Dislocation Density-based Model Research Articles

A crystal plasticity phase-field model for microstructure sensitive fatigue crack growth in a superalloy

In this work, a dislocation density-based crystal plasticity phase-field model (CP-PFM) is developed to simulate fatigue crack growth in nickel-based superalloys. Through normalization validation, the plastic dissipation work and crystallographic work are shown to be consistent with the fatigue indicator factors (FIPs), cumulative equivalent plastic strain and cumulative shear strain, and the two energies are computed as the main driving forces of the phase field. Both driving force models are able to obtain fatigue crack growth in close approximation to the experimental rate. However, the model with crystallographic work as the main driving force obtains crack growth paths that are in better agreement with electron backscattering pattern (EBSD) observations, which is attributed to its greater ability to characterize the microstructural susceptibility of fatigue crack growth. Specifically, the model is able to capture the tendency of cracks to crack along the close-packed planes and the hindering effect of grains with large misorientation angles on fatigue crack growth, which together contribute to the curved morphology of fatigue cracks. The combination of large grains or grains with small misorientation angles favors persistent slip band (PSB) formation and leads to softening of the crack tip, which results in lower fatigue crack growth rates.

Tensile properties and work hardening in Al0.3CoCrFeNi: The role of L12 precipitates and grain size

In this study, we investigate the FCC-Al0.3CoCrFeNi high entropy alloy fabricated via spark plasma sintering of atomized powders, focusing on its mechanical and work-hardening properties across three distinct microstructures: coarse-grained, fine-grained, and fine-grained with L12 nano-precipitates. Using a dislocation density-based model, we analyze the effects of grain size and L12 precipitates on these properties, achieving quantitative agreement between model predictions and experimental tensile and work-hardening behaviors. This exploration highlights the underlying deformation mechanisms at room temperature and their contributions to the strength/ductility trade-off. Significantly, our analysis reveals that twinning in HEAs manifests differently from that observed in steels. Furthermore, the incorporation of L12 precipitates emerges as a critical factor enhancing the alloy's mechanical attributes. Our findings underscore the essential roles of microstructural parameters in tailoring the mechanical properties of HEAs, offering insights that could guide the design of advanced alloys with optimized performance.

Modeling local deformation, damage distribution, and phase transformation in zirconia particle-reinforced TRIP steel composites

This study investigates tensile deformation, damage analysis, and the effect of microtexture on a 10 % vol. zirconia particle-reinforced TRIP steel composite. In situ tensile tests and simulations were conducted, with sequential SEM images captured during tensile loading tests and electron backscatter diffraction (EBSD) providing initial crystal orientation data. SEM images were utilized in digital image correlation (DIC) analysis to calculate the local plastic strain distribution using VEDDAC software. Initial micrographs and EBSD data were transformed into geometry files for simulations, employing crystal plasticity with a dislocation density-based model and a newly proposed phase field approach-based ductile-brittle damage criterion for both austenite and ceramic particles. MATLAB's image processing functions were used to compare and validate experimentally observed damage instances with simulation predictions. The effect of microtexture combined with various physical quantities was explored using MTEX. The results show that the simulations accurately predict strain concentration around ceramic particles, with a higher strain observed in the middle region compared to the DIC observations. The simulated strain in the austenitic matrix is found to be 1.4 times higher than the DIC measurements. Quantitative analysis of damage pixels reveals that the proposed critical plastic strain value aligns closely with experimental results, showing an error of - 0.25 % compared to 1.121 % for a smaller strain case. This comprehensive analysis provides insight into the effect of microstructural attributes on material performance and behavior under tensile deformation, leading to a validated microstructural-informed damage-inclusive crystal plasticity model.

Crystal Plasticity Finite Element Modeling of the Influences of Ultrafine-Grained Austenite on the Mechanical Response of a Medium-Mn Steel

Medium manganese (medium-Mn) steel, one of the third-generation advanced high-strength steels (AHSS), delivers impressive mechanical properties such as high yield strength, ultimate tensile strength, and uniform elongation. One notable feature of medium-Mn steels is the presence of ultrafine-grained (UFG) austenite, achieved through phase transformation from the parent martensite phase during intercritical annealing. While, in general, UFG is considered a strengthening mechanism, the impact of UFG austenites in medium-Mn steel has not been fully studied. In this manuscript, we advance our previous work on crystal plasticity simulation based on the Taylor model to consider fully resolved high-fidelity microstructures and systematically study the influence of the UFG austenites. The original microstructure with UFG is reconstructed from a set of serial electron backscatter diffraction (EBSD) scans, where the exact grain morphology, orientation, and phase composition are preserved. This microstructure was further analyzed to identify the UFG austenites and recover them to their parent martensite before the intercritical annealing. These two high-fidelity microstructures are used for a comparative study using dislocation density-based crystal plasticity finite modeling to understand the impact of UFG austenites on both the local and overall mechanical responses.

