Journal of Materials Science

, Volume 46, Issue 18, pp 6118–6123 | Cite as

Three-dimensional fractal analysis of fracture surfaces in titanium–iron particulate reinforced hydroxyapatite composites: relationship between fracture toughness and fractal dimension

  • Q. Chang
  • D. L. ChenEmail author
  • H. Q. Ru
  • X. Y. Yue
  • L. Yu
  • C. P. Zhang


Fractal dimension has been considered as a measure of fracture surface roughness of materials. Three-dimensional (3D) surface analysis is anticipated to provide a better evaluation of fracture surface toughness and fractal dimension. The objective of this study was to quantify the fracture surfaces and identify a potential relationship between fracture toughness and fractal dimension in a new type of core–shell titanium–iron particulate reinforced hydroxyapatite matrix composites using SEM stereoscopy coupled with a 3D surface analysis. The obtained results showed that both fracture surface roughness and fractal dimension increased with increasing amount of core–shell Ti–Fe reinforcing particles. The fractal dimension was observed to be a direct measure of fracture surface roughness. The fracture toughness of the composites increased linearly with the square root of fractal dimensional increment (i.e., followed the Mecholsky–Mackin equation well) due to the presence of Ti–Fe particles along with the effect of porosity in brittle materials. The 3D fractal analysis was suggested to be a proper tool for quantifying the fracture surfaces and linking the microstructural parameter to fracture toughness.


Fracture Surface Fracture Toughness Fractal Dimension Flexural Strength Brittle Material 



The authors would like to thank the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC). Q.C. is also to acknowledge the financial support provided by China Scholarship Council and the Fundamental Research Funds for the Central Universities (N090602001) and D.L.C. is also grateful for the financial support by the Premier’s Research Excellence Award (PREA), Canada Foundation for Innovation (CFI), and Ryerson Research Chair (RRC) program. The authors would also like to thank Q. Li, A. Machin, J. Amankrah and R. Churaman for their assistance in the experiments. Professor N. Zhang is also gratefully acknowledged for her continuous encouragement while performing this investigation.


