27 research outputs found

    An Exact Solution of the Binary Singular Problem

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    Singularity problem exists in various branches of applied mathematics. Such ordinary differential equations accompany singular coefficients. In this paper, by using the properties of reproducing kernel, the exact solution expressions of dual singular problem are given in the reproducing kernel space and studied, also for a class of singular problem. For the binary equation of singular points, I put it into the singular problem first, and then reuse some excellent properties which are applied to solve the method of solving differential equations for its exact solution expression of binary singular integral equation in reproducing kernel space, and then obtain its approximate solution through the evaluation of exact solutions. Numerical examples will show the effectiveness of this method

    An Exact Solution of the Binary Singular Problem

    No full text
    Singularity problem exists in various branches of applied mathematics. Such ordinary differential equations accompany singular coefficients. In this paper, by using the properties of reproducing kernel, the exact solution expressions of dual singular problem are given in the reproducing kernel space and studied, also for a class of singular problem. For the binary equation of singular points, I put it into the singular problem first, and then reuse some excellent properties which are applied to solve the method of solving differential equations for its exact solution expression of binary singular integral equation in reproducing kernel space, and then obtain its approximate solution through the evaluation of exact solutions. Numerical examples will show the effectiveness of this method

    Analysis of microstructures and XRD of a gradient boron alloyed composite material

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    A new type of gradient boron alloyed composite material, containing boron alloyed core layers and stainless steel coatings around the core, were designed and prepared by composite casting and hot rolling. The evolution of microstructures, phases and precipitations, as well as their influence on hot rolling process and performance are investigated. A mixture of austenitic matrix and uniformly distributed borides are obtained in the hot rolled stainless steel with 2-2.5 % boron, while massive borides are in the length of 80-120 μm together with micro gaps at the interface between the borides, and the matrix is remained after hot rolling for the core layers with higher boron contents. Hot deformation would be hindered since more precipitations of these orthorhombic or tetragonal phases occur with an increase of the boron concentration in the core layers. © (2013) Trans Tech Publications, Switzerland

    Effect of Zn addition on microstructure and mechanical properties of an Al–Mg–Si alloy

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    In the present work, an Al–0.66Mg–0.85Si–0.2Cu alloy with Zn addition was investigated by electron back scattering diffraction (EBSD), high resolution electron microscopy (HREM), tensile and Erichsen tests. The mechanical properties of the alloy after pre-aging met the standards of sheet forming. After paint baking, the yield strength of the alloy was improved apparently. GP(II) zones and ηʹ phases were formed during aging process due to Zn addition. With the precipitation of GP zones, β″ phases, GP(II) zones and ηʹ phases, the alloys displayed excellent mechanical properties

    Macro-micro dynamic behaviors and fracture modes of roll bonded 7A52/7A01/7B52 aluminum laminates in high velocity deformation

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    Damage tolerance improvement in the lamination of aluminum alloy plates and composites has been reported in many studies. In the present study the macro-micro dynamic deformation behavior and related mechanisms of 7A52/7A01/7B52 laminated plates processed by hot roll bonding of 7A52, 7A01 and 7B52 plate at high strain rates and quasi-static compression has been investigated. The microstructure of the laminated plates was examined with backscatter diffraction (EBSD) techniques, scanning and transmission electron microscopies. The results showed that with increased strain rate, obvious strain rate hardening was observed in the single layer specimens. The peak flow stress of the multilayer samples was slightly higher than that of the 7A52 monolayer samples and much lower than that of the 7B52 monolayer samples at the same strain rate. Beyond the peak stress state, the strain hardening was replaced by thermal softening in the 7A52 layer, leading to low resistance of deformation and high tendency to facilitate deformation-induced adiabatic shear bands (ASBs) that consist of dynamic recrystallized grains. ASBs in laminated samples were deflected and bifurcated at the interface of 7A52 layer. In addition, bifurcated ASBs converged at the interface between the 7A52 and 7A01 layers. The high deformation resistance observed in the laminate under dynamic loading was a consequence of the high capacity for strain hardening in the 7A01 layer. This hardening effectively overcame the influences of thermal softening and dynamic recovery during dynamic loading. This study provides an understanding of the laminate's microstructure evolution during dynamic deformation and its relevance to the fracture modes of a multilayer structure under dynamic loading. Keywords: Laminated plate, Fracture mode, Microstructure, Aluminum alloy, Mechanical propertie

    Influence of Zn Addition on the Aging Precipitate Behavior and Mechanical Properties of Al-Cu-Li Alloy

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    In the present work, the effect of Zn on the aging precipitates and mechanical properties of Al-Cu-Li alloys was investigated by Vickers hardness, tensile tests, transmission electron microscopy (TEM) and differential scanning calorimetry (DSC). The results indicated that the addition of Zn reduced the activation energy of the T1 phase and makes it easier to precipitate. The activation energy of the T1 phase, which was 107.02 ± 1.8 KJ/mol, 94.33 ± 1.7 KJ/mol, 90.33 ± 1.7 KJ/mol and 90.28 ± 1.6 KJ/mol for 0Zn, 0.4Zn, 0.8Zn and 1.2Zn alloy, respectively. The area number density of the T1 precipitate ranged from 97.0 ± 4.4 pcs/μm2 to 118.2 ± 2.8 pcs/μm2 as the Zn content increased from 0 to 1.2 wt.%. Consequently, the addition of Zn promoted the precipitation of the T1 phase. Therefore, the peak hardness and tensile strength of the alloy also increased with the increase in the Zn content, and the hardness of the alloy with Zn content of 1.2 wt.% increased by 16.5 ± 1.4 HV; meanwhile, the ultimate tensile strength increased by 46.5 ± 2.5 MPa. Therefore, the area number density of precipitates increased and improved the strength of the Zn-containing alloy

