Issue 42

G. Testa et alii, Frattura ed Integrità Strutturale, 42 (2017) 315-327; DOI: 10.3221/IGF-ESIS.42.33 318 2 2 (1 ) 3(1 2 ) 3 m eq R                  (6) R  accounts for stress triaxiality, defined as the ratio of the mean and equivalent Von Mises stress, and  is the Poisson ratio. Because plastic flow can occur without damage and, similarly, damage can occur without noticeable macroscopic plastic flow, it can be assumed that the dissipation potential for plastic deformation and damage are independent,     , ; ; , p ij k D F f A T f Y T D     (7) where T is the temperature. From the generalized normality rule, the following expression for damage evolution law is obtained, D f F D Y Y          (8) The BDM differentiates from similar CDM formulations for the following additional assumptions. a) The damage rate depends on the “active plastic strain” rate defined as: p ˆ m eq eq p       (9) Here, p eq   is the rate of the equivalent plastic strain, 〈… 〉 is the Heaviside function that is equal to 1 when the stress triaxiality is positive and 0 otherwise. Under compressive state of stress, damage cannot accumulate and its effects are temporarily restored ( 0 & 0 D D    ). b) The damage dissipation potential depends on the total accumulated active plastic strain. The following expression was proposed,     1 2 0 0 1 2 1 ˆ cr D D D S Y f S D p                                (10) where, S 0 is a material constant,  is the damage exponent,  = (2+n)/n and n is the hardening exponent. This assumption implies that the damage dissipation depends on the deformation history, which leads to a nonlinear evolution of damage with the active plastic strain for constant stress triaxiality load paths. c) In the experiments, it is impossible to separate plasticity (hardening) and damage (softening) effects. If performed correctly, damage measurement shown that the critical damage at rupture is very small and no larger than 0.1 for pure metals and alloys. Consequently, it is convenient to assume that damage effect on the material plastic flow are already accounted for in the mathematical expression of the material flow curve identified in uniaxial tensile tests [22],   0 p eq y f p      (11) This assumption, which is also justified by the fact that damage process are highly localized in the material microstructure and therefore their detrimental effects are overcome by hardening at macroscopic scale, eliminates softening in the expression of the flow curve with the advantage to avoid mesh dependence effect in finite element applications.