tesi A. Caggiano.pdf - EleA@UniSA - Università degli Studi di Salerno

Chapter 3. Zero-thickness interface model for FRCCwith p i alternatively equals to χ, c or t anφ. Last equation defines the typical degradationlaw of the internal parameters from their maximum or initial values, p i = p 0i , tothe residual ones, p i = r p p 0i , in terms of the scaling function S[ξ pi ], whereS[ξ pi ] =e −α p i ξ pi1 + (e −α p i − 1)ξ pi(3.23)in which the parameter α pi controls the decay form of the internal parameter as shownin Figure 3.7, while the non-dimensional variable ξ pi introduces the influence of theratio between the current fracture work spent and the available fracture energy, in thedecay function Eq. (3.22) asξ pi1.00cosine-based Serie6 lawSerie7 linear law0.800.600.400.200.000 0.2 0.4 0.6 0.8 1w cr /G fI or w cr /G fIIaFigure 3.8: Cosine-based vs. linear law related to the ratio between the work spent w cr and theavailable fracture energies G I f or G I I a .f⎧⎨ξ χ =⎩[ (121 − cosπw crG I f)]if w cr ≤ G I f1 otherwise(3.24)⎧⎨ξ c = ξ tanφ =⎩[ (121 − cosπw crG I I af)]if w cr ≤ G I I af1 otherwise(3.25)according to the C 1 continuity function proposed in Caballero et al. [2008]. Figures3.8 and 3.9 show typical curves obtained with Eqs. (3.24) or (3.25) and the respectivederivates compared to the original linear proposal in Carol et al. [1997].54