Abstract
We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.
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References
Bernadou, M., Haenel, C.: Modelization and numerical approximation of piezoelectric thin shells. I. The continuous problems. Comput. Methods Appl. Mech. Eng. 192(37–38), 4003–4030 (2003)
Ciarlet, P.: Mathematical Elasticity, Volume II: Theory of Plates. North-Holland, Amsterdam (1997)
Collard, C., Miara, B.: Two-dimensional models for geometrically nonlinear thin piezoelectric shells. Asymptot. Anal. 31(2), 113–151 (2003)
Eringen, A.C., Maugin, G.: Electrodynamics of Continua, Volume I: Foundations and Solid Media. Springer, Berlin Heidelberg New York (1990)
Green, A.E., Zerna, W.: Theoretical Elasticity, 2nd edn. Oxford, New York (1968)
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations, Lectures Notes in Mathematics 749. Springer, Berlin Heidelberg New York (1979)
Figueiredo, I.N., Leal, C.F.: A piezoelectric anisotropic plate model. Asymptot. Anal. 44(3–4), 327–346 (2005)
Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, London, UK (1990)
Maugin, G.A., Attou, D.: An asymptotic theory of thin piezoelectric plates. Q. J. Mech. Appl. Math. 43, 347–362 (1990)
Rahmoune, M., Benjeddou, A., Ohayon, R.: New thin piezoelectric plate models. J. Intell. Mater. Syst. Struct. 9, 1017–1029 (1998)
Raoult, A., Sene, A.: Modelling of piezoelectric plates including magnetic effects. Asymptot. Anal. 34(1), 1–40 (2003)
Sene, A.: Modelling of piezoelectric static thin plates. Asymptot. Anal. 25(1), 1–20 (2001)
Trabucho, L., Viaño, J.M.: Mathematical modelling of rods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 4, pp. 487–974. North-Holland, Amsterdam (1996)
Young, E.C.: Vector and Tensor Analysis, Monographs and Textbooks in Pure and Applied Mathematics 48. Marcel Dekker, New York (1978)
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Narra Figueiredo, I.M., Franco Leal, C.M. A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model. J Elasticity 85, 85–106 (2006). https://doi.org/10.1007/s10659-006-9072-2
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DOI: https://doi.org/10.1007/s10659-006-9072-2