Dislocation Density Evolution Research Articles

Shock resistance capability of multi-principal elemental alloys as a function of lattice distortion and grain size

This article aims to study the shock resistance capability of multi-element alloys. In this study, we utilized nonequilibrium molecular dynamics-based simulations with an embedded atom method potential to predict the deformation governing mechanism in a multi-elemental alloy system subjected to shock loading. The evolution of shock front width, longitudinal stress, shear stress, and dislocation density were investigated for different polycrystalline multi-element systems containing different mean grain sizes of 5, 10, and 18 nm, respectively. In order to quantify the effect of lattice distortion, average atom (A-atom) potential for quinary (high entropy) and ternary (medium entropy) configurations was also developed in this work. The random composition of multi-element alloys was replaced with single atom-based A-atom arrangements to study the effect of lattice distortion on shock resistance capabilities of high entropy alloy and medium entropy alloy. It was predicted from simulations that a higher value of lattice distortion component in the CoCrCuFeNi alloy leads to provide superior resistance against shock wave propagation as compared to the ternary alloy CrFeNi. In nanocrystalline configurations, dislocations, and stacking faults, only dislocations governed the deformation mechanics in monocrystalline configurations. The simulations indicate that grain size significantly affects the rates of generation of secondary/partial dislocations, hence affecting the stresses and the deformation mechanism of the structures.

Simulation of dislocation evolution in microparticle impacts over a wide range of impact velocities

The dynamic loading environment created by high velocity microparticle impacts is very difficult to model. The extremely high strain rates, large deformations, and high temperatures affect the flow stress of the material in ways that standard flow stress models cannot capture. Plastic deformation and strain hardening in metals is controlled by the motion of dislocations. Dislocations can be nucleated, stored as forest dislocations, or be annihilated as loading progresses. A comprehensive accounting of dislocation density change is needed to accurately describe rate and temperature dependent dislocation glide and evolution across the wide rage of loading conditions present in microparticle impact problems. Therefore, we implement and apply a newly proposed flow stress model (Hunter and Preston, 2015, 2022) that is then coupled with mobile and immobile dislocation density evolution equations. This model is implemented in Los Alamos National Laboratory’s hydrodynamics code, FLAG, to model copper-on-copper microparticle impacts. This new strength model allows for accurate simulation of particle rebound and flattening across a wide range of impact velocities.

Load partitioning behavior in a metastable austenitic stainless steel subjected to pre-deformation below and above Md temperature: Experimental and modeling studies

A physics-based model coupling the evolution of dislocation density in the austenite and strain-induced martensite phases with composite flow strength and kinetics of martensite transformation was developed to describe ambient temperature uniaxial tensile deformation behavior of mill annealed and pre-deformed 304 SS. The applicability of the model was evaluated by fitting the experimental stress-strain data and validated with the experimentally estimated dislocation density in the austenite phase of the steel. The modeling results revealed that the initial microstructure of the steel significantly influences the dislocation accumulation and annihilation rates as well as the load partitioning between austenite and martensite phases. Despite a lower initial dislocation density than the pre-deformed conditions, the mill annealed condition revealed a higher capacity (approximately two orders of magnitude) for dislocation accumulation in the austenite phase. The initial volume fraction of strain-induced martensite phase and the initial dislocation density of pre-deformed 304 SS are governed by the degree of pre-deformation and pre-deformation temperature, either below or above Md temperature (where Md is defined as the temperature above which plastic deformation does not result in austenite to martensite transformation). It is observed that storage, as well as annihilation rates in the austenite phase, are higher for the steel having a higher initial volume fraction of martensite and dislocation density. Though the pre-deformation altered the evolution of martensite volume fraction, the steel exhibited a minor variation in dislocation accumulation capability in the martensite phase for different conditions.

