## 3 Results  
Here, the thermal behavior of the elastomers under elastic deformations is considered to be a case study to validate the approach proposed in Section 2. In this analysis, a real mechanical component is loaded with a stress variation. The stress of the elastomer, as a function of time, is expressed by the phenomenological one-dimensional relation [25]:  
\[\sigma(\text{Fo})=\sigma_{eq}+\frac{A}{1+B\ln(\text{Fo})}\qquad\text{with Fo}\geq 1 \tag{22}\]  
where \(A\), \(B\) and \(\sigma_{eq}\) (\(\sigma_{eq}\) is the equilibrium value [25]) are constant, and Fo is the Fourier number. To obtain the temperature variation of the elastomers under elastic deformation, the relation (21) must be used. To do that, the values of the applied stress at \(\text{Fo}=1\) and \(\text{Fo}\rightarrow\infty\):  
\[ \begin{cases}\sigma_{0}=\sigma(1)=A\\ \sigma_{eq}=\lim_{\text{Fo}\rightarrow\infty}\sigma\,\text{Fo}\end{cases}  \tag{23}\]The first relation of the equation's system (23) allows us to obtain the physical meaning of the constant \(A\), which results in the initial stress value. Now, introducing these values in Equation (21), the thermal behavior of elastomers can be obtained:  
\[\ln\left(\frac{T}{T_{0}}\right)=\left[k_{e}-\left(\Gamma_{1}+\Gamma_{2}\right) \sigma_{eq}\right]A-\frac{1}{2}\left(\Gamma_{1}+\Gamma_{2}\right)A^{2} \tag{24}\]  
This last relation represents the temperature variation of elastomer as a consequence of stress variation under elastic deformation. To plot the shape of this relation, it is possible to write it in the following way:  
\[\vartheta=\exp\left(\left(\Gamma_{1}+\Gamma_{2}\right)\left(\delta A-\frac{ 1}{2}\,A^{2}\right)\right) \tag{25}\]  
where \(\vartheta=T/T_{0}\) and \(\delta=\left(\Gamma_{1}+\Gamma_{2}\right)^{-1}k_{e}-\sigma_{eq}\). The quantity \(\delta\) is related to some material properties (density \(\rho\), specific heat \(c\) and linear expansion coefficient \(\alpha\)) but also to the temperature variation of the Poisson's ratio \(\
u\) and Young's modulus \(E\); consequently, this quantity represents the thermoelastic property of the material concerning the ratio between the elastic deformation and its rigidity. The qualitative shape of the relation (25) can be plotted, by numerical simulations, assuming the normalization condition \(\Gamma_{1}+\Gamma_{2}=1\):  
* In Figure 1, it has been plotted with positive \(\delta\) values. In this case, the elastomer is also a thermal conductor, and the shape shows that, as a consequence of the applied external stress, its temperature demonstrates a considerable increase;
* In Figure 2, it has been plotted with negative \(\delta\) values. In this case, the elastomer is not a thermal conductor, and the shape shows that, as a consequence of the applied external stress, its temperature presents a considerable decrease;
* In Figure 3, it has been plotted with zero \(\delta\) values. This is a particular case in which the thermal behavior compensates for the elastic stress. In this case, the elastomer has similar thermal and elastic behavior for slow values of the applied external stress. In all three cases, there exists an upper limit of the applied external stress, which represents the physical limit of the elastic behavior of the elastomer: when the applied external stress is greater than this limit, the elastomer structure can be mechanically damaged. This upper limit depends on the kind of elastomer. Figure 1: Dimensionless temperature \(\vartheta\) shape as a function of the applied stress \(A\) [Pa] with different positive coefficient, \(\delta\) [Pa\({}^{-1}\)], of the first power in the argument of the exponential function in Equation (25). In this scenario, the elastomer acts as a thermal conductor, and the shape shows that, due to the external stress applied, there is a significant rise in its temperature. Now, we need to compare our results with experimental data. To do so, we consider the experiment developed by [PERSON] et al. [26].