Research status and prospects of the fractal analysis of metal material surfaces and interfaces

Qinjin Dai; Xuefeng Liu; Xin Ma; Shaojie Tian; Qinghe Cui

doi:10.1007/s12613-024-2961-1

Article Contents

Article Navigation > International Journal of Minerals, Metallurgy and Materials > 2025 > In press, Uncorrected proof

Qinjin Dai, Xuefeng Liu, Xin Ma, Shaojie Tian, and Qinghe Cui, Research status and prospects of the fractal analysis of metal material surfaces and interfaces, Int. J. Miner. Metall. Mater.,(2025). https://doi.org/10.1007/s12613-024-2961-1

Cite this article as:

Qinjin Dai, Xuefeng Liu, Xin Ma, Shaojie Tian, and Qinghe Cui, Research status and prospects of the fractal analysis of metal material surfaces and interfaces, Int. J. Miner. Metall. Mater.,(2025). https://doi.org/10.1007/s12613-024-2961-1

Citation:

PDF( 8932 KB)

Review

Research status and prospects of the fractal analysis of metal material surfaces and interfaces

+ Author Affiliations

Corresponding author:
Xuefeng Liu E-mail: liuxuefengbj@163.com
Received: 21 March 2024; Revised: 24 June 2024; Accepted: 25 June 2024; Available online: 26 June 2024

Abstract

Abstract

As a mathematical analysis method, fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties. Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions. Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces. However, fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials. From the perspective of two-dimensional planes and three-dimensional curved surfaces, this paper reviews the current research status of the fractal analysis of metal precipitate interfaces, metal grain boundary interfaces, metal-deposited film surfaces, metal fracture surfaces, metal machined surfaces, and metal wear surfaces. The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized. Starting from three perspectives of fractal analysis, namely, research scope, image acquisition methods, and calculation methods, this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed. It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.
- metal material;
- surfaces and interfaces;
- fractal analysis;
- fractal dimension;
- homogeneity

FullText(HTML)

Supplements(0)

References(86)

