Page 26 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 13
Equations (1.32) are solved for the strains and

Because this is a state of uniaxial stress, the only variable is x, and therefore
the displacement about this direction can be calculated as:

under the assumption that the strain is constant about the microbar’s length.
By combining now the first of Eqs. (1.33) with Eq. (1.34) results in the
following stress about the x-direction (which is also the tensile residual
stress):

where the subscript r indicates residual. The numerical value of the residual
stress is:
The work done quasi-statically by the normal stress on the volume
element of Fig. 1.6 (a) is equal to because the intensity of the stress
increases gradually from zero to its actual value Similarly, the work
performed by the shear stress on the element of Fig. 1.6 (b) is Since
the two elements are in static equilibrium, the external work fully converts
into strain (elastic) energy under ideal conditions. The potential strain energy
which is stored in a body that deforms elastically, such as the element in Fig.
1.7, comprises contributions from all the stresses and strains, namely:

The total strain energy can be expressed either in terms of stresses as:

or in terms of strains as: