Issue 41

A. Mardaliazad et alii, Frattura ed Integrità Strutturale, 41 (2017) 504-523; DOI: 10.3221/IGF-ESIS.41.62 516 deviatoric plane. This model also takes advantage of the associated inelastic flow and hardening (or softening) rules, and different dilation and friction angles. The general form of the failure surface in the Haigh-Westergaard coordinate system is given by Eq. (21). tan ( , , ) ( , ) ( ) 3 p f f t d             (21) The effect of the Lode angle θ is considered by ( , ) t   according to the Eq. (22). 3 1 1 ( , ) (1 ) (1 ) cos3 8 t K K                (22) where the material parameter K is defined as the ratio of yield stresses at triaxial tension to triaxial compression, which means it controls the shape of the linear Drucker-Prager yield function at the deviatoric plane. As can be seen in Fig. 10b, if K=1.0 , a circular failure function presents in the deviatoric plane, which is the same as the conventional Drucker-Prager. It is proved that in order to verify the convexity of the yield function, the value of parameter K should be in the range of 0.778 to 1.0. By means of an approach to match the parameters of the Mohr-Coulomb and linear Drucker-Prager (for materials with low friction angles), K can be defined according to Eq. (23). 3 sin 3 sin K      (23) where, φ is the internal friction angle used in Mohr-Coulomb model. The parameter β is commonly termed as the friction angle (in case of dealing with the Drucker-Prager theory), which can be computed as the slope of the yield function in the t-p meridian plane (see Fig. 10a. By a similar computing approach for K , it is possible to derive β according to Eq. (24). 6sin arctan( ) 3 sin      (24) It is worth mentioning that both the parameter can be defined in ABAQUS as a function of temperature and other field variables. However, they are considered as constant variables in this study because of the negligible effect of the other fields. The parameter d , called the cohesion of the material (in case of dealing with the Drucker-Prager model), is defined automatically by ABAQUS according to Eq. (25). 1 ( ) (1 tan ) 3 p u c p d       (25) where, u c  is the unconfined (uniaxial) compression stress. An isotropic hardening is implemented for this model in ABAQUS by a sub-option called “Drucker-Prager Hardening” at the property module. Therefore, the tabular data from the compression post-yield stress-strain diagram can be used to define u c  (and accordingly d ) as a function of the absolute plastic strain p  . The plastic flow of the linear Drucker-Prager is considered by means of a flow potential function LDP G given by Eq. (26). tan LDP G t p    (26) The parameter ψ , called dilation angle, specifies the direction of the plastic strain flow. As can be seen in Fig. 10a, the plastic flow is associative if ψ = β , otherwise it is non-associative. The parameters to identify the material model are reported in Tab. 6. The software automatically computes the cohesion by giving the yield stress in compression as a function of the plastic strain allowing the isotropic hardening of the yield function.