Geometrically Necessary Dislocation Density Research Articles

The Role of Geometrically Necessary Dislocations in Cantilever Beam Bending Experiments of Single Crystals.

The mechanical behavior of single crystalline, micro-sized copper is investigated in the context of cantilever beam bending experiments. Particular focus is on the role of geometrically necessary dislocations (GNDs) during bending-dominated load conditions and their impact on the characteristic bending size effect. Three different sample sizes are considered in this work with main variation in thickness. A gradient extended crystal plasticity model is presented and applied in a three-dimensional finite-element (FE) framework considering slip system-based edge and screw components of the dislocation density vector. The underlying mathematical model contains non-standard evolution equations for GNDs, crystal-specific interaction relations, and higher-order boundary conditions. Moreover, two element formulations are examined and compared with respect to size-independent as well as size-dependent bending behavior. The first formulation is based on a linear interpolation of the displacement and the GND density field together with a full integration scheme whereas the second is based on a mixed interpolation scheme. While the GND density fields are treated equivalently, the displacement field is interpolated quadratically in combination with a reduced integration scheme. Computational results indicate that GND storage in small cantilever beams strongly influences the evolution of statistically stored dislocations (SSDs) and, hence, the distribution of the total dislocation density. As a particular example, the mechanical bending behavior in the case of a physically motivated limitation of GND storage is studied. The resulting impact on the mechanical bending response as well as on the predicted size effect is analyzed. Obtained results are discussed and related to experimental findings from the literature.

Influence of Noise-Generating Factors on Cross-Correlation Electron Backscatter Diffraction (EBSD) Measurement of Geometrically Necessary Dislocations (GNDs)

Studies of dislocation density evolution are fundamental to improved understanding in various areas of deformation mechanics. Recent advances in cross-correlation techniques, applied to electron backscatter diffraction (EBSD) data have particularly shed light on geometrically necessary dislocation (GND) behavior. However, the framework is relatively computationally expensive-patterns are typically saved from the EBSD scan and analyzed offline. A better understanding of the impact of EBSD pattern degradation, such as binning, compression, and various forms of noise, is vital to enable optimization of rapid and low-cost GND analysis. This paper tackles the problem by setting up a set of simulated patterns that mimic real patterns corresponding to a known GND field. The patterns are subsequently degraded in terms of resolution and noise, and the GND densities calculated from the degraded patterns using cross-correlation ESBD are compared with the known values. Some confirmation of validity of the computational degradation of patterns by considering real pattern degradation is also undertaken. The results demonstrate that the EBSD technique is not particularly sensitive to lower levels of binning and image compression, but the precision is sensitive to Poisson-type noise. Some insight is also gained concerning effects of mixed patterns at a grain boundary on measured GND content.

The depth dependent hardness of bicrystals with dislocation transmission through grain boundaries: A theoretical model

In this paper, a dislocation density based theoretical model is proposed to explain the depth dependent hardness of bicrystals. The density of geometrically necessary dislocations (GNDs) is affected by the grain boundary (GB) through the consideration of the truncation and limited expansion of the plasticity affected region (PAR) when the indenter is in the vicinity of the GB. The latter is found to be the dominant reason for the increase of hardness when the GB impedes the movement of GNDs and prevents the slip transfer behavior. Once the accumulated stress originating from the piled up dislocations triggers the transmission of dislocations through the GB interface, the density of GNDs under the indenter decreases leading to the reduction of hardness. Based on this model, the dislocation transmission through a dedicated GB interface can be quantitatively analyzed, which involves the evolution of the density of GNDs and PAR in adjacent grains during the slip transfer process. The rationality and accuracy of the proposed model are validated by comparing the fitted theoretical results with experimental data through nano-indentation (NI). Moreover, the prediction of the model can match well with experimental results.

Modelling the transport of geometrically necessary dislocations on slip systems: application to single- and multi-crystals of ice

A model based on the elastic theory of continuously distributed dislocations, accounting for the transport of geometrically necessary dislocations (GND) on slip systems is developed. It allows keeping the crystallographic nature of glide by allocating velocities specific to slip systems to GND. At grain boundaries, the dislocation transport equation is resolved between a specific system in a grain and a specific system in the adjacent grain. It is used to simulate a compression creep test followed by unloading of a multiple slip deforming multi-crystal of ice during which kink band formation, grain boundary migration and localized grain nucleation are observed. The model predictions are compared to 2D strain fields obtained by digital image correlation and show a good agreement. Besides, the kink band position corresponds very well with an area of strong lattice misorientation predicted by the model and is also bounded by opposite densities of edge dislocations, in agreement with kink banding theory and characterization. Furthermore, the grain boundary migration is observed to happen from predicted low dislocation density area towards high dislocation ones—also in agreement with the theory. Lastly, the triple junctions where nucleation is observed are also characterized by high GND density and especially strong gradient of elastic energy density. These different features show the relevance of using a continuum theory of polarized dislocations per slip system to study the onset of relaxation mechanisms like kink banding, grain boundary migration and grain nucleation and possibly to propose nucleation and migration criteria.

