Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 517 Figure 10: The Linear Drucker-Prager criterion; (a) t-p plane, (b) deviatoric plane. E [MPa] PR  [°] K [-]  [°] 15374 0.28 65.38 0.778 16.35 Table 6 : The Drucker-Prager plasticity parameters. In this study, the failure criterion is intended as a yield surface, i.e. when Eq. (21) is satisfied, plastic flow occurs. Hardening of the yield function follows until the maximum strength is reached, afterwards failure is modeled by softening. Hardening (intended as hardening followed by softening) of the yield function (see Fig. 11) is described by the equation proposed by Lubliner et al., [42] for concrete as defined in Eq. (27). 0 [(1 ) exp( ) exp( 2 )] p p a f a b b              (27) Eq. (27) expresses the flow stress  in a uniaxial test as a function of the axial plastic strain p  . The elastic properties are taken from [36]. The Mohr-Coulomb friction angle is obtained through the procedure described in [43] using the properties reported in Tab. 7. To assure convexity of the yield function, the value of the flow-stress ratio is limited by a minimum value of 0.778. Since the obtained value is lower, the minimum value allowable is used instead resulting in the criterion obtained not being equivalent to the starting Mohr-Coulomb criterion but being its best approximation [30]. UCS, m f [MPa] Tensile splitting strength [MPa] Yield stress, 0 f [MPa] p  67.99 5.66 53.79 0.00057 Table 7. The experimental data of the Pietra-Serena sandstone. A good approximation of the dilation angle for rock showing brittle behavior is equal to 1/4 times the friction angle [44]. In Tab. 7, the parameters to define the curve of Eq. (27) are reported. The parameters a and b , which are obtained using Eqs. (28) and (29), are determined equal to 2.68 and 665.32, respectively. It is worth mentioning that the plastic strain at Eq. (29) (to define the parameter b ) is determined when the derivative of Eq. (27) is equal to zero. This plastic strain corresponds to the experimental data at which the maximum stress is reached. 2 0 0 0 ( ) 1 2 ( ) ( ) m m m f f f a f f f     (28) 1 ln p a b a           (29) (a) (b)