During the softening regime, i.e. when most of the dissipation occurs, σ* decreases from the maximum value σu* to 0, while ε* grows up to εc*. In the meantime, the non-damaged parts of the bar undergo elastic unloading. We call the σ*-ε* diagram the fractal cohesive law, which is shown in Figure 4b. Contrarily to the classical cohesive law, which is sensitive to the structural size, this curve is scale-independent. Experimental tests by van Mier & van Vliet [7] have shown that, with increasing the specimen size, the peak of the curve decreases whereas the tail rises, i.e., tensile strength decreases while critical displacement increases. More in detail, wc varies more rapidly than σu does. Therefore, an increase of the area beneath the cohesive law, i.e. of the fracture energy, is observed. Thus, the experimental trends of σu, G F and wc confirm the assumptions of the fractal model. Figure 5. Size-effect tests [7]: stress-strain diagrams (a), cohesive curves (b) and fractal cohesive law (c). The fractal model has been eventually compared with the uniaxial tensile tests described in [8]. The attention was focused on the size-effect on the ultimate tensile stress and on the fracture energy and their values interpreted by means of fractal assumptions. The exponents of the scaling laws were deduced by fitting the experimental results. In particular, they found the values dσ = 0.14 and dG = 0.38. The nominal σ - ε and σ - w diagrams are reported in Figure 5a and 5b. Here, w is the displacement localized in the damage band, obtained by subtracting from the total one the displacement due to elastic and inelastic pre-peak deformation. In addition, the value dε = 0.48 is provided by Eqn. (2), so that the fractal cohesive laws can be represented as in Figure 5c. As expected, all the curves related to the single sizes tend to merge in a unique, scale-independent cohesive law. ACKNOWLEDGEMENTS Supports by the Italian Ministry of University and Scientific Research and by the EC-TMR Contract N° ERBFMRXCT 960062, are gratefully acknowledged. REFERENCES 1. A. Carpinteri (1994) Mechanics of Materials 18, 89. 2. P. Cornetti (1999) Fractals and Fractional Calculus in the Mechanics of Damaged Solids, Ph.D. Thesis (in Italian). Politecnico di Torino, Torino. 3. T. Kleiser and M. Bocek (1986) Zeitschrift für Metallkunde 77, 582. 4. K.M. Kolwankar and A.D. Gangal (1996) Chaos 6, 505. 5. A. Carpinteri, B. Chiaia and P. Cornetti (2001) Computer Methods in Applied Mechanics and Engineering, in print. 6. A. Carpinteri and B. Chiaia (1995) Material and Structures 28, 435. 7. J.G.M. van Mier and M.R.A. van Vliet (1999) International Journal of Fracture 94, 195. 8. A. Carpinteri and G. Ferro (1994) Materials and Structures 27, 563. 0 1 2 3 4 5 0 0.0002 0.0004 strain stress (MPa) 0 1 2 3 4 5 0 50 100 150 200 localized displacement ( µm) stress (MPa) 0 2 4 6 8 0 0.004 0.008 0.012 fractal strain (mm/mm^0.52) fractal stress (N/mm^1.86) (b) (a) (c) b=50 b=100 b=200 b=200 b=100 b=50
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