Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 512 tensile meridian, respectively. The deviatoric plane of a Willam-Warnke failure model is indicated in Fig. 8. In this figure, the TXC, TXE and SHR stand for triaxial compression, triaxial tension and pure shear respectively. Figure 8: Deviatoric section proposed by Willam-Warnke model The ˆ[ ( ), ] r p   , which is the ratio between the current radius of the failure surface ( ) r  and the distance of the failure surfaces from the hydrostatic axis at the compressive meridian c r , is computed by means of the Eq. (15). This equation was obtained by dividing both sides of Eq. (14) by c r . In order to present the term ( ) p  , which is a strength index of brittle material related to the confining pressure that a material is subjected to it and equal to t c     (in KCC model also equal to t c r r ), both the numerator and the denominator of the right-hand side of equation are divided by 2 c r . 2 2 2 2 2 2 2 2(1 )cos (2 1) 4(1 )cos 5 4 ( ) ˆ[ ( ), ] 4(1 )cos (1 2 ) c r r p r                         (15) The fact that ˆ r is just a function of ( ) p  and θ , and the lode angle can be determined based on the loading conditions, implies the role of ( ) p  for computational purposes. It means that the implementation of the three invariant failure surfaces is completed by means of this parameter ( ) p  . This parameter generally depends on the hydrostatic stress and can be obtained empirically. Malvar et al. in [28] defined this parameter as a linear piecewise function on the full range of pressure according to Eq. (16). 0 1 2 1 , 0 2 1 2 3 2 , 3 ( ) , 2 3 2 3 2 3 0.753, 3 1, 8.45 t t c c c c c c c c f p f f p f f p p f f a a a f p f p f                                      (16) Where c f  is the unconfined compression strength, t f is the principal tensile strength and α is an experimental parameter related to the biaxial compression test. According to the Eq. (16), ( ) p  varies from 1⁄2 to 1, which is in accordance with the experimental data previously obtained. It also indicates that 8.45 c p f   is the transition point in which the compression meridian is equal to tension one, and accordingly from this point onwards, there is a circular failure surface on the deviatoric