The prefix “piezo” (pronounced pie-ease-o) comes from the Greek word for pressure or mechanical force. Piezoelectricity refers to the linear coupling between mechanical stress and electric polarization (the direct piezoelectric effect) or between mechanical strain and applied electric field (the converse piezoelectric effect). The equivalence between the direct and converse effects was established earlier using thermodynamic arguments (Section 6.2). The principal piezoelectric coefficient, d, relates polarization, P, to stress, X, in the direct effect (P = dX) and strain, x, to electric field E (x = dE). Thus the units of d are [C/N] or [m/V] which are equivalent to one another. Typical sizes for useful piezoelectric materials range from about 1 pC/N for quartz crystals to about 1000 pC/N for PZT (lead zirconate titanate) ceramics. To understand how the piezoelectric effect varies with direction and how it is affected by symmetry, it is necessary to determine how piezoelectric coefficients transform between coordinate systems. Since polarization is a vector and stress a second rank tensor, the physical property relating these two variables must involve three directions: . . . Pj = djklXkl . . . . In the new coordinate system . . . P'i = aijPj = aijdjklXkl . . . . Transforming the stress to the new coordinate system gives . . . P'i= aijdjklamkanlX'mn = d'imnX 'mn. . . . Thus piezoelectricity transforms as a polar third rank tensor. . . . d'imn = aijamkanldjkl . . . . In general there are 33 = 27 tensor components, but because the stress tensor is symmetric (Xij = Xji), only 18 of the components are independent. Therefore the piezoelectric effect can be described by a 6 × 3 matrix.
An optical technique based mark tracking for electrical field induced mechanical strain measurement in thin polymer films