## 3 A Brief Background of the GTN Model  
As already mentioned, the J-R curves were calculated using the GTN constitutional model, which is generally accepted as enabling a description of the ductile failure of metals. The plasticity condition in the GTN model is expressed by Equation (3):  
\[g=\left(\frac{\sigma_{e}}{\sigma_{0}}\right)^{2}+2q_{1}f\cos h\left(\frac{3q_{ 2}\sigma_{m}}{2\sigma_{0}}\right)-\left(1+q_{3}f^{2}\right)=0 \tag{3}\]  
where \(\sigma_{\text{e}}\) is the equivalent Huber-Mises-Hencky stress;  
\(\sigma_{0}\) is the yield stress;  
\(\sigma_{\text{m}}=(\sigma_{x}+\sigma_{y}+\sigma_{z})/3\) is the hydrostatic stress;  
\(f\) is the relative volume of voids;  
and \(q_{1}\), \(q_{2}\), \(q_{3}\) are material constants.  
In many cases, the constant \(q_{3}\) can be expressed by the constant \(q_{1}\):  
\[q_{3}={q_{1}}^{2}\]  
Condition (3) should be supplemented by an evolution law for the growth of voids. For the elementary increment df, the following is valid:  
\[df=df_{growth}+df_{nucleation}=(1-f)de_{p}+Ad\varepsilon_{0}{}^{p} \tag{4}\]  
The member \(df_{growth}\) corresponds to the growth of existing voids, whilst the member \(df_{nucleation}\) expresses the formation (nucleation) of new voids in a material. The quantity \(de_{p}\) is an increment of the effective macroscopic plastic strain and \(\varepsilon_{0}{}^{p}\) is the effective plastic strain of the matter, i.e., the material which encloses the void. The letter \(A\) in Equation (4) stands for the intensity of void nucleation and it can be expressed in various ways. In some cases, it is supposed that \(A\) is a constant; however, a statistical model is often used.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{3}{c}{**Static Characteristics**} & \multicolumn{3}{c}{**Ramberg–Osgood Parameters**} \\ \hline R\({}_{\text{p}0.2}\) & R\({}_{\text{m}}\) & A\({}_{5}\) & \(\alpha\) & n & \(\sigma_{0}\) \\ (MPa) & (MPa) & (\%) & (1) & (1) & (MPa) \\ \hline
439 & 587 & 32 & 2.0575 & 10.15 & 439 \\ \hline \end{tabular} Note: the symbol A\({}_{5}\) in Table 1 stands for ductility.  
\end{table}
Table 1: Basic tensile properties and Ramberg–Osgood parameters of the steel used.  
According to this model, the intensity of void nucleation is expressed by Equation (5), as shown, for example, in [12].  
\[A=\frac{f_{N}}{s_{N}\sqrt{2\tau}}\exp\left[-\frac{1}{2}\left(\frac{\varepsilon_{0 }{}^{p}-\varepsilon_{N}}{s_{N}}\right)^{2}\right] \tag{5}\]  
where \(f_{N}\) is the relative volume of inclusions from which a void could be formed, and \(\varepsilon_{N}\) and \(s_{N}\) are statistical parameters. The value of \(\varepsilon_{N}\) corresponds to the strain \(\varepsilon_{0}{}^{p}\) at which 50% of all voids are nucleated.  
It is seen that the model requires that parameters \(q_{1}\), \(q_{2}\), and \(q_{3}\) are determined. In this investigation, the J-R curve for the CT specimens was used to determine the parameters. The response of the numerical model (J-R curve) was compared with the experimental data. The magnitudes of the parameters \(q_{1}\), \(q_{2}\), and \(q_{3}\) which provided the best fit of the calculated J-R curve to the experimental one for the CT specimens were used further to calculate the J-R curve for the SENT specimens. All calculations were performed by the FEM programme WARP3D [13].