## 3 Stress Integration Algorithms of the e-HAH Model for Elastic-Plastic FE Simulations
### Stress Integration Algorithms for the e-HAH Model: Proposed Models
#### 3.4.2 Fully Implicit Euler Backward Method (EBM)
\begin{array}{l}\mathrm{R}^{(k)}_{2+i,n+1}-\left(\frac{\mathrm{dg}_{i,n+1} }{\mathrm{d}\overline{\varepsilon}}\right)\cdot\delta(\Delta\overline{ \varepsilon}_{n+1})^{(k+1)}-\partial_{\sigma}\Big{(}\frac{\mathrm{dg}_{i,n+1} }{\mathrm{d}\overline{\varepsilon}}\Big{)}\cdot\delta(\Delta\sigma_{n+1})^{(k+ 1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}\\ -\partial_{\mathrm{g}_{i}}\Big{(}\frac{\mathrm{dg}_{i,n+1}}{\mathrm{d} \overline{\varepsilon}}\Big{)}\cdot\delta\Big{(}\Delta\mathrm{g}_{i,n+1} \Big{)}^{(k+1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}-\partial_{\hat{ \mathbf{h}}^{\mathbf{s}}}\Big{(}\frac{\mathrm{dg}_{i,n+1}}{\mathrm{d} \overline{\varepsilon}}\Big{)}\cdot\delta\Big{(}\Delta\hat{\mathbf{h}}^{ \mathbf{s}}_{n+1}\Big{)}^{(k+1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}=0 \\ \end{array}  \tag{50}\]  
\[ \begin{array}{l}\mathrm{R}^{(k)}_{10,n+1}+\Delta\hat{\mathbf{h}}^{\mathbf{ s}(k)}_{n+1}-\left(\frac{\mathrm{dg}^{\mathbf{s}}_{n+1}}{\mathrm{d} \overline{\varepsilon}}\right)^{(k)}\cdot\delta(\Delta\overline{\varepsilon}_{n +1})^{(k+1)}-\partial_{\sigma}\Big{(}\frac{\mathrm{dg}^{\mathbf{s}}_{n+1}}{ \mathrm{d}\overline{\varepsilon}}\Big{)}^{(k)}\cdot\delta(\Delta\sigma_{n+1})^{( k+1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}\\ -\partial_{\mathrm{g}_{i}}\Big{(}\frac{\mathrm{dg}^{\mathbf{s}}_{n+1}}{ \mathrm{d}\overline{\varepsilon}}\Big{)}^{(k)}\cdot\delta\Big{(}\Delta\xi_{i,n +1}\Big{)}^{(k+1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}-\partial_{ \hat{\mathbf{h}}^{\mathbf{s}}}\left(\frac{\mathrm{dg}^{\mathbf{s}}_{n+1}}{ \mathrm{d}\overline{\varepsilon}}\right)^{(k)}:\delta\Big{(}\Delta\hat{\mathbf{h} }^{\mathbf{s}}_{n+1}\Big{)}^{(k+1)}\cdot\Delta\overline{\varepsilon}^{(k)}_{n+1}=0 \end{array} \,. \tag{51}\]  
The solutions of the above linearized equations are then obtained for \(\delta(\Delta\overline{\varepsilon}_{n+1})^{(k+1)}\),  
\[ \begin{array}{l}\delta(\Delta\sigma_{n+1})^{(k+1)},\,\,\delta\Big{(}\Delta \mathrm{g}_{i,n+1}\Big{)}^{(k+1)}\,,\,\,\mathrm{and}\,\,\delta\Big{(} \Delta\hat{\mathbf{h}}^{\mathbf{s}}_{n+1}\Big{)}^{(k+1)}\,\,\mathrm{by}\, \,\mathrm{solving}\,\mathrm{the}\,\mathrm{following}\,\mathrm{matrix}\\ \end{array} \]  
\[\begin{array}{l}\left\{ \begin{array}{l}\delta(\Delta\overline{\varepsilon}_{n +1})^{(k+1)}\\ \delta(\Delta\sigma_{n+1})^{(k+1)}\\ \delta\Big{(}\Delta\mathrm{g}_{i,n+1}\Big{)}^{(k+1)}\\ \delta\Big{(}\Delta\hat{\mathbf{h}}^{\mathbf{s}}_{n+1}\Big{)}^{(k+1)}\end{array}  \right\}=-\left[\begin{array}{l}\left(\frac{\partial\mathrm{R}_{1}}{\partial \Delta\overline{\varepsilon}}\right)^{k}_{n+1}\,\,\left(\frac{\partial\mathrm{R}_{1}}{ \partial\Delta\overline{\varepsilon}}\right)^{k}_{n+1}\,\,\left(\frac{\partial \mathrm{R}_{1}}{\partial\Delta\overline{\varepsilon}}\right)^{k}_{n+1}\,\,\left( \frac{\partial\mathrm{R}_{1}}{\partial\Delta\overline{\varepsilon}}\right)^{k}_{n+1} \,\,\left(\frac{\partial\mathrm{R}_{1}}{\partial\Delta\overline{\varepsilon}}\right)^{k}_{n+ 1}\,\,\left(\frac{\partial\mathrm{R}_{1}}{\partial\Delta\overline{\varepsilon}}\right)^{k}_{all state variables are updated for the \((\mathrm{k}+1)\)th iteration, and this process continues until the residuals satisfy the following criteria:  
\[ \begin{split}\|\mathrm{R}^{(\mathrm{k})}_{1+\mathrm{i}\mathrm{n}+1} \|<\overline{\sigma}(\overline{\epsilon}_{\mathrm{n}})\cdot\mathrm{Tol},\;\| \mathrm{R}^{(\mathrm{k})}_{2,n+1}\|<\|\Delta\epsilon_{\mathrm{n}+1}\|\,\cdot \mathrm{Tol},\\ \|\mathrm{R}^{(\mathrm{k})}_{2+\mathrm{i}\mathrm{n}+1}\|<\mathrm{ Tol},\;\mathrm{and}\,\|\mathrm{R}^{(\mathrm{k})}_{10,n+1}\|<\mathrm{H}^{-\frac{1}{2}} \cdot\mathrm{Tol}\end{split}  \tag{53}\]  
In this study, the tolerance, Tol, is set to be \(10^{-6}\) for all the residuals with adequate normalizations.