Continuum approximation of the Peach-Koehler force on dislocations in a slip plane

We derive a continuum model for the Peach-Koehler force on dislocations in a slip plane. To represent the dislocations, we use the disregistry across the slip plane, whose gradient gives the density and direction of the dislocations. The continuum model is derived rigorously from the Peach-Koehler force on dislocations in a region that contains many dislocations. The resulting continuum model can be written as the variation of an elastic energy that consists of the contribution from the long-range elastic interaction of dislocations and a correction due to the line tension effect.     

Influence of single crystal orientation on homogeneous dislocation nucleation under uniaxial loading

Atomistic simulations are used to investigate how the stress required for homogeneous nucleation of partial dislocations in single crystal copper under uniaxial loading changes as a function of crystallographic orientation. Molecular dynamics is employed based on an embedded-atom method potential for Cu at 10 and 300 K. Results indicate that non-Schmid parameters are important for describing the calculated dislocation nucleation behavior for single crystal orientations under tension and compression. A continuum relationship is presented that incorporates Schmid and non-Schmid terms to correlate the nucleation stress over all tensile axis orientations within the stereographic triangle. Simulations investigating the temperature dependence of homogeneous dislocation nucleation yield activation volumes of ≈0.5- and activation energies of . For uniaxial compression, full dislocation loop nucleation is observed, in contrast to uniaxial tension. One of the main differences between uniaxial tension and compression is how the applied stress is resolved normal to the slip plane on which dislocations nucleate—in tension, this normal stress is tensile, and in compression, it is compressive. Last, the tension-compression asymmetry is examined as a function of loading axis orientation. Orientations with a high resolved stress normal to the slip plane on which dislocations nucleate have a larger tension-compression asymmetry with respect to dislocation nucleation than those orientations with a low resolved normal stress. The significance of this research is that the resolved stress normal to the slip plane on which dislocations nucleate plays an important role in partial (and full) dislocation loop nucleation in FCC Cu single crystals.     

An elasto-plastic damage constitutive theory and its prediction of evolution of subsequent yield surfaces and elastic constants

Based on pair functional potentials, Cauchy-Born rule and slip mechanism, a material model assembling with spring-bundle components, a cubage component and slip components is established to describe the elasto-plastic damage constitutive relation under finite deformation. The expansion/shrink, translation and distortion of yield surfaces can be calculated based on the hardening rule and Bauschinger effect defined on the slip component level. Both kinematic and isotropic hardening are included. Numerical simulations and predictions under tension, torsion, and combined tension-torsion proportional/non-proportional loading are performed to obtain the evolution of subsequent yield surfaces and elastic constants and compare with two sets of experimental data in literature, one for a very low work hardening aluminum alloy Al 6061-T6511, and another for a very high work hardening aluminum alloy annealed 1100 Al. The feature of the yield surface in shape change, which presents a sharp front accompanied by a blunt rear under proportional loading, is described by the latent hardening and Bauschinger effect of slip components. Further, the evolution law of subsequent yield surfaces under different proportional loading paths is investigated in terms of their equivalence. The numerical simulations under non-proportional loading conditions for annealed 1100 Al are performed, and the subsequent yield surfaces exhibit mixed cross effect because the kinematic hardening and isotropic hardening follow different evolution tendency when loading path changes. The results of non-proportional loading demonstrate that the present model has the ability to address the issue of complex loading due to the introduction of state variables on slip components. Moreover, as an elasto-plastic damage constitutive model, the present model can also reflect the variation of elastic constants through damage defined on the spring-bundle components.     

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.     

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.     

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.     

Computational nanoscale plasticity simulations using embedded atom potentials

In determining structure–property relations for plasticity at different size scales, it is desired to bridge concepts from the continuum to the atom. This raises many questions related to volume averaging, appropriate length scales of focus for an analysis, and postulates in continuum mechanics. In a preliminary effort to evaluate bridging size scales and continuum concepts with descritized phenomena, simple shear molecular dynamics simulations using the Embedded Atom Method (EAM) potentials were performed on single crystals. In order to help evaluate the continuum quantities related to the kinematic and thermodynamic force variables, finite element simulations (with different material models) and macroscale experiments were also performed. In this scoping study, various parametric effects on the stress state and kinematics have been quantified. The parameters included the following: crystal orientation (single slip, double slip, quadruple slip, octal slip), temperature (300 and 500 K), applied strain rate (10⁶–10¹² s⁻¹), specimen size (10 atoms to 2 μm), specimen aspect ratio size (1:8–8:1), deformation path (compression, tension, simple shear, and torsion), and material (nickel, aluminum, and copper). Although many conclusions can be drawn from this work, which has provided fodder for more studies, several major conclusions can be drawn.