References

[1]	K. Chu and C.C. Jia, Enhanced strength in bulk graphene–copper composites, Phys. Status Solidi A, 211(2014), No. 1, p. 184. doi: 10.1002/pssa.201330051
[2]	Z.M. Lai, Y.J. Mai, H.Y. Song, J.J. Mai, and X.H. Jie, Heterogeneous microstructure enables a synergy of strength, ductility and electrical conductivity in copper alloys, J. Alloys Compd., 902(2022), art. No. 163646. doi: 10.1016/j.jallcom.2022.163646
[3]	B.M. Luo, D.X. Li, C. Zhao, Z. Wang, Z.Q. Luo, and W.W. Zhang, A low Sn content Cu–Ni–Sn alloy with high strength and good ductility, Mater. Sci. Eng. A, 746(2019), p. 154. doi: 10.1016/j.msea.2018.12.120
[4]	F.L. Wang, Y.P. Li, K. Wakoh, Y. Koizumi, and A. Chiba, Cu–Ti–C alloy with high strength and high electrical conductivity prepared by two-step ball-milling processes, Mater. Des., 61(2014), p. 70. doi: 10.1016/j.matdes.2014.04.034
[5]	S.H.S. Ebrahimi, K. Dehghani, J. Aghazadeh, M.B. Ghasemian, and S. Zangeneh, Investigation on microstructure and mechanical properties of Al/Al–Zn–Mg–Cu laminated composite fabricated by accumulative roll bonding (ARB) process, Mater. Sci. Eng. A, 718(2018), p. 311. doi: 10.1016/j.msea.2018.01.130
[6]	M. Tayyebi, D. Rahmatabadi, A. Karimi, M. Adhami, and R. Hashemi, Investigation of annealing treatment on the interfacial and mechanical properties of Al5052/Cu multilayered composites subjected to ARB process, J. Alloys Compd., 871(2021), art. No. 159513. doi: 10.1016/j.jallcom.2021.159513
[7]	B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, New York, 1982.
[8]	R.A. Maniyara, D. Rodrigo, R.W. Yu, et al., Tunable plasmons in ultrathin metal films, Nat. Photonics, 13(2019), p. 328. doi: 10.1038/s41566-019-0366-x
[9]	Y.G. Bi, Y.F. Liu, X.L. Zhang, et al., Ultrathin metal films as the transparent electrode in ITO-free organic optoelectronic devices, Adv. Opt. Mater., 7(2019), No. 6, art. No. 1800778. doi: 10.1002/adom.201800778
[10]	Z.W. Feng, H.Y. Zhao, C.W. Tan, et al., Effect of laser texturing on the surface characteristics and bonding property of 30CrMnSiA steel adhesive joints, J. Manuf. Process., 47(2019), p. 219. doi: 10.1016/j.jmapro.2019.09.046
[11]	Y.C. Guan, F.F. Luo, G.C. Lim, M.H. Hong, H.Y. Zheng, and B.J. Qi, Fabrication of metallic surfaces with long-term superhydrophilic property using one-stop laser method, Mater. Des., 78(2015), p. 19. doi: 10.1016/j.matdes.2015.04.021
[12]	B.B. Mandelbrot, D.E. Passoja, and A.J. Paullay, Fractal character of fracture surfaces of metals, Nature, 308(1984), p. 721. doi: 10.1038/308721a0
[13]	J. Li, Q. Du, and C.X. Sun, An improved box-counting method for image fractal dimension estimation, Pattern Recognit., 42(2009), No. 11, p. 2460. doi: 10.1016/j.patcog.2009.03.001
[14]	N. Sarkar and B.B. Chaudhuri, An efficient differential box-counting approach to compute fractal dimension of image, IEEE Trans. Syst. Man Cybern., 24(1994), No. 1, p. 115. doi: 10.1109/21.259692
[15]	D.A. Russell, J.D. Hanson, and E. Ott, Dimension of strange attractors, Phys. Rev. Lett., 45(1980), No. 14, p. 1175. doi: 10.1103/PhysRevLett.45.1175
[16]	R. Lopes and N. Betrouni, Fractal and multifractal analysis: A review, Med. Image Anal., 13(2009), No. 4, p. 634. doi: 10.1016/j.media.2009.05.003
[17]	S. Lou, X.Q. Jiang, and P.J. Scott, Application of the morphological alpha shape method to the extraction of topographical features from engineering surfaces, Measurement, 46(2013), No. 2, p. 1002. doi: 10.1016/j.measurement.2012.09.015
[18]	Ş. Ţălu, S. Stach, J. Zaharieva, M. Milanova, D. Todorovsky, and S. Giovanzana, Surface roughness characterization of poly(methylmethacrylate) films with immobilized Eu(III) β-Diketonates by fractal analysis, Int. J. Polym. Anal. Charact., 19(2014), No. 5, p. 404. doi: 10.1080/1023666X.2014.904149