A general steady-state creep model incorporating dislocation static recovery for pure metallic materials

In this work, a mechanistic steady-state creep model is proposed for pure metallic materials to characterize the evolution of macroscopic creep strain rate as a function of the testing temperature and applied stress. Dislocation-dominated and diffusion-dominated creep are both addressed in the developed creep model, which is able to effectively characterize the phenomena of “first-power-law” creep, “five-power-law” creep and “power-law-breakdown” creep. Thereinto, the dislocation-dominated creep behavior is systematically analyzed by considering the evolution of dislocations, which includes dislocation multiplication, strain-rate dependent dynamic recovery and time-related static recovery. Main attentions are focused on the description of dislocation static recovery that covers the annihilation of dislocations induced by the creep mechanisms of dislocation climb and thermally related dislocation glide during the long term plastic deformation. A novel form of the dislocation mobility is deduced that not only considers the effect of dislocation climb and glide, but also takes into account the contribution of mechanical work on atomic diffusion. Moreover, the latter is noticed to be the dominant reason resulting in the transition from “five-power-law” creep to “power-law-breakdown” creep. In order to verify the developed model, creep data of six metallic materials with different crystalline structures is considered to compare with the theoretical results. Good agreement is achieved for all these data over a wide range of temperature and stress, which indicates that the model can well characterize the deformation behavior during the steady-state creep stage. In addition, the contribution of dislocation multiplication, dynamic recovery and static recovery to the evolution of dislocation density is further discussed at different temperatures and stresses, which can facilitate the comprehension of the fundamental creep mechanisms of metallic materials.

A coupled crystal-plasticity and phase-field model for understanding fracture behaviors of single crystal tungsten

The brittle-ductile transition (BDT) of tungsten depends heavily on temperature-dependent dislocation mobility. To well understand the underlying mechanism of tungsten fracture behavior, the dislocation activities need to be well described, but the related model remains limited. In this work, a coupled crystal-plasticity and phase-field model (CP-PFM) is developed based on a unified thermodynamic framework, which considers the thermal activated kink-pair mechanism of screw dislocation motion, the evolution of dislocation density, and the coupling effects between the plasticity and the crack propagation. The developed model can well predict the experimental results and is used to study the BDT process of tungsten. Four typical fracture processes are disclosed depending on the temperature, including the brittle fracture, semi-brittle fracture, micro-ductile fracture and ductile fracture. They exhibit different micro-cleavage crack features, induced by the competition between the shielding effect of the plastic zone and the crack propagation near the crack tip zone. In addition, the fracture process of tungsten is found to be significantly influenced by non-Schmid effect in the low and medium temperature regimes.

Coupled quantitative modeling of microstructural evolution and plastic flow during continuous dynamic recrystallization

Continuous dynamic recrystallization (cDRX) dominates microstructural evolution during the hot working of metallic materials with high stacking fault energy (SFE), such as aluminum alloys. However, in reality, a lack of quantitative and visual modeling of the process hinders its widespread application in the hot working process. In this study, using a recently developed multilevel cellular automaton (MCA) that integrates the newly established cell switching rules and topology deformation technique, a novel mesoscale MCA-cDRX model was constructed to investigate the evolution of both microstructures and macroscopic mechanical response in the hot working of AA7075 aluminum alloy. By considering the evolution of dislocation density and the orientation angle of the local cells as the primary clues, the plastic flow, recrystallization kinetics, features of subgrain size and high-angle grain boundaries, and influence of initial matrix characteristics on the cDRX mechanism were analyzed. The model predictions are consistent with the experimental data. Quantitative analysis confirms that the incubation time for the initiation of subgrain formation is significantly short. The fine-grain matrix and high initial volume fraction of low-angle grain boundaries can significantly accelerate the progress of cDRX owing to a stronger accumulation of dislocations in the dislocation cell walls through the climb and cross-slip mechanisms in the deformed aluminum alloy. The subgrain size is dependent on the Zener-Hollomon parameter. The developed simulation framework offers an effective means to allow the visualization of the cDRX.

Crystal plasticity modeling of strain-induced martensitic transformations to predict strain rate and temperature sensitive behavior of 304 L steels: Applications to tension, compression, torsion, and impact

