A fractal approach to indentation size effect

Engineering Fracture Mechanics

Volumn 73, Issue 15, 2006, Pages 2110-2122

  Carpinteri, Alberto ^a   Puzzi, Simone ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

Fractals; Hardness; Microindentation; Nanoindentation; Size scale effects

Indexed keywords

DISLOCATIONS (CRYSTALS); FRACTALS; HARDNESS; PLASTIC DEFORMATION; POLYCRYSTALLINE MATERIALS; SINGLE CRYSTALS; STRAIN HARDENING;

INDENTATION SIZE EFFECT; MICROINDENTATION; NANOINDENTATION; SIZE-SCALE EFFECTS;

INDENTATION;

DISLOCATIONS (CRYSTALS); FRACTALS; HARDNESS; INDENTATION; PLASTIC DEFORMATION; POLYCRYSTALLINE MATERIALS; SINGLE CRYSTALS; STRAIN HARDENING;

EID: 33746813116     PISSN: 00137944     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.engfracmech.2006.04.020     Document Type: Article

References (68)

1. Mushkhelishvili V.I. Some basic problems of the mathematical theory of elasticity (1953), P. Noordhoof Ltd., Groningen, Holland
2. Dundurs J., and Lee M.S. Stress concentrations at a sharp edge in contact problems. J Elast 2 (1972) 109-112
3. Gdoutos E.E., and Theocaris P.S. Stress concentrations at the apex of a plane indenter acting on an elastic half plane. J App Mech (ASME) 42 (1975) 688-692
4. Tabor D. The hardness of metals (1951), Oxford University Press, Oxford
5. Xue Z., Huang Y., Hwang K.C., and Li M. The influence of indenter tip radius on the microindentation hardness. J Engng Mater Tech 124 (2002) 371-379
6. Stelmashenko N.A., Walls M.G., Brown L.M., and Milman Y.V. Microindentation on W and Mo oriented single crystals: an STM study. Acta Metall Mater 41 (1993) 2855-2865
7. Nix W.D., and Gao H.J. Indentation size effects in crystalline materials: a law for strain gradient plasticity. Mech Phys Solids 46 (1998) 411-425
8. Ma Q., and Clarke D.R. J Size dependent hardness in silver single crystals. Mater Res 10 (1995) 853-863
9. Aifantis E.C. On the microstructural origin of certain inelastic models. J Engng Mater Tech 106 (1984) 326-330
10. Aifantis E.C. Strain gradient interpretation of size effects. Int J Fract 95 (1999) 299-314
11. De Guzman MS, Neubauer G, Flinn P, Nix WD. The role of indentation depth on the measured hardness of materials. In: Materials Research Society Symposium - Proceedings, vol. 308, 1993. p. 613-8.
12. Fleck N.A., Muller G.M., Ashby M.F., and Hutchinson J.W. Strain gradient plasticity: theory and experiment. Acta Metall Mater 42 (1994) 475-487
13. Fleck N.A., and Hutchinson J.W. Strain gradient plasticity. Adv Appl Mech 33 (1997) 295-361
14. Arsenlis A., and Parks D.M. Crystallographic aspects of geometrically-necessary and statistically stored dislocation density. Acta Mater 47 (1999) 1597-1611
15. Busso E.P., Meissonnier F.T., and O'Dowd N.P. Gradient-dependent deformation of two-phase single crystals. J Mech Phys Solids 48 (2000) 2333-2361
16. Gao H., and Huang Y. Taylor-based nonlocal theory of plasticity. Int J Solids Struct 38 (2001) 2615-2637
17. Al-Rub R.K.A., and Voyiadjis G.Z. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. Int J Plast 20 (2004) 1139-1182
18. Elmustafa A.A., and Stone D.S. Indentation size effect in polycrystalline FCC metals. Acta Mater 50 (2002) 3641-3650
19. Elmustafa A.A., and Stone D.S. Nanoindentation and the indentation size effect: kinetics of deformation and strain gradient plasticity. J Mech Phys Solids 351 (2003) 57-381
20. Gao H., and Huang Y. Geometrically necessary dislocation and size-dependent plasticity. Scr Mater 48 (2003) 113-118
21. Gao H., Huang Y., Nix W.D., and Hutchinson J.W. Mechanism-based strain gradient plasticity - I. Theory. J Mech Phys Solids 47 (1999) 1239-1263
22. Gao H., Huang Y., and Nix W.D. Modeling plasticity at the micrometer scale. Naturwissenschaften 86 (1999) 507-515
23. Kroner E. Dislocations and continuum mechanics. Appl Mech Rev 15 (1962) 599-606
24. Ashby M.F. The deformation of plastically non-homogenous alloys. Philos Mag 21 (1970) 399-424
25. Voyiadjis G.Z., and Al-Rub R.K.A. Gradient plasticity theory with a variable length scale parameter. Int J Solids Struct 42 (2005) 3998-4029
26. Gerberich W.W., Tymiak N.I., and Grunlan J.C. Interpretation of indentation size effects. J Appl Mech 69 (2002) 433-442
27. Zhang T.Y., and Xu W.H. Surface effects on nanoindentation. J Mater Res 17 (2002) 1715-1720
28. Mandelbrot B.B., Passoja D.E., and Paullay A.J. Fractal character of fracture surfaces of metals. Nature 308 (1984) 721-722
29. Kleiser T., and Bocek M. The fractal nature of slip in crystals. Zeitschrift für Metallkunde 77 (1986) 582-587
30. Hähner P., Bay K., and Zaiser M. Fractal dislocation patterning during plastic deformation. Phys Rev Lett 81 (1998) 2470-2473
31. Sžekely F., Groma I., and Lendvai J. Static properties of dislocation structures investigated by X-ray diffraction. Mater Sci Engng A 309-310 (2001) 352-355
32. Bouchaud E., Lapasset G., and Planés J. Fractal dimension of fractured surfaces: a universal value?. Europhys Lett 13 (1990) 73-79
33. Imre A., Pajkossy T., and Nyikos L. Electrochemical determination of the fractal dimension of fractured surfaces. Acta Metall Mater 40 (1992) 1819-1826
34. Mourot G., Morel S., Bouchaud E., and Valentin G. Anomalous scaling of mortar fracture surfaces. Phys Rev E 71 (2005) 016136
35. Carpinteri A., Chiaia B., and Invernizzi S. Three-dimensional fractal analysis of concrete fracture at the meso-level. Theor Appl Fract Mech 31 (1999) 163-172
36. Mandelbrot B.B. The fractal geometry of nature (1982), Freeman, New York
37. Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater 18 (1994) 89-101
38. Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. Int J Solids Struct 31 (1994) 291-302
39. Carpinteri A., Chiaia B., and Cornetti P. On the mechanics of quasi-brittle materials with a fractal microstructure. Engng Fract Mech 70 (2003) 2321-2349
40. Carpinteri A., Chiaia B., and Cornetti P. A scale-invariant cohesive crack model for quasi-brittle materials. Engng Fract Mech 69 (2002) 207-217
41. Carpinteri A., and Chiaia B. Multifractal nature of concrete fracture surfaces and size effects on nominal fracture energy. Mater Struct 28 (1995) 435-443
42. Carpinteri A., and Chiaia B. Size effects on concrete fracture energy: dimensional transition from order to disorder. Mater Struct 29 (1996) 259-266
43. Carpinteri A, Chiaia B, Ferro G. Multifractal scaling law: an extensive application to nominal strength size effect of concrete structures. Atti del Dipartimento di Ingegneria Strutturale, Politecnico di Torino, 1995, N. 51.
44. Borri-Brunetto M., Carpinteri A., and Chiaia B. Scaling phenomena due to fractal contact in concrete and rock fractures. Int J Fract 95 (1999) 221-238
45. Carpinteri A., and Paggi M. Size-scale effects on the friction coefficient. Int J Solids Struct 42 (2005) 2901-2910
46. Carpinteri A., and Spagnoli A. A fractal analysis of size effect on fatigue crack growth. Int J Fatigue 26 (2004) 125-133
47. Carpinteri A., Chiaia B., and Invernizzi S. Applications of fractal geometry and renormalization group to the Italian seismic activity. Chaos Solitons Fractals 14 (2002) 917-928
48. Lu C., Mai Y.-W., and Xie H. A sudden drop of fractal dimension: a likely precursor of catastrophic failure in disordered media. Philos Mag Lett 85 (2005) 33-40
49. Lu C., Mai Y.-W., and Bai Y. Fractals and scaling in fracture induced by microcrack coalescence. Philos Mag Lett 85 (2005) 67-75
50. Lu C., and Mai Y.-W. Influence of aspect ratio on barrier properties of polymer-clay nanocomposites. Phys Rev Lett 95 (2005) 088303
51. Carpinteri A., and Pugno N. Are scaling laws on strength of solids related to mechanics or to geometry?. Nature Mater 4 (2005) 421-423
52. Gil Sevillano J. Patterns in heavily deformed metals. Phys Scr T 49 (1993) 405-411
53. Gil Sevillano J., Bouchaud E., and Kubin L.P. The fractal nature of gliding dislocation lines. Scr Metall Mater 25 (1991) 355-360
54. Zaiser M., Marras S., Nadgorny E., Strunk H.P., and Aifantis E.C. Fractal dislocation patterning in plastically deformed NaCl polycrystals. Physica Status Solidi 185 (2001) R4-R5
55. Zaiser M., Bay K., and Hähner P. Fractal analysis of deformation-induced dislocation patterns. Acta Mater 47 (1999) 2463-2476
56. Miguel M.-C., Vespignani A., Zapperi S., Weiss J., and Grassok J.-R. Intermittent dislocation flow in viscoplastic deformation. Nature 410 (2001) 667-671
57. Uchic M.D., Dimiduk D.M., Florando J.N., and Nix W.D. Sample dimensions influence strength and crystal plasticity. Science 305 (2004) 986-989
58. Zaiser M., and Hähner P. The flow stress of fractal dislocation arrangements. Mater Sci Engng A 270 (1999) 299-307
59. Hähner P., and Zaiser M. Dislocation dynamics and work hardening of fractal dislocation. Mater Sci Engng A 272 (1999) 443-454
60. Hähner P. Statistical mechanics approach to dislocation cell patterning. Scr Mater 47 (2002) 705-711
61. Kubin LP, Devincre B. From dislocation mechanisms to dislocation microstructures and strain hardening. In: Bilde Soerensen JB et al., editors. Deformation-induced microstructures: analysis and relation to properties. Risø Natl Lab: Roskilde; 1999. p. 61-83.
62. Wilson K.G. Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys Rev B 4 (1971) 3174-3183
63. Barenblatt G.I. Scaling, self-similarity and intermediate asymptotics (1996), Cambridge University Press, Cambridge
64. Richeton T., Weiss J., and Louchet F. Breakdown of avalanche critical behaviour in polycrystalline plasticity. Nature Mater 4 (2005) 465-469
65. Swadener J.G., George E.P., and Pharr G.M. The correlation of the indentation size effect measured with indenters of various shapes. J Mech Phys Solids 50 (2002) 681-694
66. Kocks U.F. The relation between polycrystal deformation and single crystal deformation. Metall Trans 1 (1970) 1121-1143
67. Weiss J., and Marsan M. Three-dimensional mapping of dislocation avalanches: clustering and space/time coupling. Science 299 (2003) 89-92
68. Hasegawa T, Shioya T, Okada T. Size effect on splitting tensile strength of concrete. In: Proceedings of the Japan concrete institute, 7th conference, 1985. p. 309-12.