In this paper, a Timoshenko dielectric beam model with the consideration of the direct flexoelectric effect is developed by using the variational principle. The governing equations and the corresponding boundary conditions are naturally derived from the variational principle. The effect of flexoelectricity on the Timoshenko dielectric beams is analytically investigated. The developed beam model recovers to the classical Timoshenko beam model when the flexoelectricity is not taken into account. To illustrate this model, the deflection and rotation of Timoshenko dielectric nanobeam under two different boundary conditions are calculated. The numerical results reveal that both the deflection and rotation predicted by the current model are smaller than those predicted by the classical Timoshenko beam model. Moreover, the discrepancies of the deflection and rotation between the values predicted by the two models are very large when the beam thickness is small. The current model can also reduce to the Bernoulli–Euler dielectric beam model wherein the shear deflection is not considered. This work may be helpful in understanding the electromechanical response at nanoscale and designing dielectric devices.

A Timoshenko dielectric beam model with flexoelectric effect

This paper represents the second part of the work that considers the identification and sizing of machined defects in a pipeline. This study deals with the torsional-mode reflection from defects and structural singularities in an industrial pipeline in order to perform the defect sizing. The wave finite element method (WFEM) is used to construct a numerical database of reflection coefficients from rectangular defects by varying thickness, axial and circumferential extents. Calculation is made depending on the frequency. The approximation of defects' sizes is carried out by sweeping the numerical database to find the suitable combination of dimensions for a given defect. The axial and circumferential extents are evaluated by fixed intervals for each possible thickness. Reflections from structural singularities (elbows, concrete blocks, clamps and welds) are also treated by comparing reflection coefficients obtained by the WFEM to those evaluated experimentally. Results show a good agreement for most of the structural singularities but not for the others.

Pipeline Inspection Using a Torsional Guided-Waves Inspection System. Part 2: Defect Sizing by the Wave Finite Element Method

Flexoelectricity becomes remarkable in nanoscale dielectrics where strong strain gradients are expected. The effect of flexoelectricity on elastic plane waves propagating in nanoscale layered phononic crystals is discussed in the current work. The fundamental governing equations and boundary conditions are derived from the virtual work principle. Detailed calculations are performed for nanoscale two-layered and three-layered phononic crystals using the transfer matrix method. Numerical results indicate that phononic crystals possess frequency pass bands and stop bands. For nanoscale layered phononic crystals, flexoelectricity increases the middle frequency regardless of the thickness ratio, whereas the flexoelectric effect on the bandwidth depends on the thickness ratio, which implies that there is an optimal thickness ratio to maximize the bandwidth. In addition, the middle frequency and bandwidth decrease with an increase in the unit cell thickness if the thickness ratio is fixed. Hence, considering the flexoelectric effect on band structures of nanoscale phononic crystals may provide guidance in manipulating elastic wave propagation and facilitate potential applications in phononic crystal devices.

On band structures of layered phononic crystals with flexoelectricity

The present paper is devoted to study the Love wave propagation in a fiber-reinforced medium laying over a nonhomogeneous half-space. The upper layer is assumed as reinforced medium and we have taken exponential variation in both rigidity and density of lower half-space. As Mathematical tools the techniques of separation of variables and Whittaker function are applied to obtain the dispersion equation of Love wave in the assumed media. The dispersion equation has been investigated for three different cases. In a special case when both the media are homogeneous our computed equation coincides with the classical equation of Love wave. For graphical representation, we used MATLAB software to study the effects of reinforced parameters and inhomogeneity parameters. It has been observed that the phase velocity increases with the decreases of nondimensional wave number. We have also seen that the phase velocity decreases with the increase of reinforced parameters and inhomogeneity parameters. The results may be useful to understand the nature of seismic wave propagation in fiber reinforced medium.

Propagation of love wave in fiber-reinforced medium over a nonhomogeneous half-space

Metamaterials (MMs) are composites that are artificially engineered to have unconventional mechanical properties that stem from their microstructural geometry rather than from their chemical compos...

Optimization of Viscoelastic Metamaterials for Vibration Attenuation Properties

The assumption of quasi-static electric field in the problem of wave reflection in piezoelectric half-plane results in missing an independent electric wave mode at the piezoelectric boundary, which leads to oversimplified solutions of reflected waves in a strong piezoelectric medium if only elastic bulk wave boundary conditions are considered. The paper presents a novel solution to address the issue by using the inhomogeneous wave theory and introducing a virtual reflection wave mode in addition to the elastic bulk wave modes. The virtual wave is assumed to satisfy the Snell's law as well as the piezoelectric boundary condition and can be treated in the same way as the elastic bulk waves. The analysis results show that this virtual wave always propagates along the boundary for any incident angle and can be treated as a pseudo surface wave. The energy transmission analysis reveals that this surface wave transmits zero energy and does not violate the energy conservation between the incident and the reflected elastic bulk waves. In addition, the analysis also reveals an interesting result that the quasi-transverse, not the quasi-longitudinal, incident wave will be fully reflected and no quasi-longitudinal reflected wave will be generated if the incident angle is beyond a critical angle.

Xiaoguang Yuan

Papers

Wave reflection in piezoelectric half-plane

A generalized enriched finite element method for blunt cracks