- #1
etotheipi
For a nice cubic non-centrosymmetric crystal like quartz/##\mathrm{SiO_2}## we can imagine that on application of stress the tetrahedral coordination polyhedra become distorted, and the central ion is displaced by a fraction ##\lambda## of the cell parameter ##a##. The total polarisation ##\mathbf{P} = \mathbf{P}_1 + \mathbf{P}_2## is the sum of two contributions: ##\mathbf{P}_1 \propto q (\lambda a)/a^3 \boldsymbol{e}_1## due to the 'frozen-in' displacement of the central ions in each unit cell, and ##\mathbf{P}_2 = \varepsilon_0 (\kappa - 1) \mathbf{E}## due to polarisation of the rest of the crystal (approximated to be linear and isotropic) due to the net field ##\mathbf{E}##. With the Gauss relation ##\nabla \cdot \mathbf{E} = \rho / \varepsilon_0## applied in integral form to a pillbox at the surface then we can show for a linear piezoelectric we get something like$$V = \frac{\sigma Ld}{\kappa \varepsilon_0}$$where ##\sigma## is the magnitude of stress, ##L## the width of the crystal and ##d## a piezoelectric coefficient. But for the general piezoelectric we have the constitutive equations$$\begin{align*}
\sigma_{ij} &= c_{ijkl} S_{kl} - e_{kij} E_k \\
D_k &= e_{kij} S_{ij} + \epsilon_{ki} E_i\end{align*}$$with ##\sigma_{ij}## the stress tensor, ##S_{ij}## the strain tensor, ##\epsilon_{ij}## the dielectric tensor, ##c_{ijkl}## elastic constants, ##e_{jik}## piezoelectric constants and ##E_i## & ##D_i## as usual. Does anyone have reference to derive these equations? I don't know anything about elasticity theory. Thanks
\sigma_{ij} &= c_{ijkl} S_{kl} - e_{kij} E_k \\
D_k &= e_{kij} S_{ij} + \epsilon_{ki} E_i\end{align*}$$with ##\sigma_{ij}## the stress tensor, ##S_{ij}## the strain tensor, ##\epsilon_{ij}## the dielectric tensor, ##c_{ijkl}## elastic constants, ##e_{jik}## piezoelectric constants and ##E_i## & ##D_i## as usual. Does anyone have reference to derive these equations? I don't know anything about elasticity theory. Thanks