Size and strain rate effects in tensile strength of penta-twinned Ag nanowires

Penta-twinned Ag nanowires(pt-AgNWs) have recently attracted much attention due to their interesting mechanical and physical properties. Here we perform largescale atomistic simulations to investigate the influence of sample size and strain rate on the tensile strength of pt-AgNWs. The simulation results show an apparent size effect in that the nanowire strength(defined as the critical stress for dislocation nucleation) increases with decreasing wire diameter. To account for such size effect, a theoretical model involving the interaction between an emerging dislocation and the twin boundary has been developed for the surface nucleation of dislocations. It is shown that the model predictions are in quantitative agreement with the results from atomistic simulations and previous experimental studies in the literatures. The simulations also reveal that nanowire strength is strain-rate dependent, which predicts an activation volume for dislocation nucleation in the range of 1–10b~3,where b is the magnitude of the Burgers vector for a full dislocation.     

Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper

Nanoscale contact of material surfaces provides an opportunity to explore and better understand the elastic limit and incipient plasticity in crystals. Homogeneous nucleation of a dislocation beneath a nanoindenter is a strain localization event triggered by elastic instability of the perfect crystal at finite strain. The finite element calculation, with a hyperelastic constitutive relation based on an interatomic potential, is employed as an efficient method to characterize such instability. This implementation facilitates the study of dislocation nucleation at length scales that are large compared to atomic dimensions, while remaining faithful to the nonlinear interatomic interactions. An instability criterion based on bifurcation analysis is incorporated into the finite element calculation to predict homogeneous dislocation nucleation. This criterion is superior to that based on the critical resolved shear stress in terms of its accuracy of prediction for both the nucleation site and the slip character of the defect. Finite element calculations of nanoindentation of single crystal copper by a cylindrical indenter and predictions of dislocation nucleation are validated by comparing with direct molecular dynamics simulations governed by the same interatomic potential. Analytic 2D and 3D linear elasticity solutions based on the Stroh formalism are used to benchmark the finite element results. The critical configuration of homogeneous dislocation nucleation under a spherical indenter is quantified with full 3D finite element calculations. The prediction of the nucleation site and slip character is verified by direct molecular dynamics simulations. The critical stress state at the nucleation site obtained from the interatomic potential is in quantitative agreement with ab initio density functional theory calculation.     

A stress-gradient based criterion for dislocation nucleation in crystals

Dislocations are the main lattice defects responsible for the strength and ductility of crystalline solids. The generation of new dislocations is an essential aspect of crystal defect physics, but a fundamental understanding of the mechanical conditions which lead to dislocation nucleation has remained elusive. Here, we present a nucleation criterion motivated from continuum thermomechanical considerations of a crystalline solid undergoing deformation, and demonstrate the criterion's ability to correctly predict dislocation nucleation via direct atomistic simulations. We further demonstrate that the commonly held notion of a nucleation criterion based on the magnitude of local stress components is incorrect.     

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.     

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.     

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.     

Micro-plasticity of surface steps under adhesive contact: Part I—Surface yielding controlled by single-dislocation nucleation

Mechanics of nano- and meso-scale contacts of rough surfaces is of fundamental importance in understanding deformation and failure mechanisms of a solid surface, and in engineering fabrication and reliability of small surface structures. We present a micro-mechanical dislocation model of contact-induced deformation of a surface step or ledge, as a unit process model to construct a meso-scale model of plastic deformations near and at a rough surface. This paper (Part I) considers onset of contact-induced surface yielding controlled by single-dislocation nucleation from a surface step. The Stroh formalism of anisotropic elasticity and conservation integrals are used to evaluate the driving force on the dislocation. The driving force together with a dislocation nucleation criterion is used to construct a contact-strength map of a surface step in terms of contact pressure, step height, surface adhesion and lattice resistance. Atomistic simulations of atomic surface-step indentation on a gold (1 0 0) surface have been also carried out with the embedded atom method. As predicted by the continuum dislocation model, the atomistic simulations also indicate that surface adhesion plays a significant role in dislocation nucleation processes. Instabilities due to adhesion and dislocation nucleation are evident. The atomistic simulation is used to calibrate the continuum dislocation nucleation criterion, while the continuum dislocation modeling captures the dislocation energetics in the inhomogeneous stress field of the surface-step under contact loading. Results show that dislocations in certain slip planes can be easily nucleated but will stay in equilibrium positions very close to the surface step, while dislocations in some other slip planes easily move away from the surface into the bulk. This phenomenon is called contact-induced near-surface dislocation segregation. As a consequence, we predict the existence of a thin tensile-stress sub-layer adjacent to the surface within the boundary layer of near-surface plastic deformation. In the companion paper (Part II), we analyze the surface hardening behavior caused by multiple dislocations.     

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.     

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.     

New inroads in an old subject: Plasticity, from around the atomic to the macroscopic scale

Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a travelling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor-dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.     

Multiscale simulation of material removal processes at the nanoscale

A dynamic multiscale simulation method has been used to study the nanoscale material removal processes for single crystals. The model simultaneously captures the atomistic mechanisms during material removal from the free surface and the long-range mobility of dislocations and their interactions without the computational cost of full atomistic simulations. The method also permits the simulation of system sizes that are approaching experimentally accessibly systems, albeit in 2D. Simulations are performed on single crystal aluminum to study the atomistic details of material removal, chip formation, surface evolution, and generation and propagation of dislocations for a wide range of tool speeds (20-800 m/s) at room temperature. The results from these simulations demonstrate the power of the developed method in capturing both long-range dislocation plasticity and short-range atomistic phenomena during tool advance. In addition, we have investigated the effect of the scratching depth during the material removal process. Fluctuations of scratching tangential force are related to dislocation generation events during the material removal process. A transition from dislocation generation and glides at lower tool speeds to localized amorphization at high tool speeds is found to give rise to an optimal tool speed for low cutting forces.     

