Issue 10

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 10 (2009) 3-11 ; DOI: 10.3221/IGF-ESIS.10.01 5 softening branch and a catastrophical event occurs if the loading process is deflection-controlled. Such indenting branch is not virtual only if the loading process is controlled by a monotonically increasing function of time (Biolzi et al. [21]). Figure 1 : Constitutive laws of the cohesive crack model: (a) undamaged material; (b) process zone. In the case of the cracked beam, on the contrary, the initial crack makes the specimen behaviour more ductile; for the set of S E numbers considered in Fig. 2b, the snap-back does not occur. By varying the initial crack depth, it is possible to describe the gradual transition from simple fold catastrophe (softening) to bifurcation or cusp catastrophe (snap-back instability), generating an entire equilibrium surface, or the catastrophe manifold. Figure 2 : Dimensionless load vs. deflection diagrams by varying the brittleness number S E , initially uncracked (a) and cracked (b) specimen T HE FRACTAL INTERPRETATION OF THE SIZE - SCALE EFFECT he second topic is concerned with the size-scale effects on the mechanical properties of heterogeneous disordered materials that can be interpreted synthetically through the use of fractal sets. Fractal sets are characterized by non- integer dimensions (Mandelbrot [22]) . For instance, the dimension α of a fractal set in the plane can vary between 0 and 2. Accordingly, increasing the measure resolution, its length tends to zero if its dimension is smaller than 1 or tends to infinity if it is larger. In these cases, the length is a nominal, useless quantity, since it diverges or vanishes as the measure resolution increases. A finite measure can be achieved only using noninteger units, such as meters raised to α  l. Fractals sets can be profitably used to describe the size-scale effects on the parameters of the cohesive crack model. As shown in the previous section, this model captures the ductile-brittle transition occurring by increasing the size of the structure. On the other hand, uniaxial tensile tests on dog-bone shaped specimens [23,24] have shown that the three material parameters defining the cohesive law are size dependent: increasing the specimen size, the tensile strength  u , tends to decrease, whilst the fracture energy G F and the critical displacement w c increase. In order to overcome the original cohesive crack model drawbacks, a scale-independent (fractal) cohesive crack model has been proposed recently by the first Author [25]. This model is based on the assumption of a fractal-like damage localization, suggested by experimental evidence [26,27]. T  u  u F u c 1 2 w   G