## 2 Materials and Methods  
In this Section, we develop the theoretical bases of the thermophysical model proposed. To do so, we consider a continuum elastic body [13,14,21]. It has been assumed that the bounded regular region of space, occupied by the body in a fixed reference configuration, is, respectively, closed in the case of mass transfer and opened in the case of energy transfer. This body is defined as a thermodynamic closed system. According to [PERSON] [20], the local form of the First Law of Thermodynamics can be written as:  
\[\rho\,\dot{e}=\mathbf{\hat{S}}\cdot\mathbf{\hat{F}}-\
abla\cdot\mathbf{q}+\rho\,r \tag{1}\]  
where \(\rho\) is the mass density, \(r\) represents the heat supply per unit mass and unit time, \(\mathbf{q}\) represents the heat flux vector per unit surface area and unit time, \(e\) represents the internal specific energy, \(\mathbf{\hat{S}}=\mathbf{\hat{S}}(\mathbf{F},T)\) represents the first Piola-Kirchhoff stress tensor [20; 22] and \(\mathbf{F}\) is the deformation gradient, defined as [20; 22]:  
\[\mathbf{F}=\mathbf{1}+\
abla\mathbf{u} \tag{2}\]  
with \(\mathbf{u}\) displacement. Now we introduce the definition of finite strain tensor \(\mathbf{D}\) as follows [20; 22]:  
\[\mathbf{D}=\frac{1}{2}\left(\mathbf{F}^{+}\,\mathbf{F}-\mathbf{1}\right)= \mathbf{E}+\frac{1}{2}\,\
abla\mathbf{u}^{+}\,\
abla\mathbf{u} \tag{3}\]  
where \(\mathbf{1}\) is the unit tensor, \(\mathbf{F}^{+}\) is the transposed matrix of \(\mathbf{F}\), and \(\mathbf{E}\) is the infinitesimal strain tensor, defined by the following relation:  
\[\mathbf{E}=\frac{1}{2}\left(\
abla\mathbf{u}^{+}+\
abla\mathbf{u}\right) \tag{4}\]  
Considering that:  
\[\rho\,\dot{e}-\mathbf{\hat{S}}\cdot\mathbf{\dot{F}}=\rho\,c\,\dot{T}-T\,\frac {\partial\mathbf{S}}{\partial T}\cdot\mathbf{\dot{D}} \tag{5}\]  
obtained in Refs. [13; 14] for an isotropic continuous body [23] under the restriction \(\mathbf{\hat{S}}(\mathbf{F},T)=\mathbf{F}\,\mathbf{S}(\mathbf{D},T)\)[20; 22], Equation (1) can be rewritten as follows [13]:  
\[\rho\,c\,\dot{T}=T\,\frac{\partial\mathbf{S}}{\partial T}\cdot\mathbf{\dot{D }}-\
abla\cdot\mathbf{q}+\rho\,r \tag{6}\]  
where \(c\) is the specific heat. Now we introduce the constitutive equation for an isotropic body [24]:  
\[\mathbf{S}=2\mu\mathbf{E}+\left[\lambda\,\text{tr}\mathbf{E}-\beta\left(T-T_ {0}\right)\right]\mathbf{1}  
* The thermodynamic process is adiabatic without any internal heat source: it leads to the following mathematical condition: \[\
abla\cdot\mathbf{q}-\rho\,r=0\] (11)
* The term \(\left(\partial\beta/\partial T\right)\left(T-T_{0}\right)\) is negligible with respect to \(\beta\), as a consequence of both experimental results [20; 24] and the evaluation of the order of magnitude [20],  
Consequently, Equation (10) becomes:  
\[\rho\,c\,\frac{\hat{T}}{T}=2\,\frac{\partial\mu}{\partial T}\,\mathbf{E}\cdot \hat{\mathbf{E}}+\left(\frac{\partial\lambda}{\partial T}\,\text{tr}\mathbf{E} -\beta\right)\,\text{tr}\hat{\mathbf{E}} \tag{12}\]  
which represents the differential thermoelastic relationship between temperature and strain. The constitutive equation, Equation (7) can be inverted, obtaining:  
\[\mathbf{E}=\frac{1}{2\mu}\,\mathbf{S}-\left[\frac{\lambda}{2\mu\left(3\lambda +2\mu\right)}\,\text{tr}\mathbf{S}+\alpha\left(T-T_{0}\right)\right]\mathbf{1} \tag{13}\]  
Considering Equation (13) and the following relations among the Lame constants, \(\lambda\) and \(\mu\), the Poisson's ratio \(\
u\) (Poisson's ratio is a measure of the Poisson effect, i.e., the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression [22]) and the Young's modulus \(E\) (Young's modulus is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise [22]):  
\[ \begin{cases}\lambda=\frac{E\,\
u}{\left(1+\
u\right)\left(1-2\
u\right)}\\ \mu=\frac{E\,\
u}{2\left(1+\
u\right)}\end{cases}