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: f$_{1133}$ = √ 1 a$_{33}$ [ c$_{11}$ h 2 12 − P$_{∗}$l 3 3 bhw$_{max}$. ] ω $^{2}$ϱ ( A − Ik $^{2}$) − k $^{4}$( c$_{11}$I − d$_{31}$I a$_{33}$ − f 2 1133 a$_{33}$ A ) = 0 (18) u$_{1}$ ( x, y, z, t ) ≈ − zψ ( x, t ) , u$_{2}$ ( x, y, z, t ) ≈ 0 , u$_{3}$ ( x, y, z, t ) ≈ w ( x, t ) , (19) ε$_{x}$ = − zψ $^{′}$, ε$_{y}$ = 0 , ε$_{z}$ = 0 , γ$_{xy}$ = γ$_{yz}$ = 0 , γ$_{xz}$ = w ′ − ψ. (20) − E$_{x}$ = a$_{11}$P$_{x}$ + d$_{15}$γ$_{xz}$ + f$_{1111}$u$_{,xx}$ − E$_{z}$ = a$_{33}$P$_{z}$ + d$_{31}$ε$_{x}$ + f$_{1133}$u$_{,xz}$ + f$_{1313}$u$_{,zx}$ + f$_{3113}$w$_{,xx}$, − E$_{x}$ = a$_{11}$P$_{x}$ + d$_{15}$ ( w ′ − ψ ) − f$_{1111}$zψ ′′ − E$_{z}$ = a$_{33}$P$_{z}$ − d$_{31}$zψ ′ − f$_{1133}$ψ ′ + f$_{1313}$ ( w ′′ − ψ $^{′}$) , It is obvious that the dispersive relation is where ω is a frequency and k is a wavenumber. In this model we assume that the displacements are presented as 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 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 8 IV. TIMOSHENKO BEAM MODEL