Influences of high strain rate, low temperature, and deformation direction on microstructural evolution and mechanical properties of copper

Step-wise deformations are applied to pure copper along uniaxial direction or multiaxial directions in three conditions of speed and temperature: static and room temperature, dynamic and room temperature, or dynamic and low temperature. Through examinations of total 36 deformation stages, the influences of strain rate, temperature, and deformation direction on microstructural evolution and mechanical properties are investigated. According to the results, the deformation mechanism at extreme conditions (high strain rate and low temperature) changes from only dislocation slip to dislocation slip and deformation twinning. Although deformation-direction change has no significant effect on the final mechanical properties and total dislocation densities measured from X-ray diffraction results, the evolutions of grain-boundary misorientation and geometrically necessary dislocations (GNDs) are clearly influenced. The trend of microstructural evolution, i.e. larger grain size and lower GND densities by multi-axial deformation than the uniaxial case, is reversed when twinning occurred. Based on the results, we suggest four evolution models: uniaxial or multi axial cases with dislocation slip, or with dislocation slip and deformation twining. Even if the homogenous and almost fully twinned microstructure producing improved mechanical strength was obtained by applying the extreme deformations, poor ductility of the processed specimen were observed.

Correlation of microstructure and strain hardening behavior in the ultrafine-grained Nb-bearing dual phase steels

Ultrafine-grained dual phase (DP) steels with different Nb contents (0.00, 0.06 and 0.12wt%) were produced by cold-rolling followed by intercritical annealing of ferrite/martensite starting microstructure at 770°C for different holding times. Scanning electron microscopy, equipped with electron backscattered diffraction (EBSD) detector, nanoindentation and tensile testing were used to characterize microstructural evolutions and their correlations to the strain hardening and fracture behavior. EBSD results confirmed the retardation effect of Nb on recrystallization. It was found that the strains stored in the grains and density of geometrically necessary dislocations (GNDs) were increased with the addition of Nb. Strain hardening analysis showed that plastic deformation of the DP steels occurred in three distinct stages, which based on the EBSD results, nanoindentation and fracture analysis, were controlled by microstructural features such martensite volume fraction and size, density of GNDs and individual ferrite and martensite tensile properties.

Determination of geometrically necessary dislocations in large shear strain localization in aluminum

In this paper, a systematic approach is presented to quantifying shear band evolution by quantifying geometrically necessary dislocations (GND) associated with morphological anisotropy in 7039-aluminum alloy using the compact forced-simple shear (CFSS) design. A statistically motivated approach, i.e. the line averaged GND density profile, has been developed to investigate the GND density near heavily deformed, shear band regions. Our study shows that: i) line average GND density profiles for the Al samples machined in the A-direction (transverse to pancake-shaped grains), B-direction (parallel to longitudinal pancake-shaped grains, shearing in through-thickness direction), C-direction (parallel to pancake-shaped grains, shearing in the in-plane direction) and D-direction (parallel and through the pancake-shaped grains) are nominally similar; ii) apart from 7039-aluminum alloy C-direction that has a uniform GND distribution in the direction normal to shear due to a grain-sliding mechanism, GND profiles for other samples decrease steadily away from the shear band as plastic strain diminishes, in agreement with Ashby's theory of work hardening, iii) anisotropy in damage evolution and shear-stress shear-strain response of 7039-aluminum alloy is associated with the grain structure of the material, i.e. morphological anisotropy creating variations in grain boundary interactions; iv) microbands formation in D-direction is associated with local GND peaks; v) stress-relief crack propagating along grain boundaries due to the presence of voids or inclusions generates a ‘shielding effect’ on neighboring grains; and vi) the line average GND density profile within a single grain usually varies inversely with the width of the grain for A-, B- and D-directions, leading to generally pronounced higher GND density near triple junctions.

Geometrically necessary dislocations favor the Taylor uniform deformation mode in ultra-fine-grained polycrystals

The density of geometrically necessary dislocations (GND) obtained from the lattice curvature was studied in commercially pure copper up to extreme large strains (von Mises strain of 63). Its evolution shows an increase to a maximum at a strain of about 2, then decreases until reaching the stationary limiting stage of grain refinement at a von Mises strain of about 14. At the same time, the total dislocation density is also decreasing. It is shown that the variation in the GND density correlates with the difference between the correlated (first neighbor grains) and the non-correlated (random neighbor) misorientation angle distributions. The low quantity of GND at extreme large strains is a consequence of the near Taylor-type homogeneous behavior of the polycrystalline ultrafine-grained structure.

Geometrically necessary dislocation densities in olivine obtained using high-angular resolution electron backscatter diffraction

Dislocations in geological minerals are fundamental to the creep processes that control large-scale geodynamic phenomena. However, techniques to quantify their densities, distributions, and types over critical subgrain to polycrystal length scales are limited. The recent advent of high-angular resolution electron backscatter diffraction (HR-EBSD), based on diffraction pattern cross-correlation, offers a powerful new approach that has been utilised to analyse dislocation densities in the materials sciences. In particular, HR-EBSD yields significantly better angular resolution (<0.01°) than conventional EBSD (~0.5°), allowing very low dislocation densities to be analysed. We develop the application of HR-EBSD to olivine, the dominant mineral in Earth's upper mantle by testing (1) different inversion methods for estimating geometrically necessary dislocation (GND) densities, (2) the sensitivity of the method under a range of data acquisition settings, and (3) the ability of the technique to resolve a variety of olivine dislocation structures. The relatively low crystal symmetry (orthorhombic) and few slip systems in olivine result in well constrained GND density estimates. The GND density noise floor is inversely proportional to map step size, such that datasets can be optimised for analysing either short wavelength, high density structures (e.g. subgrain boundaries) or long wavelength, low amplitude orientation gradients. Comparison to conventional images of decorated dislocations demonstrates that HR-EBSD can characterise the dislocation distribution and reveal additional structure not captured by the decoration technique. HR-EBSD therefore provides a highly effective method for analysing dislocations in olivine and determining their role in accommodating macroscopic deformation.

Influence of plastic deformation heterogeneity on development of geometrically necessary dislocation density in dual phase steel

The present investigation examined the evolution of geometrically necessary dislocation (GND) structure following tensile deformation in a commercially produced dual phase steel, DP 590. GND measurements were made using electron back scatter diffraction (EBSD). The average GND density increased with imposed macroscopic strain, however the rate of increased slowed with increasing strain. GND density was found to be influenced by the ferrite grain size and orientation of the ferrite grains. Small ferrite grains generally had a higher GND density. For this steel the highest GND density was measured for {011}[111] orientations. Analysis of these data using the classical Ashby model for GND content shows that GND density is increasing linearly with strain. The discrepancy between measured and predicted GND density is attributed to the plastic deformation of martensite reducing the requirement of compatibility between ferrite and martensite and dynamic recovery of the dislocation structures decreasing the rate of GND storage with strain.

Physical mechanisms of nanocrystallization of a novel Ni-based alloy under uniaxial compression at cryogenic temperature

A novel Ni-based alloy, which possesses a nominal chemical composition of Ni52Cu20Cr15Mo5Ti2Al1Nb5 (wt.%), is designed and fabricated in order to be used for cryogenic environments. The Ni-based alloy exhibits high plasticity and high strength in the case of uniaxial compression at −150°C. Nanocrystalline phases, in particular, are observed in the matrix of Ni-based alloy subjected to plastic deformation at −150°C. Statistically stored dislocation (SSD) density and geometrically necessary dislocation (GND) density are found to play a significant role in nanocrystallization of Ni-based alloy. SSD density and GND density increase with the increase in the plastic strain. There appears a critical dislocation density above which the critical resolved shear stress (CRSS) for dislocation slip was greater than the CRSS for deformation twinning. Consequently, deformation twinning occurs instead of dislocation slip with increasing plastic strain. Under the simultaneous action of dislocation slip and deformation twinning, dislocation wall, dislocation cell, deformation twin, low angle grain boundary and high angle grain boundary occurs in such an alternative way that nanocrystalline grains are formed in the end.

The effect of length scale on the determination of geometrically necessary dislocations via EBSD continuum dislocation microscopy

Electron backscatter diffraction (EBSD) dislocation microscopy is an important, emerging field in metals characterization. Currently, calculation of geometrically necessary dislocation (GND) density is problematic because it has been shown to depend on the step size of the EBSD scan used to investigate the sample. This paper models the change in calculated GND density as a function of step size statistically. The model provides selection criteria for EBSD step size as well as an estimate of the total dislocation content. Evaluation of a heterogeneously deformed tantalum specimen is used to asses the method.

