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.     

Orientation dependence of the plastic slip near notches in ductile FCC single crystals

Results from experiments conducted on copper FCC single crystals are reported. Two symmetric crystallographic orientations and four nonsymmetric crystallographic orientations were tested. The slip line fields that form near a pre-existing notch in these specimens were observed. The changes in these patterns as the orientation of the notch in the crystal is rotated in an {101} plane are discussed. Sectors of similar slip line patterns are identified and the type of boundaries between these sectors are discussed. A type of sector boundary called mixed kink is identified. Specimen orientations that differ by 90° are found to have different slip line patterns, contrary to the predictions of perfectly plastic slip line theory. The locations of the first slip lines to form are compared to the predictions obtained using anisotropic linear elastic stress field solutions and the initial plane-strain yield surfaces. It is found that comparison of these surface slip line fields to plane strain crack tip solutions in the annular region between 350 and is justified. The differences in anisotropic elastic solutions for orientations that are 90° apart explain the lack of agreement with perfectly plastic slip line theory.     

Plasticity size effects in voided crystals

The shear and equi-biaxial straining responses of periodic voided single crystals are analysed using discrete dislocation plasticity and a continuum strain gradient crystal plasticity theory. In the discrete dislocation formulation, the dislocations are all of edge character and are modelled as line singularities in an elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and annihilation are incorporated through a set of constitutive rules. Over the range of length scales investigated, both the discrete dislocation and strain gradient plasticity formulations predict a negligible size effect under shear loading. By contrast, under equi-biaxial loading both plasticity formulations predict a strong size dependence with the flow strength approximately scaling inversely with the void spacing. Excellent agreement is obtained between predictions of the two formulations for all crystal types and void volume fractions considered when the material length scale in the non-local plasticity model is chosen to be (about 10 times the slip plane spacing in the discrete dislocation models).     

Stochastic simulation of dislocation glide in tantalum and Ta-based alloys

We employ a kinetic Monte Carlo algorithm to simulate the motion of -oriented screw dislocation on a -slip plane in body centered cubic Ta and Ta-based alloys. The dislocation moves by the kink model: double kink nucleation, kink migration and kink-kink annihilation. Rates of these unit processes are parameterized based upon existing first principles data. Both short-range (solute-dislocation core) and long-range (elastic misfit) interactions between the dislocation and solute are considered in the simulations. Simulations are performed to determine dislocation velocity as a function of stress, temperature, solute concentration, solute misfit and solute-core interaction strength. The dislocation velocity is shown to be controlled by the rate of nucleation of double kinks and the dependence of the double kink nucleation rate on stress and temperature are consistent with existing analytical predictions. In alloys, dislocation velocity depends on both the short- and long-range solute dislocation interactions as well as on the solute concentration. The short-range solute-core interactions are shown to dominate the effects of alloying on dislocation mobility. The present simulation method provides the critical link between atomistic calculations of fundamental dislocation and solute properties and large scale dislocation dynamics that typically employ empirical equations of motion.     

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.     

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.     

Deformation of FCC nanowires by twinning and slip

We present atomistic simulations of the tensile and compressive loading of single crystal face-centered cubic (FCC) nanowires with and orientations to study the propensity of the nanowires to deform via twinning or slip. By studying the deformation characteristics of three FCC materials with disparate stacking fault energies (gold, copper and nickel), we find that the deformation mechanisms in the nanowires are a function of the intrinsic material properties, applied stress state, axial crystallographic orientation and exposed transverse surfaces. The key finding of this work is the first order effect that side surface orientation has on the operant mode of inelastic deformation in both and nanowires. Comparisons to expected deformation modes, as calculated using crystallographic Schmid factors for tension and compression, are provided to illustrate how transverse surface orientations can directly alter the deformation mechanisms in materials with nanometer scale dimensions.     

Geometrical properties of the vorticity vector derived using large-eddy simulation

In this paper, the geometrical properties of the resolved vorticity vector derived from large-eddy simulation are investigated using a statistical method. Numerical tests have been performed based on a turbulent Couette channel flow using three different dynamic linear and nonlinear subgrid-scale stress models. The geometrical properties of have a significant impact on various physical quantities and processes of the flow. To demonstrate, we examined helicity and helical structure, the attitude of with respect to the eigenframes of the resolved strain rate tensor and negative subgrid-scale stress tensor -τ_ij, enstrophy generation, and local vortex stretching and compression. It is observed that the presence of the wall has a strong anisotropic influence on the alignment patterns between and the eigenvectors of , and between and the resolved vortex stretching vector. Some interesting wall-limiting geometrical alignment patterns and probability density distributions in the form of Dirac delta functions associated with these alignment patterns are reported. To quantify the subgrid-scale modelling effects, the attitude of with respect to the eigenframe of -τ_ij is studied, and the geometrical alignment between and the Euler axis is also investigated. The Euler axis and angle for describing the relative rotation between the eigenframes of -τ_ij and are natural invariants of the rotation matrix, and are found to be effective for characterizing a subgrid-scale stress model and for quantifying the associated subgrid-scale modelling effects on the geometrical properties of .     

