Keywords: analytical models, finite element analysis, piezoelectric bimorphs, extension bimorphs, shear bimorphs, extension-shear bimorphs.

With the growing of the relatively new field of adaptive structures, piezoelectric bimorphs have been applied as sensors and actuators for high precision aerospace and medical applications as well as in shape, vibration and noise control. In particular, they are often used to validate experimentally, analytically or numerically new piezoelectric adaptive formulations. This has led to a very large research effort into closed-form one-dimensional (1D), analytical two-dimensional (2D) and three-dimensional (3D) exact solutions for multilayer adaptive beams. In this contribution, several bimorph-type configurations (extension, shear and extension-shear bimorphs) are studied with analytical and numerical approaches. More precisely, some simple well known (extension) and new (shear and extension-shear) analytical formulas, as well as the thermal analogy approach, with their common assumption of a through-the-thickness constant electric field, are evaluated against 2D plane strain and 3D piezoelectric finite elements available in some commercial codes (ABAQUS, ANSYS).

First, a number of well established analytical [1] and numerical [2,3] modelling techniques of classical extension bimorphs are assessed and evaluated. This allows the clarification of several confusing common validation practices, using well known experimental data [4] and analytical formulas. In many cases, this is due to the non explicitly mentioned applied electric load (potential, difference of potentials or electric field), connectivity (series or parallel) and interface electrodes (present or not, grounded or not).

In order to discuss the validity domain of analytical and numerical approaches, several materials (a piezoelectric polymer, Kynar, and a piezoelectric ceramics, PZT-5A), geometries (length-to-thickness ratio), mechanical boundary conditions (Clamped-Free, Clamped-Clamped, Simply Supported) and finite element types are considered. Discrepancies above 10% are found. The results are discussed in details. Focus is especially laid in the discussion on the limitations resulting from the assumption of a constant electric field through the layer thickness. This assumption is made in analytical models or in the numerical models, if the thermomechanical analogy is used. By neglected the shear effect, most of the analytical and exact reference solutions are limited to thin bimorphs with relatively high length-to-thickness ratio. Most of these solutions are based on equivalent single layer (ESL) formulations; thus, they do not really account for the multilayer aspect of the bimorph. To represent the piezoelectric effect, simplified solutions use either induced-strain or equivalent load approaches, both of which assume a constant electric field in the piezoelectric layer, thus neglecting the so-called induced electric potential; hence, they take only partially the piezoelectric coupling into account. It is shown that another reason for the discrepancy between analytical and 3D FE results lies in the transverse effect. Due to the free edges and the contraction and elongation in the transverse direction, the shape of the bimorph far from the clamped edge warped in both longitudinal and transverse directions. This leads to a "drain channel" effect which depends on the bimorph width, length-to-thickness ratio, material properties, and mechanical boundary conditions. It is suggested, and confirmed by 3D FE analyses, that this "drain channel" effect can be reduced by only partially poling the piezoelectric bimorph.

Second, analytical formulas are presented and evaluated for the first time for the relatively new shear bimorph concept and a more general extension-shear one [5], resulting from the combination of the extension- and shear-piezoelectric modes via an off-axes polarisation. It is shown that, although shear bimorphs lead to smaller maximum deflection, they develop much smaller and homogeneous stresses in the bimorphs interface. The corresponding new simple analytical formulas accurately reproduced the three-dimensional piezoelectric finite element results, confirming their usefulness for preliminary designs. A detailed comparison between several approaches underlines the importance of an appropriate model choice. Especially, the plane strain assumption from [5] or from 2D plane strain piezoelectric FE is discussed under the light of the numerical results obtained with a 3D model.

1

Smits, J.G., Dalke, S.I., Cooney, T.K., "The constituent equations of piezoelectric bimorphs", Sensors and Actuators A, 28, 41-61, 1991. doi:10.1016/0924-4247(91)80007-C

2

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

3

Poizat, C., Benevolenski, O., Thielicke, B., "From bending actuators to a peristaltic micropump : modeling and design analysis", In "Proceedings of HERCMA", Athens, Sept. 24-27, 2003.

4

Lee, C.K., Moon, F.C., "Laminated piezopolymer plates for torsion and bending sensors and actuators", Journal of the Acoustical Society of America, 85, 2432-2439, 1989. doi:10.1121/1.397792

5

Vel, S.S., Batra, R.C., "Analysis of piezoelectric bimorphs and plates with segmented actuators", Journal of Thin-Walled Structures, 39, 23-44, 2001. doi:10.1016/S0263-8231(00)00052-5

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