Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

Nanoscale rotational deformation effect on dislocation emission from an elliptically blunted crack tip in nanocrystalline materials

A grain size-dependent model is theoretically established to describe the effect of a special physical micromechanism of plastic flow on the dislocation emission from an elliptical blunt crack tip in nanocrystalline solids. The micromechanism represents the fast nanoscale rotational deformation (NRD) occurring through collective events of ideal nanoscale shear near crack tips, which as a stress source is approximately equivalent to a quadrupole of wedge disclinations. By the complex variable method, the grain size-dependent criterion for the dislocation emission from an elliptical blunt crack tip is derived. The influence of the grain size and the features of NRD on the critical stress intensity factors for dislocation emission is evaluated. The results indicate that NRD releases the high stresses near the crack tip region and thereby enhances the critical stress intensity factor for dislocation emission. The NRD has great influence on the most probable angle for dislocation emission. The critical stress intensity factor will increase with the increment of the grain size, which means the emission of the dislocation becomes more difficult for larger grain size due to the effect of NRD.     

Analysis of dislocation nucleation from a crystal surface based on the Peierls-Nabarro dislocation model

Dislocation nucleation from a stressed crystal surface is analyzed based on the Peierls-Nabarro dislocation model. The variational boundary integral approach is used to obtain the profiles of the embryonic dislocations in various three-dimensional nucleation configurations. The stress-dependent activation energies required to activate dislocations from their stable to unstable saddle point configurations are determined. Compared to previous analyses of this type of problem based on continuum elastic dislocation theory, the present analysis eliminates the uncertain core cutoff parameter by allowing for the existence of an extended dislocation core as the embryonic dislocation evolves. Moreover, atomic information can be incorporated to reveal the dependence of the nucleation process on the profile of the atomic interlayer potential as compared to continuum elastic dislocation theory in which only elastic constants and Burgers vector are relevant. Finally, the presented methodology can also be readily used to study dislocation nucleation from the surface heterogeneities such as cracks, steps, and quantum structures of electronic devices.     

Configurational force on a lattice dislocation and the Peierls stress

The solution for a crystalline edge dislocation is presented within a framework of continuum linear elasticity, and is compared with the Peierls–Nabarro solution based on a semi-discrete method. The atomic disregistry and the shear stress across the glide plane are discussed. The Peach–Koehler configurational force is introduced as the gradient of the strain energy with respect to the dislocation position between its two consecutive equilibrium positions. The core radius is assumed to vary periodically between equilibrium positions of the dislocation. The critical force is expressed in terms of the core radii or the energies of the stable and unstable equilibrium configurations. This is used to estimate the Peierls stress for both wide and narrow dislocations.     

Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics

A new numerical dislocation climb model based on incorporating the pipe diffusion theory (PDT) of vacancies with 3D discrete dislocation dynamics (DDD) is developed. In this model we hold that the climb rate of dislocations is determined by the gradient of the vacancy concentration on the segment, but not by the mechanical climb force as traditionally believed. The nodal forces on discrete dislocation segments in DDD simulation are transferred to PDT to calculate the vacancy concentration gradient. This transfer establishes a bridge connecting the DDD and PDT. The model is highly efficient and accurate. As verifications, two typical climb-involved examples are predicted, e.g. the activation of a Bardeen-Herring source as well as the shrinkage and annihilation of prismatic loops. Finally, the model is applied to study the breakup process of an infinite edge dislocation dipole into prismatic loops. This coupling methodology provides us a useful tool to intensively study the evolution of dislocation microstructures at high temperatures.     

Modelling of the internal stress in dislocation cell structures

The nonuniform distribution of dislocations in metals gives rise to material anisotropy and internal stresses that determine the mechanical response. This paper proposes a micromechanical model of a dislocation cell structure that accounts for the material inhomogeneity and incorporates the internal stresses in a physically-based manner. A composite model is employed to describe the material with its dislocation cell structure. The internal stress is obtained as a natural result of plastic deformation incompatibility and incorporated in the composite model. Applications of this model enable the prediction of the mechanical behavior of metals under various nonuniform deformations. The implementation of the model is relatively straightforward, allowing easy use in macroscopic engineering computations.     

A study of conditions for dislocation nucleation in coarser-than-atomistic scale models

We perform atomistic simulations of dislocation nucleation in defect free crystals in 2 and 3 dimensions during indentation with circular (2D) or spherical (3D) indenters. The kinematic structure of the theory of Field Dislocation Mechanics (FDM) is shown to allow the identification of a local feature of the atomistic velocity field in these simulations as indicative of dislocation nucleation. It predicts the precise location of the incipient spatially distributed dislocation field, as shown for the cases of the Embedded Atom Method potential for Al and the Lennard–Jones pair potential. We demonstrate the accuracy of this analysis for two crystallographic orientations in 2D and one in 3D. Apart from the accuracy in predicting the location of dislocation nucleation, the FDM based analysis also demonstrates superior performance than existing nucleation criteria in not persisting in time beyond the nucleation event, as well as differentiating between phase boundary/shear band and dislocation nucleation. Our analysis is meant to facilitate the modeling of dislocation nucleation in coarser-than-atomistic scale models of the mechanics of materials.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.     

Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories

This paper focuses on the unification of two frequently used and apparently different strain gradient crystal plasticity frameworks: (i) the physically motivated strain gradient crystal plasticity models proposed by Evers et al. [2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids 52, 2379-2401; 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209-5230] and Bayley et al. [2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structure 43, 7268-7286; 2007. A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine 87, 1361-1378] (here referred to as Evers-Bayley type models), where a physical back stress plays the most important role and which are further extended here to deal with truly large deformations, and (ii) the thermodynamically consistent strain gradient crystal plasticity model of Gurtin (2002-2008) (here referred to as the Gurtin type model), where the energetic part of a higher order micro-stress is derived from a non-standard free energy function. The energetic micro-stress vectors for the Gurtin type models are extracted from the definition of the back stresses of the improved Evers-Bayley type models. The possible defect energy forms that yield the derived physically based micro-stresses are discussed. The duality of both type of formulations is shown further by a comparison of the micro-boundary conditions. As a result, this paper provides a direct physical interpretation of the different terms present in Gurtin's model.     

Influence of misfit stresses on dislocation glide in single crystal superalloys: A three-dimensional discrete dislocation dynamics study

In the characteristic $\gamma /\gamma \prime$ microstructure of single crystal superalloys, misfit stresses occur due to a significant lattice mismatch of those two phases. The magnitude of this lattice mismatch depends on the chemical composition of both phases as well as on temperature. Furthermore, the lattice mismatch of γ and $\gamma \prime$ phases can be either positive or negative in sign. The internal stresses caused by such lattice mismatch play a decisive role for the micromechanical processes that lead to the observed macroscopic athermal deformation behavior of these high-temperature alloys. Three-dimensional discrete dislocation dynamics (DDD) simulations are applied to investigate dislocation glide in γ matrix channels and shearing of $\gamma \prime$ precipitates by superdislocations under externally applied uniaxial stresses, by fully taking into account internal misfit stresses. Misfit stress fields are calculated by the fast Fourier transformation (FFT) method and hybridized with DDD simulations. For external loading along the crystallographic [001] direction of the single crystal, it was found that the different internal stress states for negative and positive lattice mismatch result in non-uniform dislocation movement and different dislocation patterns in horizontal and vertical γ matrix channels. Furthermore, positive lattice mismatch produces a lower deformation rate than negative lattice mismatch under the same tensile loading, but for an increasing magnitude of lattice mismatch, the deformation resistance always diminishes. Hence, the best deformation performance is expected to result from alloys with either small positive, or even better, vanishing lattice mismatch between γ and $\gamma \prime$ phase.     

Simulation of edge dislocation emission from a crack under in-plane shear

Considered is the tandem emission of dislocations and dislocation dipoles from a crack under in-plane shear in one slip system as well as multiple slip systems. Effective stress intensity factors are determined by considering zones of local distortion similar to that in macro-plasticity. The dislocation free zone (DFZ) is also obtained which is analogous to the core region in fracture mechanics. Studied are effects of dislocation emission or development of plastic zone in front of the crack tip on the potential crack propagation based on the strain energy density factor criterion.     

Travelling wave solutions for a quasilinear model of field dislocation mechanics

We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this setting. Two classes of drag coefficient functions are considered, namely those with linear growth near the origin and those with constant or more generally sublinear growth there. A mathematical characterisation of all possible equilibria of these screw wall microstructures is given. We also prove the existence of travelling wave solutions for linear drag coefficient functions at low wave speeds and rule out the existence of nonconstant bounded travelling wave solutions for sublinear drag coefficients functions. It turns out that the appropriate concept of a solution in this scalar case is that of a viscosity solution. The governing equation in the static case is not proper and it is shown that no comparison principle holds. The findings indicate a short-range nature of the stress field of the individual dislocation walls, which indicates that the nonlinearity present in the model may have a stabilising effect. We predict idealised dislocation-free cells of almost arbitrary size interspersed with dipolar dislocation wall microstructures as admissible equilibria of our model, a feature in sharp contrast with predictions of the possible non-monotone equilibria of the corresponding Ginzburg-Landau phase field type gradient flow model.     