Multi-scale plasticity modeling: Coupled discrete dislocation and continuum crystal plasticity

A hierarchical multi-scale model that couples a region of material described by discrete dislocation plasticity to a surrounding region described by conventional crystal plasticity is presented. The coupled model is aimed at capturing non-classical plasticity effects such as the long-range stresses associated with a density of geometrically necessary dislocations and source limited plasticity, while also accounting for plastic flow and the associated energy dissipation at much larger scales where such non-classical effects are absent. The key to the model is the treatment of the interface between the discrete and continuum regions, where continuity of tractions and displacements is maintained in an average sense and the flow of net Burgers vector is managed via “passing” of discrete dislocations. The formulation is used to analyze two plane strain problems: (i) tension of a block and (ii) crack growth under mode I loading with various sizes of the discrete dislocation plasticity region surrounding the crack tip. The computed crack growth resistance curves are nearly independent of the size of the discrete dislocation plasticity region for region sizes ranging from to . The multi-scale model can reduce the computational time for the mode I crack analysis by a factor of 14 with little or no loss of fidelity in the crack growth predictions.     

Discrete dislocation plasticity analysis of the wedge indentation of films

The plane strain indentation of single crystal films on a rigid substrate by a rigid wedge indenter is analyzed using discrete dislocation plasticity. The crystals have three slip systems at ±35.3^° and 90^° with respect to the indentation direction. The analyses are carried out for three values of the film thickness, 2, 10 and , and with the dislocations all of edge character modeled as line singularities in a linear elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and dislocation annihilation are incorporated through a set of constitutive rules. Over the range of indentation depths considered, the indentation pressure for the 10 and thick films decreases with increasing contact size and attains a contact size-independent value for contact lengths . On the other hand, for the films, the indentation pressure first decreases with increasing contact size and subsequently increases as the plastic zone reaches the rigid substrate. For the 10 and thick films sink-in occurs around the indenter, while pile-up occurs in the film when the plastic zone reaches the substrate. Comparisons are made with predictions obtained from other formulations: (i) the contact size-independent indentation pressure is compared with that given by continuum crystal plasticity; (ii) the scaling of the indentation pressure with indentation depth is compared with the relation proposed by Nix and Gao [1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 43, 411-423]; and (iii) the computed contact area is compared with that obtained from the estimation procedure of Oliver and Pharr [1992. An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7, 1564-1583].     

Thin film delamination: A discrete dislocation analysis

Interface delamination during indentation of micron-scale ceramic coatings on metal substrates is modeled using discrete dislocation (DD) plasticity to elucidate the relationships between delamination, substrate plasticity, interface adhesion, elastic mismatch, and film thickness. In the DD method, plasticity in the metal substrate occurs directly via the motion of dislocations, which are governed by a set of physically based constitutive rules for nucleation, motion and annihilation. A cohesive law with peak stress characterizes the traction-separation response of the metal/ceramic interface. The indenter is a rigid flat punch and plane strain deformation is assumed. A continuum plasticity model of the same problem is studied for comparison. For low interface strengths (e.g. ), DD and continuum plasticity results are quantitatively similar, with delamination being nearly independent of interface strength, and easier for thinner, lower-modulus films. For higher interface strengths (), continuum plasticity predicts no delamination up to very high loads while the DD model shows a smooth increase in the critical indentation force for delamination with increasing interface strength. Tensile delamination in the DD model is driven by the accumulation of dislocations, and their associated high stresses, at the interface upon unloading. The DD model is thus capable of predicting the nucleation of cracks, and its dependence on material parameters, in realms of realistic constitutive behavior and/or small length scales where conventional continuum plasticity fails.     