This paper advances crystallographically-based Olson-Cohen (direct γ → α’) and deformation mechanism (indirect γ→ε→α’) phase transformation models for predicting strain-induced austenite to martensite transformation. The advanced transformation models enable predictions of not only strain-path sensitive, but also of strain-rate and temperature sensitive deformation of polycrystalline stainless steels (SSs). The deformation of constituent grains in SSs is modeled as a combination of anisotropic elasticity, crystallographic slip, and phase transformation, while the hardening is based on the evolution of dislocation density and explicit shifts in phase fractions. Such grain-scale deformation is implemented within the meso‑scale elasto-plastic self-consistent (EPSC) homogenization model, which is coupled with the implicit finite element (FE) method to provide a constitutive response at each FE integration point for solving boundary value problems at the macro-scale. Parameters pertaining to the hardening and transformation models within FE-EPSC are calibrated and validated on a suite of data including flow curves and phase fractions for monotonic compression, tension, and torsion as a function of strain-rate and temperature for wrought and additively manufactured (AM) SS304L. To illustrate the potential and accuracy of the integrated multi-level FE-EPSC simulation framework, geometry, mechanical response, phase fractions, and texture evolution are simulated during gas-gun impact deformation of a cylinder and quasi-static tension of a notched specimen made of AM SS304L. Details of the simulation framework, comparison between experimental and simulation results, and insights from the results are presented and discussed.

Comparison of the effect of ECAP and SSE on microstructure, texture, and mechanical properties of magnesium

Microstructure, crystallographic texture, and mechanical properties of pure Mg, as a model hcp metal, were compared after processing by the two severe plastic deformation (SPD) techniques of equal channel angular pressing (ECAP) and simple shear extrusion (SSE). Both processes were performed on extruded bars at 250 °C for up to four passes, where the minimum grain sizes of 13.7 and 9.8 µm were achieved in the ECAP and SSE processes, respectively. The fraction of dynamically recrystallized (DRX) grains, high angle grain boundaries (HAGBs), and the evolution of dislocation density with equivalent strain experienced the same trend in both processes with higher values after ECAP. The textural evolutions were completely different during ECAP and SSE despite their similar deformation modes. A conventional shear texture was developed after ECAP, while after SSE basal planes were aligned parallel to the processing direction. In the ECAP-processed material, texture softening and lower dislocation density counterbalanced the strengthening effect of grain refinement, resulting in a decrease in shear yield stress (SYS) and ultimate shear strength (USS), while in SSE the shear strength increased constantly with increasing number of passes. It can be deduced from the experimental results that SSE was more effective in achieving a fine-grained homogenous microstructure with high shear strengths as compared to ECAP. • Compare microstructure, and strengths after simple shear extrusion (SSE) and equal channel angular pressing (ECAP) of Mg. • SSE was found more efficient in grain refinement compared to ECAP. • Higher fractions of DRX and high angle grain boundaries (HAGBs) were achieved in ECAP than in SSE. • Develop different shear texture components after ECAP and SSE despite having similar deformation modes. • In contrast to ECAP, the shear strengths after SSE increases by increasing the number of passes.

Microplasticity in inhomogeneous alloys

We report on a preliminary modelling effort to understand the influence of compositional inhomogeneity on alloy microplasticity from a dislocation dynamics perspective. We tackle this problem by multiscale simulations in three steps: (1) analysis of the 3D composition morphology in alloys with tendency to undergo spinodal instability both thermally and under irradiation, with bcc FeCrAl alloys as a model system, (2) atomistic simulation of the dislocation mobility as a function of the local alloy composition, and (3) using dislocation dynamics simulations to understand the impact of composition inhomogeneity on microplasticity. The dislocation dynamics model takes into consideration the coherency stress associated with composition inhomogeneity when computing the forces driving the dislocation motion and on cross slip. Our preliminary investigation shows that the stress-strain response of the alloy and the dislocation density evolution depend on the wavelength of the composition fluctuations. Our investigation also shows that the alloy inhomogeneity may alter the cross-slip activity, which, in turn, influences the dislocation density evolution. The dependence of the dislocation mobility and coherency stress on local composition and its variation, as well as the altered cross slip rates, cause the dislocation microstructure to differ relative to that in the homogeneous alloy of the same average composition.

Creep curve modelling of Austenitic Steel 316LN

Physical based creep models that elucidate the creep deformation behaviour with ongoing microstructural evolution can be a useful tool for the components life assessment as well as the design of improved materials, deployed at high temperature and pressure. In this research work, a creep model that is a combination of physical based model and CDM approach is employed to predict the creep curves of steel 316LN. The microstructure based variables those are different dislocation densities (mobile and forest) are the input parameters. The model provides a provision for the assessment of each microstructural variable, in each time step of the creep deformation. Consequently, other than creep curves, the model also demonstrates the evolution of dislocation density (mobile and forest), dislocation velocity, dislocation mobility and mean free path. Initial values of input parameters (various dislocation densities and mean free path) are obtained from the literature for initializing the model. It can be observed that the predicted creep curves are in reasonable agreement with experimental ones and the evolution of other involved parameters is discussed thoroughly.

