Issue 44

G. Testa et alii, Frattura ed Integrità Strutturale, 44 (2018) 161-172; DOI: 10.3221/IGF-ESIS.44.13 164 2 ˆ 3 p p eq eq p T d d dt     (6) is the Heaviside function that is equal to 1 when the stress triaxiality is positive and 0 otherwise. The unilateral condition for damage states that under compressive state of stress, damage does not accumulate and its effects are temporarily restored ( 0 & 0 D D    ). The second term on the right hand side of eqn. (2) accounts for dissipation associated with void sheeting under shear dominated stress state. Here,  is the parameters, bounded between 0 and 1, that accounts for the for the influence of the third invariant of the deviatoric stress tensor J 3 on material ductility, which is defined as a function of the Lode parameter L as, 2 1 L    (7) where 3 3 27 2 J L    (8) The damage evolution law is obtained from the generalized normality rule     1/ 1 1 ˆ 1 ˆ 1 ln k cr D cr f f th D D F p D D D R D D D p Y p                                           (9) where   is the plastic multiplier equal to the equivalent plastic strain rate scaled by damage effect,   1 p D      (10) For constant  and T (>0) deformation process, the damage rate equation can be integrated to obtain the following expression for damage evolution, 1 1 ln 1 1 ln k th cr f th f p D D R p                                                (11) Here,  , D cr ,  th ,  f ,  f ,  , k are the material damage parameters:  th is the plastic strain threshold under uniaxial state of stress at which damage process is initiated,  f is the failure strain under constant stress triaxiality equal to 1/3,  is the damage exponent that defines the shape of damage evolution law for NAG,  and k are the damage exponents for shear controlled damage contribution, and  f is the critical strain for pure shear. These parameters can be determined performing selected experimental tests. In particular,  th and  f can be determined by fitting on the stress triaxiality vs failure strain plot of fracture data obtained with round notched bar in tension for which w=0, using the expression for the failure locus 1 R f f th th p            (12) Similarly,  f and k can be determined on the same plot fitting fracture data in the negative stress triaxiality regime with the following expression of the fracture locus,