Composite Materials with Thermal Expansion

Mathematical Model:

\[\begin{split} \begin{align*} &\text{Matrix Phase:} \quad \alpha_{\text{matrix}} \frac{\partial T_{\text{matrix}}}{\partial t} - \nabla \cdot (\beta_{\text{matrix}} T_{\text{matrix}} \mathbf{I}) = \mathbf{0} \\ &\text{Fiber Phase:} \quad \alpha_{\text{fiber}} \frac{\partial T_{\text{fiber}}}{\partial t} - \nabla \cdot (\beta_{\text{fiber}} T_{\text{fiber}} \mathbf{I}) = \mathbf{0} \end{align*} \end{split}\]

Weak Formulation:

\[\begin{split} \begin{align*} &\text{Matrix Phase:} \quad \int_{\Omega_{\text{matrix}}} \alpha_{\text{matrix}} \frac{\partial T_{\text{matrix}}}{\partial t} \phi_{\text{matrix}} \,d\Omega_{\text{matrix}} - \int_{\Omega_{\text{matrix}}} \nabla \cdot (\beta_{\text{matrix}} T_{\text{matrix}} \mathbf{I}) \cdot \phi_{\text{matrix}} \,d\Omega_{\text{matrix}} = 0 \\ &\text{Fiber Phase:} \quad \int_{\Omega_{\text{fiber}}} \alpha_{\text{fiber}} \frac{\partial T_{\text{fiber}}}{\partial t} \phi_{\text{fiber}} \,d\Omega_{\text{fiber}} - \int_{\Omega_{\text{fiber}}} \nabla \cdot (\beta_{\text{fiber}} T_{\text{fiber}} \mathbf{I}) \cdot \phi_{\text{fiber}} \,d\Omega_{\text{fiber}} = 0 \end{align*} \end{split}\]