Saint-Venant’s Problem for Homogeneous Piezoelectric Beams

[+] Author and Article Information
Vladimir Rovenski

Department of Mathematics, University of Haifa, Haifa, 31905, Israelrovenski@math.haifa.ac.il

Eugene Harash

 Faculty of Aerospace Engineering, Technion-I.I.T., Haifa, 32000, Israeleharash@techunix.technion.ac.il

Haim Abramovich

 Faculty of Aerospace Engineering, Technion-I.I.T., Haifa, 32000, Israelhaim@aerodyne.technion.ac.il

J. Appl. Mech 74(6), 1095-1103 (Dec 19, 2006) (9 pages) doi:10.1115/1.2722315 History: Received August 10, 2006; Revised December 19, 2006

This paper is devoted to the linear analysis of a slender homogeneous piezoelectric beam that undergoes tip loading. The solution of the Saint-Venant’s problem presented in this paper generalizes the known solution for a homogeneous elastic beam. The analytical approach in this study is based on the Saint-Venant’s semi-inverse method generalized to electroelasticity, where the stress, strain, and (electrical) displacement components are presented as a set of initially assumed expressions that contain tip parameters, six unknown coefficients, and three pairs of auxiliary (torsion/bending) functions in two variables. These pairs of functions satisfy the so-called coupled Neumann problem (CNP) in the cross-sectional domain. In the limit “elastic” case the CNP transforms to the Neumann problem, for a beam made of a poled piezoceramics the CNP is decomposed into two Neumann problems. The paper develops concepts of the torsion/bending functions, the torsional rigidity and shear center, the tip coupling matrix for a piezoelectric beam. Examples of exact and numerical solutions for elliptical and rectangular beams are presented.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

Notation for a slender beam

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Figure 4

Torsional function φ

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Figure 5

Bending function χ1

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Figure 3

Piezoelectric beam deformation for Mz=1

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Figure 6

Electrical bending function φ4

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Figure 7

Stress component σ4 (the tip load Mz=1)

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Figure 8

Torsional warping function w

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Figure 9

Electric displacement D1 (the tip load Mz=1)



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