[19]	C.S. Pande, L.E. Richards, N. Louat, B.D. Dempsey, and A.J. Schwoeble, Fractal characterization of fractured surfaces, Acta Metall., 35(1987), No. 7, p. 1633. doi: 10.1016/0001-6160(87)90110-6
[20]	A. Thomas and T.R. Thomas, Digital analysis of very small scale surface roughness, J. Wave Mater. Interact., 3(1988), p. 341.
[21]	T. Babadagli and K. Develi, Fractal characteristics of rocks fractured under tension, Theor. Appl. Fract. Mech., 39(2003), No. 1, p. 73. doi: 10.1016/S0167-8442(02)00139-8
[22]	S.R. Ge and S.F. Suo, The computation methods for the fractal dimension of surface profiles, Tribology, 17(1997), No. 4, p. 354.
[23]	C.Q. Yuan, J. Li, X.P. Yan, and Z. Peng, The use of the fractal description to characterize engineering surfaces and wear particles, Wear, 255(2003), No. 1-6, p. 315. doi: 10.1016/S0043-1648(03)00206-0
[24]	R.F. Voss, Random fractal forgeries, [in] R.A. Earnshaw, ed., Fundamental Algorithms for Computer Graphics, Springer Berlin, Heidelberg, 1991, p. 805.
[25]	R.F. Voss, Random fractals: Characterization and measurement, [in] R. Pynn and A. Skjeltorp, eds., Scaling Phenomena in Disordered Systems, Springer, New York, 1991, p. 1.
[26]	A.S. Balghonaim and J.M. Keller, A maximum likelihood estimate for two-variable fractal surface, IEEE Trans. Image Process., 7(1998), No. 12, p. 1746. doi: 10.1109/83.730389
[27]	Y. Liu, L.Y. Chen, H.M. Wang, et al., An improved differential box-counting method to estimate fractal dimensions of gray-level images, J. Vis. Commun. Image Represent., 25(2014), No. 5, p. 1102. doi: 10.1016/j.jvcir.2014.03.008
[28]	L. Yang, D.S. Zhang, X.N. Zhang, and A.F. Tian, Surface profile topography of ionic polymer metal composite based on fractal theory, Surf. Interfaces, 22(2021), art. No. 100834. doi: 10.1016/j.surfin.2020.100834
[29]	A. Akrami, N. Nasiri, and V. Kulish, Fractal dimension analysis of Mg2Si particles of Al–15%Mg2Si composite and its relationships to mechanical properties, Results Mater., 7(2020), art. No. 100118. doi: 10.1016/j.rinma.2020.100118
[30]	B.Y. Casas, J.C. Carranza, I.A. Figueroa, et al., Fractal and conventional analysis of Cu content effect on the microstructure of Al–Si–Cu–Mg alloys, Mater. Res., 23(2020), No. 4, art. No. e20190666. doi: 10.1590/1980-5373-mr-2019-0666
[31]	A. Wakai, A. Das, J. Bustillos, and A. Moridi, Effect of solidification pathway during additive manufacturing on grain boundary fractality, Addit. Manuf. Lett., 6(2023), art. No. 100149. doi: 10.1016/j.addlet.2023.100149
[32]	Z. Wang, X.F. Liu, Y. He, and J.X. Xie, A fractal-based model for the microstructure evolution of silicon bronze wires fabricated by dieless drawing, Int. J. Miner. Metall. Mater., 17(2010), No. 6, p. 770. doi: 10.1007/s12613-010-0387-4
[33]	M. Tarafder, P. Sinha, A. Kundu, M. Strangwood, and C. Davis, Fractal based correlations for Nb microalloyed steel undergoing static recrystallization, Mater. Charact., 85(2013), p. 92. doi: 10.1016/j.matchar.2013.08.013
[34]	R. Colás, On the variation of grain size and fractal dimension in an austenitic stainless steel, Mater. Charact., 46(2001), No. 5, p. 353. doi: 10.1016/S1044-5803(01)00105-X
[35]	N. Afrin, D.L. Chen, X. Cao, and M. Jahazi, Microstructure and tensile properties of friction stir welded AZ31B magnesium alloy, Mater. Sci. Eng. A, 472(2008), No. 1-2, p. 179. doi: 10.1016/j.msea.2007.03.018
[36]	C.K. Kaithwas, P. Bhuyan, and S. Mandal, Assessing the potential of sparsely nucleated recrystallized grains to lead grain boundary engineering during extending annealing in Alloy 600H, Mater. Charact., 168(2020), art. No. 110538. doi: 10.1016/j.matchar.2020.110538
[37]	T.S. Prithiv, P. Bhuyan, S.K. Pradhan, V. Subramanya Sarma, and S. Mandal, A critical evaluation on efficacy of recrystallization vs. strain induced boundary migration in achieving grain boundary engineered microstructure in a Ni-base superalloy, Acta Mater., 146(2018), p. 187. doi: 10.1016/j.actamat.2017.12.045
