Page 240 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 227

The first subscript of a term of Eq. (4.104) represents the direction of
application of the electric field, whereas the second subscript indicates the
direction of measuring the strain. An example will be studied next in order to
better understand the physical meaning of the amounts introduced in Eqs.
(4.99) through (4.104).

Example 4.11
Determine the total strain about the thickness direction for the fixed-free
piezoelectric plate of Fig. 4.42, which is subject to a force in the
presence of an electric field (the two vectors are parallel). The
material has a Young’s modulus of E = 48 GPa, and a charge constant
The area of the cross-section normal to the external force is

Solution:
The only mechanical stress is the one generated by the force F, about the
direction z (or 3). As a consequence, the matrix Eq. (4.99) reduces to a single
algebraic equation, namely:

Both terms are compressive as the mechanical load and the electrical field (in
conjunction with the piezoelectric poling field) generate deformations
(strains) about the negative direction of axis 3. The first term in the right-
hand side of Eq. (4.105) is the mechanical strain, which in this case can be
calculated as:

By combining Eqs. (4.105) and (4.106) and by using the given numerical
values, the total strain about the direction 3 becomes:
An equation similar to Eq. (4.99) can be written to express the
piezoelectric effects as: