Keywords: finite elements, axisymmetric shells, 3D elasticity theory, piezoelectric, cylindrical shells, bending, vibrations.

Piezoelectric materials are the more promising materials to use in structural mechanics in way to develop adaptive structures. This is due to their coupled mechanical and electrical properties and other properties, such as them being lightweight and easy to shape. In addition, piezoelectric patches can be bounded to the structures without significantly changing the overall mechanical properties. Piezoelectric cylindrical structures (hollow or solid) have a wide range of applications such as resonators, fuel injectors, high precision telescopes, electro-optics, hydro-acoustics and others.

Physically, the piezoelectric effect was discovered when certain crystals produce electric charges by application of pressure, defined as the direct effect. This properties where discovered on late XIX century by Curie brothers. Allik and Hughes [1] presented a numerical method, using the finite element method, for three-dimensional analysis for piezoelectric media.

Recently, Kapuria, et al. [2] developed a three-dimensional solution for simply supported piezoelectric cylindrical shells subjected to axisymmetric electromechanical load. Mitchell and Reddy [3] made a study of the static behavior using an approximate theory. Pinto Correia, et al. [4] developed a semi-analytical axisymmetric shell finite element model with embedded and, or surface bonded piezoelectric rings actuators and, or sensors for active damping vibration control of the structure. Santos, et al. [5] developed a general semi-analytical finite element model for bending, free vibration and buckling analysis of shells of revolution made of laminated anisotropic elastic material. The 3D elasticity theory was used and the equations of motion were obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, . The primary variables are represented by:

where , , are radial, circumferential and axial displacements, respectively, and is the electric potential. The directions of mechanical displacements are represented, respectively, by , and , which are the radial, circumferential and axial coordinates, is the time and is the total number of harmonics. In this model, , , , and , , , are the amplitudes of the symmetric and skew-symmetric components of the primary variables respectively.

This paper addresses the bending and free vibrations of multilayered cylindrical shells with piezoelectric properties using a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.

1

Allik H., Hughes T.J.R. "Finite element method for piezoelectric vibration", International Journal Numerical Methods in Engineering. 2: 151-157, 1970. doi:10.1002/nme.1620020202

2

Kapuria S., Sengupta S., Dumir P.C. "Three-dimensional solution for simply-supported piezoelectric cylindrical shell for axisymmetric load", Computational Methods in Applied Mechanics and Engineering. 140: 139-155, 1997. doi:10.1016/S0045-7825(96)01075-4

3

Mitchell J.A., Reddy J.N. "A refined hybrid plate theory for composite laminates with piezoelectric laminae", International Journal of Solids and Structures. 32(16): 2345-2367, 1995. doi:10.1016/0020-7683(94)00229-P

4

Pinto Correia I.F., Mota Soares C.M., Mota Soares C.A., Herskovits J. "Active control of axisymmetric shells with piezoelectric layers: a mixed laminated theory with a high order displacement field", Computers and Structures. 80: 2265-2275, 2002. doi:10.1016/S0045-7949(02)00239-0

5

Santos H., Mota Soares C.M., Mota Soares C.A., Reddy J.N. "A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: bending, free vibration and buckling", Composite Structures. 71(3-4): 273-281, 2005. doi:10.1016/j.compstruct.2005.09.006

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