## 1 Introduction
### Damage Models
#### 1.1.1.1. Gurson-Tvergaard-Needleman Model  
[PERSON]'s damage model (GTN model) [1] is an analytical model that predicts ductile fracture on the basis of nucleation, growth and coalescence of voids in materials. The model is defined as:  
\[\phi=\frac{\sigma_{e}^{2}}{\sigma_{M}^{2}}+2q_{1}f^{*}\cosh\left[\frac{tr\sigma }{2\sigma_{M}}\right]-1-\left(q_{1}f^{*}\right)^{2}=0 \tag{1}\]  
In which \(q_{1}\) is the material constant, \(tr\sigma\) is the sum of principal stresses, \(\sigma_{M}\) is the equivalent flow stress and \(f^{*}\) is the ratio of voids effective volume to the material volume ratio defined as follows:  
\[f^{*}(f)=f_{c}\qquad if\qquad f\leq f_{c}, \tag{2}\]  
\[f^{*}(f)=f_{c}+\frac{(1/q_{1})-f_{c}}{f_{f}-f_{c}}(f-f_{c})\;\;if\qquad f>f_{c}, \tag{3}\]  
where \(f\) is the voids' volume ratio, \(f_{c}\) is the voids' volume ratio at the beginning of nucleation and \(f_{f}\) is the voids' volume ratio when fracture occurs. The equivalent flow stress (\(\sigma_{M}\)) is obtained from the following work hardening relation:  
\[\sigma_{M}\left(e_{M}^{pl}\right)=\sigma_{y}\left(\frac{e_{M}^{pl}}{\varepsilon _{y}}+1\right)^{n}, \tag{4}\]  
In which \(n\) is the strain-hardening exponent and \(e_{M}^{pl}\) is the equivalent plastic strain. The voids' growth rate is the sum of existing voids growth \(\dot{f}_{g}\) and the new voids' nucleation \(\dot{f}_{n}\).  
\[\dot{f}=\dot{f}_{n}+\dot{f}_{g}, \tag{5}\]  
where the components are further formulated as follows:  
\[\dot{f}_{g}=(1-f)tr\dot{\varepsilon}^{pl}, \tag{6}\]  
\[\dot{f}_{n}=A\dot{\varepsilon}_{M}^{pl}, \tag{7}\]  
\[A=\frac{f_{n}}{S_{n}\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{e_{M}^{pl}- \varepsilon_{N}}{s_{N}}\right)\right], \tag{8}\]  
In which \(tr\dot{\varepsilon}^{pl}=\left(\dot{\varepsilon}_{x}+\dot{\varepsilon}_{y}+ \dot{\varepsilon}_{z}\right)\) is the volume plastic strain rate, \(s_{N}\) is the voids' nucleation mean quantity, \(f_{n}\) is volume ratio of the second phase particles (responsible for the voids' nucleation) and \(\varepsilon_{N}\) is mean strain at the time of voids' nucleation. So, GTN model involves ten parameters which can be defined in a vector form by:  
\[\phi=\phi\left(\sigma_{y}\;\varepsilon_{y}\;n\;q_{1}\;f_{0}\;f_{c}\;f_{f}\;f_ {n}\;\varepsilon_{N}\;S_{N}\right)\]  
In this model, the effect of hydrostatic pressure on the voids' growth is considered and the shear stress effect is ignored which in general cases makes the results questionable. Thus, although, this model can perfectly predict damage in tensile stress, it is not as accurate in shear stress tests. To overcome this shortcoming, an extension of the Gurson model was proposed by [PERSON] and [PERSON] [2] that incorporates damage growth under low triaxial straining for shear-dominated states. [PERSON] et al. [3] identified material parameters for Gurson-type and Lemaitre-type constitutive models for low alloy steel based on a hybrid global-local optimization technique. [PERSON] et al. [4] used experiment and FE simulations for smooth and notched tensile bars,to calibrate the parameters in the GTN model. [PERSON] et al. (2017) undertook a numerical comparative study based on GTN original model and two recent enhancements that included shear mechanisms, employing mathematical and numerical strategies to calibrate the material parameters.