Numerical Analysis of Piezoelectric Nanobeams with Flexoeletric Effect

Cutting edge composite materials now play an essential role in micro/nano-structural systems. Micro/ Nano particles and/ or structural systems are embedded into a material matrix in order to enhance material properties, and make them a "smart material" . A widely used type of smart material is piezoelectric materials, which have the property of piezoelectricity an electromechanical effect that occurs in some micronanostructural materials. A further electromechanical effect called the flexoeletric effect is also induced due to the fact that strain gradient on the micro/nano scale provides an electrical polarization. In this work, the Finite Element Method (FEM) formulation for beam theories is used to investigate materials including both piezoelectric and flexoelectric effects. The variational form of 2D piezoelectric nanobeams with flexoelectric effect is also presented and solved using FEM. The results from beam theories and 2D planar beams will be compared with analytical solutions and other numerical techniques.


Introduction
With the advancement of electro-mechanic technology, the nano-structure devices have attracted many researchers' enthusiasm and size-dependent effects of materials have become one of the hottest topics.
Piezoelectricity, first discovered in 1880 by French physicists Jacques and Pierre Curie, is the electrical polarization in response to applied mechanical stress. In 2009, Maranganti and Sharma [ 1] gave a more explicit definition of piezoelectricity: Upon application of a uniform strain, internal sub-lattice shifts within the unit cell of a noncentrosymmetric dielectric crystal result in the appearance of a net dipole moment. It has also been observed that the piezoelectric effect is symmetric in the sense that charge separation occurs due to stress (the direct piezoelectric effect) and stress and strain are induced when an electric field is applied (inverse piezoelectric effect).
The flexoelectric effect is a further electromechanical effect in insulating materials in which gradients in the strain induce an electrical polarisation. The similarity between piezoelectricity and flexoelectricity is that both produce polarizations, however they differ in that piezoelectricity is a macroscopic effect while flexoelectricity is a microscopic one. Therefore, theories of strain-gradient elasticity are needed to predict the response of a micro/nano beam when the flexoelectric effect is involved. Toupin [4] put forward a variational principle for a linear piezoelectric material. The polarization can be also induced by a strain gradient, which is the well known " flexoelectric effect" . The contributed polarization in response to macroscopic strain gradient can be written as x is the direction of the gradient.
A lot of pioneering work investigating flexoelectric effect has been undertaken. Yan and Jiang [5] acquired the electromechanical coupling behavior of piezoelectric nanowires with Bernoulli-Euler Beam theory. Two years after, they investigated the flexoelectric effect on the bending of piezoelectric nano-beams by different boundary conditions with different beams theories. In 2014, Liang et al. [6] incorporated the flexoelectric and surface effect into his Bernoulli-Euler beam model, and solved them analytically.
In this paper, the one dimensional beam theories will be used to simulate the beam' s behavior along with piezoelectricity and flexoelectricity. The new 2D beam models for piezoelectric nanobeams with the flexoelectric effect will be expressed.

Formulation
The development of a continuum electromechanical coupling framework by Hu et al. [7] has resulted in a great deal of interest in nanobeam modelling including the flexoelectric effect. Taking the flexoelectric effect into account, the bulk electric Gibbs free energy has been proposed as a hybrid of the internal energy of elastic deformation and electrostatic field.
where ij a , ijkl c , ijk d and ijkl f denote second-order permittivity, four-order elastic, piezoelectric and flexoelectric tensors, respectively. The other terms are the strain tensor, ij  , the strain gradient tensor, jkl  , and the electric field tensor, i E . Here, the constitutive equations [6] are where ij  is the classical stress tensor, i D is the electrical displacement vector and jkl  is the high-order stress tensor.
Substituting these into Eq. 2, the final form of the bulk electric Gibbs energy can be rewritten as and these can be expressed in the variational form as: Written with these variables, the variational form of Eq.7 appears similar to that of the strain gradient elasticity theory of the strain gradient elasticity theory [ 8] with the addition of an electric term. Eq. 2 can be written as in variational form as [8]: For 2D beam models in the x-z plane, the internal energy of elastic deformation can be written as [9] 1 1 13 13 113 113 0 3 3 3 3 As shown through this section the variational form was derived from the Principal of the Principal of Virtual Work and these can be solved using well known simulation tools, such as Finite element methods. Where the subscripts and superscripts represent the beam theories used: Euler-Bernoulli beam (EB), Timoshenko beam (TB) and 2D planar beam.  Table 1 shows the deflection of beam with the aspect ratio of 5. The "Classical" column represents the beam without the size-dependent effect.  Table 2 shows the deflection of beam with the aspect ratio of 10. It is noted that the displacements calculated from EB theory are unchanged though the aspect ratio under the same loads, whereas the deflections calculated by TB theory changed. This may be due to the fact that TB theory considers the effect of shear deformation in beams, resulting in the observation of a big difference between the results from 1D(EB) and 1D(TB).

Numerical Results
In order to illustrate the results more evidently, the data from Table 1 and 2 is presented in Figure 1, where the horizontal axis represents the aspect ratio ( L= h) and the vertical axis represents the percentage error.

Figure 1. Errors for clamped-ends beams VS aspect ratio
After looking into the tables deeply, the contribution of electrical load is relatively small for CC beams, comparing with the SS beams and CT beams. This shows the analysis of a piezoelectric nanobeam indicating the flexoelectric effect.

Conclusions
The results show that the flexoelectric effect plays a more essential role in the behavior of materials at micro/nano scale than the piezoelectric effect. Under a purely mechanical load (Vn=0), flexoelectricity acted to stiffen the beams, as can be seen from the reduced deflection. Under an external electrical load (Vn=1), with cantilever beams subjected to a positive voltage, flexoelectricity plays an essential role in preventing bending, greatly reducing the beam displacement.