Analytical modeling of the apparent d(33) piezoelectric coefficient determined by the direct quasistatic method for different boundary conditions

The longitudinal piezoelectric coefficient d(33) measured by the quasistatic (Berlincourt-type) method is strongly dependent on the sample aspect ratio (thickness/lateral dimension) and the type of contacts used to collect the charge and apply the pressure on the sample. In this study, we present an analytical model in which apparent (measured) d(33) is a function of the aspect ratio, the mechanical properties of contacts, and the properties of piezoelectric material. Using derived relations, it is possible to obtain the true value of d(33) in the wide range of boundary conditions. The analytical results are compared with simulations using finite element modeling (FEM) and with experimental data; and good correlation is found among them. Effect of the geometry of the contacts on apparent d(33) value is specifically addressed.