# The Fractal-Statistical Nature of Size-Scale Effects on Material Strength and Toughness

• Alberto Carpinteri
• Simone Puzzi
Conference paper
Part of the Iutam Bookseries book series (IUTAMBOOK, volume 10)

## Abstract

The size-scale effects on the mechanical properties of materials are a very important topic in engineering design. Three different modeling approaches have been proposed and analyzed at least, i.e. the statistical, the energetical and the fractal one. Aim of this paper is to revisit the fractal approach and to reject the most recurrent criticisms against it. Moreover, we will show that it is wrong to set the fractal approach to size-scale effects against the statistical one, since they are deeply connected. More in detail, by analyzing a fractal distribution of micro-cracks in the framework of Extreme Value theory, we will obtain a scaling law for tensile strength characterized, in the bi-logarithmic plot, by the slope -1/2. Conversely, by considering a fractal grain size distribution in a grained material, we will obtain a scaling law or fracture energy characterized, in the bi-logarithmic plot, by the positive slope 1/2. These slopes are the natural consequence of perfect self-similarity of the flaw (or grain) size distribution. Eventually, the theoretical results regarding the link between fractals and statistics will be confirmed by numerical simulations.

## Keywords

Size-scale effects self-similarity fractals

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Weibull W. “A statistical theory of the strength of materials”, Proceedings of the Royal Swedish Institute of Engineering research, vol. no. 151, pp. 1–45, 1939.Google Scholar
2. 2.
Bažant ZP. “Size effect in blunt fracture: Concrete, rock, metal”, Journal of Engineering Mechanics (ASME), vol. no. 110, pp. 518–535, 1984.
3. 3.
Carpinteri A. “Fractal nature of material microstructure and size effects on apparent mechanical properties”, Mechanics of Materials, vol. no. 18, pp. 89–101, 1994a. Internal Report, Laboratory of Fracture Mechanics, Politecnico di Torino, N. 1/92, 1992.
4. 4.
Carpinteri A. “Scaling laws and renormalization groups for strength and toughness of disordered materials”, International Journal of Solids and Structures, vol. no. 31, pp. 291–302, 1994.
5. 5.
Carpinteri A, Cornetti P. “Size effects on concrete tensile fracture properties: an interpretation of the fractal approach based on the aggregate grading”, Journal of the Mechanical Behavior of Materials, vol. no. 13, pp. 233–246, 2002.Google Scholar
6. 6.
Carpinteri A, Cornetti P, Puzzi S. “A stereological analysis of aggregate grading and size effect on concrete tensile strength”, International Journal of Fracture, vol. no. 128, pp. 233–242, 2004.
7. 7.
Carpinteri A, Cornetti P, Puzzi S. “Scale effects on strength and toughness of grained materials: an extreme value theory approach”, Strength, Fracture and Complexity, vol. no. 3, pp. 175–188, 2005.Google Scholar
8. 8.
Carpinteri A, Cornetti P, Puzzi S. “Size effect upon grained materials tensile strength: the increase of the statistical dispersion at the smaller scales”, Theoretical and Applied Fracture Mechanics, vol. no. 44, pp. 192–199, 2005.
9. 9.
Mandelbrot BB, Passoja DE, Paullay AJ. “Fractal character of fracture surfaces of metals”, Nature, vol. no. 308, pp. 721–722, 1984.
10. 10.
Bouchaud E. “Scaling properties of cracks”, Journal of Physics: Condensed Matter, vol. no. 9, pp. 4319–4344, 1997.
11. 11.
Williford RE. “Similarity analysis of fracture”, American Society of Mechanical Engineers, Aerospace Division (Publication) AD, vol. no. 12, pp. 39–44, 1987.Google Scholar
12. 12.
Mu ZQ, Lung CW. “Studies on the fractal dimension and fracture toughness of steel”, Journal of Physics D: Applied Physics, vol. no. 21, pp. 848–50, 1988.
13. 13.
Heping X. “Fractal effect of irregularity of crack branching on the fracture toughness of brittle materials”, International Journal of Fracture, vol. no. 41, pp. 267–274, 1989.
14. 14.
Borodich FM. “Fracture energy in a fractal crack propagating in concrete or rock”, Doklady Akademii Nauk (Russia), vol. no. 325, pp. 1138–1141, 1992. (English transl. in: Transl. Russian Akademy of Sciences: Earth Science Sections, vol. no. 327, pp. 36–40).Google Scholar
15. 15.
Borodich FM. “Fracture energy of brittle and quasi-brittle fractal cracks”, IFIP Transactions: A. Computer Science and Technology, vol. no. 41, pp. 61–68, 1994.Google Scholar
16. 16.
Borodich FM. “Some fractal models of fracture”, Journal of the Mechanics and Physics of Solids, vol. no. 45, pp. 239–259, 1997.
17. 17.
Weiss J. “Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy”, International Journal of Fracture, vol. no. 109, pp. 365–381, 2001.
18. 18.
Carpinteri A, Chiaia B. “Power scaling laws and dimensional transitions in solid mechanics”, Chaos, Solitons Fractals, vol. no. 7, pp. 1343–1364, 1996.
19. 19.
Carpinteri A, Ferro G. “Size effects on tensile fracture properties: a unified explanation based on disorder and fractality of concrete microstructure”, Materials & Structures (RILEM), vol. no. 27 pp. 563–571, 1994.
20. 20.
Carpinteri A, Chiaia B, Ferro G. “Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder”, Materials & Structures (RILEM), vol. no. 28, pp. 311–317, 1995.
21. 21.
Carpinteri A, Chiaia B. “Multifractal nature of concrete fracture surfaces and size effects on nominal fracture energy”, Materials and Structures, vol. no. 28, pp. 435–443, 1995.
22. 22.
Carpinteri A, Chiaia B. “Size effects on concrete fracture energy: dimensional transition from order to disorder”, Materials and Structures, vol. no. 29, pp. 259–266, 1996.
23. 23.
Carpinteri A, Chiaia B, Cornetti P. “On the mechanics of quasi-brittle materials with a fractal microstructure”, Engineering Fracture Mechanics, vol. no. 70, pp. 2321–2349, 2003.
24. 24.
Carpinteri A, Cornetti P, Puzzi S. “Scaling laws and multiscale approach in the mechanics of heterogeneous and disordered materials”, Applied Mechanics Reviews, vol. no. 59, pp. 283–305, 2006.
25. 25.
Bažant ZP. “Scaling of quasibrittle fracture and the fractal question”, Journal of Materials and Technology (ASME), vol. no. 117, pp. 361–367, 1995.
26. 26.
Bažant ZP. “Scaling of quasibrittle fracture: Hypotheses of invasive and lacunar fractality, their critique and Weibull connection”, International Journal of Fracture, vol. no. 83, pp. 41–65, 1997.Google Scholar
27. 27.
Bažant ZP. “Statistical and fractal aspects of size effect in quasibrittle structures”. In: N. Shiraishi et al. (eds.), Structural Safety and Reliability, 1998, pp. 1255–1262. Rotterdam: Balkema.Google Scholar
28. 28.
Borodich FM. “Fractals and fractal scaling in fracture mechanics”, International Journal of Fracture, vol. no. 95, pp. 239–259, 1999.
29. 29.
Bažant ZP et al. “Quasibrittle fracture scaling and size effect”, Materials and Structures (RILEM), vol. no. 37, pp. 547–568, 2004.Google Scholar
30. 30.
Bažant ZP, Yavari A. “Is the cause of size effect on structural strength fractal or energetic – statistical?”, Engineering Fracture Mechanics, vol. no. 72, pp. 1–31, 2005.
31. 31.
Saouma VE, Fava G. “On fractals and size effects”, International Journal of Fracture, vol. no. 137, pp. 231–249, 2006.
32. 32.
Carpinteri A. Mechanical Damage and Crack Growth in Concrete: Plastic Collapse to Brittle Fracture, Dordrecht, Martinus Nijhoff Publishers, 1986.
33. 33.
Carpinteri A. “Decrease of apparent tensile and bending strength with specimen size: two different explanations based on fracture mechanics”, International Journal of Solids and Structures, vol. no. 25, pp. 407–429, 1989.
34. 34.
Carpinteri A, Ferro G, Invernizzi S. “The nominal tensile strength of disordered materials: a statistical fracture mechanics approach”, Engineering Fracture Mechanics, vol. no. 58, pp. 421–435, 1997.
35. 35.
Newman MEJ. “Power laws, Pareto distributions and Zipf’s law”, Contemporary Physics, vol. no. 46, pp. 323–351, 2005.
36. 36.
Williams ML. “Stress singularities resulting from various boundary conditions in angular corners of plates in extension”, Journal of Applied Mechanics, vol. no. 19, pp. 526–528, 1952.Google Scholar
37. 37.
Leicester RH. Effect of Size on the Strength of Structures, Division of Building Research, Forest Products Laboratory, C.S.I.R.O., Melbourne, 1973.Google Scholar
38. 38.
Schwartz MM. Joining of composite-matrix materials, ASM International, 1994.Google Scholar
39. 39.
Huang J, Li VC. “A meso-mechanical model of the tensile behaviour of concrete. Part I: modelling of the pre-peak stress-strain relation”, Composites, vol. no. 20, pp. 361–378, 1989.
40. 40.
Wolinski S, Hordijk DA, Reinhardt HW, Cornelissen HAW. “Influence of aggregate size on fracture mechanics of parameters of concrete”, International Journal of Cement Composites and Lightweight Concrete, vol. no. 9, pp. 95–103, 1987.
41. 41.
Li Q, Duan Y, Wang G. “Behaviour of large concrete specimens in uniaxial tension”, Magazine of Concrete Research, vol. no. 54, pp. 385–391, 2002.
42. 42.
Carpinteri A, Chiaia B, Cornetti P. “A scale-invariant cohesive crack model for quasi-brittle materials”, Engineering Fracture Mechanics, vol. no. 69, pp. 207–217, 2002.
43. 43.
Wittmann FH, Mihashi H, Nomura N. “Size effect on fracture energy of concrete”, Engineering Fracture Mechanics, vol. no. 35, pp. 107–115, 1990.

## Authors and Affiliations

• Alberto Carpinteri
• 1
• Simone Puzzi
1. 1.Department of Structural Engineering and GeotechnicsPolitecnico di Torino Corso Duca degli Abruzzi 24Italy