## 2 Mathematical Model
### Structural Model  
The governing equation for the structural problem is expressed by:  
\[\sigma_{ij,i}+X_{i}=0 \tag{5}\]  
where \(\sigma_{ij}\) is the Cauchy stress tensor, \(X_{i}\) the body force per unit volume [14]. In order to solve the structural problem, Equation (5) must be coupled with the compatibility equations, the constitutive laws and the relationship between the thermal strain tensor and temperature variation \(\left(T-T_{ref}\right)\), where \(T_{ref}\) represents the reference temperature, which is the temperature at which no thermal strainsare envisaged. Furthermore, in order to take into account the nonlinear behavior, the total mechanical strain \(\varepsilon\) is decomposed into elastic \(\varepsilon_{ij}^{el}\) and plastic \(\varepsilon_{ij}^{pl}\) components [23; 24; 25]:  
\[\varepsilon_{ij}=\varepsilon_{ij}^{el}+\varepsilon_{ij}^{pl} \tag{6}\]  
The nonlinear model identified adopts the [PERSON] yield criterion, the bilinear kinematic hardening rule, and the Prandtl-Reuss flow rule; a further description of the present model can be found in [26; 27; 28; 29].  
The yield criterion defines the limit of elasticity, namely, it delimits the surface that separates the elastic stress field, inside the yield surface, from the plastic stress field which lies outside the yield surface [29]. Plastic strains will be detected when the equivalent stress \(\sigma_{e}\) equals the yield value \(\sigma_{y}\). The yield surface can be expressed as follows:  
\[F(\sigma,k,\boldsymbol{\alpha})=0 \tag{7}\]  
where \(k\) is the plastic work, and \(\boldsymbol{\alpha}\) the back stress tensor modelling the translation of the yield surface. The plastic work is formulated as:  
\[k=\int\ \boldsymbol{\sigma}^{T}[M]\boldsymbol{d}\boldsymbol{\varepsilon}^{pl} \tag{8}\]  
where \([M]\) is a diagonal matrix [30]:  
\[[M]=\left[\begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&2&0&0\\ 0&0&0&0&2&0\\ 0&0&0&0&0&2\end{array}\right]\]  
The back stress tensor \(\boldsymbol{\alpha}\) is:  
\[\boldsymbol{\alpha}=\int\ C\boldsymbol{d}\boldsymbol{\varepsilon}^{pl} \tag{9}\]  
where \(C\) is a material parameter [10].  
For a one-dimensional plasticity model, the stress-strain relations, when considering a bilinear kinematic hardening model (see also Figure 2), are expressed as:  
\[\sigma=\left\{\begin{array}{rl}\ E\varepsilon\ if\ \gamma=0\\ \frac{E\kappa}{E+K}\varepsilon\ if\ \gamma>0\end{array}\right. \tag{10}\]  
where \(\gamma\) is the absolute value of the plastic strain rate and \(K\) is the plastic modulus [28].  
Figure 2: Stress/strain—Kinematic hardening.  
Finally, the flow rule specifies the interdependence occurring between the plastic strain increase and the deviator stresses. With regard to the Prandtl-Reuss flow rule, a linear relationship between the plastic strain increment and the deviatoric stresses is considered [31]:  
\[d\varepsilon_{ij}=S_{ij}d\lambda \tag{11}\]  
where \(d\varepsilon_{ij}\) are the plastic strain increments, \(S_{ij}\) the deviatoric stresses, and \(\lambda\) the plastic multiplier obtained by imposing that the stress state lies on the yield surface during plastic flow. The elastic-plastic model here described is applied only for the copper alloy liner structure which is expected to undergo plastic strains since it is exposed to high thermal loads in the inner surface of the thrust chamber.