Piezoelectric Material

Mathematical Model:

\[\begin{split} \begin{align*} &\text{Mechanical Deformation:} \quad \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - \nabla \cdot (\boldsymbol{\sigma}) = \mathbf{0} \\ &\text{Piezoelectricity:} \quad \nabla \cdot \mathbf{D} = \rho_e, \quad \frac{\partial \mathbf{D}}{\partial t} = \mathbf{d} \frac{\partial T}{\partial t} \end{align*} \end{split}\]

Weak Formulation:

\[\begin{split} \begin{align*} &\text{Mechanical Deformation:} \quad \int_{\Omega} \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \cdot \mathbf{v} \,d\Omega - \int_{\Omega} \nabla \cdot (\boldsymbol{\sigma}) \cdot \mathbf{v} \,d\Omega = 0 \\ &\text{Piezoelectricity:} \quad \int_{\Omega} \nabla \cdot \mathbf{D} \phi \,d\Omega = \int_{\Omega} \rho_e \phi \,d\Omega, \quad \int_{\Omega} \frac{\partial \mathbf{D}}{\partial t} \cdot \boldsymbol{\psi} \,d\Omega \\ &\quad = \int_{\Omega} \mathbf{d} \frac{\partial T}{\partial t} \cdot \boldsymbol{\psi} \,d\Omega \end{align*} \end{split}\]