• The yield stress is a function of a size scale parameter (volume-per-surface area) that was determined from atomistic simulations coupled with experiments. As the size decreases, the yield stress increases.

• Although the thermodynamic force (stress) varies at different size scales, the kinematics of deformation appears to be very similar based on atomistic simulations, finite element simulations, and physical experiments.

Atomistic simulations, that inherently include extreme strain rates and size scales, give results that agree with the phenomenological attributes of plasticity observed in macroscale experiments. These include strain rate dependence of the flow stress into a rate independent regime; approximate Schmid type behavior; size scale dependence on the flow stress, and kinematic behavior of large deformation plasticity.     

Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations

The Bauschinger and size effects in the thinfilm plasticity theory arising from the defect-energy of geometrically necessary dislocations (GNDs) are analytically investigated in this paper. Firstly, this defect-energy is deduced based on the elastic interactions of coupling dislocations (or pile-ups) moving on the closed neighboring slip plane. This energy is a quadratic function of the GNDs density, and includes an elastic interaction coefficient and an energetic length scale L. By incorporating it into the work- conjugate strain gradient plasticity theory of Gurtin, an energetic stress associated with this defect energy is obtained, which just plays the role of back stress in the kinematic hardening model. Then this back-stress hardening model is used to investigate the Bauschinger and size effects in the tension problem of single crystal Al films with passivation layers. The tension stress in the film shows a reverse dependence on the film thickness h. By comparing it with discrete-dislocation simulation results, the length scale L is determined, which is just several slip plane spacing, and accords well with our physical interpretation for the defect- energy. The Bauschinger effect after unloading is analyzed by combining this back-stress hardening model with a friction model. The effects of film thickness and pre-strain on the reversed plastic strain after unloading are quantified and qualitatively compared with experiment results.     

Nucleation of partial dislocations at a crack and its implication on deformation mechanisms of nanostructured metals

Nucleation of partial dislocations at a crack is analyzed based a multiscale model that incorporates atomic information into continuum-mechanics approach. The crack and the slip plane as the extension of the crack are modeled as a surface of displacement discontinuities embedded in an elastic medium. The atomic potential between the adjacent atomic layers along the slip plane is assumed to be the generalized stacking fault energy, which is obtained based on atomic calculations. The relative displacements along the slip plane, corresponding to the configurations of partial dislocations and stacking faults, are solved through the variational boundary integral method. The energetics of partial dislocation nucleation at the crack in FCC metals Al and Cu are comparatively studied for their distinctive difference in the intrinsic stacking fault energy. Compared with nucleation of perfect dislocations in previous studies, several new features have emerged. Among them, the critical stress and activation energy for nucleation of partial dislocations are markedly lowered. Depending on the value of stacking fault energy and crack configuration, the saddle-point configurations of partial dislocations can be vastly different in terms of the nucleation sequence and the size of the stacking fault. These findings have significant implication for identifying the fundamental dislocation and grain-boundary-mediated deformation mechanisms, which may account for the limiting strength and the high strain rate sensitivity of nanostructured metals.     

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.     

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〉.     

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.     

Theoretical latent hardening in crystals—II. BCC crystals in tension and compression

The general latent hardening law of single slip derived in the first paper of this series (Havner, Baker and Vause, 1979) is applied to an analysis of “overshooting” phenomena in bcc crystals in tension and compression. This new law, which predicts anisotropic hardening of latent slip systems, is based upon the simple theory of finite distortional crystal hardening introduced by Havner and Shalaby (1977).Because of historical ambiguities regarding identification of the slip plane in bcc metals, parallel analyses are presented corresponding to two separate criteria: (i) slip on {110}, {112} and {123} crystallographic planes only; and (ii) slip on the plane of maximum resolved shear stress containing a 〈111〉 direction. It is established that the new hardening law is a theory of “overshooting” in bcc crystals according to either identification of the slip plane.A qualitative comparison between theoretical results and two experimental papers on Fe crystals is included. The general difficulties in making comparisons with the experimental literature on finite distortional latent hardening are briefly discussed.     