Analysis of heterogeneous deformation and dislocation dynamics in single crystal micropillars under compression

The size dependent deformation of Cu single crystal micropillars with thickness ranging from 0.2 to 2.5 μm subjected to uniaxial compression is investigated using a Multi-scale Dislocation Dynamics Plasticity (MDDP) approach. MDDP is a hybrid elasto-viscoplastic simulation model which couples discrete dislocation dynamics at the micro-scale (software micro3d) with the macroscopic plastic deformation. Our results show that the deformation field in these micropillars is heterogeneous from the onset of plastic flow and is confined to few deformation bands, leading to the formation of ledges and stress concentrations at the surface of the specimen. Furthermore, the simulation yields a serrated stress–strain behavior consisting of discrete strain bursts that correlates well with experimental observations. The intermittent operation and stagnation of discrete dislocation arms is identified as the prominent mechanism that causes heterogeneous deformation and results in the observed macroscopic strain bursts. We show that the critical stress to bow an average maximum dislocation arm, whose length changes during deformation due to pinning events, is responsible for the observed size dependent response of the single crystals. We also reveal that hardening rates, similar to that shown experimentally, occur under relatively constant dislocation densities and are linked to dislocation stagnation due to the formation of entangled dislocation configuration and pinning sites.     

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations resulting from a piecewise quadratic Peierls potential; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles and lattice friction, resulting in hardening, path dependency and hysteresis. A chief advantage of the present theory is that it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. In particular, no numerical grid is required in calculations. The phase-field representation enables complex geometrical and topological transitions in the dislocation ensemble, including dislocation loop nucleation, bow-out, pinching, and the formation of Orowan loops. The theory also permits the consideration of obstacles of varying strengths and dislocation line-energy anisotropy. The theory predicts a range of behaviors which are in qualitative agreement with observation, including: hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect, whereby reverse yielding gradually eliminates the influence of previous loading; the evolution of the dislocation density under cycling loading, leading to characteristic ‘butterfly’ curves; and others.     

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.     

Line-integral representations for the elastic displacements,stresses and interaction energy of arbitrary dislocation loops in transversely isotropic bimaterials

The elastic displacements, stresses and interaction energy of arbitrarily shaped dislocation loops with general Burgers vectors in transversely isotropic bimaterials (i.e. joined half-spaces) are expressed in terms of simple line integrals for the first time. These expressions are very similar to their isotropic full-space counterparts in the literature and can be easily incorporated into three-dimensional (3D) dislocation dynamics (DD) simulations for hexagonal crystals with interfaces/surfaces. All possible degenerate cases, e.g. isotropic bimaterials and isotropic half-space, are considered in detail. The singularities intrinsic to the classical continuum theory of dislocations are removed by spreading the Burgers vector anisotropically around every point on the dislocation line according to three particular spreading functions. This non-singular treatment guarantees the equivalence among different versions of the energy formulae and their consistency with the stress formula presented in this paper. Several numerical examples are provided as verification of the derived dislocation solutions, which further show significant influence of material anisotropy and bimaterial interface on the elastic fields and interaction energy of dislocation loops.     

Burst event detection in wall turbulence by WVITA method

Wavelet Variable Interval Time Average (WVITA) is introduced as a method incorporating burst event detection in wall turbulence. Wavelet transform is performed to unfold the longitudinal fluctuating velocity time series measured in the near wall region of a turbulent boundary layer using hot-film anemometer. This unfolding is both in time and in space simultaneously. The splitted kinetic of the longitudinal fluctuating velocity time series among different scales is obtained by integrating the square of wavelet coefficient modulus over temporal space. The time scale that related to burst events in wall turbulence passing through the fixed probe is ascertained by maximum criterion of the kinetic energy evolution across scales. Wavelet transformed localized variance of the fluctuating velocity time series at the maximum kinetic scale is put forward instead of localized short time average variance in Variable Interval Time Average (VITA) scheme. The burst event detection result shows that WVITA scheme can avoid erroneous judgement and solve the grouping problem more effectively which is caused by VITA scheme itself and can not be avoided by adjusting the threshold level or changing the short time average interval. The project supported by the National Natural Science Foundation of China (19732005) and the National Climbing Project of China     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part II

In Part I of this set of two papers, a model of mesoscopic plasticity is developed for studying initial-boundary value problems of small scale plasticity. Here we make qualitative, finite element method-based computational predictions of the theory. We demonstrate size effects and the development of strong inhomogeneity in simple shearing of plastically constrained grains. Non-locality in elastic straining leading to a strong Bauschinger effect is analyzed. Low shear strain boundary layers in constrained simple shearing of infinite layers of polycrystalline materials are not predicted by the model, and we justify the result based on an examination of the no-dislocation-flow boundary condition. The time-dependent, spatially homogeneous, simple shearing solution of PMFDM is studied numerically. The computational results and an analysis of continuous dependence with respect to initial data of solutions for a model linear problem point to the need for a nonlinear study of a stability transition of the homogeneous solution with decreasing grain size and increasing applied deformation. The continuous-dependence analysis also points to a possible mechanism for the development of spatial inhomogeneity in the initial stages of deformation in lower-order gradient plasticity theory. Results from thermal cycling of small scale beams/films with different degrees of constraint to plastic flow are presented showing size effects and reciprocal-film-thickness scaling of dislocation density boundary layer width. Qualitative similarities with results from discrete dislocation analyses are noted where possible.We discuss the convergence of approximate solutions with mesh refinement and its implications for the prediction of dislocation microstructure development, motivated by the notion of measure-valued solutions to conservation laws.