In the article
Ehrenfest Relations for Ultrasound Absorption in Sr2RuO4, Sigrist M., Progress of Theoretical Physics, Vol. 107, No. 5, May 2002
A superconductor with proposed p-wave pairing and order parameter of the structure $\vec{d}(\vec{k})=\hat{z}(k_x\pm i k_y)$ is discussed. The article lists the "standard" multi-components Ginzburg-Landau free energy, as well as the elastic energy density: $$ f_{el}=\frac{1}{2}\left[c_{11}(\epsilon_{xx}^2+\epsilon_{yy}^2)+c_{33}\epsilon_{zz}^2+2c_{12}\epsilon_{xx}\epsilon_{yy}+4c_{66}\epsilon_{xy}^2+2c_{13}(\epsilon_{xx}+\epsilon_{yy})\epsilon_{zz}+4c_{44}(\epsilon_{xz}^2+\epsilon_{yz}^2)\right] $$ And coupling of strain to order parameter(s): $$ f_c=(r_1(\epsilon_{xx}+\epsilon_{yy})+r_2\epsilon_{zz})|\vec{\eta}|^2+r_3\epsilon_{xy}(\eta_x^*\eta_y+\eta_x\eta_y^*)+r_4(\epsilon_{xx}-\epsilon_{yy})(|\eta_x|^2-|\eta_y|^2) $$
with $c_{ij}$ the elastic constants (in Voigt notation) determined by second derivative of free energy in respects to strain, $r_i$ coupling coefficents. The article gives no explenation into why the structure of the coupling term is as given, except to say that it is due to symmetry consideration.
How is the structure of the coupling term determined? How is the elastic energy density determined? What would be the coupling term for an order parameter belonging to the $D_{3h}$ point group, as opposed to $D_{4h}$?
