Dislocation Density-based Constitutive Model Research Articles

Advanced Computational Analysis of Cobalt-Based Superalloys through Crystal Plasticity.

This study introduces an advanced computational method aimed at accelerating continuum-scale processes using crystal plasticity approaches to predict mechanical responses in cobalt-based superalloys. The framework integrates two levels, namely, sub-grain and homogenized, at the meso-scale through crystal plasticity finite element (CPFE) platforms. The model is applicable across a temperature range from room temperature up to 900 °C, accommodating various dislocation mechanisms in the microstructure. The sub-grain level explicitly incorporates precipitates and employs a dislocation density-based constitutive model that is size-dependent. In contrast, the homogenized level utilizes an activation energy-based constitutive model, implicitly representing the γ' phase for efficiency in computations. This level considers the effects of composition and morphology on mechanical properties, demonstrating the potential for cobalt-based superalloys to rival nickel-based superalloys. The study aims to investigate the impacts of elements including tungsten, tantalum, titanium, and chromium through the homogenized constitutive model. The model accounts for the locking mechanism to address the cross-slip of screw dislocations at lower temperatures as well as the glide and climb mechanism to simulate diffusions at higher temperatures. The model's validity is established across diverse compositions and morphologies, as well as various temperatures, through comparison with experimental data. This advanced computational framework not only enables accurate predictions of mechanical responses in cobalt-based superalloys across a wide temperature range, but also provides valuable insights into the design and optimization of these materials for high-temperature applications.

A rationale for modelling hydrogen-induced softening in fcc single crystals

Several atomistic-scale mechanisms have been proposed to explain the phenomenon of hydrogen-enhanced localized plasticity (HELP). However, the relative contributions of each of these atomistic mechanisms in determining the constitutive behaviour of fcc single crystals undergoing HELP are yet to be fully understood. In this study, we address this issue by presenting a constitutive model of H-induced softening for fcc single crystals. Here, we first compute H concentrations in the trapped state close to the dislocation cores and in the normal interstitial lattice sites (NILS), which are then used to determine the efficacy of different HELP mechanisms contingent on them. The H in NILS forms atmospheres around dislocations and shields the long-range elastic forces acting on them, while the H in deep traps around dislocations reduces the activation barrier associated with short-range obstacles and enhances the ease of dislocation nucleation. These mechanisms are incorporated in a coarse-grained form into a dislocation density-based constitutive model of fcc single crystals, and the constitutive response is simulated for each one of the underlying mechanisms. Our simulations reveal that the mechanism of enhancement of dislocation nucleation in the presence of H is the most potent to cause softening of the constitutive response. In comparison, the flow softening caused by the mechanisms of reduction of the activation barrier of short-range obstacles and the shielding effect of H appears to be insignificant. The presented constitutive model can be extended to simulate H-induced softening in polycrystalline fcc materials as well.

Constitutive modeling hot tensile behavior of Ti6Al4V alloy by considering phase transformation and damage mechanisms

The hot tensile and fracture behavior, as well as damage mechanisms of Ti6Al4V alloy are investigated by the hot tension tests and microstructure observation experiments. For comprehensively characterizing the hot tensile behaviors, a developed dislocation density-based constitutive model was established by coupling the flow softening, phase transformation and damage mechanisms. The results show that the flow stresses under the relatively high strain rates (0.01–1 s−1) easily peak at low strains and then decrease with the further straining. However, under the low strain rate (0.001 s−1), the flow stress tends to a continuously increasing trend until fractured. After hot deformation, the fraction of β phase greatly increases compared to that under the initial state. Meanwhile, β-phase fraction increases with the strain rate decreasing or the deformation temperature increasing. The great input thermal energy by the high temperature and the long deformation period under low strain rates promote the α-β phase transformation. Furthermore, the necking diffusion caused by dislocation motion enhances the homogeneous deformation under low strain rate or high temperature. The cavitations preferentially nucleate at the α/β phase interface due to the inconsistent strain. The comparison of the measured and calculated results confirms the excellent prediction ability of the established model in reproducing the hot tensile behavior, β-phase fraction and damage evolution under different processing parameters.

