Dependence Of Dislocation Density Research Articles

Revisiting the strain rate sensitivity of the flow stress of copper: Theory and experiment

One of the most important issues related to the strength of metals is the strain rate sensitivity of the flow stress. In this study, an analytical model of the flow stress as a function of strain rate is derived theoretically. The model can reproduce the strain rate sensitivity of the flow stress of copper over a wide range of strain rates (up to 109 s−1) quantitatively. Our theoretical derivations indicate that the strain rate sensitivity of the flow stress, especially that above 103 s−1, is a result of both the variation of the dislocation mobility mechanism with stress and the particular stress dependence of dislocation density but is not a result of each single mechanism. In particular, the stress dependence of the dislocation density and the initial dislocation density are critical to the quantitative relation of the flow stress–strain rate at high strain rate and the strain rate threshold, under which the upturn of the flow stress occurs, respectively. Moreover, experiments with copper of different initial dislocation densities at moderate and high strain rate are performed. The strain rate threshold of the flow stress upturn observed in the experiments grows considerably as initial dislocation density increases, which is in accordance with theoretical prediction by our model.

Investigation of grain size and geometrically necessary dislocation density dependence of flow stress in Mg-4Al by using nanoindentation

Models for grain size dependence of flow stress typically ignore contributions from geometrically necessary dislocations (GNDs) in the grain boundary regions. Ashby had proposed the following equation for flow stress (τ) as a function of grain size (d) and effective plastic strain (ε¯): τ=τ0+C′Gbε¯d using an unknown empirical coefficient, C’, In this study, the coefficient, C’, and equation for the grain size dependence of flow stress were investigated in Mg-4Al using nanoindentation. C’ ranges from 1.4 to 7.11 that is related to the hardness difference (∆H0) between the adjacent grains as C′=0.62ΔH0. Ashby equation was used to predict uniaxial stress-strain curves in grain sizes of 55 µm, 187 µm, and 333 µm. The results showed lower flow stress and lower hardening rate compared to experimentally measured tensile stress-strain response. The discrepancy seems to be a result of hardening effect of SSDs. We propose a modified Ashby equation that takes into account the hardening effect of SSDs and GNDs separately, resulting in a better agreement between calculated and experimentally measured values. This analysis demonstrates estimation of uniaxial stress-strain response using nanoindentation measurements that captures effects of grain size, and strain hardening from SSDs and GNDs. The combination of the revised equation and nanoindentation method will enable rapid exploration of stress-strain response of a diverse range of single-phase polycrystalline alloys.

Strain-rate dependent crystal plasticity model and aluminum softening/hardening transition

Hardening or softening associated with the increase of dislocation density is correlated with strain rate and dislocation density. Taking aluminum as a model material, we investigate the strain rate and dislocation density dependence of the strength and the accompanying hardening or softening effect upon a dislocation increase. This investigation is enabled by employing a newly developed crystal plasticity model, which is validated against the mechanical behaviors in an extremely wide strain rate range, and featured by a parameter-free mobile dislocation fraction law based on the exponential distribution of dislocation link lengths and an averaged dislocation velocity model that considers both thermally-activated waiting mechanism and drag-dominated running mechanism. Results show that at different strain rates, the material strength varies with a universal feature that it first softens as the dislocation density increases from a sufficiently low initial value, and then hardens after reaching the minimum strength, where the critical dislocation density separating the softening from hardening regimes increases with the strain rate. Microscopically, the strength dependence on dislocation density and strain rate originates from their influence on the two competing effects of a dislocation increase, where the hardening effect comes from the higher resistance to dislocation slip by increased forest dislocations while the softening effect comes from the larger quantity of mobile dislocations. These two competing effects are quantitatively evaluated by the crystal plasticity model and are further cast into the competition between the partial derivatives of normalized dislocation velocity and of normalized mobile dislocation density, and the governing equation for the critical dislocation density is thus furnished by balancing these two competing effects. Further, a complete map in the entire space of strain rate and dislocation density (ε̇,ρ) is constructed for the zones of dislocation hardening or softening, where the zone-boundary curve for critical dislocation density exhibits a two-stage character and it agrees well with the available experimental data. Such a map is expected to help predict the trend for strength evolution under different serving conditions and also guide the material design according to the requirements for strength variation in service.

Dislocation theory of steady and transient creep of crystalline solids: Predictions for olivine

In applications critical to the geological, materials, and engineering sciences, deformation occurs at strain rates too small to be accessible experimentally. Instead, extrapolations of empirical relationships are used, leading to epistemic uncertainties in predictions. To address these problems, we construct a theory of the fundamental processes affecting dislocations: storage and recovery. We then validate our theory for olivine deformation. This model explains the empirical relationships among strain rate, applied stress, and dislocation density in disparate laboratory regimes. It predicts the previously unexplained dependence of dislocation density on applied stress in olivine. The predictions of our model for Earth conditions differ from extrapolated empirical relationships. For example, it predicts rapid, transient deformation in the upper mantle, consistent with recent measurements of postseismic creep.

A Quantitative Investigation of Dislocation Density in an Al Matrix Composite Produced by a Combination of Micro-/Macro-Rolling

An aluminum matrix composite with dispersed carbon nanotubes (CNTs) was produced via flake powder metallurgy using a micro-rolling process and vacuum hot pressing (VHP), followed by conventional rolling using a macro-rolling process. The microstructure and mechanical properties of the produced composites were studied. In addition, a new quantitative model was introduced to study the dislocation density based on the microstructural parameters. The results revealed that the distribution characteristics of the CNTs in the Al matrix and the Al-CNT interfaces were the two main governing parameters of dislocation density. Moreover, the dependence of dislocation density on the geometry of the grains and crystallographic texture was shown in this model. The microstructural evolution revealed that a lamellar grain structure had been achieved, with a high capacity for the storage of dislocation. A uniform distribution of CNTs with high bonding quality was also seen in the final microstructure.

Measurement of Dislocation Density in SiC Wafers Using Production XRT

X-ray topography (XRT) presents itself as an attractive non-destructive method to replace industry-standard destructive KOH etching used to measure dislocation density. However, a production-line-compatible XRT has to employ a low scan speed in order to work well with automated image analysis, which makes it impractical for a high-volume manufacturing to scan an entire wafer. We introduce the “radial band” approach to sampling the entire wafer’s area with a single-pass 16 mm tall scan band. Such a band spans the entire range of radii and thus captures the typically strong radial dependence of dislocation density over the entire range, while mostly ignoring the typically weak angular dependence of dislocation density and averaging the inevitable noise over the 16 mm band height. The XRT scan time savings for this approach are roughly 15-fold and 20-fold for 150mm and 200mm wafers respectively.

Strain rate dependency of dislocation plasticity

Dislocation glide is a general deformation mode, governing the strength of metals. Via discrete dislocation dynamics and molecular dynamics simulations, we investigate the strain rate and dislocation density dependence of the strength of bulk copper and aluminum single crystals. An analytical relationship between material strength, dislocation density, strain rate and dislocation mobility is proposed, which agrees well with current simulations and published experiments. Results show that material strength displays a decreasing regime (strain rate hardening) and then increasing regime (classical forest hardening) as the dislocation density increases. Accordingly, the strength displays universally, as the strain rate increases, a strain rate-independent regime followed by a strain rate hardening regime. All results are captured by a single scaling function, which relates the scaled strength to a coupling parameter between dislocation density and strain rate. Such coupling parameter also controls the localization of plasticity, fluctuations of dislocation flow and distribution of dislocation velocity.

Compositional complexity dependence of dislocation density and mechanical properties in high entropy alloy systems

This study focuses on a quantitative analysis of dislocation accumulation after cold plastic deformation and mechanical properties of FeNiCoCrMn and TiNbHfTaZr high entropy alloys (HEAs) which are single phase fcc and bcc solid solutions, respectively. In order to study the role of compositional complexity from unary to quinary compositions on dislocation accumulation and mechanical properties after plastic deformation, the single solid solution phase forming sub-alloys of the two HEAs were investigated. All studied samples revealed a large plastic deformability under cold-rotary swaging process by 85–90% area reduction without intermediate annealing. The dislocation density of all studied samples, determined by Williamson-Hall method on synchrotron X-ray diffraction patterns, were between 1014 - 1015 m−2 dependent on the alloy composition. The level of dislocation density after plastic deformation is not only affected by the number of constituent element but the lattice distortion and intrinsic properties in terms of stacking fault energy, modulus misfit, and melting point also impact the dislocation storage. The level of dislocation density determines the level of mechanical properties because of a resistance to dislocation motions. The hardness and yield compressive strength of the studied samples are proportional to the level of dislocation density.

