Page 24 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 11
plane. Two particular cases of the general state of deformations described
above are the state of plane stress and the state of plane strain. In a state of
plane stress, as the name suggests, the stresses are located in a plane (such as
the middle plane that is parallel to the xy plane in Fig. 1.7). The following
stresses are zero:

Figure 1.8 Plane state of stress/strain
Thin plates, thin bars and thin beams that are acted upon by forces in their
plane, are examples of MEMS components that are in a plane state of stress.
For thicker components, the cross-sections of shafts in torsion are also in a
state of plane stress. In a state of plane strain, the stress perpendicular to the
plane of interest does not vanish, but all other stresses in Eqs. (1.31)
are zero. Microbeams that are acted upon by forces perpendicular to the
larger cross-sectional dimension are in a state of plane strain for instance.
Figure 1.8 illustrates both the state of plane stress and the state of plane strain.

Example 1.2
A thin microcantilever, for which t << w, can be subject to a force as
shown in Fig. 1.9 (a) or to a force as pictured in Fig. 1.9 (b). Decide on the
state of stress/strain that is setup in each of the two cases.

Solution:
The loading and geometry of Fig. 1.9 (a) show that the stresses and
strains will be planar because of the thin condition of the microcantilever (t
<< w). However, because the load is perpendicular to the plane xy, the stress
about the z-direction does not vanish, and therefore, according to the
definition introduced previously, the microcantilever is in a state of plane
strain. In the case pictured in Fig. 1.9 (b), the force is located in the xy