Elasticity with Thermal Expansion#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Elasticity:} \quad \nabla \cdot \boldsymbol{\sigma} = \mathbf{0} \\
&\text{Thermal Expansion:} \quad \alpha \frac{\partial T}{\partial t} - \nabla \cdot (\beta T \mathbf{I}) = \mathbf{0}
\end{align*}
\end{split}\]
Weak Formulation:
\[\begin{split}
\begin{align*}
&\text{Elasticity:} \quad \int_{\Omega} \boldsymbol{\sigma} : \boldsymbol{\varepsilon} \,d\Omega = 0 \\
&\text{Thermal Expansion:} \quad \int_{\Omega} \alpha \frac{\partial T}{\partial t} \phi \,d\Omega - \int_{\Omega} \nabla \cdot (\beta T \mathbf{I}) \cdot \phi \,d\Omega = 0
\end{align*}
\end{split}\]