I want to solve the thermal expansion of solid by using FEM approach. When I developed the model based on the the principle the minimum potential energy, the solutions for thermal expansion are not correct, although the results of static bending problems under external mechanical loads shows good results. The basic equations I employed as follows
$\delta U = \int\limits_V {\delta {{\bf{\varepsilon }}^T}\left( {{\bf{C\varepsilon }} + {\bf{\bar \sigma }}} \right)dV} = \int\limits_V {\left( {\delta {{\bf{\varepsilon }}^T}{\bf{C\varepsilon }} + \delta {{\bf{\varepsilon }}^T}{\bf{\bar \sigma }}} \right)dV} =0 $
where C is the constituve matrix,
${\bf{\sigma }} = \left\{ \begin{array}{l} {\sigma _{xx}}\\ {\sigma _{yy}}\\ {\sigma _{zz}}\\ {\sigma _{yz}}\\ {\sigma _{xz}}\\ {\sigma _{xy}} \end{array} \right\}$
${\bf{\varepsilon }} = \left\{ \begin{array}{l} {\varepsilon _{xx}}\\ {\varepsilon _{yy}}\\ {\varepsilon _{zz}}\\ {\gamma _{yz}}\\ {\gamma _{xz}}\\ {\gamma _{xy}} \end{array} \right\}$
${\bf{\bar \sigma }} = - \frac{{E\alpha \Delta T}}{{1 - 2\nu }}\left\{ \begin{array}{l} 1\\ 1\\ 1\\ 0\\ 0\\ 0 \end{array} \right\}$
I saw that most of the references dealing with thermal bending problems are developed based on equilibrium equations $\frac{{\partial {\sigma _{xx}}}}{{\partial x}} + \frac{{\partial {\sigma _{xy}}}}{{\partial y}} + \frac{{\partial {\sigma _{xz}}}}{{\partial z}} = 0$
$\frac{{\partial {\sigma _{xy}}}}{{\partial y}} + \frac{{\partial {\sigma _{yy}}}}{{\partial y}} + \frac{{\partial {\sigma _{yz}}}}{{\partial z}} = 0$
$\frac{{\partial {\sigma _{xz}}}}{{\partial x}} + \frac{{\partial {\sigma _{yz}}}}{{\partial y}} + \frac{{\partial {\sigma _{zz}}}}{{\partial z}} = 0$
I'm not sure that I might have missed something in the energy equation or is there a condition that the problem must be dealt with basis of equilibrium equation. Could someone help me out?