## 2 Plasticity Formulation
### Return Mapping Algorithm
#### 2.2.1 Return-Mapping Scheme for the 1D Model
\right.\, \tag{15}\]

the increment of the \(j\) principal stress is then written as:

\[d\sigma_{j}=\left\{\mathbf{D}-\frac{4}{Det_{1}}\Big{[}a\Big{(}\mathcal{V}_{pm} \cdot\mathcal{V}_{cm}^{T}+\mathcal{V}_{pr}\cdot\mathcal{V}_{cr}^{T}\Big{)}-b_ {1}\Big{(}\mathcal{V}_{pm}\cdot\mathcal{V}_{cr}^{T}+\mathcal{V}_{pr}\cdot \mathcal{V}_{cm}^{T}\Big{)}\Big{]}\right\}d\epsilon_{j} \tag{16}\]

The consistent tangent operator in the main base for the right edge \(\big{(}D_{p,r}^{ep}\big{)}\) obtained analytically is therefore:

\[\mathbf{D}_{p,r}^{ep}=\mathbf{D}-\frac{4}{Det_{1}}\Big{[}a\Big{(}\mathcal{V}_{pm} \cdot\mathcal{V}_{cm}^{T}+\mathcal{V}_{pr}\cdot\mathcal{V}_{cr}^{T}\Big{)}-b_ {1}\Big{(}\mathcal{V}_{pm}\cdot\mathcal{V}_{cr}^{T}+\mathcal{V}_{pr}\cdot \mathcal{V}_{cm}^{T}\Big{)}\Big{]} \tag{17}\]

_Left edge_

and

\[d\sigma_{j}=\mathbf{D}d\epsilon_{j}-2\ d\Delta\gamma^{q2}\ \mathcal{V}_{pm}-2\ d \Delta\gamma^{b2}\ \mathcal{V}_{pl} \tag{17}\]

In addition, taking into account the differentiation of the consistency conditions given in Equations (28) and (44) and defined by

\[\left\{ \begin{array}{c}d\mathcal{I}_{1}^{tr}\left(\sigma^{tr},c\right)=2 \mathcal{V}_{cm}d\epsilon_{j}\\ d\mathcal{I}_{2}^{trial}\left(\sigma^{trial},c\right)=2\mathcal{V}_{cl}d \epsilon_{j}\end{array} \right. \tag{18}\]

with

\[\mathcal{V}_{cl}=\left\{ \begin{array}{c}\left(K-\frac{2\zeta}{3}\right)\sin \phi\\ \left(K+\frac{\zeta}{3}\right)\sin\phi+G\\ \left(K+\frac{\zeta}{3}\right)\sin\phi-G\end{array} \right\} \tag{19}\]

and plastic multipliers given as:

\[\left\{ \begin{array}{c}d\Delta\gamma^{q2}=\frac{a\ d\mathcal{I}_{1}^{trial -b_{2}}\ d\mathcal{I}_{2}^{trial}}{Det_{2}}\\ d\Delta\gamma^{b2}=\frac{a\ d\mathcal{I}_{2}^{trial-b_{2}}\ d\mathcal{I}_{2}^{ trial}}{Det_{2}}\end{array} \right.\, \tag{20}\]

the increment of the main stress \(j\) is then written as:

\[d\sigma_{j}=\left\{\begin{array}{c}D-\frac{4}{Det_{2}}\Big{[}a\Big{(} \mathcal{V}_{pm}\cdot\mathcal{V}_{cm}^{T}+\mathcal{V}_{pl}\cdot\mathcal{V}_{ cl}^{T}\Big{)}-b_{2}\Big{(}\mathcal{V}_{pm}\cdot\mathcal{V}_{cl}^{T}+ \mathcal{V}_{pl}\cdot\mathcal{V}_{cm}^{T}\Big{)}\Big{]}\right\}d\epsilon_{j} \tag{21}\]

Finally, the consistent tangent operator in the principal basis and for the left edge \(\big{(}\mathcal{D}_{p,m}^{ep}\big{)}\) obtained analytically is:

\[\mathcal{D}_{p,l}^{ep}=\mathbf{D}-\frac{4}{Det_{2}}\Big{[}a\Big{(}\mathcal{V} _{pm}\cdot\mathcal{V}_{cm}^{T}+\mathcal{V}_{pl}\cdot\mathcal{V}_{cl}^{T}\Big{)} -b_{2}\Big{(}\mathcal{V}_{pm}\cdot\mathcal{V}_{cl}^{T}+\mathcal{V}_{pl}\cdot \mathcal{V}_{cm}^{T}\Big{)}\Big{]} \tag{22}\]

_(d) Apex_

By evaluating the pressure and strain equations Equations (15) and (50), respectively, we find:

\[d\sigma_{j}=\mathbf{0} \tag{23}\]

Thus, the consistent tangent operator in the principal basis for the Apex \(\big{(}\mathcal{D}_{p,\mathbf{A}}^{ep}\big{)}\) is

\[\mathbf{D}_{p,\mathbf{A}}^{ep}=\mathbf{0} \tag{24}\]

**Remark A1**. _It should be noted that the analytical calculation of the elastoplastic tangent tensor in this case is only valid for constant parameters (without any material non-linearity as in the Duncan-Chang hyperbolic constitutive law [22] or the HS law [6])._

## References

* [PERSON] et al. (2008) [PERSON]; [PERSON]; [PERSON] _Computational Methods for Plasticity: Theory and Applications_; John Wiley & Sons: Hoboken, NJ, USA, 2008. * [PERSON] et al. (2021) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Implicit numerical integration of highly nonlinear plasticity models. _Comput. Geotech._**2021**, _132_, 103961. [CrossRef]
* [PERSON] (2017) [PERSON] A return mapping algorithm for isotropic and anisotropic plasticity models using a line search method. _Comput. Methods Appl. Mech. Eng._**2017**, _317_, 526-553. [CrossRef]
* [PERSON] and [PERSON] (2006) [PERSON]; [PERSON] _Computational Inelasticity_; Springer: Berlin/Heidelberg, Germany, 2006; Volume 7. * [PERSON] and [PERSON] (1985) [PERSON]; [PERSON] Consistent tangent operators for rate-independent elastoplasticity. _Comput. Methods Appl. Mech.