## Appendix A Analytical Derivatives of e-HAH Model
### The Second Derivatives of Equivalent Stress
\[\frac{\partial^{2}\Phi}{\partial\sigma\partial\sigma}=\tfrac{1}{\mathrm{q}} \Big{(}\tfrac{1}{\mathrm{q}}-1\Big{)}\Phi^{2-\mathrm{q}}(\mathbf{A}\otimes \mathbf{A})+\tfrac{1}{\mathrm{q}}\Phi^{1-\mathrm{q}}\tfrac{\partial\mathbf{A} }{\partial\sigma}\]  
with  
\[\mathbf{A}=\mathrm{q}\Phi^{\mathrm{q}-1}\tfrac{\partial\Phi}{ \partial\sigma}\quad\mathbf{B}=2\Big{\{}\psi(\mathbf{s})\frac{\partial\psi( \mathbf{s})}{\partial\sigma}+\psi(\mathbf{s}_{\mathrm{p}})\frac{\partial\psi( \mathbf{s}_{\mathrm{p}})}{\partial\sigma}\Big{\}} \tag{15}\] \[\frac{\partial\mathbf{A}}{\partial\sigma}=\tfrac{9}{2}\big{(}\tfrac {9}{2}-1\big{)}\Big{\{}\psi(\mathbf{s})^{2}+\psi(\mathbf{s}_{\mathrm{p}})^{2} \Big{\}}^{\frac{q}{2}-2}\mathbf{B}\otimes\mathbf{B}\] \[\qquad+\tfrac{9}{2}\Big{\{}\psi(\mathbf{s})^{2}+\psi(\mathbf{s}_{ \mathrm{p}})^{2}\Big{\}}^{\frac{q}{2}-1}\tfrac{\partial\mathbf{B}}{\partial \sigma}+\ell_{2}^{\mathrm{q}}\mathrm{q}(\mathrm{q}-1)\big{(}2\hat{\mathbf{h}}^{ \mathbf{s}}:\tfrac{\partial\mathbf{s}}{\partial\sigma}\big{)}\otimes\Big{(}2 \hat{\mathbf{h}}^{\mathbf{s}}:\tfrac{\partial\mathbf{s}}{\partial\sigma}\Big{)} \quad,\] \[\frac{\partial\mathbf{B}}{\partial\sigma}=2\bigg{\{}\frac{\partial \psi(\mathbf{s})}{\partial\sigma}\otimes\frac{\partial\psi(\mathbf{s})}{ \partial\sigma}+\frac{\partial\psi(\mathbf{s}_{\mathrm{p}})}{\partial\sigma} \otimes\frac{\partial\psi(\mathbf{s}_{\mathrm{p}})}{\partial\sigma\sigma}+\psi( \mathbf{s}_{\mathrm{p}})\frac{\partial^{2}\psi(\mathbf{s})}{\partial\sigma \sigma\sigma}\big{\}}\] \[\frac{\partial^{2}\psi(\mathbf{s})}{\partial\sigma\partial\sigma}= \Big{(}\tfrac{\partial\mathbf{s}^{\mathbf{s}}}{\partial\mathbf{s}}:\tfrac{ \partial\mathbf{s}}{\partial\sigma}\Big{)}^{\mathrm{T}}\cdot\Big{(}\tfrac{ \partial^{2}\Phi}{\partial\delta\delta^{\mathbf{s}}}:\tfrac{\partial\mathbf{s}}{ \partial\sigma}\Big{)}\cdot\Big{(}\tfrac{\partial\mathbf{s}^{\mathbf{s}}}{ \partial\delta\delta}\Big{)}\] \[\frac{\partial^{2}\psi(\mathbf{s}_{\mathrm{p}})}{\partial\sigma \partial\sigma}=\Big{(}\tfrac{\partial\mathbf{s}^{\mathbf{s}}}{\partial\delta \delta}:\tfrac{\partial\mathbf{s}}{\partial\sigma}\Big{)}^{\mathrm{T}}\cdot\Big{(} \tfrac{\partial^{2}\Phi}{\partial\delta\delta^{\mathbf{s}}}:\tfrac{\partial \mathbf{s}^{\mathbf{s}}}{\partial\delta\delta}\Big{)}\cdot\Big{(}\tfrac{ \partial\mathbf{s}^{\mathbf{s}}}{\partial\delta\delta}\Big{)}\]\[\begin{array}{l}\frac{\partial^{2}\Phi}{\partial\sigma\partial\delta^{\mathbf{s}} }=(1-\mathrm{q})\Phi^{-1}\frac{\partial\Phi}{\partial\sigma}\otimes\frac{ \partial\Phi}{\partial\mathbf{h}^{\mathbf{s}}}+\frac{1}{\mathrm{q}}\Phi^{1- \mathrm{q}}\frac{\partial\mathbf{A}}{\partial\mathbf{h}^{\mathbf{s}}}\\ \text{with}\\ \frac{\partial\mathbf{A}}{\partial\mathbf{h}^{\mathbf{s}}}=\frac{\mathrm{q}}{2 }\big{(}\frac{\mathrm{q}}{2}-1\big{)}\Big{\{}\psi(\mathbf{s})^{2}+\psi( \mathbf{s}_{\mathrm{p}})^{2}\Big{\}}^{\frac{\mathrm{q}}{2}-2}\mathbf{B} \otimes\mathbf{B}^{\mathbf{s}^{\mathbf{s}}}+\frac{\mathrm{q}}{2}\Big{\{}\psi( \mathbf{s})^{2}+\psi(\mathbf{s}_{\mathrm{p}})^{2}\Big{\}}^{\frac{\mathrm{q}}{ 2}-1}\cdot\frac{\partial\mathbf{B}}{\partial\mathbf{h}^{\mathbf{s}}}\\ +\mathrm{f}_{\mathrm{d}}^{2}\mathrm{q}\big{(}2\hbar^{\mathbf{s}}:\mathbf{s}^{ \prime}\big{)}^{\mathrm{q}-1}\Big{(}\frac{\partial\mathbf{s}}{\partial \mathbf{s}}\Big{)}^{\mathrm{T}}+\mathrm{f}_{\mathrm{d}}^{2}\mathrm{q}(\mathrm{ q}-1)\Big{(}2\hbar^{\mathbf{s}}:\mathbf{s}\Big{)}^{\mathrm{q}-2}\Big{(}2\hbar^{ \mathbf{s}}:\frac{\partial\mathbf{s}}{\partial\sigma}\Big{)}\otimes 2\mathbf{s}\\ \mathbf{B}^{\mathbf{\hat{h}}^{\mathbf{s}}}=2\Big{\{}\psi(\mathbf{s})\frac{ \partial\psi(\mathbf{s})}{\partial\mathbf{h}^{\mathbf{s}}}+\psi(\mathbf{s}_{ \mathrm{p}})\frac{\partial\psi(\mathbf{s}_{\mathrm{p}})}{\partial\mathbf{h}^{ \mathbf{s}}}\Big{\}}\\ \frac{\partial\mathbf{B}}{\partial\mathbf{\hat{h}}^{\mathbf{s}}}=2\bigg{\{} \frac{\partial\mathrm{p}(\mathbf{s})}{\partial\sigma}\otimes\frac{\partial \psi(\mathbf{s}_{\mathrm{p}})}{\partial\mathbf{h}^{\mathbf{s}}}+\frac{ \partial\psi(\mathbf{s}_{\mathrm{p}})}{\partial\sigma}\otimes\frac{\partial \psi(\mathbf{s}_{\mathrm{p}})}{\partial\mathbf{h}^{\mathbf{s}}}+\psi(\mathbf{ s}_{\mathrm{p}})\frac{\partial^{2}\psi(\mathbf{s}_{\mathrm{p}})}{\partial \sigma\partial\mathbf{h}^{\mathbf{s}}}\bigg{\}}\\ \frac{\partial^{2}\psi(\mathbf{s})}{\partial\sigma\partial\mathbf{h}^{\mathbf{ s}}}=\Big{(}\frac{\partial\mathbf{s}}{\partial\sigma}\Big{)}^{\mathrm{T}}:\Big{\{} \Big{(}1-\frac{1}{\mathrm{g}_{\mathrm{L}}}\Big{)}\mathrm{H}\Big{(}\frac{ \partial\mathbf{s}}{\partial\sigma}:\mathbf{h}^{\mathbf{s}}\Big{)}\mathbf{I}_ {\mathrm{4}}+\mathbf{\hat{h}}^{\mathbf{s}}\otimes\frac{\partial\mathbf{s}}{ \partial\sigma^{\prime}}\Big{\}}+\Big{(}\frac{\partial\mathbf{s}^{\prime}}{ \partial\mathbf{s}}:\frac{\partial\mathbf{s}}{\partial\sigma}\Big{)}^{\mathrm{ T}}+\frac{\partial^{2}\Phi}{\partial\mathbf{w}^{\prime}\partial\mathbf{w}^{ \prime}}:\frac{\partial\mathbf{w}^{\prime}}{\partial\mathbf{w}^{\prime}}\\ \frac{\partial^{2}\psi(\mathbf{s}_{\mathrm{p}})}{\partial\sigma\partial\mathbf{h }^{\mathbf{s}}}=\Big{(}\frac{\partial\mathbf{s}}{\partial\sigma}\Big{)}^{ \mathrm{T}}:\bigg{\{}-\Big{(}\frac{4(1-\mathrm{g}_{\mathrm{L}})}{\mathrm{g}_{ \mathrm{L}}}\Big{)}\mathrm{H}\Big{(}\frac{\partial\mathbf{s}}{\partial \mathbf{w}^{\prime}}:\mathbf{\hat{h}}^{\mathbf{s}}\big{)}\mathbf{I}_{\mathrm{4} }+\mathbf{\hat{h}}^{\mathbf{s}}\otimes\frac{\partial\mathbf{s}}{\partial \mathbf{w}^{\prime}}\bigg{\}}\\ +\bigg{(}\frac{\partial\mathbf{s}^{\prime}}{\partial\mathbf{s}}:\frac{\partial \mathbf{s}}{\partial\sigma}\Big{)}^{\mathrm{T}}:\bigg{\{}\frac{\partial^{2} \Phi}{\partial\mathbf{s}^{\prime}}\mathbf{p}\otimes\frac{\partial\mathbf{s}^{ \prime}}{\partial\mathbf{h}^{\mathbf{s}}}\] (A7)  
\[