Strain Dependence of Longitudinal Piezoelectric, Elastic, and Dielectric Constants ofX-Cut Quartz

Abstract

Experimental measurements of the piezoelectric current from impact-loaded $X$-cut quartz are utilized to determine the finite-strain piezoelectric and elastic constitutive relations for uniaxial strain ${\eta }_1$ from 2.4×${10}^{-3\mathrm{}}$ to 4.3×${10}^{-2\mathrm{}}$. The measurements are analyzed to provide values for the linear piezoelectric stress constant $e_{11}$, the direct-effect nonlinear constant $\frac{\partial e_{11}}{\partial {\eta }_1}$, the strain dependence of the permittivity, and the second-, third-, and fourth-order longitudinal elastic constants. It is found that consitutive relations developed to describe nonlinear responses at small strains describe the response of $X$-cut quartz for the large strains employed in the present investigation. The results show that $e_{11}=(0.1711\mathrm{}\pm 0.00094)$ C ${\mathrm{m}}^{-2\mathrm{}}$, $\frac{\partial e_{11}}{\partial {\eta }_1}=-2.64\mathrm{}\pm 0.048$ C ${\mathrm{m}}^{-2\mathrm{}}$, and ${\epsilon }_{11}^{-1\mathrm{}}\frac{\partial {\epsilon }_{11}}{\partial {\eta }_1}=-0.46\mathrm{}\pm 0.13$. The elastic constants are $c_{11}=(0.868\mathrm{}\pm 0.0095)\times {10}^{12}$ dyn ${\mathrm{cm}}^{-2\mathrm{}}$, $c_{111}=(-3.0\mathrm{}\pm 0.3)\times {10}^{12}$ dyn ${\mathrm{cm}}^{-2\mathrm{}}$, and $c_{1111}=+(75\mathrm{}\pm 25)\times {10}^{12}$ dyn ${\mathrm{cm}}^{-2\mathrm{}}$. Data from previous authors are analyzed to obtain values for the nonlinear piezoelectric constants at 573 and 79 °K.