A Multiphysics Model for Predicting Microstructure Changes and Microhardness of Machined AerMet100 Steel.

The machined-surface integrity plays a critical role in corrosion resistance and fatigue properties of ultra-high-strength steels. This work develops a multiphysics model for predicting the microstructure changes and microhardness of machined AerMet100 steel. The variations of stress, strain and temperature of the machined workpiece are evaluated by constructing a finite-element model of the orthogonal cutting process. Based on the multiphysics fields, the analytical models of phase transformation and dislocation density evolution are built up. The white layer is modeled according to the phase-transformation mechanism and the effects of stress and plastic strain on real phase-transformation temperature are taken into consideration. The microhardness changes are predicted by a model that accounts for both dislocation density and phase-transformation evolution processes. Experimental tests are carried out for model validation. The predicted results of cutting force, white-layer thickness and microhardness are in good agreement with the measured data. Additionally, from the proposed model, the correlation between the machined-surface characteristics and processing parameters is established.

Revealing the Geometrically Necessary Dislocation Density Evolution during Hot Compression of AA7050

Dislocations play a critical role in metal forming processes, and accurate values of dislocation density are important in modelling these processes. However, direct determination of the dislocation density is challenging. In this study, electron backscatter diffraction is used to estimate the evolution of geometrically necessary dislocation density as a function of plastic strain, strain rate and temperature in hot compression of AA7050 alloy. The geometrically necessary dislocation density was found to increase at a higher strain rate and lower temperature; the higher dislocation density in these samples promoted continuous dynamic recrystallisation leading to grain refinement. At lower strain rates and higher temperatures, the dislocation densities were lower and dislocations formed into walls, channels and cells. These observations agree with accepted theories of dislocation evolution and demonstrate the capability of electron backscatter diffraction to provide representative dislocation density values as well as comprehensive information linking plastic flow with microstructural evolution.Keywords: electron backscatter diffraction (EBSD), geometrically necessary dislocations (GNDs), hot deformation, AA7050

A Unified Constitutive Model of Stress Relaxation of Ti-6Al-4V Alloy with Different Temperatures from Elastic to Plastic Loading

The effects of temperature and pre-strain levels on the stress relaxation behavior and corresponding microstructural evolutions of Ti-6Al-4V alloys have been investigated experimentally and numerically in this study. A series of tests (stress relaxation (SR) and repeated stress relaxation (RSR)) and microstructural observations (scanning electron microscope) have been performed, based on which the deformation-related variables, i.e., stress component and activation energy, as a function of the testing time are calculated according to the classical thermal activation theories. The experimental SR behavior and the obtained thermal related variables show that at lower temperatures (700 °C and 750 °C), a large number of dislocations introduced by plastic loading enhance dislocation slip/climb creep, giving rise to rapid relaxation compared with those with elastic loading conditions at the same temperature. At higher temperatures (800 °C and 850 °C), a similar SR phenomenon has been observed at both elastic and plastic loading conditions, which is due to the severe interaction between diffusion creep and dislocation creep after the loading stage. Based on these results, a unified constitutive equation has been proposed to successfully predict the behavior of the whole stress relaxation process composed of the loading stage and subsequent SR stage. The model considering the continuous evolution of internal variables, e.g., dislocation density and lamellar width, in two stages can predict the stress response and microstructure variation with different temperatures from elastic to plastic loading and provide a foundation to effectively optimize the hot forming process combining pre-deformation and stress relaxation.