    Influence of enhanced Li content on the as-cast eutectic phase features and the evolution during homogenization of Al-Cu-Li alloys

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    The microstructure of Al-Cu-Li alloys is influenced by the alloying contents, which in turn affect the ultimate performance. This study examined the microstructure characteristics of Al-Cu-Li alloys with varying Li contents (1.05, 1.30, and 1.66 wt%) in as-cast state, as well as microstructure evolution during homogenization treatments. The investigation employed a range of analytical techniques, including optical microscopy (OM), scanning electron microscopy (SEM), energy dispersive spectrometry (EDS), X-ray diffraction (XRD), differential scanning calorimeter (DSC), focused ion beam (FIB), transmission electron microscopy (TEM), and thermodynamic calculation software. The findings indicate that the eutectic phase exhibited significant variations with the Li content changing. Three distinct low-melting phases were identified in 1.05Li alloy, including Ag-containing Al2CuMg, Al7Cu4Li, and Al2Cu. As Li content was elevated to 1.30 wt%, there was a noticeable drop in the quantities of Ag-containing Al2CuMg and Al7Cu4Li phases. Conversely, the Al2Cu phase increased, and a few Al2CuLi phases emerged. When the Li raised to 1.66 wt%, an increased Al2CuLi phase was discovered, which existed along the dendritic edges. However, there were only little Ag-containing Al2CuMg phase, and no Al7Cu4Li phase detected. The initial melting temperatures for the four soluble eutectic phases were systematically arranged in the following order: Ag-containing Al2CuMg < Al7Cu4Li < Al2Cu < Al2CuLi. The elevated Li content resulted in a challenge of dissolution of the eutectic phases. Following 500 °C/12h homogenization process, the containing-Ag Al2CuMg phase dissolved. Additionally, the Al7Cu4Li and Al2Cu phases were eliminated through subsequent 515 °C/12h homogenization. In addition, the Al2CuLi phase was still existing

    Machine learning assisted design of aluminum-lithium alloy with high specific modulus and specific strength

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    Advanced aluminum-lithium alloys are the key structural materials urgently needed for the development of light-weighted aircraft in the aerospace field. In this study, we employ a machine learning approach accompanied by domain knowledge to realize the accelerated design of aluminum-lithium alloy with high specific modulus and specific strength by identifying an optimal combination of key features through a three-step feature filtering of datasets containing 145 alloys. The maximum specific modulus in the experimental alloys that screened from the predicted results increases by 4 % compared with the maximum specific modulus in the comparative dataset. The specific modulus of the designed alloy with the best comprehensive performance increased by 12.6 % compared with the widely used 2195-T8 alloy while maintaining a similar specific strength. Machine learning shows appealing feasibility and reliability in the field of materials design

    Pre-aging on early-age behavior and bake hardening response of an Al-0.90Mg-0.80Si-0.64Zn-0.23Cu alloy

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    Pre-aging on early-age behavior and bake hardening response of an Al-0.90Mg-0.80Si-0.64Zn alloy was investigated by differential scanning calorimetry (DSC), high resolution transmission electron microscopy (HRTEM), 3-dimensional atom probe (3DAP), Erichsen test and tensile test. The results indicated that pre-aged alloy exhibited excellent formability and bake-hardening response, while bake hardening response was poor in samples with natural aging. Clustering behavior during natural aging was inhibited by pre-aging. Numerous GP zones formed in pre-aged samples. GP zones were the nuclei of β′′ precipitates or directly transformed β′′ phases during paint baking process. A large number of β′′ phases were observed in pre-aged samples after paint bake treatment. There was no sign to indicate that β′′ phase precipitated in natural aged samples after bake hardening treatment

    Machine Learning Phase Prediction of Light-Weight High-Entropy Alloys Containing Aluminum, Magnesium, and Lithium

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    With the development of society, there is an increasingly urgent demand for light-weight, high-strength, and high-temperature-resistant structural materials. High-entropy alloys (HEAs) owe much of their unusual properties to the selection among three phases: solid solution (SS), intermetallic compound (IM), and mixed SS and IM (SS and IM). Therefore, accurate phase prediction is crucial for guiding the selection of element combinations to form HEAs with desired properties. Light high-entropy alloys (LHEAs), as a significant branch of HEAs, exhibit excellent performance in terms of specific strength. In this study, we employ a machine learning (ML) method to realize the design of light-weight high-entropy alloys based on solid solutions. We determined the Gradient Boosting Classifier model as the best machine learning model through a two-step feature and model selection, in which its accuracy and F1_Score achieve 0.9166 and 0.8923. According to the predicted results, we obtained Al28Li35Mg15Zn10Cu12 LHEAs, which are mainly composed of 90% solid solution. This alloy accords with the prediction results of machine learning. But it is made up of a two-phase solid solution. In order to obtain a light-weight high-entropy alloy dominated by a single solid solution, we designed Al24Li15Mg26Zn9Cu26 LHEAs on the basis of machine learning prediction results accompanied by expert experience. Its main structure includes a single-phase solid solution. Our work provides an alternative approach to the computational design of HEAs and provides a direction for future exploration of light-weight high-entropy alloys
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