Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 511   1 2 3 ( 2 ) 3 2 2 3( ) cos 3 1 a l a l I J                    (10) Therefore, the equations of the three-failure surfaces for the axisymmetric state of stress can be written as the Eqs. (11), (12) and (13). 0 1 2 0.55 ( ) [ ( ) ; ( ) 0.45( ) ] 3 y m m y y y y p a p p a a p                  (11) 0 1 2 ( ) m p a a a p      (12) 1 2 ( ) r f f p a a p     (13) where, the i a - parameters are the user-defined input parameters to define the failure surfaces in the compressive meridian. These parameters can be calibrated based on the experimental data of the triaxial compression test by means of a curve- fitting approach, e.g. a genetic algorithm. The parameter p is the pressure (positive in compression) or the mean stress equal to 1 2 3 ( ) 3      . The meridian plane defined by means of the above-mentioned equations is shown schematically in Fig. 7. Figure 7 : Schematic representation of the KCC failure surfaces in the compressive meridian. As written in the Eq. (6), the KCC material model considers the effect of the third invariant, i.e. the Lode angle θ , by means of the function ˆ[ ( ), ] r p   . This function was derived from the Eq. (14), which is the shape of the failure criterion in the deviatoric plane, proposed by Willam and Warnke [35]. 2 2 2 2 2 2 2 2 2 2 2 ( )cos (2 ) 4( )cos 5 4 ( ) 4( )cos (2 ) c c t c t c c t t t c c t t c r r r r r r r r r r r r r r r r               (14) where, ( ) r  determines the distance of the failure surface at the deviatoric section by considering the effect of the Lode angle θ . The parameters c r and t r express the distances of failure surfaces from the hydrostatic axis at the compressive and