Mesh free model of nanobeam integrated with a flexoelectric actuator layer
Introduction
Research on nanostructures has been extensively being carried out during the last decade for developing micro or nano-electromechanical systems. For example, Peddieson et al. [1] analyzed the bending of nanobeams using the concept of nonlocal elasticity. Nilsson et al. [2] fabricated and characterized ultrashort nanocantilevers. Employing the conventional approach of finite element analysis, Bhusan and Agrawal [3] confirmed that an elastic model can be used for the analysis of nanobeams. McFarland and Jonathan [4] investigated the effect of microstructure on the stiffness of microcantilever. Lu et al. [5] carried out static and dynamic analysis of plate like thin-film structure using a generalized thin plate theory. Reddy [6] presented the reformulation of existing beam theories for analyzing nanobeams. Extensive research is being reported on the analysis of nanobeams based on the nonlocal theory of elasticity [7], [8], [9], [10], [11], [12], [13]. Liu and Rajapakse [14] developed continuum models for investigating the effect of surface energy on the static and dynamic response of nanoscaled beams.
The most widely known electro-elastic coupling phenomenon is the piezoelectric effect exhibited by the non-centrosymmetric dielectric materials at macro-scale level. These dielectric piezoelectric materials are electrically polarized because of the applied mechanical stress or strain and are conversely deformed due to the applied electric field. An exclusive type of the electro-elastic coupling phenomenon was first theoretically observed by Mashkevich and Tolpygo [15] while deriving equations of motion for lattice dynamics in crystals. They observed that the electric polarization is induced in the crystalline solids due to the presence of the strain gradient in the solid. A phenomenological theory explaining that the electron-phonon coupling in the centrosymmetric crystals is attributed to the strain gradient was presented by Kogan [16]. Haris [17] observed the presence of the electric polarization in the centrosymmetric solid due to the strain gradient caused by the shock wave. Askar et al. [18] first attempted to compute the electro-elastic coupling coefficients of cubic crystals governing the coupling between the electric field and the strain gradient. Bursian and Trunov [19] carried out a thermodynamic analysis demonstrating that the ferroelectric materials undergo bending due to the electric polarization gradient while the strain gradient induces the electric polarization. Indenbon et al. [20] followed the terminology used to describe the polarization in the nonpiezoelectric liquid crystals [21] and termed the electro-elastic coupling in the centric solid crystals due to the strain gradient and the polarization gradient as the flexoelectric effect in solids. Tagantsev [22] theoretically confirmed that the flexoelectric effect in the crystalline solids is different from the piezoelectric effect and derived a simple model for computing the flexoelectric coefficients. Similar to the definitions of the direct and the converse piezoelectric effects, the phenomenon of inducing the electric polarization due to the strain gradient has been defined as the direct flexoelectric effect while the phenomenon of developing induced mechanical stress or strain due to the electric field gradient has been termed as the inverse or the converse flexoelectric effect. Ma and Cross [23], [24], [25], [26], [27] experimentally investigated the direct flexoelectric effects in different ferroelectric ceramics. Fu et al. [28] carried out experimental studies of the converse flexoelectric effect induced by the inhomogeneous electric field in barium strontium. Cross [29] also conducted number of experiments for measuring different flexoelectric coefficients. Electromechanical coupling in nonpiezoelectric materials due to nonlocal size effects at nano scale has been studied by Maranganti et al. [30]. Majdoub et al. [31] demonstrated that the flexoelectricity enhances the apparent size dependent elastic modulus of naobeam. Maranganti and Sharma [32] derived an atomistic model to predict the flexoelectric coefficient tensor of crystalline dielectrics. Gharbi et al. [33] theoretically and experimentally estimated the flexoelectric properties of ferroelectrics. Jiang et al. [34] presented an extensive review on the consideration of flexoelectricity for developing different flexoelectric structures. They also demonstrated that the performance of the piezoelectric nanogenerators can be enhanced by exploiting the flexoelectric effect. Hu et al. [35] studied the performance of a thin flexoelectric layer as a sensor for modal analysis of circular rings. State-of-the-art reviews on the flexoelectricity in solids have been reported by Yudin and Tagantsev [36] and Zubko et al. [37]. Recently, Ray [38] investigated the static flexoelectric responses of flexoelectric nanobeam. The foregoing literature review indicates that the research on flexoelectricity in solids has been gaining serious attention as an emerging area of research and provides ample scope for challenging research work. Harnessing the flexoelectric effect may lead to the development of high performance smart nano sensors and nano actuators. However, the analysis of the flexoelectric materials as the potential actuator materials of smart nanostructures has not yet been extensively studied. Most recently, author [39] presented an exact solution for the nanobeam integrated with a flexoelectric layer acting as the distributed nanoactuator. It is known that the exact solutions are not possible for arbitrary loading and boundary conditions. Hence, the derivation of the numerical model is essential for any new structural problem. Although finite element (FE) analysis has been established as the efficient