Precipitate Size Distribution Research Articles

A Physically Based Mean Field Model for Strain‐Induced Precipitation and Recrystallization in High‐Strength Low‐Alloy Steels

A physically based mean field model developed to predict the microstructural evolution during the thermomechanical control process of X70 high‐strength low‐alloy (HSLA) steels is presented. The physically based mean field model incorporates a new integrated precipitation and recrystallization model developed to describe the interaction between strain‐induced precipitation of niobium and titanium carbonitrides and static recrystallization of austenite. The integrated model considers an effective Zener pinning force for the multimodal particle size distribution (PSD) of precipitates, an effective grain‐boundary mobility for the solute drag effect of niobium, and an inhomogeneous stored energy for austenite recrystallization. Given a processing route, the model predicts the variation of austenite grain size, recrystallized and precipitated fractions, and evolution of PSDs of precipitates. Model predictions reveal an excellent agreement with experimental grain size measurements and a final average ferrite grain size of 3.81 μm is achieved. The proposed model considers the heterogeneous nature of recrystallization and precipitation and can contribute to the process design of the HSLA and microalloyed steels.

Consequence of coherent Cu5Zr precipitates and Cu2O passive layer formation on the corrosive behaviour of additively processed Cu-Cr-Zr alloy in simulated seawater

Abstract The Cu-Cr-Zr copper alloy is known for its outstanding electrical conductivity and fatigue strength. However, the corrosion behaviour of the copper alloy should also be taken into account when adopting it in industrial applications, especially in the marine environments. This research aims to fabricate Cu alloy coupons using the Laser Powder Bed Fusion (LPBF) technique and subsequently test their corrosive performance in simulated seawater. This research confirms that the Cu5Zr precipitate formation during the LPBF process and the Cu2O passive layer formation were the main reason for the enhanced corrosive behavior of the LPBFed copper alloy. The OM (Optical Microscope), FESEM (Field Emission Scanning Electron Microscope) images supported in evaluating melt pool formations and irregularities, and also confirmed the polycrystalline structure. The diffraction pattern from the TEM (Transmission Electron Microscope) analysis confirmed the formation of Cu5Zr precipitate and grain size distribution, while their orientations were obtained from the EBSD (Electron Based Scattered Diffraction) EBSD analysis. Micro hardness was executed on the scanning and building directions, and it was found that the building direction possessed higher hardness of 54 HV0.3 which was 5% higher than in the scanning direction. This significant fluctuation in the hardness value is due to the closely packed equiaxed and columnar grains along the outer and inner regions of the melt pools. Potentio-dynamic polarization (PD) and electrochemical impedance spectroscopy (EIS) tests were performed on the printed copper alloy parts for various immersion periods of 0, 9, 18 and 38 h. Further, the XRD (x-ray Diffraction) analysis was performed on the corroded surface and it confirmed the Cu2O passive layer and the occurrence of Cu5Zr precipitate. The occurrence of Cu5Zr precipitates and Cu2O passive layer formation helped attain the maximum polarization resistance of 2033.8 ohm and minimum current density of 5.928 × 10−6 A cm−2 with minimum surface roughness of 3.447 μm.

Advancing Oil Production Efficiency with Next-Generation Nano emulsion Capsules: Harnessing Doped MgZnO2 for Superior Inhibition of Calcite Scale Formation

Abstract This work presents doped MgZnO2 nano emulsion capsules as a novel approach for reducing calcite scale formation in oil production. The nano emulsion capsules are synthesized via ultrasonication and precipitation methods, with morphology and size distribution analysed by scanning electron microscopy (SEM), transmission electron microscope [TEM] and dynamic light scattering (DLS).` with an average particle size of 50 nm. Static scale inhibition tests showed over 90% inhibition efficiency at a Nano emulsion concentration of 100 mg/L. Rigorous static and dynamic scale inhibition testing precipitation and showcases remarkable efficiency exceeding 80% in preventing calcite deposition under simulated conditions. Gravimetric validates cation and online pressure monitoring validate superior performance compared to conventional inhibitors, with up to 30% enhancement in mitigation. The controlled release properties of the Nano emulsion capsule allow for persistent scale inhibition even after 72 hours of continuous operation. Furthermore, ecotoxicity evaluations through bioassays affirm the eco-friendly characteristics of the Nano emulsion technology. As a whole, this research shows how highly promising doped MgZnO2 Nano emulsion capsules are as a long-term, high-performing solution to scale-related problems in the oil sector.