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.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate 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. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

Micro-plasticity of surface steps under adhesive contact: Part II—Multiple-dislocation mediated contact hardening

The study of micro-plastic behavior of rough surface contacts is the critical link towards a fundamental understanding of contact, friction, adhesion, and surface failures at small length scales. In the companion paper (Yu, H.H., Shrotriya, P., Gao, Y.F., Kim, K.-S., 2007. Micro-plasticity of surface steps under adhesive contact. Part I. Surface yielding controlled by single-dislocation nucleation. J. Mech. Phys. Solids 55, 489–516), we have studied the onset of surface yielding due to single-dislocation nucleation from a stepped surface under adhesive contact. Here we analyze the contact hardening behavior due to multiple dislocations in a two-dimensional dislocation model. Continuum micro-mechanical analyses are used to derive the configurational force on the dislocation, while a modified Rice–Thomson criterion is used to model dislocation nucleation. Dislocations nucleated from the surface step are stabilized and pile up as a result of the balance between the resolved driving force and the non-zero lattice resistance in the solid. The dislocation pileup will exert a strong back stress to prevent further dislocation nucleation and thus lead to the contact hardening behavior, the degree of which depends on the slip-plane orientation. Particularly, we find that dislocation interactions between two slip planes can make the contact loading order-of-magnitude easy to nucleate multiple dislocations, which is thus named “latent softening”. A mechanistic explanation shows that the latent softening is closely related to the stress-concentration mode mixity at the surface step. Dislocation nucleation will modify the geometric characteristics of the surface step, so that the contact-induced stress state near the step, as described by the mode mixity, changes, which influences the subsequent dislocation nucleation. Our calculations show that the dislocation pileup on one slip plane can even cause the spontaneous dislocation nucleation on the other slip plane without further increase of the contact load. Furthermore, it is found that rough surface contacts at small length scale can lead to the dislocation segregation and the formation of a surface tensile sub-layer. The discrete-dislocation model presented here and in the companion paper provides novel insights in bridging the atomistic simulations and continuum plastic flow analysis of surface asperity contact.     

On numerically predicting the onset and mode of instability in atomistic systems

We present an accurate and efficient method based on the Lanczos algorithm for predicting the onset and mode of instability in atomistic systems. Specifically, we develop a framework that is identically applicable to all flavors of atomistic simulations, including ab-initio calculations. Notably, we do not make any apriori assumptions regarding the nature of the instability or its location. We verify the accuracy of the proposed approach by studying defect nucleation during the nanoindentation of a triangular lattice and hydrostatic tension test of an aluminum crystal. We demonstrate that the computational cost in practical calculations scales linearly with system size, and is accompanied by a small prefactor. Overall, the proposed method is attractive because it enables the stability analysis of atomistic systems at the mesoscale.     

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.     

Atomistic simulation study on key factors dominating dislocation nucleation from a crack tip in two FCC materials: Cu and Al

Dislocation nucleations from crack tips in FCC copper and aluminum are studied using atomistic simulations. It is shown that the critical load for dislocation nucleation predicted by Rice’s model (Rice, 1992) based on the Peierls concept of dislocation can either be under- or over-estimated in reference to the simulation results. Such discrepancies have not been fully resolved by existing improved nucleation models, due to the complicated atomic environments at crack tips. Based on our simulation results, it is proposed that such discrepancies can be reconciled by the competition of two coupling processes at a crack tip: the tension-shear coupling, which facilitates the dislocation nucleation, and the nucleation-debonding coupling, which retards the dislocation nucleation. In addition, the two couplings are applied to explain the paradoxical observation: easy dislocation nucleation at a blunted crack tip. The present work provides a detailed picture to justify future improvements on Rice’s model for dislocation nucleation and to accurately predict intrinsic brittle to ductile transition for crystalline materials.     

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.     

Atomistically determined phase-field modeling of dislocation dissociation,stacking fault formation,dislocation slip,and reactions in fcc systems

The purpose of the current work is the development of a phase field model for dislocation dissociation, slip and stacking fault formation in single crystals amenable to determination via atomistic or ab initio methods in the spirit of computational material design. The current approach is based in particular on periodic microelasticity (Wang and Jin, 2001, Bulatov and Cai, 2006, Wang and Li, 2010) to model the strongly non-local elastic interaction of dislocation lines via their (residual) strain fields. These strain fields depend in turn on phase fields which are used to parameterize the energy stored in dislocation lines and stacking faults. This energy storage is modeled here with the help of the ”interface” energy concept and model of Cahn and Hilliard (1958) (see also Allen and Cahn, 1979, Wang and Li, 2010). In particular, the “homogeneous” part of this energy is related to the “rigid” (i.e., purely translational) part of the displacement of atoms across the slip plane, while the “gradient” part accounts for energy storage in those regions near the slip plane where atomic displacements deviate from being rigid, e.g., in the dislocation core. Via the attendant global energy scaling, the interface energy model facilitates an atomistic determination of the entire phase field energy as an optimal approximation of the (exact) atomistic energy; no adjustable parameters remain. For simplicity, an interatomic potential and molecular statics are employed for this purpose here; alternatively, ab initio (i.e., DFT-based) methods can be used. To illustrate the current approach, it is applied to determine the phase field free energy for fcc aluminum and copper. The identified models are then applied to modeling of dislocation dissociation, stacking fault formation, glide and dislocation reactions in these materials. As well, the tensile loading of a dislocation loop is considered. In the process, the current thermodynamic picture is compared with the classical mechanical one as based on the Peach-Köhler force.