Dislocation Fields Research Articles

Continuous modeling of the structure of symmetric tilt boundaries

In polycrystals, the discontinuity of lattice rotation occurring across symmetric tilt boundaries is accommodated by the periodic arrangement of atoms in structural units. A crossover between this atomistic description and a continuous representation of tilt boundaries is carried out by designing periodic arrays of appropriately chosen smooth disclination dipoles. A comprehensive description of the boundary structure in terms of elastic strain, curvature and energy fields is then derived from a continuous theory of dislocation and disclination density fields, by allowing the initial distributions to relax in their own stress/couple stress fields. The resulting fields are obtained at nanoscale from finite element approximations of the theory. They compare remarkably well with predictions from molecular statics and experimental data. Beyond this description of grain boundaries as continua, the theory naturally provides a basis for coarse-grained spatio-temporal continuous descriptions of polycrystals.

Displacement field of a screw dislocation in a〈011〉Cu nanowire: An atomistic study

By performing atomistic calculations with a tight-binding potential, we study the displacement field induced by a screw dislocation lying along a free $\ensuremath{\langle}011\ensuremath{\rangle}$ Cu cylindrical nanowire. For this anisotropic orientation that is often encountered experimentally, we show that the displacement field ${u}_{z}$ along the nanowire can be seen as the superposition of three different fields: the screw dislocation field in an infinite medium, the warping displacement field caused by the so-called Eshelby twist, and an additional image field induced by the free surfaces. A Fourier series analysis of this latter image displacement and stress fields is given. For a circular cross section of the wire, this image field corresponds mainly to an additional warping displacement ${u}_{z}\ensuremath{\propto}xy$. The dissociation mechanism of the dislocation into partials and the surface stress effects being also captured in our simulations, the present study enables one to quantify the various contributions to the formation of the x-ray diffractograms.

Phase ordering kinetics of second-phase formation near an edge dislocation

The time-dependent Ginzburg-Landau (TDGL) equation for a single component non-conservative structural order parameter is used to study the spatio-temporal evolution of a second phase in the vicinity of an edge dislocation in an elastic crystalline solid. A symmetric Landau potential of sixth-order is employed. Dislocation field and elasticity modify the second-order and fourth-order coefficients of the Landau polynomial, respectively, where the former makes the coefficient singular at the origin. The TDGL equation is solved numerically using a finite volume method, where a wide range of parameter sets is explored. Computations are made for temperatures both above and below the transition temperature of a defect-free crystal . In both cases, the effects of the elastic properties of the solid and the strength of interaction between the order parameter and the displacement field are examined. If the system is quenched below , a steady state is first reached on the compressive side of the dislocation. On the tensile side, the growth is held back. The effect of thermal noise term in the TDGL equation is studied. We find that if the dislocation is introduced above , thermal noise supports the nucleation of the second phase, and a steady state will be attained earlier than if the thermal noise was absent. For a dislocation-free solid, we have compared our numerical computations for a mean-field (spatially averaged) order parameter versus time with the late time growth of the ensemble-averaged order parameter, calculated analytically, and find that both results follow upper asymptotes of sigmoid curves.

Resonant dislocation motion in NaCl crystals in the EPR scheme in the Earth’s magnetic field with pulsed pumping

Resonant dislocation motions in NaCl(Ca) crystals under the simultaneous action of the Earth’s magnetic field BEarth (∼66 μT) and a pulsed pump field \(\tilde B\) of sufficient amplitude \(\tilde B_m \) and certain duration τ have been detected and studied. The measured dislocation path peaks l(τ) have a maximum at τ = τr ≈ 0.53 μs. The resonance criterion has been found to be the ordinary EPR condition in which the g-factor is close to 2 and the optimum inverse pulse duration τr−1 is used instead of the harmonic pump field frequency νr. The largest peak l(τ) height is reached at mutually orthogonal dislocation (L) and magnetic field (BEarth and \(\tilde B\)) orientations. Pulsed field rotation to the position \(\tilde B\) ‖ BEarth significantly decreases but does not “kill” the effect. For dislocations parallel to the Earth’s field (L ‖ BEarth), the resonance almost disappears even at \(\tilde B\) ⊥ BEarth. In the optimum geometry of experiments, as the pump field amplitude \(\tilde B_m \) decreases from 17.6 to 10 μT, the path peak height lr = l(τr) decreases only by 7.5%, remaining at the level of lr ∼ 102 μm, and at a \(\tilde B_m \) further fall-off to 4 μT, it rapidly decreases to background values. In this case, the relative density of mobile dislocations similarly decreases from ∼90 to 40%. Possible physical mechanisms of the observed effect have been discussed.

From systems of discrete dislocations to a continuous field description: stresses and averaging aspects

Metal plasticity is governed by the motion of dislocations, and predicting the interactions and resulting collective motion of dislocations is a major task in understanding and modeling plastically deforming materials. This task has, despite all the efforts and advances of the last few decades, not yet been fully accomplished. The reason for this is that discrete models which describe the dislocation system with high accuracy are only computationally feasible for small systems, small strains, and high strain rates. Classical continuum models do not suffer from these restrictions but lack sufficiently detailed information about dislocation microstructure. In this paper we present the steps that are needed for averaging systems of discrete dislocations toward a continuous and hence more efficient representation. Our main emphasis lies on investigating the effects of averaging on the description of stress fields and dislocation interactions. We show how the evolution of continuous dislocation fields can then be appropriately described by a dislocation density-based model and validate our results by comparison with discrete dislocation dynamic simulations.

Nanoindentation cracking in gallium arsenide: Part I. In situ SEM nanoindentation

Abstract

Transformation Mechanism of Preprecipitation NiMn Nanostructures

The transformation mechanism of constrained b.c.c NiMn (β phase) clusters coherently precipitated during isothermal aging of Fe-Ni-Mn maraging alloy at 753 K (480 °C) has been studied. Preprecipitation of β-NiMn phase was demonstrated using conventional and high-resolution transmission electron microscopy and confirmed by flanking of electron diffraction spots, X-ray diffraction sidebands, and the invariance of lattice parameter and peak breadths in the earlier stages of age hardening. The β-NiMn clusters were recognized with interfacial dislocation dipoles lying on {011}α planes and Burgers vectors of a0\( \left\langle {01\overline{1} } \right\rangle_{\alpha } \) type. During further aging, the β-NiMn clusters transformed martensitically into fct NiMn (θ phase) precipitates as envisaged by the characteristic microtwinning texture. Considering the anomalous elastic softness of β-NiMn alloy, the transformation was plausibly assumed to take place by the localized soft-mode mechanism in which the structural instability of clusters is stimulated by the strain field of the interfacial dislocations. Computerized calculations in accordance with the phenomenological theory of martensite crystallography indicated that microtwinning of martensitic θ precipitates occurs in a self-accommodating manner. Transformation of very small β clusters at earlier stages of aging is thermodynamically hampered by the interfacial energy barrier.

Phase Field Simulation of Precipitation on Edge Dislocation

The second phase nucleation and precipitation around an edge dislocation are studied using phase field method. A new free energy function including the elastic strain energy is established to describe the dislocation interaction with the solute. The results give the distribution of strain field of dislocation and show the precipitates grow up according to the power growth law with the index 2. The simulated results are in good agreement with that of theory of non-uniform nucleation.

Unprecedented grain size effect on stacking fault width

Using an atomistic-phase field dislocation dynamics model, we isolate and investigate grain size and stress effects on the stacking fault width created by partial dislocation emission from a boundary. We show that the nucleation stress for a Shockley partial is governed by size of the boundary defect and insensitive to grain size. We reveal a grain size regime in which the maximum value the stacking fault width attains increases with grain size.

Disclination mediated plasticity in shear-coupled boundary migration

The shear-coupled boundary migration of 〈001〉 symmetric tilt boundaries is investigated within the framework of an elasto-plastic theory of disclination and dislocation fields. The tilt boundaries are built from periodic partial wedge-disclination dipole arrays, on the basis of their atomistic topography. Non-locality of the elastic response of the adjacent crystals stems from the defected structure of their boundary. Upon applying a shear strain to the bicrystal, couple stresses are generated, which set the disclination dipole array into motion normal to the boundary. In the process, edge dislocation densities with partial Burgers vector lying along the boundary are nucleated, whose glide parallel to the boundary and annihilation produces plastic shear. The misorientation dependence of the shear coupling factor predicted by the model is in full agreement with data from atomistic simulations and experiments. It is found to depend on the polarity and the magnitude of the wedge disclination dipoles composing the grain boundary.

