Issue 42

G. Testa et alii, Frattura ed Integrità Strutturale, 42 (2017) 315-327; DOI: 10.3221/IGF-ESIS.42.33 319 From Eq. (8) and Eq. (10) together with the definition of Y , and assuming a power law expression for the material flow curve, the following expression for the kinetic law of damage evolution can be obtained,   1 ˆ ˆ cr p D A R D D p            (12) where   1/ ln cr f th D A     (13) A detailed derivation of Eq. (12) can be found elsewhere [23]. Under the assumption of proportional loading, Eq. (12) can be integrated analytically,     ˆ ln / 1 1 ln / th cr f th p p D D R                          (14) At failure, for D=D cr , assuming that the strain threshold for damage initiation is pressure independent - which is reasonable at relatively low stress triaxiality while it is not true in general [16, 24, 25] - the following expression can be obtained, 1 ˆ R f f th th p            (15) This expression provides the relationship between the equivalent “active” plastic strain at failure and stress triaxiality and can be used to build the LSD. Damage model parameters identification The model requires four material parameters to be determined: the uniaxial threshold strain at which the damage initiates ɛ th ; the failure strain for stress triaxiality equal to 1/3, ɛ f , the damage at failure D cr and the damage exponent α. Under quasi- static loading condition, the damage exponent does not affect the failure condition. It does have an effect on the damage rate that becomes relevant in time dependent deformation processes. The critical damage defines the maximum reduction of the elastic modulus and the released damage strain energy. For several classes of metals and alloys this value is usually less than 0.1, [26]. The remaining damage parameters can be identified by means of experiments and finite element simulation. Firstly, uniaxial tensile tests on round notched bar samples (RNB) with at least three different notch radii are performed. The notch radius is selected to ensure that rupture will occurs at the specimen center (cup-cone type rupture). At least three samples for each notch radius shall be tested in order to have indication on the experimental scatter. From these tests, the diameter at fracture is measured and used to determine the average plastic strain at rupture according to the Bridgman expression, 0 ˆ 2 ln f R p          (16) Successively, finite element simulation of each notched geometry is performed and the stress triaxiality versus plastic strain at the specimen center is obtained. The stress triaxiality does not remain constant during the traction; therefore, the average stress triaxiality for the selected sample is defined as follow,     ˆ 1 ˆ ˆ f m th f th eq p T dp p                  (17)