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An important feature of the time series in the real world is that its distribution has different degrees of asymmetry, which is what we call irreversibility. In this paper, we propose a new method named permutation pattern (PP) to calculate the Kullback–Leibler divergence ( $${D}_{\mathrm{KL}}$$ ) and the Jensen–Shannon divergence ( $${D}_{\mathrm{JS}}$$ ) to explore the irreversibility of time series. Meanwhile, we improve $${D}_{\mathrm{JS}}$$ and obtain a complete mean divergence ( $${D}_{\mathrm{m}}$$ ) through averaging $${D}_{\mathrm{KL}}$$ of a time series and its inverse time series. The variation trend of $${D}_{\mathrm{m}}$$ is similar to $${D}_{\mathrm{JS}}$$ , but the value of $${D}_{\mathrm{m}}$$ is slightly larger and the description of irreversibility is more intuitive. Furthermore, we compare $${D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ calculated by PP with those calculated by the horizontal visibility graph, and discuss their respective characteristics. Then, we investigate the advantages of $$\mathrm{D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ through length variation, dynamic time variation, multiscale and so on. It is worth mentioning that we introduce Score and variance to analyze the practical significance of stock irreversibility.

A bimodal microstructure composed of large primary α-Al phases (PPs) and small secondarily solidified grains (SSGs) is fabricated in 6061 aluminum alloy by powder thixoforming (PTF) method after solution treatment. Artificial aging treatment introduces nanosized precipitates into the bimodal microstructure, in which the number and growth rate of precipitates are different: the number in PPs is larger than that in SSGs due to the higher density of dislocations and lattice distortions in PPs, while the growth rate in SSGs is higher than that in PPs owing to the higher solute concentration in SSGs. As the number and size of precipitates are optimized in the bimodal microstructure, i.e., in peak-aged state, remarkable increments of 56.1% and 130.3% in ultimate tensile strength and yield strength while a moderate increase of 6.3% in ductility are achieved compared with the as-fabricated state, which overcomes the trade-off in strength and ductility of conventional aging-treated 6061 alloys with unimodal microstructure. The strengthening comes from the geometrically necessary dislocations induced by the deformation incompatibility of bimodal microstructure and from precipitate strengthening by nanosized precipitates, while the toughening is from the enhanced dislocation accumulation in this alloy.

A microstructure-based numerical simulation is performed to understand the mechanical properties and fracture of a Ti-Al3Ti core-shell structured particulate reinforced A356 composite ((Ti-Al3Ti)cs/A356). A series of two-dimensional (2D) representative volume element (RVE) models are generated automatically by embedding Ti-Al3Ti core-shell structured particulates in an A356 matrix. Microstructure-based 2D RVE of monolithic Al3Ti particulate reinforced A356 composite (Al3Tip/A356) is also simulated for comparison. The ductile fracture of both Ti core and A356 matrix as well as the brittle fracture of the Al3Ti shell are considered. The simulation confirms that the high elongation of the (Ti-Al3Ti)cs/A356 composite is attributed to the uniform distribution of the overall ductile globular reinforcing particulates, which prevent a premature failure effectively by reducing local stress concentration both on and inside the core-shell structured particulates. The surrounding ductile phases of the Al3Ti shell blunt the crack tips effectively and, therefore, restricting the propagation of the cracks in a nominal strain range of 2.2%–6.1%. For both (Ti-Al3Ti)cs/A356 and Al3Tip/A356 composites, the simulation results are in good agreement with microstructural observations during an in-situ tensile test in a scanning electron microscope.