A dislocation density-based crystal plasticity constitutive model: comparison of VPSC effective medium predictions with ρ-CP finite element predictions

This work presents a dislocation density-based crystal plasticity constitutive model for glide kinetics, strengthening and dislocation density evolution, implemented in the effective medium-based visco-plastic self consistent (VPSC) framework and the spatially resolved, ρ-CP crystal plasticity finite element framework. Additionally, a distribution of intragranular stresses is introduced in the VPSC framework, instead of the conventionally used mean value of grain stress for effective medium calculations. The ρ-CP model is first calibrated to predict the mechanical response of a bcc ferritic steel with an initial rolled texture. The same set of constitutive model parameters are then used in VPSC to predict the aggregate stress–strain response and total dislocation densities. For these VPSC simulations, the interaction parameter governing the interaction between the grain and the effective medium in the Eshelby inclusion formalism, and a scalar parameter representative of the distribution of intragranular stresses within a grain, are used to calibrate the VPSC predictions in order to match the predictions of the ρ-CP model. A parametric study is performed to understand the effect of these two parameters on the VPSC predictions. Further, simulations are also performed for a random untextured polycrystal to identify the corresponding VPSC simulation parameters for predicting a similar response as the ρ-CP model. The novelty of the work is in the same set of constitutive models and associated parameters have been implemented in VPSC and ρ-CP to predict similar aggregate stress–strain response and total dislocation densities. This finite element-calibrated effective medium crystal plasticity approach reduces the computational time by at least two orders of magnitude and represents an advance towards the development of multiscale crystal plasticity modeling tools.

Assessment of phenomenological models for Indentation Size Effect (ISE) through physically based dislocation density evolution

The depth dependent hardness, referred to as indentation size effect (ISE) has gained considerable attention. ISE is attributed to the strain gradient caused by the inhomogeneous strain distribution beneath the indenter tip. The depth dependence of hardness is phenomenologically accounted using an appropriate length scale parameter. The Unified model is one such model that is built on Nix–Gao’s model and focuses on integrating various length scale parameters resulting from both microstructural and geometrical effects. One aspect of the current work is to assess the applicability of the Unified model to a material like bulk Nickel with a history of cold work. Also, the physics of ISE in metals can be related to the dislocation mean free path influenced by strengthening mechanisms. Past attempts are scarce to quantify the ISE through the microstructure variables. In the present work, an attempt has been made to simulate the nanoindentation test using a dislocation density based constitutive model. The objective of this study is to establish a correlation between the equivalent plastic strain resulting from deep depth indentation (which does not exhibit a size effect) and an equivalent dislocation density. The dislocation density here serves as a state variable that describes the microstructure-dependent mechanical behaviour. The modified KME, a dislocation density-based model, is incorporated as a user subroutine in a commercial finite element solver. It has been proven to accurately predict the indentation behaviour at substantially large depths, which aligns well with the predictions of the Unified model or experimental observations. The good correlation between the value of effective dislocation density and equivalent plastic strain in the plastic zone underneath the indenter assures to extend the use of dislocation density as a state variable to predict ISE.

Simulating shot peening based on a dislocation density-based model with a novel time integration algorithm

Shot peening has been widely used in processing various components since it can bring in residual compressive stress and effectively refine the grain size of impacted area. To simulate grain refinement induced by shot peening, the dislocation density-based model has recently been introduced, however, the existing time integration algorithm is not stable and usually leads to divergent solutions in iterations. In this paper, a novel time integration algorithm is proposed for the dislocation density-based model. Based upon the algorithm, numerical studies on multi-shot AISI4340 steel are carried out with different coverages, velocities, shot diameters, and peening angles. It is shown that the method converges faster than the two-level iteration method, and the predicted dislocation cell structure sizes after shooting are consistent with experimental results. Besides that, increasing coverage can refine the size of a dislocation cell, which is closely dependent on the shot diameter, impact velocity, and angle. Thus, to achieve the desired grain size or the depth of refinement, it is necessary to take the shot diameter and velocity into account simultaneously.

