This paper discusses the principle and mathematical method to measure the phase fractions of multiphase flows by using a dual-energy gamma-ray system. The dual-energy gamma-ray device is composed of radioactive isotop...This paper discusses the principle and mathematical method to measure the phase fractions of multiphase flows by using a dual-energy gamma-ray system. The dual-energy gamma-ray device is composed of radioactive isotopes of 241Am and 137Cs with emission energies of 59.5 keV and 662 keV respectively. A rational method to calibrate the absorption coefficient was introduced in detail. The statistical error has been analyzed on the basis of the accurate absorption coefficient which enables determination phrase fractions almost independent of the flow regime. Improvement has been achieved on the measurement accuracy of phase fractions.展开更多
In this article, principle and mathematical method of determining the phase fractions of multiphase flows by using a dual-energy γ -ray system have been described. The dual-energy γ -ray device is composed of radioa...In this article, principle and mathematical method of determining the phase fractions of multiphase flows by using a dual-energy γ -ray system have been described. The dual-energy γ -ray device is composed of radioactive isotopes of 241Am and 137Cs with γ -ray energies of 59.5 and 662 keV, respectively. A rational method to calibrate the absorption coefficient was introduced in detail. The modified arithmetic is beneficial to removing the extra Compton scattering from the measured value. The result shows that the dual-energy γ -ray technique can be used in three-phase flow with average accuracy greater than 95%, which enables us to determine phase fractions almost independent of the flow regime. Improvement has been achieved on measurement accuracy of phase fractions.展开更多
To remove the scalar ambiguity in conventional blind channel estimation algorithms, totally blind channel estimation （TBCE） is proposed by using multiple constellations. To estimate the unknown scalar, its phase is ...To remove the scalar ambiguity in conventional blind channel estimation algorithms, totally blind channel estimation （TBCE） is proposed by using multiple constellations. To estimate the unknown scalar, its phase is decomposed into a fractional phase and an integer phase. However, the maximum-likelihood （ML） algorithm for the fractional phase does not have closed-form solutions and suffers from high computational complexity. By ex- ploring the structures of widely used constellations, this paper proposes a low-complexity fractional phase estimation algorithm which requires no exhaustive search. Analytical expressions of the asymptotic mean squared error （MSE） are also derived. The theo- retical analysis and simulation results indicate that the proposed fractional phase estimation algorithm exhibits almost the same performance as the ML algorithm but with significantly reduced computational burden.展开更多
The measurement of void fraction is of importance to the oil industry and chemical industry. In this article, the principle and mathematical method of determining the void fraction of horizontal gas-liquid flow by usi...The measurement of void fraction is of importance to the oil industry and chemical industry. In this article, the principle and mathematical method of determining the void fraction of horizontal gas-liquid flow by using a sin- gle-energy γ-ray system is described. The γ-ray source is the radioactive isotope of 241Am with γ-ray energy of 59.5 keV. The time-averaged value of the void fraction in a 50.0-mm i.d. transparent horizontal pipeline is measured under various combinations of the liquid flow and gas flow. It is found that increasing the gas flow rate at a fixed liquid flow rate would increase the void fraction. Test data are compared with the predictions of the correlations and a good agreement is found. The result shows that the designed γ-ray system can be used for measuring the void fraction in a horizontal gas-liquid two-phase flow with high accuracy.展开更多
This paper analyses robotic signals in the perspective of fractional dynamics and the pseudo phase plane （PPP）.It is shown that the spectra of several experimental signals can be approximated by trend lines whose sl...This paper analyses robotic signals in the perspective of fractional dynamics and the pseudo phase plane （PPP）.It is shown that the spectra of several experimental signals can be approximated by trend lines whose slope characterizes their fractional behavior.For the PPP reconstruction of each signal,the time lags are calculated through the fractal dimension.Moreover,to obtain a smooth PPP,the noisy signals are filtered through wavelets.The behavior of the spectra reveals a relationship with the fractal dimension of the PPP and the corresponding time delay.展开更多
We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic condu...We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.展开更多
The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governi...The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials(FGM)(i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach.The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.展开更多