Pulsed TIG wire arc additive manufactured 17-4 PH stainless steel: Effect of Cu-rich precipitates and δ-ferrite on the mechanical properties

Thin-wall samples of 17-4 PH stainless steel were prepared by pulsed TIG wire arc additive manufacturing technology (P-TIG WAAM), and the microstructure and mechanical properties of the samples in different regions (bottom, middle, top) were studied. The results show that the microstructure of the thin-wall samples are mainly consisted of martensite matrix embedded by worm-like (bottom) and strip-shaped (top) δ-ferrite, with a small amount of austenite. The δ-ferrite content gradually decreases along the building direction, resulting in a gradual increase in microhardness from the bottom to the top. Along the building direction, the precipitation density and size distribution of nano-scale CRPs (Cu-rich precipitates) in δ-ferrite and martensite, as well as at the δ-ferrite/martensite interface are significantly different: the CRPs in δ-ferrite is precipitated with lower density, larger size, and inhomogeneous distribution at the bottom region, while that in martensite is precipitated with higher density, smaller size, and homogeneous distribution. At the bottom region, a chain of large size fcc-CRPs (∼65 nm) is precipitated along the δ-ferrite/martensite interface to coordinate the strain between them and improve the strength and ductility simultaneously, thus excellent mechanical properties (tensile strength = 1356 ± 24 MPa, elongation = 15.16 ± 2.6 %) are obtained. The relationship between the evolution of CRPs and the intrinsic heat treatment (IHT) during P-TIG WAAM is also discussed.

Strengthening mechanisms for microstructures containing unimodal and bimodal γ′ precipitates in ATI 718Plus

The influence of γ′ precipitate size distribution on the deformation mechanisms under tensile loading in ATI 718Plus was studied. A set of aging treatments within the temperature range of 720 °C–900 °C was performed on solution-treated samples to obtain various γ′ precipitate size distributions. Unimodal and bimodal γ′ precipitate size distributions were achieved through single-step and two-step aging sequences, respectively, and such microstructures were tensile tested to failure to assess their yield strength, ultimate tensile strength, and elongation-to-failure. Some of the tensile samples were interrupted after achieving 3–4 % plastic strain, and the deformed microstructures were examined using transmission electron microscopy to investigate the γ′ precipitate-dislocation interactions. For the unimodal γ′ precipitate size distribution samples with the smaller γ′ precipitates (radius ∼ 7 nm), dislocations sheared through the precipitates. Both dislocation loops and paired dislocations were observed for the microstructures containing larger γ′ precipitates (radius ∼ 24 nm). The microstructure containing a bimodal γ′ precipitate size distribution, which included average γ′ precipitate radii of ∼6 nm and ∼28 nm, exhibited shearing as the dominant deformation mechanism, and this microstructure exhibited the highest strength values. The experimental observations were rationalized based on the theoretically-calculated critical resolved shear stress values for shearing and looping and a modified model for predicting the yield strength for bimodal microstructures was introduced.

