Thermal Conductivity of Composite Materials#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Microstructural Behavior:} \quad \nabla \cdot (\mathbf{k}_{\text{micro}} \nabla T_{\text{micro}}) = 0 \\
&\text{Homogenized Behavior:} \quad \nabla \cdot (\mathbf{k}_{\text{eff}} \nabla T_{\text{eff}}) = 0
\end{align*}
\end{split}\]
Weak Formulation:
\[\begin{split}
\begin{align*}
&\text{Microstructural Behavior:} \quad \int_{\Omega_{\text{micro}}} \nabla \cdot (\mathbf{k}_{\text{micro}} \nabla T_{\text{micro}}) \phi_{\text{micro}} \,d\Omega_{\text{micro}} = 0 \\
&\text{Homogenized Behavior:} \quad \int_{\Omega_{\text{eff}}} \nabla \cdot (\mathbf{k}_{\text{eff}} \nabla T_{\text{eff}}) \phi_{\text{eff}} \,d\Omega_{\text{eff}} = 0
\end{align*}
\end{split}\]