Differential detection of continuous phase modulation suffers from significant intersymbol interference. To reduce bit error rate, multi-branch fractional multi-bit differential detection (MFMDD) with decision feed-ba...Differential detection of continuous phase modulation suffers from significant intersymbol interference. To reduce bit error rate, multi-branch fractional multi-bit differential detection (MFMDD) with decision feed-back is proposed. By introducing decision feedback in multi-bit differential detected signals, severe inter-symbol interference can be removed. Simulation results show that the proposed structure can greatly im-proves the performance compared with MFMDD without decision feedback, and the performance of 9 FMDD is very near to the performance of the coherent detection.展开更多
The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonaliz...The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.展开更多
Nanostructure of magnetically hard and soft materials is fascinating for exploring next-generation ul-trastrong permanent magnets with less expensive rare-earth elements.However,the resulting hard/soft nanocomposites ...Nanostructure of magnetically hard and soft materials is fascinating for exploring next-generation ul-trastrong permanent magnets with less expensive rare-earth elements.However,the resulting hard/soft nanocomposites often exhibit a low remanence/energy product due to the challenge in obtaining ideal phase components and appropriate soft phase fraction.In this work,a novel microstructure of multiple phases consisting of 1:5 phase and 1:3 phase as main hard phase,and a high ratio of Fe(Co)(27 wt.%-48 wt.%)as soft phase was obtained in Sm-Co(Fe)/Fe nanocomposite magnet.The grain size of both hard and soft phases below 15 nm was observed.The optimal energy product for Sm-Co(Fe)/Fe(Co)nanocom-posite is 2.1 times(an increment of 107%)of the corresponding single-hard-phase powders without soft phase.It reports that the isotropic nanocomposite powders exhibit a record of magnetic energy product larger than 25 MGOe(the highest value is 28.6 MGOe).The high performance and the microstructure achieved in this work for the isotropic powders will shed light on and provide a good premise for syn-thesizing high performance anisotropic bulk nanocomposite magnets.展开更多
In this study, the effect of temperatures and cooling rates of heat treatment on the microstructure of a powder metallurgy (PM) Ti-46Al-2Cr-2Nb-(B,W) (at.%) alloy was studied. Depending on the cooling rate and tempera...In this study, the effect of temperatures and cooling rates of heat treatment on the microstructure of a powder metallurgy (PM) Ti-46Al-2Cr-2Nb-(B,W) (at.%) alloy was studied. Depending on the cooling rate and temperature, the different structures were obtained from the initial near-γ (NG) microstructures by heat treatment in the α+γ field. The results show that the microstructures of samples after furnace cooling (FC) consist primarily of equiaxed γ and α 2 grains, with a few grains containing lamellae. Duplex microstructures consist mainly of γ grains and lamellar colonies were obtained in the quenching into another furnace at 900°C (QFC) samples. However, further increasing of the cooling rate to air cooling (AC) induces the transformation of α→α_2 and results in a microstructure with equiaxed γ and α_2 grains, and no lamellar colonies are found.展开更多
Tensile tests of GH4169 alloy were performed at room temperature. Different fractions, distributions and shapes of δ phase was prepared by aging treated at 880 ℃, 930 ℃ and 980 ℃ for 5 h or 10 h. The effect of δ ...Tensile tests of GH4169 alloy were performed at room temperature. Different fractions, distributions and shapes of δ phase was prepared by aging treated at 880 ℃, 930 ℃ and 980 ℃ for 5 h or 10 h. The effect of δ phase on the mechanical pro- perties of GH4169 alloy was investigated. The results show that 0.2% yield strength and ultimate tensile strength of GH4169 alloy increase by 61 MPa and 78 MPa respectively when the fraction of δ phase increases from 2.20% to 5.21%. Then, the ultimate tensile strength remains at 1 012 MPa even when the fraction of δ phase reaches 7.56%. The fraction effect of δ phase on the strength improvement of GH4169 alloy is more significant than morphology, and the critical fraction value is 5.21%. In addition, the elongation decreases by 14.1% when the fraction of δ phase increases from 2.20% to 7.56%. Excessive needle or short rod shaped 8 ohase is resoonsible for the reduction of eloneation.展开更多
Most image saliency detection models are dependent on prior knowledge and demand high computational cost. However, spectral residual（SR） and phase spectrum of the Fourier transform（PFT） models are simple and fast ...Most image saliency detection models are dependent on prior knowledge and demand high computational cost. However, spectral residual（SR） and phase spectrum of the Fourier transform（PFT） models are simple and fast saliency detection approaches based on two-dimensional Fourier transform without the prior knowledge. For seismic data, the geological structure of the underground rock formation changes more obviously in the time direction. Therefore, one-dimensional Fourier transform is more suitable for seismic saliency detection. Fractional Fourier transform（FrFT） is an improved algorithm for Fourier transform, therefore we propose the seismic SR and PFT models in one-dimensional FrF T domain to obtain more detailed saliency maps. These two models use the amplitude and phase information in FrFT domain to construct the corresponding saliency maps in spatial domain. By means of these two models, several saliency maps at different fractional orders can be obtained for seismic attribute analysis. These saliency maps can characterize the detailed features and highlight the object areas, which is more conducive to determine the location of reservoirs. The performance of the proposed method is assessed on both simulated and real seismic data. The results indicate that our method is effective and convenient for seismic attribute extraction with good noise immunity.展开更多