Formation of molten metal films during metal-on-metal slip under extreme interfacial conditions

The present paper describes results of plate-impact pressure-shear friction experiments conducted to study time-resolved growth of molten metal films during dry metal-on-metal slip under extreme interfacial conditions. By employing tribo-pairs comprising hard tool-steel against relatively low melt-point metals such as 7075-T6 aluminum alloys, interfacial friction stress ranging from 100 to and slip speeds of approximately have been generated. These relatively high levels of friction stress combined with high slip-speeds generate conditions conducive for interfacial temperatures to approach the melting point of the lower melt point metal (Al alloy) comprising the tribo-pair.A Lagrangian finite element code is developed to understand the evolution of the thermo-mechanical fields and their relationship to the observed slip response. The code accounts for dynamic effects, heat conduction, contact with friction, and full thermo-mechanical coupling. At temperatures below the melting point the material is described as an isotropic thermally softening elastic-viscoplastic solid. For material elements with temperatures in excess of the melt point a purely Newtonian fluid constitutive model is employed.The results of the hybrid experimental-computational study provides new insights into the thermoelastic-plastic interactions during high speed metal-on-metal slip under extreme interfacial conditions. During the early part of frictional slip the coefficient of kinetic friction is observed to decrease with increasing slip velocity. During the later part transition in interfacial slip occurs from dry metal-on-metal sliding to the formation of molten Al films at the tribo-pair interface. Under these conditions the interfacial resistance approaches the shear strength of the molten aluminum alloy under normal pressures of approximately 1- and shear strain rates of . The results of the study indicate that under these extreme conditions molten aluminum films maintain a shearing resistance as high as .Scanning electron microscopy of the slip surfaces reveal molten aluminum to be smeared on the tribo-pair interface. Knoop hardness measurements in 7075-T6 Al alloy at various depths from the slip interface indicate that the hardness increases approximately linearly with depth and reaches a plateau at approximately from the surface.     

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.     

Indentation Schmid factor and orientation dependence of nanoindentation pop-in behavior of NiAl single crystals

Instrumented nanoindentation techniques have been widely used to characterize the small-scale mechanical behavior of materials. The elastic-plastic transition during nanoindentation is often indicated by a sudden displacement burst (pop-in) in the measured load-displacement curve. In defect-free single crystals, the pop-in is believed to be the result of homogeneous dislocation nucleation because the maximum shear stress corresponding to the pop-in load approaches the theoretical strength of the materials and because the statistical distribution of pop-in stresses is consistent with what is expected for a thermally activated process of homogeneous dislocation nucleation. This paper investigates whether this process is affected by crystallography and stress components other than the resolved shear stress. A Stroh formalism coupled with the two-dimensional Fourier transformation is used to derive the analytical stress fields in elastically anisotropic solids under Hertzian contact, which allows the determination of an indentation Schmid factor, namely, the ratio of maximum resolved shear stress to the maximum contact pressure. Nanoindentation tests were conducted on B2-structured NiAl single crystals with different surface normal directions. This material was chosen because it deforms at room temperature by {1 1 0}〈0 0 1〉 slip and thus avoids the complexity of partial dislocation nucleation. Good agreement is obtained between the experimental data and the theoretically predicted orientation dependence of pop-in loads based on the indentation Schmid factor. Pop-in load is lowest for indentation directions close to 〈1 1 1〉 and highest for those close to 〈0 0 1〉. In nanoindentation, since the stress component normal to the slip plane is typically comparable in magnitude to the resolved shear stress, we find that the pressure sensitivity of homogeneous dislocation nucleation cannot be determined from pop-in tests. Our statistical measurements generally confirm the thermal activation model of homogeneous dislocation nucleation. That is, the extracted dependence of activation energy on resolved shear stress is almost the same for all the indentation directions considered in this study, except for those close to 〈0 0 1〉. Because very high pop-in loads are measured for orientations close to 〈0 0 1〉, which implies a large contact area at pop-in, there is a higher probability of activating pre-existing dislocations in these orientations, which may explain the discrepancy near 〈0 0 1〉.     

The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity

We consider finite plasticity based on the decomposition F=F^eF^p of the deformation gradient F into elastic and plastic distortions F^e and F^p. Within this framework the macroscopic Burgers vector may be characterized by the tensor field . We derive a natural convected rate for G associated with evolution of F^p and as our main result show that, for a single-crystal,

•

temporal changes in G—as characterized by its convected time derivative—may be decomposed into temporal changes in distributions of screw and edge dislocations on the individual slip systems.

We discuss defect energies dependent on the densities of these distributions and show that corresponding thermodynamic forces are macroscopic counterparts of classical Peach-Koehler forces.     

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.