Abstract
The paper presents a generic solution methodology for a quasi-static homogeneous monoclinic piezoelectric beam under axially distributed electric and mechanical surface loads and body forces expressed as polynomials of degree K≥ 0 of the axis variable. (In the absence of any electrical loading, this problem is known as the Almansi–Michell problem). The stress and the electrical displacement components are presented as a set of polynomials of degree ≤K+2 of the axis variable (“solution hypothesis”) containing 4K unknown tip loading constants and 3K stress functions of two variables. The cases K=0,1 stand for uniform or linear distributed loads in the axis direction. The analysis is initiated by the Kth level and continues down to lower levels. The main result of this work generalizes the “elastic” solution given recently by O. Rand and the first author (2005). Examples of solutions for axially uniform distributed loads (K=0), and equilibrium in which the stress and the electrical displacement do not depend on the axis variable, are presented. The applications to constant body loads and a hydrostatic pressure are considered.
Similar content being viewed by others
References
Almansi, E.: Sopra la deformazione dei cilindri sollecitati lateralmente. Atti della Academia Nazionale dei Lincei, 10, Nota I: 333–338; Nota II: 400–408 (1901)
Bors, C.: Almansi–Michell problem for an elastic orthotropic cylinder. Atti R. Accad. Naz. Lincei 54(3), 441–446 (1973) (Roma)
Dzhanelidze, G.Yu.: The Almansi problem. In: Proceedings of Leningrad Polytechnic Institute, vol. 210, pp. 25–38, (1960) (in Russian)
Iesan, D.: On the theory of uniformly loaded cylinders. J. Elast. 16(4), 375–382 (1989)
Khatiashvili, G.: The Almansi–Michell problems for homogeneous and composite bodies. Metsniereba. Tbilisi pp. 237, I (1983), pp. 185, II (1985) (in Russian)
Ruchadze, A., Berekašvili, R.: On a generalized Almansi problem. Soobshch. Akad. Nauk Gruzin. SSR 100(3), 561–564 (1980) (in Russian)
Wang, M.Z., Xu, X.S.: A generalization of Almansi’s theorem and its application. Appl. Math. Model. 14(5), 275–279 (1990)
Varisov, M.-Z.: The Almansi problem for a cylindrically anisotropic body. Trudy Inst. Vychisl. Mat. Akad. Nauk Gruzin. SSR 27(1), 3–11 (1987) (in Russian)
Rovenski, V., Harash, E., Abramovich, H.: St.Venant’s problem for homogeneous piezoelectric beams. J. Appl. Mech. 74, 1–10 (2007)
Batra, R., Yang, J.: Saint-Venant’s principle in linear piezoelectricity. J. Elast. 38(2), 209–218 (1995)
Kudryavtsev, B.A., Parton, V.Z., Senik, N.A.: Electromagnetoelasticity. Applied Mechanics: soviet reviews, vol. 2, pp. 1–230. Hemisphere Publ. Co. (1990)
Rand, O., Rovenski, V.: Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools. Birkhauser, Boston, USA (2005)
St. Venant, B.: Memoire sur la flexion des prismes. J. math, pures et appl. (Liouville). Ser II, 1, 89–189 (1856)
St. Venant, B.: Memoire sur la torsion des prismes, Memoires presentes par divers savants a l’academie des sciences. Sciences Math. et Phys. Paris 14, 233–560 (1856)
Michell, J.H.: The theory of uniformly loaded beams. J. Math. 32, 28–42 (1901)
Cady, W.G.: Piezoelectricity – An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, vol. 1. Dover Publ. New-York (1964)
Rovenski, V., Abramovich, H.: Almansi–Michell Problem for Homogeneous Piezoelectric Beams. TAE report No. 969, 1–108 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rovenski, V., Abramovich, H. Behavior of Piezoelectric Beams under Axially Non-uniform Distributed Loading. J Elasticity 88, 223–253 (2007). https://doi.org/10.1007/s10659-007-9132-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-007-9132-2