The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but ba...The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].展开更多
Tow-phase flow mixed variational formulations of evolution filtration problems with seawater intrusion are analyzed. A dual mixed fractional flow velocity-pressure model is considered with an air-fresh water and a fre...Tow-phase flow mixed variational formulations of evolution filtration problems with seawater intrusion are analyzed. A dual mixed fractional flow velocity-pressure model is considered with an air-fresh water and a fresh water-seawater characterization. For analysis and computational purposes, spatial decompositions based on nonoverlapping multidomains, above and below the sea level, are variationally introduced with internal boundary fluxes dualized as weak transmission constraints. Further, parallel augmented and exactly penalized duality algorithms, and proximation semi-implicit time marching schemes, are established and analyzed.展开更多
This study characterizes the mechanical properties and volume fractions of the different phases in precision annealed GCr15 steel using nanoindentation technology. Experimental results indicate that the nanoindentatio...This study characterizes the mechanical properties and volume fractions of the different phases in precision annealed GCr15 steel using nanoindentation technology. Experimental results indicate that the nanoindentation hardness of cementite grains is between 14.15 GPa and 17.61 GPa,with a mean value of 15.40 GPa. This hardness is much higher than the hardness of ferrite grains. The nanoindentation hardness of ferrite is between 2.78 GPa and 4.89 GPa, with a mean value of 3.35 GPa. The volume fractions of the different phases were also determined using nanoindentation technology, and the volume fraction of cementite in the steel was identified as 15%.展开更多
文摘This paper discusses the principle and mathematical method to measure the phase fractions of multiphase flows by using a dual-energy gamma-ray system. The dual-energy gamma-ray device is composed of radioactive isotopes of 241Am and 137Cs with emission energies of 59.5 keV and 662 keV respectively. A rational method to calibrate the absorption coefficient was introduced in detail. The statistical error has been analyzed on the basis of the accurate absorption coefficient which enables determination phrase fractions almost independent of the flow regime. Improvement has been achieved on the measurement accuracy of phase fractions.
基金Supported by National Natural Science Foundation of China (No.10572143) and Joint Project between the Royal Society and the Chinese Academy of Sciences (No.15933).
文摘In this article, principle and mathematical method of determining the phase fractions of multiphase flows by using a dual-energy γ -ray system have been described. The dual-energy γ -ray device is composed of radioactive isotopes of 241Am and 137Cs with γ -ray energies of 59.5 and 662 keV, respectively. A rational method to calibrate the absorption coefficient was introduced in detail. The modified arithmetic is beneficial to removing the extra Compton scattering from the measured value. The result shows that the dual-energy γ -ray technique can be used in three-phase flow with average accuracy greater than 95%, which enables us to determine phase fractions almost independent of the flow regime. Improvement has been achieved on measurement accuracy of phase fractions.
基金supported by the National Science and Technology Major Project of China(2013ZX03003006-003)
文摘To remove the scalar ambiguity in conventional blind channel estimation algorithms, totally blind channel estimation （TBCE） is proposed by using multiple constellations. To estimate the unknown scalar, its phase is decomposed into a fractional phase and an integer phase. However, the maximum-likelihood （ML） algorithm for the fractional phase does not have closed-form solutions and suffers from high computational complexity. By ex- ploring the structures of widely used constellations, this paper proposes a low-complexity fractional phase estimation algorithm which requires no exhaustive search. Analytical expressions of the asymptotic mean squared error （MSE） are also derived. The theo- retical analysis and simulation results indicate that the proposed fractional phase estimation algorithm exhibits almost the same performance as the ML algorithm but with significantly reduced computational burden.
基金Supported by National Natural Science Foundation of China (No. 10572143)
文摘The measurement of void fraction is of importance to the oil industry and chemical industry. In this article, the principle and mathematical method of determining the void fraction of horizontal gas-liquid flow by using a sin- gle-energy γ-ray system is described. The γ-ray source is the radioactive isotope of 241Am with γ-ray energy of 59.5 keV. The time-averaged value of the void fraction in a 50.0-mm i.d. transparent horizontal pipeline is measured under various combinations of the liquid flow and gas flow. It is found that increasing the gas flow rate at a fixed liquid flow rate would increase the void fraction. Test data are compared with the predictions of the correlations and a good agreement is found. The result shows that the designed γ-ray system can be used for measuring the void fraction in a horizontal gas-liquid two-phase flow with high accuracy.
