# Failure Prediction for the Tearing of a Pin-Loaded Dual Phase Steel (DP980) Adjusting Guide
## 2 Damage Model and Local Strain Measurement
### Triaxility Failure Diagram and GISSMO  
GISSMO is an isotropic phenomenological damage model and can be regarded as a very straightforward and pragmatic approach towards the prediction of damage, and accounts for shear-dominated failure [9]. In the GISSMO framework, the equivalent strains at the onset of necking, \(\varepsilon_{u}\), and at fracture, \(\varepsilon_{f}\), are treated as stress triaxiality \(\eta\)-dependent weight functions in the calculation of the forming intensity, \(F\), and damage parameter, \(D\), as follows:  
\[F_{i+1}=\int_{0}^{\varepsilon_{sq}}\frac{m}{\varepsilon_{u}(\eta_{i})}F_{i}^{ 1-1/m}d\varepsilon_{eq} \tag{1}\]  
\[D_{i+1}=\int_{0}^{\varepsilon_{eq}}\frac{m}{\varepsilon_{f}(\eta_{i})}D_{i}^{ 1-1/m}d\varepsilon_{eq} \tag{2}\]  
where \(i\) and \(i+1\) denote the previous and the current FE analysis step, respectively. In LS-Dyna, the initial \(F\) is set to \(10^{-20}\), so as to allow for the evolution of \(F\). \(\varepsilon_{eq}\) denotes the equivalent plastic strain (\"equivalent\" in a von Mises sense), and \(\eta\) represents the stress state. For plane stress conditions, as is the case for thin sheets, \(\eta\) is limited to (\(-2/3\), \(2/3\)), and is defined as follows:  
\[\eta=\frac{\sigma_{m}}{\sigma_{eq}}=\frac{\sqrt{2}}{3}\frac{\sigma_{1}+\sigma _{2}}{\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\sigma_{1}^{2}+\sigma_{2}^{ 2}}} \tag{3}\]  
where \(\sigma_{m}\), \(\sigma_{eq}\), \(\sigma_{1}\), and \(\sigma_{2}\) are the mean stress, equivalent stress (here in a von Mises sense), and maximum and minimum principal in-plane stresses, respectively. Note that while \(\varepsilon_{u}\) and \(\varepsilon_{f}\) are triaxiality-dependent material properties, there was only one set of \(m\) and \(f\) that was found. The variations of \(\varepsilon_{u}\) and \(\varepsilon_{f}\) with \(\eta\) were provided by the TFD, as schematically shown in Figure 2. It is worth noting that the shapes of the curves are material-dependent; the run between two points is unknown. A further parameter, \(m\) (\(\geq\)1), the damage evolution exponent, governs the shape of the damage accumulation law. For \(m=1\), Equation (1) reduces to the linear damage evolution proposed by [PERSON] and [PERSON], and with increasing \(m\), we approach the [PERSON] model.  
Figure 1: Acquisition of load and local deformations by universal testing machine (UTM) and GOM Aramis.  
Once \(F_{new}\) reaches unity, which is tantamount to the onset of diffuse necking, the corresponding \(D_{new}\), calculated by Equation (2), is assigned (\(D_{c}\)), and upon further loading, the stress tensor, \(\sigma\), is reduced according to the following:  
\[\widetilde{\sigma}=\sigma\left[1-\left(\frac{D_{new}-D_{c}}{1-D_{c}}\right)^{ \prime}\right]\,\text{for}\,\,D\,\geq\,D_{c} \tag{4}\]  
Equation (4) implies that the material weakens equally in all directions, and thus remains isotropic. For further fitting and mesh regularization, which are necessary in order to tackle the problem of mesh-dependence in the post-critical region, the fading exponent (\(f\)) was introduced. For \(f=1\) and \(D_{c}=0\), we arrived at the classical [PERSON] definition of the effective stress tensor. As can be seen from Equation (2), \(D\) starts becoming non-zero with the onset of yielding, and although \(D=(0,1)\) still holds, \(D>0\) no longer necessarily means that the material's performance has deteriorated (but \(D>D_{c}\)). However, \(D=1\) still indicates failure. To mitigate the problem of mesh-dependence, the GISSMO parameters were made characteristic element length (\(e\))-dependent through the regularization of the energy dissipation in the post-critical region [14]. For this, tensile tests were simulated with various \(e\), and the numerical force-displacement (FD) results were fitted to the experimental FD data.