Abstract
The solution of the Saint-Venant’s Problem for a slender compound piezoelectric beam presented in this paper generalizes the recent solution by the authors and E. Harash (J. Appl. Mech. 11:1–10, 2007) for a homogeneous piezoelectric beam and the solution for a compound elastic beam developed by O. Rand and the first author (Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Boston, 2005). Justification for this approximation emerges from the St. Venant’s Principle. The stress, strain and (electrical) displacement components (“solution hypothesis”) are presented as a set of initially assumed expressions involving twelve tip loading parameters, six unknown weight coefficients, and three pairs of torsion/bending functions of two variables. Each pair of functions satisfies the so-called coupled non-homogeneous Neumann problem (CNNP) in the cross-sectional domain. The work develops concepts of the torsion/bending functions, the torsional rigidity and piezoelectric shear center, the tip coupling matrix, for a compound piezoelectric beam. Examples of exact and approximate solutions for rectangular laminated beams made of transtropic materials are presented.
Similar content being viewed by others
References
Rand, O., Rovenski, V.: Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools. Birkhauser, Boston (2005)
Bisegna, P.: The Saint-Venant problem in the linear theory of piezoelectricity. Atti Conv. Lincei, Accad. Naz. Lincei, Rome 140, 151–165 (1998)
Bisegna, P.: The Saint-Venant Problem for Monoclinic piezoelectric cylinders. Z. Angew. Math. Mech. 78(3), 147–165 (1998)
Davh, F.: Saint-Venant’s problem for linear piezoelectric bodies. J. Elast. 43, 227–245 (1996)
dell’Isola, F., Rosa, L.: Saint-Venant’s problem in linear piezoelectricity. Math. Control Smart Struct. 2715, 399–409 (1996)
Batra, R., Yang, J.: Saint-Venant’s principle in linear piezoelectricity. J. Elast. 38, 209–218 (1995)
Batra, R.: Saint-Venant’s principle for linear elastic porous materials. J. Elast. 39, 265–271 (1995)
Batra, R.: Saint-Venant’s principle for a helical piezoelectric body. J. Elast. 43(1), 69–79 (1996)
Migórski, S.: Weak solvability of a piezoelectric contact problem. Eur. J. Appl. Math. 20, 145–167 (2009)
Fernández, J.: Numerical analysis of an elasto-piezoelectric problem with damage. Int. J. Numer. Methods Eng. 77(2), 261–284 (2008)
Borrelli, A.: Saint-Venant’s principle for antiplane shear deformations of linear piezoelectric materials. SIAM J. Appl. Math. 62(6), 2027–2044 (2002)
Borrelli, A., Patria, M.: Saint-Venant’s principle for a piezoelectric body. SIAM J. Appl. Math. 59(3), 1098–1111 (1999)
Borrelli, A., Horgan, C., Patria, M.: Saint-Venant end effects for plane deformations of linear piezoelectric solids. Int. J. Solids Struct. 43(5), 943–956 (2006)
Borrelli, A., Horgan, C., Patria, M.: Saint-Venant end effects in anti-plane shear for classes of linear piezoelectric materials. J. Elast. 64(2–3), 217–236 (2001)
Borrelli, A., Horgan, C., Patria, M.: Saint-Venant’s principle for anti-plane shear deformations of linear piezoelectric materials. J. Appl. Math. 62, 2027–2044 (2002)
Borrelli, A., Horgan, C., Patria, M.: End effects for pre-stressed and pre-polarized piezoelectric solids in anti-plane shear. Z. Angew. Math. Phys. (ZAMP) 54, 797–806 (2003)
Borrelli, A., Horgan, C., Patria, M.: Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials. Proc. R. Soc. Lond. Ser. A 460, 1193–1212 (2004)
Davi, F.: Saint-Venant’s problem for linear piezoelectric bodies. J. Elast. 43, 227–245 (1996)
Lakes, R.: Saint Venant end effects for materials with negative Poisson’s ratios. J. Appl. Mech. 59, 744–746 (1992)
Vidoli, S., Batra, R., dell’Isola, F.: Saint-Venant’s problem for a second-order piezoelectric prismatic bar. Int. J. Eng. Sci. 38, 21–45 (2000)
Tarn, J.-Q., Huangb, L.-J.: Saint-Venant end effects in multilayered piezoelectric laminates. Int. J. Solids Struct. 39(19), 4979–4998 (2002)
Xu, X.S., Zhong, W.X., Zhang, H.W.: The Saint-Venant problem and principle in elasticity. Int. J. Solids Struct. 34(22), 2815–2827 (1997)
Rovenski, V., Abramovich, H.: Saint-Venant’s problem for non-homogeneous piezoelectric beams. TAE Rep. 975, 1–90 (2007)
Rovenski, V., Harash, E., Abramovich, H.: Saint-Venant’s problem for homogeneous piezoelectric beams. J. Appl. Mech. 11, 1–10 (2007)
Saint-Venant, B.: Memoire sur la flexion des prismes. J. Math. Pures Appl. (Liouville). Ser. II 1, 89–189 (1856)
Saint-Venant, B.: Memoire sur la torsion des prismes, Memoires presentes par divers savants a l’academie des sciences. Sci. Math. Phys., Paris 14, 233–560 (1856)
Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Body. Moscow, Mir Publ. (1981)
Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Groningen, Noordhoff (1953)
Kudryavtsev, B.A., Parton, V.Z., Senik, N.A.: Electromagnetoelasticity. In: Applied Mechanics: Soviet Reviews, vol. 2, pp. 1–230. Hemisphere, Washington (1990)
Cady, W.G.: Piezoelectricity—An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals. Dover, New York (1964), in two volumes
Yang, J.: An Introduction to the Theory of Piezoelectricity. Advances in Mech. and Math., vol. 9. New York, Springer (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rovenski, V., Abramovich, H. Saint-Venant’s Problem for Compound Piezoelectric Beams. J Elasticity 96, 105–127 (2009). https://doi.org/10.1007/s10659-009-9201-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-009-9201-9