文摘This paper analyses robotic signals in the perspective of fractional dynamics and the pseudo phase plane （PPP）.It is shown that the spectra of several experimental signals can be approximated by trend lines whose slope characterizes their fractional behavior.For the PPP reconstruction of each signal,the time lags are calculated through the fractal dimension.Moreover,to obtain a smooth PPP,the noisy signals are filtered through wavelets.The behavior of the spectra reveals a relationship with the fractal dimension of the PPP and the corresponding time delay.
基金supported by the National Natural Science Foundation of China(Grant Nos.11102102,11472161,and 91130017)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2014AQ015)the Independent Innovation Foundation of Shandong University,China(Grant No.2013ZRYQ002)
文摘We build a fractional dual-phase-lag model and the corresponding bioheat transfer equation, which we use to interpret the experiment results for processed meat that have been explained by applying the hyperbolic conduction. Analytical solutions expressed by H-functions are obtained by using the Laplace and Fourier transforms method. The inverse fractional dual-phase-lag heat conduction problem for the simultaneous estimation of two relaxation times and orders of fractionality is solved by applying the nonlinear least-square method. The estimated model parameters are given. Finally, the measured and the calculated temperatures versus time are compared and discussed. Some numerical examples are also given and discussed.
文摘The present work is concerned with the solution of a problem on thermoelastic interactions in a functional graded material due to thermal shock in the context of the fractional order three-phase lag model. The governing equations of fractional order generalized thermoelasticity with three-phase lag model for functionally graded materials(FGM)(i.e., material with spatially varying material properties) are established. The analytical solution in the transform domain is obtained by using the eigenvalue approach.The inversion of Laplace transform is done numerically. The graphical results indicate that the fractional parameter has significant effects on all the physical quantities. Thus, we can consider the theory of fractional order generalized thermoelasticity an improvement on studying elastic materials.
文摘Differential detection of continuous phase modulation suffers from significant intersymbol interference. To reduce bit error rate, multi-branch fractional multi-bit differential detection (MFMDD) with decision feed-back is proposed. By introducing decision feedback in multi-bit differential detected signals, severe inter-symbol interference can be removed. Simulation results show that the proposed structure can greatly im-proves the performance compared with MFMDD without decision feedback, and the performance of 9 FMDD is very near to the performance of the coherent detection.
文摘The phase diagrams and phase transitions of a typical bilayer fractional quantum Hall (QH) system with filling factor ν = 2/3 at the layer balanced point are investigated theoretically by finite size exact-diagonalization calculations and an exactly solvable model. We find some basic features essentially different from the bilayer integer QH systems at ν = 2, reflecting the special characteristics of the fractional QH systems. The degeneracy of the ground states occurs depending on the difference between intralayer and interlayer Coulomb energies, when interlayer tunneling energy (ΔSAS) gets close to zero. The continuous transitions of the finite size systems between the spin-polarized and spin-unpolarized phases are determined by the competition between the Zeeman energy (ΔZ) and the electron Coulomb energy, and are almost not affected by ΔSAS.
基金supported by the National Natural Science Foundation of China (Nos.52171184,51771220,51771095)Zhejiang Provincial Natural Science Foundation of China (No.LD19E010001).
文摘Nanostructure of magnetically hard and soft materials is fascinating for exploring next-generation ul-trastrong permanent magnets with less expensive rare-earth elements.However,the resulting hard/soft nanocomposites often exhibit a low remanence/energy product due to the challenge in obtaining ideal phase components and appropriate soft phase fraction.In this work,a novel microstructure of multiple phases consisting of 1:5 phase and 1:3 phase as main hard phase,and a high ratio of Fe(Co)(27 wt.%-48 wt.%)as soft phase was obtained in Sm-Co(Fe)/Fe nanocomposite magnet.The grain size of both hard and soft phases below 15 nm was observed.The optimal energy product for Sm-Co(Fe)/Fe(Co)nanocom-posite is 2.1 times(an increment of 107%)of the corresponding single-hard-phase powders without soft phase.It reports that the isotropic nanocomposite powders exhibit a record of magnetic energy product larger than 25 MGOe(the highest value is 28.6 MGOe).The high performance and the microstructure achieved in this work for the isotropic powders will shed light on and provide a good premise for syn-thesizing high performance anisotropic bulk nanocomposite magnets.
基金supported by the National Natural Science Foundation of China (No. 51101003)
文摘In this study, the effect of temperatures and cooling rates of heat treatment on the microstructure of a powder metallurgy (PM) Ti-46Al-2Cr-2Nb-(B,W) (at.%) alloy was studied. Depending on the cooling rate and temperature, the different structures were obtained from the initial near-γ (NG) microstructures by heat treatment in the α+γ field. The results show that the microstructures of samples after furnace cooling (FC) consist primarily of equiaxed γ and α 2 grains, with a few grains containing lamellae. Duplex microstructures consist mainly of γ grains and lamellar colonies were obtained in the quenching into another furnace at 900°C (QFC) samples. However, further increasing of the cooling rate to air cooling (AC) induces the transformation of α→α_2 and results in a microstructure with equiaxed γ and α_2 grains, and no lamellar colonies are found.
基金Sponsored by the Fund of China State Key Laboratory of Advanced Processing and Recycling of Non-ferrous Metals,Lanzhou University of Technology(SKLAB02014001)
文摘Tensile tests of GH4169 alloy were performed at room temperature. Different fractions, distributions and shapes of δ phase was prepared by aging treated at 880 ℃, 930 ℃ and 980 ℃ for 5 h or 10 h. The effect of δ phase on the mechanical pro- perties of GH4169 alloy was investigated. The results show that 0.2% yield strength and ultimate tensile strength of GH4169 alloy increase by 61 MPa and 78 MPa respectively when the fraction of δ phase increases from 2.20% to 5.21%. Then, the ultimate tensile strength remains at 1 012 MPa even when the fraction of δ phase reaches 7.56%. The fraction effect of δ phase on the strength improvement of GH4169 alloy is more significant than morphology, and the critical fraction value is 5.21%. In addition, the elongation decreases by 14.1% when the fraction of δ phase increases from 2.20% to 7.56%. Excessive needle or short rod shaped 8 ohase is resoonsible for the reduction of eloneation.
基金supported by the National Natural Science Foundation of China (Nos.61571096,61775030,41274127,41301460,and 40874066)
文摘Most image saliency detection models are dependent on prior knowledge and demand high computational cost. However, spectral residual（SR） and phase spectrum of the Fourier transform（PFT） models are simple and fast saliency detection approaches based on two-dimensional Fourier transform without the prior knowledge. For seismic data, the geological structure of the underground rock formation changes more obviously in the time direction. Therefore, one-dimensional Fourier transform is more suitable for seismic saliency detection. Fractional Fourier transform（FrFT） is an improved algorithm for Fourier transform, therefore we propose the seismic SR and PFT models in one-dimensional FrF T domain to obtain more detailed saliency maps. These two models use the amplitude and phase information in FrFT domain to construct the corresponding saliency maps in spatial domain. By means of these two models, several saliency maps at different fractional orders can be obtained for seismic attribute analysis. These saliency maps can characterize the detailed features and highlight the object areas, which is more conducive to determine the location of reservoirs. The performance of the proposed method is assessed on both simulated and real seismic data. The results indicate that our method is effective and convenient for seismic attribute extraction with good noise immunity.
文摘The classical Gerchberg-Saxton algorithm is introduced into the image recovery in fractional Fourier domain after adaptation. When this algorithm is applied directly, its performance is good for smoothed image, but bad for unsmoothed image. Based on the diversity of fractional Fourier transform on its orders, this paper suggests a novel iterative algorithm, which extracts the information of the original image from amplitudes of its fractional Fourier transform at two orders. This new algorithm consists of two independent Gerchberg-Saxton procedures and an averaging operation in each circle. Numerical simulations are carried out to show its validity for both smoothed and unsmoothed images with most pairs of orders in the interval [0, 1].
文摘Tow-phase flow mixed variational formulations of evolution filtration problems with seawater intrusion are analyzed. A dual mixed fractional flow velocity-pressure model is considered with an air-fresh water and a fresh water-seawater characterization. For analysis and computational purposes, spatial decompositions based on nonoverlapping multidomains, above and below the sea level, are variationally introduced with internal boundary fluxes dualized as weak transmission constraints. Further, parallel augmented and exactly penalized duality algorithms, and proximation semi-implicit time marching schemes, are established and analyzed.
文摘This study characterizes the mechanical properties and volume fractions of the different phases in precision annealed GCr15 steel using nanoindentation technology. Experimental results indicate that the nanoindentation hardness of cementite grains is between 14.15 GPa and 17.61 GPa,with a mean value of 15.40 GPa. This hardness is much higher than the hardness of ferrite grains. The nanoindentation hardness of ferrite is between 2.78 GPa and 4.89 GPa, with a mean value of 3.35 GPa. The volume fractions of the different phases were also determined using nanoindentation technology, and the volume fraction of cementite in the steel was identified as 15%.