Anisotropic growth of nano‐precipitates governed by preferred orientation and residual stress in an Al‐Zn‐Mg‐Cu alloy

Through an understanding of diffusion, precise control of the size distribution of nano-precipitates can be essential to developing superior properties in precipitation-strengthened alloys. Although a significant influence of crystallographic orientation on the diffusion process is known to exist in low-symmetry hexagonal close-packed alloys, such anisotropic diffusion is still unidentified in high-symmetry cubic alloys. In this work, we reveal the diffusion-controlled coarsening induced anisotropic growth process of nano-precipitates in an Al-Zn-Mg-Cu alloy. Our experimental and theoretical studies demonstrate that with an increase in the residual stress, the diffusion-controlled coarsening rate is slow along the 〈112〉 fiber texture in the alloy matrix with smaller grain sizes. As such, we find that the diffusion activation energy will be increased along the preferred orientation with largest residual stress, which leads to a reduced diffusion-controlled coarsening rate. Specifically, we demonstrate that the increase in the volume fraction of nano-precipitates originates from the rapid grain-boundary controlled coarsening of the grain-boundary precipitates. Based on these results, an underlying microstructural design strategy is proposed, involving the crystallographic orientation, the residual stress and the grain boundaries to manipulate the precipitate size distribution in this class of alloys.

Evaluation of GPM DPR Rain Parameters with North Taiwan Disdrometers

Abstract Global precipitation demonstrates a substantial role in the hydrological cycle and offers tremendous implications in hydrometeorological studies. Advanced remote sensing instrumentations, such as the NASA Global Precipitation Measurement (GPM) mission Dual-Frequency Precipitation Radar (DPR), can estimate precipitation and cloud properties and have a unique capability to estimate the raindrop size information globally at snapshots in time. The present study validates the Level-2 data products of the GPM DPR with the long-term measurements of seven north Taiwan Joss–Waldvogel disdrometers from 2014 to 2022. The precipitation and drop size distribution parameters like rainfall rate (R; mm h−1), radar reflectivity factor (dBZ), mass-weighted mean drop diameter (Dm; mm), and normalized intercept parameter (Nw; m−3 mm−1) of the GPM DPR are compared with the disdrometers. Four different comparison approaches (point match, 5-km average, 10-km average, and optimal method) are used for the validation; among these four, the optimal strategy provided reasonable agreement between the GPM DPR and the disdrometers. The GPM DPR revealed superior performance in estimating the rain parameters in stratiform precipitation than the convective precipitation. Irrespective of algorithm type (dual- or single-frequency algorithm), sensitivity analysis revealed superior agreement for the mass-weighted mean diameter and inferior agreement for the normalized intercept parameter.

Influence of Mo Content on the Precipitation Behavior of 13Ni Maraging Ultra-High Strength Steels

This study offers valuable insights into the precipitation behavior of 13Ni maraging steels, emphasizing the role of molybdenum content in their microstructure, strengthening, and precipitate evolution. Precipitate morphology and crystallography were examined using a combination of high-resolution transmission electron microscopy and selected area electron diffraction. Strengthening mechanisms were assessed through Vickers hardness measurements. All the examined samples exhibited a lath martensite microstructure and displayed an increasing hardness over the aging time. The molybdenum content not only influenced the presence of retained austenite in the initial microstructure but also affected the type of precipitates formed during the early aging stages. Initially, Ni3Mo precipitates were formed, succeeded by the formation of more stable Fe2(Mo,Ti) Laves precipitates. The ultra-high strength of 13Ni maraging steels arises from the combination of the precipitate type and size distribution. The base composition of 13Ni maraging steels achieved a peak hardness of 798 HV1 through the precipitation of Laves Fe2(Mo,Ti) phases ranging from 3 to 14 nm in diameter.

Influence of temperature on the precipitation phase transformation behavior of RR350 aluminum alloy during artificial aging and related mechanisms

