Why would the dielectric constant at zero strain be different than the dielectric constant at zero stress? 


Hello. My question is about the last equation, where the conversion is between dielectric constant at zero strain and the dielectric constant at zero stress. What's the difference between them? Why would the dielectric constant differ when taking stress equal to zero, than when taking strain equal to zero?
 A: Zero stress and zero strain may sound essentially equivalent, but the constraints are very different: either (1) we apply no loads at all and allow the system to deform freely, or (2) we apply sufficient loads to keep the system from deforming at all.
It's typically only the case of very simple elastic systems with constant conditions and no other physical phenomena occurring that the two conditions can be equivalent (as in isothermal Hooke's Law $\sigma=E\varepsilon$, where $\sigma=0$ and $\varepsilon=0$ give the compatible result $0=0$).
As a slightly more sophisticated situation, consider a material held in place between two relatively rigid supports. Any temperature change will causes a stress (compressive or tensile, depending on the sign of the thermal expansion coefficient) even as the strain along the connection axis remains zero. We can expect other phenomena such as electrical behavior to behave somewhat differently under this stress state compared to that under the stress-free state.
In general, we shouldn't expect equal behavior (or equal associated constants such as the dielectric constant) when different conjugate variables are held constant, e.g., constant temperature vs. constant entropy, constant pressure vs. constant volume, constant electric field vs. constant polarization, or constant magnetic field vs. constant magnetization. The same applies to constant stress vs. constant strain.
