Page 20 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 7
on the right face of the element shown in Fig. 1.6 (a), while the opposite face
is fixed, the elastic body will deform linearly by a quantity such that the
final length about the direction of deformation will be The ratio of
the change in length to the initial length is the linear strain:

If an elementary area dA is isolated from the face that has translated, one can
define the normal stress on that surface as the ratio:

Figure 1.6 Element stresses: (a) normal; (b) shearing

where is the elementary force acting perpendicularly on dA. For small
deformations and elastic materials, the stress-strain relationship is linear, and
in the case of Fig. 1.6 (a) the normal stress and strain are connected by means
of Hooke’s law:

where E is Young’s modulus, a constant that depends on the material under
investigation.
When the distributed load acts on the upper face of the volume element
and is contained in that face, as sketched in Fig. 1.6 (b), while the opposite
face is fixed, the upper face will shear (rotate) with respect to the fixed
surface. The relevant deformation here is angular, and the change in angle
is defined as the shear strain in the form: