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

Chapter 8. Elasto-plastic microplane formulation for FRCCaimed at controlling the evolution of the model parameters by means of the followingexpression[ (p mi ci= 1 − 1 − rpmi c)]S[ξ pmi c ] p mi ci0i(8.20)where p mi c alternatively equals to χ mi c , c mi c , t an(φ mi c ) (hyperbolic function) andiσ N ,C AP (ellipse). These model parameters vary from their maximum values p mi c to0ithe corresponding residual ones rpmi c p mi c . The parameter α0i pi controls the decay formof the internal [ parameter ] as shown in Figure 8.4, while the non-dimensional variablew cr /G # introduces the influence of the ratio between the current fracturefξ # = ξ pmi ciwork spent and the available fracture energy as follows⎧⎨ξ χmi c =⎩[ (121 − cosπw crG I f)]if w cr ≤ G I f1 otherwise(8.21)⎧⎨ξ cmi c = ξ tan(φmi c ) = ξ σ N ,C AP=⎩[ (121 − cosπw crG I I af)]if w cr ≤ G I I af1 otherwise(8.22)according to the function proposed in Caballero et al. [2008]. Figure 8.5 shows thetypical curves obtained with Eqs. (8.21) or (8.22).S 1.0αpi = - 20.8αpi = - 10.60.4αpi = 0αpi = + 10.20.0αpi = + 20.0 0.2 0.4 0.6 0.8 1.0Figure 8.4: S[ξ # ] scaling function of the microplane model in terms of the ξ # parameter, wherethe # symbol = χ mi c ,c mi c , tan(α mi c ) or σ N ,C AP .162