## 3 Stress Integration Algorithms of the e-HAH Model for Elastic-Plastic FE Simulations
### Stress Integration Algorithms for the e-HAH Model: Review of Existing Models
#### 3.3.2 Euler Backward Method (EBM)  
Previous works have focused on the development of the EBM--based stress integration algorithm of the distortional anisotropic hardening laws. However, these algorithms simplified the set of equations in the EBM scheme by reducing the number of residuals associated with the state variables. Residuals on the stress tensor and equivalent plastic strain were then required to satisfy the consistency condition in elastic-plasticity under the associated flow rule. However, this method is affected by similar problems as the CPM--based algorithm [49] since a portion of the state variables should be treated as functions of independent variables when the number of residuals is less than that of the independent variables in the e-HAH model. This may cause difficulty in calculating the first and second derivatives of the yield function during its evolution as a function of plastic deformation. The EBM algorithm developed by [PERSON] et al. [56] introduced the following residuals:  
\[\mathrm{R}^{(\mathrm{k})}_{1,n+1}=\Phi\Big{(}\sigma^{(\mathrm{k})}_{n+1},\, \mathcal{S}^{(\mathrm{k})}_{1,n+1},\,\mathcal{\bar{\mathrm{R}}}^{(\mathrm{k} )}_{i,n+1}\Big{)}-\overline{\sigma}\Big{(}\overline{\epsilon}^{(\mathrm{k})} _{n+1}\Big{)}\,,\,(i=1-7,\,5=\mathrm{L},\,6=\mathrm{S},\,7=\mathrm{R}), \tag{33}\]  
\[\mathrm{R}^{(\mathrm{k})}_{2,n+1}=-\Delta\epsilon^{\mathrm{P}(\mathrm{k})}_{n +1}+\Delta\gamma\,\partial_{\sigma}\mathcal{O}^{(\mathrm{k})}_{n+1} \tag{34}\]  
To apply the multi-variable Newton-Raphson method, the residuals are linearized according to the following:  
\[\mathrm{R}^{(\mathrm{k})}_{1,n+1}-\partial_{\overline{\epsilon}}\overline{ \sigma}\Big{(}\overline{\epsilon}^{(\mathrm{k})}_{n+1}\Big{)}\cdot\mathcal{S }(\Delta\overline{\epsilon}_{n+1})^{(\mathrm{k}+1)}+\partial_{\sigma} \mathcal{O}^{(\mathrm{k})}_{n+1}:\mathcal{S}(\Delta\sigma_{n+1})^{(\mathrm{k} +1)}=0\,, \tag{35}\]  
\[\mathrm{R}^{(\mathrm{k})}_{2,n+1}+\partial_{\sigma}\mathcal{O}^{(\mathrm{k})}_ {n+1}\cdot\mathcal{S}(\Delta\overline{\epsilon}_{n+1})^{(\mathrm{k}+1)}+ \Xi^{-1}_{n+1}:\mathcal{S}(\Delta\sigma_{n+1})^{(\mathrm{k}+1)}=0. \tag{36}\]  
The linearized equations can be calculated with respect to \(\mathcal{S}(\Delta\overline{\epsilon}_{n+1})^{(\mathrm{k})}\) and \(\mathcal{S}(\Delta\sigma_{n+1})^{(\mathrm{k})}\) by solving the following system:  
\[\left\{ \begin{array}{l}\mathcal{S}(\Delta\overline{\epsilon}_{n+1})^{( \mathrm{k}+1)}\\ \mathcal{S}(\Delta\sigma_{n+1})^{(\mathrm{k}+1)}\end{array} \right\}=-\left[ \begin{array}{l}\left(\frac{\partial\mathcal{R}_{1}}{\partial\Delta\overline {\epsilon}}\right)^{\mathrm{k}}_{n+1}&\left(\frac{\partial\mathcal{R}_{1}}{ \partial\Delta\sigma}\right)^{\mathrm{k}}_{n+1}\\ \left(\frac{\partial\mathcal{R}_{2}}{\partial\Delta\overline{\epsilon}}\right) ^{\mathrm{k}}_{n+1}&\left(\frac{\partial\mathcal{R}_{2}}{\partial\Delta\sigma} \right)^{\mathrm{k}}_{n+1}\end{array} \right]^{-1}\cdot\left\{ \begin{array}{l} \mathrm{R}^{(\mathrm{k})}_{1,n+1}\\ \mathbf{R}^{(\mathrm{k})}_{2,n+1}\end{array} \right\}. \tag{37}\]  
Upon reaching convergence for the solutions of Equation (37), the other variables are updated explicitly using Equations (25)-(27). The iterative process stops when the following conditions are satisfied:  
\[\|\mathrm{R}^{(\mathrm{k})}_{1+i\mathrm{n}+1}\|<\overline{\sigma}(\tau_{\mathrm{ n}})\cdot\mathrm{Tol},\,\|\mathbf{R}^{(\mathrm{k})}_{2,n+1}\|<\mathrm{Tol}\,. \tag{38}\]  
As previously mentioned, the major difficulty in this algorithmic procedure lies in the calculation of the first and second derivatives, due to the anisotropically evolving yield function. Additionally, this evolution is closely associated with a multiple number of dependent variables in the e-HAH model. To overcome this issue, [PERSON] et al. [55] employed the multi-step Newton-Raphson method by subdividing the variables \(\mathcal{S}_{1\alpha\mathrm{2}}\).