High Crystallinity and Highly Relaxed Al0.60Ga0.40N Films Using Growth Mode Control Fabricated on a Sputtered AlN Template with High‐Temperature Annealing

Herein, a method for growing AlGaN using an intermediate AlN molar fraction on a sputtered AlN template with high‐temperature annealing via an AlN homoepitaxial layer is investigated. The growth mode dependence of AlGaN on the amount of Ga doping in the AlN homoepitaxial layer is examined in detail. As a result, homoepitaxial growth of AlN on sputtered AlN improve the surface flatness. In addition, it is found that the macroscopic flatness is improved by increasing the amount of Ga doping, but this decreases the microscopic flatness. Moreover, when Al0.60Ga0.40N is grown on homoepitaxial AlN layers with different amounts of Ga doping, a difference in the growth mode is observed, and this reveal a remarkable difference in the dislocation density of Al0.60Ga0.40N. The dislocation density dependence of the lasing threshold power density in a UV–B optical pumped laser is also investigated. As a result, the lasing threshold power density is reduced from ≈220 to 50 kW cm−2 by reducing the dislocation density from 1.6 × 109 to 8.0 × 108 cm−2. It is confirmed that it is, therefore, essential to reduce the dislocation density to reduce the threshold power density of the UV–B laser.

Flow stress and electrical resistivity in plastically deformed Al subjected to intermittent annealing

The behaviour of the flow stress and electrical resistivity has been studied in pure Al polycrystals deformed at 4K with intermittent annealing at temperatures up to 298K. Electrical resistivity measurements permit discrimination between regimes of plastic flow determined by dislocation–dislocation interactions and dislocation–debris interactions. Continued deformation at 4K occurs under dynamic quasi–equilibrium controlled by dislocation–dislocation interactions. This regime is represented by a Taylor–type stress – dislocation density relationship, τ≈0.2μbρ. The onset of immediate post–yielding deformation in annealed samples is dominated by dislocation–debris interactions and is characterized by a high rate of dislocation production and a low rate of flow stress increase than determined by dislocation–forest interactions. Upon annealing to 298K, Al shows two regimes of correlated stress and resistivity recovery, controlled by the amount of energy release during annealing. The relationship between flow stress and density of dislocations for the component of recoverable and unrecoverable defects has been assessed from the experimental measurement data for two different scaling laws. It is found that while Taylor–type equation, τ∝ρ, gives good estimates of the flow stress for the density of dislocations determined from resistivity results in continuously deformed samples, it distorts considerably stress – dislocation density dependence in samples subjected to intermittent annealing.

Modeling the mechanical response and microstructure evolution of magnesium single crystals under c-axis compression

Atomistically informed multiscale dislocation dynamics plasticity (MDDP) framework is used to investigate the mechanical response and microstructure evolution in bulk magnesium single crystals subjected to c-axis compression. The MDDP framework was modified to account for the 〈c+a〉 dislocation slip on the pyramidal I and pyramidal II planes. Several aspects of the 〈c+a〉 dislocation reactions such as the transition of pyramidal I near edge and pyramidal II pure edge segments to basal plane, and the thermally activated cross slip are considered. Additionally, a generalized mobility law and anisotropic frictional stress models are implemented based on the findings of several molecular dynamics studies. MDDP predictions of the yielding and strain hardening behaviors, as well as the maximum stress prior to failure are in good agreement with the reported experimental values. Moreover, detailed study of the dislocation microstructure evolution and the dislocation density dependence on the applied stress are presented and compared with available experimental results.

Pattern formation in a minimal model of continuum dislocation plasticity

The spontaneous emergence of heterogeneous dislocation patterns is a conspicuous feature of plastic deformation and strain hardening of crystalline solids. Despite long-standing efforts in the materials science and physics of defect communities, there is no general consensus regarding the physical mechanism which leads to the formation of dislocation patterns. In order to establish the fundamental mechanism, we formulate an extremely simplified, minimal model to investigate the formation of patterns based on the continuum theory of fluxes of curved dislocations. We demonstrate that strain hardening as embodied in a Taylor-type dislocation density dependence of the flow stress, in conjunction with the structure of the kinematic equations that govern dislocation motion under the action of external stresses, is already sufficient for the formation of dislocation patterns that are consistent with the principle of similitude.