[38]	S. Kobayashi, T. Maruyama, S. Tsurekawa, and T. Watanabe, Grain boundary engineering based on fractal analysis for control of segregation-induced intergranular brittle fracture in polycrystalline nickel, Acta Mater., 60(2012), No. 17, p. 6200. doi: 10.1016/j.actamat.2012.07.065
[39]	X. Wei, M.J. Xu, J.Q. Chen, et al., Fractal analysis of Mo and Nb effects on grain boundary character and hot cracking behavior for Ni–Cr–Fe alloys, Mater. Charact., 145(2018), p. 65. doi: 10.1016/j.matchar.2018.08.024
[40]	M.J. Xu, J.M. Chen, H. Lu, J.J. Xu, C. Yu, and X. Wei, Effects of residual stress and grain boundary character on creep cracking in 2.25Cr–1.6W steel, Mater. Sci. Eng. A, 659(2016), p. 188. doi: 10.1016/j.msea.2016.02.025
[41]	S. Kobayashi, R. Kobayashi, and T. Watanabe, Control of grain boundary connectivity based on fractal analysis for improvement of intergranular corrosion resistance in SUS316L austenitic stainless steel, Acta Mater., 102(2016), p. 397. doi: 10.1016/j.actamat.2015.08.075
[42]	H. Fu, W. Wang, X.J. Chen, G. Pia, and J.X. Li, Grain boundary design based on fractal theory to improve intergranular corrosion resistance of TWIP steels, Mater. Des., 185(2020), art. No. 108253. doi: 10.1016/j.matdes.2019.108253
[43]	W. Feng, Z. Wang, Q. Sun, Y.Q. He, and Y.X. Sun, Effect of thermomechanical processing via rotary swaging on grain boundary character distribution and intergranular corrosion in 304 austenitic stainless steel, J. Mater. Res. Technol., 19(2022), p. 2470. doi: 10.1016/j.jmrt.2022.06.032
[44]	H. Fu, X.J. Chen, W. Wang, G. Pia, J.L. Zhang, and J.X. Li, Statistical study on the effects of heterogeneous deformation and grain boundary character on hydrogen-induced crack initiation and propagation in twining-induced plasticity steels, Corros. Sci., 192(2021), art. No. 109796. doi: 10.1016/j.corsci.2021.109796
[45]	F.M. Mwema, E.T. Akinlabi, and O.P. Oladijo, Fractal analysis of hillocks: A case of RF sputtered aluminum thin films, Appl. Surf. Sci., 489(2019), p. 614. doi: 10.1016/j.apsusc.2019.05.340
[46]	A. Roy, B. Sundaravel, R. Batabyal, and B.N. Dev, Fractal pattern formation in thermal grooving at grain boundaries in Ag films on Si(111) surfaces, Thin Solid Films, 520(2012), No. 15, p. 5086. doi: 10.1016/j.tsf.2012.03.011
[47]	A. Arman, Ş. Ţălu, C. Luna, A. Ahmadpourian, M. Naseri, and M. Molamohammadi, Micromorphology characterization of copper thin films by AFM and fractal analysis, J. Mater. Sci. Mater. Electron., 26(2015), No. 12, p. 9630. doi: 10.1007/s10854-015-3628-5
[48]	Y. Wang and K.W. Xu, Characterization of surface morphology of copper tungsten thin film by surface fractal geometry and resistivity, Thin Solid Films, 468(2004), No. 1-2, p. 310. doi: 10.1016/j.tsf.2004.05.132
[49]	Ş. Ţălu, S. Abdolghaderi, E.P. Pinto, R.S. Matos, and M. Salerno, Advanced fractal analysis of nanoscale topography of Ag/DLC composite synthesized by RF-PECVD, Surf. Eng., 36(2020), No. 7, p. 713. doi: 10.1080/02670844.2019.1710937
[50]	Ş. Ţălu, C. Luna, A. Ahmadpourian, et al., Micromorphology and fractal analysis of nickel–carbon composite thin films, J. Mater. Sci. Mater. Electron., 27(2016), No. 11, p. 11425. doi: 10.1007/s10854-016-5268-9
[51]	Ş. Ţălu, R.S. Matos, E.P. Pinto, S. Rezaee, and M. Mardani, Stereometric and fractal analysis of sputtered Ag–Cu thin films, Surf. Interfaces, 21(2020), art. No. 100650. doi: 10.1016/j.surfin.2020.100650
[52]	Ş. Ţălu, M. Bramowicz, S. Kulesza, et al., Gold nanoparticles embedded in carbon film: Micromorphology analysis, J. Ind. Eng. Chem., 35(2016), p. 158. doi: 10.1016/j.jiec.2015.12.029
[53]	B. Venkatesh, D.L. Chen, and S.D. Bhole, Three-dimensional fractal analysis of fracture surfaces in a titanium alloy for biomedical applications, Scripta Mater., 59(2008), No. 4, p. 391. doi: 10.1016/j.scriptamat.2008.04.010
[54]	O.A. Hilders, M. Ramos, N.D. Peña, and L. Sàenz, Fractal geometry of fracture surfaces of a duplex stainless steel, J. Mater. Sci., 41(2006), No. 17, p. 5739. doi: 10.1007/s10853-006-0102-z
[55]	B.D. Venkatesh, D.L. Chen, and S.D. Bhole, Effect of heat treatment on mechanical properties of Ti–6Al–4V ELI alloy, Mater. Sci. Eng. A, 506(2009), No. 1-2, p. 117. doi: 10.1016/j.msea.2008.11.018
[56]	O. Hilders and N. Zambrano, The effect of aging on impact toughness and fracture surface fractal dimension in SAF 2507 super duplex stainless steel, J. Microsc. Ultrastruct., 2(2014), No. 4, art. No. 236. doi: 10.1016/j.jmau.2014.07.001
[57]	L. Girelli, M. Giovagnoli, M. Tocci, et al., Evaluation of the impact behaviour of AlSi10Mg alloy produced using laser additive manufacturing, Mater. Sci. Eng. A, 748(2019), p. 38. doi: 10.1016/j.msea.2019.01.078
[58]	W. Macek, Fractal analysis of the bending-torsion fatigue fracture of aluminium alloy, Eng. Fail. Anal., 99(2019), p. 97. doi: 10.1016/j.engfailanal.2019.02.007
[59]	W. Macek, Post-failure fracture surface analysis of notched steel specimens after bending-torsion fatigue, Eng. Fail. Anal., 105(2019), p. 1154. doi: 10.1016/j.engfailanal.2019.07.056
[60]	W. Macek, R. Branco, M. Korpyś, and T. Łagoda, Fractal dimension for bending–torsion fatigue fracture characterisation, Measurement, 184(2021), art. No. 109910. doi: 10.1016/j.measurement.2021.109910
[61]	W. Macek, D. Rozumek, and G.M. Królczyk, Surface topography analysis based on fatigue fractures obtained with bending of the 2017A-T4 alloy, Measurement, 152(2020), art. No. 107347. doi: 10.1016/j.measurement.2019.107347
[62]	Z.Y. Zhang, Quantitative characterization on fatigue fracture features of A6005 aluminum alloy welded joints, Eng. Fail. Anal., 129(2021), art. No. 105687. doi: 10.1016/j.engfailanal.2021.105687
[63]	W. Macek, D. Rozumek, S. Faszynka, R. Branco, S.P. Zhu, and R.M. Nejad, Fractographic-fractal dimension correlation with crack initiation and fatigue life for notched aluminium alloys under bending load, Eng. Fail. Anal., 149(2023), art. No. 107285. doi: 10.1016/j.engfailanal.2023.107285
[64]	Z.D. Jiang, H.R. Wang, and B. Fei, Research into the application of fractal geometry in characterising machined surfaces, Int. J. Mach. Tools Manuf., 41(2001), No. 13-14, p. 2179. doi: 10.1016/S0890-6955(01)00085-2
[65]	P.V. Kuznetsov, V.E. Panin, and J. Schreiber, Fractal dimension as a characteristic of deformation stages of austenite stainless steel under tensile load, Theor. Appl. Fract. Mech., 35(2001), No. 2, p. 171. doi: 10.1016/S0167-8442(00)00058-6
[66]	Q. Chen, Y.D. Wang, J.J. Zhou, Y.M. Wu, and H. Song, Research on characterization of anisotropic and isotropic processing surfaces by characteristic roughness, J. Mater. Process. Technol., 275(2020), art. No. 116277. doi: 10.1016/j.jmatprotec.2019.116277
[67]	Ş. Ţălu, S. Kulesza, M. Bramowicz, H. Sağlam, and R. Kus, Fractal geometry of internal thread surfaces manufactured by cutting tap and rolling tap, Manuf. Lett., 23(2020), p. 34. doi: 10.1016/j.mfglet.2019.12.001
[68]	D. Xu, Q. Yang, F. Dong, and S. Krishnaswamy, Evaluation of surface roughness of a machined metal surface based on laser speckle pattern, J. Eng., 2018(2018), No. 9, p. 773.
[69]	M.A. Mahmood, T.Y. Tsai, Y.J. Hwu, et al., Effect of fractal parameters on optical properties of cold rolled aluminum alloy strips with induced surface deflection: Simulations and experimental correlations, J. Mater. Process. Technol., 279(2020), art. No. 116554. doi: 10.1016/j.jmatprotec.2019.116554
[70]	M.O. Qadri and H. Namazi, Fractal-based analysis of the relation between surface finish and machine vibration in milling operation, Fluct. Noise Lett., 19(2020), No. 1, art. No. 2050006. doi: 10.1142/S0219477520500066