繰り返し変形における圧縮変形が転位組織の発達に及ぼす影響

Slip deformation in tricrystal models subjected to cyclic deformation is investigated by a finite element crystal plasticity analysis code. Accumulation of geometrically necessary dislocations (GNDs) are studied in detail. Results of the analysis shown that non-uniform deformation and the high density of GNDs accumulated in the form of band around the grain boundary triple junction, this dislocation pattern formation shown that develop during increase of number of cyclic deformation and extend of development of dislocation pattern depend on the strain amplitude. Mechanism of dislocation pattern formation and extend of development of dislocation pattern are discussed from the viewpoint of influence of compressive deformation and strain amplitude on dislocation interaction of primary and secondary slip systems.

Lattice strains at cracks in single crystal titanium: Elastic distortion and GND contributions

There is evidence from diffraction experiments that significant peak broadening is measured local to crack tips and this has been attributed to the development of geometrically necessary dislocations (GNDs) which are retained upon unloading. This is reasonable due to the stress singularity found locally at the crack which is expected to activate slip on favourably oriented slip systems, potentially resulting in plastic strain gradients and geometrically necessary dislocation development. Hence, a systematic study is presented here to ascertain the contributions of both elastic distortional strain and GND density to lattice deformation local to the crack at loaded and subsequently unloaded states. The results show that whilst elastic strains dominate lattice distortion in comparison to GNDs at the loaded state i.e. at the peak load applied, these strains are largely recovered upon unloading and the contribution from GND development subsequently dominates the broadening seen. Two initial crystallographic configurations were considered. In the example in which the crystal c-axis was oriented parallel to the loading direction, the <c+a> pyramidal systems contributed most to slip with <a> basal slip system contributing to a lesser extent. However, in the example where the c-axis was oriented perpendicular to the loading direction, the <c+a> pyramidal and <a> prismatic systems were the more significant contributors to slip and the <a> basal contributing to a lesser extent. However, basal, prismatic and c+a pyramidal slip systems were found to be active in both examples and this was attributed to significant lattice rotation driven by locally high stresses which enabled otherwise badly oriented slip systems to become favourable for slip. Finally, increases in GND density were seen upon unloading for c-axis orientation parallel with loading. This was attributed to the influence of <c+a> pyramidal slip on reverse plasticity leading to diffuse GND density distributions and significant resulting lattice strains compared to that from <a> prism slip loading

Assessment of geometrically necessary dislocation levels derived by 3D EBSD

Existing alternatives for the calculation of geometrically necessary dislocation (GND) densities from orientation fields are discussed. Importantly, we highlight the role of reference frames and consider different sources of error. A well-controlled micro cantilever bending experiment on a copper bicrystal has been analyzed by 3-dimensional electron back scatter diffraction (3D EBSD). The GND density is determined experimentally by two different approaches and assessed theoretically, assuming a homogeneous bending of the cantilever. Experiment and theory agree very well. It is further shown that the deformation is accommodated mainly by GNDs, which carry and store lattice rotation, and not (only) by mobile dislocations that leave a crystal portion inspected, without lattice rotations. A detailed GND analysis reveals a local density minimum close to the grain boundary and a distinct difference in edge to screw ratios for both grains.

On the mechanistic basis of fatigue crack nucleation in Ni superalloy containing inclusions using high resolution electron backscatter diffraction

A series of interrupted three-point bend low-cycle fatigue tests were carried out on a powder metallurgy FHG96 nickel superalloy sample containing non-metallic inclusions. High resolution electron backscatter diffraction (HR-EBSD) was used to characterise the distribution and evolution of geometrically necessary dislocation (GND) density, residual stress and total dislocation density near a non-metallic inclusion. A systematic study of room temperature cyclic deformation is presented in which slip localisation, cyclic hardening, ratcheting and stabilisation occur, through to crack formation and microstructurally-sensitive propagation. Particular focus is brought to bear at the inclusion–matrix interface. Complex inhomogeneous deformation structures were directly observed from the first few loading cycles, and these structures were found not to vary significantly with increasing number of cycles. A clear link was observed between crack nucleation site and microstructurally-sensitive growth path and the spatially-resolved sites of extreme values of residual stress and GND density.