Micro-crack tip fracture of commercial grade aluminum under mixed mode loading

In situ tensile tests were made in a scanning electron microscope (SEM) to investigate the deformation and micro-fracture in the immediate vicinity of a micro-crack tip in commercial pure aluminum with large-size crystal. Examined are the slip line field, stress intensity factor, strain energy density factor and crack tip opening displacement (CTOD) for mixed mode loading. Blunting and sharpening effects are observed. The latter is controlled by localized slip while the former by uniformed slip of the operating slip system with the highest crack tip Schmid factor. The operating slip system depends on the crystallographic orientation of crystal containing micro-cracks.The damage and fracture take place in the blunted region and depend on the coarsening and spacing of uniformed slip lines. The mixed mode micro-crack propagates along the direction where the voids grow and coalesce into the micro-crack. The direction also depends on the orientation of the applied loading. This suggests that the formation of macro-fracture mechanics could be applied. In particular, the minimum strain energy density criterion is suitable for determining the direction of micro-crack instability in the mixed mode. The in situ data were used to yield a nearly constant critical, minimum strain energy density factor for onset of micro-cracking.     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

Effet de l'anisotropie élastique cristalline sur la distribution des facteurs de Schmid à la surface des polycristauxInfluence of crystalline elasticity anisotropy on Schmid factor distribution at the free surface of polycrystals

Experimental studies of the plasticity mechanisms of polycrystals are usually based on the Schmid factor distribution supposing crystalline elasticity isotropy. A numerical evaluation of the effect of crystalline elasticity anisotropy on the apparent Schmid factor distribution at the free surface of polycrystals is presented. Cubic elasticity is considered. Order II stresses (averaged on all grains with the same crystallographic orientation) as well as variations between averages computed on grains with the same crystallographic orientation but with different neighbour grains are computed. The Finite Element Method is used. Commonly studied metals presenting an increasing anisotropy degree are considered (aluminium, nickel, austenite, copper). Concerning order II stresses in strongly anisotropic metals, the apparent Schmid factor distribution is drifted towards small Schmid factor values (the maximum Schmid factor is equal to 0.43 instead of 0.5) and the slip activation order between characteristic orientations of the crystallographic standard triangle is modified. The computed square deviations of the stresses averaged on grains with the same crystallographic orientation but with different neighbour grains are a bit higher than the second order ones (inter-orientation scatter). Our numerical evaluations agree quantitatively with several observations and measures of the literature concerning stress and strain distribution in copper and austenite polycrystals submitted to low amplitude loadings. Hopefully, the given apparent Schmid factor distributions could help to better understand the observations of the plasticity mechanisms taking place at the free surface of polycrystals. To cite this article: M. Sauzay, C. R. Mecanique 334 (2006).     

The effect of biaxial loading on the critical resolved shear stress of zinc single crystals

This paper reports an experimental investigation to test the hypothesis that the critical resolved shear stress is independent of the resolved normal stress on the active slip system plane. The resolved normal stress is varied independently for crystals of constant orientation by applying a uniform biaxial tension stress through slitted rubber strips glued to a flat tension sample. It is found that the critical resolved shear stress is constant until the resolved normal stress on the active slip system becomes twice the resolved shear stress; and then, for further increases in the resolved normal stress, the critical resolved shear stress begins to decrease.     

Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源,从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程.本文基于晶体塑性理论,从滑移系的受力和变形出发研究材料的细观损伤,建立了考虑滑移面分解正应力的细观损伤模型,为晶体材料解理断裂的分析提供了新方法.首先,在晶体弹塑性变形构型的基础上引入损伤变形梯度...     

Cyclic behavior of extruded magnesium: Experimental, microstructural and numerical approach

The present study aims at determining the influence of cyclic straining on the behavior of pure extruded magnesium. For this purpose, tensile, compressive and cyclic tests are performed (small plastic strains are applied (Δε^p/2 = 0.1% and 0.4%). Deformation mechanisms (slip and twin systems) have been observed by TEM and the different critical resolved shear stress (CRSS) have been determined. Based on microscopic observations, a crystal-plasticity-based constitutive model has been developed. The asymmetry between tensile and compressive loadings mainly results from the activation of hard slip systems in tension (such as 〈a〉 pyramidal and prismatic and 〈c + a〉 pyramidal glides) and twinning in compression. It is shown that basal slip is very easy to activate even for small Schmid factors. Numerical simulations reveal that untwinning in tension subsequent to compression must be considered to correctly fit the experimental S-shaped hysteresis curves. TEM observations indicate also intense secondary slips or twins inside the mother twins under cyclic conditions, so that twinning in compression and dislocation glide in tension are affected by cycling. The polycrystalline model allows to predict slip activities and twin volume fraction evolutions.