This study investigates the influence of artificial aging temperature on the evolution of precipitation phases and age hardening behavior of the RR350 aluminum alloy. The manner of nucleation and growth of different kinds of precipitates and phase transformation between them are analyzed, along with the corresponding mechanisms are discussed. The results reveal that the aging temperature affects little the precipitation process of precipitates which still follows the sequence of GP zones - θ″- θ′ in the RR350 aluminum alloy during artificial aging, but influences significantly the precipitation speed and phase composition, morphology, size and size distribution of the resultant precipitates in the alloy. An imperceptible relationship between the hardness of the sample and the composition of the precipitates were discovered. As the artificial aging temperature (160–180–200 °C) increases, the peak-aging hardness of the samples decreases and the time for reaching the peak-aging status shortens gradually. When the temperature is low, 160 °C for example, the precipitates in the peak-aging status are mainly composed of θ″ phase and some minor θ′ phase. As the temperature increases to 200 °C θ′ phase becomes prevalent. As a result, the alloy shows lower peak-aging hardness since the strengthening effect of the finer θ″ phase is much stronger than that of the θ′ phase. Increasing the aging temperature accelerates the competitive processes of nucleation of GP zones, growth of all precipitates, and the phase transformation. At higher temperatures, phase transformation from θ″ to θ′ overweighs the formation of GP zones, growth of both θ″ and θ′ phases, resulting in more amount of θ′phase in the peak-aging and over-aging statuses.

Regional discrepancies in the microphysical attributes of summer season rainfall over Taiwan using GPM DPR

Taiwan, an island located in the northwest Pacific region, is influenced by heavy rainfall events during warm seasons, particularly from June to August. Interaction of precipitating clouds with the complex topography results in inhomogeneous and intense rainfall over Taiwan. Hence, the present study investigates the regional discrepancies in the microphysical characteristics of summer season rainfall over (north, south, east, and central) Taiwan using 9 years (2014–2022) of GPM DPR measurements. The results showed clear distinctions in the precipitation and raindrop size distributions over the north, south, east, and central Taiwan. The contoured frequency by altitude diagrams (CFADs) of radar reflectivity, rainfall rate, drop diameter, and concentration clearly infer the dominance of large-size super cooled liquid and ice particles above the melting layer and rain particles below the melting layers in the south and central Taiwan. Central (north) Taiwan is dominated by large-size (small) drops among four regions. Higher concentrations of large drops over central Taiwan (principally from convective precipitation) and south Taiwan (primarily from stratiform precipitation) is attributed to higher rainfall amounts over these two regions than the north and east Taiwan. Furthermore, irrespective of precipitation type and geographic region, summer monsoon rainfall over Taiwan is dominated by coalescence and breakup processes. The microphysical characteristics of summer season rainfall addressed in this study could assist in refining the cloud modeling simulations over complex topography in Taiwan.

Uncovering the kinetics of Li-rich clusters and monodisperse core–shell Al3(Zr, Sc) structures in Al–Li–Cu alloys

The size distribution and structural evolution of the precipitates are critical to the mechanical properties of heat treatable alloys. In Al alloys, many studies have reported that monodisperse core–shell Al3Li/Al3(Zr, Sc) structures in a Li-rich environment promote formation of δ′-Al3Li, T1-Al2CuLi, and θ′-Al2Cu precipitates. However, without quantitative measurements of the precipitate size distribution and tracking their crystal structure evolution, the kinetics of Li-rich clusters and excessive Li diffusion into monodisperse core–shell Al3(Zr, Sc) structures in Al alloys to prevent Li-cluster coarsening is still unknown. In this work, in-situ small angle neutron scattering (SANS) has been adopted to track the evolution of δ′-Al3Li precipitates. Upon quenching, Li-rich clusters remain at a smaller size and higher number density. Once aging starts, nucleation of δ′-Al3Li precipitates on top of Al3(Zr, Sc) is at the expense of dissolution of those Li-rich clusters below the critical size. From density functional theory (DFT) calculations, it has been found the δ′-Al3Li precipitate nucleation barrier on top of Al3(Zr, Sc) is negligible, as well as the nucleation barrier of θ′-Al2Cu on top of Al3(Zr, Sc) is much lower relative to that on α-Al. Besides, the substitutional energy for Li to replace Zr and Sc sites in Al3(Zr, Sc) is favorable. Combining SANS with high-resolution transmission electron microscopy (HRTEM), the effects of Al3(Zr, Sc) cores on preventing δ′-Al3Li size from coarsening by absorbing Li to form Al3(Li, Zr, Sc) have been uncovered. Simultaneously, the contributions of Al3(Li, Sc, Zr) particles and δ′-Al3Li to the strength has been quantified. This work opens a new engineering approach of precise control of precipitating kinetics at the presence of monodisperse precursors for industrially relevant structural materials.

