Abstract
The size effects on the mean values of the mechanical properties of condensed matter and on the related variances are analysed by means of a unified approach based on the multiscale character of energy dissipation. In particular, the scaling law for fragmentation energy density is obtained taking into account the self-similarity of fragments. It is based on a generalization of the three classical comminution laws that has been performed to evaluate the energy dissipation, computing volume and surface area of the particles for one- two- and three-dimensional fragmented objects. The result is general and can be applied to different fractal energy dissipation mechanisms, e.g., plasticity. Based on this approach, the scaling laws for mean and standard deviation values of the main mechanical properties of materials can be derived, like Young's and shear elastic moduli, ultimate normal and shear stresses and strains, fracture energy and toughness.
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References
Béla Beke, D. (1964). Principles of Comminution. Publishing House of the Hungarian Academic of Sciences, Budapest.
Bond, F.C. (1952). The third theory of comminution. Mining. Enginee. 4, 484-494.
Bouchaud, E., Lapasset, G. and Planes, G. (1990). Fractal dimension of fractured surfaces: a universal value? Europhys. Lett. 13, 73-79.
Carpinteri, A. and Pugno, N. (2002a). One-two and three-dimensional universal laws for fragmentation due to impact and explosion. J. Appl. Mech. 69, 854-856.
Carpinteri, A. and Pugno, N. (2002b). A fractal comminution approach to evaluate the drilling energy dissipation. Numerical and Analytical Methods in Geomechanics 26, 499-513.
Carpinteri, A. and Pugno, N. (2002c). Fractal fragmentation theory for shape effects of quasi-brittle materials in compression. Mag. of Concrete Research 54(6), 473-480.
Carpinteri, A. and Pugno, N. (2003). A multifractal comminution approach for drilling scaling laws. Powder Technology 131(1), 93-98.
Carpinteri, A. (1997). Structural Mechanics — A Unified Approach E & FN Spon.
Carpinteri, A. (1994a). Scaling laws and renormalization groups for strength and toughness of disordered materials. Int. J. Solid Struct. 31, 291-302.
Carpinteri, A. (1994b). Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech. Mater. 18, 89-101.
Carpinteri, A., Chiaia, B. and Cornetti P. (2002). A scale-invariant cohesive crack model for quasi-brittle materials. Enginee. Fract. Mech. 69(2), 207-217.
Engleman, R., Rivier, N. and Jaeger, Z. (1988). Size distribution in sudden breakage by the use of entropy maximization. J. Appl. Phys. 63, 4766-4768.
Feder, J. (1988). Fractals. Plenum Publishing Corporation, New York.
Griffith, A.A. (1921). The phenomenon of rupture and flow in solids. Philosoph. Trans. of the Royal Society, London A221, 163-198.
Hahner, P., Bay, K. and Zaiser, M. (1988) Fractal dislocation patterning during plastic deformation. Physical Review Letters 81, 2470-2473.
Kendall, K. (1978). The impossibility of comminuting small particles by compression. Nature 272, 711-712.
Kick, F. (1985). Das Gesetz der Proportionalen Widerstände. Leipzig.
Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. Freeman, New York.
Mandelbrot, B. (1967). How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156, 636-638.
Nakamura, A. and Fujiwara, A. (1991). Icarus 92, 132.
Novozhilov, V. (1969). On a necessary and sufficient criterion for brittle strength. Prik. Mat. Mek. 33, 212-222.
Smekal, A. (1937). Physikalisches und technisches Arbeitsgesetz der Zerkleinerung. Zeitschr. VDI, Beiheft Verfahrenstechnik.
Simonov, I.V. (2002). The feasibility of using large impact to destroy a dangerous asteroid. Int. J. of Impact Engineering 27, 293-315.
Treacy, M.M., Ebbesen, T.W. and Gibson, J.M. (1996). Exceptionally high Young's modulus observed for individual carbon nanotubes. Nature 381, 678-680.
Turcotte, D.L. (1992). Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge.
von Rittinger, P.R.(1867). Lehrbuch der Aufbereitungskunde. Berlin.
Weiss, J. (2001). Fracture and fragmentation of ice: a fractal analysis of scale invariance. Enginee. Fract. Mech. 68, 1975-2012.
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Carpinteri, A., Pugno, N. Scale-effects on mean and standard deviation of the mechanical properties of condensed matter: an energy-based unified approach. International Journal of Fracture 128, 253–261 (2004). https://doi.org/10.1023/B:FRAC.0000040988.61253.05
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DOI: https://doi.org/10.1023/B:FRAC.0000040988.61253.05