  1. 1.
    Mandelbrot BB (2000) The fractal geometry of nature, revised edn, 19th printing. W.H. Freeman and Company, New YorkGoogle Scholar
  2. 2.
    Gomory R (2010) Nature 468:378CrossRefGoogle Scholar
  3. 3.
    Peitgen H-O (2010) Science 330:926CrossRefGoogle Scholar
  4. 4.
    Coster M, Chermant JL (1983) Int Metals Rev 28:228Google Scholar
  5. 5.
    Mandelbrot BB, Passoja DE, Paullay AJ (1984) Nature 308:721CrossRefGoogle Scholar
  6. 6.
    Mandelbrot BB (2006) Int J Fract 138:13CrossRefGoogle Scholar
  7. 7.
    Cahn RW (1989) Nature 338:201CrossRefGoogle Scholar
  8. 8.
    Lakes R (1993) Nature 361:511CrossRefGoogle Scholar
  9. 9.
    Schaefer DW (1989) Science 243:1023CrossRefGoogle Scholar
  10. 10.
    Meakin P (1991) Science 252:226CrossRefGoogle Scholar
  11. 11.
    Wang ZG, Chen DL, Jiang XX, Ai SH, Shih CH (1988) Scripta Metall 22:827CrossRefGoogle Scholar
  12. 12.
    Williford RE (1990) Scripta Metall Mater 24:455Google Scholar
  13. 13.
    Charkaluk E, Bigerelle M, Iost A (1998) Eng Fract Mech 61:119CrossRefGoogle Scholar
  14. 14.
    Yang AM, Xiong YH, Liu L (2001) Sci Technol Adv Mater 2:101CrossRefGoogle Scholar
  15. 15.
    Kotowski P (2006) Int J Fract 141:269CrossRefGoogle Scholar
  16. 16.
    Venkatesh B, Chen DL, Bhole S (2008) Scripta Mater 59:391CrossRefGoogle Scholar
  17. 17.
    Tanaka M, Ono J, Sakashita M, Kato R (2009) ISIJ Int 49:1229CrossRefGoogle Scholar
  18. 18.
    Chappard D, Degasne I, Huré G, Legrand E, Audran M, Baslé MF (2003) Biomaterials 24:1399CrossRefGoogle Scholar
  19. 19.
    Tanaka M (1995) J Mater Sci 30:3668. doi: CrossRefGoogle Scholar
  20. 20.
    Hilders OA, Ramos M, Pena ND, Saenz L (2006) J Mater Sci 41:5739. doi: CrossRefGoogle Scholar
  21. 21.
    Chen CT, Runt J (1989) Polym Commun 30:334CrossRefGoogle Scholar
  22. 22.
    Kozlov HV, Burya OI, Aloev VZ (2004) Mater Sci 40:491CrossRefGoogle Scholar
  23. 23.
    Du PH, Xue B, Song YH, Lu SJ, Yu J, Zheng Q (2010) Polym Bull 64:185CrossRefGoogle Scholar
  24. 24.
    Blacher S, Maquet V, Schils F, Martin D, Schoenen J, Moonen G et al (2003) Biomaterials 24:1033CrossRefGoogle Scholar
  25. 25.
    Mecholsky JJ, Passoja DE, Feinberg-Ringel KS (1989) J Am Ceram Soc 72:60CrossRefGoogle Scholar
  26. 26.
    Wasen J, Heier E, Hansson T (1998) Scripta Mater 38:953CrossRefGoogle Scholar
  27. 27.
    Chen Z, Mecholsky JJ, Joseph T, Beatty CL (1997) J Mater Sci 32:6317. doi: CrossRefGoogle Scholar
  28. 28.
    Mechtcherine V (2009) Cem Concr Res 39:620CrossRefGoogle Scholar
  29. 29.
    Jiang MQ, Meng JX, Gao JB, Wang XL, Rouxel T, Keryvin V et al (2010) Intermetallics 18:2468CrossRefGoogle Scholar
  30. 30.
    Chen DL, Pang DX, Yang ZJ, Kong S, Wang LT, Yang K et al (1988) J Phys C 21:271CrossRefGoogle Scholar
  31. 31.
    Fratini M, Poccia N, Ricci A, Campi G, Burghammer M, Aeppli G et al (2010) Nature 466:841CrossRefGoogle Scholar
  32. 32.
    Zaanen J (2010) Nature 466:825CrossRefGoogle Scholar
  33. 33.
    Rishabh A, Joshi MR, Balani K (2010) J Appl Phys 107:123532. doi: CrossRefGoogle Scholar
  34. 34.
    Liang JZ, Wu CB (2010) Mater Sci Technol 18:178Google Scholar
  35. 35.
    Liang JZ, Wu CB (2009) J Mater Eng 10:53Google Scholar
  36. 36.
    Liang JZ, Wu CB (2008) J Appl Polym Sci 109:3763CrossRefGoogle Scholar
  37. 37.
    Cantor GJ, Brown CA (2009) Wear 266:609CrossRefGoogle Scholar
  38. 38.
    Briones V, Aguilera JM, Brown C (2006) J Food Eng 77:776CrossRefGoogle Scholar
  39. 39.
    Dougherty G, Henebry GM (2001) Med Eng Phys 23:369CrossRefGoogle Scholar
  40. 40.
    Wolski M, Podsiadlo P, Stachowiak GW (2009) Proc IMechE H 223:211CrossRefGoogle Scholar
  41. 41.
    Majumder SR, Mazumdar S (2007) Physica A 377:559CrossRefGoogle Scholar
  42. 42.
    Gentile F, Tirinato L, Battista E, Causa F, Liberale C, di Fabrizio EM et al (2010) Biomaterials 31:7205CrossRefGoogle Scholar
  43. 43.
    Wang P, Li L, Zhang C, Lei QF, Fang WJ (2010) Biomaterials 31:6201CrossRefGoogle Scholar
  44. 44.
    Borodinsky LN, Fiszman ML (2001) Methods 24:341CrossRefGoogle Scholar
  45. 45.
    Mecholsky JJ (2009) Key Eng Mater 409:145CrossRefGoogle Scholar
  46. 46.
    Bulpakdi P, Taskonak B, Yan J, Mecholsky JJ (2009) Dental Mater 25:634CrossRefGoogle Scholar
  47. 47.
    Carpinteri A, Paggi M (2010) Int J Fract 161:41CrossRefGoogle Scholar
  48. 48.
    Mecholsky JJ (2006) Mater Lett 60:2485CrossRefGoogle Scholar
  49. 49.
    Carpinteri A, Pugno N, Puzzi S (2009) Chaos Solitons Fractals 39:1210CrossRefGoogle Scholar
  50. 50.
    Spagnoli A (2004) Chaos Solitons Fractals 22:589CrossRefGoogle Scholar
  51. 51.
    Tanaka M, Kimura Y, Oyama N, Kato R (2006) J Mater Sci 41:6181. doi: CrossRefGoogle Scholar
  52. 52.
    Drummond JL, Thompson M, Super BJ (2005) Dental Mater 21:586CrossRefGoogle Scholar
  53. 53.
    Della Bona A, Hill TJ, Mecholsky JJ (2001) J Mater Sci 36:2645. doi: CrossRefGoogle Scholar
  54. 54.
    Carpinteri A, Chiaia B, Invernizzi S (1999) Theor Appl Fract Mech 31:163CrossRefGoogle Scholar
  55. 55.
    Zhou HW, Xie HP (2003) Surf Rev Lett 10:751CrossRefGoogle Scholar
  56. 56.
    Tanaka M, Kimura Y, Kayama A, Taguchi J, Kato R (2005) J Mater Sci 40:6291. doi: CrossRefGoogle Scholar
  57. 57.
    Ruzicka S, Hausild P (2010) Eng Fract Mech 77:744CrossRefGoogle Scholar
  58. 58.
    Adachi K, Chung SH, Friedrich H, Buseck PR (2007) J Geophys Res 112:D14202. doi: CrossRefGoogle Scholar
  59. 59.
    Elfallagh F, Inkson BJ (2009) J Eur Ceram Soc 29:47CrossRefGoogle Scholar
  60. 60.
    Chang Q, Chen DL, Ru HQ, Yue XY, Yu L, Zhang CP (2010) Biomaterials 31:1493CrossRefGoogle Scholar
  61. 61.
    Kruzic J, Ritchie RO (2003) J Am Ceram Soc 86:1433CrossRefGoogle Scholar
  62. 62.
    ANSI/ASME B46.1-2002, Surface texture (Surface roughness, waviness and lay). American Society of Mechanical Engineers, 2002Google Scholar
  63. 63.
    Kruzic JJ, Satet RL, Hoffmann MJ, Cannon RM, Ritchie RO (2008) J Am Ceram Soc 91:1986CrossRefGoogle Scholar
  64. 64.
    Kumar R, Prakash KH, Cheang P, Khor KA (2005) Acta Mater 53:2327CrossRefGoogle Scholar
  65. 65.
    Mecholsky JJ, Mackin TJ (1988) J Mater Sci Lett 7:1145CrossRefGoogle Scholar
  66. 66.
    Ritchie RO, Dauskardt RH, Yu W, Brendzel AM (1990) J Biomed Mater Res 24:189CrossRefGoogle Scholar
  67. 67.
    Fett T, Munz D (2006) Arch Appl Mech 76:667CrossRefGoogle Scholar
  68. 68.
    Ponton CB, Rawlings RD (1989) Mater Sci Tech 5:865CrossRefGoogle Scholar
  69. 69.
    Scherrer SS, Denry IL, Wiskott HWA (1998) Dental Mater 14:246CrossRefGoogle Scholar
  70. 70.
    Kruzic JJ, Ritchie RO (2004) Ceramic Transactions 156:83Google Scholar
  71. 71.
    Gatto A (2006) J Mater Proc Tech 174:67CrossRefGoogle Scholar
  72. 72.
    Denry IL, Holloway JA (2004) Dental Mater 20:213CrossRefGoogle Scholar
  73. 73.
    Imbeni V, Kruzic JJ, Marshall GW, Marshall SJ, Ritchie RO (2005) Nat Mater 4:229CrossRefGoogle Scholar
  74. 74.
    Merkel I, Messerschmidt U (1992) Mater Sci Eng A 151:131CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Q. Chang
    • 1
    • 2
  • D. L. Chen
    • 1
    Email author
  • H. Q. Ru
    • 2
  • X. Y. Yue
    • 2
  • L. Yu
    • 2
  • C. P. Zhang
    • 2
  1. 1.Department of Mechanical and Industrial EngineeringRyerson UniversityTorontoCanada
  2. 2.Department of Materials Science and Engineering, School of Materials and MetallurgyNortheastern UniversityShenyangChina

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