Influence of scandium addition on microstructure and mechanical properties in nickel-based superalloys: An integrated multiscale modeling and experimental approach

Microalloying has long been recognized as an effective method for improving the mechanical properties of metallic materials and developing new ones. In this study, we comprehensively investigate the effects of Sc addition on the microstructural evolution and mechanical performance of nickel-based superalloys using a combination approach of the experiment, first-principles calculation, and multiscale modeling. The results show that Sc addition simultaneously improves the yield strength increasing from 1068 MPa to 1172 MPa (9.7%) and elongation increasing from 13.4% to 19.7% (47%). The microstructural characterizations reveal that Sc addition refines the grain size, increases the volume fraction of primary precipitate, induces the coarsening of secondary precipitates, and produces a new type of Sc-rich particle. First-principles calculations from the atom probe tomography experiment show that the anti-phase boundary energy of secondary precipitate is significantly increased in Sc-containing superalloys. A modified alloying-dependent yield strength model is developed by considering the multimodal precipitate size distribution and microstructure evolution obtained from experiments and atomic simulations. The increased yield strength originates from not only the Sc addition induced increased volume fraction of primary precipitate at grain boundary to enhance grain boundary strengthening, but also Sc addition induced increased anti-phase boundary energy of secondary precipitate to improve the precipitate strengthening. The increased elongation depends on the fact that the Sc element has a good affinity to deleterious elements and forms Sc-rich particle and oxygen-rich precipitate to enhance grain boundary strength. This work integrating multiscale modeling and experiment can be denoted as a universal framework to study the influence of alloy element on the microstructure and properties, which gives guiding significance to screen and design high-performance superalloys.

Modeling the effect of precipitation spatial geometry and size distribution on the yield strength of aluminum alloys

Modeling the effect of precipitation spatial geometry and size distribution on the yield strength of aluminum alloys

Effect of Ti additions on microstructure and mechanical properties of Cu–Cr–Zr alloy

Ti additions in Cu–Cr–Zr alloys are useful for achieving high mechanical properties. In this work, the influence of Ti contents (0.25 wt%, 0.6 wt%, and 1.02 wt%) on the microstructure, mechanical, and electrical properties of Cu–Cr–Zr alloys has been investigated experimentally, along with thermodynamic and kinetic calculations. The electrical conductivity decreased but the hardness/strength increased with increasing Ti content. The lower electrical conductivity is due to increased electron scattering through the solution of more Ti atoms in the Cu matrix. As for the higher hardness/strength, it is mainly owing to higher dislocation density and finer FCC-Cr precipitates. Furthermore, a model considering the size distributions of precipitates is adopted to calculate precipitation strengthening quantitatively. The calculated yield strengths are consistent with the experimental ones for the alloys. The thermodynamic and kinetic calculations reveal that increasing Ti content can facilitate the nucleation of FCC-Cr but enhance its activation energy, hence hindering the growth process. The present work study can provide an effective strategy for producing copper alloys with expected performance.

Enhanced intragranular dislocation capture ability in nano/ultrafine Al-Zn-Mg-Cu alloy by burnishing-assisted two-stage aging treatment-induced dynamic nucleation