Effect of dislocation structure evolution on low-angle grain boundary formation in 7050 aluminum alloy during aging

The effect of dislocation structure evolution on low-angle grain boundary formation in 7050 aluminum alloy during aging was studied by using optical microscopy, transmission electron microscopy, and electron backscatter diffraction analysis of misorientation angle distribution, cumulative misorientation and geometrically necessary dislocation (GND) density. Experimental results indicate that coarse spindle-shaped grains with the dimension of 200 µm × 80 µm separate into fine equiaxed grains of 20 µm in size as a result of newborn low-angle grain boundaries formed during the aging process. More specifically, the dislocation arrays, which are rearranged and formed due to scattered dislocations during earlier quenching, transform into low-angle grain boundaries with aging time. The relative frequency of 3°-5° low-angle grain boundaries increases to over 30%. The GND density, which describes low-angle grain boundaries with the misorientation angle under 3°, tends to decrease during initial aging. The inhomogeneous distribution of GNDs is affected by grain orientation. A decrease in GND density mainly occurs from 1.83 × 1013 to 4.40 × 1011 m−2 in grains with 〈111〉 fiber texture. This is consistent with a decrease of unit cumulative misorientation. Precipitation on grain boundaries and the formation of a precipitation free zone (PFZ) are facilitated due to the eroding activity of the Graff etchant. Consequently, low-angle grain boundaries could be readily viewed by optical microscopy due to an increase in their electric potential difference.

About quantitative EBSD analysis of deformation and recovery substructures in pure Tantalum

The aim of this work is to present a quantitative analysis of features involved in recovery during annealing of deformed Tantalum. In pure metals where crystalline defects usually have high mobility, dislocation annihilation and rearrangement occur to a great extent prior to recrystallization. Therefore a complete understanding of recrystallization cannot be accomplished without an advanced knowledge of the recovery phenomenon. Depending on whether dislocations induce a measurable curvature in the crystal lattice or not, they are called Geometrically Necessary Dislocations (GNDs) or Statistically Stored Dislocations (SSDs) respectively. In the present work only GNDs are considered. For this purpose electron backscatter diffraction (EBSD) is an advantageous technique to obtain statistically representative results when compared to Transmission Electron Microscopy (TEM). However, a quantitative analysis of GNDs from EBSD data is not straightforward. Since local misorientations are induced by the curvature of the crystal lattice caused by GNDs, GNDs analysis can be done using local misorientations. However the values obtained from this analysis are step size dependent and influenced by the measurement noise. Reasoning on the basis that when the step size tends to zero, local misorientation should also tend to zero, measurement noise can be estimated [1]. The measurement noise appears to notably be very much dependent on the amplitude of local misorientations, which must be considered in the perspective of GND density calculation.

Evolution of intragranular stresses and dislocation densities during cyclic deformation of polycrystalline copper

We have used the cross-correlation based high resolution electron backscattering diffraction (HR-EBSD) technique to evaluate at high spatial resolution the spatial patterning of the type III intragranular residual stresses and geometrically necessary dislocation (GND) density within polycrystalline Cu after cyclic deformation. Oxygen free high conductivity (OFHC) polycrystalline copper samples were cyclically deformed under stress-control at a load ratio of 0.1 and EBSD measurements were made at points throughout the early stages of fatigue when strain amplitudes and cyclic creep rates are changing most significantly, namely at 0 cycles, 2 cycles, 200 cycles and 2000 cycles. Statistical analysis is presented showing that moderate correlations exist between stored GND density, residual intragranular stress and distance from the nearest grain boundaries and/or triple junctions.

A dislocation-based model for deformation and size effect in multi-phase steels

In this work, we investigate the mechanical behavior of multi-phase steels using a continuum dislocation dynamic model (CDD) coupled with a viscoplastic self-consistent (VPSC) model that accounts for both the effect of dislocations evolution inside the grain as well as grain–grain interactions. Because the conventional viscoplasticity theory does not capture the grain size effect, we introduce an intrinsic length scale within the concepts of geometrically necessary dislocations (GND) by means of the Nye's dislocation tensor. The effect of the GND density is implemented into the model for the mean free path of dislocations and is shown to contribute to strain hardening. As a validation of this multiscale model, we investigate the mechanical behavior of various dual phase steels. The stress-strain response obtained from this approach is compared to experimental data found in the literature and reveal good agreement between experimental results and predictions. The model also predicts the evolution of dislocation densities in each phase and suggests the connection between underlying deformation mechanisms and macroscopic material hardening. The relation between flow stress and grain size is also investigated. The model predictions follow the Hall–Petch relation of strength versus grain size for grains larger than one micron meter but deviates from this relation for grains in the order of tens of nanometers.