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The coupling effects between the mechanical and electric properties of piezoelectric materials have drawn significant attention for their potential applications as sensors and actuators. In this investigation, two piezoelectric actuators are symmetrically surface bonded on a cross-ply composite laminate. Electric voltages with the same amplitude and opposite sign are applied to the two symmetric piezoelectric actuators, resulting in the bending effect on the laminated plate. The bending moment is derived by using the classical laminate theory and piezoelectricity. The analytical solution of the flexural displacement of the simply supported composite plate subjected to the bending moment is solved by using the plate theory. The analytical solution is compared with the finite element solution to show the validation of present approach. The effects of the size and location of the piezoelectric actuators on the response of the composite laminate are presented through a parametric study. A simple model incorporating the classical laminate theory and plate theory is presented to predict the deformed shape of the simply supported laminate plate.

Piezoelectric materials with the advantages of quick response, low power consumption and high linearity have drawn much attention in the past decade. Piezoelectric devices are of great interest in structural engineering with applications to shape control, vibration suppression and noise reduction [

The present work investigated the load transfer between surface bonded piezoelectric actuators and the host structure. The proposed method is an extension of the one dimensional beam embedded with PZT derived by Crawley and de Luis [

Consider two piezoelectric actuators symmetrically surface bonded on a cross-ply composite laminate. The polarized direction is along the z-axis. For an unconstrained thin piezoelectric actuator, equal strains in both _{31}, applied voltage _{pe}

The two actuators are activated by applying a voltage of equal magnitude and opposite sign to the opposing actuators. The opposite directions of the surface tractions at the interfaces between the actuator and plate cause the uniform bending moments along the actuator boundaries as shown in

The symmetry of the actuators with respect to the midplane (

The strains across the thickness of the composite laminate due to the bending moment can be expressed as follows:
_{x}_{y}_{xy}

The bending stresses in the k-th layer of the composite laminate are:
_{1} and _{2} are the Young’s moduli along the fiber direction and normal to the fiber direction, respectively. _{12} is the shear modulus, _{12} and _{21} are the Poisson’s ratios,

In the case of cross-ply (_{16} and _{26} are equal to zero. The bending stresses in the k-th layer can be reduced to:

The bending stresses in the piezoelectric actuator are:
_{pe}_{pe}

The bending moments per unit length _{x}_{y}_{x}dz_{y}dz_{k}_{k−1} represent the positions of the top and bottom surfaces of the k-th layer in the composite laminate, respectively.

The curvatures κ_{x}_{y}_{x}_{y}

A rectangular composite laminate plate with simply supported boundary conditions is considered in this work. The location of the surface bonded actuator is shown in

Using the classical plate theory, the equilibrium equation for the plate can be written in terms of the plate internal moments _{x}_{y}_{xy}_{x}, m_{y}

The internal moments _{x}_{y}_{xy}_{x}, m_{y}_{11})_{p}_{22})_{p}_{66})_{p}

Substituting

For a simply supported rectangular plate, the flexural displacement

Substituting _{mn}

The finite element method is a widely used and powerful tool for analyzing complex structures. It is capable of dealing with the piezoelectrical materials. Many researchers have modelled the piezoelectric actuation using the finite element method. A commercially available finite element software ANSYS has the ability to analyze the piezoelectrical materials. In this study, the finite element software ANSYS is adopted to investigate the deflection of a simply supported plate induced by the surface bonded piezoelectric actuators. To perform the ANSYS finite element analysis for the piezoelectric actuator bonded structure, SOLID 45 elements and SOLID 5 were used in the composite plate and piezoelectric actuators, respectively.

A typical three dimensional finite element mesh is shown in

In the following numerical examples, the composite material is carbon/epoxy with stacking sequence [0/90/90/0].The composite material properties of carbon/epoxy are listed in _{p}_{pe}_{pe}=_{pe}^{2}, piezoelectric constant _{31}^{−10} V/m and thickness _{pe}

Two piezoelectric actuators are surface bonded on the top and bottom surfaces of the composite plate. Three different sizes of piezoelectric actuators with the dimensions of 0.06 m × 0.04 m, 0.08 m × 0.06 m and 0.1 m × 0.08 m, respectively, bonded on the central area of the composite plate as shown in

To study the capability of control the deflection shape of the plate, actuators are placed at various locations. In this example, the piezoelectric actuators are surface bonded at three different locations, central, right and top region of the plate, respectively, as shown in

Piezoelectric materials are often used as strain actuators and shape control of smart structures, as they are compact and response quickly. In this investigation, two piezoelectric actuators are symmetrically surface bonded on a composite laminate plate. Electric voltages with the same amplitude and opposite sign are applied to the two symmetric piezoelectric actuators, resulting in the bending effect on the plate. Theoretical model of the bending moment is derived by using the theory of elasticity to represent the interaction of the actuator and the host plate. Following the classical plate theory, the deflection of a simply supported plate subjected to the bending moment can be obtained. The analytical solutions are validated with the finite element results. The effects of size and location of actuators on the responses of the plate are presented through the parametric study. Utilization of the laminate theory and plate theory, the deformed shape of the laminate plate can be predicted analytically. The methodology proposed in this paper is easy to employ, and provides an alternate way of solving this complicated problem analytically with accurate results.

Bending moment acting on the composite laminate induced by the lead zirconate titanate PZT actuators.

Strain distribution across the thickness of the composite laminate.

Surface bonded actuator on the composite laminate plate.

3-D finite element mesh.

Three different sizes of PZT actuators.

Deformation of the composite plate induced by the PZT actuator 0.06 m × 0.04 m obtained by (a)

Displacements of the composite plate induced by PZT actuators with three different sizes.

Three different locations of the actuator.

deformation of the composite plate induced by the PZT actuator bonded on the right region calculated by (a)

deformation of the composite plate induced by the PZT actuator bonded on the top region calculated by (a)

Flexural displacement of the composite plate obtained by ANSYS and

Material properties of carbon/epoxy.

_{1} |
_{2} |
_{12} |
_{23} |
_{12} |
_{23} |
---|---|---|---|---|---|

108 GPa | 10.3 GPa | 7.13 GPa | 4.02 GPa | 0.28 | 0.28 |

maximum deflection of the composite plate induced by the PZT actuators with three different sizes.

PZT |
1.68 × 10^{−3} mm |
1.59 × 10^{−3} mm |
5.4 |

PZT |
2.90 × 10^{−3} mm |
2.73 × 10^{−3} mm |
6.3 |

PZT |
4.26 × 10^{−3} mm |
3.98 × 10^{−3} mm |
6.8 |

maximum deflection of the composite plate induced by the PZT actuators at three different locations.

PZT at central region | 1.68 × 10^{−3} mm |
1.59 × 10^{−3} mm |
5.4 |

PZT at right region | 8.62 × 10^{−4} mm |
8.00 × 10^{−4} mm |
7.7 |

PZT at top region | 1.27 × 10^{−3} mm |
1.34 × 10^{−3} mm |
4.9 |