Issue 10

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11; DOI: 10.3221/IGF-ESIS.10.01 6 Let us consider fractal geometries for both the resistant cross section at maximum load (Fig. 3a) and the dissipation domain (Fig. 3c) [25]. Hence we can compute the maximum load F , the critical displacement w c and the total dissipated energy W as: * * u 0 u res A A F     (1a) ε 1- * ε ε d c c c w b b   (1b) * F 0 F dis A A * W   G G (1c) These quantities are size-dependent. The true scale-independent quantities are the right hand side ones, i.e. the fractal strength  u *, the f ractal critical strain ε c * and the fractal fracture energy G F *. They show non-integer physical dimensions: [F][L] –(2– d σ) for  u *, [L] ( d ε) for w c and [FL][L] –(2+ dG ) for G F *. Because of the measure of the resistant cross section A res and the dissipation domain A dis , from Eqs. (1) the scaling laws for strength, critical displacement and fracture energy can be obtained: σ * u u d b     (2a) ε 1- * ε d c c w b  (2b) F F d * b   G G G (2c) Figure 3 : A concrete specimen subjected to tension. Fractal localization of the resistant cross section (a); fractal localization of the strain (b) and the energy dissipation inside the damaged band (c). The three size effect laws (2) of the cohesive law parameters are not completely independent of each other. In fact, there is a relation among the scaling exponents that must be always satisfied. In order to get this relation, the simplest path is to consider the damage domain in Fig. 3c as the cartesian product of those i n Figs. 3a and 3b. As a result, we obtain: σ ε 1 d d d    G (3) According to these definitions, we call the  *  ε* diagram the fractal or scale-independent cohesive law. Contrarily to the classical cohesive law, which is experimentally sensitive to the structural size, this curve is an exclusive property of the material since it is able to capture the fractal nature of the damage process. The area below the softening fractal stress- strain diagram represents the fractal fracture energy G F *. In order to validate the model, it has been applied to the data obtained in 1994 by Carpinteri and Ferro [23,24] for tensile tests on dog-bone shaped concrete specimens of various sizes under controlled boundary conditions (Fig. 4a). They