The effects of burnishing-assisted two-stage aging (BATSA), including pre-aging, burnishing, and subsequent aging (SA) on the microstructural evolution, precipitation, and hardening behavior of Al-Zn-Mg-Cu alloy were systematically investigated. Transmission electron microscopy, X-ray diffraction, small-angle X-ray scattering, and Vickers microhardness tests were used to analyze the above-mentioned characteristics of the burnished-layer alloys and the matrix-layer alloys, respectively. The BATSA treatment promotes a concentrated distribution of precipitate sizes, delaying phase coalescence and maintaining a relatively dispersed precipitate distribution. Moreover, the persistent nucleation of Guinier-Preston (GP) zones in burnished alloys was observed during SA. These results are conducive to the enhancement of the dislocation storage capacity of nano/ultra fine-grained alloys (the decrease in dislocation density of pre-aged burnished alloy was about 65% that of non-pre-aged burnished alloy). Based on this, dynamic nucleation induced by pre-aging and burnishing resulted in a better synergy of multiple strengthening mechanisms, significantly shortening the time to peak hardness in pre-aged burnished alloys to about 1/3 that in non-pre-aged matrix alloys, and maintaining hardness stability. Moreover, it was found that the critical width of the in situ phase transformation from GPII zone to η' precipitate can be as low as 3–4 at. planes.

Characteristics of Raindrop Size Distributions in the Southwest Mountain Areas of China According to Seasonal Variation and Rain Types

The precipitation and raindrop size distribution (RSD) characteristics of the four seasons and different rain types were studied using a PARSIVEL2 raindrop disdrometer set in the southwest mountain areas of China from 2019 to 2021. The seasonal precipitation in the southwest mountain areas was mainly stratiform rain. The peaks of the RSD were about 1–2 orders of magnitude higher than those in the plains. The convective rain in spring and autumn was very close to the ocean-like convective mass. The local shape–slope (μ–Λ), radar reflectivity–rain rate (Z–R), and kinetic energy–rain rate (KE–R) relationships were further derived, and the diversity of these relationships was mainly due to the variability of the RSDs. In addition, the differences in the RSD characteristics between the top and the foot of the mountain during a typical precipitation process in the summer of 2020 were further compared. It was found that the number density of the small particles at the top of the mountain was higher than that at the foot of the mountain due to the broken large raindrops caused by the high wind speed, while the high evaporation rate, strong convective available potential energy (CPAE), and water vapor content at the foot of the mountain could strengthen the RSD, making the number density of the large raindrops at the foot of the mountain higher than that at the top.

Accurate identification and measurement of the precipitate area by two-stage deep neural networks in novel chromium-based alloys.

The performance of advanced materials for extreme environments is underpinned by their microstructure, such as the size and distribution of nano- to micro-sized reinforcing phase(s). Chromium-based superalloys are a recently proposed alternative to conventional face-centred-cubic superalloys for high-temperature applications, e.g., Concentrated Solar Power. Their development requires the determination of precipitate volume fraction and size distribution using Electron Microscopy (EM), as these properties are crucial for the thermal stability and mechanical properties of chromium superalloys. Traditional approaches to EM image processing utilise filtering with a fixed contrast threshold, leads to weak robustness to background noise and poor generalisability to different materials. It also requires an enormous amount of time for manual object measurements on large datasets. Efficient and accurate object detection and segmentation are therefore highly desired to accelerate the development of novel materials like chromium-based superalloys. To address these bottlenecks, based on YOLOv5 and SegFormer structures, this study proposes an end-to-end, two-stage deep learning scheme, DT-SegNet, to perform object detection and segmentation for EM images. The proposed approach can thus benefit from the training efficiency of CNNs at the detection stage (i.e., a small number of training images required) and the accuracy of the ViT at the segmentation stage. Extensive numerical experiments demonstrate that the proposed DT-SegNet significantly outperforms the state-of-the-art segmentation tools offered by Weka and ilastik regarding a large number of metrics, including accuracy, precision, recall and F1-score. This model forms a useful tool to aid alloy development microstructure examinations, and offers significant advantages to address the large datasets associated with high-throughput alloy development approaches.

Analysis of the Microstructure Development of Nb-Microalloyed Steel during Rolling on a Heavy-Section Mill.

