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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 7
On Analytical and Numerical Modelling of Piezoelectric Bimorphs C. Poizat+ and A. Benjeddou*
+FraunhoferInstitute for Mechanics of Materials, Freiburg, Germany
C. Poizat, A. Benjeddou, "On Analytical and Numerical Modelling of Piezoelectric Bimorphs", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 7, 2004. doi:10.4203/ccp.79.7
Keywords: analytical models, finite element analysis, piezoelectric bimorphs, extension bimorphs, shear bimorphs, extensionshear bimorphs.
Summary
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 closedform onedimensional (1D), analytical twodimensional (2D) and
threedimensional (3D) exact solutions for multilayer adaptive beams.
In this contribution, several bimorphtype configurations (extension, shear and
extensionshear bimorphs) are studied with analytical and numerical approaches.
More precisely, some simple well known (extension) and new (shear and
extensionshear) analytical formulas, as well as the thermal analogy approach, with their
common assumption of a throughthethickness 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, PZT5A), geometries (lengthtothickness ratio), mechanical boundary conditions (ClampedFree, ClampedClamped, 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 lengthtothickness 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 inducedstrain or equivalent load approaches, both of which assume a constant electric field in the piezoelectric layer, thus neglecting the socalled 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, lengthtothickness 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 extensionshear one [5], resulting from the combination of the extension and shearpiezoelectric modes via an offaxes 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 threedimensional 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. References
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