## 2 Theoretical Background  
### Mathematical Formalization of CKH Model  
In this paragraph, the aim is to derive the differential equation about the CKH model for a uniaxial loading case, according to [4]. In a plain specimen under uniaxial stress, the stress tensor is given by (1), where \(x\) indicates the loading direction. \[\mathbf{S}= \begin{pmatrix}\sigma_{xx}&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}  \tag{1}\]  
The total strain tensor is obtained by adding the plastic strain tensor and the elastic strain tensor, as in Equation (2). \[\varepsilon^{\mathbf{tot}}=\varepsilon^{\mathbf{el}}+\varepsilon^{\mathbf{ pI}} \tag{2}\]  
When creating an elastic-plastic analysis, the first component is the yield law, which, in this case, is the common Von Mises yield criterion. The evolution of the yield surface according to this criterion is formalized in Equation (3). \[\mathrm{F}=\sqrt{\frac{3}{2}(\mathbf{S_{D}}-\boldsymbol{\chi}):(\mathbf{S_{D} }-\boldsymbol{\chi})}-\sigma_{\mathrm{Y}}=0 \tag{3}\]  
In this latter equation, \(\mathbf{S_{D}}\) is the deviatoric component of the stress tensor \(\mathbf{S}\), \(\sigma_{\mathrm{Y}}\) is the elastic limit of the material and \(\boldsymbol{\chi}\) is the backstress tensor. The second fundamental part is the plastic flow rule. Prandtl-Reuss is the plastic flow rule associated with the Von Mises yield function and is presented in Equation (4). This is not the only plastic flow rule suitable for an elastic-plastic analysis. \[ \begin{cases}\mathrm{d}\varepsilon_{xx}^{\mathrm{PI}}=\begin{bmatrix}\sigma_{ xx}-0.5(\sigma_{yy}+\sigma_{zz})\end{bmatrix}\mathrm{d}\lambda\\ \mathrm{d}\varepsilon_{yy}^{\mathrm{PI}}=\begin{bmatrix}\sigma_{yy}-0.5( \sigma_{xx}+\sigma_{zz})\end{bmatrix}\mathrm{d}\lambda\\ \mathrm{d}\varepsilon_{zz}^{\mathrm{PI}}=\begin{bmatrix}\sigma_{zz}-0.5( \sigma_{xx}+\sigma_{yy})\end{bmatrix}\mathrm{d}\lambda\\ \mathrm{d}\varepsilon_{yy}^{\mathrm{PI}}=\begin{bmatrix}\sigma_{xy}\mathrm{d} \lambda\\ \mathrm{d}\varepsilon_{xz}^{\mathrm{PI}}=\begin{matrix}3\\ \sigma_{xz}\end{matrix}\mathrm{d}\lambda\\ \mathrm{d}\varepsilon_{yz}^{\mathrm{PI}}=\begin{matrix}3\\ \sigma_{yz}\end{matrix}\mathrm{d}\lambda\end{cases}  \tag{4}\]  
The plastic flow rule allows for us to understand which of the plastic strain tensor components are not null, but it is not able to calculate their values, as \(\mathrm{d}\lambda\) remains unknown in Equation (4). To accomplish this, it is necessary to introduce a hardening rule. The CKH model is described by the differential Equation (5):  
\[\mathrm{d}\boldsymbol{\chi}_{i}=\frac{2}{3}C_{i}\mathrm{d}\varepsilon^{ \mathbf{PI}}-\gamma_{i}\boldsymbol{\chi}_{i}\mathrm{d}p \tag{5}\]In Equation (5), \(C_{i}\) and \(\gamma_{i}\) are the contants of the CKH model and \(\mathrm{d}p\) is defined in Equation (6). \[\mathrm{d}p=\sqrt{\frac{2}{3}\mathrm{d}\varepsilon^{\mathrm{pl}}:\mathrm{d} \varepsilon^{\mathrm{pl}}} \tag{6}\]  
Focusing on only a plain specimen under uniaxial loading, the incremental plastic strain tensor is given in Equation (7), whereas the backstress tensor is reported in Equation (8) for each backstress component. \[\mathrm{d}\varepsilon^{\mathrm{pl}}=\mathrm{d}\varepsilon_{xx}^{\mathrm{pl}} \begin{pmatrix}1&0&0\\ 0&-0.5&0\\ 0&0&-0.5\end{pmatrix}  \tag{7}\]  
\[\chi_{i}=\frac{2}{3}\chi_{i} \begin{pmatrix}1&0&0\\ 0&-0.5&0\\ 0&0&-0.5\end{pmatrix}  \tag{8}\]  
Equation (7) is consistent with the Prandtl-Reuss plastic flow rule, which states that the incremental plastic strain tensor and the deviatoric component of the stress tensor are parallel. Equation (8) provides the backstress tensor form used for each backstress component. After substituting Equations (1) and (8) into Equation (3), it is possible to obtain the relationship between the stress and the backstress for an uniaxial loading case (Equation (9)), where \(\mu\) is equal to 1 during positive loading (tending-to-tensile stress), while it is equal to \(-1\) during negative loading (tending to compression).