Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory

Electromechanical Analysis of Flexoelectric Nanosensors Based on Nonlocal Elasticity Theory

“…The boundary condition in Equation 21indicates that nonlocal and flexoelectric effects act as a concentrated bending moment at the free end, which would change the shape of the deflection curve, especially near the free end. Hence, Figure 3 is presented to investigate the normalized deflection The boundary condition in Equation (21) indicates that nonlocal and flexoelectric effects act as a concentrated bending moment at the free end, which would change the shape of the deflection curve, especially near the free end. Hence, Figure 3 is presented to investigate the normalized deflection w(L)/w 0 (L) at the free end of sensors versus the normalized thickness h/h 0 for the different nonlocal parameters, in which h 0 = 12µ 2 13 / c 11 κ 33 + e 2 311 denotes an intrinsic thickness for the maximum induced electric potential of flexoelectric beams [15].…”

“…When the uniformly distributed load 0 has been considered, the expression of d 2 ( 1 )/d 1 2 = 0 should be satisfied in Equation (21). Here, a concentrated force at the free end and transverse From Equation 23, it can be clearly found that neglecting the nonlocal effect causes the induced electric potential to disappear as /( 0 ) = −1/3.…”

“…When the uniformly distributed load q 0 has been considered, the expression of d 2 q(x 1 )/dx 2 1 = 0 should be satisfied in Equation (21). Here, a concentrated force F at the free end and transverse distributed loads q = q 0 sin(nπx 1 /L) are applied on the nano-cantilever beams, where n = 1, 2, 3, .…”

“…In that paper, the piezoelectric effect played a leading role in the induced large-scale electric potential, while the flexoelectric and strain gradient elastic effects dominated the induced electric potential at the nanoscale. Based on the general modified strain gradient theory, Chu et al [21] established a mathematical framework to analyze bending and vibration responses of functionally graded piezo-flexoelectric nanobeams. The authors verified that the variation of the polarization density field along the thickness direction is fully dependent on the functionally graded model.…”

Size-dependent analysis of a functionally graded piezoelectric micro-cylinder based on the strain gradient theory with the consideration of flexoelectric effect: plane strain problem

“…Therefore, multiscale modeling of microstructures [27][28][29] and higher-order continuum elasticity theories including the non-local [30][31][32], couple stress [33][34][35] and strain gradient theories [36][37][38][39], which contain additional material length scale parameters for capturing the effect of size in small-scale structures, have been proposed. Many studies have been conducted on the basis of the SGT and non-local Eringen's theory to examine the size-dependent response of micro-and nanostructures with the consideration of surface and flexoelectric effects [40][41][42][43][44][45][46][47][48][49]. Lazopoulos and Lazopoulos [50] proposed another form of strain energy density function with surface energy to develop a new microscale Bernoulli-Euler beam model.…”