A dynamic discrete dislocation plasticity method for the simulation of plastic relaxation under shock loading

In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (10 6 s −1 or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of time-dependent plasticity during shock loading is outlined here and will be presented in detail elsewhere.

Mode I crack analysis in single crystals with anisotropic discrete dislocation plasticity: I. Formulation and crack growth

Analyses of monotonic loading of a plane-strain mode I crack in an fcc single crystal under small-scale yielding are carried out using discrete dislocation plasticity (DDP) incorporating anisotropic elasticity. Two crystallographically symmetric crack orientations are considered where plane-strain plastic deformation is achieved by the nucleation and glide of edge dislocations on three effective slip systems. A cohesive surface ahead of the initial crack tip, endowed with the universal binding law, allows for the crack to grow. Simulations are carried out in an incremental way, using the superposition of the singular fields of dislocations in an anisotropic half-space and image fields that enforce boundary conditions and the cohesive properties. Considering a wide range of DDP parameters, i.e. source and obstacle densities, the dependence of fracture characteristics on these parameters is demonstrated. The crack growth resistance is found to be orientation-dependent and slightly smaller if the elastic anisotropy of the crystal is taken into consideration. Investigations of plastic dissipation and dislocation density evolution reveal that the dislocation structure evolving around the crack tip plays a key role in the fracture process.

Geometry of stress function surfaces for an asymmetric continuum

A two-dimensional stress field of dislocation or fault is geometrically studied for an asymmetric continuum. For geometric surfaces of the stress and couple-stress functions, the mean and Gaussian curvatures are derived. The mean curvature of couple-stress function surface is connected with the asymmetric of stress tensor. Moreover, the Gaussian curvature of stress function surface is characterized by both the stress and couple-stress. On the other hand, the mean curvature of stress function surface is not affected by the asymmetry of stress. Based on these geometric expressions, the Coulomb’s failure criterion and the friction coefficient are expressed by the curvatures of couple-stress function surface. Moreover, geometric structures of stress and couple stress function surfaces are shown for edge and wedge dislocations as faults. The curvatures of these surfaces show that the effect of couple-stress is constrained around the dislocations only.

Dislocations and inclusions in prestressed metals

The effect of prestress on dislocation (and inclusion) fields in nonlinear elastic solids is analysed by extending previous solutions by Eshelby and Willis. Using a plane-strain constitutive model (for incompressible incremental nonlinear elasticity) to describe the behaviour of ductile metals ( J 2 -deformation theory of plasticity), we show that when the level of prestress is high enough that shear band formation is approached, strongly localized strain patterns emerge, when a dislocation dipole is emitted by a source. These may explain cascade activation of dislocation clustering along slip band directions.

Full-Field Strain Mapping at a Ge/Si Heterostructure Interface.

The misfit dislocations and strain fields at a Ge/Si heterostructure interface were investigated experimentally using a combination of high-resolution transmission electron microscopy and quantitative electron micrograph analysis methods. The type of misfit dislocation at the interface was determined to be 60° dislocation and 90° full-edge dislocation. The full-field strains at the Ge/Si heterostructure interface were mapped by using the geometric phase analysis (GPA) and peak pairs analysis (PPA), respectively. The effect of the mask size on the GPA and PPA results was analyzed in detail. For comparison, the theoretical strain fields of the misfit dislocations were also calculated by the Peierls-Nabarro and Foreman dislocation models. The results showed that the optimal mask sizes in GPA and PPA were approximately three tenths and one-tenth of the reciprocal lattice vector, respectively. The Foreman dislocation model with an alterable factor a = 4 can best describe the strain field of the misfit dislocation at the Ge/Si heterostructure interface.