Modeling the hot deformation behavior of micro-alloyed steel in the single and two-phase fields

Continuous casting is a well-established process to produce steel slabs. However, the surfaces of the slabs may crack during processing. The occurrence of cracks is related to different phenomena, i.e., ferrite formation, presence of precipitates, and microstructural modifications during manufacturing processing. In the present work, we developed a physically based mesoscale model to describe the plastic deformation of micro-alloyed steel in the single and two-phase fields, as well as the microstructure evolution. We implemented a dislocation density-based constitutive model to calculate the strain hardening and the stress softening produced by dynamic recovery using the Kocks-Mecking (KM) formalism for the austenite and ferrite. We coupled the KM model with the Johnson-Mehl-Avrami-Kolmogorov model to consider the dynamic recrystallization of the austenitic phase. The nucleation and growth of the recrystallized grains, driven by the stored energy, compete with the annihilation of dislocations due to dynamic recovery. In the two-phase domain, the iso-work condition for the load partition is implemented. We calculated the ferrite volume fraction by fitting the data obtained using JmatPro software. Moreover, an Arrhenius-type equation correlates the yield stress to the Zener-Hollomon parameter. We validated the model with isothermal uniaxial compression tests of microalloyed steel over the temperature range of 650 °C-1100 °C and strain rates of 10−2 s−1 and 10−3 s−1 using a Gleeble® 3800 thermomechanical simulator.

Numerical Study on Laser Shock Peening of Pure Al Correlating with Laser Shock Wave.

Laser shock peening (LSP) is an innovative and promising surface strengthening technique of metallic materials. The LSP-induced plastic deformation, the compressive residual stresses and the microstructure evolution are essentially attributed to the laser plasma-induced shock wave. A three-dimensional finite element model in conjunction with the dislocation density-based constitutive model was developed to simulate the LSP of pure Al correlating with the LSP-induced shock wave, and the predicted in-depth residual stresses are in reasonable agreement with the experiment results. The LSP-induced shock wave associated with the laser spot diameter of 8.0 mm propagates in the form of the plane wave, and attenuates exponentially. At the same time, the propagation and attenuation of the LSP-induced shock wave associated with the laser spot diameter of 0.8 mm are in the form of the spherical wave. The reflection of the LSP-induced shock wave at the bottom surface of the target model increases the plastic deformation of the target bottom, resulting in the increase of dislocation density and the decrease of dislocation cell size accordingly. Reducing the target thickness can significantly increase the reflection times of the LSP-induced shock wave at the bottom and top surfaces of the target model, which is considered to be conductive to the generation of the compressive residual stress field and grain refinement.

Multi-Scale Crystal Plasticity Model of Creep Responses in Nickel-Based Superalloys.

The current study focuses on the modeling of two-phase - nickel-based superalloys, utilizing multi-scale approaches to simulate and predict the creep behaviors through crystal plasticity finite element (CPFE) platforms. The multi-scale framework links two distinct levels of the spatial spectrum, namely, sub-grain and homogenized scales, capturing the complexity of the system responses as a function of a tractable set of geometric and physical parameters. The model considers two dominant features of morphology and composition. The morphology is simulated using three parameters describing the average size, volume fraction, and shape. The sub-grain level is expressed by a size-dependent, dislocation density-based constitutive model in the CPFE framework with the explicit depiction of - morphology as the building block of the homogenized scale. The homogenized scale is developed as an activation energy-based crystal plasticity model reflecting intrinsic composition and morphology effects. The model incorporates the functional configuration of the constitutive parameters characterized over the sub-grain - microstructural morphology. The developed homogenized model significantly expedites the computational processes due to the nature of the parameterized representation of the dominant factors while retains reliable accuracy. Anti-Phase Boundary (APB) shearing and, glide-climb dislocation mechanisms are incorporated in the constitutive model which will become active based on the energies associated with the dislocations. The homogenized constitutive model addresses the thermo-mechanical behavior of nickel-based superalloys for an extensive temperature domain and encompasses orientation dependence as well as the loading condition of tension-compression asymmetry aspects. The model is validated for diverse compositions, temperatures, and orientations based on previously reported data of single crystalline nickel-based superalloy.

Dislocation Density-Based Constitutive Model and Processing Map for T2 Copper During Isothermal and Time-Variant Deformation

Dislocation Density-Based Constitutive Model and Processing Map for T2 Copper During Isothermal and Time-Variant Deformation

Size-dependent plasticity of hetero-structured laminates: A constitutive model considering deformation heterogeneities

Deformation mechanism-based constitutive modeling can serve as a flexible tool to optimize the microstructure of hetero-structured metals for achieving better mechanical performance. In this work, a dislocation density-based constitutive model, which considers geometrically necessary dislocations (GNDs) and the back stress induced by internal heterogeneous deformation, is developed for describing the deformation of hetero-structured laminate materials. The heterogeneous laminates are considered to consist of alternating coarse-grained and nano-grained layers separated by extended interfaces (interface affected zones, IAZs). The respective deformation mechanisms of each phase are established based on experimental information. The evolution laws of back stress and GNDs for the coarse-grained layers, nano-grained layers and IAZs are formulated within a unified framework based on dislocation pile up theory. The simulated tensile responses are in quantitative agreement with experimental data for heterogeneous copper/bronze laminates with different interface spacings and different volume fractions of the nanostructured layers. The model also allows to represent the back stress and strain-hardening rate accurately. It is found that smaller interface spacing, implying a higher volume fraction of the IAZ, leads to a higher density of GNDs and more significant back stress. The IAZ is demonstrated to be the key to understanding the strength-ductility combination of laminates. The developed model can be readily applied to other kinds of metallic materials with architectured microstructures.

Dynamic stress propagation induced transition of stress state and microstructure characteristics during high-speed cutting of OFHC copper

High-speed cutting (HSC) is a popular technique applied in manufacturing as it can significantly improve machining efficiency and surface quality. However, when the cutting speed is increased to extremely high, the deformation process of the material will be changed due to the alteration of microstructures and loading conditions; as a result, some deeper understandings of the mechanisms are still required. In this study, a dislocation density-based constitutive model is implemented to describe the material behavior of OFHC (oxygen-free high conductivity) copper during high-speed cutting process, which is then used to obtain the characteristics of dynamic stress propagation through physical parameters of the material, so that the relationship between dynamic stress propagation and microstructure evolution can be directly described. The results show that the propagation of plastic deformation will be significantly changed when cutting speed is increased to extremely high, and the transformation of stress states between loading and unloading will lead to the change of gradient distribution of strains and microstructures. This research can be used to understand the stress state variation during dynamic deformation in the cutting process through microstructure characterization in future investigations.

A dislocation density-based model and processing maps of Ti-55511 alloy with bimodal microstructures during hot compression in α+β region

Hot compression features of Ti-55511 alloy are investigated by high-temperature compression tests in α+β region. It is found that the flow stress and softening mechanisms are obviously influenced by deformation conditions. The true stress decreases with the reduced strain rate or the raised temperature. The spheroidization of α phase and dynamic recrystallization (DRX) of β phase easily occur at low temperatures such as 973, 1003 and 1033 K, while the dynamic recovery (DRV) of β phase mainly occurs at high temperatures such as 1063 K because of the transformation from α phase to β phase at relatively high temperatures A dislocation density-based constitutive model, which is associated with DRV, work hardening mechanisms and the spheroidization of α phases, is established and validated to describe flow behavior. The correlation coefficient (R) and average absolute relative error (AARE) of the established model are 0.9924 and 6.8%, respectively. 3D power dissipation efficiency maps and processing maps are established to determine the appropriate processing window, i.e., too low temperatures (lower than 973 K) or too high strain rates (higher than 1 s−1) easily induce flow instability. Therefore, the medium temperature (1003–1063 K) and the low strain rate (0.001–0.1 s−1) are applicable for thermal compression of the studied titanium alloy.

Micromechanical modeling of work hardening for coupling microstructure evolution, dynamic recovery and recrystallization: Application to high entropy alloys

A dislocation density-based multiscale constitutive model is developed to describe the hot deformation of high entropy alloys. The work hardening process of high entropy alloy is divided into several stages according to the change of hardening rate. As crystalline material, high entropy alloys are regarded as two-phase materials. In the current study, a physical-based constitutive model with three key microstructural state variables, cell mobile dislocation density, cell immobile dislocation density and wall immobile dislocation density, is proposed to simulate the phases deformations. The physical mechanisms including strain hardening, dynamic recovery, and recrystallization are considered in describing the evolution of dislocation densities. It is noted that the competition and interaction among the three mechanisms result in a typical hardening behavior of high entropy alloys. The dislocation density-based constitutive model is embedded in the modified Mori-Tanaka scheme to describe the hardening behaviors of two-phase materials with an elastic phase cell wall and an elastic-inelastic phase cell interior. The model is applied for simulating the hardening curves of CoCrFeMnNiC0.5 high entropy alloy at different temperatures and strain rates. Compared with experimental data, it shows that the developed model can describe hot deformation of high entropy alloys with reasonable accuracy. The macroscopic inelastic strain in the Mori-Tanaka scheme is calculated by a classical plasticity method. The simulation results show that the modified method can give a more accurate prediction compared to the average inelastic strain of all phases.

Dislocation-based study on the influences of shot peening on fatigue resistance

Shot peening technology is widely used to enhance the fatigue resistance of mechanical parts exposed to the cyclic loadings. The residual stress, grain refinement, work hardening and surface roughness are the four major results of shot peening, which play the important role in enhancing effects of fatigue performances. A framework linking the dislocation-based finite element simulation to the Navarro-Rios model (N-R model) was developed to evaluate the fatigue resistance of shot-peened materials. The parametric three-dimensional finite element models of multiple shot impacts, embedded with the dislocation density-based constitutive model, were developed firstly. The predicted residual stresses, grain refinements, surface topographies and surface roughnesses are all in good agreement with the experimental results. The simulation results of shot peening were then imported into the N-R model to analytically evaluate the stresses for fatigue crack arrest and corresponding values of crack tip opening displacement (CTOD). The computation results show that the S110 shots can effectively improve the stresses for the surface crack arrest, while the more grains within the shot-peened surface and subsurface are endowed with the larger stresses for crack arrest by S230 shots, and the double shot peening (S230 + S110 shots) takes the advantages of the cases of S230 and S110 shots.

Dislocation density-based study of grain refinement induced by laser shock peening

Laser shock peening (LSP) is an innovative surface processing technique. Grain refinement induced by LSP has been proved to be feasible to improve the surface properties of materials and prolong the service life of metallic components. The three-dimensional finite element model, which incorporates a dislocation density-based constitutive model and the temporal-spatial distribution of laser shock wave, was adopted to simulate the process of grain refinement induced by LSP. The predicted dislocation cell sizes, dimple fabrications induced by the repetitive LSP of copper are in good agreement with experimental results, which confirms the validity of the dislocation density-based three-dimensional finite element model. The effects of laser spot overlap ratio and laser power density (peak laser shock wave pressure) on LSP-induced grain refinement were investigated in detail based on the numerical simulations of multiple LSP of copper and CP-Ti.

Dislocation Density-Based Model for Flow Behavior of a Near-α Titanium Alloy Considering Effects of Initial Lamellar Thickness

A dislocation density-based constitutive model that considered the effects of initial lamellar thickness was proposed to predict the flow behavior of a near-α high-temperature Ti-5.4Al-3.7Sn-3.3Zr-0.5Mo-0.4Si alloy. The evolution equations of dislocation density were improved by introducing the influence of the dynamic globularization of lamellar microstructure with different initial thicknesses. The evolution equations and Hall-Petch behavior were coupled into the constitutive model, which was composed of athermal and thermal stresses. According to the established constitutive equations, the variation and contributions of corresponding stress components were analyzed. The calculated dislocation densities were compared with the experimental values obtained by electron backscattered diffraction data to verify the model. Moreover, the flow stresses of the studied alloy with different initial lamellar microstructures under various deformation conditions were predicted. Good agreements between the predicted and experimental results were obtained. Additionally, steady-state flow behavior was further analyzed and the relationship between the calculated steady-state flow stress and the equiaxed α-phase size was quantitatively obtained. The results indicated the validity of the proposed constitutive model for predicting the flow behavior of the alloy with different initial lamellar microstructures over a range of deformation temperatures and strain rates.

Tailoring tensile ductility of thin film by grain size graded substrates

Extensive experiments have shown that thin metal or metallic glass films usually rupture under a small tensile strain, which is extremely unfavorable for their engineering applications. The introduction of a highly deformable substrate with homogeneous microstructures can enhance the ductility of the films. However, the strong mechanical contrast between the substrate and the film usually induces interface debonding and premature cracks at the interface. Another disadvantage of the above film/substrate system is its significantly reduced strength due to the low strength of the homogeneous substrate. Recently, experiments have shown that a heterogeneous substrate with a grain size graded microstructure is able to provide an exceptional strength-ductility synergy in the film/substrate system. However, why does the graded substrate render such outstanding properties remains unsolved due to its complex microstructure. In order to answer this question, here a dislocation density-based constitutive model was incorporated into a finite element scheme to investigate the deformation of a thin film attached to a graded substrate. The deformation of the film on a normal substrate with a homogeneous grain microstructure was also analyzed for comparison. A V-shaped notch was adopted in the film to model its necking behavior based on a necking criterion. The necking strain is adopted as the tensile ductility. The results show that three typical deformation behaviors can be triggered by tailoring the microstructure of the graded substrate including the grain size of the topmost surface layer and the thickness of the grain size graded region. The three modes correspond to very different necking strains. Interestingly, the graded substrate with an optimal microstructure can provide the film with much higher necking strain than the homogeneous coarse-grained substrate could although the former itself is less ductile than the latter. In addition, the film/graded-substrate system possesses much better strength-ductility synergy than the film/homogeneous-substrate system. The enhancement of the tensile ductility results from the strong strain delocalization effect induced by the graded substrate. The predicted stress-strain responses are also compared with available experimental measurements.

A Dislocation Density-Based Viscoplasticity Model for Cyclic Deformation: Application to P91 Steel

To describe the viscoplastic behavior of materials under cyclic loading, a dislocation density-based constitutive model is developed based on the unified constitutive theory in which both the creep and plastic strain are integrated into an inelastic strain tensor. The stress evolution during cyclic deformation is caused by the mutual competition and interaction between hardening and recovery. To incorporate the physical mechanisms of cyclic deformation, the change of mobile dislocation density is associated with inelastic stain in the proposed model. The evolution of immobile dislocation density induced by strain hardening, dynamic recovery, static recovery and strain-induced recovery are simulated separately. The deterioration of yield strength following the hardening in tension (or compression) and subsequently in compression (or tension) is described by the Bauschinger effect and reduction of immobile dislocation density, the latter is induced by static- and strain-induced recovery. A kinematic hardening law based on dislocation density is proposed, both isotropic hardening and softening are described by determining the evolution of hardening parameters. The experimental data of P91 steel under different strain rates and temperatures are adopted to verify the proposed model. In general, the numerical predictions agree well with the experimental results. It is demonstrated that the developed model can accurately describe the hardening rate change, the yield strength deterioration and the softening under cyclic loading.