numerical methods for structural analysis, development of finite element model (FEM) for the analysis of flexoelectric structure may be difficult as the electric field gradient and the strain gradient are to be modeled. Also the generation of the appropriate mesh for the development of the FEM for the structure with complicated shape is not an easy task and often poses disadvantages. Beyond the FEM for numerical analysis, Belytschko et al. [40] first proposed an advanced computational method for structural analysis and the method is being popularized as the mesh free method. The mesh free method offers some of the important attributes of numerical method like upper bounds of the solution which adjudge a computational model as a better one than the existing one. Mesh less methods have been extensively used for the analysis of linear elastic problems. For example, Krysal and Belytscho [41], [42] analyzed thin plates and shells using mesh free Galerkin method. Atluri and Zhu [43] developed a new mesh less method for linear potential problems and the method has been named as the mesh-less petrov galerkin (MLPG) method. Liu and Chen [44] used the mesh free method for static and free vibration analyzes of thin plates of complicated shape. Analysis of free vibrations of laminated composites with complicated shape using a mesh free method was reported by Chen et al. [45]. Linear and nonlinear analyzes of corrugated plates using a mesh free method were carried out by Liew et al. [46], [47]. Post buckling analysis of functionally graded cylindrical shells has been carried out by using mesh free methods [48]. Recently, Liew et al. [49] developed a mesh free model of laminated composite plates integrated with the piezoelectric sensors/actuators based on the first order shear deformation theory. Zang and Liew [50] developed a new mesh free method based on the Ritz method for two-dimensional elasticity problems. Zhang et al. [51] used such mesh free method based on the Ritz method for the geometrically nonlinear analysis of carbon nanotube reinforced functionally graded composite plates with elastically restrained edges and internal supports. The mesh free method based on the Ritz method has also been implemented for solving three-dimensional wave equations [52]. The analysis of flexural strength and free vibration of functionally graded carbon nanotube reinforced cylindrical panels has also been presented by Zang et al. [53].
To the authors’ best knowledge, the use of the mesh free method for numerical modeling of flexoelectric smart composite structure is not yet available in the open literature. Hence, as a first numerical model, author intends to develop a mesh free model of nanobeam integrated with a flexoelectric layer acting as the distributed nanoactuator of the beam. This paper is concerned with the presentation of this mesh free model. Both axial and transverse shear actuations by the flexoelectric layer have been taken into consideration for deriving the model.
Section snippets
Basic equations
Three dimensional constitutive relations for the direct and the converse flexoelectric effects are, respectively given by [39]in which is the electric polarization, is the electric field, is the stress tensor, is the strain tensor, is the elastic coefficient tensor, is the flexoelectric coefficient tensor and is the dielectric susceptibility tensor of the flexoelectric layer. In Eq. (1), the coma
Mesh free models
The element free Galerkin method being one of the mesh free methods [40] is implemented to derive the mesh free model (MFM) of the overall smart flexoelectric composite nanobeam. According to the moving least squares (MLS) approximation, the generalized displacement at any point (x) in the beam can be approximated aswhere is a column vector constructed from a basis function and is a column vector consisting of the coefficients of the basis function. The displacement fields
Axial and transverse shear stresses
Once the nodal generalized parameters are computed by solving Eq. (26), the constitutive relations are used to compute the axial stress at any point in the substrate nanobeam as follows:
Using Eq. (28) in the governing equilibrium equation and satisfying the surface traction at the bottom surface of the substrate, the transverse shear stress at any point in the substrate nanobeam can be computed as follows:
Weight and Shape functions
In the present model the quartic type weight function is chosen as follows [44]:in which and is the size of the domain of influence of the th node. It may be observed that the first and second derivatives of with respect to x are necessary for computing the derivatives of the shape function matrix given by Eq. (19) and that of the matrices , given by Eq. (17). These derivatives are executed as follows:
Results and discussions
In this section, numerical results are presented to investigate the efficacy of the MFM model derived in the previous section. The flexoelectric layer is made of barium titanate (BaTiO3) which falls into the centrosymmetric point group possessing cubic symmetry [27]. Unless otherwise mentioned, the thickness of the substrate nanobeam is considered as 50 nm while the thickness of the flexoelectric layer is taken as 10 nm. The elastic material properties of the flexoelectric layer are used as
Conclusions
In this paper, a mesh free model (MFM) has been developed for the static analysis of smart nanobeams integrated with a flexoelectric layer at its top surface which acts as the distributed actuator of the substrate nanobeam. For activating the the flexoelectric layer its top surface is subjected to the distributed electric potential while its bottom surface being in contact with the top surface of the substrate nanobeam is grounded. The model is derived based on a layer wise displacement theory.
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