*2.3. Gurson-Tvergaard-Needleman (GTN) Damage Model*

Ductile crack growth in metals is a result of nucleation, growth and coalescence of micro voids. In order to obtain the fracture resistance curves of different models, the finite element method (FEM) simulation based on Gurson-Tvergaard-Needleman (GTN) damage model was used in this study. There are nine parameters in the GTN damage model: the constitutive parameters *q1*, *q2* and *q3*, the void nucleation parameters *εN*, *SN* and *fN*, the initial void volume fraction *f0*, the critical void volume fraction *fC* and the final failure parameter *fF*. The void coalescence occurs when the void volume fraction reaches the critical value *fC*, and the fracture occurs when the void volume fraction reaches the final value *fF*. These parameters have been obtained and listed in Table 1 [26].

**Table 1.** The GTN parameters of different materials. Reprinted by permission from Springer, Copyright 2017 [26].



**Table 1.** *Cont.*

This GTN damage model has been implemented in the ABAQUS code (6.14, Dassault Systèmes group company, Shanghai, China), and is widely used to simulate the crack propagation process and calculate the *J*-resistance curve. During the finite element analysis, the 3D eight-node isoperimetric element with reduced integration (C3D8R) are used [13,14]. The typical finite element mesh for the "121" model with *W*52Mb = 16 mm is illustrated in Figure 4a, the minimum size of mesh in the crack growth region is 0.1 mm × 0.1 mm [27], as shown in Figure 4b. This typical model contains 75,872 elements and 87,849 nodes. In addition, the surface-to-surface contact (explicit) interaction type was used in the model. Moreover, the sliding formulation is finite sliding, the mechanical constraint formulation is kinematic contact method.

**Figure 4.** The whole mesh of the typical model (**a**) and the mesh in the crack growth region (**b**).

The load versus load-line displacement curve can be obtained from the FEM simulation. With instantaneous crack lengths obtained at each loading point, a crack growth resistance curve can be determined, as specified in ASTM (American Society for Testing and Materials) E1820 [28].
