Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells J.K. Nath, S. Kapuria ⇑ Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110 016, India article info Article history: Available online 30 December 2011 Keywords: Zigzag theory Third order theory Piezoelectric shells Dynamics Smart cylindrical shells abstract The improved third order zigzag theory and its smeared counterpart (without the zigzag effect), recently developed by the authors for static analysis of piezoelectric laminated cylindrical shells, are extended to dynamics. The piezoelectric layers are considered as radially polarized to make use of the extension actu- ation mechanism that is best suited for effective actuation and sensing. The zigzag theory accounts for the layerwise variation of inplane displacements and includes the transverse normal extensibility under elec- tric field, and also satisfies the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces of the shell. Yet, the number of primary displacement variables is only five, same as its smeared counterpart. The two theories are critically assessed for their accuracy by direct compar- ison with the three dimensional piezoelasticity solutions for free and forced vibration response of simply supported smart angle-ply infinite-length and cross-ply finite-length shells, with a variety of heteroge- neous composite and sandwich laminates. It is shown that the zigzag theory, in spite of being computa- tionally efficient, is very accurate even for shells with highly inhomogeneous laminates. In contrast, the smeared third order theory is grossly inadequate for smart shells made of inhomogeneous composite and sandwich substrates. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Multilayered composite structures integrated with piezoelectric materials offer the possibility of combining the superior properties of composite materials such as the high specific strength and high specific stiffness, with the ability of sensing, actuation, and control. Due to the discontinuity of mechanical, thermal and electrical properties at the layer interfaces, accurate modeling of these smart piezolaminated structures poses a considerable challenge. Cylin- drical shells are widely used as primary structural components in various structures in aerospace, automobile, naval, petrochemicals and refinery industries. Radially polarized piezoelectric layers with the electric field applied or measured across the layer thickness are best suited (among other configurations of piezoelectric materials) for effective sensing and actuation of such shell-type structures. The most accurate response of hybrid multilayered structures can be obtained by the exact analytical solution of the equations of three dimensional (3D) piezoelasticity, wherein no adhoc assumptions are made on the variations of the field variables across the thickness. Exact solutions of 3D piezoelasticity equa- tions have been presented for axisymmetric vibration of infinitely long, radially polarized, single and multilayered piezoelectric cylinders [1–4]. Semianalytical 3D piezoelasticity solutions for non-axisymmetric vibration of infinitely long [5] and finite-length [6,7] orthotropic hybrid laminated cylindrical shells have been pre- sented using a finite element approximation in the radial direction. Santos et al. [8] presented the bending and free vibration analysis of hybrid cylindrical shells using a semi-analytical axisymmetric 3D finite element model. Recently, the authors have presented exact analytical 3D piezoelasticity solutions for vibration of sim- ply-supported infinite-length angle-ply [9] and finite-length cross-ply [10] hybrid cylindrical shells with radially polarized piezoelectric layers. Such analytical 3D solutions are possible only for specific geometries and boundary conditions, while the 3D finite element (FE) solutions [11] become computationally too in- volved and inefficient for practical design. This necessitates the development of accurate and efficient two-dimensional (2D) theo- ries with appropriate a priori assumptions on the variations of field variables across the thickness. Validity of these assumptions, how- ever, must be checked in comparison with 3D solutions. A 2D laminate theory for smart piezolaminated structures should ideally have the following attributes [12]: (a) be computa- tionally efficient (number of primary variables being independent of the number of layers), (b) be accurate and robust (valid for any laminate configurations), (c) consider two-way electromechanical coupling, (d) consider the slope discontinuity in the inplane dis- placements at layer interfaces, (e) consider the transverse normal deformability in presence of electric field, and (f) satisfy the 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.12.015 ⇑ Corresponding author. Fax: +91 11 26581119. E-mail addresses: jkniter@gmail.com (J.K. Nath), kapuria@am.iitd.ac.in (S. Kapuria). Composite Structures 94 (2012) 1886–1899 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct