The fractal-statistical approach to the size-scale effects on material strength and toughness

The size-scale effects on the mechanical properties of materials are a very important topic in engineering design. In recent years, a great deal of research on size-scale effects has been carried out in order to gain a precise description of this phenomenon and to highlight the physical mechanisms that lie behind it. Three different approaches have been proposed or at least analyzed. These include the statistical [Weibull W. A statistical theory of the strength of materials. Proceedings of the Royal Swedish Institute of Engineering Research 1939;151:1–45], the energetical [Bažant ZP. Size effect in blunt fracture: Concrete, rock, metal. Journal of Engineering Mechanics (ASME) 1984;110:518–35. [2]] and the fractal approach [Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials 1994;18:89–101. Internal Report, Laboratory of Fracture Mechanics, Politecnico di Torino, N. 1/92, 1992; Carpinteri A. Scaling laws and renormalization groups for strength and toughness of disordered materials. International Journal of Solids and Structures 1994;31:291–302].The fractal approach, which exploits the fractal nature of fracture [Molosov AB, Borodich FM. Fractal fracture of brittle bodies during compression. Soviet Physics-Doklady 1992;37:263–5. [5]], has been a matter of intense debate, particularly in the papers by Bažant [Scaling of quasibrittle fracture and the fractal question. Journal of Materials and Technology (ASME) 1995;117:361–7; Scaling of quasibrittle fracture: Hypotheses of invasive and lacunar fractality, their critique and Weibull connection. International Journal of Fracture 1997;83:41–65; Statistical and fractal aspects of size effect in quasibrittle structures. In: Shiraishi, editor. Structural safety and reliability. Rotterdam: Balkema; 1998. p. 1255–62], Borodich [Fractals and fractal scaling in fracture mechanics. International Journal of Fracture 1999;95:239–59], Bažant and Yavari [Is the cause of size effect on structural strength fractal or energetic-statistical? Engineering Fracture Mechanics 2005;72:1–31] and, more recently, by Saouma and Fava [On fractals and size effects. International Journal of Fracture 2006;137:231–49], who question its validity and even argue that it lacks sound physical and mathematical basis. In this long standing controversy about the interpretation of scaling laws on material strength [Carpinteri A, Pugno N. Are scaling laws on strength of solids related to mechanics or to geometry? Nature Materials 2005;4:421–3. [12]], the fractal approach has been counterposed to the energetical approach at first and to the so-called energetical-statistical one only more recently.The 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, as shown in several papers [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 2002; 13:233–46. [13]; Carpinteri A, Cornetti P, Puzzi S. A stereological analysis of aggregate grading and size effect on concrete tensile strength. International Journal of Fracture 2004;128:233–42; Carpinteri A, Cornetti P, Puzzi S. Scale effects on strength and toughness of grained materials: An extreme value theory approach. Strength, Fracture and Complexity 2005;3:175–88; 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 2005;44:192–9]. By analyzing in detail 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 for 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. And finally, the theoretical results regarding the link between fractals and statistics will be confirmed by numerical simulations.