PSI - Issue 13

David Lenz et al. / Procedia Structural Integrity 13 (2018) 2239–2244 Author name / Structural Integrity Procedia 00 (2018) 000–000

2241

In contrast to the original BW model, the modified Bai-Wierzbicki model does not assume that material damage will occur from the beginning of the plastic strain. This takes into account that materials can withstand a certain plastic deformation without damage occurring. The new critical strain ̅ , also called damage initiation strain, describes the point at which damage can be detected in the material. The flow criterion from equation (1) of the BW model is changed in the modified BW model, as listed in equation (6) by Lian et al. (2012):   1 0 yld D         (6) Here the flow condition is extended by the damage parameter D (coupled model), whereby the damage is directly taken into account in the calculation. If there is no material damage, � � 0 . With the onset of ductile damage, the damage parameter develops linearly. is the stress at the initiation of ductile damage. Furthermore, stands for the energy necessary to initiate a crack opening. 0 p     In addition to ductile damage, cleavage fracture plays a major role, especially at low temperatures. Since this does not follow the mechanisms of ductile damage, a different criterion is used for this case. The cleavage fracture is triggered when the maximum local principal normal stress exceeds the material-specific cleavage fracture stress according to Orowan (1949). If this happens, damage parameter D is immediately set to one and the local material point fails. 2.2. GTN The GTN model was derived from Gurson in 1977 based on micromechanical observations of the behavior of a pore in a circular continuum (Gurson 1977). Tvergaard and Needleman added empirical fit parameters and a phenomenological failure specification that describes the final stages of void coalescence in order to better adapt the simulation results obtained with the model to the results of experimental investigations (Tvergaard 1981) (Tvergaard 1984). In addition, Chu and Needleman formulated a law that describes the formation of secondary pores phenomenologically as a normal distribution depending on the plastic comparative strain (Chu and Needleman 1980). The flow potential in equation (8) determines whether plastic flow takes place.   2 * * 1 2 1 3 2 cosh * 1 0 2 e h y y q f q q f                               (8) The damage variable ∗ , which corresponded to the void volume in the original Gurson model, is contained in the flow potential and thus makes it a coupled model. In addition to the hydrostatic and von Mises stress, the empirical fit parameters are taken into account in the flow potential. Starting from the assumed initial void volume 0 , the void volume fraction, or the damage variable , develops according to the law of evolution in equation (9):   0 1 p ij ij pl kk e s f f k f                (9) The first term of this equation describes the growth of voids by volumetric strains, whereas the second term includes the shear damage depending on the stress state proposed by Nielsen and Tvergaard (2010). The void coalescence is phenomenologically as shown in equation (10) by assuming an acceleration of the void growth with the acceleration factor κ from a critical void content . i  i yi p       i f f p    cr f D d G D           (7)