The powder thixoforming method was used to fabricate 10 vol% silicon carbide particle (SiCp) reinforced 6061 Al matrix composites with high mechanical performances successfully. Here, we demonstrated with proof that proper solution treatment could not only enhance tensile strength of the composite: its ultimate tensile strength and yield strength increased from 230 to 128 MPa in the as-fabricated state to 275 and 212 MPa solutionized at 808 K for 6 h but also improve composite’s tensile elongation significantly with an increment of 161.5% from 2.6% to 6.8%. Corresponding toughening mechanisms are mainly investigated from the perspective of both microstructure examination and total strain to failure calculation through a modified model. The theoretical predictions are in reasonably good agreement with the experimental data. This work may provide a practical way to alleviate the inverse strength–ductility relationship of particulate reinforced metal matrix composites and provide reference for the SF calculation of similar composites subjected to solution treatment.

The trade-off between strength and ductility of particulate reinforced metal matrix composites (PRMMCs) has been a longstanding puzzle. Here we propose an effective strategy to surmount the inverse relationship between strength and ductility of an A356 Al alloy based PRMMC by in situ synthesizing novel reinforcing particulates with a special core-shell (CS) structure. Such structure features a Ti core inside a dual-layer shell: the inner layer has a nano-grained (~130 nm) heterogeneous structure, and the outer layer possesses a composite structure composed of a (Al,Si)3Ti substrate with dense dispersion of nanoparticles. As a result, the obtained composite reinforced with such CS reinforcing particulates (CS composite) achieves an unprecedented tensile elongation to failure of 8.3 ± 0.8% and a uniform elongation of 7.1 ± 0.6%, which nearly triples that of the same alloy based composite reinforced with monolithic (Al,Si)3Ti particulates (monolithic composite) and equivalent to corresponding matrix alloy while maintaining high ultimate tensile strength of 373 ± 8.8 MPa and yield strength of 268 ± 7.9 MPa, equivalent to monolithic composite simultaneously. This special architecture of shell renders itself a high capability of stress bearing and good toughness, and the nanoparticles in outer layer further slower crack development, which significantly postpone crack formation in shell. Subsequent propagation of cracks in Ti core is also constrained remarkably by the transformation-induced plasticity effect occurred ahead of crack tips resulting from stress-induced phase transformation of hcp-Ti into fcc-Ti. These factors lead to highest work hardening rate that undergoes a long plateau and thus overcome the strength-ductility trade-off of A356 alloy based PRMMC.

6061 Al-based composites reinforced with 10% volume fractions of 6.94 μm SiC particles (SiCp) have been fabricated by powder thixoforming, and the effects of solution treatment on their tensile properties and strengthening mechanisms were investigated. The obtained results indicate that the addition of SiCp significantly improved the tensile strength of a 6061 Al matrix alloy, but reduced its ductility. The ultimate tensile and yield strengths of the alloy increased from 180 MPa and 99 MPa to 230 MPa and 128 MPa, respectively, while its elongation decreased from 8.0% to 2.6%. However, the solution treatment at 535 °C for 6 h significantly alleviated ductility losses and further enhanced the composite tensile strength; as a result, its ultimate tensile strength, yield strength, and elongation were increased by 19.6%, 65.2%, and 161.5% up to 275 MPa, 212 MPa, and 6.8%, respectively. Due to the presence of SiCp, the resulting composite solution rate was noticeably decreased compared to that of the matrix alloy. The relationship between the solution time and the composite yield strength was discussed theoretically by using a modified micromechanical model, which took into account the SiCp failure. The obtained theoretical results were in good agreement with the experimental ones.

Permutation entropy (PE) has been proposed to reflect the complexity of time series in recent years, and it has a wide range of applications. In this paper, we extend PE to the fractional order through the parameter α, and then obtain fractional permutation entropy (FPE). FPE can increase the sensitivity of the complex system by adjusting the fractional order. We combine FPE with fractional Jensen–Shannon divergence (FJSD) to construct the entropy plane and discover that the entropy plane can classify various time series. In particular, it can distinguish the stock indices of developing countries and developed countries. We introduce FPE and FJSD to multiscale, and gain multiscale fractional permutation entropy (MFPE) and multiscale fractional Jensen–Shannon divergence (MFJSD). We employ simulated data and financial stock data to explore the relationship of MFPE and MFJSD under different scales, and the results show that some data are similar to direct proportion relationship and others data are sharply increasing curves, which portrays the internal mechanism of the stock market excellently.