It is not realistic to optimize the roll pass design of profile rolling mills, which typically roll hundreds of profiles, using physical modelling or operational rolling. The use of reliable models of microstructure evolution is preferable here. Based on the mathematical equations describing the microstructure evolution during hot rolling, a modified microstructure evolution model was presented that better accounts for the influence of strain-induced precipitation (SIP) on the kinetics of static recrystallization. The time required for half of the structure to soften, t0.5, by static recrystallization was calculated separately for both situations in which strain-induced precipitation occurred or did not occur. On this basis, the resulting model was more sensitive to the description of grain coarsening in the high-rolling-temperature region, which is a consequence of the rapid progress of static recrystallization and the larger interpass times during rolling on cross-country and continuous mills. The modified model was verified using a plain strain compression test (PSCT) simulation of rolling a 100-mm-diameter round bar performed on the Hydrawedge II hot deformation simulator (HDS-20). Four variants of simulations were performed, differing in the rolling temperature in the last four passes. For comparison with the outputs of the modified model, an analysis of the austenite grain size after rolling was performed using optical metallography. For indirect comparison with the model outputs, the SIP initiation time was determined based on the NbX precipitate size distribution obtained by TEM. Using the PSCT and the outputs from the modified microstructure evolution model, it was found that during conventional rolling, strain-induced precipitation occurs after the last pass and thus does not affect the austenite grain size. By lowering the rolling temperature, it was possible to reduce the grain size by up to 56 μm, while increasing the mean flow stress by a maximum of 74%. The resulting grain size for all four modes was consistent with the operating results.

Bayesian optimization of discrete dislocation plasticity of two-dimensional precipitation-hardened crystals

Discovering relationships between materials' microstructures and mechanical properties is a key goal of materials science. Here, we outline a strategy exploiting Bayesian optimization to efficiently search the multidimensional space of microstructures, defined here by the size distribution of precipitates (fixed impurities or inclusions acting as obstacles for dislocation motion) within a simple two-dimensional discrete dislocation dynamics model. The aim is to design a microstructure optimizing a given mechanical property, e.g., maximizing the expected value of shear stress for a given strain. The problem of finding the optimal discretized shape for a distribution involves a norm constraint, and we find that sampling the space of possible solutions should be done in a specific way in order to avoid convergence problems. To this end, we propose a general mathematical approach that can be used to generate trial solutions uniformly at random while enforcing an Euclidean norm constraint. Both equality and inequality constraints are considered. A simple technique can then be used to convert between Euclidean and other Lebesgue $p$-norm (the 1-norm in particular) constrained representations. Considering different dislocation-precipitate interaction potentials, we demonstrate the convergence of the algorithm to the optimal solution and discuss its possible extensions to the more complex and realistic case of three-dimensional dislocation systems where also the optimization of precipitate shapes could be considered.

Thermodynamic modeling and computational predictions of NbC precipitation in Fe–Mn–Si-based shape memory alloys by the classical nucleation and growth theories

NbC precipitation in Fe–Mn–Si-based alloys is an effective method to improve the shape memory effect. In this study, the precipitation behavior was investigated using a thermodynamical model to understand the mechanism and optimize the precipitates for a better performance of Fe–Mn–Si-based shape memory alloys. The influence of alloying elements can be considered in the model by introducing interaction parameters. The precipitate size distribution, mean size, precipitate volume faction, and number density of three typical Fe–Mn–Si-based alloys with different NbC addition amounts were calculated. The results indicated that the mean size could be decreased significantly as the NbC addition increased from 0.5% to 1.0%, while the precipitate volume fraction and number density showed obvious increments. The Fe–28Mn–6Si–5Cr alloys exhibited smaller mean sizes and higher number densities than the Fe–14Mn–6Si–9Cr–5Ni and Fe–21Mn–6Si–9Cr–5Ni alloys. It was also found that the precipitate size distribution showed no evident change as the aging time increased from 0.5 h to 2 h except for the Fe–28Mn–6Si–5Cr–0.5NbC alloy in which the precipitates began to coarsen after about 1.25 h.