Page 255 - Mechanics of Microelectromechanical Systems

242 Chapter 4

Equation (4.138) has to be substituted into Eq. (4.134) in order to determine
the curvature radius of a thermal bimorph.
A more realistic case is when both layers of a bimorph are exposed to the
same temperature variation. For instance, when the lower layer has a
tendency to expand more than the upper layer and this situation is equivalent
to only decreasing the temperature of the upper layer. This design was
analyzed as early as 1925 by S. Timoshenko [11], who called the thermal
bimorph a bi-metal thermostat since the materials of the two layers were
metals. By following a procedure similar to the one already presented in the
introduction to this sub-section, it can be shown that Eq. (4.134) remains
valid by taking:

Example 4.14
Compare the bending performance of two physically-identical thermal
bimorphs, when for one of them the lower layer is heated by a temperature
whereas for the other bimorph both layers are heated by the same
temperature. Assume that

Solution:
Equation (4.134) gives the curvature radius for both bimorph designs by
means of the induced strain of either Eq. (4.138) – for the design with one
heated layer, or Eq. (4.139) – for the configuration with both layers heated.
The ratio of the two radii is simply:

Although the ratio of Eq. (4.140) is larger than 1 only when
and therefore the radius of curvature of the bimorph with one heated layer is
greater than the radius of the similar bimorph with both layers heated. When
the ratio of Eq. (4.140) is less than 1, which indicates that

8.3 Piezoelectric (PZT) Bimorph

When one of the layers forming the bimorph (for instance the upper one)
is made up of a piezoelectric material, the free strain of this layer can be
expressed, according to Eq. (4.99), as: