## 2 Materials and Methods
 \tag{14}\]  
it is possible to obtain the following equation:  
\[\frac{\hat{T}}{\hat{T}}=\left(\Gamma_{1}\,\text{tr}\mathbf{S}-k_{e}\right) \,\text{tr}\mathbf{S}+\Gamma_{2}\,\mathbf{S}\cdot\hat{\mathbf{S}} \tag{15}\]  
where  
\[ \begin{cases}k_{e}=\frac{\alpha}{\rho\,c}\\ \Gamma_{1}=\frac{1}{\rho\,c}\left(\frac{1}{E}\,\frac{\partial\
u}{\partial T}- \frac{\
u}{E^{2}}\,\frac{\partial E}{\partial T}\right)=\frac{1}{\rho\,c}\, \frac{\partial}{\partial T}\left(\frac{\
u}{E}\right)\\ \Gamma_{2}=\frac{1}{\rho\,c}\left(\frac{1+\
u}{E^{2}}\,\frac{\partial E}{ \partial T}-\frac{1}{E}\,\frac{\partial\
u}{\partial T}\right)=-\frac{1}{\rho \,c}\,\frac{\partial}{\partial T}\left(\frac{1+\
u}{E}\right)\end{cases}  \tag{16}\]  
The quantity \(k_{e}\) is related to some material properties (density \(\rho\), specific heat \(c\), and linear expansion coefficient \(\alpha\)) and represents the ratio between the elastic property and the energy storage capacity of the material. The \(\Gamma\) quantities are related to the thermal variation of the ratio between the elastic deformation and its rigidity weighted on the energy storage capacity of the material. Now, introducing the usual mathematical-physical notation  
\[ \begin{cases}\sigma=\text{tr}\mathbf{S}\\ \sigma=\text{tr}\mathbf{\dot{S}}\\ \sigma_{i}=(\mathbf{S})_{i}\\ \sigma_{i}=(\mathbf{S})_{i}\\ \sigma_{i}=(\mathbf{S})_{i}\end{cases}  \tag{17}\]where \((S)_{i}\) are the principal stresses with respect to an orthogonal basis \(\{e_{i}\}\), and remembering that [20; 22]:  
\[\mathbf{S}\cdot\hat{\mathbf{S}}=\sum_{i=1}^{3}(\mathbf{S})_{i}(\hat{\mathbf{S}}) _{i} \tag{18}\]  
it is possible to obtain:  
\[\frac{\hat{T}}{T}=\left(\Gamma_{1}\sigma-k_{e}\right)\hat{\sigma}+\Gamma_{2} \,\sum_{i=1}^{3}\sigma_{i}\sigma_{i} \tag{19}\]  
which is the thermoelastic relation between temperature and stress. To obtain a relationship between temperature and stress, the differential Equation (19) must be considered with the following hypotheses:  
* The mass density \(\rho\) remains constant throughout the thermodynamic process;
* The partial derivatives of the Poisson's ratio \(\
u\) and the Young's modulus \(E\) with respect to temperature \(T\) remain constant over time and temperature. By means of the above assumptions and using the following notation:  
\[ \begin{cases}\sigma=\sigma_{1}+\sigma_{2}+\sigma_{3}\\ f=\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+\sigma_{1}\sigma_{3}\end{cases}  \tag{20}\]  
with \(\sigma\) and \(f\), respectively, the first and second invariant of the stress tensor, it is possible to integrate the differential temperature-stress Equation (19), obtaining:  
\[\ln\left(\frac{T}{T_{0}}\right)=-k_{e}\left(\sigma-\sigma_{0}\right)+\frac{ \Gamma_{1}+\Gamma_{2}}{2}\left(\sigma^{2}-\sigma_{0}^{2}\right)-\Gamma_{2} \left(f-f_{0}\right) \tag{21}\]  
with \(\sigma_{0}\) and \(f_{0}\) representing the initial values of \(\sigma\) and \(f\). The result obtained (Equation (21)) is the integral and global temperature-stress relation, in which the right member exhibits a nonlinear dependence on both the first and second invariants of stress, while showing a linear dependence on the second invariant. By assuming that the Poisson's ratio and [PERSON]'s modulus of elasticity vary with temperature, we can derive a connection between changes in temperature and changes in stress. This relationship includes coefficients that not only denote the thermoelastic constant, as in the Kelvin equation, but also vary based on the average stress, as well as the first and second stress invariants and the modulus of elasticity.