Advances in Applied Mechanics

8.3.2 Thermodynamic Potential for a Linear Thermoelastic Composite

First, by taking the local thermal stress coefficient ${\mathrm{\Gamma }}_{ij}=\overline{{\mathrm{\Gamma }}_{ij}}+{\mathrm{\Gamma }}_{ij}^{\prime }$ (with $\overline{{\mathrm{\Gamma }}_{ij}^{\prime }}=0$) and strain ${\varepsilon }_{ij}=\overline{{\varepsilon }_{ij}}+{\varepsilon }_{ij}^{\prime }$ (with $\overline{{\varepsilon }_{ij}^{\prime }}=0$), the volume integration of (165) gives

$\begin{array}{l}\hfill \mathrm{\Psi }=\frac{1}{2}\overline{{\varepsilon }_{ij}}\overline{{\sigma }_{ij}}+\frac{1}{2}\overline{{\mathrm{\Gamma }}_{ij}}\overline{{\varepsilon }_{ij}}\mathrm{\Delta }T+\frac{1}{2V}{\int }_V\left({\mathrm{\Gamma }}_{ij}^{\prime }{\varepsilon }_{ij}^{\prime }\mathrm{\Delta }T\right)dV+\frac{1}{2}\overline{c_v}\frac{\mathrm{\Delta }{T}^2}{T_0}.\end{array}$

  (170)

Here we used the fact that the Hill–Mandel condition (