# Ductile Fracture Behavior of Mild and High-Tensile Strength Shipbuilding Steeds
## 2 Ductile Fracture Model  
### Model Formulation  
The preliminary information characterizing the stress state and adopted plasticity model is given in Appendix A. Following [PERSON] et al. [37], a ductile fracture model in association with non-porous plasticity is formulated by combining the normalized maximum shear stress criterion (MSS) and maximum principal tensile stress criterion (modified version of CL criterion by [PERSON] et al. [13]) in a weighted sum format  
\[\left(C_{1}\left(\frac{\tau_{max}}{\sigma}\right)+\left(1-C_{1}\right)\left( \frac{\left\langle\sigma_{1}\right\rangle}{\sigma}\right)\right)^{C_{2}}\varepsilon _{p}=C_{3} \tag{1}\]  
where \(C_{1}\in[0,1]\) is the weighting factor between the normalized maximum principal tensile stress and the normalized maximum shear stress. \(C_{2}\) is an exponent that represents the stress state-dependency of the ductility. \(C_{3}\) is a constant that determines the overall ductility of the material. The operator used for the maximum principal stress, i.e., \(\left\langle\right\rangle\), is the Macaulay bracket, which yields zero for negative values. Using the transformation of stress states given in Equation (A11) the proposed fracture model can be expressed as follows  
\[\frac{\tau_{max}}{\sigma}=\frac{1}{\sqrt{3}}\cos\left[\frac{\pi}{6}\bar{\theta}\right] \tag{2}\]  
\[\frac{\left\langle\sigma_{1}\right\rangle}{\bar{\sigma}}=\frac{\left\langle 3 \eta+2\cos\left[\frac{\pi}{6}\left(1-\bar{\theta}\right)\right]\right\rangle}{ 3} \tag{3}\]  
\[\left(C_{1}\left(\frac{1}{\sqrt{3}}\cos\left[\frac{\pi}{6}\bar{\theta}\right] \right)+\left(1-C_{1}\right)\left(\frac{\left\langle 3\eta+2\cos\left[\frac{\pi}{6} \left(1-\bar{\theta}\right)\right]\right\rangle}{3}\right)\right)^{C_{2}} \varepsilon_{p}=C_{3} \tag{4}\]  
The fracture model can be expressed as the macroscopic equivalent plastic strain at the instant of first localization under strictly proportional loading paths as follows  
\[\varepsilon_{f}^{pr}\left[\eta,\bar{\theta}\right]=C_{3}\left(C_{1}\left( \frac{1}{\sqrt{3}}\cos\left[\frac{\pi}{6}\bar{\theta}\right]\right)+\left(1-C _{1}\right)\left(\frac{\left\langle 3\eta+2\cos\left[\frac{\pi}{6}\left(1-\bar{ \theta}\right)\right]\right\rangle}{3}\right)\right)^{-C_{2}} \tag{5}\]  
For general cases, i.e., non-proportional loading paths, the fracture indicator framework is adopted. In this case, the strain to fracture under proportional loading serves as a stress-state-dependent weighting function. The fracture indicator, \(D\in[0,1]\), is defined as an integration of the weighted equivalent plastic strain increment along the strain path as follows  
\[dD=\frac{d\varepsilon_{p}}{\bar{\varepsilon}_{f}^{pr}\left[\eta,\bar{\theta}\right]} \tag{6}\]  
Note that \(D\) is not coupled with the material's elasto-plastic behavior, such that it does not cause any irreversible weakening (softening) with increasing plastic deformation. For finite element analysis of the implentation of the model in the software package Abaqus/Explicit, a user-defined material subroutine (VUMAT) is developed. The user subroutine adopts a standard return-mapping algorithm for plasticity calculations, where equivalent plastic strain increment is determined. Then, it is used in Equation (6) together with the proposed fracture model as the weighing factor to calculate the increment of fracture indicator. An element is assumed to have failed and deleted if all integration points satisfy the condition \(D=1\).