High temperature nanoindentation of iron: Experimental and computational study

Application of reduced activation ferritic/martensitic (RAFM) steels as the structural material in future fusion reactors requires the knowledge of their mechanical properties under relevant operational conditions i.e. temperatures and irradiation by fast neutrons. Execution of the neutron irradiation and post irradiation examination is expensive and lengthy, therefore experimental and computational solutions to ease the characterization of as-irradiated materials are in the scope of interests of nuclear materials scientific community. Moreover, ion irradiation is considered as one possible way to surrogate high flux neutron irradiation damage. The extraction of the mechanical properties after ion irradiation primarily relies on the nanoindentation techniques and its subsequent post-processing to extract engineering relevant information, although some innovative techniques such as compression micropillars and micro-tensile testing also exist. In this work, we have performed nanoindentation on BCC iron, as the basis material for ferritic steels, by using a new Bruker stage developed for high temperature operation. The obtained results were analyzed by means of crystal plasticity finite element method (CPFEM), whereas the constitutive laws of the material were derived and established by using tensile deformation data, thus providing an interconnection of material's behavior under compressive and tensile deformations. The microstructural features such as indentation pile-up formation or dislocation density evolution were obtained by using transmission and scanning electron microscopy, and were compared with the predictions derived by the developed CPFEM model. It is demonstrated that a good agreement between the CPFEM and experimental data set, including tensile and compressive loads as well as associated microstructural changes, is obtained at room temperature and elevated temperature tests.

Effect of Long-Term Thermal Aging on Microstructure Evolution and Creep Deformation Behavior of a Novel 11Cr-3W-3Co Martensite Ferritic Steel.

This paper focused on the microstructure evolution under different thermal aging times at 650 °C and its effect on creep behavior in 11Cr-3W-3Co heat-resistant steel. After short-term thermal aging at 650 °C (>750 h), a Laves phase was found in the regions adjacent to the PAG boundaries, martensitic lath boundaries, and M23C6 carbides, and gradually swallowed adjacent M23C6 carbides with the aging time increased. Higher contents of Si and P are good promoters of the nucleation of the Laves phase during long-term aging. In addition, the coarsening behavior of the Laves phase, M23C6, and MX were investigated. As the aging time increases, the coarsening behavior among precipitated phases in the above-mentioned example exhibits remarkable variability, which is discussed in detail in this paper, and the evolution of the subgrain size was also analyzed in detail. The increasing rate of subgrain size is, in general, consistent with that of the M23C6 carbide size. The evolution of dislocation density in different aging times shows an obvious difference, and the decreasing rate of dislocation density is significantly affected by the precipitated phase after long-term aging time. The creep performance of the material decreases significantly as the aging time increases, which is closely related to the coarsening of the precipitates such as M23C6 carbides and subgrain during long-term aging.

Microstructure evolution and deformation behavior during stretching of a compositionally inhomogeneous TWIP-TRIP cantor-like alloy by laser powder deposition

A TWIP-TRIP high-entropy alloy with inhomogeneous chemical composition was prepared by laser powder deposition (LPD). The microstructure and properties of the alloy were quite different from those of compositionally homogeneous alloy with the same composition produced via electric arc melting + hot forging (EAM-HF) technique. During stretching, deformation twinning occurred under low strain, while FCC-HCP transformation was observed under high strain. The yield strength, tensile strength, and elongation were 425 MPa, 715 MPa, and 83%, respectively. For comparison, a sample with the same chemical composition was fabricated by EAM-HF. While twinning was detected under high strain during stretching, no FCC-HCP transformation was observed. Its yield strength, tensile strength, and elongation were found to be 69.6%, 86.9%, and 90% of those of LPD sample, respectively. In order to discuss the evolution of hardening behavior, a constitutive model based on dislocation density evolution was established, taking the contribution from twinning and FCC-HCP transformation into account. Since deformation twinning could be generated at a moderate speed in a wide tensile strain range and FCC-HCP transformation occurred under high strains, the high strain hardening rate could be guaranteed under high strain, thus avoiding strain localization, which endowed the LPD sample with good plasticity. It was shown to be possible to adjust the mechanical behaviors of alloys by increasing their stacking fault energy span through LPD.

Line profile analysis and rocking curve evaluation of 3D diffraction data reveal a strain softening mechanism

Diffraction patterns from ∼100 individual grains of a solutionized and quenched metastable β-Ti alloy were obtained by high energy synchrotron diffraction during in-situ tensile deformation experiments. The diffraction patterns of select grains were analyzed per an established single-crystal line profile analysis technique to assess the dislocation density evolution on individual slip systems. Further, a new technique to estimate the geometrically necessary dislocation (GND) density from rocking curves is introduced. The results provide a powerful complement to previously published comparisons between measured and crystal plasticity simulated internal elastic strains (and stresses). In particular, they reveal there is no preference for ½<111> Burgers vector dislocations to reside on a particular glide plane, since they have similar densities on {110} and {112} planes. In addition, an explanation for the observation of strain softening in some of the grains is hypothesized as form of “plastic buckling.” A select number of strain softening grains exhibit higher lattice curvature (GND density) than other grains, indicating that the grains have “broken up” into smaller domains which are deforming in distinct ways from one another, and more easily than they would have together. On the other hand, no significant differences in total dislocation density increment were observed, and examination of diffraction data from a larger subset of grains shows no strong correlations between total or GND densities with hardening/softening behavior. The results of this study reveal that GND density does not correlate with higher grain-level stress or more rapid strain hardening response.