A new method, powder thixoforming, has been proposed to fabricate an in situ Al3Tip/2024Al composite. During partial remelting, the microstructural evolution of the bulk alloy prepared by cold pressing of the Ti, Al, 2024Al powder mixture was investigated, and the formation mechanism of the Al3Ti particles produced by the reaction between the Ti powder and the Al alloy melt is also discussed in detail. The results indicate that the microstructural evolution of the 2024 alloy matrix can be divided into three stages: a rapid coarsening of the powder grains; a formation of primary α-Al particles surrounded with a continuous liquid film; and a slight coarsening of the primary α-Al particles. Simultaneously, a reaction layer of Al3Ti can be formed on the Ti powder surface when the bulk is heated for 10 min at 640 °C The thickness (X) of the reaction layer increases with the time according to the parabolic law of \(X = -0.43t^{2} + 4.21t + 0.17\). The stress generated in the reaction layer due to the volume dilatation can be calculated by using the equationσ \(\sigma_{Al_{3}Ti} = -\frac{ E_{Al_{3}Ti} }{6(1-v{Al_{3}Ti})} \frac{ t^{3}_{Al_{3}Ti} }{t_{Ti}} \left(\frac{1}{R} - \frac{1}{R_{0}} \right) \). Comparing the obtained data with the results of the drip experiment, the reaction rate for the Ti powder and Al powder mixture is greater than that for the Ti plate and Al alloy mixture, respectively.

In this work, a novel processing method called powder thixoforming was proposed to prepare composites reinforced with 50 vol % of SiC particles (SiCp) that were used for electronic packaging in order to investigate the effects of remelting duration on its microstructure and properties. Optical Microscope (OM), Scanning Electron Microscope (SEM), X-ray Diffraction (XRD) and Transmission Electron Microscope (TEM) methods were applied for the material characterization and the corresponding physical and mechanical properties were examined in detail. The obtained results indicate that the remelting duration exerted a large effect on the microstructure as well as the SiCp/Al interfacial reaction. The density and hardness of the composite continuously increased with increasing remelting duration. The thermal conductivity (TC) and bending strength (BS) first increased during the initial 90 min and then decreased. The remelting duration exerted a limited influence on the coefficient of thermal expansion (CTE). The optimal TC, BS, and hardness of these composites were up to 135.79 W/(m·K), 348.53 MPa, and 105.23 HV, respectively, and the CTE was less than 6.5 ppm/K after the composites were remelted at 600 °C for 90 min. The properties of the composites could thus be controlled to conform to the application requirements for electronic packaging materials.

Microstructural and mechanical characterization of 10 vol% SiC particles (SiCp) reinforced 6061 Al-based composite fabricated by powder thixoforming (PTF) was investigated in comparison with the PTF and permanent mold cast (PMC) 6061 monolithic alloys. The results reveal that the microstructure of the PMC alloy consists of coarse and equiaxed α dendrites and interdendritic net-like eutectic phases. However, the microstructure of the PTF composite, similar to that of the PTF alloy, consists of near-spheroidal primary particles and intergranular secondarily solidified structures except SiCp, which are distributed in the secondarily solidified structures. The eutectics amount in the PTF materials is distinctly lower than that in the PMC alloy, and the microstructures of the former materials are quite compact while that of the latter alloy is porous. Therefore, the PTF alloy shows better tensile properties than the PMC alloy. Owing to the existence of the SiC reinforcing particles, the PTF composite attains an ultimate tensile strength and yield strength of 230 MPa and 128 MPa, representing an enhancement of 27.8% and 29.3% than those (180 MPa and 99 MPa) of the PTF alloy. A modified model based on three strengthening mechanisms was proposed to calculate the yield strength of the PTF composite. The obtained theoretical results were quite consistent with the experimental data.