塑性変形オーステナイト鋼中の転位密度の定量化

Grain size dependence of dislocation density and its character in austenitic stainless steels subjected to plastic strain was evaluated quantitatively by means of X-ray diffraction and Electron Back Scattering Diffraction (EBSD). Various amounts of the plastic strain up to 15% were given by a tensile instrument at 650℃ and three kinds of grain size (17, 46 and 87 μm) were employed. Misorientation in crystal direction in each grain was determined by two parameters, GOS (Grain Orientation Spread) and KAM (Kernel Average Misorientation). GN (Geometrically Necessary) dislocation density was estimated from KAM parameter. GOS parameter increased with a linear relationship to the amount of plastic strain regardless of the grain size, whereas KAM parameter increased with increasing plastic strain but showed the grain size dependence. Change in total dislocation density with plastic strain showed a similar trend in KAM parameter. It is found that the difference in the total dislocation density at a plastic strain was consistent exactly with the difference in GN dislocation density depending on the grain size. These results were confirmed by TEM observations using several kinds of specimens used in X-ray experiments.

Dislocation density dependence of stimulated emission characteristics in AlGaN/Al multiquantum wells

AbstractWe investigated dislocation density dependence on stimulated emission characteristics of AlGaN/AlN multiquantum wells (MQWs: emission wavelength of approximately 285 nm) on AlN templates with various dislocation densities. We found that the stimulated emission characteristics of the deep UV MQWs were strongly dependent on the dislocation densities. A reduction of the dislocation densities is very important in order to realize good stimulated emission characteristics of deep UV MQWs. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of Dislocation Density in Pb(1−x)Sn x Se Grown on ZnTe/Si by MBE

The growth of (211) Pb(1−x)SnxSe on Si is achieved with a thick ZnTe buffer layer. The obtained films are specular, but contain widely dispersed void defects. Because the lattice misfit between Pb(1−x)SnxSe and ZnTe is small, dislocation density values on the order of 106/cm2 in the Pb(1−x)SnxSe are obtained. The variation of the dislocation density as a function of Pb(1−x)SnxSe thickness, h, is analyzed in terms of dislocation annihilation. The analysis predicts an inverse quadratic dependence of dislocation density on h, and quantitative agreement with experimental measurements of the dislocation density is obtained.

Relation between the internal stress measured in creep and the stress generated by the dislocation structure in the fcc metals

From the steady-state creep rate data treated as a function of the applied and the measured effective stress and temperature, a phenomenological dependence of the internal stress on the applied stress and temperature was derived. The result determined the expected character of the applied stress- and temperature dependences of the dislocation density, which was considered the microstructure parameter of the internal stress σ i = α MGbρ 1/2 (σi is the internal stress, α the dislocation interaction factor, M the Taylor factor, G the shear modulus, b the Burgers vector length and ρ the dislocation density). A scaling of the expected dislocation density by fitting it to measured dislocation density data yielded reasonable values of the parameter α in the Taylor formula, but the experimental data indicated a weaker applied stress dependence of the measured dislocation density than that of the expected dependence. An admission of an empirical formula fitting the dependence of dislocation density on the applied stress and temperature leads to a suggestion that the parameter α might be dependent on applied stress and temperature.

Evaluation of the jogged-screw model of creep in equiaxed γ-TiAl: identification of the key substructural parameters

On the basis of microstructural observation of 1/2[1 1 0]-type jogged-screw dislocations, it has been previously proposed that the creep deformation mechanism in equiaxed γ-Ti–48Al alloys is controlled by the climb of the jogs on these dislocations. This is validated by predictions from a creep model based on these observations. However, several assumptions made in the model were not fully substantiated by experiment or theory. The aim of this study is to verify and validate the parameters and functional dependencies assumed in the model. In addition, the original solution has been reformulated to take into account the finite height of the moving jog. The substructural model parameters have been further investigated in light of this reformulation. The stress dependence of dislocation density, jog spacing and jog height has been evaluated via simulations, analytical modeling and experimental observations. Combining all of these parameters and dependencies in a reformulated model leads to an excellent prediction of creep rates and stress exponents.