Estimation of Dislocation Distribution at Mid Thickness for 1050 Al

Abstract The current study reports three different techniques to estimate the distribution of dislocation density at the mid thickness of 1050 Al alloy. It is well known that the strain distribution is inhomogeneous through the thickness of rolled materials, which affects the evolution of dislocation density during the process of deformation. In this study, the number of dislocations was calculated experimentally using indentation technique in 46.8% cold rolled 1050 Al sheet and the result was verified by two numerical methods.

Experimental characterization and crystal plasticity modeling of dual-phase steels subjected to strain path reversals

This paper is concerned with monotonic and load reversal deformation of four dual-phase (DP) 590, 780, 980, 1180 and one martensitic (MS) 1700 steel sheets. While the monotonic data are presented for all steels, the load reversal data are provided for DP 590, DP 780, and DP 1180. Particularities pertaining to the reversal deformation including the decreasing hardening rate during forward tension, a linear and then a non-linear unloading, followed by the Bauschinger effect, and a shift in the hardening rate during continuous straining were quantified and discussed as a function of loading history. Moreover, parameters such as the reloading stress differential, reloading softening stress, ratcheting strain, and unloading deviation stress were determined and discussed as a function of martensite fraction. The data were interpreted and predicted using an elasto-plastic self-consistent (EPSC) crystal plasticity model incorporating anisotropic elasticity, a dislocation density based hardening law, and a slip system backstress law. The model parameters associated with the slip strengths of ferrite and martensite and backstress were established. This work demonstrated the ability of crystal plasticity modeling to account for the co-dependent nature of crystallographic slip in ferrite and martensite and the sources of hardening caused by history-dependent dislocation density evolution and backstress to predict not only monotonic but also hysteresis in plastic response during the forward-reversal cycles. The combination of comprehensive experimental data and modeling results allowed us to infer that the tradeoff between the magnitude of backstress per phase and the volume fraction of ferrite versus martensite per steel governs the unloading and subsequent yielding per steel, while the dissolution of dislocations facilitates capturing the hardening rates during load reversal deformation.

Experimental characterization and crystal plasticity modeling for predicting load reversals in AA6016-T4 and AA7021-T79

The detailed contribution of microstructural-level phenomena, such as dislocation structure development and annihilation, as well as inter-granular and intra-granular backstress fields, to reverse loading behavior in metal alloys remains an area of active research and debate. The ability to predict unloading nonlinearities, the Bauschinger effect (BE), and changes in hardening rates during reverse loading is necessary for accurate modeling of deformation and springback in forming operations that involve strain path changes. This paper applies a recently developed elasto-plastic self-consistent (EPSC) crystal plasticity model to predict and interpret reverse loading in two commercially sensitive aluminum alloys (AA): 6016-T4 and 7021-T79. Model calibration and verification was enabled by an extensive experimental campaign of cyclic loading applied to the two alloys. The experimental data included hardening rates during monotonic tension, linear followed by non-linear unloading, the BE, and hardening rate changes during reverse loading that induce permanent softening. By considering anisotropic elasticity, dislocation density-based hardening, intra-granular slip system-level backstress fields, and inter-granular stress fields, the model predicted and quantified the contribution of different micro-scale phenomena to the observed behavior. The ability of the model to capture contrasting characteristics of the two alloys, particularly the distinct permanent softening and reloading yield stresses, demonstrated its ability to account for the co-dependent nature of crystallographic glide and the sources of hardening originating from the deformation history-dependent dislocation density evolution and backstress fields. Comparison of the experimental and modeling results revealed that the unloading behavior is primarily driven by backstress, the BE is governed by backstress and inter-granular stresses, and the hardening rates upon load reversals are controlled primarily by the strain-path sensitive evolution of dislocation density.