Keywords: modelling, piezoelectric, adaptive structures, beams, plates, shells.

Piezoelectric materials have been widely used during the last two-decades for shape, noise and vibration control of aerospace and mechanical engineering structural elements such as beams, plates and shells. Either surface-mounted or embedded in laminates, they have greatly contributed to the development and rapid growth of the so-called smart, intelligent or adaptive materials, systems, structures and technologies. Their modelling and analysis require:

• An electromechanical coupling representation,
• A mechanical model,
• An electric model,
• A finite element (FE) model and,
• A piezoelectric actuation/sensing mechanism.

Regarding these aspects, recent literature surveys[1,2] indicate that:

• The piezoelectric effect was often represented through the popular pin-force, thermal analogy and engineering (actuation-only) models, although they neglect the stiffness and mass effects of the attached piezoelectric patches. Also, these models do not consider the electrostatic charge equation. That is why they are seen as uncoupled electromechanical models.
• The multilayer aspect of the adaptive structures was mostly handled through the classical equivalent single-layer (ESL) mechanical model. The latter assumes a unique displacement field through the laminate thickness. Hence, it does not represent correctly the interaction between the host structure and the piezoelectric layers or patches.
• The electric potential of the piezoelectric layers was generally considered in-plane uniform and through-the-thickness linear. Hence, only the transverse components of the electric field and displacement are usually retained in the electric model. Thus, the electrostatic charge equation is no longer verified.
• In FE modelling of adaptive beams and plates, the piezoelectric effect has been often represented implicitly, i.e. without electric degrees of freedom (DOF). In consequence, the stiffening effect, due to the direct piezoelectric effect, cannot be taken into account if the electrostatic charge equation is neglected, as is often the case.
• Most literature on piezoelectric adaptive structures has been limited to parallel initial polarisation and applied electric field. Hence, only the membrane strains can be actuated, leading to the conventional extension actuation mechanism (EAM).

This paper aims to present some of the author and his co-workers developments and results in the field of modelling piezoelectric adaptive beams, plates and shells with regards to the above research concerns:

• Full electromechanical coupling is achieved by considering the electrostatic charge equation either explicitly, by retaining the electric potential as a fundamental variable, or implicitly, through the so-called induced electric potential. The latter represents the quadratic term in the electric potential. It is expressed either in terms of the mechanical displacements or as a "bubble" potential function to be condensed using the electrical equilibrium equations.
• The proposed multilayer and sandwich mechanical models of piezoelectric adaptive beams, plates and shells consider explicitly the interaction between the host structures and the attached piezoelectric actuators or sensors.
• The proposed electric models are based on higher-order a priori electric assumptions for the electric potential, field and displacement. They result from mathematical analyses or specially developed exact solutions.
• Four special sandwich beam and shell FE are proposed for the modelling of piezoelectric adaptive beams and axisymmetric shells.
• Beside the EAM, the less-known piezoelectric shear actuation mechanism (SAM) will be presented and evaluated theoretically and numerically for piezoelectric adaptive beams, plates and shells.

Selected numerical approximate and exact analyses results focus on the influence of the correct representation of the electromechanical coupling, multilayer/sandwich aspect and the shell curvatures on the free-vibration characteristics and static EAM and SAM based actuation and sensing of piezoelectric adaptive beams, plates and shells. Current and future research interests are also outlined in a closure section.

1

A. Benjeddou, "Advances in piezoelectric finite element modelling of adaptive structural elements: a survey", Computers and Structures, 76(1-3), 347-363, 2000. doi:10.1016/S0045-7949(99)00151-0

2

A. Benjeddou, "Advances in hybrid active-passive vibrations and noise control via piezoelectric and viscoelastic constrained layer treatments", Journal of Vibration and Control, 7(4), 565-602, 2001. doi:10.1177/107754630100700406

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