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An efficient spectral element (SE) with electric potential degrees of freedom (DOF) is proposed to investigate the static electromechanical responses of a piezoelectric bimorph for its actuator and sensor functions. A sublayer model based on the piecewise linear approximation for the electric potential is used to describe the nonlinear distribution of electric potential through the thickness of the piezoelectric layers. An equivalent single layer (ESL) model based on first-order shear deformation theory (FSDT) is used to describe the displacement field. The Legendre orthogonal polynomials of order 5 are used in the element interpolation functions. The validity and the capability of the present SE model for investigation of global and local responses of the piezoelectric bimorph are confirmed by comparing the present solutions with those obtained from coupled 3-D finite element (FE) analysis. It is shown that, without introducing any higher-order electric potential assumptions, the current method can accurately describe the distribution of the electric potential across the thickness even for a rather thick bimorph. It is revealed that the effect of electric potential is significant when the bimorph is used as sensor while the effect is insignificant when the bimorph is used as actuator, and therefore, the present study may provide a better understanding of the nonlinear induced electric potential for bimorph sensor and actuator.

Piezoelectric materials generate electric potentials in response to mechanical stresses, and conversely, produce mechanical movements in response to electric potentials. Therefore, piezoelectric materials can be used both as actuators and sensors, they transform electrical energy into mechanical energy, and

There have been many theories and models developed for analyzing piezoelectric bimorph structures with emphasis on approximating the mechanical displacement and electric potential. By carrying out exact 3-D analytical solutions for the simply supported piezoelectric plate [

Generally, accurately simulation of the local responses of the piezoelectric bimorph structures would inevitably lead to a very dense FE mesh when using the FE method. Hence, conventional FE simulation becomes computationally very inefficient. A more efficient method is the spectral element (SE) method which combines the geometric flexibility of FE method with the high accuracy of the pseudo spectral method. This method was first presented by Patera in the mid 1980s [

For the purpose of accurately representing the mechanical displacement and the electric potential, a reasonable choice is to use the ESL model for the mechanical variables and the layer-wise theory or the sublayer theory for the electric variables. In the present work, we attempt to combine the merits of the SE method and the sublayer model. More specifically, the mechanical variables,

A piezoelectric bimorph made of two identical PZT-4 piezoelectric layers, which has been investigated by Fernandes [_{x} σ_{y} σ_{z} τ_{yz} τ_{zx} τ_{xy}^{T} and _{x} ε_{y} ε_{z} γ_{yz} γ_{zx} γ_{xy}^{T} represent stress vector and strain vector, respectively. _{x} E_{y} E_{z}^{T}, the electric field vector, _{x} D _{y} D _{z}^{T}, the electric displacement vector,

An ESL model adopting the FSDT is adopted to describe the mechanical displacement. The displacement field of a piezoelectric bimorph based on FSDT takes on the form [

We define:

The infinitesimal strain components associated with the displacements are given by:

The Legendre polynomials based SE method can be described as follows: the bimorph is firstly discretized using a set of non-overlapping rectangular elements, as in the traditional FE method. Each rectangular element, denoted by Ω^{e}, is then mapped to a reference element, denoted by Ω^{ref}:

Subsequently, a set of nodes, denoted by _{i}_{j}^{ref} as roots of the following polynomial expression:
_{N}^{ref}, as depicted in

The 1-D shape functions at the 1-D GLL points _{i}_{ij}

Coordinates ^{e} may be uniquely related to _{ij}_{ij}_{i}_{j}^{ref} are discretized by the 2-D shape functions as:
_{ij}_{ij}_{ij}_{ij}_{ij}_{ij} is the displacement vector of the node _{i}_{j}

Substituting _{u}_{5×5} is a 5×5 identity matrix. Substituting _{u}

For the purpose of accurately modeling the distribution of the electric potential across thickness, each layer of the piezoelectric bimorph is subdivided mathematically into _{i}_{i}_{−1}, respectively. In each sublayer, the distribution of the electric potential ^{i}^{i}

In this way, the assumption of linear distribution of electric potential across the thickness is used not in the whole piezoelectric layer, but in each sublayer instead. As a result, the electric potential is approximated as piecewise linear across the thickness and it is expected that the quadratic distribution of the electric potential across the bimorph thickness can be approached with more sublayers adopted. As mentioned before, the mechanical displacement field is approximated using ESL model based on FSDT and the piezoelectric bimorph is discretized using 2-D mesh. To keep the compatibility, each sublayer of the piezoelectric layer is also discretized using the same mesh. Consequently, an element potential vector ^{e} is then introduced in the spectral plate finite element, which is defined as:

The surface potential of the sublayer, _{i}_{0}, _{1}, ⋯ _{2}_{n}^{i}

By applying Hamilton's principle, the elementary dynamic equations for the piezoelectric bimorph plate can be obtained:
_{s}_{s}

The GLL integration rule is then used to calculate the characteristic matrices and the nodal force vector in _{0} = 1,000N/m^{2} applied to the upper surface, and _{0} = 50V applied to the top and bottom surface of the bimorph. By applying the electric boundary conditions, the DOFs for the electric potential are condensed out such that _{uu}_{uu}_{p}_{u}_{a}

In this section, the derived SE model is converted into a numerical code and case studies are carried out to validate the effectiveness and the capability of the present model for predicting both the global responses and the local responses,

The length _{0} is taken as _{0} = 10^{10}V/m. For the purpose of comparison, a coupled 3-D analysis is carried out using 20-noded hexahedral 3-D piezoelectric elements (C3D20RE) with a mesh size of 40×20×10 in ABAQUS and the results from the coupled 3-D FE analysis are taken as accurate.

For this case a uniform pressure load of _{0} = 1,000N/m^{2} is applied to the upper surface and the bimorph is used as a sensor with the top and bottom surfaces grounded. The variations of both the deflection

To achieve practically meaningful actuation capabilities and guarantee that the piezoelectric material behaves linearly, an electric potential of _{0} = 50V is applied to the top and bottom surfaces of the bimorph with intermediate electrode grounded. The through-the-thickness variations of

Once again, the present model based on FSDT cannot predict accurately the through-the-thickness distribution of

The present work aims to develop an efficient SE model with electric potential DOFs for the static electromechanical response of a piezoelectric bimorph. The approach is the combination of an ESL model based on FSDT for the mechanical displacement with a sublayer model based on the piecewise linear approximation for the electric potential. 2-D GLL shape functions are used to discretize the displacements and then the governing equation of motion is derived following the standard SEM procedure. By applying the electric boundary conditions, the DOFs for the electric potential are condensed out such that the present model will not result in a large number of potential DOFs.

Numerical simulations based on the present model are carried out for two different load cases,

The research was supported by National Science Fund for Distinguished Young Scholars (grant number 11125209) and Natural Science Foundation of China (grant number 10702039).

The authors declare no conflict of interest.

Geometry of a piezoelectric bimorph.

Discretization of a plate and an example of spectral element.

Selected shape functions for a 36-node spectral element. (_{32}(_{45}(

A sublayer model for a piezoelectric bimorph.

Bimorph sensor of

Bimorph sensor of

Bimorph actuator of

Bimorph actuator of