Temperature dependent deformation behavior and texture evolution in AA6082 aluminum alloy: An integrated experimental and crystal plasticity simulation approach

This research provides a comprehensive analysis of the texture and temperature dependent deformation behavior of the aluminum alloy AA6082. The study is performed using a combination of experimental deformation tests and computational simulations using a crystal plasticity (CP) framework. The primary objective is to identify the critical influence of temperature on local stress states and dislocation density within the material during tensile tests. From an experimental perspective, the study employs ex-situ deformation tests and subsequent electron backscatter diffraction (EBSD) data in combination with energy dispersive x-ray spectrometer (EDS) analysis at 200 °C and 400 °C. The experimental EBSD data are adopted, preprocessed and converted into a geometry file for the numerical simulations. On the computational front, a dislocation density-based material model is adopted for CP simulations. The physical and fitting parameters of the model are calculated, adopted from the literature, or calibrated by comparing the global simulation results with experimental stress-strain observations under uniaxial tensile load at room temperature, 200 °C and 400 °C. Empirical functions for solid solution strengthening, dislocation density, fitting parameters controlling mean free path and dislocation annihilation have been derived that can be used to quickly interpolate them for any intermediate temperature. These functions combined with other model parameters can now be used for temperature-dependent CP modeling of AA6082. The isothermal grain-scale simulation and ex-situ experimental results confirmed a noteworthy texture transformation at higher temperatures, characterized by a reduction in the primary Cube orientation and its transition into a Copper orientation due to the stretching process. The correlation between the experimental results and the simulations on macro- and micro-scales is reasonable, indicating the accuracy and effectiveness of the CP approach in predicting the temperature dependent deformation behavior and texture evolution in aluminum alloys.

Eulerian finite element implementations of a dislocation density-based continuum model

In Eulerian finite element simulations, the mesh moves relative to the material. After every change of position between the mesh and the material, the state variables are interpolated to the new mesh position, which is referred to as advection. Large strain crystal plasticity models are based on the multiplicative decomposition of the total deformation gradient. The stress is evaluated as a function of the thermoelastic strain, temperature, and other state variables. Advection of tensor quantities, such as the strain, is coupled with possibly significant advection errors. In an effort to reduce the advection errors, we develop two rate forms of an established dislocation density-based continuum model. To that end, we replace the multiplicative decomposition of the deformation gradient with the additive decomposition of the velocity gradient, and define the stress rate instead of the total stress. The Eulerian implementation is compared with Lagrangian calculations, and two numerical examples with severe deformation levels are presented.

Temperature-dependent behavior of CP-Ti interpreted via self-consistent crystal plasticity simulation

A commercially-pure titanium sample was investigated using an elasto-visco-plastic self-consistent polycrystal model. Temperature-dependent plastic deformation behavior was described by a dislocation density-based hardening model. The model reasonably reproduced various experimental behaviors, including stress vs. strain curves, diffraction lattice strains, twin volume fraction, and crystallographic texture. The temperature-dependent behavior was interpreted based on the dislocation evolution rules comprised of dislocation generation and recombination in accordance with the Kocks-Mecking rule. The modeling results successfully provided an in-depth interpretation of mechanical behavior of α-titanium under a wide range of temperatures.

Surface microstructural alterations and grain refinement mechanisms of tungsten heavy alloys by thermo-mechanical coupling: A laser-assisted milling case

Ultrafine-grained (UFG) tungsten heavy alloys possess potential applications in precision instruments and nuclear protection. In the present study, a comprehensive investigation of machined surface microstructural alterations and grain refinement mechanisms in laser-assisted milling (LAM) 95W-3.5Fe-1.5Ni alloy was demonstrated by recognizing its surface microstructures and mechanical properties under different combinations of the laser power and cutting depth. Graded microstructural characteristics along the surface degenerating layer and subsurface deformed layer of tungsten heavy alloy were identified for the first time, dramatically changing from workpiece bulk. Results showed that the adaptability between the thickness of the laser heat-affected layer and the depth of cut greatly influenced surface recrystallization layer microstructures regarding the thermal-mechanical coupling effect. An ordered, continuous, and equiaxed refined grains distribution layer was recognized when a particular machining condition was adopted, resulting in a substantial improvement of mechanical hardness. It could be indicated that a redundant heat-affected area would induce strain accumulation and promote recrystallization in LAM. Moreover, a dislocation density-based recrystallization model was applied to complement the experimental results considering the temporal transformation of grain refinement and recovery phenomena. Clarifying the above aspects is crucial to control the tungsten heavy alloys' surface microstructures and service performance during high-performance machining operations.

Dislocation density-based modeling of the yield drop phenomenon in nickel-based single crystal superalloy

The phenomenon of yield drop, characterized by a decrease in flow stress after initial yield, has been observed in various nickel-based superalloys. Despite numerous proposed physical mechanisms, there is still a lack of a meso-mechanism-based constitutive model to explain this phenomenon. In this study, the tensile behavior of a nickel-based single crystal superalloy (DDX), was investigated at different strain rates and a temperature of 900 °C. It was observed that the yield drop phenomenon in DDX became more pronounced with increasing strain rate. To predict the yield drop phenomenon during tensile processing, an improved strength law based on continuum dislocation density theory was considered in the crystal plasticity framework. The proposed constitutive model was implemented using nonlinear iteration and incorporated into a finite element analysis software. The simulation results exhibited a good agreement between the experimental data and the stress–strain curve in the vicinity of the yield drop region, affirming the predictive aptitude of the proposed model in elucidating the yield drop phenomenon at various strain rates.

A modified dislocation density-based model of 30CrMnSiA considering the coupling effect of electroplasticity and grain refinement during the pulsed current-assisted plane strain deformation

A modified dislocation density-based model of 30CrMnSiA considering the coupling effect of electroplasticity and grain refinement during the pulsed current-assisted plane strain deformation

Modelling mechanical behaviour of a gradient-microstructured material obtained by surface mechanical attrition treatment accounting for residual stresses

In this work, an austenitic 316 L steel was processed by surface mechanical attrition treatment (SMAT), and the induced gradient microstructure was highlighted by experimental measurements. A combined dislocation density-based and grain size-dependent constitutive model is developed to describe the mechanical behaviour of the gradient-microstructured material. In addition, this model also incorporates the grain size-dependent initial twin distribution and evolution of deformation twinning. Residual stress, initial dislocation density and twins are considered to reconstruct the depth-dependent residual fields induced by SMAT. Furthermore, the evolution of dislocation density and twin volume fraction is described during uniaxial tensile loading. Particular attention is devoted to investigate the effect of residual stress and deformation twinning on mechanical behaviour. Results of finite element simulation showed that the gradient microstructure enhances the yield strength, which is in agreement with previous experimental observations. It was revealed that residual stress significantly weakens the yield strength at the initial deformation stage, whereas it has little influence on the plastic behaviour at large deformation. The assumed grain size-dependent deformation twinning model allows describing the evolution of deformation twinning on gradient microstructure. The deformation twinning increases in relation to increased strain hardening during tensile loading. However, the evolution of twin volume fraction shows almost no twin and no variation of twinning within the nanocrystalline grain region.

Dispersed strain bands promote the ductility of gradient nano-grained material: A strain gradient constitutive modeling considering damage effect

Gradient nano-grained (GNG) metals have achieved superior strength-ductility synergy than their homogeneous counterparts. The high strength is usually attributed to the grain size effect and hetero-deformation-induced strengthening. However, incommensurate work on ductility leads to an incomplete understanding of the strength-ductility combination. In this work, a dislocation density-based strain gradient plasticity model coupling with a damage model is developed to describe the strain hardening and softening behavior of GNG material. A grain size-dependent back stress model derived from the generation of dislocation pileups is invoked to describe the widely-concerned back stress hardening. Finite element implementation of the model quantitatively predicts the tensile response of GNG nickel with various degrees of grain size gradient. The results reveal that dispersed strain bands propagate stably in the nano-grained surface layer of GNG material, which is totally different from those occurring in a freestanding nano-grained material. The stabilization of dispersed strain bands enables the nano-grained layer to deform uniformly, thus premature failure of the whole GNG material is suppressed and improved ductility is achieved. Furthermore, increasing the grain size gradient renders the strain bands more stable, leading to enhanced ductility. The method developed in this work is helpful for understanding the strength-ductility synergy of GNG materials and for optimizing the microstructure gradient in GNG materials.

Modeling of microstructural workpiece rim zone modifications during hard machining

Surface integrity determines, among other factors such as tool material and wear, the chemical composition and microstructure of the workpiece material as well as selected cutting parameters, the performance of machined parts. The aggressive nature of the cutting process influences the surface integrity and causes altered microstructure on the workpiece surface. Most popular example is the White Layer formation, which reduces the fatigue resistance of components. To study the influence of cutting parameters on White Layer thickness for a steel of grade AISI 4140, in this paper, a physics-based dislocation density-based model is implemented to predict the White Layer formation under the assumption of dynamic recrystallization. As a material model, a modified coupled damage mechanics model based on Johnson–Cook is implemented, which considers in comparison to the original one, damage induced softening using damage initiation and fracture indicators as well as strengthening mechanisms through grain refinement and dislocation damping at high strain rates. Dry orthogonal cutting is modeled via Coupled-Eulerian–Lagrangian method in Abaqus/Explicit. The developed model is validated based on experimental data regarding grain size and White Layer thickness. The model overestimated the grain size on the machined workpiece rim zone by a factor of 2–4, while the White Layer thickness was predicted with an error of max. 0.36 μm.

Metal Crystal/Polycrystal Plasticity and Strengths

A brief historical sketch is given of Taylor’s dislocation density-based model description, leading to the prediction of a parabolic, tensile, stress–strain curve for the plastic deformation of aluminum. The present focus is on additional results or analyses obtained on the subject for crystal/polycrystal strain hardening. Our current understanding of such material behavior is attributed to post-Taylor descriptions of sequential deformation stages in stress–strain measurements that are closely tied to specific dislocation interaction and reaction mechanisms. A schematic comparison is given for individual face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp) crystal curves and to related strength properties determined for individual crystals and polycrystalline material. For the fcc case, an example sessile dislocation reaction is described based on a stereographic projection. Then, quantitative constitutive-relation-based assessments are presented for the tensile strain hardening leading to the plastic instability behaviors of copper and tantalum materials.

Physics-Based Flow Stress Model for Alloy 718

A dislocation density-based model for alloy 718 in the annealed state is proposed in order to accurately describe the deformation behavior of this alloy for a wide range of thermo-mechanical loadings. The model accounts for numerous microstructural mechanisms, including strain hardening, grain size effect, dynamic strain aging (DSA), solid solution strengthening, as well as phonon and electron drag which affects dislocation movements at high strain rates. Two types of recovery mechanisms are also included: recovery due to dislocation glide and recovery associated with cross-slip of screw dislocations. The model is calibrated using experimentally determined stress–strain curves for both low and high strain rates in the order of 10–3 to 103 s−1, and for temperatures in the range 20 °C to 800 °C. The stress–strain data computed with the model are in good agreement with the experimental data. The inclusion of DSA is found to be effective in the combination of temperatures and strain rates corresponding to experimental observations. The solid solution strengthening contribution increases with decreasing temperature and increasing strain rate. The drag effect in the model proves to be significant only for deformation at high strain rate (~ 103 s−1).

Numerical and Experimental Analysis of the Anisotropy Evolution in Aluminium Alloys Processed by Asymmetric Rolling

One of the most important characteristics of the sheet metal is its anisotropy. Asymmetric rolling (ASR) shows to be an adequate process to change the material anisotropy by increasing the normal anisotropy and decreasing the planar anisotropy. In this work, it is analysed the relationship between anisotropy and texture evolution using experimental and numerical approaches. Experimentally, the texture is modified by rolling, involving symmetric (SR), asymmetric rolling continuous (ARC) and asymmetric reverse (ARR) routes and different reductions per pass. The numerical analysis was performed through the visco-plastic self-consistent model where two hardening laws were considered, namely the Voce-type (V) and the dislocation density-based model (DDR). The main objective of the numerical method was to test the performance of the VPSC model for large plastic deformation. The Lankford coefficients decrease in RD and increase in TD with the increase in the total thickness reduction. This trend observed experimentally is well captured by the VPSC model, however, in terms of R-value, an overestimation is observed in both cases with better results for Voce-type law.

A dislocation density-based model considering dynamic strain aging for annealed and water-quenched aluminum alloys under low-temperature conditions

Low-temperature forming is a novel manufacturing method for aluminum alloy components. To better understand the flow characteristics and deformation behavior of aluminum alloys at low temperatures, uniaxial tensile tests were conducted with an initial strain rate range of 2.5 × 10−4–1 × 10−2 s−1 and a temperature range of 77–298 K for annealed (O) and water-quenched (WQ) 2060 Al–Cu–Li alloys. The results showed that the two tempered alloys exhibited excellent work hardening at lower temperatures, resulting in improved ductility, especially in the WQ-tempered alloy. Dynamic strain aging (DSA) caused by interactions between solute atoms and moving dislocations occurred at higher temperatures and lower strain rates, which resulted in serrated flow. Because moving dislocations were additionally pinned by solute clusters, the flow stress increased. The stress increase and negative strain rate caused by DSA could not be captured using the typical dislocation density-based K-M model. Thus, a modified K-M model that considered DSA was proposed to describe the tensile behavior of the studied alloys, and the stress-strain values predicted by the proposed model agreed well with the experimental ones.