Experimental verification of segregation of carbon and precipitation of carbides due to deep cryogenic treatment for tool steel by internal friction method

This work presents an internal frictional behavior of cold work tool steel subjected to different heat treatment schedules to get insight related to segregation of carbon and refinement of carbide particles due to deep cryogenic treatment. The temperature dependence of internal friction was used to describe the variation of carbon concentration in solid solution of the martensite matrix in successive tempering steps. The results indicate that the carbon atoms segregated to nearby defects forming atomic clusters producing strong interactions, including interstitial carbon atoms themselves and between the interstitial carbon atoms with time-dependent strain field of dislocations because of lattice shrinking and thermodynamic instability of martensite during the deep cryogenic treatment. The clusters act as and grow into nuclei for the formation of fine carbide particle on subsequent tempering that was verified by analyses of TEM micrographs.

Effect of anisotropy, SIA orientation, and one-dimensional migration mechanisms on dislocation bias calculations in metals

Swelling in metals exposed to neutron irradiation has long been known to be enhanced by the preferential absorption of interstitials rather than vacancies, by dislocations. A common measure of this preference is called the dislocation bias factor and is computed from the ratio of the capture efficiency of interstitials to that of vacancies, by dislocations. Bias factors calculated through analytical and numerical means have been known to be approximately an order of magnitude larger than those expected by empirical swelling data. While this discrepancy has been justified by some in the past, it has remained an issue of debate among many. A major issue that has not been explored, however, is the effect of the numerous assumptions and approximations in previous studies. In this study, we explore the effect of three major assumptions of past studies. First, anisotropy in the displacement fields of both point-defects and dislocations is accounted for in our calculations. This relieves the assumption that self-interstitial atoms (SIAs) are centers of dilatation in isotropic media. Secondly, since SIAs have energetically-preferred orientations in dislocation stress fields, we use a spatially dependent elastic dipole tensor that accounts for preferred orientations. Lastly, since SIAs are known to undergo fast one-dimensional (1-D) migration in the Burgers direction in the vicinity of dislocations, we specify a fraction of SIAs to migrate one-dimensionally using a modified SIA diffusion tensor. These attributes are implemented into a combined finite-element method (FEM) rate-theory (RT) approach to calculate dislocation bias factors in body-centered cubic iron and face-centered cubic copper. In copper, we observe a significant increase in the bias when anisotropy and preferred SIA orientations are considered. In iron, our results show a drastic decrease in the bias when anisotropy and 1-D SIA motion are implemented, but these effects are countered when preferred SIA orientations are considered.

Thermal response of grain boundaries in graphene sheets under shear strain from atomistic simulations

Thermal transport properties of pristine and defective graphene sheets under shear strains are investigated by molecular dynamics simulations. It is found that the graphene symmetric tilt grain boundary with a higher (lower)-dislocation density has a higher (lower) shear failure strength. This counter-intuitive result is attributed to the mutual cancelation of strain field of grain boundary dislocations when they are close to each other. In particular, we compute the thermal conductivity across the graphene grain boundaries with different grain boundary dislocation densities. The applied shear strain ε is between 0 and 0.2 to avoid rupture of the defective graphene. In this range of strains, the thermal conductivity of polycrystalline graphene with lower-dislocation density can be modulated up to 23% (0<ε<0.12), which is significantly greater than that of the pristine graphene (below 10% in the same range), thereby opening the possibility for thermomutability by using the defective graphene under shear strain.

Dependence of equilibrium stacking fault width in fcc metals on the γ-surface

A phase field dislocation dynamics model that can model widely extended dislocations is presented. Through application of this model, we investigate the dependence of equilibrium stacking fault width (SFW) on the material γ-surface in fcc metals. This phase field model includes a direct energetic dependence on a parametrization of the entire γ-surface, which is directly informed by density functional theory. A wide range of materials are investigated and include both very low stacking fault energy (SFE) materials, such as silver, and high SFE materials, such as palladium. Additionally, analysis shows that by accounting for the unstable stacking fault energy, γU, we can better describe the material dependence of the equilibrium SFW rather than using only the intrinsic SFE, γI. Specifically, we see a direct dependence of the stable SFW between partial dislocations on the energy difference (γU − γI), which describes the energy barrier that partial dislocations must overcome in order to widen the stacking fault.

Crystal dynamics of forming ε-martensite with {334}α habitus planes in titanium

Formation of crystals with {334}α habitus planes often observed in titanium during the BCC-HCP (α–e) reorganization is considered in the context of the dynamic theory of martensitic transformations. It is demonstrated that the fast wave growth of the crystals is started in elastic fields of edge dislocations with lines α and Burgers vectors α.