Homogenization of Composite Structures

Mathematical Model:

\[\begin{split} \begin{align*} &\text{Microstructural Behavior:} \quad \nabla \cdot (\mathbf{C}_{\text{micro}} : \boldsymbol{\varepsilon}_{\text{micro}}) = \mathbf{0} \\ &\text{Homogenized Behavior:} \quad \nabla \cdot (\mathbf{C}_{\text{eff}} : \boldsymbol{\varepsilon}_{\text{eff}}) = \mathbf{0} \end{align*} \end{split}\]

Weak Formulation:

\[\begin{split} \begin{align*} &\text{Microstructural Behavior:} \quad \int_{\Omega_{\text{micro}}} \nabla \cdot (\mathbf{C}_{\text{micro}} : \boldsymbol{\varepsilon}_{\text{micro}}) \cdot \boldsymbol{\phi}_{\text{micro}} \,d\Omega_{\text{micro}} = 0 \\ &\text{Homogenized Behavior:} \quad \int_{\Omega_{\text{eff}}} \nabla \cdot (\mathbf{C}_{\text{eff}} : \boldsymbol{\varepsilon}_{\text{eff}}) \cdot \boldsymbol{\phi}_{\text{eff}} \,d\Omega_{\text{eff}} = 0 \end{align*} \end{split}\]