content_image/354177.jpg content_image/354178.jpg content_image/354179.jpg content_image/354180.jpg content_image/354181.jpg content_image/354182.jpg content_image/354183.jpg content_image/354184.jpg
<img src='content_image/354177.jpg'>

where Q el z and M el z are the electric transverse shear force and the electric bending moment, respectively: 

with the associated boundary conditions at x = 0 

at x = l 

<img src='content_image/354178.jpg'>

<img src='content_image/354179.jpg'>

<img src='content_image/354180.jpg'>

<img src='content_image/354181.jpg'>

and at z = ± h/ 2 

where P$_{∗}$ is a force applied to the right edge of the cantilever. 

<img src='content_image/354182.jpg'>

The equations (13), (14) and the boundary conditions (15) - (17) couple the mechanical displacement w and the electric potential ϕ . 

From the equation (13) it is seen that piezoelectric and flexoelectric effects increase the bending rigidity of the beam ( a$_{33}$ < 0) and for the beams with large cross-section area and consequently large its moment of inertia, the term Af 2 $_{1133}$/a$_{33}$ plays insignificant role. However, for small cross-sectional dimensions, when 

<img src='content_image/354183.jpg'>

where h is the thickness of the beam, the flexoelectric effect becomes noticeable. 

The normalized Young modulus is 

<img src='content_image/354184.jpg'>