A dislocation climb/glide coupled crystal plasticity constitutive model and its finite element implementation

The glide and climb of dislocations are two important plastic deformation mechanisms of the metallic crystals at elevated temperatures. In this work, a new dislocation density-based constitutive model for single crystal plasticity with explicit consideration of both dislocation glide and climb is presented. Three contributions of dislocation climb to the plastic deformation are involved: the kinematics, the climb-enhanced dislocation mobility and the climb-induced dislocation annihilation. A fully implicit time-integration scheme for this model is given and implemented by a user material subroutine in software ABAQUS. Then, the compression tests of 〈110〉 single crystalline aluminum at high temperature and low strain rate are simulated, showing good agreements with the experimental results. Moreover, this model is used to predict the creep deformation of single crystalline aluminum at different temperatures and applied stress levels. The results show that the present model can capture the power law creep behavior of single crystalline aluminum and the predicted creep exponent falls within the range suggested by earlier research.

Optimization and validation of a dislocation density based constitutive model for as-cast Mg-9%Al-1%Zn

A dislocation density-based constitutive model, including effects of microstructure scale and temperature, was calibrated to predict flow stress of an as-cast AZ91D (Mg-9%Al-1%Zn) alloy. Tensile stress-strain data, for strain rates from 10-4 up to 10-1s-1 and temperatures from room temperature up to 190°C were used for model calibration. The used model accounts for the interaction of various microstructure features with dislocations and thereby on the plastic properties. It was shown that the Secondary Dendrite Arm Spacing (SDAS) size was appropriate as an initial characteristic microstructural scale input to the model. However, as strain increased the influence of subcells size and total dislocation density dominated the flow stress. The calibrated temperature-dependent parameters were validated through a correlation between microstructure and the physics of the deforming alloy. The model was validated by comparison with dislocation density obtained by using Electron Backscattered Diffraction (EBSD) technique.

Modeling the effect of strain reversal on grain refinement and crystallographic texture during simple shear extrusion

This paper is concerned with the modeling of the effect of strain reversal on crystallographic texture and microstructure evolution in simple shear extrusion (SSE) processing of commercial purity copper. To reach this goal, the evolution of microstructure is predicted by a dislocation density-based constitutive model in which the effect of strain reversal is considered. This dislocation-based model is embedded in a crystal plasticity finite element (CPFE) model. In FE models, a representative volume element of the polycrystal at the central region of the billet is subjected to deformation history experienced by the SSE process with backpressure. By comparison the predicted crystallographic texture with the published data, it is concluded that the CPFEM well predicts a simple shear texture after a complete pass. The predicted results show 5–10% increase in the rate of dynamic recovery of the second half of deformation zone in comparison with the first half. The modified dislocation-based model predicts that the cell size of a complete pass is more than that of 0.5 pass sample as much as 27%.

A dislocation density based constitutive model for as-cast Al-Si alloys: Effect of temperature and microstructure

The flow stress of an as-cast Al-Si based alloy was modeled using a dislocation density based model. The developed dislocation density-based constitutive model describes the flow curve of the alloy with various microstructures at quite wide temperature range. Experimental data in the form of stress-strain curves for different strain rates ranging from 10−4 to 10−1s−1 and temperatures ranging from ambient temperature up to 400°C were used for model calibration. In order to model precisely the hardening and recovery process at elevated temperature, the interaction between vacancies and dissolved Si was included. The calibrated temperature dependent parameters for different microstructure were correlated to the metallurgical event of the material and validated. For the first time, a dislocation density based model was successfully developed for Al-Si cast alloys. The findings of this work expanded the knowledge on short strain tensile deformation behaviour of these type of alloys at different temperature, which is a critical element for conducting a reliable microstructural FE-simulation.