13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 1988) [14]. In macroscopic, a critical value of equivalent plastic strain is always used as a measurement of material ductility. Therefore, critical equivalent plastic strain can be used to denote material failure by void coalescence. To implement this concept, a possible approach is to establish a failure criterion based on equivalent plastic strain at the location where failure is most likely to initiate. In practical, for axisymmetric round tensile bars, failure always initiates in the center of the minimum section in specimens, where corresponds to the site with the highest stress triaxiality. In addition, stress triaxiality is often used as the sole parameter to characterize the effect of the triaxial stress states on ductile fracture. So the critical equivalent plastic strain can be established as a function of the stress trixiality ratioT. The power law form of the equivalent plastic strain criterion for API X65 steel proposed by Oh et al. (2007) [15] can be written as: 0.10 3.29 1.54 + = − T ef e E , (14) with q p T =− , (15) where ef E represents the critical equivalent plastic strain. This criterion is also used to develop extended damage model in this paper. 2.3. Post coalescence response The * f function, introduced by Tvergaard and Needleman (1984) [5], is adopted, to account for the effects of rapid void coalescence at failure. After void volume fraction reaches critical value determined by voids coalescence criteria, f is replaced by * f in the extended damage models. ( ) * * c u c c c c F c f for f f f f f f f f for f f f f < ⎧ ⎪ = − ⎨ + − ≥ ⎪ − ⎩ , (16) where cf is the critical void volume fraction at which voids begin coalesce; * 1 1/ uf q = is the * f value at zero stress; and Ff denotes the void volume fraction at final complete failure. 3. Finite element applications API X65 steel which is main pipe material largely utilized in gas transportation networks is discussed in this paper. To investigate the effect of triaxial stress states on tensile ductility of the material, three tensile round bar specimens with different notch root radii are analyzed, see Fig. 1. These specimens are also analyzed by Oh et al. (2007) [15] and the present simulation results are compared to their experimental data. The extended damage models described in the previous section are implemented in ABAQUS via a user defined material subroutine (UMAT). Two critical numerical procedures are involved in the finite element implementations. The integration of the rate form constitutive equations is following the backward Euler method by Aravas (1987) [16]. In an implicit code, the linearization modulus is needed to construct the stiffness matrix (Jacobin) for Newton scheme which is used to solve the global equilibrium equations. The explicit consistent tangent modulus based on a return mapping
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