Issue 47

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 47 (2019) 394-400; DOI: 10.3221/IGF-ESIS.47.29 399 The softening curve is defined by three main ingredients: the decohesion strength, f decoh , which is the value of the softening curve for null crack opening; the area comprised between the curve and the abscissa axis, which represents the specific fracture energy of the material, G F ; and the shape of the softening curve, which depends on the way in which the material is damaged as the crack is opened. In the numerical simulations presented in this paper, the CZM was implemented in combination with the embedded crack approach. Further details about the implementation can be found in [13, 14]. When applying the CZM to quasi-brittle materials, the decohesion strength is usually set equal to the tensile strength of the material measured through a splitting tensile test [15], which according to the experimental campaign previously detailed was equal to f t =7.1MPa (Young modulus was taken as E = 75 GPa). About the area under the softening curve, it was set equal to the specific fracture energy determined in the three-point-bending tests under static conditions, that is to say, G F = 60 N/m, considering the results of two experimental tests reported in Fig. 2. Unfortunately, up to date no general procedure has been set to determine the shape of the softening curve for a given material. There are some standard curves, such as the bi-linear curve for plain concrete [16], or the rectangular curve for steels [17] but to the authors' knowledge, no standard has been set in the case of marble. For this reason, a trial-and-error procedure was made by numerical simulation of the three-point-bending tests conducted for the determination of the specific fracture energy, trying different softening curve shapes until achieving a good agreement with the experimental results. To facilitate this trial-and-error process, according to [18], the following family of softening curves was used:     * 1 1 1 n n f e e            , where f * is the non-dimensional stress transferred across the crack lips (stress divided by the decohesion strength),  is the non-dimensional crack opening (crack opening divided by the critical crack opening) and n is a material constant. With this family of curves it is possible to modify the shape of the softening curve by playing with the n parameter ( n = 0 leads to a linear softening curve). In the case of the marble analyzed in this paper, it was found that a value of n =2 leads to a reasonable agreement between the experimental behaviour and the numerical prediction, as is shown in Fig. 2. The numerical simulation of the three-point-bending static tests allowed us to provide a meso-mechanics insight on the results of cyclic loading tests. In quasi-brittle materials, it has been demonstrated that for samples having dimensions of the order of magnitude of the size of the Fracture Process Zone (FPZ), the peak load is reached after such an FPZ has been largely developed ahead of the notch tip of the specimen. Thanks to the CZM, it is possible to analyze the initiation and evolution of the FPZ, since it is equivalent to the cohesive zone. Fig. 6 shows the length of the cohesive zone (FPZ) as a function of the nominal stress intensity factor, K I , applied during the simulation of the three-point bending static test. In this plot the K I required to initiate the FPZ is marked, which is considerably lower than the K I corresponding to the peak load. In the same plot, the  K I corresponding to the fatigue propagation threshold is also highlighted. It can be seen how the latter is mostly inside the region of FPZ generation, revealing that a significant level of damage (corresponding to a large FPZ) ahead of the notch tip is required for fatigue crack propagation. Actually, according to this analysis, those  K I ranges below the threshold represented in the figure would not lead to crack propagation in spite of causing appreciable damage (FPZ) at the notch tip. It must be reminded that this result is solely based on a numerical simulation, requiring further experimental data to confirm the hypothesis. Figure 6 : Fracture process zone (FPZ) size in terms of % of the ligament as a function of the stress intensity K I applied. The figure also shows the  K I corresponding to the fatigue crack propagation threshold. 0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 1.2 FPZ size FPZ size (% of ligament) K I (MPaꞏm 1/2 ) K I required for FPZ initiation K I max in fatigue propagation threshold K I min in fatigue propagation threshold K I corresponding to peak load  K I for fatigue propagation threshold