content_image/355894.jpg content_image/355893.jpg content_image/355895.jpg content_image/355896.jpg content_image/355897.jpg content_image/355898.jpg content_image/355899.jpg content_image/355900.jpg
It is seen that at h → 0 the flexoelectric term plays the dominant role in the bending rigidity. 

The classical formula for the maximum delection of a cantilever beam w$_{max}$ = P$_{∗}$l $^{3}$/ 3 Y may be used to find the coefficient f$_{1133}$ from experimental results: 

It is obvious that the dispersive relation is 

where ω is a frequency and k is a wavenumber. 

## IV. TIMOSHENKO BEAM MODEL 

In this model we assume that the displacements are presented as 

<img src='content_image/355894.jpg'>

<img src='content_image/355893.jpg'>

<img src='content_image/355895.jpg'>

where w is the transverse displacement of the points of the centroidal axis( y = z = 0), and ψ is the rotation of the beam cross-section about the positive y − axis. 

From (19), the strains are equal to 

<img src='content_image/355896.jpg'>

Since the narrow beam deflects in x − z plane, we consider only two components of the electric field E$_{x}$ and E$_{z}$ . From (5) they can be expressed as 

or 

<img src='content_image/355897.jpg'>

<img src='content_image/355898.jpg'>

<img src='content_image/355899.jpg'>

<img src='content_image/355900.jpg'>

