and constrained tensile deformation

Molecular dynamics simulations are used to study the mechanisms by which Shockley partial dislocations are nucleated from 1 1 0 copper grain boundaries which contain the E structural unit. Simulations in this work indicate that the natural conformation of the interface porosity with respect to the primary dislocation slip systems is responsible for the easy emission of Shockley partial dislocations during a tensile deformation. Furthermore, it is found that tensile stresses parallel to the interface plane can diminish the severity of the E structural unit on the dislocation nucleation process.     

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.     

Atomistic insights into dislocation-based mechanisms of void growth and coalescence

One of the low-temperature failure mechanisms in ductile metallic alloys is the growth of voids and their coalescence. In the present work we attempt to obtain atomistic insights into the mechanisms underpinning cavitation in a representative metal, namely Aluminum. Often the pre-existing voids in metallic alloys such as Al have complex shapes (e.g. corrosion pits) and the defromation/damage mechanisms exhibit a rich size-dependent behavior across various material length scales. We focus on these two issues in this paper through large-scale calculations on specimens of sizes ranging from 18 thousand to 1.08 million atoms. In addition to the elucidation of the dislocation propagation based void growth mechanism we highlight the observed length scale effect reflected in the effective stress-strain response, stress triaxiality and void fraction evolution. Furthermore, as expected, the conventionally used Gurson's model fails to capture the observed size-effects calling for a mechanistic modification that incorporates the mechanisms observed in our (and other researchers') simulation. Finally, in our multi-void simulations, we find that, the splitting of a big void into a distribution of small ones increases the load-carrying capacity of specimens. However, no obvious dependence of the void fraction evolution on void coalescence is observed.     

High strain gradient plasticity associated with wedge indentation into face-centered cubic single crystals: Geometrically necessary dislocation densities

Experimental studies on indentation into face-centered cubic (FCC) single crystals such as copper and aluminum were performed to reveal the spatially resolved variation in crystal lattice rotation induced due to wedge indentation. The crystal lattice curvature tensors of the indented crystals were calculated from the in-plane lattice rotation results as measured by electron backscatter diffraction (EBSD). Nye's dislocation density tensors for plane strain deformation of both crystals were determined from the lattice curvature tensors. The least L²-norm solutions to the geometrically necessary dislocation densities for the case in which three effective in-plane slip systems were activated in the single crystals associated with the indentation were determined. Results show the formation of lattice rotation discontinuities along with a very high density of geometrically necessary dislocations.     

Atomistic simulations and continuum modeling of dislocation nucleation and strength in gold nanowires

The strength of true metallic nanowires and nanopillars (diameters below 100 nm) is known to be higher than the strength of bulk metals and is most likely controlled by dislocation nucleation from free surfaces. Dislocation nucleation is a thermally activated process that is sensitive to both temperature and strain rate. However, most simulations rely on high strain rate molecular dynamics to investigate strength and nucleation, which is limited by short molecular dynamics time scales. In this work, the energetics of dislocation nucleation in gold nanowires are computed using atomistic simulations, and transition state theory is used to estimate the strength at experimental strain rates revealing detailed information outside the realm accessible to molecular dynamics simulations. This allows investigation into the competition between thermally activated dislocation nucleation and other failure mechanisms such as elastic and structural instabilities. Additionally, the mechanisms of dislocation nucleation are compared against analytical continuum models which allow a better understanding of the nucleation process including the effects of the wire surfaces. This study helps clarify and consolidate our understanding of the nature of dislocation nucleation in small structures.     

Neural networks for tip correction of spherical indentation curves from bulk metals and thin metal films

Spherical indentation is widely used to determine a variety of important mechanical properties from small volumes. However, the available nanoindenter tips mostly deviate from the perfect spherical shape making the application of analysis methods developed for perfect spheres uncertain. In this paper, neural network-based methods are presented that are used to correct force-depth curves measured with such indenter tips. Finite element simulations for imperfect and perfect spherical tips with varying material behaviour are used to train the neural networks, which solve the inverse problem of mapping the true tip shape and the measured force-depth curve to one that corresponds to a perfect spherical indenter. Solutions are provided for bulk materials and thin films. The method has been verified experimentally on nanocrystalline nickel and a copper film on a titanium substrate for different spherical tips.     

Sample boundary effect in nanoindentation of nano and microscale surface structures

Nanoindentation is a widely used technique to characterize mechanical properties of materials in small volumes. When the sample size is comparable to the indent size, the indentation-induced plastic zone can be affected by the sample boundary which may cause inaccurate interpretation of the mechanical properties. In this study, the sample boundary effect is investigated by performing experiments and atomistic simulations of nanoindentation into nano- and micro-scale Au pillars and bulk Au (0 0 1) surfaces. In experiments, a more compliant deformation is observed in pillar indentations compared to bulk Au. The elastic modulus decreases with increasing indent size over sample size ratio. Atomistic simulations are performed to gain insights on the mechanisms of pillar deformation and pillar boundary effect. The reduced modulus has a similar trend of decrease with increasing indent size over sample size ratio. Significantly different dislocation activities and dislocation interactions with the pillar boundary contribute to the lower value of the reduced modulus in the pillar indentation. The presence of the free surface would allow the dislocations to annihilate, causing a higher plastic recovery during the pillar unloading process.     

The structure and energetics of, and the plasticity caused by, Eshelby dislocations

The structure of coaxial, or Eshelby, dislocations are computed using isotropic elasticity for arrays of up to 500 dislocations. The energies of these arrays are determined in order to predict the lowest energy configuration and multiple meta-stable configurations are often found. The energy from these elasticity predictions shows good agreement with molecular statics simulations of aluminum. From these simulations, the torque-twist curves are predicted and compared with molecular dynamics simulations.     

The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals

The mechanical response of engineering materials evaluated through continuum fracture mechanics typically assumes that a crack or void initially exists, but it does not provide information about the nucleation of such flaws in an otherwise flawless microstructure. How such flaws originate, particularly at grain (or phase) boundaries is less clear. Experimentally, “good” vs. “bad” grain boundaries are often invoked as the reasons for critical damage nucleation, but without any quantification. The state of knowledge about deformation at or near grain boundaries, including slip transfer and heterogeneous deformation, is reviewed to show that little work has been done to examine how slip interactions can lead to damage nucleation. A fracture initiation parameter developed recently for a low ductility model material with limited slip systems provides a new definition of grain boundary character based upon operating slip and twin systems (rather than an interfacial energy based definition). This provides a way to predict damage nucleation density on a physical and local (rather than a statistical) basis. The parameter assesses the way that highly activated twin systems are aligned with principal stresses and slip system Burgers vectors. A crystal plasticity-finite element method (CP-FEM) based model of an extensively characterized microstructural region has been used to determine if the stress–strain history provides any additional insights about the relationship between shear and damage nucleation. This analysis shows that a combination of a CP-FEM model augmented with the fracture initiation parameter shows promise for becoming a predictive tool for identifying damage-prone boundaries.     

Non-slipping adhesive contact between mismatched elastic spheres: A model of adhesion mediated deformation sensor

A generalized JKR model is established for non-slipping adhesive contact between two dissimilar elastic spheres subjected to a pair of pulling forces and a mismatch strain. We discuss the full elastic solution to the problem as well as the so-called non-oscillatory solution in which tension and shear tractions along the contact interface is decoupled from each other. The model indicates that the mismatch strain has significant effect on the contact area and the pull-off process. Under a finite pulling force, a pair of adhering spheres is predicted to break apart spontaneously at a critical mismatch strain. This study suggests an adhesion mediated deformation sensing mechanism by which cells and molecules can detect mechanical signals in the environment via adhesive interactions.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.