Application of a Modified Jogged-Screw Model for Creep of Titanium Aluminides: Evaluation Of The Key Substructural Parameters

AbstractA modification of the jogged-screw model has been adopted recently by the authors to explain observations of 1/2[110]-type jogged-screw dislocations in equiaxed Ti-48Al under creep conditions. The aim of this study has been to verify and validate the parameters and functional dependencies that have been assumed in this previous work. The original solution has been reformulated to take into account the finite length of the moving jog. This is a better approximation of the tall jog. The substructural model parameters have been further investigated in light of the Finite Length Moving Line (FLML) source approximation. The original model assumes that the critical jog height (beyond which the jog is not dragged) is inversely proportional to the applied stress. By accounting for the fact that there are three competing mechanisms (jog dragging, dipole dragging, dipole bypass) possible, we can arrive at a modified critical jog height. The critical jog height was found to be more strongly stress dependent than assumed previously. The original model assumes the jog spacing to be invariant over the stress range. However, dynamic simulation using a line tension model has shown that the jog spacing is inversely proportional to the applied stress. This has also been confirmed by TEM measurements of jog spacings over a range of stresses. Taylor's expression assumed previously to provide the dependence of dislocation density on the applied stress, has now been confirmed by actual dislocation density measurements. Combining all of these parameters and dependencies, derived both from experiment and theory, leads to an excellent prediction of creep rates and stress exponents. The further application of this model to other materials, and the important role of atomistic and dislocation dynamics simulations in its continued development is also discussed.

Application of a Modified Jogged-Screw Model for Creep of Titanium Aluminides: Evaluation Of The Key Substructural Parameters

AbstractA modification of the jogged-screw model has been adopted recently by the authors to explain observations of 1/2[110]-type jogged-screw dislocations in equiaxed Ti-48Al under creep conditions. The aim of this study has been to verify and validate the parameters and functional dependencies that have been assumed in this previous work. The original solution has been reformulated to take into account the finite length of the moving jog. This is a better approximation of the tall jog. The substructural model parameters have been further investigated in light of the Finite Length Moving Line (FLML) source approximation. The original model assumes that the critical jog height (beyond which the jog is not dragged) is inversely proportional to the applied stress. By accounting for the fact that there are three competing mechanisms (jog dragging, dipole dragging, dipole bypass) possible, we can arrive at a modified critical jog height. The critical jog height was found to be more strongly stress dependent than assumed previously. The original model assumes the jog spacing to be invariant over the stress range. However, dynamic simulation using a line tension model has shown that the jog spacing is inversely proportional to the applied stress. This has also been confirmed by TEM measurements of jog spacings over a range of stresses. Taylor's expression assumed previously to provide the dependence of dislocation density on the applied stress, has now been confirmed by actual dislocation density measurements. Combining all of these parameters and dependencies, derived both from experiment and theory, leads to an excellent prediction of creep rates and stress exponents. The further application of this model to other materials, and the important role of atomistic and dislocation dynamics simulations in its continued development is also discussed.

A Revised Jogged-Screw Model For Creep Of Equiaxed γ-TiAl: Identification Of The Key Substructural Parameters.

ABSTRACTA modification of the classic jogged-screw model has been previously adopted to explain observations of 1/2[110]-type jogged-screw dislocations in equiaxed Ti-48Al under creep conditions. The aim of this study has been to verify and validate the parameters and functional dependencies that have been assumed in that model. The original solution has been reformulated to take into account the finite length of the moving jog. This is a better approximation of the tall jog. The substructural model parameters have been further investigated in light of the Finite Length Moving Line (FLML) source approximation. The original model assumes that the critical jog height (beyond which the jog is not dragged) is inversely proportional to the applied stress. By accounting for the fact that there are three competing mechanisms (jog dragging, dipole dragging, dipole bypass) possible, we can arrive at a modified critical jog height. The critical jog height was found to be more strongly stress dependent than assumed previously. The original model assumes the jog spacing to be invariant over the stress range. However, dynamic simulation using a line tension model has shown that the jog spacing is inversely proportional to the applied stress. This has also been confirmed by TEM measurements of jog spacings over a range of stresses. Taylor's expression assumed previously to provide the dependence of dislocation density on the applied stress, has now been confirmed by actual dislocation density measurements. Combining all of these parameters and dependencies, derived both from experiment and theory, leads to an excellent prediction of creep rates and stress exponents.