Keywords: hybrid beams, patch actuators, finite element model, zigzag theory, piezoelectricity.

Finite element modelling of adaptive structural elements consisting of laminated elastic substrate with surface-bonded or embedded piezoelectric layers have been recently reviewed by Benjeddou [1]. Due to inhomogeneity in the mechanical properties across the thickness and the presence of electric heterogeneity caused by the embedded or surface bonded piezoelectric layers, accurate analysis of these hybrid laminates require efficient and accurate approximation of the displacement and the potential field across the thickness in order to yield an accurate description of the in-plane stress, transverse shear stress, induced electric potentials and warping and strains of the normal to the midplane. A coupled layer-wise theory with layer-wise approximation of the displacements and potentials has been presented by Saravanos and Heyliger [2]. This theory yields accurate results but the computational effort increases with the number of layers and hence it is not suitable for practical applications specially for dynamic and control problems. Equivalent single layer theory approximations such as first order shear deformation theory (FSDT) and refined third order theory (TOT) for the mechanical field variables and a layer-wise approximation of the electric field have been developed [3,4,5], but the same global variation of displacements across the thickness in these theories fails to account for the zigzag nature of variations of in-plane displacements as observed from exact three-dimensional piezoelasticity solutions. Recently, Kapuria et al. [6] developed an efficient coupled zigzag theory for piezoelectric hybrid beams, which combines global cubic variation of in-plane displacements with local piecewise linear layer-wise distribution. The deflection field accounts explicitly for the transverse strain due to the piezoelectric constant without introducing additional variables. The theory satisfies the shear traction free conditions at the top and bottom surface and the transverse shear stress continuity conditions at the layer interfaces in the presence of in-plane and transverse electric fields. The number of displacement variables is three which is the same as in a first order or third order equivalent single layer theory. A two noded finite element model for static electromechanical analysis of hybrid beam has also been developed [7] based on the new zigzag theory. Each node has four degrees of freedom for the displacements and a variable number of degrees of freedom for the electric potential depending on the discretisation of electric potential across the thickness of the element. Hermitian cubic interpolation is used for the deflection and electric potential and linear interpolation is used for axial displacement and shear rotation.

The objective of present work is to model and analyse hybrid beams with segemental/patch piezoelectric sensors/actuators under static electromechanical loading using the finite element model based on the coupled zigzag theory. Physical modelling of piezoelectric patches using variable number of degrees of freedom for the potential variables have been discussed. The accuracy of such a modelling is assessed in comparison with a coupled two-dimensional finite element solution obtained using ABAQUS. Results are presented for hybrid composite and sandwich beams with piezoelectric patches under mechanical and electric potential loads for various span-to-thickness ratios. It is observed that the present finite element model yields quite accurate results for the stress and deflection for moderately thick hybrid beam with patch sensors and actuators. The effect of piezoelectric patch location is investigated. The piezoelectric actuator patch is most effective in controlling tip deflection of a cantilever beam, when placed nearest to the fixed end.

1

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

2

D.A. Saravanos, P.R. Heyliger, "Coupled layerwise analysis of composite beams with embedded piezoelectric sensors and actuators", Journal of Intelligent Material Systems and Structures, 6, 350-363, 1995. doi:10.1177/1045389X9500600306

3

D.A. Saravanos, "Mixed Laminate Theory and Finite Element for Smart Piezoelectric Composite Shell Structures", AIAA Journal, 35, 1327-33, 1997. doi:10.2514/2.264

4

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

5

V.M.F. Correia, M.A.A. Gomes, A. Suleman, C.M.M Soares, "Modelling and design of adaptive composite structures", Computational Methods in Applied Mechanics and Engineering, 185, 325-346, 2000. doi:10.1016/S0045-7825(99)00265-0

6

S. Kapuria, P.C. Dumir, A. Ahmed, "An efficient coupled layerwise theory for static analysis of piezoelectric sandwich beams", Archive of Applied Mechanics, 73, 147-159, 2003. doi:10.1007/s00419-003-0277-6

7

S. Kapuria, P.C. Dumir, A. Ahmed, N. Alam, "Finite element model of efficient zigzag theory for static analysis of hybrid piezoelectric beams", Computational Mechanics. In press. doi:10.1007/s00466-004-0592-y

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