content_image/360139.jpg content_image/360138.jpg content_image/360140.jpg content_image/360141.jpg content_image/360135.jpg
Here k = 5 / 6 is the shear correction factor and 

Now, let us introduce new quantities 

Then, the dispersion relation can be written as 

or 

<img src='content_image/360139.jpg'>

<img src='content_image/360138.jpg'>

<img src='content_image/360140.jpg'>

<img src='content_image/360141.jpg'>

The equation (22) presents the frequencies for two wave modes. The lower frequency relates to flexural wave and the higher frequency relates to shear wave. 

From the formulas (22) and (18) it is seen that the eigenfrequencies increase if we take into account the piezoelectric and/or flexoelectric effects. The results of the comparison between the eigenfrequencies calculated within the framework of the Euler-Bernoulli and the Timoshenko beam models coincide with the conclusions related to single-walled nanotubes presented in [8]. 

## V. TWO-DIMENSIONAL MODEL 

In this model, we consider two components of the displacement u = u$_{1}$ and w = u$_{3}$ and only three components of strain tensor acting in x and z direction 

<img src='content_image/360135.jpg'>