[71]	C. Xu, T.H. Wu, Y.W. Huo, and H.B. Yang, In-situ characterization of three dimensional worn surface under sliding-rolling contact, Wear, 426-427(2019), p. 1781. doi: 10.1016/j.wear.2018.12.045
[72]	S.Y. Chen, H.D. Wang, G.Z. Ma, J.J. Kang, and B.S. Xu, Fractal properties of worn surface of Fe-based alloy coatings during rolling contact process, Appl. Surf. Sci., 364(2016), p. 96. doi: 10.1016/j.apsusc.2015.12.107
[73]	W. Macek, A. Tomczyk, R. Branco, M. Dobrzyński, and A. Seweryn, Fractographical quantitative analysis of EN-AW 2024 aluminum alloy after creep pre-strain and LCF loading, Eng. Fract. Mech., 282(2023), art. No. 109182. doi: 10.1016/j.engfracmech.2023.109182
[74]	M.F. Zawrah, H.A. Zayed, R.A. Essawy, A.H. Nassar, and M.A. Taha, Preparation by mechanical alloying, characterization and sintering of Cu–20wt.% Al₂O₃ nanocomposites, Mater. Des., 46(2013), p. 485. doi: 10.1016/j.matdes.2012.10.032
[75]	Y.L. Deng, J.J. Xu, J.Q. Chen, and X.B. Guo, Effect of double-step homogenization treatments on the microstructure and mechanical properties of Al–Cu–Li–Zr alloy, Mater. Sci. Eng. A, 795(2020), art. No. 139975. doi: 10.1016/j.msea.2020.139975
[76]	O. Ghaderi, M.R. Toroghinejad, and A. Najafizadeh, Investigation of microstructure and mechanical properties of Cu–SiC_P composite produced by continual annealing and roll-bonding process, Mater. Sci. Eng. A, 565(2013), p. 243. doi: 10.1016/j.msea.2012.11.004
[77]	S. Amirkhanlou, R. Jamaati, B. Niroumand, and M.R. Toroghinejad, Fabrication and characterization of Al/SiCp composites by CAR process, Mater. Sci. Eng. A, 528(2011), No. 13-14, p. 4462. doi: 10.1016/j.msea.2011.02.037
[78]	L. Huang, Z.S. Cui, X.P. Meng, et al., Effects of microelements on the microstructure evolution and properties of ultrahigh strength Cu–Ti alloys, Mater. Sci. Eng. A, 823(2021), art. No. 141581. doi: 10.1016/j.msea.2021.141581
[79]	N. Ye, X.P. Ren, and J.H. Liang, Microstructure and mechanical properties of Ni/Ti/Al/Cu composite produced by accumulative roll bonding (ARB) at room temperature, J. Mater. Res. Technol., 9(2020), No. 3, p. 5524. doi: 10.1016/j.jmrt.2020.03.077
[80]	L.L. Yuan, M.X. Guo, Y. Wang, Y. Wang, and L.Z. Zhuang, Synergistic effect of gradient Zn content and multiscale particles on the mechanical properties of Al–Zn–Mg–Cu alloys with coupling distribution of coarse–fine grains, Int. J. Miner. Metall. Mater., 31(2024), No. 6, p. 1392. doi: 10.1007/s12613-024-2871-2
[81]	E. Safary, R. Taghiabadi, and M.H. Ghoncheh, Mechanical properties of Al–15Mg₂Si composites prepared under different solidification cooling rates, Int. J. Miner. Metall. Mater., 29(2022), No. 6, p. 1249. doi: 10.1007/s12613-020-2244-4
[82]	S.L. Semiatin, R.C. Kramb, R.E. Turner, F. Zhang, and M.M. Antony, Analysis of the homogenization of a nickel-base superalloy, Scripta Mater., 51(2004), No. 6, p. 491. doi: 10.1016/j.scriptamat.2004.05.049
[83]	L.H. Zhao, Y. Tan, Y.L. Wang, et al., Homogenization behavior of IN 718 superalloy prepared by electron beam layered solidification technology, J. Mater. Res. Technol., 13(2021), p. 1567. doi: 10.1016/j.jmrt.2021.05.109
[84]	S.Y. Yuan, H. Xu, and H.C. Gu, Fractal analysis of polypropylene composite filled with nano-calcium carbonate, J. Appl. Polym. Sci., 110(2008), No. 4, p. 1955. doi: 10.1002/app.27718
[85]	R. De Rosa, P. Donato, and G. Ventura, Fractal analysis of mingled/mixed magmas: An example from the Upper Pollara eruption (Salina Island, southern Tyrrhenian Sea, Italy), Lithos, 65(2002), No. 3-4, p. 299. doi: 10.1016/S0024-4937(02)00197-4
[86]	T. Liu, X.N. Zhang, Z. Li, and Z.Q. Chen, Research on the homogeneity of asphalt pavement quality using X-ray computed tomography (CT) and fractal theory, Constr. Build. Mater., 68(2014), p. 587. doi: 10.1016/j.conbuildmat.2014.06.046